On a principle of virtual work for thermo-elastic bodies with couple stresses

A principle of virtual work is proposed for thermo-elastic bodies with couple stresses. From it are derived the equations of motion, the stress principles of Cauchy and the Cosserats, and the stress, couple stress and entropy relations. The principle is also used to analyze the effects of an internal constraint on the response of the body.     

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.     

On Saint-Venant's principle in linear elastodynamics

We investigate the spatial behaviour of the steady state and transient elastic processes in an anisotropic elastic body subject to nonzero boundary conditions only on a plane end. For the transient elastic processes, it is shown that at distance x ₃ >ct from the loaded end, (c is a positive computable constant and t is the time), all the activity in the body vanishes. For x ₃ , an appropriate measure of the elastic process decays with the distance from the loaded end, the decay rate of end effects being controlled by the factor % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaKazaaiacaGGOaGccaaIXaGaaeiiaiabgkHiTiaabccadaWcaaqa% amXvP5wqonvsaeHbfv3ySLgzaGqbciab-Hha4naaBaaaleaacaqGZa% aabeaaaOqaaiaabogacaqG0baaaKazaakacaGGPaaaaa!4BB0!\[(1{\text{ }} - {\text{ }}\frac{{x_{\text{3}} }}{{{\text{ct}}}})\]. Next, it is shown that for isotropic materials, in the case of a steady state vibration, an analogue of the Phragmén-Lindelöf principle holds for an appropriate cross-sectional measure. One immediate consequence is that in the class of steady state vibrations for which a quasi-energy volume measure is bounded, this measure decays at least algebraically with the distance from the loaded end.     

On heat flux in simple media

For any simple body we prove that there is an infinite number of mathematical functions which can be added to the response functional for the heat flux vector without affecting the balance laws, the entropy inequality, and the boundary conditions on the normal heat flux. The presence of this indetermination cannot be detected by usual physical experiments, namely by experiments in which cuts of the body are not taken into account. The maximal class of this indetermination is fully characterized in the thermoelastic case.This work has been performed within the activity sphere of the Consiglio Nazionale delle Ricerche, Group no 3, in the academic year 1988/9.     

Two illustrations of rapid crack growth in a pre-stressed compressible neo-Hookean material

An unbounded isotropic compressible neo-Hookean solid is initially in equilibrium under uniform tensile (possibly large) pre-stress. In one case, plane strain conditions generate slit crack growth at a constant sub-critical rate; in the other, axial symmetry produces penny-shaped crack growth. The procedure of superposing infinitesimal deformations upon those that are large is carried out in terms of tractable exact full-field solutions.These solutions are examined apart from a specific fracture mechanics model, nevertheless, they show that pre-stress induces, in addition to the expected anisotropy, a critical value above which a negative Poisson effect occurs. It is also found that dilatational, rotational and Rayleigh wave speeds decrease, and that the decrease is greater for the plane strain state associated with slit crack growth than for the axially symmetric state of the penny-shaped crack.Dynamic stress intensity factors are also extracted, and found to fall below those for a linear isotropic solid at the same pre-stress and crack growth rate. Moreover, the range of growth rates for sub-critical crack propagation is also decreased.     

On the Saint-Venant principle in the case of infinite energy

Estimates on the distribution of the elastic energy in a cylindrical domain in the context of linear elasticity are obtained. The estimates remain valid when the total elastic energy is infinite, and they can be used to establish Saint-Venant's principle without an assumption about finiteness of the total energy.Examples of boundary conditions resulting in infinite energy are constructed in the context of both linear elastostatics and special finite elastostatics, where a quadratic strain energy density function is assumed. The examples show that estimates of the type obtained are sometimes necessary.The results obtained are valid with obvious modifications in a space of any dimension n2.The results in this paper represent a partial fulfilment of the requirements for the degree of Doctor of Philosophy at Tel Aviv University by Y.S. under the guidance of J.J.R.     

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.     

On Stroh orthogonality relations: An alternative proof applicable to Lekhnitskii and Eshelby theories of an anisotropic body

This paper establishes that the Stroh orthogonality relations for an anisotropic body are a direct consequence of the fact that the system of equations of equilibrium is self-adjoint and positive definite. It is demonstrated that, assuming a complex representation of displacements and boundary tractions, the Betti theorem of reciprocity implies the orthogonality, and positive definiteness of strain energy implies the full rank of the normalization matrix, in the Stroh orthogonality relations. The presented proof is applicable to both the Lekhnitskii and Eshelby theories of an anisotropic body.     

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.     

Uniqueness and continuous dependence of solutions to a multidimensional thermoelastic contact problem

Uniqueness and continuous dependence on the initial temperature are established for the solution of a multidimensional, quasistatic thermoelastic contact problem. The proof of this result does not depend on the ability to decouple the system of governing equations as required in the technique used by Shi and Shillor [European J. Appl. Math., 1990, 371–387] in the one dimensional analogue of this problem. Some extensions to other contact problems are suggested.     

Well-posedness of the equations of generalized thermoelasticity

In this paper the local existence, uniqueness and continuous dependence for smooth solutions to the initial value problem for a class of generalized (dependent on the time derivative of temperature) thermoelastic materials is proved. The field equations are written as a quasilinear hyperbolic system and the known results by Hughes, Kato and Marsden are applied.     

A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation

In the context of the linear theory of thermoelasticity without energy dissipation for homogeneous and isotropic materials, the uniqueness of solution of a natural initial, mixed boundary value problem is proved. The proof depends on an equation of energy balance formulated entirely in terms of temperature and velocity fields.     

Solution of a Riemann problem for elasticity

This paper describes a numerical algorithm for the Riemann solution for nonlinear elasticity. We assume that the material is hyperelastic, which means that the stress-strain relations are given by the specific internal energy. Our results become more explicit under further assumptions: that the material is isotropic and that the Riemann problem is uniaxial. We assume that any umbilical points lie outside the region of physical relevance. Our main conclusion is that the Riemann solution can be obtained by the iterative solution of functional equations (Godunov iterations) each defined in one- or two-dimensional spaces.Supported in part by AFOSR-88-0025.     

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 minimal representations for constitutive equations of anisotropic elastic materials

The problem of determining minimal representations for anisotropic elastic constitutive equations is proposed and investigated. For elastic constitutive equations in any given case of anisotropy, it is shown that there exist generating sets consisting of six generators and such generating sets are minimal in all possible generating sets. This fact implies that most of the established results for representations of elastic constitutive equations are not minimal and remain to be sharpened. For elastic constitutive equations in some cases of anisotropy, including orthotropy, transverse isotropy, the trigonal crystal class S ₆, and the classes C _2mh , m=1, 2, 3,..., etc., representations in terms of minimal generating sets are presented for the first time.     

On the Rate of Decay of the Oseen Semigroup in Exterior Domains and its Application to Navier–Stokes Equation

We prove L_p-L_q estimates of the Oseen semigroup in n-dimensional exterior domains which refine and improve those obtained by Kobayashi and Shibata [15]. As an application, we give a globally in time stability theory for the stationary Navier–Stokes flow whose velocity at infinity is a non-zero constant vector. We thus extend the result of Shibata [21]. In particular, we find an optimal rate of convergence of solutions of the non-stationary problem to those of the corresponding stationary problem.     

Navier–Stokes Equations: Controllability by Means of Low Modes Forcing

We study controllability issues for 2D and 3D Navier–Stokes (NS) systems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of differential geometric/Lie algebraic control theory are used to establish global controllability of finite-dimensional Galerkin approximations of 2D and 3D NS and Euler systems, global controllability in finite-dimensional projection of 2D NS system and L2-approximate controllability for 2D NS system. Beyond these main goals we obtain results on boundedness and continuous dependence of trajectories of 2D NS system on degenerate forcing, when the space of forcings is endowed with so called relaxation metric.     

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.     

On the Galerkin vector and the Eshelby solution in linear elasticity

Displacement potentials in linear static elasticity consist of three functions which can be regarded as the three components of a vector, e.g., the Galerkin vector. This research note gives an explanation as to why the biharmonic equations govern these functions in isotropic elasticity as opposed to the sixth-order partial differential equations that govern them in anisotropic elasticity. This note also shows that the Eshelby solution in two-dimensional anisotropic elasticity can be derived from the method of displacement potentials.     

On the separation of stress-induced and texture-induced birefringence in acoustoelasticity

In this paper we develop a simple micromechanical model of a prestressed polycrystalline aggregate, in which the texture-induced and stress-induced anisotropies of the aggregate are precisely defined; here the word texture always refers to the texture of the aggregate at the given prestressed configuration, not to that of a perhaps fictitious natural state of the aggregate. We use this model to derive, for a prestressed orthotropic aggregate of cubic crystallites, a birefringence formula which shows explicitly the effects of the orthotropic texture on the acoustoelastic coefficients. From this formula we observe that, generally speaking, we cannot separate the total birefringence into two distinct parts, one reflecting purely the influence of stress on the birefringence, and the other encompassing all the effects of texture. The same formula, on the other hand, provides for each material specific quantitative criteria under which the separation of stress-induced and texture-induced birefringence would become meaningful in an approximate sense.