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# Arash_Yavari's blog

## The mathematical foundations of anelasticity: Existence of smooth global intermediate configurations

Tue, 2020-12-01 08:31 - Arash_YavariA central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground.

## The Anelastic Ericksen Problem: Universal Deformations and Universal Eigenstrains in Incompressible Nonlinear Anelasticity

Thu, 2020-09-17 09:23 - Arash_YavariEricksen's problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold.

## Transformation Cloaking in Elastic Plates

Sun, 2020-09-13 10:20 - Arash_YavariIn this paper we formulate the problem of elastodynamic transformation cloaking for Kirchhoff-Love plates and elastic plates with both in-plane and out-of-plane displacements. A cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic plate (virtual problem) to that of an anisotropic and inhomogeneous elastic plate with a hole surrounded by a cloak that is to be designed (physical problem). For Kirchhoff-Love plates, the governing equation of the virtual plate is transformed to that of the physical plate up to an unknown scalar field.

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## Nonlinear Mechanics of Thermoelastic Accretion

Fri, 2020-03-20 11:52 - Arash_YavariIn this paper, we formulate a theory for the coupling of accretion mechanics and thermoelasticity. We present an analytical formulation of the thermoelastic accretion of an infinite cylinder and of a two-dimensional block.

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## Universal Displacements in Linear Elasticity

Fri, 2019-11-08 08:53 - Arash_YavariIn nonlinear elasticity, universal deformations are the deformations that exist for arbitrary strain-energy density functions and suitable tractions at the boundaries. Here, we discuss the equivalent problem for linear elasticity. We characterize the universal displacements of linear elasticity: those displacement fields that can be maintained by applying boundary tractions in the absence of body forces for any linear elastic solid in a given anisotropy class.

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## Riemannian and Euclidean Material Structures in Anelasticty

Sat, 2019-10-05 15:18 - Arash_YavariIn this paper we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to nonlinear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold.

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## Applications of Algebraic Topology in Elasticity

Mon, 2019-09-16 17:40 - Arash_YavariIn this book chapter we discuss some applications of algebraic topology in elasticity. This includes the necessary and sufficient compatibility equations of nonlinear elasticity for non-simply-connected bodies when the ambient space is Euclidean. Algebraic topology is the natural tool to understand the topological obstructions to compatibility for both the deformation gradient **F** and the right Cauchy-Green strain **C**. We will investigate the relevance of homology, cohomology, and homotopy groups in elasticity.

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## Compatible-Strain Mixed Finite Element Methods for 3D Compressible and Incompressible Nonlinear Elasticity

Mon, 2019-08-26 09:04 - Arash_YavariA new family of mixed finite element methods --- *compatible-strain mixed finite element methods* (CSFEMs) --- are introduced for three-dimensional compressible and incompressible nonlinear elasticity. A Hu-Washizu-type functional is extremized in order to obtain a mixed formulation for nonlinear elasticity. The independent fields of the mixed formulations are the displacement, the displacement gradient, and the first Piola-Kirchhoff stress. A pressure-like field is also introduced in the case of incompressible elasticity.

## The 55th Meeting of the Society for Natural Philosophy (support for graduate students and postdoctoral researchers)

Sun, 2019-05-19 18:00 - Arash_YavariThe 55th Meeting of the Society for Natural Philosophy: Microstructure, defects, and growth in mechanics will be from September 13-15, 2019 at Loyola University Chicago.

http://webpages.math.luc.edu/55SNP.html

A very limited number of openings to give Roundtable (25 min) talks are available. Special consideration will be given to young researchers. Two nights of lodging will be funded for these speakers. If you are interested in giving a Roundtable talk, you must submit an abstract.

## The 55th Meeting of the Society for Natural Philosophy Call for Roundtable Talks

Sun, 2019-04-07 11:27 - Arash_YavariThe 55th Meeting of the Society for Natural Philosophy: Microstructure, defects, and growth in mechanics

September 13-15, 2019, Loyola University Chicago

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## Nonlinear and Linear Elastodynamic Transformation Cloaking

Tue, 2019-03-26 14:48 - Arash_YavariIn this paper we formulate the problems of nonlinear and linear elastodynamic transformation cloaking in a geometric framework. In particular, it is noted that a cloaking transformation is neither a spatial nor a referential change of frame (coordinates); a cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem).

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## Nonlinear Mechanics of Accretion

Tue, 2019-01-08 09:08 - Arash_YavariWe formulate a geometric nonlinear theory of the mechanics of accretion. In this theory the reference configuration of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material is added to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body.

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## Faculty Opening at GA Tech: Space Habitat Systems

Sat, 2018-11-10 20:17 - Arash_YavariThe Daniel Guggenheim School of Aerospace Engineering and the School of Civil and Environmental Engineering at the Georgia Institute of Technology are seeking applications for a tenure-track faculty position in the area of space habitat systems. The position is expected to be a joint appointment between both schools. Multidisciplinary collaboration with related research groups and colleges at Georgia Tech is highly encouraged.

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## Line and Point Defects in Nonlinear Anisotropic Solids

Wed, 2018-05-16 15:18 - Arash_YavariIn this paper, we present some analytical solutions for the stress fields of nonlinear anisotropic solids with distributed line and point defects.

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## Compatible-Strain Mixed Finite Element Methods for Incompressible Nonlinear Elasticity

Tue, 2018-01-30 16:58 - Arash_YavariWe introduce a new family of mixed finite elements for incompressible nonlinear elasticity — *compatible-strain mixed finite element methods* (CSFEMs). Based on a Hu-Washizu-type functional, we write a four-field mixed formulation with the displacement, the displacement gradient, the first Piola-Kirchhoff stress, and a pressure-like field as the four independent unknowns. Using the Hilbert complexes of nonlinear elasticity, which describe the kinematics and the kinetics of motion, we identify the solution spaces of the independent unknown fields.

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## CEE Department Chair Opening at GA Tech

Fri, 2017-06-02 10:49 - Arash_YavariThe College of Engineering at the Georgia Institute of Technology is seeking nominations and applications for the position of the Karen and John Huff Chair of the School of Civil and Environmental Engineering (CEE).

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## Nonlinear Elastic Inclusions in Anisotropic Solids

Mon, 2017-04-10 14:47 - Arash_YavariIn this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic.

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## On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

Thu, 2016-12-22 19:24 - Arash_YavariIn 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.

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

Mon, 2016-11-07 18:51 - Arash_YavariThe 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.

## Small-on-Large Geometric Anelasticity

Wed, 2016-10-12 10:19 - Arash_YavariIn 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

Mon, 2016-10-10 15:32 - Arash_YavariWe 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)

Sun, 2016-10-09 18:27 - Arash_YavariDear 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

Wed, 2016-09-28 16:20 - Arash_YavariIn 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 Mechanics of Surface Growth for Cylindrical and Spherical Elastic Bodies

Thu, 2016-08-25 18:10 - Arash_YavariThis paper is dedicated to the memory of my friend Professor Bill Klug whose life was tragically cut short on June 1, 2016.

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## Nonlinear Elasticity in a Deforming Ambient Space

Fri, 2016-05-27 13:02 - Arash_YavariIn 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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