Generalized Finsler geometry and the anisotropic tearing of skin

A continuum mechanical theory with foundations in generalized Finsler geometry describes the complex anisotropic behavior of skin. A fiber bundle approach, encompassing total spaces with assigned linear and nonlinear connections, geometrically characterizes evolving configurations of a deformable body with microstructure. An internal state vector is introduced on each configuration, describing subscale physics. A generalized Finsler metric depends on position and the state vector, where the latter dependence allows for both direction (i.e., as in Finsler geometry) as well as magnitude. Equilibrium equations are derived using a variational method, extending concepts of finite-strain hyperelasticity coupled to phase-field mechanics to generalized Finsler space. For application to skin tearing, state vector components represent microscopic damage processes (e.g., fiber rearrangements and ruptures) in different directions with respect to intrinsic orientations (e.g., parallel or perpendicular to Langer's lines). Nonlinear potentials, motivated from soft-tissue mechanics and phase-field fracture theories, are assigned with orthotropic material symmetry pertinent to properties of skin. Governing equations are derived for one- and two-dimensional base manifolds. Analytical solutions capture experimental force-stretch data, toughness, and observations on evolving microstructure, in a more geometrically and physically descriptive way than prior phenomenological models.


Introduction
Finsler geometry and its generalizations suggest the possibility of enriched descriptions of numerous phenomena in mathematical physics, albeit at the likely expense of greater complexity of analysis and calculations compared to Riemannian geometry. Fundamentals of Finsler geometry, aptly credited to Finsler [1], are discussed in the classic monograph of Rund and the more recent text of Bao et al. [2,3]. See also the overview article by Eringen [4]. Extensions to pseudo-and generalized Finsler geometries are described in the monograph of Bejancu [5] and research cited therein [6][7][8], as well as several more recent works [9,10]. Generalized Finsler geometry is predominantly used herein, since strict classical Finsler geometry is insufficient to describe all phenomena pertinent to the present class of materials physics. Applications of (generalized) Finsler geometry in the broad physical sciences are vast and diverse; a thorough review is outside the scope of the present article. Available books discuss applications in optics, thermodynamics, and biology [11] and spinor structures and other topics in modern physics [12]. Finsler geometry and its generalizations have also been used for describing anisotropic spacetime, general relativity, quantum fields, gravitation, electromagnetism, and diffusion [13][14][15][16][17][18][19]. The current work implements a continuum mechanical theory for the behavior of solid materials.
In the classical Riemannian context, a continuous material body is viewed as a differentiable manifold M of dimension n, parameterized by coordinate chart(s) {X A } (A = 1, . . . , n). A Riemannian metric is introduced on M : in components, G AB = G AB (X) populate a symmetric and positive-definite n × n tensor field, where argument X implies functional dependence on the X A [22]. A covariant derivative ∇ (i.e., the linear connection on M ) completes the geometric description. The corresponding linear connection coefficients most generally consist of n 3 independent field components Γ A BC = Γ A BC (X). For usual solid bodies, n = 3, but other dimensions are permissible. Coordinate descriptions may be framed in terms of holonomic or anholonomic bases, where the latter do not correspond to continuous curves parameterizing the body [31,32]; anholonomic coordinates emerge for a multiplicative decomposition of the deformation gradient [33,34].
In Finsler geometry and its generalizations, a base manifold M , parameterized by one or more charts {X A } (A = 1, . . . , n), is again introduced. A fiber bundle (Z , M , Π , U ) of total (generalized Finsler) space Z of dimension n+m is constructed. The total space in Finsler geometry is typically identified with the slit tangent bundle, i.e., Z → TM \0 [3] with m = n, but this is not essential in more general formulations [5,9,10]. Auxiliary coordinates {D K } (K = 1, . . . , m) cover each fiber U , such that Z is parameterized by {X A , D K }. Particular transformation laws are assigned for changes of coordinates associated with X and D. Nonlinear connection coefficients define nonholonomic bases that transform conventionally on T M under X-coordinate changes. Furthermore, at least two, and up to four [5,35], additional fields of connection coefficients are needed to enable horizontal and vertical covariant derivatives with respect to X and D on Z [2,5,10].
The generalized pseudo-Finsler metric tensor is of the form G AB = G AB (X, D), always symmetric. The metric is positive definite for Finsler geometry, but not always so for the pseudo-Finsler case [10]. For strict Finsler geometry (but not necessarily its generalizations [5,6,18,36]), G AB are second derivatives of a (squared) fundamental function 1 2 F 2 with respect to D A [2,3]. In Finsler geometry, F is positively homogeneous of degree one in D; the resulting metric is homogeneous of degree zero in D [2,3] and thus should not depend only on the magnitude of a vector comprised of components {D A }. In (generalized) Finsler space, the (X, D)-coordinate dependencies of the metric and various linear and nonlinear connections enter other geometric objects and tensorial operations: torsion and curvature forms, volume and area forms, the divergence theorem [37], etc.
Motivation for Finsler geometry is description of detailed physics via the set of auxiliary coordinates {D A } attached to each position X in real physical space. In solid mechanics, the idea can be interpreted as an extension of micropolar, micromorphic, or Cosserat-type theory [38][39][40][41][42][43] from Riemannian (and more often, Euclidean) space to generalized Finsler space. However, in classical micromorphic theories, a Riemannian, rather than Finsler, metric is used, the material domain with microstructure is fully parameterized by the {X A }; basis vectors and coordinate transformation laws are those of classical continuum mechanics, as are integral theorems. The director triads of micromorphic theories enter the balance laws and constitutive functions, but they do not affect geometric objects and covariant derivatives in the same way as D of generalized Finsler space.

Prior work
The first application of Finsler geometry in the context of continuum mechanics of solids appears to be the treatment of ferromagnetic elastic-plastic crystals of Amari [44]. Conservation laws and field theories, with application to ferromagnetics, were further developed by Ikeda [45][46][47][48]. Bejancu [5] gives a generalized Finsler treatment of kinematics of deformable bodies. More contemporary theories includes those of Saczuk and Stumpf [49][50][51], with underpinnings in a monograph [52]. Different physical phenomena (e.g, different physical meanings of {D K } [51]) are encompassed by their models that include kinematics, balance laws, and thermodynamics, but their focus is most often on mechanics of elastic-plastic crystals and dislocations [49,50,52]. See also recent theory [53] that applies generalized Finsler geometry to topological defects and the comprehensive review [54] of prior works on generalized Finsler geometry in continuum physics.
A new theory of Finsler-geometric continuum mechanics was developed for nonlinear elastic solids with evolving microstructure, first published in the article [55] with a preliminary version in a technical report [56]. This variational theory was extended to allow for explicit inelastic deformation and applied to study phase transitions and shear localization in crystalline solids [55,57]. The theory has also been broadened for dynamics and shock waves [58,59], and most recently has been used to describe ferromagnetic solids [54], enriching the governing equations of Maugin and Eringen [60,61] with pertinent aspects arising from Finsler geometry [44,48] .
Prior to this theory [54,55], pragmatic solutions of boundary value problems using continuum mechanical models incorporating generalized Finsler geometry appeared intractable due to complexity of governing equations and unwieldy parameterizations (e.g., uncertain constitutive functions and material constants). Most aforementioned work [5, 44-49, 51, 53] presented purely theoretical constructions without attempt to formulate or solve physical boundary value problems. A material response was calculated by Saczuk and Stumpf [50,52], but motion and internal state coordinates were prescribed a priori, without apparent solution of governing conservation laws for macroscopic and microscopic momentum and energy. In contrast, the present theory [55,56] appears to be the first Finsler geometry-based continuum mechanics theory for which analytical and numerical solutions to the governing equations have been found, as evidenced by solutions to numerous problems for (non)linear elastic materials with evolving microstructure (e.g, fractures, twinning, phase transitions, dislocations), as evidenced in those and subsequent works [54][55][56][57][58][59]62].
All prior applications considered stiff crystalline solids or generic materials. The current research newly applies the theory to soft biological tissues, specifically the skin. Furthermore, prior applications in fracture and cavitation [54,55,59,62] were limited to either locally isotropic damage or to local material separation on a single cleavage plane. The current treatment advances the description to anisotropic fractures or ruptures on multiple material surfaces at a single point X. Most cited prior applications invoked only a single non-trivial state vector component in D (an exception being a multi-component D for twinning and fracture [59]) and most often conformal Weyl-type rescaling of G AB with canonically vanishing nonlinear connection (with a few exceptions studied [57,62]). The current research incorporates an anisotropic generalized Finsler metric for multi-dimensional problems and non-trivial nonlinear connections to show utility by example.

Purpose and scope
The scope of this paper covers two primary purposes: • Demonstration of utility of the generalized Finsler geometric theory for describing anisotropic elasticity and anisotropic structural rearrangements in soft biological tissue; • Consolidation and refinement of the theory for the equilibrium (i.e., quasi-static) case. The first item furnishes the first known application of Finsler geometry-based continuum theory to analyze finite-strain mechanics of soft biological tissue. Prior work of others [63,64] used ideas from Finsler geometry to reproduce nonlinear stress-strain to failure responses of biologic solids, but that work used a discrete, rather than continuum, theory with material points represented as vertices linked by bonds; interaction potentials comprised bonding energies within a Hamiltonian. In that approach [65][66][67], a Finsler metric for bond stretch depends on orientation of local microstructure entities (e.g., molecular chains or collagen fibers) described by the Finsler director vector field D. Instead, the current continuum theory considers, in a novel way, effects of microstructure on anisotropy (elastic and damage-induced) in both a geometric and constitutive sense. The second item includes a renewed examination of Rund's divergence theorem [37] in the context of an osculating Riemannian metric. It is shown that certain choices of metric and connection coefficients, with possible addition of a source term to the energy conservation law, can recover governing equations for biologic tissue growth [30] in the quasi-static limit (Appendix B).

Soft tissue and skin mechanics
Most soft tissues have inherent directionality due to their collagen fiber-based and/or aligned cellular microstructures [68,69], toward which tools of analysis from Finsler geometry might be anticipated to aptly apply. The mechanics of skin deformation [68,70,71], degradation [72,73], and tearing [73,74] are investigated herein. Like most biological materials, microstructure of skin is complex. The respective middle and outer layers of skin are the dermis and epidermis, with elastin and collagen fibers and cells embedded in a ground matrix. Underlying hypodermis (i.e., adipose) can be labeled an inner layer of the skin. The microstructure dictates nonlinear, anisotropic, viscoelastic, and tearing behaviors [74][75][76]. Mechanical behavior at small strains is primarily controlled by the elastin and ground substance, whereby the collagen fibers are coiled or slack [75]. Under increasing tensile stretch, the collagen fibers straighten and tighten, supporting most of the load, and compliance decreases. Under more severe stretching, fibers slide, delaminate, and rupture, leading to reduced stiffness, strain softening, and material failure [72][73][74]77].
Experiments indicate that skin elasticity has orthotropic symmetry [68,70,71,75]. Orthotropy arises from preferred arrangements of the collagen fibers, leading to greater stiffness along direc-tions along which more fibers are aligned. In the plane of the dermis, fibers tend to be dispersed about a primary axis along which stiffness is greatest. In vivo, resting skin tension is greatest along this axis, parallel to Langer's lines [75]. In typical uniaxial and biaxial tests [68,70,71,74], extracted skin is unstretched initially, but the greater stiffness along the primary direction persists, with differences in stiffness also emerging between orthogonal in-plane and out-of-plane directions [70,75]. As might be expected, damage processes are also anisotropic due to fiber degradation that differs with respect to direction of loading relative to the microstructure [73,74].
Skin, as is most biological tissue, is simultaneously nonlinear elastic, viscoelastic, and poroelastic [68,76,78,79]; pertinence of mechanisms depends on the time scale of loading. The present application considers only monotonic loading at a constant rate (e.g, no cycling or rate fluctuations). Loading rates are assumed much slower or faster than viscous relaxation times. Thus, the pseudo elastic approach is justified to study these experiments [68], whereby hyperelastic models are deemed reasonable [71,[80][81][82][83], albeit noting that different elastic constants (e.g., static and dynamic moduli) are needed to fit data at vastly different limiting low and high loading rates [84,85]. In future applications to problems with time dependence, internal state variables can be extended, leading to kinetic laws with explicit viscous dissipation [78,86]. The current study is limited to relatively small samples, tested in vitro, under uniaxial or biaxial extension [68,70,74,87]. The material is modeled as unstressed initially and homogeneous with regard to elastic properties. In the future, the current theory can be extended to study residual stress due to growth or heterogeneous material features, as well as heterogeneous elastic properties. Residual stresses can be addressed, in the context of Riemannian manifolds, using a material metric having a non-vanishing Riemann-Christoffel curvature of its Levi-Civita connection [27,30] or an anholonomic multiplicative term in the deformation gradient [29,88]. These ideas may be extended to generalized Finsler space (e.g., invoking the current fiber bundle approach) in future.
An early nonlinear elastic model described orthotropic symmetry using a phenomenological pseudo-strain energy potential [89]. Another early model delineated contributions of elastin and collagen fibers [79]. More recently, a class of nonlinear elastic models accounting for anisotropy from fiber arrangements using structure tensors has been successful for representing many soft tissues, including arterial walls [80,90], myocardium [82,91], and skin [71]. Polyconvex energy potentials can be incorporated for stability and to facilitate existence of (unique) solutions to nonlinear elastic problems [81,90]. Fiber dispersion can be incorporated to modulate the degree of anisotropy [71,92]. To date, most damage models accounting for softening and failure have been phenomenological, whether implemented at the macroscopic scale (either isotropic or along preferred fiber directions) or at the scale of individual fibers and their distributions [73,77,90,93]. These damage models, with a basis in continuum damage mechanics [94], are thermodynamically consistent in the sense that damage is dissipative, but their particular kinetic laws and (often numerous) parameters are calibrated to experimental data without much physical meaning. In contrast, the phase-field approach has been recently implemented for soft-tissue fracture or rupture, incorporating relatively few parameters with physical origin (e.g., surface energy) and regularization facilitating unique solutions to problems involving material softening [95,96]. The kinetic law or equilibrium equation for damage is derived from fundamental principles [97] and drives material to a local minimum-energy state, in contrast to ad hoc equations simply selected to match data.

Overview of the current work
Implementation of the present generalized Finsler theory consists of four key elements: definition of the internal state D, assignment of the metric tensor, assignment of the linear and nonlinear connections, and prescription of the local free energy potential. For soft tissue mechanics, the state vector represents the fiber rearrangements. Damage anisotropy is monitored via its direction, with different components of D reflecting fiber reorganization and rupture with respect to orientations of microstructure features [73,74]; the magnitude of each component of D measures local intensity of damage in a given material direction. The metric tensor with components G AB (X, D) depends on position X as well as direction and magnitude of D in generalized Finsler space; novel D-dependence captures rescaling of the material manifold as damage entities open, close, or rearrange in different directions [54,62]. The preferred linear connection is that of Chern and Rund [3], ensuring compatibility with the divergence theorem used to derive the Euler-Lagrange equations [54,55]. The generalized Finslerian D-dependence of both the metric and linear connection explicitly affect the governing equations. Roles of nonlinear connections are newly examined; a non-trivial prescription is shown to influence the fracture energy and stress-strain response.
The free energy density consists of nonlinear elastic contribution and an internal structure contribution. The nonlinear elastic potential enriches the orthotropic theory of Holzapfel, Ogden, Gasser, and others [71,80,82,83,92] with implicit contributions from the generalized Finsler metric as well as anisotropic degradation from D. The structural contribution is motivated from phasefield mechanics [95,98]. A previous model for arterial dissection [95] accounted for fiber-scale damage anisotropy using a scalar order parameter. The current theory invokes a more physically descriptive, vector-valued order parameter (i.e., normalized D) of generalized Finsler type. With regard to skin experiments, solutions obtained for the current model are shown to admirably match extension and failure data, including stress-strain behavior and fracture toughness [73,74,99] with parameters having physical or geometric origins. The general theory is thus potentially more physically realistic, and considered more descriptive from a geometric perspective, than past models based on phenomenological damage mechanics [90,94,100,101]. This paper is organized as follows. Mathematical preliminaries (e.g., notation and definitions for objects in referential and spatial configurations) are provided in §2. The Finsler-geometric theory of continuum mechanics is presented in §3, including kinematics of finite deformation and equilibrium equations derived with a variational approach. The next two sections specialize the theory to model soft tissue, specifically skin. In §4, a one-dimensional (1-D) model for the base manifold M is formulated. Analytical and semi-numerical solutions are obtained for uniaxial extension and compared to experimental data. In §5, a two-dimensional (2-D) model for M is formulated, whereby the skin has orthotropic symmetry; solutions are obtained for biaxial extension with anisotropic damage in orthogonal material directions. Conclusions follow in §6.

Generalized Finsler space
Content of §2 consolidates a more thorough exposition given in a recent review [54], from which notation is adopted. Other extensive texts include those of Rund, Bejancu, and Bao et al. [2,3,5].
A new contribution in the present §2 is interpretation of the divergence theorem [37,54] using an osculating Riemannian metric, whereby for the further simplifying assumption of vanishing nonlinear connection, a representation akin to that of classical Riemannian geometry is obtained.

Reference configuration
The very general fiber bundle approach of Bejancu [5] encompasses geometric fundamentals of the theory. The reference configuration is linked to a particular instant in time at which a deformable solid body is undeformed relative to some intrinsic state. A differential manifold M of dimension n is physically identified with a body embedded in ambient Euclidean space of dimension N ≥ n.
Remark 2.1.1. 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 §2.1.5, can generally be embedded in Euclidean space [2,102,103].
Let X ∈ M denote a material point or particle, and let {X A }(A = 1, 2, . . . , n) denote a coordinate chart that may partially or completely cover M . Attached to each material point is a vector D; chart(s) of secondary coordinates {D K }(K = 1, 2, . . . , m) are assigned over M . Fields {D K } are smooth over M : D is as many times continuously differentiable with respect to {X A } as needed. Define Each fiber is a vector space of dimension n; (Z , Π , M ) constitutes a vector bundle. Let M ′ ⊂ M be an open neighborhood of any X ∈ M , Φ an isomorphism of vector spaces, and P 1 a projection operator onto the first factor. Then the following diagram is commutative [5]: Given (2.1), { ∂ ∂ X A } and {dD K } do not transform as conventional basis vectors on Z . Define [5,9] The set { δ δ X A , ∂ ∂ D K } are used as a convenient local basis on T Z , and the dual set {dX A , δ D K } on T * Z [3,9]. The N K B (X, D) are the nonlinear connection coefficients; N K B are presumed differentiable with respect to (X, D). These do not obey coordinate transformation rules for linear connections nor always correspond to a covariant derivative with the properties of a linear connection. For (2.7) to hold under coordinate transformations X →X [3,5], (2.8) The geometry of tangent bundle T Z with nonlinear connection admits an orthogonal decomposition T Z = V Z ⊕ HZ into a vertical vector bundle V Z with local field of frames { ∂ ∂ D A } and a horizontal distribution HZ with local field of frames { δ δ X A } [5]. Fibers of V Z and HZ are of respective dimensions m and n. Henceforth, vertical and horizontal subspaces are of the same dimension: m = n. Indices J, K, . . . can thus be replaced with A, B, . . . in the summation convention, which runs from 1 to n. In (2.1), let A formal way of achieving (2.9) via soldering forms is given by Minguzzi [10]. Coordinate differentiation operations are expressed as follows, with f a differentiable function of arguments (X, D): The special cases f → X and f → D are written [54,55] (2.11)

Length, area, and volume
The Sasaki metric tensor [3,35,104] enables a natural inner product of vectors over Z : (2.13) Components of G andǦ are equal and simply hereafter referred to as G AB , but their bases span orthogonal subspaces. Components G AB and inverse components G AB lower and raise indices in the usual manner, and G denotes the determinant of the n × n non-singular matrices of components of G orǦ: (2.14) When interpreted as a block diagonal 2n × 2n matrix, the determinant of G G G is [49,50,52] (2.15) Let dX denote a differential line element of M referred to non-holonomic horizontal basis { δ δ X A }, and let dD denote a line element of U referred to vertical basis { ∂ ∂ D A }. Squared lengths are The respective volume element dV and volume form dΩ of the n-dimensional base manifold M , and the area form Ω for its boundary ∂M , are defined by [37] Local coordinates on the (n − 1)-dimensional oriented hypersurface ∂M are given by parametric equations

Covariant derivatives
Horizontal gradients of basis vectors are determined by generic affine connection coefficients H A BC and K A BC , where ∇(·) is the covariant derivative: Analogously, vertical gradients employ generic connection coefficients V A BC and Y A BC : (2.20) For example, let V = V A δ δ X A ∈ HZ be a vector field. Then the (total) covariant derivative of V is   [4,22,31,98] and is the transpose of others [2,5,34]. For symmetric connections, it is inconsequential.
Components of the horizontal covariant derivative of metric tensor G = G AB dX A ⊗ dX B (i.e., the horizontal part of G G G ) are The following identity is also noted for G = det(G AB ), a scalar density [37]: Christoffel symbols of the second kind for the Levi-Civita connection are γ A BC , Cartan's tensor is C A BC , and horizontal coefficients of the Chern-Rund and Cartan connections are Γ A BC . All are torsion-free (i.e., symmetric): Nonlinear connection coefficients N A B (X, D) admissible under (2.1) and (2.8) can be obtained in several ways. When T Z is restricted to locally flat sections [3,10] (2.28) Remark 2.1.7. Let G AB (X, D) be positively homogeneous of degree zero in D. Then G A below are components of a spray [3,10], and canonical nonlinear connection coefficients N A B = G A B that obey (2.8) are Then a complete generalized Finsler connection is the set (N

A divergence theorem
Let M be a manifold of dimension n having (n − 1)-dimensional boundary ∂M of class C 1 , a positively oriented hypersurface. Stokes' theorem for a Proof. The proof, not repeated here, is given in the review article [54], implied but not derived explicitly in an earlier work [55]. The proof of (2.31) [54] extends that of Rund [37]-who specified a Finsler space F n with Cartan connection (G A B ,Γ A BC ,C A BC ) and metric acquired from a Finsler (Lagrangian) function F ( §2.1.5)-to a generalized Finsler space with arbitrary positive-definite metric G AB (X, D) and arbitrary nonlinear connection N A B (X, D). □ Remark 2.1.10. A different basis and its dual over M could be prescribed for V and N given certain stipulations [54]. However, geometric interpretation of covariant differentiation on the left side of (2.31) suggests { δ δ X A } should be used for V, by which dual basis {dX B } should be used for N to ensure invariance: As assumed in Theorem 2.1.1 [37,54,55], C 1 functions D = D(X) must exist over all X ∈ M . Relations of generalized Finsler geometry [5] still apply, but additional relations emerge naturally when metric G AB is interpreted as an osculating Riemannian metric [2,44]. Specifically, an alternative representation of (2.31) is newly proven in the following. where the vectorṼ A (X) = V A (X, D(X)), unit normalÑ A (X) = N A (X, D(X)), and covariant deriva- Proof. The right of (2.32) is identical to the right of (2.31) given the change of variables. In the left of (2.32), from chain-rule differentiation, vanishing (2.23), and (2.27), (2.34) Adding (2.33) to (2.34) with canceling ±N B A∂ B V A terms then produces Integrands on the left sides of (2.31) and (2.32) are thus verified to match, completing the proof. □ Remark 2.1.11. Coefficients of the Levi-Civita connection ofG AB satisfy the symmetry and metric-compatibility requirements used to prove (2.32):

Pseudo-Finsler and Finsler spaces
Preceding developments hold for generalized Finsler geometry, by which the metric tensor components need not be derived from a Lagrangian [5,6,36]. Subclasses of generalized Finsler geometry do 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 → 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 ∞ is often assumed [3], but C 5 is usually sufficient [10]). Then (M , L ) is a pseudo-Finsler space when the n × n matrix of components G AB is both non-singular over Z and obtained from Lagrangian L : [2,3], the fundamental scalar Finsler function F (X, D) is introduced, positive homogeneous of degree one in D: In Finsler geometry [2,3,5], it follows that L = L and G A = G A in (2.28) and (2.29), and that

Spatial configuration
A description on a fiber bundle analogous to that of §2.1 is used for the spatial configuration (i.e., current configuration) of a body. A differential manifold m of dimension n represents a (deformed) physical body, with base space embedded in ambient Euclidean space of dimension N ≥ n. Let x ∈ m denote the spatial image of a body particle or point with {x a }(a = 1, 2, . . . , n) being a coordinate chart on m. At each spatial point is a vector d, and chart(s) of secondary coordinates {d k }(k = 1, 2, . . . , m) are assigned over m. Define z = (z, π, m, u) as a fiber bundle of total space z (dimension n + m), where π : z → 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, referred to herein as the motion. The global mapping from referential to current total spaces is the set Ξ = (ϕ, θ ), where in general ϕ(X, D) : M → m and Ξ (X, D) : Z → z. Functional forms of ϕ(X, D) and Ξ (X, D) vary in the literature [54]; details are discussed in §3.1. Mappings and field variables can be made time (t) dependent via introduction of independent parameter t [50,58,59]. Explicit time dependence is excluded from the current theoretical presentation that focuses on equilibrium configurations [55,62]. The following diagram commutes [5]:

Basis vectors and nonlinear connections
Coordinate transformations from {x, d} to {x,d} on z are of the general Finsler form Tangent bundle T z with nonlinear connection admits an orthogonal decomposition into vertical vector bundle and horizontal distribution: T z = V z ⊕ Hz. The transformation law of the spatial nonlinear connection isÑ Subsequently, take m = n. Indices j, k, . . . → a, b, . . ., summation over duplicate indices is from 1 to n, and in (2.40), the d a transform like components of a contravariant vector field over M : Spatial coordinate differentiation is described by the compact notation (2.45)

Length, area, and volume
The Sasaki metric tensor that produces an inner product of vectors over z is [104] Denote by dx a differential line element of m referred to non-holonomic horizontal basis { δ δ x a } and dd a differentiable line element of u referred to vertical basis { ∂ ∂ d a }. Their squared lengths are The scalar volume element and volume form of m, where dim m = n, and the area form of ∂m, the (n − 1)-dimensional boundary of a compact region of m, are respectively

Covariant derivatives
Denote by ∇ the covariant derivative. Horizontal gradients of basis vectors are determined by coefficients H a bc and K a bc , and vertical gradients by V a bc and Y a bc : By example, covariant derivative operations over z are invoked like (2.21) for V = V a δ δ x a ∈ Hz: (2.52) Herein (·) |a and (·)| b denote horizontal and vertical covariant differentiation with respect to coordinates x a and d b . Let γ a bc be coefficients of the Levi-Civita connection on z, C a bc coefficients of the Cartan tensor on z, and Γ a bc horizontal coefficients of the Chern-Rund and Cartan connections on z:

A divergence theorem
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 ∂m. Let α α 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 δ δ x a ∈ Hz. Denote the field of components for positive-definite metric tensor on the horizontal subspace by g ab (x, d) with g = det(g ab ) > 0. Assign horizontal connection H a bc = H a cb such that ( √ g) |a = 0 (e.g., H a bc = Γ a bc ) , and assume that C 1 functional relations d = d(x) exist for representation of the vertical fiber coordinates ∀x ∈ m. Then in a chart {x a }, with volume and area forms given in (2.49), with n a the unit outward normal on ∂m, Proof matches that of Theorem 2.1.1 upon changes of variables; a corollary akin to Corollary 2.1.1 also holds.

Finsler-geometric continuum mechanics
The original theory of Finsler-geometric continuum mechanics [55,56] accounts for finite deformations under conditions of static equilibrium for forces conjugate to material particle motion and state vector evolution. Subtle differences exist among certain assumptions for different instantiations, incrementally revised in sucessive works. Most differences are explained in a review [54].

Motion and deformation
Particle motion ϕ : M → m and its inverse Φ : m → M are the one-to-one and C 3 -differentiable functions x a = ϕ a (X), Remark 3.1.1. Vector field D and its spatial counterpart d are referred to as internal state vector fields or director vector fields, but neither vector must be of unit length. These are assigned physical interpretations pertinent to the specific class of mechanics problem under consideration [54].
Motions of state vectors are defined as C 3 functions: Extension for m ̸ = n is conceivable [5,45]. However, setting m = n enables a more transparent physical interpretation of the vertical vector bundle, and it allows use of (2.9) and (2.43) that simplify notation and calculations. For usual three-dimensional solid bodies, n = 3 as implied in parts of prior work [54], but other dimensions are permissible (e.g., two-dimensional membranes (n = 2) and one-dimensional rods (n = 1)).

From (3.1) and (3.2), transformation formulae for partial differentiation operations between configurations of a differentiable function
Remark 3.1.3. Unlike Chapter 8 of Bejancu [5], basis vectors need not convect from T Z to T z with the motion Ξ . Rather, as in classical continuum field theories of mechanics [20,105], basis vectors-as well as metric tensors and connection coefficients-can be assigned independently for configuration spaces Z and z. As such, by which δ A (·) = F a A δ a (·) simply relates the delta derivative across configurations. As implied in (3.4), deformation gradient field F : HZ → Hz is defined as the two-point tensor field with (3.1) used in the rightmost equality. The inverse deformation gradient f : Hz → HZ is defined as the following: (3.6) Usual stipulations on regularity [22] of motions (3.1) apply such that det(F a A ) > 0 and det( f A a ) > 0. Transformation equations relating differential line elements of (2.16) and (2.48) follow: Advancing (3.7), with definition of the determinant, (2.17), and (2.49), volume elements and forms, respectively, transform between reference and spatial representations on M and m, with J = det(F a A ) g/G and j = 1/J = J −1 > 0, via (e.g., [22,98] Strain can be quantified using symmetric Lagrangian deformation tensor C = C AB dX A ⊗ dX B : Then from the first of (2.50) and (3.4) [56,62], Similarly, the second of (2.50) gives , though this is not needed later.

Particular assumptions 3.2.1 Director fields
The divergence theorem (2.31) is used to derive Euler-Lagrange equations for equilibrium of stress and state vector fields in §3.3.3. Its derivation [37,54] requires existence of functional relations where the second of (3.12) is implied by the first under a consistent change of variables per (2.56) and (3.1). Existence of the following functional forms emerges from (3.1), (3.2), and (3.12): In some prior work [55,56], alternative representations of particle motions incorporating state vector fields as arguments have been posited. These likely more complex alternatives are admissible but inessential [54]. The current theory, like some others [44,50,52], does not always require θ or Θ be specified explicitly, though use of the former is implied later in §5.

Connections and metrics
Use of (2.31) for any admissible G AB (X, D) necessitates a symmetric linear connection horizontally compatible with G AB , meaning H A BC = Γ A BC , with Γ A BC Chern-Rund-Cartan coefficients of (2.26). The simplest admissible choice of vertical coefficients is V A BC = 0, corresponding to the Chern-Rund connection [2,3,106]. The canonical choice N A B = G A B of (2.29) also corresponds to the Chern-Rund connection, but it is inessential for generalized Finsler geometry. Choices K A BC = H A BC [55,56] and Y A BC = V A BC are logical given (2.9), but these are not mandatory. Setting K A BC = 0, providing compatibility with Cartesian metric δ AB , may also be of utility [54].
Given a Sasaki metric G G G of (2.12) with G AB of (2.13), pragmatic connection coefficients over Z are summarized in (3.15); complementary connections over z given Sasaki metric g g g with g ab (x, d) of (2.47) follow thereafter: If the fields G AB (X, D) and g ab (x, d) are known, liinear connection coefficients in (3.15) can be calculated from definitions in §2.1 and §2.2. Metric G AB need not be homogeneous of degree zero with respect to D, but it can be. Components G AB need not be derived, as in §2.1.5, from a Lagrangian L or more specifically a fundamental Finsler function F , but they can be. Dependence of G G G on X and D is based on symmetry and physics pertinent to the particular problem of study. Similar statements describe the spatial metric g g g and components g ab .
A decomposition of G AB into a Riemannian partḠ AC and a director-dependent partĜ C B is useful for describing fundamental physics and for solving boundary value problems [55][56][57]62]: More specific functional forms in (3.16) are advocated herein, as implied by past applications [54]: Remark 3.2.5. Components ofḠ AB are chosen to best represent symmetry of the physical body; in elasticity, often a Riemannian metric for rectilinear, cylindrical, or spherical coordinates on M . Components ofĜ C B are assigned based on how microstructure D affects measured lengths of material elements with respect to an observer in total generalized Finsler space Z (i.e., the space of the physical body enriched with microstructure geometry) [54,55,62].
Ideas apply analogously to spatial metric g ab (x, d) with X replaced by x, and with D replaced by d. For example, the spatial analog of (3.17) is All metrics in (3.17) and (3.18) are assumed invertible with positive determinants. A symmetric tensorC [62] and volume ratioJ > 0 are defined to exclude internal state-dependence of strain:

Variational principle
A variational principle [54][55][56] is implemented. Let Ψ denote the total energy functional for a compact domain M ′ ⊂ M with positively oriented boundary ∂M ′ , and let ψ be the local free energy density per unit reference volume of material: Denote surface forces as p = p a dx 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. Denote a generic local, vector-valued volumetric source term conjugate to structure variations by R = R A δ D A , extending prior theory [54][55][56] to accommodate more possible physics [30,107] (Appendix B). A variational principle for Finsler-geometric continuum mechanics, holding X fixed but with x = ϕ(X) and D variable, is In coordinates, with variation of D in parentheses to distinguish from non-holonomic basis {δ D A }, Results used in §3.3.3 are now noted, with α = 1 or α = 2 (derived in Appendix A using (3.15)):

General energy density
As evident in (3.22), independent variables entering total free energy density, per unit reference volume, function ψ are the deformation gradient, the internal state vector, the horizontal gradient of the internal state vector, and the reference position of the material particle: Dependence on F accounts for bulk elastic strain energy. Dependence on D generally accounts for effects of microstructure on stored energy. Energy from heterogeneity of microstructure (e.g., internal material surfaces) is captured by dependence on the internal state gradient: Dependence on X permits heterogeneous properties. Prior work [54,55] adds motivation for (3.26). .15), provides no information, so it is excluded from the arguments of energy density in (3.26).
Expansion of the integrand on the left in (3.23), with δ X A = 0 by definition, is Denoted by P is the mechanical stress tensor (i.e., the first Piola-Kirchhoff stress, a two-point tensor, generally non-symmetric), Q an internal force vector conjugate to D, and Z a micro-stress tensor conjugate to the horizontal gradient of D.

Euler-Lagrange equations
Connection coefficients in (3.15) are employed along with (3.1), (3.11), (3.12), (3.24), and (3.25). Insertion of (3.29) into the left side of (3.23), followed by integration by parts and use of (2.31) of Euler-Lagrange equations consistent with any admissible variations δϕ ϕ ϕ and δ D locally at each X ∈ M ′ , as well as natural boundary conditions on ∂M ′ are obtained as follows. Steps follow those outlined in the original works [55,56] with minor departures [54]. The first of these culminating Euler-Lagrange equations is the macroscopic balance of linear momentum, derived by setting the first integral on the right-hand side of (3.30) equal to zero, consistent with the right side of (3.23). Localizing the outcome and presuming the result must hold for any admissible variation δ ϕ a , The second Euler-Lagrange equation is the balance of micro-momentum (i.e., director momentum or internal state equilibrium). It is derived by setting the second integral on the right side of (3.30) equal to the rightmost term in (3.23) and then localizing, giving for any admissible variation δ (D C ), (3.32) Natural boundary conditions on ∂M ′ are derived by setting the second-to-last and last boundary integrals in (3.30) equal to the remaining, respective first and second boundary integrals on the right side of (3.23) and localizing the results, yielding for any admissible variations δ ϕ a and δ (D C ), With natural boundary conditions (3.33) or essential boundary conditions (i.e., prescribed ϕ ϕ ϕ(X) and D(X) for X ∈ ∂M ′ ) and local force density vector R(X) for each X ∈ M ′ , (3.31) and (3.32) comprise 2n coupled PDEs in 2n degrees-of-freedom x a = ϕ a (X) and D A (X) at any X ∈ M ′ , and by extension, any X ∈ M .
Remark 3.3.3. Consider the simplified case when Riemannian metrics are used: no D-dependence of G and no d-dependence of g. Then Γ A BC = γ A BC , Γ a bc = γ a bc , and C A BC = 0. The right side of (3.31) vanishes, and (3.31) is of the form of the static momentum balance of classical continuum mechanics with null body force [22,23,33]. Further taking N A B and K A BC independent of D, (3.32) is similar to equilibrium equations for gradient materials [108] as in phase-field mechanics [97,109].
In some prior work [55], G AB (X, D) was an argument of ψ, extending (3.26), and D-dependence of the metric manifested in a distinct thermodynamic force, rather than entering implicitly in Q A . The present approach is favored for brevity [54], but the former is admissible. Proposition 3.3.1. Euler-Lagrange equations can be expressed in the following alternative way:

(3.35)
Proof. From (2.10) and (2.27), , in all forms of the Euler-Lagrange equations is also a distinctive feature. This term, of course, vanishes when G AB is independent of D (i.e., a Riemannian rather than Finslerian metric).

Spatial invariance and material symmetry
First consider rotations of the spatial frame of reference, given by orthonormal transformation q a b in (2.40) whereby det(q a b ) = 1 andq a b = g ac q d c g bd (i.e., q −1 = q T [22]). Since F → qF under such coordinate changes, ψ in (3.26) should obey more restricted forms to maintain proper observer independence. Two possibilities are , first Piola-Kirchhoff stress P A a of (3.29) is calculated using the chain rule: The resulting Cauchy stress tensors with spatial components σ ab andσ ab obey symmetry rules consistent with the classical local balance of angular momentum [20,22,33]: Now consider changes of the material frame of reference, given by transformation Q A B of (2.1) and (2.9) with inverseQ B A . Under affine changes of coordinates Energy densities ψ,ψ, andψ should be invariant under all transformationsQ A B (e.g., rotations, reflections, inversions) belonging to the symmetry group Q of the material [33,61,81,110] (e.g., ψ → ψ). The present focus is on polynomial invariants [81,110] with basis P of invariant functions with respect toQ ∈ Q and energy offsetsψ 0 = constant,ψ 0 = constant: P = {I 1 , I 2 , . . . , I υ }; I α = I α (C, D, ∇D),ψ =ψ(I 1 , I 2 , . . . , I υ , X) +ψ 0 ; (3.42) The total number of applicable invariants is υ or ζ for (3.37) or (3.38). Stress of (3.40) becomes Remark 3.3.6. A thorough and modern geometric treatment of material symmetry, uniformity, and homogeneity in continuous media is included in a recent monograph [111].

Geometry and kinematics
Let X = X 1 . Considered is a reference domain {M : X ∈ [−L 0 , L 0 ]}, where the total length relative to a Euclidean metric is 2L 0 , and boundary ∂M is the endpoints X = ±L 0 . The referential internal state vector reduces to the single component D = D 1 , which is assumed to have physical units, like X, of length. The spatial coordinate is x = x 1 , and the spatial state component is d = d 1 . A normalization constant (i.e., regularization length) l is introduced, and the physically meaningful domain for internal state is assumed as D ∈ [0, l]. The associated order parameter is with meaningful domain ξ ∈ [0, 1], and where (3.12) and (3.14) are invoked. For generic f and h differentiable in their arguments, let  This assumption (4.4), used henceforth in §4, may be relaxed in future applications to address residual stress (e.g., from growth [30]; see Appendix B), especially for n = dim M > 1.
Henceforth in §4, functional dependence on D or d is replaced with that on ξ . Then (4.5) The following functional forms are assumed for referential nonlinear connection N A B and linear connection K A BC , with N 0 = constant andK(X) both dimensionless: Spatial coefficients K a bc do not affect the governing equations and thus are left unspecified. Conditions (3.15) apply in 1-D, leading to, with (4.1)-(4.6), The deformation gradient, deformation tensor, Jacobian determinant, and director gradient are

Governing equations
A generic energy density is assigned and equilibrium equations are derived for the 1-D case given prescriptions of §4.1.

Energy density
In 1-D, C AB consists of a single invariant C, and D A and D A |B likewise. Dependencies in (3.26) are suitably represented by F, ξ , and (ξ ′ , X) with (4.1) and (4.11). SinceC = C = F 2 , all energy densities ψ of (3.26) in (3.37) -(3.39) are expressed simply as ψ = ψ(C, ξ , ξ ′ , X). (4.12) Denote by µ 0 a constant, later associated to an elastic modulus, with units of energy density. Denote by ϒ 0 a constant, related to surface energy, with units of energy per unit (2-D fixed crosssectional) area. Let W be strain energy density and Λ energy density associated with microstructure. Denote by w a dimensionless strain energy function, y a dimensionless interaction function (e.g., later representing elastic degradation from microstructure changes), λ a dimensionless phase energy function, and ι a dimensionless gradient energy function assigned a quadratic form. Free energy density (4.12) is then prescribed in intermediate functional form as follows: (4.14) Note ι(0, X) = 0. For null ground-state energy and stress, ψ(1, 0, 0, X) = 0 and ∂ ψ ∂C (1, 0, 0, X) = 0: The third of (4.15) ensures convexity of w. Thermodynamic forces originating in (3.29) are derived as The volumetric source term in (3.22) is prescribed as manifesting from changes in energy density proportional to changes of the local referential volume form (e.g., physically representative of local volume changes from damage/tearing, similar to effects of tissue growth on energy (Appendix B)):

Linear momentum
The macroscopic momentum balance, (3.31) This is a separable first-order ordinary differential equation (ODE) that can be integrated directly: The integration limit on G(ξ (X)) is G 0 = G(0), and P 0 is a constant stress corresponding to ξ = 0.
where the value of P 0 , constant for a given static problem, depends on the boundary conditions.

Micro-momentum
DefineK(X) = lK(X). Then the microscopic momentum balance, (3.32) or (3.35), is, upon use of relations in §4.1 and §4.2.1 and dividing by 2ϒ 0 (1 + N 0 ), This is a nonlinear and non-homogeneous second-order ODE with variable coefficients. General analytical solutions are not feasible. However, the following assumption is made henceforth in §4 to reduce the nonlinearity (second term on left side) and render some special solutions possible: Applying (4.24) with notation of (4.2), (4.23) reduces to the form studied in the remainder of §4: This is a linear second-order ODE, albeit generally non-homogeneous with variable coefficients.
For the special case that ϒ 0 (1 + N 0 ) = 0, terms on the left of (4.23) all vanish, and equilibrium demands

Constitutive model
The framework is applied to a strip of skin loaded in tension along the X-direction.
Remark 4.4.1. A 1-D theory cannot distinguish between uniaxial strain or uniaxial stress conditions, nor can it account for anisotropy. Thus, parameters entering the model (e.g., µ 0 , ϒ 0 ) are particular to those loading conditions and material orientations from experiments to which they are calibrated (e.g., uniaxial stress along a preferred fiber direction).
The nonlinear elastic potential of §4.4.2 specializes a 3-D model [71,82,83,92] to 1-D. The internal structure variable ξ = D/l accounts for local rearrangements that lead to softening and degradation under tensile load [72][73][74]77]: fiber sliding, pull-out, and breakage of collagen fibers, as well as rupture of the elastin fibers and ground matrix.

Remark 4.4.2.
Specifically, D is a representative microscopic sliding or separation distance among microstructure constituents, and l is the value of this distance at which the material can no longer support tensile load. In the context of cohesive theories of fracture [73,112,113], D can be interpreted as a crack opening displacement.
Remark 4.4.3. Some physics represented by the present novel theory, not addressed by nonlinear elastic-continuum damage [73,90] or phase-field [95,114] approaches, are summarized as follows. The Finslerian metrics G(ξ ) = g(ξ ) account for local rescaling of material and spatial manifolds M and m due to microstructure changes (e.g, expansion due to tearing or cavitation). Nonlinear connection N 0 rescales the quadratic contribution of the gradient of ξ to surface energy by a constant, and linear connectionK rescales the linear contribution of the gradient of ξ to surface energy by a continuous and differentiable function of X, enabling a certain material heterogeneity.

Metrics
From ( Herein, the metric is assigned an exponential form frequent in generalized Finsler geometry [7,55] and Riemannian geometry [27,30]: For ξ ∈ [0, 1], two constants are k, which is positive for expansion, and r > 0. Physically, this rescaling arises from changes in structure associated with degradation, to which measure 1 2 ln G(ξ ) is interpreted as a contributor to remnant strain. For Riemannian metrics, G = G =ḡ = g = 1, in which case (4.36) is independent of ξ and this remnant strain always vanishes.
The ratio of constants is determined by the remnant strain contribution at failure:ε = k r = 1 2 ln G(1). Since ξ ∈ [0, 1], smaller r at fixed k r gives a sharper increase in 1 2 ln G versus ξ . Values of k and r are calibrated to data in §4.5; choices of N 0 andK are explored parametrically therein. Nonlinear connection N 0 = constant and linear connectionK(X) =K(X)/l affect the contribution of state gradient ξ ′ to surface energy ι via (4.13) and (4.14). Constraint N 0 > −1 is applied to avoid model singularities and encompass trivial choice N 0 = 0. The value of N 0 uniformly scales the contribution of (ξ ′ ) 2 to ι and ψ. FunctionK scales, in a possibly heterogeneous way, the contribution of ξ ′ to ι and ψ. Even when ξ ′ vanishes, N 0 andK can affect solutions.

Nonlinear elasticity
Strain energy density W in (4.13) is dictated by the normalized (dimensionless) function w(C): where dimensionless constants are a 1 ≥ 0 and b 1 > 0, and µ 0 > 0 is enforced along with ϒ 0 > 0 in (4.13). This adapts prior models for collagenous tissues [71,82,83,92] to the 1-D case. The first term on the right, linear in C, accounts for the ground matrix and elastin. The second (exponential) term accounts for the collagen fibers, which, in the absence of damage processes, stiffen significantly at large C. Such stiffening is dominated by the parameter b 1 , whereas a 1 controls the fiber stiffness at small stretch √ C ≈ 1 [71]. The elastic degradation function y(ξ ) and independent energy contribution λ (ξ ) in (4.13) are standard from phase-field theories [95,114], where ϑ ∈ [0, ∞) is a constant with ϑ = 2 typical for brittle fracture and ϑ = 0 → y = 1 for purely elastic response: When ϑ > 0, y(1) = 0: no strain energy W or tensile load P are supported at X when D(X) = l. Verification of (4.15) for prescriptions (4.38) and (4.39) is straightforward [81,82]. Stress P conjugate to F = √ C and force Q conjugate to D = lξ are, from (4.16), (4.17), (4.38), and (4.39):
Stress P is shown in Fig. 2(a), first assuming N 0 = 0 and K ′ 0 = 0 for simplicity. The Finsler model, with A 0 = 8.5 × 10 −2 corresponding to baseline parameters given in Table 1, successfully matches experimental 1 data [74]. The value of µ 0 is comparable to the low-stretch tensile modulus in some experiments [71,75], acknowledging significant variability in the literature.
Remark 4.5.1. The ideal elastic solution (ξ = 0) is shown for comparison. Excluding structure evolution corresponding to collagen fiber rearrangements, sliding, and breakage, the model is too stiff relative to this data for which such microscopic mechanisms have been observed [74]. The ideal elastic model is unable to replicate the linearizing, softening, and failure mechanisms with increasing stretch √ C reported in experiments on skin and other soft tissues [63,64,68,74,87].
In Fig. 2(b), effects of ϑ on P are revealed forε = 0.1, r = 2, N 0 = 0, and K ′ 0 = 0, noting ϑ = 0 produces the ideal nonlinear elastic solution ξ H = 0 in (4.42). Peak stress increases with decreasing ϑ ; the usual choice from phase-field theory ϑ = 2 provides the close agreement with data in Fig. 2(a). In Fig. 2(c), effects of Finsler metric scaling factorsε = k r and r on stress P are demonstrated, where at fixed r, peak stress increases (decreases) with increasing (decreasing)ε and k. Baseline choicesε = 0.1 and r = 2 furnish agreement with experiment in Fig. 2(a). A remnant strain of 0.1 is the same order of magnitude observed in cyclic loading experiments [72,78]. Complementary effects on evolution of structure versus stretch are shown in Fig. 2(e): modest changes in ξ produce significant changes in P. In Fig. 2(d), effects of connection coefficients N 0 and K ′ 0 are revealed, holding material parameters at their baseline values of Table 1. For this homogeneous problem, maximum P decreases with increasing N 0 and K ′ 0 . Corresponding evolution of ξ is shown in Fig. 2(f). When K ′ 0 < 0, a viable solution ξ H ∈ [0, 1] exists only for √ C > 1. The total energy per unit cross-sectional area of the specimen isΨ , found upon integration of ψ(C H , ξ H ) in (4.13) over M with local volume element dV = G(ξ H ) dX:

Geometry and kinematics
Reference coordinates are Cartesian (orthogonal): where the total area relative to a Euclidean metric is 4L 0 W 0 , and boundary ∂M is the edges (X 1 , X 2 ) = (±L 0 , ±W 0 ). The referential internal state vector has coordinates {D 1 , D 2 }, both with physical units of length. Spatial coordinates are Cartesian {x 1 , x 2 } and {d 1 , d 2 }. A normalization constant (i.e., regularization length) is l, with physically meaningful domain assumed as D A ∈ [0, l] (A = 1, 2). With notation f (X, D) = f (X A , D B ), dimensionless order parameters are, with (3.12) and (3.14) invoked, Herein, the following constraint is imposed: making m and M isometric when φ a (X) = δ a A X A +c a 0 ⇔ F a A = δ a A regardless of {ξ , η} at x = ϕ(X). Though other non-trivial forms are admissible (e.g., §4.1), assume nonlinear N A B and linear K A BC connections vanish: The K a bc do not affect the governing equations to be solved later, so they are unspecified. Applying (3.15) and (5.1)-(5.4), The deformation gradient, deformation tensor, Jacobian determinant, and director gradient are Unless F a A and G AB are diagonal, C andC can differ. From (3.19) and (3.20),

Governing equations
A generic energy density is chosen and equilibrium equations are derived for the 2-D case of §5.1.

Energy density
For the present case, dependencies on D A and D A |B are suitably represented by (ξ , η) and (∂ A ξ , ∂ A η) of (5.1) and (5.8). The functional form of (3.38) is invoked without explicit X dependency, whereby (5.10) Henceforth in §5, the over-bar is dropped from ψ to lighten the notation. Denote by µ 0 a constant, later associated to a shear modulus, with units of energy density. Denote by ϒ 0 a constant related to surface energy with units of energy per unit (e.g., 2-D fixed cross-sectional) area, and by γ ξ and γ η two dimensionless constants. Let W be strain energy density and Λ energy density associated with microstructure. Denote by w a dimensionless strain energy function (embedding possible degradation), λ and ν dimensionless phase energy functions, ι a dimensionless gradient energy function assigned a sum of quadratic forms, and ∇ 0 (·) = ∂ ∂ X (·) the partial material gradient. Free energy (5.10) is prescribed in intermediate functional form as Note ι(0, 0) = 0. Therefore, for null ground-state energy density ψ and stress P A a , Convexity and material symmetry are addressed in §5.4.2. Thermodynamic forces of (3.29) are, applying (3.40), 14) The source term in (3.22) manifests from changes in energy proportional to changes of the local referential volume form (e.g., local volume changes from damage, treated analogously to an energy source from tissue growth (Appendix B)):

Homogeneous fields
Examine cases for which ξ (X) → ξ H = constant and η(X) → η H = constant at all points X ∈ M ; the constants may differ: ξ H ̸ = η H in general. Apply the notation f H (X) = f (X, ξ H , η H ). Restrict µ 0 > 0. Then (5.14) and (5.18) reduce to This should be satisfied for any homogeneous F a A = (F H ) a A for which ∂ 2 ϕ a /∂ X A ∂ X B = 0. Micromomentum conservation laws (5.19) and (5.20)

Stress-free states
Consider cases whereby P A a = 0 ∀X ∈ M . Linear momentum conservation laws (3.31), (3.34), and (5.18) are trivially satisfied. Restrict µ 0 > 0. Since F a A is non-singular, (5.14) requires ∂ w/∂C AB = 0. This is obeyed atC AB = δ AB via (5.13); thus assume rigid body motion (i.e., ϕ a = Q a A X A + c a 0 , with Q a A constant and proper orthogonal and c a 0 constant) whereby w = 0 vanishes as well by (5.13).

Constitutive model
The framework is applied to a rectangular patch of skin loaded in the X 1 -X 2 plane. A 2-D theory (i.e., membrane theory) cannot distinguish between plane stress and plane strain conditions [115], nor can it account for out-of-plane anisotropy. Nonetheless, 2-D nonlinear elastic models are widely used to represent soft tissues, including skin [68,89]. Thus, parameters entering the model (e.g., µ 0 , ϒ 0 ) are particular to loading conditions and material orientations from experiments to which they are calibrated (e.g., here, plane stress).
Remark 5.4.1. In a purely 2-D theory, incompressibility often used for 3-D modeling of biological tissues [68,71,80,82], cannot be assumed since contraction under biaxial stretch is not quantified in a 2-D theory. Incompressibility is also inappropriate if the material dilates due to damage.
The skin is treated as having orthotropic symmetry, with two constant orthogonal directions in the reference configuration denoted by unit vectors n 1 and n 2 : Remark 5.4.2. The collagen fibers in the plane of the skin need not all align with n 1 or n 2 , so long as orthotropic symmetry is respected. For example, each n i can bisect the alignments of two equivalent primary families of fibers in the skin whose directions are not necessarily orthogonal [71,92]. In such a case, n 1 is still a unit vector orthogonal to n 2 ; planar orthotropy is maintained with respect to reflections about both unit vectors n i .
Remark 5.4.3. The internal structure variables ξ = D 1 /l and η = D 2 /l account for mechanisms that lead to softening and degradation under tensile load: fiber sliding, pull-out, and breakage of collagen fibers, and rupture of the elastin fibers and ground matrix. Each D A (A = 1, 2) is a representative microscopic sliding or separation distance in the n i δ i A direction, with l the distance at which the material can no longer support tensile load along that direction. Finslerian metrics G AB (ξ , η) = δ a A δ b B g ab (ξ , η) of §5.4.1 anisotropically rescale material and spatial manifolds M and m due to microstructure changes in different directions. In the absence of damage, the nonlinear elastic potential of §5.4.2 specializes a 3-D model [71,82,83,92] to 2-D.
The Cartesian coordinate chart {X A } is prescribed such that n A i = δ A i in (5.24); thus n 1 and n 2 are parallel to respective X 1 -and X 2 -directions on M . Rescaling arises from changes in structure associated with degradation and damage in orthogonal directions, to which remnant strain contributions 1 2 ln[G 11 (ξ )] and 1 2 ln[G 22 (η)] can be linked. The metric tensor G AB is hereafter assigned specific exponential terms, generalizing the 1-D form of §4.4.1 to an anisotropic 2-D form appropriate for orthotropic symmetry:

Nonlinear elasticity
The nonlinear elasticity model generalizes that of §4.4.2 to a 2-D base space M with anisotropic Finsler metric depending on two structure variable components, ξ and η in normalized dimensionless form. For the 2-D case, material symmetry of §3.3.4 requires careful consideration. Here, the skin is treated as a planar orthotropic solid [68,75,89]. Viewing the D A as components of a material vector field, orthotropic symmetry suggests invariants ξ 2 and η 2 . For physically admissible ranges ξ ∈ [0, 1] and η ∈ [0, 1], these can be replaced with ξ and η. Viewing the D A |B similarly, orthotropic symmetry permits a more general functional dependence than the sum of quadratic forms in ι of (5.11) and (5.12). However, the chosen form of ι in (5.12) allows for partial anisotropy, not inconsistent with orthotropy, when γ ξ and γ η differ. Thus, the structure-dependent contribution to ψ, Λ l = ϒ 0 (γ ξ λ + γ η ν + ι), more specifically here is consistent with material symmetry requirements. Strain energy density W in (5.11) is dictated by dimensionless function w(C AB , ξ , η). Per the above discussion, ξ and η are treated as scalar invariant arguments. A partial list of remaining invariants [82,91] of (3.43) for orthotropic symmetry of a 2-D material entering w (and thus ψ =ψ) is then, applying n A i = δ A i in (5.24), Remark 5.4.7. As n 1 and n 2 are orthonormal,Ī 1 =Ī 3 +Ī 4 , so one ofĪ 1 ,Ī 3 ,Ī 4 in (5.29) is redundant. Since J ≥ 1, dependence onĪ 2 =C 11C22 − (C 12 ) 2 can be replaced by J (or by (C 12 ) 2 givenĪ 3 ,Ī 4 ).
The Euclidean metricḠ AB = δ AB , rather than Finsler metric G AB , is used for scalar products in (5.28) and (5.29), consistent with (5.9). In 2-space,Ī 1 andĪ 2 are the complete set of isotropic invariants ofC. Two orthotropic invariants areĪ 3 andĪ 4 ; several higher-order invariants are admissible [82,91] but excluded here since (5.29) is sufficient for the present application. The dimensionless strain energy function entering (5.11) is prescribed specifically as Remark 5.4.8. Potential w in (5.30) extends prior models for collagenous tissues [71,82,83,92] to include anisotropic structure changes. The first term on the right, linear inĪ 1 /J and independent of volume change, accounts for isotropic shearing resistance of ground matrix and elastin. The second term on the right accounts for resistance to volume (area) change, k 0 being a dimensionless bulk (area) modulus finite for a 2-D model; the dimensional bulk modulus κ 0 = k 0 µ 0 . Exponential terms account for stiffening from collagen fibers in orthogonal directions n i . Heaviside functions prevent fibers from supporting compressive load [82,116] since they would likely buckle.
Remark 5.4.12. An isotropic version of the theory can be obtained, if along with m = k in (5.26), the following choices are made instead of (5.31): Collagen fiber contributions to strain energy are removed such that w now only depends on isotropic invariants ofC. Equilibrium equations (5.18), (5.19), and (5.20) are identical under the change of variables ξ ↔ η, implying η(X) = ξ (X) if identical boundary conditions on D A or z A are applied for each field on ∂M . In this case, one of (5. 19), and (5.20) is redundant and replaced with η = ξ .

Specific solutions
Possible inputs to the 2-D model are seventeen constants l > 0, k, m, r > 0, Values of l and ϒ 0 are taken from the analysis in §4.5.2 of complete tearing of a 1-D specimen of skin to a stress-free state. This is appropriate given that 1-D and 2-D theories are applied to describe surface energy and material length scale pertinent to the same experiments [73,74,99,113], and since stress-free solutions in §5.5.3 perfectly parallel those of §4.5.2. The remaining parameters are evaluated, in §5.5.1, by applying the constitutive model of §5.4 to the general solutions for homogeneous fields derived in §5.3.1 to uniaixal-stress extension of 2-D skin specimens along the material X 1 -and X 2 -directions, respectively aligned perpendicular and parallel to Langer's lines.
Remark 5.5.1. Collagen fibers of the microstructure in the dermis are aligned predominantly along Langer's lines and are more often pre-stretched in vivo along these directions [75]. In vivo or in vitro, elastic stiffness at finite stretch tends to be larger in directions along Langer's lines (i.e., parallel to X 2 and n 2 ) than in orthogonal directions (e.g., parallel to n 1 ). Degradation and failure behaviors are also anisotropic: rupture stress tends to be larger, and failure elongation lower, for stretching in the stiffer n 2 -direction [74,75,87].
In §5.5.2, model outcomes are reported for planar biaxial extension [68,70,115] of 2-D specimens, highlighting simultaneous microstructure degradation perpendicular and parallel to Langer's lines. Lastly, in §5.5.3, stress-free states analogous to those modeled in a 1-D context in §4.5.2 are evaluated for the 2-D theory. In Consistent with (4.24) for N 0 = 0 [55,56,62], β = α − 2 is assumed in (5.37) and (5.38), reducing the number of requisite parameters to fifteen; α and β enter the governing equations only through their difference. Boundary conditions on internal state, are, for homogeneous conditions, Alternative conditions to (5.36)-(5.39) are considered for heterogeneous stress-free states in §5.5.3.
Values of all baseline parameters are listed in Table 1. Identical values of those constants shared among 1-D and 2-D theories are found to aptly describe the experimental data for stretching along n 1 , in conjunction with natural choice γ ξ = 1. The 2-D theory features additional parameters to account for orthotropic anisotropy (e.g., stiffer response along n 2 , with peak stress occurring at lower stretch) as well as an areal bulk modulus κ 0 absent in the 1-D theory.
Model outcomes for non-vanishing stress components and internal state vector components are presented in respective Fig. 4(a) and Fig. 4(b). Experimental P 1 1 versus λ 1 data for loading along n 1 , with λ 1 ≥ 0 prescribed in the corresponding model calculations, are identical to P versus √ C data depicted using the 1-D theory in §4.5.1. These data [74] are for relatively high-rate extension of rabbit skin along a longitudinal direction, parallel to the backbone of the torso and perpendicular to Langer's lines. Nonlinear elastic parameters should be viewed as instantaneous dynamic moduli in a pseudo elastic representation [68,84,85], since loading times are brief relative to stress relaxation times [74]. Single-experiment data of similar fidelity for transverse extension, parallel to Langer's lines, to complete load drop were not reported, but a range of maximum stress and strain were given for extension along n 2 [74]. A representative peak stress P 2 2 and corresponding stretch λ 2 based on such data [74] are included in Fig. 4(a). According to such data [74], the material is stiffer, and ruptures at a higher stress (≈ 4 3 ×) but lower strain (≈ 2 3 ×), in the transverse n 2 -direction. Remark 5.5.3. For loading along n 1 , ξ → 1 and η → 0 for λ 1 ≳ 3.5, meaning most internal structure evolution correlates with degradation in this direction, with small transverse effects of η. Analogously, loading along n 2 gives η → 1 and ξ → 0 for λ 2 ≳ 3. The rate of increase of η with λ 2 > 1 is more rapid than the rate of increase of ξ with λ 1 > 1, since the skin degrades sooner and fails at a lower strain for stretching parallel to Langer's lines. The present diffuse model is an idealization characteristic of experiments when there is no sharp pre-crack [64,68,72,74,87]. Fig. 4(c) and Fig. 4(d) are predictions at modest stretch along n 1 or n 2 under uniaxial stress conditions identical to those of Fig. 4(a) as well as uniaxial strain, whereby λ 2 = 1 or λ 1 = 1 is enforced using the scheme of §5.5.2 rather than respective P 2 2 = 0 or P 1 1 = 0. Predictions for the ideal elastic case (ϑ = ς = 0 ⇒ ξ = η = 0) are shown for comparison. Results are stiffer for the ideal elastic case since degradation commensurate with structure change is omitted. In agreement with other data [70], skin is elastically stiffer in uniaxial strain relative to uniaxial stress. Choosing a higher value of k 0 = κ 0 /µ 0 > 1 in (5.30) would further increase this difference if merited

Biaxial extension
Now consider homogeneous biaxial-stress extension in the X 1 -and X 2 -directions. From symmetry, P 1 2 = 0, P 2 1 = 0, F 1 2 = 0, F 2 1 = 0, andC 12 =C 21 = 0. The homogeneous deformation fields are Stretch ratios are λ 1 > 0 and λ 2 > 0, constants over M . Mechanical boundary conditions are With λ 1 and λ 2 prescribed by (5.44), equilibrium equations (5.37) and (5.38) are solved simultaneously for ξ and η as functions of λ 1 , λ 2 , giving homogeneous values of fields consistent with (5.39). Then P 1 1 and P 2 2 are obtained afterwards with (5.36) for a = A = 1 and a = A = 2. Model predictions for equi-biaxial stretching, λ 1 = λ 2 , are produced using the baseline material parameters of Table 1, obtained for the 2-D theory in §5.5.1. In Fig. 5(a), stresses also include those (a) axial stress  Figure 4 Uniaxial extension and tearing of skin for imposed axial stretch λ 1 ≥ 1 or λ 2 ≥ 1, 2-D model: (a) stress P 1 1 or P 2 2 (baseline parameters, Table 1) with representative experimental data [74] (see text §4.5.1 for consistent definition of experimental stretch accounting for pre-stress) for straining perpendicular or parallel to Langer's lines (b) normalized internal structure components ξ and η (baseline parameters) (c) stress P 1 1 for moderate extension λ 1 ≤ 2.1 under uniaxial stress (P 2 2 = 0) or uniaxial strain (λ 2 = 1) conditions for Finsler model (baseline parameters) and ideal elastic model (ϑ = ς = 0) (d) stress P 2 2 for moderate extension λ 2 ≤ 2.0 under uniaxial stress (P 1 1 = 0) or uniaxial strain (λ 1 = 1) conditions for Finsler model (baseline) and ideal elastic model (ϑ = ς = 0)  for the ideal elastic case (ϑ = ς = 0 ⇒ ξ = η = 0) that are noticeably higher for λ 1 > 1.5 and increase monotonically with stretch. For the Finsler theory, under this loading protocol (λ 1 = λ 2 ), P 2 2 increases more rapidly than P 1 1 with increasing λ 1 , reaching a slightly lower peak value at significantly lower stretch. Elastic stiffness during the lower-stretch loading phase is higher in the n 2 -direction due to the preponderance of aligned collagen fibers, but degradation associated with internal structure evolution is more rapid due to the lower toughness of skin when torn in this direction. The latter phenomena is evident in Fig. 5 Remark 5.5.4. Experimental data on failure skin focus on uniaxial extension [74,75]. Known biaxial data (e.g, [68,70]) do not report stretch magnitudes sufficient to cause tearing, so direct validation does not appear possible. Should skin prove to be more stiff and damage tolerant in equibiaxial stretch experiments, w of (5.30) can be modified so the tangent bulk modulus proportional to k 0 increases more strongly with J and does not degrade so severely with structure evolution.
Remark 5.5.6. All parameters in Table 1 have clear physical or geometric origins; none are ad hoc. Constant l is the critical fiber sliding distance or crack opening displacement for rupture. Ratios k r and m r are associated with remnant strain contributions in orthogonal n 1 -and n 2 -directions along primary initial fiber directions (e.g., perpendicular and parallel to Langer's lines). The isotropic shear modulus and bulk modulus for the matrix, consisting of ground substance and elastin, are µ 0 and κ 0 . Nonlinear elastic constants a 1 and b 1 control stiffening due to collagen fiber elongation in the n 1 -direction, while a 2 and b 2 control stiffening due to fiber elongation in the n 2 -direction. Loss of elastic stiffness due to fiber rearrangements and damage processes in matrix, fibers, and their interfaces, in respective n 1 -and n 2 -directions, is modulated by ϑ and ς . Isotropic surface energy is ϒ 0 , with factors γ ξ and γ η scaling the fracture toughness in respective n 1 -and n 2 -directions.

Conclusion
A theory of finite-deformation continuum mechanics with a basis in generalized Finsler geometry has been developed and refined. Elements of an internal state vector represent evolving microstructure features and can be interpreted as order parameters. Dependence of the material metric on internal state affects how distances are measured in the material manifold and how gradients (i.e., covariant derivatives) are resolved. A new application of the theory to anisotropic soft-tissue mechanics has been presented, whereby the internal state is primarily associated with collagen fiber rearrangements and breakages. The material metric contains explicit contributions from sliding or opening modes in different material directions. Solutions to boundary value problems for tensile extension with tearing in different directions agree with experimental data and microscopic observations on skin tissue, providing physical and geometric insight into effects of microstructure.
Funding: This research received no external funding.

Conflicts of interest:
The author declares no conflicts of interest. Data availability statement: Not applicable; this research produced no data.

Appendix A: Variational derivatives
The variational derivative δ (·) of §3.3.1 invokes (ϕ a , D A ) with a, A = 1, 2, . . . , n as the total set of 2n varied independent parameters or degrees-of-freedom.
Denote by f (X, D) a generic differentiable function of arguments {X A , D A } in a coordinate chart on Z . The variation of f (X, D) is defined by the first of the following: where (·)| A is the vertical covariant derivative (e.g., as in (2.21)). For the choices V A BC = 0 and Y A BC = 0 of (3.15), f (X, D)| A =∂ A f (X, D) and the rightmost form is obtained, consistent with prior definitions [54,55]. This is used with (3.28) to obtain the second of (3.24): where it is assumed per (3.12) that∂ C δ [D A (X)] =∂ C [δ (D A )(X)] = 0 on M and Z .

A.2 Volume form
Two definitions have been set forth in prior work for the variation of the volume form dΩ (X, D).
The first quoted here sets [54] δ where the first equality is a definition and (2.27) [54] and implied in a prior numerical implementation [59].
The second definition quoted here was used in the original theoretical derivations [55,56]: Remark B.1.1. Terms on the left side of (B.1) are standard for nonlinear elasticity theory [22]. If the free energy ψ does not depend on D A or D A |B , then the stress P A a = ∂ ψ/∂ F a A is also conventional, presuming ψ is such that in the undeformed state C AB = G AB ⇒ P A a = 0. In that case, when the right side of (B.1) vanishes, the body manifold M should not contain residual stresses when F a A = ∂ A ϕ a for regular motions ϕ a (X) (e.g., in the absence of topological changes).
Remark B.1.2. Departures from classical nonlinear elasticity arise when (i) ψ has dependencies on D A or D A |B , (ii) when P A a or G depends on D A along with heterogeneous state field ∂ A D B ̸ = 0, or (iii) when a different connection than the Levi-Civita connection is used for Γ c ba (i.e., Γ c ba ̸ = γ c ba due to d-dependence of spatial metric g ab ). Each of these departures could potentially induce stresses P A a ̸ = 0 in a simply connected body externally unloaded via p a = P A a N A = 0 everywhere on its oriented boundary ∂M (i.e., residual stresses).
Analysis of a particular version of the general theory offers more insight. First assume in (3.18) thatĝ a b → δ a b such that g ab (x, d) → g ab (x) =ḡ ab (x): the spatial metric tensor g is Riemannian rather than Finslerian. Then γ a bc = Γ a bc . Now use the osculating Riemannian interpretation of the Finslerian material metric G offered by Corollary 2. Remark B.1.3. Expression (B.5) has the standard appearance for static equilibrium in classical continuum mechanics, but stressP A a and connectionγ A BC both implicitly depend on internal state D A , and the former possibly by its gradient D A |B as well if appearing in ψ. Coefficientsγ A BC are those of the Levi-Civita connection ofG AB via (2.36). Now neglect dependence on internal state gradient in the energy density, require D-dependence to arise only through G AB , and assume the body is homogeneous (with mild abuse of notation): ψ = ψ(F a A , D A ) = ψ(F a A , G AB (X, D)) =ψ(F a A ,G AB (X)) =ψ(C AB (F a A , g ab ),G AB (X)). (B.6) Recall from (3.10) that C AB = F a A g ab F b B . As a simple example, take, where n = dim M , and where µ 0 > 0 is a constant (e.g., an elastic shear modulus). Now assume that spatial manifold m is Euclidean [27,30] such that the Riemann-Christoffel curvature tensor from γ a bc (and thus derived from g ab ) vanishes identically.
Remark B.1.4. In this case, (B.5), the last of (B.6), and the example (B.7) are consistent with the geometric theory of growth mechanics of Yavari [30] in the setting of quasi-statics. Incompressibility can be addressed by appending linear momentum to include contribution from an indeterminant pressure to be determined by boundary conditions under isochoric constraint J = 1 [22]. Otherwise,ψ can be augmented with term(s) to ensure C A B → δ A B ⇒P A a = 0 (e.g., (4.38) for n = 1).
The Riemann-Christoffel curvature tensor fromγ A BC (and thusG AB ) need not vanish in general: Remark B.1.5. In Riemannian geometry,γ A BC are symmetric, differentiable, and obey (2.36); (B.8) has 1 12 n 2 (n 2 − 1) independent components [31]. For n = 3,R A BCD contains six independent components, determined completely by the metric and Ricci curvatureR A ABC [30,98]. For n = 2, R A BCD contains only one independent component, determined completely by the scalar curvaturẽ κ = 1 2 R ABG AB . For n = 1,R A BCD always vanishes (i.e., a 1-D manifold is always flat in this sense).
WhenR A BCD is nonzero over a region of M , then no compatible deformationF A a (X) exists that can push-forwardG AB to match the Euclidean metric g ab (φ (X)) that would render corresponding regions of M and m isometric. In other words, the push-forward g ab =F A aG ABF B b whereF A a = ∂ a ζ A does not exist, ζ A being (nonexistent) Euclidean coordinates on M . In such cases, M would always have to be deformed (e.g., strained) to achieve its spatial representation m, since no isometry exists between the two configurations.
Remark B.1.6. If an intrinsically curved body manifold in the reference state M is stress-free per the constitutive prescription (e.g., (B.7) or any other standard elasticity model), then the intrinsically flat body in the current state m would be necessarily strained and stressed, even if external traction p a vanishes along its boundary. Thus, this particular rendition of the generalized Finsler theory supplies residual stress from a non-Euclidean material metric tensorG AB in a manner matching other works that use Riemannian geometry [27,30].
In the full version of the generalized Finsler theory [54,55], as discussed following (B.1), residual stresses could emerge from additional sources to those discussed under the foregoing assumptions of a Euclidean spatial metric, a conventional hyperelastic energy potential, and an osculating Riemannian material metric with non-vanishing curvature. A number of different curvature forms can be constructed from the various connections and derivatives of Finsler geometry and its generalizations [3,5]. Further analysis, beyond the present scope, is needed to relate these geometric objects to physics in the continuum mechanical setting, including residual stresses.
Remark B.1.7. Deformation gradient F a A could be decomposed into a product of two mappings [62]: F a A (X) = ∂ A ϕ a (X) = (F E ) a α (X)(F D ) α A (D(X)). In this case, the strain energy potential is written to emphasize the elastic deformation F E , with the state-dependent deformation F D accounting explicitly for inelastic deformation mechanisms, including growth [29,107]. In this setting, residual stresses can arise if (F E ) −1 and thus F D do not fulfill certain integrability conditions: neither two-point tensor (F E ) −1 nor F D is always integrable to a vector field [98].

B.2 Micro-momentum and growth
Now consider the internal state-space equilibrium equation, (3.35), first under the foregoing assumptions used to derive (B.5). Furthermore, take N A B = N A B (X) K A BC = γ A BC (X), and α = 1. Then, with these assumptions, in the osculating Riemannian interpretation of Corollary 2. where (B.10) follows from (3.29). Use energy density ψ of (B.6), soZ A B = 0 identically. Choose the volumetric source term R C = ψ∂ C (ln √ G), which here represents the local change in energy density per unit reference volume due to effects of growth on the local volume form dΩ (X, D), since now, per (A.4) of Appendix A, ψδ (dΩ ) = ψ[∂ C (ln √ G)δ (D C )]dΩ = R C δ (D C )dΩ .