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).     

Gleanings of radial cavitation

It is shown that the constitutive character inducing radial cavitation of elastic balls is carried by the liquid-like portion, if any, of their stored-energy density.     

A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain

Necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials were first given by Knowles and Sternberg [3,4] by means of a lengthy calculation. Since then Aubert and Tahraoui [1] have shown the necessity of these conditions and simplified one of them using a different but still complicated method. The purpose of this note is to show how the conditions can be derived in a very simple way.     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Saint-Venant's problem for linear piezoelectric bodies

We seek a solution for a piezoelectric cylinder acted on the end faces by applied tractions and charges, under the hypothesis that both the stress and electric displacement fields depend linearly on the axial coordinate. The analysis, restricted to monoclinic materials of crystallographic class 2, leads to an explicit solution in terms of the strain and electric fields, which depend on the stress and charge resultants and on two scalar functions determined by the solution of a plane piezoelectric problem.     

Coupled shear-torsional motion of a rubber support system

The finite amplitude, coupled shear-torsional motion of a circular disk supported between identical rubber spring cylinders is studied. The material of the springs is assumed to be an incompressible elastic material. The oscillatory motion oscillatory of the disk is studied for two different cases. In the first case, the material of the spring is assumed to be an incompressible elastic material whose response functions are constants. Typical examples include the Mooney-Rivlin model. The motion of the disk in this case is governed by two independent equations whose closed form solutions are noted. For the second case, the material of the spring is assumed to be an incompressible quadratic material. The motion oscillatory of the disk in this case is governed by two coupled nonlinear differential equations. The stability properties of small shearing oscillation superimposed on finite torsion and small torsional oscillation superimposed on finite shearing are studied.     

The spreading of nonlinear elastic surface waves

A theory for the lateral spreading of a beam of nonlinear surface acoustic waves across the surface of an arbitrary, homogeneous, elastic half-space is developed. The resulting evolution equation generalizes that obtained for uni-directional waves by replacing an ordinary derivative by a diffusion operator of Schrödinger type. The coefficients arising in the evolution equation are related to partial derivatives of the dispersion relation for linearized surface waves on the half space. Details are given for isotropic materials and for two special cases of beams travelling along axes of high elastic symmetry.     

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.     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.Work supported in part by National Science Foundation Grant INT-9106519.     

On a principle of virtual work for thermo-elastic bodies

A principle of virtual work is proposed for thermo-elastic bodies. From it are derived the equations of motion, the Cauchy stress principle and the Gibbs relations. The principle is also used to analyse the response of internally constrained bodies.     

On the plane problem of orthotropic quasi-static thermoelasticity

The plane displacement boundary value problem of quasi-static linear orthotropic thermoelasticity is discussed. The thermoelastic system on a bounded simply-connected domain is decoupled. The decoupled temperature equation is investigated by using an accurate estimate and the contractive mapping principle. Representation of solution of the field equation is obtained, and some solvability results are proved. The results are of both theoretical and numerical interest.     

Cavitation for incompressible anisotropic nonlinearly elastic spheres

In this paper, the effect ofmaterial anisotropy on void nucleation and growth inincompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid sphere composed of an incompressible homogeneous nonlinearly elastic material which is transversely isotropic about the radial direction. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Closed form analytic solutions are obtained for a specific material model, which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials. In contrast to the situation for a neo-Hookean sphere, bifurcation here may occur locally either to the right (supercritical) or to the left (subcritical), depending on the degree of anisotropy. In the latter case, the cavity has finite radius on first appearance. Such a discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon of structural mechanics. Such dramatic cavitational instabilities were previously encountered by Antman and Negrón-Marrero [3] for anisotropiccompressible solids and by Horgan and Pence [17] forcomposite incompressible spheres.     

On wave propagation in inextensible elastic bodies

In this paper we study the propagation of acceleration waves in inextensible elastic bodies. ¹ While the computations are but an exercise, the results are interesting and quite unlike the corresponding results for unconstrained bodies. Indeed, a wave travelling in the direction of inextensibility must necessarily be transverse, and, when the reaction stress is compressive and sufficiently large, the corresponding speed of propagation becomes non-real, so that even transverse waves fail to exist.We also study (infinitesimal) progressive waves and we find the corresponding propagation condition to be the same as that for acceleration waves. Here, however, non-real speeds of propagation have a definite physical meaning: they imply exponential growth of the wave. Thus, in particular, when the reaction stress is compressive and sufficiently large, a transverse progressive wave travelling in the direction of inextensibility grows without bound. We conjecture that this indicates the presence of local buckling. ²     

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J₂-deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H²(Ω) by using monotone operator theory and Browder-Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H²-norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.     

Remarks on ellipticity for the generalized Blatz-Ko constitutive model for a compressible nonlinearly elastic solid

One of the most widely used constitutive models for compressible isotropic nonlinearly elastic solids is the generalized Blatz-Ko material for foam-rubber and its various specializations. For this model, a unified derivation of necessary and sufficient conditions for ellipticity of the governing three-dimensional displacement equations of equilibrium is provided. When the parameterf occurring in the generalized Blatz-Ko model is in the range 0f<1, it is shown that ellipticity is always lost at sufficiently large stretches, while forf=1, the equilibrium equations are globally elliptic. The implications of these results for a variety of physical problems are discussed.     

Saint-Venant's problem with Voigt's hypotheses for anisotropic solids

We seek for a solution of Saint-Venant's problem for inhomogeneous and anisotropic materials under the assumptions, introduced by Voigt, that the stress is either constant along the axis of the cylinder or depends linearly on the axial coordinate. We first prove the uniqueness of the solution in terms of resultants, then we exhibit an explicit formula for such a solution; we show finally how Clebsch's hypothesis, that the stress vector on axial planes is parallel to the axis, is compatible with Voigt's hypotheses provided that the symmetry group of the material comprising the cylinder contains the reflections on the cross-section.     

Injections of bodies into bodies are continuous

In this paper we show that if we assume that a deformation carries bodies into bodies, then the continuity of the deformation follows as a consequence.     

Elastic waves in bodies with initial stresses

Conclusions An analysis of research available on the problem under consideration shows that at this time: a) Wave propagation in unbounded, bounded, and composite laminar bodies with homogeneous initial states has been studied sufficiently extensively; b) methods to determine the third-order elastic constants have been developed on the basis of existing theories; c) an ultrasonic nondestructive method to determine the stress in solids has been developed, where the stresses averaged over the bulk of the body are determined.In light of the above, in the authors' opinion, the following should be considered the most urgent questions for further investigations on the problem: a) the investigation of wave propagation regularities in bodies with inhomogeneous initial states (it is hence important to execute quantitative and qualitative analyses of the phenomenon); b) development of an ultrasonic nondestructive method of determining the stress in the near surface layers of a solid, which will afford a possibility of determining not only the membrane stresses but also the bending stresses; c) an investigation of the wave propagation regularities in fibrous composites with initial stresses.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 15, No. 4, pp. 3–23, April, 1979.