Energy release rates for a misfitted spherical inclusion under far-field mechanical and uniform thermal loads

• Spherically symmetric misfitted inclusion.
• Far-field triaxial load, σ∞, and uniform thermal load, ΔT.
• Path-independent J-, L- and M-integrals.
• Effect of interactions of loadings on the M-integral.

A misfitted spherical inclusion embedded in an infinite matrix under far-field triaxial mechanical and uniform thermal loads is analyzed in three-dimensional linear elasticity. Closed-form expressions for each loading case have been obtained separately utilizing proper continuity and far-field loading conditions. The principle of superposition has been utilized to obtain the complete solution from these three individual solutions. The solutions are discussed in terms of stress distribution, magnitude and locations of stress concentrations. Closed-form expressions for energy release rates have been derived based on the stress and strain solutions by employing the path-independent integrals, J, L and M considering the inclusion as a defect. The measure of self-similar expansion energy release rate provided by the M-integral brings some interesting insight which can be applicable to many practical cases such as phase transformations in metals and nanoscale defects. The fundamental work presented here will help to better understand the behavior of materials with a misfitted inclusion subjected to various loading environments.