Thermal stress intensity factors of interface cracks in bimaterials

Transcription

1 International Journal of Fracture 57: , Kluwer Academic Publishers. Printed in the Netherlands. 381 Thermal stress intensity factors of interface cracks in bimaterials M. MEYER and S. SCHMAUDER Max-Planck-lnstitut ffir Metallforschung, Institut fiir Werkstoffwissenschaft, Seestrasse 92, D-7000 Stuttgart 1, Germany Received 20 June 1991; accepted 23 March 1992 Abstract. In this communication numerical results for thermal stress intensity factors (TSIFs) of interface cracks are presented for arbitrary material combinations which are characterized by Dundurs' parameters a and ft. It is shown that TSIFs are linear in crack length ratio a/w and quadratic in ct and depend also on ft. The local phase angle at the interface crack tip is a linear function of cc The striking feature of residual thermal stresses is their strong mode II character at the tip of an interface crack. In the framework of linear elasticity these TSIFs can be linearly superimposed on the stress intensity factors (SIFs) from applied loads for interface cracks in composites. 1. Introduction Composites develop thermal residual stresses during cooling from processing temperatures to room temperature. Bimaterials with relatively large thermal expansion mismatch are therefore often prone to interfacial failure. Particularly in the case of brittle components thermal residual stresses cannot be neglected. They may contribute strongly to the energy release rate of interface cracks, thus reducing the apparent toughness of bimaterials. However, a complete analysis of TSIFs for interface cracks between brittle components is lacking. In this paper, an extension of an existing virtual crack extension based method for obtaining TSIFs will be described. Furthermore, preliminary results of a systematic study for TSIFs of interface cracks are presented and discussed. 2. Problem formulation Interface toughness values Kc are usually calibrated by tabulated Yk functions, which are given for a wide range of test geometries. Unfortunately these kinds of calibrations do not take into account inherent residual stresses. Their application will consequently lead to different Kc values for bimaterials with identical elastic properties and crack length ratios but different residual stress states. However, by definition K~ must be taken as a material characterizing parameter. The main objective of this section is therefore to determine a general formulation of TSIFs and to calculate thermal calibration functions y[os for arbitrary material combinations and crack length ratios. A bimaterial with a crack of length 'a' along the interface between materials '1' and '2' is considered (Fig. 1). The nominal residual stresses are given as 0 "res = E*ACtAT. (1) From Hooke's law of elasticity, the difference in thermal expansion coefficients is defined as Act f(1 + V1)Ct 1 -- (1 + V2)Ct 2 for plane strain (ctl- ct2 for plane stress" (2)

2 382 M. Meyer and S. Schmauder E1,G,a~,AT - - a,_[ W E 2, v2,a,,at Fig. I. Analyzed geometry of a bimaterial with an interface crack. The temperature difference AT between service temperature To and processing temperature Tp is given through AT = To- T,, (3) E* is a mean Young's modulus of the bimaterial and is defined as [1] /~, - ~ +, (4) where = (Ed(1-v 2) for plane strain (i= 1,2) (5) E+ (Ei for plane stress and E + are the respective Young's moduli of the components. The four elastic constants (Young's moduli El (i = 1, 2) and Poisson's ratios vl (i = 1, 2)) can be contracted to the two well-known Dundurs' parameters [2] k(~x + 1) - (K2 + 1) = (6a) k(~c~ + 1) + (x2 + 1)' k(k1 -- 1)- (K2 -- 1) 13 = k(~:l + 1) + (~2 + 1)' (6b)

3 Interface cracks in bimaterials 383 where k = #2/#1 is the ratio of the shear moduli. Muskhelishvili's constant ~:i (i = l, 2) is given by 4vi for plane strain ~'3 (i -- 1, 2). (7) x~ = ((3 - vi)/(1 + Vi) for plane stress It is well known [3] that most technically important material combinations can be approximately described through the relation fl = a/4 and are within the limits (Fig. 2) 0~<c ~<0.6; cq4-0.1 ~<fl~<cq4+0.l. (8) These ~- and fl-values change signs when material 'l' and '2' are exchanged. Residual stresses as well as applied stresses following (1) produce a local stress field at the tip of an interface crack (0 = 0), which is given as l i~ (aoo + izro)o=o = = K r, (9) ~/2~r 'r' is the radial distance from the crack tip and the bimaterial constant e is only a function of fl 1 (1 + fl'] (10) e = ~ In \i-~j' The complex stress intensity factor K has the generic form K = K1 + ikz = YkTx/~a -~ e io, (11) a is the crack length and T is the representative stress amplitude. By definition O is the phase of Ka ~ where ~b can be interpreted as the phase of the tractions at r = a, assuming that (9) still ~=a/4+o.l ~ J ~ llass/epoxy 0.2 Al~,,/Ai ~ - - j J o.1 ~yz. ~.o_~/c. ~ ~'e ~- 'J~"- ~ ~ o.o. ~ ~ ~..~._qs ~ ~ ~=a/ Fig. 2. Distribution of typical material combinations in the ~-fl-diagram according to [3].

4 384 M. Meyer and S. Schmauder holds at this distance ahead of the crack tip. Y~ is a dimensionless geometric function of material properties, loading conditions and crack length ratios. According to Rice [4] a global SIF for interface cracks may therefore be defined in the usual manner, if the distance r is chosen as a fixed length quantity r = f. Due to this substitution we can rearrange (11) to Kf i~ = Kl(f) + ikn(f) = Yk Tx/~(f/a) i~ e i, (12) where Kl(f) = K1 cos(e In(i)) - K 2 sin(e In(i)), Kn(f) = K1 sin(e In(i)) + K 2 cos(/~ In(1)). (13a) (13b) These global SIFs Ki(f) (i = I, II) have the usual dimensions (MPa,v/-m) and may be interpreted in the conventional manner according to (9). For our subsequent treatment of thermal stress intensity factors we incorporate at this point Ki(f) - Ki (i = I, II) and f = a. The fact that f = a obviously lies outside the zone of K-dominance is of no consequence as long as f is recorded along with the results of ~ and as long as one is familiar with the ff transformation given through ~kr2 = ~brl + eln(r2/rl). (14) Analogous to the SIF definition in the case of applied loads, we may now express the complex TSIF through the residual stresses using the relation Kres = y~es o.res N/-~aa-ie ei,ros, (15) where y~,es is the dimensionless geometric function of Dundurs' parameters ~,/~ and normalized crack length a/w. Thus the calibration of a bimaterial crack geometry under thermal loading is reduced to determining Y~eS and q;re~ for the interesting range of material combinations and crack length ratios. Using (15) a correct K calibration can be performed by superimposing single mode SIFs due to applied loads (app) and residual stresses (res) according to Ki = K app q- K[ es (i = I, II). (16) It should be kept in mind that both the K app values and a proper fracture criterion must be available to apply (16) [9]. 3. Method An effective numerical method for calculating SIFs Ki (i = 1, 2) for interface cracks is given by Matos et al. [5]. This procedure is based on the evaluation of the J-integral of Rice [6] using the

5 Interface cracks in bimaterials 385 finite element method. If there are two different displacement fields u a and u b, each representing a solution to the two different boundary value problems for the crack, the values of the energy release rate G associated with them are G, and Gb. When the displacement fields u" and u b are summed to give u c, the value of G resulting is Gc = Ga + Gb + M1, (17) where M 1 is a J-like path independent integral [7], characterizing the energetic interaction of the two displacement fields. Using (17) and the relation 1 - fie G - E----Z--(K 2 + K2), (18) the interaction energy can be expressed as M1-2(1 ~g/2)(klaklb + K2aK2b). (19) From this the Kla (K2a) -- SIFs can easily be extracted, if special Kb-values are chosen, i.e. Kxb = AKlb and K2b = 0 (Klb = 0 and K2b = AK2b). Following Hellen [8], the energy release rate G can be evaluated numerically by a virtual crack extension method (VCE) G = - (~ U/c~a) ~ - ½{u. } r {OS/Oa} {u. }, (20) where U is the potential energy of the body, a is the crack length, {u.} contains the element displacement vectors and [S] is the stiffness matrix for the mesh of elements used to solve the problem. To apply the VCE-method for thermal stress problems, (20) must be modified since the initial strains due to the temperature difference have to be taken into account additionally. The effective displacement field {u, } for use in (20) is then given by {Un} = {Un}ac t -- {Un}ini, (21) where {Un}a t is the actual displacement field of the cracked body and {Un}in i is the initial displacement field of the uncracked structure. Two finite element calculations are therefore necessary to provide {u.}act and {u.}ini (Fig. 3). In a post processing step the crack is virtually extended by rigidly moving a core of elements around the tip in order to determine a new stiffness matrix IS] (Fig. 4). Then only the distorted ring of elements contributes with non-zero {OS/Oa}-values to the energy release rate G in (20). Now a second displacement field u b- {Au.}I is superimposed on u "= {Un} for which K1 = AK1 and K2 = 0, i.e. AK1/~- e ~ Au{ - 2ttj ~/~ (1 + e2~) fl(r' 0, e, xj), (22)

6 386 M. Meyer and S. Schmauder AaAT ia~t / rigid distorted {Un}ac t - {Un}in i = {Un} Fig. 3. Schematic for calculating the effective displacement field {u,}. Fig. 4. A typical ring of elements around the crack tip to be distorted in the J-calculations. where fl is given in 1-5] and j = 1, 2 is the material index. The calculation in (20) is now repeated with the vector {u,} + {Au,}l used instead of {u,} alone. The result of this calculation, G + AG1, can be shown from (17), (18) and (19) to be such that E* AG1 K1-2(1 - ]~2~ AK1 ½AK,. (23) To obtain K 2 the procedure can be repeated for an added vector u b =- {AUn} 2 such that K 1 = 0 and K2 = AK2 using Au~ AK2 ~ e ~ - 2#s X/2n (1 ~--~2~)f2(r, 0, e, xs), (24) or alternatively using (18) to determine K2. A more detailed description of computing interfacial stress intensities can be found in Matos et al. [5]. 4. Results and discussion In the following, results are presented for the geometry function of thermal residual stress intensity factors y~es and the corresponding mode mixity ~,res for use in (15). The dependence of geometry function Y~y on ~ is shown in Fig. 5. Obviously, Y~y is a decreasing function of crack length but increases with increasing ~- and fl-values. For short cracks (a/w ~ 0.1) there is a distinct effect of the elastic mismatch ~, fl on Y~y, whereas long interface cracks (a/w ~ 0.5) produce a modest change in this geometry function. Similarly, the local phase angle ~k r~ develops significant mode I fractions for short cracks and increasing ~- and ~-values. For the range of moderate a-values (c~ ~< 0.5, Fig. 2) and typical notch lengths of 0.2 ~< a/w <~ 0.5 residual thermal stresses result in a nearly pure shear loading at the tip of the interface crack (Fig. 6).

7 ~=a/4-o.lo a-e B=a/4-O.05 B=a/4+O.05 O=a/4+O.lO Interface cracks in bimaterials a/w=o. I 387 a/w=0.2 res Yk 0.22, a/w= a/w= a/w=0.5 ' I ' I ' (3( Fig. 5. Correction function for TSIF as a function of crack length and elastic mismatch ,0 7 a/w-~02 85 ',/,,=05 _...~~" "X~~..~ ~/,,-0.~ 80 o. ~/.=oa a/.-o.a a/w=0.2 ~/* 65 o. 70. "/" '~xn E~o.Z]-~*o"~a / "/'= ",75.,,/,,=o z,~. / 80 ' a/.=o a I a/.=o ~ ~.-~ Fig. 6. Local phase angle as a function of crack length and elastic mismatch. For engineering purposes the following approximations may be used for 0.1 ~< a/w <<, 0.5 and fl = ~/4. y~,es = { ~ z } - { O.195otZ}(a/w), ~res = { (a/w) + lo06.1(a/w) (a/w) 3 }~. (25a) (25b)

8 388 M. Meyer and S. Schmauder It is worth mentioning that mode mixity is fully determined by the elastic mismatch and the crack length: for non-zero Aa-values, the phase angle ~k res is independent of the thermal mismatch Aa and the cooling interval AT. Only the absolute amount of K res changes with E*A~AT following (15). In the presence of applied stresses, e.g. when testing the toughness of real bimaterials, TSIFs can significantly increase the mode mixity and the effective SIF at the tip of interface cracks. 5. Conclusion A modified virtual crack extension technique for evaluating thermal stress intensity factors in interfacial fracture has been presented. In summary, it may be concluded that stress intensity factors and geometry functions of thermally stressed bimaterials with interface cracks have been presented for the first time in a systematic manner as function of crack length and elastic mismatch. Simple analytical expressions of these functions were presented. It was found that residual thermal stresses result in a dominating shear component at the interface crack tip. The exact knowledge of TSIFs is important for all systems with residual thermal stresses. In the framework of linear elasticity TSIFs may be superimposed to applied stress intensity factors [9J. References 1. D.R. Mulville, P.W. Mast and R.N. Vaishnav, Engineering Fracture Mechanics 8 (1976) J. Dundurs, Journal of Applied Mechanics 36 (1969) T. Suga, G. Elssner and S. Schmauder, Journal of Composite Materials 22 (1988) J.R. Rice, Journal of Applied Mechanics 55 (1988) P.P. Matos, R.M. McMeeking, P.G. Charalambides and M.D. Drory, International Journal of Fracture 40 (1989) J.R. Rice, Journal of Applied Mechanics 35 (1968) F.H.K. Chen and R.T. Shield, Zeitschriftfar angewandte Mathematik und Mechanik 28 (1977) T.K. Hellen, International Journal for Numerical Methods in Engineering 9 (1975) M. Meyer, S. Schmauder and G. Elssner, work in progress.