Nonlinear elasticity

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.

Amit Acharya, Gui-Qiang Chen, Siran Li, Marshall Slemrod, and Dehua Wang

(To appear in Archive for Rational Mechanics and Analysis)

We are concerned with underlying connections between fluids,
elasticity, isometric embedding of Riemannian manifolds, and the existence of
wrinkled solutions of the associated nonlinear partial differential equations. In
this paper, we develop such connections for the case of two spatial dimensions,
and demonstrate that the continuum mechanical equations can be mapped into
a corresponding geometric framework and the inherent direct application of
the theory of isometric embeddings and the Gauss-Codazzi equations through
examples for the Euler equations for fluids and the Euler-Lagrange equations
for elastic solids. These results show that the geometric theory provides an
avenue for addressing the admissibility criteria for nonlinear conservation laws
in continuum mechanics.

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.

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.

 By :Payam SOLTANI, J SABERIAN, R BAHRAMIAN, A FARSHIDIANFAR

In this Paper, the nonlinear free and force vibration of a single-walled carbon nanotube (SWCNT) with simply supported ends is 

investigated based on von Karman’s geometric nonlinearity. The SWCNT described as an individual shell and the Donnell’s 

equations of cylindrical shells are used to obtain the governing equations. The Galerkin's procedure is used to discretized partial 

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.

Applications are invited for the position of
Post‐doctoral Research
Fellow as part of a joint research project between Masdar Institute of Science
& Technology and Massachusetts Institute of Technology (MIT). Details of the position are given in the attachement.

Compatibility equations of elasticity are almost 150 years old. Interestingly they do not seem to have been rigorously studied for non-simply-connected bodies to this date. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible even if the standard compatibility equations ("bulk" compatibility equations) are satisfied.

Abstract

Payam Soltani, J. Saberian, R. Bahramian, and A. Farshidianfar

http://fundamentaljournals.org/ijfps/archive.html#A14 

In this Paper, the nonlinear free and force vibration of a single-walled carbon nanotube (SWCNT) with simply supported ends is

investigated based on von Karman’s geometric nonlinearity. The SWCNT described as an individual shell and the Donnell’s

equations of cylindrical shells are used to obtain the governing equations. The Galerkin's procedure is used to discretized partial

In the post

 node/9509

The Mathematical Institute at Oxford is advertising three post-doctoral positions within the Oxford Centre for Collaborative Applied Mathematics for(ideally) a 1st April 2011 start. The project titles (with links for more information) are:

1. Brain Mechanics, Cortex Folding, and Pattern Formation in Growing Tissues.
http://www.maths.ox.ac.uk/node/1405

Qiang Lu and Rui Huang

Department of Aerospace Engineering and Engineering mechanics, University of Texas, Austin,
TX 78712, USA

Electro-sensitive (ES) elastomers form a class of smart materials whose mechanical properties can be changed rapidly by the application of an electric field. These materials have attracted considerable interest recently because of their potential for providing relatively cheap and light replacements for mechanical devices, such as actuators, and also for the development of artificial muscles. In this paper we are concerned with a theoretical framework for the analysis of boundary-value problems that underpin the applications of the associated electromechanical interactions. We confine attention to the static situation and first summarize the governing equations for a solid material capable of large electroelastic deformations. The general constitutive laws for the Cauchy stress tensor and the electric field vectors for an isotropic electroelastic material are developed in a compact form following recent work by the authors. The equations are then applied, in the case of an incompressible material, to the solution of a number of representative boundary-value problems. Specifically, we consider the influence of a radial electric field on the azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation characteristics of the same tube under either a radial or an axial electric field (or both fields combined), and the effect of a radial field on the deformation of an internally pressurized spherical shell.