Khanh Chau Le

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.

Nonlinear continuum theory of dislocations

Within the framework of mechanics of generalized continua, a principle of virtual work and a set of static equations are formulated for a body with continuously distributed dislocations at finite strain. The stored energy is supposed to depend on the elastic deformation as well as on the dislocation density. The internal dynamics of dislocations are shown to be described by the model of oriented media. Governing and constitutive equations for such material are proposed with respect to the initial and current description, using absolute tensor notation. Special attention is focused to the link with the engineering plasticity.

Plastic Deformation of Bicrystals Within Continuum Dislocation Theory

Within continuum dislocation theory the plastic deformation of bicrystals under plane strain constrained shear is considered. An analytical solution is found in the symmetric case (for twins) which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. Similar features hold true also for the numerical solution in the general case.

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.

Thermodynamic dislocation theory of high-temperature deformation in aluminum and steel

The statistical-thermodynamic dislocation theory developed in previous papers is used here in an analysis of high-temperature deformation of aluminum and steel. Using physics-based parameters that we expect theoretically to be independent of strain rate and temperature, we are able to fit experimental stress-strain curves for three different strain rates and three different temperatures for each of these two materials. Our theoretical curves include yielding transitions at zero strain in agreement with experiment. We find that thermal softening effects are important even at the lowest temperatures and smallest strain rates.

Estimation of crack density due to fragmentation of brittle ellipsoidal inhomogeneities embedded in a ductile matrix

SummaryAn estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ellipsoidal shape embedded in a ductile matrix under remote static loading. In the state of completed fragmentation the inhomogeneity is replaced by a void with zero stiffness. Thus, the problem of estimating the energy release reduces to the eigenstrain problem solved by Eshelby. The energy release calculated for prolate spheroidal inhomogeneities is used in the balance of energy to determine the crack density. The application to the geological system of garnet inhomogeneities embedded in a quartz matrix is considered.

Variational problems of crack equilibrium and crack propagation

Let us start with the well-known principle of minimum energy in linear elastostatics. For simplicity we restrict ourselves to the 2-D plane strain problems by considering a homogeneous elastic body of cylindrical shape, whose generator is directed along the x 3-axis. Let the cross section of the body occupy a 2-D regular open region B of the (x 1, x 2)-plane, and assume that all quantities we are looking for depend only on x 1 and x 2. The boundary of B, ∂B, is decomposed into two curves ∂ ω and ∂ τ such that

Autonomous Single Oscillator

\partial B = {\partial _\omega } \cup {\partial _\tau },{\partial _\omega } \cap {\partial _\tau } = \emptyset