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

Finite deformations of an electroelastic circular cylindrical tube

In this paper the theory of nonlinear electroelasticity is used to examine deformations of a pressurized thick-walled circular cylindrical tube of soft dielectric material with closed ends and compliant electrodes on its curved boundaries. Expressions for the dependence of the pressure and reduced axial load on the deformation and a potential difference between, or uniform surface charge distributions on, the electrodes are obtained in respect of a general isotropic electroelastic energy function. To illustrate the behaviour of the tube, specific forms of energy functions accounting for different mechanical properties coupled with a deformation independent quadratic dependence on the electric field are used for numerical purposes, for a given potential difference and separately for a given charge distribution. Numerical dependences of the non-dimensional pressure and reduced axial load on the deformation are obtained for the considered energy functions. Results are then given for the thin-walled approximation as a limiting case of a thick-walled cylindrical tube without restriction on the energy function. The theory described herein provides a general basis for the detailed analysis of the electroelastic response of tubular dielectric elastomer actuators, which is illustrated for a fixed axial load in the absence of internal pressure and fixed internal pressure in the absence of an applied axial load

The effect of deformation dependent permittivity on the elastic response of a finitely deformed dielectric tube

In this paper, the influence of a radial electric field generated by compliant electrodes on the curved surfaces of a tube of dielectric electroelastic material subject to radially symmetric finite deformations is analyzed within the framework of the general theory of nonlinear electroelasticity. The analysis is illustrated for two constitutive equations based on the neo-Hookean and Gent elasticity models supplemented by an electrostatic energy term with a deformation dependent permittivity

Extension, inflation and torsion of a residually-stressed circular cylindrical tube

In this paper, we provide a new example of the solution of a finite deformation boundary-value problem for a residually stressed elastic body. Specifically, we analyse the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress, the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress

Bifurcation of finitely deformed thick-walled electroelastic cylindrical tubes subject to a radial electric field

This paper is concerned with the bifurcation analysis of a pressurized electroelastic circular cylindrical tube with closed ends and compliant electrodes on its curved boundaries. The theory of small incremental electroelastic deformations superimposed on a finitely deformed electroelastic tube is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect cylindrical shape of the tube. First, prismatic bifurcations are examined and solutions are obtained which show that for a neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This result contrasts with that for the purely elastic case for which prismatic bifurcation modes were found only for an externally pressurized tube. Second, axisymmetric bifurcations are analyzed, and results for both neo-Hookean and Mooney–Rivlin electroelastic energy functions are obtained. The solutions show that in the presence of a moderate electric field the electroelastic tube becomes more susceptible to bifurcation, i.e., for fixed values of the axial stretch axisymmetric bifurcations become possible at lower values of the circumferential stretches than in the corresponding problems in the absence of an electric field. As the magnitude of the electric field increases, however, the possibility of bifurcation under internal pressure becomes restricted to a limited range of values of the axial stretch and is phased out completely for sufficiently large electric fields. Then, axisymmetric bifurcation is only possible under external pressure

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