DEVELOPMENT OF A VARIABLE-ORDER SINGULAR BOUNDARY ELEMENT WITH APPLICATION TO IC PACKAGE ANALYSIS

Abstract

The present development of a variable-order singular field element for use with the Boundary Element Method (BEM) is initiated by the need to analyze the stress singularity arising at sharp corners of Integrated-Circuit (IC) packages. Stress singularities are known to occur at corners and crack tips where discontinuities in geometry and material properties are present. Such singular stresses can be quantified using the concept of Stress Intensity Factors (SIFs) which characterize the state of stress at the singularity point. These quantities are useful in determining the possibility of delamination initiation and propagation by comparing them against some critical values reflecting the strength of the material system. For two-dimensional stress analysis using the BEM, a specially-developed variable-order singular field element provides better accuracy and efficiency in evaluating the SIF than standard elements. In addition, this element also provides the versatility in modeling different orders of stress singularities arising in different material system configurations. This element is derived from the standard quadratic one-dimensional line element by modification of the displacement and traction shape functions to incorporate the appropriate singular field arising at the corner. This singular stress field consisting of an order of singularity and a mode shape is obtained by solving an eigenvalue problem formulated for the singularity existing in an infinite plane. The BEM utilizing this singular field element is implemented numerically and several test cases are conducted on stress singularity problems such as those at crack tips and notches. Results obtained with this new element are compared and verified against those reported in the literature, and the present method is shown to be useful and efficient in obtaining the SIF for stress singularities. Using the same routine, a two-dimensional stress analysis is performed on an IC package subjected to thermal loading. The SIFs at corners and crack tips of the package are obtained. These quantities can be used in predicting the initiation and propagation of delamination in the IC package. Finally, a study on the effect of size and position of the delamination in the package on the SIP values shows that catastrophic failure of the package results once any delamination starts to propagate.