Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Incompressible Elastic Bodies with Non-Convex Energy under Dead-Load Surface Tractions

In this paper we study the equilibrium deformations of an incompressible elastic body with a non-convex strain energy function which is subjected to a homogeneous distribution of dead-load tractions. To determine the stable solutions we consider the mixtures of the phases which minimize the total energy density. In the special case of a trilinear material we discuss the stability of the equilibrium phases in detail. Finally, we show that multiphase solutions are possible when the surface loads correspond to a critical simple shear and we sketch their possible forms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stress-Induced Phase Transformations of an Isotropic Incompressible Elastic Body Subject to a Triaxial Stress State

The present paper concerns the stable multiphase isochoric deformations for an isotropic elastic body subject to a surface traction of uniform Piola stress with two equal principal forces which are opposite to the third. To model the occurrence of such deformations, we consider a strain energy density function which depends on the first principal invariant of deformation through a non-convex function and which has an added linear dependence on the second invariant. We establish existence conditions for equilibrium multiphase deformations which give restrictions on the morphology of the connecting phases as well as on the orientation of the flat interfaces between the phases. Finally, by considering a special, but representative, form for the non-convex strain energy function, we show that there exists a “critical” value of the external load which allows for the emergence of stable coexistent deformation fields.     

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.     

General stress-strain relations in non-linear theory of viscoelasticity for large deformations

Conclusion General phenomenoligical stress-strain relations in non-linear theory of visco-elasticity for large deformations have been presented.In the first place, according to V. V. Novozhilov 1 we express the generalized equilibrium equation for large deformations in the Lagrange representation, and we apply the generalized Hamilton's principle to the equation of energy conservation, which denotes that the sum of the elastic energy and the dissipative energy is equal to the work done by the body force and the surface on the substance; so that we obtain the required general stress-strain relations in comparison with the above two equations.On the condition that the elastic potential is a function only of the strain, and the dissipation function is a function of the rate of strain and of strain; such a substance is reduced to the Voigt material necessarily, and the stresses which act on the substance are given by the sum of elastic- and viscous stresses, and the stress-strain relations are reduced to the so-called Lagrangian form.If elongations, shears and angles of rotation are small and also the strains and rates of strain are sufficiently small, the stress-strain relations are expressed by a linear Voigt model constituting a Hookian spring in parallel with a Newtonian dashpot.Non-linearity in the theory is classified into two groups i. e. the geometrical non-linearity and the physical non-linearity. The former is introduced into the theory through the definition of the generalized strain and of the generalized stress and through the equilibrium equation for large deformation, and the latter through the general stress-strain relations.The main result of this paper is that the general stress-strain relations in viscoelasticity are deduced necessarily from the physically appropriate assumptions.     

A Characterisation of the Boundary Displacements Which Induce Cavitation in an Elastic Body

In this paper we present numerical and theoretical results for characterising the onset of cavitation-type material instabilities in solids. To model this phenomenon we use nonlinear elasticity to allow for the large, potentially infinite, stresses and strains involved in such deformations. We give a characterisation of the set of linear displacement boundary conditions for which energy minimising deformations produce a single isolated hole inside an originally perfect elastic body, based on a notion of the derivative of the stored energy functional with respect to hole-producing deformations. We conjecture that, for many stored energy functions, the critical linear boundary conditions which cause an isolated cavity to form correspond to the zero set of this derivative. We use this characterisation to propose a numerical procedure for computing these critical boundary displacements for general stored energy functions and give numerical examples for specific materials. For a degenerate stored energy function (with spherically symmetric boundary deformations) and for an elastic fluid, we show that the vanishing of the volume derivative gives exactly the critical boundary conditions for the onset of this type of cavitation.     

A gradient approach to localization of deformation. I. Hyperelastic materials

By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre- and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.     

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.     

Non-elliptic elastic materials and the modeling of elastic-plastic behavior for finite deformation

For illustrative purposes this paper treats a special problem in the theory of finite deformations of elastic materials whose associated displacement equations of equilibrium do not remain elliptic at all strains. The typical deformation arising in this problem possesses a discontinuous gradient, so that quasi-static motions involving such equilibrium states may be dissipative. For a special class of such “non-elliptic” elastic materials, it is shown that the macroscopic response in the problem treated may be precisely of the form associated with elastic—perfectly plastic behavior. The counterparts of yield, plastic strain and plastic strain rate are determined by the underlying elastic strain energy function.     

Some energetic properties of smooth solutions in rate-type viscoelasticity

A free energy function compatible with the second law of thermodynamics is constructed for the semilinear rate-type viscoelasticity. Under physically acceptable conditions this energy function is a positive and convex function of stress and strain.

It is shown that for a class of one dimensional initial-boundary value problems, the total energy at any time is bounded by the sum of the total energy of the initial data and the energy supplied to the body by nonvanishing body forces. A Maxwell type viscosity approach to non-linear elasticity for an isolated body is also discussed.

The stability in total energy and uniqueness of the smooth solutions of some initial-boundary value problems is discussed for the one dimensional case as well as for the three dimensional case of small deformations.     

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

The theory of a Cosserat point has been used to formulate a new 3-D finite element for the numerical analysis of dynamic problems in nonlinear elasticity. The kinematics of this element are consistent with the standard tri-linear approximation in an eight node brick-element. Specifically, the Cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations. For hyperelastic response, the constitutive equations for the director couples are determined by derivatives of a strain energy function. Restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test. It is shown that the Cosserat balance laws are in one-to-one correspondence with those obtained using a Bubnov–Galerkin formulation. Nevertheless, there is an essential difference between the two approaches in the procedure for obtaining the strain energy function. Specifically, the Cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data. In contrast, the Bubnov–Galerkin approach determines these constitutive coefficients by integrating the 3-D strain energy function using the kinematic approximation. It is shown that the resulting Cosserat equations eliminate unphysical locking, and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions.     

On Compressible Materials Capable of Sustaining Axisymmetric Shear Deformations. Part 4: Helical Shear of Anisotropic Hyperelastic Materials

Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.     

Energy-minimizing deformations of elastic sheets with bending stiffness

Necessary conditions for energy-minimizing deformations are derived for a theory of sheets in which the strain energy function depends on the second derivatives of the deformation as well as its first derivatives. All of these conditions are extensions of well-known necessary conditions in classical calculus of variations. The interpretation of some of these conditions as material stability conditions is explained.     

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.     

A theory of amorphous viscoelastic solids undergoing finite deformations with application to hydrogels

We consider a hydrogel in the framework of a continuum theory for the viscoelastic deformation of amorphous solids developed by Anand and Gurtin [Anand, L., Gurtin, M., 2003. A theory of amorphous solids undergoing large deformations, with application to polymeric glasses. International Journal of Solids and Structures, 40, 1465–1487.] and based on (i) a system of microforces consistent with a microforce balance, (ii) a mechanical version of the second law of thermodynamics and (iii) a constitutive theory that allows the free energy to depend on inelastic strain and the microstress to depend on inelastic strain rate. We adopt a particular (neo-Hookean) form for the free energy and restrict kinematics to one dimension, yielding a classical problem of expansion of a thick-walled cylinder. Considering both Dirichlet and Neumann boundary conditions, we arrive at stress relaxation and creep problems, respectively, which we consider, in turn, locally, at a point, and globally, over the interval. We implement the resulting equations in a finite element code, show analytical and/or numerical solutions to some representative problems, and obtain viscoelastic response, in qualitative agreement with experiment.     

The Attainable Region of strain-invariant space for elastic materials

Within the theory of isothermal isotropic non-linear elasticity, the selection of the appropriate form for the strain energy function W in terms of the strain invariants is still an issue. The purpose of this paper is to introduce ideas and techniques which it is hoped will contribute to the task of finding an appropriate form for the strain energy. Three principal ideas are developed in this paper. Firstly, not all of invariant-space corresponds to real deformations. Constitutive equations only need to match real behaviour over a restricted part of invariant space, called the Attainable Region, bounded by states of deformation corresponding to uniaxial and equi-biaxial extension. Secondly, examples are given of how to exploit the fact that the Attainable Region is restricted. Mapping a deformation onto this region allows visualization of how close the deformation is to the well-understood uniaxial, equi-biaxial and simple shear deformations, and how this varies in space or time. Thirdly, acceptable strain invariants do not have to be obviously symmetric functions of the principal stretches. The ordered principal stretches are themselves invariants, and explicit unique algebraic expressions can be given through which the greatest, middle or least stretch can be calculated in terms of the usual invariants. Thus invariants can be chosen which are apparently non-symmetric functions of the ordered stretches.     

Optimization of the stored energy and coaxiality of strain and stress in finite elasticity

The strain energy density of a hyperelastic anisotropic body which is rotated before being subjected to a given but arbitrary deformation is viewed as a smooth function defined on the group of rotations, parametrized by the deformation gradient. It is shown that the critical points of this function correspond to rotations which, when composed with the prescribed deformation, yield a total strain tensor which is coaxial with the corresponding stress. For any type of material symmetry, there are at least two such rotations. Coaxiality of stress and strain for all deformations is shown to be a sufficient condition for the isotropicity of hyperelastic materials.Research supported by GNFM of CNR (Italy).     

Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

横观各向同性介质中椭圆夹杂受非弹性剪切变形引起的弹性场的确定

本文求解了横观各向同性介质中椭圆夹杂内受非弹性剪切变形引起的弹性场。采用各向异性弹性力学平面问题的复变函数解法,结合保角变换,获得夹杂内应变能和基体内边界的应力分布和相应的应变能的表达式。进一步,根据最小应变能原理,获得表征夹杂平衡边界的两个特征剪切应变,从而得到了弹性场的解析解。通过应力转换关系,验证了应力解满足夹杂边界上法向正应力和剪应力的连续条件,表明了该解的正确性。本文解可用于复合材料断裂强度的分析中。     

A class of similarity solutions for radial motions of compressible hyperelastic spheres and cylinders

A class of similarity solutions is obtained for radial motions of spherical and cylindrical bodies made of a certain type of compressible hyperelastic materials. The equations satisfied by the infinitesimal generators of the symmetry group of the unified governing first order field equations for spheres and cylinders are found. It is shown that these equations admit a special class of solutions which generate a five-parameter group of transformations. The form of the strain energy function corresponding to the resulting symmetry group is evaluated. The similarity variable is determined and ordinary differential equations satisfied by similarity solutions are obtained. Numerical solutions are given for a Ko material which falls into the class of admissible materials.