Shearing Deformation in Partial Areal Arrays: Analytical Results

Abstract

Simple, closed-form expressions, based on elasticity theory, are derived for determining the location and magnitude of the maximum shearing displacement in a partial areal array of solder joints. Both uniform and nonuniform thermal loadings are considered, as is the heterogeneity of the component, which often arises due to different values of coefficient of thermal expansion (CTE) and elastic properties among the module’s constituent materials. The model is based on the following assumptions: (a) the square geometry of the array and component may be replaced with an equivalent axisymmetric geometry; and (b) the stiffness of the solder joints is negligible with respect to that of the component and substrate. The “soft joint” assumption corresponds to low-modulus solders or to thermal excursions occurring at high temperatures or low frequencies, for which significant stress relaxation occurs in the solder. For arrays exhibiting higher stiffness characteristics, the model yields conservative estimates of shearing displacement. Results indicate that, unlike homogeneous-component models under uniform temperature changes, the critical joints are not necessarily at the outer corners of the array. Other candidate locations predicted by the model (and observed in experimental and numerical studies) include the inner corner joints and any joints positioned beneath the die corners. The analytical results, also presented graphically, are found to depend on only three dimensionless parameters: the ratio of inner to outer array dimension, the ratio of die size to outer array dimension, and a “mismatch parameter,” which depends on the material, geometry, and loading characteristics of the problem. The results can be used to quickly determine the location and magnitude of peak shearing displacement in the array, possibly minimizing or eliminating the need to perform expensive and time-consuming finite element macroanalyses on entire assemblies involving hundreds of joints. Thus, the analyst may proceed directly to a detailed finite element microanalysis of the critical joint for fatigue life estimation, using the calculated shearing displacement as a required boundary condition in the finite element model.