Analytical Expressions for Shear and Axial Joint Deformations in Area-Array Assemblies Due to Global CTE Mismatch

In a previous study the authors derived an analytical expression for calculating the maximum shear deformation, (Δu)_max, in an area array of solder joints under global CTE mismatch loading. The result was expressed in the form (Δu)_max = β_sh(Δu)₀, where β_sh fish is the “shear correction factor” and (Δu)₀ is the commonly used and easily calculated estimate of shear deformation, which is based on the free thermal expansion of component and substrate. A key assumption in the previous model was that warping of the assembly was neglected. In the present work the companion problem of assembly warpage is treated, the results of which are analytical expressions for the maximum axial deformation in the array, (Δw)_max. The present results are cast in the form of an “axial correction factor,” β_ax, to be applied to the same convenient reference deformation: (Δw)_max = β_ax(Δu)₀. Exact solutions are presented for several cases of practical interest: (1) an assembly in which the component and substrate have the same plan dimensions, (2) a rigid-substrate assembly, and (3) a rigid-component assembly. In addition, approximate expressions are presented in simple analytical form for the case in which the array is relatively flexible in comparison with the component and substrate. When combined with the previous solution for array shear deformation, the present results may be viewed as furnishing the complete solution to the thermal deformation problem for area-array assemblies of the types considered. The analytical results clearly indicate the relationship between the correction factors and the physical parameters of the problem: (a) the dimensions and material properties (elastic and thermal) of the component and substrate; (b) the material properties of the interconnect material (effective Young’s modulus and Poisson’s ratio); (c) the array size and population; and (d) the geometric parameters of the individual joints. An interesting and potentially useful conclusion is that the sign of the maximum axial deformation (tension or compression) in the corner joints can be related to the sign of a very simple expression that depends on the assembly dimensions and the material properties of the component and substrate. This result could be used to design more reliable assemblies, as it enables one to minimize the amount of axial deformation in the corner joints or, for a given thermal loading condition, to create a desirable compression in the corner joints to inhibit shear-driven fatigue cracking. The solution is based on a theoretical model of two circular elastic disks connected by an elastic layer whose distributed axial and shear stiffnesses are related to the joint/array characteristics by means of the authors’ previously derived stiffnesses for a single joint.