An Elastoplastic Beam Model for Column-Grid-Array Solder Interconnects

Abstract

An analytical model is developed and implemented to analyze the deformation of solder columns in column-grid-array(CGA) assemblies. Each solder column is modeled as a prismatic beam of circular cross section subjected to end shearing deflections caused by thermal mismatch between the module and the circuit board. The solder is idealized as an elastic-perfectly plastic material whose yielding is governed by the von Mises criterion. Since the columns are relatively short by beam standards, transverse shear deformation has been incorporated into the beam model. The results generated with the model indicate the following: 1. Yielding is governed by bending for slenderness ratios (height-to-diameter) of h/d ≥ 1/3. 2. The nonlinear stiffness relationship for a sheared column, presented in dimensionless form, reduces to a single curve, which is valid for arbitrary values of slenderness ratio (≥1/3) and material parameters. 3. The normalized relationship between maximum shear strain (in the Tresca sense) and the relative end deflection depends on Poisson’s ratio but is independent of the other material parameters and the slenderness ratio. The peak shear strain can easily be decomposed into elastic and plastic portions. 4. The maximum value of the total or the plastic shear strain in the solder column can be expressed in terms of a correction factor to be applied to the nominal shear strain (shear displacement divided by the column height). The correction factors are quite sensitive to the column slenderness and the load level. Since correction factors less than unity are possible, one should not interpret the nominal shear strain as the “average” shear strain in the column. Regardless of the load level, the nominal strain underestimates the peak strain by the greatest amount at a slenderness ratio (height/diameter) of 1.7. The nonlinear stiffness results presented in the paper may be used to create more efficient finite element models of entire assemblies by replacing each column with a single nonlinear spring element. When used in conjunction with an appropriate Coffin-Manson relationship, the maximum shear strain results presented herein may be utilized to increase column fatigue life.