Conservation properties for plane deformations of anisotropic linearly elastic curved strips

Plane deformations of a curved strip, composed of an homogeneous cylindrically anisotropic linearly elastic material, are considered. The strip is in equilibrium under the action of end loads, with the lateral sides traction-free. Two conservation properties for certain cross-sectional stress measures are established, generalizing previously known results for the case of a rectangular strip. Such conservation properties are useful in assessing the influence of material anisotropy on Saint-Venant's principle, as well as in establishing convexity properties for cross-sectional stress measures. In particular, it is anticipated that the results should be useful in determining the extent of edge effects in the testing of anisotropic and composite curved strips.     

End effects for plane deformations of an elastic anisotropic semi-infinite strip

In the linear theory of elasticity, Saint-Venant's principle is used to justify the neglect of edge effects when determining stresses in a body. For isotropic materials, the validity of this is well established. However for anisotropic and composite materials, experimental results have shown that edge effects may persist much farther into the material than for isotropic materials and as a result cannot be neglected. This paper further examines the effects of material anisotropy on the exponential decay rate for stresses in a semi-infinite elastic strip. A linearly elastic semi-infinite strip in a state of plane stress/strain subject to a self-equilibrated end load is considered first for a specially orthotropic material and then for the general anisotropic material. The problem is governed by a fourth-order elliptic partial differential equation with constant coefficients. In the former case, just a single dimensionless material parameter appears, while in the latter, only three dimensionless parameters are required. Energy methods are used to establish lower bounds on the actual stress decay rate. Both analytic and numerical estimates are obtained in terms of the elastic constants of the material and results are shown for several contemporary engineering materials. When compared with the exact stress decay rate computed numerically from the eigenvalues of a fourth-order ordinary differential equation, the results in some cases show a high degree of accuracy. In particular, for strongly orthotropic materials, an asymptotic estimate provides extremely accurate estimates for the decay rate. Results of the type obtained here have several important practical applications. For example, they provide physical insight into the mechanical testing of anisotropic and laminated composite structures (including the off-axis tension test), are useful in assessing the influence of fasteners, joints, etc. on the behavior of composite structures and allow for tailoring a material with specific properties to ensure that local stresses attenuate at a desired rate.     

End effects for anti-plane shear deformations of sandwich structures

The purpose of this research is to further investigate the effects of material inhomogeneity and the combined effects of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects. Saint-Venant decay rates for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The problem is governed by a second-order, linear, elliptic, partial differential equation with discontinuous coefficients. The most general anisotropy consistent with a state of anti-plane shear is considered, as well as a variety of boundary conditions. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement which depends only on the in-plane coordinates. They can be thought of as complementary deformations to those of plane elasticity. While these deformations have received little attention compared with the plane problems of linear elasticity, they have recently been investigated for anisotropic and inhomogeneous linear elasticity. In the context of linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that quickly decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Self-equilibrated loads are a class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity significantly affects the practical application of Saint-Venant's principle to sandwich structures.     

Coaxiality of strain and stress in anisotropic linear elasticity

For a linearly elastic anisotropic body there are at least two rotations of the principal axes of strain such that the stress and strain tensors become coaxial. These rotations correspond to critical points for the stored energy, viewed as a function of the relative orientation between the body and the strain tensor.Supported by Gruppo Nazionale per la Fisica Matematica of C.N.R. (Italy).     

Conservation properties for plane deformations of isotropic and anisotropic linearly elastic strips

Plane deformations of a rectangular strip, composed of an homogeneous fully anisotropic linearly elastic material, are considered. The strip is in equilibrium under the action of end loads, with the lateral sides traction-free. Two conservation properties for certain cross-sectional stress measures are established, generalizing previously known results for the isotropic case. It is noteworthy that in the first of these conservation laws only one of the off-axis elastic constants appears explicitly while in the second only the opposite off-axis constant appears explicitly. Such conservation properties are useful in assessing the influence of material anisotropy on Saint-Venant's principle, as well as in establishing convexity properties for cross-sectional stress measures. In particular, it is anticipated that the results should be useful in determining the extent of edge effects in the off-axis testing of anisotropic and composite materials.     

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 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.     

Asymmetric vibrations of an elastic sphere

Historically, the vector Navier equation governing the dynamic response of an elastic, homogeneous, isotropic sphere has been solved using the Helmholtz decomposition of the displacement vector. Further, many of the problems in the literature have been restricted to ones involving axisymmetric geometry. In this presentation, the time-dependent Navier equation is solved using a set of vector spherical harmonics which, previously, has been used primarily in quantum mechanics studies but which seems particularly useful in solving asymmetric problems with nonconservative body forces. Expressions for the displacements, strains, and stresses and a discussion of the vibrations of an elastic sphere are given.Part of the material presented here was developed while the author was on a Developmental Leave at the University of Texas at Austin.     

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 influence of cohesive stress-separation laws on elastic stress singularities

The nature of the stress field occurring at the vertex of an angular elastic plate under in-plane loading is reconsidered. An additional boundary condition is introduced. This boundary condition reflects the action of cohesive stress-separation laws. Companion asymptotic analysis proceeds routinely on employing coupled eigenfunction expansions. Results show that a number of configurations that had previously contained stress singularities become singularity free.     

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.     

Effects of curvilinear anisotropy on radially symmetric stresses in anisotropic linearly elastic solids

It has been known for some time that certain radial anisotropies in some linear elasticity problems can give rise to stress singularities which are absent in the corresponding isotropic problems. Recently related issues were examined by other authors in the context of plane strain axisymmetric deformations of a hollow circular cylindrically anisotropic linearly elastic cylinder under uniform external pressure, an anisotropic analog of the classic isotropic Lamé problem. In the isotropic case, as the external radius increases, the stresses rapidly approach those for a traction-free cavity in an infinite medium under remotely applied uniform compression. However, it has been shown that this does not occur when the cylinder is even slightly anisotropic. In this paper, we provide further elaboration on these issues. For the externally pressurized hollow cylinder (or disk), it is shown that for radially orthotropic materials, the maximum hoop stress occurs always on the inner boundary (as in the isotropic case) but that the stress concentration factor is infinite. For circumferentially orthotropic materials, if the tube is sufficiently thin, the maximum hoop stress always occurs on the inner boundary whereas for sufficiently thick tubes, the maximum hoop stress occurs at the outer boundary. For the case of an internally pressurized tube, the anisotropic problem does not give rise to such radical differences in stress behavior from the isotropic problem. Such differences do, however, arise in the problem of an anisotropic disk, in plane stress, rotating at a constant angular velocity about its center, as well as in the three-dimensional problem governing radially symmetric deformations of anisotropic externally pressurized hollow spheres. The anisotropies of concern here do arise in technological applications such as the processing of fiber composites as well as the casting of metals.     

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.     

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.     

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.     

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.     

A new formula for the C-matrix in the Somigliana identity

By making use of a convenient decomposition of the fundamental tractions, a new formula for the C-matrix in the Somigliana identity for a three- or two-dimensional elastic isotropic body is derived. This kind of formula is more advantageous for analytical and computational C-matrix evaluations than the currently well-known formula. A general closed analytical formula of the C-matrix for the case of any finite number of tangent planes to the boundary of the body at a non-smooth boundary point, presented in the final section of this paper, demonstrates the usefulness of the new formula.     

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.