(304a) Surface Tractions on Rigid Ellipsoidal Inclusion in Elastic Media

Using models of microstructure with simplified rules for the motion of the inclusions, a modeling framework for material processing has already been established (Phan-Thien and Kim. Microstructures in elastic media: principles and computational methods. Oxford University, 1994). In this work, we consider the micromechanics: the analysis of the stress and strain in the elastic material resulting from the relative displacement of the rigid inclusions. Particularly, we have derived the traction field for a rigid ellipsoidal inclusion in uniform, rotational, and linear strain fields.

Our study is motivated by the recent weighted ellipsoidal metric space (WEMS) theorem in ellipsoidal microhydrodynamics (Industrial & Engineering Chemistry Research 54.42 (2015): 10549-10551). The WEMS theorem is based on the linear operator theory originated from the integral representation, which represents the disturbance velocity around a rigid ellipsoid. The theorem proves a simple relationship between ambient flow field and ellipsoidal particle surface traction field, which fundamentally resolves the 140-year fluid mechanics enigma.

Although the integral representation also applies to the problem of a rigid inclusion in elastic media, we find that, unlike the microhydrodynamics, the corresponding weighted linear operator in micromechanics is not likely self-adjoint. Next, we examined the relationships between the traction fields and the inclusion motions in uniform, rotational, and linear strain fields, respectively. Among these three scenarios, the WEMS theorem applies in both uniform and linear strain fields, but not the rotational field. This interesting finding leads us to develop a subspace WEMS. The goal of our research is to develop a unified WEMS theorem for both microhydrodynamics and micromechanics.