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# differential geometry

## Prescribing Surface Strains to change Gauss curvature

Sun, 2016-08-28 08:08 - NarasimhamPrescribing Surface Strains to change Gauss curvature

To change Gauss curvature K of a surface we need to strain each differential shell element by virtue of Egregium theorem ( K is invariant if strain is zero in isometry mappings).

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## The metric-restricted inverse design problem

Thu, 2015-01-08 19:42 - Amit AcharyaAmit Acharya Marta Lewicka Mohammad Reza Pakzad

In Nonlinearity, 29, 1769-1797

We study a class of design problems in solid mechanics, leading to a variation on the

classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new

context, we derive a necessary and sufficient existence condition, given through a system of total

differential equations, and discuss its integrability. In the classical context, the same approach

yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor.

In the present situation, the equations do not close in a straightforward manner, and successive

differentiation of the compatibility conditions leads to a more sophisticated algebraic description

of integrability. We also recast the problem in a variational setting and analyze the infimum value

of the appropriate incompatibility energy, resembling "non-Euclidean elasticity". We then derive a

Γ-convergence result for the dimension reduction from 3d to 2d in the Kirchhoff energy scaling

regime. A practical implementation of the algebraic conditions of integrability is also discussed.

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## Differential Complexes in Continuum Mechanics

Wed, 2014-09-24 11:50 - Arash_YavariWe 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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## Non-Metricity and the Nonlinear Mechanics of Distributed Point Defects

Thu, 2014-05-01 12:09 - Arash_YavariWe 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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## Affine Development of Closed Curves in Weitzenbock Manifolds and the Burgers Vector of Dislocation Mechanics

Fri, 2012-09-14 13:47 - Arash_YavariIn 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.

## Minisymposium on "Applications of computational geometry in analysis" within ECCOMAS 2012

Tue, 2011-11-29 07:25 - Fehmi Cirak## A Geometric Theory of Thermal Stresses

Mon, 2009-11-30 13:00 - Arash_YavariIn this paper we formulate a geometric theory of thermal stresses.

Given a temperature distribution, we associate a Riemannian

material manifold to the body, with a metric that explicitly

depends on the temperature distribution. A change of temperature

corresponds to a change of the material metric. In this sense, a

temperature change is a concrete example of the so-called

referential evolutions. We also make a concrete connection between

our geometric point of view and the multiplicative decomposition

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## Energy Balance Invariance for Interacting Particle Systems

Wed, 2008-05-28 15:01 - Arash_YavariThis paper studies the invariance of balance of

energy for a system of interacting particles under groups of

transformations. Balance of energy and its invariance is first

examined in Euclidean space. Unlike the case of continuous media,

it is shown that conservation and balance laws do not follow

from the assumption of invariance of balance of energy under

time-dependent isometries of the ambient space. However, the

postulate of invariance of balance of energy under arbitrary

diffeomorphisms of the ambient (Euclidean) space, does yield

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