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This first chapter of the volume on Constitutive Modelling of Solid Continua sets out briefly the main concepts of general continuum mechanics without reference to specific material behaviour as a backdrop for the detailed descriptions of different types of material behaviour that are contained in the remaining chapters. The focus is therefore on the fundamental ideas of kinematics of a continuum (deformation and motion), the global balance equations governing the motion of a general continuum, stress and the energy balance equation and the derivation of the local governing equations, including mass conservation and the equation of motion. For the most part, detailed proofs are not provided and reference is made to standard texts for more details since this volume is aimed at researchers who have a reasonable background in continuum mechanics and wish to widen the scope of their knowledge in different areas of solid mechanics.

This chapter provides the framework for the development of constitutive theories of solids by focusing on constitutive laws for nonlinearly elastic solids. These exemplify the general principles of constitutive theory that should be applied to all types of material behaviour, in particular, the notions of objectivity and material symmetry, including the important symmetries of isotropy, transverse isotropy and orthotropy based in part on deformation invariants. Details are given for the various general stress–deformation relations for each case of symmetry in respect of hyperelastic materials (which are characterized by a strain-energy function), with or without an internal constraint such as incompressibility, and these are illustrated by particular prototype models. The notion of residual stress (in an unloaded configuration) is discussed and the form of strain-energy function required to accommodate residual stress in the material response is developed.

In this paper the theory of nonlinear electroelasticity is used to examine radial deformations of a thick-walled spherical shell of soft dielectric material with compliant electrodes on its inner and outer surfaces combined with an internal pressure. A general expression for the dependence of the pressure on the deformation and a potential difference between the electrodes, or uniform surface charge distributions on the electrodes, is obtained in respect of a general incompressible isotropic electroelastic energy function. To illustrate the behavior of the shell specific forms of energy functions are used for numerical purposes, for either fixed potential difference or fixed charge distribution. Explicit general results are given for the limiting case of a thin-walled (or membrane) shell. Depending on the elastic part of the energy function (as distinct from that part which involves the electric field) different behaviors are obtained, including non-monotonicity of the pressure–radius relationship reminiscent of that known in the purely elastic situation, but moderated by the presence of an electric field.

In the last few years, it has been recognized that the large deformation capacity of elastomeric materials that are sensitive to electric fields can be harnessed for use in transducer devices such as actuators and sensors. This has led to the reassessment of the mathematical theory that is needed for the description of the electromechanical (in particular, electroelastic) interactions for purposes of material characterization and prediction. After a review of the key experiments concerned with determining the nature of the electromechanical interactions and a discussion of the range of applications to devices, we provide a short account of the history of developments in the nonlinear theory. This is followed by a succinct modern treatment of electroelastic theory, including the governing equations and constitutive laws needed for both material characterization and the analysis of general electroelastic coupling problems. For illustration, the theory is then applied to two simple representative boundary-value problems that are relevant to the geometries of activation devices; in particular, (a) a rectangular plate and (b) a circular cylindrical tube, in each case with compliant electrodes on the major surfaces and a potential difference between them. In (a), an electric field is generated normal to the major surfaces and in (b), a radial electric field is present. This is followed by a short section in which other problems addressed on the basis of the general theory are described briefly.

In this paper the theory of nonlinear electroelasticity is used to examine diffuse modes of instability for two problems involving a thin dielectric plate subject to large deformations. The analysis is based on the equations governing linearized incremental deformations and electric displacements and accompanying boundary conditions superimposed on a known finitely deformed configuration in the presence of a known electric field (the underlying configuration). For each problem the underlying deformation is taken to correspond to an equibiaxial stretch with the electric field normal to the major faces of the plate. In the first problem the electric field is generated by equal and opposite charges applied to flexible electrodes on the major surfaces, while in the second problem there are no electrodes and the field is applied externally. Numerical results are obtained in respect of a simple model of an electroelastic material by way of illustration. The critical stretch corresponding to loss of stability of the uniform underlying configuration is obtained as a function of a dimensionless measure of the initial thickness of the plate for a series of values of the electric (displacement) field and of the parameters included in the material model. For the first problem, the results obtained for the critical stretch are compared with the results based on the so-called Hessian approach, which provides only a limited analysis of stability. It neither takes account of the plate thickness nor allows for non-homogeneous deformations, and significant differences in the predictions of the two methods are identified.

Two-dimensional wave propagation is studied in an isothermal linear isotropic elastic material with voids rotating with constant angular velocity based on a theory of elastic material with voids developed by Iesan (1986) in the thermoelastic context. It is found that there exist three coupled plane waves propagating with distinct phase speeds. The presence of voids and the rotation of the medium are responsible for this coupling. In the absence of voids, the classical longitudinal and transverse waves are found to be coupled through the rotation of the medium. At very large frequency or when the angular rotation is very small relative to the wave frequency the waves are decoupled and propagate with distinct phase speeds. These are (i) a longitudinal wave, (ii) a transverse wave and (iii) a longitudinal wave corresponding to the change in void volume fraction. The first two correspond to the waves of classical elasticity, while the third is new and arises from the presence of the voids. The results are illustrated graphically.

In this paper, we review critically some basic models derived from the statistics of long-chain molecules and then discuss the status of such models within the three-dimensional nonlinear theory of elasticity. We draw attention to some deficiencies of certain worm-like chain (WLC) models when viewed within the three-dimensional continuum mechanics framework. Modifications of the WLC models motivated by such considerations and avoiding these deficiencies are then discussed and shown to correspond well with data generated by the exact WLC model.

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.

This chapter provides a detailed summary of the background from the nonlinear theory of continuum mechanics that is required in the modeling of the elastic properties of soft biological tissues. In particular, it highlights methods for including the fibrous structure of such tissues within the constitutive description of the material properties at the macroscopic level. Of particular relevance in this connection are the so-called preferred directions associated with fibers and the structure tensors and associated deformation invariants that are needed for taking these fibers and their dispersed directions into consideration. These are incorporated into the material models and the effect of fiber structure on the material response is then illustrated with several basic examples. Generalizations of structure tensors are also used for including within the theory the important residual stresses that are evident in unloaded tissues such as arteries and the myocardium, and the influence of residual stresses on the material response is illustrated by considering the extension and inflation of a thick-walled circular cylindrical tube.

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney–Rivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein–Gulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.

Elastic filaments play an important role in the behaviour of cells and biological tissues. In this paper a two-dimensional nonlinear elastic framework, incorporating both bending and stretching, for the behaviour of biopolymer filaments treated as one-dimensional continua is developed. Explicit formulas for the extension-force relationship are obtained which include dependence on the initial end-to-end distance of the filament, unlike some existing models in the literature of, for example, the worm-like chain. The approach adopted allows treatment of both flexible and semi-flexible filaments and has the flexibility to accommodate different degrees of approximation. A key ingredient in the application of the model is inclusion of a body force term in the equilibrium equation. This is essential for finding non-trivial solutions of the governing equations and boundary conditions for filaments under tension. This highlights certain inconsistencies in the mechanics evident in the biophysics literature. Since the behaviour of individual filaments has a strong influence on the behaviour of networks of filaments the theory developed here can serve as a basis for analyzing the elasticity of networks such as actin and other filamentous biopolymer networks.

In this paper, the coupling between a finite deformation and an electric field is examined with particular reference to the propagation of small amplitude waves in a non-linear electroelastic material based on the quasi-electrostatic approximation. The general equations governing the linearized response of electroelastic solids superimposed on a state of finite deformation in the presence of an electric field are derived along with incremental forms of the electroelastic constitutive laws and boundary conditions. Both unconstrained and incompressible materials are considered. Without restriction on the electroelastic constitutive law, the theory is first applied to the analysis of plane waves propagating in a homogeneously deformed material with an underlying uniform electric field and illustrated in the case of an isotropic material. The general equations governing 2D incremental motions are then derived and applied to the study of surface waves in a homogeneously deformed half-space of incompressible isotropic material with the electric field normal to the surface of the half-space. The dependence of the wave speed on the deformation, the electric field and the electromechanical coupling parameters is illustrated for a prototype electroelastic constitutive law.

In this paper we first provide an overview of the recently formulated nonlinear constitutive framework for the quasi-static response of electroelastic solids and its isotropic specialization. The general theory exhibits a strong nonlinear coupling between electric and mechanical effects. The main part of the paper focuses on the governing equations describing the linearized response of electroelastic solids superimposed on a state of finite deformation in the presence of an electric field for independent incremental changes in the electric displacement and the deformation within the material. The associated incremental changes in the stress and the electric field within the material and the surrounding space and the incremental boundary conditions are derived for mechanically unconstrained and constrained electroelastic solids and in the isotropic specialization. By way of illustration of the incremental theory, we specialize the constitutive law to an electroelastic neo-Hookean material, and consider the stability of a half-space subjected to pure homogeneous deformation in the presence of an applied electric field normal to its surface. We show that stability is crucially dependent on the magnitudes of the electromechanical coupling parameters in the constitutive equation.

We review in detail two concepts that appear in the theory that governs the interaction between electric fields and deformable media, namely the ‘Maxwell stress’ and the ‘total stress’. We focus on the quasi-static context, based on the force, couple and energy balance equations, with particular reference to boundary conditions. Equivalent formulations of the governing equations expressed in terms of different measures of stress are highlighted and connections with a principle of virtual work and a variational principle are noted.

In the context of nonlinear anisotropic elasticity the model plane-strain problem of azimuthal shear of a circular cylindrical tube is considered. In particular, the effect of anisotropy associated with preferred material directions on the emergence and disappearance of non-uniqueness of solution is examined. The non-smooth character of the global energy-minimizing solution of the boundary-value problem in which the inner boundary is fixed and the outer boundary is subject to a prescribed shear traction is highlighted and illustrated graphically.

The equations describing the interaction of an electromagnetic sensitive elastic solid with electric and magnetic fields under finite deformations are summarized, both for time-independent deformations and, in the non-relativistic approximation, time-dependent motions. The equations are given in both Eulerian and Lagrangian form, and the latter are then used to derive the equations governing incremental motions and electromagnetic fields superimposed on a configuration with a known static finite deformation and time-independent electromagnetic field. As a first application the equations are specialized to the quasimagnetostatic approximation and in this context the general equations governing time-harmonic plane-wave disturbances of an initial static configuration are derived. For a prototype model of an incompressible isotropic magnetoelastic solid a specific formula for the acoustic shear wave speed is obtained, which allows results for different relative orientations of the underlying magnetic field and the direction of wave propagation to be compared. The general equations are then used to examine two-dimensional motions, and further expressions for the wave speed are obtained for a general incompressible isotropic magnetoelastic solid.

Different formulations of the constitutive laws and governing equations for nonlinear electroelastic solids are reviewed and two new variational principles for electroelastostatics are introduced. One is based on use of the electrostatic scalar potential and one on the vector potential, combined with the deformation function. In each case Lagrangian forms of the electric variables are used. Their connections with several formulations of nonlinear electroelasticity in the literature are established and some differences highlighted.

The mechanical testing of anisotropic nonlinearly elastic solids is a topic of considerable and increasing interest. The results of such testing are important, in particular, for the characterization of the material properties and the development of constitutive laws that can be used for predictive purposes. However, the literature on this topic in the context of soft tissue biomechanics, in particular, includes some papers that are misleading since they contain errors and false statements. Claims that planar biaxial testing can fully characterize the three-dimensional anisotropic elastic properties of soft tissues are incorrect. There is therefore a need to clarify the extent to which biaxial testing can be used for determining the elastic properties of these materials. In this paper this is explained on the basis of the equations of finite deformation transversely isotropic elasticity, and general planar anisotropic elasticity. It is shown that it is theoretically impossible to fully characterize the properties of anisotropic elastic materials using such tests unless some assumption is made that enables a suitable subclass of models to be preselected. Moreover, it is shown that certain assumptions underlying the analysis of planar biaxial tests are inconsistent with the classical linear theory of orthotropic elasticity. Possible sets of independent tests required for full material characterization are then enumerated.

In the context of nonlinear magnetoelasticity theory very few boundary-value problems have been solved. The main problem that arises when a magnetic field is present, as compared with the purely elastic situation, is the difficulty of meeting the magnetic boundary conditions for bodies with finite geometry. In general, the extent of the edge effects is unknown a priori, and this makes it difficult to interpret experimental results in relation to the theory. However, it is important to make the connection between theory and experiment in order to develop forms of the magnetoelastic constitutive law that are capable of correlating with the data and can be used for making quantitative predictions. In this paper the basic problem of a circular cylindrical tube of finite length that is deformed by a combination of axial compression (or extension) and radial expansion (or contraction) and then subjected to an axial magnetic field is examined. Such a field cannot be uniform throughout, since the boundary conditions on the ends and the lateral surfaces of the tube would be incompatible in such circumstances. The resulting axisymmetric boundary-value problem is formulated and then solved numerically for the case (for simplicity of illustration) in which the deformation is not altered by the application of the magnetic field. The distribution of the magnetic-field components throughout the body and the surrounding space is determined in order to quantify the extent of the edge effects for both extension and compression of the tube.

It is known from Shield’s theorem that the equilibrium equations for the inverse deformation in finite elasticity are equivalent to those for the primary deformation. We review this result for compressible elastic solids and then use it to provide a simple proof that strong ellipticity in the material setting is equivalent to that in the spatial setting. The result is then extended to incompressible materials.

In this paper we first summarize the basic constitutive equations for (nonlinear) magnetoelastic solids capable of large deformations. Equivalent formulations are given using either the magnetic induction vector or the magnetic field vector as the independent magnetic variable in addition to the deformation gradient. The constitutive equations are then specialized to incompressible, isotropic magnetoelastic materials in order to determine universal relations. A universal relation, in this context, is an equation that relates the components of the stress tensor and the components of the magnetic field and/or the components of the magnetic induction that holds independently of the specific choice of constitutive law for the considered class or subclass of materials. As has been shown previously for the case in which the magnetic induction is the independent magnetic variable, in the general case there exists only one possible universal relation. We show that this is also the case if the magnetic field is taken as the independent variable and that the universal relations resulting from the two cases are equivalent. A number of special cases are found for certain specializations of the constitutive equations. These include some connections between the deformation, the magnetic field and magnetic induction that do not involve the components of the stress tensor. Universal relations are then examined for some representative homogeneous and inhomogeneous universal solutions.

In this paper we analyze some aspects of the mechanics of anisotropic non-linearly elastic solids for which the anisotropy is associated with two preferred directions, thereby modelling the effect of two families of fiber reinforcement. Particular attention is focused on the role of the invariant I 8 , which couples the fiber kinematics in the constitutive law, and results are illustrated for a model strain–energy function that includes this invariant. The distinctive effects of this invariant on the stress–strain behaviour of the material and on the failure of ellipticity are discussed separately for pure homogeneous deformation and for a simple shear deformation. For pure homogeneous deformation it is shown that ellipticity fails with an associated weak surface of discontinuity that bisects the fiber directions. Moreover, ellipticity fails close to the reference configuration in either tension or compression. For simple shear it is found that loss of ellipticity occurs both for positive and negative shearing, and the orientation of the weak surface relative to the fiber directions is characterized in each case. Again ellipticity fails close to the reference configuration, and inclusion of the invariant I 8 in the constitutive law therefore has a pronounced destabilizing influence.

Electro-sensitive elastomers are materials that can support large elastic deformations under the influence of an electric field. There has been growing interest recently in their applications as so-called "smart materials". This paper is devoted to the derivation of universal relations in the context of the nonlinear theory of electroelasticity that underpins such applications. Universal relations are equations relating the components of the stress, the electric variables and the deformation that are independent of the constitutive law for a family of materials. For the general constitutive equations of an isotropic electroelastic material derived from a free energy function and for some special cases of these equations, we obtain universal relations, the word "universal" being relative to the considered class or subclass of constitutive laws. These universal relations are then applied to some controllable states (homogeneous and non-homogeneous) in order to highlight some examples that may be useful from the point of view of experimental characterization of the material properties. Additionally, we examine the (non-controllable) problem of helical shear of a circular cylindrical tube in the presence of a radial electric field, and we find that a nonlinear universal relation that has been obtained previously for an elastic material also holds when the electric field is applied.

Electro-sensitive (ES) elastomers form a class of smart materials whose mechanical properties can be changed rapidly by the application of an electric field. These materials have attracted considerable interest recently because of their potential for providing relatively cheap and light replacements for mechanical devices, such as actuators, and also for the development of artificial muscles. In this paper we are concerned with a theoretical framework for the analysis of boundary-value problems that underpin the applications of the associated electromechanical interactions. We confine attention to the static situation and first summarize the governing equations for a solid material capable of large electroelastic deformations. The general constitutive laws for the Cauchy stress tensor and the electric field vectors for an isotropic electroelastic material are developed in a compact form following recent work by the authors. The equations are then applied, in the case of an incompressible material, to the solution of a number of representative boundary-value problems. Specifically, we consider the influence of a radial electric field on the azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation characteristics of the same tube under either a radial or an axial electric field (or both fields combined), and the effect of a radial field on the deformation of an internally pressurized spherical shell.

For incompressible isotropic elastic heat conducting solids we examine the classes of universal deformations that are controllable under homothermal conditions and, in respect of a general law of heat conduction, determine the members of these classes that can support non-homothermal temperature fields. It is shown, in particular, that in some cases the values of the components of the (Cauchy) stress tensor T necessary for the satisfaction of the equilibrium equation ▿·T = 0 in the homothermal problem are retained in the corresponding non-homothermal problem to within an additive constant. These observations have implications for the design of experiments in which a material is subject to a temperature gradient since it is of value to know in advance that a specific deformation can be controlled in the presence of a suitable temperature gradient. The temperature may then be obtained as the solution of the steady state heat equation.

In this paper we examine the influence of magnetic fields on the static response of magnetoelastic materials, such as magneto-sensitive elastomers, that are capable of large deformations. The analysis is based on a simple formulation of the mechanical equilibrium equations and constitutive law for such materials developed recently by the authors, coupled with the governing magnetic field equations. The equations are applied in the solution of some simple representative and illustrative problems, with the focus on incompressible materials. First, we consider the pure homogeneous deformation of a slab of material in the presence of a magnetic field normal to its faces. This is followed by a review of the problem of simple shear of the slab in the presence of the same magnetic field. Next we examine a problem involving non-homogeneous deformations, namely the extension and inflation of a circular cylindrical tube. In this problem the magnetic field is taken to be either axial (a uniform field) or circumferential. For each problem we give a general formulation for the case of an isotropic magnetoelastic constitutive law, and then, for illustration, specific results are derived for a prototype constitutive law. We emphasize that in general there are significant differences in the results for formulations in which the magnetic field or the magnetic induction is taken as the independent magnetic variable. This is demonstrated for one particular problem, in which restrictions are placed on the admissible class of constitutive laws if the magnetic induction is the independent variable but no restrictions if the magnetic field is the independent variable.

In this paper we use a recently developed concise general theory of nonlinear magnetoelasticity to analyze the mechanical response of (a) a circular cylindrical tube under steady rotation about its axis in an azimuthal magnetic field, and (b) a solid circular cylinder also under steady rotation about its axis in an axial magnetic field. It is found that for problem (a) the magnetic field can either enhance or counteract the effect of rotation, while for problem (b) the magnetic field reinforces the effect of rotation.

Loss of ellipticity and associated failure in fiber-reinforced non-linearly elastic solids is examined for uniaxial plane deformations. We consider separately fiber reinforcement that either endows the material with additional stiffness only in the fiber direction or introduces additional stiffness under shear deformations. In the first case it is shown that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and requires concavity of the Cauchy stress–stretch curve. For the second example loss of ellipticity occurs after the nominal stress maximum and prior to a turning point of the Cauchy stress.

Recently, a formula for the Rayleigh wave speed in an isotropic elastic half-space has been given by Malischewsky and a detailed derivation given by the present authors. This study deals with the generalization of this formula to orthotropic elastic materials and Malischewsky’s formula is recovered as a special case. The formula is obtained using the theory of cubic equations and is expressed as a continuous function of three dimensionless material parameters.

In the light of recent and growing interest in the applications of magneto-sensitive elastomers and the corresponding theoretical analysis of their properties, this paper is devoted to the derivation of universal relations for these materials, that is connections between the components of a stress tensor and the components of the magnetic (induction) field vector that hold independently of the choice of constitutive law within a considered class of such laws. Here, attention is focussed on isotropic magnetoelastic materials. In particular, within this framework, it is shown that in general there is only one possible universal relation for these materials, but for particular classes of constitutive laws or for special deformations there can be more than one. The theory is exemplified by application to the problem of homogeneous triaxial deformation combined with a simple shear.

In this paper we first summarize, in a simple form, the equilibrium equations for a solid material capable of large magnetoelastic deformations. Such equations are needed for the analysis of boundary-value problems for elastomers endowed with magnetic properties by the embedding of distributions of ferrous particles. The general constitutive law for an isotropic material in the presence of a magnetic field is described and expressed in a compact form, with either the magnetic field or the magnetic induction as the independent magnetic variable. The equations are applied, in the case of an incompressible material, to the solution of representative problems involving circular cylindrical geometry, specifically the helical shear of a circular cylindrical tube, its specializations to axial and azimuthal shear, and the problem of extension and torsion of a solid circular cylinder. For each problem a general formulation is afforded without specialization of the (isotropic) constitutive law, and then specific results are discussed briefly for special choices of such laws. It is noted, in particular, that certain restrictions may be placed on the class of constitutive laws for a considered combination of deformation and magnetic field to be admitted.

Magneto-sensitive (MS) elastomers are “smart materials” whose mechanical properties may be changed rapidly by the application of a magnetic field. Such materials typically consist of micron-sized ferrous particles dispersed within an elastomeric matrix. The equations governing deformations of these materials were discussed in a recent paper by the present authors and applied in a particular specialization of the constitutive model to the problem of axial shear of a circular cylindrical tube subject to a radial magnetic field. In the present paper we develop the governing equations for a more general form of constitutive model and provide alternative forms of the equations, including a Lagrangian formulation. To illustrate the theory the problem of azimuthal shear of a circular cylindrical tube is formulated and then solved for a specific constitutive law with a magnetic field that is initially radial. The results, which show the stiffening of the azimuthal shear stress/strain response with increasing magnetic field strength, are illustrated graphically.

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.

A phenomenological model based on the three-dimensional theory of nonlinear elasticity is developed to describe the phenomenon of overstretching in the force-extension curve for dsDNA. By using the concept of a material with multiple reference configurations a single formula is obtained to fit the force-extension curve., Comment: 13 pages, 4 figures

International audience; In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.

This paper illustrates the influence of pure homogeneous strain on the reflection and transmission of plane waves at the boundary between two half-spaces of incompressible isotropic elastic material. In general, the half-spaces consist of different material and are subjected to different deformations but with their principal axes aligned. For a certain class of constitutive laws it is shown that a homogeneous plane (SV) wave incident on the boundary from one half-space gives rise to a reflected wave (with angle of reflection equal to the angle of incidence) together with an interfacial wave in the same half-space, while in the other half-space two possibilities arise depending on the angle of incidence, the material properties and the magnitudes of the deformations in the two half-spaces. Either (a) there is a transmitted (homogeneous plane) wave accompanied by an interfacial wave, or (b) there are two interfacial waves with equal speeds of propagation but different rates of (spatial) decay away from the boundary. For a second class of constitutive laws similar behaviour is found for certain combinations of angle of incidence, material properties and deformations, but additional possibilities also arise. In particular, there may be two reflected waves instead of one reflected wave and an interfacial wave, coupled with either possibility (a) or (b) in the second half-space. Equally, there may be two transmitted waves for each of the possible combinations of reflected and interfacial waves in the first half-space. For each class of materials conditions under which the incident wave is not refracted are identified and the dependence of the amplitudes of the reflected, transmitted and interfacial waves on the angle of incidence, the states of deformation and the materials is illustrated graphically.

In a recent paper, Ogden and Roxburgh (Proc. R. Soc. London A 455 (1999a) 2861) developed a theory of pseudo-elasticity for the description of damage-induced stress softening in rubberlike solids (the Mullins effect), and the theory was modified to incorporate residual strains by Ogden and Roxburgh (in: A. Dorfmann, A. Muhr (Eds.), Proceedings of the First European Conference on Constitute Models for Rubber, Vienna, 1999, Balkema, Rotterdam, pp. 23–28.). In the present paper this theory is applied to a problem involving non-homogeneous deformation, namely the (plane strain) azimuthal shear of a thick-walled circular cylindrical tube of incompressible material. Loading, effected by application of a specified rotation of the outer surface of the tube relative to the inner one, is described by an isotropic elastic strain-energy function. Unloading, associated with reduction in the applied shearing stress on the outer boundary, is described by a different isotropic elastic strain-energy function, which is inhomogeneous and dependent locally on the extent of the initial loading. It is shown that if the maximum applied shear stress on the outer boundary is below a certain critical value, then there is no residual strain after the shearing stress is removed, while if it is greater than a second critical value then there is residual strain throughout the tube. In the intermediate situation there is residual strain only within a certain radius. Outside this radius the final state of deformation corresponds to a rigid rotation. The results are specialized in respect of a particular material model, which allows the residual strain to be calculated explicitly. Also, the total residual (non-recoverable) energy due to the loading–unloading cycle is calculated. Numerical calculations are used to illustrate the stress softening effect by comparison of the shear stress on unloading with that on loading.

This chapter begins with a summary of the equations of magnetostatics following which there is a discussion of the magnetic forces and couples acting on the material. The main aim of the chapter is to combine the nonlinear theory of elasticity with the magnetostatic theory of magnetizable materials in order to provide a general theory of nonlinear magnetoelastic interactions. Towards this end the modifications of the mechanical balance equations required to incorporate magnetic influences are derived. The balance of energy is then used to catalogue a variety of forms of the general constitutive law for a magnetoelastic material, first in Eulerian form. Lagrangian versions of the magnetic field and magnetic induction field vectors are then introduced, leading to particularly simple forms of the constitutive laws. Material symmetry is then discussed in general form, and explicit representations for the constitutive equations of a nonlinear isotropic magnetoelastic material are obtained, followed by a brief discussion of their linear specializations.

In this chapter we summarize the main ingredients of continuum mechanics and their application to nonlinear elasticity with a view to the subsequent development of the nonlinear theories of electroelastic and magnetoelastic interactions. The setting is purely mechanical without reference to any electromagnetic effects. In particular, we review the necessary kinematics of deformation and motion; some required integral theorems involving tensors; the balance equations of mass, linear and angular momentum; and the associated stress tensors. Balance of energy is then used to introduce the strain-energy function of a hyperelastic material, following which we discuss the notions of objectivity and material symmetry as applied to the constitutive equations, and we provide some simple examples of constitutive equations for isotropic and transversely isotropic materials. We then give a general formulation of boundary-value problems in nonlinear elasticity, which is applied to some representative problems involving non-homogeneous deformations that will also feature in later chapters dealing with electroelastic and magnetoelastic boundary-value problems.

In this chapter we bring together the theory of electrostatics and the nonlinear theory of elasticity in order to build a theory of nonlinear electroelastic interactions. For this purpose it is necessary to distinguish between different measures of stress, and this is reflected in different choices of potential energy in considering the equation of energy balance. We then introduce Lagrangian counterparts of the usual Eulerian electric and electric displacement field vectors, and this facilitates the construction of electroelastic constitutive equations that have compact general forms. Material symmetry considerations lead to the definition of an isotropic electroelastic material and explicit representations for the total Cauchy stress and the electric displacement field (if the electric field is the independent electric variable), or the electric field (if the electric displacement is the independent electric variable), in terms of invariants of the independent kinematic and electric variables. In the final section of the chapter we consider the linear specialization of the general nonlinear theory and discuss briefly the linear theory of piezoelectricity.

In this paper the axial shear deformation of a thick-walled right circular cylindrical tube of compressible isotropic elastic material is considered. Under the assumption that the deformation is isochoric the resulting two governing ordinary differential equations of equilibrium are combined to yield a necessary condition on the strain-energy function for the material to admit pure axial shear (or anti-plane shear). For any strain-energy function satisfying this condition further conditions are required to guarantee the existence of solutions. Such conditions are discussed and used to obtain explicit solutions for several forms of strain-energy function. Existing results are recovered as a special case and new solutions are obtained for some specific strain-energy functions. It is then shown how the results may be used to generate solutions for the axial shear problem in incompressible materials.

In a recent paper the authors have incorporated the notion of discontinuous internal variables into the theory of finite elasticity in order to model certain coexistence of phases phenomena encountered in the deformation of solids. For example, phase changes in polymeric solids may be associated with changes in material properties induced by damage, in which case the internal variables are interpreted as damage variables. In the present paper the inflation of a thick-walled spherical shell of incompressible isotropic elastic material with a single internal variable is examined on the basis of this theory. The development of two-phase solutions is demonstrated for the loading process, and, for unloading, an example in which a residual deformation remains after removal of the pressure is given and the corresponding residual stress is calculated.

A theory for three–dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described. The Gurtin–Murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance. This modification yields a model that can be used to describe equilibrium deformations in the presence of compressive–surface stress fields. An associated variational theory is given and material symmetry considerations are discussed. The theory is illustrated by examples.

The effect of pre-stress on the propagation and reflection of plane waves in an incompressible isotropic elastic half-space has been examined recently by the authors (Ogden & Sotiropoulos, 1997). In the present paper the corresponding analysis for compressible materials is detailed. In the two-dimensional context considered for incompressible materials the (homogeneous) plane waves were necessarily shear waves. By contrast, in the compressible context pure shear waves can propagate only in specific directions in the considered principal plane and, in a general direction, a quasi-shear wave may be accompanied by a quasi-longitudinal wave, as is the case in the anisotropic linear theory. The dependence of the (in-plane) slowness section on the pre-stress (and finite deformation) and on the choice of constitutive law is elucidated. This information is used to determine the reflection coefficients for reflection of either a (quasi-) shear wave or a (quasi-) longitudinal wave from the boundary of the half-space and to characterize the different cases which arise depending on the geometry of the slowness section. The theoretical results are illustrated by numerical calculations for the range of possible types of behaviour with reference to different forms of strain-energy function and different states of finite deformation and to the question of stability of the half-space.

In this paper the plane strain character of the finite azimuthal shear of a circular cylindrical annulus of compressible isotropic elastic material is used to express the strain energy as a function of two independent deformation invariants. The resulting formulation of the equilibrium equations is applied to the special case of pure azimuthal shear (an isochoric deformation), and explicit necessary and sufficient conditions on the strain-energy function for the material to admit this deformation are obtained. These conditions are examined for two classes of strain-energy functions and in each case complete solutions of the equilibrium equations are obtained. Some known results are recovered as special cases and some new results for particular strain-energy functions are determined and discussed. Numerical results showing the dependence of the applied shear stress on the resulting relative rotation of the boundaries are given for different strain-energy functions for comparison. The theory is then used to show how corresponding solutions can be deduced for incompressible materials. It is pointed out that the method of considering isochoric deformations in compressible elastic materials provides a means of generating classes of strain-energy functions for which closed-form solutions can be found for incompressible materials.

In this paper, a nonlinear theory of elastic boundary coating (or reinforcement) of an elastic solid is developed for plane strain deformations. The coating consists of a material curve endowed with intrinsic elastic properties associated with extensibility and bending stiffness bonded to part, or all, of the bounding curve of the elastic body. The equations describing the equilibrium of the coated body when subject to finite deformation are derived using a variational method. The incremental equations describing a small departure from an equilibrium configuration are then derived and used to investigate the stability of a deformed configuration and the possibility of bifurcation. The theory is applied to the analysis of the equilibrium of a finitely deformed half-plane consisting of compressible elastic material coated along its edge. The influence of the coating on the bifurcation behaviour of the half–plane is assessed against known results for an uncoated half–plane. Numerical results are used to illustrate the influence of certain material parameters on bifurcation.

On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with the initial stress and in general, for a compressible material, it requires 10 invariants, reducing to 9 for an incompressible material. Expressions for the Cauchy and nominal stress tensors in a finitely deformed configuration are given along with the elasticity tensor and its specialization to the initially stressed undeformed configuration. The equations governing infinitesimal motions superimposed on a finite deformation are then used to study the combined effects of initial stress and finite deformation on the propagation of homogeneous plane waves in a homogeneously deformed and initially stressed solid of infinite extent. This general framework allows for various different specializations, which make contact with earlier works. In particular, connections with results derived within Biot's classical theory are highlighted. The general results are also specialized to the case of a small initial stress and a small pre-deformation, i.e. to the evaluation of the acoustoelastic effect. Here the formulas derived for the wave speeds cover the case of a second-order elastic solid without initial stress and subject to a uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are consistent with results for an undeformed solid subject to a residual stress [Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for acoustic evaluation of the second- and third-order elasticity constants and of the residual stresses. The results are further illustrated in respect of a prototype model of nonlinear elasticity with initial stress, allowing for both finite deformation and nonlinear dependence on the initial stress., 23 pages

In this paper, we analyze the effect of a combined pure homogeneous strain and simple shear in a principal plane of the latter on the propagation of surface waves for an incompressible isotropic elastic half-space whose boundary is normal to the glide planes of the shear. This generalizes previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of materials, the secular equation is obtained in explicit form and then specialized to recover results obtained previously for the two cases mentioned above. A method for obtaining the secular equation for a general form of strain–energy function is then outlined. In general, this is very lengthy and the result is not listed, but, for the case in which there is no normal stress on the half-space boundary, the result is given, for illustration, in respect of the so-called generalized Varga material. Numerical results are given to show how the surface wave speed depends on both the underlying pure homogeneous strain and the superimposed simple shear. Further numerical results are provided for the Gent model of limiting chain extensibility.

International audience; This paper is concerned with numerical simulations of three-dimensional finite deformation of a thick-walled circular elastic tube subject to internal or external pressure and zero displacement on its ends. We formulate the system of equations that can accommodate large strain and displacement for the incompressible isotropic neo-Hookean material. The fully non-linear governing equations are solved using the Cþþ based object-oriented finite element library libMesh. A Lagrangian mesh is used to discretize the governing equations, and a weighted residual Galerkin method and Newton iteration solver are used in the numerical scheme. To overcome the sensitivity of the fully non-linear system to small changes in the iterations, the analytical form of the Jacobian matrix is derived, which ensures a fast and better numerical convergence than using a numerically approximated Jacobian matrix. Results are presented for different parameters in terms of wall thickness/radius ratio, and length/radius ratio, as well as internal/external pressure. Validation of the model is achieved by the excellent agreement with the results obtained using the commercial package Abaqus. Comparison is also made with the previous work on axisymmetric version of the same system (Zhu et al., 2008 [34]; Zhu et al. 2010 [43]), and interesting fully three-dimensional post-buckling deformations are highlighted. The success of the current approach paves the way for fluid–structure interaction studies with potential application to collapsible tube flows and modeling of complex physiological systems.