On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials II: Axially Symmetric Deformations

In Part I of this article, we have formulated the general structure of the equations governing small plane strain deformations which are superimposed upon a known large plane strain deformation for the perfectly elastic incompressible 'modified' Varga material, and assuming only that the initial large plane deformation is a known solution of one of three first integrals previously derived by the authors. For axially summetric deformations there are only two such first integrals, one of which applies only to the single term Varga strain-energy function, and we give here the corresponding general equations for small superimposed deformations. As an illustration, a partial analysis for the case of small deformations superimposed upon the eversion of a thick spherical shell is examined. The Varga strain-energy functions are known to apply to both natural and synthetic rubber, provided the magnitude of the deformation is restricted. Their behaviour in both simple tension and equibiaxial tension, and in comparison to experimental data, is shown graphically in Part I of this paper [1]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Superposition of Generalized Plane Strain on Anti-Plane Shear Deformations in Isotropic Incompressible Hyperelastic Materials

The purpose of this research is to investigate the basic issues that arise when generalized plane strain deformations are superimposed on anti-plane shear deformations in isotropic incompressible hyperelastic materials. Attention is confined to a subclass of such materials for which the strain-energy density depends only on the first invariant of the strain tensor. The governing equations of equilibrium are a coupled system of three nonlinear partial differential equations for three displacement fields. It is shown that, for general plane domains, this system decouples the plane and anti-plane displacements only for the case of a neo-Hookean material. Even in this case, the stress field involves coupling of both deformations. For generalized neo-Hookean materials, universal relations may be used in some situations to uncouple the governing equations. It is shown that some of the results are also valid for inhomogeneous materials and for elastodynamics.     

Reduced equilibrium equations for perfectly elastic materials

Sufficient conditions are given on the coordinate systems which enable reduced equilibrium equations to be derived for perfectly elastic materials involving deformations which depend in an essential way only on two of the three coordinates. Reduced equilibrium equations given previously for plane and axially symmetric deformations are special cases of the equations given here. These equations considerably reduce the calculations involved in investigating possible solutions of finite elasticity, either exact semi-inverse solutions or approximate perturbation solutions. Moreover a formula for the pressure function appearing in the reduced equilibrium equations is given which relates to the corresponding pressure function associated with the inverse deformation. This formula is similar to one given previously for fully three dimensional deformations.     

Asymptotic Axially Symmetric Deformations for Perfectly Elastic Neo-Hookean and Mooney Materials

For axially symmetric deformations of the perfectly elastic neo-Hookean and Mooney materials, formal series solutions are determined in terms of expansions in appropriate powers of 1/R, where R is the cylindrical polar coordinate for the material coordinates. Remarkably, for both the neo-Hookean and Mooney materials, the first three terms of such expansions can be completely determined analytically in terms of elementary integrals. From the incompressibility condition and the equilibrium equations, the six unknown deformation functions, appearing in the first three terms can be reduced to five formal integrations involving in total seven arbitrary constants A, B, C, D, E, H and k ², and a further five integration constants, making a total of 12 integration constants for the deformation field. The solutions obtained for the neo-Hookean material are applied to the problem of the axial compression of a cylindrical rubber tube which has bonded metal end-plates. The solution so determined is approximate in two senses; namely as an approximate solution of the governing equations and for which the stress free and displacement boundary conditions are satisfied in an average manner only. The resulting load-deflection relation is shown graphically. The solution so determined, although approximate, attempts to solve a problem not previously tackled in the literature.      

Plane Deformations in Incompressible Nonlinear Elasticity

The substantially general class of plane deformation fields, whose only restriction requires that the angular deformation not vary radially, is considered in the context of isotropic incompressible nonlinear elasticity. Analysis to determine the types of deformations possible, that is, solutions of the governing systems of nonlinear partial differential equations and constraint of incompressibility, is developed in general. The Mooney-Rivlin material model is then considered as an example and all possible solutions to the equations of equilibrium are determined. One of these is interpreted in the context of nonradially symmetric cavitation, i.e., deformation of an intact cylinder to one with a double-cylindrical cavity. Results for general incompressible hyperelastic materials are then discussed. The novel approach taken here requires the derivation and use of a material formulation of the governing equations; the traditional approach employing a spatial formulation in which the governing equations hold on an unknown region of space is not conducive to the study of deformation fields containing more than one independent variable. The derivation of the cylindrical polar coordinate form of the equilibrium equations for the nominal stress tensor (material formulation) for a general hyperelastic solid and a fully arbitrary cylindrical deformation field is also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Change of Accumulation Mechanisms of Irreversible Deformations of Materials in an Example of Viscometric Deformation

Within the framework of the model of large deformations, the deformation of a material exhibiting elastic, viscous, and plastic properties and placed between two rigid cylinders is investigated when turning the internal cylinder. The accumulation of irreversible deformations prior to the onset of plastic flow and upon its termination is associated with creep. Reversible and irreversible deformations according to the model in question are determined by differential transport equations. To calculate the displacement fields, stresses, and reversible, irreversible, and complete deformations, a system of partial differential equations is obtained, for which a finite-difference scheme is constructed.     

Anti-plane Shear Deformations in Compressible Transversely Isotropic Materials

Anti-plane shear deformations in a compressible, transversely isotropic hyperelastic material are under investigation. The displacement is assumed to be along the direction of the symmetry axis and is independent of the axial position. The resulting equations of equilibrium form an overdetermined system of partial differential equations for which solutions do not exist in general. Necessary and sufficient conditions are derived for such materials to sustain anti-plane shear deformations in the sense that every solution to the axial equation automatically satisfies the other two in-plane equations. Comparison is made with results for isotropic materials. A weaker version of the conditions specialized to axisymmetric anti-plane shear deformations is also obtained.     

The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney-Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.     

On Non-Symmetric Deformations of Neo-Hookean Solids

Deformations possible (i.e., those satisfying the governing three-dimensional equations of equilibrium and the incompressibility constraint) within a class of non-symmetric deformations for a neo-Hookean nonlinearly elastic body were determined in [1], where it was found that only three special cases of the class of deformation fields considered could be solutions. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation contained within a family of universal deformations. In this paper, the results reported in [1] are shown to hold for a substantially broadened deformation field. This revised version was published online in July 2006 with corrections to the Cover Date.     

Hypoplasticity theory for granular materials—II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.     

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.     

Incremental Magnetoelastic Deformations,with Application to Surface Instability

In this paper the equations governing the deformations of infinitesimal (incremental) disturbances superimposed on finite static deformation fields involving magnetic and elastic interactions are presented. The coupling between the equations of mechanical equilibrium and Maxwell’s equations complicates the incremental formulation and particular attention is therefore paid to the derivation of the incremental equations, of the tensors of magnetoelastic moduli and of the incremental boundary conditions at a magnetoelastic/vacuum interface. The problem of surface stability for a solid half-space under plane strain with a magnetic field normal to its surface is used to illustrate the general results. The analysis involved leads to the simultaneous resolution of a bicubic and vanishing of a 7×7 determinant. In order to provide specific demonstration of the effect of the magnetic field, the material model is specialized to that of a “magnetoelastic Mooney–Rivlin solid”. Depending on the magnitudes of the magnetic field and the magnetoelastic coupling parameters, this shows that the half-space may become either more stable or less stable than in the absence of a magnetic field.      

Transversal forced vibrations of an axially moving sandwich belt system

Based on the author’s previously published results for transversal free vibrations of axially moving sandwich belts described by coupled partial differential equations, which are derived and analytically solved, this paper contains new analytical results, for forced vibrations of the same system excited by transversal external excitation. The transversal forced vibrations of the axially moving sandwich belts are described by the coupled partial nonhomogeneous differential equations. The partial differential equations are analytically solved. Bernoulli’s method of particular integrals and Lagrange’s method of the variations of the constants are used.     

Pairwise Deformations of an Incompressible Elastic Body under Dead-Load Tractions

In this paper we obtain necessary conditions for the existence of pairwise deformations of an incompressible, isotropic elastic body subjected to a homogeneous distribution of dead-load tractions. Explicit restrictions on the boundary loads and on the surface of discontinuity between the phases are determined. For hyperelastic bodies with stored energy depending only on the first invariant of strain, we show that pairwise deformations under examination are necessarily (within a rigid rotation) plane deformations.     

Nonlinear Plane Waves in Finite Deformable Infinite Mooney Elastic Materials

For infinite perfectly elastic Mooney materials, nonlinear plane waves are examined in both two and three dimensions. In two dimensions, longitudinal and shear plane waves are examined, while in three dimensions, longitudinal and torsional plane waves are considered. These exact dynamic deformations, applying to the incompressible perfectly elastic Mooney material, can be viewed as extensions of the corresponding static deformations first derived by Adkins [1] and Klingbeil and Shield [2]. Furthermore, the Mooney strain-energy function is the most general material admitting nontrivial dynamic deformations of this type. For two dimensions the determination of plane wave solutions reduces to elementary mathematical analysis, while in three dimensions an integral of the governing system of highly nonlinear ordinary differential equations is determined. In the latter case, solutions corresponding to particular parameter values are shown graphically. This revised version was published online in June 2006 with corrections to the Cover Date.     

Plane strain and generalized plane stress problems for fibre-reinforced materials

In this paper the basic equations governing the plane strain or generalized plane stress deformations of a linear elastic material reinforced by a single family of parallel inextensible fibres are deduced. It is found that a single system of equations will cover all cases. The solutions for plane, half-plane and strip problems are evaluated and compared with those for an ideal fibre-reinforced material.     

The bending moment and springback in pure bending of anisotropic sheets

Finite deformation rigid plastic and elastic–plastic analyses of plane strain pure bending of a plastically anisotropic sheet is presented. An efficient method for finding the exact solution is proposed by extending the previously developed method to the stage of unloading. Using this method the solutions are obtained in closed form or reduced to a numerical treatment of ordinary integrals, or an ordinary differential equation, or transcendental equations. An effect of plastic anisotropy and elastic properties on the bending moment is analyzed. The distribution of residual stresses is illustrated and an effect of material and process parameters on springback is investigated.     

Direct Stiffness Analysis of Lateral Buckling

Abstract

A direct stiffness method of analyzing the elastic ftexural-torsional buckling of rigid-jointed plane frames composed of l-section members and subjected to in-plane loads is presented. The in-plane stiffness matrix and the fixed-end resultants are obtained from the member stiffness matrices derived from the in-plane differential equations. These member stiffness matrices are assembled and solved, and their solutions are used to linearize the flexural-torsional buckling equations. The out-of-plane member stiffness matrices are then obtained numerically from the buckling equations by the method of finite integrals. The out-of-plane frame -stiffness matrix is assembled, and the critical loads are obtained when its determinant is zero. A computer program is developed which carries out either a first- or second-order in-plane analysis, and then determines the flexural-torsional buckling loads. The effects of in-plane deformations prior to buckling can be included. Very good agreement is obtained between the results computed by this program and known solutions, and its ability to analyze large complex frames is demonstrated.     

Finite Deformations and Motions of Radially Inextensible Hollow Spheres

Radial inflation–compaction and radial oscillation solutions are presented for hollow spheres of isotropic elastic material that are radially inextensible. The solutions for radial inflation–compaction and radial oscillation are obtained also for everted radially inextensible hollow spheres of isotropic elastic material. The static and dynamic results for everted and uneverted radially inextensible hollow spheres are then compared. Harmonic and compressible Varga materials are used to demonstrate the solutions.      

On the stability of elastic annular rods

The stability of equilibrium of non-linearly elastic rods, whose deformations obey the classical Kirchhoff’s equations, is considered. A variational formulation of the equilibrium problem is given, and the equilibrium equations for infinitesimal deformations superimposed to a finite transformation of a rod are deduced. The stability of annular rings, in which the twisting strain is non-null, is investigated by study of the second variation of the energy functional.