A study is made of a class of singular solutions to the equations of nonlinear elastostatics in which a spherical cavity forms at the centre of a ball of isotropic material placed in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the stored-energy function W for large strains and is consistent with strong ellipticity of W . Under appropriate hypotheses it is shown that a singular solution bifurcates from a trivial (homogeneous) solution at a critical value of the surface traction or displacement, at which the trivial solution becomes unstable. For incompressible materials both the singular solution and the critical surface traction are given explicitly, and the stability of all solutions with respect to radial motion is determined. For compressible materials the existence of singular solutions is proved for a class of strongly elliptic materials by means of the direct method of the calculus of variations, an important step in the analysis being to show that the only radial equilibrium solutions without cavities are homogeneous. Work of Gent & Lindley (1958) shows that the critical surface tractions obtained agree with those observed in the internal rupture of rubber.

Discontinuous Equilibrium Solutions and Cavitation in Nonlinear Elasticity

The effect of growth in the stability of elastic materials is studied. From a stability perspective, growth and resorption have two main effects. First a change of mass modifies the geometry of the system and possibly the critical lengths involved in stability thresholds. Second, growth may depend on stress but also it may induce residual stresses in the material. These stresses change the effective loads and they may both stabilize or destabilize the material. To discuss the stability of growing elastic materials, the theory of finite elasticity is used as a general framework for the mechanical description of elastic properties and growth is taken into account through the multiplicative decomposition of the deformation gradient. The formalism of incremental deformation is adapted to include growth effects. As an application of the formalism, the stability of a growing neo-Hookean incompressible spherical shell under external pressure is analyzed. Numerical and analytical methods are combined to obtain explicit stability results and to identify the role of mechanical and geometric effects. The importance of residual stress is established by showing that under large anisotropic growth a spherical shell can become spontaneously unstable without any external loading.

http://www.phys.ens.fr/~benamar/paper/2005-JMPS(growth).pdf

Growth and instability in elastic tissues

B ifurcations 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.

https://hal.archives-ouvertes.fr/hal-01302310/document

Bifurcation of inflated circular cylinders of elastic material under axial loading—II. Exact theory for thick-walled tubes

1 Introduction 2 Elements of tensor algebra and analysis 3 Solid mechanics at finite strains 4 Isotropic nonlinear hyperelasticity 5 Solutions of simple problems in finitely deformed nonlinear elastic solids 6 Constitutive equations and anisotropic elasticity 7 Yield functions with emphasis on pressure-sensitivity 8 Elastoplastic constitutive equations 9 Moving discontinuities and boundary value problems 10 Global conditions of uniqueness and stability 11 Local conditions for uniqueness and stability 12 Bifurcation of elastic solids deformed incrementally 13 Applications of local and global uniqueness and stability criteria to non-associative elastoplasticity 14 Wave propagation, stability and bifurcation 15 Post-critical behaviour and multiple shear band formation 16 A perturbative approach to material instability

/pdf/nonlinear-solid-mechanics-bifurcation-theory-and-material-3aoukzbrog.pdf

Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability

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.

/pdf/nonlinear-electroelastic-deformations-4x8l374hg2.pdf

Nonlinear Electroelastic Deformations

Bifurcations of spherical elastic shells subsequent to a finite radially-symmetric inflation are analysed. In the case of a membrane, necessary and sufficient conditions for the initiation of asymmetric deformation modes are obtained in respect of an arbitrary (incompressible, isotropic) form of elastic strain-energy function. The method used differs from that used anywhere before, and the results obtained generalize, extend and elucidate those given previously for this problem. In particular, we find that aspherical bifurcations first become possible when the surface tension reaches a maximum. The corresponding analysis is given for thick shells, also for an arbitrary form of the strain-energy function. Despite the inhomogeneity of the deformation in this case, some analytical results are derived which have certain features in common with the membrane results. However, the bifurcation criteria for thick shells are not equivalent to those for the membrane. This is illustrated for particular forms of the strain-energy function. The results depend very much on the shell thickness. In particular, aspherical bifurcations are not possible for shells whose thickness is beyond some critical value. Nevertheless, membrane theory provides a good prediction of the behaviour of shells of moderate finite thickness.

On the incremental equations in non-linear elasticity — II. Bifurcation of pressurized spherical shells

A new formulation of the equations of membrane theory in non-linear elasticity is described. It is based on the consistent use of certain conjugate variables averaged through the (undeformed) thickness of the thin shell which the membrane approximates. The deformation gradient is taken as the basic measure of deformation, and its average value as the membrane measure of deformation. It is shown that the average elastic strain energy can be regarded as a function of the average deformation gradient to within an error which is of the second order in a certain small parameter. Moreover, to the same order, the average strain energy is a potential function for the average nominal stress. This means that the averages of the conjugate variables (nominal stress and deformation gradient) are also conjugate. In terms of the average conjugate variables, the membrane equilibrium equations are obtained by averaging from the equilibrium equations of the full three-dimensional theory. Discussion of the order of magnitude of the errors involved in the membrane approximation is a feature of the analysis. The corresponding incremental equations are also derived as a prelude to their application in certain bifurcation problems. One such problem is examined in the companion paper (Part II) in which results for thick shells and membranes are compared.

On the incremental equations in non-linear elasticity — I. Membrane theory

Bifurcation of finitely deformed rotating elastic cylinders

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.

Bifurcation of rotating thick-walled elastic tubes

Bifurcation from a finitely deformed circular cylindrical configuration of a rotating circular cylindrical elastic membrane is examined. It is found (for a physically realistic choice of elastic strain-energy function) that the angular speed attains a maximum followed by a minimum relative to the increasing radius of the cylinder for either a fixed axial extension or fixed axial force. At fixed axial extension ( a ) a prismatic mode of bifurcation (in which the cross-section of the cylinder becomes uniformly non-circular) may occur at a maximum of the angular speed provided the end conditions on the cylinder allow this; ( b ) axisyim-metric modes may occur before, at or after the angular speed maximum depending on the length of the cylinder and the magnitude of the axial extension; ( c ) an asymmetric or ‘wobble’ mode is always possible before either ( a ) or ( b ) as the angular speed increases from zero for any length of cylinder or axial extension. Moreover, ‘wobble’ occurs at lower angular speeds for longer cylinders. At fixed axial force the results are similar to ( a ), ( b ) and ( c ) except that an axisym-metric mode necessarily occurs between the turning points of the angular speed.

D.M. Haughton

Papers

On the incremental equations in non-linear elasticity — II. Bifurcation of pressurized spherical shells

On the incremental equations in non-linear elasticity — I. Membrane theory

Bifurcation of finitely deformed rotating elastic cylinders

Bifurcation of rotating thick-walled elastic tubes

Bifurcation of rotating circular cylindrical elastic membranes