Interfacial Cracks in Bimaterials under Thermal and Applied Stresses,.

Transcription

1 Interfacial Cracks in Bimaterials under Thermal and Applied Stresses,. Summary S. Schmauder, M. Meyer, G. Eissner Max-Planck-Institut für Metallforschung, Für Werkstoffwissenschaft, Seestraße 92, Stuttgart, Germany Residual stresses provide strong mode 11contributions while four-point-bending loading leads to dominant mode I components at the tip of interface cracks in bimaterials. Thermal stress intensity factors (TSIFs) primarily depend on crack length ratio a/w and Dundurs' first parameter a. SIFs due to bending are mainly dependent on crack length whereas Dundurs' second parameter ß plays a minor role in both cases. Effective SIFs for combined loading are derived and discussed.' Introduction Differences in thermal expansion coefficients often lead to interfacial failure in composites and bimaterials. EspeciaIly, in the case of brittle components thermal residual stresses should not be neglected. ~J.'heymay strongly contribute to the driving energy of interface cracks [1]. At the tip of interface cracks, normal as weil as shear stresses are acting simultaneously. Thus interface cracks often tend to kink out of their initial plane, even if pure mode I loading is applied externally [1,2]. Whether or not interface cracks remain at the interface is strongly dependent on the mixed mode loading conditions at the crack' tip [2]. For these reasons it is necessary to quantify the local mode mixity at the crack tip as a function of elastic properties, crack length ratios as weil as externally and internally loading conditions. In this paper, results of a systematic study on the influence of thermal and applied stresses on interface cracks are presented and briefly discussed.,bimaterials Since their discovery, Dundurs' parameters a and ß have often proven their usefulness in characterizing elastic properties of bimaterials and other material joints (1-3]. Elastic solutions for interface crack problems depend only on a and ß 42

2 [4] wh ich are contractions of the four elastic constants (Young's moduli Ei and Poisson's ratios Vi) according to (plane strain) [5] E2/(1-v2) - E1/(1-v1) a=. (1) E2/(1-v2) + E/(1-v1) ß 4 I-v =![(1-2v1 1 I-v + 1-2v2) 2 I-v I-v a+ (1-2v1_1 1-2v2J~ 2 These parameters are limited within the following parallelogram, -l.o$;a$;loand -O.25+a/ 4$;ß$;O.25+a/4. The relationship ß=a/4 is derived from eqn. (2) when v1=v2 =1/3. As the sign of a and ß is reverted when materials are interchanged it is sufficient to examine one half of the valid a-ß-regime, e.g., a ~ o. For many material combinations: I al$; 0.6 and ß$;a/4 (Pig. 1). In several studies, Dundurs' second parameter was even set to zero arguing that ß is sufficiently small [6] ß Cu/A1203.Ti/A1203.Nb/A~03 Al/A~03 (2) ~8 Cl Fig. 1: Dundurs' parameters for a number of metal/ceramic combinations. Model In the following, we consider a bima terial under external mode I (ben ding) or thermalloading containing an interfacial edge crack with cra(,:ktip coordinates as shown in Fig. 2. Thermal expansion coefficients are chosen such thatai~a2. A 'cut and paste'-technique [7] is used to calculate TSIFs and a virtual crack extension method based on' Rice's J-integral concept is applied for deriving SIFs from applied loads in case of interface cracks. Theory The local stress field at the tip of an interface cra(,:kin real material joints is ci con- 43

3 F/2 e Thickness t w[.==:~~ F/2... Crack tip mesh Fig. 2: FE model of a four-point bending specimen with an interfacial edge crack and crack tip coordinates. sequence of residual stresses as weil as applied stresses. In the vicinity of the crack tip the local stress Held along the interface (8=0) is given by KriE (cree + i'tre)e=o =,,21tr _~ (3) The complex stress intensity factor K in eqn. (3) has the generie form K = K} + ikz = YP -{; a-ie ei'l' (4) with crack length a, and representative stress amplitude P. By definition, '" is the phase of KaiEwhere '" can be interpreted as the phase of the tractions at r=a, assuming that eqn. (4) still holds at this distance ahead of the crack tip. Y is a dimensionless geometrie function of material properties, loading conditions and crack length ratios. According to Rice [8], aglobai SIF for interface cracks may, therefore, be defined in the usual manner, if the radial distance from the craek tipi-r, is chosen as a fixed length quantity, r=~. Due to this substitution we can rearrange eqn. (4) to /\. /\ /\ r JE. KrlE= K1(r )+ ikn(r ) = YP {; (/\) a. el'l' where KI(~) = K} co~(eln(~»- Kz sin(eln(~» KII(~) = K} sin(eln(~» + Kz cos(eln(~» (5) (6a) (6b) These global SIFs Ki(~) (i=i,ii) have the usual dimensions (MPa~ m) and may be interpreted in the conventional manner according to eqn. (6). A detailed discussion of stress oscillation and the effect of length quantities can be found in [8]. For our subsequent treatment of interfacial stress intensity factors we incorporate at this point Ki(~) == Ki 0=1,11)and ~=a. The fact that ~=a lies obviously outside the zone of K-dominance is of no consequence as long as ~ is recorded along with the 44

4 results of "'~ = '" a and as long as one is familiar with the ",-transformation distance a to r given through from "'r = "'a + E ln(~) (7) Interface toughness values Kc are usually calibrated by tabulated yapp-functions, which are given for a wide range of test geometries. As these kinds of calibrations do not take into account inherent stresses their application will lead to different Kc values for bimaterials with identical elastic properties and crack length ratios but different residual stress states. However, by definition Kc must be taken as a material characterizing parameter. A correct K calibration can be performed by superimposing single mode SIFs due to applied loads (app) and residual stresses (res) according to Kj = y,app 1\.j + Kj res (I. = 1, 2) In analogy to the SIF definition in the case of applied loads, eqn. (4), we may now express the complex TSIF through the nominal residual stresses, ares=e*.1u.1t, using the relation K res= Yres ares \J _,- a a-1e.. eh!, res (9) where yres is the dimensionless geometrie function of Dundurs' parameters u and ß as weil as normalized crack length a/w. The difference in thermal expansion coefficients in plane strain is obtained from Hooke's law as.1u= (1+v 1)u1 (1+v2)u2. The cooling interval is determined by the difference between room temperature Ta and processing temperature Tp,.1T=Ta- Tp while the mean Young's modulus is given by 1/E*=«1-v1)/E1+(1-v2)/E2)/2. According to eqn. (9), the thermal calibration of abimaterial interface crack geometry is thus reduced to determining yres and ",res for the interesting range of material combinations and crack length ratios. (8) The main objective of the following section is, to combine thermal and mechanical correction functions as weil as phase angles to effective correction functions and effective phase angles (eff) in order to discuss the influence of thermal stresses on the effective toughness values for interfacial failure in bimaterials Keffa je = yeff crapp _'- -\Jae'l' j\lfeff (10) Results and Discussion Calculated thermal and mechanical correction functions as weil as phase angles, are shown in Figs. 3 and 4 for different a-values and crack lengths for bimaterials 45

5 with ß=O(a significantß-dependence of results has not been found in this study). Effective yeff_ and ",eff-values for combined loading conditions and one crack length are depicted in Fig. 5. Normalization of yeff is done with respect to Yo, the correction function for the case of a straight interface crack in a homogeneous material under pure bending. SIFs from bending loading are found to be identical for all bimaterials and, therefore, yapp may be represented by the usual correction function for homogeneous materials (Fig. 3). In contrast, TSIFs show some dependence on 0., decrease with crack length and are smaller by a factor of 3 to 11 depending on crack length compared to SIFs from bending. This fact manifests a neglectable influence of yres on effective correction functions yeff over 'a wide range of loading ratios (O<~<O.4)in Fig. 5, where ~=crres/(crres+crapp). +6 Bending loading provides small phase angles, ",app""oo,for all bimaterials and are even identical for a/w=o.3. In opposite, ",res""goofor homogeneous bi materials, but deviations from this value above and below this value are observed for elastically non-homogeneous materials. Thus, effective phase angles reflect a dependente on 0. for high thermal stress contributions, ~>O.4.It should be emphasized that the sign of ",res changes when the mismatch is inverted to a.l~a.2. 0 x.,., t-o U=-0.6 'V I , ß=O -SO 'V Cl= o u= ~ -SO ~ <> +2 r I _4-2" Ir ~ ß = 0-0- u= x- Cl= u= Cl= _6._ u= Cl= alw Fig. 3: Correction functions yapp and phase angles ",app for bimaterials four-point bending loading. under But the influence of Dundurs' first parameter 0. on the effective corre~tion function is seen to be limited. Moreover, effective SIFs are notably higher compared to pure mechanical SIFs, only for large values of nominal residual stresses (~>O.5). However, apparent SIFs of bimaterials obtained by neglecting TSIFs can be underestimates of more than 25%. 46

6 0.6,". A a= L~ a= - 0.4,". ---:... ß=O 0 a=-0.6 I t... ß=O -:>-a= '-..., a = ± 0.0,... ''.:.::'" a= ;.. )...0.4~ ~'" 90 ~'.:: uuh a= r a= 0 O.3~,.<...~ ~85 80 r ~ ~ -"'-a= ' _ -...-a= , 0.2, 0.3, 0.4 I WW WW Fig_ 4: Correction functions yrcs and phase angles 'I/res for bimaterials. / 1250 l!l B~l! iif "., o o -0.4 u=-o ~ ~ 0 u= ~.0.2 " I u=o.o )...0 ~ 0.75Ot- ~, V ~, ~ Fig. 5: yeff and \Vefffor birnaterials with interface cracks under combined thermal resid ual stresses and four-poin t ben ding, a / w=o.3 (~=ares / (ares+aapp)). \ Acknowledgement The financial support by the DFG (project EI 53/13-1) is gratefully acknowledged. References 1.Dreier, G., Schrnauder, S./ Elssner, G., Eur Strct Int Soc/ Salisbury, UK, 185/ Meyer, M., Schrnauder, 5./ Elssner, G./ Eur Strct Int Soc/ Salisbury, UK, 303/ Bogy, D.B., Int J Sol Struct, 6/ 1287/ Dundurs, J.W./ J Appl Mech, 36/ 650/ Schrnauder, 5., Meyer, M., Z Metallkd, 83/ 524/ Hutchinson, J.W./ Acta Scr Met Proc, 4, Pergarn Press, Oxford, UK, 295, 'Dowd, N.P., Shih, c.f., and Stout, M.G., Int J Solids Structures, 29/ 571, Rice, J.R./ J Appl Mech, 55, 98,

7 FRACTURE MECHANICS Proceedings ofthe Indo-German Workshop March 1994 Editors A.V. KRISHNA MURTY F.-G. BUCHHOLZ Department of Aerospace Engineering Indian Institute of Science Bangalore , INDIA INTERLINE PUBLISHING. Bangalore, India - L

8 Copyright 1994 Department 01 Aerospace Engineering Indian Institute 01 Science, Bangalore. This book or any part thereol may not be reproduced in any lorm without the written permission 01 the publisher. ISBN: Published by Interline Publishing 40/H, 5th Cross, Wilson Garden, Bangalore Typeset and Printed by Pan Media 43/1-2, Obalesh Complex, 6th Cross, Wilson Garden, Bangalore Printed in India