Correlation Between Dundurs' Parameters and Elastic Constants

Transcription

1 Carl Hanser Verlag, München 1992 Siegfried Schmauder and Markus Meyer (Max-Planck-Institutfür Metallforschung, Institutfür Werkstoffwissenschaft, Seestraße 92, D-7000 Stuttgart 1) Correlation Between Dundurs' Parameters and Elastic Constants Dedicated to Professor Dr. phil., tekn. dr. h. c., Dr. techno E. h. Hellmut F. Fischmeister on the occasion of his 65th birthday The correlation between the four elastic constants of two adjoint isotropic materials and Dundurs ' parameters a and ß is elucidated. Whereas ais a function of elastic moduli alone, ß depends on Poisson's ratios in addition to the elastic moduli. The linear dependency of ß on ais explicitly derived in this work. From these a- ß relationships it can be determined whether or not the assumption of negligible ß values is applicable für a given material combination. These figures are powerful means for the design of composites and provide information about which elastic properties of the components are required für obtaining certain Dundurs ' parameters. The method is demonstrated for two metal/ceramic systems. Korrelation zwischen Dundurs-Parametern und elastischen Konstanten In dieser Arbeit wird der Zusammenhang zwischen den vier elastischen Konstanten zweier aneinandergrenzender Materialien und den dazugehörigen Dundurs-Parametem a und ß beleuchtet. Während a nur eine Funktion der elastischen Moduln ist, war schon immer bekannt, daß ß zusätzlich von den Poissonzahlen abhängt. In dieser Arbeit wird zum ersten Mal die lineare Abhängigkeit zwischen a und ß explizit gezeigt. Graphische Darstellungen zeigen, ob für eine gegebene Materialkombination die Annahme vemachlässigbarer ß-Werte zutrifft oder nicht. Die Bilder stellen ein wichtiges Hilfsmittel zum Design von Verbundwerkstoffen dar und verschaffen Informationen darüber, welche elastischen Eigenschaften der Komponenten zur Erzielung bestimmter Dundurs-Parameter erforderlich sind. Die Methode wird an zwei Metall/Keramik-Systemen demonstriert. 1 Introduction Since their discovery, Dundurs ' parameters a and ß have been proven useful in characterizing the elastic properties of bimaterials and other material joints [1 to 13]. Using the original notation ofdundurs [1] andbogy [2], one can write a and ß as a = k(1\:1 + 1) - (1\:2 + 1) (1) k(1\:1 + 1) + (1\:2 + 1) ß = k(1\:1-1)- (1\:2-1) (2) k(1\:j + 1) + (1\:2 + 1) In Eqs. (1) and (2), k is the ratio ofthe shear moduli,uz/,ul, 1\:i is Muskhelishwili's constant [14] and is equal to (3-4vi) für plane strain and (3-v ;)/(1+vi) forplane stress, where Viis Pois- 524 son's ratio, and the index Ci = 1,2) refers to material "I" and "2", respectively. Although the Dundurs' parameters are applicable for a variety of bonded materials and joints, we shall refer to these materials as simply a "bimaterial" for convenience. Bogy [2] has shown that the upper and lower limits of the constant a of any bimaterial are - ::;;; a::;;; and that the constant ß is bounded for the plane strain and plane stress cases, respectively % - 5 ::;;;ß ::;;;% + 5 ~ a - 25 ::;;;ß ::;;;~ a + 25 One useful property of the Dundurs ' parameters is that only the sign of a and ß is reversed when materials 1 and 2 are exchanged; thus, it is sufficient to examine a single case (e.g. a;;;; 0) to fully understand the elastic properties of the bimaterial. Schmauder [3] has shown that the expression for parameter a can be simplified in terms of the Young's moduli of the phases E!-Et a=w 2 + E+ I (6) In Eq. (6), Et is given by CI. Fig. 1. Dependence of a on ratio of Young's moduli. (3) (4) (5) Z.Metallkd.83 (1992) 7

2 ß 0"... 1_.",. " "..1~:99.11l1... >Oe Q - - o Al / Ceramic,Mo,W. B,C o Cu/ Ceramic,Mo.W.B,C 6 Ni / Ceramic. Mo.W,B.C Ti,Zr,Hf,V,Nb /Ceramic.Mo.W Ta.Cr.Mo.W /Ceramic '-44 / Plastics /Others Fig. 2. a - ß diagram for bimaterials of isotropie eomponents [9]. Er = {EJ(1- E I vt) andejare Young's moduli ofthe components (i = 1,2). Thus, a is easily obtained using the ratio of the plane moduli Et and E2 (Fig. I). Because of the importance of Young's moduli, ais of primary interest in studies on the e1astic response of composites. For typical material combinations, the elastic properties are often within a range given by 025 < Et < 4. = E:fj= Thus, the value of Dundurs' parameter ais limited to lai;;;; (7) (8) (9) This statement is supported by a systematic investigation by Suga [4, 9]. A few examples of different material joints are shown in Fig. 2. In addition, Meyer and Schmauder [7] have noted that many technically relevant material combinations follow ~-OI<ß<~ = =4+. (10) in agreement with Eq. (4). Equation (10) is sometimes simplified to a ß= (11) 4 to reduce the number of parameters [10]. It is interesting to note that when VI = V2 = 1/3, Eq. (11) is exact. Equation (11) is also valid for many composites with components which have unequal Poisson's ratios. Hutchinson [11] has argued that Dundurs ' second parameter may be ignored in many practical cases since ß is small according to Eqs. (9) and (11). Nevertheless, except for the case of identical Poisson's ratios, the correlation between ß and elastic constants of involved phases remained unc1ear. The relationship between ß and the elastic constants (Eq. (2)) provides the basis for resolving these uncertainties and for correlating aand ßin a predictive sense for arbitrary Poisson's ratios. The role of ß is discussed in detail in the following section. 2 Dundurs' Second Parameter ß The relationship in Eqs. (1) and (2) between the four elastic constants and ß can be rewritten as (e. g. in [12] for plane strain deformation) E:fj I- 2vI _ Et I- 2V2 I I - VI 1- V2.. ß = 2: r.+. r.+ (plane stram) (12) ß =! E:fj (1- VI) - Et (1- V2) 2 Et+E2 (13) We discuss the cases of identical and unequal Poisson' s ratios in sections 2.1 and 2.2, respectively. Z. Metallkd. 83 (1992) 7 525

3 fyex i3 Fig. 3. Variation of ß/a für identical Poisson's ratios v = v I = V2. v a) b) Fig. 4. Linear a - ß dependence für identical Poisson's ratios v = v I = V2 under (a) plane strain, and (b) plane stress deformation. ex 2.1 Identical Poisson's Ratios ß=!!. 3 (16) For identical Poisson's ratios (v = v 1= V2), it follows from Eqs. (6), (12) and (13) that 1 (1-2v). ß =:2 /1 \ a (plane stram) (14) (15) Equations (14) and (15) are evaluated in Fig. 3 and Eq. (11) is derived from Eq. (14) by setting Poisson's ratio to v = 1/3. However, one mayaiso observe that many other linear relationships between a and ß may be derived by changing v. Under plane stress assumptions a relationship similar to Eq. (11) holds for v = 1/3 526 It is also interesting to note that for the case of identical Poisson's ratios v = VI = V2 ;;;; and for positive a values which fulfill Eq. (9), Eq. (14) leads to a narrow range of possible ß values (Fig. 4) given by 0;:;:;ß;:;:; 5 (17) 2.2 Unequal Poisson's Ratios For the more realistic case of unequal Poisson's ratios, the following expressions can be derived from Eqs. (6), (12) and (13) Z. Metallkd. 83 (1992) 7

4 -' [-;ane st;~~ steeljceramic plane strain I I: AlJceramic n ~ :~{.C/Co WC/Co!.~ o.ooo~ O.S a. a. ~ ~ Fig. 7. Distribution of typical metal/ceramic systems with different metals in the a - ß diagram : (a) steel, (b) A!. - ß =![(1-4 2vI VI + 1-2V2) a + ( 1-2vI VI _ 1-2V2)] 3 Application to Metal/Ceramic Systems 4 Concluding Remarks (18) When both materials possess the same Poisson's (19) ratio Eqs. (18) and (19) can be simplified to Eqs. (14) and (15). The existence of a linear relationship between a and ß was first noted by Bogy [13]. However, its form was not given explicitly until now. It can be seen that the range of possible ßvalues is limited to a narrow regime when a and Poisson' s ratio of one material are fixed. The form of Eqs. (18) and (19) provides one with a simple method of predicting Dundurs ' second parameter ß for technically important composites with Poisson's ratios limited as follows: ;;;;V], V2 ;;;; ; this is shown in Fig. 5. In combination with Eq. (6), ß can be graphically presented as a function of elastic constants of involved materials (Fig. 6). Steel and Al are frequently used as the metal part in metall ceramic systems (Figs. 7a and 7b). From the discussion ofthe previous sections resulting Dundurs' parameters and the scattering of the data in the a - ß-diagram are now weil understood. As shown in Figs. 7a and 7b, a large scatter is present for smail ß values in systems with small mismatch of Young's moduli; this is basically due to the mismatch in Poisson's ratios. Conversely, one may expect larger ß values and less scatter in systems where I al ;;;; (Fig. 7b). Generally, small ß values can be achieved in typical metall ceramic systems with I a I ;;;; when ceramics with smail Poisson's ratios are used. new composites with pre-defined elastic material characteristics. Thus, only certain Dundurs' parameters a and ß can be achieved for composites where the elastic constants of one of the two adjoining materials are already fixed. Figure 5 together with Eqs. (18) and (19) provide the means to determine the range of admissible Young's moduli and Poisson's ratios of materials to be joined. The trends discussed in this paper for a and ß are typical for many metallceramic systems. This work was supported by the German Research Foundation (DFG project EI 53/13-1); the support is gratefully acknowledged. Thanks are due to Dr. Steve Rozeveld for valuable suggestions in improving the manuscript. Literature 1. Dundurs, J. W: J. App!. Mech. 36 (1969) Bogy, D. B.: Int. So!. Struct. 6 (1979) Schmauder, S.: CF! (Ceramic Forum International) (1987) 4. Suga, T.: Ph. D. Thesis, University 5. Schmauder, S.: in "Metal-Ceramic of Stuttgart Interfaces", (1983). Acta Scripta Metallurgica Proceedings, Se ries 4, M. Rühle, A. G. Evans, M. F. Ashby, J. P. Hirth (eds.), Pergamon Press, Oxford, England (1990) Dreier, G.; Schmauder, S.; Elssner, G.: Int. J. Fatigue Fract. Eng. Mater. Struct. (1992) in press. 7. Meyer, M.; Schmauder, S.: Int. J. Fract. (1992) accepted. 8. O'Dowd,N. P.; Shih, C. F; Stout,M. G.: Int. J. Solids Structures 29 (1992) Suga, T.; ElssneJ; G.; Schmauder, 917. S.: J. Comp. Mat. 22 (1988) 10. Schmauder, S.: CFI (Ceramic Forum International) 3 (1988) Hutehinson, J. W: in "Metal-Ceramic Interfaces, Acta Scripta Metallurgica Proceedings" Series 4, M. Rühle, A. G. Evans, M. F. Ashby, J. P. Hirth (eds.), Pergamon Press, Oxford, England (1990) Jensen, H. M.: Eng. Fract. Mech. (1992) submitted. 13. Bogy, D. B.: Int. J. Solids Structures 6 (1970) Muskelishwili, N. I.: "Some Basic Problems of the Mathematical Theory of Elasticity" (trans!. by J. R. M. Radok), R. Noordhoff Pub!. Company, Groningen, Netherlands (1953). For any given material combination a is obtained from Eq. (6). The simple graphical presentations of the linear interrelation between a and ß provide a powerful tool to determine ß in a simple manner. These graphs mayaiso be applied to design Z. Metallkd. 83 (1992) 7 (Reeeived Oetober 22, 1991) 527

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