On the problem of eversion for incompressible elastic materials

The paper contains a discussion on when eversion of cylindrical tubes and spherical shells is possible. The analysis shows that eversion of a cylindrical tube of every isotropic incompressible elastic material with no applied forces is possible assuming only the E-inequalities. This is not always true for spherical shells. Conditions are given as to when this is possible and when it is not possible.     

An existence theorem for the Dirichlet problem in the elastodynamics of incompressible materials

Communicated by M. E. Gurtin     

The slowness surfaces of incompressible and nearly incompressible elastic materials

The slowness surface of a compressible elastic material has three sheets whilst that of an incompressible elastic material has only two sheets. The explanation for this qualitative difference is found to be that as the material approaches an incompressible limit the inmost sheet becomes a small sphere collapsing to the origin whilst the other two sheets tend to the two sheets of the limiting incompressible solid. The theory of nearly incompressible materials is developed here because of its important applications to rubberlike solids. Some results on the wave polarisations and on the convexity of the slowness surfaces are also given.     

Non-linear stress-strain equations for incompressible transversely isotropic elastic materials

Non-linear stress-strain equations for incompressible, transversely isotropic elastic materials are developed. In order to obtain these equations, the expressions for a strain energy function is found. The derivation of the strain energy function follows a geometrical approach and a method suggested by Mooney. These stress-strain relations are expressed in terms of three principal stretches to the sixth order.     

Dynamic universal solutions for fiber-reinforced incompressible isotropic elastic materials

We consider the effect of a fiber-reinforcement on the dynamic universality of the following families of motions: bending and shearing of a rectangular block; straightening and shearing of a sector of a circular tube; inflation, eversion, extension, bending and shearing of a sector of a circular tube; inflation, extension, bending and azimuthal shearing of a sector of a circular tube.

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Résumé Nous avons considéré l'effet d'un renforcement par fibre sur l'universalité dynamique des familles de mouvement suivant: courbage et cisaillement d'un bloc rectangulaire; redressement et cisaillement d'un secteur de tube circulaire; gonflement, retournement, allongement, courbage et cisaillement d'un secteur de tube circulaire; gonflement, allongement, courbage et cisaillement azimuthal d'un secteur de tube circulaire.

     

Plane strain problem for an incompressible nonlinearly elastic solid

The plane strain of an incompressible body is studied with geometrical and physical nonlinearity and potential forces taken into account. A nonlinear system of equations for strains is obtained in actual variables, and conditions of its ellipticity are derived in terms of the elastic potential. Boundary conditions for strains are found from specified loads. Analytical solutions of the boundary problem in strains and their corresponding stress fields are found for the case of identical elongations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 217–225, March–April, 2009     

Universal deformations in incompressible isotropic elastic materials

One is concerned with the problem of determining the static deformations which can be produced in every isotropic, homogeneous, incompressible elastic body by the action of surface tractions alone. It is shown that any new solution cannot have more than one of the proper vectors of the deformation tensorc determining a vector field of constant non-zero abnormality.     

An integral representation of the surface-impedance tensor for incompressible elastic materials

A fundamental result in anisotropic elasticity and surface-wave theory is the integral representation for the surface-impedance tensor first derived by Barnett and Lothe in 1973. However, this representation is only valid for compressible materials but not valid for incompressible materials. In this paper the corresponding integral representation for the surface-impedance tensor valid for incompressible materials is derived and is used to establish the uniqueness of surface-wave speed and to obtain an expression for the tensor Green's function for the infinite space. Mathematics subject classifications (2000) 74B05, 74B15, 74B20, 74J15     

On iterative methods for the incompressible Stokes problem

In this paper, we discuss various techniques for solving the system of linear equations that arise from the discretization of the incompressible Stokes equations by the finite‐element method. The proposed solution methods, based on a suitable approximation of the Schur‐complement matrix, are shown to be very effective for a variety of problems. In this paper, we discuss three types of iterative methods. Two of these approaches use the pressure mass matrix as preconditioner (or an approximation) to the Schur complement, whereas the third uses an approximation based on the ideas of least‐squares commutators (LSC). We observe that the approximation based on the pressure mass matrix gives h‐independent convergence, for both constant and variable viscosity. Copyright © 2010 John Wiley & Sons, Ltd.     

Plane contact problem for a prestressed incompressible elastic layer clamped along the base

The problem of a rigid punch penetration into the upper face of a layer is considered in the case of a homogeneous field of initial stresses. The model of isotropic incompressible nonlinearly elastic material determined by the Mooney potential is used. The case of rigid clamping of the layer along its lower face is considered under the assumption that the additional stresses caused by the penetrating punch are small compared with the initial ones. This assumption allows one to linearize the problem of determining the additional stresses. This problem is then reduced to solving an integral equations of the first kind with a difference kernel which allows one to determine the pressure in the contact region. An asymptotic solution is constructed for large values of the parameter characterizing the relative thickness of the layer. Amodified Multhopp-Kalandiyamethod is also used to obtain a solution for a wider range of the parameter.     

Equilibrium shocks in plane deformations of incompressible elastic materials

This paper is concerned with piecewise smooth plane deformations in an isotropic, incompressible elastic material. An explicit necessary and sufficient condition for the existence of piecewise homogeneous equilibrium states is established, and the set of all such states is precisely characterized. A particularly simple expression is derived for the driving traction on a surface of discontinuity in the deformation gradient.     

Moderate deformations in extension-torsion of incompressible isotropic elastic materials

It has been previously shown by anand (1979) that the classical strain energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain energy function is replaced by the Hencky or logarithmic measure of finite strain. The basis in Anand's paper for relating Hencky's strain energy function to experiment was data from experiments on metals and rubbers in uniaxial strain, simple tension and compression, and pure shear. Here, to test further the validity of this strain energy function for moderate deformations, its predictions for the twisting moment and the axial force in simple torsion and combined extension-torsion of solid cylinders of incompressible materials are calculated and shown to be in good agreement with data from the classical experiments of Rivlin and Saunders (1951) on vulcanized natural rubber. Indeed, the predictions from Hencky's strain energy function are in better accord with experiment than the predictions from the widely used Mooney (or Mooney-Rivlin) strain energy function.     

On Saint-Venant's problem for elastic dielectrics

The equilibrium theory of linear piezoelectricity is considered. Saint-Venant's problem for a homogeneous and anisotropic piezoelectric cylinder is studied.     

Note on a class of controllable deformations of incompressible elastic materials

A controllable deformation is one which is possible in every material by the application of suitable surface tractions. In this note we give a direct complex variable method of obtaining a three parameter family of deformations due to Singh and Pipkin.     

Exact solutions to a problem in terms of stresses for incompressible elastic conical bodies

Exact solutions to the elasticity theory problem in terms of stresses for an incompressible conical body of arbitrary shape under the action of a given concentrated force applied at its vertex are given and analyzed. A solution in terms of stresses with a singularity whose order is higher by one than that in the classical solution is discussed. The surface load at the boundary of the conical body corresponding to such a solution is obtained.     

Semi-inverse solution for Saint-Venant's problem in the theory of porous elastic materials

The purpose of this research is to study the Saint-Venant's problem for right cylinders with general cross-section made of inhomogeneous anisotropic elastic materials with voids. We reformulate the quasi-static equilibrium equations with the axial variable playing the role of a parameter. Two classes of semi-inverse solutions to Saint-Venant's problem are described in terms of five generalized plane strain problems. These classes are used in order to obtain a semi-inverse solution for the relaxed Saint-Venant's problem. An application of this results in the study of extension, bending, torsion and flexure of right circular cylinders in the case of isotropic materials is presented.     

On the Dirichlet problem for quasi-linear elliptic equation with a small parameter

In this paper we deal with the Dirichlet problem for quasilinear elliptic equation with a small parameter at highest derivatives. In case the characteristics of the degenerated equation are curvilinear and the domain, where the problem is defined, is a bounded convex domain, we offer a method to construct the uniformly valid asymptotic solution of this problem, and prove that the solution of this problem really exists, and being uniquely determined as the small parameter is sufficiently small.     

On the radial oscillations of incompressible, isotropic, elastic and limited elastic thick-walled tubes

The finite amplitude, free radial oscillations of a thick-walled circular cylindrical tube are studied for an arbitrary incompressible, isotropic and homogeneous rubber-like material having limiting molecular chain extensibility. First, based on classical results for hyperelastic tubes, some results for thick-walled Mooney-Rivlin tubes are described graphically in the phase plane. Then the periodicity of the finite amplitude, free oscillations of a general limited elastic, thick-walled tube is studied; and some analytical results for the Gent model are illustrated in several numerical examples. Results for thick-walled Gent tubes are compared with those for corresponding Mooney-Rivlin tubes; and the motion of thin-walled Gent tubes is illustrated in the phase plane. Physical conclusions are presented. The period of small amplitude oscillations of an arbitrary elastic or limited elastic tube is derived from relations obtained by a linearization of a general class of equations of which the tube problem is a special case. Classical results of the linear theory are thereby recovered and compared with results for Mooney-Rivlin and Gent tubes.