Wave curves for the riemann problem of plane waves in isotropic elastic solids

A modified Riemann problem in which the initial and boundary conditions are constants is considered for plane waves in a half space occupied by an elastic solid. The governing quasilinear differential equations form a system of hyperbolic conservation laws which possesses three wave speeds c₁ ≥ c₂ ≥ c₃. The system is genuinely nonlinear with respect to c₁ and c₃ and linearly degenerate with respect to c₂. Thus it is sufficient to study a two-wave-speed system with c₁ and c₃. Wave curves for simple waves and shock waves are used to construct the solution. Second-order hyperelastic materials which contain four material constants are considered and the solution in the form of wave curves is obtained for all possible combinations of initial and boundary conditions. With a proper nondimensionalization, the wave curves depend only on one material parameter k. The solutions are thermodynamically correct because entropy effects do not come into the picture until the third-order terms in stresses are included in the constitutive laws. The two-wave-speed system has one umbilic point at which c₁ = c₃ and hence the system is not totally hyperbolic (or not strictly hyperbolic). Several interesting and unexpected results are obtained due to the existence of the umbilic point. In one example, we find that a shock wave satisfies the Lax stability condition for a V₁ shock as well as a V₃, shock. In another, a shock wave which involves only one stress component does not satisfy the Lax stability condition for either a V₁ shock or a V₃ shock. However, it satisfies the Lax stability condition if we consider it under the context of a one-wave-speed system. Finally we consider the effects on the solution when the third-order terms are included. We show that although the entropy affects the shock wave solution, it does not appear in the simple wave solution until the fourth-order terms are included. With the third-order terms, there may be as many as three umbilic points, one of which may be an umbilic line.     

On thermal stresses in plane strain of isotropic micropolar elastic solids

Summary We consider the static problem of plane strain in the linear theory of micropolar thermoelasticity assuming that heat sources are absent. We give a method for the reduction of the thermoelastic problem to an isothermal one with certain known boundary conditions. In the case of a multiply connected region this leads to the notion of dislocations.

---

Über die Wärmespannungen bei ebener Verzerrung elastischer mikropolarer isotroper Festkörper

Zusammenfassung Das statische Problem der ebenen Verzerrung in der linearen Theorie der mikropolaren Thermoelastizität wird betrachtet unter der Voraussetzung, daß keine Wärmequellen vorhanden sind. Es wird eine Methode zur Reduzierung des Problems auf ein isothermes mit bekannten Randbedingungen gegeben.

     

Acceleration waves through an isotropic mixture of two non-linear elastic solids

Summary In this paper we study the propagation of acceleration waves through an isotropic isothermal mixture of two non-linear elastic solids. After giving the constitutive equations of the mixture, we calculate the possible normal speeds of propagation. Then we state that, in general, it is possible to distinguish between longitudinal and transverse acceleration waves. Finally, we establish the evolution law of the discontinuities along the normal trajectories associated with the wave front.Work performed under the auspices of C.N.R. (GNFM) and supported by M.P.I. of Italy.     

The propagation of plane waves in bilevel anisotropic elastic solids with nonlocal polar constitution

In analyzing problems involving material behavior from the standpoint of generalized continuum mechanics, one is often faced with different forms of anisotropy at different levels of microscopic and macroscopic aggregates within the same material. In this article, a continuum theory incorporating nonlocal effects within the microstructure of anisotropic solids is developed. In order to illustrate the mathematical development of the theory in practical applications, the theory is applied to the case of materials possessing orthotropy on the nonlocal micropolar level and transverse isotropy on the local micropolar level. This case may apply to materials such as wood and wood composites. The resulting field equations are solved for the propagation of plane waves in a bilevel, anisotropic, nonlocal, micropolar elastic solid.     

Boundary condition treatment in 2×2 systems of propagation equations

Numerical modelling of the water hammer phenomenon involves solving a 2×2 system of propagation equations numerically. In the present paper, this system is solved using the Piecewise Parabolic Method (PPM) Scheme, a higher-order extension of the Godunov Method. To reach high-order discretization accuracy, the PPM scheme uses six points in space to solve the advection equation. Hence, treatment of boundary conditions—which proves to be of importance to water hammer modelling—is not straightforward. Several options for the handling of boundary conditions are presented herein, and only one combination among nine is shown to provide good results. This shows that even very accurate numerical schemes may be of poor help in problem solving if boundary conditions are not handled properly. Results given by the PPM scheme are compared with those given by other solution techniques (Method Of Characteristics—MOC), proving the superior accuracy–efficiency relations used by the PPM over the usual approximations of the MOC. © 1998 John Wiley & Sons, Ltd.     

Thermoelastic plane waves in a transversely isotropic body

Summary Plane waves in a linear, homogeneous and transversely isotropic thermoelastic body are discussed on the basis of a unified system of governing equations. It is found that the motion influenced by the thermal field takes place in three coupled modes. Explicit expressions for the phase velocities and attenuation coefficients of these modes in the cases of high and low frequencies are obtained. Results valid in the conventional and generalized thermoelasticity theories are recovered as particular cases. Comparison with the corresponding results obtained in earlier works is made.     

Thermoelastic plane waves in a rotating isotropic medium

Summary We discuss the theory of thermoelastic wave propagation in a rotating isotropic material. In any given direction there are four waves. In general, all of these waves are attenuated, and none of them is purely dilatational or transverse. Some earlier published results are found to be false.     

Propagation of acceleration waves in micropolar elastic solids

The propagation and growth of acceleration waves of arbitrary form propagating into a deformed micropolar elastic solid are investigated. The speeds of propagation are obtained for isotropic micropolar elastic materials and the growth equation is established. The growth equation of acceleration wave is integrated in the case of principle waves for isotropic materials and the decay conditions are examined for both macro- and micro-amplitudes.     

Spatio-temporal symmetries in SN- and SN × 2-equivariant systems

This paper investigates the spatio-temporal symmetries of periodic trajectories in dynamical systems with S_N and S_N × ₂ symmetry. It turns out that trajectories in S_N-equivariant systems cannot exhibit spatio-temporal symmetries beyond the trivial symmetry of all periodic orbits. More complex symmetries in the trajectories require additional constraints on the dynamics. The possibilities offered by S_N × ₂ symmetric systems are considered and a specific S₃ × ₂-equivariant system is investigated numerically.     

On finite amplitude waves in heterogeneous elastic solids

The perturbation technique of the ‘stretching of the coordinates’ is used to obtain first and second order perturbation solutions of finite amplitude plane waves which propagate into an elastic half-space whose material property varies in the direction of the propagation. The interaction between nonlinearity and heterogeneity is discussed, and the results are illustrated by means of two examples: the longitudinal waves propagating in an elastic half-space with harmonic heterogeneity; the shear wave in a half-space whose property varies as A[1 + ?(X/L)ⁿ], where A, ? ? 1, L, and n are constants, and X denotes the initial particle position measured normal to the plane boundary.     

Acceleration waves in elastic solids near the melting point

By adopting the self-consistent Einstein model for elastic solids and by using the method of singular surface, adiabatic and isothermal acceleration waves in elastic solids in the vicinity of the melting point are studied theoretically. The propagation velocities of the waves are determined and differential equations which govern the variation of amplitudes with time are obtained. It is shown that properties of solid materials near the melting point reflect sensitively on the wave propagations. The following facts are also shown:

• 1.(1) The variation of amplitudes may have a universal character irrespective of elastic solids.
• 2.(2) There exist singularities in the temperature dependences of the wave propagations just at the melting point.

     

Thermo elastic waves with thermal relaxation in isotropic micropolar plate

In the present investigation, we have discussed about the features of waves in different modes of wave propagation in an infinitely long thermoelastic, isotropic micropolar plate, when the generalized theory of Lord–Shulman (L–S) is considered. A more general dispersion equation is obtained. The different analytic expressions in symmetric and anti-symmetric vibration for short as well as long waves are obtained in different regions of phase velocities. It is found that results agree with that of the existing results predicted by Sharma and Eringen in the context of various theories of classical as well as micropolar thermoelasticity.     

An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer

In this paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half-space is assumed to be welded. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate secular equation of fourth order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate secular equation has high accuracy. From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established.     

Effect of surface stresses on surface waves in elastic solids

This paper deals with the propagation of surface waves in homogeneous, elastic solid media whose free surfaces or interfaces of separation are capable of supporting their own stress fields. The general theory for the propagation of surface waves in a medium which supports surface stresses is first deduced, and then this theory is employed to investigate the particular cases of surface waves, viz. (a) Rayleigh waves, (b) Love waves and (c) Stoneley waves. It is seen that the Rayleigh waves become dispersive in nature; and, in case of low frequency with residual surface tension, a critical wavelength exists, below which the propagation of Rayleigh waves is not possible. This critical wave length is directly proportional to the surface tension. Some numerical calculations have been made in the case of Love waves and conclusions have been drawn.     

A complex variable Green's function representation of plane inelastic deformation in isotropic solids

Summary A Green's function representation of the plane inelastic deformation in isotropic solids is given using a complex variable method of Muskhelishvili. Based on the fact that the inelastic deformation in a plane infinitesimal region (which we call a plastic source) can be represented by a double couple, its Green's functions are derived in terms of the complex potential functions; these Green's functions, then, are used as the kernel functions in an area integral representation of the complex potential functions for the inelastic deformation of a finite extent. Emphasis is placed on deriving the area integral representation of the two basic complex potential functions (i.e., and in Muskhelishvili's notation); once they are obtained, any physical quantities such as the displacement, the stress, and the traction can be calculated by simply following the formulae of Muskhelishvili.With 4 Figures     

Interaction of antiplane cracks with elastic waves in transversely isotropic materials

Summary The interaction of plane time-harmonic SH-waves with micro-cracks in transversely isotropic materials is investigated. Elastic wave scattering by a single micro-crack is first analyzed. The scattered displacement is expressed as a Fourier integral containing the crack opening displacement. By using this representation formula and by invoking the traction-free boundary condition on the faces of the crack, a boundary integral equation for the unknown crack opening displacement is obtained. Expanding the crack opening displacement into a series of Chebyshev polynomials and adopting a Galerkin method, the boundary integral equation is converted into an infinite system of inear algebraic equations for the expansion coefficients which is solved numerically. Numerical results are presented for the elastodynamic stress intensity factors, the scattered far-field and the scattering cross section of a single crack. Then, propagation of plane time-harmonic SH-waves in a transversely isotropicmaterial permeated by a random and dilute distribution of micro-cracks is investigated. The effects of the micro-crack density on the attenuation coefficient and the phase velocity are analyzed by appealing to a simple energy consideration and by using Kramers-Kronig relations.     

Analysis of intrinsic stability criteria for isotropic third-order Green elastic and compressible neo-Hookean solids

Internal stability of isotropic nonlinear elastic materials under homogeneous deformation is studied. Results provide new insight into various intrinsic stability measures, first proposed elsewhere, for generic nonlinear elastic solids. Three intrinsic stability criteria involving three different tangent elastic stiffness matrices are considered, corresponding to respective increments in strain measures conjugate to thermodynamic tension, first Piola–Kirchhoff stress, and Cauchy stress. Primary deformation paths of interest include spherical (i.e., isotropic) deformation, uniaxial strain, and simple shear; unstable modes are not constrained to remain along primary deformation paths. Effects of choices of second- and third-order elastic constants on intrinsic stability are systematically studied for physically realistic ranges of constants. For most cases investigated here, internal stability according to strain increments conjugate to Cauchy stress is found to be the most stringent criterion. When third-order constants vanish, internal stability under large compression tends to decrease as Poisson’s ratio increases. When third-order constants are nonzero, a negative (positive) pressure derivative of the shear modulus often promotes unstable modes in compression (tension). For large shear deformation, larger magnitudes of third-order constants tend to result in more unstable behavior, regardless of the sign of the pressure derivative of the shear modulus. A compressible neo-Hookean model is generally much more intrinsically stable than second- and third-order elastic models when Poisson’s ratio is non-negative.