On non-uniqueness in the traction boundary- value problem for a compressible elastic solid

For a compressible isotropic elastic solid local and global non-uniqueness of the homogeneous deformation resulting from prescribed dead-load boundary tractions is examined. In particular, for the plane-strain problem with equibiaxial in-plane tension, equations governing the paths of deformation branching from the bifurcation point on a deformation path corresponding to in-plane pure dilatation are derived. Explicit calculations are given for a specific strain-energy function and the stability of the branches is discussed. Some general results are then given for an arbitrary form of strain-energy function

Principal stress and strain trajectories in non-linear elastostatics

The Maxwell-Lame equations governing the principal components of Cauchy stress for plane deformations are well known in the context of photo-elasticity, and they form a pair of coupled first-order hyperbolic partial differential equations when the deformation geometry is known. In the present paper this theme is developed for non-linear isotropic elastic materials by supplementing the (Lagrangean form of the) equilibrium equations by a pair of compatibility equations governing the deformation. The resulting equations form a system of four first-order partial differential equations governing the principal stretches of the plane deformation and the two angles which define the orientation of the Lagrangean and Eulerian principal axes of the deformation. Coordinate curves are chosen to coincide locally with the Lagrangean (Eulerian) principal strain trajectories in the undeformed (deformed) material.Coupled with appropriate boundary conditions these equations can be used to calculate directly the principal stretches and stresses together with their trajectories. The theory is illustrated by means of a simple example

Local and global bifurcation phenomena

Bifurcation, global non-uniqueness and stability of solutions to the plane-strain problem of an incompressible isotropic elastic material subject to in-plane dead-load tractions are considered. In particular, for loading in equibiaxial tension, bifurcation from a configuration in which the in-plane principal stretches are equal is shown to occur at a certain critical value of the tension (which depends on the form of strain-energy function). Results concerning the global invertibility of the elastic stress- deformation relations are obtained and then used to derive an equation governing the deformation paths branching from this critical value. The stability of each branch is also examined. The analysis is carried through for a general form of strain-energy function and the results are then illustrated for a particular class of strain-energy functions

On Eulerian and Lagrangean objectivity in continuum mechanics

In continuum mechanics the commonly—used definition of objectivity (or frame-indifference) of a tensor field does not distinguish between Eulerian, Lagrangean and two—point tensor fields. This paper highlights the distinction and provides a definition of objectivity which reflects the different transformation rules for Eulerian, Lagrangean and two- point tensor fields under an observer transformation. The notion of induced objectivity is introduced and its implications examined

On large bending deformations of transversely isotropic rectangular elastic blocks

In this paper we examine the classical problem of finite bending of a rectangular block of elastic material into a sector of a circular cylindrical tube in respect of compressible transversely isotropic elastic materials. More specifically, we consider the possible existence of isochoric solutions. In contrast to the corresponding problem for isotropic materials, for which such solutions do not exist for a compressible material, we determine conditions on the form of the strain-energy function for which isochoric solutions are possible. The results are illustrated for particular classes of energy function

Non-linear optimization of the material constants in Ogden's strain-energy function for incompressible isotropic elastic materials

The Levenberg—Marquardt non—linear least squares optimization algorithm is adapted to compute the material constants in Ogden' s strain—energy function for incompressible isotropic elastic materials. In previous papers, three terms have been included in the strain-energy function. In the present paper, four terms are used and it is shown that the optimal values of the eight material constants, which are determined using the Levenberg—Marquardt algorithm, give a much closer fit to experimental data than the strain-energy function with three terms

Hyperelastic cloaking theory: Transformation elasticity with pre-stressed solids

Transformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting the similarity between transformation elasticity and the theory of incremental motion superimposed on finite pre-strain it is shown that the constitutive parameters of transformation elasticity correspond to the density and moduli of small-on-large theory. The formal equivalence indicates that transformation elasticity can be achieved by selecting a particular finite (hyperelastic) strain energy function, which for isotropic elasticity is semilinear strain energy. The associated elastic transformation is restricted by the requirement of statically equilibrated pre-stress. This constraint can be cast as \tr {\mathbf F} = constant, where $\mathbf{F}$ is the deformation gradient, subject to symmetry constraints, and its consequences are explored both analytically and through numerical examples of cloaking of anti-plane and in-plane wave motion.Comment: 20 pages, 5 figure