arXiv subjects

Arash Yavari

Publications and source records attributed to Arash Yavari.

At least 19 recordsLinked to original sources

arXiv2026

On Physical Components of Tensors in Elasticity and Inelasticity

Widely used in mechanics and mathematical physics, physical components remove the inherent coordinate-dependent scaling in curvilinear coordinates, yielding components of consistent physical dimension. In orthogonal coordinates, they are constructed by normalizing the coordinate frame and coframe; in general coordinates, however, their construction requires additional, non-trivial choices. In this paper, we extract physical components of arbitrary tensors on arbitrary Riemannian manifolds by orthonormalization of the coordinate frame. We further formulate a general normalization framework distinguishing three requirements: dimensional consistency, dual frame-coframe compatibility, and unit normalization. We show that dimensional consistency alone leaves independent general linear gauge freedoms for the contravariant and covariant components. Further requiring dual compatibility locks these into a single general linear gauge; a dual-compatible frame and coframe are both of unit length if and only if they are orthonormal. Thus, the choice of orthonormal transformations emerges as the only physical-components framework that satisfies all three requirements. We apply this framework to nonlinear elasticity and inelasticity, referring broadly to constitutive responses involving internal distortions, of which we study anelasticity, viscoelasticity, and visco-anelasticity. We examine the deformation gradient, inelastic distortions, strain measures, and stress tensors. We conclude by arguing that physical components remain neither intrinsic nor unique.

math-ph↗

arXiv2026

Dual Variational Principles for Curl Forces

Curl forces are position-dependent, non-conservative, and non-dissipative forces that, in general, cannot be derived from an ordinary potential energy. Consequently, their equations of motion do not, in general, follow from a standard variational principle. In this paper, we present a dual variational formulation for particle dynamics under curl forces. By introducing variables dual to position and velocity and an auxiliary function, we construct a pre-dual action in which the equations of motion act as constraints. Stationarity with respect to the primal variables defines a dual-to-primal mapping, whose substitution into the pre-dual action gives an action expressed entirely in terms of the dual variables. The Euler--Lagrange equations of the dual action recover both the original equations of motion and their prescribed initial conditions. We also introduce an auxiliary dual Hamiltonian that is conserved along stationary dual trajectories, although it does not represent the physical energy. The formulation is illustrated using two nonlinear curl force fields in two and three dimensions and the classical Ziegler column. These examples demonstrate that non-conservative curl-force dynamics can admit variational descriptions even in the absence of an ordinary potential energy or a conventional Lagrangian.

math-ph↗

arXiv2026

First-Order Compatible-Strain Mixed Quadrilateral Finite Elements for 2D Nonlinear Elasticity

Compatible-strain mixed finite elements (CSMFEs) use the differential complex of nonlinear elasticity to construct discretizations that preserve the underlying topological structure. Existing CSMFEs have focused on simplicial meshes for compressible and incompressible nonlinear elasticity. In this paper, we develop compatible-strain mixed formulations for quadrilateral elements applicable to both compressible and incompressible solids. For general quadrilateral elements, the Piola transformation preserves vector fields tangent and normal to element edges only for special geometries, such as rectangles, parallelograms, and trapezoids, and therefore cannot be used to construct compatible shape functions. To overcome this limitation, the shape functions are computed directly in the physical space using numerical integration. Nedelec shape functions of the first kind are employed to interpolate the displacement gradient, and a new class of stress shape functions compatible with the displacement and displacement-gradient discretizations is introduced. The compressible formulation follows the framework previously developed for simplicial CSMFEs, whereas a new incompressible formulation employing element-level condensation of the pressure field is proposed, thereby avoiding additional global degrees of freedom. These developments extend the compatible-strain mixed finite element framework from simplicial to general quadrilateral meshes for both compressible and incompressible nonlinear elasticity. Numerical examples demonstrate that the proposed quadrilateral elements can solve problems that require second-order simplicial CSMFEs while using fewer degrees of freedom.

math.NA↗

arXiv2026

Nonlinear Anisotropic Visco-Anelasticity

We formulate a nonlinear geometric theory of visco-anelasticity that unifies viscoelastic and anelastic responses within a single thermodynamic framework. At each material point, the total deformation gradient is multiplicatively decomposed into elastic, viscous, and anelastic distortions, thereby generalizing the Bilby-Kr\"oner-Lee decomposition to visco-anelasticity. The theory explicitly incorporates the material metric, which encodes the evolving natural configuration of the solid, the transformed structural tensors, and provides a consistent formulation of the constitutive equations, the balance laws, the thermodynamic potentials, and the kinetic equations. The first and second laws of thermodynamics are systematically applied to derive the constitutive and evolution equations without invoking observer invariance. Anisotropy is treated in full generality through structural tensors. As illustrative examples, we specialize the general framework to isotropic and transversely isotropic visco-anelastic solids. Two examples within the class of universal deformations, which admit closed or partially closed form solutions, show how the proposed framework can be used to model the coupled viscous and anelastic response of incompressible anisotropic solids with distributed eigenstrains and the associated residual stresses. This geometric framework unifies nonlinear viscoelasticity and anelasticity by coupling time-dependent and eigenstrain-driven effects within a single, fully consistent geometric formulation. In particular, the proposed framework clarifies the geometric structure of the elastic, viscous, and anelastic distortions and resolves ambiguities associated with intermediate configurations in existing formulations of nonlinear viscoelasticity and viscoplasticity.

math-ph↗

arXiv2026

Universal deformations and universal residual stresses in incompressible isotropic Cauchy elasticity

We study universal deformations in incompressible isotropic Cauchy elastic solids with residual stress, without assuming any specific origin for the residual stress. Starting from the constitutive representation of the Cauchy stress as an isotropic tensor-valued function of strain and residual stress, we derive the universality constraints for residually-stressed incompressible isotropic Cauchy elastic solids. We show that for the six known families of universal deformations the set of universal deformations is identical to that of incompressible isotropic elasticity in the absence of residual stress. We also show that residual stress does not enlarge the space of universal deformations. We then determine the universal residual stress fields corresponding to the six known families of universal deformations. Assuming that the residual stress field has the same symmetry as the corresponding universal deformation, the universality constraints reduce to systems of ordinary differential equations that can be solved explicitly. The resulting universal residual stress fields are characterized and discussed for each family.

math-ph↗

arXiv2026

Rational Mechanics of Material Strength in Brittle Solids

Material strength is a classical concept with renewed importance in fracture mechanics, particularly in crack nucleation in brittle solids. We formulate material strength in finite elasticity and examine its geometric, constitutive, and symmetry-theoretic foundations. Spatial covariance requires a strength function to depend on both stress and the corresponding strain measure, so that strength is governed by the pair (stress,strain), not stress alone, and only then can representations based on different stress measures be consistently related, with classical stress-based criteria recovered as a special case. We analyze covariance under spatial diffeomorphisms and relate formulations based on the first Piola--Kirchhoff, second Piola--Kirchhoff, and Cauchy stresses. For stress-based criteria, we define the strength hypersurface as a subset of the constitutively admissible stress manifold and study the associated safe domain. Under standard regularity assumptions and the requirement that sufficiently large stresses are inadmissible, the strength surface is a smooth compact hypersurface of this manifold. For isotropic solids, we show that the safe domain is star-shaped under a proportional-reduction hypothesis. We extend the formulation to anelastic brittle solids, showing that residual stresses and eigenstrains modify the strength surface through the material metric, and discuss anisotropic strength via material symmetry.

cond-mat.mtrl-sci↗

arXiv2026

Universal Displacements in Linear Strain-Gradient Elasticity

We study universal displacement fields in three-dimensional linear strain-gradient elasticity within the Toupin-Mindlin first strain-gradient theory. Building on the approach of Yavari (2020), we derive, for each material symmetry class, the universality PDEs obtained by requiring the equilibrium equations (in the absence of body forces) to hold for any material in that class, and we determine the complete set of universal displacements. Using the full symmetry classification together with compact matrix representations of the elasticity tensors, we provide explicit characterizations for all 48 strain-gradient symmetry classes, including centrosymmetric and chiral classes. For several high-symmetry classes, the strain-gradient universality PDEs impose no additional restrictions beyond the classical ones, so the universal displacement families coincide with those of classical linear elasticity (for example, the isotropic classes SO(3) and O(3)). For lower symmetry classes, the strain-gradient universality PDEs can be stricter than their classical counterparts, so the universal displacements form proper subsets of the classical universal displacement families due to additional higher-order differential conditions.

physics.class-ph↗

arXiv2025

A Geometric Theory of Surface Elasticity and Anelasticity

In this paper we formulate a geometric theory of elasticity and anelasticity for bodies containing material surfaces with their own elastic energies and distributed surface eigenstrains. Bulk elasticity is written in the language of Riemannian geometry, and the framework is extended to material surfaces by using the differential geometry of hypersurfaces in Riemannian manifolds. Within this setting, surface kinematics, surface strain measures, surface material metric, and the induced second fundamental form follow naturally from the embedding of the material surface in the material manifold. The classical theory of surface elasticity of Gurtin and Murdoch (1975) is revisited and reformulated in this geometric framework, and then extended to anelastic bodies with anelastic material surfaces. Constitutive equations for isotropic and anisotropic material surfaces are formulated systematically, and bulk and surface anelasticity are introduced by replacing the elastic metrics with their anelastic counterparts. The balance laws are derived variationally using the Lagrange-d'Alembert principle. These include the bulk balance of linear momentum together with the surface balance of linear momentum, whose normal component gives a generalized Laplace's law. As an application, we obtain the complete solution for a spherical incompressible isotropic solid ball containing a cavity filled with a compressible hyperelastic fluid, where the cavity boundary is an anelastic material surface with distributed surface eigenstrains. The analytical and numerical results quantify the effects of surface and fluid eigenstrains on the pressure-stretch response and residual stress.

math-ph↗

arXiv2025

On Universal Deformations and Material Preferred Directions in Anisotropic Cauchy Elasticity

In this paper we study universal deformations in anisotropic Cauchy elasticity. We show that the universality constraints of hyperelasticity and Cauchy elasticity for transversely isotropic, orthotropic, and monoclinic solids are equivalent. This implies that for each of these symmetry classes the universal deformations and the corresponding universal material preferred directions of hyperelastic and Cauchy elastic solids are identical. This is consistent with previous findings for isotropic solids. Universal deformations and material preferred directions are therefore independent of the existence or absence of a strain energy function.

physics.class-ph↗

arXiv2025

Nonlinear Mechanics of Arterial Growth

In this paper, we formulate a geometric theory of the mechanics of arterial growth. An artery is modeled as a finite-length thick shell that is made of an incompressible nonlinear anisotropic solid. An initial radially-symmetric distribution of finite radial and circumferential eigenstrains is assumed. Bulk growth is assumed to be isotropic. A novel framework is proposed to describe the time evolution of growth, governed by a competition between the elastic energy and a \emph{growth energy}. The governing equations are derived through a two-potential approach and using the Lagrange-d'Alembert principle. An isotropic dissipation potential is considered, which is assumed to be convex in the rate of growth function. Several numerical examples are presented that demonstrate the effectiveness of the proposed model in predicting the evolution of arterial growth and the intricate interplay among eigenstrains, residual stresses, elastic energy, growth energy, and dissipation potential. A distinctive feature of the model is that the growth variable is not constrained by an explicit upper bound; instead, growth naturally approaches a steady-state value as a consequence of the intrinsic energetic competition.

cond-mat.soft↗

arXiv2025

Universal Deformations in Compressible Isotropic Cauchy Elastic Solids with Residual Stress

We investigate universal deformations in compressible isotropic Cauchy elastic solids with residual stress, without assuming any specific source for the residual stress. We show that universal deformations must be homogeneous, and the associated residual stresses must also be homogeneous. Since a non-trivial residual stress cannot be homogeneous, it follows that residual stress must vanish. Thus, a compressible Cauchy elastic solid with a non-trivial distribution of residual stress cannot admit universal deformations. These findings are consistent with the results of \citet{YavariGoriely2016}, who showed that in the presence of eigenstrains, universal deformations are covariantly homogeneous and in the case of simply-connected bodies the universal eigenstrains are zero-stress (impotent).

math-ph↗

arXiv2025

On Universal Deformations of Compressible Cauchy Elastic Solids Reinforced by Inextensible Fibers

Universal deformations are those that can be maintained in the absence of body forces and with boundary tractions alone, for all materials within a given constitutive class. We study the universal deformations of compressible isotropic Cauchy elastic solids reinforced by a single family of inextensible fibers. We consider straight fibers parallel to the Cartesian Z-axis in the reference configuration and derive the associated universality constraints, which depend explicitly on the geometry of the deformed fibers. We study universal deformations in two cases: (i) deformed fibers are straight lines, and (ii) deformed fibers have non-vanishing curvature. For case (i), we provide a complete classification. The universality constraints reduce to geometric restrictions on the orthogonal surfaces, which must be planes, circular cylinders, or spheres. This gives one inhomogeneous universal deformation family: the non-isochoric Family Z1 of combined bending and stretching deformations. In addition, Family 0Z consists of homogeneous deformations that respect the inextensibility constraint. We further show that if all principal invariants are constant and deformed fibers remain straight, then only homogeneous universal deformations are possible. For case (ii), when deformed fibers have non-vanishing curvature, the universality constraints become significantly more complex. The existence of universal deformations in this case remains an open problem. In particular, we demonstrate that Family 5 universal deformations of incompressible elasticity, when restricted to satisfy the inextensibility constraint, are no longer universal in fiber-reinforced solids. Finally, we prove that the universal deformations of Cauchy and hyperelastic solids with the same fiber reinforcement coincide.

math-ph↗

arXiv2025

The Darboux Classification of Curl Forces

We study particle dynamics under curl forces. These forces are a class of non-conservative, non-dissipative, position-dependent forces that cannot be expressed as gradient of a potential function. We show that the fundamental quantity of particle dynamics under curl forces is a work $1$-form. By using the Darboux classification of differential $1$-forms on $\mathbb{R}^2$ and $\mathbb{R}^3$, we establish that any curl force in two dimensions has at most two generalized potentials, while in three dimensions, it has at most three. These potentials generalize the single potential of conservative systems. For any curl force field, we introduce a corresponding conservative force field -- the conservative auxiliary force. The Hamiltonian of this conservative force is a conserved quantity of motion for the dynamics of a particle under the curl force, although it is not the physical energy.

math-ph↗

arXiv2025

Second-Order Compatible-Strain Mixed Finite Elements for 2D Compressible Nonlinear Elasticity

In recent years, a new class of mixed finite elements -- compatible-strain mixed finite elements (CSMFEs) -- has emerged that uses the differential complex of nonlinear elasticity. Their excellent performance in benchmark problems, such as numerical stability for modeling large deformations in near-incompressible solids, makes them a promising choice for solving engineering problems. Explicit forms exist for various shape functions of first-order CSMFEs. In contrast, existing second-order CSMFEs evaluate shape functions using numerical integration. In this paper, we formulate second-order CSMFEs with explicit shape functions for the displacement gradient and stress tensor. Concepts of vector calculus that stem from exterior calculus are presented and used to provide efficient forms for shape functions in the natural coordinate system. Covariant and contravariant Piola transformations are then applied to transform the shape functions to the physical space. Mid-nodes and pseudo-nodes are used to enforce the continuity constraints for the displacement gradient and stress tensor over the boundaries of elements. The formulation of the proposed second-order CSMFEs and technical aspects regarding their implementation are discussed in detail. Several benchmark problems are solved to compare the performance of CSMFEs with first-order CSMFEs and other second-order elements that rely on numerical integration. It is shown that the proposed CSMFEs are numerically stable for modeling near-incompressible solids in the finite strain regime.

math.NA↗

arXiv2024

Nonlinear Cauchy Elasticity

Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyper-elasticity. However, in its original form the theory of elasticity does not assume the existence of a strain-energy function. In this case, called Cauchy elasticity, stresses are directly related to the strains. Since the emergence of modern elasticity in the 1940s, research on Cauchy elasticity has been relatively limited. One possible reason is that for Cauchy materials, the net work performed by stress along a closed path in the strain space may be nonzero. Therefore, such materials may require access to both energy sources and sinks. This characteristic has led some mechanicians to question the viability of Cauchy elasticity as a physically plausible theory of elasticity. In this paper, motivated by its relevance to recent applications, such as the modeling of active solids, we revisit Cauchy elasticity in a modern form.

physics.class-ph↗

arXiv2024

Controllable Deformations in Compressible Isotropic Implicit Elasticity

For a given material, \emph{controllable deformations} are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, \emph{universal deformations} are those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress $\boldsymbolσ$ and the left Cauchy-Green strain $\mathbf{b}$, have the implicit form $\boldsymbol{\mathsf{f}}(\boldsymbolσ,\mathbf{b})=\mathbf{0}$. We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is permissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations.

cond-mat.mtrl-sci↗

arXiv2024

Universal Deformations and Inhomogeneities in Isotropic Cauchy Elasticity

For a given class of materials, \emph{universal deformations} are those deformations that can be maintained in the absence of body forces and by applying solely boundary tractions. For inhomogeneous bodies, in addition to the universality constraints that determine the universal deformations, there are extra constraints on the form of the material inhomogeneities -- \emph{universal inhomogeneity constraints}. Those inhomogeneities compatible with the universal inhomogeneity constraints are called \emph{universal inhomogeneities}. In a Cauchy elastic solid, stress at a given point and at an instance of time is a function of strain at that point and that exact moment in time, without any dependence on prior history. A Cauchy elastic solid does not necessarily have an energy function, i.e., Cauchy elastic solids are, in general, non-hyperelastic (or non-Green elastic). In this paper we characterize universal deformations in both compressible and incompressible inhomogeneous isotropic Cauchy elasticity. As Cauchy elasticity includes hyperelasticity, one expects the universal deformations of Cauchy elasticity to be a subset of those of hyperelasticity both in the compressible and incompressible cases. It is also expected that the universal inhomogeneity constraints to be more strict than those of hyperelasticity, and hence, the set of universal inhomogeneities to be smaller than that of hyperelasticity. We prove the unexpected result that the sets of universal deformations of isotropic Cauchy elasticity and isotropic hyperelasticity are identical, in both the compressible and incompressible cases. We also prove that their corresponding universal inhomogeneities are identical as well.

physics.class-ph↗

arXiv2024

Nonlinear mechanics of phase-change-induced accretion

In this paper, we formulate a continuum theory of solidification within the context of finite-strain coupled thermoelasticity. We aim to fill a gap in the existing literature, as the existing studies on solidification typically decouple the thermal problem (the classical Stefan's problem) from the elasticity problem, and often limit themselves to linear elasticity with small strains. Treating solidification as an accretion problem, with the growth velocity correlated with the jump in the heat flux across the boundary, it presents an initial boundary-value problem (IBVP) over a domain whose boundary location is a priori unknown. This IBVP is solved numerically for the specific example of radially inward solidification in a spherical container. Several parametric studies are conducted to compare the numerical results with the rigid cases in the literature and gain insights into the role of elastic deformations in solidification.

cond-mat.mtrl-sci↗