Viscous inclusions in anisotropic materials: Theoretical development and perspective applications

Abstract

Theories and numerical solutions for a viscous ellipsoid in an infinite anisotropic viscous medium subjected to far-field homogeneous deformation lie at the heart of self-consistent homogenization models and multiscale simulations of texture and fabric development in Earth's lithosphere. There is considerable literature on ellipsoid inclusions, focused on anisotropic elastic materials, published in multi-disciplinary fields. To make this body of work more accessible as well as to advance viscous inclusion studies, an effort is made here to summarize recent advances and to further develop formally more explicit and, where possible, analytic solutions for incompressible viscous materials. The point-force concept and equivalent inclusion method of Eshelby are used together with the Green function approach. This leads to generalized equations for ellipsoid inclusion behaviors in anisotropic materials. In the particular case of isotropic materials, the new mathematical development here enables the use of existing methods for elastic materials to get solutions for corresponding viscous inclusion problems efficiently and accurately. A 2D formulation is also presented for elliptic cylinders in plane-straining flows of anisotropic materials, using the same Green function method that is adopted for 3D inclusions. The 2D formulation is benchmarked with existing analytic solutions. A reconnaissance investigation to compare the behaviors of 2D elliptic inclusions and triaxial ellipsoids in a matrix of planar anisotropy undergoing far-field plane-straining flows suggests that conclusions drawn from 2D cannot be applied to 3D in anisotropic cases.The application of the viscous inclusion theory to the rheologically heterogeneous and non-linear lithosphere is discussed. By regarding a heterogeneous element as an ellipsoidal inclusion embedded in a hypothetical homogeneous equivalent matrix whose effective rheology is obtained through micromechanical homogenization and assuming that the conditions of scale separation are satisfied, the viscous inclusion theory forms the backbone of multiscale models for natural deformation and the accompanying texture and fabric development.