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.     

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.     

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 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.     

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.     

Vibration of a transversely isotropic,simply supported and layered rectangular plate

The propagator matrix method is used in this paper to study the vibration of a transversely isotropic, simply supported and layered rectangular plate. A new system of vector functions is constructed to deal with general surface loading, and general solutions and layer matrices of exact closed form are obtained in this system. The particular solution for forced vibration, and the characteristic equations for free vibration of various surface conditions are then obtained by simple multiplication of layer matrices. These results are presented in such a way that the dilatational and distortional modes of vibration are separated. As a special case of the layered plate, results for the corresponding homogeneous thick plate are also derived. They are presented in a very simple form, and contain the previous results for the static transversely isotropic and the dynamic isotropic plates.     

An exact solution for transversely isotropic,simply supported and layered rectangular plates

Levinson's solution for the problem of a simply supported rectangular plate of arbitrary thickness by normal surface loads is extended to the transversely isotropic and layered case. The exact closed form solution is obtained by using the propagator matrix method in a system of vector functions. As a special case of the layered medium, the normal displacement or deflection of a homogeneous plate of arbitrary thickness by normal surface loads is also given. It is shown that it approaches the classical solution for the transversely isotropic thin plate as the thickness approaches zero on the one hand, and on the other hand reduces to the thick plate expression as given by Levinson when the medium is isotropic.     

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.     

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.     

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.     

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.     

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.     

The theory of Kirchhoff rods as an exact consequence of three-dimensional elasticity

A one-dimensional model of a linearly elastic thin rod is deduced from three-dimensional elasticity by regarding the Kirchhoff hypotheses as internal constraints prevailing in a three-dimensional tubular region. It follows from such an assumption that the displacement and the strain fields are linear in the cross-sectional coordinates. A constitutive relation that exhibits the maximal symmetry compatible with the assumed constraints is chosen and the equilibrium equations in terms of displacements are obtained.     

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.     

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.     

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.     

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.     

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.     

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.