Geometric mechanics

In this paper we revisit the mathematical foundations of nonlinear viscoelasticity. We study the underlying geometry of viscoelastic deformations, and in particular, the intermediate configuration. Starting from the multiplicative decomposition of deformation gradient into elastic and viscous parts F=FeFv, we point out that Fv can be either a material tensor (Fe is a two-point tensor) or a two-point tensor (Fe is a spatial tensor).

In this paper we formulate a geometric nonlinear theory of the mechanics of accreting-ablating bodies. This is a generalization of the theory of accretion mechanics of Sozio and Yavari (2019). More specifically, we are interested in large deformation analysis of bodies that undergo a continuous and simultaneous accretion and ablation on their boundaries while under external loads.

In this paper a geometric field theory of dislocation dynamics and finite plasticity in single crystals is formulated. Starting from the multiplicative decomposition of the deformation gradient into elastic and plastic parts, we use Cartan's moving frames to describe the distorted lattice structure via differential 1-forms. In this theory the primary fields are the dislocation fields, defined as a collection of differential 2-forms. The defect content of the lattice structure is then determined by the superposition of the dislocation fields.

In this paper we formulate and solve the initial-boundary value problem of accreting circular cylindrical bars under finite extension. We assume that the bar grows by printing stress-free cylindrical layers on its boundary cylinder while it is undergoing a time-dependent finite extension. Accretion induces eigenstrains, and consequently residual stresses. We formulate the anelasticity problem by first constructing the natural Riemannian metric of the growing bar. This metric explicitly depends on the history of deformation during the accretion process.

In this paper we formulate the initial-boundary value problem of accreting circular cylindrical bars under finite torsion. It is assumed that the bar grows as a result of printing stress-free cylindrical layers on its boundary while it is under a time-dependent torque (or a time-dependent twist) and is free to deform axially. In a deforming body, accretion induces eigenetrains, and consequently residual stresses. We formulate the anelasticity problem by first constructing the natural Riemannian metric of the growing bar.

The recent literature of finite eignestrains in nonlinear elastic solids is reviewed, and Eshelby's inclusion problem at finite strains is revisited. The subtleties of the analysis of combinations of finite eigenstrains for the example of  combined finite radial, azimuthal, axial, and twist eigenstrains in a finite circular cylindrical bar are discussed. The stress field of a spherical inclusion with uniform pure dilatational eigenstrain in a radially-inhomogeneous spherical ball made of arbitrary incompressible isotropic solids is analyzed.

In this paper we analyze the stress field of a solid torus made of an incompressible isotropic solid with a toroidal inclusion that is concentric with the solid torus and has a uniform distribution of pure dilatational finite eigenstrains. We use a perturbation analysis and calculate the residual stresses to the first order in the thinness ratio (the ratio of the radius of the generating circle and the overall radius of the solid torus). In particular, we show that the stress field inside the inclusion is not uniform.

The elastic Ericksen's problem consists of finding deformations in isotropic hyperelastic solids that can be maintained for arbitrary strain-energy density functions.  In the compressible case, Ericksen showed that only homogeneous deformations are possible. Here, we solve the anelastic version of the same problem, that is we determine both the deformations and the eigenstrains such that a solution to the anelastic problem exists for arbitrary strain-energy density functions. Anelasticity is described by finite eigenstrains.

In this paper we are concerned with finding exact solutions for the stress fields of nonlinear solids with non-symmetric distributions of defects (or more generally finite eigenstrains) that are small perturbations of symmetric distributions of defects with known exact solutions. In the language of geometric mechanics this corresponds to finding a deformation that is a result of a perturbation of the metric of the Riemannian material manifold. We present a general framework that can be used for a systematic analysis of this class of anelasticity problems.

We introduce some Hilbert complexes involving second-order tensors on flat compact manifolds with boundary that describe the kinematics and the kinetics of motion in nonlinear elasticity. We then use the general framework of Hilbert complexes to write Hodge-type and Helmholtz-type orthogonal decompositions for second-order tensors.

In this paper we formulate a nonlinear elasticity theory in which the ambient space is evolving. For a continuum moving in an evolving ambient space, we model time dependency of the metric by a time-dependent embedding of the ambient space in a larger manifold with a fixed background metric. We derive both the tangential and the normal governing equations. We then reduce the standard energy balance written in the larger ambient space to that in the evolving ambient space.

In this paper we formulate a geometric theory of nonlinear thermoelasticity that can be used to calculate the time evolution of the temperature and thermal stress fields in a nonlinear elastic body. In particular, this formulation can be used to calculate residual thermal stresses. In this theory the material manifold (natural stress-free configuration of the body) is a Riemannian manifold with a temperature-dependent metric. Evolution of the geometry of the material manifold is governed by a generalized heat equation.

The geometric formulation of continuum mechanics provides a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a non-elastic component arising from defects, thermal stresses, growth effects, or other effects leading to residual stresses. The central idea is to assume that the material manifold, prescribing the reference configuration for a body, has an intrinsic, non-Euclidean, geometric structure. Residual stresses then naturally arise when this configuration is mapped into Euclidean space.

We discuss the relevance of non-metricity in a metric-affine manifold (a manifold equipped with a connection and a metric) and the nonlinear mechanics of distributed point defects. We describe a geometric framework in which one can calculate analytically the residual stress field of nonlinear elastic solids with distributed point defects. In particular, we use Cartan's machinery of moving frames and construct the material manifold of a finite ball with a spherically-symmetric distribution of point defects.

I am looking for a new Ph.D. student to work on discretization of nonlinear elasticity using geometric and topological ideas. Requirements for this position are a strong background in solid mechanics and some background in differential geometry and analysis. If interested please email me your CV.

In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized elastic body modeled by a simplicial complex. After characterizing the configuration manifold of volume-preserving discrete deformations, we use Hamilton's principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers.

In the theory of dislocations, the Burgers vector is usually defined by referring to a crystal structure. Using the notion of affine development of curves on a differential manifold with a connection, we give a differential geometric definition of the Burgers vector directly in the continuum setting, without making use of an underlying crystal structure.

This paper presents a geometric theory of the mechanics of growing bodies.

This paper is dedicated to the memory of Professor Jim Knowles.

http://lanl.arxiv.org/abs/0911.4671