Lallit Anand

Crystallographic texture evolution in bulk deformation processing of FCC metals

Abstract A Taylor-type polycrystalline model, together with a new fully-implicit time-integration scheme has been developed and implemented in a finite element program to simulate the evolution of crystallographic texture during bulk deformation processing of face centered cubic metals deforming by crystallographic slip. The constitutive equations include a new equation for the evolution of slip system deformation resistance which leads to macroscopic strain hardening behavior that is in good accord with experiments performed on OFHC copper. The good predictive capabilities of the constitutive equations and the time-integration procedure for simulating the stress-strain behavior and the evolution of texture under both homogeneous and non-homogeneous deformation conditions are demonstrated by comparing numerical simulations against experimental measurements in simple shear and a simple plane-strain forging experiment on copper.

An internal variable constitutive model for hot working of metals

Abstract A set of internal variable type constitutive equations which model large elastic-viscoplastic deformations of metals at high temperatures is formulated. For a first-order representation of large deformation constitutive behavior we use a scalar as an internal variable to represent the isotropic resistance to plastic flow offered by the internal state of the material. Hot compression tests on an iron-2% silicon alloy and a commercially pure aluminum have been conducted. Based on these experiments, specific viscoplastic constitutive functions are proposed, and the values for the materials parameters appearing in these functions are determined. The constitutive equations with these functions and material parameters accurately reproduce the basic tests and also accurately predict the response of the metal to some critical experiments which were not used to determine the material parameters in the constitutive functions. Although the use of a single scalr internal variable limits the constitutive equations to describing the deformation behavior of initially isotropic metals up to deformation levels where significant internal damage and polycrystal texturing has not developed, the constitutive equations presented here should be useful for obtaining an improved analysis of various hot deformation processing operations of metals.

Finite deformation constitutive equations and a time integrated procedure for isotropic hyperelastic—viscoplastic solids

Abstract Constitute equations for finite deformation, isotropic, elastic-viscoplastic solids are formulated. The concept of a multiplicative decomposition of the deformation gradient into an elastic and a plastic part is used. The constitutive equation for stress is a hyperelastic relation in terms of the logarithmic elastic strain. Since the material is assumed to be isotropic in every local configuration determined by the plastic part of deformation gradient, the internal variables are necessarily scalars. We use a single scalar as an internal variable to represent the isotropic resistance to plastic flow offered by the internal state of the material. The constitutive equation for stress is often expressed in a rate form, and for metals it is common to approximate this rate equation, under the assumption of infinitesimal elastic strains, to arrive at a hypoelastic equation for the stress. Here, we do not express the stress constitutive equation in a rate form, nor do we make this approximative assumption. For the total form of the stress equation we present a new implicit procedure for updating the stress and other relevant variables. Also, the principle of virtual work is linearized to obtain a consistent, closed-from elasto-viscoplastic tangent operator (the ‘Jacobian’) for use in solving for global balance of linear momentum in implicit, two-point, deformation driven finite element algorithms. The time integration algorithm is implemented in the finite element program ABAQUS. To check the accuracy and stability of the algorithm, some representative problems involving large, pure elastic and combined elastic-plastic deformations are solved.

Constitutive equations for hot-working of metals

Abstract Elevated temperature deformation processing - “hot-working,” is an important step during the manufacturing of most metal products. Central to any successful analysis of a hot-working process is the use of appropriate rate and temperature-dependent constitutive equations for large, interrupted inelastic deformations, which can faithfully account for strain-hardening, the restoration processes of recovery and recrystallization and strain rate and temperature history effects. In this paper we develop a set of phenomenological, internal variable type constitutive equations describing the elevated temperature deformation of metals. We use a scalar and a symmetric, traceless, second-order tensor as internal variables which, in an average sense, represent an isotropic and an anisotropic resistance to plastic flow offered by the internal state of the material. In this theory, we consider small elastic stretches but large plastic deformations (within the limits of texturing) of isotropic materials. Special cases (within the constitutive framework developed here) which should be suitable for analyzing hot-working processes are indicated.

A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy AZ31B

Abstract A crystal-mechanics-based constitutive model, which accounts for both slip and twinning, has been developed for polycrystalline hcp materials. The model has been implemented in a finite-element program. The constitutive model is evaluated for the room-temperature deformation of the magnesium alloy AZ31B. By using comparisons between model predictions and macroscopically-measured stress-strain curves and texture evolution, we have deduced information about the dominant slip and twinning systems active at room temperature, and the values of the single-crystal parameters associated with slip and twin system deformation resistances. Our calculations show that the two main crystallographic mechanisms: (i) slip on basal (0001) 〈11 2 0〉, prismatic {10 1 0} 〈11 2 0〉, and pyramidal {10 1 1} 〈11 2 0〉 systems, and (ii) twinning on pyramidal {10 1 2} 〈 1 011〉 systems, play the dominant role in the deformation of magnesium at room temperature. However, to match the observed stress-strain curves, it is found necessary to account for non-crystallographic grain boundary related effects. We approximately account for these grain-boundary region accommodation effects by adding a suitably-weighted isotropic term to the flow rule. The isotropic plasticity term serves the important function of bounding the stress levels in the numerical calculations; it does not contribute to the crystallographic texture evolution. Overall, we show that a simple non-hardening crystal-mechanics-based constitutive model is able to reproduce the experimentally-measured stress–strain curves and crystallographic texture evolution in simple tension and compression on specimens made from an initially-textured rod, as well as plane strain compression experiments on specimens made from an initially-textured plate.

Polycrystalline plasticity and the evolution of crystallographic texture in FCC metals

A Taylor-type model for large deformation polycrystalline plasticity is formulated and evaluated by comparing the predictions for the evolution of crystallographic texture and the stress-strain response in simple compression and tension, plane strain compression, and simple shear of initially ‘isotropic’ OFHC copper against (a) corresponding experiments, and (b) finite element simulations of these experiments using a multitude of single crystals with accounting for the satisfaction of both compatibility and equilibrium. Our experiments and calculations show that the Taylor-type model is in reasonable first-order agreement with the experiments for the evolution of texture and the overall stress-strain response of single-phase copper. The results of the finite element calculations are in much better agreement with experiments, but at a substantially higher computational expense.

A computational procedure for rate-independent crystal plasticity

In the rate-independent theory of crystal elasto-plasticity there have been three long-standing problems. The first is to determine which slip systems are active, and the second is to determine the increments of shear on the active slip systems. Third, because of the typical multiplicity of slip systems in ductile crystals, the selection of slip systems required to produce an arbitrary deformation increment is not necessarily unique. The purpose of this paper is to present a robust calculation scheme which determines a unique set of active slip systems and the corresponding shear increments in a rate-independent theory. We show by comparing the predictions from our computational procedure for the rate-independent theory against corresponding predictions from a procedure for a similar but rate-dependent theory (with a low value of the rate sensitivity parameter) that the results from the two procedures are essentially indistinguishable.

Polycrystalline shape-memory materials: effect of crystallographic texture

Abstract A crystal-mechanics-based constitutive model for polycrystalline shape-memory materials has been developed. The model has been implemented in a finite-element program. In our finite-element model of a polycrystal, each element represents one crystal, and a set of crystal orientations which approximate the initial crystallographic texture of the shape-memory alloy are assigned to the elements. The macroscopic stress–strain responses are calculated as volume averages over the entire aggregate. Pseudoelasticity experiments in tension, compression, and shear have been performed on an initially textured polycrystalline Ti–Ni alloy. In order to determine the material parameters for Ti–Ni, the stress–strain results from a finite-element calculation of a polycrystalline aggregate subjected to simple tension have been fit to corresponding results obtained from the physical experiment. Using the material parameters so determined, the predicted pseudoelastic stress–strain curves for simple compression and thin-walled tubular torsion of the initially textured Ti–Ni are shown to be in good accord with the corresponding experiments. Our calculations also show that the crystallographic texture is the main cause for the observed tension–compression asymmetry in the pseudoelastic response of Ti–Ni. The predictive capability of the model for the variation of the pseudoelastic behavior with temperature is shown by comparing the calculated stress–strain response from the model against results from experiments of Shaw and Kyriakides (J. Mech. Phys. Solids 43 (1995) 1243) on Ti–Ni wires at a few different temperatures. By performing numerical experiments, we show that our model is able to qualitatively capture the shape-memory effect by transformation. We have also evaluated the applicability of a simple Taylor-type model for shape-memory materials. Our calculations show that the Taylor model predicts the macroscopic pseudoelastic stress–strain curves in simple tension, simple compression and tubular torsion fairly well. Therefore, it may be used as a relatively inexpensive computational tool for the design of components made from shape-memory materials.

A theory of amorphous solids undergoing large deformations, with application to polymeric glasses

Abstract This paper develops a continuum theory for the elastic–viscoplastic deformation of amorphous solids such as polymeric and metallic glasses. Introducing an internal-state variable that represents the local free-volume associated with certain metastable states, we are able to capture the highly non-linear stress–strain behavior that precedes the yield-peak and gives rise to post-yield strain softening. Our theory explicitly accounts for the dependence of the Helmholtz free energy on the plastic deformation in a thermodynamically consistent manner. This dependence leads directly to a backstress in the underlying flow rule, and allows us to model the rapid strain-hardening response after the initial yield-drop in monotonic deformations, as well as the Bauschinger-type reverse-yielding phenomena typically observed in amorphous polymeric solids upon unloading after large plastic deformations. We have implemented a special set of constitutive equations resulting from the general theory in a finite-element computer program. Using this finite-element program, we apply the specialized equations to model the large-deformation response of the amorphous polymeric solid polycarbonate, at ambient temperature and pressure. We show numerical results to some representative problems, and compare them against corresponding results from physical experiments.

Initiation of localized shear bands in plane strain

Abstract This study is concerned with the initiation of localized shear bands in plane-strain tension and compression. A theoretical framework which views the initiation of such shear bands as a bifurcation phenomenon from a homogeneous equilibrium field in an elastic-plastic body is first briefly reviewed, and then the predictions of the theory are compared with some experimental observations on an aged maraging steel. The experiments support the physical relevance of the theory within the framework of continuum mechanics. Specifically, the comparison between theory and experiment is concerned with the critical strains to localization and the orientation of the shear bands relative to the load axis. The theoretical predictions are only in qualitative agreement with the experimental observations. A better agreement is obtained with use of the constitutive equations corresponding to the classical deformation theory (a simple vertex model) than with use of the constitutive equations of the classical flow theory. It is concluded that better constitutive equations for elastic-plastic materials are needed before theoretical predictions can be obtained which might be expected to be in closer quantitative accord with experiment.