Propagation of weak waves in elastic-plastic and elastic-viscoplastic solids with interfaces

A numerical technique based on the method of singular surfaces has been developed for the computation of wave propagation in solids exhibiting rate-independent elastic-plastic or rate-dependent elastic-viscoplastic behavior. The von Mises yield condition and associated flow rule is taken to represent the rate-independent behavior, while the Perzyna dynamic overstress model is taken to represent the rate-dependent behavior. For 1100-0 Al, a good empirical fit with published experimental data was found to be: $J_2{}^{1/2}?\kappa \left(W^p\right)=\left({\tau }_0/{\gamma }_0\right)\left(\stackrel{0}{W^p}/J_2{}^{1/2}\right)$ where:J₂ is the second invariant of the stress deviator;_k(W^p) is the static hardening curve;W^p is the plastic work and the parameter (τ₀/γ₀) = 0 (rate-independent model) or (80)^?1 to (70)^?1 MPa · s. In the numerical technique, the “connection equations” which provide relations between discontinuities in space and time derivatives lend themselves naturally to finite difference representations. A five-point space-time grid (center point coincident with the instantaneous location of the singular surface) is sufficient for the differenced form of the connection equations and suggests a natural marching scheme for the calculation of all necessary variables at each time step. Supplementing these equations which hold in the interior of the specimen are interface equations which assure continuity in stress and velocity across boundaries which separate materials with dissimilar properties. Application of the technique is made to wave propagation in pure shear for the purpose of comparing numerical predictions with relevant experimental data. The measurements of Duffyet al.[10] which are obtained from the torsional Kolsky apparatus (one dimensional torsional shear wave propagation in a thin-walled tube) were compared with predictions obtained numerically. By using the experimental input pulse history and the constitutive equation reported above, excellent agreement between the predicted and observed histories of reflected and transmitted pulses was obtained when the viscoplastic model was used. Poorer agreement was observed when the rate-independent model (τ₀/γ₀=0) was used. It is concluded that the Perzyna model gives good results for the behavior of 1100-0 Al at high rates of strain.     

On shock waves in elastic-plastic solids

The behavior of shock waves in an elastic-plastic material is investigated with systematic reference to the theory of shocks in fluids. The classical hydrodynamic theory and the notions of the Hugoniot curve and of the Hugoniot contour are first briefly reviewed. Then, it is shown that continuous adiabatic compression is not isentropic and that, in general, the Hugoniot curve cannot be obtained by the classical rate independent elastic-plastic behavior. Two methods are proposed in order to overcome this difficulty. The second one, which is physically more satisfactory, requires the introduction of rate effects. It is shown that when the shock structure is composed of a purely elastic jump followed by a continuous profile, the Hugoniot curve can be defined independently of the precise formulation of the law for the rate effects.     

Propagation of weak shock waves

An asymptotic method is developed in order to treat the evolution of weak shock waves. One obtains a geometrical theory according to which weak shock waves propagate along rays and satisfy a transport law.     

Propagation of steady shock waves in non-linear thermoviscoelastic solids

This paper is concerned with the dynamic response of a class of thermoviscoelastic solids. In particular a specific one-dimensional model, consistent with the laws of continuum thermodynamics, is proposed and applied to the problem of the propagation of steady shock waves. The governing equations are written in terms of material response functions which can be determined from shock wave, thermophysical, and bulk response data. The results of the analysis are compared with experimental steady wave studies involving the solid polymer, polymethyl methacrylate.     

Propagation of acceleration waves in incompressible saturated porous solids

Within the framework of the incompressible porous media model, the propagation properties of acceleration waves in liquid-filled porous solids is discussed. The incompressibility of the two constituents in the model forces the amplitudes of the longitudinal waves in the skeleton and in the liquid to satisfy a certain relation. The two propagation speeds are presented by examination for the existence of acceleration waves and only longitudinal and transverse waves are realizable in the incompressible two-phase porous materials.     

Propagation of weak shock waves in a magnetic field

This paper gives a solution of the problem of the propagation of weak shock waves in an inhomogeneous conducting medium in the presence of a magnetic field. The width of the perturbed region is taken to be small compared with the characteristic dimensions of the problem. The magnetic Reynolds number is also assumed small, which allows one to neglect the induced magnetic field. The method of solution employed is similar to that used in [1–3],The author is grateful to B. I. Zaslavskii for useful advice and for discussing the paper.     

应力波在变截面体中的传播特性

对应力波在变截面体中的传播特性进行了理论研究和数值分析。以杆中一维纵波波动理论和谐波分析法为基础，研究截面变化所导致的应力波的波形弥散和波幅变化。推导了与截面变化相关的应力波演化因子，并对由于截面变化所造成的几何弥散等二维效应进行了分析，同时计算了变截面体的几何特征参数和截面变化等因素影响应力波演化规律的特点。     

Propagation and interaction of non-linear elastic plane waves in soft solids

Using the perturbation method of weakly non-linear asymptotics we analyze the propagation and interaction of elastic plane waves in a model of a soft solid proposed by Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44]. We derive the evolution equations for the wave amplitudes and find analytical formulas for all interaction coefficients of quadratically non-linear interacting waves. We show that in spite of the assumption of almost incompressibility used in Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44], the model behaves essentially like that of a compressible isotropic material. Both the structure of the equations and the interaction patterns are similar. The models differ, however, in the elastic constants that characterize them, and hence the values of the coefficients in the evolution equations and the values of the interaction coefficients differ.     

Numerical method for elastic-plastic waves in cracked solids,Part 1: anti-plane shear problem

Summary The paper deals with a numerical method combining a characteristic-based scheme and Zwas' method in order to solve the hyperbolic PDE's of elastic and elastic-plastic anti-plane shear waves in two space dimensions. First, the need of new physically reasonable numerical methods for stress waves in solids is demonstrated by numerical applications to problems with impulsive loading, where defects of some standard methods are shown. Then, the new secon-dorder accurate method is derived. A suitable procedure to model the elastic-plastic behaviour of materials by simple waves is included. The capability of the methods is demonstrated by application to several examples. Additionally, for comparison with numerical results, a similarity solution for the semi-infinite crack undergoing an elastic-plastic shock loading is derived in the appendix.

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Numerisches Verfahren für elastisch-plastische Wellen in gerissenen Festkörpern, Teil 1: Das dynamische Querschubproblem

Übersicht In der Arbeit wird eine numerische Methode vorgestellt, in der das Charakteristiken erfahren mit der Methode von Zwas kombiniert wurde, um die hyperbolischen, partiellen Differentialgleichungen der elastisch-plastischen Wellenausbreitung für das dynamische Querschubproblem zu lösen. Um die Notwendigkeit neuer, physikalisch brauchbarer Methoden für Spannungswellenprobleme zu begründen, werden zunächst an Beispielen typische Schwächen einiger Standardmethoden gezeigt. Danach wird die neue Methode dargestellt. Dabei wird auch ein geeigneter Weg im Spannungsraum angegeben, um elastisch-plastische Wellen durch sog. einfache Wellen zu beschreiben. Die Fähigkeiten der Methode werden an einigen Beispielen überprüft. Ferner wird im Anhang, um mit numerischen Ergebnissen zu vergleichen, eine Ähnlichkeitslösung hergeleitet, die für den halbunendlichen Riß bei stoßartig aufgebrachter Belastung und Beanspruchung bis in den elastisch-plastischen Bereich gilt.

     

Numerical method for elastic-plastic waves in cracked solids,Part 2: plane strain problem

Summary In this paper, we first give some results for an elastic solid with a crack which is loaded dynamically, by using Zwas' finite difference method introduced in gasdynamics. Then, an elastic-plastic loading path in the stress space is proposed to model the plastic yield phenomenon in solids. Based on this stress path and the ideas of Godunov and Zwas for the formulation of finite difference schemes, a finite difference method is developed to treat the elastic-plastic wave motion in solids under plane strain. The method can be applied to the mode I and mode II crack problems in order to determine the shape of plastic zones and their time history. Besides several results for test examples which were calculated for validation, one result for a finite crack affected by a mode I loading is presented which demonstrates a repeated plastic yielding and elastic unloading caused by Rayleigh waves running along the crack edges and interacting with the two crack tips.

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Numerisches Verfahren für elastisch-plastische Wellen in gerissenen Festkörpern, Teil 2: ebener Verzerrungszustand

Übersicht Zunächst werden für einen dynamisch beanspruchten, rißbehafteten Festkörper mit elastischem Materialverhalten Ergebnisse aus Berechnungen mit der von Zwas für gasdynamische Probleme entwickelten Finite-Differenzen-Methode gezeigt. Davon ausgehend wird, zur Modellierung des Phänomens der dynamischen Plastifizierung, ein elastisch-plastischer Belastungspfad im Spannungsraum vorgeschlagen. Auf diesem Belastungspfad aufbauend wird, unter Nutzung der Lösungsideen von Godunov und Zwas, ein Differenzenverfahren zur Beschreibung der elastisch-plastischen Wellenausbreitung in Festkörpern bei ebenem Verzerrungszustand entwickelt. Die Methode kann auf Rißprobleme angewendet werden, um die plastische Zone und deren zeitliche Entwicklung zu bestimmen. Außer einigen Testbeispielen zur Validierung wird ein Resultat für einen nach Modus I belasteten Riß präsentiert, das ein wiederholtes plastisches Fließen und elastisches Entlasten anzeigt, das von Rayleigh-Wellen verursacht wird, die an den Rißufern entlanglaufen und mit den Rißspitzen wechselwirken.

     

Propagation of waves in a piezoelectric layer with a weak horizontal nonhomogeneity

The process of wave propagation along a piezoelectric layer is considered. It is assumed that the properties of the medium vary slowly along the horizontal directions, and the boundaries of the layer are slightly bent. The propagation of the wave along the layer is studied by the ray method. The transport equations, which describe change of the intensity along the rays, are solved.     

Stability conditions for elastic-plastic prandtl-reuss solids

Summary The algebraic condition ensuring the validity of Hadamard's criterion of local stability in a Prandtl-Reuss solid is examined. Four fundamental situations depending on the possible transitions of state are considered.

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Sommario Si esaminano le condizioni algebriche assicuranti la validità del criterio di Hadamard di stabilità locale in un solido di tipo Prandtl-Reuss. Quattro situazioni fondamentali dipendenti dalle possibili permanenze o conversioni di stato sono considerate.

     

Numerical modelling of shocks in solids with elastic-plastic conditions

Results are presented for a range of one- and two-dimensional shock-wave problems in elastic-plastic and hydrodynamic metals. These problems were solved numerically using the Flux-Corrected Transport (FCT) technique which achieves high resolution without non-physical oscillations, especially at shock fronts, and has not been used before in elastic-plastic solids. The two-dimensional problems were solved using both operator- and non-operator-split techniques to highlight the significant differences between these techniques when solving shock-wave problems in elastic-plastic solids. Comparisons of the elastic-plastic solutions with the hydrodynamic solutions are made and illustrate the importance of including elastic-plastic conditions when modelling the behaviour of solids. Also, the errors in these solutions that are due to the initial conditions are discussed in detail.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Propagation of unsteady weak shock waves in a vibrationally nonequilibrium gas subjected to external radiation

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 26–35, September–October, 1990.     

Dispersive waves in microstructured solids

The wave motion in micromorphic microstructured solids is studied. The mathematical model is based on ideas of Mindlin and governing equations are derived by making use of the Euler–Lagrange formalism. The same result is obtained by means of the internal variables approach. Actually such a model describes internal fields in microstructured solids under external loading and the interaction of these fields results in various physical effects. The emphasis of the paper is on dispersion analysis and wave profiles generated by initial or boundary conditions in a one-dimensional case.     

Propagation of plane waves at the imperfect boundary of elastic and electro-microstretch generalized thermoelastic solids

The problem of reflection and transmission of plane waves incident on the contact surface of an elastic solid and an electro-microstretch generalized thermoelastic solid is discussed. It is found that there exist five reflected waves, i.e., longitudinal displacement （LD） wave, thermal （T） wave, longitudinal microstretch （LM） wave and two coupled transverse displacement and microrotational （CD（I） and CD（II）） waves in the electro-microstretch generalized thermoelastic solid, and two transmitted waves, i.e., longitudinal （P） and transverse （SV） waves in the elastic solid. The amplitude ratios of different reflected and transmitted waves are obtained for an imperfect boundary and deduced for normal force stiffness, transverse force stiffness, and perfect bonding. The variations of amplitude ratios with incidence angles have been depicted graphically for the LD wave and the CD（I） wave. It is noticed that the amplitude ratios of reflected and transmitted waves are affected by the stiffness, electric field, stretch, and thermal properties of the media. Some particular interest cases have been deduced from the present investigations.     

Restrictions on dynamically propagating surfaces of strong discontinuity in elastic-plastic solids

For dynamic three-dimensional deformations of elastic-plastic materials, we elicit conditions necessary for the existence of propagating surfaces of strong discontinuity (across which components of stress, strain or material velocity jump). This is accomplished within a small-displacement-gradient formulation of standard weak continuum-mechanical assumptions of momentum conservation and geometrical compatibility, and skeletal constitutive assumptions which permit very general elastic and plastic anisotropy including yield surface vertices and anisotropic hardening. In addition to deriving very explicit restrictions on propagating strong discontinuities in general deformations, we prove that for anti-plane strain and incompressible plane strain deformations, such strong discontinuities can exist only at elastic wave speeds in generally anisotropic elastic-ideally plastic materials unless a material's yield locus in stress space contains a linear segment. The results derived seem essential for correct and complete construction of solutions to dynamic elastic-plastic boundary-value problems.