Stress analysis in three-dimensional joints with a crack at the vertex of the interface

Abstract

A crack initiates frequently at a vertex in three-dimensional joints under an external load and a thermal load. In the present paper, the stress distributions near a very small crack occurring at the vertex of the interface in a three-dimensional joint are analyzed under a tensile load using a boundary element method, and the stress intensity factor of mode II is investigated along the crack front. The joint model is composed of silicon and resin, which is modeled on a material combination in electronic devices. Three kinds of crack shape, triangular, quarter circular, and concave shapes, are supposed as a crack shape. First, the stress distributions near the vertex in the model without a crack are obtained and are used for normalizing the singular stress at the front of the crack. Dimensionless stress intensity factor for an interface crack is defined and determined from the distribution of the normalized stress. Next, the stress distribution near the intersection point of the crack front and the side surface is precisely investigated. An eigenanalysis at the intersection point is conducted, and eigenvalues yielding the stress singularity are obtained. Then, it is found that there are two values yielding the stress singularity. The stress distributions near the intersection point are expressed using the angular functions for each value yielding the singularity. Finally, it is shown that the stress intensity factor for mode II along the crack front varies following the summation of functions composed of the distance from the intersection point with the power indices of ( $${0.5 - {\lambda _1}}$$ ) and ( $${0.5 - {\lambda _2}}$$ ), where $${\lambda _1}$$ and $${\lambda _2}$$ are the orders of stress singularity at the intersection point.