The physical basis of computationally tractable models of crystalline plasticity is reviewed. A statistical mechanical model of dislocation motion through forest dislocations is formulated. Following Franciosi and co-workers (1980-88) the strength of the short-range obstacles introduced by the forest dislocations is allowed to depend on the mode of interaction. The kinetic equations governing dislocation motion are solved in closed form for monotonic loading, with transients in the density of forest dislocations accounted for. This solution, coupled with suitable equations of evolution for the dislocation densities, provides a complete description of the hardening of crystals under monotonic loading. Detailed comparisons with experiment demonstrate the predictive capabilities of the theory. An adaptive finite element formulation for the analysis of ductile single crystals is also developed. Calculations of the near-tip fields in Cu single crystals illustrate the versatility of the method.

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Computational modelling of single crystals

A recently proposed model for a continuum with microstructure is further substantiated by identifying the microstructure with dislocations. In particular, the continuum is viewed as a superimposed state composed of a perfect lattice state, an immobile dislocation state, and a mobile dislocation state. It is assumed that each state evolves continuously in space-time and transitions from one state to another take place spontaneously according to the balance laws of effective mass and momentum. When the constitutive equations are subjected to the requirements of invariance, familiar statements from dislocation dynamics are deduced. When plastic strain and yield are identified in terms of the parameters characterizing the dislocation states, familiar flow rules and yield surfaces are produced. The capability of the model to predict not only Tresca and Von-Mises plastic behavior but also phenomena such as kinematic hardening, different responses in tension and compression, latent hardening, and the Bauschinger effect, is shown. Finally, the appropriateness of our equations to model creep, cyclic plasticity, and fatigue, is illustrated.

On a proposal for a continuum with microstructure

The contribution of plastic strain evolution of mobile and forest dislocation densities to the thermal and athermal components of polycrystalline metals flow stress is investigated in this work. The thermomechanical response is characterized here for body centered cubic (bcc), face centered cubic (fcc) and hexagonal close-packed (hcp) structures of metals at low and high strain rates and temperatures. Consequently, the simulation of the plastic flow stress for these metals is developed based on the concept of thermal activation energy, the additive decomposition of the flow stress, dislocations interaction mechanisms and the role of dislocations dynamic in crystals. The material parameters of the proposed modeling are physically defined and related to the nano- and micro-structure quantities. On the other hand, the hardening parameters of each kind of metal structures are presented in two different forms; physically based definition which is developed, based on the aforementioned concepts and empirical relation which is used by several authors and is based on experimental observations. Several experimental data obtained by different authors for Niobium, Tantalum, Vanadium, Oxygen Free High Conductivity (OFHC) Copper, and Titanium are used in evaluating the proposed models. Good correlation is observed between the proposed models predictions and the experimental observations. Moreover, the predicted results show that the effect of mobile and forest dislocation densities evolution with plastic strain on the thermal stress of bcc metals is almost negligible and pertained totally to the athermal stress part, whereas the plastic strain evolution of these dislocation densities play crucial roles in determining the plastic thermal flow stress of most fcc metals. The thermal and athermal flow stresses for hcp metals, however, show a behavior that is a combination of that for both bcc and fcc plastic deformation models.

http://am.ippt.pan.pl/am/article/viewFile/v57p299/pdf

Effect of dislocation density evolution on the thermomechanical response of metals with different crystal structures at low and high strain rates and temperatures

The three-dimensional near-tip fields in copper single crystals loaded in four-point bending are characterized numerically. For comparison purposes, the corresponding plane-strain fields are also computed numerically and their asymptotic behavior determined semi-analytically. On the basis of these analyses, we investigate: (i) the dependence of the fields on the hardening law; (ii) the degree of correlation between surface and interior fields in finite specimens; and (iii) the degree of correlation between plane-strain and three-dimensional fields. In particular, we endeavor to ascertain the extent to which surface observations of near-tip fields in single crystals, such as those obtained by Moire interferometry, are representative of interior fields, and the extent to which these are representative of plane-strain fields. Our calculations reveal marked differences in the pattern of slip activity in the interior and on the surface of the specimen. These discrepancies, in turn, result in significant variations in the state of stress and strain. These observations suggest that, for the test geometries under consideration, surface observations provide only an indirect measure of conditions in the interior, and point to a need for the development of experimental techniques enabling the direct observation of interior fields.

Three-dimensional crack-tip fields in four-point-bending copper single-crystal specimens

The plasticity and viscoplasticity of polycrystalline materials are studied analytically in terms of lattice dislocations, with the principal effects attributed to non-extended obstacles. Non-equilibrium statistical mecha­nics is used to describe the evolution of the dislocation structures during loading and unloading processes. A plausible variation in the probability density function for mobile dislocations for such processes is suggested. The proposed material model is in good qualitative agreement with several observed phenomena that previously could not be quantified on the basis of the dislocation theory. Numerical examples illustrate the effect of the rate of loading, the variations in the recovery effect as it relates to the extent of load reversal, and a means for treating materials that exhibit a yield plateau. In the limit, the proposed model yields results for inviscid plasticity.

A statistical theory of polycrystalline plasticity

A set of constitutive equations for polycrystalline plasticity is derived using arguments based directly on the dislocation processes involved. Distributed glide-plane orientations and Burgers-vector directions facilitate handling of the polycrystalline structure, and they yield equations involving probability distributions for variables that are directly related to measurable dislocation quantities. When the motion of the dislocations is isochoric, the tensorial character of the plastic strain rate is shown to be entirely determined by a second-rank symmetric tensor directly related to ordinary elements of crystallographic glide. This same tensor is also shown to determine the resolved shear stress acting on a dislocation in the direction of its Burgers vector, a quantity critical to the determination of the dislocation speed. Evolutionary equations for the dislocation density and the mobile fraction of dislocations are developed to complete the material description. The resulting theory, which allows for the production and interaction of non-uniform dislocation distributions, can model such phenomena as the development of anisotropy with plastic deformation, and material hardening or softening.

A probabilistic approach to polycrystalline plasticity part I: theory

Using our constitutive theory for polycrystalline plasticity, we have calculated characteristic yield conditions for several sample materials in which the dependence of the dislocation velocity on stress is given by an empirical power function. The shape of the yield surfaces for these materials varies resembling that of von Mises when the stress exponent is small and that of Tresca when the exponent is large. Additional examples illustrate the ability of the proposed theory to model such phenomena as the development of anisotropy during plastic deformation, material hardening and softening, the occurrence of upper and lower yield points, and a Bauschinger effect.

Steven J. Sackett

Papers

A probabilistic approach to polycrystalline plasticity part I: theory

A probabilistic approach to polycrystalline plasticity part II: applications