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The Anelastic Ericksen's Problem: Universal Eigenstrains and Deformations in Compressible Isotropic Elastic Solids

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.

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Small-on-Large Geometric Anelasticity

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.

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Hilbert Complexes of Nonlinear Elasticity

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.

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Gérard Maugin (December 2, 1944 - September 22, 2016)

Dear Friends:

As was also mentioned by another colleague (http://imechanica.org/node/20391), Prof. Gérard Maugin passed away on September 22, 2016.

The following is a message that my good friend Prof. Marcelo Epstein sent me and a few other colleagues. He has kindly given me permission to share it with you.

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Compatible-Strain Mixed Finite Element Methods for 2D Compressible Nonlinear Elasticity

In this paper, using the Hilbert complexes of nonlinear elasticity, the approximation theory for Hilbert complexes, and the finite element exterior calculus, we introduce a new class of mixed finite element methods for 2D nonlinear elasticity -- compatible-strain mixed finite element methods (CSFEM). We consider a Hu-Washizu-type mixed formulation and choose the displacement, the displacement gradient, and the first Piola-Kirchhoff stress tensor as independent unknowns. We use the underlying spaces of the Hilbert complexes as the solution and test spaces.

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Nonlinear Elasticity in a Deforming Ambient Space

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.

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The Role of Mechanics in the Study of Lipid Bilayers

The following summer school on the role of mechanics in the study of lipid bilayers may be of interest to some of you.

www.cism.it/courses/C1608/

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Finite Eigenstrains in Nonlinear Elastic Wedges

Eigenstrains are created as a result of anelastic effects such as defects, temperature changes, bulk growth, etc., and strongly affect the overall response of solids. In this paper, we study the residual stress and deformation fields of an incompressible, isotropic, infinite wedge due to a circumferentially-symmetric distribution of finite eigenstrains. In particular, we establish explicit exact solutions for the residual stresses and deformation of a neo-Hookean wedge containing a symmetric inclusion with finite radial and circumferential eigenstrains.

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A Geometric Theory of Nonlinear Morphoelastic Shells

We formulate a geometric theory of nonlinear morphoelastic shells that can model the time evolution of residual stresses induced by bulk growth. We consider a thin body and idealize it by a representative orientable surface. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell (material manifold). We consider the evolution of both the first and second fundamental forms in the material manifold by considering them as dynamical variables in the variational problem.

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The Twist-Fit Problem: Finite Torsional and Shear Eigenstrains in Nonlinear Elastic Solids

Eigenstrains in nonlinear elastic solids are created through defects, growth, or other anelastic effects. These eigenstrains are known to be important as they can generate residual stresses and alter the overall response of the solid. Here, we study the residual stress fields generated by finite torsional or shear eigenstrains. This problem is addressed by considering a cylindrical bar made of an incompressible isotropic solid with an axisymmetric distribution of shear eigenstrains.

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The Weak Compatibility Equations of Nonlinear Elasticity and the Insufficiency of the Hadamard Jump Condition for Non-Simply Connected Bodies

We derive the compatibility equations of L2 displacement gradients on non-simply-connected bodies. These compatibility equations are useful for non-smooth strains such as those associated with deformations of multi-phase materials. As an application of these compatibility equations, we study some configurations of different phases around a hole and show that, in general, the classical Hadamard jump condition is not a sufficient compatibility condition.

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On the origins of the idea of the multiplicative decomposition of the deformation gradient

Usually the multiplicative decomposition of deformation gradient in finite plasticity is (incorrectly) attributed to Lee and Liu (1967). This short note discusses the origins of this idea, which go back to the late 1940s. We explain that the first explicit mention of this decomposition appeared a decade earlier in the work of Bilby, et al. (1957) and Kröner (1959). While writing this note I found out that Bruce Bilby passed away a couple of years ago at the age of 91.

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On the Compatibility Equations of Nonlinear and Linear Elasticity in the Presence of Boundary Conditions

We use Hodge-type orthogonal decompositions for studying the compatibility equations of the displacement gradient and the linear strain with prescribed boundary displacements. We show that the displacement gradient is compatible if and only if for any equilibrated virtual first-Piola Kirchhoff stress tensor field, the virtual work done by the displacement gradient is equal to the virtual work done by the prescribed boundary displacements. This condition is very similar to the classical compatibility equations for the linear strain.

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A new paper on Hencky-logarithmic strain by Prof. Neff

Dear Colleagues:

I thought the following recent paper by Prof. Neff may be of interest to some of you.

http://arxiv.org/abs/1505.02203

This paper discusses the natural appearance of the  Hencky-logarithmic strain tensor together with the Hencky strain energy, which can be motivated from some purely geometrical (kinematical) arguments based on the geodesic distance on the general linear group of all invertible tensors GL(n).

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On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities

The problems of singularity formation and hydrostatic stress created by an inhomogeneity with eigenstrain in an incompressible isotropic hyperelastic material are considered. For both a spherical ball and a cylindrical bar with a radially-symmetric distribution of finite possibly anisotropic eigenstrains, we show that the anisotropy of these eigenstrains at the center (the center of the sphere or the axis of the cylinder) controls the stress singularity.

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Geometric nonlinear thermoelasticity and the time evolution of thermal stresses

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.

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Differential Complexes in Continuum Mechanics

We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motions of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of the displacement gradient and the existence of stress functions on non-contractible bodies.

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Geometry, topology, and solid mechanics

Differential geometry in simple words is a generalization of calculus on some curved spaces called manifolds. An n-manifold is a space that locally looks like R^n but globally can be very different. The first significant application of differential geometry happened to be in Einstein’s theory of general relativity.

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Three interesting, recent books

Dear friends:

I’d like to bring to your attention three recently published books that I have read. The following are a few words about each book in the order that I read them.

1) “Vito Volterra” by A. Guerraggio and G. Paoloni
http://www.amazon.com/Vito-Volterra-Angelo-Guerraggio/dp/3642272622/ref=...

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The Geometry of Discombinations and its Applications to Semi-Inverse Problems in Anelasticity

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.

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A recent paper on Logarithmic Strain by Prof. Patrizio Neff

Dear friends:

Prof. Patrizio Neff has asked me to post the attached recent paper on Logarithmic Strain. It looks quite interesting (I haven’t read it yet).

Regards,
Arash

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Non-Metricity and the Nonlinear Mechanics of Distributed Point Defects

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.

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PhD Position in Geometric Mechanics at Georgia Tech

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.

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Nonlinear elastic inclusions in isotropic solids

We introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space.

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