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In Finsler geometry, each point of a base manifold can be endowed with coordinates describing its position as well as a set of one or more vectors describing directions, for example. The associated metric tensor may generally depend on direction as well as position, and a number of connections emerge associated with various covariant derivatives involving affine and nonlinear coefficients. Finsler geometry encompasses Riemannian, Euclidean, and Minkowskian geometries as special cases, and thus it affords great generality for describing a number of phenomena in physics. Here, descriptions of finite deformation of continuous media are of primary focus. After a review of necessary mathematical definitions and derivations, prior work involving application of Finsler geometry in continuum mechanics of solids is reviewed. A new theoretical description of continua with microstructure is then outlined, merging concepts from Finsler geometry and phase field theories of materials science.

Mechanical behavior of homogeneous isotropic elastic solids can be described by constitutive models that depend only on local deformation, for example, some metric or strain tensor that may generally vary with position in a body. Materials with microstructure require more elaborate constitutive models, for example, describing lattice orientation in anisotropic crystals, dislocation mechanisms in elastic-plastic crystals, or cracks or voids in damaged brittle or ductile solids. In conventional continuum mechanics approaches, such models typically assign one or more time- and position-dependent vector(s) or higher-order tensor(s), in addition to total deformation or strain, that describe physical mechanisms associated with evolving internal structure.

Mathematically, in classical continuum physics [

Finsler geometry, first attributed to Finsler in 1918 [

The use of Finsler geometry to describe continuum mechanical behavior of solids was perhaps first noted by Kr

This paper is organized as follows. In Section

Notation used in the present section applies to a referential description, that is, the initial state; analogous formulae apply for a spatial description, that is, a deformed body.

Denote by

The fundamental function

the fundamental tensor

Consider now a coordinate transformation to another chart on

Christoffel symbols of the second kind derived from the symmetric fundamental tensor are

Focusing again on the Chern-Rund connection

Nonholonomicity (i.e., nonintegrability) of the horizontal distribution is measured by [

The first application of Finsler geometry to finite deformation continuum mechanics is credited to Ikeda [

Let

Let

The second known application of Finsler geometry towards finite deformation of solid bodies appears in Chapter 8 of the book of Bejancu [

Define

Let

Let

Saczuk et al. [

Define

Restricting attention to the time-independent case, a Lagrangian is posited of the form

In Section

Deformation twinning involves shearing and lattice rotation/reflection induced by mechanical stress in a solid crystal. The usual elastic driving force is a resolved shear stress on the habit plane, in the direction of twinning shear. Twinning can be reversible or irreversible depending on material and loading protocol; the physics of deformation twinning is described more fully in [

As in Section

Consider a crystal with a single potentially active twin system. Applying (

A more general and potentially powerful approach would be to generalize fundamental function

Finsler geometry and its prior applications towards continuum physics of materials with microstructure have been reviewed. A new theory, in general considering a deformable vector bundle of Finsler character, has been posited, wherein the director vector of Finsler space is associated with a gradient of a scalar order parameter. It has been shown how a particular version of the new theory (Minkowskian geometry) can reproduce governing equations for phase field modeling of twinning in initially homogeneous single crystals. A more general approach allowing the fundamental function to depend explicitly on material coordinates has been posited that would offer enriched description of interfacial mechanics in polycrystals or materials with multiple phases.

The author declares that there is no conflict of interests regarding the publication of this paper.