Thermo-mechanical stresses for a triple junction of dissimilar materials: Global-local finite element analysis

Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

Thermo-mechanical stresses for a triple junction of dissimilar materials: Global-local ®nite element analysis E. Madenci *, S. Shkarayev, B. Sergeev Department of Aerospace and Mechanical Engineering and Center for Electronics Packaging Research, The University of Arizona, Tucson, AZ 85721, USA

Abstract Finite element analysis with conventional elements fails to provide convergent stresses in regions where a free edge with a bimaterial interface or a junction of dissimilar materials exists. However, these regions are characteristic of electronic devices, and they are the most critical locations for failure. A ®nite element analysis with global (special) and local (conventional) elements has been developed to provide an accurate description of the stress ®eld at these locations. The global elements capture the singular nature of the stresses arising from geometric and material discontinuities. With this method, the designer can accurately evaluate the thermo-mechanical integrity of various electronic devices. Ó 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Understanding the nature of interfacial thermomechanical stresses in electronic packages is critical to achieving reliable components subjected to thermal fatigue loading. Electronic packages usually consist of bonded materials with di€erent thermal and mechanical properties. The bonding interfaces of such structures near the free edge or near the junction of dissimilar materials su€er high stress gradients due to the presence of thermal and sti€ness mismatch of the bonded materials. Reliable predictions of the gross thermo-mechanical response of the device cannot be made accurately unless a precise description of the interface through which the mechanical, electrical, and thermal integrity of the device is achieved. Inter-

* Corresponding author. Fax: 1 520 621 8191; e-mail: [email protected].

faces of dissimilar materials are prone to crack initiations, leading to delaminations. Such defects may result in shorting between layers at di€erent potentials, causing electrical degradation of the device. Numerous analytical and numerical studies have been conducted to characterize the interfacial stresses in bonded dissimilar materials subjected to thermal loading. The majority of the analytical models employed the concept of force equilibrium in calculating thermal stresses. Models of this type [1±3] captured the e€ect of di€erent geometrical and material parameters on bond failure. However, they could not enforce the traction-free boundary conditions along the edges of the strip. Although the edge stresses obtained from these models are concentrated, they do not possess the characteristic singular behavior. Also, it was reported [4] that such models entailed signi®cant shortcomings in predicting accurate interlayer peeling and shear stresses near the free edges. As

0167-8442/98/$ ± see front matter Ó 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 8 ) 0 0 0 4 7 - 0

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E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

an extension of these models, a solution was presented [5] for a ®nite-length, bimetallic thermostat with free edges at both ends and appropriate boundary conditions. High stress was shown to be concentrated near the free edges. Presented in [6] was an analytical solution to determine the thermal stresses at the free edge of a semi-in®nite, bimetallic strip subjected to uniform heating or cooling. This analysis also failed to capture the exact nature of the stress ®eld near the free edge. Extending the work of [7±9], investigation was made [10±12] for the stresses due to a change in uniform temperature near the free edge of a junction of dissimilar materials. For an in®nite wedge consisting of two dissimilar materials, they determined the stresses to be the sum of one or two singular terms and a regular term within the theory of elasticity. Although this solution provides a precise description of the stress ®eld near the junction of two dissimilar materials, it is not capable of capturing the e€ect of complex geometric con®gurations with ®nite dimensions and the presence of a junction with more than two dissimilar materials. In order to account for the e€ect of ®nite geometry, conventional ®nite element analysis [13] was made. It was concluded that the ®nite element analysis would not guarantee a convergent peak stress, even with continued re®nement of the mesh with conventional elements near the free edge. Later, the transient energy release rate was computed [14] for a crack along the interface between two dissimilar materials by using standard elements. This analysis also su€ered from inaccurate convergence. In order to capture the exact nature of the stress ®eld and to minimize the intensive computations arising from the re®nement of the mesh, an iterative scheme in conjunction with ®nite element analysis was introduced [15±17] to determine the exact strength of the singular stress ®eld without the use of a special element. This approach is e€ective for a bimaterial interface with or without cracks. However, it su€ers from the iterations required for convergence and the inability to enforce the continuity of traction components across the interface. In modeling a free edge with a bimaterial interface, an element was developed [18] with ap-

propriate interpolation functions built in to account for the singularity at the free edge of a bimaterial interface crack subjected to mechanical loading only. Based on this concept, a hybrid element with an appropriate stress ®eld was introduced [19] to investigate the transient thermal stresses in multilayered devices with ®nite dimensions. However, this hybrid element is limited to speci®c geometry where the free edge is perpendicular to the bimaterial interface. In the presence of a re-entry corner, this element fails. Also, it is not applicable for capturing the exact nature of the stress ®eld at the junction of three di€erent materials. Based on the foregoing discussion, an analysis capability is lacking for determining the thermomechanical stresses in electronic devices with complex geometric con®gurations and a junction involving three dissimilar materials as shown in Fig. 1. In order to eliminate this shortcoming, a global ®nite element with appropriate interpolation functions that can capture the appropriate singular behavior arising from material and geometric discontinuities under both mechanical and uniform thermal loading has been developed. Implementation of the global ®nite element(s) into a ®nite element program with conventional ®nite elements permits the investigation of the thermomechanical stresses in complex electronic devices.

Fig. 1. Geometry for the junction of three dissimilar materials.

E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

2. Solution method

where

The present analysis is based on the concept of a global-local ®nite element method. This concept, employing both conventional (local) and special (global) elements, was introduced in [20] and extended [21±24]. Development of the global element sti€ness matrix is similar to that of a local element, except for the interpolation functions. In this study, these functions are established by solving for the stress and displacement ®elds in a region consisting of three wedge-shaped sectors of material, as shown in Fig. 1. Each material is assumed to be elastic, homogeneous, and isotropic, with Young's modulus, Ei , Poisson's ratio, mi , and a thermal expansion coecient, ai . The interfaces among the adjacent materials speci®ed by angles hi are assumed to be perfectly bonded. As suggested in [25] and later extended in [12], for combined mechanical and thermal loading, the stress ®eld in each sector of the region is represented by

r2 ˆ

k k …h† ‡ fab …h†; rkab ˆ rk Fab

a; b ˆ r; h; k ˆ 1; 2; 3: …1†

The origin of the polar coordinate system (r, h) coincides with the junction of the vertices. The parameter k, dependent on the material properties and the geometric con®gurations, indicates the strength of the stress ®eld near the junction. The K K …h† and fab parameter k and the functions Fab constitute the form of the interpolation functions. The presence of regular stress terms arising from either mechanical or thermal loading is re¯ected K …h†. The signi®cance of the regular through fab stress term in relation to a cracked plate under biaxial loading was discussed in [26]. Determination of these functions and the strength of the stress ®eld begins with satisfying the two-dimensional equilibrium and compatibility equations of elasticity: 1 1 rrr;r ‡ rrh;h ‡ …rrr ÿ rhh † ˆ 0; r r 1 2 …2† rrh;r ‡ rhh;h ‡ rrh ˆ 0; r r r2 …rrr ‡ rhh † ˆ 0;

105

o2 1 o 1 o2 ‡ ‡ : or2 r or r2 oh2

Substituting for the stress components from Eq. (1) in these equations results in a system of ordinary di€erential equations for each sector of the region: h i k …h† ÿ Fhhk …h† rk …1 ‡ k†Frrk …h† ‡ Frh;h h i k ‡ frh;h …h† ‡ frrk …h† ÿ fhhk …h† ˆ 0; h i k rk …2 ‡ k†Frhk …h† ‡ Fhh;h …h† …3† h i k k ‡ fhh;h …h† ‡ 2frh …h† ˆ 0;   
ÿ 2 k k r k Frr …h† ‡ Fhhk …h† ‡ Frrk …h† ‡ Fhhk …h† hh   ‡ frrk …h† ‡ fhhk …h† hh ˆ 0: The general solution form for the unknown functions are obtained as: 1 Frrk …h† ˆ 2 ÿ k‰Ak cos kh ‡ Bk sin khŠ 4 ÿ ‰Ck cos…2 ‡ k†h ‡ Dk sin…2 ‡ k†hŠ; 1 Fhhk …h† ˆ 2 ‡ k‰Ak cos kh ‡ Bk sin khŠ 4 ‡ ‰Ck cos…2 ‡ k†h ‡ Dk sin …2 ‡ k†hŠ; 1 Frhk …h† ˆ ‰Ak sin kh ÿ Bk cos khŠ 4 ‡ ‰Ck sin…2 ‡ k†h ÿ Dk cos…2 ‡ k†hŠ; 1 1 frrk …h† ˆ ak cos 2h ‡ bk sin 2h ‡ ck h ‡ dk ; 2 2 1 1 k fhh …h† ˆ ÿak cos 2h ÿ bk sin 2h ‡ ck h ‡ dk ; 2 2 1 k …4† frh …h† ˆ ÿak sin 2h ‡ bk cos 2h ÿ ck ; 4 in which Ak , Bk , Ck , Dk , ak , bk , ck and dk are the unknown constants. After substituting for the K K …h† and Fab …h†, in Eq. (1), unknown functions, Fab in conjunction with the stress±strain relations under plane stress conditions given by 1 …rrr ÿ mrhh † ‡ aT ; E 1 ehh ˆ …rhh ÿ mrrr † ‡ aT ; E 2…1 ‡ m† crh ˆ rrh ; E err ˆ

…5†

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E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

with T being the change in temperature, and integrating the strain± displacement relations err ˆ ur;r ; 1 ehh ˆ …ur ‡ uh;h †; …6† r   1 uh ; crh ˆ ur;h ‡ r r r r lead to the displacement components in each region as  rk‡1 …2 ÿ k† ÿ …2 ‡ k†mk k ur …r; h† ˆ Ek …1 ‡ k† 4 …Ak cos kh ‡ Bk sin kh† ÿ…1 ‡ mk †‰Ck cos…2 ‡ k†h  ‡Dk sin…2 ‡ k†hŠ  r …1 ‡ mk †…ak cos 2h ‡ bk sin 2h† ‡ Ek  1 ÿ mk …ck h ‡ dk † ‡ 2 Zr ‡ ak Tk …s; h† ds ‡ Wk …h†;

ukh …r; h†

rk‡1 1 ˆ Ek 1 ‡ k



‡

9 =

Wk …g† dg ‡ W0k h : ;

…8†

In Eqs. (7) and (8), the unknown functions Wk …h† and Uk …h† arise as a result of inde®nite integrations. Under uniform temperatures, Tk …r; h† ˆ T0 ; and the condition that limrÿ0 ukr …r; h† ˆ 0, the expressions for Wk …h† and UK …r† become ck Wk …h† ˆ 0 and Uk …r† ˆ ek r ÿ r ln r …9† Ek

uak‡1 …r; hk †; uakÿ2 …r; hk †;

ˆ ˆ a; b ˆ r; h:

r …1 ‡ mk †…ak sin 2h ÿ bk cos 2h† Ek Zh Zr Zh Tk …s; g† ds dg ‡ rak Tk …r; g† dg ÿ ak Zh

1

‰ak Tk;r …r; g† dg ÿ rTk …r; g† dg Zh

uka …r; hk † uka …r; hk †

ÿ

with the additional equation

‡

k ˆ 1; 2; rkab …r; hk † ˆ rk‡1 ab …r; hk †; k kÿ2 rab …r; hk † ˆ rab …r; hk †; k ˆ 3; a; b ˆ r; h with a ˆ b 6ˆ r;

‰k…1 ‡ mk † ‡ 4Š 4

Wk …g† dg ‡ Uk …r†

Zh

in which ek are unknown constants. The unknown constants in the expressions for the stress and displacement ®elds are determined by requiring the continuity of traction and displacement components along the interface to be given by

…Ak sin kh ÿ Bk cos kh† ‡…1 ‡ mk †‰Ck sin…2 ‡ k†h  ÿDk cos…2 ‡ k†hŠ

ÿ

8 r < Z 1 ÿe 1 k U0k …r† ÿ Uk …r† ˆ ÿ ak Tk;h …s; h† ds Ek r r:

…7†

1

Prime denotes di€erentiation with respect to its argument.

…10†

k ˆ 1; 2; k ˆ 3;

Imposing these conditions results in two decoupled systems of algebraic equations. The ®rst system is homogeneous and composed of 12 equations involving 12 of the unknown constants Ak , Bk , Ck , and Dk (k ˆ 1, 2, 3). In the presence of thermal loading, the second system is non-homogeneous and involves the remaining unknown constants, ak , bk , ck , dk , and ek (k ˆ 1, 2, 3). Under mechanical loading only, the second system becomes homogeneous, requiring its determinant to vanish for non-trivial solutions. In matrix form, these two systems of equations are expressed as: 2 38 9 P11 P12 0 > < p1 > = 6 7 4 0 P22 P23 5 p2 ˆ 0; > : > ; P31 0 P33 p3

E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

2

Q11

6 4 0 Q31

Q12 Q22 0

8 9 38 9 > > < q1 > = < R1 > = 7 ˆ Q23 5 q2 R2 : > > : > ; : > ; Q33 q3 R3

107

0

…11†

The explicit forms of the submatrices Pij and Qij and the vectors Ri are given in Appendix A. The vectors pi and qi contain the unknown constants in the form pi ˆ fAi ; Bi ; Ci ; Di g

T

and

T

qi ˆ fai ; bi ; ci ; di ; ei g : …12†

A non-trivial solution to the homogeneous system exists for values of k that cause the determinant of the coecient matrix P to vanish. Roots of the characteristic equation |P| ˆ 0 provide the eigenvalues, ki . Depending on the material properties, the eigenvalues may be complex. Substituting each eigenvalue back into the equation permits the expression of the unknown constants in terms of a single constant, A1 . This undetermined coecient corresponding to the ith eigenvalue is denoted by xi . Solving for the remaining unknown constants in the second system permits the expression of the stress and displacement components as 2 rab ˆ

N X

xi Fab …r; h; ki † ‡ xN ‡1 fab …r; h†;

iˆ1

a; b ˆ r; h; Ni X ua ˆ xi Ga …r; h; ki † ‡ xN ‡1 ga …r; h†;

…13†

iˆ1

a; b ˆ r; h with xN ‡1 ˆ 1 in the presence of thermal and mechanical loading. If there is no temperature change, xN ‡1 is either 0 or is determined as part of the solution, depending on whether |Q| ¹ 0 or |Q| ˆ 0, respectively. The number of eigenvalues whose magnitude falls in the interval of )1 < Re (k) < 1 establishes the value for N. The generalized coecients, xi , can be determined in terms of the nodal displacements by enforcing the continuity of nodal displacements at

2 Since the values for r and h specify a sector of the region, the superscript utilized to designate a speci®c sector is not retained in the remaining expressions.

Fig. 2. Interface nodes between the global element and the surrounding local elements.

the interface nodes between the global element and the surrounding local elements. As illustrated in Fig. 2, the global element with M interface nodes requires the imposition of continuity conditions given by 8 9 ur …r1 ; h1 † > > > > > > > > > uh …r1 ; h1 † > > > > > < = .. . > > > > > > > > > u …r ; hM † > r M > > > > : ; uh …rM ; hM † 2 Gr …r1 ; h1 ; k1 † 6 G …r ; h ; k 6 h 1 1 1 6 6 .. ˆ6 . 6 6 4 Gr …rM ; hM ; k1 † Gh …rM ; hM ; k1 †

Gr …r1 ; h1 ; k2 †


Gh …r1 ; h1 ; k2 †


.. .. . . Gr …rM ; hM ; k2 †


Gh …rM ; hM ; k2 †


8 9 gr …r1 ; h1 † > 8 9 > > > > > x1 > > > > > gh …r1 ; h1 † > > > > > > > > > > > < x2 = < = . .  ‡ x N ‡1 . . .. > > > > > > > > > > > > > > : > > ; > gr …rM ; hM † > > > > > xN : ; gh …rM ; hM †

3 Gr …r1 ; h1 ; kN † Gh …r1 ; h1 ; kN † 7 7 7 7 .. 7 . 7 7 Gr …rM ; hM ; kN † 5 Gh …rM ; hM ; kN †

…14†

or fug ˆ ‰GŠfxg ‡ fgg:

…15†

In general, the number of equations, 2M, exceeds the number of unknown coecients, N, resulting in an overdetermined system. Therefore, the unknown coecients are expressed in terms of nodal displacements based on the least squares minimization procedure as follows:

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E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

Fig. 3. Description and ®nite element discretization of the bimetal thermostat [19].

Minimizing the strain energy with respect to the nodal displacements associated with the global element results in the system of equations

fxg ˆ ‰ZŠffug ÿ fggg with

h iÿ1 T T ‰ZŠ ˆ ‰GŠ ‰GŠ ‰GŠ :

…16†

Determination of these coecients permits the expression of the stress and displacement components in terms of the nodal displacements, and the strain energy in the global element becomes 3 Z o 1 n T T T T Uˆ fug ÿ fgg ‰ZŠ fFab g 2 S  ‡fab …fGa g‰ZŠffug ÿ ‰gŠg ‡ ga †gb ds; a; b ˆ r; h; …17† where gb are the components of the unit normal to the surface S of the global element. The vectors fFab g and {Ga } are de®ned as fFab g ˆ fFab …r; h; k1 †Fab …r; h; k2 †


Fab …r; h; kN †; fGa g ˆ fGa …r; h; k1 † Ga …r; h; k2 †


Ga …r; h; kN †g: …18† 3

Repeated subscripts imply summation.

‰kŠfug ˆ ff g;

…19†

where the global sti€ness matrix [k] and the load vector {f} are de®ned as: Z  1 T T ‰ZŠ fFab g fGa g ‰kŠ ˆ 2 S  T ‡fGa g fFab g ‰ZŠgb dS; …20† ff g ˆ

1 2

Z

 ‰ZŠT fFab gT fGa g

S

 ‡fGa gT fFab g ‰ZŠfggT gb dS Z   1 T T T ÿ ‰zŠ fFab g ga ‡ fGa g fab gb dS: 2 S

…21† The global and local element equations having the same unknown displacements are assembled to establish the system equilibrium equations as

E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

‰KŠfdg ˆ fF g;

…22†

where [K] is the system sti€ness matrix and the vectors {d} and {F} include the total nodal displacement and force components, respectively. This process led to the development of a ®nite element program incorporating both global and local elements. The local elements consist of quadrilateral and triangular elements whose interpolation functions can be found in any textbook on the elementary ®nite element method. The shape of

109

the global element can be an n-sided polygon, depending on the details of the mesh surrounding the global element. The number of interface nodes and the size of the global element were established based on convergence requirements. 3. Numerical results The validity of this solution method has been established by solving the two bimetal thermostat

Fig. 4. Description and ®nite element discretization of the bimetal thermostat [10].

110

E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

con®gurations considered previously [10,19]. These two con®gurations and their ®nite element representation are shown in Figs. 3 and 4. The comparison of the results from this analysis to those published in [10] and [19] are presented Figs. 5 and 6, respectively. As observed in the ®gures, the behaviors of the results for these two simple geometric and material con®gurations are similar to those obtained from a special case of the present global-local analysis. The variations in results are possibly due to the coarse ®nite element mesh used in the previous studies. The eigenvalues retained in the analysis of these problems are computed to be the same as those given in [10,19]. Their numerical values are provided in Figs. 5 and 6. A typical die/die attach/substrate con®guration subjected to a uniform temperature change of 100°C and with the geometric dimensions and material properties shown in Figs. 7 and 8 is also considered in order demonstrate the simplicity of this analysis method. In this con®guration, three global elements, along with 568 conventional elements, are used in the construction of the ®nite element mesh. The eigenvalues retained in the construction of the interpolation functions for

Fig. 6. Variations of the peeling stress along the interface near the free edge in a bimetal thermostat [10].

each global element are given in Figs. 9±14. These ®gures illustrate the behavior of the thermal stresses in each of the global elements. The behavior of the peeling and shear stresses is singular near the free edge with the bimaterial interface and the junction of two dissimilar materials. Although the results presented in Figs. 9±14 capture the e€ect of the variation in modulus for the die-attach material and the gradient of the stress ®eld near the junction, they do not provide the information required for failure prediction. Based on the concept of the stress intensity factors, the generalized stress intensity factors, KI and KII [27,28] were introduced for adhesive failure prediction of a bi-material interface as: kI ˆ lim rÿkm rhh …r; h ˆ h †; rÿ0

kII ˆ lim rÿkm rrh …r; h ˆ h †;

…23†

rÿ0

Fig. 5. Variations of the stress components along the interface near the free edge in a bimetal thermostat [19].

in which h indicates the angular position of the interface. The parameters, KI and KII , and the order of the singularity, km , can be determined through least-squares approximation using the stress ®eld given by the ®rst of Eq. (13) within the global element. Based on this criterion, the bimaterial joint always fails along the interface when

Fig. 7. Description and ®nite element representations of a die%/die attach%/substrate con®guration common to electronic devices.

E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117 111

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E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

Fig. 8. Location of the global elements as part of the local ®nite element representation.

the generalized stress intensity factors reach their experimentally measured critical values for speci®ed material combinations. However, this type of criterion lacks acceptable physical interpretation because the critical values for the generalized stress intensity factors, KIC and KIIC , are dependent on material properties and the geometry. In other words, the generalized stress intensity factors do not have a simple interpretation, as in the homogeneous case. A more meaningful parameter for this case may be based on the energy rather than the stress ®eld. Therefore, the concept of a strain energy density criterion [29] can be applied to predict failure initiation in a region composed of dissimilar materials. According to this criterion, the crack in a homogeneous material propagates in the direction for which the strain energy density function, dW/dV (r,h), is minimum and that crack

growth occurs when it reaches its critical value, (dW/dV)cr . This critical value can be related to the critical energy release rate of the homogeneous material. Although this criterion has been applied to various problems concerning homogeneous materials exhibiting linear and non-linear behavior, it has not been applied to predict failure in a region consisting of dissimilar materials with discrete interfaces. The angular variation of the strain energy density around the junction point in each of the global elements is shown in Figs. 15±17 for a speci®ed distance of r ˆ 0.004 mm from the junction. The strain energy density is not a continuous function. It exhibits a jump across the interface because of the discontinuous normal stresses parallel to the interface. In this case, the strain energy density function representing the interface can be

E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

113

Fig. 9. Variations of the tangential stresses along the interface near the free edge in global element 1.

Fig. 11. Variations of the tangential stresses along the vertical interface near the junction in global element 2.

de®ned as the average value belonging to both sides of the interface,     dW 1 dW dW …r; h‡ † ‡ …r; hÿ † : ˆ …24† dV cr 2 dV dV

The superscripts denote the upper and lower part of the interface, respectively. The failure initiation at the interface may occur when this averaged value reaches its critical value, (dW/dV)int , deter-

Fig. 10. Variations of the shear stresses along the horizontal interface near the junction in global element 1.

Fig. 12. Variations of the shear stresses along the interface near the free edge in global element 2.

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E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

Fig. 15. Variations of strain energy density around the junction in global element 1.

Fig. 13. Variations of the tangential stresses along the horizontal interface near the free edge in global element 3.

Fig. 16. Variations of strain energy density around the junction in global element 2.

Fig. 14. Variations of the shear stresses along the horizontal interface near the free edge in global element 3.

Fig. 17. Variations of strain energy density around the junction in global element 3.

mined experimentally for a particular material combination. Based on this criterion, the results presented in Figs. 15±17 indicate that failure may initiate in the substrate in global element 1 near a location de-

®ned by h ˆ )70° and in the chip in global element 2 at h ˆ )20° where local minima occur. In global element 3, failure initiation may be in both the chip de®ned by h ˆ 85° and in the die-attach de®ned by h ˆ 280°.

E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

Based on the average strain energy density values for the interfaces between the chip and the die-attach in global elements 2 and 3 as de®ned by Eq. (24), failure is more likely to initiate along the interface in global element 2 because global elements 2 and 3 involve the same material combinations and the critical value of the strain energy density for the interface remains the same regardless of the geometry. This prediction is consistent with the experimental observation presented in [30]. It is worth noting that the variation of the stresses, shown in Figs. 9±14, or the highest order of the stress singularity associated with these junctions points would not predict this experimental observation.

The global-local ®nite element analysis provides an accurate description of the stress ®eld at critical locations in a complex electronic device. This capability, in conjunction with a strain energy density failure criterion, may serve as a design tool to identify the e€ects of geometric and material parameters on potential failure sites and failure loading (thermal and mechanical). Appendix A The submatrices Pij are de®ned as q

P1j ˆ P1j ‰h1 ; mjÿ1 ; vj ; …ÿ1† 1 ; kŠ for j ˆ 1; 2; q1 ˆ 1 ‡ j; q P2j ˆ P2j ‰h2 ; mjÿ2 ; vj ; …ÿ1† 2 ; kŠ for j ˆ 2; 3; q2 ˆ 2 ‡ j; q P3j ˆ P3j ‰h3 ; m3=2…jÿ1† ; vj ; …ÿ1† 3 ; kŠ for j ˆ 1; 3; q3 ˆ …3 ‡ j†2 ; m0 ˆ 1; E1 …1 ‡ v2 † ; m1 ˆ E2 …1 ‡ v1 † E2 …1 ‡ v3 † ; m2 ˆ E3 …1 ‡ v2 † E1 …1 ‡ v3 † m3 ˆ ; E3 …1 ‡ v1 †

The explicit form of Pij is given by Pij ˆ …ÿ1†qi 2 ni ti 6 6 n t 6 i‡1 i‡1 6 6 6 ms ni‡2 ti 4 ms ni‡3 ti‡1

ni ti‡1

ti‡2

ÿni‡1 ti

ti‡3

ms ni‡2 ti‡1

ÿms ti‡2

ÿms ni‡3 ti

ms ti‡3

ti‡3

3

7 ÿti‡2 7 7 7; 7 ÿms ti‡3 7 5 ÿms ti‡2

where s ˆ j ) 1, j ) 2, and 3/2(j ) 1) for i ˆ 1, 2, and 3, respectively. The expressions for ti and ni are de®ned as: ti ˆ cos…k hi †; ti‡1 ˆ sin…k hi †;

4. Conclusions

in which

115

ti‡2 ˆ cos‰…2 ‡ k†hi Š; ti‡3 ˆ sin‰…2 ‡ k†hi Š; 1 ni ˆ 2 ‡ k; 4 1 ni‡1 ˆ k; 4 1 ni‡2 ˆ 2 ÿ k ‡ 4mi ; 4 1 ni‡3 ˆ 4 ‡ k ÿ 4mi : 4 The submatrices Qij are de®ned as Qij ˆ Qij ‰hi ; Ei ; mi ; Ej ; mj Š for i ˆ j ˆ 1; 2; i ˆ 3; j ˆ 1 with its explicit form Qij

8 ÿ cos…2hi † ÿ sin…2hi † > > > > > > > ÿ sin…2hi † cos…2hi † > > > < ˆ sin…2hi † cos…2hi † > > > > > > > > ÿ sin…2hi † cos…2hi † > > : 0 0

hi 2

1 2

ÿ 14

0

1ÿmi 1‡mi

Ei h2i

1ÿmi Ei 1‡mi 2

0

0

Ej

0

9 0 > > > > > > 0 > > > > = 0 > > > Ei > > > 1‡mi > > > > ; 0:

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E. Madenci et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 103±117

For i ˆ 1 and j ˆ 2, and i ˆ 2, 3 and j ˆ 3, the function form of Qij becomes Qij ˆ Qij ‰hj ; Ei ; mi ; Ej ; mj Š with its implicit form Qij

ˆ

8 > > > > > > > <

cos…2hi † sin…2hi †

ÿci cos…2hi † > > > > ci sin…2hi † > > > : 0

ÿci sin…2hi †

ÿhj 2 1 4 1ÿmi ÿ 1‡mi Ei h2i

i Ei ÿ 1ÿm 1‡mi 2

ÿ cos…2hi † 0

0 ÿEi

0 0

sin…2hi † ÿ cos…2hi †

ÿ 12 0

0 0

9 > > > > > > > =

0 > > E > > ÿ 1‡mj i > > > ; 0:

in which ci ˆ

Ei …1 ‡ mj † : Ej …1 ‡ m1 †

The explicit forms of Ri are given by 9 8 0 > > > > > > > > > > 0 > > = < Ei Ri ˆ …ai‡1 ÿ ai †DT 1‡mi : > > > > > > 0 > > > > > > ; : 0

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