Finsler differential geometry in continuum mechanics: Fundamental concepts,history,and renewed application to ferromagnetic solids

Abstract

Finsler differential geometry enables enriched mathematical and physical descriptions of the mechanics of materials with microstructure. The first propositions for Finsler geometry in solid mechanics emerged some six decades ago. Ideas set forth in these early works are reviewed, along with subsequent literature culminating in contemporary theories of Finsler-geometric continuum mechanics. Concepts unique to generalized Finsler spaces, in the context of continuum mechanical applications, are highlighted. Capabilities afforded by physical models in generalized Finsler spaces are contrasted with those of standard approaches in affinely connected spaces. Theory and several examples of reduced dimensionality are reported for boundary value problems of fracture and phase transformations, showing how simultaneously novel, physical, and pragmatic model predictions can be obtained from Finsler-type continuum field theory. Lastly, the modern theory is newly applied to describe nonlinear elastic ferromagnetic solids in the magnetically saturated state. A variational approach is used to derive Euler–Lagrange equations for macroscopic and microscopic, i.e., respective electromechanical and electronic continuum, equilibrium states. For a representative generalized Finsler metric depending on material symmetry, augmented conservation laws of macroscopic momentum and electronic spin angular momentum naturally emerge.

Keywords

Continuum mechanics differential geometry ferromagnetism field theory Finsler geometry

1. Introduction

Differential geometry and continuum physics are inextricably related. In the context of solid mechanics, geometric methods provide mathematical insight into physical phenomena and establish rigorous foundations for construction of models of kinematics and constitutive behaviors of real materials. Perhaps most prominently known are compatibility conditions for deformation fields in nonlinear elastic bodies, eloquently derived and explained via Riemannian geometry [1 –3].

Another well-known class of applications of non-Euclidean geometry in continuum solid mechanics is the description of imperfect crystalline materials, namely inelasticity, dislocations, and other kinds of lattice defects. The torsion tensor of the Weitzenböck connection corresponding to gradients of lattice vectors, or to gradients of lattice distortion or plastic deformation, depending on model kinematics, can be associated with a density of geometrically necessary dislocations [4 –8]. The geometry of the connection afforded in this case is non-Riemannian. More general affine connections have been used to describe disclinations, point defects, fractures, and other lattice anomalies [9 –15]. When defects induce residual or internal stresses, the curvature tensor associated with the metric of the distorted lattice does not vanish: the Levi-Civita connection is one of Riemannian geometry in this case, but is torsion-free by construction [4, 5]. Constitutive frameworks incorporating geometric concepts in the context of higher-grade elasticity and elasto-plasticity include [7, 8, 16 –20]. Other geometric descriptions applicable to deformable media with microstructure invoke $G$ -structures [21, 22].

In the above-mentioned examples, none of which incorporates Finsler geometry, a material body in the continuum limit can be viewed as a differentiable manifold $M$ of dimension $n$ , parameterized by one (or more, depending on topology) coordinate charts ${X^{A}}$ ( $A = 1, \dots, n$ ). A Riemannian metric, by definition a symmetric quadratic differential form, is introduced on $M$ for the inner product of vectors. In components, $G_{AB} = G_{AB} (X)$ populate a symmetric and positive-definite $n \times n$ tensor field, where argument $X$ herein implies functional dependence on the $X^{A}$ [3]. The geometric description is complete upon designation of covariant derivative ∇, i.e., the linear connection on $M$ . In the present case, given the ${X^{A}}$ and corresponding bases, the connection is sufficiently determined by the choice of connection 1-forms or linear, i.e., affine, connection coefficients, most generally consisting of $n^{3}$ independent field components $Γ_{BC}^{A} = Γ_{BC}^{A} (X)$ . For usual solid bodies, $n = 3$ , but other dimensions are permissible. Coordinate descriptions of vector- and tensor-type fields can be framed in terms of holonomic or anholonomic, i.e., non-holonomic, bases, where the latter do not correspond to continuous curves parameterizing the body [23, 24]. Anholonomic coordinates emerge naturally when a multiplicative decomposition of deformation gradient is imposed as a kinematic assumption [25, 26].

In contrast to approaches with geometry based on a Riemannian metric structure and affine connection over $M$ , Finsler geometry offers a more general and, thus, potentially richer description, albeit at the expense of considerably higher complexity. Fundamentals of Finsler geometry, credited by name to origins in [27], are discussed in [28, 29]. Other geometers that made pioneering contributions to the topic, for example via introduction of aptly named linear connections, include Berwald [30], Cartan [31], Chern [32], and Rund [28]. Extensions to pseudo- and generalized Finsler spaces are described in [33 –35]. The latter are often applied to mechanics as reviewed and developed in this paper, as strict Finsler geometry has certain characteristics that prove overly restrictive.

In Finsler geometry and its generalizations, a base manifold $M$ , parameterized by one or more charts ${X^{A}}$ ( $A = 1, \dots, n$ ), is again introduced. A fiber bundle $(Z, M, Π, U)$ of total (generalized pseudo-Finsler) space $Z$ of dimension $n + m$ is then constructed. In Finsler geometry, the total space is typically identified with the slit tangent bundle, i.e., $Z \to T M \ 0$ [29] with $m = n$ , but these assumptions are not essential in more general formulations [33 –35]. Auxiliary coordinates ${D^{K}}$ ( $K = 1, \dots m$ ) cover each fiber $U$ , such that $Z$ is parameterized by the paired set ${X^{A}, D^{K}}$ . Particular transformation laws are assigned for changes of coordinates associated with $X$ and $D$ . Nonlinear connection coefficients are used to define non-holonomic bases that transform conventionally on $T M$ under $X$ -coordinate changes. Furthermore, at least two, and up to four [33, 36], additional fields of connection coefficients $H_{BC}^{A} (X, D), V_{IJ}^{K} (X, D)$ , etc., are needed to enable covariant differentiation of tensor fields with respect to $X$ and $D$ , corresponding to respective horizontal and vertical derivatives on $Z$ [28, 33, 35].

Components of the (generalized) pseudo-Finsler metric are of the form $G_{AB} = G_{AB} (X, D)$ . The metric tensor is always symmetric; it is always positive-definite for Finsler geometry, but not always so for the pseudo-Finsler case [35]. Furthermore, for strict Finsler geometry (but not necessarily its generalizations [33, 37, 38]), $G_{AB}$ are obtained as second derivatives of a (squared) fundamental scalar function $\frac{1}{2} F^{2}$ with respect to $D^{A}$ [28, 29]. In Finsler geometry, $F$ is positively homogeneous of degree one in $D$ ; the resulting metric is homogeneous of degree zero in $D$ [28, 29] and thus cannot depend only on the magnitude of a vector comprised of components ${D^{A}}$ . In Finsler space and its generalizations, the $(X, D)$ -coordinate dependencies of the metric and definitions for covariant differentiation enter other geometric objects and tensorial operations: torsion and curvature forms, volume and area forms, and the divergence theorem [39], for example.

When used to model physical phenomena, the essential motivation for Finsler geometry and its generalizations is description of detailed physics via the set of independent field parameters or auxiliary coordinates ${D^{A}}$ attached to each position $X$ in real physical space. For strict Finsler geometry, directional-dependent phenomena are of prime interest, owing to the homogeneity properties of $F$ and its derivatives such as $G_{AB}$ . In solid mechanics, the idea can be interpreted as an extension of micropolar, micromorphic, or Cosserat-type theory [40 –44] from Riemannian (and, more often, Euclidean) space to Finsler space. In micromorphic-type models [40 –42], a triad of vectors is introduced pointwise to represent material microstructure that generally deforms differently than the macroscopic continuum. One could associate $D$ of generalized Finsler space with one (or more, if $m > n$ ) such director vectors. However, in classical micromorphic theories, a Riemmannian, rather than Finsler, metric is used, the material domain with microstructure is fully parameterized by the ${X^{A}}$ , and basis vectors and coordinate transformation laws are those of standard continuum mechanics [2, 3, 45], as are integral theorems such as those of Stokes and Gauss. The director triads of micromorphic theory enter the balance laws and constitutive functions, but they do not explicitly affect geometric objects and covariant derivative operations, in contrast to $D$ of Finsler space.

In addition to applications in continuum mechanics of solids, to be reviewed later in this work, Finsler differential geometry has witnessed diverse use in other physical sciences since its inception over a century ago. These include broad field-theoretic descriptions of particles, space–time, and gravitation [46 –50]. Generalized treatments of classical mechanics with Finsler geometry include [51, 52]. Heat flow and fluid flow (e.g., rheology) have also been characterized with tools from Finsler geometry and its generalizations [53 –55]. Wave propagation in seismology [56, 57] and stochastic processes in biology [58] comprise several other recent applications. Contemporary theories with a basis in Finsler geometry of spinor structures and other topics in modern physics are discussed in [59] and references therein. A complete and detailed review of this vast application space, however, is outside the present scope that is focused on continuum mechanical applications.

This paper is structured as follows. Essential mathematical preliminaries on generalized Finsler spaces are given in Section 2. A review of historical applications in continuum physics of solids (1962–1990) is undertaken in Section 3. Contemporary literature, specifically that appearing in the last three decades (1991–present), is reviewed in Section 4. Sections 2–4 significantly update an earlier review [60], in both scope and rigor. Finally, in Section 5, a modern implementation of generalized Finsler-geometric continuum theory [61, 62] is newly advanced in a reformulation of a historic theory of Finsler geometry [63] of ferromagnetic crystals, whereby ideas from the variational approach of [64] framed in Euclidean space are herein extended to generalized Finsler space.

2. Generalized Finsler space

Fundamental aspects are tersely explained in the following, albeit in sufficient rigor and detail to support applications in mechanics discussed subsequently. The current presentation improves upon [61, 62]. Brief summaries with historical context are given in [65, 66]. Extensive treatments include [28, 29, 33].

2.1. Material representation via a fiber bundle

The fiber bundle approach of [33] for generalized Finsler spaces is general enough to encompass all aspects of the present description. In usual continuum mechanical settings, the reference configuration is identified with a particular instant in time at which a deformable solid body is considered undeformed. A differential manifold $M$ of spatial dimension $n$ is physically identified with a deformable solid body embedded in ambient Euclidean space of dimension $N \geq n$ . Such an embedding only applies to base manifold $M$ . Neither the total space of the fiber bundle $Z$ , to be introduced in what follows, nor its specialization to a Finsler space $F_{n}$ discussed in Section 2.1.5, can generally be embedded in Euclidean space, as discussed in Section 2.1.6.

Let $X \in M$ denote a material point or a material particle, and let ${X^{A}} (A = 1, 2, \dots, n)$ denote a coordinate chart that, depending on topology, may partially or completely cover $M$ . Attached to each material point is a vector $D$ , such that chart(s) of secondary coordinates ${D^{K}} (K = 1, 2, \dots, m)$ are assigned over $M$ . Fields ${D^{K}}$ are presumed smooth over $M$ , meaning $D$ is as many times continuously differentiable with respect to ${X^{A}}$ as needed.

Define $Z = (Z, Π, ℳ, U)$ as a fiber bundle of total space $Z$ (dimension $n + m$ ), where $Π : Z \to M$ is the projection and $U = Z_{X} = Π^{- 1} (X)$ is the fiber at $X$ . A chart covering a region of $Z$ is ${X^{A}, D^{K}}$ . Each fiber is a vector space of dimension $n$ ; the set ( $Z, Π, M)$ constitutes a vector bundle. Let $M' \subset M$ denote an open neighborhood of any $X \in M$ , $Φ$ an isomorphism of vector spaces, and $P_{1}$ a projection operator onto the first factor. Then the following diagram is commutative [33]:

2.1.1. Basis vectors and nonlinear connections

In (generalized) Finsler geometry, coordinate transformations from ${X, D}$ to another chart ${\tilde{X}, \tilde{D}}$ on $Z$ are of the form

{\tilde{X}}^{A} = {\tilde{X}}^{A} (X), {\tilde{D}}^{J} (X, D) = Q_{K}^{J} (X) D^{K},

(2.1)

where $Q_{K}^{J}$ is non-singular and differentiable, with inverse obeying ${\tilde{Q}}_{K}^{I} Q_{J}^{K} = δ_{J}^{I}$ . The holonomic basis for the tangent bundle $T Z$ is the field of frames ${\frac{\partial}{\partial X^{A}}, \frac{\partial}{\partial D^{K}}}$ . The holonomic basis for the cotangent bundle $T^{*} Z$ is ${d X^{A}, d D^{K}}$ . From (2.1), under a change of coordinates $(X, D) \to (\tilde{X}, \tilde{D})$ on $Z$ induced by a change of coordinates $X \to \tilde{X}$ on $M$ , holonomic basis vectors on $T Z$ transform as [33, 35]

\frac{\partial}{\partial {\tilde{X}}^{A}} = \frac{\partial X^{B}}{\partial {\tilde{X}}^{A}} \frac{\partial}{\partial X^{B}} + \frac{\partial D^{K}}{\partial {\tilde{X}}^{A}} \frac{\partial}{\partial D^{K}} = \frac{\partial X^{B}}{\partial {\tilde{X}}^{A}} \frac{\partial}{\partial X^{B}} + \frac{\partial {\tilde{Q}}_{J}^{K}}{\partial {\tilde{X}}^{A}} {\tilde{D}}^{J} \frac{\partial}{\partial D^{K}},

(2.2)

\frac{\partial}{\partial {\tilde{D}}^{J}} = \frac{\partial X^{B}}{\partial {\tilde{D}}^{J}} \frac{\partial}{\partial X^{B}} + \frac{\partial D^{K}}{\partial {\tilde{D}}^{J}} \frac{\partial}{\partial D^{K}} = {\tilde{Q}}_{J}^{K} \frac{\partial}{\partial D^{K}} .

(2.3)

Similarly, for the holonomic basis on $T^{*} Z$ ,

d {\tilde{X}}^{A} = \frac{\partial {\tilde{X}}^{A}}{\partial X^{B}} d X^{B} + \frac{\partial {\tilde{X}}^{A}}{\partial D^{K}} d D^{K} = \frac{\partial {\tilde{X}}^{A}}{\partial X^{B}} d X^{B},

(2.4)

d {\tilde{D}}^{J} = \frac{\partial {\tilde{D}}^{J}}{\partial X^{B}} d X^{B} + \frac{\partial {\tilde{D}}^{J}}{\partial D^{K}} d D^{K} = \frac{\partial Q_{K}^{J}}{\partial X^{B}} D^{K} d X^{B} + Q_{K}^{J} d D^{K} .

(2.5)

Clearly, the ${\frac{\partial}{\partial X^{A}}}$ and ${d D^{K}}$ do not transform as conventional basis vectors under coordinate changes on $Z$ . A remedy is use of the non-holonomic basis vectors ${\frac{δ}{δ X^{A}}}$ and ${δ D^{K}}$ defined as [33, 34]

\frac{δ}{δ X^{A}} = \frac{\partial}{\partial X^{A}} - N_{A}^{K} \frac{\partial}{\partial D^{K}}, δ D^{K} = d D^{K} + N_{B}^{K} d X^{B} .

(2.6)

By construction [35],

\frac{δ}{δ {\tilde{X}}^{A}} = \frac{\partial X^{B}}{\partial {\tilde{X}}^{A}} \frac{δ}{δ X^{B}}, δ {\tilde{D}}^{J} = Q_{K}^{J} δ D^{K}; 〈 \frac{δ}{δ X^{B}}, d X^{A} 〉 = δ_{B}^{A}, 〈 \frac{\partial}{\partial D^{K}}, δ D^{J} 〉 = δ_{K}^{J} .

(2.7)

The set ${\frac{δ}{δ X^{A}}, \frac{\partial}{\partial D^{K}}}$ thus serves as a convenient local basis on $T Z$ , and the dual set ${d X^{A}, δ D^{K}}$ on $T^{*} Z$ [29, 34]. The $N_{B}^{K} (X, D)$ are called nonlinear connection coefficients. These do not obey the coordinate transformation rules for linear connections, nor do they always correspond to a covariant derivative with the properties of a linear connection. However, special forms do enable construction of a covariant derivative like that of a linear connection [35]. For (2.7) to hold, following transformation rule is essential for arbitrary coordinate changes $X \to \tilde{X}$ :

{\tilde{N}}_{A}^{J} = (Q_{K}^{J} N_{B}^{K} - \frac{\partial Q_{K}^{J}}{\partial X^{B}} D^{K}) \frac{\partial X^{B}}{\partial {\tilde{X}}^{A}} .

(2.8)

The $N_{B}^{K}$ are presumed differentiable with respect to $(X, D)$ . The geometry of tangent bundle $T Z$ with nonlinear connection admits an orthogonal decomposition $T Z = V Z \oplus H Z$ into a vertical vector bundle $V Z$ with local field of frames ${\frac{\partial}{\partial D^{A}}}$ and a horizontal distribution $H Z$ with local field of frames ${\frac{δ}{δ X^{A}}}$ [33]. Fibers of $V Z$ and $H Z$ are of respective dimensions $m$ and $n$ . The Lie bracket of horizontal basis vector fields is strictly vertical:

[\frac{δ}{δ X^{A}}, \frac{δ}{δ X^{B}}] = (\frac{δ N_{B}^{K}}{δ X^{A}} - \frac{δ N_{A}^{K}}{δ X^{B}}) \frac{\partial}{\partial D^{K}} = R_{BA}^{K} \frac{\partial}{\partial D^{K}} = - R_{AB}^{K} \frac{\partial}{\partial D^{K}} .

(2.9)

The horizontal subspace is thus not involutive unless (2.9) vanishes identically, and $H Z$ is integrable iff $R_{AB}^{K} = 0$ . The object with components $R_{BA}^{K}$ has been called curvature [35] or torsion [33]. When $N_{A}^{K} = 0$ in one system $(X, D)$ , ${\tilde{R}}_{BA}^{K} = 0$ in any other system $(\tilde{X}, \tilde{D})$ .

Subsequent presentation assumes vertical and horizontal subspaces of the same dimension: $m = n$ . As such, indices $J, K, \dots$ can be replaced with $A, B, \dots$ in the summation convention, which runs from 1 to $n$ . Furthermore, in (2.1), let

Q_{B}^{A} = \frac{\partial {\tilde{D}}^{A}}{\partial D^{B}} = \frac{\partial {\tilde{X}}^{A}}{\partial X^{B}} .

(2.10)

Specialization in (2.10) is standard in Finsler geometry [29, 33] and not overly restrictive for subsequent applications. A formal way of achieving (2.10) using soldering forms is given in [35], whereby ${\frac{\partial}{\partial D^{A}}}$ and ${\frac{\partial}{\partial X^{A}}}$ are chosen to coincide for any given local coordinate chart.

Finally, coordinate differentiation operations are denoted by the following condensed notation:

\partial_{A} f (X, D) = \frac{\partial f (X, D)}{\partial X^{A}}, {\bar{\partial}}_{A} f (X, D) = \frac{\partial f (X, D)}{\partial D^{A}}; δ_{A} (\cdot) = \frac{δ (\cdot)}{δ X^{A}} = \partial_{A} (\cdot) - N_{A}^{B} {\bar{\partial}}_{B} (\cdot) .

(2.11)

2.1.2. Metric tensors, lengths, areas, and volumes

The Sasaki metric tensor [29, 36, 67], symmetric by definition, enables a natural inner product of vectors over $Z$ :

G (X, D) = G (X, D) + Ǧ (X, D) = G_{AB} (X, D) d X^{A} \otimes d X^{B} + Ǧ_{AB} (X, D) δ D^{A} \otimes δ D^{B};

(2.12)

G_{AB} = G_{AB} = G (\frac{δ}{δ X^{A}}, \frac{δ}{δ X^{B}}) = Ǧ_{AB} = Ǧ (\frac{\partial}{\partial D^{A}}, \frac{\partial}{\partial D^{B}}) = Ǧ_{BA} = G_{BA} = G_{BA} .

(2.13)

Components of $G$ and $Ǧ$ are equal and henceforth referred to as $G_{AB}$ , but their bases span orthogonal subspaces. Components $G_{AB}$ and inverse components $G^{AB}$ are used to lower and raise indices in the usual manner ( $G^{AB} G_{BC} = δ_{C}^{A}$ ), and $G (X, D) = det [G_{AB} (X, D)]$ . For example, let $V = V^{A} \frac{δ}{δ X^{A}} \in H Z$ be a vector field over $Z$ ; its magnitude at point $(X, D)$ is then $| V | = 〈 V, G V 〉^{1 / 2} = | V \cdot V |^{1 / 2} = | V^{A} G_{AB} V^{B} |^{1 / 2} = | V^{A} V_{A} |^{1 / 2}$ where components $V^{A}$ and $G_{AB}$ are evaluated at $(X, D)$ .

Let $d X$ denote a differential line element of $M$ referred to non-holonomic horizontal basis ${\frac{δ}{δ X^{A}}}$ , and let $d D$ denote a line element of $U$ referred to vertical basis ${\frac{\partial}{\partial D^{A}}}$ . Squared lengths of these elements with respect to (2.12) are

| d X |^{2} = 〈 d X, G d X 〉 = G_{AB} d X^{A} d X^{B}, | d D |^{2} = 〈 d D, G d D 〉 = G_{AB} d D^{A} d D^{B} .

(2.14)

The scalar volume element and the volume form of $n$ -dimensional base manifold $M$ are, respectively,

dV = \sqrt{G} d X^{1} d X^{2} \dots d X^{n}, d Ω = \sqrt{G} d X^{1} \land d X^{2} \land \dots \land d X^{n} .

(2.15)

The second of definitions (2.15) is consistent with [39]; others exist [68, 69], e.g., those representing volume elements and forms on total space $Z$ . The embedding of an $(n - 1)$ -dimensional oriented hypersurface in $M$ is represented by local parametric equations $X^{A} = X^{A} (U^{α}) (α = 1, \dots, n - 1)$ , $B_{α}^{A} = \frac{\partial X^{A}}{\partial U^{α}}$ , and $B = det (B_{α}^{A} G_{AB} B_{β}^{B})$ . The area form corresponding to a such a hypersurface, which can include boundary $\partial M$ having externally directed outward normal to any compact region $M$ , is [39]

Ω = \sqrt{B} d U^{1} \land \dots \land d U^{n - 1} .

(2.16)

2.1.3. Linear connections and covariant derivatives

Let ∇ denote the covariant derivative. Horizontal gradients of basis vectors are determined by generic affine connection coefficients $H_{BC}^{A}$ and $K_{BC}^{A}$ :

\nabla_{δ / δ X^{B}} \frac{δ}{δ X^{C}} = H_{BC}^{A} \frac{δ}{δ X^{A}}, \nabla_{δ / δ X^{B}} \frac{\partial}{\partial D^{C}} = K_{BC}^{A} \frac{\partial}{\partial D^{A}} .

(2.17)

Vertical gradients are specified by generic connection coefficients $V_{BC}^{A}$ and $Y_{BC}^{A}$ :

\nabla_{\partial / \partial D^{B}} \frac{\partial}{\partial D^{C}} = V_{BC}^{A} \frac{\partial}{\partial D^{A}}, \nabla_{\partial / \partial D^{B}} \frac{δ}{δ X^{C}} = Y_{BC}^{A} \frac{δ}{δ X^{A}} .

(2.18)

As in a prior example, let $V = V^{A} \frac{δ}{δ X^{A}} \in H Z$ . Covariant derivative operations are deconstructed as

\begin{matrix} \nabla V = \nabla_{δ / δ X^{B}} V \otimes d X^{B} + \nabla_{\partial / \partial D^{B}} V \otimes δ D^{B} \\ = (δ_{B} V^{A} + H_{BC}^{A} V^{C}) \frac{δ}{δ X^{A}} \otimes d X^{B} + ({\bar{\partial}}_{B} V^{A} + Y_{BC}^{A} V^{C}) \frac{\partial}{\partial D^{A}} \otimes δ D^{B} \\ = V_{| B}^{A} \frac{δ}{δ X^{A}} \otimes d X^{B} + V^{A} |_{B} \frac{\partial}{\partial D^{A}} \otimes δ D^{B}, \end{matrix}

(2.19)

where $(\cdot)_{| A}$ and $(\cdot) |_{B}$ denote horizontal and vertical covariant derivatives with respect to local coordinates of charts ${X^{A}}$ and ${D^{B}}$ , respectively. The sequence of covariant indices on connection coefficients follows [3, 23, 70, 71] and is the transpose of [26, 28, 33]; this is inconsequential for symmetric connections. Components of the horizontal covariant derivative of the metric tensor $G = G_{AB} d X^{A} \otimes d X^{B}$ are

G_{AB | C} = δ_{C} G_{AB} - H_{CA}^{D} G_{DB} - H_{CB}^{D} G_{AD} = \partial_{C} G_{AB} - N_{C}^{D} {\bar{\partial}}_{D} G_{AB} - H_{CA}^{D} G_{DB} - H_{CB}^{D} G_{DA} .

(2.20)

The following identity is also noted for $G = det (G_{AB})$ , a scalar density:

(\sqrt{G})_{| A} = \partial_{A} (\sqrt{G}) - N_{A}^{B} {\bar{\partial}}_{B} (\sqrt{G}) - \sqrt{G} H_{AB}^{B} .

(2.21)

Particular linear connections used often in Finsler geometry are discussed next. Christoffel symbols of the second kind for the Levi-Civita connection on $Z$ are derived in the usual way:

γ_{BC}^{A} = \frac{1}{2} G^{AD} (\partial_{C} G_{BD} + \partial_{B} G_{CD} - \partial_{D} G_{BC}) = G^{AD} γ_{BCD} .

(2.22)

Cartan’s tensor is defined as

C_{BC}^{A} = \frac{1}{2} G^{AD} ({\bar{\partial}}_{C} G_{BD} + {\bar{\partial}}_{B} G_{CD} - {\bar{\partial}}_{D} G_{BC}) = G^{AD} C_{BCD} .

(2.23)

Horizontal coefficients of the Chern–Rund and Cartan connections are defined as

Γ_{BC}^{A} = \frac{1}{2} G^{AD} (δ_{C} G_{BD} + δ_{B} G_{CD} - δ_{D} G_{BC}) = G^{AD} Γ_{BCD} .

(2.24)

Finally, the Berwald linear connection coefficients are defined on $Z$ as

N_{BC}^{A} = {\bar{\partial}}_{B} N_{C}^{A} .

(2.25)

Individually, (2.22), (2.23), and (2.24) are torsion-free, i.e., symmetric. The Chern–Rund–Cartan connection coefficients are metric-compatible for horizontal covariant differentiation of $G$ : $H_{BC}^{A} = Γ_{BC}^{A} \Rightarrow G_{AB | C} = 0$ in (2.20). Cartan’s tensor is metric-compatible for vertical covariant differentiation of $G$ : $Y_{BC}^{A} = C_{BC}^{A} \Rightarrow G_{AB} |_{C} = 0$ . Furthermore, from (2.23) and (2.24),

{\bar{\partial}}_{A} (\ln \sqrt{G}) = \frac{1}{2} G^{BC} {\bar{\partial}}_{A} G_{CB} = C_{AB}^{B}, δ_{A} (\ln \sqrt{G}) = \frac{1}{2} G^{BC} δ_{A} G_{CB} = Γ_{AB}^{B} .

(2.26)

Then $H_{BC}^{A} = Γ_{BC}^{A} \Rightarrow G_{| A} = 2 G (\ln \sqrt{G})_{| A} = 0$ and $Y_{BC}^{A} = C_{BC}^{A} \Rightarrow G |_{A} = 2 G (\ln \sqrt{G}) |_{A} = 0$ .

With the metric tensor now introduced, different sets of nonlinear connection coefficients $N_{B}^{A} (X, D)$ , always admissible under transformation conditions (2.8), can be obtained in several ways. When $T Z$ is restricted to locally flat sections, $N_{B}^{A} = 0$ is one possibility. In more general cases, a differentiable real Lagrangian function $L (X, D)$ can be introduced, from which $N_{B}^{A} = G_{B}^{A}$ , where [33]

G_{B}^{A} = {\bar{\partial}}_{B} G^{A} = {\bar{\partial}}_{B} [G^{AE} (D^{C} {\bar{\partial}}_{E} \partial_{C} L - \partial_{E} L)] .

(2.27)

Let $G_{AB} (X, D)$ be positively homogeneous of degree zero in $D$ . Then $G^{A}$ defined next are components of a spray [29, 35], and canonical nonlinear connection coefficients $N_{B}^{A} = G_{B}^{A}$ that obey (2.8) follow as

G^{A} = \frac{1}{2} γ_{BC}^{A} D^{B} D^{C}, G_{B}^{A} = {\bar{\partial}}_{B} G^{A} .

(2.28)

For characterization purposes, let $K_{BC}^{A} = H_{BC}^{A}$ and $Y_{BC}^{A} = V_{BC}^{A}$ . Then a complete (generalized) Finsler connection is defined by the set $(N_{B}^{A}, H_{BC}^{A}, V_{BC}^{A})$ . In this context, the Chern–Rund connection is described by $(G_{B}^{A}, Γ_{BC}^{A}, 0)$ . The Cartan connection is described by $(G_{B}^{A}, Γ_{BC}^{A}, C_{BC}^{A})$ . The Berwald connection is described by $(G_{B}^{A}, G_{BC}^{A}, 0)$ , where $G_{BC}^{A} = N_{BC}^{A} = {\bar{\partial}}_{B} {\bar{\partial}}_{C} G^{A}$ .

2.1.4. Stokes’ theorem on the base manifold

Let $M$ be a manifold of dimension $n$ , with $(n - 1)$ -dimensional boundary $\partial M$ of class $C^{1}$ , a positively oriented hypersurface. Stokes’ theorem in terms of a class $C^{1}$ differentiable $(n - 1)$ form $α$ on $M$ is

\int_{M} d α = \int_{\partial M} α .

(2.29)

Theorem 2.1. Let $M$ , $\dim M = n$ , be the base manifold of a generalized Finsler bundle of total space $Z$ with positively oriented boundary $\partial M$ of class $C^{1}$ and $\dim \partial M = n - 1$ . Let $α (X, D) = V^{A} (X, D) N_{A} (X, D) Ω (X, D)$ be a differentiable $(n - 1)$ -form, and let $V^{A}$ be contravariant components of vector field $V = V^{A} \frac{δ}{δ X^{A}} \in H Z$ . Denote the field of metric tensor components for the horizontal subspace by $G_{AB} (X, D)$ with $G = det (G_{AB})$ . Assign a symmetric horizontal linear connection $H_{BC}^{A} = H_{CB}^{A}$ such that $(\sqrt{G})_{| A} = 0$ , and assume that $C^{1}$ functional relations $D = D (X)$ exist for representation of the vertical fiber coordinates at each $X \in M$ . Then in a coordinate chart ${X^{A}}$ , (2.29) is explicitly, with volume and area forms given in (2.15) and (2.16),

\int_{M} [V_{| A}^{A} + (V^{A} C_{BC}^{C} + {\bar{\partial}}_{B} V^{A}) D_{; A}^{B}] d Ω = \int_{\partial M} V^{A} N_{A} Ω,

(2.30)

where $N_{A}$ is the unit outward normal on $\partial M$ , $V_{| A}^{A} = δ_{A} V^{A} + V^{A} H_{BA}^{B}$ , and $D_{; A}^{B} = \partial_{A} D^{B} + N_{A}^{B}$ .

Proof. The current proof, implied but not derived explicitly in [61], extends that originally derived by Rund [39], who specified a Finsler space $F_{n}$ with Cartan connection $(G_{B}^{A}, Γ_{BC}^{A}, C_{BC}^{A})$ and metric acquired from a Finsler (Lagrangian) function $F$ (see Section 2.1.5), newly here to a generalized Finsler space with arbitrary positive-definite metric $G_{AB} (X, D)$ and arbitrary nonlinear connection $N_{B}^{A} (X, D)$ .

First, the integrand on the right of (2.30), i.e., the $(n - 1)$ form in (2.29), is transformed as [39]

\begin{matrix} α = 〈 V, N 〉 Ω = V^{A} N_{A} \sqrt{B} d U^{1} \land \dots \land d U^{n - 1} = V^{A} δ_{A}^{j} p_{j} \sqrt{G} d U^{1} \land \dots \land d U^{n - 1} \\ = \sum_{j = 1}^{n} {(- 1)}^{j - 1} V^{A} δ_{A}^{j} \sqrt{G} d X^{1} \land \dots \land d X^{j - 1} \land d X^{j + 1} \land \dots \land d X^{n}, \end{matrix}

(2.31)

where $p_{j} = p δ_{j}^{A} N_{A} = (- 1)^{j + 1} \frac{\partial (X^{1}, \dots, X^{j - 1}, X^{j + 1}, \dots, X^{n})}{\partial (U^{1}, \dots, U^{n - 1})}$ is the outward normal to $M$ with magnitude $p = | G^{AB} δ_{A}^{i} δ_{B}^{j} p_{i} p_{j} |^{1 / 2}$ and direction $N = N_{A} \frac{δ}{δ X^{A}}$ when referred to the same basis as $V$ . The exterior derivative of (2.31) is, under the stated presumption on existence of $C^{1}$ functions $D = D (X) \forall X \in M$ ,

\begin{matrix} d α = \sum_{j = 1}^{n} {(- 1)}^{j - 1} d {V^{A} δ_{A}^{j} \sqrt{G}} \land d X^{1} \land \dots \land d X^{j - 1} \land d X^{j + 1} \land \dots \land d X^{n} \\ = \sum_{j = 1}^{n} {(- 1)}^{j - 1} δ_{A}^{j} {\partial_{B} (V^{A} \sqrt{G}) + {\bar{\partial}}_{C} (V^{A} \sqrt{G}) \partial_{B} D^{C}} d X^{B} \land d X^{1} \land \dots \land d X^{j - 1} \land d X^{j + 1} \land \dots \land d X^{n} \\ = \sum_{j = 1}^{n} {(- 1)}^{j - 1} δ_{A}^{j} {\partial_{B} (V^{A} \sqrt{G}) + {\bar{\partial}}_{C} (V^{A} \sqrt{G}) \partial_{B} D^{C}} (- {1)}^{j - 1} δ_{j}^{B} d X^{1} \land \dots \land d X^{n} \\ = {\partial_{A} (V^{A} \sqrt{G}) + {\bar{\partial}}_{B} (V^{A} \sqrt{G}) \partial_{A} D^{B}} d X^{1} \land \dots \land d X^{n} . \end{matrix}

(2.32)

Using (2.21) with $G_{| A} = 0$ ,

\begin{matrix} \partial_{A} (V^{A} \sqrt{G}) = \sqrt{G} [\partial_{A} V^{A} + V^{A} (H_{AB}^{B} + N_{A}^{B} C_{BC}^{C})] \\ = \sqrt{G} [V_{| A}^{A} + N_{A}^{B} {\bar{\partial}}_{B} V^{A} + V^{A} ({H_{AB}^{B} - H_{BA}^{B}} + N_{A}^{B} C_{BC}^{C})], \end{matrix}

(2.33)

and then using the first of (2.26),

\begin{matrix} {\bar{\partial}}_{B} (V^{A} \sqrt{G}) \partial_{A} D^{B} & = \sqrt{G} [{\bar{\partial}}_{B} V^{A} + V^{A} C_{BC}^{C}] \partial_{A} D^{B} . \end{matrix}

(2.34)

Then noting that $H_{BC}^{A} = H_{CB}^{A}$ , with symmetry and $G_{| A} = 0$ fulfilled (uniquely) by $H_{BC}^{A} = Γ_{BC}^{A}$ for (arbitrary) $G$ , the sum of (2.33) and (2.34) is inserted into (2.32):

d α = [V_{| A}^{A} + (V^{A} C_{BC}^{C} + {\bar{\partial}}_{B} V^{A}) D_{; A}^{B}] \sqrt{G} d X^{1} \land \dots \land d X^{n} = V_{| | A}^{A} d Ω,

(2.35)

where condensed notation is $(\cdot)_{| | A} = (\cdot)_{| A} + [(\cdot) C_{BC}^{C} + {\bar{\partial}}_{B} (\cdot)] D_{; A}^{B}$ (see [72]). Result (2.35) is verified as the integrand on the left-hand side of (2.29) and (2.30), completing the proof.

Remark 2.1. Under the stipulations of Stokes’ theorem inherent in (2.29), (2.30) holds when $M$ and $\partial M$ are replaced with any compact region of $M' \subset M$ and positively oriented boundary of that region.

Remark 2.2. When functions $D = D (X)$ exist, metric components can be treated as absolute functions of $X$ , i.e., $G_{AB} (X, D) \to G_{AB} (X, D (X)) = G_{AB} (X)$ over the domain of integration. In this setting, the generalized Finsler metric can be classified as a locally osculating Riemannian metric [28, 63]. Additional relationships naturally emerge, but equations of generalized Finsler geometry still apply.

Remark 2.3. A different basis and its dual over $M$ could be used for $V$ and $N$ , so long as $H_{BC}^{A}$ and $(\cdot)_{| B}$ are treated as the same operators in (2.33)–(2.35) with requisite symmetry and $G$ -compatibility.

2.1.5. Pseudo-Finsler and Finsler spaces

Developments to this point apply for generalized Finlser geometry, wherein the metric tensor components need not be derived from a Lagrangian [33, 37, 38]. Particular subclasses of generalized Finsler geometry require such a Lagrangian function, denoted by $L$ .

Let $Z = T M \ 0$ , i.e., the tangent bundle of $M$ excluding zero section $D = 0$ . Let $L (X, D) : Z \to R$ be positive homogeneous of degree two in $D$ , and as many times differentiable as needed with respect to ${X^{A}}$ and ${D^{A}}$ ( $C^{\infty}$ is often assumed [29], but $C^{5}$ is usually sufficient [35]). Then $(M, L)$ is a pseudo-Finsler space when the $n \times n$ matrix of components (with associated Lagrangian $L$ )

G_{AB} (X, D) = {\bar{\partial}}_{A} {\bar{\partial}}_{B} L (X, D), L = \frac{1}{2} G_{AB} D^{A} D^{B},

(2.36)

is non-singular over $Z$ . A Finsler space $(M, F)$ , also denoted by $F_{n}$ where $n = \dim M$ , is a pseudo-Finsler space for which $G_{AB} (X, D)$ is always positive-definite over $Z$ . In that case, the fundamental scalar Finsler function $F (X, D)$ is introduced, positive homogeneous of degree one in $D$ :

\begin{matrix} F (X, D) = \sqrt{2 L (X, D)} = | G_{AB} (X, D) D^{A} D^{B} |^{1 / 2} \\ \leftrightarrow L (X, D) = \frac{1}{2} F^{2} (X, D); F (X, D) > 0 \forall D \neq 0 . \end{matrix}

(2.37)

In Finsler geometry [28, 29, 33], it follows that $L = L$ and $G^{A} = G^{A}$ in (2.27) and (2.28), and that

G_{AB} = \frac{1}{2} {\bar{\partial}}_{A} {\bar{\partial}}_{B} (F^{2}), G_{B}^{A} = γ_{BC}^{A} D^{C} - C_{BC}^{A} γ_{DE}^{C} D^{D} D^{E} = Γ_{BC}^{A} D^{C}, C_{ABC} = \frac{1}{4} {\bar{\partial}}_{A} {\bar{\partial}}_{B} {\bar{\partial}}_{C} (F^{2}) .

(2.38)

2.1.6. Reductions and embeddings

A (strict) $n$ -dimensional Finsler space $F_{n}$ has been defined in Section 2.1.5. The following reductions then apply under certain stated conditions [28, 29, 35]:

F_n → M_n, where $M_{n}$ is a (local) Minkowski space, if $F (X, D) \to F (D)$ , i.e., if (local) coordinates $(X, D)$ exist on $T M \ 0$ such that the fundamental function and derived metric $G_{AB} = G_{AB} (D)$ are independent of $X$ ;

F_n → V_n, where $V_{n}$ is a (local) Riemannian space, if $F^{2} (X, D) = G_{AB} (X) D^{A} D^{B}$ , i.e., if the fundamental function is quadratic in $D$ such that metric $G_{AB} = G_{AB} (X)$ is (locally) independent of $D$ (this implies (local) vanishing of Cartan’s torsion, i.e., $C_{ABC} = 0$ , and vice versa);

F_n → E_n, where $E_{n}$ is a (local) Euclidean space, if (local) coordinates $(X, D)$ exist on $T M \ 0$ such that $F^{2} (X, D) = δ_{AB} D^{A} D^{B} \Rightarrow G_{AB} = δ_{AB}$ .

Notions regarding embeddings of Finsler spaces in Riemannian spaces are summarized from [28, 73, 74]:

a Finsler space $F_{n}$ cannot generally be embedded in a Riemannian space without torsion;

a Finsler space $F_{n}$ can be regarded as a non-holonomic subspace of a Riemannian space, with torsion, of dimension $2 n - 1$ , denoted by $V_{2 n - 1}$ , where a procedure exists [73] to determine the metric and torsion tensors of this $V_{2 n - 1}$ from a given $F_{n}$ under certain restrictions on the torsion;

a Finsler space $F_{n}$ can be regarded as a non-holonomic subspace of a Riemannian space, with torsion, of dimension $2 n$ , denoted by $V_{2 n}$ , whose torsion satisfies less restrictive conditions than those set forth in [73], which can produce an induced Cartan connection on $F_{n}$ [74].

2.2. Spatial representation via a fiber bundle

An analogous description on a fiber bundle is used for the spatial, i.e., current, configuration of a body. A differential manifold $m$ of spatial dimension $n$ represents the physical body in the current configuration, embedded in ambient Euclidean space of dimension $N \geq n$ . Let $x \in m$ denote the spatial image of a body particle or a spatial point, and let ${x^{a}} (a = 1, 2, \dots, n)$ be a coordinate chart on $m$ . Attached to each spatial point is a vector $d$ , with chart(s) of secondary coordinates ${d^{k}} (k = 1, 2, \dots, m)$ thereby assigned over $m$ . Define $z = (z, π, m, u)$ as a fiber bundle of total space $z$ (dimension $n + m$ ), where $π : z \to m$ is the projection and $u = z_{x} = π^{- 1} (x)$ is the fiber at $x$ . A chart covering a region of $z$ is ${x^{a}, d^{k}}$ . Each fiber is an $n$ -dimensional vector space, so ( $z, π, m)$ constitutes a vector bundle.

The global mapping from referential to spatial base manifolds is $φ$ , most often termed the motion in continuum mechanics. A global mapping from referential to current total spaces is denoted by the set $Ξ = (φ, θ)$ , where $φ (X, D) : M \to m$ and $Ξ (X, D) : Z \to z$ . Functional forms of $φ (X, D)$ and $Ξ (X, D)$ vary widely in the literature, with details to be discussed in Sections 3 and 4. Furthermore, mappings and field variables can be made time-dependent via introduction of independent parameter $t$ . For brevity, explicit time dependence is postponed for discussion in the reviewed content of Section 4.2. The following diagram commutes [33]:

Definitions in the remainder of Section 2.2 fully parallel those of Section 2.1, where lowercase indices and symbols, with the exception of connections, are used to distinguish current-configurational quantities.

2.2.1. Basis vectors and nonlinear connections

Coordinate transformations from ${x, d}$ to ${\tilde{x}, \tilde{d}}$ on $z$ are of the form

{\tilde{x}}^{a} = {\tilde{x}}^{a} (x), {\tilde{d}}^{j} (x, d) = q_{k}^{j} (x) d^{k},

(2.39)

where $q_{k}^{j}$ is non-singular and differentiable, with inverse obeying ${\tilde{q}}_{k}^{i} q_{j}^{k} = δ_{j}^{i}$ . The holonomic basis for the tangent bundle $T z$ is ${\frac{\partial}{\partial x^{a}}, \frac{\partial}{\partial d^{k}}}$ , and the holonomic basis for the cotangent bundle $T^{*} z$ is ${d x^{a}, d d^{k}}$ . Non-holonomic basis vectors ${\frac{δ}{δ x^{a}}}$ and ${δ d^{k}}$ that transform traditionally under a coordinate change $x \to \tilde{x}$ are defined as

\frac{δ}{δ x^{a}} = \frac{\partial}{\partial x^{a}} - N_{a}^{k} \frac{\partial}{\partial d^{k}}, δ d^{k} = d d^{k} + N_{b}^{k} d x^{b} .

(2.40)

The set ${\frac{δ}{δ x^{a}}, \frac{\partial}{\partial d^{k}}}$ is used as a local basis on $T z$ , and ${d x^{a}, δ d^{k}}$ on $T^{*} z$ . Tangent bundle $T z$ with nonlinear connection admits an orthogonal decomposition into vertical vector bundle and horizontal distribution: $T z = V z \oplus H z$ . The transformation law of the nonlinear connection and the curvature $R_{ab}^{k}$ entering the Lie bracket of non-holonomic basis follow as

{\tilde{N}}_{a}^{j} = (q_{k}^{j} N_{b}^{k} - \frac{\partial q_{k}^{j}}{\partial x^{b}} d^{k}) \frac{\partial x^{b}}{\partial {\tilde{x}}^{a}},

(2.41)

[\frac{δ}{δ x^{a}}, \frac{δ}{δ x^{b}}] = (\frac{δ N_{b}^{k}}{δ x^{a}} - \frac{δ N_{a}^{k}}{δ x^{b}}) \frac{\partial}{\partial d^{k}} = R_{ba}^{k} \frac{\partial}{\partial d^{k}} .

(2.42)

Subsequently, take $m = n$ . Indices $j, k, \dots \to a, b, \dots$ , summation over duplicate indices is from 1 to $n$ , and in (2.39), the $d^{a}$ transform-like components of a contravariant vector field over $M$ :

q_{b}^{a} = \frac{\partial {\tilde{d}}^{a}}{\partial d^{b}} = \frac{\partial {\tilde{x}}^{a}}{\partial x^{b}} .

(2.43)

Spatial coordinate differentiation is described by the compact notation

\partial_{a} f (x, d) = \frac{\partial f (x, d)}{\partial x^{a}}, {\bar{\partial}}_{a} f (x, d) = \frac{\partial f (x, d)}{\partial d^{a}}; δ_{a} (\cdot) = \frac{δ (\cdot)}{δ x^{a}} = \partial_{a} (\cdot) - N_{a}^{b} {\bar{\partial}}_{b} (\cdot) .

(2.44)

2.2.2. Metric tensors, lengths, areas, and volumes

The Sasaki metric tensor [67] providing an inner product of vectors over $z$ is

g (x, d) = g (x, d) + ǧ (x, d) = g_{ab} (x, d) d x^{a} \otimes d x^{b} + ǧ_{ab} (x, d) δ d^{a} \otimes δ d^{b};

(2.45)

g_{ab} = g_{ab} = g (\frac{δ}{δ x^{a}}, \frac{δ}{δ x^{b}}) = ǧ_{ab} = ǧ (\frac{\partial}{\partial d^{a}}, \frac{\partial}{\partial d^{b}}) = ǧ_{ba} = g_{ba} = g_{ba} .

(2.46)

Let $d x$ denote a differential line element of $m$ referred to non-holonomic horizontal basis ${\frac{δ}{δ x^{a}}}$ , and let $d d$ denote a differentiable line element of $u$ referred to vertical basis ${\frac{\partial}{\partial d^{a}}}$ . Then

| d x |^{2} = 〈 d x, g d x 〉 = g_{ab} d x^{a} d x^{b}, | d d |^{2} = 〈 d d, g d d 〉 = g_{ab} d d^{a} d d^{b} .

(2.47)

The scalar volume element and volume form of $m$ , where $\dim m = n$ , are

d v = \sqrt{g} d x^{1} d x^{2} \dots d x^{n}, d ω = \sqrt{g} d x^{1} \land d x^{2} \land \dots \land d x^{n} .

(2.48)

The embedding of an $(n - 1)$ -dimensional hypersurface in $M$ is represented by local parametric equations $x^{a} = x^{a} (u^{α}) (α = 1, \dots, n - 1)$ , $b_{α}^{a} = \frac{\partial x^{a}}{\partial u^{α}}$ , and $b = det (b_{α}^{a} g_{ab} b_{β}^{b})$ . The associated area form of $\partial m$ , the boundary of a compact region of $m$ , is

ω = \sqrt{b} d u^{1} \land \dots \land d u^{n - 1} .

(2.49)

2.2.3. Linear connections and covariant derivatives

Let ∇ denote the covariant derivative. Horizontal gradients of basis vectors are determined by generic affine connection coefficients $H_{bc}^{a}$ and $K_{bc}^{a}$ , and vertical gradients by $V_{bc}^{a}$ and $Y_{bc}^{a}$ :

\nabla_{δ / δ x^{b}} \frac{δ}{δ x^{c}} = H_{bc}^{a} \frac{δ}{δ x^{a}}, \nabla_{δ / δ x^{b}} \frac{\partial}{\partial d^{c}} = K_{bc}^{a} \frac{\partial}{\partial d^{a}};

(2.50)

\nabla_{\partial / \partial d^{b}} \frac{\partial}{\partial d^{c}} = V_{bc}^{a} \frac{\partial}{\partial d^{a}}, \nabla_{\partial / \partial d^{b}} \frac{δ}{δ x^{c}} = Y_{bc}^{a} \frac{δ}{δ x^{a}} .

(2.51)

Let $V = V^{a} \frac{δ}{δ x^{a}} \in H z$ . Covariant derivative operations over $z$ are invoked as follows:

\begin{matrix} \nabla V = \nabla_{δ / δ x^{b}} V \otimes d x^{b} + \nabla_{\partial / \partial d^{b}} V \otimes δ d^{b} \\ = (δ_{b} V^{a} + H_{bc}^{a} V^{c}) \frac{δ}{δ x^{a}} \otimes d x^{b} + ({\bar{\partial}}_{b} v^{a} + Y_{bc}^{a} V^{c}) \frac{\partial}{\partial d^{a}} \otimes δ d^{b} \\ = V_{| b}^{a} \frac{δ}{δ x^{a}} \otimes d x^{b} + V^{a} |_{b} \frac{\partial}{\partial d^{a}} \otimes δ d^{b} . \end{matrix}

(2.52)

Operations $(\cdot)_{| a}$ and $(\cdot) |_{b}$ are horizontal and vertical covariant differentiation with respect to coordinates $x^{a}$ and $d^{b}$ . Denote by $γ_{bc}^{a}$ coefficients of the Levi-Civita connection on $z$ , by $C_{bc}^{a}$ coefficients of the Cartan tensor on $z$ , and by $Γ_{bc}^{a}$ horizontal coefficients of the Chern–Rund and Cartan connections on $z$ :

γ_{bc}^{a} = \frac{1}{2} g^{ad} (\partial_{c} g_{bd} + \partial_{b} g_{cd} - \partial_{d} g_{bc}) = g^{ad} γ_{bcd},

(2.53)

C_{bc}^{a} = \frac{1}{2} g^{ad} ({\bar{\partial}}_{c} g_{bd} + {\bar{\partial}}_{b} g_{cd} - {\bar{\partial}}_{d} g_{bc}) = g^{ad} C_{bcd},

(2.54)

Γ_{bc}^{a} = \frac{1}{2} g^{ad} (δ_{c} g_{bd} + δ_{b} g_{cd} - δ_{d} g_{bc}) = g^{ad} Γ_{bcd} .

(2.55)

The Berwald linear connection on $z$ is

N_{bc}^{a} = {\bar{\partial}}_{b} N_{c}^{a} .

(2.56)

2.2.4. Stokes’ theorem on the base manifold

Let $m$ , $\dim m = n$ , be the base manifold of a generalized Finsler bundle of total space $z$ with positively oriented $(n - 1)$ -dimensional $C^{1}$ boundary $\partial m$ . Let $α (x, d) = V^{a} (x, d) n_{a} (x, d) ω (x, d)$ be a differentiable $(n - 1)$ -form, and let $V^{a}$ be contravariant components of vector field $V = V^{a} \frac{δ}{δ x^{a}} \in H z$ . Denote the field of metric tensor components for the horizontal subspace by $g_{ab} (x, d)$ with $g = det (g_{ab})$ . Assign a symmetric horizontal linear connection $H_{bc}^{a} = H_{cb}^{a}$ such that $(\sqrt{g})_{| a} = 0$ , and assume that $C^{1}$ functional relations $d = d (x)$ exist for representation of the vertical fiber coordinates $\forall x \in m$ . Then in a coordinate chart ${x^{a}}$ , (2.29) is, with volume and area forms given in (2.48) and (2.49),

\int_{m} [V_{| a}^{a} + (V^{a} C_{bc}^{c} + {\bar{\partial}}_{b} V^{a}) d_{; a}^{b}] d ω = \int_{\partial m} V^{a} n_{a} ω,

(2.57)

where $n_{a}$ is the unit outward normal on $\partial m$ , $V_{| a}^{a} = δ_{a} V^{a} + V^{a} H_{ba}^{b}$ , and $d_{; a}^{b} = \partial_{a} d^{b} + N_{a}^{b}$ .

2.2.5. Pseudo-Finsler and Finsler spaces

Definitions in Section 2.1.5 carry over directly from $Z$ to $z$ .

2.2.6. Reductions and embeddings

Remarks in Section 2.1.6 transfer directly to the spatial representation.

3. Historical applications in field theory and mechanics of continua

The potential utility of Finsler geometry for continuum mechanics applications was recognized by several prominent theoreticians of the mid-twentieth century. Kondo [75] posited trial ideas towards a yielding criterion for plasticity based on deviations from geodesics of a Finsler space with a curvature tensor derived from the Chern–Rund–Cartan horizontal connection coefficients $Γ_{BC}^{A}$ . Kröner [76], citing [63], noted how Finsler geometry can be used to describe continua comprised of particles with inherent spin-type degrees of freedom. Eringen [70] summarized concepts from Cartan’s treatise [31] on Finsler geometry, with the stated intent of raising interest among researchers of continuum physics on the topic for applications to ferromagnetic materials, oriented media, and spin waves.

However, none of the aforementioned researchers (i.e., Kondo, Kröner, nor Eringen) appear to have developed a continuum mechanical theory, i.e., a description of kinematics and balance laws of continuous bodies, based on Finsler geometry. Rather, the first continuum mechanical theory with a basis in Finlser geometry is apparently due to Amari [31], dealing with elastic–plastic–ferromagnetic crystals. Generalized Finsler-type treatments of solid bodies, with a focus on kinematics, were subsequently formulated by Ikeda [77] and Bejancu [33]. Key features of each of these works [33, 63, 77] are reviewed in what follows next. Choices of metric tensors, connection coefficients, motions, and tangent mappings between material and spatial configurations are highlighted in each case. None of these works contain solutions to any boundary value problems; hence, predictive capabilities of these early theories cannot be fully evaluated.

3.1. Amari’s theory of mechanics of ferromagnetic substances

The first theory (known to the present author) describing geometry, kinematics, and balance laws of solid mechanics with a basis in Finsler geometry is due to Amari [63]. The general kinematical theory admits finite deformation; however, derivations of balance laws invoke geometric linearization. A spatial, rather than material, description is adopted for implementation of Finsler concepts.

3.1.1. Geometry and kinematics

The geometric formalism of Section 2.2 of the present work applies to Amari’s theory, albeit with several extensions. The base manifold $m$ is parameterized by charts ${x^{a}}$ , while fiber coordinate charts are ${d^{a}}$ , with $a = 1, 2, 3$ in each case. Total fiber bundle space $z$ has coordinate fields $(x, d)$ , with holonomic bases ${\frac{\partial}{\partial x^{a}}, \frac{\partial}{\partial d^{a}}}$ for $T z$ and ${d x^{a}, d d^{a}}$ for $T^{*} z$ . Amari does not introduce a nonlinear connection or non-holonomic basis vectors, but nothing in the exposition of [63] precludes their existence.

Vector field $d$ is associated physically with the spin direction of the magnetic moment per unit mass of a ferromagnetic crystal. It is assumed in [63] that $d$ is a unit vector over all of $z$ when the physical body $m$ is immersed in an Euclidean vector space $E_{3}$ with Cartesian metric $δ_{ab}$ :

| d |_{E_{3}}^{2} = d^{a} d^{b} δ_{ab} = 1 \Rightarrow δ_{ab} d^{a} \partial_{c} d^{b} = 0, δ_{ab} d^{a} d d^{b} = 0 .

(3.1)

Such an immersion is not possible, in contrast, for the total generalized Finsler space $z$ .

A reference configuration in the form of a generalized Finsler bundle, as in Section 2.1, is not introduced in the theory of [63]. Rather, a locally unloaded or relaxed configuration of the crystal (Figure 1), with base space denoted by $\bar{m}$ , is achieved by altering the lengths and directions of lattice vectors such that all continuum stress fields are degenerate over $\bar{m}$ . The tangent bundle of this “natural” configuration is the target space of the global field mapping $A : T m \to T \bar{m}$ :

A = A_{a}^{α} e_{α} \otimes d x^{a}, A_{a}^{α} = A_{a}^{α} (x, d), (α = 1, 2, 3),

(3.2)

where $e_{α}$ are Cartesian basis vectors in $T \bar{m}$ such that $〈 e_{α}, e_{β} 〉 = δ_{α β}$ . Tangent map $A$ is required to be invertible and at least $C^{2}$ -differentiable with respect to $x$ and $d$ , but it is generally not integrable (hence, anholonomic), because conditions $\partial_{a} A_{b}^{α} = \partial_{b} A_{a}^{α}$ are not essential. Let $d \bar{x} = d {\bar{x}}^{α} e_{α}$ denote a differential line element of the crystal in the locally relaxed state. Then the following Pfaffian relation applies:

d \bar{x} = A d x \leftrightarrow d {\bar{x}}^{α} = A_{a}^{α} d x^{a} .

(3.3)

Figure 1.

Ferromagnetic crystalline solid in spatial configuration (left) and locally relaxed, natural configuration (right), based on Figure 3 of [63]. The spin direction vector $d$ is fixed relative to the crystal lattice within each local material element during the relaxation process, described mathematically by anholonomic tangent map $A$ .

In analogy to the elastic–plastic decomposition of deformation gradient $F = F^{E} F^{P}$ of finite crystalline plasticity theory [5, 19, 25, 26], $A$ is similar to $F^{E - 1}$ .

The spatial metric components $g_{ab} (x, d)$ of metric $g = g_{ab} d x^{a} \otimes d x^{b}$ are defined from

| d \bar{x} |^{2} = 〈 d \bar{x}, d \bar{x} 〉 = A_{a}^{α} δ_{α β} A_{b}^{β} d x^{a} d x^{b} = g_{ab} d x^{a} d x^{b} = 〈 d x, g d x 〉 \Rightarrow g_{ab} = A_{a}^{α} δ_{α β} A_{b}^{β} .

(3.4)

The spatial metric is expanded in (even) powers of $d$ under the restriction $g_{ab} (d^{c}) = g_{ab} (- d^{c})$ , then truncated at order two:

g_{ab} (x, d) = {\bar{g}}_{ab} (x) - \frac{1}{2} m_{abcd} d^{c} d^{d} - \dots \approx {\bar{g}}_{ab} (x) - \frac{1}{2} m_{abcd} d^{c} d^{d},

(3.5)

where the fourth-order tensor of magnetostriction constants is $m_{abcd} = m_{bacd} = m_{abdc} = m_{badc}$ . Note that $g_{ab} (x, d)$ contains terms quadratic in $d$ and, thus, not homogeneous of order zero, so this metric is not strictly Finslerian. However, rescaling of $d$ is not admitted due to constraint (3.1). When equipped with a global Cartesian metric $δ_{ab}$ as in (3.1), the strain state $ϵ$ of the physical body (deformed lattice) is described in [63] by Eulerian field components

ϵ_{ab} (x, d) = \frac{1}{2} [δ_{ab} - g_{ab} (x, d)] = \frac{1}{2} [δ_{ab} - {\bar{g}}_{ab} (x)] + \frac{1}{4} m_{abcd} d^{c} d^{d} = e_{ab} (x) + {\hat{e}}_{ab} (d),

(3.6)

where $e$ is elastic lattice strain and $\hat{e}$ accounts for magnetostriction.

No nonlinear connection $N_{b}^{a}$ or decomposition of $T z$ into horizontal distribution and vertical vector bundle are introduced in [63]. A covariant derivative with two sets of linear connection coefficients is introduced, on the other hand. Parallel transport of a (lattice) vector with respect to this connection is used to describe the structure of the deformed crystal lattice in $z$ , similarly to the lattice structure so obtained from a Weitzenböck connection of non-Riemannian geometry for crystals with dislocations [4, 5, 12]. Let $V^{a} (x, d)$ denote contravariant components of a vector field on $z$ . The covariant differential of $V^{a}$ is the 1-form [63]

\hat{\nabla} V^{a} = (\partial_{b} V^{a} + {\hat{γ}}_{bc}^{a} V^{c}) d x^{b} + ({\bar{\partial}}_{b} V^{a} + {\hat{C}}_{bc}^{a} V^{c}) d d^{b} .

(3.7)

For a generalized Finsler space with distant parallelism, affine connections are prescribed as

{\hat{γ}}_{bc}^{a} (x, d) = (A^{- 1})_{α}^{a} (x, d) \partial_{b} A_{c}^{α} (x, d), {\hat{C}}_{bc}^{a} (x, d) = (A^{- 1})_{α}^{a} (x, d) {\bar{\partial}}_{b} A_{c}^{α} (x, d) .

(3.8)

This connection has non-vanishing torsion tensors, but it is metric compatible with $g_{ab}$ , i.e., $\hat{\nabla} g_{ab} = 0$ . In the most general form of the theory [63], coefficients ${\hat{γ}}_{bc}^{a}$ are left unspecified, admitting non-vanishing curvature with respect to differentiation by $x^{a}$ . A decomposition of torsion tensors into contributions from conventional lattice dislocations (owing to skew components of ${\hat{γ}}_{bc}^{a}$ ) and from a dislocation cloud of ferromagnetism (owing to torsion arising from ${\hat{C}}_{bc}^{a}$ ) is presented in [63].

Other specializations of the geometric theory explored in [63] include reduction to an osculating non-Riemannian space, whereby $d^{a} = d^{a} (x) \Rightarrow d d^{a} = \partial_{b} d^{a} d x^{b}$ , and a multiplicative decomposition $A (x, d) = \hat{A} (d) \bar{A} (x)$ into an elastic term $\bar{A}$ and a magnetostriction term $\hat{A}$ . Regarding the latter, a similar decomposition was used in [78] to split total deformation into finite elastic and thermal parts, albeit in the context of Riemannian rather than Finslerian geometry.

3.1.2. Equilibrium conditions

Balance laws of momentum conservation are derived in [63] under the conditions of small deformations and additive separation of lattice displacement gradient:

β_{b}^{a} (x) = δ_{b}^{a} - {\bar{A}}_{b}^{a} (x), \frac{1}{4} δ^{α b} m_{abcd} d^{c} d^{d} = δ_{a}^{α} - {\hat{A}}_{a}^{α} (d), ω_{ab} (x) = \frac{1}{2} [β_{ab} (x) - β_{ba} (x)],

(3.9)

ϵ_{ab} (x, d) = \frac{1}{2} [β_{ab} (x) + β_{ba} (x)] + \frac{1}{4} m_{abcd} d^{c} d^{d} = e_{ab} (x) + {\hat{e}}_{ab} (d) .

(3.10)

Elastic lattice distortion is $β_{ab}$ , elastic strain is $e_{ab}$ , and lattice rotation is $ω_{ab}$ . A total energy functional $I$ (i.e., the negative of a Lagrangian in the static case) is defined as the integral of total potential energy density $I$ over a fixed, closed, and simply connected region $m' \subset m$ whose volume is contained within the physical space $E_{3}$ , with oriented boundary $\partial m'$ :

I [u, ω, d] = \int_{m'} I (u, e, ω, \partial ω / \partial x, d, \partial d / \partial x) d v, d v = d x^{1} d x^{2} d x^{3} .

(3.11)

Total potential energy density depends on displacement $u$ (to account for conservative body forces), elastic strain, lattice rotation and its gradient, and magnetization spin and the spin gradient. Cartesian coordinates are used, and the traditional volume element of Cartesian space $dv$ does not include dependence upon $\sqrt{g (x, d)}$ as in (2.48).

Euler–Lagrange equations are obtained from a variational principle with independent variations $δ u$ , $δ ω$ , and $δ d$ , in which inelastic contributions to strain and rotation are held fixed, as in [17]:

δ e_{ab} = \frac{1}{2} (\partial_{a} δ u_{b} + \partial_{b} δ u_{a}), δ \partial_{c} ω_{ab} = \partial_{c} δ [\frac{1}{2} (\partial_{a} u_{b} - \partial_{b} u_{a})], δ \partial_{b} d^{a} = \partial_{b} δ (d^{a}) .

(3.12)

The variational principle of [63] is as follows, where $\hat{λ}$ is a Lagrange multiplier used to enforce (3.1):

δ I = δ \int_{m'} (I + \frac{1}{2} \hat{λ} {| d |_{E_{3}}^{2} - 1}) d v = \oint_{\partial m'} 〈 p, δ u 〉 d a,

(3.13)

with $p$ being the mechanical traction vector acting on area element $da (x)$ with externally directed normal $n_{a} (x)$ . Application of the classical divergence theorem in Euclidean space (not the divergence theorem of generalized Finsler space (2.57)) produces, for all $x \in m'$ , a macroscopic balance of linear momentum consisting of three partial differential equations (PDEs),

\partial_{b} (\partial I / \partial e_{ab}) - \partial I / \partial u_{a} = 0,

(3.14)

a macroscopic balance of angular momentum consisting of three independent PDEs because $ω_{ab} = - ω_{ba}$ ,

\partial_{c} (\partial I / \partial (\partial_{c} ω_{ab})) - \partial I / \partial ω_{ab} = 0,

(3.15)

and a microscopic force balance in the spin domain, consisting of three PDEs subject to constraint (3.1),

\partial_{b} (\partial I / \partial (\partial_{b} d^{a})) - \partial I / \partial d^{a} - \hat{λ} δ_{ab} d^{b} = 0 .

(3.16)

Free natural boundary conditions on $\partial m'$ are assumed for angular and spin momentum conservation, whereas traction boundary conditions for linear momentum emerge from the right-hand side of (3.13):

[\partial I / \partial e_{ab}] n_{b} = p^{a}, [\partial I / \partial (\partial_{c} ω_{ab})] n_{c} = 0, [\partial I / \partial (\partial_{b} d^{a})] n_{b} = 0 .

(3.17)

A particular form of energy density $I$ is introduced in [63] with pertinent material coefficients; stress $\partial I / \partial e$ is obviously symmetric, but other details are beyond the present scope. It is notable that generalized Finslerian features of the theory do not affect the balance laws and boundary conditions so derived, because the domain of their application is taken as Euclidean space, rather than generalized Finsler space, and because dependence of the volume element on internal state $d$ is ignored.

In a subsequent refinement of the theory in [63], a different covariant differential and corresponding modified connection coefficients have been suggested [79] such that the Euclidean length of the directors given by $δ_{ab} d^{a} d^{b}$ remains constant under parallel transport. Using a theorem of [80] on a method to derive a metric-compatible connection from non-metric connection, coefficients of the modified connection are obtained from ${\hat{γ}}_{bc}^{a}$ and ${\hat{C}}_{bc}^{a}$ of [63] (or from $γ_{bc}^{a}$ and $C_{bc}^{a}$ of [31]) compatible with $g_{ab}$ but not $δ_{ab}$ [79]. Homogeneity of orders 0 and $- 1$ on $g_{ab}$ and $C_{bc}^{a}$ , respectively, is not required in [79].

3.2. Ikeda’s theory of directors in the mechanics of oriented media

Similarly to the fiber bundle approach in Section 2, denote by $M$ the base manifold of a generalized geometric space, representative of an $n$ -dimensional material body with microstructure, where $n = 3$ in [77]. Let $X \in M$ be a material point, and introduce coordinate chart(s) ${X^{A}} (A = 1, 2, 3)$ to cover the body in its undeformed state. The description of Section 2 is extended in [77] to accommodate up to $p$ vectors at each $X$ , similar to micropolar or micromorphic models [40 –42]. These vectors, called director vectors in [77] though not necessarily of unit length, are herein denoted by $D_{(α)}$ , where $(α = 1, 2, \dots, p)$ . In component form, directors are written ${D_{(α)}^{A}}$ , where only the contravariant index is linked to a basis. In the context of Section 2.1.5, the theory of [77] involves more degrees of freedom when $p > 1$ , no Lagrangian $L$ or fundamental Finsler function $F$ is necessarily introduced, and no restrictions on homogeneity of the metric tensor or connection coefficients exist. Explicit time dependence of fields is not addressed.

Let $x$ denote the spatial location in a deformed body of a point initially at $X$ , and let $d_{(α)}$ denote a deformed director vector. The globally defined motions for the base coordinates and fiber coordinates are the diffeomorphisms $φ : M \to m$ and $θ : Z \to z$ . In local coordinates $(a = 1, 2, 3)$ ,

x^{a} = φ^{a} (X^{A}), d_{(α)}^{a} = θ_{(α)}^{a} [D_{(β)}^{A} (X^{B})] = θ_{(α)}^{a} (X^{B}) .

(3.18)

Components of the deformation gradient $F$ and its inverse $f$ are

F_{A}^{a} = \partial x^{a} / \partial X^{A}, f_{a}^{A} = (F^{- 1})_{a}^{A} = \partial X^{A} / \partial x^{a} .

(3.19)

The following transformations are posited in [77]:

d x^{a} = F_{A}^{a} d X^{A}, d_{(α)}^{a} = B_{A}^{a} D_{(α)}^{A},

(3.20)

with $B_{A}^{a} = B_{A}^{a} (X)$ differentiable and invertible. In [77], the dual basis vectors ${d X^{A}}$ at $X$ are taken to map from $T_{X}^{*} M$ to $T_{x}^{*} m$ similarly to differential line element components $d X^{A}$ . In other words, no distinction is made between $d X^{A}$ and an element of length $d X^{A}$ . This transformation of dual basis vectors is a kind of convected coordinate representation [26, 71].

The external derivative of the second of (3.20) is calculated as [77]

{dd}_{(α)}^{a} = E_{(α) A}^{a} d X^{A} = (B_{B : A}^{a} D_{(α)}^{B} + B_{B}^{a} D_{(α) : A}^{B}) d X^{A},

(3.21)

where $(\cdot)_{: A}$ denotes the total covariant derivative of a two-point tensor [71, 81] with respect to $X^{A}$ , affine connection coefficients $H_{BC}^{A}$ , and affine coefficients $H_{bc}^{a}$ and/or $Γ_{(β) (γ)}^{(α)}$ . Coefficients $H_{BC}^{A}$ and $Γ_{(β) (γ)}^{(α)}$ of linear connections associated with respective continuum and director fields in the reference configuration are related by

H_{BC}^{A} = D_{(α)}^{A} D_{B}^{(β)} D_{C}^{(γ)} Γ_{(β) (γ)}^{(α)} + D_{(α)}^{A} \partial_{B} D_{C}^{(α)}, \nabla_{\partial / \partial X^{B}} \frac{\partial}{\partial X^{C}} = H_{BC}^{A} \frac{\partial}{\partial X^{A}},

(3.22)

with $D_{(α)}^{A} D_{B}^{(α)} = δ_{B}^{A}$ and $D_{(β)}^{A} D_{A}^{(α)} = δ_{β}^{α}$ . Nonlinear connection coefficients $N_{B (α)}^{A}$ are not introduced (i.e., $N_{B}^{A} = 0$ identically), so a horizontal distribution $H Z$ is not formally defined for $T Z$ . The covariant derivative of a referential vector field $V (X, D_{(α)}) = V^{A} (X, D_{(α)}) \frac{\partial}{\partial X^{A}}$ is defined in [77] as the 1-form

\nabla V^{A} = V_{| B}^{A} d X^{B} + V^{A} |_{B}^{(α)} {dD}_{(α)}^{B} = (\partial_{B} V^{A} + H_{BC}^{A} V^{C}) d X^{B} + \frac{\partial V^{A}}{\partial D_{(α)}^{B}} {dD}_{(α)}^{B},

(3.23)

implying $\nabla_{\partial / \partial D_{(α)}^{B}} \frac{\partial}{\partial X^{C}} = Y_{BC}^{A (α)} \frac{\partial}{\partial X^{A}} = 0$ .

The treatment of the spatial configuration in [77], in contrast to the reference configuration, admits a generalized nonlinear connection. Dual basis vectors on $V^{*} z$ are defined as follows:

δ d_{(α)}^{a} = N_{b (α)}^{a} χ_{c}^{b (β)} {dd}_{(β)}^{c} + N_{b (α)}^{a} d x^{b}, (Q^{- 1})_{b (α)}^{a (β)} = N_{c (α)}^{a} χ_{b}^{c (β)} .

(3.24)

The covariant derivative of a spatial vector field of components $V^{a} (x, d_{(α)})$ is defined in [77] as the 1-form

\begin{matrix} \nabla V^{a} = V_{| b}^{a} d x^{b} + V^{a} |_{b}^{(α)} δ d_{(α)}^{b} \\ = (\partial_{b} V^{a} - N_{b (β)}^{c} Q_{c (α)}^{d (β)} \frac{\partial V^{a}}{\partial d_{(α)}^{d}} + H_{bc}^{a} V^{c}) d x^{b} + (\frac{\partial V^{a}}{\partial d_{(β)}^{c}} + Y_{cd}^{a (β)} V^{d}) Q_{b (β)}^{c (α)} δ d_{(α)}^{b} . \end{matrix}

(3.25)

This expression reduces to (2.52) when $p = 1$ and $Q_{b (β)}^{a (α)} = δ_{b}^{a} δ_{(β)}^{(α)}$ .

Let $G_{AB} (X, D_{(α)})$ denote metric tensor components in the horizontal product space of the reference configuration, where $G = G_{AB} d X^{A} \otimes d X^{B}$ . Similarly, the spatial metric tensor is $g (x, d_{(α)}) = g_{ab} (x, d_{(α)}) d x^{a} \otimes d x^{b}$ . Squared arc lengths are defined as $(dS)^{2} = d X^{A} G_{AB} d X^{B}$ and $(ds)^{2} = d x^{a} g_{ab} d x^{b}$ . Then from (3.20), covariant components of deformation tensors $C$ and $c$ follow:

(ds)^{2} = C_{AB} d X^{A} d X^{B}, C_{AB} = F_{A}^{a} g_{ab} F_{B}^{b}; (dS)^{2} = c_{ab} d x^{a} d x^{b}, c_{ab} = f_{a}^{A} G_{AB} f_{b}^{B} .

(3.26)

To complete the geometric description, connection coefficients $(H_{BC}^{A}, H_{bc}^{a}, Y_{bc}^{a (α)}, N_{b (α)}^{a}, χ_{b}^{a (α)})$ need to be supplied along with the metrics $(G_{AB}, g_{ab})$ , while deformations in (3.18) are imposed for kinematics. These connections and metrics are unspecified in [77]. One could logically take $H_{BC}^{A} = Γ_{BC}^{A}$ and $H_{bc}^{a} = Γ_{bc}^{a}$ , but no direct propositions from Section 2 are apparent for those remaining with free index $(α)$ . Ikeda [77] implies that fundamental variables entering a field theory for directed media should include the set $(F, B, E, L, M)$ , where $L$ and $M$ are Euler–Schouten tensors derived from gradients of $F$ and $E$ .

3.3. Bejancu’s theory of geometry and deformations of oriented media

Another rather early application of Finsler geometry towards finite deformation of solid bodies comprises Chapter 8 of the monograph [33]. Fiber bundle descriptions of Sections 2.1 and 2.2 for $Z$ and $z$ , respectively, are used for a continuum body in reference and spatial configurations. The contribution of the present section is presentation of the prescribed relationships of [33] between entities defined on each configuration, notably the mappings $Ξ = (φ, θ)$ leading to convected basis vectors, metric tensors, and connections for $z$ , given the analogous quantities for $Z$ .

Recall from Section 2.1 that coordinate fields on $Z$ are $(X, D)$ , and from Section 2.2, coordinate fields on $z$ are $(x, d)$ . In the theory of [33], motion is dictated by the following diffeomorphisms in local coordinates:

x^{a} = x^{a} (X^{A}), d^{k} = B_{K}^{k} (X) D^{K} D^{K}, (a, A = 1, 2, \dots, n; k, K = 1, 2, \dots, m) .

(3.27)

Components of the deformation gradient and its inverse are, as in (3.19),

F_{A}^{a} = \partial x^{a} / \partial X^{A}, f_{a}^{A} = (F^{- 1})_{a}^{A} = \partial X^{A} / \partial x^{a} .

(3.28)

Holonomic basis vectors ${\frac{\partial}{\partial X^{A}}, \frac{\partial}{\partial D^{K}}}$ , locally defined at each $(X, D) \in Z$ , are assumed in [33] to map from $T Z$ to $T z$ under a change of coordinates in the same way as they would under a change of coordinates $(X, D) \to (\tilde{X}, \tilde{D})$ in the absence of deformation. From (3.27), for a differentiable function $h (x, d) : z \to R$ , similarly to Finsler-type changes of basis vectors in (2.2) and (2.3),

\frac{\partial (h ° Ξ)}{\partial X^{A}} = F_{A}^{a} \frac{\partial h}{\partial x^{a}} + \frac{\partial B_{K}^{k}}{\partial X^{A}} D^{K} \frac{\partial h}{\partial d^{k}}, \frac{\partial (h ° Ξ)}{\partial D^{K}} = B_{K}^{k} \frac{\partial h}{\partial d^{k}} .

(3.29)

Accordingly, “deformed” nonlinear connection coefficients $N_{a}^{k}$ for $z$ are obtained from $N_{A}^{K}$ for $Z$ as follows, leading to the implied transformation law for the deformed, i.e., convected, non-holonomic basis vectors:

N_{a}^{k} F_{A}^{a} = N_{A}^{K} B_{K}^{k} - \frac{\partial B_{K}^{k}}{\partial X^{A}} D^{K}, \frac{δ (h ° Ξ)}{δ X^{A}} = F_{A}^{a} \frac{δ h}{δ x^{a}} .

(3.30)

As derived in [33] under assumptions inherent in (3.27)–(3.30), linear connection coefficients are pushed forward from reference ((2.17) and (2.18)) to current ((2.50) and (2.51)) configurations through deformation $Ξ$ as

H_{cb}^{a} F_{B}^{b} F_{C}^{c} = H_{CB}^{A} F_{A}^{a} - \partial_{B} F_{C}^{a}, K_{ak}^{j} F_{A}^{a} B_{K}^{k} = K_{AK}^{J} B_{J}^{j} - \partial_{A} B_{K}^{j},

(3.31)

V_{kj}^{i} B_{J}^{j} B_{K}^{k} = V_{KJ}^{I} B_{I}^{i}, Y_{aj}^{k} B_{J}^{j} F_{A}^{a} = Y_{AJ}^{K} F_{K}^{k} .

(3.32)

Convected covariant metric tensor components for $g (x, d)$ are obtained from those of $G (X, D)$ (or vice versa) through the respective push-forward (or pull-back) operations arising in [33]:

g_{ab} F_{A}^{a} F_{B}^{b} = G_{AB}, ǧ_{ij} B_{I}^{i} B_{J}^{j} = Ǧ_{IJ} .

(3.33)

When $m = n$ , the treatment of [33] admits, but does not require, $G_{AB} (X, D) = Ǧ_{AB} (X, D)$ as in (2.13), and similarly for the spatial metric tensors. Using the above transformations, a complete set of basis vectors, connection coefficients, and metric tensors can be obtained for $z$ from reference quantities on $Z$ if the deformation functions in (3.27) are prescribed. The treatment of finite deformation of a fiber bundle in Chapter 8 of [33] does not consider balance laws for mechanical forces, nor does it address energy conservation. However, a gauge theory for general physical processes on a fiber bundle, in the absence of deformation, is constructed elsewhere in [33]. Extending the latter construction, presumably a Lagrangian can be posited, and Euler–Lagrange equations can be derived from a variational principle where the action integral is taken over either the reference ( $Z$ ) or deformed ( $z$ ) total space.

4. Contemporary applications in nonlinear mechanics and materials physics

Applications of generalized Finsler geometry to continuum mechanical problems remain scarce. Using a pseudo-Finsler fiber bundle approach similar to that outlined in [33] and Section 3.3, the theory of [82] associates $D$ and $d$ with respective referential and deformed director vectors of an oriented area element that undergoes damage processes such as fracture or void growth. This provides an enriched description over classical theory with damage tensors derived from load-supporting area reduction concepts [83]. In [84], a local balance of linear momentum was derived for a Finsler space using an undisclosed version of Stokes’ theorem that appears to be different from (2.30). Complementary jump conditions across a surface of discontinuity representative of a crack face are presented. Governing equations for a Finsler geometric description of crack propagation were elaborated on in [85]. Generalized Finsler space was used to describe topological singularities arising from non-integrability of micropolar field variables in [86]. However, none of the previously quoted treatments attempted a complete continuum mechanical theory that includes kinematics, necessary balance laws, and constitutive behaviors enabling solutions of boundary value problems, nor do they contain any problem solutions, analytical or numerical.

The first known application of generalized Finsler geometry to calculate a material response in the context of mechanics of solids with microstructure was reported in the monograph [87], and more concisely summarized in [68, 88] with extensions in [89, 90]. In that work, reviewed in detail in Section 4.1, a complete continuum mechanical theory was developed, and the constitutive response was calculated numerically for a polycrystalline bar element undergoing combinations of homogeneous extension and shear. A more recently developed, complete theory of generalized Finsler-geometric continuum mechanics, originally published in [61, 91], contains the first known (semi-)analytical solutions to equilibrium equations for boundary value problems using a model of this class. Further theoretical advancements and solutions to many other physical problems, both analytical and numerical, were reported in [62, 72, 92 –96]. Ideas from this body of work are reviewed in Section 4.2. New example problems in Section 4.2 demonstrate features of model predictions for a material body framed in generalized Finsler space absent for a material body framed in Euclidean space. Several improvements that alleviate inessential assumptions in the original framework [61, 91] are newly advocated.

4.1. Saczuk’s generalized Finsler theory of oriented media

Saczuk et al. [68, 87 –90] invoked Finsler geometry to construct finite-deformation mechanics models of solids with microstructure. Director $D$ , also called an internal state vector [87, 88], is associated with various microscopic physical phenomena. For example, the internal state vector describes a microscopic displacement similar to that of micromorphic models in [87, 89], a dislocation slip system in [88], and an effective Burgers vector of a density of one or more crystal dislocations in [68]. The present review is focused on content of [68], which addresses geometry, kinematics, balance laws, and thermodynamics in a systematic fashion. Minor differences among theoretical features exist within the remaining body of that work [87 –90].

4.1.1. Governing equations

The generalized fiber bundle-type description of reference and current configurations of a material body given in Sections 2 and 3 applies. Recall that $Z$ and $z$ are reference and current total spaces of generalized Finsler base manifolds $M$ and $m$ , respectively. Decompositions of the tangent bundle of each total space into vertical vector bundles and horizontal distributions hold: $T Z = V Z \oplus H Z$ and $T z = V z \oplus H z$ . Though it was remarked in [68] that the general theory admits a Lagrangian function $L (X, D)$ not homogeneous with respect to $D$ , the primary treatment therein invokes a Finlser function $F = \sqrt{2 L}$ , positively homogeneous of degree one in $D$ . Specifically,

F^{2} (X, D) = G_{AB} (X, D) D^{A} D^{B}

(4.1)

supplies the matching Sasaki metric tensor components $G_{AB} = G_{AB} = Ǧ_{AB}$ of (2.13)–(2.38). Particular choices of connection coefficients were made in [68]. Specifically, the Cartan connection was used over $Z$ , where $H_{BC}^{A} = Γ_{BC}^{A}$ for horizontal affine coefficients, $V_{BC}^{A} = C_{BC}^{A}$ for vertical affine coefficients, and $N_{B}^{A} = G_{B}^{A}$ for nonlinear coefficients.

Coordinates of material points $(X, D) \in Z$ move to $(x, d) \in z$ by functions $Ξ (t) = (φ (t), θ (t))$ :

x^{a} = φ^{a} (X^{A}, D^{A}, t), d^{a} = θ^{a} (X^{A}, D^{A}, t), (a, A = 1, 2, 3),

(4.2)

where $t$ is the independent time variable. A total deformation gradient $F (X, D, t) : T_{(X, D)} Z \to T_{(x, d)} z$ is defined as a sum of tangent mappings for horizontal and vertical subspaces:

F = F^{H} + F^{V} = (F^{H})_{A}^{a} \frac{δ}{δ x^{a}} \otimes d X^{A} + (F^{V})_{A}^{a} \frac{\partial}{\partial d^{a}} \otimes δ D^{a} .

(4.3)

In component form, horizontal $(\cdot)^{H}$ and vertical $(\cdot)^{V}$ parts are given by respective horizontal and vertical covariant derivatives of $φ^{a} (X, D)$ , each defined as follows [68, 87, 88]:

(F^{H})_{A}^{a} = φ_{| A}^{a} = δ_{A} x^{a} + Γ_{Ab}^{a} x^{b} = \frac{\partial x^{a}}{\partial X^{A}} - N_{A}^{B} \frac{\partial x^{a}}{\partial D^{B}} + Γ_{AC}^{B} δ_{b}^{C} δ_{B}^{a} x^{b},

(4.4)

(F^{V})_{A}^{a} = φ^{a} |_{A} = {\bar{\partial}}_{A} x^{a} + C_{Ab}^{a} x^{b} = \frac{\partial x^{a}}{\partial D^{A}} + C_{AC}^{B} δ_{B}^{a} δ_{b}^{C} x^{b} .

(4.5)

Mixed-configurational components of linear connections are attained by shifting $Γ_{BC}^{A}$ and $C_{BC}^{A}$ via $δ_{B}^{a}$ and $δ_{b}^{A}$ . Similarly, the nonlinear connection coefficients for $z$ are acquired from reference configuration coefficients by $N_{b}^{a} = δ_{A}^{a} δ_{b}^{B} N_{B}^{A} ° Ξ^{- 1}$ . Total lengths of differential line elements are defined as the sums of inner products

(d S)^{2} = | d X |^{2} + | d D |^{2}, (d s)^{2} = | d x |^{2} + | d d |^{2},

(4.6)

where it is assumed that deformations of line elements obey

d x = d x^{a} \frac{δ}{δ x^{a}} = F^{H} d X, d d = d d^{a} \frac{\partial}{\partial d^{a}} = F^{V} d D .

(4.7)

Lagrangian strain tensors $E^{H} = (E^{H})_{AB} d X^{A} \otimes d X^{B}$ and $E^{V} = (E^{V})_{AB} δ D^{A} \otimes δ D^{B}$ are defined from deformation gradient components as

(E^{H})_{AB} = \frac{1}{2} [(F^{H})_{A}^{a} g_{ab} (F^{H})_{B}^{b} - G_{AB}], (E^{V})_{AB} = \frac{1}{2} [(F^{V})_{A}^{a} ǧ_{ab} (F^{V})_{B}^{b} - Ǧ_{AB}] .

(4.8)

Differences between squared total line lengths follow from (4.6)–(4.8) as

(d s)^{2} - (d S)^{2} = 2 (〈 d X, E^{H} d X 〉 + 〈 d d, E^{V} d d 〉) .

(4.9)

Balance laws were postulated as follows in [68, 87, 88], restricting current attention to the quasi-static, isothermal, and non-dissipative case. A Lagrangian function $I$ is defined as the integral of potential energy density $I$ over a fixed, closed, and simply connected region $Z' \subset Z$ contained within the total fiber bundle space, with oriented boundary $\partial Z'$ :

I [x, X, D] = \int_{Z'} I (x, X, D, F^{H}, F^{V}) d V, d V = \sqrt{G} d X^{1} d X^{2} d X^{3} d D^{1} d D^{2} d D^{3},

(4.10)

with $G (X, D) = G (X, D) \cdot Ǧ (X, D) = G^{2} (X, D)$ [87]. The variation of displacement field $δ (\cdot)$ is assigned horizontal and vertical parts in vector form:

δ x = δ x^{H} + δ x^{V} = δ φ^{a} (\frac{δ}{δ x^{a}} + \frac{\partial}{\partial d^{a}}),

(4.11)

where a similar description of $x$ itself is posited in [87, 88, 90]. The following virtual work principle is then applied, where $p^{H}$ and $p^{V}$ are traction 1-forms conjugate to horizontal and vertical displacement increments on $\partial Z'$ :

δ I = δ \int_{Z'} I d V = \oint_{\partial Z'} (〈 p^{H}, δ x^{H} 〉 + 〈 p^{V}, δ x^{V} 〉) d A,

(4.12)

with $d A$ being an oriented area element of $\partial Z'$ . Local equations of momentum conservation are obtained from (4.12) under the presumptions of $δ X = 0$ , $δ D = 0$ , and availability of a divergence theorem on the fiber bundle space $Z$ , which necessarily differs from that of (2.30) on (a compact region of) base manifold $M$ .

The local macroscopic linear momentum balance is the following, with $P^{H}$ the two-point stress tensor and $f^{H}$ a body force covector [68]:

δ_{A} (P^{H})_{a}^{A} - (P^{H})_{b}^{A} Γ_{Aa}^{b} + (f^{H})_{a} = 0; (P^{H})_{a}^{A} = \partial I / \partial (F^{H})_{A}^{a}, (f^{H})_{a} = - \partial_{a} I .

(4.13)

The double summation on components of $Γ_{Aa}^{b}$ differs notably from the usual form of total covariant derivative of first Piola–Kirchhoff stress in the momentum balance of nonlinear elasticity on a Riemannian manifold [3, 25]. Similarly, the local microscopic linear momentum balance is, with $P^{V}$ the two-point microstress tensor and $f^{V}$ another body force covector,

{\bar{\partial}}_{A} (P^{V})_{a}^{A} - (P^{V})_{b}^{A} C_{Aa}^{b} + (f^{V})_{a} = 0; (P^{V})_{a}^{A} = \partial I / \partial (F^{V})_{A}^{a}, (f^{V})_{a} = - {\bar{\partial}}_{a} I .

(4.14)

Traction boundary conditions on $\partial Z'$ are obtained from (4.12) as

〈 P^{H}, N^{H} 〉 = p^{H}, 〈 P^{V}, N^{V} 〉 = p^{V},

(4.15)

with $N^{H}$ and $N^{V}$ unit normal covectors with components referred to respective horizontal and vertical dual bases on $H^{*} Z$ and $V^{*} Z$ . Angular momentum balances at locations $(X, D) \in Z'$ are stated in [68] as

(F^{H})_{A}^{a} (P^{H})^{bA} = (F^{H})_{A}^{b} (P^{H})^{aA}, (F^{V})_{A}^{a} (P^{V})^{bA} = (F^{V})_{A}^{b} (P^{V})^{aA};

(4.16)

these are perfectly analogous to those of classical continuum mechanics [2, 45], whereby corresponding Cauchy stress(es) are symmetric. Rate dependence, temperature, and dissipation were also discussed in [68, 90]; details are beyond the scope of this review. Furthermore, extensions to address viscoelasticity, damage mechanics, and gradients of $F$ in the energy density were offered in [90]. No boundary value problems are solved using the governing equations derived for these extended theories, however.

4.1.2. Material response problem

In the earliest known application of Finsler geometry to predict a material response in the context of mechanics of solids with microstructure, Saczuk [87] and Stumpf and Saczuk [68] used the quasi-static theory reviewed in Section 4.1.1 to study deformation of a material element representative of a bar loaded in tension and/or shear, with $D^{A}$ specifying a preferred material direction of (the Burgers vector for) slip. In the absence of body forces, the Lagrangian density $I \to ψ$ , with $ψ$ the free energy density per unit reference volume, is chosen of the following form with resulting stress components:

ψ = ψ (E^{H}, E^{V}); (P^{H})^{aA} = (F^{H})_{B}^{a} \frac{\partial ψ}{\partial {(E^{H})}_{AB}}, (P^{V})^{aA} = (F^{V})_{B}^{a} \frac{\partial ψ}{\partial {(E^{V})}_{AB}} .

(4.17)

More specifically, $ψ$ is taken as a quadratic form in each of $E^{H}$ and $E^{V}$ , with isotropic elastic response for both dependencies, where stiffness moduli depend only on the classical Lamé parameters $μ$ and $λ$ . The metric components $G_{AB} (X, D)$ entering Finsler function $F$ of (4.1) are prescribed as the following homogeneous functions of degree zero in $D$ :

G_{11} = \frac{2 X^{1} D^{1}}{{[(D^{1})^{2} + {(D^{2})}^{2}]}^{1 / 2}}, G_{22} = 2 X^{2}, G_{33} = 2 X^{3}; G_{AB} = 0 \forall A \neq B .

(4.18)

Heuristic arguments were given in [68, 87] that associate $L = \frac{1}{2} F^{2}$ with the stored energy of dislocations, but formal derivations in this context are absent. A way of connecting $G_{AB}$ to dislocation density connections of [7] was posited in [87] but is not used in calculations.

A single material element corresponding to a bar of length $L$ is assigned the following homogeneous (i.e., linear with respect to $(X, D)$ ) motion function [68, 87]:

x^{1} (X, D, t) = a (t) X^{1} + D^{1} + γ (t) D^{2}, x^{2} (X, D, t) = X^{2} + D^{2}, x^{3} = X^{3} + D^{3},

(4.19)

where $a$ and $γ$ are scalar functions of $t$ , with the latter simply a load parameter. Coordinates $D^{A}$ appear as initial conditions that dictate a preferred slip direction, presumably dependent on $X$ but fixed in $t$ . Prescription of the director deformation $θ (X, D, t)$ of (4.2) is neither given nor seems necessary to solve the problem. The spatial metric components $g_{ab} (x, d)$ are also unstated; presumably $g_{ab} = δ_{ab}$ . Different choices of $a$ and $γ$ are used to imposed different loading programs, with $t$ increased incrementally.

At each increment, the metric tensor, deformation gradient, linear and nonlinear connection coefficients, strain measures, and stresses are calculated. The stress fields so obtained should satisfy (4.13)–(4.16). Whether this consistency is simply assumed, or if the governing PDEs with boundary conditions are solved numerically, is unclear from the text [68, 87].

Computed results show features qualitatively similar to those of polycrystalline metals undergoing strain softening. Even though the free energy density $ψ$ is analogous to that of isotropic elasticity, path dependence and anisotropic response depend on orientation of the microstructure with respect to loading direction. Results that would not arise for a purely elastic analysis are attributed to the choice of generalized Finsler-type motion (4.19), initial orientation field $D (X)$ , and Finsler metric (4.18). Such results are claimed noteworthy [68, 87] because physically realistic behavior can be predicted without recourse to a bifurcation analysis, and without use of yield and flow functions of phenomenological plasticity.

4.2. Finsler-geometric continuum mechanics

The present author used concepts from generalized Finsler geometry to construct a complete variational theory for nonlinear elastic bodies with microstructure. The original theory [61, 91] accounts for finite deformations under conditions of static equilibrium for forces conjugate to material particle motion and state vector evolution. Time dependence does enter this theory, which is reviewed in Sections 4.2.1–4.2.3, with several demonstrative problems solved in Section 4.2.4. Extensions to explicit time dependence, dynamics, and dissipative processes are noted in Section 4.2.5.

4.2.1. Motions and deformations

Particle motion $φ : M \to m$ and its inverse $Φ : m \to M$ are respectively described by the following one-to-one and $C^{3}$ -differentiable functions:

x^{a} = φ^{a} (X), X^{A} = Φ^{A} (x), (a, A = 1, 2, 3),

(4.20)

with $(Φ ° φ) (X) = X$ . Recall that total motion is $Ξ : Z \to z$ , where $Ξ = (φ, θ)$ . Vector field $D$ and its spatial counterpart $d$ are referred to as internal state vector fields or director vector fields, but neither vector need be of unit length. These are assigned physical interpretations pertinent to the specific class of mechanics problem under consideration, to be discussed in Sections 4.2.4 and 4.2.5. Motions of director vectors are defined as the $C^{3}$ functions

d^{a} = θ^{a} (X, D), D^{A} = Θ^{A} (x, d), (a, A = 1, 2, 3) .

(4.21)

Note that the fiber dimensions are chosen as $m = \dim U = \dim u = \dim m = \dim M = n$ . Extension of the theory to allow for $m \neq n$ is conceivable, as in [33, 77]. However, setting $m = n$ enables a more transparent physical interpretation of the vertical vector bundle, and it allows use of (2.10) and (2.43), among other relations in Section 2, that streamline tensor calculations and the index notation.

From (4.20) and (4.21), the following transformation formulae apply for partial differentiation operations between configurations of a differentiable function $h (x, d) : z \to R$ :

\frac{\partial (h ° Ξ)}{\partial X^{A}} = \frac{\partial h}{\partial x^{a}} \frac{\partial φ^{a}}{\partial X^{A}} + \frac{\partial h}{\partial d^{a}} \frac{\partial θ^{a}}{\partial X^{A}}, \frac{\partial (h ° Ξ)}{\partial D^{A}} = \frac{\partial h}{\partial d^{a}} \frac{\partial θ^{a}}{\partial D^{A}} .

(4.22)

However, unlike the theory of [33] reviewed in Section 3.3, basis vectors need not convect from $T Z$ to $T z$ with the motion $Ξ$ . Rather, as in classical continuum field theories of mechanics [2, 45, 81], basis vectors, as well as metric tensors and connection coefficients, can be assigned independently for configurations $Z$ and $z$ . In other words, $(\frac{δ}{δ x^{a}}, \frac{\partial}{\partial d^{a}}, g_{ab}, H_{bc}^{a}, K_{bc}^{a}, V_{bc}^{a}, Y_{bc}^{a}, N_{b}^{a})$ need not be obtained from $(\frac{δ}{δ X^{A}}, \frac{\partial}{\partial D^{A}}, G_{AB}, H_{BC}^{A}, K_{BC}^{A}, V_{BC}^{A}, Y_{BC}^{A}, N_{B}^{A})$ using push-forward operations of $Ξ$ . However, choosing $N_{b}^{a}$ as the push-forward of $N_{B}^{A}$ as in (3.30) is beneficial because

N_{a}^{b} \frac{\partial φ^{a}}{\partial X^{A}} = N_{A}^{B} \frac{\partial θ^{b}}{\partial D^{B}} - \frac{\partial θ^{b}}{\partial X^{A}} \Rightarrow \frac{δ (h ° Ξ)}{δ X^{A}} = \frac{δ h}{δ x^{a}} \frac{\partial φ^{a}}{\partial X^{A}} = \frac{δ h}{δ x^{a}} \frac{δ φ^{a}}{δ X^{A}} = \frac{δ h}{δ x^{a}} F_{A}^{a},

(4.23)

meaning $δ_{A} (\cdot) = F_{A}^{a} δ_{a} (\cdot)$ simply relates the delta derivative across configurations.

As implied in (4.23), the deformation gradient field $F : H Z \to H z$ is defined as the two-point tensor field

F = \frac{δ φ}{δ X} = \frac{δ φ^{a}}{δ X^{A}} \frac{δ}{δ x^{a}} \otimes d X^{A} = \frac{\partial φ^{a}}{\partial X^{A}} \frac{δ}{δ x^{a}} \otimes d X^{A} .

(4.24)

The inverse deformation gradient $f : H z \to H Z$ is defined as follows, noting $F_{A}^{a} (X) f_{b}^{A} (x (X)) = δ_{b}^{a}$ and $F_{A}^{a} (X) f_{a}^{B} (x (X)) = δ_{A}^{B}$ :

f = \frac{δ Φ}{δ x} = \frac{δ Φ^{A}}{δ x^{a}} \frac{δ}{δ X^{A}} \otimes d x^{a} = \frac{\partial Φ^{A}}{\partial x^{a}} \frac{δ}{δ X^{A}} \otimes d x^{a} .

(4.25)

Usual stipulations on regularity [3] of motions (4.20) apply such that $det (F_{A}^{a}) > 0$ and $det (f_{a}^{A}) > 0$ .

Transformation equations relating differential line elements of (2.14) and (2.47) follow as

d x = d x^{a} \frac{δ}{δ x^{a}} = F_{A}^{a} d X^{A} \frac{δ}{δ X^{A}} = F d X, d X = d X^{A} \frac{δ}{δ X^{A}} = f_{a}^{A} d x^{a} \frac{δ}{δ x^{a}} = f d x .

(4.26)

Applying (4.26), definitions of the determinant, and (2.15) and (2.48), volume elements and volume forms, respectively, transform between reference and spatial coordinate systems on $M$ and $m$ as

d v = J d V = [det (F_{A}^{a}) \sqrt{g / G}] d V, d V = j d v = [det (f_{a}^{A}) \sqrt{G / g}] d v,

(4.27)

φ^{*} d ω = Jd Ω, Φ^{*} d Ω = jd ω .

(4.28)

Lengths of deformed and initial line elements can be compared using the Lagrangian deformation tensor $C = C_{AB} d X^{A} \otimes d X^{B}$ :

| d x |^{2} = F_{A}^{a} g_{ab} F_{B}^{b} d X^{A} d X^{B} = C_{AB} d X^{A} d X^{B} = 〈 d X, C d X 〉 .

(4.29)

It follows that $det (C_{AB}) = J^{2} G$ . Then from the first of (2.50) and (4.23) [62, 91],

\nabla_{δ / δ X^{A}} \frac{δ}{δ x^{c}} = \frac{δ x^{a}}{δ X^{A}} \nabla_{δ / δ x^{a}} \frac{δ}{δ x^{c}} = δ_{A} φ^{a} H_{ac}^{b} \frac{δ}{δ x^{b}} = F_{A}^{a} H_{ac}^{b} \frac{δ}{δ x^{b}} .

(4.30)

A similar procedure with the second of (2.50) yields $\nabla_{δ / δ X^{A}} \frac{\partial}{\partial d^{c}} = F_{A}^{a} K_{ac}^{b} \frac{\partial}{\partial d^{b}}$ . An equivalent result to (4.30) was proven in Theorem 2 of [61], albeit using a different definition of $F_{A}^{a}$ . Conditions (4.23), not stated explicitly in [61, 62, 91], should be imposed for arbitrary $H_{bc}^{a}$ and arbitrary motions consistent with the forms in (4.20) and (4.21).

4.2.2. Pragmatic assumptions

Stokes’ theorem 1 and (2.30), which extends the divergence theorem of Rund [39] to the base manifold $M$ of a generalized Finsler fiber bundle space $Z$ , is used to derive Euler–Lagrange equations for equilibrium of stress and state vector fields, as shown in Section 4.2.3. Its application requires existence of functional relations

D^{A} = D^{A} (X), d^{a} = d^{a} (x),

(4.31)

where the second of (4.31) is implied by the first under a consistent change of variables. Existence of the following functional forms emerges from (4.20), (4.21), and (4.31):

d^{a} = θ^{a} (X, D) = {\hat{θ}}^{a} (X, D (X)) = {\bar{θ}}^{a} (X), D^{A} = Θ^{A} (x, d) = {\hat{Θ}}^{A} (x, d (x)) = {\bar{Θ}}^{A} (x) .

(4.32)

Remark 4.1. In several prior works, alternative representations of particle motions incorporating state vector fields as arguments have been posited in conjunction with (4.31):

x^{a} = {\hat{φ}}^{a} (X, D (X)) = φ^{a} (X), X^{A} = {\hat{Φ}}^{A} (x, d (x)) = Φ^{A} (x) .

These functional forms are not inconsistent with (4.20): if functions (4.31) are known, then $φ$ and $Φ$ can be determined uniquely from $\hat{φ}$ and $\hat{Φ}$ . However, (4.20) and (4.31) are not alone sufficient to uniquely determine $\hat{φ}$ and $\hat{Φ}$ . From functional forms of motion immediately above, different definitions of $F_{A}^{a}$ have been analyzed, including $δ_{A} {\hat{φ}}^{a}$ [91] and $\partial_{A} {\hat{φ}}^{a}$ [61]. Definition $F_{A}^{a} = δ_{A} φ^{a} = \partial_{A} x^{a}$ is consistent with the proposition in [62], as well as that in [33], and it is recommended following (4.23) and (4.30).

The present theory, similar to those in [63, 68, 87], does not require that $θ$ or $Θ$ be specified explicitly. The simplest canonical choice for $θ (X, D)$ , given the field $D^{A} (X)$ , is [62]

d = D ° Φ \Leftrightarrow d (x) = D (Φ (x)) \Rightarrow θ^{a} (D (X)) = D^{A} (X) 〈 δ d^{a}, \frac{\partial}{\partial D^{A}} 〉 = D^{A} (X) δ_{A}^{a},

(4.33)

where $δ_{A}^{a}$ is a shifter between $V z$ and $V Z$ . In this case, $\partial_{A} θ^{a} (X, D (X)) = 0$ and ${\bar{\partial}}_{A} θ^{a} (X, D (X)) = δ_{A}^{a}$ , such that (4.23) reduces to $N_{b}^{a} = N_{B}^{A} f_{b}^{B} δ_{A}^{a}$ , and conveniently, $N_{B}^{A} = 0 \Leftrightarrow N_{b}^{a} = 0$ .

Another requirement for use of (2.30) for arbitrary admissible $G_{AB} (X, D)$ , noted in Theorem 1, is choice of symmetric linear connection horizontally compatible with $G_{AB}$ , meaning $H_{BC}^{A} = Γ_{BC}^{A}$ , where $Γ_{BC}^{A}$ are the Chern–Rund–Cartan coefficients of (2.24). The simplest admissible choice of vertical linear connection coefficients is vanishing of $V_{BC}^{A}$ , which corresponds to the Chern–Rund connection [28, 29, 32]. The canonical choice of $N_{B}^{A}$ is given by $G_{B}^{A}$ of (2.28), which also corresponds to the Chern–Rund connection. The prescriptions $K_{BC}^{A} = H_{BC}^{A}$ and $Y_{BC}^{A} = V_{BC}^{A}$ are logical given (2.10), but are not essential. The former is used for applications in Section 4.2.4, whereas the alternative choice $K_{BC}^{A} = 0$ , providing compatibility with a Cartesian metric $δ_{AB}$ , is useful for the application in Section 5. Thus, given a referential Sasaki metric $G$ of (2.12) with components $G_{AB}$ of (2.13), recommended linear and nonlinear connection coefficients over $Z$ are

H_{BC}^{A} = Γ_{BC}^{A}, K_{BC}^{A} = Γ_{BC}^{A} or K_{BC}^{A} = 0, V_{BC}^{A} = Y_{BC}^{A} = 0, N_{B}^{A} = G_{B}^{A} .

(4.34)

If metric $G_{AB} (X, D)$ is not positively homogeneous of degree zero with respect to $D$ , then (2.8) is not necessarily ensured for arbitrary changes of coordinates, so transformation properties of $N_{B}^{A}$ should be checked for consistency. Complementary affine connection coefficients are consistently recommended for $z$ following from Sasaki metric $g$ with components $g_{ab} (x, d)$ of (2.46):

H_{bc}^{a} = Γ_{bc}^{a}, K_{bc}^{a} = Γ_{bc}^{a} or K_{bc}^{a} = 0, V_{bc}^{a} = Y_{bc}^{a} = 0, N_{b}^{a} = G_{B}^{A} f_{b}^{B} δ_{A}^{a},

(4.35)

while the preferred choice of nonlinear connection $N_{b}^{a}$ presumes (4.33) is invoked along with (4.23) and the last of (4.34).

Once the field of referential generalized Finsler metric components $G_{AB} (X, D)$ is prescribed, then all connection coefficients in (4.34) can be calculated from definitions in Section 2.1. The metric $G_{AB}$ need not be homogeneous of degree zero with respect to $D$ , but it can be. Furthermore, components $G_{AB}$ need not be derived, as in Section 2.1.5, from a Lagrangian function $L$ or more specifically a fundamental Finsler function $F$ , but they can be. Dependence $G$ on $X$ and $D$ should be chosen based upon the symmetry and physics pertinent to the particular problem of study.

The following decomposition of $G_{AB}$ into a Riemannian part ${\bar{G}}_{AC}$ and a director-dependent part ${\hat{G}}_{B}^{C}$ has been proposed and used often for solving boundary value problems in prior work [61, 62, 91, 94]:

G_{AB} (X, D) = {\bar{G}}_{AC} (X) {\hat{G}}_{B}^{C} (X, D) .

(4.36)

It proves convenient to choose ${\bar{G}}_{AC}$ that best represents the symmetry of the physical body under consideration; in elasticity theory, this is most often a Riemannian metric for rectilinear, cylindrical, or spherical coordinates in base space $M$ . Then ${\hat{G}}_{B}^{C}$ is assigned based on how microstructure, as represented by $D$ , affects measured lengths of material elements with respect to an observer in total space $Z$ , i.e., the space of the physical body enriched with microstructure geometry. For example, if a variation of $D$ is associated with a microstructure change that causes isotropic expansion or contraction (e.g., cavitation [61] or densification under a phase transformation [92]), then a spherical form ${\hat{G}}_{B}^{C} (X, D) = δ_{B}^{C} β (X, D)$ , where $β \in (0, \infty)$ is a scalar function, is in order. An exponential form of $β (X, D) = \exp [2 σ (X, D)]$ provides conformal Weyl-type rescaling of the Riemannian metric ${\bar{G}}_{AC}$ . This type of rescaling, analyzed for generalized Finsler space in [97], has been applied to nonlinear elastic problems in [61, 92] wherein $σ (X, D) = \tilde{σ} (D (X))$ , and elsewhere applied to a gyroscope problem in gravitation [98].

Statements of the prior paragraph for $G_{AB} (X, D)$ apply analogously to the spatial metric tensor $g$ of (2.45), with components $g_{ab} (x, d)$ , with $X$ replaced by $x$ , and $D$ by $d$ , where linear connection coefficients of (4.35) are found from definitions in Section 2.2. Following from (4.31), $G_{AB} (X, D) = G_{AB} (X, D (X)) = {\tilde{G}}_{AB} (X)$ , where the ${\tilde{G}}_{AB}$ comprise an osculating Riemannian metric [28].

4.2.3. Energy functional and static equilibrium equations

A variational principle is invoked, where $Ψ$ is the total energy functional for a compact domain $M' \subset M$ with positively oriented boundary $\partial M'$ , and $ψ$ is the local energy density per unit reference volume:

δ Ψ [φ, D] = \oint_{\partial M'} (〈 p, δ x 〉 + 〈 z, δ D 〉) Ω, Ψ [φ, D] = \int_{M'} ψ (F_{A}^{a}, D^{A}, D_{| B}^{A}) d Ω .

(4.37)

Surface forces are $p = p_{a} d x^{a}$ , a mechanical load vector (force per unit reference area), and $z = z_{A} δ D^{A}$ , a thermodynamic force conjugate to the internal state vector. In coordinates, this becomes

δ \int_{M'} ψ d Ω = \oint_{\partial M'} [p_{a} δ x^{a} + z_{A} δ (D^{A})] Ω,

(4.38)

where the variation of $D$ is enclosed in parentheses to avoid confusion with a non-holonomic basis vector $δ D^{A}$ .

Independent variables entering $ψ$ are the deformation gradient, the internal state vector, and the horizontal gradient of the internal state vector. Dependence on $F$ accounts for bulk elastic strain energy. Dependence on $D$ generally accounts for effects of microstructure features on stored energy. Dependence on the gradient of internal state $D_{| B}^{A} = \partial_{B} D^{A} + K_{BC}^{A} D^{C}$ accounts for heterogeneity of microstructure. Further motivation is obtained from micropolar and micromorphic theories [40], generalized continuum theories of media with microstructure [99], and phase field theory [93, 100], whereby $D$ can be treated like a vector-valued order parameter [60]. The state vector gradient dependence should enact a regularizing effect on solutions, which may simplify (e.g., numerical) computations relative to those of sharp-interface type theories. Vertical gradient $D^{A} |_{B} = δ_{B}^{A}$ with $V_{BC}^{A} = 0$ according to (4.34) provides no constitutive information so is absent in arguments of $ψ$ . Explicit dependence of $ψ$ on $X$ is omitted; this could be added for materials with heterogeneous properties without affecting forthcoming governing equations.

Expansion of the integrand on the left-hand side of (4.38) is written

δ ψ = \frac{\partial ψ}{\partial F_{A}^{a}} δ F_{A}^{a} + \frac{\partial ψ}{\partial D^{A}} δ (D^{A}) + \frac{\partial ψ}{\partial D_{| B}^{A}} δ D_{| B}^{A} = P_{a}^{A} δ F_{A}^{a} + Q_{A} δ (D^{A}) + Z_{A}^{B} δ D_{| B}^{A} .

(4.39)

Here $P$ is mechanical stress, $Q$ is an internal force conjugate to $D$ , and $Z$ is micro-stress conjugate to the horizontal gradient of $D$ . The following calculations are noted, where variation $δ (\cdot)$ is performed with $X$ fixed but $D$ variable:

δ F_{A}^{a} = δ_{A} (δ φ^{a}), δ D_{| B}^{A} = [δ (D^{A})]_{| B} - ({\bar{\partial}}_{C} N_{B}^{A} - {\bar{\partial}}_{C} K_{BD}^{A} D^{D}) δ (D^{C}),

(4.40)

δ (d Ω) = \frac{1}{2} G^{AB} {\bar{\partial}}_{C} G_{AB} δ (D^{C}) d Ω = C_{CA}^{A} δ (D^{C}) d Ω .

(4.41)

Choices of connection coefficients in (4.34) and (4.35) are invoked, noting $K_{BC}^{A} = Γ_{BC}^{A}$ particularly herein, along with (4.20), (4.30), and (4.31). Then insertion of (4.39) into (4.38), followed by repeated integration by parts and use of (2.30) of Theorem 1 produces the Euler–Lagrange equations that are consistent with any admissible variations $δ φ$ and $δ D$ locally at each $X \in M'$ , as well as natural boundary conditions on $\partial M'$ .

The culminating Euler–Lagrange equations consist of the macroscopic balance of linear momentum:

\partial_{A} P_{a}^{A} + {\bar{\partial}}_{B} P_{a}^{A} \partial_{A} D^{B} + P_{a}^{B} Γ_{AB}^{A} - P_{c}^{A} Γ_{ba}^{c} F_{A}^{b} = - P_{a}^{A} C_{BC}^{C} (\partial_{A} D^{B} + N_{A}^{B})

(4.42)

and the balance of micro-momentum (i.e., director momentum or internal state equilibrium):

\begin{matrix} \partial_{A} Z_{C}^{A} + {\bar{\partial}}_{B} Z_{C}^{A} \partial_{A} D^{B} + Z_{C}^{B} Γ_{AB}^{A} - Z_{B}^{A} Γ_{AC}^{B} - Q_{C} \\ = ψ C_{CA}^{A} - Z_{A}^{B} [{\bar{\partial}}_{C} N_{B}^{A} - {\bar{\partial}}_{C} Γ_{BD}^{A} D^{D} + δ_{C}^{A} C_{ED}^{D} (\partial_{B} D^{E} + N_{B}^{E})] . \end{matrix}

(4.43)

The natural boundary conditions derived on $\partial M'$ are

p_{a} = P_{a}^{A} N_{A}, z_{A} = Z_{A}^{B} N_{B} .

(4.44)

Imposition of invariance under rigid rotations of spatial coordinate frames leads to the restricted form of energy density:

ψ = ψ [C (F, g), D, \nabla D] = ψ (C_{AB}, D^{A}, D_{| B}^{A}),

(4.45)

from which the first Piola–Kirchhoff stress $P_{a}^{A}$ and Cauchy stress $σ^{ab}$ obey the symmetry rules consistent with the classical local balance of angular momentum [2, 45]:

P_{a}^{A} = 2 g_{ab} F_{B}^{b} \frac{\partial ψ}{\partial C_{AB}}, σ^{ab} = j g^{ac} P_{c}^{A} F_{A}^{b} = 2 {jF}_{A}^{a} F_{B}^{b} \frac{\partial ψ}{\partial C_{AB}} = σ^{ba} .

(4.46)

Given natural boundary conditions (4.44) and/or essential boundary conditions (prescribed $φ (X), D (X)$ for $X \in \partial M'$ ), (4.42) and (4.43) comprise six coupled nonlinear PDEs in six degrees of freedom $x^{a} (X)$ and $D^{A} (X)$ at any point $X \in M'$ , and by extension, for any $X \in M$ .

Remark 4.2. When a Riemannian metric is used (i.e., no $D$ -dependence of $G$ ), then $Γ_{BC}^{A} = γ_{BC}^{A}$ , $Γ_{bc}^{a} = γ_{bc}^{a}$ , $N_{B}^{A} = 0$ , and $C_{BC}^{A} = 0$ . The right-hand sides of (4.42) and (4.43) vanish accordingly. Equation (4.42) becomes the static linear momentum balance of classical continuum mechanics in the absence of body forces, and (4.43) the static equilibrium equation for a gradient material as in phase field mechanics.

Remark 4.3. In the original derivation of (4.42)–(4.44) reported in Theorem 3 of [61], the following differences are noted. First, $G_{AB} (X, D)$ was included as an argument of $ψ$ , such that effects of $D$ -dependence of the metric manifested in a distinct thermodynamic force, rather than entering implicitly in $Q_{A}$ as derived herein. Second, (4.34) and (4.35) were not imposed explicitly in [61], though $K_{BC}^{A} = H_{BC}^{A}$ was assumed for expansion of $Z_{B | A}^{A}$ for physical consistency with reduction of the theory to Riemannian geometry. Finally, motion of the form $\hat{φ}$ in (4.33) was used rather than the simpler (4.20) advocated in [62], leading in [61] to a different representation of $F_{A}^{a}$ and additional terms from ${\bar{\partial}}_{A} {\hat{φ}}^{a}$ .

4.2.4. Simple example problems

One-dimensional (1D) problems on base manifold $M$ demonstrate utility of the generalized Finsler theory and differences from standard continuum approaches in Riemannian (and, more specifically, Euclidean) geometry. A nonlinear elastic bar of length $L_{0}$ is the material manifold ${M : X \in [0, L_{0}]}$ , where $X = X^{1}$ . The state vector consists only of non-vanishing component $D (X) = D^{1} (X^{1})$ . Orthogonal coordinates $X^{2}, X^{3}, D^{2}, D^{3}$ do not enter the description.

Herein, $D (X)$ represents a microscopic motion of elements of microstructure, distinct from $φ (X)$ . Specifically considered in what follows are problems involving tensile fracture, whereby $D$ is associated with a crack opening displacement, and phase transformation, where $D$ is associated with atomic (inner) displacement in a crystal lattice, namely, a change in lattice parameter. For both classes of problems, i.e., fracture and phase changes, the geometry, kinematics, and reduced governing equations are essentially the same. Differences among energy functions and material parameters produce different results.

The formulation of Sections 4.2.2 and 4.2.3 reduces as follows for the 1D case. Spatial coordinates are $x = x^{1}$ and $d = d^{1}$ . Motions and generalized Finsler metrics are

x = φ (X), d (x (X)) = D (X); G = G_{11} (D), g = g_{11} (d); g (d (x (X))) = G (D (X)) .

(4.47)

Noting that $G$ and $g$ do not depend explicitly on $X$ or $x$ , spaces $Z$ and $z$ are locally Minkowskian. From (4.47) with (2.22), (2.24), (2.28), (2.53), (2.55), and (4.35), the Levi-Civita, Chern–Rund, and nonlinear connection coefficients all vanish identically in both configurations:

γ_{BC}^{A} = Γ_{BC}^{A} = 0, N_{B}^{A} = G_{B}^{A} = 0; γ_{bc}^{a} = Γ_{bc}^{a} = 0, N_{b}^{a} = δ_{A}^{a} N_{B}^{A} f_{b}^{B} = 0 .

(4.48)

However, the Cartan tensor $C_{BC}^{A} (X, D)$ of (2.23) and $C_{bc}^{a} (x, d)$ of (2.54) have potentially non-vanishing component $C_{11}^{1}$ :

C_{11}^{1} (D) = \frac{1}{2 G (D)} \frac{d G (D)}{d D} = χ (D), C_{11}^{1} (d) = \frac{1}{2 g (d)} \frac{d g (d)}{d x} = χ (d) .

(4.49)

The pertinent component of deformation gradient and horizontal director gradient are, respectively,

F (X) = F_{1}^{1} (X) = \frac{d φ (X)}{d X}, D' (X) = D_{| 1}^{1} = \frac{d D (X)}{d X} .

(4.50)

The deformation tensor has axial component $C_{1}^{1} = F_{1}^{1} F_{1}^{1} G^{11} g_{11} = (F)^{2}$ . For convenience, assume $D$ , like $X$ , has a physical dimension of length. A dimensionless representation of internal state, and its corresponding gradient, are defined as follows, where $l$ is a constant regularization length:

ξ (X) = D (X) / l, ξ' (X) = d ξ (X) / d X = D' (X) / l .

(4.51)

The value of $l$ is selected based on physics of the problem and the material, to be discussed later. With $l$ suitably chosen, variable $ξ$ is interpreted similarly to an order parameter of phase field theory.

Energy density entering (4.37) and (4.45) is specialized as follows, noting $F = (C_{1}^{1})^{1 / 2}$ , $D = l ξ$ , and $\nabla D = l ξ'$ :

ψ (F, ξ, ξ') = W (F, ξ) + A [f_{0} (ξ) + (l ξ')^{2}] .

(4.52)

Function $W$ is elastic strain energy, function $f_{0}$ is normalized (dimensionless) phase energy, and $A$ is a material constant with units of energy density. Dependence on $ξ'$ is quadratic, the usual assumption in phase field mechanics.

Macroscopic and microscopic stresses are

P = \frac{\partial W}{\partial F}, Q = \frac{1}{l} (\frac{\partial W}{\partial ξ} + A \frac{d f_{0}}{d ξ}), Z = 2 Al ξ' .

(4.53)

Macroscopic linear momentum balance (4.42) reduces to

\frac{d P (X, D)}{d X} + P (X, D) \cdot χ (D) \cdot \frac{d D}{d X} = 0,

(4.54)

where $d P (X, D (X)) / dX = \partial P (X, D) / \partial X + (\partial P (X, D) / \partial D) (\partial D / \partial X)$ . Substituting from (4.49), the general analytical solution of momentum balance (4.54) is derived as

P (D (X)) = P_{0} \sqrt{G_{0} / G (D (X))}; P_{0} = P (0), G_{0} = G (0) .

(4.55)

Here, $P_{0}$ is the constant stress and $G_{0}$ the constant metric corresponding to $D = 0$ . For a Riemannian metric, $G = G_{0}$ so $P = P_{0} = constant$ , i.e., axial stress per unit reference cross-sectional area is spatially constant for quasi-static tension or compression of a homogeneous 1D bar. In contrast, for a generalized Finslerian metric $G (D)$ , $P$ can vary with $X$ in a quasi-static 1D problem if $D$ varies with $X$ . Microscopic momentum balance (4.43) reduces to

\frac{d Z (X, D)}{d X} - Q (X, D) = χ (D) [ψ (X, D) - Z (X, D) \frac{d D}{d X}] .

(4.56)

Let $E = λ + 2 μ$ be the longitudinal elastic modulus, where $λ, μ$ are the Lamé parameters. Define the dimensionless ratio $α = \frac{E}{2 A}$ , and define $ϒ = A \cdot l$ as the surface energy. Let $\hat{χ} = χ / l = \frac{1}{2 G} \frac{d G}{d ξ}$ . Take

W (F, ξ) = \frac{1}{2} E w (F, ξ) = α Aw (F, ξ),

(4.57)

where $w$ is a dimensionless strain energy density. Then (4.56) becomes, in dimensionless form,

l^{2} ξ ″ - \frac{1}{2} [α \frac{\partial w}{\partial ξ} + \frac{d f_{0}}{d ξ}] = \frac{\hat{χ}}{2} [α w + f_{0} - {(l ξ')}^{2}],

(4.58)

where $ξ ″ = d^{2} ξ / d X^{2}$ . Different dimensionless functions $w (F, ξ)$ , $f_{0} (ξ)$ , and $G (ξ)$ then enable study of different physical problems, with proper boundary conditions for (4.54) and (4.58), in the 1D case.

Fracture. First consider a tensile fracture problem. Let $D = l \Leftrightarrow ξ = 1$ represent the condition when the crack opening displacement attains a critical magnitude such that the material supports no load. A logarithmic nonlinear elastic potential $w$ [101, 102] is used, with quadratic degradation due to increasing $ξ$ , and a quadratic form of $f_{0}$ is chosen. The latter two prescriptions are standard among phase field theories of fracture [103, 104]. A conformal Weyl-type transformation [94, 97, 105] is chosen for the dependence of $G$ on $ξ$ . When measured according to such a generalized Finsler metric, a material element expands with increasing $ξ$ , physically associated with cavitation, void growth, and/or bulking that arise during progression of local degradation processes.

Summarizing, fracture prescriptions are

w (F, ξ) = [(\ln F) (1 - ξ)]^{2}, f_{0} (ξ) = ξ^{2}, G (ξ) = \exp [k f_{0} (ξ)] = \exp [k ξ^{2}],

(4.59)

where parameter $k > 0$ for microstructure expansion. Stress $P$ in (4.55) and normalized Cartan tensor component $\hat{χ} (ξ)$ are, respectively,

P / E = (\ln F / F) (1 - ξ)^{2} = (P_{0} / E) \exp (- \frac{1}{2} k ξ^{2}), \hat{χ} = k ξ .

(4.60)

Microscopic momentum balance (4.58) is then

l^{2} ξ ″ - [ξ - α {(\ln F)}^{2} (1 - ξ)] = \frac{1}{2} k ξ [α {(\ln F)}^{2} {(1 - ξ)}^{2} + ξ^{2} - {(l ξ')}^{2}] .

(4.61)

Non-zero $k$ corresponds to a generalized Finsler space. When $k = 0$ , the present problem reduces to a standard quasi-static phase field representation of 1D fracture in Euclidean space.

Two specific boundary value problems are solved. The first invokes boundary conditions at $X = 0, X = L_{0}$ of

ξ (0) = ξ (L_{0}) = ξ_{H}; φ (0) = 0, φ (L_{0}) = φ_{L} .

(4.62)

Sought are solutions of the form $ξ (X) = ξ_{H} = constant$ . Accordingly, the first expression of (4.60) demands

\begin{matrix} P / E = (\ln F / F) (1 - ξ_{H})^{2} = (P_{0} / E) \exp (- \frac{1}{2} k ξ_{H}^{2}) = constant \\ \Rightarrow F = constant = φ_{L} / L_{0} \Rightarrow φ (X) = FX . \end{matrix}

(4.63)

Balance (4.61), with $ξ ″ = 0$ and $ξ' = 0$ , can then be solved analytically for $F$ with $ξ_{H}$ imposed:

\ln F = \pm {[\frac{ξ_{H} (1 + \frac{k}{2} ξ_{H}^{2})}{α {(1 - ξ_{H}) - \frac{k}{2} ξ_{H} {(1 - ξ_{H})}^{2}}}]}^{1 / 2} .

(4.64)

For tensile loading, the positive root applies. Solutions for $ξ_{H} \in [0, 1]$ are obtained because (4.59) is physically valid only for this domain of $ξ$ . A baseline choice of $α = 10^{2}$ corresponds to realistic physical parameters $E = 200$ GPa, $ϒ = 1$ J/m ², and $l = 1$ nm [104, 106]. Simple choice $k = 1$ demonstrates effects of the generalized Finsler metric, where $\sqrt{G} = e^{1 / 2} = 1.648 \dots$ is maximum volume scaling at $ξ = 1$ .

Solutions derived in (4.63) and (4.64) are given in Figure 2, with total energy and domain length with respect to $G$ of

Ψ / E = \int_{0}^{L_{0}} \frac{1}{2} [(\ln F)^{2} (1 - ξ_{H})^{2} + ξ_{H}^{2} / α] \exp (\frac{1}{2} k ξ_{H}^{2}) dX, L = L_{0} \exp (\frac{1}{2} k ξ_{H}^{2}) .

(4.65)

Figure 2.

Solutions to homogeneous fracture problem (4.64) for Finsler ( $k = 1$ ) and Euclidean ( $k = 0$ ) metrics.

Figure 3.

Solutions to stress-free fracture problem (4.67) for Finsler ( $k = 1$ ) and Euclidean ( $k = 0$ ) metrics.

As $α$ increases, ductility measured by the stretch $F$ at peak load decreases. Choice $k = 1$ produces less accumulation of damage $ξ$ relative to choice $k = 0$ at the same value of $F$ . Correspondingly, $P$ is larger for the generalized Finsler case relative to the Euclidean case with $k = 0$ , and the stretch at peak load is also larger. Normalized total energy $Ψ$ is likewise slightly larger for $k = 1$ and the same choice of $α$ . The conclusion is that setting $k > 0$ offers a one-parameter means, with geometric foundation, of modeling increasing strength and ductility associated with microscopic dilatation mechanisms that would inhibit extension of (e.g., blunt) an otherwise sharp crack and absorb energy in its vicinity.

The second fracture problem models complete fracture at $X = 0$ , and surmises null damage at $ξ = L_{0}$ . Boundary conditions are

ξ (0) = 1, ξ (L_{0}) = 0; P (0) = P (L_{0}) = 0 .

(4.66)

Stress-free conditions require, from (4.60), $(\ln F) (1 - ξ) = 0$ . Balance law (4.61) becomes the nonlinear ordinary differential equation (ODE)

l^{2} ξ ″ - ξ = \frac{1}{2} k ξ [ξ^{2} - {(l ξ')}^{2}] .

(4.67)

For Riemmannian geometry ( $k = 0$ ), the ODE is linear, and an exact analytical solution is easily derived [61]. An analytical solution is not obvious for $k \neq 0$ ; hence, a second-order implicit finite difference method is used to solve (4.67) numerically, with an iterative method to account for nonlinear terms on the right-hand side.

Profiles of $ξ$ for different ratios $l / L_{0}$ are shown in Figure 2, where the larger the value of $l$ , the more diffuse the crack emanating from $X = 0$ . Cases with $k = 0$ and $k = 1$ are nearly indistinguishable for the same $l / L_{0}$ . Define the normalized energy per unit cross-sectional area $\bar{γ}$ as the line integral of $ψ / ϒ$ :

\bar{γ} = \int_{0}^{L_{0}} [ξ^{2} / l + l {(ξ')}^{2}] [1 + c_{0} \cdot (\sqrt{G} - 1)] d X,

(4.68)

where $c_{0} = 0$ for integration over $M$ with Euclidean volume element $dX$ and $c_{0} = 1$ for the Finsler volume element $d Ω = \sqrt{G} d X$ , noting $k = 0 \Rightarrow G = 1$ . Results in Table 1 for $k = 0$ show how $\bar{γ}$ approaches unity as $l / L_{0}$ decreases, as in phase-field fracture mechanics, whereby the Griffith energy for a sharp crack surface is $\bar{γ} = 1$ [103]. Energy $\bar{γ}$ is essentially identical for $k = 0$ or $k = 1$ when $c_{0} = 0$ , with $\approx 1$ % discrepancy from the Griffith solution at $l / L_{0} = 0.1$ in each case. When $c_{0} = 1$ is chosen as the metric for averaging fracture energy, this energy per unit are a $\bar{γ}$ is amplified by $\approx 30$ % for $k = 1$ .

Table 1.

Normalized surface energy $\bar{γ}$ of (4.68) for stress-free fracture.

Regularization	$(\bar{γ} - 1) \times 10^{2}$	$(\bar{γ} - 1) \times 10^{2}$	$(\bar{γ} - 1) \times 10^{2}$
length $l / L_{0}$	$k = c_{0} = 0$	$k = 1, c_{0} = 0$	$k = c_{0} = 1$
0.1	0.911	0.911	30.568
0.5	3.839	3.846	33.737

Phase transformation. Now consider a stress-driven, solid–solid phase change problem. Let $D (X) = 0$ denote the local indication of a material element at $X$ in the parent phase. Let $D (X) = l \Leftrightarrow ξ (X) = 1$ represent the condition that the material element at $X$ has fully transformed. Dimensionless field variable $ξ$ thus serves as an order parameter. Denote by $ϵ_{0}$ the eigenstrain due to a locally complete phase change, here assumed positive such that tensile loading drives transformation. The atomic displacement for complete transformation is $δ_{0} = a \cdot ϵ_{0}$ , with $a$ the lattice parameter of the parent phase. The ground state energy of the two phases at $ξ = 0, 1$ is assumed the same, and this energy is assumed to vanish when $(F, ξ) = (1, 0)$ or $(F, ξ) = (1 + ϵ_{0}, 1)$ .

The eigenstrain for intermediate states $0 < ξ < 1$ , physically corresponding to partially transformed zones in phase boundaries, is interpolated by function $ϕ (ξ)$ with derivative denoted by $\overset{\cdot}{ϕ} = d ϕ / d ξ$ . The following conditions are imposed on $ϕ (ξ)$ : $ϕ (0) = 0$ , $ϕ (1) = 1$ , $ϕ (1 - ξ) = 1 - ϕ (ξ)$ , and $\overset{\cdot}{ϕ} (0) = \overset{\cdot}{ϕ} (1) = 0$ . The simplest polynomial satisfying these requirements is

ϕ (ξ) = 3 ξ^{2} - 2 ξ^{3}, \overset{\cdot}{ϕ} (ξ) = 6 ξ (1 - ξ) .

(4.69)

Define $Λ^{- 1} (ξ)$ as the stretch associated purely with phase transformation, such that $\bar{F} = F Λ$ is the elastic stretch:

\bar{F} (F, ξ) = F \cdot Λ (ξ); Λ (ξ) = 1 / [1 + ϵ_{0} ϕ (ξ)], \overset{\cdot}{Λ} (ξ) = - ϵ_{0} Λ^{2} (ξ) \overset{\cdot}{ϕ} (ξ) .

(4.70)

A logarithmic nonlinear elastic potential $w$ [101, 102] is used, with dependence on $\bar{F}$ rather than $F$ . A double-well potential [107, 108] is used for $f_{0}$ , where the transformation energy barrier is $\frac{A}{16}$ at $ξ = \frac{1}{2}$ . A conformal transformation [94, 97, 105] is used for $G (ξ)$ . The material is assumed to expand in atomically disordered zones in the vicinity of phase boundaries, as has been likewise suggested for dislocations, point defects, stacking faults, and grain boundaries arising from anharmonicity [25, 109]. As measured by generalized Finsler metric $G (ξ) = G (D / l)$ , the material in a boundary region deviates microscopically from close packing of a perfect lattice, distinctly from the eigenstrain measured macroscopically by $Λ^{- 1}$ . Such deviation should be maximal at $ξ = 0.5$ , and it should vanish in the parent and fully transformed states, such that $G (ξ = 0) = G (ξ = 1) = 1$ .

Summarizing the above modeling assumptions, phase transformation prescriptions are

w (\bar{F} (F, ξ)) = (\ln \bar{F})^{2}, f_{0} (ξ) = ξ^{2} (1 - ξ)^{2}, G (ξ) = \exp [k f_{0} (ξ)] = \exp [k ξ^{2} (1 - ξ)^{2}],

(4.71)

where parameter $k > 0$ for microstructure expansion in phase boundary zones. Stress in (4.55) and Cartan tensor component are calculated as

P / E = (\ln \bar{F} / \bar{F}) Λ = (P_{0} / E) \exp [- \frac{1}{2} k ξ^{2} {(1 - ξ)}^{2}], \hat{χ} = k ξ (1 - ξ) (1 - 2 ξ) .

(4.72)

Microscopic momentum balance (4.58) is, with $Λ = [1 + ϵ_{0} ξ^{2} (3 - 2 ξ)]^{- 1}$ ,

\begin{matrix} l^{2} ξ ″ - [ξ (1 - ξ) (1 - 2 ξ) - 6 α ϵ_{0} ξ (1 - ξ) Λ \ln (F Λ)] \\ = \frac{1}{2} k ξ (1 - ξ) (1 - 2 ξ) [α {\ln (F Λ)}^{2} + ξ^{2} {(1 - ξ)}^{2} - {(l ξ')}^{2}] . \end{matrix}

(4.73)

As was the case for the fracture example, non-zero $k$ corresponds to generalized Finsler space. When $k = 0$ , the description reduces to a phase field model of expansive phase change in 1D Euclidean space.

One specific boundary value problem is solved, with boundary conditions at $X = 0, X = L_{0}$ given by

ξ (0) = ξ (L_{0}) = ξ_{H}; φ (0) = 0, φ (L_{0}) = φ_{L} .

(4.74)

Homogeneous solutions of the form $ξ (X) = ξ_{H} = constant$ are sought. The first expression of (4.72) requires

\begin{matrix} P / E = (\ln \bar{F} / \bar{F}) Λ = (P_{0} / E) \exp [- \frac{1}{2} k ξ_{H}^{2} {(1 - ξ_{H})}^{2}] = constant \\ \Rightarrow F = constant = φ_{L} / L_{0} \Rightarrow φ (X) = FX . \end{matrix}

(4.75)

Microscopic balance (4.73) becomes, with $ξ = ξ_{H}$ , $Λ (ξ) = Λ (ξ_{H}) = Λ_{H}$ , $ξ' = 0$ , and $ξ ″ = 0$ ,

\begin{matrix} 6 α ϵ_{0} ξ_{H} (1 - ξ_{H}) Λ_{H} \ln (F Λ_{H}) - ξ_{H} (1 - ξ_{H}) (1 - 2 ξ_{H}) \\ = \frac{1}{2} k ξ_{H} (1 - ξ_{H}) (1 - 2 ξ_{H}) [α {\ln (F Λ)}^{2} + ξ_{H}^{2} {(1 - ξ_{H})}^{2}] . \end{matrix}

(4.76)

A closed-form solution for $ξ_{H} (F)$ or $F (ξ_{H})$ is not apparent, but the solution to (4.76) can be obtained at each prescribed value of $F$ by simple numerical iteration over physically admissible domain $ξ_{H} \in [0, 1]$ . For tensile loading, take $F \geq 1$ . Note also that $ξ_{H} = 0$ and $ξ_{H} = 1$ are trivial solutions to (4.76) for any $F$ . When multiple solutions exist, then $ξ_{H}$ is chosen such that $Ψ$ is minimal among those solutions, some of which may be metastable.

The transformation strain is chosen as $ϵ_{0} = 0.1$ for demonstration purposes. An order-of-magnitude choice $α = 10^{2}$ is likewise relevant [106, 108]. Prescription $k = 10$ demonstrates effects of the generalized Finsler metric, where $\sqrt{G} = e^{5 / 16} = 1.366 \dots$ is maximum volume scaling at $ξ = \frac{1}{2}$ . Solutions to (4.76) are reported in Figure 4, where total energy and domain length with respect to $G$ are

\frac{Ψ}{E} = \int_{0}^{L_{0}} \frac{1}{2} [(\ln \bar{F})^{2} + \frac{1}{α} ξ_{H}^{2} (1 - ξ_{H}^{2})] \sqrt{G (ξ_{H})} dX, \frac{L (ξ_{H})}{L_{0}} = \sqrt{G (ξ_{H})} = \exp [\frac{1}{2} k ξ_{H}^{2} {(1 - ξ_{H})}^{2}] .

(4.77)

From Figure 4(a), as $α = \frac{E}{2 A}$ increases, the stretch $F$ at which transformation $ξ > 1$ commences decreases, but the stretch $F$ at which transformation is complete ( $ξ \to 1$ ) increases. Stress fluctuates between tension ( $P > 0$ ) and compression ( $P < 0$ ) as $ξ : 0 \to 1$ with increasing $F$ . After complete transformation ( $ξ = 1$ ), loading is purely elastic and stress increases monotonically with $F$ thereafter. At fixed $α$ , stress in Figure 4(b) and energy in Figure 4(c) are identical for $k = 0$ or $k = 10$ when $ξ = 0$ , $ξ = 1$ , or $ξ = 0.5$ .

Figure 4.

Homogeneous phase change solutions of (4.76) for Finsler ( $k = 10$ ) and Euclidean ( $k = 0$ ) metrics.

Note from Figure 4(c) that energy attains a local maximum at $F = 1 + 0.5 ϵ_{0}$ , whereby $ξ = 0.5$ and $P = 0$ , corresponding to an unstable equilibrium state. For $0 < ξ < 0.5$ and $0.5 < ξ < 1$ , normalized stress magnitude $| P α / E |$ is often noticeably larger for the Finsler model ( $k = 10$ ), and normalized total energy $Ψ \cdot α / (E L)$ tends to be slightly larger, for the same choice of $α$ , versus cases with $k = 0$ . Particular results for $F = 1 + \frac{1}{4} ϵ_{0}$ and $F = 1 + \frac{3}{4} ϵ_{0}$ in Table 2 confirm this observation for $α = 10^{2}$ ; results for other $α$ show the same trends. The conclusion is that by setting single parameter $k > 0$ , a traditional phase-field type model can be enriched, with a basis in Finsler geometry, to account for stress and energy amplification within an externally stressed boundary region due to atomic-scale disorder and commensurate dilatation. Experimental methods to obtain analogous stress and energy data at the nanoscale may not yet exist, but quantum simulations could be sought to verify predictions [62].

Table 2.

Stress $P$ of (4.75) and energy $Ψ$ of (4.77) for homogeneous phase boundary solution ( $α = 10^{2}$ ).

Strain	$P α / E$	$P α / E$	$Ψ α / (E L)$	$Ψ α / (E L)$
$F - 1$	$k = 0$	$k = 10$	$k = 0$	$k = 10$
$ϵ_{0} / 4$	0.796	1.073	0.02176	0.02205
$3 ϵ_{0} / 4$	−0.859	−1.021	0.02134	0.02143

4.2.5. Extensions and applications

Recent advances in Finsler-geometric continuum theory consider the following:

Nonlinear elastic bodies undergoing shear failure, with $D$ a microscopic displacement for slip or sliding, or cavitation, with $D$ a radial microscopic displacement for material opening, in [61].

Additive or multiplicative decompositions [5, 63] of $F$ used, respectively, to account for residual deformation from tensile crack opening or anisotropic contraction under phase transformation [62].

Multiplicative decompositions of $F$ , and of $G$ as in [97], accounting for spherical inelastic dilatation commensurate with free volume changes under shear and tensile loads [92].

Boundary value problems for simultaneous torsion, axial stretch, and/or radial contraction, with time-dependent field $D (X, t)$ for microscopic shear displacement due to dislocations, twist disclinations, or sliding fractures driven by Ginzburg–Landau kinetics motivated by (4.43) [94].

Numerical simulations of polycrystals with $D$ measuring displacement of twinning dislocations within a plane and crack opening displacement orthogonal to this plane [93]. Solutions for static equilibrium are obtained via finite element discretization with degrees-of-freedom $argmin Ψ [φ, D]$ .

Fully dynamic fields $φ (X, t)$ and $D (X, t)$ and thermodynamics of adiabatic processes [72]. Linear momentum balance (4.42) is extended to account for body forces and inertia; however, application of (2.30) in [72] presumes existence of Euclidean enveloping space for integrating vector fields referred to basis ${d x^{a}}$ over $T^{*} m$ [3], thus restricting the geometry of the spatial manifold.

A transport theorem and jump conditions for mass, momentum, and energy exchange across a moving planar surface of discontinuity are derived in [72, 96]. Solutions to the jump conditions are obtained for crystals under shock compression undergoing shear on pyramidal planes [72, 95, 96].

Problems solved in [61, 62, 72, 92 –96] have considered various crystalline materials (e.g., ice, metals, ceramics) with realistic properties and comparison with experiments and/or quantum mechanical results.

5. Ferromagnetic crystals revisited

In this final section, the generalized Finsler-geometric theory of Section 2 (mathematical foundations) and Section 4.2 (variational continuum mechanics) is adapted to describe ferromagnetic materials, revisiting a topic addressed by the apparent first continuum mechanical theory based on Finsler geometry [63]. The present theory essentially extends the nonlinear quasi-magnetostatic theory of Maugin and Eringen [64, 110] from its original setting on a Riemannian (and, more specifically, Eucildean) material manifold to a generalized Finsler manifold $M$ for the reference configuration. The present theory is limited to static mechanical equilibrium (i.e., no inertia), null electric fields and null electric currents (i.e., magnetostatics [64, 111, 112]), and isothermal elastic bodies, with no dislocations, plastic deformation, or other sources of dissipation. As in [63, 64, 110], the ferroelectric crystalline solid occupies a magnetically saturated state, whereby the magnetization vector is of constant magnitude over the body, but may vary in direction. Temperature is presumed far below the Curie point. A temporally fixed spatial frame of reference is invoked in the context of Maxwell’s equations; issues related to Lorentz invariance are not considered, nor is reconciliation among different representations of electrodynamics of finitely deforming continua [111, 113 –115] attempted.

Notable differences between the present approach and that of Amari [63] reviewed in Section 3.1 are as follows:

lattice deformations $F_{A}^{a}$ are integrable in the present model, unlike $(A^{- 1})_{α}^{a}$ in Section 3.1;

finite deformations are considered in the variational formulation, unlike the linearized treatment in Section 3.1.2;

a spatial-rotationally objective constitutive theory is constructed, unlike the formulation of Section 3.1.2 that includes explicit potential energy dependence on lattice rotations and lattice rotation gradients;

the divergence theorem for generalized Finsler space, (2.30) of Theorem 1, is used to derive local equilibrium equations, unlike the treatment of Section 3.1.2 that invokes the classical divergence theorem on Euclidean space with a Cartesian metric;

the director vector of Finsler geometry is of constant magnitude with respect to an Euclidean metric, but it is not necessarily of unit length as in Section 3.1;

a different, and more general, form of Finsler metric is admitted in the present theory to capture various interactions between the magnetization direction and atomic-scale structure, as opposed to that of Section 3.1.1 accounting only for magnetostriction.

Fundamental aspects of this new theory of Finsler-geometric continuum mechanics of magnetically saturated, nonlinear elastic solids are defined and derived in Sections 5.1–5.5. A representative generalized Finsler metric, with conceivable physical origin, is chosen in Section 5.6 to demonstrate resulting differences from the original theory based in Euclidean space [64, 110].

5.1. Director vectors and fiber coordinates

The director vector field $d (x)$ on total space $z$ is identified with the local magnetic moment per unit mass $μ$ , which in turn is of fixed magnitude $μ_{S}$ , i.e., the saturation magnetization, when measured with respect to the Euclidean metric $δ_{ab}$ :

d^{a} (x) = μ^{a} (x), d_{a} d^{a} = δ_{ab} d^{a} d^{b} = μ_{S}^{2} = constant; M^{a} (x) = ρ (x, d (x)) \cdot μ^{a} (x) .

(5.1)

Covariant components $d_{a} = δ_{a b} d^{b}$ are obtained via the Cartesian metric, distinguished from $d_{a} = g_{ab} d^{b}$ obtained from generalized Finsler (Sasaki) metric of (2.45). The mass density of the solid at $x \in m$ is $ρ (x, d (x))$ , and the spatial magnetization vector, as entering Maxwell’s equations of conventional form in Section 5.5, is $M$ .

Fiber coordinates on $Z$ are obtained from inversion of the suggestion of (4.33) in Section 4.2.4:

D^{A} (X) = d^{a} (φ (X)) δ_{a}^{A}, D_{A} D^{A} = μ_{S}^{2}; D_{A} = δ_{A B} D^{B} = δ_{A B} δ_{b}^{B} δ^{b a} d_{a} = δ_{A}^{a} d_{a} .

(5.2)

Clearly, $D$ and $d$ are parallel at coincident point $x = φ (X)$ and of the same magnitude $μ_{S}$ . Unit vectors parallel to local magnetization are defined via consistent normalization:

ξ^{a} (x) = δ_{A}^{a} ξ^{A} (Φ (x)) = \frac{1}{μ_{S}} d^{a} (x) = \frac{1}{μ_{S}} δ_{A}^{a} D^{A} (Φ (x)); ξ^{a} δ_{ab} ξ^{b} = ξ^{A} δ_{AB} ξ^{B} = 1 .

(5.3)

Subsequent calculations are vastly simplified if connection coefficients for horizontal gradients of basis vectors on $V Z$ and $V z$ defined in (2.17) and (2.50) vanish, ensuring that the following horizontal covariant derivatives on $Z$ likewise vanish:

K_{BC}^{A} = 0, K_{bc}^{a} = 0 \Rightarrow δ_{A | B}^{a} = 0, δ_{a | B}^{A} = 0, δ_{AB | B} = 0, δ_{ab | B} = 0 .

(5.4)

Analogous spatial horizontal covariant derivatives $(\cdot)_{| a}$ of the Cartesian shifters and Cartesian metrics all vanish. A more general formulation admitting curvilinear shifters $g_{A}^{a} (x, X)$ [71, 81] and non-zero $K_{bc}^{a}, K_{BC}^{A}$ is conceivable, but calculations are beyond the present scope. Affine connection coefficients $H_{bc}^{a}$ and $H_{BC}^{A}$ need not vanish; for example, curvilinear coordinates ${x^{a}}$ and ${X^{A}}$ are allowed on $m$ and $M$ , respectively. It follows from the second expression of (4.22), (5.2), and (5.4) that

{\bar{\partial}}_{A} (\cdot) = δ_{A}^{a} {\bar{\partial}}_{a} (\cdot), d_{| A}^{a} = δ_{B}^{a} D_{| A}^{B}, \partial (\cdot) / \partial d_{| A}^{a} = δ_{a}^{B} \partial (\cdot) / \partial D_{| A}^{B} .

(5.5)

Consequences of the constant magnitude constraints in (5.1) and (5.2) with (5.4) include

d_{a} d_{| B}^{a} = d_{a} \partial_{B} d^{a} = 0 . D_{A} D_{| B}^{A} = D_{A} \partial_{B} D^{A} = 0 .

(5.6)

Similar constraints are derived as follows for a variational differential $δ d$ of $d$ , with components $δ (d^{a})$ :

δ (μ_{S}^{2}) = δ (d^{a} δ_{a b} d^{b}) = 2 d_{a} \cdot δ (d^{a}) = 0 \Rightarrow d_{a} \cdot δ (d^{a}) = 〈 d, δ d 〉 = 0 .

(5.7)

As $d$ is fixed in magnitude, it can only rotate during an incremental variation. Denoting × the cross product and $e^{abc}$ the alternator symbols, a dimensionless angular variation $δ ω$ can be introduced [64]:

δ d = - d \times δ ω \leftrightarrow δ (d^{a}) = - e^{a b c} d_{b} δ ω_{c}; μ_{S}^{2} δ ω = d \times δ d + 〈 d, δ ω 〉 d .

(5.8)

5.2. Energy density and variational principle

A local energy density $ψ$ , measured per unit reference volume, of the following form is posited:

ψ = ψ (F, d, \nabla d) = ψ (F_{A}^{a}, d^{a}, d_{| A}^{a}) .

(5.9)

Dependence on deformation gradient $F$ accounts for elastic strain energy and coupling between strain and magnetic field (e.g., piezomagnetism and magnetostriction). Dependence on magnetic moment $d = μ$ accounts for magnetic anisotropy energy and aforementioned coupling with strain. Dependence on the material gradient of magnetization $\nabla d$ accounts for exchange energy arising from spin interactions. Microscopic origins of these energetic contributions are discussed in [63, 64, 110 –112]. Explicit dependence on $X$ , as arising in a heterogeneous material in its reference state, is omitted for brevity.

Let $δ (\cdot)$ denote the variation of a quantity at a fixed base material particle at $X$ . Expanding (5.9),

δ ψ = \frac{\partial ψ}{\partial F_{A}^{a}} δ F_{A}^{a} + \frac{\partial ψ}{\partial d^{a}} δ (d^{a}) + \frac{\partial ψ}{\partial d_{| A}^{A}} δ d_{| A}^{a} = P_{a}^{A} δ F_{A}^{a} + Q_{a} δ (d^{a}) + Z_{a}^{A} δ d_{| A}^{a} .

(5.10)

Here, $P$ is a macroscopic stress, $Q$ is an internal force conjugate to $d$ , and $Z$ is a micro-stress conjugate to the horizontal gradient of $d$ . Similarly to (4.40) and (4.41) and applying (5.4),

δ F_{A}^{a} = δ_{A} (δ φ^{a}), δ d_{| A}^{a} = [δ (d^{a})]_{| A} - δ_{B}^{a} {\bar{\partial}}_{c} N_{A}^{B} δ (d^{c}), δ (d Ω) = δ_{c}^{B} C_{BA}^{A} δ (d^{c}) .

(5.11)

Following [64], the following variational principle is posited, here in the absence of macroscopic mechanical inertia:

δ \hat{Ψ} = δ W_{T} + δ W_{B} + δ W_{S} .

(5.12)

The left-hand side of (5.12) accounts for the change in stored energy of the body subject to the constraint that $d$ is of constant length in Euclidean space, and terms on the right-hand side account for work done by surface tractions, body forces, and electronic spin angular momentum, respectively. Specifically, with $\hat{λ} (X)$ a Lagrange multiplier and $M'$ a compact region of $M$ with oriented boundary $\partial M'$ ,

\hat{Ψ} [φ, d] = \int_{M'} ψ + \frac{1}{2} \hat{λ} (d_{a} d^{a} - μ_{S}^{2}) d Ω, δ W_{T} = \oint_{\partial M'} (〈 p, δ x 〉 + 〈 z, δ d 〉) Ω,

(5.13)

where $p$ and $z$ are macroscopic and microscopic traction covectors conjugate to particle motion and magnetic moment, respectively, on the surface of the material region. Macroscopic body forces per unit mass consist of the usual mechanical body force $f$ and the ponderomotive force $\nabla B \cdot μ$ , with $B$ denoting Maxwell’s magnetic flux density [64, 111]. And as in [64, 110], the microscopic body force is magnetic flux density $B$ itself:

δ W_{B} = \int_{M'} ρ_{0} [(f_{a} + d^{b} B_{b | a}) δ φ^{a} + B_{a} δ (d^{a})] d Ω, ρ_{0} = ρ J = constant .

(5.14)

Reference mass density is $ρ_{0}$ , assumed constant for homogeneous reference domain $M'$ . Denote $\overset{\cdot}{m} = \overset{\cdot}{ω} \times μ$ as the local magnetization rate from angular velocity $\overset{\cdot}{ω}$ , where the latter is generally not the time derivative of a vector function. Define $Δ$ as the gyromagnetic ratio, with $e$ and $m_{e}$ charge magnitude and mass of an electron and $c$ velocity of light. The electronic spin contribution to the virtual work balance in (5.12) is defined as [64, 110]

δ W_{S} = \int_{ℳ'} \frac{ρ_{0}}{Δ} {\dot{d}}^{a} δ ω_{a} d Ω, Δ = - \frac{e}{m_{e} c}, (\dot{d} = \dot{μ}) .

(5.15)

5.3. Euler–Lagrange equations and electromechanical boundary conditions

Equations (5.10), (5.11), and (5.13)–(5.15) are substituted into (5.12). Theorem 1, i.e., (2.30), is used with integration by parts to convert volume to surface integrals. Independent variations are $δ φ$ and $δ ω$ , recalling $δ d = δ ω \times d$ from (5.8). Horizontal connection coefficients are $H_{BC}^{A} = Γ_{BC}^{A}$ and $H_{bc}^{a} = Γ_{bc}^{a}$ .

Assuming (5.12) must hold for all admissible variations $δ φ$ , macroscopic mechanical equilibrium equations $\forall X \in M'$ are derived as

\begin{matrix} \partial_{A} P_{a}^{A} + {\bar{\partial}}_{b} P_{a}^{A} \partial_{A} d^{b} + P_{a}^{B} Γ_{AB}^{A} - P_{c}^{A} Γ_{ba}^{c} F_{A}^{b} + ρ_{0} (f_{a} + d^{b} B_{b | a}) = - P_{a}^{A} C_{BC}^{C} (δ_{b}^{B} \partial_{A} d^{b} + N_{A}^{B}) . \end{matrix}

(5.16)

Expression (5.16) is identical to (4.42) with the exception of body force contributions. Natural mechanical boundary conditions obtained $\forall X \in \partial M'$ are standard as in (4.44), with $N_{A}$ the unit outward normal covector to $\partial M'$ :

p_{a} = P_{a}^{A} N_{A} .

(5.17)

Remark 5.1. Terms on the left-hand side of (5.16) match the formulation of [64, 110] when the generalized Finsler metrics $G$ and $g$ degenerate to classical Riemannian metrics such that $Γ_{BC}^{A} \to γ_{BC}^{A}$ , $Γ_{bc}^{a} \to γ_{bc}^{a}$ are Levi-Civita connection coefficients and the covariant derivative of $B$ is taken with respect to $γ_{bc}^{a}$ . Terms on the right-hand side of (5.16) arise from Cartan’s tensor $C_{BC}^{A}$ and nonlinear connection coefficients $N_{B}^{A}$ , both of which vanish identically in Riemannian geometry, but not necessarily in Finsler geometry.

The remaining terms in volume integrals from (5.12), localized to a point $X \in M'$ , comprise

\begin{matrix} [\partial_{A} Z_{c}^{A} + {\bar{\partial}}_{b} Z_{c}^{A} \partial_{A} d^{b} + Z_{c}^{B} Γ_{AB}^{A} - Q_{c} - \hat{λ} d_{c} + ρ_{0} B_{c}] δ (d^{c}) + [(ρ_{0} / Δ) {\overset{\cdot}{d}}^{a}] δ ω_{a} \\ = [ψ δ_{c}^{C} C_{CA}^{A} - Z_{d}^{B} δ_{A}^{d} {{\bar{\partial}}_{c} N_{B}^{A} + δ_{c}^{A} δ_{e}^{E} C_{ED}^{D} (\partial_{B} d^{e} + δ_{F}^{e} N_{B}^{F})}] δ (d^{c}), \end{matrix}

(5.18)

which is a microscopic work balance. Applying (5.8) to eliminate $δ (d^{c})$ in favor of $δ ω_{a}$ ,

\begin{matrix} [\partial_{A} Z_{c}^{A} + {\bar{\partial}}_{b} Z_{c}^{A} \partial_{A} d^{b} + Z_{c}^{B} Γ_{AB}^{A} - Q_{c} - \hat{λ} d_{c} + ρ_{0} B_{c}] e^{caf} d_{f} δ ω_{a} + [(ρ_{0} / Δ) {\overset{\cdot}{d}}^{a}] δ ω_{a} \\ = [ψ δ_{c}^{C} C_{CA}^{A} - Z_{a}^{B} δ_{A}^{a} {{\bar{\partial}}_{c} N_{B}^{A} + δ_{c}^{A} δ_{e}^{E} C_{ED}^{D} (\partial_{B} d^{e} + δ_{F}^{e} N_{B}^{F})}] e^{caf} d_{f} δ ω_{a} . \end{matrix}

(5.19)

As $e^{c a f} d_{c} d_{f} = 0$ , the Lagrange multiplier $\hat{λ}$ is superfluous, and the microscopic momentum balance, denoted in [64, 10] as the balance of magnetic or electronic spin angular momentum, is obtained for admissible variations $δ ω$ :

\begin{matrix} e^{acf} [\partial_{A} Z_{c}^{A} + {\bar{\partial}}_{b} Z_{c}^{A} \partial_{A} d^{b} + Z_{c}^{B} Γ_{AB}^{A} - Q_{c} + ρ_{0} B_{c}] d_{f} - (ρ_{0} / Δ) {\overset{\cdot}{d}}^{a} \\ = e^{acf} [ψ δ_{c}^{C} C_{CA}^{A} - Z_{a}^{B} δ_{A}^{a} {{\bar{\partial}}_{c} N_{B}^{A} + δ_{c}^{A} δ_{e}^{E} C_{ED}^{D} (\partial_{B} d^{e} + δ_{F}^{e} N_{B}^{F})}] d_{f} . \end{matrix}

(5.20)

Remark 5.2. Terms on the left-hand side of (5.20) are consistent with [64, 110] when a Riemannian metric $G_{AB} = G_{AB} (X)$ is used such that $Γ_{BC}^{A} \to γ_{BC}^{A}$ . Terms on the right-hand side emerge from nonlinear connection coefficients $N_{B}^{A}$ and Cartan’s tensor $C_{BC}^{A}$ that can exist only when $M$ is the base manifold of a generalized Finsler space, rather than a Riemannian space.

Remark 5.3. If energy per unit reference volume $ψ$ is independent of the gradient of magnetization $\nabla d = \nabla μ$ , spin inertia is neglected ( $m_{e} c \to 0 \Rightarrow Δ \to \infty$ ), and terms of Finsler origin on the right-hand side all vanish, then the constitutive relation $B = ρ_{0}^{- 1} Q = ρ_{0}^{- 1} \partial ψ / \partial μ = \partial ψ / \partial (J M)$ is inferred from (5.20).

Finally, natural boundary conditions for the electronic spin continuum are derived from (5.12) and (5.13) as follows, in agreement with (4.44) and [44]:

[z_{a} - Z_{a}^{A} N_{A}] δ (d^{a}) = e^{abc} [z_{a} - Z_{a}^{A} N_{A}] d_{c} δ ω_{b} = 0 \Rightarrow z_{a} = Z_{a}^{A} N_{A} .

(5.21)

Remark 5.4. In [64, 110], it is assumed that $z_{a} = 0$ $\forall X \in \partial M'$ .

5.4. Objectivity and macroscopic angular momentum

Energy density $ψ$ should be objective, i.e., unaffected by orthogonal transformations of spatial basis vectors ${\frac{δ}{δ x^{a}}, \frac{\partial}{\partial d^{a}}}$ . Let $R : H z \to H z$ and $\hat{R} : V z \to V z$ be constant, orthogonal, mixed-variant tensors enabling such coordinate transformations, whereby

F \to RF, d \to \hat{R} d, \nabla d \to \hat{R} \nabla d, (R_{\cdot c}^{a} R_{b}^{\cdot c} = {\hat{R}}_{\cdot d}^{a} {\hat{R}}_{b}^{\cdot d} = δ_{b}^{a}) .

(5.22)

Here the field variables $\bar{C} = {\bar{C}}_{AB} d X^{A} \otimes d X^{B}$ (generally differing from $C$ of (4.29) when $g$ depends on $d$ ), $Π = Π_{A} d X^{A}$ , and $Λ = Λ_{AB} d X^{a} \otimes d X^{B}$ , all of which are invariant under action of $R$ and $\hat{R}$ are introduced:

{\bar{C}}_{AB} = F_{A}^{a} {\bar{g}}_{ab} F_{B}^{b}, Π_{A} = F_{A}^{a} {\tilde{g}}_{a}^{b} d_{b}, Λ_{AB} = d_{| A}^{a} d_{a | B} .

(5.23)

Note the symmetries ${\bar{C}}_{AB} = {\bar{C}}_{BA}$ and $Λ_{AB} = Λ_{BA}$ . Shifter ${\tilde{g}}_{a}^{b} (x, d (x)) = 〈 \frac{δ}{δ x^{a}}, δ d^{b} 〉 (x, d (x))$ converts the local fiber basis at $d (x)$ to the horizontal basis at $x$ and transforms as ${\hat{g}}_{a}^{b} \to R_{c}^{\cdot a} {\hat{g}}_{a}^{b} {\hat{R}}_{\cdot b}^{d}$ . According to assumption (2.43), a uniform base space rotation $R$ induces a uniform fiber space rotation $\hat{R}$ with the same components ${\hat{R}}_{\cdot b}^{a} = R_{\cdot b}^{a}$ . However, quantities in (5.23) are rotationally invariant even when ${\hat{R}}_{\cdot b}^{a} \neq R_{\cdot b}^{a}$ .

The spatial Sasaki metric $g (x, d)$ of (2.45) and (2.46) is decomposed into a Riemannian part $\bar{g}$ and a director-dependent part $\hat{g} = ǧ^{- 1}$ :

g_{ab} (x, d) = {\bar{g}}_{ac} (x) {\hat{g}}_{b}^{c} (x, d), g^{ab} (x, d) = {\bar{g}}^{ac} (x) ǧ_{c}^{b} (x, d), ({\bar{g}}_{ab} = {\bar{g}}_{ba}, {\hat{g}}_{c}^{a} ǧ_{b}^{c} = δ_{b}^{a}) .

(5.24)

This is analogous to the decomposition of $G (X, D)$ in (4.36).

The set of 15 quantities $({\bar{C}}_{AB}, Π_{A}, Λ_{AB}$ ), where only 14 are independent due to the constraint arising from $d^{a} d_{a} = μ_{S}^{2} = constant$ , comprises a sufficient minimal integrity basis for $ψ$ [110]:

ψ = ψ [\bar{C} (F), Π (d, F), Λ (\nabla d)] = ψ ({\bar{C}}_{AB}, Π_{A}, Λ_{AB}); μ_{S}^{2} = δ^{bc} ({\tilde{g}}^{- 1})_{b}^{a} ({\tilde{g}}^{- 1})_{c}^{d} f_{d}^{B} f_{a}^{A} Π_{A} Π_{B} .

(5.25)

From (5.10) and (5.23), the thermodynamic force $Q$ conjugate to magnetization per unit mass $μ^{a} = d^{a}$ is

Q_{a} = \frac{\partial ψ}{\partial Π_{B}} \frac{\partial Π_{B}}{\partial d^{a}} = F_{B}^{b} Y_{ab} \frac{\partial ψ}{\partial Π_{B}}, Y_{ab} = δ_{ac} {\tilde{g}}_{b}^{c} + d_{c} {\bar{\partial}}_{a} {\tilde{g}}_{b}^{c} .

(5.26)

The micro-stress $Z$ conjugate to magnetization gradient $d_{| A}^{a}$ is

Z_{a}^{A} = \frac{\partial ψ}{\partial Λ_{BC}} \frac{\partial Λ_{BC}}{\partial d_{| A}^{a}} = 2 d_{a | B} \frac{\partial ψ}{\partial Λ_{AB}} .

(5.27)

First Piola–Kirchhoff stress $P$ conjugate to $F_{A}^{a}$ is likewise, letting $y^{ac} Y_{cb} = δ_{b}^{a}$ and recalling from (4.25) that $f_{a}^{A} = (F^{- 1})_{a}^{A}$ ,

P_{b}^{B} = \frac{\partial ψ}{\partial {\bar{C}}_{AC}} \frac{\partial {\bar{C}}_{AC}}{\partial F_{B}^{b}} + \frac{\partial ψ}{\partial Π_{A}} \frac{\partial Π_{A}}{\partial F_{B}^{b}} = 2 {\bar{g}}_{ba} F_{C}^{a} \frac{\partial ψ}{\partial {\bar{C}}_{BC}} + f_{c}^{B} y^{ca} {\tilde{g}}_{b}^{e} Q_{a} d_{e} .

(5.28)

Recall from (4.46) that the Cauchy stress tensor is $σ = σ^{ab} \frac{δ}{δ x^{a}} \otimes \frac{δ}{δ x^{b}}$ ; for saturated magnetoelastic solids, this tensor can be split according to (5.28), with $j = 1 / J = det (f_{a}^{A}) \sqrt{G / g}$ , as

σ^{ab} = j g^{ac} P_{c}^{A} F_{A}^{b} = {\bar{σ}}^{ab} + {\hat{σ}}^{ab}; {\bar{σ}}^{ab} = 2 j \frac{\partial ψ}{\partial {\bar{C}}_{AB}} F_{A}^{b} F_{B}^{d} ǧ_{d}^{a}, {\hat{σ}}^{ab} = j g^{ac} {\tilde{g}}_{c}^{f} y^{be} Q_{e} d_{f} .

(5.29)

The term $\bar{σ}$ arises from elastic strain $\bar{C}$ as in classical nonlinear elasticity, whereas $\hat{σ}$ arises from interaction of the local magnetic moment $d$ with its work conjugate $Q$ . The local macroscopic angular momentum balance is defined [64, 110] as the skew part of (5.29):

e_{abc} σ^{ab} = e_{abc} {\bar{σ}}^{ab} + e_{abc} {\hat{σ}}^{ab} = j e_{abc} [2 \frac{\partial ψ}{\partial {\bar{C}}_{AB}} F_{A}^{b} F_{B}^{d} ǧ_{d}^{a} + g^{ac} {\tilde{g}}_{c}^{f} y^{be} Q_{e} d_{f}] .

(5.30)

None of $σ^{ab}$ , ${\bar{σ}}^{ab}$ , or ${\hat{σ}}^{ab}$ is generally symmetric. However, if ${\hat{g}}_{b}^{a} (x, d) = β (x, d) δ_{b}^{a}$ is spherical, then $ǧ_{b}^{a} = β^{- 1} δ_{b}^{a} \Rightarrow {\bar{σ}}^{ab} = {\bar{σ}}^{ba}$ and the only contribution to (5.30) is from ${\hat{σ}}^{ab}$ , as in [64, 110].

Remark 5.5. Theory of [64, 110] is recovered if ${\tilde{g}}_{b}^{a} = {\hat{g}}_{b}^{a} = ǧ_{b}^{a} = δ_{b}^{a}$ whereby $Y^{ab} = δ^{ab}$ , $y_{ab} = δ_{ab}$ .

5.5. Maxwell’s equations and electromagnetic boundary conditions

Local versions of Maxwell’s equations for non-polar continua in the magnetostatic approximation (i.e., no electric field, electric current, or electric polarization) are

\begin{matrix} \nabla \times H = 0 \Rightarrow \nabla \times B = \nabla \times (ρ μ) \leftrightarrow e^{abc} \partial_{b} ({\tilde{g}}_{c}^{d} B_{d}) = e^{abc} \partial_{b} (ρ {\tilde{g}}_{c}^{d} d_{d}); \\ \nabla \cdot B = 0 \leftrightarrow B_{| a}^{a} = \partial_{a} B^{a} + K_{ab}^{a} B^{b} = 0 . \end{matrix}

(5.31)

Denoted by $H$ is the magnetic field or magnetic intensity, and $B$ the magnetic induction or magnetic flux density. Recall also that $M = ρ μ = ρ d$ is the magnetization per spatial volume, related in Heaviside–Lorentz units [111] to magnetic field and magnetic flux by $M = B - H$ . Equations (5.31) are assumed to apply for any spatial point $x$ , both inside and outside spatial body manifold $m$ , whereby $μ (x) = 0$ in free space. At any point $x = ϕ (X)$ along $\partial m$ with unit outward normal $n = n_{a} (x) d x^{a}$ and [[·]] denoting the jump in a quantity across $m$ , corresponding magnetostatic boundary conditions are [64, 110]

n \times [[B - ρ μ]] = 0 \leftrightarrow e^{abc} n_{b} [[{\tilde{g}}_{c}^{d} (B_{d} - ρ d_{d})]] = 0; 〈 n, [[B]] 〉 = 0 \leftrightarrow n_{a} [[({\tilde{g}}^{- 1})_{b}^{a} B^{b}]] = 0 .

(5.32)

Remark 5.6. Equations (5.16) , ( 5.20 ), and the first expression of (5.31) constitute nine coupled PDEs in nine degrees of freedom ${φ^{a} (X), d^{a} (X), B^{a} (φ (X))}$ , $\forall X \in M$ , with constraints $| d |_{E_{3}}^{2} = constant$ and $B$ solenoidal.

5.6 Example metric with reduced governing equations

A particular, relatively simple, example is used to highlight differences between the generalized Finsler model of ferromagnetic solids formulated in Sections 5.1–5.5 and the theory of [64, 110] framed in Euclidean space.

5.6.1. Coordinates and metrics

Respective sets of Cartesian coordinates ${X^{1}, X^{2}, X^{3}} = {X, Y, Z}$ and ${x^{1}, x^{2}, x^{3}} = {x, y, z}$ cover $M$ and $m$ . Riemannian metrics are ${\bar{G}}_{AB} = δ_{AB}$ in (4.36) and ${\bar{g}}_{ab} = δ_{ab}$ in (5.24). Generalized Finsler metrics are prescribed as the spherically symmetric functions

G_{AB} (D) = δ_{AB} β (D) = δ_{AB} \exp [k f_{0} (D)], g_{ab} (d) = δ_{ab} β (d) = δ_{ab} \exp [k f_{0} (d)] .

(5.33)

Here, $k$ is a constant controlling the magnitude of the rescaling of the Euclidean metric, and $f_{0}$ is a dimensionless and differentiable function of its argument. The metrics in (5.33) are examples of those analyzed geometrically in [97]. A particular choice of $f_{0}$ , positive homogeneous of degree zero with respect to $d$ or $D$ , is assigned later in Section 5.6.4 with physical rationale.

5.6.2. Connection coefficients

Recall from Section 4.2.2 that vertical affine coefficients of the Chern–Rund connection vanish by definition, and from Section 5.1 that horizontal gradients of basis vectors on fiber spaces vanish. For convenience, take referential holonomic basis vectors on $H Z$ to be uniform and to coincide with those on $V Z$ at all $(X, D (X))$ , and similarly for spatial basis vectors. As nonlinear connection coefficients vanish as derived in (5.35), it follows for these assumptions that

V_{BC}^{A} = Y_{BC}^{A} = K_{BC}^{A} = 0, V_{bc}^{a} = Y_{bc}^{a} = K_{bc}^{a} = 0; {\tilde{g}}_{b}^{a} = δ_{b}^{a}, Y_{ab} = δ_{ab}, y^{ab} = δ^{ab} .

(5.34)

Coefficients $H_{BC}^{A} = Γ_{BC}^{A}$ , $H_{bc}^{a} = Γ_{bc}^{a}$ , $N_{B}^{A} = G_{B}^{A}$ , and $N_{b}^{a}$ are calculated from (2.23), (2.28), (2.54), and (4.35) as vanishing identically because $G_{AB} (D (X))$ and $g_{ab} (d (x))$ in (5.33) do not depend on $X$ or $x$ independently from $D$ or $d$ :

γ_{BC}^{A} = 0 \Rightarrow N_{B}^{A} = 0 \Rightarrow Γ_{BC}^{A} = 0; γ_{bc}^{a} = 0, N_{b}^{a} = δ_{A}^{a} N_{B}^{A} f_{b}^{B} = 0 \Rightarrow Γ_{bc}^{a} = 0 .

(5.35)

In contrast, Cartan tensors (2.23) and (2.54) do not always vanish since metrics $G$ or $g$ in (5.33) do generally depend on $D$ or $d$ :

C_{AB}^{D} = \frac{1}{2} k (δ_{A}^{D} {\bar{\partial}}_{B} f_{0} + δ_{B}^{D} {\bar{\partial}}_{A} f_{0} - δ^{DC} δ_{AB} {\bar{\partial}}_{C} f_{0}), C_{AB}^{B} = \frac{3}{2} k {\bar{\partial}}_{A} f_{0};

(5.36)

C_{ab}^{d} = \frac{1}{2} k (δ_{a}^{d} {\bar{\partial}}_{b} f_{0} + δ_{b}^{d} {\bar{\partial}}_{a} f_{0} - δ^{dc} δ_{ab} {\bar{\partial}}_{c} f_{0}), C_{ab}^{b} = \frac{3}{2} k {\bar{\partial}}_{a} f_{0};

(5.37)

noting ${\bar{\partial}}_{C} G_{AB} = k G_{AB} {\bar{\partial}}_{C} f_{0}$ and ${\bar{\partial}}_{c} g_{ab} = k g_{ab} {\bar{\partial}}_{c} f_{0}$ from (5.33).

5.6.3. Reduced governing equations

Applying (5.5) and (5.33)–(5.37) to (5.16), (5.30), (5.20), and (5.31), reduced forms of the macroscopic balance of linear momentum, macroscopic balance of angular momentum, balance of electronic spin momentum, and Maxwell’s equations are obtained, respectively:

\begin{matrix} \partial_{A} P_{a}^{A} + {\bar{\partial}}_{b} P_{a}^{A} \partial_{A} d^{b} + ρ_{0} (f_{a} + d^{b} \partial_{a} B_{b}) = - \frac{3}{2} {kP}_{a}^{A} {\bar{\partial}}_{b} f_{0} \partial_{A} d^{b}; \end{matrix}

(5.38)

e_{abc} σ^{ab} - j e_{abc} d^{a} δ^{bd} Q_{d} = 0;

(5.39)

e^{acf} [\partial_{A} Z_{c}^{A} + {\bar{\partial}}_{b} Z_{c}^{A} \partial_{A} d^{b} - Q_{c} + ρ_{0} B_{c}] d_{f} - (ρ_{0} / Δ) {\overset{\cdot}{d}}^{a} = \frac{3}{2} k e^{acf} [ψ {\bar{\partial}}_{c} f_{0} - Z_{c}^{B} {\bar{\partial}}_{e} f_{0} \partial_{B} d^{e}] d_{f};

(5.40)

\begin{matrix} e^{abc} \partial_{b} B_{c} - e^{abc} \partial_{b} (ρ d_{c}) = 0, \partial_{a} B^{a} = 0 . \end{matrix}

(5.41)

Left-hand sides of each equation in (5.38)–(5.41) agree with [64, 110]. Non-vanishing terms on right-hand sides of (5.38) and (5.40) are the only differences from the theory of [64, 110] for the macroscopically quasi-static case with spin inertia. When $k = 0$ , metric tensors (5.33) become Riemannian (more specifically, Cartesian) rather than Finslerian, and the governing equations of the present theory reduce exactly to those of [64, 110] in Cartesian coordinates.

5.6.4. Magneto-elastic energy and Finsler metric scaling

The model is complete upon specification of energy density $ψ (\bar{C}, Π, Λ)$ of (5.25) and function $k f_{0}$ entering the metric tensors (5.33). Admissible descriptions of the former accounting for elasticity, piezomagnetism, magnetostriction, magnetic anisotropy, and exchange energy are given in [110] for several magnetic material symmetry groups (see also [111, 112]).

The key new contribution of the present theory is implementation of $k f_{0}$ , whereby the internal state space locally expands or contracts according to local changes in direction of magnetic moment per unit mass $d$ . Though not essential, the form of $f_{0}$ proposed here is positive homogeneous of degree zero in $d$ and $D$ , a condition ensured by allowing $f_{0}$ to depend on $d^{a} = δ_{A}^{a} D^{A}$ only through its unit direction $ξ^{a} = μ_{S}^{- 1} d^{a} = δ_{A}^{a} ξ^{A}$ defined in (5.3). (Zeroth-order positive homogeneity is demonstrated for any real number $r > 0$ , $d^{a} \to r d^{a} \Rightarrow μ_{S} \to r μ_{S} \Rightarrow ξ^{a} \to r^{0} ξ^{a} = ξ^{a}$ .) Note also identities $ξ_{a} \partial_{b} ξ^{a} = 0$ and $ξ_{a} {\bar{\partial}}_{b} ξ^{a} = 0$ following from $ξ^{a} δ_{ab} ξ^{b} = 1$ . Thus,

f_{0} (d) = f_{0} (ξ (d)), f_{0} (D) = f_{0} (ξ (D)); {\bar{\partial}}_{a} f_{0} = \frac{1}{μ_{S}} \frac{\partial f_{0}}{\partial ξ^{a}} .

(5.42)

Consider first a cubic ferromagnetic crystal with lattice directions [100], [010], [001] aligned parallel to ${\frac{\partial}{\partial x^{1}}, \frac{\partial}{\partial x^{2}}, \frac{\partial}{\partial x^{3}}}$ , such that $ξ^{a}$ are direction cosines of the normalized magnetization $μ_{S}^{- 1} μ^{a}$ with respect to the edges of the unit cell in the global spatial coordinate frame. Taken is a representative choice of $f_{0}$ proportional to the first two non-zero terms, respectively orders four and six in $μ^{a}$ due to material and inversion [ $f_{0} (ξ) = f_{0} (- ξ)$ ] symmetries, of the magnetic anisotropy energy in the small deformation approximation [111, 112]:

k \cdot f_{0} = k \cdot {[(ξ^{1})^{2} (ξ^{2})^{2} + (ξ^{2})^{2} (ξ^{3})^{2} + (ξ^{3})^{2} (ξ^{1})^{2}] + κ \cdot [(ξ^{1})^{2} (ξ^{2})^{2} (ξ^{3})^{2}]} .

(5.43)

Dimensionless material constants $k \propto μ_{S}^{4}$ and $κ \propto μ_{S}^{2}$ . With this choice of $f_{0}$ , metrics $g_{ab} (ξ (d))$ and $G_{AB} (ξ (D))$ account for microscopic lattice distortion associated with the direction of (e.g., easy versus hard) magnetization, a phenomenon reported in [112]. A more general treatment would also allow for potential shape changes via anisotropic metric tensors $g$ and $G$ , such as invoked in [62] for phase transformations, but such extensions are outside the scope of the present demonstration limited to spherical microscopic dilatation or contraction. Differentiating (5.43), with each of $a, k, l = 1, 2, 3$ ,

μ_{S}^{2} {\bar{\partial}}_{a} f_{0} (d (ξ)) = 2 d_{a} \cdot {[(ξ^{k})^{2} + (ξ^{l})^{2}] + κ [(ξ^{k} ξ^{l})^{2}]} |_{a \neq l \neq k}^{a \neq k} = 2 d_{a} \cdot f_{1} (ξ),

(5.44)

where $f_{1}$ is a dimensionless function of magnetization direction. Then (5.38) and (5.40) become

\begin{matrix} \partial_{A} P_{a}^{A} + {\bar{\partial}}_{b} P_{a}^{A} \partial_{A} d^{b} + ρ_{0} (f_{a} + d^{b} \partial_{a} B_{b}) = - 3 k μ_{S}^{- 2} f_{1} P_{a}^{A} d_{b} \partial_{A} d^{b} = 0, \end{matrix}

(5.45)

\begin{matrix} e^{acf} [\partial_{A} Z_{c}^{A} + {\bar{\partial}}_{b} Z_{c}^{A} \partial_{A} d^{b} - Q_{c} + ρ_{0} B_{c}] d_{f} - (ρ_{0} / Δ) {\overset{\cdot}{d}}^{a} \\ = 3 k μ_{S}^{- 2} f_{1} e^{acf} [ψ d_{c} - Z_{c}^{B} d_{e} \partial_{B} d^{e}] d_{f} = 0 . \end{matrix}

(5.46)

Thus, for prescription (5.43) of a cubic crystal, all balance laws revert to those of Riemannian geometry.

Finally, consider a uniaxial (e.g., hexagonal) ferromagnetic crystal with c-axis [0001] oriented along $\frac{\partial}{\partial x^{3}}$ . In this case, the leading term in magnetic anisotropy energy is quadratic in sole component $ξ^{3}$ , and (5.43)–(5.46) become, with $k \propto μ_{S}^{2}$ , the following:

k \cdot f_{0} = - k \cdot (ξ^{3})^{2},

(5.47)

μ_{S}^{2} {\bar{\partial}}_{1} f_{0} (d (ξ)) = μ_{S}^{2} {\bar{\partial}}_{2} f_{0} (d (ξ)) = 0, μ_{S}^{2} {\bar{\partial}}_{3} f_{0} (d (ξ)) = - 2 d_{3};

(5.48)

\begin{matrix} \partial_{A} P_{a}^{A} + {\bar{\partial}}_{b} P_{a}^{A} \partial_{A} d^{b} + ρ_{0} (f_{a} + d^{b} \partial_{a} B_{b}) = 3 k μ_{S}^{- 2} P_{a}^{A} d_{3} \partial_{A} d^{3}, \end{matrix}

(5.49)

\begin{matrix} e^{a c f} [\partial_{A} Z_{c}^{A} + {\bar{\partial}}_{b} Z_{c}^{A} \partial_{A} d^{b} - Q_{c} + ρ_{0} B_{c}] d_{f} - (ρ_{0} / Δ) {\dot{d}}^{a} \\ = - 3 k μ_{S}^{- 2} [ψ e^{a 3 f} d_{3} - e^{acf} Z_{c}^{B} d_{3} \partial_{B} d^{3}] d_{f} . \end{matrix}

(5.50)

Finsler-originated contributions on the right-hand sides of (5.49) and (5.50) ( $k \neq 0$ ) to linear and electronic spin angular momentum manifest, respectively, as an effective body force (proportional to $P ξ^{3}$ ) and effective electronic torque (contributions proportional to $ψ ξ_{3}$ and $Z ξ_{3}$ ). These can be non-negligible only when $ξ^{3} (X) \neq 0$ , i.e., when magnetization $μ$ is partially or fully aligned with [0001] in the crystal, the direction normal to the basal plane.

6. Concluding remarks

Early and contemporary theories of continuum mechanics of solids with foundations in generalized Finsler geometry have been analyzed. A modern theory of Finsler-geometric continuum mechanics has been refined, with new analytical and numerical solutions reported for 1D problems involving fractures and phase transformations. The modern theory has also been newly adapted to model ferromagnetic solids in the magnetically saturated state. Differences among governing equations, and problem solutions where applicable, among theories based in Finsler space versus those based in affine Riemannian space have been highlighted. Results demonstrate how the Finsler approach might enrich descriptions of physical phenomena, requiring, minimally, only a single additional parameter entering Sasaki metric tensors.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

ORCID iD

John D. Clayton,

References

Sokolnikoff

Tensor Analysis. New York: John Wiley & Sons, 1951.

Truesdell

Toupin

. The classical field theories. In Flugge

(ed.) Handbuch der Physik, volume III/1. Berlin: Springer-Verlag, 1960, pp. 226–793.

Marsden

Hughes

Mathematical Foundations of Elasticity. Englewood Cliffs NJ: Prentice-Hall, 1983.

Bilby

Bullough

Smith

Continuous distributions of dislocations: A new application of the methods of non-Riemannian geometry. Proc Roy Soc A 1955; 231: 263–273.

Kröner

Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch Rat Mech Anal 1960; 4: 273–334.

Kondo

On the analytical and physical foundations of the theory of dislocations and yielding by the differential geometry of continua. Int J Eng Sci 1964; 2: 219–251.

Wang

CC.

On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch Rat Mech Anal 1967; 27: 33–94.

Cleja-Ţigoiu

Non-local elasto-viscoplastic models with dislocations in finite elasto-plasticity. Part I: Constitutive framework. Math Mech Solids 2013; 18: 349–372.

Bilby

Gardner

Grinberg

, et al. Continuous distributions of dislocations VI. Non-metric connexions. Proc Roy Soc Lond A 1966; 292: 105–121.

10.

Marcinkowski

The differential geometry of fracture. Acta Mech 1978; 1978: 175–195.

11.

De Wit

. A view of the relation between the continuum theory of lattice defects and non-Euclidean geometry in the linear approximation. Int J Eng Sci 1981; 19: 1475–1506.

12.

Clayton

Bammann

McDowell

A geometric framework for the kinematics of crystals with defects. Philos Mag 2005; 85: 3983–4010.

13.

Clayton

Defects in nonlinear elastic crystals: differential geometry, finite kinematics, and second-order analytical solutions. Z angew Math Mech 2015; 95: 476–510.

14.

Yavari

Goriely

The geometry of discombinations and its applications to semi-inverse problems in anelasticity. Proc R Soc Lond A 2014; 470: 0403.

15.

Roychowdhury

Gupta

Non-metric connection and metric anomalies in materially uniform elastic solids. J Elasticity 2017; 126: 1–26.

16.

Noll

Materially uniform simple bodies with inhomogeneities. Arch Rat Mech Anal 1967; 27: 1–32.

17.

Teodosiu

. Contributions to the continuum theory of dislocations and initial stresses. I. Rev Roum Sci Tech Mec Appl 1967; 12: 961–977.

18.

Teodosiu

. Contributions to the continuum theory of dislocations and initial stresses. III. Rev Roum Sci Tech Mec Appl 1967; 12: 1291–1308.

19.

Stumpf

Nonlinear continuum theory of dislocations. Int J Eng Sci 1996; 34: 339–358.

20.

Steigmann

Remarks on second-grade elasticity in plastically deformed crystals. Mech Res Commun 2020; 105: 103517.

21.

Epstein

de Leon

Geometrical theory of uniform Cosserat media. J Geom Phys 1998; 26: 127–170.

22.

Epstein

The Geometrical Language of Continuum Mechanics. Cambridge: Cambridge University Press, 2010.

23.

Schouten

Ricci Calculus. Berlin: Springer-Verlag, 1954.

24.

Clayton

On anholonomic deformation, geometry, and differentiation. Math Mech Solids 2012; 17: 702–735.

25.

Clayton

Nonlinear Mechanics of Crystals. Dordrecht: Springer, 2011.

26.

Steinmann

Geometrical Foundations of Continuum Mechanics. Berlin: Springer, 2015.

27.

Finsler

Uber Kurven und Flachen in allgemeiner Raumen. Dissertation, University of Gottingen, 1918.

28.

Rund

The Differential Geometry of Finsler Spaces. Berlin: Springer-Verlag, 1959.

29.

Bao

Chern

Shen

An Introduction to Riemann–Finsler Geometry. New York: Springer-Verlag, 2000.

30.

Berwald

Uber parallelubertragung in Raumen mit allgemeiner Maßbestimmung. Jahresber Deut Math Ver 1926; 34: 213–220.

31.

Cartan

Les Espaces de Finsler. Paris: Hermann, 1934.

32.

Chern

SS.

Local equivalence and Euclidean connections in Finsler spaces. Sci Rep Nat Tsing Hua Univ Ser A 1948; 5: 95–121.

33.

Bejancu

Finsler Geometry and Applications. New York: Ellis Horwood, 1990.

34.

Bejancu

Farran

Geometry of Pseudo-Finsler Submanifolds. Dordrecht: Kluwer, 2000.

35.

Minguzzi

The connections of pseudo-Finsler spaces. Int J Geom Meth Mod Phys 2014; 11: 1460025.

36.

Hushmandi

Rezaii

On the curvature of warped product Finsler spaces and the Laplacian of the Sasaki–Finsler metrics. J Geom Phys 2012; 62: 2077–2098.

37.

Watanabe

Ikeda

On metrical Finsler connections of a metrical Finsler structure. Tensor NS 1982; 39: 37–41.

38.

Miron

Metrical Finsler structures and metrical Finsler connections. J Math Kyoto Univ 1983; 23: 219–224.

39.

Rund

A divergence theorem for Finsler metrics. Monatsh Math 1975; 79: 233–252.

40.

Toupin

Theories of elasticity with couple stress. Arch Rat Mech Anal 1964; 17: 85–112.

41.

Mindlin

Microstructure in linear elasticity. Arch Rat Mech Anal 1964; 16: 51–78.

42.

Eringen

. Mechanics of micromorphic continua. In Kröner

(ed.) Mechanics of Generalized Continua. Berlin: Springer, 1968, pp. 18–35.

43.

Regueiro

On finite strain micromorphic elastoplasticity. Int J Solids Struct 2010; 47: 786–800.

44.

Eremeyev

Lebedev

Altenbach

Foundations of Micropolar Mechanics. Heidelberg: Springer, 2013.

45.

Eringen

Nonlinear Theory of Continuous Media. New York: McGraw-Hill, 1962.

46.

Randers

On an asymmetrical metric in the four-space of general relativity. Phys Rev 1941; 59: 195–199.

47.

Takano

. Theory of fields in Finsler spaces. I. Prog Theoret Phys 1968; 40: 1159–1180.

48.

Ikeda

On the conservation laws in the theory of fields in Finsler spaces. J Math Phys 1981; 22: 1211–1214.

49.

Ikeda

On the theory of fields in Finsler spaces. J Math Phys 1981; 22: 1215–1218.

50.

Brandt

Differential geometry of spacetime tangent bundle. Int J Theoret Phys 1992; 31: 575–580.

51.

Grifone

Structure presque-tangente et connexions I. Ann Inst Fourier 1972; 22: 287–334.

52.

Grifone

Structure presque-tangente et connexions II. Ann Inst Fourier 1972; 22: 291–338.

53.

Ikeda

A geometrical construction of the physical interaction field and its application to the rheological deformation field. Tensor NS 1972; 24: 60–68.

54.

Yajima

Nagahama

Differential geometry of viscoelastic models with fractional-order derivatives. J Phys A Math Theoret 2010; 43: 385207.

55.

Ohta

Sturm

KT.

Non-contraction of heat flow on Minkowski spaces. Arch Rat Mech Anal 2012; 204: 917–944.

56.

Miklashevich

Energy storage and geo-massif fracture. Nonlin Anal Real World Applicat 2009; 10: 2939–2944.

57.

Yajima

Nagahama

Finsler geometry of seismic ray path in anisotropic media. Proc Roy Soc Lond A 2009; 465: 1763–1777.

58.

Antonelli

Zastawniak

. Stochastic calculus on Finsler manifolds and an application in biology. In: Antonelli

Lackey

(eds.) The Theory of Finslerian Laplacians and Applications. Dordrecht: Springer, 1998, pp. 63–88.

59.

Vacaru

Stavrinos

Gaburov

, et al. Clifford and Riemann–Finsler Structures in Geometric Mechanics and Gravity. Geometry Balkan Press, 2005.

60.

Clayton

On Finsler geometry and applications in mechanics: review and new perspectives. Adv Math Phys 2015; 828475.

61.

Clayton

Finsler geometry of nonlinear elastic solids with internal structure. J Geom Phys 2017; 112: 118–146.

62.

Clayton

Generalized Finsler geometric continuum physics with applications in fracture and phase transformations. Z angew Math Phys 2017; 68: 9.

63.

Amari

A theory of deformations and stresses of ferromagnetic substances by Finsler geometry. In: Kondo

(ed.) RAAG Memoirs, Vol. 3. Tokyo: Gakujutsu Bunken Fukyu-Kai, 1962, pp. 257–278.

64.

Maugin

Eringen

Deformable magnetically saturated media. I. Field equations. J Math Phys 1972; 13: 143–155.

65.

Vargas

Torr

Finslerian structures: The Cartan–Clifton method of the moving frame. J Math Phys 1993; 34: 4898–4913.

66.

Chern

SS.

Finsler geometry is just Riemannian geometry without the quadratic restriction. Not Amer Math Soc 1996; 43: 959–963.

67.

Yano

Ishihara

Tangent and Cotangent Bundles: Differential Geometry. New York: Marcel Dekker, 1973.

68.

Stumpf

Saczuk

A generalized model of oriented continuum with defects. Z angew Math Mech 2000; 80: 147–169.

69.

The variation formulas of Finsler submanifolds. J Geom Phys 2011; 61: 890–898.

70.

Eringen

Tensor analysis. In: Eringen

(ed.) Continuum Physics, Vol. I. New York: Academic Press, 1971.

71.

Clayton

Differential Geometry and Kinematics of Continua. Singapore: World Scientific, 2014.

72.

Clayton

Finsler-geometric continuum dynamics and shock compression. Int J Fract 2017; 208: 53–78.

73.

Deicke

Finsler spaces as non-holonomic subspaces of Riemannian spaces. J Lond Math Soc 1955; 30: 53–58.

74.

Yano

Davies

On the connection in Finsler space as an induced connection. Rend Circ Mat Palermo 1954; 3: 409–417.

75.

Kondo

Non-Riemannian and Finslerian approaches to the theory of yielding. Int J Eng Sci 1963; 1: 71–88.

76.

Kröner

. Interrelations between various branches of continuum mechanics. In: Kröner

(ed.) Mechanics of Generalized Continua. Berlin: Springer, 1968, pp. 330–340.

77.

Ikeda

A physico-geometrical consideration on the theory of directors in the continuum mechanics of oriented media. Tensor, NS 1973; 27: 361–368.

78.

Stojanovitch

On the stress relation in non-linear thermoelasticity. Int J Non-Lin Mech 1969; 4: 217–233.

79.

Ikeda

On the covariant differential of spin direction in the Finslerian deformation theory of ferromagnetic substances. J Math Phys 1981; 22: 2831–2834.

80.

Kawaguchi

Beziehung zwishen einer metrishen linearen ubertragung und einer nicht-metrishen in einem allgemeinen metrishen Raum. Akad Wetensch Amsterdam Proc 1937; 40: 596–601.

81.

Ericksen

Tensor fields. In: Flugge

(ed.) Handbuch der Physik, Vol. III/1. Berlin: Springer-Verlag, 1960, pp. 794–858.

82.

Saczuk

Stumpf

On fibre bundle approach to a damage analysis. Int J Eng Sci 1998; 36: 1741–1762.

83.

Murakami

Mechanical modeling of material damage. J Appl Mech 1988; 55: 280–286.

84.

Miklashevich

Influence of the defect structure of the material on the fracture and crack motion. Russ Phys J 2002; 45: 1165–1169.

85.

Miklashevich

Geometric characteristics of fracture-associated space and crack propagation in a material. J Appl Mech Tech Phys 2003; 44: 255–261.

86.

Yajima

Nagahama

Connection structures of topological singularity in micromechanics from a viewpoint of generalized Finsler space. Ann Phys 2020; 532: 2000306.

87.

Saczuk

Finslerian Foundations of Solid Mechanics. Gdansk, Poland: Polskiej Akademii Nauk, 1996.

88.

Saczuk

On the role of the Finsler geometry in the theory of elasto-plasticity. Rep Math Phys 1997; 39: 1–17.

89.

Saczuk

Continua with microstructure modelled by the geometry of higher-order contact. Int J Solids Struct 2001; 38: 1019–1044.

90.

Saczuk

Hackl

Stumpf

Rate theory of nonlocal gradient damage-gradient viscoinelasticity. Int J Plasticity 2003; 19: 675–706.

91.

Clayton

Finsler-geometric continuum mechanics. Technical Report ARL-TR-7694, DEVCOM Army Research Laboratory, Aberdeen Proving Ground, MD, May 2016.

92.

Clayton

Finsler-geometric continuum mechanics and the micromechanics of fracture in crystals. J Micromech Mol Phys 2016; 1: 1640003.

93.

Clayton

Knap

Continuum modeling of twinning, amorphization, and fracture: Theory and numerical simulations. Continuum Mech Thermodyn 2018; 30: 421–455.

94.

Clayton

Generalized pseudo-Finsler geometry applied to the nonlinear mechanics of torsion of crystalline solids. Int J Geom Meth Mod Phys 2018; 15: 1850113.

95.

Clayton

Shock compression of metal single crystals modeled via Finsler-geometric continuum theory. AIP Conf Proc 2018; 1979: 180001.

96.

Clayton

Nonlinear Elastic and Inelastic Models for Shock Compression of Crystalline Solids. Cham: Springer, 2019.

97.

Watanabe

Ikeda

On a metrical Finsler connection of a generalized Finsler metric . Tensor NS 1983; 39: 97–102.

98.

Aryngazin

Precession of a gyroscope for the nonregular generalized Finslerian metric . Sov Phys J 1989; 32: 398–401.

99.

Capriz

Continua with Microstructure. New York: Springer, 1989.

100.

Emmerich

The Diffuse Interface Approach in Materials Science: Thermodynamic Concepts and Applications of Phase-field Models. Berlin: Springer, 2003.

101.

Anand

On H. Hencky’s approximate strain-energy function for moderate deformations. J Appl Mech 1979; 46: 78–82.

102.

Clayton

Analysis of shock compression of strong single crystals with logarithmic thermoelastic–plastic theory. Int J Eng Sci 2014; 79: 1–20.

103.

Borden

Verhoosel

Scott

, et al. A phase-field description of dynamic brittle fracture. Comput Meth Appl Mech Eng 2012; 217: 77–95.

104.

Clayton

Knap

A geometrically nonlinear phase field theory of brittle fracture. Int J Fract 2014; 189: 139–148.

105.

Weyl

Space-Time-Matter. 4th ed. New York: Dover, 1952.

106.

Jafarzadeh

Levitas

Farrahi

, et al. Phase field approach for nanoscale interactions between crack propagation and phase transformation. Nanoscale 2019; 11: 22243–22247.

107.

Levitas

Levin

Zingerman

, et al. Displacive phase transitions at large strains: Phase-field theory and simulations. Phys Rev Lett 2009; 103: 025702.

108.

Clayton

Knap

A phase field model of deformation twinning: Nonlinear theory and numerical simulations. Physica D 2011; 240: 841–858.

109.

Holder

Granato

Thermodynamic properties of solids containing defects. Phys Rev 1969; 182: 729–741.

110.

Maugin

Eringen

Deformable magnetically saturated media. II. Constitutive theory. J Math Phys 1972; 13: 1334–1347.

111.

Maugin

Continuum Mechanics of Electromagnetic Solids. Amsterdam: North-Holland, 1988.

112.

Landau

Lifshitz

Pitaevskii

Electrodynamics of Continuous Media. 2nd ed. Oxford: Pergamon, 1982.

113.

Tiersten

Tsai

On the interaction of the electromagnetic field with heat conducting deformable insulators. J Math Phys 1972; 13: 361–378.

114.

Lax

Nelson

Maxwell equations in material form. Phys Rev B 1976; 13: 1777–1784.

115.

Weile

Hopkins

Gazonas

, et al. On the proper formulation of Maxwellian electrodynamics for continuum mechanics. Continuum Mech Thermodyn 2014; 26: 387–401.