# Stress Relief in Soldered Assemblies by Using Inhomogeneous Bonds

*“The only real voyage of discovery consists not in seeing new landscapes, but in having new eyes."*

—Marcel Proust, French author and critic

## Background/Incentive

Solder materials, providing mechanical support to surface-mounted devices and subjected to elevated thermally induced interfacial shearing stresses, are prone to inelastic deformations. This considerably shortens their fatigue lifetime. Therefore, there exists a crucial need for stress reduction in solder joint interconnections. One effective way for doing that is the use of inhomogeneous bonding systems. Various assemblies with inhomogeneous bonding layers were addressed in application to assemblies bonded at the ends [1-10] to predict the size of an inelastic zone in ball- grid array (BGA) assemblies [11, 12], to explain a paradoxical situation when stiffer midportions of compliant bonds could result in an appreciably lower stresses at the assembly ends [13], to assess the possible stress relief when solder joints with elevated stand-off heights are employed [14-17], to evaluate the peculiarities associated with the use of identical adherends [18-20], the possibility of using the model developed for a homogeneously bonded assembly (of the type suggested in [21]) for assemblies of the BGA or column-grid array (CGA) type, in which the solder supports are “spaced” at certain distances from each other [22], and in a number of other electronic and photonic reliability-related problems (see, for example, [23, 24]).

Identical bonded components and “piecewise-continuous” bonding layers were considered in application to a new generation of low-cost memory storages, in which the bonding layer played the role of the memory storing medium [18-20]. The emphasis was on the evaluation of the conditions that could lead to plane boundaries between the “pieces” of the bonding layer, rather than to low level stresses. It has been recently shown [14-17] that significant stress relief can be obtained owing to the application of BGAs with the increased stand-off of the solder joint interconnections, not to mention the use of CGAs.

The objective of this chapter is to show that significant thermal stress relief can be achieved in an optimized design of an inhomogeneously bonded bimaterial assembly, if its bonding system is designed and fabricated in such a way that the interfacial shearing stress at the ends of the high-modulus and high-bonding temperature midportion of the assembly at its boundary with the low-modulus and low-bonding temperature peripheral portion is made equal to the stress at the assembly ends. In such an assembly, the interfacial shearing stress increases first from zero at the assembly mid-cross section to its maximum value at the end of the high-modulus and/or high-fabrication temperature midportion at its boundary with the low- modulus and/or low-fabrication temperature peripheral portion(s) of the assembly, drops to a low value at this boundary and then increases again to the same maximum value at the assembly end(s). Each of these maximum values is well below the maximum interfacial stress in a regular, homogeneously bonded, assembly.

## Assembly’s Midportion

Consider first the midportion of an in-homogeneously bonded assembly (having different bonding materials at the midportion and the peripheral portions of the assembly), experiencing the given change *At,,,* in temperature and subjected, because of its mismatch with the peripheral portions, to external forces *T* applied to the midportion in a symmetric fashion (Figure 6.1). The interfacial longitudinal displacements of the assembly components can be sought, in an approximate analysis, using the concept of the interfacial compliances (see Chapter 2), as follows:

Here, a, and a_{2} are the coefficients of thermal expansion (CTE) of the component materials, *At,,,* is the change in temperature (say, between the bonding temperature and the room or testing temperature),

FIGURE 6.1 The midportion of an inhomogeneously bonded bimaterial assembly.

are the axial compliances of the assembly components, /г, and *h _{2}* are the component thicknesses, £, and

*E*are Young’s moduli of the materials, v, and v

_{2}_{2}are Poisson’s ratios of materials,

are the distributed forces acting in the *x* cross section of the assembly, x(x) is the interfacial shearing stress, *T* is the thus far unknown force applied to the midportion from the peripheral portions of the assembly, / is half the assembly length, tC[ and k_{2} are the interfacial compliances of the assembly components, and w,(x) and *w _{2}(x) *are the components’ deflections, so that their derivatives are angles of rotation. The origin of the coordinate

*x*is in the mid-cross section of the assembly.

The first terms in (6.1) are stress-free thermal contractions. The second terms determine the displacements caused by the induced thermal forces. These displacements are evaluated in accordance with Hooke’s law and do not consider that the interfacial displacements are somewhat larger than the displacements of the inner points of the cross section. The third terms account for that. They are, in effect, corrections to the displacements evaluated in accordance with the second terms. It is assumed that these corrections can be evaluated as the product of the interfacial compliances [15]

of the bonded component of interest and the interfacial shearing stress acting in this cross section.

Here,

are the shear moduli of the bonded component materials. The fourth terms in (6.1) are due to bending. Since the case of cooling is considered here, and the CTE of component #1 is assumed to be lower than that of component #2, the interfacial surface of component #1 is configured in the concave fashion and the surface of component #2 is configured in the convex fashion. This circumstance is reflected by the signs in front of the corresponding displacement-related terms.

The condition of the compatibility of the interfacial displacements (6.1) can be written, in the presence of a compliant bond, if any, as

where

is the longitudinal interfacial compliance of the bonding layer, and

is the shear modulus of the bonding material. The second term in the right part of condition (6.6) is due to the interfacial compliance of the bond.

Introducing formulas (6.1) into condition (6.6), the following equation for the interfacial shearing stress function x(x) can be obtained:

Here, к = K_{0} + K( + K_{2} is the total interfacial compliance of the assembly, and Да = а_{2} - a! is the difference in the CTEs of the component materials. By differentiating equation (6.9) with respect to the coordinate *x* we have

The curvatures *w"(x)* and *w"(x)* of the assembly components can be determined from the equilibrium equations

where

are the flexural rigidities of the assembly components treated here as elongated rectangular plates, and *p(x)* is the peeling stress. The left parts of equations (6.11) are elastic bending moments, the first terms in the right parts are the bending moments caused by the induced thermal forces *T(x),* and the second terms are the bending moments due to the induced peeling stress *p(x).*

Solving equations (11) for the component curvatures, we have

Introducing these expressions into equation (6.10), we obtain

Here,

is the axial compliance of the assembly with consideration of the finite flexural rigidities of the assembly components, and

is the factor that considers the role of the components’ geometrical dissimilarity. This role can be neglected in an approximate analysis, so that equation (6.14) can be replaced with the following simplified equation:

in which the shearing stress only is considered.

The force *T(x)* should be symmetric with respect to the mid-cross section of the assembly and could be sought as

Introducing this solution into equation (17) and considering that, in accordance with formula (6.3),

we conclude that equation (6.17) is fulfilled, if the relationships

take place. The constant *C _{2}* can be found from the boundary condition

as

Then, solution (6.18) results in the following expression for the induced force *T(x*):

The first term in this expression is due to the local thermal mismatch of the assembly components, and the second term is caused by the external force *T* applied from the peripheral portions of the assembly. The interfacial shearing stress can be determined from (6.23) by differentiation

The shearing stress is zero at the mid-cross section and increases to at the end of the midportion.

## Assembly’s Peripheral Portion(s) and Forces at the Boundaries

Consider now the peripheral portion of the assembly (Figure 6.2). Unlike the midportion, the peripheral portion is subjected to an external loading applied to only one side of the assembly, but, even more importantly, its modulus and the application (soldering, curing) temperature could be much different from those in the midportion. The boundary conditions for the induced forces are also different:

To satisfy these two conditions, the force *T(x)* should be sought in the form

This solution contains two constants of integration and is not symmetric with respect to its mid-cross section. Introducing the sought solution (6.27) into equation (6.17), we conclude that formulas (20) are still valid, provided, of course, that the appropriate

FICURE 6.2 The peripheral portion of an inhomogeneously bonded bimaterial assembly subjected to thermal loading.

change in temperature is considered *(At,,* instead of *At _{m}).* The constants of integration obtained from the two conditions in (6.26) are

Introducing the second formula in (6.20) and formulas (6.28) into sought solution (6.27), we obtain the following formula for the induced force:

The first (“thermal”) term in this expression is not different of the first term in (6.23), but the second term is quite different, because of the differently applied thermomechanical loading.

The interfacial shearing stress can be found from (6.29) by differentiation It changes from

at the left end of the assembly to

at its right end. Formula (6.32) particularly indicates that if the peripheral portion is characterized by a large enough product of the parameter of the interfacial shearing stress and half the assembly length, the force at the inner end of the assembly where the external force is applied will not affect the shearing stress at the outer (free) end of the assembly, and this stress can be found as

Formulas (6.25) and (6.31) yield

where the notation in the first formula has been changed to account, because for the possibly different parameter of the interfacial shearing stress *(K* instead of *k)* and to indicate that half the length *L- 21* of the midportion of the assembly can be found as the difference between half the assembly length *L* and the length *21* of one of its peripheral portions. The force *T* at the boundary between the midportion and the peripheral portions of the assembly with an inhomogeneous bonding layer can be determined from the condition of the compatibility of the longitudinal interfacial displacements of the two portions of the assembly at their boundary. Since the interfacial displacements can be found as products of the interfacial compliances and the interfacial shearing stresses, the condition of interest can be written for a long enough midportion (for such a midportion, the hyperbolic tangent in the first formula in (6.34) can be put equal to one), as

This condition yields

where the notations

and the formulas

are used.