First-order hyperbolic form of velocity-stress equations for waves in elastic solids with hexagonal symmetry

This paper reports a theoretical framework to analyze wave propagation in elastic solids of hexagonal symmetry. The governing equations include the equations of motions and partial differentiation of elastic constitutive relations with respect to time. The result is a set of nine, first-order, fully-coupled, hyperbolic partial differential equations with velocities and stress components as the unknowns. The equation set is then cast into a vector form with three 9 × 9 coefficient (or Jacobian) matrices. Physics of wave propagation are fully described by the eigen structure of these matrices. In particular, the eigenvalues of the Jacobian matrices are the wave speeds and a part of the left eigenvectors represents the wave polarization. Without invoking the plane wave solution and the Christoffel equation, two- and three-dimensional slowness profiles can be calculated. As an example, slowness profiles of a cadmium sulfide crystal are presented.     

Simulations of waves in elastic solids of cubic symmetry by the Conservation Element and Solution Element method

This paper reports a new numerical approach to simulate waves in elastic solids with cubic symmetry. The governing equation includes the equation of motion and the constitutive relation of the elastic medium. With velocity and stress components as the unknowns, the equations are a set of nine, first-order, hyperbolic partial differential equations. To aid numerical simulation, the characteristic form of the equations is derived. By using the Schur complement in linear algebra, the one-dimensional equations are shown to be equivalent to the Christoffel equations without using the harmonic plane-wave solution. To solve the governing equations by the Conservation Element and Solution Element (CESE) method, we first use Gauss' theorem to recast the equations into a space-time integral form. By integrating the integral equation, space-time flux conservation is imposed over Conservation Elements (CEs). Numerical integration is aided by using prescribed linear discretization of the unknowns in Solution Elements (SEs). A convergence test shows that the CESE method employed is second-order accurate. To demonstrate the capabilities of the present approach, reported numerical results include one-dimensional resonant waves, collinear impact of two blocks, and two-dimensional wave expansion from a point source. Additional results of waves interacting with an interface separating two media with different lattice orientations are also reported. Results compared well with the available analytical solutions. All results show salient features of waves in solids of cubic symmetry.     

Strong ellipticity for tetragonal system in linearly elastic solids

The present paper studies the strong ellipticity for all crystal classes of tetragonal system in a linearly elastic material. Explicit conditions characterizing the strong ellipticity of the elasticity tensor are established for the tetragonal system with six elasticities (that is tetragonal–scalenohedral, ditetragonal–pyramidal, tetragonal–trapezohedral, ditetragonal–dipyramidal crystal classes) as well as for the tetragonal system with seven elasticities (tetragonal–disphenoidal, tetragonal–pyramidal and tetragonal–dipyramidal crystal classes).     

A multiscale thermomechanical model for cubic to tetragonal martensitic phase transformations

We develop a multiscale thermomechanical model to analyze martensitic phase transformations from a cubic crystalline lattice to a tetragonal crystalline lattice. The model is intended for simulating the thermomechanical response of single-crystal grains of austenite. Based on the geometrically nonlinear theory of martensitic transformations, we incorporate microstructural effects from several subgrain length scales. The effective stiffness tensor at the grain level is obtained through an averaging scheme, and preserves crystallographic information from the lattice scale as well as the influence of volumetric changes due to the transformation. The model further incorporates a transformation criterion that includes a surface energy term, which takes into account the creation of interfaces between martensite and austenite. These effects, which are often neglected in martensitic transformation models, thus appear explicitly in the expression of the transformation driving force that controls the onset and evolution of the transformation. In the derivation of the transformation driving force, we clarify the relations between different combinations of thermodynamic potentials and state variables. The predictions of the model are illustrated by analyzing the response of a phase-changing material subjected to various types of deformations. Although the model is developed for cubic to tetragonal transformations, it can be adapted to simulate martensitic transformations for other crystalline structures.     

Material symmetry and constitutive equations

Summary Constitutive equations which are relations between tensors are considered. It is shown how the restrictions which are imposed on the form of such relations by symmetry of the material can be obtained in canonical form.

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übersicht Stoffgleichungen werden als Tensorbeziehungen betrachtet. Es wird gezeigt, wie die aus der Forderung nach Symmetrie des Materials folgenden Einschr?nkungen in kanonischer Form erhalten werden k?nnen.

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Dedicated to Professor Hans Ziegler on the occasion of his 70th birthday     

Riccati equations for wave propagation in planarly-stratified solids

Wave propagation and scattering are investigated in three-dimensional, dissipative, anisotropic bodies with one-dimensional inhomogeneity. The constitutive equations are taken to be linear. The incidence is allowed to be oblique. Three different procedures are set up which involve the impedance matrix and two local reflection matrices. All three matrix functions are shown to satisfy appropriate Riccati equations and jump conditions (at discontinuities). The operative aspects are examined to solve reflection–transmission problems for a layer sandwiched between two half spaces. The unknown field within the layer is determined through the impedance or the reflection matrices.     

On constitutive equations for fiber-reinforced nonlinearly viscoelastic solids

In this paper, we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation. These invariants are analyzed, and we obtain restrictions such as positivity of some of them.     

Analytical solutions for tapering quadratic and cubic rat-holes in highly frictional granular solids

New exact analytical solutions are presented for both stress and velocity fields for a Coulomb–Mohr granular solid assuming non-dilatant double-shearing theory. The solutions determined apply to highly frictional materials for which the angle of internal friction φ is assumed equal to 90°. This major assumption is made primarily to facilitate exact analytical solutions, and it is discussed at length in the Introduction, both in the context of real materials which exhibit large angles of internal friction, and in the context of using the solutions derived here as the leading term in a regular perturbation solution involving powers of 1−sinφ. The analytical velocity fields so obtained are illustrated graphically by showing the direction of the principal stress as compared to the streamlines. The stress solutions are also exploited to determine the static stress distribution for a granular material contained within vertical boundaries and a horizontal base, which is assumed to have an infinitesimal central outlet through which material flows until a rat-hole of parabolic or cubic profile is obtained, and no further flow takes place. A rat-hole is a stable structure that may form in storage hoppers and stock-piles, preventing any further flow of material. Here we consider the important problems of two-dimensional parabolic rat-holes of profile y=ax², and three-dimensional cubic rat-holes of profile z=ar³, which are both physically realistic in practice. Analytical solutions are presented for both two and three-dimensional rat-holes for the case of a highly frictional granular solid, which is stored at rest between vertical walls and a horizontal rigid plane, and which has an infinitesimal central outlet. These solutions are bona fide exact solutions of the governing equations for a Coulomb–Mohr granular solid, and satisfy exactly the free surface condition along the rat-hole surface, but approximate frictional conditions along the containing boundaries. The analytical solutions presented here constitute the only known solutions for any realistic rat-hole geometry, other than the classical solution which applies to a perfectly vertical cylindrical cavity.     

Elastic interaction of point defects in crystals with cubic symmetry

The energy of elastic mechanical interaction between point defects in cubic crystals is analyzed numerically. The finite-element complex ANSYS is used to investigate the character of interaction between point defects depending on their location along the crystallographic directions 〈100〉, 〈110〉, 〈111〉 and on the distance from the free boundary of the crystal. The numerical results are compared with the results of analytic computations of the energy of interaction between two point defects in an infinite anisotropic medium with cubic symmetry. The interaction between compressible and incompressible defects of general type is studied. Conditions for onset of elastic attraction between the defects, which leads to general relaxation of the crystal elastic energy, are obtained.     

General solution for the coupled equations of transversely isotropic magnetoelectroelastic solids

IntroductionMechanicsandphysicsofmediapossessingsimultaneouslypiezoelectric ,piezomagneticandmagnetoelectriceffects ,namely ,magnetoelectroelasticsolids,haveattractedmoreandmoreattentionduetotheirgreatpotentialapplicationsinthetechnologiesofsmartandadaptivematerialsystem[1] .Sometheoreticalinvestigationsappearedintheliteratureinclude :1)Theexistenceproblemofsurfacewavesinsemi_infiniteanisotropicmagnetoelectroelasticmediawithvariousboundaryconditions[2 ,3 ] ;2 )Green’sfunctions[4～ 7] ;3)Inho…     

Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

A set of transport equations for the growth or decay of theamplitudes of shock waves along an arbitrary propagation directionin three-dimensional nonlinear elastic solids is derived using theLagrangian coordinates.The transport equations obtained showthat the time derivative of the amplitude of a shock wave alongany propagation ray depends on (i) an unknown quantity immediatelybehind the shock wave,(ii) the two principal curvatures of theshock surface,(iii) the gradient taken on the shock surface ofthe normal shock wave speed and (iv) the inhomogeneous term.whichis related to the motion ahead of the shock surface.vanisheswhen the motion ahead of the shock surface is uniform.Severalchoices of the propagation vector are given for which the tran-sport equations can be simplified.Some universal relations,which relate the time derivatives of various jump quantities toeach other but which do not depend on the constitutive equationsof the material,are also presented.     

General irreducible representations for constitutive equations of elastic crystals and transversely isotropic elastic solids

By means of the combined invariance restrictions due to material frame-indifference and material symmetry, the present paper provides general reduced forms for non-polynomial elastic constitutive equations of all 32 classes of crystals and transversely isotropic solids.Project supported by National Natural Science Foundation and National Postdoctoral Science Foundation of China.     

Two approximate analytical solutions for the kinematically determined velocity equations for granular solids

For axially symmetric flows of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming dilatant double shearing a closed set of three first-order partial differential equations are obtained. These supposedly simple equations are deceptive, because although they are simple in appearance, the determination of exact solutions is non-trivial. For one of the known families of solutions which has not been studied previously, the authors present the non-linear ordinary differential equation for the stress angle ψ and determine two small ψ approximations. Furthermore, the stream function and streamlines are obtained for ψ determined numerically and from the two small ψ approximations. For purposes of comparison, the streamlines for three further known exact solutions are also presented. In addition, we briefly examine the circumstances for which solutions of the velocity equations satisfy the principle of non-negative plastic work. For example, we are able to establish that in the case when the velocity equations are derived from a plastic potential, the solutions always satisfy the principle when the material has no cohesion.     

Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids

The propagation of a three-dimensional shock wave in an elastic solid is studied. The material is assumed to be a simple elastic solid in which the Cauchy stress depends on the deformation gradient only. It is shown that the growth or decay of a discontinuity ψ depends on (i) an unknown quantity φ^? behind the shock wave, (ii) the two principal curvatures of the shock surface, (iii) the gradient on the shock surface of the shock wave speeds and (iv) the inhomogeneous term which depends on the motion ahead of the shock surface and vanishes when the motion ahead of the shock surface is uniform. If a proper choice is made of the propagation vectorb along which the growth or decay of the discontinuity is measured, the dependence on item (iii) can be avoided. However,b assumes different directions depending on the choice of discontinuity ψ with which one is concerned and the unknown quantity φ^? behind the shock wave on which one chooses to depend. As in the case of one-dimensional shock waves, the growth (or decay) of one discontinuity may not be accompanied by the growth (or decay) of other discontinuities. A universal equation relating the growth or decay of discontinuities in the normal stress, normal velocity and specific volume is also presented.     

Differential characteristic set algorithm for the complete symmetry classification of partial differential equations

In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations （PDEs） with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu＇s method, in differential equations.