H. Stumpf

A MODEL OF ELASTOPLASTIC BODIES WITH CONTINUOUSLY DISTRIBUTED DISLOCATIONS

The kinematics of elastoplastic bodies at finite strain based on the multiplicative decomposition of the deformation gradient is developed taking into account the motion of continuously distributed dislocations. In comparison with the macro-theories of finite elastoplasticity additional degrees of freedom are introduced through Cartans torsion having the physical meaning of the dislocation density. The set of balance equations has to be enlarged to account also for the internal motion of dislocations. Constitutive equations for such bodies are proposed in consistency with the entropy production inequality.

The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity

Abstract The exact formulae for the plastic and the elastic spin referred to the deformed configuration are derived, where the plastic spin is a function of the plastic strain rate and the elastic spin a function of the elastic strain rate. With these exact formulae we determine the macroscopic substructure spin that allows us to define the appropriate corotational rate for finite elastoplasticity. Plastic, elastic and substructure spin are considered and simplified for various sub-classes of restricted elastic-plastic strains. It is shown that for the special cases of rigid-plasticity and hypoelasticity the proposed corotational rate is identical with the Green-Naghdi rate, while the ZarembaJaumann rate yields a good approximation for moderately large strains. To compare our exact plastic spin formula with the constitutive assumption for the plastic spin introduced by Dafalias and others, we simplify our result for small elastic-moderate plastic strains and introduce a simplest evolution law for kinematic hardening leading to the Dafalias formula and to an exact determination of its unknown coefficient. It is also shown that contrary to statements in the literature the plastic spin is not zero for vanishing kinematic hardening. For isotropic-elastic material with induced plastic flow undergoing isotropic and kinematic hardening constitutive and evolution laws are proposed. Elastic and plastic Lagrangean and Eulerian logarithmic strain measures are introduced and their material time derivatives and corotational rates, respectively, are considered. Finally, the elastic-plastic tangent operator is derived. The presented theory is implemented in a solution algorithm and numerically applied to the simple shear problem for finite elastic-finite plastic strains as well as for sub-classes of restricted strains. The results are compared with those of the literature and with those obtained by using other corotational rates.

Theoretical and computational aspects in the shakedown analysis of finite elastoplasticity

Abstract A fully nonlinear shakedown analysis is considered for structures undergoing large elastic-plastic strains. The underlying kinematics of finite elastoplasticity are based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. It is Shown that the notion of a fictitious, self-equilibrated residual stress field of Melans linear shakesdown theorem has to be replaced by the notion of real, self-equilibrated residual state. Path-dependent and path-independent shakedown theorems are presented that can be realized in an incremental step-by-step procedure using Finite Element codes. The numerical implementation is considered for highly nonlinear truss structures undergoing large cyclic deformations with ideal-plastic, isotropic and kinematic hardening material behavior. Path-dependency of the residual states in the case of non-adaptation and path-independency in the case of shakedown are shown, and the shakedown domain is determined taking into account also the stability boundaries of the structure.

Rate theory of nonlocal gradient damage-gradient viscoinelasticity

Abstract A general concept for the analysis of damage evolution in heterogeneous media is proposed. Since macroscopic failure is governed by physical mechanisms on two different length-scale levels, the macro- and mesolevel, we introduce a 6-dimensional kinematical model with manifold structure accounting for discontinuous fields of microcracks, microvoids and microshear bands. As point of departure a variational functional is introduced with a Lagrangian density depending on macro- and microdeformation gradients and of a damage variable representing scalar-, vector- and/or tensor-type quantities. To derive the equations of motion for viscoinelastic damage evolution on macro- and mesolevel, we introduce into the Lagrangian the macro- and microdeformation gradients, damage variable and also their gradients and time rates. The equations of motion on macro- and mesolevel are derived for non-equilibrium states. We assume that the Lagrangian can be split into two contributions, a time-independent and a time-dependent one which can be identified with the Helmholtz free energy and a dissipation potential. This split of the Lagrangian can be used to decompose the stresses and forces into reversible and irreversible ones. The latter can be considered as dissipative driving stresses and driving forces, respectively, on defects. The model presented in this paper can be considered as a framework, which enables to derive various nonlocal and gradient, respectively, damage theories by introducing simplifying assumptions. As special cases a scalar damage and a solid-void model are considered.

Buckling equations for elastic shells with rotational degrees of freedom undergoing finite strain deformation

Abstract A rigorous theory of small deformation superimposed on finite deformation is developed within a fully general theory of elastic shells. The mathematical structure of the configuration space and its associated tangent space is examined for the underlying shell model. Essential features of the theory are examined in the context of applications to the buckling analysis of specific problems.

Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-brittle materials

Abstract The aim of this paper is to develop a thermodynamically consistent micromechanical concept for the damage analysis of viscoelastic and quasi-brittle materials. As kinematical damage variables a set of scalar-, vector-, and tensor-valued functions is chosen to describe isotropic and anisotropic damage. Since the process of material degradation is governed by physical mechanisms on levels with different length scale, the macro- and mesolevel, where on the mesolevel microdefects evolve due to microforces, we formulate in this paper the dynamical balance laws for macro- and microforces and the first and second law of thermodynamics for macro- and mesolevel. Assuming a general form of the constitutive equations for thermo-viscoelastic and quasi-brittle materials, it is shown that according to the restrictions imposed by the Clausius–Duhem inequality macro- and microforces consist of two parts, a non-dissipative and a dissipative part, where on the mesolevel the latter can be regarded as driving forces on moving microdefects. It is shown that the non-dissipative forces can be derived from a free energy potential and the dissipative forces from a dissipation pseudo-potential, if its existence can be assured. The micromechanical damage theory presented in this paper can be considered as a framework which enables the formulation of various weakly nonlocal and gradient, respectively, damage models. This is outlined in detail for isotropic and anisotropic damage.

STRAIN MEASURES, INTEGRABILITY CONDITION AND FRAME INDIFFERENCE IN THE THEORY OF ORIENTED MEDIA*

This paper is concerned with the appropriate choice of state variables within the continuum model or oriented media. It is shown that residual deformation, strain and wryness can be considered as such quantities. The compatibility conditions for them are derived, which make the inverse problem of determining the displacement and director triad fields well-posed. The principle of frame indifference justifies the use of these quantities as state variables in the free energy density. Governing and constitutive equations are studied in detail. A comparison with the continuum model of crystals with continuously distributed dislocations is provided.

On a general concept for the analysis of crack growth and material damage

Abstract For finite strain dynamics a variational model of crack evolution is formulated within the generalized oriented continuum methodology. In this approach position- and direction-dependent deformation and strain measures are used to describe the (macro)motion of the body with defects, which may evolve relative to the moving body. The inelastic behaviour of continua with evolving defects is represented by phenomenological equations including the transversal crack evolution. A strain-induced crack propagation criterion is defined by the difference between the strain energy release rate and the rate of the surface energy of the crack. A possible nucleation of microcracks in terms of the average drag coefficient of the crack configuration is proposed. Based on the crack growth criterion presented in this paper, the kinking of cracks is investigated using variational concepts. A constitutive damage model of Kachanovs type accounting for the crack density is derived in terms of the free energy functional and a dissipation potential.

Dissipative driving force in ductile crystals and the strain localization phenomenon

Abstract This paper is concerned with crystals undergoing large plastic deformations. The free energy per unit volume of the reference crystal is supposed to depend on the elastic distortion as well as on constant tensors characterizing the crystal symmetry. The dissipative driving force is shown to be equal to the Eshelby stress tensor relative to the reference crystal. For single crystals obeying Taylor’s equation the driving force reduces to the Eshelby resolved shear stress, which is power-conjungate to the slip rate. The Schmid law formulated with respect to the latter is used to determine the critical hardening rate at the onset of the shear band formation.

A continuum model accounting for defect and mass densities in solids with inelastic material behaviour

In this paper the analysis of structures with inelastic material behaviour is considered taking into account the evolution of defects and changes in mass density. The underlying kinematical concept of an oriented continuum is general enough to describe the micro- and macrobehaviour of material bodies appropriately. Based on the logical and consistent variational arguments for a Lagrangian functional the dynamic balance laws, boundary and transversality conditions, all related to the evolution of defect density and mass changes, are derived for macro- and microstresses of deformational as well as of configurational type. The adopted procedure, which formally leaves the balance laws unaltered, leads to the additional balance law for changes in defect density and additional boundary conditions for the changes in mass and defect densities. Driving forces or affinities, associated with the evolution of defect and mass densities, and a generalization of the J-integral representing the thermodynamic forces on defects are obtained. A nonlocal constitutive model accounting for changes in the defect density is presented.