The relation between the Valanis-Landel and classical strain-energy functions

Necessary and sufficient conditions are derived for the strain-energy function of an isotropic elastic solid, regarded as a function of the strain invariants, to be expressible in the Valanis-Landel form, both when the material is compressible and when it is incompressible. In the case when the Valanis-Landel strain-energy function is a polynomial in squares of the principal extension ratios, explicit representations as polynomials in the basic isotropic strain invariants are obtained.     

On SH waves in a pre-stressed layered half-space for an incompressible elastic material

The propagation of Love waves along the boundary between a half-space and a layer of different pre-stressed material is examined for incompressible isotropic elastic materials. The secular equation is obtained for a general strain-energy function and analysed for particular deformations and materials. For the neo-Hookean strain-energy function, numerical results are obtained to illustrate the dependence of the wavespeed on the wave number and on the deformation.     

Bifurcation conditions for a thick elastic plate under thrust

A thick rectangular plate of incompressible isotropic elastic material is subjected to a pure homogeneous deformation by tensile forces or thrusts applied to a pair of opposite faces. The theory of small deformations superposed on finite deformations is applied to determine the critical conditions under which bifurcation solutions (i.e. adjacent equilibrium positions) can exist. The adjacent equilibrium positions considered are those for which the superposed deformation is two-dimensional and is coplanar with the loading force and the thickness direction of the plate, the faces of the plate normal to its thickness being force-free. A number of theorems relating to the critical conditions for superposed deformations of the flexural and barreling types are derived under conditions on the strain-energy function more general than those employed in earlier work. It is also shown how these results can be applied to the determination of the bifurcation conditions corresponding to any specified strain-energy function.     

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 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.     

Limiting Chain Extensibility Constitutive Models of Valanis–Landel Type

A new general constitutive model in terms of the principal stretches is proposed to reflect limiting chain extensibility resulting in severe strain-stiffening for incompressible, isotropic, homogeneous elastic materials. The strain-energy density involves the logarithm function and has the general Valanis–Landel form. For specific functions in the Valanis–Landel representation, we obtain particular strain-energies, some of which have been proposed in the recent literature. The stress–stretch response in some basic homogeneous deformations is described for these particular strain-energy densities. It is shown that the stress response in these deformations is similar to that predicted by the Gent model involving the first invariant of the Cauchy–Green tensor. The models discussed here depend on both the first and second invariants.      

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers

A popular model for the finite element simulation of slightly compressible solid rubber-like materials assumes that the strain-energy function can be additively decomposed into a volumetric part and a deviatoric part. Based on mathematical convenience, the volumetric part is usually assumed to be a finite polynomial in the volume change. Experimental evidence suggests that for solid rubbers in compression, this polynomial can be taken to be a simple quadratic for moderate deformations and that this function also adequately models the volume change and the stress/stretch relation for materials in simple tension, up to stretches of order 100%. For larger tensile deformations, however, experimental data suggest that the Cauchy stress-volume change relation has an increasingly large slope and therefore a truncated Taylor series expansion is not the most appropriate. A rational function approach is proposed here as an alternative.     

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.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D², which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16]. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the failure of ellipticity of the equations for finite elastostatic plane strain

Summary In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.     

On the restrictions imposed by non-affine material symmetry groups for elastic rods: application to helical substructures

Wire ropes, DNA strands and helical springs are among those bodies which can be modeled as an elastic rod with a helical substructure. The resulting form of the strain-energy function is a matter of material symmetry. This symmetry is explored using a novel treatment which combines non-affine transformations and a relabeling of the material coordinates. The restrictions this treatment imposes on the strain-energy function include a periodic dependency on torsional strain. In addition, comparisons are made with results from a recent treatment of helical symmetry by Healey. Finally, conclusions applicable to material symmetry restrictions for other polar elastic continua are presented.     

Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power-law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.     

Bifurcation of rotating thick-walled elastic tubes

The deformation of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthai deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (Haughton and Ogden, 1980) for membrane tubes.Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function.Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.     

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.     

Nonlinear stability analysis of pre-stressed elastic bodies

This article is concerned with the nonlinear analysis of the stability of thick elastic bodies subjected to finite elastic deformations. The analysis is based on the theory of small elastic deformations superimposed on a finite elastic deformation. Attention is drawn to methods developed in the stability analysis of fluids and of thin shells and plates which are readily applicable to the present circumstances. The state of development of the nonlinear stability analysis of thick elastic bodies is summarized in order to provide a basis for subsequent studies, and some new results relating to the stability of an elastic plate subjected to a pre-stress associated either with uniaxial thrust or with simple shear in the presence of all-round pressure are discussed. Near-critical modes in the neighbourhood of so-called critical configurations are considered to depend on, for example, a slow time variable, and nonlinear evolution equations for the mode amplitudes are derived both in the case of a monochromatic mode and for a resonant triad of modes. The crucial role of the ‘nonlinear coefficient’ in such an equation in the analysis of stability, imperfection sensitivity and localization is highlighted. An efficient (virtual work) method for the determination of this coefficient is described together with an alternative method based on the calculation of the total energy of a monochromatic near-critical mode. The influence of the boundary conditions and of the form of the pre-stress is examined and explicit calculation of the nonlinear coefficient is provided for the two representative pre-stress conditions mentioned in the above paragraph. It is shown, in particular, that a resonant triad of modes has an effect similar to that generated by the presence of a geometrical imperfection. The Appendices gather together for reference certain expressions which are used in the body of the article. These include expressions, not given previously in the literature, for the components of the tensor of third-order instantaneous elastic moduli in terms of the principal stretches of the deformation in respect of a general form of incompressible isotropic elastic strain-energy function. Received November 9, 1998     

Some New Implications of Certain Constitutive Inequalities in Plane Isotropic Elasticity

In plane isotropic elasticity a strengthened form of the Ordered–Forces inequality is shown to imply that the restriction of the strain-energy function to the class of deformation gradients which share the same average of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformations in this class. For boundary conditions of place, this property (together with a certain version of the Pressure–Compression inequality) is then used (i) to show that the plane radial conformal deformations are stable with respect to all radial variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with arbitrary plane radial deformations. For the same type of boundary conditions and together with a different version of the Pressure–Compression inequality, an analogous property in plane isotropic elasticity (established in [3] under the assumption that the material satisfies a strengthened form of the Baker–Ericksen inequality and according to which the restriction of the strain-energy function to the class of deformation gradients which share the same determinant is bounded from below by the strain energy corresponding to the conformal deformations in that class) is used (i) to show that the plane radial conformal deformations are stable with respect to all variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

Post-bifurcation of perfect and imperfect spherical elastic membranes

When a spherical elastic membrane is inflated it is well known that it may bifurcate into an aspherical mode after the pressure maximum is reached. Upon further inflation the spherical configuration is regained. Here we follow the developing aspherical solution path, for specific forms of strain-energy function, using a simple numerical method. For a realistic strain-energy function it is shown that the post-bifurcation solution curve connects the two bifurcation points. We also consider the inflation of imperfect spherical membranes and show that bifurcation may still occur. For the class of Ogden materials we investigate the asymptotic shape of arbitrary axisymmetric membranes.