Applications of Two-State M-Integral to a Triple Junction Vertex Under Thermal Loading

The two-state M-integral, which is derived from the M-integral and is applicable for two elastic states, is applied for computing an intensity of a singular near-tip field around the vertex of a triple junction of three different materials under thermal loading, often encountered in electronic packaging. Numerically we verify that a simple auxiliary field associated with every eigenfunction for the composite wedge under consideration exists in the form of the conjugate solution in the sense of the M-integral. The auxiliary field is then employed for superposition with the elastic field under consideration, and the associated two-state M-integral is computed via the domain integral technique. This enables us to extract the intensity for a leading singular eigenfunction. The proposed computational scheme enjoys the efficiency and simplicity in that the near-field information for a singular elastic boundary layer is extracted from the domain integral representation without resort to singular finite elements for the wedge vertex.