2.
Communications the entire loading-unloading period (Fig. 1): Wd Wt 2 We . (4) The work performed during loading (Wt ) and that re- gained during unloading (We ) can be calculated by the integration of Eqs. (2) and (3), respectively. It was found that in spite of the linear terms appear- ing in Eqs. (2) and (3) for a broad variety of materials and in a wide load range, the following relationship is with good accuracy valid (see Fig. 2): r We c3 . (5) Wt c3 The parameter c3 characterizes the resistance of the material against the elastic-plastic deformation.11 In the case of ideally plastic materials, the load-depth function FIG. 1. Schematic picture of an indentation cycle. is purely quadratic: P c3 h2 ,1,12 and in this case there is no elastic relaxation, the d 7 ? hm equation betweenTABLE I. Compositions and densities of Si3 N4 ceramic samples. the diagonal d and the maximum indentation depth hm 90 wt. %, Si3 N4 , 4 wt. % 90.9 wt. %, Si3 N4 , 3 wt. % which is the consequence of the geometry of the Vickers Al2 O3 , 6 wt. % Y2 O3 Al2 O3 , 6.1 wt. % Y2 O3 pyramid is exactly satisﬁed [Fig. 3(a)]. Consequently the Meyer hardness of ideally plastic materials can be givenSample Density (g cm3 ) Sample Density (g cm3 ) in the following form: 1 2.03 6 2.697 2 2.11 7 2.823 P P H 2 a1 ? c3 (6) 3 2.34 8 2.935 d2 2 24.5hm 4 2.54 9 2.954 5 2.70 10 3.032 with a1 0.0408 for the Vickers geometry. 11 3.115 If the material is not ideally plastic then with in- 12 3.161 creasing elastic contribution, the elastic deﬂection under the indenter is increasing [Fig. 3(b)]. Consequently 7hm becomes increasingly larger than d, and as a result of(sy E is large) then a signiﬁcant portion of the contact this a1 ? c3 will be less than H. This can be taken intoarea at maximum depth is due to elastic deformation. account introducing a relationship between H and a1 ? c3Consequently, the residual projected area is smaller than of the form:the projected contact area at maximum penetration, and Wtthe conventional Meyer hardness is larger than the mean H a1 ? c3 ? . (7)contact pressure. Wd The load-penetration depth function can be de-scribed with quadratic polinoms (Fig. 1): P c2 h 1 c3 h2 , (2) P c2 h 2 h0 1 c3 h 2 h0 2 , (3)both in the loading and in the unloading periods, respec-tively, where P is the load, h is the penetration depth,and h0 is the residual indentation depth after removingthe punch; c2 , c3 , c2 , and c3 are ﬁtting parameters.The total indentation work, Wt , is the integral of theload versus the indentation depth, i.e., the area underthe load-penetration depth curve during the unloadingperiod. Upon unloading a part of this work, We , can beregained; it equals to the area under the load-indentationdepth curve for this latter period. The difference of these FIG. 2. The ratio of the elastic and total work versus the parameterstwo quantities, Wd , gives the net work expended during of the indentation curves. J. Mater. Res., Vol. 11, No. 12, Dec 1996 2965
3.
Communications Sneddon’s elastic theory13 and empirical results of Oliver and Pharr,3 the contact depth can be given as: Pm hc hm 2 e , (9) S where S is the slope of the initial part of the unloading curve (Fig. 1) and e 0.75 for the case of Vickers indenters. To compare the new hardness formula (7) with the mean contact pressure as expressed in Eqs. (8) and (9), Pm and hc are expressed with the c3 and c3 parameters and the indentation works. If only the quadratic terms existed in Eqs. (2) and (3), Pm and hc could be expressed easily with the parameters c3 and c3 . Because of the existence of the linear terms, some approximations areFIG. 3. Schematic picture showing the behavior of various materials used in the considerations.(a –c) during Vickers indentation. According to Eq. (3) S can be given as dP ÅIt is obvious that this new deﬁnition of H gives back S c2 1 2c3 hm 2 h0 . (10)Eq. (6) for ideally plastic materials and it increases dh hmwith increasing elasticity, and in the limiting case of The second term in Eq. (10) may be expressed as aan ideally elastic material it becomes inﬁnite, because in fraction of S:this case there is no residual deformation after unloading[Fig. 3(c)]. Figure 4 shows that the H values determined kS 2c3 hm 2 h0 (11)from DSI measurements according to Eq. (7) agree wellwithin the experimental errors with the conventionally and similarly the quadratic term of the load-depth func-determined hardness. tion of the unloading curve as a fraction of the maximum The most frequently used hardness determination load:method from DSI measurements introduced by Oliver k Pm c3 hm 2 h0 2 . (12)and Pharr3 is based on the mean contact pressure atthe maximum indentation depth with the following With Eqs. (11) and (12) Pm S can be given asexpression: Pm Pm Pm Pm k k hm 2 h0 Hm a1 , (8) . (13) A 2 24.5hc 2 hc S k k 2 From Eqs. (3) and (10) –(13), hc can be given in thewhere A is the projected contact area and hc is the following form:contact depth at the maximum load [Fig. 3(b)]. Using 2 e hc hm 2 h m 2 h0 . (14) 11k 2 Expressing the quadratic term in Eq. (2) as a fraction of the maximum load: 2 kPm c3 hm , (15) the Hm mean contact pressure in Eq. (8) can be written as follows: 1 Hm a 1 c3 ≥ p p q ¥. 2 k e c3 (16) k2 11k 2 c3 According to our measurements for the different materials investigated with different loads, k and k vary pFIG. 4. The conventionally determined hardness number versus the between 0.64 and 1. If k > 0.64 then 2 k 11quantity calculated on the basis of Eq. (7). k > 0.98; therefore, this quantity can be taken as 1.2966 J. Mater. Res., Vol. 11, No. 12, Dec 1996
4.
CommunicationsUsing Eq. (5) the following relationship is obtained for A new formula is proposed for the characterizationHm : of the hardness of materials from DSI tests. The val- 1 ues of the hardness calculated by this equation agree Hm a 1 c3 ≥ p ¥ . (17) well with those measured by the conventional method e We 2 k2 2 Wt for a broad variety of materials. Comparing the new hardness formula with that based on the mean contactThis can be compared with the new hardness formula pressure, the former correctly describes the behavior ofEq. (7), which using Eq. (4) can be written as: the conventional hardness both in the ideally plastic and 1 ideally elastic limiting cases, while the latter deviates H a 1 c3 We . (18) from the conventional hardness number in the ideally 12 Wt elastic limit. At the same time in the range where optical The difference between H and Hm can be seen in measurements can also be applied (0 < We Wt < 0.7),Fig. 5 in which the two hardness numbers divided by the difference between the two DSI evaluation methodsa1 c3 are shown as a function of We Wt . For Hm a1 c3 p is within the experimental error. A technical advantagethree curves are shown as k equal to 0.8 or 0.9 or of the new evaluation method is that the parameters used1 (k 0.64 or 0.81 or 1). We Wt equals zero for the in the hardness formula can be determined at a highideally plastic limiting case; it increases with increasing accuracy from the registered indentation curves. Furtherelastic deformation in the indentation, and it is 1 for the measurements are planned on materials and for loadingideally elastic material. By the new hardness formula conditions falling in the region where the new hardness[Eqs. (7) or (18)], the behavior of the conventional hard- number is rapidly increasing (0.8 < We Wt < 1). Thisness number (and consequently the plastic properties of regime, however, is not easy to obtain experimentally.materials) is better characterized than by the mean con-tact pressure, because while the former tends to inﬁnity ACKNOWLEDGMENTSthe latter gives a ﬁnite value in the ideally elastic limitingcase. As it can be seen in Fig. 5 in the region 0 < The authors are grateful to P. Arat´ for providing oWe Wt < 0.7, there is no signiﬁcant difference between the Si3 N4 ceramic samples. This work was supportedthe two hardness numbers. The We Wt values for the ma- by the Hungarian National Scientiﬁc Fund in Contractterials investigated here are in this region; consequently Nos. T-017637 and T-017639.the hardness numbers obtained by both methods agreewell with the values of the Meyer hardness obtained by REFERENCESoptical observation after unloading (Fig. 5). 1. M. Sakai, Acta Metall. Mater. 41, 1751 (1993). 2. G. M. Pharr, W. C. Oliver, and F. R. Brotzen, J. Mater. Res. 7, 613 (1992). 3. W. C. Oliver and G. M. Pharr, J. Mater. Res. 7, 1564 (1992). 4. G. M. Pharr and W. C. Oliver, MRS Bull. 17, 28 (1992). 5. A. Juh´ sz, G. V¨ r¨ s, P. Tasn´ di, I. Kov´ cs, I. Somogyi, and a oo a a J. Sz¨ ll¨ si, Colloque C7, suppl´ ment au J. de Phys. 3, 1485 o o e (1993). 6. F. R. Brotzen, Int. Mat. Rev. 39, 24 (1994). 7. A. Juh´ sz, M. Dimitrova-Luk´ cs, G. V¨ r¨ s, J. Gubicza, a a oo P. Tasn´ di, P. Luk´ cs, and A. Kele, Fortchrittsberichte der a a Deutschen Keramischen Gesellschaft 9, 87 (1994). 8. J. Gubicza, Key Eng. Mater. 103, 217 (1995). 9. J. Gubicza, A. Juh´ sz, P. Arat´ P. Tasn´ di, and G. V¨ r¨ s, a o, a oo J. Mater. Sci. 31, 3109 (1996). 10. D. Tabor, Hardness of Metals (Clarendon Press, Oxford, 1951). 11. F. Fr¨ hlich, P. Grau, and W. Grellmann, Phys. Status Solidi (a) o 42, 79 (1977).FIG. 5. The hardness numbers divided by a1 c3 as a function of 12. B. R. Lawn and V. R. Howes, J. Mater. Sci. 16, 2745 (1981).We Wt . 13. I. N. Sneddon, Int. J. Engng. Sci. 3, 47 (1965). J. Mater. Res., Vol. 11, No. 12, Dec 1996 2967
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