mechanical-properties-and-working-of-metals-and-alloys_compress

132 Depth and hardness 3 Hardness indicating dial Fig. 3.4 One model of a Levers Rockwell hardness tester Indenter under cover (Courtesy Prof. K. Biswas, IIT Anvil Load Kharagpur) selector Hand wheel to elevate anvil Adjustable weight under cover Handle to apply load produced by action of an indenter on the test piece under the above the shaft to which the indenter is attached. Various applications of two static loads. One of the static loads is a loads and indenters are used depending on the conditions of minor load, and the other is a major load. The minor load is test, as standardized by ASTM E18. applied initially to cause penetration of the indenter up to a certain depth, where a zero datum is established. The major 3.8.2 Loads load is then applied for a definite length of time, which produces an additional depth of penetration of the indenter Loads during testing are applied in two stages: beyond the zero datum point previously set up by the minor load. Without removing the minor load, the major load is (a) Initially, a minor static load of 10 kg is applied to form released after the definite holding period, which causes the a very shallow indentation on the surface of the speci- indentation to undergo elastic recovery. Now, the Rockwell men through compression of a calibrated coiled spring hardness value is read from the difference in depth from the placed within the machine between the indenter shaft zero datum position resulting from the withdrawal of the and the dial. The purpose of applying minor load is as applied major load while the minor load is still in position. follows: Thus, the resulting Rockwell hardness represents the elasti- cally recovered indentation depth due to the major load, • To eliminate the error that may arise due to variable which is equal to the difference between the total indentation contacts between the indenter and the surface of the depth upon withdrawal of the applied major load with the specimen. minor load still applied and the indentation depth due to the minor load. The hardness number found is related inversely • To set the indenter on the specimen and hold it in to this depth difference. The hardness value is a purely position. arbitrary number having no units, unlike the BHN and the VHN having units of kg per mm2 and is read directly from a • To eliminate the error that may arise due to slight specially graduated hardness dial having two pointers, per- surface imperfections, i.e. to minimize the surface manently located on a specially designed machine directly preparation of the specimen. • To reduce the tendency for ‘ridging’ or ‘sinking in’ caused by the indenter.

3.8 Rockwell Hardness 133 (b) Finally, a major static load of either 50, 90 or 140 kg is load for each of the selected steel ball indenters. In general, applied on the surface of the specimen through a system the smallest ball, that can properly be used, should be of weights and levers by means of an operating handle selected because of the loss of sensitivity to small differences in the machine that enlarges the initially formed in hardness with the increase in ball diameter. An exception indentation under minor load. The total static load to this is when soft non-homogeneous material is to be applied for indentation on the test piece is the sum- tested, in which case it may be preferable to use a larger mation of the minor load and the major load, which is diameter ball indenter because the relatively large size of the equal to either 60 or 100 or 150 kg. indentation will include local heterogeneities in the material and provide its average hardness. 3.8.3 Indenters 3.8.4 Direct-Reading Hardness Dial The indenter or ‘penetrator’ is either made of hardened steel The direct-reading hardness dial of a Rockwell hardness with shape of a spherical ball or made of diamond with tester is shown in Figs. 3.5, 3.6 and 3.7. The dial is designed shape of a cone having a spherical tip, called the ‘Brale’ to indicate the increment of penetration due to increment in (trade name). The spherical tip of the Brale indenter removes load between the minor and total loads applied and does not the sharpness at the apex of the cone and thus helps to indicate the total indentation depth made by the indenter. minimize damage of the indenter under load, otherwise the The dial face of the machine is inscribed with 100 equal chipped or cracked or blunted indenter would introduce division marks. Each division represents one unit in the errors in the hardness measurements. Sometimes, when the Rockwell hardness number, which increases in the clock- Brale indenter is damaged, reconditioning is required to wise direction and is equivalent to an indentation depth of repair it. 0.002 mm caused by the vertical motion of the indenter into the test piece. The dial has two sets of figures, one set printed The diameter of the steel ball indenter is normally in red and designated by letter ‘B’ that stands for ball, the 1=16 in: (1.588 mm) but the steel ball indenters of larger other in black and designated by letter ‘C’, which stands for diameter such as 1=8; 1=4; or 1=2 in: (3.175, 6.35, or cone. The red figures are used for reading the hardness 12.7 mm) are available that may be used for soft materials. numbers in the tests obtained with ball indenters regardless These larger size indenters are particularly employed for of the size of the ball or the magnitude of the major load. determining the hardness of plastic materials, cast iron with The black figures are used only for the Brale indenter under large graphite flakes, soft bearing alloys, solders, etc. The any major load to read the hardness numbers in the tests. spheroconical diamond tip of the Brale indenter is manu- factured to very close dimensional tolerances. The included The dial divisions for ball indenter differ from those for angle of this diamond cone indenter is 120°, with the radius cone indenter not only by letter designation and colour but of the spherical tip being 0.2 mm, which is lapped tangent to also in the precise location of specific numbers. The dial was the cone. It is advisable to check the ball indenters regularly designed to accommodate the first standardized and the most to see that they do not become flattened and the Brale to see widely used B and C scales (to be discussed subsequently), that it has not become blunted or chipped. which differ by 30 hardness numbers. The red scale for ball indenter is shifted purposely 30 divisions in a counter- Out of the several indenters and major loads used in the clockwise direction, so that B 30 corresponds to the same Rockwell hardness test, proper combinations of indenter and position on the dial face as C 0, which is the ‘set’ point for major load will serve to measure the hardness of nearly all the large pointer of the dial. Two advantages were gained by engineering materials. For harder materials, the combination this shifting of the zero points in the red scale: of the Brale indenter and the highest major load of 140 kg should be used. As the hardness of material decreases, the • Negative hardness readings were avoided on soft mate- test is carried out with the Brale indenter but with a gradual rials, such as brass, when tested on the Rockwell B scale, decrease in the major load. Thereafter, the Brale indenter and must be changed to the steel ball indenter of 1=16 in: diameter under the application of the highest major load, • This established Rockwell B 100 at the upper practical which must decrease gradually for softer material. Further limit of hardness that could be tested with the 100 kg softer materials require the application of a gradual increase total load and the 1=16 in: diameter ball without seri- in the steel ball diameter and a gradual decrease in the major ously deforming the ball penetrator.

134 3 Hardness Fig. 3.5 Positions of the large and small pointers on Rockwell SET dial, when the minor load is fully POINT applied but the major load is yet to be applied 90 0 80 C0 Fig. 3.6 Rockwell dial showing rotational movement of the large 70 C90 B30 C10 pointer in the anticlockwise B40 direction from the set-point mark B20 and the small pointer in clockwise B50 direction from the index mark, 60 1 09 C20 when the major load is applied C80 28 B60 C30 B10 7 B70 3 56 50 4 C40 C70 B0 B80 40 C50 B90 30 C60 20 10 0 SET POINT 90 0 80 C0 70 C90 B30 C10 B40 B20 B50 60 1 09 C20 C80 28 B60 C30 B10 7 B70 3 56 50 4 C40 C70 B0 B80 40 C50 B90 30 C60 20 10 0

3.8 Rockwell Hardness 135 Fig. 3.7 Rockwell dial showing SET clockwise movement of the large POINT pointer and anticlockwise movement of the small pointer, 90 0 when the major load is withdrawn 80 C0 70 C90 B30 C10 B40 B20 B50 60 1 09 C20 C80 28 B60 C30 B10 7 B70 3 56 50 4 C40 C70 B0 B80 40 C50 B90 30 C60 20 10 0 3.8.5 Hardness Scale such as annealed low- and medium-carbon steels. The working range of this scale is from B 0 to B 100, the reasons Rockwell hardness value cannot be designated solely by a of which are as follows: number, which is read from the position of large indicator on the dial face, because the same number may be obtained for • The test piece harder than RB100 will tend to flatten the materials of different hardness when different combinations steel ball indenter elastically during test that causes to of total or major load and indenter are employed during the decrease the indentation depth and increase the hardness tests. Therefore, it is necessary to indicate which indenter value. and total or major load have been employed in making the test. So, for each combination of the indenter used and the • Furthermore, the shape of the steel ball indenter makes it magnitude of total or major load applied during the test, a less sensitive to small differences in the hardness of hard scale symbol arbitrarily represented by a capital letter is specimens, whereas the Brale indenter maintains its employed. As the minor load of 10 kg remains always sensitivity at higher hardness. constant, the three values of total or major load and five numbers of indenter constitute fifteen different combinations • If the test piece is softer than RB0; the indenter will sink of total or major load and indenter. Thus, there are arbitrarily deeper into it and may cause a damage of the cap holding established 15 standard Rockwell hardness scales, as shown the ball indenter due to its contact with the surface of the in Table 3.5. The Rockwell hardness number is expressed by specimen or the weight arm may descend too far and rest the capital letter ‘R’ with subscript of a scale symbol fol- on its stop pin. lowed by the dial reading. For example, if the indenter used is 1=8 in: diameter steel ball and the magnitude of total load • Further, below the hardness value of RB0; the ball applied is 100 kg in a test, the Rockwell hardness scale will indenter, owing to its shape, becomes supersensitive and be ‘E’ and if the dial reading in that test is found to be 50, the the hardness readings are erratic. Rockwell hardness number will be expressed as RE50: The hardness test employing the Brale indenter under a The hardness test employing 1=16 in:-diameter steel ball total load of 150 kg is the Rockwell C scale that is the one under a total load of 100 kg is the Rockwell B scale that is most commonly used for materials harder than RB100: The normally applied for testing materials of medium hardness hardness of the hardest steel is about RC70: The useful range of this scale is from Rockwell C 20 to about Rockwell C 70 or slightly higher. The Brale penetrator should never be

136 3 Hardness Table 3.5 Standard Rockwell hardness scales and prefix lettersa Scale symbol Indenter Total static Dial Typical applications of scales and working ranges or prefix load (kg) numerals letter Bb 1 in:-diameter ball 100 (major Red Copper alloys, annealed low- and medium-carbon content steels, malleable 16 iron, aluminium alloys, rolled sheet stock, etc. Useful range from B0 to Cb load = 90 kg) B100. At higher hardness, ball deforms and becomes less sensitive to small changes in hardness Brale 150 (major Black Materials harder than B100; such as fully hardened steel, quenched and load = 140 kg) tempered steel, hard cast iron, pearlitlc malleable iron, deep case-hardened steel. Useful range from C20 to about C70 A Brale 60 (major Black Cemented carbides, thin shallow case-hardened steel surfaces, hard thin materials, etc. Also used under circumstances where the diamond may chip load = 50 kg) or be otherwise damaged if used under 150 kg total load D Brale 100 (major Black Thin steel, medium case-hardened steel surfaces. Useful for applications where major load is desired intermediate between those required for A and load = 90 kg) C scales E 1 in:-diameter ball 100 (major Red Cast iron, aluminium and magnesium alloys, bearing alloys. Useful for 8 load = 90 kg) measuring hardness of very soft materials (lower than RB0) F 1 in:-diameter ball 60 (major Red Annealed copper alloys, thin soft sheet metals 16 load = 50 kg) G 1 in:-diameter ball 150 (major Red Phosphor bronze, beryllium copper, malleable iron. Useful for some 16 load = 140 kg) materials slightly harder than RB100 H 1 in:-diameter ball 60 (major Red Aluminium, lead, zinc 8 load = 50 kg) K 1 in:-diameter ball 150 (major Red Bearing metal and other very soft or thin materials 8 load = 140 kg) L 1 in:-diameter ball 60 (major Red Useful for special-purpose testing of very soft materials 4 load = 50 kg) Use smallest ball and heaviest load that does not give anvil effect M 1 in:-diameter ball 100 (major Red 4 load = 90 kg) P 1 in:-diameter ball 150 (major Red 4 load = 140 kg) R 1 in:-diameter ball 60 (major Red 2 load = 50 kg) S 1 in:-diameter ball 100 (major Red 2 load = 90 kg) V 1 in:-diameter ball 150 (major Red 2 load = 140 kg) Minor load = 10 kg aBased on ASTM E18 bCommonly used scales and indenters employed to measure the hardness below RC20; which is than at the apex of the cone. So, the test employing Rock- equivalent to RB97; due to the following reason: well C scale would introduce errors in the hardness values of materials having hardness lower than RC20: As the manufacturing of the diamond indenter in the shape of a cone with an apex angle of 120° having a 3.8.6 Method of Testing spherical tip of 0.2 mm radius is quite costly, the manu- facturer does not care to check the accuracy of the indenter Initially, the specimen to be tested is made flat by grinding during its shaping at penetration depths greater than that and then roughly polished because any surface irregularities equivalent to a hardness value of RC20; because the hardness will be taken care of by the minor load. The application of of materials lower than RC20 can be obtained with the the minor load becomes effective when the surface of the Rockwell B scale. Moreover, during the manufacturing specimen kept on the anvil is brought in contact with the process if there is any inaccuracy in the shaping of the diamond indenter, it will be more pronounced at the base

3.8 Rockwell Hardness 137 indenter by rotating the anvil elevating wheel. By further hardness reading. Upon withdrawal of the major load, the elevating the specimen, the indenter is forced into the large pointer on the dial face moves in the clockwise direction specimen and the minor load of 10 kg is slowly applied. before it will come to a rest position and the small pointer After raising the specimen sufficiently high when the small moves in the anticlockwise direction, which is shown in pointer, located in the upper left-hand quadrant of the dial Fig. 3.7. The displacement of the large pointer occurring face, becomes coincident with an index mark, usually a red during the elastic recovery of the test piece is used to measure mark, it indicates that the minor load of 10 kg has been fully the elastic recovery of the metal under test. For example, if the applied. During the minor load application, the large pointer clockwise movement of the large pointer on the dial face of the dial will rotate and after the minor load is fully during the elastic recovery of the test piece is from C32 or applied, the large pointer will stop in a near vertical position. B62 (Fig. 3.6) to C47 or B77 (Fig. 3.7), the depth recovered After full application of the minor load, the large pointer elastically on the release of the major load will be, respec- must coincide with the set-point mark of the dial scale, tively, 0:002 mm  ð47 À 32Þ ¼ 0:03 mm; or 0:002 mm  otherwise to achieve it the bezel of the dial has to be rotated. ð77 À 62Þ ¼ 0:03 mm; since each division of Rockwell Figure 3.5 shows positions of the large and small pointers on hardness number is equivalent to an indentation depth of Rockwell dial, when the minor load is fully applied but the 0.002 mm. However, upon withdrawal of the applied major major load is yet to be applied. load, Rockwell hardness number is read from the rest posi- tion of the large pointer on the scale of the dial while the Under the application of the major load, the vertical minor load is still in position. According to Fig. 3.7, the motion of the indenter into the test piece produces a rotational Rockwell hardness value is 47 for cone indenter and 77 for motion of the large pointer on the face of the Rockwell dial in ball indenter. The method of Rockwell hardness testing the anticlockwise direction from the set-point mark and the operation has been illustrated in Fig. 3.8. small pointer in clockwise direction from the index mark, which is shown in Fig. 3.6. It is to be noted that the dial face When the large pointer after release of the applied major is inscribed with hardness numbers in the clockwise direction, load coincides with the set-point mark of the dial scale, that is, the scale of the dial is reversed with respect to the which is usually designated by C0 in black scale or B 30 in motion of the large pointer. As soon as the rotational move- red scale, the hardness reading will be either 100 for the ment of the large pointer stops, it indicates that the major load Brale indenter or 130 for the steel ball indenter. Under the has been fully applied. It is specified that the period of full application of the major load, the movement of the large application of the major load should not be more than 2 s. pointer in the anticlockwise direction from the set point of After this specified interval without disturbing the application the dial causes the decrement of hardness number with of minor load, the operating handle should be brought back increment of indentation depth. Upon withdrawal of the immediately to release the major load that will cause elastic major load, if the large pointer rests below the zero reading recovery of the metal under test and does not enter the crossing minimum 100 divisions in black scale or 130 0 0 0 0 0 75 25 75 25 75 25 75 25 75 25 50 50 50 50 50 Minor Major load Major Major load not yet load load not yet applied applied withdrawn applied Minor Minor Minor load load load applied left withdrawn applied Steel ball AA AB indenter B D Test piece Test piece now has CD B a firm seating due C to minor load Elevating screw Fig. 3.8 Method of Rockwell hardness testing operation (schematic). metal under test upon withdrawal of major load. This does not enter the AB = Depth of hole made by minor load. AC = Depth of hole made by hardness reading. BD = Difference in depth, Dh; of holes major load and minor load combined. DC = Elastic recovery of the made = Rockwell hardness number

138 3 Hardness divisions in red scale from the set point of the dial, the h the depth of indentation upon withdrawal of the material is softer showing negative hardness value. Under applied major load with the minor load still applied such circumstances, either smaller major load or/and larger diameter steel ball indenter must be used to reduce the hm the depth due to application of minor load and indentation depth resulting in a positive hardness value. Δh the depth difference between indentation upon When the steel ball indenter is used and the large pointer in the red hardness scale rests at B 0 (a shift of only 30 divi- withdrawal of the applied major load with the minor sions in a counterclockwise direction from the set point of load still applied and indentation made by the minor the dial) after release of the applied major load, then the load ¼ h À hm ¼ ðht À hrÞ À hm ¼ ðht À hmÞ À hr ¼ hardness value must be read as 100. Dht À hr; in which hr ¼ the depth recovered elastically upon withdrawal of the applied major load The depth difference between the indentation upon with- drawal of the applied major load with the minor load still For example, if after application of the major load, the applied and the indentation made by the minor load can be large pointer rotates in an anticlockwise direction from set measured from the number of divisions in a counterclockwise point to 32 on the dial in black scale for Brale indenter or 62 direction from the set point on the dial to the rest position of in red scale for ball indenter, as shown in Fig. 3.6, the depth the large pointer indicating the hardness number. Since the due to application of the major load, Dht ¼ AC À AB (in Rockwell hardness number on the dial face are reversed with Fig. 3.8) will be, respectively, ð100 À 32Þ Â 0:002 mm ¼ respect to the movement of the large pointer on the dial face, 0:136 mm; or ð130 À 62Þ Â 0:002 mm ¼ 0:136 mm: If after a shallow indentation caused by the major load will indicate a withdrawal of the major load, the large pointer rotates in a high Rockwell hardness number, and a deeper indentation clockwise direction from numbers 32 to 47 on the dial in will indicate a correspondingly lower hardness number. The black scale for Brale indenter or 62 to 77 in red scale for ball Rockwell readings before elastic recovery of metal under test indenter, as shown in Fig. 3.7, then the depth difference and the Rockwell hardness numbers for steel ball and Brale Dh ¼ BD (in Fig. 3.8) will be, respectively, ð100 À 47Þ Â indenters are shown by the following expressions: 0:002 mm ¼ 0:106 mm; or ð130 À 77Þ Â 0:002 mm ¼ 0:106 mm: The above difference Dh corresponds to the For the steel ball indenter; Rockwell reading Rockwell hardness number 47 for Brale indenter and 77 for ball indenter. Hence, elastically recovered depth in Fig. 3.8 before elastic recovery of metal ð3:14aÞ is hr ¼ DC ¼ ðAC À ABÞ À BD ¼ 0:136 À 0:106 ¼ 0:30 mm; which is same as the earlier derived value. ¼ 130 À Dht mm ¼ 130 À ðht À hmÞ mm 0:002 mm 0:002 mm However, the maximum depths of penetration for steel ball and Brale indenters are obtained when the Rockwell Rockwell hardness number ¼ 130 À Dh mm hardness number = 0. 0:002 mm Therefore, the maximum depth of penetration for the steel ¼ 130 À ðh À hmÞ mm ball indenter is: 0:002 mm ð130 Â 0:002 þ hmÞ mm ¼ ð0:26 þ hmÞ mm: ð3:14bÞ And the maximum depth of penetration for the Brale For the Brale indenter; Rockwell reading indenter is: before elastic recovery of metal ð3:15aÞ ð100 Â 0:002 þ hmÞ mm ¼ ð0:20 þ hmÞ mm: ¼ 100 À Dht mm ¼ 100 À ðht À hmÞ mm At the beginning of the test, it is always better to check 0:002 mm 0:002 mm the accuracy of Rockwell hardness tester using special test blocks that are available for all ranges of hardness. If the Rockwell hardness number ¼ 100 À Dh mm error of the tester is more than ±2 hardness numbers, 0:002 mm reconditioning of the tester to bring it into proper adjustment is required. Further, when comparing test results, it is nec- ¼ 100 À ðh À hmÞ mm essary to establish the speed and time of application of the 0:002 mm major load. The machine should be so adjusted that the operating handle will complete its travel in 4–5 s under the ð3:15bÞ application of the major load of 90 kg without any specimen on the anvil. It has already been specified that the period of where ht the total depth of indentation upon application of the major and minor loads combined,

3.8 Rockwell Hardness 139 full application of the major load, while indenting specimen, the regular Rockwell. For hardness measurements on thin should not be more than 2 s. sheets of the softer metals, the superficial tester employs the steel ball indenters of the same diameters as used in the 3.8.7 Advantages regular Rockwell test. A Brale indenter, very similar to that used in the regular Rockwell, is employed in the superficial • Its advantage over the Brinell test is that it can measure test for harder materials, such as hardened steel surfaces, the hardness of harder materials that is beyond the scope except that a closer tolerance limit is maintained in the of the Brinell test. shaping of the spherical end of the superficial Brale. • It is faster because arbitrary hardness values can be read Similar to the regular one, the superficial direct-reading directly from the dial of the machine. Due to the rapidity dial face is also inscribed with 100 equal division marks and of the test, it is widely used in industrial work. each division represents one unit in the Rockwell superficial hardness number. The outermost figures inscribed on the left • It differs from the Brinell test in that the indenters and the half of the dial face shown in Figs. 3.5, 3.6 and 3.7 are used loads are smaller, and hence, the resulting indentation is for measurements of hardness and depth of indentation in smaller and shallower, which is less objectionable in the Rockwell superficial hardness test. However, the superficial finished parts. dial differs in that it has only one set of figures from 0 to 100, and each division is equivalent to an indentation depth of • Due to application of minor load, the surface preparation 0.001 mm caused by the vertical motion of the indenter into of the specimen is minimized in comparison to the Bri- the test piece. So the superficial tester has a more sensitive nell as well as Vickers hardness tests, to be discussed in depth-measuring system because of the more shallow Sect. 3.10. Only rough grinding of specimen surface may indentations made by it. Similar to the Rockwell hardness be adequate for Rockwell hardness test. number, the Rockwell superficial reading before elastic recovery and the Rockwell superficial hardness number for 3.9 Rockwell Superficial Hardness the indenters of steel ball and Brale are given by the fol- lowing relation: The Rockwell superficial hardness tester is a special-purpose machine, designed for indentation hardness test under a For the steel ball and the Brale indenter constant static load, where Rockwell superficial reading before elastic recovery • Only very shallow indentation is permissible. • It is desired to know the surface hardness of the ¼ 100 À Dht mm ¼ 100 À ðht À hmÞ mm 0:001 mm 0:001 mm specimen. • It might require light loads for testing for some other ð3:16aÞ reasons. Rockwell superficial hardness number As the depth of indentation is usually less than ¼ 100 À Dh mm ¼ 100 À ðh À hmÞ mm ð3:16bÞ 0.125 mm, the superficial tester is applied particularly to test 0:001 mm 0:001 mm thin sheets of brass, bronze and steel, such as razor blades, and to determine the surface hardness of nitrided steel and All the terms in (3.16) have been defined before in rela- lightly carburized or decarburized steel, etc. tion to (3.14) and (3.15). The accuracy of the superficial tester is checked by means of special test blocks, similar to those used in the regular Rockwell test. 3.9.1 Principle of Operation 3.9.2 Superficial Hardness Scale The principle of operation of the Rockwell superficial The hardness scales for the regular tester cannot be used for hardness tester is the same as that of the regular Rockwell the superficial tester due to difference in the magnitudes of tester, but the superficial tester differs from the regular one their applied loads. So just like the regular Rockwell scales, with respect to hardness scales, the magnitude of applied arbitrary superficial hardness scales have been developed load, sensitiveness of depth-measuring system and numerals depending on the major load and the type of indenter used. of direct-reading dial. The superficial tester applies a minor Since the Brale in the superficial machine is intended load of 3 kg and total loads of 15, 30 or 45 kg, instead of the especially for use on surface of ‘nitrided’ steels and the 10-kg minor load and the 60-, 100- or l50-kg total loads of 1=16 in:-diameter steel ball for testing ‘thin’ sheets of the

140 3 Hardness Table 3.6 Rockwell superficial hardness scalesa Total static Scale symbol or prefix letter load (kg) N scale T scale W scale X scale Y scale Type of N Brale or 1 in:-diameter ball 1 in:-diameter ball 1 in:-diameter ball 1 in:-diameter ball indenter diamond cone 16 8 4 2 15 15 N 15 T 15 W 15 X 15 Y 30 W 30 X 30 Y 30 30 N 30 T 45 W 45 X 45 Y 45 45 N 45 T Minor load = 3 kg aBased on ASTM E 18 softer metals, such as brasses, bronzes and unhardened steel, 3.10 Vickers Hardness the letters N and T have been selected, respectively, for scale designations of the above indenters. So, the particular Brale The Vickers hardness test, used to determine quantitatively used in the superficial tester is designated as the ‘N Brale’. the indentation hardness of material under the application of The standard superficial hardness scales are shown in a constant static load, is a widely accepted method for Table 3.6. research work because it is capable of measuring hardness from very soft materials to extremely hard materials without To report Rockwell superficial hardness number, it is changing the load or indenter unlike the Brinell or Rockwell essential to prefix each dial reading with the appropriate hardness test. So the hardness of different materials can be hardness scale symbol depending on each combination of compared with a continuous scale of Vickers hardness under the indenter used and the magnitude of major load applied a constant load, or if required by varying the applied load, during the test, just as in the case of reporting regular since the Vickers hardness is normally independent of load. Rockwell hardness numbers because the dial reading alone Further, the test can be used to measure the hardness of thin has no meaning. sheets and superficial hardened surfaces. A photograph of one model of Vickers hardness tester is displayed in Fig. 3.9. For example, if the indenter used is 1=8 in: diameter steel ball and the magnitude of total load applied is 30 kg in a test, 3.10.1 Indenters and Loads the Rockwell superficial hardness scale will be ‘30 W’ and if the dial reading in that test is found to be 50, the Rockwell superficial hardness number will be reported as 30 W-50. 3.9.3 Merits and Demerits The standard indenter of the Vickers hardness test is made of diamond in the form of a square-based pyramid with an The superficial test, employing lower values of the applied included angle of 136 between opposite faces. Such design loads than the regular test, forms extremely small and of this Vickers indenter was specified by the British Stan- shallow indentation on the test surface particularly by the dards Institution and accepted universally. A Vickers 1=16 in: diameter ball or the Brale indenter. So, the condi- indenter is shown in Fig. 3.10a, and indentation by Vickers tion of the test surface in the superficial test must be producing square indentation on specimen surface is shown smoother and less irregular than that required in the regular in Fig. 3.10b. test. The superficial test has the following merits and de- merits over the regular hardness test with respect to the The angle of 136 between opposite faces of pyramid condition of the test surface, although the advantages of both shaped indenter was purposely chosen because it approxi- the above tests over other test methods are very much mates the most desirable ratio of indentation diameter,d; to similar. ball indenter diameter, D; i.e. d=D ¼ 0:375 in the Brinell hardness test. If tangents are drawn to the edge of a Brinell • The merit is that the extremely small and shallow indentation, where d=D ¼ 0:375; the included angle indentation is much less objectionable in the finished between the tangents will be 136 (discussed in Sect. 3.6), parts. which is the same as the included angle between opposite faces in the Vickers indenter. • The demerit is that a careful surface preparation of the test piece is required. Surface polishing of the test piece is In some model of Vickers hardness tester, the testing desired for accurate hardness measurement because of facility with steel balls indenters of 1 and 2 mm diameters is formation of the extremely small and shallow indentation. available, which can also be used to determine the Brinell hardness under light load using this above machine.

3.10 Vickers Hardness 141 Indentation Micrometer indentation. For most cases, an appropriate applied load is focusing screw 30 or 50 kg, although it is desirable to apply further higher optical loads for the hardness measurement of a heterogeneous screen Levers structure, like cast iron and certain non-ferrous alloys, if the under average hardness of such structure is desired. cover Timer Start push 3.10.2 Principle of Operation indicator button Diamond Convex lens The basic principles of operation of Vickers hardness test is indenter in front of the same as that of the Brinell hardness test, and the halogen Vickers hardness scale is identical with that of the Brinell. Anvil lamp Because of the precise shape of the Vickers indenter, the Vickers hardness number is frequently called as the dia- Hand wheel Objective mond pyramid hardness number. Similar to the definition of to elevate lens the Brinell hardness number, the diamond pyramid hardness anvil number (DPH or DPN), or the Vickers hardness number Weight (VHN, or VPH), expressed in units of kilograms per square selection millimetre, is defined as the ratio of the applied load, P in knob kilograms, to the surface area of the elastically recovered pyramidal indentation, AS in square millimetres. Like the Brinell hardness, the Vickers hardness can be expressed as follows: Fig. 3.9 One model of a Vickers hardness tester (Courtesy Prof. VHN or DPH ¼ P kgmm2 ð3:17Þ K. Biswas, IIT Kharagpur) AS The loads that can be applied to the indenter in Vickers where, hardness tester are 1, 2.5, 5, 10, 20, 30, 50, 100 and 120 kg through appropriate selection of weights. The largest possi- P the applied load in kg and ble load should be selected for a given test depending on the As the lateral area of elastically recovered pyramidal purpose of the test, the relative hardness of the test piece and its dimension, particularly its thickness, provided that the indentation in the test piece, sq mm image of the indentation is not bigger than the measuring area of the image viewing screen of the instrument. This will The lateral area, AS; is calculated from microscopic reduce the personal error arising from the measurement of measurements of the lengths of the diagonals of the square diagonal lengths of a smaller sized square-shaped base of the pyramidal indentation. Different geometrical parameters of a schematic Vickers indentation are shown in Figs. 3.11a, b. From these figures, we can write Fig. 3.10 a A Vickers indenter. (a) P b Indentation by Vickers producing square indentation on (b) specimen surface Surface of h test piece 136° a D a Square indentation

142 3 Hardness AS ¼ 4  area of each lateral triangular face, Substituting the value of AS from (3.19) into (3.17), a that constitutes1=4 th part of lateral area working formula to determine VHN or DPH is obtained as follows: of the pyramidal indentation  ¼ 4  1  base  altitude of each lateral triangular face P P 2  VHN or DPH ¼ D2=1:854 ¼ 1:854  D2 ð3:20Þ ¼4 1  a  LS ¼ 2aLS where 2 P the applied load in kg and ð3:18Þ D the average diagonal length of square indentation in mm where Accurate measurement of the lengths of two diagonals of the square indentation, D1 and D2; for each indentation a the length of each arm of the square base of the should be taken and the arithmetic average of D1 and D2 pyramidal indentation that is the same as the base of must be put in place of the average diagonal length of the each triangular face, mm and indentation, ‘D’, in the above (3.20) for calculation of DPH. LS the slant height of the pyramidal indentation that In practice to eliminate the calculation of DPH with is same as the altitude of each lateral triangular (3.20), DPH corresponding to the measured average diago- face, mm nal length may be determined directly from the computed table based on any applied load, say P0 kg. For an applied Suppose, D ¼ average diagonal length of the square base load of P kg, DPH may be determined using the above table of the pyramidal indentation, mm. based on P0 kg in the following manner: For a perfectly square indentation, the relation between D and a is given by D2 ¼ 2a2; or; a ¼ pDffiffi : (a) Measure the average diagonal length of the indentation, 2 ‘D’ in mm. As the included angle between opposite faces of the (b) Multiply ‘D’ by P0 and obtain an adjusted diagonal pyramid indentation is 136; so from Fig. 3.11b, the slant length D0 mm. height LS may be expressed as (c) Determine DPHP0 kg corresponding to À pffiffiÁ D0 mm½¼ DðmmÞ Â P0Š; from a table computed for a AC a=2 D= 2 2 load of P0 kg. sin 68 sin 68 sin 68 LS ¼ AO ¼ ¼ ¼ (d) Get the correct value of DPHPkg by multiplying the If the value of LS is substituted in (3.18), the total DPHP0kg [determined in step (c) from the table], with lateral area of the indentation, AS; can be obtained as follows: product of the actual applied load ‘P’ and the table load P0 i.e. with ðP  P0Þ: À pffiffiÁ D2 Justification of the above is as follows: D= 2 2 2 pDffiffi D sin 68 From a table computed for a load of P0 kg, AS ¼ 2a sin 68 ¼ 2 pffiffi sin ¼ 22 68 2 1:854  P0 1:854  P0 kgmm2: ¼ D2 ¼ D2 ð3:19Þ DPHP0kg ¼ D02 ¼ ðD  P0Þ2  0:927 1:854 2 Fig. 3.11 a Vickers indentation (a) (b) B (not to scale), showing different h geometrical parameters. D a b Section ACBOA in a is redrawn A 2C for derivation of Vickers hardness a number B 68° LS C O A h 136º O

3.10 Vickers Hardness 143 For an applied load of P kg, the correct value of Vickers 1000 hardness will be: 800 DPHPkg ¼ 1:854 Â P0 Â ðP Â P0Þ ¼ 1:854 Â P kgmm2: Vickers diamond indenter D2 Â P02 D2 Brinell hardness values 600 3.10.3 VHN Versus BHN The Vickers and Brinell hardness numbers are practically 400 identical up to the hardness value of about 300 kg/mm2, beyond which the Vickers hardness numbers give the true 10 mm steel ball indenter of indication of the hardness values, whereas the Brinell Brinell, with 3000 kg load hardness numbers become lower than the Vickers hardness numbers, as illustrated in Fig. 3.12. For harder materials, the 200 Brinell indenter made of steel suffers from distortion which produces a lower value of Brinell hardness number. The 0 distortion of the steel ball indenter increases with the rise of 0 200 400 600 800 1000 hardness of materials and causes further decrease in BHN. It Vickers hardness values was observed that materials harder than about 600 Brinell would result remarkably lower hardness values when mea- Fig. 3.12 Illustrating approximate divergence of Brinell from Vickers sured with the Brinell than with the Vickers, because of hardness number at high hardness values (Kehl 1949) insignificant amount of deformation of the diamond indenter of the Vickers test performed at the high hardness level. For to inertia effects and sudden application of the entire load. example, the Vickers hardness of 1300 is equivalent to the The load is automatically removed after the preset period of Brinell hardness of about 850. As shown in Fig. 3.12, the full load application which varies between 10 and 30 s, where Vickers hardness curve remains practically straight at all the most common is 20 s. Operation of a foot or hand lever hardness values, whereas at hardness above 300 kg/mm2 the resets the machine. After an indentation has been formed on Brinell hardness curve starts bending to result in a lower the surface under test, the test piece along with its supporting value and it diverges remarkably causing an unacceptable anvil is first lowered by means of a hand wheel and a hardness value above 600 Brinell, where the Vickers test is metallurgical-type microscope attached to the side of the considered to be a reliable hardness measurement. instrument is swung over the indentation formed on the specimen surface. The eyepiece attached to the microscope 3.10.4 Operational Method can be rotated through 90° or more about its principal axis and thus enables to measure both the diagonals of the square The Vickers hardness instrument is semi-automatic in oper- indentation with an accuracy up to 0.001 mm. ation that normally requires care, rather than skill on the part of the operator. Because of the very small indentation nor- In some models of the Vickers hardness tester, the spec- mally produced, it is recommended that the test surface must imen surface is not brought into contact with the indenter but be polished like a metallographic specimen. For conducting a is brought close to the point of the indenter until the image of test with Armstrong-Vickers hardness tester, the polished test the polished surface of the test piece is focused on a viewing piece placed on the anvil is raised by means of a hand wheel screen mounted above the instrument. After the formation of or a screw and the test surface is brought into contact with the an indentation on the surface of the test piece due to appli- indenter. By tripping the starting lever, a loading beam, with cation of the selected load to the indenter for some definite a usual ratio of 20:1, is unlocked and weight attached thereon preset length of time, the test piece is not shifted from its causes to apply the selected load to the indenter for some earlier focused position on the anvil. A magnified square definite preset duration. The preselected load is applied image of the indentation is projected and focused on the slowly and at a decreasing rate for the last part of the loading above viewing screen, which can be rotated through 90° and and thus, eliminates practically the errors that might arise due facilitate to measure both the diagonal lengths of the square indentation. In comparison to the Armstrong-Vickers hard- ness tester, this above model saves time particularly for routine hardness testing in production.

144 3 Hardness 3.10.5 Minimum Thickness of Test Section square indentation become the convex ones. The measured diagonal length of such indentation is found to be lower than As discussed previously in Sect. 3.3, the minimum thickness the true diagonal length of a perfect square indentation and a of the test section should be at least ‘10 Â h;’ where ‘h’ high hardness value would result. This type of behaviour is is the depth of the indentation. For Vickers hardness test, it is frequently observed with cold-worked materials having little recommended that the minimum thickness of the test piece ability to strain-harden. It has been proposed (Crowe and be at least 1.5 times the diagonal length of the indentation, Hinsely 1946) to correct this effect empirically. ‘D’, i.e. at least ‘1:5 Â D’. It can be shown that 1:5 Â D % 10 Â h; which is as follows: 3.10.7 Advantages and Disadvantages With reference to Fig. 3.11b, AC ¼ a=2 ¼ tan 68; or ; a ¼ 2:475 Â h; The Vickers hardness test has the following advantages: CO h 2 • The Vickers hardness test is widely accepted for research pffiffi work because it is capable of measuring hardness from ) Minimum thickness ¼ 1:5 Â Dpffi¼ffi 1:5 Â 2a very soft materials with a DPH of 5 to extremely hard materials with a DPH of 1,500 without changing the load ¼ 1:5 Â 2 Â fð2 Â 2:475Þhg or indenter unlike the Brinell or Rockwell hardness test. So, the hardness of different materials can be compared ¼ 10:5 Â h: with a continuous scale of Vickers hardness under a constant load, or if required by varying the applied load, 3.10.6 Anomalous Behaviour as the Vickers hardness is normally independent of load. The reason for such load-independency is that the A perfect Vickers indenter would form a perfect square impressions made by the pyramid indenter are geomet- indentation, as shown in Fig. 3.10b. However, two types of rically similar, no matter what their sizes are. This is anomalous behaviour similar to those described previously generally found to be the case, except at very light loads, for Brinell indentation can frequently occur with a Vickers usually less than 300 g. indenter because of localized deformation of the material at the indentation, as illustrated in Figs. 3.13a, b. The sketch in • The test is rapid than the Brinell test. Fig. 3.13a is the ‘pincushion indentation’ which results due • The test provides accurate results for hardness as high as to ‘sinking in’ or formation of depressed surface of the material around the flat faces of the Vickers indenter. Hence, 1300 Vickers. the straight arms of the square indentation are changed to the • The Vickers test may be performed on the finished parts, concave ones. The measured diagonal length of such inden- tation is greater than the true diagonal length of a perfect because compared to the Brinell indentation Vickers square indentation and would result in a low hardness value. indentation is smaller and shallower, which is less This type of behaviour is commonly found in annealed objectionable in the finished parts. materials having a high rate of strain hardening. The • The test can be applied for measuring hardness on test ‘barrel-shaped indentation’, as shown in Fig. 3.13b, is the piece as thin as 0.15 mm as well as hardness of super- result of ‘ridging’ or ‘piling up’ of the material around the flat ficial hardened surfaces. faces of the Vickers indenter, where the straight arms of the • A much more accurate reading can be made of the diagonal measurement of a square indentation in this test (a) (b) than can be made of the diameter measurement of a circular indentation in the Brinell test, because the mea- surement must be made between two tangents to the circle in a Brinell impression. In spite of these advantages, the Vickers hardness test is not widely accepted for routine hardness testing in produc- tion because of its following disadvantages: Fig. 3.13 Anomalous behaviour with a Vickers indenter. a Pincushion • The test is slower than the Rockwell test. indentation due to sinking in; b Barreled indentation due to ridging or • Careful surface preparation of the test piece is required. pilling up Surface polishing of the test piece is desired for accurate hardness measurement because of the very small size indentation normally produced.

3.10 Vickers Hardness 145 • Chance for personal error in the measurement of the with base rhombic in shape. It has an included longitudinal diagonal length of Vickers indentation is involved, which angle, opposite to the longer or longitudinal diagonal of the is absent in the Rockwell test. base, of 172300 and an included transverse angle, opposite to the shorter or transverse diagonal of the base, of 13000: 3.11 Microhardness (Knoop Hardness) The indentation produced by the Knoop indenter, when viewed normal to the specimen surface as shown in To determine quantitatively the indentation hardness of Fig. 3.14b, is rhombic in shape with the longitudinal and materials over a very small area under the application of a transverse diagonals being normal to each other and in the constant static load, the National Bureau of Standards approximate ratio of 7:1; as shown in the following (3.24), introduced a diamond indenter known as the ‘Knoop’ that results in a state of plane strain in the deformed region. indenter (Knoop et al. 1939; Natl. Bur. 1946) that would produce an indentation smaller than that produced by the The Tukon tester is capable of applying loads down to Vickers indenter and developed the Tukon tester for the 25 g, and depending on the different models of the instru- purpose of microhardness testing that may be used as a ment, the upper limit of the load applied to the indenter may routine laboratory procedure. Although the Tukon tester is vary from 10 kg to even 50 kg, to extend the measurement normally supplied with the Knoop indenter, it can easily be of macro-Vickers hardness at or above 1 kg load with the used with the standard Vickers 136 diamond pyramid use of the standard Vickers indenter. Various microhardness indenter. Hence, Knoop as well as Vickers indenter, both can testing machines other than the Tukon tester, such as Leco be used for microhardness measurements. In this section, microhardness tester, Buehler Micromet of various models, measurement of microhardness with Knoop indenter, i.e. are nowadays available in market. The available loads in Knoop hardness has been considered. some microhardness testers are from 10 g to 1 kg or more, and some microhardness testers can apply loads as low as 1 The Tukon tester, attached permanently with a gm. A photograph of one model of microhardness tester is metallurgical-type microscope, is fully automatic in making displayed in Fig. 3.15. the indentation. The operator selects the point of indentation on the test piece under high microscopic magnification, Generally, the applied load should produce an indentation places the selected area to a position directly below the as large as possible provided that the image of the indenta- indenter and applies the selected load for an effective period tion is not bigger than the measuring area of the image of about 20 s and finally relocates the specimen under the viewing screen of the instrument. This will reduce the per- microscope after release of the load for reading the length of sonal error arising from the measurement of diagonal length the diagonal of the indentation from which the Knoop of a smaller sized rhombic-shaped indentation. An appro- hardness number is calculated. priate load is selected by trial depending on the purpose of the test, the relative hardness of the test piece and its 3.11.1 Penetrators and Loads dimension, particularly its thickness. The depth of the indentation need not exceed about 1 micron that can be The Knoop indenter, which is shown schematically in achieved by appropriate selection of loads. Note that the Fig. 3.14a, is made of diamond in the form of a pyramid depth of Knoop indentation is: h ffi l=30; as shown in the following (3.23), where l is the length of longitudinal or longer diagonal of the indentation. In microhardness tests, the most commonly used loads suitable for material layers that are thicker than about 3 mm range from 100 to 500 gmf. Fig. 3.14 a Schematic shape (a) (b) (not to scale) of the Knoop indenter, showing included l w longitudinal and transverse angles to be, respectively, 172 300 and 130 00: b Associated shape of Knoop indentation on surface of test piece 172.5º h 130º l = 7.114w h≅ l 30

146 3 Hardness Fig. 3.15 One model of a Load selector knob Microhardness tester (Courtesy Prof. K. Biswas, IIT Kharagpur) Ocular Halogen Indentation lamp value input house switch Measurement Objective knob lens Indenter Cross-travel Up / down stage handle Loading Display start unit switch Levelling screws 3.11.2 Principle of Operation ratio of the longitudinal to the transverse diagonal is about 7:1, as shown in the following (3.24), which causes the The Knoop hardness number (KHN), expressed in units of elastic recovery of the projected indentation to occur mainly kilograms per square millimetre, is defined as the ratio of the along the transverse or shorter diagonal, rather than along applied load, P in kilograms, to the elastically unrecovered the longitudinal or longer diagonal. As the elastically unre- projected area of the indentation, Aup in square millimetres. covered projected area of the indentation, Aup; is more pre- That is, cisely related to the elastically unrecovered longitudinal or longer diagonal length, l, than to the elastically recovered Knoop hardness number or KHN ¼ P Kg=mm2 ð3:21Þ diagonal length, w, it is required to measure l and not w. So it Aup is necessary to express Aup in terms of l only as follows, so that the Knoop hardness number or KHN can be determined where in terms of l only. With reference to Fig. 3.16, we get P applied load in kg and The depth of Knoop indentation, Aup projected area of elastically unrecovered indentation h ¼ l=2 ¼ w=2 in the test piece, sq mm tanð172300=2Þ tanð13000=2Þ It must be noted that for the Knoop hardness, the area of the indentation is the elastically unrecovered projected area Or, h ¼ l=2 ¼ w=2 or; h ¼ l ¼ w 15:257 2:1445 30:514 4:289 and neither the elastically recovered projected area as in the Meyer hardness nor the elastically recovered surface area as ð3:23Þ in the Vickers and Brinell hardness tests. The area Aup for From (3.23) Knoop hardness is determined from the measurement of elastically unrecovered longitudinal or longer diagonal length l ¼ 30:514  w ¼ 7:114  w ð3:24Þ 4:289 of Knoop indentation after the release of the applied load. For rhombic base of Knoop indentation, as shown in Or, Fig. 3.14b, suppose w ¼ length of transverse or shorter diagonal of the base, mm, and l ¼ length of longitudinal or 1 7:114 longer diagonal of the base, mm. w ¼  l ¼ 0:14056  l ð3:25Þ As l and w are normal to each other,  Substituting the value of w in terms of l from (3.25) in 1Âw l ¼ wl (3.22), the elastically unrecovered projected area of the ) Aup ¼ 4  222 2 ð3:22Þ indentation will be expressed as It has been mentioned earlier that there is always some Aup ¼ ð0:14056  lÞl ¼ 0:07028  l2 ¼ Cpl2 ð3:26Þ elastic recovery of the indentation after the release of the 2 applied load. Owing to the shape of the Knoop indenter the

3.11 Microhardness (Knoop Hardness) 147 (a) (b) l w 2 2 86.25º h 65º Fig. 3.16 Vertical sections of Knoop indentation along a the longitudinal diagonal and b the transverse diagonal of the rhombic base where Cp ¼ 0:07028 ¼ theoretical Knoop indenter constant place the rhombic Knoop indentations much closer relating the longer diagonal l to the elastically unrecovered together than the square Vickers indentations. projected area, Aup: • Hardness of very hard and brittle materials, such as glass, porcelain, metallic carbides, because in a brittle Combining (3.21) and (3.26), we get the working equa- material the tendency for fracture is proportional to the tion for the Knoop hardness number or KHN as follows: volume of the stressed material, i.e. the volume of the zone of deformation, which is much less for a Knoop Knoop hardness number or KHN indentation. • Hardness on the finished parts because smaller and ¼ P ¼ P kgmm2 ð3:27Þ shallower Knoop indentation is less objectionable in the Cpl2 0:07028 finished parts. Â l2 Similar to the Vickers test, the other advantages of the Equation (3.27) is applicable for an indentation formed by Knoop test are: a theoretically perfect Knoop indenter. The equation for KHN is also valid for an acceptable Knoop indenter, which has an • The Knoop test is rapid than the Brinell test. indenter constant within 1%of the theoretical value that is the • A much more accurate reading can be made of the constant value must be within the range of Cp Æ 1% Â Cp: diagonal measurement of a rhombic indentation in this Hardness measurements can be made on etched, as well test than can be made of the diameter measurement of a as unetched specimens and also on smaller sized objects that circular indentation in the Brinell test. are mounted in conventional metallographic plastics. The disadvantages of the Knoop hardness test are: 3.11.3 Advantages and Disadvantages • The test is slower than the Rockwell test. For the same diagonal lengths, as the depth and projected • Metallographic polishing of the surface of the test piece area of a Knoop indentation are, respectively, 23 and 14% of those of a Vickers indentation, (derived in Prob. 3.18.6), the is essential to make the surface free from defects and volume of the deformed zone will be lower in the Knoop scratches for accurate measurement of hardness because hardness test than in the Vickers hardness test. So, compared the indentation formed by the Knoop indenter, particu- to the Vickers test, the Knoop test is advantageous for larly under very light loads, is extremely small in size. measurement of • Chance for personal error in the measurement of the diagonal length of the very small-sized Knoop indenta- • Hardness over a very small area, such as (a) individual tion is involved, which is absent in the Rockwell test. As constituents of a microstructure, i.e. structural phases in the size of the indentation decreases, the chance for alloyed metals; (b) microscopic areas of segregation; erroneous measurement increases. (c) delicate watch gear; (d) single crystals; (e) tips of • Careful grinding of the back side of the specimen or cutting tools; (f) small wires. specimen mount and accurate checking for parallelism with the prepared surface by means of a measuring • Hardness of very thin sheet materials and of surface micrometer may be required in order to place the speci- layers, such as (a) thin electroplated materials; (b) car- men surface normal to the vertical axis of the indenter burized, decarburized and nitrided surfaces. and to avoid errors in the final hardness number. • A steep hardness gradient at a carburized surface or adjacent to a critical surface, because it is possible to

148 3 Hardness 3.12 Monotron Hardness load. The Monotron hardness is same as the pressure, having units of kilograms per square millimetre, necessary to give a The Monotron hardness test also operates on the indentation fixed indentation depth of 0.0018 in. (0.04572 mm). principle; but in this test the depth of the indentation is fixed or predetermined under the application of variable static Monotron hardness number ¼ P kgmm2 loads during hardness measurements of different materials, Aup whereas different sizes of the indentations are formed under kgmm2 an applied constant load in other hardness tests, discussed ¼ 4P ð3:28Þ previously. pdu2p The Monotron hardness tester is a constant-depth where direct-reading pressure instrument, in which the specimen is placed on the anvil and brought into contact with the P applied load in kg, indenter. The load is applied to the indenter for penetration into the test piece up to a fixed depth by means of a side arm, Aup elastically unrecovered projected area of indentation where it is difficult to control the load precisely. Thereafter, in the test piece, mm2, and the hardness is read from a dial attached to this devise. dup diameter of indentation prior to release of the applied 3.12.1 Indenters and Hardness Scales load, mm. The standard Monotron indenter is made of diamond in the There is a hardness registering dial, which is inscribed shape of hemisphere with a diameter of 0.75 mm. When the with an inner set and an outer set of hardness numbers standard depth of indentation is maintained, the hardness increasing in a clockwise direction. The inner set is equally measurements obtained with this standard Monotron inden- divided into 160 divisions, and each division represents a ter are referred to M-1 scale. The term ‘Monotron hardness’ pressure of 1 kg per mm2, i.e. one unit in the Monotron indicates the hardness values with reference to the M-1 scale. hardness number. So the Monotron hardness number in M-1 There are three more scales; they are M-2, M-3 and M-4. scale can be read directly from the hardness registering dial When hemispherical diamond tip of 1 mm diameter is used in terms of kilograms per square millimetre while the load is as an indenter it is referred to as M-2 scale but it is seldom still applied to the indenter. The outer set is numbered used because the M-l scale normally meets the purpose that directly in Brinell hardness numbers, which may be read would be served by the M-2 scale. When soft materials are directly from this dial only when the standard test conditions tested with the standard indenter of 0.75 mm diameter, the are maintained. small differences in their hardness values are difficult to determine. So for testing on soft materials, the standard In M-1 scale, as dup = 0.36 mm, so the applied load will indenter is replaced by larger sized indenters made of be: tungsten carbide that are available in diameters of either 1.53 or 2.5 mm and the hardness values obtained with these PðkgÞ ¼ p ð0:36 mmÞ2ÂMonotron Hardness NumberÀkgmm2 Á indenters are referred to as M-3 and M-4 scales, respectively. ; 4 3.12.2 Principle of Operation read from the dial: Depending on the hardness of the material under test, such a load is applied to force the indenter into the test piece that the Therefore, in the standard test with M-1 scale, the load indenter penetrates up to a fixed indentation depth of 0.0018 required to force the standard penetrator to a depth of 0.0018 in. (0.04572 mm) under standard testing condition. This in. can be calculated from the reading indicated on the standard depth corresponds to an indentation depth of 6% of hardness registering dial. the 0.75-mm-diameter diamond indenter and produces an indentation with diameter of 0.36 mm (derived in Prob. There is also an indentation depth indicating dial, which 3.18.7). The Monotron hardness number is defined as the consists of an inner set and an outer set of number scales. Each ratio of the applied load to the projected area of the elastically scale is divided into 100 equal numbered divisions, and each ‘unrecovered’ indentation, prior to the release of applied division represents an indentation depth of 0.0002 in. (0.005 mm) caused by the vertical motion of the indenter into the test piece. The number of the inner set scale increases in a clockwise direction and is utilized for standard constant-depth Monotron hardness testing, whereas the number of the outer set scale increases in an anticlockwise direction and is used to measure the hardness in terms of flow and ductility under selected constant loads. Resetting of the dial indicator to zero value is not necessary at the beginning of each test due to presence of twin indicators that help to get readings on the dial. Releasing of the applied load causes elastic recovery of

3.12 Monotron Hardness 149 the metal under test during which the pointer on the dial face • The method of partial penetration, as described above, moves in the counterclockwise direction towards zero number may be used where it is desired to have a smaller sized before the pointer will come to a rest position. Similar to that indentation. of the Rockwell tester, the displacement of the pointer on the dial face can be used to measure the elastic recovery of the test • The test surface either in a prepared or an unprepared piece and the number in the dial scale corresponding to the condition may be used for testing with any of the rest position of the pointer can be used to measure the depth of indenters mentioned above. permanent indentation. • The test is suitable to use over the entire hardness range An unprepared surface of the material, consisting of a of materials. scaly, rough or decarburized layer, may be used for testing if a prepressure is applied to penetrate the indenter through this The disadvantages of the Monotron hardness test are: unwanted surface layer. To apply a prepressure, the pointer of the hardness registering dial is first set at 10 or 20 or more • Indentations of exactly the same sizes are difficult to divisions below the zero mark. Then such a load is applied produce with the same indenter on the same material that the pointer moves to the zero mark and at that instant the because the depth indicating dial is not so sensitive. pointer of the depth indicating dial is set at zero value. Now the test is carried out as usual. • It is difficult to control the applied load precisely in this device. The standard depth of indentation in thin hard cases may produce deeper penetrations and result in lower hardness 3.13 Shore Scleroscope Hardness values, due to the presence of the underlying softer base metal. In such cases, it is desired to have a shallower The most common example of a dynamic hardness tester is indentation. Hence to determine the hardness on thin car- the Shore scleroscope, invented by A. F. Shore in 1907, that burized or nitrided surfaces, a method of partial penetration is, probably at the present time, the most widely used has been recommended. In this method, such a load is dynamic hardness measurement device. It operates on the applied that the indenter penetrates into the test piece up to a indentation principle to determine quantitatively the hard- fraction of the standard depth of indentation and the standard ness of materials under the application of a constant dynamic hardness value is obtained by multiplying the pressure load. In the Shore scleroscope tester, a diamond-tipped reading with the reciprocal of the fractional value. For hammer, acting as an indenter, is usually dropped by its own example, if one-half or one-third or one-fourth of the stan- weight from a definite height onto the surface of the test dard depth is allowed to penetrate, the standard hardness piece, and the hardness is measured by noting the height to value is obtained by multiplying the pressure reading, which the indenter rebounds from the surface, i.e. in terms of respectively, with 2 or 3 or 4. the rebound height of the indenter. As the height of rebound, rather than the size or depth of indentation, is taken as the Hardness test of wood is also a constant-depth test, measure of hardness, the hardness measured by this instru- similar to Monotron hardness test. In wood-hardness testing, ment is often called as ‘rebound hardness’, which is pro- a steel ball of 0.444 in. (11.28 mm) diameter is forced to portional to the rebound height of the indenter. It should be penetrate into the wood up to a fixed depth, which is equal to noted that the scleroscope hardness value is a purely arbi- half of the ball diameter. The load required for this pene- trary number having no unit, just like the Rockwell hardness tration is reported as the hardness of wood. This hardness number. This hardness test is best used on finished surfaces value is used for comparison purposes only. where large permanent indentations cannot be tolerated, such as surface of a hardened steel roll. 3.12.3 Advantages and Disadvantages There are two types of scleroscope instruments: The Monotron hardness test has the following advantages: (a) ‘Vertical Scale’ type • The test is quite rapid in its operation, like the Rockwell hardness test. The instrument consists of a drop tube, usually made of glass, through which a small pointed hammer indenter is allowed to • The instrument is well suited to determine the hardness fall downward from a height of 10 in. (254 mm) on the sur- of thin materials or case-hardened surfaces. face of the specimen. After forming an indentation on the test surface, the hammer rebounds upward and Scleroscope • The indentation mark is almost entirely invisible to the hardness number is expressed by the rebound height of the unaided eye and so it is not objectionable for most fin- hammer. The drop tube is accurately aligned in a vertical ished surfaces because the standard depth of indentation produces an indentation of 0.36 mm in diameter.

150 3 Hardness position that is ensured by plumb rod provide in the side of the lesser will be the energy absorbed in forming the smaller sized tester, otherwise hammer might strike the drop tube and cause indentation. So the greater amount of energy will be available a lower hardness reading due to loss of energy in the collision. for the indenter to rebound that will result in the higher The standard hammer is a cylindrical plug with tapering at one rebound height of the indenter and the corresponding higher end and weighs approximately 2.6 g (Kehl 1949). The hardness of the material. For example, the energy absorbed for cylindrical plug is approximately 1=4 in: (6.35 mm) in plastic deformation in lead is 98% of the total energy available diameter and 3=4 in: (19.05 mm) long, and its tapered end is from the striking indenter in comparison to only 20% in the fitted with a ground and polished diamond striking tip case of martensitic high-carbon steel (Kehl 1949). Hence, the rounded to a 0.01-in. (0.254 mm) radius (Davis et al. 1964). rebound height of striking indenter for martensitic The diamond tip, serving as the actual indenter, prevents high-carbon steel is higher than that for lead, showing that the distortion of the hammer end while striking the test piece. As former has higher hardness than the latter. It must be noted that the rebound height of the regular hammer is limited when in addition to the resistance to permanent deformation, the testing soft materials, small differences in their hardness damping capacity of the material is an important factor on values are difficult to detect immediately. So for use on soft which the rebound height of the indenter will depend. Apart materials, the regular hammer may be replaced by a magnifier from energy expended for permanent deformation, the rest of hammer, having a larger point area than the standard ham- the energy is not fully utilized in the rebound of the hammer in mer. The magnifier hammer gives a higher rebound height, those materials having high damping capacity because some thus magnifying small but significant variations in hardness. energy is lost in the form of internal friction in such materials. So the scleroscope hardness numbers between two dissimilar The indenter is lifted to the top position of the tube by an materials such as rubber having high and steel having low air-suction device and held there by means of a mechanical damping capacity cannot be compared. Obviously, rebound catch. The indenter is released from the mechanical catch by hardness can be compared only in case of similar materials. If air pressure, developed from squeezing a rubber air bulb, Shore hardness measurements are carried out only for mate- and allowed to fall on the test surface whenever required for rials of low damping capacity, the loss of energy due to hardness testing. The Shore hardness scale, to be discussed internal friction will be negligible and the rebound hardness subsequently, is inscribed within the drop tube, and the will become virtually an indentation hardness measurement as hardness value is read from the scale by noting the top of the the rebound height would depend only on the energy absorbed first rebound height of the hammer indenter. in forming the indentation. (b) ‘Dial recording’ type The rebound height of the hammer from fully hardened high-carbon steel usually varies between Shore scleroscope The principle and the energy relationship of the dial numbers of 95 and 105. So the average rebound height of recording instrument is the same as those of the 100, being equivalent to hardness of martensitic high-carbon vertical-scale-type tester. However, they differ from each steel, is divided into 100 equal divisions on the Shore other with respect to the fall heights and weights of the hardness scale. The area of contact between the indenter and indenters, hardness measuring systems, their appearance and the test piece for martensitic high-carbon steel is only about design. The hammer indenter weighs approximately 37 g 0.0004 sq. in. (0.258 mm2), and the stress exerted on the and the distance through which it falls is only about 0.75 in. surface of the test piece momentarily exceeds 4 Â 105 psi (Kehl 1949). The rebound height of the indenter in the dial (2758 MPa) (Davis et al. 1964). In spite of the lightness of recording type is automatically recorded on a dial attached to the indenter, the stresses developed in all cases are suffi- the top of the instrument. The dial is calibrated directly in ciently high to overcome the resistance offered by the sur- scleroscope hardness numbers, and the dial hand moves to a face of the hardest engineering materials. Most often the certain hardness value and rests there depending on the scleroscope hardness scale is extended to 140 divisions to rebound height of the indenter until it is reset. make the hardness measurement possible on very hard and brittle materials, such as glass, porcelain, metallic carbides. 3.13.1 Principle of Testing When the hammer indenter falls from the definite height its 3.13.2 Mass Effect of Test Piece potential energy is transformed into kinetic energy, which is consumed partly for plastic deformation, i.e. in the formation The mass of the test piece should be such that it can over- of the indentation and partly in the rebound of the hammer. In come inertia effects or shock of the striking hammer. It has general, the harder the test piece, the lesser will be the plastic been seen that a cubic-shaped block of hardened steel deformation caused by the striking indenter and as a result the weighing about 0.45 kg is sufficient to overcome inertia

3.13 Shore Scleroscope Hardness 151 effects. As the mass decreases, the amount of energy avail- values will be somewhat lower than that obtained by using able for rebound of the hammer also decreases due to the the more accurate table-type standard hardness testers. shock of the striking hammer on the test piece. This causes to decrease the rebound height of the hammer and thereby The Poldi impact hardness tester is probably the cheapest Shore hardness. Hence, the mass of test piece influences the and the lightest indentation hardness measuring instrument. reproducibility of hardness values. Further, it has been seen Brinell hardness numbers and tensile strengths of steel and that the decrease in hardness with decrease in mass is more BHN of other various non-ferrous metals can be found from pronounced in soft metals than in hard metals. For example, the tables of a booklet supplied with this device but neither if the mass is reduced from 0.45 to 0.225 kg or to 0.028 kg, the standard Brinell hardness tester nor the tensile testing the rebound height of the hammer will be reduced by machine can be replaced by the Poldi tester. approximately 1 and 20%, respectively, for the hardened steel and by about 2 and 40%, respectively, for a metal The instrument consists of a plunger pressed by a spring whose hardness is 1=5 of the hardness of the hardened steel within a cylinder, and a square slot is provided beneath the (Kehl 1949). If the mass of a specimen is not sufficient, the plunger. A steel ball indenter of 10 mm in diameter is fixed specimen should be appropriately clamped between the anvil in such a way at the base of the slot on the diametrically and the drop tube of the instrument to avoid inertia effects. If opposite side of the bottom face of the plunger that half the test piece is properly clamped to the anvil, an audible portion of the ball indenter remains inside the space of the dull sound will be heard when the specimen (even when in slot and half portion is projected outside and below the slot. the form of sheet material) is struck by the falling hammer A standard test bar made of steel with square cross-section during the test. A similar sound would be produced on having dimensions of 10 mm  10 mm and usually 6 in. in striking the anvil directly by the hammer. If the specimen is length is supplied with this device. The standard test bar is not appropriately clamped, a higher pitched (shrill) sound tapered at one end to facilitate its insertion into the space of will be heard when the hammer strikes the specimen. Hence, the slot between the ball indenter and the plunger. The it is essential to listen carefully the sound produced during dimensions of the slot will be such that the square the test when the test surface is struck by the falling hammer. cross-section of the standard test bar is just fitted into it, If a shrill sound is noted during the test, clamping is not where the top and bottom surfaces of the standard test bar adequate, which must be rectified and the test has to be touch, respectively, the bottom face of the plunger and the repeated until the characteristic dull sound is heard. top point of the ball indenter. Normally, the hardness value of the standard test bar in BHN along with its tensile strength value is supplied by the manufacturer. 3.13.3 Advantages 3.14.1 Principle of Testing • The indentations, produced even on a soft surface, are so The tapered end of the standard test bar is inserted into the minute that they do not become objectionable for a fin- space of the slot between the ball indenter and the plunger, ished surface. For example, the depths of penetration on and it is pushed further inside the slot beyond the tapered specimen surfaces are 0.025 mm for mild steel and about end such that a clean surface of the bar touches the top point 0.013 mm for hardened tool steel (Vander Voort 1999). of the fixed ball indenter. The standard bar will thus be firmly gripped between the ball indenter and the plunger. • Each variety of the instruments can be used as a portable The surface of the test piece is properly ground or filed and hardness tester when the scleroscope unit is detached polished. The tester along with the inserted standard test bar from its base. is placed vertically on the prepared surface of the test piece so that it is touched by the ball indenter. Now, the top end of • The hardness testing is quite rapid. the plunger is struck with a hammer that applies an arbitrary • Generally, the determination of hardness on round or unknown load P on the indenter to force it to penetrate into the surfaces of the standard test bar as well as the test piece. irregularly shaped surfaces does not involve any error, Obviously, there will be two indentations—one on the sur- unless the hammer, on striking the test surface, rebounds face of the standard test and another on the surface of test to one side. piece. The striking with the hammer should be as far as possible vertical and not be such hard that the diameter of 3.14 Poldi Impact Hardness indentation on the standard bar or on the test piece crosses 4.2 mm. The diameters of these two indentations are mea- The Poldi test provides a method for quantitative determi- sured accurately with a low-power microscope. Two read- nation of the indentation hardness of materials under impact ings of the diameter of each nearly circular indentation at (dynamic) loading, although the accuracy of the hardness

152 3 Hardness right angles should usually be taken and their arithmetic Brinell hardness number of the test specimen; BHpNTffiffiffi¼ffiffiffiffiffiAffiAffiffiTffiSffiffiffiBffiiHNS average must be used to determine the hardness of the test h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h piece as discussed below. If a non-circular indentation forms on the surface of test piece, the indentation diameters will be fðpDÞ=2g D À pffiDffiffiffi2ffiffiffiÀffiffiffiffiffidffiffiS2ffiffii D À pffiDffiffiffi2ffiffiffiÀffiffiffiffiffidffiffiS2ffiffii measured in four directions, roughly 45° apart and their h h arithmetic average must be considered to obtain the average ¼ BHNS ¼ BHNS hardness of the material. fðpDÞ=2g D À D2 À dT2 D À D2 À dT2 Suppose, the hardness of the standard test bar is HS and ð3:31aÞ that of the test piece is HT and the areas of the indentations on the surfaces of the standard test bar and the test piece are, Similarly, the Meyer hardness of the specimen can be respectively, AS and AT. As the same load P is applied to the standard test bar and the test piece through the same determined in the following way, if AS and AT are considered indenter, so, as the projected areas of the two indentations and the value of HS is known or determined in the Meyer hardness num- ber. Assuming HT ¼ ðpmÞT and HS ¼ ðpmÞS; Meyer hard- ness number of the test specimen is: HS ¼ P and HT ¼ P ðpmÞT¼ AS ðpmÞS¼ À Á ðpmÞS¼  2 ð3:31bÞ AS AT AT pdS2 4 dS ðpmÞS ðpdT2 Þ 4 dT ð3:29Þ HT P=AT ¼ AS ; AS ) HS ¼ P=AS AT or; HT ¼ AT HS If the standard test bar is not available or lost, a steel bar with square cross-section having correct dimensions of If the areas of the two indentations, AS and AT, are cal- 10 mm  10 mm can be used as a standard test bar provided culated from the average values of the measured diameters the hardness of the bar HS in BHN or in Meyer hardness is of indentations on the standard bar and on the test piece, the determined by using the table-type standard Brinell hardness hardness of the test piece, HT, can be determined as the tester. hardness of the standard test bar, HS, is known or supplied to us. Let, 3.14.2 Use of Supplied Table to Determine BHN dS the average value of the measured diameters of The supplied table is used to know the hardness value of the indentation on the standard bar, standard test bar HS in BHN, as supplied by the manufac- turer as well as Brinell hardness numbers and tensile dT the average value of the measured diameters of strengths of steel test piece and BHN of test pieces made indentation on the test piece and from other various non-ferrous metals and thus, the calcu- lation of BHN using the above (3.31a) can be avoided. D the diameter of the steel ball indenter, usually 10 mm In the table, there are column for diameter of indentation Normally the value of HS in BHN is supplied by the on standard test bar and row for diameter of indentation on manufacturer. In such case, AS and AT will be taken as the test piece. The values of the tensile strength and BHN of the surface areas of the two indentations, which according to supplied standard test bar can be obtained, respectively, from the upper and lower figures at any of the diagonal sites of the (3.2c), are expressed as follows: table where any value of indentation diameter on the stan- dard test bar in the vertical column intersects the same value Surface area of indentation on the standard bar, of indentation diameter on the test piece in the horizontal row. The standard bar is produced by the manufacturer with AS ¼ pD qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ð3:30aÞ an aim to maintain its tensile strength and BHN values same 2 D À D2 À dS2 as those provided in the table, but in production it is very difficult to achieve the above. So each standard bar after its and surface area of indentation on the test piece, production is properly tested and a multiplying factor is inscribed on it to take care of the variations in the strength AT ¼ pD qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ð3:30bÞ 2 D À D2 À dT2 From (3.29), (3.30a) and (3.30b), the hardness of the test piece, HT, can be determined in BHN, assuming HT ¼ BHNT and HS ¼ BHNS; as follows:

3.14 Poldi Impact Hardness 153 and hardness values, whereas the computed table based on • The heavy specimens, which cannot be taken to a the aimed values of the tensile strength and BHN of the table-type hardness tester for testing, can easily be tested standard bar remains unaltered. with this instrument. After formation of indentations, the measured value of • The tester can easily be handled, whereas most table-type diameter for the indentation on the standard test bar is first hardness testers require greater skill to operate them selected in the column of the table and that on the specimen properly. is then selected in the row of the table. The respective values of the tensile strength and Brinell hardness number of steel • The tester is much cheaper than any standard table-type test piece or only the BHN value of other non-ferrous metal hardness tester. test piece can be read from the upper and lower figures or from only one figure provided in the site of the table where The disadvantages of the Poldi impact hardness test are: the selected vertical column and horizontal row intersect. Most often the value of the tensile strength is supplied in the • The accuracy of the Poldi hardness values will be unit of ton per square inch that may be converted into the somewhat lower and may vary to the extent of ±10% as unit of kg per square mm or MPa by multiplying, respec- against the more accurate table-type standard hardness tively, with 1.575 or 15.4448. All these strength and hard- testers. ness values of the test piece read from the table are based on those values of the standard bar obtained from the table. If • Reconditioning or replacement of the plunger of this some multiplying factor is inscribed on the standard bar, all instrument may be required when the protruding end of the strength and hardness values of the test piece read from the plunger gets deformed after some blows with the the table have to be multiplied with this factor in order to get striking hammer. the correct values of the tensile strength and BHN of the test piece, as shown in the following example. 3.15 The Herbert Pendulum Hardness Example The Herbert pendulum hardness test (Herbert 1926) may be considered in the class of dynamic tests, although no impact Suppose, diameter of indentation on the standard test bar = load is involved in the test. The tester is mainly a special 1.8 mm and research instrument and used to measure hardness as well as to study the work-hardening characteristics of metals. The Diameter of indentation on steel specimen ¼ 1.7 mm. hardness of very hard materials, which would not be appropriate to determine by the Brinell hardness test due to Tensile strength of steel specimen read from the supplied attendant deformation of the ball indenter, can be success- table ¼ 50:8 ton in:2 ¼ 80:01 kg mm2 ¼ 784:59 MPa: fully measured by the Herbert pendulum test. The test can be employed readily to measure quite accurately the hardness of Brinell hardness number of steel specimen read from the excessively thin test sections, wires, carburized and nitrided supplied table ¼ 225 kg mm2: cases, etc. Since the test generally produces a deformation mark of microscopic size on the test piece, it can also be Brinell hardness number of thestandard test bar obtained applied to breakable objects or finished surfaces, without from the supplied table ¼ 197 kg mm2: causing any visible damage to them. This test can also be performed above and below ambient temperature. Multiplying factor is inscribed on the supplied standard test bar ¼ 1.03. The instrument consists of an arched metal frame of 12 in. (304.8 mm) in length which acts as pendulum and is So the correct Brinell hardness number of the supplied capable to stretch over flat surfaces of 6 in. (152.4 mm) standard test bar ¼ 197 Â 1:03 ¼ 202:91 kg mm2: width or cylindrical articles of 8 in. (203.2 mm) diameter. Under the centre of the arch, a suitable chuck holds a Hence, the correct Brinell hardness number of steel test 1-mm-diameter hardened steel or diamond ball which makes piece ¼ 225 Â 1:03 ¼ 231:75 kg mm2; and a contact at a point on the test surface. The centre of gravity of the pendulum is set to a predetermined distance, usually The correct tensile strength of steel test piece ¼ 50:8 Â 0.1 mm, below the centre of the ball, which can be con- 1:03 ¼ 52:324 ton in:2 ¼ 82:41 kg mm2 ¼ 808:13 MPa: trolled by means of an adjustable weight. On the top of the frame, there is a curved bubble tube which consists of a scale 3.14.3 Advantages and Disadvantages graduated from 0 to 100, with the 50 mark directly over the ball. The standard pendulum weighs 4 kg, but pendulums The Poldi impact hardness test has the following advantages: • The Poldi impact hardness tester is a small-sized portable instrument, whereas most table-type hardness testers are not portable or small in size.

154 3 Hardness weighing either 2 or 24 kg and equipped with 1-mm- time test is performed and continued, until the time hardness diameter hardened steel, ruby, or diamond balls may be reaches a maximum value. The maximum induced time available that are used for special testing purposes. hardness produced in the test specimen is a measure of work-hardening capacity of the metal. The hardness is measured by swinging the pendulum and to allow the free motion of the ball during the swing. The 3.15.4 Scale Work-Hardening Test test piece should be prepared to have reasonably smooth surface without any surface imperfections. In making a test, The increase in work-hardening of the test specimen is the ball is placed at the point on the surface of the test piece measured in this test by repeated rolling of the ball on the where it is required to determine the hardness. The pendu- same spot of the test specimen. Five scale tests are per- lum is made to swing and, by noting the oscillations of the formed successively by tilting the pendulum alternately in pendulum when swung, the following various types of opposite directions. The work-hardening capacity of the hardness measurements may be obtained: metal is then expressed by the difference between the orig- inal scale hardness number and the average of the other four 3.15.1 Time Test successively determined scale hardness numbers. The ‘time hardness number’ is determined by noting the 3.16 Nanohardness time in seconds required for 10 single swings (5 over and 5 back) of the pendulum through a small arc. It is a measure of Depending on the magnitudes of loads applied for indenta- indentation hardness of the metal which can be converted to tions, hardness tests are classified into three categories: Brinell hardness numbers. In the case of hardened steels, the Brinell number is obtained directly by multiplying by 100 • Macro-hardness tests, in which the magnitudes of the number of seconds required for one single swing. Her- bert obtained a time hardness value of 3 for lead, 85 for applied loads are 1 kgf (9.8 N) or greater. As discussed hardened steel and 100 for glass. Indentation tests show that hardened steels have higher hardness than glass. in earlier sections, the applied loads may vary from 1 to 3.15.2 Scale Test 120 kgf in Vickers test, from 15 to 150 kgf in Rockwell The pendulum is tilted until the bubble arrives at zero mark tests and from 6.25 kgf, which may be occasionally used, on the scale and then released for angular oscillation. The ‘scale hardness number’ is determined by noting the scale to 3000 kgf in Brinell tests. number in the bubble level at the end of one swing. Since the magnitude of the angle reached at the end of one swing of • Microhardness tests, performed with either Vickers the pendulum, which is recorded as the hardness, is depen- dent upon the energy absorbed at the point of contact indenter or Knoop indenter, use smaller loads that vary between the ball and specimen surface, the scale hardness number is said to measure the resistance of specimen (metal) from 1 gmf (9.8 mN) to 1 kgf (9.8 N). The loads com- to deformation. Herbert obtained a scale hardness value of 0 for lead and 100 for glass. The flow hardness of a material is monly used range from 100 to 500 gmf, and materials expressed by the scale-time ratio, which is obtained by dividing the scale hardness number by the time hardness thicker than about 3 mm is suitable for microhardness number. measurements. 3.15.3 Time Work-Hardening Test • Nanohardness tests or nanoindentation tests, in which the magnitudes of applied forces are usually in the milli-newton ð10À3NÞ range, may be as low as 0.1 mN. The forces involved are measured with a resolution of a few nanonewtons ð10À9NÞ: The test depends on the simultaneous measurement of the load and the depth of indentation produced by load. The measures of indenta- tion dmepictrhoanrseÀi1n0tÀh6emraÁnogre of nanometres ð10À9mÞ; rather than millimetres ð10À3mÞ; the latter being common, respectively, in conventional micro- hardness or macro-hardness tests. If the pendulum ball is rolled over the test specimen by Majority of nanoindentation tests aim to obtain Young’s moving the pendulum first to the extreme right and then to modulus along with measurement of hardness of the speci- the extreme left, the hardness of the material under the ball is men material from the load–displacement data obtained in changed due to strain hardening. After each two passes of tests. Determination of elastic modulus from nanoindenta- the ball over the test specimen as described just above, a tion tests is beyond the scope of the text, and for this, the

3.16 Nanohardness 155 widely used method developed by Oliver and Pharr is examine ion-implanted and laser heat-treated surfaces, thin referred (Oliver and Pharr 2004). Working of the nanoin- films and coatings, small pieces of materials with thicknesses dentation is carried out by forcing an indenter, usually of the order of several tens of nanometres and materials with Berkovich indenter (Berkovich 1951), under a prescribed nano-sized structural features. load into a specimen at the selected location. The readings of load and displacement of the indenter (depth of indentation) The equipment used to perform instrumented indentation are recorded during loading and unloading, which gives testing consists of three basic components: characteristics load versus displacement curve, which has been shown subsequently in Fig. 3.19. The principal 1. An indenter of specific geometry usually mounted on a mechanical property determined in nanoindentation test is stiff but lightweight shaft through which the load is the hardness, HNano; which is defined as the applied maxi- imparted. mum load divided by the projected area of contact surface of the indenter under load, i.e. 2. An actuator to apply the load, which can be performed in different ways. Small loads can be generated electro-  ð3:32Þ statically using a capacitor or by an electromagnetic coil HNano ¼ Pmax Apc or the expansion of a piezoelectric element. where Pmax is the applied maximum load and Apc is the 3. A sensor to measure displacements of the indenter, which projected area of contact surface at that load. Since this are measured by a variety of means. These include a definition of hardness is based on the projected contact area changing capacitance, linear variable differential trans- under load, and the traditional hardness is determined from formers (LVDTs) and laser interferometers. the projected contact area of the residual plastic impression left in the specimen upon the removal of load, the Most nanoindentation instruments are load controlled, nanohardness defined by (3.32) may deviate from the tra- that is, a commanded force is applied and the resulting ditional hardness if there is significant elastic recovery dur- displacement is read. ing unloading. In nanohardness test, since the depth of indentation beneath the specimen surface produced by load 3.16.1 Indenters is measured simultaneously with the measurement of the load, the size of the projected contact area under full load Although nanohardness test or IIT testing uses a variety of can be determined from the depth of impression and the indenters made from a variety of materials, diamond is known geometry (angle or radius) of the indenter. Since probably the most frequently used material for indenter due from the measurement of the depth of indentation and the to its high hardness and elastic modulus so that the contri- known geometry of the indenter the projected contact area is bution to the measured displacement from the indenter itself determined, the nanohardness test is sometimes called depth- remains to a minimum level. Indenters made of less-stiff sensing indentation test. materials, such as tungsten carbide, sapphire or hardened steel, may be used but the elastic displacements of the Nanoindentation test is also called ultra-low-load inden- indenter must be removed from the total measured dis- tation test, and this test is carried out using the technique of placement, as in the case of the machine compliance. instrumented indentation testing (IIT), also known as con- tinuous-recording indentation test. A high-resolution Nanoindentation hardness tests are generally made with instrument used in IIT can continuously control and record either spherical or pyramidal indenters, but the pyramidal the loads and displacements of an indenter as it is forced into Berkovich indenter is used most frequently in IIT testing. and drawn back from the material. In a nanoindentation test, Berkovich indenter, shown in Fig. 3.17a, is an equilateral the indentation size upon unloading is often only a few triangular-based pyramidal-shaped indenter made of dia- microns, which is very difficult to measure using optical mond. This indenter is comparable to Vickers indenter in the techniques. With the above instrument, loads as small as sense that both of them make sharp contacts and produce 1 nN can be applied and displacements as small as 0.1 nm geometrically similar indentations. The centre-line-to-face (1 Å) can be measured. Another important advantage of IIT angle for an ideal Berkovich indenter with a three-sided is that several mechanical properties, such as hardness, pyramid is 65.27°, whereas that for the four-sided Vickers Young’s modulus, yield stress, strain-hardening character- pyramid indenter is 68°. For small-scale testing, i.e. for istics of metal, the activation energy and stress exponent for nanohardness measurements, the geometry of Berkovich creep, can be determined using load–displacement data indenter is preferred to that of Vickers indenter, because the without the requirement to image the impressions formed by edges of a three-sided pyramid can be ground to meet at a the indenter. Because of formation of only a few nanometres single sharp point, while a four-sided pyramid ends at a deep indentations, nanoindentation test can be applied to ‘chisel edge’ (a line) rather than a point. The point apex of Berkovich indenter maintains its self-similar geometry to

156 (a) 3 Hardness Fig. 3.17 a A Berkovich (b) indenter. b Indentation parameters for Berkovich Apc θ hc very small scales but the chisel edge defect even for the best could be more easily measured for shallower depths of Vickers indenter has a length of about a micron, which residual impression. Now, further discussions in this section causes its small-scale geometry to differ from that at larger will be mainly concentrated on Berkovich indenter, which is scales, although Vickers indenters could be used for used routinely for measurement of nanohardness, because its large-scale testing like microhardness measurements. apex is a sharper point and thus ensures a more precise Besides Berkovich and Vickers indenters, other indenter’s control over the indentation process. shapes, which may be used for nanoindentation tests, include spherical and conical or sphero-conical indenter, cube corner 3.16.2 Derivation for Berkovich Hardness indenter with three-sided pyramid having centre-line-to-face angle of 35.26° (similar to but sharper than Berkovich The Berkovich indenter normally used for nanoindentation indenter) and rhombic-based four-sided pyramidal-shaped tests has a centre-line-to-face angle of 65.27°, which gives Knoop indenter. The use of spherical indenters at the micron the same ratio of projected area to indentation depth as the scale is restricted due to difficulties in obtaining high-quality Vickers indenter. The tip radius for a typical Berkovich spheres made of hard, rigid materials. Conical indenters have indenter is of the order of 50–100 nm, although the tip very little application in the small-scale work because of radius usually increases to about 200 nm with use. Berko- difficulty in manufacturing conical diamonds with sharp tips. vich indenter is a geometrically self-similar sharp indenter On the other hand, the sharper cube corner indenter displaces having well defined tip geometry. When penetrated into the more than three times the volume of the Berkovich indenter surface, it causes well defined plastic deformation. It is good at the same load and thereby develops much higher stresses for measuring values of hardness and elastic modulus. The and strains in the vicinity of the contact. This produces very disadvantage is that the elastic–plastic transition is not clear. small well defined cracks around hardness impression in brittle materials, and the indenter is thus ideal for estimating Indentation parameters for Berkovich indenter are shown fracture toughness at relatively small scale using such in Fig. 3.17b, where the apex semi-angle h ¼ 65:27: The cracks. The four-sided pyramidal Knoop indenter with two mean contact pressure, which is a measure of hardness, different face angles is generally used to investigate aniso- HNano; is usually determined by (3.32) from a measure of the tropy of the surface of the specimen from the measurements contact depth of penetration, hc; which is related to the of the unequal lengths of the diagonals of the residual projected area of the contact, Apc; as derived below: impression. This indenter was originally developed for the testing of very hard materials where a longer diagonal length For a Berkovich indentation, as shown in Fig. 3.18, suppose Fig. 3.18 a Equilateral (a) (b) triangular impression formed by Berkovich indenter on surface of 2L L specimen. b Vertical section of 33 Berkovich indentation along the median line of the triangular θ =65.27° impression L hc L 3 60° a

3.16 Nanohardness 157 a length of each arm of the equilateral triangular Originally, the Berkovich indenter was constructed with a impression formed on the surface of specimen, centre-line-to-face angle of 65.03°, which gives the same actual surface area-to-depth ratio as a Vickers indenter. L perpendicular distance from the base to the apex Since the hardness in nanoindentation is defined by the mean of the triangular impression formed on the surface contact pressure, the original face angle of Berkovich of specimen, indenters used in nanoindentation work has been modified to 65.27° to have the same projected area-to-depth ratio as the hc contact depth of penetration beneath the specimen Vickers indenter. The equivalent semi-angle of cone inden- surface under load, and ter, which gives the same projected area-to-depth ratio, is 70.296°. h centre-line-to-face angle of Berkovich indentation ¼ 65:27: 3.16.3 Determination of Contact Depth of Penetration From Fig. 3.18a, the projected area of contact is: In a typical test, the load is applied from zero to some pre- ÀÁ 1 a  L ¼ 1 a  a tan 60 ¼ pffiffi a2 ð3:33Þ determined maximum value to drive the indenter into the test Apc Berkovich¼ 2 2 2 3 piece and then the load is released from the maximum value 4 back to zero while the indenter is withdrawn from the test piece. The applied load and displacement of the indenter are In the nanohardnessÀ testÁ, since hc is measured, so the recorded simultaneously during loading and unloading per- projected contact area, Apc Berkovich; has to be expressed in iod, as shown in Fig. 3.19, and by an analysis of the load– terms of hc; which is given below: displacement data, the contact depth of penetration is determined. From Fig. 3.18b, As the sample is indented, both elastic and plastic L ¼ hc tan h; or; L ¼ a tan 60 ¼ 3hc tan h; deformation takes place, forming a hardness impression that 3 2 conforms to the shape of the indenter to some contact depth, pffiffi hc: When the indenter is withdrawn, only the elastic part of ) a ¼ 2Â3 hc tan h ¼ 2 3hc tan h ð3:34Þ the displacement is recovered, which facilitates one to sep- tan 60 arate the elastic properties of the material from the plastic. Figure 3.19 shows schematically a typical indentation load, Substituting for a from (3.34) into (3.33) and remem- P; versus displacement (relative to the initial undeformed bering that h ¼ 65:27; we get surface), h; data obtained during one full cycle of loading ÀÁ pffiffi  pffiffi 2 pffiffi tan2 65:27 Apc Berkovich 3 h¼ 3 3hc2 ¼ 4 2 3hc tan ¼ 24:494h2c % 24:5h2c ð3:35Þ Hence, the mean contact pressure or Berkovich hardness according to (3.32) is: HBerkovich ¼ Pmax ð3:36Þ Pmax 24:5hc2 For comparison purpose, let us consider nanoindentation Load, P Loading Slope = Mcs Unloading test using Vickers indenter, where the apex semi-angle h ¼ Possible 68: If a is the length of each arm of Vickers square range for impression formed on the surface of specimen and contact hc depth of Vickers indentation beneath the specimen surface under load is h ¼ hc; then from Fig. 3.11b, a=2 ¼ hc tan h: Hence, the projected area of contact will be ÀÁ a2 ¼ ð2hc tan hÞ2¼ 4h2c tan2 68 ¼ 24:504hc2 hf Apc Vickers¼ % 24:5h2c hmax hc for χ = 1 hc for χ = 0.72 ð3:37Þ Displacement, h And the Vickers nanohardness according to (3.32) is: Fig. 3.19 Schematic illustration of a typical indentation load–dis- placement (relative to the initial undeformed surface) data during one HVickers ¼ Pmax ð3:38Þ complete cycle of loading and unloading, showing important measured 24:5hc2 parameters (Oliver and Pharr 1992)

158 3 Hardness and unloading, also known as compliance curves for loading unloading part of the load–displacement data rather than by and unloading, where deformation during loading is imaging, as used in conventional hardness testing. assumed to be both elastic and plastic in nature and only the elastic displacement is assumed to be recovered during The most widely method used to establish the projected unloading. The method of analysis based on the elastic area of contact surface was developed by Oliver and Pharr nature of the unloading curve is not applicable to materials (1992). They found that the unloading curve (Fig. 3.19) is in which plasticity reverses during unloading. However, it usually not linear as suggested by Doerner and Nix (1986), has been shown (Pharr and Bolshakov 2002) that reverse but is usually well approximated by the power law relation: plastic deformation is usually negligible. P ¼ Bðh À hf Þm0 ð3:39Þ The important quantities in Fig. 3.19 are the maximum load,Pmax; the maximum displacement, hmax; the final per- where B and m0 are empirically determined power law fitting manent depth of penetration after complete unloading, hf, constants. The power law exponents, m0 in (3.39) are always the slope of the upper portion of the unloading curve during the initial stages of unloading, Mcs ¼ dP=dh: The parameter greater than 1, varying in the range of 1.25–1.51 from Mcs is called the elastic contact stiffness, or simply contact stiffness and has the dimension of force per unit distance. material to material (Oliver and Pharr 1992). The contact Hardness and elastic modulus can be derived from mea- surements of these four important quantities from the P À h stiffness, Mcs; is then evaluated by differentiating (3.39) at curve in Fig. 3.19. In Fig. 3.19, hc is the contact depth of the maximum depth of indentation, h ¼ hmax; which gives penetration of the indenter under the maximum load. The the following (3.40): values of hc vary depending on the geometry of indenter used for penetration and the two values of hc for two values  of v, i.e. for v ¼ 1 and v ¼ 0:72 ; have been shown dP schematically in Fig. 3.19, where v is a constant that Mcs ¼ dh ¼ Bm0ðhmax À hf Þm0À1 ð3:40Þ depends on the indenter geometry. For determination of nanohardness, it is required to know the value of hc under h¼hmax the applied maxiÀmumÁ load, from which the projected area of contact surface, Apc Berkovich; at that load and nanohardness Experience has shown that the entire unloading curve is for Berkovich indenter can be calculated bÀy usÁing, respec- not always adequately described by (3.39). To avoid unac- tively, (3.35) and (3.36). Similarly, Apc Vickers and ceptable errors in computing the contact stiffness from nanohardness for Vickers indenter can be calculated by (3.40), it is recommended to determine the contact stiffness using, respectively, (3.37) and (3.38). The contact depth of by fitting only the upper portion of the unloading data. penetration, hc; is determined from an analysis of the The deformation pattern of an elastic–plastic sample during and after indentation (Oliver and Pharr 1992) is shown schematically in Fig. 3.20, in which the behaviour of the Berkovich indenter is assumed to represent by a conical indenter with an equivalent half-included angle, / ¼ 70:296; which gives the same projected area-to-depth ratio. The basic assumption in this model is that ‘sinking in’ of material at the contact periphery of the indenter occurs and it P Indenter profile max Indenter profile Initial surface hs hf hmax hc Surface profile after unloading ϕ Surface profile under load Fig. 3.20 Schematic representation of deformation pattern of an elastic–plastic sample during and after indentation showing parameters characterizing the contact geometry (Oliver and Pharr 1992)

3.16 Nanohardness 159 is shown in Fig. 3.20. ‘Sink-in’ is the movement of the of the machine and the test piece. The contribution from the indented material around the indenter below the original machine compliance that may be significant must be cali- surface plane. This assumption restricts the method to brated and subtracted from the total measured displacement. account for the ‘pile-up’ of material at the contact periphery For the purpose of calibration, a calibration material with that occurs in some elastic–plastic materials. ‘Pile-up’ is the known elastic properties is required to be indented. Typically, movement of the indented material around the indenter fused quartz is considered as the calibration material, whose above the original surface plane. However, assuming the elastic modulus, E ¼ 72 GPa and Poisson’s ratio, m ¼ 0:17: ‘pile-up’ to be negligible, the amount of ‘sink-in’, hs; is The most important reason to choose fused quartz is that it given by (Oliver and Pharr 2004): does not pile-up because of its high ratio of hardness to elastic modulus. Further, it is relatively inexpensive, available in hs ¼ v Pmax ð3:41Þ highly polished form producing repeatable results with little Mcs scatter, highly isotropic due to its amorphous nature, not susceptible to oxidation and its near-surface properties are where v is a constant that depends on the geometry of similar to those of the bulk. To determine the machine indenter. For example, a conical indenter has v ¼ 0:72; and compliance, a convenient procedure is to assume that the a Berkovich as well as a Vickers indenter has v ¼ 0:75 indenter does not deviate from its perfect geometric shape at (Oliver and Pharr 1992; Fischer-Cripps 2011). Since the large depths of indentations and the machine compliance is a constant, independent of load. Thus, indentations are made on vertical displacement of the contact periphery can be fused quartz at several large depths for which ideal area of the approximated from (3.41), the contact depth, hc; can be indenter is expected to apply and the machine compliance is estimated from the geometry in Fig. 3.20 as follows: determined by using the load–displacement data from fused quartz in a manner described below. hc ¼ hmax À hs ¼ hmax À v Pmax ð3:42Þ Mcs A geometry independent relation (a very general relation that applies to all axial-symmetric indenter, from which the The above analysis is based on an elastic solution and elastic modulus is determined) among contact stiffness, projected contact area and elastic modulus is given below works well when ‘sink-in’ pre-dominates and ‘pile-up’ is (Pharr et al. 1992): negligible. Neglecting the displacement arising from the compliance of the testing machine and ‘piling up’ of the material, if any, the projected contact area for an ideally sharp Berkovich indenter is given by (3.35), which is p2ffipffiffi pffiffiffiffiffiffi Eeff Apc ÀÁ 24:494hc2: Mcs ¼ b ð3:43aÞ Apc Berkovich¼ In practical nanoindentation tests, indenters are not ide- where b is a constant that depends on the geometry of ally sharp. Therefore, an empirically determined indenter area function at the contact depth hc is evaluated to calculate indenter (Bulychev et al. 1975; Oliver and Pharr 1992), with the projected contact area. The area function relates the a value of b ¼ 1:034 for a Berkovich indenter, and Eeff is the cross-sectional area of the indenter to the distance from its effective elastic modulus, which takes into account the fact tip. An experimental procedure to determine the area func- tion is presented subsequently in Sect. 3.16.5. Once the that elastic deformations occur in both the test piece and the projected contact area is known, the nanohardness is com- indenter. Since the indenter is diamond with E ¼ 1141 GPa puted using (3.32). and m ¼ 0:07; the effective elastic modulus in above (3.43a) is Eeff ¼ 69:6 GPa; when fused quartz is used as the cali- 3.16.4 Correction for Machine Compliance bration material. The load applied on the test piece causes the frame of the Let Cm ¼ the machine compliance ¼ the inverse of testing machine to deflect. This deflection divided by the machine stiffness ðMmÞ ¼ 1=Mm; applied load is the compliance of the machine. Hence, the displacement arising from the compliance of the machine itself Ccs ¼ the elastic compliance of the specimen-indenter is included as a component in the displacement recorded in an contact ¼ the inverse of contact stiffness ðMcsÞ ¼ 1=Mcs; instrumented indentation testing system. Thus, the stiffness, and Ct ¼ the total compliance ¼ the inverse of total mea- dP=dh; measured from the unloading curve of P À h plot in sured stiffness ðMtÞ ¼ 1=Mt ¼ dh=dP: an indentation test has contributions from both the responses The unloading part of the load–displacement data from fused quartz is used to measure the total stiffness, dP=dh: To determine Cm or Mm; the machine and the specimen-indenter combination are assumed to act like springs in series, whose compliances are additive. Thus, the total measured compli- ance, Ct; is given by

160 3 Hardness Ct ¼ Cm þ Ccs ð3:44Þ 3.16.5 Indenter Shape Function From (3.43), it follows that The indenter cannot be prepared to have a perfectly sharp tip. Therefore, correction due to real geometrical shape of pffiffiffi the indenter tip is necessary, especially at shallow depth of 1 p p1ffiffiffiffiffiffi contact. Hence, calibration of indenter area function, also Ccs ¼ Mcs ¼ 2bEeff Apc ð3:43bÞ sometimes called the indenter shape function or indenter tip function, is needed to calculate the projected contact area. Substituting for Ccs from (3.43b) into (3.44) yields the following relation: Independent measurements must be made for careful calibration of the indenter area function so that deviations pffiffiffi from perfect indenter geometry are taken into account. These p p1ffiffiffiffiffiffi deviations can be quite severe near the tip of the Berkovich Ct ¼ Cm þ 2bEeff Apc ð3:45Þ indenter, where some rounding undoubtedly occurs during the grinding process. To implement the area-function cali- ÀÁ bration for a Berkovich indenter, a series of indentations is For an ideal Berkovich indenter, since Apc Berkovich¼ made on a calibration material with known elastic properties 24:494h2c (3.35), we obtain from (3.45): at depths of interest, usually from as small as possible to as large as possible to establish the area function over a wide rffiffiffiffiffiffiffiffiffiffiffiffiffi range. The most commonly used material for this purpose is dh p 11 fused quartz. After correction for machine compliance as dP ¼ Cm þ 24:494 2bEeff hc ð3:46Þ described in Sect. 3.16.4, the load–displacement data are reduced, from which the contact stiffness, Mcs; and the Machine compliance calibration is usually based on the contact depth, hc; are obtained by means of (3.40) and general relation, given by (3.45), or the relation, given by (3.42). From these quantities and the known elastic proper- (3.46) for a Berkovich indenter. Hence, the most common ties of fused silica, the projected contact areas are determined method of determining a value for the machine compliance by rewriting (3.43a) as: ðCmÞ is to make a linear plot of Ct or dh=dP versus AÀpc1=2; in general, or a linear plot of dh=dP versus 1=hc; for a Ber- Apc ¼ p  Mcs 2 ð3:48Þ kovich indenter, obtained for an elastic unloading into an 4 bEeff elastic–plastic material for a range of large indentation depths. The intercept of this linear plot gives the machine A plot of Apc versus hc then gives a graphical represen- compliance while the slope is proportional to the effective tation of the area function, which can be curve fit according elastic modulus, Eeff: The machine compliance can be to the following functional form, shown by (3.49): measured with great accuracy, when the second term on the right of (3.45) or (3.46) is small, i.e. when the contact areas Apc ¼ f ðhcÞ ¼ C0hc2 þ C1hc þ C2h1c=2 ð3:49Þ or depths are large, because extrapolation of the data to þ C3h1c=4 þ Á Á Á þ C8hc1=128 AÀpc1=2 ¼ 0 or hÀc 1 ¼ 0 is required. At shallow depths, cor- responding to small contact areas, the contact compliance is where C0 Á Á Á C8 are constants determined by curve-fitting high and dominates the total measured compliance, while as procedures. The reason for selection of this function was the contact depths (and areas) increase, the contact compli- strictly due to its ability to fit data over a wide range of ance decreases, and the machine compliance becomes the depths and is, in fact, quite convenient to describe a number more dominant factor. of important indenter geometries. The first term alone of (3.49) describes a perfect pyramid or cone, with C0 ¼ 24:5; The total displacement, ht; that occurs in the test equip- and the first two terms describes a perfect sphere of radius R; ment can be measured from the load–displacement data and with C0 ¼ Àp; and C1 ¼ 2pR: The other higher order terms is the summation of machine displacement, hm and the in (3.49) generally describe deviations from the perfect specimen displacement, h. Once machine compliance, Cm; is known, the displacement in the machine at any load P is geometry near the indenter tip due to its blunting (Oliver and given by hm ¼ CmP and the true displacement in the test piece will be obtained by subtracting machine displacement Pharr 1992) and assist to develop an area function that is hm; from the total measured displacement ht : h ¼ ht À hm ¼ ht À CmP ð3:47Þ

3.16 Nanohardness 161 accurate over several orders of magnitude in depth. Thus, for It has been shown (Bolshakov and Pharr 1998) that the extent of ‘pile-up’ or ‘sink-in’ depends on the a Berkovich indenter (3.46) becomes: work-hardening behaviour of the material and the ratio of final indentation depth, hf; to the depth of the indentation at ÀÁ ¼ 24:5hc2 þ C1hc þ C2hc1=2 ð3:50Þ the maximum load, hmax, i.e. hf=hmax; which can be easily Apc Berkovich obtained from the unloading curve in a nanoindentation þ C3hc1=4 þ Á Á Á þ C8hc1=128 experiment. Since conical and pyramidal or Berkovich indenters have self-similar geometries, hf=hmax; does not The area functions determined for three diamond inden- depend on the depth of indentation. The lower limit of ters, which are Berkovich, Vickers and a 70.3° diamond hf=hmax; is 0, which is observed for fully elastic deformation cone, are shown in Fig. 3.21 (Tsui et al. 1997), taking the and the upper limit of hf=hmax; is 1, which corresponds to same ideal area function, Apc ¼ 24:5h2c ; for each of the three rigid–plastic behaviour. It has been observed that large diamond indenters. At large depths, all three indenters tend ‘pile-up’ occurs, specifically only when hf=hmax; is close to to this ideal area function. However, at small depths, the data 1, and the degree of work hardening is small. Further that show that there is indeed tip blunting for all three indenters, when hf=hmax\\0:7; ‘pile-up’ observed is very little irre- whose area functions differ from each other due to different spective of the work-hardening behaviour of the material. degrees of tip rounding. Indenter tip rounding for the conical On the other hand, when hf=hmax [ 0:7; the accuracy of the diamond is the most and for Berkovich is the least. The data method is dependent on the amount of work hardening in the confirm that the Berkovich geometry provides the sharpest diamond indenter. material. If nanoindentation is made on material that has capacity to work harden, then no or negligible ‘pile-up’ 3.16.6 Errors Due to Pile-Up occurs, and the contact area and thereby, the nanohardness The fundamental material properties that affect pile-up are are predicted very well by the Oliver-Pharr method. On the the ratio of the yield stress to elastic modulus and the other hand, the method underestimates the contact area by as strain-hardening behaviour. In general, materials having low much as 50% for an elastic–perfectly plastic material that has ratio of the yield stress to elastic modulus and little or no no capacity to strain-harden and shows large ‘pile-up’. capacity for work hardening (i.e. ‘soft’ metals that have been cold-worked prior to indentation) exhibit large ‘pile-up’, for If from the value of hf=hmax; and/or based on other example, aluminium creates a ‘pile-up’ condition under independent knowledge of the properties of the material, elastic/plastic conditions (Pharr 1998). This ‘pile-up’ leads ‘pile-up’ is suspected, indentations should be imaged to to an underestimation of the contact area and thus, an examine the extent of the pile-up. If large pile-up occurs, the overestimation of the nanohardness. In general, ‘pile-up’ contact area deduced from the load–displacement data can- error increases as the indentation depth increases. not give the accurate measurement of nanohardness and the true area of contact should be measured from the image for computation of correct nanohardness, although method to correct for pile-up without imaging the contact impression has been developed (Oliver and Pharr 2004). 1010 109 3.16.7 Martens Hardness Area, nm2 108 Similar to nanoindentation measurements, Martens hardness 70.3° Cone values are determined from the simultaneous measurements of the applied load and the indentation depth produced 107 Vickers during the application of load. Martens hardness, HM, is Berkovich defined as the applied load divided by the actual surface area of contact of the indenter under load, i.e. 106 HM ¼ P=As ð3:51Þ 105 Ideal, A = 24.5 hc2 104 102 103 104 where P is the applied load and As is the actual contact 10 Depth (hc), nm surface area at that load. It is to be noted that As is not measured from the dimensions of residual impression Fig. 3.21 Calibrated area function for three diamond indenters (Tsui obtained after elastic recovery on unloading as performed in et al. 1997) the traditional hardness measurement. As is a function of the

162 3 Hardness contact depth, hc; of indentation and can be determined in a terms of hc for the known geometry of a given indenter. The Martens hardness was previously called ‘Universal hard- L ness’ and designated as HU: The Martens hardness value 3 HM has been defined for Vickers and Berkovich indenters and not for Knoop or spherical indenters. hc θ θ = 65.03° For a Vickers indenter, with semi-angle h ¼ 68 at the b apex of the pyramidal indentation between the height and the Fig. 3.22 Unmodified Berkovich indenter (schematic and not to scale) slant surface, the derivation of relation between As and hc is showing indentation parameters, used for derivation of Martens given below. As shown in (3.18), the actual surface area of hardness pyramidal indentation in the test piece will be ðAsÞVickers¼ 2aLS; where a is the length of each arm of the square base of the pyramidal indentation that is the same as the base of each lateral triangular face, and LS is the slant height of the pyramidal indentation that is the same as the altitude of each lateral triangular face. If contact depth of Vickers indentation beneath the specimen surface under load is h ¼ hc; then from Fig. 3.11b, we can write a=2 ¼ tan 68; or; a ¼ ð2 tan 68Þhc; and LS ¼ hc : hc cos 68 ) ðAsÞVickers¼ 2aLS ¼ 4 tan 68 hc2 ¼ 26:428hc2 % 26:43hc2 and cos 68 pffiffi ð3:52Þ L L ¼ a tan 60 ¼ 3a a=2 60; 22 Hence, Martens hardness for a Vickers indenter is given ¼ tan or; ð3:54cÞ by Equating for L from (3.54b) and (3.54c), we get ðHMÞVickers¼ P ¼ P ð3:53Þ pffiffi pffiffi ðAsÞVickers 26:43h2c 3a or, a ¼ 2 3 tan h hc 2 ¼ 3hc tan h; ð3:54dÞ Martens hardness for an unmodified Berkovich indenter, From (3.54a) and (3.54d), we can write having a centre-line-to-face angle of 65.03° (since this angle gives the same actual surface area-to-depth ratio as a Vickers ðAs ÞBerkovich ¼ 3 ab ¼ 3  pffiffi tan hhc  hc  indenter), can be found from Fig. 3.22 as follows: ¼ 23 cos h p2ffiffi 2 The actual contact surface area As is 3 times the area of 33 tan 65:03 hc2 ¼ 26:43h2c ð3:55Þ each lateral triangular face of a Berkovich indentation. If b is the slant height of the pyramidal indentation that is same as cos 65:03 the altitude of each lateral triangular face and a is the base of each triangular face, as shown in Fig. 3.22, then ) ðHMÞBerkovich¼ P ¼ P ð3:56Þ ðAsÞBerkovich¼ 3ðab=2Þ: From Fig. 3.22, ðAsÞBerkovich 26:43h2c hc ¼ cos h; or; b ¼ hc h ð3:54aÞ Usually, an indentation depth greater than 0:2 lm is b cos required for the test. The Martens hardness value is reported by the symbol HM; followed by the name of the indenter if If L is the altitude of the base triangle of the pyramidal other than Vickers indenter is used, the test load in N, the indentation, then time (in s) over which the load is applied for indentation, and the number of steps of application of the load if the load is L=3 ¼ tan h; or; L ¼ 3hc tan h ð3:54bÞ not applied continuously. For example, ‘HM (Berkovich) hc 0.5/10/25 ¼ 5000 N mm2’ means the Martens hardness value of the test piece is 5000 N mm2; determined with a Berkovich indenter by applying a load of 0.5 N for a time of 10 s in 25 steps.

3.17 Relationship to Flow Curve and Prediction of Tensile Properties 163 3.17 Relationship to Flow Curve The above Tabor relation, (3.57), has been proved [see (10.59) and Prediction of Tensile Properties in Chap. 10] using slip-line field theory under plane-strain condition (analogous to a two-dimensional hardness test), Tabor (1951a, b) has proposed a procedure by which the although the proof assumes a frictionless condition. plastic region of true stress–true strain curve, as obtainable by the conventional uniaxial tension and compression tests, Tabor suggested that similar strain distributions would be may be determined from the indentation tests. Tabor com- produced by geometrically similar indentations. To charac- pared the flow curve determined from uniaxial tension and terize the strain field, he studied the deformation at inden- compression tests with the curve obtained from the inden- tations and concluded that the true strain, e; might be tation tests performed on a number of specimens subjected to considered to be proportional to the ratio d=D; which could increasing amounts of plastic strain and found a similarity be expressed as between the shapes of the curves. This is basically an empirical method because the stress distribution in the e ¼ 0:2 d ð3:58Þ indentation test is quite complex and difficult to relate D directly with that in the tension or compression test. How- ever, a good agreement has been observed between the flow where curves obtained using this empirical method and the con- ventional ones for several ductile metals that do not show d the chordal diameter of indentation after unloading, and high anisotropy of deformation. When data with a high D the diameter of spherical indenter. degree of accuracy are required, this method is unable to substitute the conventional test methods and the flow data Hence, Tabor relation between tensile true strain and the obtained by this means must only be used in situations ratio of diameter of indentation after unloading to diameter where it is not possible to determine the flow curve by the of spherical indenter is given by (3.58). conventional destructive test methods probably because of the size or nature of the test piece or some other reason. After knowing the diameter of spherical indenter, D; and the applied load, P; and measuring the average chordal When the specimen is indented with a spherical indenter, diameter of indentation after unloading, pm and d=D can be the mean contact pressure, pm; according to Tabor, is related obtained. If tests are conducted with various values of d=D to the uniaxial flow stress (true stress), r0; at a given value of starting from the smallest one for full plasticity up to large true strain as follows: ones and the corresponding values of pm are measured, the various values of true strain and the corresponding true pm ¼ Cr0 ¼ 3r0 ð3:57Þ stresses can be determined, respectively, from various values of d=D and the corresponding values of pm by means of where pm is equal to the applied load divided by the projected (3.58) and (3.57). The true stress versus true strain curve, area  of elastically recovered indentation, i.e. pm ¼ thus determined, makes it possible at least to approximate ð4PÞ ðpd2Þ; in which P ¼ applied load, and d ¼ chordal the tensile flow curve. For mild steel and copper, Tabor observed a good agreement between the plots of true stress diameter of indentation after unloading. Since about vs. true strain obtained from uniaxial compression tests and two-thirds of the mean contact pressure in an indentation indentation tests using spherical indenters for various values hardness test is hydrostatic pressure, which cannot plastically of d=D (Tabor 1951a). Lenhart (1955) verified Tabor’s deform a material in the uniaxial tension test even if the results for duralumin and OFHC copper, but the flow curve applied stress would exceed significantly the material’s yield for magnesium could not be predicted from Tabor’s analysis, strength, so only one-third of the mean contact pressure pro- the reason of which referred by Lenhart was the high ani- duces plastic deformation. Thus, the mean pressure between sotropy of deformation in magnesium. the surface of the indenter and the indentation pm; is three times the flow stress, r0; as shown by (3.57). Hence, (3.57) The ultimate tensile strength (UTS) of a metal is related represents Tabor relation between uniaxial tensile flow stress directly to its Brinell hardness number (BHN) but the con- or true stress and mean contact pressure or Meyer hardness. stants used in the correlation differ for different metals and alloys and depend largely upon the structural condition of the metal. From an extensive statistical study after reviewing most of the early work, Greaves and Jones (1926) Table 3.7 Relation between Steel type UTS in MPa ¼ UTS (MPa) and BHN (kg mm−2) Heat-treated alloy steel (250–400 Brinell) 3:24Â Brinell hardness value Heat-treated carbon and alloy steel (Below 250 Brinell) 3:32Â Brinell hardness value for steel Medium-carbon steel (as-rolled, normalized, or annealed) 3:4Â Brinell hardness value

164 3 Hardness recommended the multiplying factors between the UTS and Subsequently, Cahoon et al. (1971) found that VHN=3 was the BHN for steels, which are presented in Table 3.7, where acceptable for steels, aluminium and brass. Later, a simpli- the UTS is expressed in units of MPa and the BHN in fied relationship between TS and VHN was published by kg mm2: Cahoon (1972): One general thumb rule for heat-treated plain carbon and TS; Su ¼ VHN h n in ð3:62Þ medium-alloy steels is to estimate approximately the UTS 2:9 0:217 from the BHN with the following multiplying factor (Bain and Paxton 1966): where n ¼ the strain-hardening coefficient ¼ n0 À 2: Cahoon et al (1971) also proposed the following relation from which Ultimate tensile strength; in pounds per square inch the 0.2% offset yield strength of carbon steel can be calcu- ¼ 500ðBHNÞ lated with good precision from the measurement of its Vickers hardness: ð3:59aÞ Or, ½S0Še0¼0:002¼ VHN ð0:1Þn0À2 ð3:63Þ 3 Ultimate tensile strength; in MPa ¼ 3:45ðBHNÞ ð3:59bÞ If we make the simplifying assumption that steel does not where ½S0Še0¼0:002¼ the 0.2% offset yield strength, kgf mmÀ2 work-harden (in fact, steel work-hardens but to a less extent than many other metals like annealed copper), then the (=9.81 MPa), Vickers hardness number,  and n0 ¼ tensile strength of steel is equal to its yield strength and an VHN ¼ the kg mm2; agreement between (3.59b) and Tabor’s result can be obtained as follows: n þ 2 ¼ the exponent in Meyer’s law. For aluminium alloys, Petty (1962) predicted the expressions for the tensile strength and the 0.2% offset yield strength in terms of the Vickers hardness number: UTS; Su ¼ 1 ¼ 0:33pm kgf mmÀ2 ¼ 3:23pm MPa À tonsin2Á; Su ¼ 0:189  VHN À 1:38 ð3:64Þ 3 pm TS in ð3:60Þ 0:2% offset yield strength À tonsin2Á; in ð3:65Þ Since the BHN is only a few percent less than the value of ½S0Še0¼0:002¼ 0:148  VHN À 1:59 pm; the multiplying factor between the UTS and the BHN The units of the tensile sftrroemngttohnasndint.h2eto0.2M%Paofbfsyetmyuieltlid- will be slightly higher than 3.23. For a metal possessing strength can be converted greater capability to work harden, such as annealed copper, the multiplying factor will have a higher value than that plying by 15.4448. employed for steels in (3.59). The multiplying factors relating tensile strength with hardness for different materials are available as a result of detailed study conducted by 3.18 Solved Problems Taylor (1942). A strength-hardness correlation has not been 3.18.1. Rockwell hardness test performed on a material with 1=8 in. diameter steel ball indenter shows that after appli- exhibited by magnesium alloy casting. cation of the total load of 60 kg, the large pointer moves to number 45 on red dial and after removal of the major load, Tabor (1951b) suggested a relation to calculate the tensile the large pointer moves to number 65 on the same dial. What is the strength (TS) in kilograms per square millimetre using the (a) Rockwell hardness of that material? Vickers hardness (VHN) and the exponent in Meyer’s law (b) Depth difference between indentation upon withdrawal (Meyer’s index), n0; for carbon and low alloy steels as of the applied major load with the minor load still applied and indentation made by the minor load? follows: (c) Difference between the depth of indentation upon application of the total load and that due to application of the À kgf mm2 Á minor load? TS in ; (d) Depth of indentation recovered elastically? ¼ VHN ½1 À ðn0 À 2ފ  12:5ðn0 À 2Þ!n0À2 ð3:61Þ 2:9 1 À ðn0 À 2Þ Su  The units of TS can be converted from kgf mm2 to MPa by multiplying by 9.81. In (3.61), Tabor recommended VHN=2:9 for steels, but VHN=3 for copper (Tabor 1951b).

3.18 Solved Problems 165 Solution (a) Now the indentation depth due to minor load, neglecting very small elastic recovery, will be: (a) Rockwell hardness of the material = RF 65. (b) If h is the depth of indentation upon withdrawal of the hm ¼ h À 0:104 mm ¼ ð0:118 À 0:104Þ mm applied major load with the minor load still applied and hm is ¼ 0:014mm: the depth of indentation under minor load of 10 kg, then from (3.14b) we get (b) Again by rearranging (3.4), we can obtain the indentation diameter dm corresponding to hm for the same indenter RF 65 ¼ 130 À ðh À hmÞ mm diameter D ¼ 3:175 mm as follows: 0:002 mm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi So; h À hm ¼ ð130 À 65 Þ Â 0:002 ¼ 0:13 mm: dm ¼ D2 À ðD À 2hmÞ2 (c) If ht the depth of indentation upon application of total qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi load of 60 kg, then from (3.14a) we can write ¼ 3:1752 À ð3:175 À 2  0:014Þ2mm ðht À hmÞmm ¼ 0:421 mm: 0:002 mm Rockwell red dial reading ¼ 45 ¼ 130 À Now, using Meyer’s law given by (3.6), we can write For major load (100 kg in E scale): So; ht À hm ¼ ð130 À 45Þ Â 0:002 ¼ 0:17 mm: (d) The elastically recovered depth is: 100 ¼ K0dn0 ¼ K0ð1:2Þn0 ð3:66Þ ht À h ¼ ðht À hmÞ À ðh À hmÞ ¼ ð0:17 À 0:13Þ mm For minor load (10 kg): ¼ 0:04 mm: 10 ¼ K0dmn0 ¼ K0ð0:421Þn0 ð3:67Þ 3.18.2. If Rockwell hardness of a metal is RE 78 and the average diameter of the corresponding Rockwell indentation Dividing (3.66) by (3.67), we get on that metal measured with an optical microscope after the removal of total load is 1.2 mm, calculate the 10 ¼  1:2 n0 2:85n0 or, n0 ¼ log 10 ¼ 2:2 0:421 ¼ log 2:85 (a) Depth due to minor load in mm, (neglect very small So the required value of strain-hardening coefficient is: elastic strain due to minor load); (b) Value of strain-hardening coefficient for the above metal. n ¼ n0 À 2 ¼ 2:2 À 2 ¼ 0:2: Solution 3.18.3. The Meyer hardness of a metal, having a strain- From (3.14b), we can write hardening coefficient value of 0.15, is 210 kg mm2; under an applied load of 3000 kg with 10 mm-diameter ball indenter. RE78 ¼ 130 À ðh À hmÞ mm ; or; If the diameter of ball indenter is changed to 7 mm and the 0:002 mm same metal is tested keeping the above applied load constant, h À hm ¼ ð130 À 78 Þ Â 0:002 ¼ 0:104 mm; what would be the Meyer hardness of the above metal? where the symbols have the same meaning as defined earlier Solution  with (3.14b). Given that Meyer hardness, pm ¼ 210 kg mm2; under an Given that the average diameter of the Rockwell inden- applied load, P ¼ 3000 kg with indenter diameter D1 ¼ tation after release of total load is d ¼ 1:2 mm: In E scale of 10 mm: Therefore from (3.5), the indentation diameter is: Rockwell, the diameter of ball indenter is D ¼ ð1=8Þ in: ¼ sffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð25:4=8Þ mm ¼ 3:175 mm: If we neglect very small elastic 4P 4  3000 d1 ¼ ppm ¼ p  210 ¼ 4:265 recovery upon withdrawal of the minor load, then h is equal mm mm to the total permanent indentation depth, which can be cal- culated from the measured d by means of (3.4). Hence, The strain-hardening coefficient of the given metal is n ¼ 0:15: Hence, the value of Meyer’s index will be: n0 % pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ 2 ¼ 2:15: If K10 is the resistance of metal to penetration h¼DÀ D2 À d2 ¼ 3:175 À 3:1752 À 1:22 in Meyer’s law for indenter diameter D1 ¼ 10 mm; then mm 22 from Meyer’s law given by (3.6) we get ¼ 0:118 mm:

166 3 Hardness K10 ¼ P ¼ 3000 ¼ 132:68 kg mmÀ2:15: Let d ¼ diameter of permanent indentation under total d1n0 4:2652:15 load of 100 kg and dm ¼ diameter of indentation under minor load of 10 kg. Now, using Meyer’s law given by Let the indentation diameter is d2; and the resistance of the metal to penetration is K20 ; when the indenter diameter is D2 ¼ (3.6), we can write 7 mm: The value of K20 is obtained using (3.10) as follows: For total load: 100 ¼ K0dn0 ¼ 125:66 Á d2; K10 Á D1ðn0À2Þ ¼ K20 Á D2ðn0À2Þ; or; Or; d2 ¼ 100 ¼ 0:7958 mm2: 125:66 132:68  10ð2:15À2Þ ¼ K20  7ð2:15À2Þ For minor load: 10 ¼ K0dmn0 ¼ 125:66 Á dm2 ; ) K20 ¼ 132:68  100:15¼ 139:97 kg mmÀ2:15: Or; dm2 ¼ 10 ¼ 0:07958 mm2: 7 125:66 Again from Meyer’s law given by (3.6), we get the value If h is the depth of permanent indentation under total load of 100 kg, then from (3.4): of d2 as  1=n0 133090:9071=2:15¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P h¼DÀ D2 À d2 ¼ 1:5875 À 1:58752 À 0:7958 d2 ¼ K20 ¼ 4:16 mm: mm 22 ¼ 0:1372 mm: Hence, the Meyer hardness of the same metal under the applied load, P ¼ 3000 kg with indenter diameter D2 ¼ If hm is the depth of indentation under minor load of 7 mm will be: 10 kg, pm ¼ 4P ¼ 4  3000 kg mmÀ2 ¼ 220:7 kg mmÀ2: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pd22 p  4:162 hm ¼ D À D2 À dm2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1:5875 À 1:58752 À 0:07958 mm ¼ 0:0126 mm: 3.18.4. If the Meyer hardness of a cold-worked metal is 2 160 kg mm−2, determine its corresponding Rockwell hard- Now, from (3.14b) Rockwell hardness of the metal in ‘B’ ness number in ‘A’ scale and ‘B’ scale. Neglect spherical scale is given by apex of the cone indenter. Solution ¼ À ðh À hmÞ mm ¼ À 0:1372 À 0:0126 0:002 mm 0:002 RB 130 130 Let P is the applied load in kg and d is the diameter of ¼ 67:7 % 68: indentation in mm. So the required hardness is RB 68. Given that, Meyer hardness number, pm ¼ ¼16016k0gÂmpmk2g¼mp4mdP22:; or; Rockwell Hardness in A Scale P 4 d2 Applied total load ¼ 60 kg, and minor load ¼ 10 kg: Dia- mond cone indenter, which has an apex cone angle of 120°, Meyer’s Law: P ¼ K0dn0 ; where Meyer’s index n0 ¼ 2 is used in ‘A’ scale. for a cold-worked metal. Let d ¼ diameter of permanent indentation under total load of 60 kg and dm ¼ diameter of indentation under minor So, for the given cold-worked metal, K0 ¼ P ¼ load of 10 kg. Now, using Meyer’s law given by (3.6) we 160  p kgmm2 ¼ 125:66 kgmm2: d2 can write 4 For total load: 60 ¼ K0dn0 ¼ 125:66 Á d2; rffiffiffiffiffiffiffiffiffiffiffiffiffi Rockwell Hardness in B Scale 60 Or; d¼ 125:66 mm ¼ 0:691 mm: Applied total load ¼ 100 kg, and minor load ¼ 10 kg: ‘B’ scale uses steel ball indenter whose diameter is: For minor load:r10ffiffiffiffi¼ffiffiffiffiffiKffiffiffi0ffidmn0 ¼ 125:66 Á dm2 ; D ¼ 1 in: ¼ 25:4 mm ¼ 1:5875 mm: Or; dm ¼ 10 mm ¼ 0:2821 mm: 16 16 125:66

3.18 Solved Problems 167 If h is the depth of permanent indentation under total load From (3.6), Meyer’s Law: P ¼ K0dn0 ; where Meyer’s of 60 kg, then from the geometry of the cone indenter we index n0 ¼ 2:5 for a fully annealed metal. If dm ¼ diameter can write of indentation under minor load of 10 kg, we can use d=2 ¼ tan 60; or; Meyer’s law as follows: h d 0:691 For total load: 100 ¼ K0dn0 ¼ K0ð1:5875Þ2:5; tan tan 60 h¼ 2 60 ¼ 2 ¼ 0:1995 mm: 100 kgmm2:5: ð1:5875Þ2:5 Or; K0 ¼ ¼ 31:49 If hm is the depth of indentation under minor load of For minor load: 10 ¼ K0dmn0 ¼ 31:49 Á dm2:5; 10 kg, similarly we get  1 dm=2 ¼ tan 60; or, 10 2:5 hm Or; dm ¼ 31:49 ¼ 0:632 mm: hm¼ 2 dm ¼ 0:2821 ¼ 0:0814 mm: If h is the depth of permanent indentation under total load tan 60 2 tan 60 of 100 kg, then from (3.4): Now, from (3.15b), Rockwell hardness of the metal in ‘A’ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi scale is given by h ¼ D À D2 À d2 ¼ 3:175 À 3:1752 À 1:58752 22 RA ¼ 100 À ðh À hmÞ mm ¼ 100 À 0:1995 À 0:0814 ¼ 0:213 mm: 0:002 mm 0:002 ¼ 40:95 % 41: If hm is the depth of indentation under minor load of So the required hardness is RA 41. 10 kg, then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hm ¼ D À D2 À dm2 ¼ 3:175 À 3:1752 À 0:6322 3.18.5. Suppose, the Rockwell direct-reading hardness dial 2 2 is not working properly and it is required to estimate the ¼ 0:032 mm: Rockwell hardness and related flow stress for a fully annealed copper specimen. To obtain the above, the copper Now from (3.14b), Rockwell hardness of copper in ‘E’ specimen is tested in the Rockwell hardness tester using the scale is given by load and the indenter of ‘E’ scale and after release of total load, the diameter of indentation formed on the specimen is RE ¼ 130 À ðh À hmÞ mm ¼ 130 À 0:213 À 0:032 measured. If the measured indentation on the copper speci- 0:002 mm 0:002 men corresponds to the true strain value of 0.1, found by Tabor relation, determine ¼ 39:5 % 40: (a) Rockwell hardness of copper in ‘E’ scale, and (a) The required hardness is RE 40. (b) Flow stress of copper at the specified true strain, using (b) From Tabor relation given by (3.57), Tabor relation. Flow stresss r0 ¼ pmðmean pressureÞ=3: Solution  Since from (3.5), mean pressure pm ¼ ð4PÞ pd2; so Let P is the applied total load in kg and d is the diameter of Flow stress of copper ¼ pm ¼ 4P permanent indentation in mm, formed under this load. 3 3pd2 In E scale of Rockwell, applied total load P ¼ 100 kg, ¼ 4 Â 100 ¼ 16:84 kg=mm2 and minor load ¼ 10 kg: ‘E’ scale uses steel ball indenter 3pð1:58752Þ whose diameter is: ¼ ð16:84 Â 9:807Þ N=mm2 D ¼ ð1=8Þ in: ¼ ð25:4=8Þ mm ¼ 3:175 mm: ¼ 165:15 N mmÀ2 or MPa: From Tabor relation (3.58), the true strain is: e ¼ 3.18.6. Assuming that the longitudinal diagonal length of the 0:2ðd=DÞ; and given value of true strain is e ¼ 0:1: So, Knoop indentation is equal to the diagonal length of Vickers indentation, compute the ratios of d ¼ e ÂD ¼ 0:1 Â 3:175 ¼ 1:5875 mm: 0:2 0:2

168 3 Hardness (a) Projected areas of Knoop to Vickers indentations. (a) The required percentage is: (b) Indentation depths of Knoop to Vickers indentations. h 0:04572 D  100 ¼ 0:75  100 ¼ 6%: Solution (b) By rearranging (3.4), we can obtain diameter dup of the standard elastically unrecovered Monotron indentation in From (3.26), projected area of Knoop inÀdentÁation in terms of its longitudinal diagonal length l is: Aup K¼ 0:07028 l2: M-1 scale as follows: Projected areÀa ofÁVickers indentation in terms of its diagonal length D is: Ap V¼ a2 ¼ D2 2; [where a is length of each qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arm of the square base of the Vickers indentation and dup ¼ qffiDffiffiffi2ffiffiffiÀffiffiffiffiffiðffiffiDffiffiffiffiÀffiffiffiffiffi2ffiffihffiffiffiÞffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 ¼ 2a2] ¼ 0:752 À ð0:75 À 2  0:04572Þ2 mm ¼ 0:3589 mm % 0:36 mm: (a) Given that l ¼ D; so the required ratio is:  (c) Since given Monotron hardness ¼ 100 kg mm2; so the ÀÁ l2 applied load P is obtained from (3.28) as follows: ÀAupÁ K 0:07028 Ap V ¼ D2=2 ¼ 0:14056 ¼ 14%: (b) From (3.23), indentation depth of Knoop indentation in P ¼ Monotron hardness  pdu2p ¼ 100  p  0:35892 kg terms of its longitudinal diagonal length l is: hK ¼ l=30:514: 44 With reference to Fig. 3.11b, indentation depth of Vickers indentation in terms of its longitudinal diagonal length D is: ¼ 10:117 kg: hV ¼ 2 a 68 ¼ pffiffi D 68 ¼ D: Hence, the BHN of the material corresponding to the tan 22 tan 7 elastically unrecovered indentation depth, h ¼ 0:04572 mm, can be computed from (3.3b): Since l ¼ D; so the required ratio is: BHN ¼ p93DP:h9k¼gpmÂm02 :%71509:Â1410k7:g04m5m72À2k:gmm2 ¼ hK l=30:514 hV ¼ D=7 ¼ 0:229 % 23%: 3.18.7. If Monotron hardness tester is used to indent a Exercise material up to the standard penetration depth using indenter of M-1 scale and Monotron hardness of the material is found 3.Ex.1. Rockwell hardness test performed on a material with to be 100 kg mmÀ2; compute the Brale indenter shows that after application of the total load of 100 kg, the large pointer moves to number 50 on black (a) Percentage of the indenter diameter that equals the depth dial and after removal of the major load, the large pointer of the standard elastically unrecovered impression. moves to number 65 on the same dial. What is the (b) Diameter of the standard elastically unrecovered impression. (a) Rockwell hardness of that material? (c) Applied load in kg and the corresponding BHN (for (b) Depth difference between indentation upon withdrawal elastically unrecovered indentation) of the material. of the applied major load with the minor load still applied and indentation made by the minor load? Solution (c) Difference between the depth of indentation upon application of the total load and that due to application of the Standard depth of penetration in Monotron hardness test is: minor load? (d) Depth of indentation recovered elastically? h ¼ 0:0018 in: ¼ ð0:0018  25:4Þ mm ¼ 0:04572 mm: The diameter of Monotron indenter in M-1 scale is 3.Ex.2. Rockwell superficial hardness test is performed on a D ¼ 0:75 mm: specimen with 1/2 in. diameter steel ball indenter under a

3.18 Solved Problems 169 major load of 30 kg. If the depth difference between 3.Ex.10. Assume that the flow curve of copper in the region indentation upon withdrawal of the applied major load with of uniform plastic deformation is described by rðMPaÞ ¼ the minor load still applied and indentation made by the 317 Á e0:54: Compute the engineering stress corresponding to minor load is found to be 0.055 mm, find the Rockwell the condition in which the angle between the tangents drawn superficial hardness of that specimen. to the edge of a Brinell indentation formed on copper is 140°. Use Tabor’s relation to find true strain. 3.Ex.3. A standard Brinell hardness test is performed on a cast iron specimen. If the angle between the tangents drawn 3.Ex.11. Suppose a depth-sensing instrument is used to to the edge of the Brinell indentation formed on the cast iron perform microhardness indentation test on a sample with specimen is the same as the angle between two opposite Vickers indenter under a load of 10 g. If the indentation faces of any ideal Vickers indentation, what will be the BHN depth after release of the applied load is found to be 1 lm; of the above cast iron specimen? what will be the Vickers hardness? If it is assumed that the same indentation depth under the same applied load is 3.Ex.4. If the Rockwell superficial hardness number of a obtained when the sample is indented with Knoop indenter, material is reported to be 30 W-45, mention the size, shape then what would be the Knoop hardness? Compare it with and material of the indenter, the value of minor load and Vickers hardness. total load used in the above test. If the average diameter of the above indentation on that metal is measured with an 3.Ex.12. optical microscope after the release of total load and found to be 0.9 mm, calculate the depth due to application of minor (a) If the Vickers microhardness of the sample estimated in load in mm, neglecting very small elastic recovery that problem 3.Ex.11 is same as the Vickers nanohardness, occurs upon withdrawal of the minor load. compute the contact depth of indentation based on the basic principle of nanoindentation. 3.Ex.5.The Meyer hardness of a fully annealed metal is (b) What would be the nanohardness with Berkovich or 140 kg mm2; under an applied load of 1000 kg with Vickers indenter if it is assumed that the contact depth of 7-mm-diameter ball indenter. If the same metal is tested with nanoindentation is 1 lm under the applied load of 10 g? 10-mm-diameter ball indenter under the same load, what would be the Meyer hardness of the above metal? 3.Ex.13. Indicate the correct or most appropriate answer from the following multiple choices: 3.Ex.6. Assume that Knoop hardness number is based on the measurement elastically unrecovered surface area instead of (a) A Brale indenter used in Rockwell hardness test has the elastically unrecovered projected area of indentation. Under following geometry and material: such condition derive the Knoop hardness number in terms of the applied load, P; and the longitudinal diagonal length, (A) Square-based diamond pyramid with 136° included l; of the Knoop indenter. angle; (B) Conical shape steel with 120° apex angle; 3.Ex.7. The respective indentation diameters on the test and (C) Conical-shaped diamond with 120° apex angle; the standard specimens are measured to be 3 and 2 mm (D) 10-mm-diameter hardened steel ball. when an indenter of 10 mm diameter is used in a Poldi hardness test. If the Meyer hardness of the standard speci- (b) If Mohs’ hardness number of mica is 3.5, it can scratch men is 200 kg/mm2, determine the following: (A) Apatite; (B) Feldspar; (C) Fluorite; (D) Calcite. (a) Meyer hardness of the test specimen. (b) BHN of the test specimen. (c) If a material scratches Feldspar and is scratched by (c) True stress and true strain of the test specimen using Quartz, the hardness of the material lies between Mohs’ Tabor’s relation. hardness numbers of 3.Ex.8. Calculate the Monotron hardness of a material in (A) 3 and 4; (B) 4 and 5; (C) 5 and 6; standard M-3 scale under an applied load of 21.32 kg. (D) 6 and 7; (E) 7 and 8. 3.Ex.9. Using Tabor’s relation, calculate the value of % true (d) With the increase of applied load, the Meyer hardness strain corresponding to the standard penetration depth on a number for a given indenter size in case of a fully annealed specimen by a Monotron hardness tester using indenter of metal M-2 scale. (A) Remains constant; (B) First increases and then remains constant; (C) First increases and then decreases; (D) Decreases.

170 3 Hardness (e) Out of the following cubic-shaped unclamped specimens, Answer to Exercise Problems the lowest Shore scleroscope hardness value will be shown by 3.Ex.1. (a) RD 65; (b) 0.07 mm; (c) 0.1 mm; (d) 0.03 mm. 3.Ex.2. 30Y − 45 (A) Copper weighing 400 g; (B) Copper weighing 4 kg; 3.Ex.3. 261:7 kg mm2: (C) Lead weighing 400 g; (D) Lead weighing 4 kg. 3.Ex.4. 1=8 inchdiameter steel ball, 3 and 30 kg; 0.01 mm. 3.Ex.5. Â121:4 kg mm2: Ã  (f) The hardness test, where the direction of load application 3.Ex.6. P ð0:07774l2Þ kg mm2: is parallel to specimen surface, is  (A) Shore scleroscope; (B) Poldi; 3.Ex.7. (a) 88:89 kg mm2; (b) 86:84 kg mm2; (C) Knoop; (D) Herbert Pendulum. (c) 290.55 MPa, and 6%. 3.Ex.8. 100 kg mm2: (g) The hardness test based on the principle of variable load 3.Ex.9. 8.36%. and fixed depth of indentation is 3.Ex.10. 69.55 MPa.   (A) Rockwell; (B) Vickers; (C) Meyer; 3.Ex.11. VHN ¼ 378:3 kg mm2; KHN ¼ 152:8 kg mm2; (D) Knoop; (E) Monotron. KHN : VHN ¼ 0:4 : 1:  3.Ex.12. (a) 1:04 lm; (b) 408:16 kg mm2: (h) With the increase of applied load, the Brinell hardness 3.Ex.13. (a) (C) Conical-shaped diamond with 1200 apex number for a given indenter size in case of a heavily cold-worked metal angle. (b) (D) Calcite. (c) (D) 6 and 7. (d) (B) First increases and then remains constant. (e) (C) Lead weighing 400 g. (A) Remains constant; (f) (D) Herbert Pendulum. (g) (E) Monotron. (h) (D) (B) First increases and then remains constant; Decreases. (i) (D) P 0:07028 l2: (j) (C) First increases and (C) First increases and then decreases; (D) Decreases. then decreases. (k) (A) Remains constant. (i) Knoop hardness number is expressed by the formula: References À  Á Bain, E.C., Paxton, H.W.: Alloying elements in steel. Am. Soc. Met. (A) P=ðwl=2Þ; (B) P 7:114w2 2 ; 225 (1966) (C) ð1:854 PÞ l2; (D) P 0:07028 l2: Berkovich, E.S.: Three-faceted diamond pyramid for micro-hardness where P ¼ applied load, l ¼ longitudinal diagonal length of testing. Ind. Diamond Rev. 11(127), 129–133 (1951) Knoop indentation and w ¼ transverse diagonal length of Knoop indentation. Bolshakov, A., Pharr, G.M.: J. Mater. Res. 13, 1049 (1998) Bulychev, S.I., Alekhin, V.P., Shorshorov, MKh, Ternovskii, A.P., (j) With the increase of applied load, the Brinell hardness number for a given indenter size in case of a fully annealed Shnyrev, G.D.: Determining young’s modulus from the indenter metal penetration diagram. Zavod. Lab. 41(9), 1137–1140 (1975) Cahoon, J.R.: An improved equation relating to ultimate strength. (A) Remains constant; Metall. Trans. 3, 3040 (1972) (B) First increases and then remains constant; Cahoon, J.R., Broughton, W.H., Kutzak, A.R.: The determination of (C) First increases and then decreases; yield strength from hardness measurements. Metall. Trans. 2, 1979– (D) Decreases 1983 (1971) Crowe, T.B., Hinsely, J.F.: J. Inst. Met. 72, 14 (1946) (k) With the increase of applied load, the Meyer hardness Davis ,H.E., Troxell, G.E., Wiskocil, C.T.: The testing and inspection number for a given indenter size in case of a heavily of engineering materials, 3rd edn, pp. 188–189, 205-206, 209. cold-worked metal McGraw-Hill Book Company, Inc., New York (1964) Doerner, M.F., Nix, W.D.: A method for interpreting the data from (A) Remains constant; depth-sensing indentation instruments. J. Mater. Res. 1(4), 601–609 (B) First increases and then remains constant; (1986) (C) First increases and then decreases; Fischer-Cripps, A.C.: (2011) Nanoindentation test standards. In: (D) Decreases. Nanoindentation, 3rd edn., p. 186. Mechanical Engineering Series I. Springer Science + Business Media, New York Greaves, R.H., Jones, J.A.: The ratio of the tensile strength of steel to the brinell hardness number. J. Iron Steel Inst. 113(1), 335–353 (1926) Herbert, E.G.: Work-hardening properties of metals. Trans. ASME 48, 705–745 (1926)

References 171 Hoyt, S.L.: The ball indentation hardness test. Trans. Am. Soc. Steel Pharr, G.M.: Measurement of mechanical properties by ultra-low load Treat. 6, 396–420 (1924) indentation. Mater. Sci. Eng., A 253, 151–159 (1998) Kehl G. L. (1949). The principles of Metallographic laboratory Pharr, G.M., Bolshakov, A.: J. Mater. Res. 17, 2660 (2002) practice, 3rd edn., pp. 226, 249–255. McGraw-Hill Book Company, Pharr, G.M., Oliver, W.C., Brotzen, F.R.: On the generality of the Inc., New York (Indian Reprint by Eurasia Publishing House Pvt. Ltd, New Delhi: 1965) relationship among contact stiffness, contact area, and the elastic modulus during indentation. J. Mater. Res. 7(3), 613–617 (1992) Knoop, F., Peters, C.G., Emerson, W.B.: A sensitive Rockwell, S.P.: The testing of metals for hardness. Trans. Am. Soc. pyramidal-diamond tool for indentation measurements. Research Steel Treat. 11, 1013–1033 (1922) paper RP1220. J. Res. Natl. Bur. Stand. 23, 39–61 (1939) Tabor, D.: The hardness of metals, pp. 67–76. Oxford University Press, New York (1951a) Lenhart, R.E.: WADC Tech. Rept. 55–114 (1955) Tabor, D.: The hardness and strength of metals. J. Inst. Met. 79(1–18), Meyer E.: Contribution to the knowledge of hardness and hardness 464–474 (1951b) Taylor, W.J.: The hardness test as a means of estimating the tensile testing. Zeits. d. Vereines Deutsch. Ingenieure 52, 645–654, strength of metals. J. R. Aeronaut. Soc. 46(380), 198–209 (1942) 740-748, 835-844 (1908) Tsui, T.Y., Oliver, W.C., Pharr, G.M.: Indenter geometry effects on the Natl. Bur. (1946). National Bureau of Standards Specifications for measurement of mechanical properties by nanoindentation with Knoop Indenters, Letter Circular LC819, Apr. 1, 1946 sharp indenters. In: Thin Films–Stresses and Mechanical Proper- O’Neill, H.: The hardness of metals and its measurement. Sherwood ties VI, MRS Symposium Proceedings, Materials Research Society, Press, Cleveland, Ohio (1935) vol. 436, pp. 147–152 (1997) O’Neill, H.: Hardness measurement of metals and alloys, 2nd edn. Vander Voort, G.F.: Metallography Principles and Practice. ASM Chapman & Hall Ltd., London (1967) International, Materials Park, Ohio, p. 370 (1999) Oliver, W.C., Pharr, G.M.: An improved technique for determining Wahlberg A.: Brinell’s method of determining hardness and other hardness and elastic modulus using load and displacement sensing properties of iron and steel. J. Iron Steel Inst. 59(I), 243–298; 60(II), indentation experiments. J. Mater. Res. 7(6), 1564–1583 (1992) 234–271 (1901) Oliver, W.C., Pharr, G.M.: Measurement of hardness and elastic Williams, S.R.: Hardness and hardness measurements. ASM, Cleve- modulus by instrumented indentation: advances in understanding land, Ohio (1942) and Refinements to methodology. J. Mater. Res. 19(1), 3–20 (2004) Zwikker, C.: Physical properties of solid materials, p. 28. Interscience Petty, E.R.: Relationship between hardness and tensile properties over a wide range of temperature for aluminium alloys. Metallurgia 65, Publishers Inc, New York (1954) 25–26 (1962)

Bending 4 Chapter Objectives • Bending stresses, flexure formula and experimental method in pure bending. • Beam design from economical and strength considerations of cross-sectional shapes. • Modulus of elastic resilience and important variables affecting modulus of rupture. • Yielding: discontinuous yielding and shape factor. • Nonlinear stress–strain relationships for materials either deforming in the plastic range or exhibiting nonlinear elastic deformation. • Shear stresses in elastically bent beam subjected to non-uniform bending moment. • Problems and solutions. 4.1 Introduction confined to the assessment of bending stresses developed in a transversely loaded beam. A beam is a suitably supported Majority of structural members used in engineering appli- structural member that has length reasonably higher than its cations are subjected to some bending. Any eccentricity in lateral dimensions and undergoes bending under the appli- load applied in the longitudinal direction of a member may cation of transverse loads. A beam under transverse loading cause its bending in the lateral direction. A member is said to will be subjected to bending moment which will induce be loaded in bending if one side of it is extended, i.e. under stresses in the beam, and these stresses are called bending tension, while the other side of it is shortened, i.e. under stresses. Bending action in beams subjected to transverse compression. The reaction of the member in bending is to loading is generally referred to as ‘flexure’. Structural develop internal stresses that will oppose its extension and members subjected to bending in actual service conditions compression. Thus, on each cross-section of the member experience mostly elastic deformation with occasional neg- subjected to bending, there will be normal stresses that will ligible plastic flow. In such conditions, there is either no change from tension on one side to compression on the change or very small changes in cross-sectional areas, which other. Further, there will be variation in the magnitude of can be neglected, i.e. the cross-sectional areas are assumed to tensile or compressive normal stresses across the remain approximately constant during deformation and so, cross-section of the member. the engineering stress, S % the true stress, r. Further, during elastic deformation, the engineering strain, e % the true In actual services, structures and members are subjected to strain, e. Hence, the stress will be designated by the term r bending that may be associated with direct stress, torsional and the strain by the term e, subsequently in this chapter, shear or transverse shear. However, to know the behaviour of although initial cross-sectional areas of the members are materials subjected to bending, the discussion will be mostly used to determine the stresses. © Springer Nature Singapore Pte Ltd. 2018 173 A. Bhaduri, Mechanical Properties and Working of Metals and Alloys, Springer Series in Materials Science 264, https://doi.org/10.1007/978-981-10-7209-3_4

174 P 4 Bending P 4.2 Pure Bending A C B The simplest form of bending is the pure bending, where the x beam is free from shear force and the bending moment a applied to the beam is constant. A beam is subjected to pure P D bending when it is loaded in such a way that two equal and a opposite couples, say ‘MB’, act at its ends, as shown in y P Fig. 4.1. Although the shearing stresses are zero and only P uniaxial tensile or compressive normal stresses are present P on the cross-section of the beam, there is a stress gradient across the cross-section with respect to these normal stresses. For convenience, the beam may be regarded as made of longitudinal elements of infinitesimal cross-section or ‘fi- bres’, each of which is in a state of uniaxial tension or compression. 4.2.1 Bending Stresses and Flexure Formula Let us consider a transversely loaded beam ‘AB’, along with Pa Pa its shear force and bending moment diagrams, as shown in Fig. 4.2. This figure shows that two transverse loads of equal Fig. 4.2 A transversely loaded beam along with its shear force and magnitudes, say ‘P’, act at equal distances, say ‘a’, from the bending moment diagrams end supports of the beam, i.e. at points ‘Ç’ and ‘D’ of the beam. Figure 4.2 also shows that the bending moment MB ¼ (4) The bending deformation is uniform in the portion P a; remains uniform along the length ‘ÇD’ and the shear ‘CD’ of the beam, i.e. this portion ‘CD’ takes the shape force is also absent there. So, the middle portion ‘CD’ of the of a circular arc, as shown in Fig. 4.3. This assumption beam is subjected to pure bending. is quite reasonable because the applied bending moment is uniform along the length ‘ÇD’ of the beam. To study the state of internal stresses caused by pure bending and the deformations taking place within the (5) Each cross-section of the beam that was initially plane material, the following assumptions are made. For more before the deformation will remain plane and normal to complicated cases of bending, the readers may consult texts the longitudinal fibres of the beam in its deformed on mechanics of materials (Popov 1990; Timoshenko 1955, configuration, not only during its elastic bending but 1956). also beyond its elastic range. For the case of pure bending, this assumption has been found to hold good, (1) The beam made of material is homogeneous and fol- as confirmed by the accurate strain measurements in the lows Hooke’s law. Its modulus of elasticity in tension, laboratory. E, is the same as that in compression, EC. (6) The method of the load application and the supports to (2) The beam made of material is prismatic, and it has an the beam are such that no constraint is introduced in the axial plane of symmetry, which is taken as plane ‘xy’, longitudinal direction of the beam. as shown in Fig. 4.2. As a result of bending deformation occurring under the (3) The loads are applied in the above-mentioned plane of application of bending moment, upper fibres on the concave symmetry so that no twisting occurs and so that side of the beam are shortened while lower fibres on the bending takes place only parallel to this plane. convex side of the beam are extended that can be visualized in Fig. 4.3. Somewhere in between the compressed top MB MB fibres and the elongated bottom fibres of the beam, there Fig. 4.1 Beam subjected to pure bending exists a bed of fibres that are neither in compression nor in tension and remain unchanged in length. This layer of fibres representing the location of zero strain is called the neutral plane or surface of the beam. All fibres on the convex side of the neutral plane are in tension while those on the concave

4.2 Pure Bending 175 Fig. 4.3 ‘CD’ portion of the beam, shown in Fig. 4.2, takes the shape of a circular arc due to the action of pure bending and as a result, cross-sections of the beam rotate side are in compression. So, it can be stated that the elon- The initial orientation of the cross-section ‘pq’ before gated fibres are located at a positive distance from the neutral deformation is indicated in Fig. 4.3 by a line ‘p′q′’ drawn plane along the positive transverse direction of y-axis and the parallel to the cross-section ‘mn’ through the point ‘b’ on the longitudinal tensile strain in ‘x’-direction experienced by neutral axis of the beam, at which the cross-section ‘pq’ them is positive, while the compressed fibres are located at a intersects. Since p0q0 k mn; the angle formed between the negative distance from the neutral plane and the longitudinal initial cross-section ‘p′q′’ and the rotated cross-section ‘pq’ compressive strain undergone by them is negative. The at their intersection point ‘b’ = the angle between the adja- intersection of this neutral plane with the axial plane of cent planes ‘mn’ and ‘pq’ at their intersection point ‘O’ ¼ symmetry ‘xy’ of the beam is called the neutral axis of the dh: From the figure, it is clear that after rotation of the initial beam, which passes through its centroid that will be proved cross-section ‘p′q′’ to ‘pq’, the segment of fibre between the subsequently. Again, the intersection of this neutral plane two cross-sections ‘mn’ and ‘p′q′’ has increased in the with the plane of any cross-section is called the neutral axis positive direction of y-axis from their initial separation ab ¼ of that cross-section. dx ¼ cd0 to ‘cd’ at any arbitrary distance ‘y’ from the neutral plane. This segment at any arbitrary distance ‘y’ is longitu- It is to be noted that the bending action causes dinally elongated in the ‘x’-direction by the amount d0d ¼ cross-sections of the beam to rotate, and Fig. 4.3 shows that ydh: Since cd0 ¼ dx and from (4.1), dh=dx ¼ 1=q; the lon- the rotation causes one extreme cross-section of the beam to gitudinal tensile strain, ex; in the ‘x’-direction is given by make an angle ‘h’ with another extreme cross-section at their intersection point ‘O’. Let us consider a short segment of the ex ¼ d0d ¼ ydh ¼ y ð4:2aÞ beam contained between the planes of two adjacent cd0 dx q cross-sections. After bending deformation, these two adja- cent cross-sections ‘mn’ and ‘pq’ rotate and intersect at point If we consider any fibre on the concave side of the neutral ‘O’, as evident from Fig. 4.3. Let the intersection point of the neutral axis of the beam with the cross-section ‘mn’ is ‘a’ plane, the longitudinal compressive strain exC ; in the and that with the cross-section ‘pq’ is ‘b’. ‘x’-direction will be given by If dh ¼ the angle between these adjacent planes ‘mn’ and exC ¼ Ày ð4:2bÞ ‘pq’ at their intersection point ‘O’; q dx ¼ the spacing between these adjacent cross-sections According to Hooke’s law, the longitudinal tensile stress ‘mn’ and ‘pq’ along the neutral axis of the beam ¼ ab; and rx; in each fibre is q ¼ the radius of curvature of the neutral axis of the rx ¼ Eex ¼ E y ð4:3aÞ beam, then q dh ¼ dx ð4:1Þ q

176 4 Bending (a) (b) (c) εxc σxc Compression (–) (–) 0 0 Neutral Neutral surface axis y (+) (+) Tension εx σx Beam cross-section Fig. 4.4 Linear variations of longitudinal fibre a strains ex in tension and exC in compression and b stresses rx in tension and rxC in compression with distance ‘ý’ from the neutral surface. c Transverse curvature of beam Fig. 4.5 Diagrams to determine (a) (b) the location of the neutral axis ‘Oz’ of cross-sections, which are c c2 a symmetrical with respect to the c1 neutral axis and b not symmetrical with respect to the neutral axis O z O z y dA y dA c yy Similarly, the longitudinal compressive stress rxC ; in each distance y from its neutral axis is denoted by dA; then the fibre is given by elemental normal force on this area in the cross-section is rxC ¼ ECexC ¼ À EC y ð4:3bÞ rxdA: Now using (4.3a), it can be written as q ZZ E Equations (4.3) and (4.2) show, respectively, that the Nx ¼ rxdA ¼ q y dA ð4:4Þ longitudinal fibre stresses rx in tension and rxC in com- pression and strains ex in tension and exC in compression vary AA linearly with distance ‘y’ from the neutral surface, as long as Hooke’s law is obeyed by the material. Such distributions of Since for pure bending there must be no resultant normal stress and strain over the depth of the beam are shown in force Nx on the cross-section of the beam, (4.4) becomes Fig. 4.4a, b, where jrxj ¼ jrxC j is taken according to the assumption, E ¼ EC: It is to be noted that the fibre stresses E Z are normal to the cross-section of the beam. q Nx ¼ y dA ¼ 0 ð4:5Þ Referring to Fig. 4.5, the location of the neutral axis ‘Oz’ A of the cross-section can be found by equating the summation Since E ¼6 0; it is concluded from (4.5) that ð4:6Þ of the elemental normal forces with the resultant normal q force, say Nx; on the cross-section of the beam. If the area of Z an element of the cross-section at an arbitrary positive y dA ¼ Ayc ¼ 0 A

4.2 Pure Bending 177 where A is the total area of the cross-section and yc is the extreme fibres in tension and compression, then from (4.9a) distance from the centroid of the cross-section to its neutral and (4.9b), we obtain axis. Again since A 6¼ 0; it is concluded from (4.6) that yc ¼ 0: Maximum bending stress; ðrxÞmax¼ MBc1 ð4:10aÞ Therefore, the neutral axis of the cross-section passes through Iz its centroid. The fibre stress acting normal to any cross-section of the stress; ðrxÞmin¼ MBc2 ð4:10bÞ Iz beam caused by the applied bending moment MB is found Minimum bending from the following consideration. The moment of the ele- mental normal force rxdA at an arbitrary positive distance Due to the imposed maximum and minimum bending y from the neutral axis of the cross-section with respect to stresses as mentioned above, there must be a maximum longitudinal tensile strain on the convex extreme face of the that axis produces the elemental bending moment, denoted beam and a maximum longitudinal compressive strain on the concave extreme face. Denoting the former by the maximum by dMB; where dMB ¼ yrxdA: The summation of these and the latter by the minimum (as it is negative) longitudinal elemental bending moments over the total area A of strain, we obtain by substituting the value of 1=q from (4.8) into (4.2a) and (4.2b), cross-section must give rise to the bending moment MB on that cross-section. Therefore, with help of (4.3a), ZZ E Z E q q MB ¼ dMB ¼ yrxdA ¼ y2dA ¼ Iz ð4:7Þ Maximum longitudinal strain; ðexÞmax¼ c1 ¼ MBc1 q EIz AA A where Iz ¼ the moment of inertia ofRthe cross-sectional area ð4:11aÞ about its neutral axis, i.e. z-axis ¼ A y2dA: From (4.7), the curvature 1=q of the neutral axis of the beam is Minimum longitudinal strain; ðexÞmin¼ c2 ¼ MBc2 q EIz 1 ¼ MB ð4:11bÞ q EIz ð4:8Þ Let Iz=c1 ¼ Z1 and Iz=c2 ¼ Z2; where Z1 and Z2 are called the section moduli. Hence, (4.10) and (4.11) can also where the quantity ‘EIz’ is called the flexural rigidity of the take the following forms: beam. It includes both the stiffness of the material as mea- sured by E and the proportions of the cross-sectional area as ðrxÞmax¼ MB ð4:12aÞ measured by Iz: Equation (4.8) shows that the curvature of Z1 the neutral axis of the beam is directly proportional to the applied bending moment and inversely proportional to the and flexural rigidity of the beam. ðrxÞmin¼ MB ð4:12bÞ By substitution of the value of 1=q from (4.8) into (4.3a), Z2 we get rx ¼ Ey MB ¼ MBy ð4:9aÞ ðexÞmax¼ MB ð4:13aÞ EIz Iz EZ1 Similarly from (4.3b), and rxC ¼ À MBy ð4:9bÞ ðexÞmin¼ MB ð4:13bÞ Iz EZ2 Equations (4.9) show that the bending stresses reach the If the cross-section of the beam is symmetrical with maximum values in those fibres farthest away from the neutral plane. So, there will be a maximum tensile stress on respect to its neutral axis, i.e. centroidal axis, such as square, the convex extreme face of the beam and a maximum rectangular or circular cross-section, then c1 ¼ c2 ¼ cðsayÞ compressive stress on the concave extreme face. Since the and Z1 ¼ Z2 ¼ ZðsayÞ; and in such cases, the bending maximum compressive stress value is negative, it is denoted stresses as well as the longitudinal strains in the extreme as the minimum bending stress. If c1 and c2 are, respectively, fibres in tension and compression are equal, as shown below: the absolute values of distance from the neutral plane to the ðrxÞmax ¼ ðrxÞmin ¼ MBc ¼ MB ð4:14Þ Iz Z

178 (b) (c) (d) 4 Bending z (a) b c2 z a a D/2 h/2 z c1 z D/2 h/2 yy y y Fig. 4.6 Figures to explain section moduli for different cross-sections ðexÞmax ¼ ðexÞmin ¼ c ¼ MBc ¼ MB ð4:15Þ lateral strain in the z-direction due to Poisson’s effect, just q EIz EZ like in uniaxial tension and compression. This lateral strain in the z-direction is given by ez ¼ Àm Á ex; and similar to ex; Each of (4.10) as well as (4.14) is called the flexure the variation of ez will be always linear across the formula. cross-section of the beam. Thus, the fibres on the concave side of the neutral surface, i.e. the compression side of the For a square cross-section of side ‘a’ shown in Fig. 4.6a, beam, expand laterally, and those on the convex side, i.e. the tension side, contract laterally. As a result, the beam c1 ¼ c2 ¼ c ¼ a ; so; Iz ¼ a4 and Z1 ¼ Z2 ¼ Z ¼ a3 becomes narrower on the tension side and wider on the 2 12 6 compression side compared to the neutral surface that remains unchanged in dimensions, as shown in Fig. 4.4c. ð4:16aÞ Thus, a transverse curvature of the beam is developed in a direction opposite to the longitudinal curvature. The above For a rectangular cross-section of depth ‘h’ and width ‘b’, two opposite curvatures interfere with each other, and this as in Fig. 4.6b, interference increases with increase of width-to-depth ratio of the beam. If the beam is made wider keeping its depth c1 ¼ c2 ¼ c ¼ h ; so; Iz ¼ bh3 and fixed, these two opposite curvatures will become incom- 2 12 patible, which will cause the transverse curvature to remain bh2 ð4:16bÞ little or vanish. 6 Z1 ¼ Z2 ¼ Z ¼ So far, the theory for the case of pure bending has been developed. But shear force is present at each cross-section of In case of a circular cross-section of diameter, ‘D’, as in the beam when the applied bending moment varies from one Fig. 4.6c, cross-section to another. This results in a non-uniform bending of the beam which induces shearing stresses in the c1 ¼ c2 ¼ c ¼ D ; so; Iz ¼ pD4 and material unlike in the case of pure bending. The non-uniform 2 64 bending of the beam combined with these shearing stresses pD3 ð4:16cÞ causes distortion of the different cross-sections so that each 32 cross-section of the beam that was initially plane before the Z1 ¼ Z2 ¼ Z ¼ deformation will not remain plane in its deformed configu- ration. An extensive analysis of this complicated problem If a beam with the trapezoidal cross-section as shown in shows that the normal bending stresses, as evaluated from Fig. 4.6d is bent convex downwards, the maximum tensile (4.9) for pure bending, do not vary much due to the exis- stress in the fibres of the convex extreme face will be lower tence of these shear stresses. Accordingly, it is reasonable to than the maximum compressive stress in the fibres of the apply the theory of pure bending for estimation of the nor- concave extreme face, as c1 < c2 and consequently, Z1 > Z2. mal bending stresses and usually adopted in practice. Since a brittle material is usually stronger in compression than in tension, the trapezoidal cross-section of the beam made of brittle materials like cast iron or concrete may prove to be advantageous. So far we have discussed about the longitudinal strain of the fibres in the x-direction, but the beam also experiences a

4.2 Pure Bending 179 4.2.2 Experimental Method the test-beam supports, although any equal location dis- tances can be used. In order to fulfil the requirement of For experimental investigation of materials’ behaviour under loading in a vertical plane of symmetry, the loading beam pure bending, the first requirement is to satisfy the condi- and the test-beam as well as the supporting rollers must be tions mentioned earlier which are: precisely machined and the rollers remain normal to the axes of all beams. The supports of the test-beam in the form of • Loading in an axial plane of symmetry. rollers provide freedom from longitudinal restraint. • Freedom from constraint in the longitudinal direction of Local stresses arise near the loading points, but they the beam. disappear at a short distance from the loading points; this • Application of uniform bending moment with zero shear distance is nearly equal to the depth of the test-beam. The minimum length for the centre region of the test-beam force in the part of the beam under investigation. should be 4 times its depth so that the region free from local stresses is long enough. A typical experimental set-up for pure bending is schematically illustrated in Fig. 4.7. A loading beam that Load and either deflection or strain are usually recorded rests on two rollers on the top of beam to be tested is used to in the test. The deflection is measured in the portion of the apply the loads. Accurate spacing of the supports and beam that is subjected to pure bending. Device for measur- loading points is necessary to provide constant bending ing deflection in pure bending is shown in Fig. 4.8. The moment with zero shear force. A load, say P; is applied to device is attached to the test-beam by pins or setscrews the loading beam accurately at the mid-point between its two within the pure-bending region at equal distances from the supporting rollers. These rollers in turn must be spaced loading points and at the neutral plane in order to eliminate accurately at equal distances from the supporting rollers for the effects of shear in the end regions. A standard calibrated the beam to be tested. If the distance between the supporting dial indicator is used to measure the deflection. If strain is to rollers of the test-beam is L; the supporting rollers of the be measured, strain gages are used and attached to the top loading beam are often located at L=3 or L=4 distances from and bottom of the test-beam. Fig. 4.7 A typical experimental P set-up (schematic) for pure Loading beam bending Test-beam Fig. 4.8 Frame to measure _P_ Loading points _P_ deflection in pure bending 2 2 Pivot Dial indicator

180 4 Bending 4.3 Beam Design in Pure Bending economical it is. Let us compare various cross-sectional shapes with respect to the economical aspect. In designing a beam subjected to pure bending, the choice of its cross-sectional shapes from the point of a satisfactory If ‘A’ denotes the cross-sectional area for any shape, the utilization of the strength of beam and the selection of its section modulus ‘Z’ of a rectangular cross-section from cross-sectional areas and shapes from an economical point of (4.16b) is view will be discussed. Z ¼ bh2 ¼ A h ð4:17aÞ (1) Strength considerations of cross-sectional shapes: 66 If strength of a material in tension is the same as that in com- Equation (4.17a) shows that as long as the section mod- pression, as usually observed in ductile metals, it will be rea- ulus ‘Z’ is kept constant, increasing the height ‘h’ of the sonable to select such cross-sectional shapes, which will be cross-section causes the area ‘A’ of the rectangular symmetrical with respect to the neutral axis of the cross-section, cross-section to decrease and make it more and more eco- i.e. the centroid of the cross-section must be at the centre of the nomical. However, the height ‘h’ can be increased up to a depth of the beam. Beam made of such materials like structural certain extent beyond which the rectangular cross-section steel, whose yield strength in tension is almost the same as in becomes so narrow that the beam collapses due to lateral compression, is designed by using the same factor of safety for buckling. For a discussion on lateral buckling of beams, the fibres in tension and for those in compression. reader may consult Timoshenko and Gere (1963). For brittle materials like cast iron or concrete, the strength From (4.16c), the section modulus ‘Z’ of a circular in tension is lower than that in compression. For them, the cross-section is preferable cross-sectional shapes are those which are not symmetrical with respect to the neutral axes of these Z ¼ pD3 ¼ A D ¼ 0:125AD ð4:17bÞ cross-sections. If c1 and c2 are, respectively, the distances 32 8 from the centroidal axis of the cross-section to the extreme fibres in tension and compression, the best cross-section to If the area ‘A’ of a square cross-section of side ‘a’ is the obtain equal strength in tension and compression must sat- isfy the condition, which is saa¼meðppffiaffiffisDÞ=t2h;aat ndoffromth(e4.16cair)c,utlhaer cross-section, then section modulus ‘Z’ of c1 ¼ strength of material in tension : c2 strength of material in compression a square cross-section will be For example, the dimensions of the bases of a trapezoidal pffiffiffi ð4:17cÞ section or those of the flange and web of a T-section may be Z ¼ a A ¼ p DA ¼ 0:148AD adjusted to obtain the centroid of the cross-section in such a location along the depth of the cross-section that the strength 6 12 of the extreme fibres of the beam in tension is equal to that in compression. Comparison of (4.17c) with (4.17b) shows that a square cross-section is more economical than a circular one. (2) Economical considerations of cross-sectional areas and shapes: We have already seen that the absolute value of the pure-bending stresses increases with distance from the neu- Along with the satisfactory fulfilment of the strength con- tral axis of the cross-section. So, it may be concluded that dition, the condition of economy in the cross-sectional area the beam design in pure bendingwill be economical, if the of the beam must also be considered. If the section modulus material of the beam is put at a distance as much as possible of the cross-sectional area is the same and the strength away from the neutral axis. If we consider a beam of a given condition is satisfied with the same factor of safety, then the cross-sectional area ‘A’ and depth ‘h’, the theoretical ideal smaller the cross-sectional area of the beam the more distribution for area ‘A’ would be to put each half of the area, i.e. A=2; at a distance h=2; from the neutral axis on both sides of it, as shown in Fig. 4.9. In this case, the moment of inertia Iz; of this cross-sectional area about its neutral axis, i.e. z-axis, will be Iz ¼ 2 Â ðA=2Þ Â ðh=2Þ2¼ ðAh2Þ=4 and the section modulus Z will be given by Z ¼ Iz ¼ 1 Ah ð4:17dÞ h=2 2

4.3 Beam Design in Pure Bending 181 _A_ 4.4 Linear Elastic Behaviour 2 The mechanical behaviour of a material is always repre- h/2 sented by some kind of variables like force and deformation z variables. In pure bending, the force variable used may be h/2 the applied bending moment MB; the applied load P; or some particular value of stress like the maximum bending _A_ stress ðrxÞmax; because the stress varies across the 2 cross-section of the beam. Similarly, the deformation vari- able used may be the maximum deflection or a particular Fig. 4.9 Theoretical ideal distribution for a given cross-sectional area value of strain like the maximum bending strain ðexÞmax: The ‘A’ and depth ‘h’ of a beam for economical design maximum bending strain is simple to measure because the bending strain varies always linearly across the cross-section Equation (4.17d) shows the limiting condition for an ideal case, which can never be achieved in practice. But this of the beam. The maximum deflection is also widely condition may be approached by an I-section with wide flange and narrow web, where most of the material is put in employed because it is easy to measure. the flange section and the minimum necessary part of the A typical bending moment–strain diagram for a ductile material is put in the web section. For such an I-section, we may have the section modulus as material subjected to pure bending is shown in Fig. 4.10. For convenience of comparison, the tensile stress–strain diagram Z ¼ 1 Ah ð4:17eÞ for the same material is included in Fig. 4.10. The scales along 3 the ordinate of this figure have been so adjusted that the loca- tion of the bending moment at the proportional limit ðMBÞpl From the comparison of (4.17e) with (4.17a), it is clear coincides with that of the tensile stress at the proportional limit that for maintaining the same value of the section modulus rpl (rpl is used instead of Spl; because of negligible change in ‘Z’, an I-section requires less area ‘A’ than a rectangular cross-sectional area during elastic deformation). This point of section of the same depth ‘h’ and so an I-section is more coincidence, ‘A’, in this figure represents the end of elastic economical than a rectangular section of the same depth. region. The extension of the linear elastic line ‘OA’ in Moreover, when the lateral buckling is concerned, the sta- Fig. 4.10 towards the point ‘B’ corresponds to the values of bility of an I-section will be more because of its wide flanges bending moment or tensile stress that would exist if the than that of a rectangular section having the same depth and material proceeded to respond elastically. If Hooke’s law is section modulus. obeyed by the material, the equation of the linear elastic line ‘OA’ in Fig. 4.10 is obtained as follows from (4.18), assuming that the beam has a symmetrical cross-section. MB ¼ EIz ðexÞmax ð4:18Þ c Fig. 4.10 Comparison of MB , σ Moment–strain diagram bending moment–strain and Stress–strain diagram tensile stress–strain diagrams for B a ductile material Y' Proportional Y limit A O ε1 εmax , ε Offset

182 4 Bending The modulus of elasticity E; used in (4.18) has the same Test results on cast-iron beams of various shapes but value as that obtained in uniaxial tension or compression. having approximately equal cross-sectional area show that the modulus of rupture is lower for beams in which a rela- For a perfectly brittle material, the flexure formula given tively greater proportion of the cross-sectional area is con- centrated near the extreme fibres, such as an I-section beam, by (4.14) can be applied all the way to rupture. If the rupture although the breaking loads for such sections are signifi- cantly larger (Withey and Washa 1954). moment, i.e. bending moment, at the point of rupture for a brittle material is ðMBÞr; then the bending stress required to Test results on both concrete and cast iron have shown cause rupture at the outermost fibre, also called the modulus that strengths are lower for beams having larger of rupture ðrmaxÞr; will be given by cross-sectional dimensions (MacKenzie 1931; Kellerman 1933; Wright 1952). Modulus of rupture; ðrmax Þr ¼ ðMB Þr c ð4:19Þ Iz In general, the effect of testing speed in the flexure test is the same as that in the tension and compression tests. If the Equation (4.19) is commonly used to evaluate the max- rate of stressing is increased from 138 to 7860 kPa (20–1140 imum bending stress required to cause rupture of brittle psi) per min, then the modulus of rupture for concrete beams plastics and other brittle materials like cast iron, ceramics, increases by about 15% (Wright 1952). concrete and wood which are not perfectly brittle and usually undergo inelastic deformation prior to their fracture, 4.4.2 Modulus of Elastic Resilience although the actual rupture stresses for these materials will be lower than the stress computed from (4.19) for a brittle elastic solid. 4.4.1 Important Variables Affecting Modulus The modulus of elastic resilience in pure bending for of Rupture materials showing linear elasticity is expressed in terms of the work done per unit volume as the applied bending The important factors that affect the modulus of rupture of moment is gradually increased from zero to the proportional brittle materials in flexure tests are: limit moment, ðMBÞpl: The total work done in pure bending is the area between the linear elastic portion of the applied • Types of loading; bending moment ðMBÞ versus the angular rotation ðhÞ dia- • Length of span; gram and the h-axis: So, the work done per unit volume is • Shape of the cross-section of a beam; equal to half the product of the proportional limit moment • Cross-sectional dimensions of a beam; and the angular rotation of the beam at the proportional limit • Rate of loading, i.e. speed of testing. divided by the volume of the beam, which is given by Modulus of resilience in pure bending, There are three common types of loading: (1) symmetri- ðUBÞMR ¼ ðMBÞplhpl ð4:20Þ cal two-point loading of simple beam, as shown in Fig. 4.2; 2AL (2) centre loading of simple beam, as shown in Fig. 4.25 related to solved problem 4.8.3; (3) single-point loading at where the free end of a cantilever beam. Flexure test results of concrete (Kellerman 1933; Goldbeck 1943; Wright 1952) ðMBÞpl the applied uniform bending moment at the have indicated the effect of the types of loading on the rel- proportional limit; ative magnitudes of the modulus of rupture. The modulus of A the cross-sectional area of the beam; rupture obtained from symmetrical two-point loading is L the length of the beam subjected to pure bending; roughly 10–25% less than the centre loading. But the centre hpl the angular rotation of one end of the beam with loading tends to give slightly lower results than cantilever respect to the other at the proportional limit loading although the difference, on the average, is not great. The most concordant results are generally obtained by the During bending, the length L of the beam remains symmetrical two-point loading method. unchanged at the neutral axis of the beam and the angle between the radii to the ends of the beam is the same as hpl; Test results on both concrete and cast iron have indicated therefore L ¼ qplhpl; where qpl ¼ the radius of curvature of that, for beams of the same cross-section, the greater the the neutral axis of the beam, when the applied moment length of span, the lower the modulus of rupture (Mathews reaches the proportional limit. At the proportional limit, the 1910; Kellerman 1933; MacKenzie and Donoho 1937). maximum bending strain ðemaxÞpl for a cross-section that is