Piotr Perzyna

Fundamental Problems in Viscoplasticity

Publisher Summary The fundamental assumption of all theories of plasticity—that of time independence of the equations of state—makes simultaneous description of the plastic and rheologic properties of a material impossible. I t is well-known that in many practical problems, the actual behaviour of a material is governed by plastic as well as by rheologic effects. It can even be said that for many important structural materials, rheologic effects are more pronounced after the plastic state has been reached. Every material shows more or less pronounced viscous properties. In some problems, the influence of viscous properties of the material may be negligible, while in others, it may be essential. Both sciences—plasticity and rheology—are concerned with the description of important mechanical properties of structural materials. Each of them has created its own methods of investigation and has developed within the framework of certain assumptions which, unfortunately, cannot always be satisfied in reality. The results of rheology are confined to cases where plastic strain is of no decisive importance.

Thermodynamic Theory of Viscoplasticity

Publisher Summary This chapter reviews the thermodynamic foundations of the theory of viscoplasticity. The essential feature of viscoplasticity is the simultaneous description of rheologic and plastic effects of a material. The necessity for simultaneous consideration of viscoelastic and plastic properties of a material is indicated by the experimental investigations of dynamic loads. The thermodynamic theory of elastic-viscoplastic materials presents for finite strains two basic difficulties. The first of these is connected with the kinematic description of plastic deformation. The second difficulty concerns the problem of choice of thermodynamic variables of state. The chapter discusses development of the thermodynamic theory of plasticity for finite strains using the rate-type theory, and generalized the inviscid theory of plasticity to nonisothermal finite deformations. The principle of material frame indifference called “invariance requirements under superposed rigid body motions,” was used and explored the thermodynamical restrictions. The chapter presents the formulation of the thermodynamic theory of a rate-sensitive plastic material within the framework of thermodynamics of a material with internal state variables. In this thermodynamic theory of an inelastic material, the deformation tensor and temperature are considered as thermodynamic state variables, while the components of the inelastic deformation tensor appear as internal state parameters.

Internal state variable description of dynamic fracture of ductile solids

Abstract The paper aims at the description of ductile fracture phenomenon in dynamic processes in inelastic solids by means of internal state variable structure. Dynamical test data for aluminum and copper have been discussed. Particular attention is given to postshot photomicrographic observations of the residual porosity and on investigation of fracture mechanisms and spalling phenomenon. Spalling process has been described as a sequence of the nucleation, growth and coalescence of microvoids. Heuristic considerations of the growth and nucleation of microvoids are presented General evolution equation for porosity parameter is postulated. This equation describes the work-hardening viscoplastic response of solid and takes also account of the interactions of microvoids. An elastic-viscoplastic model of a material with internal imperfections is proposed. Internal imperfections are generated by the nucleation, growth and coalescence of microvoids. The model describes the dynamical behavior of dissipative solids observed experimentally as well as the mechanisms of fracture. A dynamical criterion of fracture (spalling) of metals is proposed. The criterion describes the dependence of fracture phenomenon upon the evolution of the constitutive structure. As an application of the theory the dynamic fragmentation process is considered. Prediction of fragment size is based on a very simple evolution equation assumed for the porosity parameter. The procedure of the determination of the interaction material functions has been developed.

The localization of plastic deformation in thermoplastic solids

Abstract The main objective of this paper is the investigation of the influence of thermomechanical couplings and thermal softening effects on adiabatic shear band localization criteria for finite rate-independent deformation of an elastic-plastic body. The constitutive equations for thermoelastic-plastic J 2 -flow theory are formulated within a framework of the rate type covariance structure with internal state variables. Two alternative descriptions are presented. Both constitutive structures formulated are invariant with respect to diffeomorphisms and are materially isomorphic. Particular attention is focused on the coupling phenomena generated by the internal heat resulting from internal dissipation. An identification procedure has been developed which permits the determination of the exact form of the evolution equation for the internal state variable vector. A set of coupled evolution equations for the Kirchhoff stress tensor and for temperature is investigated. The assumption that the thermodynamic process considered is adiabatic permits the elimination of the rate of temperature and gives the fundamental cvolution equation for the Kirchhoff stress tensor. This important result allows the use of the standard bifurcation method in the examination of the adiabatic shear band localization criteria. For the particular clastic properties of the material and for some simplified case of the coupling effects the criteria for adiabatic shear band localization are obtained in exact analytical form. Discussions of the influence of thermomechanical couplings, thermal expansion, thermal plastic softening effects and the covariance terms on the localization criteria are presented.

Instability phenomena and adiabatic shear band localization in thermoplastic flow processes

SummaryThe main objective of the paper is the development of the viscoplastic regularization procedure valid for a broad class of thermodynamic plastic flow processes in damaged solids. The additional aim is to investigate instability phenomena and adiabatic shear band localization criteria when spatial covariance, thermomechanical coupling, strain induced anisostropy and micro-damage softening effects are taken into consideration. This investigation is based on an analysis of acceleration waves and takes advantage of a notion of the instantaneous adiabatic acoustic tensor. In the first part of the paper the formulation of an inelastic flow process is given and particular attention is focussed on the thermomechanical coupling effects. The thermodynamic theory of elastic-viscoplastic damaged solids is presented within a framework of the rate type covariance material structure with a finite set of the internal state variables. A notion of covariance is understood in the sense of invariance under an arbitrary spatial diffeomorphism. Rate sensitivity effect is introduced by the assumption of the viscoplastic overstress conception. A notion of a relaxation time has been used to control the description of mechanical as well as thermal disturbances. By the assumption that the mechanical relaxation time is equal to zero the thermo-elastic-plastic (rate independent) response of the damaged material is accomplished. In the second part of the paper the existence of a solution to the initial-boundary value problem is examined and its stability property is investigated based on the application of nonlinear semi-group methods and an analysis of continuity of evolution operators. For an adiabatic process the investigation of acceleration waves is given. The determination of eigenvalues of the appropriate acoustic tensor is presented. This helps to assess the well-posedness of the initial-value problems which describe the thermodynamic plastic flow processes. Differences for two constitutive assumptions, namely for rate dependent and rate independent responses, are examined. In the case of an adiabatic process and elastic-viscoplastic response of a material the conditions for the existence, uniqueness and well-posedness of the initial value problem have been investigated.Criteria for adiabatic shear band localization of plastic deformation are obtained by assuming that some eigenvalue of the instantaneous adiabatic acoustic tensor for rate independent response is equal to zero. Several particular cases of the constitutive model have been considered.

Localized Fracture in Inelastic Polycrystalline Solids under Dynamic Loading Processes

The main objective of the paper is the investigation of adiabatic shear band localized fracture phenomenon in inelastic solids during dynamic loading processes. This kind of fracture can occur as a result of an adiabatic shear band localization generally attributed to a plastic instability implied by micro-damage and thermal softening during dynamic plastic flow processes. The theory of thermoviscoplasticity is developed within a framework of the rate type covariance material structure with a finite set of internal state variables. The theory takes into consideration the effects of micro-damage mechanism and thermomechanical coupling. The micro-damage mechanism has been treated as a sequence of nucleation, growth, and coalesence of microcracks. The micro-damage kinetics interacts with thermal and load changes to make failure of solids a highly rate, temperature, and history dependent, nonlinear process. The dynamic failure criterion within localized shear band region is proposed. The relaxation time is used as a regularization parameter. By assuming that the relaxation time tends to zero, the rate independent micro-damage mechanism is considered. Rate dependency (viscosity) allows the spatial differential operator in the governing equations to retain its ellipticity, and the initial-value problem is well posed. The viscoplastic regularization procedure assures the stable integration algorithm by using the finite element method. Particular attention is focused on the well-posedness of the evolution problem (the initial-boundary value problem), as well as on its numerical solutions. Convergence, consistency, and stability of the discretised problem are discussed. The Lax equivalence theorem is formulated and conditions under which this theorem is valid are examined. Utilizing the finite element method and ABAQUS system for regularized elastoviscoplastic model, the numerical investigation of the three-dimensional dynamic adiabatic deformation in a particular body at nominal strain rates ranging from 10-1-104 s-1 is presented. Three particular examples have been considered; namely, a dynamic adiabatic process for a thin-walled steel tube and dynamic adiabatic and quasi-static processes for a thin steel plate. In each case, a thin shear band region of finite width which undergoes significant deformations and temperature rise has been determined. Its evolution until occurrence of final fracture has been simulated. Numerical results are compared with available experimental observation data.

The Numerical Analysis of the Intrinsic Anisotropic Microdamage Evolution in Elasto-Viscoplastic Solids

The objective of the present article is to show the formulation for elastic-viscoplastic material model accounting for intrinsic anisotropic microdamage. The strain-induced anisotropy is described by the evolution of the intrinsic microdamage process — defined by the second-order microdamage tensor. The first step of the possibility of identification procedure (calibration of parameters) are also accounted and illustrated by numerical examples.

Interaction of deformation waves and localization phenomena in inelastic solids

Abstract The main objective of this paper is the investigation of the interaction and reflection of elastic–viscoplastic waves which can lead to localization phenomena in solids. The rate type constitutive structure for an elastic–viscoplastic material with thermomechanical coupling is developed. An adiabatic inelastic flow process is considered. The Cauchy problem is investigated and the conditions for well-posedness are examined. Discussion of fundamental features of rate-dependent plastic medium is presented. This medium has dissipative and dispersive properties. Mathematical analysis of the evolution problem (the dynamical initial-boundary value problem) is presented. The dispersion property implies that in the viscoplastic medium any initial disturbance can break up into a system of group of oscillations or wavelets. On the other hand, the dissipation property causes the amplitude of a harmonic wavetrain to decay with time. In the evolution problem considered in such dissipative and dispersive medium, the stress and deformation due to wave reflections and interactions are not uniformly distributed, and this kind of heterogeneity can lead to strain localization in the absence of geometrical or material imperfections. Since the rate-independent plastic response is obtained as the limit case, when the relaxation time T m tends to zero, the theory of viscoplasticity offers the regularization procedure for the numerical solution of the dynamical initial-boundary value problems with localization of plastic deformation. Numerical examples are presented for a steel bar axisymmetric specimen subjected to tension, with the controlled displacements imposed at one or two opposite sides with different velocities. Two cases of the initial-boundary conditions are considered; (A) symmetric (double side) tension of the specimen which results in symmetric pattern of deformations; (B) asymmetric (single side) tension of the specimen with the opposite side fixed, which leads to non-symmetric deformation. For both cases of boundary conditions a set of examples is computed with different initial velocities changing between 0.5 and 20 m/s. The final states are defined by prescribed value of the total elongation of a specimen. In the numerical examples the attention is focused on the investigation of the interactions and reflections of waves and on the location of localization of plastic deformation. The distribution of plastic equivalent strain, temperature and vector plots of velocities represents the results. The computations are performed using the industrial finite element program ABAQUS (explicit method).

Adiabatic Shear Band Localization in Elastic-Plastic Single Crystals

The main objective of the paper is the investigation of shear band localization criteria for finite elastic-plastic deformations of single crystal subjected to adiabatic process. The next objective is to focus attention on temperature dependent plastic behaviour of single crystal considered. A constitutive model is developed within the thermodynamic framework of the rate type covariance constitutive structure, i.e., it is invariant with respect to diffeomorphism.

On combined isotropic and kinematic hardening effects in plastic flow processes

Abstract Attention is focused on the description of the combined isotropic and kinematic hardening effects in plastic solids. It has been found that making use of the simple geometrical relation permits us to determine the coefficient in the evolution equation describing the kinematic hardening of the Ziegler type in such a way that the evolution law is consistent with the loading criterion and satisfies the time independence requirement. On the other hand this method leaves room for the identification procedure for the material constants based on available experimental results. General constitutive and evolution equations for plastic solids are formulated. The plastic potential is assumed different than the yield criterion. Simplifications for the associated flow rule are also investigated. A new evolution law for the anisotropic hardening is discussed. This law represents the linear combination of the Prager and Ziegler kinematic hardening rules. Application of the theory developed to the description of the plastic behavior of damaged solids is given. A particular example is considered. The discussion and the interpretation of the isotropic and anisotropic hardening moduli are presented.