On Non-Symmetric Deformations of an Incompressible Nonlinearly Elastic Isotropic Sphere

A class of non-symmetric deformations of a neo-Hookean incompressible nonlinearly elastic sphere are investigated. It is found via the semi-inverse method that, to satisfy the governing three-dimensional equations of equilibrium and the incompressibility constraint, only three special cases of the class of deformation fields are possible. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation describing inflation and stretching. The implications of these results for cavitation phenomena are also discussed. In the course of this work, we also present explicitly the spherical polar coordinate form of the equilibrium equations for the nominal stress tensor for a general hyperelastic solid. These are more complicated than their counterparts for Cauchy stresses due to the mixed bases (both reference and deformed) associated with the nominal (or Piola-Kirchhoff) stress tensor, but more useful in considering general deformation fields. This revised version was published online in August 2006 with corrections to the Cover Date.     

Universal Deformations for a Class of Compressible Isotropic Hyperelastic Materials

Differential conditions are derived for a smooth deformation to be universal for a class of isotropic hyperelastic materials that we regard as a compressible variant (a notion we make precise) of Mooney–Rivlin’s class, and that includes the materials studied originally by Tolotti in 1943 and later, independently, by Blatz. The collection of all universal deformations for an incompressible material class is shown to contain, modulo a uniform dilation, all the universal deformations for its compressible variants. As an application of this result, by searching the known families of universal deformations for all incompressible isotropic materials, a nontrivial universal deformation for Tolotti materials is found. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Effects of Compressibility on Inhomogeneous Deformations for a Class of Almost Incompressible Isotropic Nonlinearly Elastic Materials

Experimental data for simple tension suggest that there is a power–law kinematic relationship between the stretches for large classes of slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Here we confine attention to a particular constitutive model for such materials that is of generalized Varga type. The corresponding incompressible model has been shown to be particularly tractable analytically. We examine the response of the slightly compressible material to some nonhomogeneous deformations and compare the results with those for the corresponding incompressible model. Thus the effects of slight compressibility for some basic nonhomogeneous deformations are explicitly assessed. The results are fundamental to the analytical modeling of almost incompressible hyperelastic materials and are of importance in the context of finite element methods where slight compressibility is usually introduced to avoid element locking due to the incompressibility constraint. It is also shown that even for slightly compressible materials, the volume change can be significant in certain situations.      

Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials

The classical problem of simple shear in nonlinear elasticity has played an important role as a basic pilot problem involving a homogeneous deformation that is rich enough to illustrate several key features of the nonlinear theory, most notably the presence of normal stress effects. Here our focus is on certain ambiguities in the formulation of simple shear arising from the determination of the arbitrary hydrostatic pressure term in the normal stresses for the case of an incompressible isotropic hyperelastic material. A new formulation in terms of the principal stretches is given. An alternative approach to the determination of the hydrostatic pressure is proposed here: it will be required that the stress distribution for a perfectly incompressible material be the same as that for a slightly compressible counterpart. The form of slight compressibility adopted here is that usually assumed in the finite element simulation of rubbers. For the particular case of a neo-Hookean material, the different stress distributions are compared and contrasted.     

Onset of Cavitation in Compressible,Isotropic, Hyperelastic Solids

In this work, we derive a closed-form criterion for the onset of cavitation in compressible, isotropic, hyperelastic solids subjected to non-symmetric loading conditions. The criterion is based on the solution of a boundary value problem where a hyperelastic solid, which is infinite in extent and contains a single vacuous inhomogeneity, is subjected to uniform displacement boundary conditions. By making use of the “linear-comparison” variational procedure of Lopez-Pamies and Ponte Castañeda (J. Mech. Phys. Solids 54:807–830, 2006), we solve this problem approximately and generate variational estimates for the critical stretches applied on the boundary at which the cavity suddenly starts growing. The accuracy of the proposed analytical result is assessed by comparisons with exact solutions available from the literature for radially symmetric cavitation, as well as with finite element simulations. In addition, applications are presented for a variety of materials of practical and theoretical interest, including the harmonic, Blatz-Ko, and compressible Neo-Hookean materials.     

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.     

Pure Axial Shear of Isotropic, Incompressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the pure axial shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Two popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Non-Symmetric Deformations of Neo-Hookean Solids

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

Saturated Elastic Porous Solids: Incompressible,Compressible and Hybrid Binary Models

Constitutive models for a general binary elastic-porous media are investigated by two complementary approaches. These models include both constituents treated as compressible/incompressible, a compressible solid phase with an incompressible fluid phase (hybrid model of first type), and an incompressible solid phase with a compressible fluid phase (hybrid model of second type). The macroscopic continuum mechanical approach uses evaluation of entropy inequality with the saturation condition always considered as a constraint. This constraint leads to an interface pressure acting in both constituents. Two constitutive equations for the interface pressure, one for each phase, are identified, thus closing the set of field equations. The micromechanical approach shows that the results of Didwania and de Boer can be easily extended to general binary porous media.     

Remarks on the Behavior of Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solids

The effect of directional reinforcing in generating qualitative changes in the mechanical response of a base neo-Hookean material is examined in the context of homogenous deformation. Single axis reinforcing giving transverse isotropy is the major focus, in which case a standard reinforcing model is characterized by a single constitutive reinforcing parameter. Various qualitative changes in the mechanical response ensue as the reinforcing parameter increases from the zero-value associated with neo-Hookean response. These include (in order): the existence of a limiting contractive stretch for transverse-axis tensile load; loss of monotonicity in off-axis simple shear; loss of monotonicity in on-axis compression; loss of positivity in the stress-shear product in off-axis simple shear; and loss of monotonicity for plane strain in on-axis compression. The qualitative changes in the simple shear response are associated with stretch relaxation in the reinforcing direction due to finite rotation. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Nonlinearly Elastic Equilibrium of a Rectangular Multilayered Plate

A three-dimensional boundary-value problem of physically nonlinear elastic theory is solved for a multilayered plate. The nonlinear relationships between the stresses and small strains are assumed to be of the Kauderer form. The solution under given conditions is constructed as series in powers of a physical dimensionless small parameter. The physically nonlinear boundary-value problem is reduced to a recurrent sequence of linear boundary-value problems. The effect of the physical nonlinearity of the material on the stress–strain state of the plate is studied.     

考虑剪切变形的各向同性,正交各向异性矩形板的屈曲和后屈曲

本文讨论考虑横向剪切变形的各向同性、正交各向异性矩形板的屈曲和后屈曲性态。应用Reissner理论,采用文[1]提供的摄动方法,给出了完善和非完善各向同性、正交各向异性矩形板的后屈曲平衡路径,并与薄板理论结果作了比较。     

On Cavitation, Configurational Forces and Implications for Fracture in a Nonlinearly Elastic Material

Experiments on polymers indicate that large tensile stress can induce cavitation, that is, the appearance of voids that were not previously evident in the material. This phenomenon can be viewed as either the growth of pre-existing infinitesimal holes in the material or, alternatively, as the spontaneous creation of new holes in an initially perfect body. In this paper our approach is to adopt both views concurrently within the framework of the variational theory of nonlinear elasticity. We model an elastomer on a macroscale as a void-free material and, on a microscale, as a material containing certain defects that are the only points at which hole formation can occur. Mathematically, this is accomplished by the use of deformations whose point singularities are constrained. One consequence of this viewpoint is that cavitation may then take place at a point that is not energetically optimal. We show that this disparity will generate configurational forces, a type of force identified previously in dislocations in crystals, in phase transitions in solids, in solidification, and in fracture mechanics. As an application of this approach we study the energetically optimal point for a solitary hole to form in a homogeneous and isotropic elastic ball subject to radial boundary displacements. We show, in particular, that the center of the ball is the unique optimal point. Finally, we speculate that the configurational force generated by cavitation at a non-optimal material point may be sufficient to result in the onset of fracture. The analysis utilizes the energy-momentum tensor, the asymptotics of an equilibrium solution with an isolated singularity, and the linear theory of elasticity at the stressed configuration that the body occupies immediately prior to cavitation. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Linear Theory of Porous Elastic Solids

The theory of porous elastic solids with large vacuous interstices, considered by Giovine like materials with ellipsoidal structure, includes, as a particular case, the nonlinear theory of Nunziato and Cowin of elastic materials with small spherical voids finely dispersed in the matrix.In this paper we propose appropriate constitutive relations and then specialize the basic balance equations of Giovine to the linear theory. Also, generalizing the developments of Cowin and Nunziato, we formulate boundary-initial-value problems and examine classical applications as responses to homogeneous deformations and small-amplitude acoustic waves.     

On Entropy Flux of Transversely Isotropic Elastic Bodies

Recently (Liu in J. Elast. 90:259–270, 2008) thermodynamic theory of elastic (and viscoelastic) material bodies has been analyzed based on the general entropy inequality. It is proved that for isotropic elastic materials, the results are identical to the classical results based on the Clausius-Duhem inequality (Coleman and Noll in Arch. Ration. Mech. Anal. 13:167–178, 1963), for which one of the basic assumptions is that the entropy flux is defined as the heat flux divided by the absolute temperature. For anisotropic elastic materials in general, this classical entropy flux relation has not been proved in the new thermodynamic theory. In this note, as a supplement of the theory presented in (Liu in J. Elast. 90:259–270, 2008), it will be proved that the classical entropy flux relation need not be valid in general, by considering a transversely isotropic elastic material body.      

Torsin of Functionally Graded Isotropic Linearly Elastic Bars

The purpose of this research is to investigate the effects of material inhomogeneity on the torsional response of linearly elastic isotropic bars. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The classic approach to the torsion problem for a homogenous isotropic bar of arbitrary simply-connected cross-section in terms of the Prandtl stress function is generalized to the inhomogeneous case. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined. It is shown that the maximum shear stress does not, in general, occur on the boundary of the rod, in contrast to the situation for the homogeneous problem. It is shown that the material inhomogeneity may increase or decrease the torsional rigidity compared to that for the homogeneous rod. Optimal upper and lower bounds for the torsional rigidity for nonhomogeneous bars of arbitrary cross-section are established. A new formulation of the basic boundary-value problem is given. The results are illustrated using specific material models used in the literature on functionally graded elastic materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Generation of Discrete Isotropic Orientation Distributions for Linear Elastic Cubic Crystals

We consider a model for the elastic behavior of a polycrystalline material based on volume averages. In this case the effective elastic properties depend only on the distribution of the grain orientations. The aggregate is assumed to consist of a finite number of grains each of which behaves elastically like a cubic single crystal. The material parameters are fixed over the grains. An important problem is to find discrete orientation distributions (DODs) which are isotropic, i.e., whose Voigt and Reuss averages of the grain stiffness tensors are isotropic. We succeed in finding isotropic DODs for any even number of grains N≥4 and uniform volume fractions of the grains. Also, N=4 is shown to be the minimum number of grains for an isotropic DOD. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite Inhomogeneous Shearing Deformations of a Transversely Isotropic Incompressible Material

The present paper deals with finite inhomogeneous shearing deformations of a slab of a special anisotropic solid. Two cases according to the directions of the anisotropic director of the medium are examined. In one case the solution reduces to a quadrature and gives an exact deformation field for particular values of the material constants. In the other case an exact solution is obtained. All such solutions reduce to the same existing solution for the corresponding isotropic elastic material. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Thermal Contact of Two Axially Symmetric Elastic Solids

A contact problem of two elastic convex and axially symmetric solids heated (or cooled) to temperatures of different values is considered. Pertinent formulae have been derived for relations between the contact pressure, geometrical characteristics of the solids and distributions of heat flux over the contacting region. We have analysed: 1. The problem of the loss of the contact between two solids pressed together with active heat fluxes. We discuss the cases for which the contact of the axially symmetric solids can take the form of a circle, or an annulus. 2. The problem of a paradox when the mathematically well posed contact problem of thermoelasticity leads to a physically unacceptable solution with a region of overlapping materials. Here we discuss a generalization of the cooled sphere paradox. The heat flux functions are continuously differentiable, of constant sign. The conditions have been derived for the cases when the paradox can be avoided.     

Finite Inhomogeneous Radial Expansion (Contraction) and Longitudinal Shearing of an Isotropic Compressible Elastic Circular Cylindrical Shell

We have first obtained that the equations of equilibrium governing the finite radial expansion (contraction) and longitudinal shearing of a circular cylindrical shell become uncoupled for a class of harmonic materials (a class of isotropic homogeneous compressible elastic materials). Next it has been assumed that the dilatation is uniform. Following this the exact solutions of the uncoupled equations of equilibrium have been obtained for a simple harmonic material which is reduced to the Neo-Hookean material for the incompressible case. The deformation is nonhomogeneous in nature. The stresses have been obtained. This revised version was published online in August 2006 with corrections to the Cover Date.