# Method of Interfacial Compliance

*“There is nothing more practical than a good theory."*

—Kurt Zadek Lewin, German American psychologist

## Introduction

Thermal loads arise during manufacturing, testing, and operation of electronic, optoelectronic, and photonic devices, assemblies, packages, and systems. Stresses and deformations caused by thermal loads are the major contributors to their finite service life. High thermal stresses and strains can lead not only to physical (mechanical) damage, but also to a functional failure of the electronic or photonic product. If the heat produced by the chip cannot readily escape then the highly localized thermal stresses in the integrated circuit (IC) can lead to the PN junction failure. Thermally induced warpage is a serious problem in today’s manufacturing of electronic packages. Low-temperature microbending in dual-coated optical fibers, although it might not be high enough to lead to appreciable bending stresses, could result nonetheless in significant added transmission losses. Thermal stresses are responsible for “curling” of optical fibers during drawing. Substantial loss in optical coupling efficiency occurs when the lateral displacement in the gap between two light-guides becomes too large because of thermal stress-related phenomena (elevated deformations, stress relaxation in laser welds, and so on). The ability to predict (model) and, if necessary, minimize the adverse consequences of thermal stresses and deformations is of obvious practical importance in electronics and photonics reliability engineering.

Pioneering work on thermal stress modeling belongs to Timoshenko [1]. He applied a structural analysis (strength of materials) approach to evaluate the stresses in, and the bow of, bimetal thermostat strips. Timoshenko did not address interfacial shearing and peeling stresses in his paper, but indicated that these stresses cannot be determined using engineering methods of structural analysis, but could be evaluated, if necessary, using methods of the elasticity theory. This was done by Aleck [2] who employed theory-of-elasticity to model thermal stresses in a rectangular plate clamped along the edge. Both approaches were extended later on by numerous investigators, including the field of electronics and photonics engineering (see, for example, [3-22]). Suhir [7-10] managed to come up with a structural analysis (strength of materials) engineering solution for interfacial shearing and peeling thermally induced stresses in thermostat-like assemblies of finite size, such as for assemblies typical in electronics and optical engineering. This has been done by introducing a concept of interfacial compliance. This concept enables to separate the roles of loading, not even necessarily thermal, and structural characteristics of the assembly, and, owing to that, obtain simple and physically meaningful closed-form solutions for the stresses and strains, including interfacial ones. No singularities appear in the obtained expressions. This is because the concept of the interfacial compliance, unlike the elasticity theory, whether analytical or finite element analysis (FEA)-based, is an approximate structural analysis (strength of materials) method and does not require that all the theory-of-elasticity equations and conditions of this theory are fulfilled in every point of the body under stress. Some of the rather numerous problems successfully solved using the method of interfacial compliance include thermal stresses in thin films [23, 24], in trimaterial assemblies [25-27], in coated optical fibers [28-31], and in assemblies with seal glass bonds, whose coefficient of thermal expansions (CTEs) have to be treated as random variables [32]; problems, in which the probabilistic design for reliability (PDfR) concept is employed [33]; assemblies with power cores sandwiched between dissimilar insulated metal substrates [34]; and delamination (interfacial fracture) problems [35]. Mishkevich and Suhir [36] established good agreement between the analytical modeling data and FEA predictions in various thermally induced stress problems, and have shown that although the interfacial shearing and peeling stresses are coupled, a simplified approach can be used, when analytical model is employed: the shearing stress can be evaluated in an approximate analysis without considering its coupling w'ith peeling, and that the peeling stresses can be evaluated w'ith sufficient accuracy (confirmed by FEA modeling) from the computed shearing stress. It has been shown, particularly, that the longitudinal gradient of the shearing stress plays the role of the excitation force when the peeling stress is evaluated.

An overview of the substance, attributes, and applications of the method of the interfacial compliance was recently published in a special issue dedicated to the 90th birthday of Prof. Richard Hetnarski, editor of the Taylor and Francis Group *Journal of Thermal Stresses* [37]. A modification of the method of interfacial compliance is set forth below. In addition to the general theory of thermal stresses in bimaterial and trimaterial assemblies, we also show', having in mind soldered assemblies, how' the concept of interfacial compliance could be used in the interfacial fracture (delamination) problem [35].

## Stresses in the Midportion of a Multimaterial Body Subjected to a Change in Temperature

Let a multimaterial assembly (body) consisting of *n* components be fabricated at an elevated temperature and subsequently cooled dowrn to a low' (room or testing) temperature. The thermally induced strain, e, at the state of equilibrium should be the same for all the components, and could be evaluated for the /th component as follows:

Here, *At* is the change in temperature, a_{(} is the CTE of the /th component’s material, *T,* is the thermally induced force acting in the component’s cross sections,

is the longitudinal (axial) compliance of the component. *E°* = -—^{1}— is the effective

Young’s modulus of the material, £, is its Young’s modulus, and v, is Poisson’s ratio. The condition

of equilibrium results in the following formula for the induced strain: where

is an effective CTE of the assembly. The first term in formula (2.1) is unrestricted thermal contraction and the second term is the elastic extension evaluated in accordance with Hooke’s law.

Formulas (2.1) and (2.4) lead to the following expression for the induced forces *T _{t }*acting in the materials’ cross sections:

The corresponding stresses are

and are component thickness independent. This stress depends on the generalized Young’s modulus of the material (i.e., Young’s modulus with consideration of its Poisson’s ratio), the thermal mismatch of the given material with the effective CTE of the assembly, and the change in temperature.

In a situation when one of the components of the assembly is significantly stiffer (thicker, and/or has a much higher Young’s modulus) than the other components (such as, say, substrate in a thin film structure), then, as follows from formula (2.5), the effective CTE, a_{(},, of the assembly is simply the CTE of the stiff component’s material. In this case, it is only the mismatch of the given material with this material that determines the induced force and stress in the material, and not its mismatch with the adjacent materials.

## Bimaterial Assembly: Interfacial Shearing Stresses

In the case of a bimaterial assembly, formulas (2.5) and (2.6) yield:

where

is the total longitudinal (axial) compliance of the assembly.

To determine how the thermally induced forces are distributed along an assembly of finite length, we seek these forces in the form

The function %(jc) has to be determined. To do that, we use the following approximate formulas for the interfacial longitudinal displacements:

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

The first terms in formulas (2.11) are stress-free thermal displacements. The second terms are determined using Hooke’s law, are due to the thermally induced forces, and reflect an assumption that the displacements of all the points in the given cross section are the same. The third terms are, in effect, corrections to this assumption and account for the fact that the interfacial displacements are somewhat larger than the displacements of the inner points of the given cross section. The structure of these terms reflects an assumption that sought corrections can be computed as a product of the interfacial compliance, к, or K_{2}, of the corresponding component and the interfacial shearing stress, *x(x),* in the given cross section. Formulas for the evaluation of the interfacial compliances have been suggested in [7-9]. These formulas are based on the theory of elasticity solution of the Ribiere problem for a long-and- narrow strip and depend on how the strip is loaded [7-9, 18]. If the strip is loaded on its long edges by equal distributed forces applied in an antisymmetric fashion (in- or

*h*

outward), its interfacial compliance is к = —, where *h* is the height (thickness) of the

*G*

strip, and *G* is the shear modulus of its material. This is typically the adhesive/solder layer. If the strip is loaded only on one of its long edges in an antisymmetric fashion, while the opposite long edge is loading free, its interfacial compliance is only *h*

к = —. These are typically the adherends in thermally mismatched assemblies.

3 *G*

The interfacial shearing stress *x(x)* is related to the forces *T _{l 2}(x)* acting in the cross sections of the bonded components as follows:

where *a* is half the assembly length. The condition of the compatibility of the displacements (2.11) can be written as

where k_{0} is the interfacial compliance of the bonding layer, if any. If no separate material is used as the bonding layer (e.g., when thermo-compression bonding is employed), then the interfacial compliance of the assembly is due only to the bonded components themselves, and the second term in the right part of formula (2.13) should be omitted.

Introducing formula (2.11) for the interfacial displacements into the condition (2.13) and considering formulas (2.10) and (2.12) for the induced forces, we obtain the following equation for the sought function %(л) that accounts for the nonuniform distribution of the forces acting in the cross sections of the assembly components:

Here,

is the parameter of the interfacial shearing stress, and

is the total interfacial compliance of the assembly. In typical adhesively bonded or soldered assemblies, in which the thickness and Young’s modulus of the bonding layer are significantly smaller than those of the bonded components, the axial compliance *X* of the assembly is due primarily to the bonded components. Its interfacial compliance, however, is due to both the adherend and the bonding materials. This is because these compliances are defined as ratios of the component thicknesses to their shear moduli, and these ratios might be very well comparable for the adherends and the adhesive.

The integral equation (2.14) can be reduced, by differentiation, to the following differential equation:

Its solution

indicates that the thermally induced forces *T, _{2}(x)* are symmetric with respect to the mid-cross section of the assembly.

The zero boundary conditions Г|._{2}(±а) = 0 for the sought forces lead to the conditions %(+«) = 0 for the function х(л'). From (2.10), (2.12), and (2.18) we obtain the following simple formula for the thermally induced interfacial shearing stress:

The maximum interfacial shearing stresses

occur at the end cross sections *x = ±a.* and, as evident from (2.20), change from
in the case of a short assembly to

in the case of a sufficiently long assembly, when tanhka can be put equal to one. As evident from equations (2.21) and (2.22), the maximum shearing stress t_{max} at the ends of a bimaterial assembly subjected to the change in temperature is proportional to the assembly length, in the case of a short assembly, and becomes assembly length independent for assemblies characterized by high *ka* values. Equations (2.21) and (2.22) also show that this stress is inversely proportional to the interfacial compliance of the assembly and is independent of its axial compliance, while, for a long and stiff assembly, it is inversely proportional to the square root of the axial and interfacial compliances.