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Wedge corner stress behaviour of bonded dissimilar materials
Theoretical and Applied Fracture Mechanics 32 (1999) 209±222
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Wedge corner stress behaviour of bonded dissimilar materials Z.Q. Qian, A.R. Akisanya * Department of Engineering, University of Aberdeen, KingÕs College, Fraser Noble Building, Aberdeen AB24 3UE, UK
Abstract A contour integral, based on BettiÕs reciprocal theorem, is used in conjunction with the ®nite element method to evaluate the magnitude of the wedge corner stress intensities associated with the higher order terms of the singular stress ®eld near the interface corner of a bi-material joint. It is shown that using a dierent auxiliary ®eld can eliminate the dependence of the wedge corner stress intensity on the integration path observed by [W.C. Carpenter, Int. J. Fracture 73 (1995) 93±108]. Finite element analysis of a typical joint geometry is used to demonstrate the path-independence of the magnitude of the stress intensities evaluated using the proposed method, and to show the eects of higher order terms on the stress state near the interface corner. Ó 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Stress singularity; Wedge corner stress intensity; Path-independent contour integral; Finite element method; Failure initiation
1. Introduction Many engineering applications involve the joining of dissimilar materials. For example, present day automobiles and aircrafts contain signi®cant number of bonded components ranging from ceramic/metal/ ceramic to composite/adhesive/metal joints. A stress singularity may develop at the corner where the interface between the bonded materials intersects the free surface when the joints are subjected to mechanical and/or thermal loading. The type of singularity depends on the choice of joint geometry and the elastic and thermal properties of the materials [2±4]. Since failure is most likely to be initiated from a singular point it is therefore important to properly characterise the singular stresses and the associated displacements so that joint geometries and material combinations can be appropriately chosen to minimise such failure. There have been a lot of studies on the analysis of stress singularities at interface corners and at wedge/ notch and interface crack tips since the pioneering work described in [2,5]. Initially these studies focussed on the in¯uence of the wedge/notch angle and of material elastic properties on the order of the singularity. However, the emphasis has changed, in the last 7 yr or so, to the determination of the complete structure of the stress ®eld near the interface corner of bonded dissimilar materials, including the evaluation of the associated corner stress intensity. This has been in¯uenced by the success of using a critical value of crack tip stress intensity factor to correlate the onset of fracture in brittle monolithic solids. There is increasing evidence that a critical wedge corner stress intensity can also be used in a similar manner to predict the
*
Corresponding author. Tel.: +44-1224-272989; fax: +44-1224-272497. E-mail address: [email protected] (A.R. Akisanya)
0167-8442/99/$ - see front matter Ó 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 9 ) 0 0 0 4 1 - 5
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onset of failure from sharp notches and interface corners, see for example, [6,7]. This, however, will depend on the availability of calibration curves for the magnitude of the stress intensities at an interface corner as a function of joint geometry (or wedge angle), material properties and applied load. Fig. 1 shows the geometrical con®guration at an interface corner A characterised by the angles h1 and h2 which the interface makes with the free surfaces of Material 1 and Material 2, respectively. The material in the region 0 6 h 6 h1 is referred to as Material 1 while Material 2 occupies the region ÿh2 6 h < 0. Both materials are assumed to be elastic, isotropic and homogeneous. In general, the stresses and displacements near the interface corner can be expressed as [8] rmij
N X k1
m Hk rkk ÿ1 fijk rmijo ;
umi
N X Hk rkk gikm umio ;
1
k1
where i; j r; h are plane polar co-ordinates centred at the interface corner; m 1; 2 is the material number; kk k 1; N are the eigenvalues of the problem; fijk and gik are non-dimensional constant functions of the material elastic properties, eigenvalue kk , the local edge geometry characterised by angles h1 and h2 , and of the polar co-ordinate h; and Hk is the wedge corner stress intensity associated with the eigenvalue kk . The second terms in (1), i.e. rmijo and umio , are the constant stress ®eld and the associated displacement near the interface corner; these terms vanish for remote mechanical loading and are ®nite for thermal loading and surface traction near the interface corner. The non-dimensional functions fijk and gik are determined analytically while the values of kk for a given geometry are obtained by numerically solving a characteristic equation. The closed form expression for these functions and the characteristic equation, for a range of joint geometries, can be found, for example, in [4,9,10]. The eigenvalue kk can be positive real, negative or complex. The expressions given in (1) are for positive real values of kk , and kk < 1 results in a stress singularity (k 0:5 for a crack in a monolithic solid). The wedge corner stress intensities Hk , which are de®ned such that fijk 1 along the interface h 0, are functions of the local edge geometry, material elastic properties and of the applied load. The dimensions of Hk , which are of the form (stress) length1ÿkk , change with the wedge angle and material properties. The need to develop a failure initiation criterion at interface corners has led to recent interest in the evaluation of the magnitude of Hk for various joint geometries and loading. However, most of the available studies have only considered the stress term associated with the smallest eigenvalue, i.e. the ®rst term of the summation in (1), and have therefore neglected the contribution from the higher order terms. For some
Fig. 1. General con®guration at the interface corner between two dissimilar materials.
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joint geometries and loading where there are more than one eigenvalues (i.e. N > 1 in Eq. (1)), the higher order terms may signi®cantly contribute to the stresses and displacements near the interface corner. There are some suggested methods in the literature about how to calculate the magnitude of the wedge corner stress intensities associated with all the eigenvalues at an interface corner based either on a contour integral [1,11] or an extrapolation method [9]. However, there is an inherent interaction between the terms in the existing methodology. As such, the accuracy of the intensities Hk evaluated by the contour integral method depends on the distance of the integration path from the interface corner while that of those evaluated by the extrapolation method depends on the outer limit of the extrapolation region. In this paper the contour integral method is used in conjunction with the ®nite element solution to evaluate the intensities Hk for higher order terms with very good accuracy and without any constraint on the location of the integration path. In the following a brief review of the existing methods of evaluating the intensities Hk associated with higher order terms are given and the limitations of the methods are discussed. The proposed method is then described and a joint geometry is used as an example to demonstrate the bene®ts of the method over the existing ones. 2. Evaluation of wedge corner stress intensities associated with higher order terms The evaluation of crack tip stress intensity factors for geometries subjected to mixed mode loading is now well established. For a crack in a monolithic solid the modes of deformation at the crack tip can easily be decoupled, for example, into a symmetric (Mode I) and anti-symmetric (Mode II) parts. However there is a coupling between the two modes for interfacial cracks, where a remote Mode I loading always result in a mixed mode deformation at the crack tip. In both cases the magnitude of the stress intensity factor associated with the individual mode can be accurately obtained by using an interaction energy method [12,13]. This method involves evaluating the J-integral for the elastic body of interest and a carefully chosen auxiliary ®eld. If the asymptotic crack tip ®eld with a known magnitude of stress intensity factor (Mode I or II) is chosen as the auxiliary ®eld, the interaction between the actual and the auxiliary ®elds can be used to evaluate the stress intensity factors of the elastic body of interest. The order of the singularity k ÿ 1 at an interface corner is in general greater than that of a crack and a ®nite wedge/notch opening exists at the interface corner. Therefore, the J-integral approach cannot be applied to the evaluation of the stress intensities associated with higher order terms at the tip of sharp interface wedges/notches. There are few studies in the open literature on the evaluation of the stress intensities associated with higher order terms at interface corners. The wedge corner stress intensities for a problem with two eigenvalues k1 and k2 have been determined by Knesl et al. and Theocaris [14,15]. The stress intensity H1 associated with the eigenvalue k1 was ®rst calculated from the stress state near the interface corner (i.e. small r) while neglecting the contribution from the higher order terms. The magnitude of H2 was then determined from the stress state at a distance further away from the interface corner, taking account of the contribution from the stress associated with k1 . One of the problems with this approach, as with conventional collocation method, is that under certain conditions the higher order terms signi®cantly contribute to the stress and displacement ®elds near the interface corner, and therefore the value of H1 evaluated using this method may be subject to a signi®cant error. In a separate study, ®nite element solutions were combined with an extrapolation method to determine the intensities Hk (also for k 1; 2) [9]. Here, the least square method was used to minimise the dierence between the asymptotic and the ®nite element solutions of the stresses along a radial line emanating from the interface corner. The minimisation was carried out with respect to the particular intensity Hk which was to be determined. The accuracy of the method depends not only on the radial direction chosen for the evaluation but also on the length of the radial line. The numerical solutions of the stresses at distances close to the interface corner are not reliable due to the stress singularity, while contribution from higher order
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terms are important at greater distances from the interface. It was shown in [9] that reasonably accurate values of the intensities H1 and H2 can be obtained provided the upper limit of the radial line used in the extrapolation is within a particular range. This range, however, varies with the eigenvalues kk and therefore a prior knowledge of the range is needed for each value of kk in order to obtain accurate value of the associated intensity Hk . A path-independent contour integral method based on BettiÕs reciprocal theorem has been used by a number of investigators to evaluate the stress intensities at the tip of a ®nite opening wedges in monolithic solids and at interface corners. The method combines the numerical stress and displacement solutions with an appropriate complementary solution so that the value of the integral gives the magnitude of the wedge corner stress intensity. Since the method proposed in this paper is based on this contour integral method, a brief description of the concepts is appropriate here. Consider for example the interface corner A of a joint geometry, as shown in Fig. 1. The stress and displacement ®elds near the corner are given by (1), where Hk are the wedge corner stress intensities to be determined. The emphasis in this paper is on the free-edge ®elds associated with remote mechanical loading, and hence rijo uio 0. In the absence of any body forces, the reciprocal work contour integral can be stated as [1,10,11] I rij ui ÿ rij ui nj ds 0; 2 R
where i; j r; h represent plane polar co-ordinates centred at the interface corner, rij and ui are the actual free-edge singular stress and displacement ®elds, (rij , ui ) are auxiliary ®elds satisfying the same boundary conditions as rij and ui , nj is the outward unit normal to the counterclockwise closed contour R, and ds is an in®nitesimal line segment of R. Since the auxiliary ®eld must satisfy the same boundary conditions as the actual ®eld, the integral in (2) vanishes at the free edges of the materials adjacent to the interface corner, i.e. along C1 and C2 as shown in Fig. 2. Thus Eq. (2) reduces to Z Z rij ui ÿ rij ui nj ds ÿ rij ui ÿ rij ui nj ds: 3 C4
C2
Fig. 2. A closed integration path R C1 C2 C3 C4 around the interface corner A.
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Previous studies [11,16,17] used the integral method to determine the magnitude of the wedge corner stress intensities at the tip of ®nite opening notches in monolithic solids. In applying the method, however, the singular stress ®eld at the tip of the notch was separated into symmetric and anti-symmetric parts. The auxiliary ®elds for both integrals in (3) were assumed to be exactly the same as the asymptotic ®elds at the tip of the notch but with an associated intensity H and eigenvalue k ÿk. The unstarred ®elds in (3) were chosen as the asymptotic ®elds at the notch tip with eigenvalue k and intensity H (which is to be determined) for the integration along C4 and as the ®nite element solutions for the integration along C2 . The integration along C4 could therefore be written as QHH , where Q is a constant whose value depends on the wedge opening, and can easily be determined analytically. In evaluating the integral along C2 the value of the intensity H of the auxiliary ®eld was chosen as H 1=Q so that the integral gave the intensity H of interest. This approach gave good results for notched monolithic solids where the asymptotic ®elds can easily be separated into symmetric and antisymmetric parts. This method was later extended to the evaluation of wedge corner stress intensities at the interface corner of bonded joint [18]. However, the auxiliary ®elds were chosen as the asymptotic singular part (i.e. all the terms in the summation part of (1)) with kk replaced by ÿkk and intensity Hk replaced by Hk . This approach results in an interaction between the auxiliary and the actual ®elds. The eects of this interaction on the magnitude of the wedge corner stress intensity can be signi®cant. The magnitude of the intensity Hk is signi®cantly aected by the stress ®eld associated with kj j < k, and also by the radius of the inner contour C4 [1]. In the current study a dierent choice of auxiliary ®eld is used in the contour integral to evaluate the magnitude of the wedge corner stress intensities associated with higher order terms with very good accuracy and without any path-dependence eects.
3. The proposed method Consider the local free-edge geometry near an interface corner as shown in Fig. 1. The singular stress and displacement ®elds are as given in Eq. (1), where rijo uio 0 for a remote mechanical loading. The material combinations and local edge geometry (characterised by the angles h1 and h2 ) are such that there are more than one eigenvalue, i.e. N > 1. The contour integral method described above is used to evaluate the wedge corner stress intensity associated with each of the eigenvalues. The evaluation of the integral in (2) over a closed contour near the interface corner (Fig. 2) gives the integral in (3), provided the auxiliary ®elds satisfy the same boundary conditions as the actual ®elds. The determination of the magnitude of the stress intensity Hk associated with the eigenvalue kk is ®rst considered. In evaluating the integral along C4 in Eq. (3), the actual ®elds (rij , ui ) are taken as the full expansion of the summation term in (1), and therefore contain as many terms as the number of eigenvalues. The ®nite element solutions of (rij , ui ) are used for evaluating the integral along C2 . However, the auxiliary ®elds rij ; ui are chosen as the asymptotic terms associated with the particular eigenvalue of interest. For example, in the determination of the intensity Hk , the auxiliary ®elds are chosen as the terms in (1) associated with kk , but with the intensity Hk replaced by Hk , and the eigenvalue kk replaced by kk ÿkk . The auxiliary ®elds for the evaluation of the wedge corner stress intensity associated with eigenvalue kk are therefore given by ; rij kk Hk rÿkk ÿ1 fijk
ui kk Hk rÿkk gik ;
4
where fijk fijk ÿkk and gik gik ÿkk . It is well known that if kk is an eigenvalue to the problem, so is ÿkk . Therefore, the formulation of the asymptotic solution in (1) ensures that these auxiliary ®elds satisfy the same boundary conditions as the actual ®elds.
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In contrast to the approach in [1,18] both the auxiliary stress and displacement ®elds contain one term each, while all the other terms in the asymptotic solutions have been neglected. The advantage of this choice of auxiliary ®eld is discussed in the next paragraph. The full form of the functions fijk kk and gik kk can be determined by using complex variable or AiryÕs stress function approach; these functions are given in Appendix A for completeness. The substitution of (1) and (4) into the left-hand side of (3) gives Z h1 0; ` 6 k kk ÿk` 5 fij kk ; hgi ÿk` ; h ÿ fij ÿk` ; hgi kk ; h dh H` Hk r Q H H ; ` k k ` k ÿh2 for any joint geometry and material combinations having positive real eigenvalues. In Eq. (5), Qk is a constant, which depends on the material properties of the materials and on the edge geometry. Therefore all the interaction terms, which in¯uenced the accuracy of the results in [1], vanish in this case. The integral along C4 (see Eq. (3)) using the auxiliary ®elds proposed in this paper can therefore be written as Qk Hk Hk , where the magnitude of Qk does not depend on the radial distance of C4 from the interface corner. In the evaluation of the integral along C2 the auxiliary ®elds are chosen as (4) but with the intensity Hk 1=Qk . This ensures the value of (3) gives the magnitude of the wedge corner stress intensity Hk . This process can be repeated to evaluate the magnitude of the stress intensities associated with other eigenvalues, by just changing the auxiliary ®elds to the stresses and displacements for that particular eigenvalue. 4. Analysis and results The method discussed in Section 3 above can be used to evaluate the wedge corner stress intensities associated with all the eigenvalues at the interface corner of a bi-material joint. An arbitrarily joint geometry is chosen to demonstrate the applicability and bene®ts of the method; this geometry is shown in Fig. 3. The angles which the interface makes with the free surface of the bonded materials are given by h1 ; h2 90° ; 120° . The elastic properties of the materials are characterised by the two elastic mismatch parameters, which are de®ned for plane strain conditions by Dundurs [19]
Fig. 3. A schematic diagram of the joint geometry considered in the paper.
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a
l1 j2 1 ÿ j1 1l2 ; l1 j2 1 j1 1l2
b
l1 j2 ÿ 1 ÿ j1 ÿ 1l2 ; l1 j2 1 j1 1l2
215
6
where lm Em =2 1 mm , Em and mm denote shear modulus, Young's modulus and Poisson's ratio for material m 1; 2, respectively, and jm 3 ÿ 4mm for plane strain. Here, the material above the interface is referred to as Material 1, while the material below the interface is referred to as Material 2. The material parameter a is positive when Material 2 is more compliant than Material 1, and it is negative when Material 2 is stier than Material 1. Both a and b vanish when the elastic properties of both materials are identical, and switching Materials 1 and 2 reverses the signs of a and b. Without loss of generality, only the results for material combinations with b a=4 are presented here. The joint is subjected to a remote tension r as shown in Fig. 3. The asymptotic stress and displacements near interface corner A are given by (1) with rijo uio 0. The eigenvalues for the joint geometry are obtained by solving the characteristic equation given in Appendix A. The values of kk a; b a=4 are shown in Fig. 4. There are three regimes: (i) in the region ÿ1 < a < ÿ0:2, there is only one eigenvalue k1 < 1 resulting in one singular stress term, (ii) there are two eigenvalues k1 < 1 and k2 > 1 in the region ÿ0:2 < a < 0:1 resulting in one singular and one non-singular stress terms, and (iii) there are three eigenvalues k1 < 1; k2 > 1, and k3 > 1 when a > 0:1, corresponding to one singular and two non-singular stress terms. Since this paper is on the evaluation of the stress intensities associated with higher order terms, the material combinations with a > 0:1 and b a=4 are considered. The values of kj j 1; 3 shown in Fig. 4 are in agreement with those given in [3,4,20]. The magnitude of the stress intensity associated with each eigenvalue is related to the material elastic parameters (a, b), the joint geometry characterised by the angles (h1 , h2 ) and the applied load r, according to Hk r h1ÿkk ak kk ; a; b; h1 ; h2 ;
7
where 2h is the distance between the two interface corners (see Fig. 4), ak is a dimensionless constant function of the angles (h1 , h2 ), and of the material elastic parameters (a, b). Elastic analysis of the above boundary value problems was carried out using the ®nite element method. The ®nite element mesh for the joint geometry is shown in Fig. 5. Only half of the joint was analysed due to the obvious symmetry in the geometry and loading. The ®nite element mesh consists of 630 eight-node quadrilateral and 43 six-node triangular, plane strain isoparametric elements. In the analysis h 1 unit and the other dimensions and boundary conditions are included in Fig. 5.
Fig. 4. The eect of the material parameters a and b a=4 on the eigenvalues kk for the joint geometry shown in Fig. 3.
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Fig. 5. The ®nite element mesh (a) for half of the joint geometry, and (b) near the interface corner of the joint.
The integration on the right-hand side of Eq. (2) (i.e. along C2 ) was performed by the domain integration method, whereby the line integral is converted to an area integral via GaussÕs theorem. Once the value of Hk has been obtained for a given material combination and a chosen domain of integration, the non-dimensional constant ak kk ; a; b; h1 ; h2 can be obtained via Eq. (7).
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217
In order to demonstrate the insensitivity of the current method to higher order terms and to the integration path, the magnitude of the non-dimensional constant ak k 1; 3 was determined using dierent domains of integration. The results for a 0:8 and b a=4 are listed in Table 1 together with the distance of the inner and outer boundaries of the domain from the interface corner. For this particular material combination, the eigenvalues are k1 0:6747, k2 1:1637 and k3 1:5938. As expected the magnitude of ak , and hence of the intensity Hk k 1; 3, does not depend on the distance of the integration path from the interface corner. The choice of the auxiliary ®elds in the present study ensures vanishing interaction terms, which originally caused the path dependence of the stress intensities determined by Carpenter [1]. The non-dimensional constant ak associated with the eigenvalue kk was determined over the full range of the material parameter a ÿ1 < a < 1 and for b a=4 using domain III, which was between a distance r 0:0869h and r 0:11h from interface corner A. The results are shown in Fig. 6. The magnitude of a1 and a3 increase with increasing value of a, while the value of a2 decreases monotonically with increasing value of a. Note that the wedge corner stress intensity Hk k 1; 3 is directly proportional to the nondimensional constant ak (see, Eq. (7)). The importance of higher order terms in accurately predicting the asymptotic free-edge stress ®eld is examined by comparing the ®nite element solution with the asymptotic prediction. The stress component rhh along various radial directions from the interface is compared with the corresponding theoretical prediction. The number of terms considered in the theoretical estimate was increased from one to the maximum number of eigenvalues. The results of the comparison are shown in Fig. 7. The asymptotic stress rhhk k 1; 3 associated with the eigenvalue kk was calculated using Eq. (1), with the values of kk shown in Fig. 4, the non-dimensional functions fhhk given in Appendix A and the intensity Hk obtained via Eq. (7) and Fig. 6. The stress component rhh in Fig. 7 has been normalised by the magnitude of the applied remote tension r, while the radial distance r has been normalised by the half width of the interface h.
Table 1 The magnitude of the non-dimensional constant ak (k 1, 3) determined using dierent domainsa Domain
I
II
III
Range a1 a2 a3
0.0053h±0.0063h 0.6301 ÿ0.3666 0.5407
0.0217h±0.0255h 0.6301 ÿ0.3673 0.5430
0.0869h±0.11h 0.6300 ÿ0.3675 0.5491
a
The material elastic mismatch parameters are: a 0.8 and b 0.2; and the eigenvalues are k1 0.6747, k2 1.1637, and k3 1.5938.
Fig. 6. The non-dimensional constant ak as a function of material parameter a.
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Fig. 7. Comparison between the ®nite element and asymptotic solutions of the stress component rhh for a 0:8 and b a=4. (a) The asymptotic rhh1 associated with k1 and the total rhh from the ®nite element analysis. (b) The asymptotic rhh1 rhh2 associated with k1 and k2 and the total rhh from the ®nite element analysis. (c) The asymptotic rhh1 rhh2 rhh3 associated with k1 , k2 and k3 and the total rhh from the ®nite element analysis.
Although the stress ®elds associated with k2 and k3 are not singular, they contribute signi®cantly to the overall stress state near the interface corner. The stress ®eld associated with k1 alone is not in agreement with the ®nite element solution along most radial directions (see Fig. 7a). The agreement between the solutions improved when the second term was considered in the asymptotic solutions (Fig. 7b) and was even better when all the three terms were considered (Fig. 7c). It is clear from the results shown in Fig. 7 that the contribution of the higher order terms needs to be considered in order to accurately model the asymptotic singular free-edge stress ®eld near an interface corner. This can only be done if the stress intensities associated with these higher order terms can be accurately determined. The eects of the stress ®eld associated with higher order terms on the overall stress state near an interface corner could be more signi®cant than that shown in Fig. 7 if the joint was subjected to a thermal loading instead of a remote tension [9].
5. Conclusions The applicability of the contour integral to the evaluation of the wedge corner stress intensities associated with higher order terms near an interface corner has been demonstrated. By a careful choice of the auxiliary ®eld it has been shown that the magnitude of the stress intensity is independent of the integration path. The theoretical prediction of the stresses near the interface corner of a bonded joint subjected to a
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remote tension is in good agreement with the ®nite element solution if the contribution of the higher order terms is included in the theoretical prediction.
Acknowledgements The authors are grateful for ®nancial support from the Engineering and Physical Sciences Research Council (UK).
Appendix A. The asymptotic ®elds near the interface corner of a bi-material joint The asymptotic free-edge stress and displacement ®elds near an interface corner are generally obtained using complex variable method or the AiryÕs stress function approach. Both methods should give the same asymptotic expressions for the stresses and the displacements. Using either of these methods, it can be shown that the stresses and displacements near the interface corner of a joint subjected to a remote mechanical load, and having the local edge geometry shown in Fig. 1, are of the form rmij
N X k1
m Hk rkk ÿ1 fijk ;
umi
N X Hk rkk gikm ;
8
k1
where i; j r; h are plane polar co-ordinates centred at the interface corner; m 1; 2 is the material number; kk k 1; N are the eigenvalues of the problem; fijk and gik are non-dimensional constant functions of the material elastic properties, eigenvalue kk , the local edge geometry characterised by angles h1 and h2 , and of the polar co-ordinate h; and Hk is the wedge corner stress intensity associated with the eigenvalue kk . The eigenvalues kk are determined by solving for the values of k which satisfy the characteristic equation F e2 b2 ÿ c2 ÿ d 2 0;
9
where e a ÿ bf cos 2kh1 ÿ cos 2kh1 ÿ 2kh2 k2 cos 2h1 ÿ cos 2h1 2h2 ÿ 1 cos 2h2 g 1 a1 ÿ cos 2kh1 ÿ 1 ÿ b1 ÿ cos 2kh2 ;
10
b a ÿ bf sin 2kh1 ÿ sin 2kh1 ÿ 2kh2 ÿ k2 sin 2h1 ÿ sin 2h1 2h2 sin 2h2 g ÿ 1 a sin 2kh1 ÿ 1 ÿ b sin 2kh2 ;
11
c kf a ÿ b cos 2kh1 ÿ cos 2kh1 2h2 cos 2kh2 ÿ cos 2kh2 ÿ 2h1 ÿ 1 cos 2h1 1 a1 ÿ cos 2h1 ÿ 1 ÿ b1 ÿ cos 2h2 g;
12
d kf a ÿ b sin 2h1 sin 2kh2 ÿ 2h1 ÿ sin 2kh1 sin 2kh1 2h2 ÿ sin 2kh2 ÿ 1 a sin 2h1 ÿ 1 ÿ b sin 2h2 g:
13
In the above equations, a and b are the material elastic mismatch parameters [19], which are de®ned in Eq. (6). For wedges/notches in monolithic materials where a b 0, the characteristic Eq. (9) reduces to that given in [2,20].
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The8functions fijm and gim associated with kk k are given by 9 1 > > > gr > > > > > > 2 3 > > gh1 > > > > > 1 0 > 1 > > > f > > rr 6 0 > > 1 7 > > 6 7 > > > fhh1 > > > 6 7 X X > > 1> " #6 31 32 7 > < frh = N 0 6X 7 X 41 42 7 y1 54 6 ; gr2 > 0 M 6 X51 X52 7 > > > 7 y2 54 6 > > 2 > >g > 6 X61 X62 7 > h > > 6 7 > > > 2> 4 X71 X72 5 > > > > f > rr > > > > > X81 X82 > 2 > > > f > hh > > > > ; : 2> frh where X31 ÿ cos 2kh1 k cos 2h1 ;
14
15
X32 sin 2kh1 ÿ k sin 2h1 ;
16
X41 sin 2kh1 k sin 2h1 ;
17
X42 cos 2kh1 ÿ k cos 2h1 ;
18
X51
a ÿ bk ÿ cos 2kh1 ÿ k cos 2h1 1 ÿ b ; 1a
19
X52
a ÿ b sin 2kh1 ÿ k sin 2h1 ; 1a
20
X61 ÿ X62 X71
a ÿ b sin 2kh1 k sin 2h1 ; 1a
1 ÿ b ÿ a ÿ bk cos 2kh1 ÿ k cos 2h1 ; 1a a ÿ b sin 2kh1 k sin 2h1 sin 2kh2 ÿ k sin 2h2 1a f1 ÿ b a ÿ bk ÿ cos 2kh1 ÿ k cos 2h1 g cos 2kh2 k cos 2h2 ; ÿ 1a a ÿ b sin 2kh1 ÿ k sin 2h1 cos 2kh2 k cos 2h2 1a f1 ÿ b ÿ a ÿ bk cos 2kh1 ÿ k cos 2h1 g sin 2kh2 ÿ k sin 2h2 ; ÿ 1a
21 22
23
X72 ÿ
X81 ÿ ÿ
24
a ÿ b sin 2kh1 k sin 2h1 cos 2kh2 ÿ k cos 2h2 1a f1 ÿ b a ÿ bk ÿ cos 2kh1 ÿ k cos 2h1 g sin 2kh2 k sin 2h2 ; 1a
25
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a ÿ b sin 2kh1 ÿ k sin 2h1 sin 2kh2 k sin 2h2 1a f1 ÿ b ÿ a ÿ bk cos 2kh1 ÿ k cos 2h1 g cos 2kh2 ÿ k cos 2h2 ; 1a
221
X82 ÿ
26
y1
cÿe ; kfk 1 ÿ cos 2kh1 ÿ k cos 2h1 c ÿ e b dg
27
y2
bd ; kfk 1 ÿ cos 2kh1 ÿ k cos 2h1 c ÿ e b dg
28
and
2
j1 ÿ k cos k ÿ 1h 6 2l1 6 6 6 j1 k sin k ÿ 1h 6 6 2l1 N 6 6 6 k 3 ÿ k cos k ÿ 1h 6 6 4 k k 1 cos k ÿ 1h
k ÿ j1 sin k ÿ 1h 2l1
ÿ cos k 1h 2l1
j1 k cos k ÿ 1h 2l1
sin k 1h 2l1
k k ÿ 3 sin k ÿ 1h k k 1 sin 1 ÿ kh
ÿ k cos k 1h k cos k 1h
k k ÿ 1 sin k ÿ 1h
k k ÿ 1 cos k ÿ 1h
k sin k 1h
3 sin k 1h 7 2l1 7 7 cos k 1h 7 7 7 2l1 7: 7 k sin k 1h 7 7 7 ÿ k sin k 1h 5
29
k cos k 1h
In the above equations, jm 3 ÿ 4mm for plane strain, and mm and lm are the PoissonÕs ratio and the shear modulus of material m, respectively. The matrix [M] is obtained by replacing j1 by j2 and l1 by l2 in the de®nition of [N]. The unknown parameter Hk associated with the eigenvalue kk is determined for the speci®c geometry and applied loading as described in the main text of the paper. References [1] W.C. Carpenter, Insensitivity of the reciprocal work contour integral method to higher order eigenvectors, Int. J. Fract. 73 (1995) 93±108. [2] M.L. Williams, Stress singularity resulting from various boundary conditions in angular corners of plates in extension, J. Appl. Mech. 19 (1952) 526±528. [3] D.B. Bogy, Edge bonded dissimilar orthogonal elastic wedge under normal and shear loading, J. Appl. Mech. 90 (1968) 460±466. [4] D.B. Bogy, Two edge-bonded elastic wedges of dierent materials and wedge angles under surface tractions, J. Appl. Mech. 38 (1971) 245±268. [5] M.L. Williams, The stresses around a fault or crack in dissimilar media, Bull. Seismological Soc. 49 (1959) 199±204. [6] E.D. Reedy, T.R. Guess, Comparison of butt tensile strength data with interface corner stress intensity factor prediction, Int. J. Solids Struct. 30 (1993) 2979±2986. [7] Z.Q. Qian, A.R. Akisanya, An experimental investigation of failure initiation in bonded joints, Acta Materialia 46 (1998) 4895± 4904. [8] Z.Q. Qian, A.R. Akisanya, Analysis of free-edge stress and displacement ®elds in scarf joints subjected to a uniform change in temperature, Fat. Fract. Engrg. Mat. Struct. 21 (1998) 687±703. [9] D. Munz, Y.Y. Yang, Stresses near the edge of bonded dissimilar materials described by two stress intensity factors, Int. J. Fract. 60 (1993) 169±177. [10] A.R. Akisanya, N.A. Fleck, Interfacial cracking from the free-edge of a long bi-material strip, Int. J. Solids Struct. 34 (1997) 1645± 1665. [11] G.B. Sinclair, M. Okajima, J.H. Grin, Path independent integral for computing stress intensity factors at sharp notches in elastic strips, Int. J. Num. Meth. Engrg. 10 (1984) 999±1008. [12] P.P.L. Matos, R.M. McMeeking, P.G. Charalambides, M.D. Drory, A method for calculating stress intensities in bi-material fracture, Int. J. Fract. 40 (1989) 235±254.
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[13] A.R. Akisanya, N.A. Fleck, The edge cracking and decohesion of thin ®lms, Int. J. Solids Struct. 31 (1994) 3175±3199. [14] Z. Knesl, A. Sramek, J. Kadourek, F. Kroupa, Stress concentration at the edge of coatings on tensile specimens, Acta Tech. CSAV 36 (1991) 574±593. [15] P.S. Theocaris, The Mesophase Concept in Composites, Springer, Berlin, 1987. [16] G.B. Sinclair, A remark on the determination of Mode I and Mode II stress intensity factors for sharp re-entrant corners, Int. J. Fract. 27 (1985) R81±R85. [17] W.C. Carpenter, Mode I and Mode II stress intensity factors for plates with cracks of ®nite opening, Int. J. Fract. 26 (1984) 201± 214. [18] W.C. Carpenter, C. Byers, A path independent integral for computing stress intensities for V-notched cracks in a bi-material, Int. J. Fract. 35 (1987) 245±268. [19] J. Dundurs, Mathematical Theory of Dislocations, American Society of Mechanical Engineers, New York, 1969. [20] G.C. Sih, J.W. Ho, Sharp notch fracture strength characterised by critical energy density, J. Theor. Appl. Fract. Mech. 16 (1991) 179±214.
www.elsevier.com/locate/tafmec
Wedge corner stress behaviour of bonded dissimilar materials Z.Q. Qian, A.R. Akisanya * Department of Engineering, University of Aberdeen, KingÕs College, Fraser Noble Building, Aberdeen AB24 3UE, UK
Abstract A contour integral, based on BettiÕs reciprocal theorem, is used in conjunction with the ®nite element method to evaluate the magnitude of the wedge corner stress intensities associated with the higher order terms of the singular stress ®eld near the interface corner of a bi-material joint. It is shown that using a dierent auxiliary ®eld can eliminate the dependence of the wedge corner stress intensity on the integration path observed by [W.C. Carpenter, Int. J. Fracture 73 (1995) 93±108]. Finite element analysis of a typical joint geometry is used to demonstrate the path-independence of the magnitude of the stress intensities evaluated using the proposed method, and to show the eects of higher order terms on the stress state near the interface corner. Ó 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Stress singularity; Wedge corner stress intensity; Path-independent contour integral; Finite element method; Failure initiation
1. Introduction Many engineering applications involve the joining of dissimilar materials. For example, present day automobiles and aircrafts contain signi®cant number of bonded components ranging from ceramic/metal/ ceramic to composite/adhesive/metal joints. A stress singularity may develop at the corner where the interface between the bonded materials intersects the free surface when the joints are subjected to mechanical and/or thermal loading. The type of singularity depends on the choice of joint geometry and the elastic and thermal properties of the materials [2±4]. Since failure is most likely to be initiated from a singular point it is therefore important to properly characterise the singular stresses and the associated displacements so that joint geometries and material combinations can be appropriately chosen to minimise such failure. There have been a lot of studies on the analysis of stress singularities at interface corners and at wedge/ notch and interface crack tips since the pioneering work described in [2,5]. Initially these studies focussed on the in¯uence of the wedge/notch angle and of material elastic properties on the order of the singularity. However, the emphasis has changed, in the last 7 yr or so, to the determination of the complete structure of the stress ®eld near the interface corner of bonded dissimilar materials, including the evaluation of the associated corner stress intensity. This has been in¯uenced by the success of using a critical value of crack tip stress intensity factor to correlate the onset of fracture in brittle monolithic solids. There is increasing evidence that a critical wedge corner stress intensity can also be used in a similar manner to predict the
*
Corresponding author. Tel.: +44-1224-272989; fax: +44-1224-272497. E-mail address: [email protected] (A.R. Akisanya)
0167-8442/99/$ - see front matter Ó 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 9 ) 0 0 0 4 1 - 5
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onset of failure from sharp notches and interface corners, see for example, [6,7]. This, however, will depend on the availability of calibration curves for the magnitude of the stress intensities at an interface corner as a function of joint geometry (or wedge angle), material properties and applied load. Fig. 1 shows the geometrical con®guration at an interface corner A characterised by the angles h1 and h2 which the interface makes with the free surfaces of Material 1 and Material 2, respectively. The material in the region 0 6 h 6 h1 is referred to as Material 1 while Material 2 occupies the region ÿh2 6 h < 0. Both materials are assumed to be elastic, isotropic and homogeneous. In general, the stresses and displacements near the interface corner can be expressed as [8] rmij
N X k1
m Hk rkk ÿ1 fijk rmijo ;
umi
N X Hk rkk gikm umio ;
1
k1
where i; j r; h are plane polar co-ordinates centred at the interface corner; m 1; 2 is the material number; kk k 1; N are the eigenvalues of the problem; fijk and gik are non-dimensional constant functions of the material elastic properties, eigenvalue kk , the local edge geometry characterised by angles h1 and h2 , and of the polar co-ordinate h; and Hk is the wedge corner stress intensity associated with the eigenvalue kk . The second terms in (1), i.e. rmijo and umio , are the constant stress ®eld and the associated displacement near the interface corner; these terms vanish for remote mechanical loading and are ®nite for thermal loading and surface traction near the interface corner. The non-dimensional functions fijk and gik are determined analytically while the values of kk for a given geometry are obtained by numerically solving a characteristic equation. The closed form expression for these functions and the characteristic equation, for a range of joint geometries, can be found, for example, in [4,9,10]. The eigenvalue kk can be positive real, negative or complex. The expressions given in (1) are for positive real values of kk , and kk < 1 results in a stress singularity (k 0:5 for a crack in a monolithic solid). The wedge corner stress intensities Hk , which are de®ned such that fijk 1 along the interface h 0, are functions of the local edge geometry, material elastic properties and of the applied load. The dimensions of Hk , which are of the form (stress) length1ÿkk , change with the wedge angle and material properties. The need to develop a failure initiation criterion at interface corners has led to recent interest in the evaluation of the magnitude of Hk for various joint geometries and loading. However, most of the available studies have only considered the stress term associated with the smallest eigenvalue, i.e. the ®rst term of the summation in (1), and have therefore neglected the contribution from the higher order terms. For some
Fig. 1. General con®guration at the interface corner between two dissimilar materials.
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211
joint geometries and loading where there are more than one eigenvalues (i.e. N > 1 in Eq. (1)), the higher order terms may signi®cantly contribute to the stresses and displacements near the interface corner. There are some suggested methods in the literature about how to calculate the magnitude of the wedge corner stress intensities associated with all the eigenvalues at an interface corner based either on a contour integral [1,11] or an extrapolation method [9]. However, there is an inherent interaction between the terms in the existing methodology. As such, the accuracy of the intensities Hk evaluated by the contour integral method depends on the distance of the integration path from the interface corner while that of those evaluated by the extrapolation method depends on the outer limit of the extrapolation region. In this paper the contour integral method is used in conjunction with the ®nite element solution to evaluate the intensities Hk for higher order terms with very good accuracy and without any constraint on the location of the integration path. In the following a brief review of the existing methods of evaluating the intensities Hk associated with higher order terms are given and the limitations of the methods are discussed. The proposed method is then described and a joint geometry is used as an example to demonstrate the bene®ts of the method over the existing ones. 2. Evaluation of wedge corner stress intensities associated with higher order terms The evaluation of crack tip stress intensity factors for geometries subjected to mixed mode loading is now well established. For a crack in a monolithic solid the modes of deformation at the crack tip can easily be decoupled, for example, into a symmetric (Mode I) and anti-symmetric (Mode II) parts. However there is a coupling between the two modes for interfacial cracks, where a remote Mode I loading always result in a mixed mode deformation at the crack tip. In both cases the magnitude of the stress intensity factor associated with the individual mode can be accurately obtained by using an interaction energy method [12,13]. This method involves evaluating the J-integral for the elastic body of interest and a carefully chosen auxiliary ®eld. If the asymptotic crack tip ®eld with a known magnitude of stress intensity factor (Mode I or II) is chosen as the auxiliary ®eld, the interaction between the actual and the auxiliary ®elds can be used to evaluate the stress intensity factors of the elastic body of interest. The order of the singularity k ÿ 1 at an interface corner is in general greater than that of a crack and a ®nite wedge/notch opening exists at the interface corner. Therefore, the J-integral approach cannot be applied to the evaluation of the stress intensities associated with higher order terms at the tip of sharp interface wedges/notches. There are few studies in the open literature on the evaluation of the stress intensities associated with higher order terms at interface corners. The wedge corner stress intensities for a problem with two eigenvalues k1 and k2 have been determined by Knesl et al. and Theocaris [14,15]. The stress intensity H1 associated with the eigenvalue k1 was ®rst calculated from the stress state near the interface corner (i.e. small r) while neglecting the contribution from the higher order terms. The magnitude of H2 was then determined from the stress state at a distance further away from the interface corner, taking account of the contribution from the stress associated with k1 . One of the problems with this approach, as with conventional collocation method, is that under certain conditions the higher order terms signi®cantly contribute to the stress and displacement ®elds near the interface corner, and therefore the value of H1 evaluated using this method may be subject to a signi®cant error. In a separate study, ®nite element solutions were combined with an extrapolation method to determine the intensities Hk (also for k 1; 2) [9]. Here, the least square method was used to minimise the dierence between the asymptotic and the ®nite element solutions of the stresses along a radial line emanating from the interface corner. The minimisation was carried out with respect to the particular intensity Hk which was to be determined. The accuracy of the method depends not only on the radial direction chosen for the evaluation but also on the length of the radial line. The numerical solutions of the stresses at distances close to the interface corner are not reliable due to the stress singularity, while contribution from higher order
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terms are important at greater distances from the interface. It was shown in [9] that reasonably accurate values of the intensities H1 and H2 can be obtained provided the upper limit of the radial line used in the extrapolation is within a particular range. This range, however, varies with the eigenvalues kk and therefore a prior knowledge of the range is needed for each value of kk in order to obtain accurate value of the associated intensity Hk . A path-independent contour integral method based on BettiÕs reciprocal theorem has been used by a number of investigators to evaluate the stress intensities at the tip of a ®nite opening wedges in monolithic solids and at interface corners. The method combines the numerical stress and displacement solutions with an appropriate complementary solution so that the value of the integral gives the magnitude of the wedge corner stress intensity. Since the method proposed in this paper is based on this contour integral method, a brief description of the concepts is appropriate here. Consider for example the interface corner A of a joint geometry, as shown in Fig. 1. The stress and displacement ®elds near the corner are given by (1), where Hk are the wedge corner stress intensities to be determined. The emphasis in this paper is on the free-edge ®elds associated with remote mechanical loading, and hence rijo uio 0. In the absence of any body forces, the reciprocal work contour integral can be stated as [1,10,11] I rij ui ÿ rij ui nj ds 0; 2 R
where i; j r; h represent plane polar co-ordinates centred at the interface corner, rij and ui are the actual free-edge singular stress and displacement ®elds, (rij , ui ) are auxiliary ®elds satisfying the same boundary conditions as rij and ui , nj is the outward unit normal to the counterclockwise closed contour R, and ds is an in®nitesimal line segment of R. Since the auxiliary ®eld must satisfy the same boundary conditions as the actual ®eld, the integral in (2) vanishes at the free edges of the materials adjacent to the interface corner, i.e. along C1 and C2 as shown in Fig. 2. Thus Eq. (2) reduces to Z Z rij ui ÿ rij ui nj ds ÿ rij ui ÿ rij ui nj ds: 3 C4
C2
Fig. 2. A closed integration path R C1 C2 C3 C4 around the interface corner A.
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Previous studies [11,16,17] used the integral method to determine the magnitude of the wedge corner stress intensities at the tip of ®nite opening notches in monolithic solids. In applying the method, however, the singular stress ®eld at the tip of the notch was separated into symmetric and anti-symmetric parts. The auxiliary ®elds for both integrals in (3) were assumed to be exactly the same as the asymptotic ®elds at the tip of the notch but with an associated intensity H and eigenvalue k ÿk. The unstarred ®elds in (3) were chosen as the asymptotic ®elds at the notch tip with eigenvalue k and intensity H (which is to be determined) for the integration along C4 and as the ®nite element solutions for the integration along C2 . The integration along C4 could therefore be written as QHH , where Q is a constant whose value depends on the wedge opening, and can easily be determined analytically. In evaluating the integral along C2 the value of the intensity H of the auxiliary ®eld was chosen as H 1=Q so that the integral gave the intensity H of interest. This approach gave good results for notched monolithic solids where the asymptotic ®elds can easily be separated into symmetric and antisymmetric parts. This method was later extended to the evaluation of wedge corner stress intensities at the interface corner of bonded joint [18]. However, the auxiliary ®elds were chosen as the asymptotic singular part (i.e. all the terms in the summation part of (1)) with kk replaced by ÿkk and intensity Hk replaced by Hk . This approach results in an interaction between the auxiliary and the actual ®elds. The eects of this interaction on the magnitude of the wedge corner stress intensity can be signi®cant. The magnitude of the intensity Hk is signi®cantly aected by the stress ®eld associated with kj j < k, and also by the radius of the inner contour C4 [1]. In the current study a dierent choice of auxiliary ®eld is used in the contour integral to evaluate the magnitude of the wedge corner stress intensities associated with higher order terms with very good accuracy and without any path-dependence eects.
3. The proposed method Consider the local free-edge geometry near an interface corner as shown in Fig. 1. The singular stress and displacement ®elds are as given in Eq. (1), where rijo uio 0 for a remote mechanical loading. The material combinations and local edge geometry (characterised by the angles h1 and h2 ) are such that there are more than one eigenvalue, i.e. N > 1. The contour integral method described above is used to evaluate the wedge corner stress intensity associated with each of the eigenvalues. The evaluation of the integral in (2) over a closed contour near the interface corner (Fig. 2) gives the integral in (3), provided the auxiliary ®elds satisfy the same boundary conditions as the actual ®elds. The determination of the magnitude of the stress intensity Hk associated with the eigenvalue kk is ®rst considered. In evaluating the integral along C4 in Eq. (3), the actual ®elds (rij , ui ) are taken as the full expansion of the summation term in (1), and therefore contain as many terms as the number of eigenvalues. The ®nite element solutions of (rij , ui ) are used for evaluating the integral along C2 . However, the auxiliary ®elds rij ; ui are chosen as the asymptotic terms associated with the particular eigenvalue of interest. For example, in the determination of the intensity Hk , the auxiliary ®elds are chosen as the terms in (1) associated with kk , but with the intensity Hk replaced by Hk , and the eigenvalue kk replaced by kk ÿkk . The auxiliary ®elds for the evaluation of the wedge corner stress intensity associated with eigenvalue kk are therefore given by ; rij kk Hk rÿkk ÿ1 fijk
ui kk Hk rÿkk gik ;
4
where fijk fijk ÿkk and gik gik ÿkk . It is well known that if kk is an eigenvalue to the problem, so is ÿkk . Therefore, the formulation of the asymptotic solution in (1) ensures that these auxiliary ®elds satisfy the same boundary conditions as the actual ®elds.
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In contrast to the approach in [1,18] both the auxiliary stress and displacement ®elds contain one term each, while all the other terms in the asymptotic solutions have been neglected. The advantage of this choice of auxiliary ®eld is discussed in the next paragraph. The full form of the functions fijk kk and gik kk can be determined by using complex variable or AiryÕs stress function approach; these functions are given in Appendix A for completeness. The substitution of (1) and (4) into the left-hand side of (3) gives Z h1 0; ` 6 k kk ÿk` 5 fij kk ; hgi ÿk` ; h ÿ fij ÿk` ; hgi kk ; h dh H` Hk r Q H H ; ` k k ` k ÿh2 for any joint geometry and material combinations having positive real eigenvalues. In Eq. (5), Qk is a constant, which depends on the material properties of the materials and on the edge geometry. Therefore all the interaction terms, which in¯uenced the accuracy of the results in [1], vanish in this case. The integral along C4 (see Eq. (3)) using the auxiliary ®elds proposed in this paper can therefore be written as Qk Hk Hk , where the magnitude of Qk does not depend on the radial distance of C4 from the interface corner. In the evaluation of the integral along C2 the auxiliary ®elds are chosen as (4) but with the intensity Hk 1=Qk . This ensures the value of (3) gives the magnitude of the wedge corner stress intensity Hk . This process can be repeated to evaluate the magnitude of the stress intensities associated with other eigenvalues, by just changing the auxiliary ®elds to the stresses and displacements for that particular eigenvalue. 4. Analysis and results The method discussed in Section 3 above can be used to evaluate the wedge corner stress intensities associated with all the eigenvalues at the interface corner of a bi-material joint. An arbitrarily joint geometry is chosen to demonstrate the applicability and bene®ts of the method; this geometry is shown in Fig. 3. The angles which the interface makes with the free surface of the bonded materials are given by h1 ; h2 90° ; 120° . The elastic properties of the materials are characterised by the two elastic mismatch parameters, which are de®ned for plane strain conditions by Dundurs [19]
Fig. 3. A schematic diagram of the joint geometry considered in the paper.
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a
l1 j2 1 ÿ j1 1l2 ; l1 j2 1 j1 1l2
b
l1 j2 ÿ 1 ÿ j1 ÿ 1l2 ; l1 j2 1 j1 1l2
215
6
where lm Em =2 1 mm , Em and mm denote shear modulus, Young's modulus and Poisson's ratio for material m 1; 2, respectively, and jm 3 ÿ 4mm for plane strain. Here, the material above the interface is referred to as Material 1, while the material below the interface is referred to as Material 2. The material parameter a is positive when Material 2 is more compliant than Material 1, and it is negative when Material 2 is stier than Material 1. Both a and b vanish when the elastic properties of both materials are identical, and switching Materials 1 and 2 reverses the signs of a and b. Without loss of generality, only the results for material combinations with b a=4 are presented here. The joint is subjected to a remote tension r as shown in Fig. 3. The asymptotic stress and displacements near interface corner A are given by (1) with rijo uio 0. The eigenvalues for the joint geometry are obtained by solving the characteristic equation given in Appendix A. The values of kk a; b a=4 are shown in Fig. 4. There are three regimes: (i) in the region ÿ1 < a < ÿ0:2, there is only one eigenvalue k1 < 1 resulting in one singular stress term, (ii) there are two eigenvalues k1 < 1 and k2 > 1 in the region ÿ0:2 < a < 0:1 resulting in one singular and one non-singular stress terms, and (iii) there are three eigenvalues k1 < 1; k2 > 1, and k3 > 1 when a > 0:1, corresponding to one singular and two non-singular stress terms. Since this paper is on the evaluation of the stress intensities associated with higher order terms, the material combinations with a > 0:1 and b a=4 are considered. The values of kj j 1; 3 shown in Fig. 4 are in agreement with those given in [3,4,20]. The magnitude of the stress intensity associated with each eigenvalue is related to the material elastic parameters (a, b), the joint geometry characterised by the angles (h1 , h2 ) and the applied load r, according to Hk r h1ÿkk ak kk ; a; b; h1 ; h2 ;
7
where 2h is the distance between the two interface corners (see Fig. 4), ak is a dimensionless constant function of the angles (h1 , h2 ), and of the material elastic parameters (a, b). Elastic analysis of the above boundary value problems was carried out using the ®nite element method. The ®nite element mesh for the joint geometry is shown in Fig. 5. Only half of the joint was analysed due to the obvious symmetry in the geometry and loading. The ®nite element mesh consists of 630 eight-node quadrilateral and 43 six-node triangular, plane strain isoparametric elements. In the analysis h 1 unit and the other dimensions and boundary conditions are included in Fig. 5.
Fig. 4. The eect of the material parameters a and b a=4 on the eigenvalues kk for the joint geometry shown in Fig. 3.
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Fig. 5. The ®nite element mesh (a) for half of the joint geometry, and (b) near the interface corner of the joint.
The integration on the right-hand side of Eq. (2) (i.e. along C2 ) was performed by the domain integration method, whereby the line integral is converted to an area integral via GaussÕs theorem. Once the value of Hk has been obtained for a given material combination and a chosen domain of integration, the non-dimensional constant ak kk ; a; b; h1 ; h2 can be obtained via Eq. (7).
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In order to demonstrate the insensitivity of the current method to higher order terms and to the integration path, the magnitude of the non-dimensional constant ak k 1; 3 was determined using dierent domains of integration. The results for a 0:8 and b a=4 are listed in Table 1 together with the distance of the inner and outer boundaries of the domain from the interface corner. For this particular material combination, the eigenvalues are k1 0:6747, k2 1:1637 and k3 1:5938. As expected the magnitude of ak , and hence of the intensity Hk k 1; 3, does not depend on the distance of the integration path from the interface corner. The choice of the auxiliary ®elds in the present study ensures vanishing interaction terms, which originally caused the path dependence of the stress intensities determined by Carpenter [1]. The non-dimensional constant ak associated with the eigenvalue kk was determined over the full range of the material parameter a ÿ1 < a < 1 and for b a=4 using domain III, which was between a distance r 0:0869h and r 0:11h from interface corner A. The results are shown in Fig. 6. The magnitude of a1 and a3 increase with increasing value of a, while the value of a2 decreases monotonically with increasing value of a. Note that the wedge corner stress intensity Hk k 1; 3 is directly proportional to the nondimensional constant ak (see, Eq. (7)). The importance of higher order terms in accurately predicting the asymptotic free-edge stress ®eld is examined by comparing the ®nite element solution with the asymptotic prediction. The stress component rhh along various radial directions from the interface is compared with the corresponding theoretical prediction. The number of terms considered in the theoretical estimate was increased from one to the maximum number of eigenvalues. The results of the comparison are shown in Fig. 7. The asymptotic stress rhhk k 1; 3 associated with the eigenvalue kk was calculated using Eq. (1), with the values of kk shown in Fig. 4, the non-dimensional functions fhhk given in Appendix A and the intensity Hk obtained via Eq. (7) and Fig. 6. The stress component rhh in Fig. 7 has been normalised by the magnitude of the applied remote tension r, while the radial distance r has been normalised by the half width of the interface h.
Table 1 The magnitude of the non-dimensional constant ak (k 1, 3) determined using dierent domainsa Domain
I
II
III
Range a1 a2 a3
0.0053h±0.0063h 0.6301 ÿ0.3666 0.5407
0.0217h±0.0255h 0.6301 ÿ0.3673 0.5430
0.0869h±0.11h 0.6300 ÿ0.3675 0.5491
a
The material elastic mismatch parameters are: a 0.8 and b 0.2; and the eigenvalues are k1 0.6747, k2 1.1637, and k3 1.5938.
Fig. 6. The non-dimensional constant ak as a function of material parameter a.
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Fig. 7. Comparison between the ®nite element and asymptotic solutions of the stress component rhh for a 0:8 and b a=4. (a) The asymptotic rhh1 associated with k1 and the total rhh from the ®nite element analysis. (b) The asymptotic rhh1 rhh2 associated with k1 and k2 and the total rhh from the ®nite element analysis. (c) The asymptotic rhh1 rhh2 rhh3 associated with k1 , k2 and k3 and the total rhh from the ®nite element analysis.
Although the stress ®elds associated with k2 and k3 are not singular, they contribute signi®cantly to the overall stress state near the interface corner. The stress ®eld associated with k1 alone is not in agreement with the ®nite element solution along most radial directions (see Fig. 7a). The agreement between the solutions improved when the second term was considered in the asymptotic solutions (Fig. 7b) and was even better when all the three terms were considered (Fig. 7c). It is clear from the results shown in Fig. 7 that the contribution of the higher order terms needs to be considered in order to accurately model the asymptotic singular free-edge stress ®eld near an interface corner. This can only be done if the stress intensities associated with these higher order terms can be accurately determined. The eects of the stress ®eld associated with higher order terms on the overall stress state near an interface corner could be more signi®cant than that shown in Fig. 7 if the joint was subjected to a thermal loading instead of a remote tension [9].
5. Conclusions The applicability of the contour integral to the evaluation of the wedge corner stress intensities associated with higher order terms near an interface corner has been demonstrated. By a careful choice of the auxiliary ®eld it has been shown that the magnitude of the stress intensity is independent of the integration path. The theoretical prediction of the stresses near the interface corner of a bonded joint subjected to a
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remote tension is in good agreement with the ®nite element solution if the contribution of the higher order terms is included in the theoretical prediction.
Acknowledgements The authors are grateful for ®nancial support from the Engineering and Physical Sciences Research Council (UK).
Appendix A. The asymptotic ®elds near the interface corner of a bi-material joint The asymptotic free-edge stress and displacement ®elds near an interface corner are generally obtained using complex variable method or the AiryÕs stress function approach. Both methods should give the same asymptotic expressions for the stresses and the displacements. Using either of these methods, it can be shown that the stresses and displacements near the interface corner of a joint subjected to a remote mechanical load, and having the local edge geometry shown in Fig. 1, are of the form rmij
N X k1
m Hk rkk ÿ1 fijk ;
umi
N X Hk rkk gikm ;
8
k1
where i; j r; h are plane polar co-ordinates centred at the interface corner; m 1; 2 is the material number; kk k 1; N are the eigenvalues of the problem; fijk and gik are non-dimensional constant functions of the material elastic properties, eigenvalue kk , the local edge geometry characterised by angles h1 and h2 , and of the polar co-ordinate h; and Hk is the wedge corner stress intensity associated with the eigenvalue kk . The eigenvalues kk are determined by solving for the values of k which satisfy the characteristic equation F e2 b2 ÿ c2 ÿ d 2 0;
9
where e a ÿ bf cos 2kh1 ÿ cos 2kh1 ÿ 2kh2 k2 cos 2h1 ÿ cos 2h1 2h2 ÿ 1 cos 2h2 g 1 a1 ÿ cos 2kh1 ÿ 1 ÿ b1 ÿ cos 2kh2 ;
10
b a ÿ bf sin 2kh1 ÿ sin 2kh1 ÿ 2kh2 ÿ k2 sin 2h1 ÿ sin 2h1 2h2 sin 2h2 g ÿ 1 a sin 2kh1 ÿ 1 ÿ b sin 2kh2 ;
11
c kf a ÿ b cos 2kh1 ÿ cos 2kh1 2h2 cos 2kh2 ÿ cos 2kh2 ÿ 2h1 ÿ 1 cos 2h1 1 a1 ÿ cos 2h1 ÿ 1 ÿ b1 ÿ cos 2h2 g;
12
d kf a ÿ b sin 2h1 sin 2kh2 ÿ 2h1 ÿ sin 2kh1 sin 2kh1 2h2 ÿ sin 2kh2 ÿ 1 a sin 2h1 ÿ 1 ÿ b sin 2h2 g:
13
In the above equations, a and b are the material elastic mismatch parameters [19], which are de®ned in Eq. (6). For wedges/notches in monolithic materials where a b 0, the characteristic Eq. (9) reduces to that given in [2,20].
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The8functions fijm and gim associated with kk k are given by 9 1 > > > gr > > > > > > 2 3 > > gh1 > > > > > 1 0 > 1 > > > f > > rr 6 0 > > 1 7 > > 6 7 > > > fhh1 > > > 6 7 X X > > 1> " #6 31 32 7 > < frh = N 0 6X 7 X 41 42 7 y1 54 6 ; gr2 > 0 M 6 X51 X52 7 > > > 7 y2 54 6 > > 2 > >g > 6 X61 X62 7 > h > > 6 7 > > > 2> 4 X71 X72 5 > > > > f > rr > > > > > X81 X82 > 2 > > > f > hh > > > > ; : 2> frh where X31 ÿ cos 2kh1 k cos 2h1 ;
14
15
X32 sin 2kh1 ÿ k sin 2h1 ;
16
X41 sin 2kh1 k sin 2h1 ;
17
X42 cos 2kh1 ÿ k cos 2h1 ;
18
X51
a ÿ bk ÿ cos 2kh1 ÿ k cos 2h1 1 ÿ b ; 1a
19
X52
a ÿ b sin 2kh1 ÿ k sin 2h1 ; 1a
20
X61 ÿ X62 X71
a ÿ b sin 2kh1 k sin 2h1 ; 1a
1 ÿ b ÿ a ÿ bk cos 2kh1 ÿ k cos 2h1 ; 1a a ÿ b sin 2kh1 k sin 2h1 sin 2kh2 ÿ k sin 2h2 1a f1 ÿ b a ÿ bk ÿ cos 2kh1 ÿ k cos 2h1 g cos 2kh2 k cos 2h2 ; ÿ 1a a ÿ b sin 2kh1 ÿ k sin 2h1 cos 2kh2 k cos 2h2 1a f1 ÿ b ÿ a ÿ bk cos 2kh1 ÿ k cos 2h1 g sin 2kh2 ÿ k sin 2h2 ; ÿ 1a
21 22
23
X72 ÿ
X81 ÿ ÿ
24
a ÿ b sin 2kh1 k sin 2h1 cos 2kh2 ÿ k cos 2h2 1a f1 ÿ b a ÿ bk ÿ cos 2kh1 ÿ k cos 2h1 g sin 2kh2 k sin 2h2 ; 1a
25
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a ÿ b sin 2kh1 ÿ k sin 2h1 sin 2kh2 k sin 2h2 1a f1 ÿ b ÿ a ÿ bk cos 2kh1 ÿ k cos 2h1 g cos 2kh2 ÿ k cos 2h2 ; 1a
221
X82 ÿ
26
y1
cÿe ; kfk 1 ÿ cos 2kh1 ÿ k cos 2h1 c ÿ e b dg
27
y2
bd ; kfk 1 ÿ cos 2kh1 ÿ k cos 2h1 c ÿ e b dg
28
and
2
j1 ÿ k cos k ÿ 1h 6 2l1 6 6 6 j1 k sin k ÿ 1h 6 6 2l1 N 6 6 6 k 3 ÿ k cos k ÿ 1h 6 6 4 k k 1 cos k ÿ 1h
k ÿ j1 sin k ÿ 1h 2l1
ÿ cos k 1h 2l1
j1 k cos k ÿ 1h 2l1
sin k 1h 2l1
k k ÿ 3 sin k ÿ 1h k k 1 sin 1 ÿ kh
ÿ k cos k 1h k cos k 1h
k k ÿ 1 sin k ÿ 1h
k k ÿ 1 cos k ÿ 1h
k sin k 1h
3 sin k 1h 7 2l1 7 7 cos k 1h 7 7 7 2l1 7: 7 k sin k 1h 7 7 7 ÿ k sin k 1h 5
29
k cos k 1h
In the above equations, jm 3 ÿ 4mm for plane strain, and mm and lm are the PoissonÕs ratio and the shear modulus of material m, respectively. The matrix [M] is obtained by replacing j1 by j2 and l1 by l2 in the de®nition of [N]. The unknown parameter Hk associated with the eigenvalue kk is determined for the speci®c geometry and applied loading as described in the main text of the paper. References [1] W.C. Carpenter, Insensitivity of the reciprocal work contour integral method to higher order eigenvectors, Int. J. Fract. 73 (1995) 93±108. [2] M.L. Williams, Stress singularity resulting from various boundary conditions in angular corners of plates in extension, J. Appl. Mech. 19 (1952) 526±528. [3] D.B. Bogy, Edge bonded dissimilar orthogonal elastic wedge under normal and shear loading, J. Appl. Mech. 90 (1968) 460±466. [4] D.B. Bogy, Two edge-bonded elastic wedges of dierent materials and wedge angles under surface tractions, J. Appl. Mech. 38 (1971) 245±268. [5] M.L. Williams, The stresses around a fault or crack in dissimilar media, Bull. Seismological Soc. 49 (1959) 199±204. [6] E.D. Reedy, T.R. Guess, Comparison of butt tensile strength data with interface corner stress intensity factor prediction, Int. J. Solids Struct. 30 (1993) 2979±2986. [7] Z.Q. Qian, A.R. Akisanya, An experimental investigation of failure initiation in bonded joints, Acta Materialia 46 (1998) 4895± 4904. [8] Z.Q. Qian, A.R. Akisanya, Analysis of free-edge stress and displacement ®elds in scarf joints subjected to a uniform change in temperature, Fat. Fract. Engrg. Mat. Struct. 21 (1998) 687±703. [9] D. Munz, Y.Y. Yang, Stresses near the edge of bonded dissimilar materials described by two stress intensity factors, Int. J. Fract. 60 (1993) 169±177. [10] A.R. Akisanya, N.A. Fleck, Interfacial cracking from the free-edge of a long bi-material strip, Int. J. Solids Struct. 34 (1997) 1645± 1665. [11] G.B. Sinclair, M. Okajima, J.H. Grin, Path independent integral for computing stress intensity factors at sharp notches in elastic strips, Int. J. Num. Meth. Engrg. 10 (1984) 999±1008. [12] P.P.L. Matos, R.M. McMeeking, P.G. Charalambides, M.D. Drory, A method for calculating stress intensities in bi-material fracture, Int. J. Fract. 40 (1989) 235±254.
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[13] A.R. Akisanya, N.A. Fleck, The edge cracking and decohesion of thin ®lms, Int. J. Solids Struct. 31 (1994) 3175±3199. [14] Z. Knesl, A. Sramek, J. Kadourek, F. Kroupa, Stress concentration at the edge of coatings on tensile specimens, Acta Tech. CSAV 36 (1991) 574±593. [15] P.S. Theocaris, The Mesophase Concept in Composites, Springer, Berlin, 1987. [16] G.B. Sinclair, A remark on the determination of Mode I and Mode II stress intensity factors for sharp re-entrant corners, Int. J. Fract. 27 (1985) R81±R85. [17] W.C. Carpenter, Mode I and Mode II stress intensity factors for plates with cracks of ®nite opening, Int. J. Fract. 26 (1984) 201± 214. [18] W.C. Carpenter, C. Byers, A path independent integral for computing stress intensities for V-notched cracks in a bi-material, Int. J. Fract. 35 (1987) 245±268. [19] J. Dundurs, Mathematical Theory of Dislocations, American Society of Mechanical Engineers, New York, 1969. [20] G.C. Sih, J.W. Ho, Sharp notch fracture strength characterised by critical energy density, J. Theor. Appl. Fract. Mech. 16 (1991) 179±214.