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.     

Small deformations superposed on large deformations of an elastic-plastic material

In the context of a purely mechanical rate-type theory of plasticity, and utilizing a strain space formulation, an infinitesimal theory is developed for motions superposed on any given motion of an elastic-plastic material. The given motion may involve all forms of elastic-plastic deformation, including both loading and unloading, and in addition, the loading conditions are allowed to change due to the superposed motion. The infinitesimal theory is properly invariant under arbitrary finite superposed rigid motions.     

A model for elastic-plastic pressure sensitive materials subjected to large deformations

Development of a finite deformation elasto-plastic model for pressure sensitive materials is presented. The chosen model, which has its roots in the MRS-Lade material model is influenced by recent developments. The thermodynamic consequences of introducing non-associative yielding (both deviatoric and volumetric) and hardening⧸softening characteristics are assessed. The consistently linearized Algorithmic Tangent Stiffness (ATS) tensor is presented. This tensor is used in the constitutive driver as a key feature of the efficient iterative procedure for satisfying equilibrium in the case of stress (or mixed) control.The chosen model is calibrated using data from experiments conducted in a Directional Shear Cell (DSC) , which has been used extensively at the University of Colorado at Boulderto investigate the behavior of pressure sensitive materials under deformations of large magnitude.     

On equilibrium stability of an elastic-viscoplastic medium

We consider the stability of an elastic-plastic medium when one part of the body is in an elastic state and the other part in a plastic state.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 86–92, September–October, 1970.     

A nonstandard approach to the micromodelling of large deformations in incompressible elastic-plastic composites

Summary The aim of this paper is to present a new homogenized model of periodic composites made of incompressible elastic-plastic Prandtl-Reuss materials with isotropioc work hardening. The nonstandard method of modelling is applied. The considerations are carried out within the large deformation theory. As an example the steady simple extension of a laminate is calculated.

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Eine nicht-klassische Methode der Mikromodellierung elastisch-plastischer Verbundwerkstoffe bei großen Formänderungen

Übersicht Beschrieben wird ein neues Modell der Homogenisierung von periodischen Verbundwerkstoffen aus inkompressiblem elastisch-plastischem Prandtl-Reuss-Material mit isotroper Verfestigung, wobei eine nicht-klassische Methode der Modellbildung verwendet wird. Die Betrachtungen erfolgen im Rahmen der Theorie großer Formänderungen. Als Beispiel wird eine einfache Streckung eines Schichtwerkstoffs berechnet.

     

On the propagation of elastic-viscoplastic waves in damage-softened polycrystalline materials

Theories for uniaxial wave propagation as, for example, along the longitudinal axis of slender rods composed of materials that behave elastically or plastically with hardening, encounter difficulty when confronted with softening material. For such theories, onset of softening causes the value of the wave speed to become complex thereby transforming the governing partial differential equations from hyperbolic to elliptic, implying no further possibility for wave-like motion in the softened material.The purpose of this paper is to show how an elastic-viscoplastic-damage type of constitutive theory together with the equation of motion produce a system of governing partial differential equations that can be shown to be hyperbolic. As an outgrowth of the calculation for the characteristics of the system, an expression relating the elastic dilatational wave speed with material damage and softening can be derived, demonstrating positive value for all phases of the material deformation including material softening that terminates in fracture. The paper also shows how experimental data from plate impact spall fracture tests can illustrate the reality of wave motion through damage-softened polycrystalline material.     

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.     

On shakedown theory for elastic-plastic materials and extensions

The idea that an elastic-plastic structure under given loading history may shake down to some purely elastic state (and hence to a safe state) after a finite amount of initial plastic deformation, can apply to many sophisticated material models with possible allowable changes of additional material characteristics, as has been done in the literature. Despite some claims to the contrary, it is shown; however, that the shakedown theorems in a Melan-Koiter path-independent sense have been extended successfully only for certain elastic-plastic hardening materials of practical significance. Shakedown of kinematic hardening material is determined by the ultimate and initial yield stresses, not the generally plastic deformation history-dependent hardening curve between. The initial yield stress is no longer the convenient one (corresponding to the plastic deformation at the level of 0.2%) as in usual elastic-plastic analysis but to be related to the shakedown safety requirement of the structure and should be as small as the fatigue limit for arbitrary high-cycle loading. Though the ultimate yield strength is well defined in the standard monotonic loading experiment, it also should be reduced to the so-called “high-cycle ratchetting” stress for the path-independent shakedown analysis. A reduced simple form of the shakedown kinematic theorem without time integrals is conjectured for general practical uses. Application of the theorem is illustrated by examples for a hollow cylinder, sphere, and a clamped disk, under variable (including quasiperiodic dynamic) pressure.     

On an axially symmetric elastic-plastic torsion problem

I.IntroductionTheelastic-plastictorsionofashaftisaclassicalmechanicsproblem.Inthemiddleofthel96os,theinventionofvariationalinequalitiesmadetheresearchofthisdirectionquiteactiveanddeep.Theresearchaboutbarswithconstantcross'sectionsisrelativelycompleteandin…     

On kinematic hardening and large plastic deformations

The model of physically linear kinematic hardening is investigated in view of its convenient generalization to large strains. This generalization implies the definition of differential operators in the form of objective tensor derivatives. It has been shown in the literature that an inconvenient choice of the time derivatives leads to a non-monotonic stress strain behavior for the case of simple shear. In this paper, it is shown that this is also true if an additional “damping term” is introduced into the evolution equation. Thus, the non-monotonic behavior should be avoided by a proper choice of variables and time derivatives.     

On buckling of axisymmetric thin elastic-plastic shells

A buckling criterion for shells with an axisymmetric middle surface and subjected to edge loads and hydrostatic surface pressure loading is formulated starting from Hill's three-dimensional continuum theory for uniqueness of deformation of inelastic solids. It turns out that a physically consistent two-dimensional set of equations may be derived for a quite general class of strain-hardening elastic-plastic solids, the only essential restriction being that of a smooth yield function. The intrinsic errors inherent in the derived rate equations, being an integral part of an eigenvalue problem, are discussed in relation to a circular cylinder under axial compression. Analytical results are given which illustrate the influence of the constitutive properties and the boundary contraints on the magnitude of the critical load.     

On an elastic-plastic transition zone in cellular metals

Summary A theory of plasticity is proposed for cellular metals to describe their elastic-plastic transition zone at small strain. Under certain conditions, only a plane strain test is necessary to determine the yield surface. The method to derive the elastic–plastic behaviour [14, 15] was originally proposed for classical metals. A simple cubic model of a cellular metal is used to demonstrate the method by the finite element method. Recommendations for the numerical simulation are given. The influence of the relative density and the hardening behaviour of the cell wall material is investigated.     

On deformations of cracked anisotropic slabs

The effect of a crack on the bending and shearing of an anisotropic elastic slab is considered. The problems considered are reduced to simultaneous Fredholm equations which may be solved numerically.     

On mean rotations in finite deformations

Summary Firstly, the classical interpretation of the mean rotation tensor of Cauchy and Novozhilov is revised. Indeed, two distinctive and defective features of this interpretation reflect a severe restriction on the class of admissible deformations. Secondly, an alternative measure of mean rotation is introduced and its explicit aspect for pure rotations, pure strains and additively pure rotations is determined.

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Sommario In questa nota viene riesaminata l'interpretazione classica del tensore di rotazione media di Cauchy e Novozhilov. Preliminarmente, si mostra come due tratti distintivi di questa interpretazione ne limitino l'ampiezza e riflettano una severa restrizione della classe delle deformazioni ammissibili. Successivamente, si perviene ad una misura alternativa di rotazione media, il cui calcolo esplicito è condotto nei casi di rotazioni pure, deformazioni pure e rotazioni additivamente pure.

     

On equations of continuity of deformations

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