Realizing the Willis equations with pre-stresses

This paper proves that the linear elastic behavior of the material with inhomogeneous pre-stresses can be described by the Willis equations. In this case, the additional terms in the Willis equations, compared with the classical linear elastic equations for homogeneous media, are related to the gradient of pre-stresses. In this way, the material length scale is naturally incorporated in the framework of continuum mechanics. All these findings also coincide with the results of transformation elastodynamics, so that they can meet the requirement of the principle of material objectivity and the principle of general invariance.     

The effect of a variation in the elastic moduli on Saint-Venant's principle for a half-cylinder

A semi-infinite prismatic cylinder composed of a linear anisotropic classical elastic material is in equilibrium under zero body force and either zero displacement or zero traction on the lateral boundary. The elastic moduli become perturbed. Under suitable conditions on the base load decay estimates are derived for the difference between corresponding quantities in the unperturbed and perturbed bodies. The amplitude in each estimate involves a multiplicative factor that tends to zero as the perturbation tends to zero. The analysis, based upon a first-order differential inequality, introduces apparently new modifications of Korn's inequalities of the first and second kind.     

Pulse propagation in heat-conducting elastic materials

The influence of second-order effects on the propagation of a weak dilatational stress pulse in a heat-conducting elastic material is investigated. As a first approximation, the problem is studied using the linear theory of thermoelasticity. It is found that thermal diffusion dominates the wave motion, and a time is reached when second-order terms must be considered. The wave motion is then found to be isentropic and the shock structure is governed by Burgers's equation. Solutions to this equation are obtained and the influence of heat-conduction on pulse propagation in elastic materials is discussed, with some numerical results being presented for copper. Also, previous work in linear thermoelasticity theory is clarified and related to known results in linear and nonlinear elastodynamics.     

Energetic bounds in finite elasticity

Summary We study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.     

On the elastic-plastic deformation of a sheet containing an edge crack

The solution of the planar tension and bending of an edge-cracked sheet of elastic-plastic material is given when the plastic deformation is represented by a Dugdale model. The analysis assumes conditions of generalized plane stress (for which the model of plastic relaxation is often a suitable one), but the usual transformation of elastic constants may be used to obtain the results also for plane-strain conditions. The method of solution involves the use of a Mellin transform and a Weiner-Hopf technique. Computed results for the size of the plastic zone and the opening at the crack tip are presented, and asymptotic results are obtained for small-scale and large-scale yielding. The results suggest that, when the material is constrained to fracture close to its ultimate tensile stress, the extra severity of a surface flaw compared with a corresponding internal crack is significantly greater than that predicted by linear elastic fracture mechanics.     

SECOND ORDER EFFECTS IN AN ELASTIC HALF－SPACE ACTED UPON BY A NON－UNIFORM NORMAL LOAD

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Local and overall extremal properties of time-independent materials and non-linear elastic comparison materials

For time-independent materials which undergo non-linear deformations from some given reference configuration two (dual) hypotheses are considered. Firstly it is supposed that the work done to a given state of deformation is bounded below and that the bound is attainable on a physically possible path; secondly that the complementary work to a given state of stress is bounded above and that this bound too is attainable on a physically possible path. The consequences of these assumptions are analysed, and the results of Ponter and Martin [1] in the linear theory are generalized to account for non-linear deformations, due attention being paid to questions of stability.A non-linear elastic comparison material is defined whose strain energy is equal to the work done on a minimum path for the time-independent material. Extremum principles for non-linear elastic materials given in [2] are then applied to the comparison elastic material, and bounds are thereby placed on the work and complementary-work functional of the time-independent material. Corresponding overall properties of the time-independent and elastic materials are then compared by defining respective overall constitutive laws and overall stress and deformation variables.Following the definition of strengthening (weakening) of a non-linear elastic solid given by Ogden[2] a time-independent material is said to be strengthened (weakened) when its comparison elastic material is strengthened (weakened). Local and overall aspects of this definition are examined.     

Elasto-plastic coupling within a thermodynamic strain space formulation of plasticity

Elasto-plastic coupling is studied within a general thermodynamic train space formulation of rate-independent plasticity by means of plastic internal variables. The strain space formulation offers a unified approach in treating both stable and unstable material behavior simultaneously. The conditions on elasto-plastic coupling are imposed by the second law of thermodynamics and the consistency equation in the strain space formulation, and are expressed by two inequalities. This is further illustrated by specific examples where explicit necessary and sufficient conditions on the elastic moduli and their change with plastic deformation are derived for the two inequalities to be satisfied.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.     

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung’s (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain–energy density function. The fluid is described through a Navier–Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.     

Parametric study of the distribution of the tensile stresses in pavilion structures constituted by four sectors of barrel shells

A semi-analytical approach is followed in order to obtain an approximate solution for an analytical model describing the static behaviour of pavilion shells. With the aim to parametrically investigate the distribution of the linear elastic tensile stresses, a linear elastic isotropic behaviour of the material is considered. Inside the pavilion shell, a family of arches and a family of straight beams can be recognized. This assumption justifies the separation of the variables which is at the basis of the proposed semi-analytical approach. The sensitivity of the behaviour of the shells to the values of the mechanical and geometrical parameters characterizing the system is investigated. Comparisons with results obtained by finite element models are performed to confirm both the validity of the semi-analytical approach and the provided results. Since only a linear elastic isotropic material has been taken into account, the results do not claim to describe the behaviour of masonry pavilion vaults.     

On the radial oscillations of incompressible, isotropic, elastic and limited elastic thick-walled tubes

The finite amplitude, free radial oscillations of a thick-walled circular cylindrical tube are studied for an arbitrary incompressible, isotropic and homogeneous rubber-like material having limiting molecular chain extensibility. First, based on classical results for hyperelastic tubes, some results for thick-walled Mooney-Rivlin tubes are described graphically in the phase plane. Then the periodicity of the finite amplitude, free oscillations of a general limited elastic, thick-walled tube is studied; and some analytical results for the Gent model are illustrated in several numerical examples. Results for thick-walled Gent tubes are compared with those for corresponding Mooney-Rivlin tubes; and the motion of thin-walled Gent tubes is illustrated in the phase plane. Physical conclusions are presented. The period of small amplitude oscillations of an arbitrary elastic or limited elastic tube is derived from relations obtained by a linearization of a general class of equations of which the tube problem is a special case. Classical results of the linear theory are thereby recovered and compared with results for Mooney-Rivlin and Gent tubes.     

On the evolution of the linear material properties of PZT during loading history—an experimental study

Ferroelectrics exhibit material behavior which is strongly affected by its loading history. Among other phenomena, the coefficients describing the linear material behavior are known to change when the state of polarization is altered. There are several approaches to modeling ferroelectric/ferroelastic behavior. However, with all models, assumptions have to be made on how the linear coefficients depend on the state of polarization. Often the elastic and dielectric coefficients are defined to be constant for the sake of simplicity. Alternatively, their evolution and that of the piezoelectric constants are described rather intuitively, while systematic experimental data are sparse. The present study explores the impact of large signal mechanical and electrical loading on the low frequency linear response of a soft PZT ceramic. This is accomplished via cyclic tests with progressively increasing maximum electrical or mechanical load. Upon load reversal, the quasi-linear response is measured. Remanent polarization and remanent strain are used as internal variables to describe the material behavior as a function of loading history. While the dielectric permittivity κ₃₃ is shown to exhibit only minor variation, Young’s modulus and the piezoelectric coefficient d₃₃ change significantly in the course of loading.     

Evaluation of the second order elastic moduli

The expressions of the apparent linear elastic moduli and their first and second derivatives, with respect to hydrostatic pressure, are obtained according to the second order elasticity theory. As a particular case when the material is hyperelastic, formulae of the first derivatives of the linear elastic moduli reduce to those obtained by Seeger and Buck.     

Steady Sliding Frictional Contact Problems in Linear Elasticity

The existence and uniqueness of an equilibrium solution to frictional contact problems involving a class of moving rigid obstacles is studied. At low friction coefficient values, the steady sliding frictional contact problem is uniquely solvable, thanks to the Lions-Stampacchia theorem on variational inequalities associated with a nonsymmetric coercive bilinear form. It is proved that the coerciveness of the bilinear form can be lost at some positive critical value of the friction coefficient, depending only on the geometry and the elastic properties of the body. An example presented here, shows that infinitely many solutions can be obtained when the friction coefficient is larger than the critical value. This result is paving the road towards a theory of jamming in terms of bifurcation in variational inequality. The particular case where the elastic body is an isotropic half-space is studied. The corresponding value of the critical friction coefficient is proved to be infinite in this case. In the particular case of the frictionless situation, our analysis incidentally unifies the approaches developed by Lions-Stampacchia (variational inequalities) and Hertz (harmonic analysis on the half-space) to contact problems in linear elasticity.     

Axial motion of a transversely isotropic elastic half-space in which the head wave and conical point arrival times coincide

The constraint which must be imposed on the elastic stiffnesses of a linear, transversely isotropic, elastic half-space in order that the arrival times of the conical points on the wave surface and the head wavefront coincide along the epicentral axis is established. Such a material, whose elastic stiffnesses approximate closely those of Zinc, is investigated in some detail. It is found that the coincident singularity travelling along the epicentral axis has order -1/4, in contrast to -1/2 for the singularity due to the conical point on the wave surface only when the arrivals are distinct, as is the case for Zinc.     

Cloaking of a circular cylindrical elastic inclusion from antiplane elastic waves and resonance effects

Cloaking of a circular cylindrical elastic inclusion embedded in a homogeneous linear isotropic elastic medium from antiplane elastic waves is studied. The transformation or change-of-variables method is used to determine the material properties of the cloak and the homogenization theory of composites is used to construct a multilayered cloak consisting of many bi-material cells. The large system of algebraic equations associated with this problem is solved by using the concept of multiple scattering with wave expansion coefficient matrices. Numerical results for cloaking of an elastic inclusion and a rigid inclusion are compared with the case of a cavity. It is found that while the cloaking patterns for the three cases are similar, the major difference is that standing waves are generated in the elastic inclusion and the multilayered cloak cannot prevent the motion inside the elastic inclusion, even though the cloak seems nearly perfect. Waves can penetrate into and cause vibrations inside the elastic inclusion, where the amplitude of standing waves depend on the material properties of the inclusion but are very much reduced when compared to the case when there is no cloak. For a prescribed mass density, the displacements inside the elastic cylinder decrease as the shear modulus increases. Moreover, the cloaking of the elastic inclusion over a range of wavenumbers is also investigated. There is significant low frequency scattering even if the cloak consists of a large number of layers. When the wavenumber increases, the multilayered cloak is not effective if the cloak consists of an insufficient number of layers. Resonance effects that occur in cloaking of elastic inclusions are also discussed.     

Motion of an Elastic Solid inside an Incompressible Viscous Fluid

The motion of an elastic solid inside an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of regularity which has left the basic question of existence open until now.In this paper, we prove the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. Our functional framework is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.This revised version was published in April 2005. The volume number has now been inserted into the citation line.     

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.     

Bounds on the Effective Anisotropic Elastic Constants

Hill [12] showed that it was possible to construct bounds on the effective isotropic elastic coefficients of a material with triclinic or greater symmetry. Hill noted that the triclinic symmetry coefficients appearing in the bounds could be specialized to those of a greater symmetry, yielding the effective isotropic elastic coefficients for a material with any elastic symmetry. It is shown here that it is possible to construct bounds on the effective elastic constants of a material with any anisotropic elastic symmetry in terms of triclinic symmetry elastic coefficients. Similarly, it is then possible to specialize the triclinic symmetry coefficients appearing in the bounds to those of a greater symmetry. Specific bounds are given for the effective elastic coefficients of cubic, hexagonal, tetragonal and trigonal symmetries in terms of the elastic coefficients of triclinic symmetry. These results are obtained by combining the approach of Hill [12] with a representation of the stress-strain relations due, in principle, to Kelvin [25,26] but recast in the structure of contemporary linear algebra. This revised version was published online in August 2006 with corrections to the Cover Date.