Paul R. Dawson

On modeling the development of crystallographic texture in bulk forming processes

Abstract A mathematical formulation is presented for modeling the evolution of deformation induced crystallographic texture in steady state bulk forming processes. The formulation treats the material response of the polycrystalline aggregate as a statistical function of the response of the individual grains. A viscoplastic relationship is assumed for deformation along the grain slip systems. Strain hardening on these slip systems is included. The viscoplastic stiffness matrix code is presented. A streamline technique has been used to integrate the evolution equation for this aggregate is derived and its implementation in a large deformation Eulerian finite element for the lattice rotation and the slip system hardness. As an application, the evolution of texture in a multipass aluminum rolling simulation has been modeled. The numerical predictions have been compared with reported experimental rolling textures.

A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking

A hybrid finite element formulation for the plastic deformation of FCC metals with anisotropy is outlined. Polycrystal plasticity theory is used to develop the constitutive response. The hybrid approach facilitates introduction of the microscale stress in the macroscopic statement of equilibrium. Convergence of the hybrid formulation is contrasted with that of a velocity-pressure formulation. It is demonstrated that the hybrid formulation is well suited for studies where significant spatial variations in constitutive response result from having only one, or a very few, crystal orientations represented in each finite element. A simulation of channel die compression is made with one crystal per finite element. The resulting texture evolution is compared with other texture evolution models and experimental data for cold rolled aluminum. It is demonstrated that the brass texture component, observed in the experimental data, is developed through shear deformations arising from grain-to-grain interactions.

Application of polycrystal plasticity to sheet forming

Abstract A methodology for including anisotropy in metal forming analyses is presented. A finite element formulation is developed for the analysis of the inhomogeneous macroscopic deformations. Anisotropic material properties are derived from a microscopic description based on polycrystal plasticity theory. Efficient computation of the microscopic variables is achieved through massive data parallel computations. A procedure is set forth for initialization of the microscopic state variables from experimental measurement of the metal texture. The feasibility of initializing (from experimental data) and evolving (through massive computations) a detailed microscopic description for a complex deformation process is demonstrated through a predictive simulation. The predicted location and height of ears in the hydroforming of aluminium sheets are in good agreement with experiment.

Three-dimensional deformation process simulation with explicit use of polycrystal plasticity models

Abstract The combination of massive parallel processing and polycrystal plasticity theory offers the potential for applying detailed microstructural models to macroscopic deformation processes. In this work the finite element method is used to solve for the three-dimensional deformation of a plastic workpiece. The elemental constitutive response is derived from the microstructural response of a polycrystal aggregate situated in the element. Crystal orientations and their respective weighted contributions to the aggregate response are selected to approximate the orientation distribution derived from experimental pole figure measurements. The interaction of the material symmetry adopted in analysis of pole figures and the boundary conditions posed in the plasticity boundary value problem are examined. Through the introduction of distinct aggregates with decreasing crystal to aggregate ratio, an inhomogenous material response is developed where: (1) the orientation distribution becomes well approximated only by a collection of spatially distinct aggregates, and (2) these aggregates experience deformation paths of increasing variation. It is shown that the use of spatially distinct aggregates in a material experiencing local kinematic inhomogeneities throughout its deformation history leads to texture predictions that compare favorably with experimental measurements.

On modelling the elasto-viscoplastic response of metals using polycrystal plasticity

A constitutive framework for the elasto-viscoplastic response of metals that utilizes polycrystal plasticity is presented together with a corresponding numerical integration procedure. The single crystal equations are written in an intermediate configuration obtained by elastically unloading the deformed crystal without rotation from the current configuration to a stress-free state. The elastic strains are assumed always to be small. The accompanying numerical integration is implicit and proceeds by decoupling the volumetric and the deviatoric crystal responses. The extended Taylor hypothesis is used to relate the response of individual crystals to that of the polycrystal. Various homogeneous deformations have been simulated using the constitutive model and the integration scheme to compute the stress response and texture development. Aggregates of either face centered cubic (FCC) or hexagonal close-packed (HCP) crystals are subjected to both monotonic and non-monotonic loading histories. Numerical results demonstrate the performance of the model as well as show the stability and accuracy of the integration procedure. The present constitutive model and corresponding numerical procedures can be used to predict elastic effects (e.g. residual stresses) during the large deformation of polycrystalline materials while accounting for texture development and the associated anisotropy.

Micromechanical modeling of powder compacts—I. Unit problems for sintering and traction induced deformation

Abstract A micromechanical model has been developed for the constitutive behavior of powder compacts based on unit models for the interaction between individual particles. In this paper, the first of two companion papers, two unit problems have been formulated for the study of inter-particle behavior. These correspond to the cases of deformation by external tractions and local surface tension forces. The contact area between individual particles is used to characterize the state of the powder compact. Specific numerical results are presented for linear viscous material properties and are compared with existing models. The choice of unit problems for sintering has been discussed at length and some ambiguities in the traditional method of modeling viscous sintering are pointed out. The results of this study are used in the companion paper [Acta metall.36, 2563 (1988)] to develop a model for the sintering and compaction of discrete packings.

Effects of grain interaction on deformation in polycrystals

The deformations of a face-centered cubic polycrystal are simulated under idealized (plane strain compression) rolling conditions. A polycrystal is constructed using 1099 rhombic dodecahedron shaped crystals, each discretized with 48 tetrahedra elements. The rhombic dodecahedron is a 12-sided, space-filling polyhedron and serves as an idealized crystal geometry. The material behavior is specified at the level of a single crystal with rate dependent slip. The numerical formulation maintains compatibility and equilibrium in a weak sense using hybrid finite element methodology. The influence of the local neighborhood on crystal deformation is examined by conducting a series of numerical experiments on the same set of crystals. Each simulation uses a different spatial mapping of orientations to effectively alter the neighborhood of each crystal, allowing the dependence of deformation on crystal orientation to be examined. Comparisons are made to earlier results obtained with brick shaped crystals and to the results obtained with a second polycrystal consisting of 172 crystals with 576 elements in each rhombic dodecahedron. Coarse crystal discretizations are adequate for modeling bulk anisotropic properties, but a detailed investigation of local neighborhood effects require a finely discretized mesh that is better able to capture gradients in the deformation field.

Polycrystal plasticity modeling of intracrystalline boundary textures

Abstract A finite element formulation is used to simulate and study the deformation response of face-centered cubic polycrystals. The polycrystals consist of rhombic dodecahedral-shaped crystals, each finely discretized with tetrahedral elements. Rhombic dodecahedra are twelve-sided, regular, space-filling polyhedra that are used to represent a microstructure with equiaxed grains. Material behavior is based on rate-dependent, crystallographic slip on a restricted number of slip systems. The numerical formulation maintains compatibility and equilibrium under the application of applied loads using an assumed-stress hybrid finite element methodology. Spatial variations in deformation arise in the polycrystal even under simple external loadings and lead to grain subdivision characterized by the formation of boundaries separating regions with differing lattice orientation. Particular attention is focused on the resulting crystallographic misorientation across these boundaries and their orientations relative to the applied loads. This evolving intragrain boundary texture is compared to published experimental data obtained using TEM and Kikuchi pattern analysis.

Texture predictions using a polycrystal plasticity model incorporating neighbor interactions

A viscoplastic model is presented for distributing the deformation applied to a polycrystal in a non-uniform fashion among the constituent crystals. Interactions with surrounding crystals are incorporated in the calculation of the deformation rate of each crystal through an appropriately defined local neighborhood. A compliance tensor is computed for each crystal based on a viscoplastic constitutive relation for deformation by crystallographic slip. The compliance of the crystal relative to that of its neighborhood provides a means for partitioning the macroscopic deformation rate among the crystals. The deviation of the crystal deformation rate from the macroscopic value is suitably scaled to obtain the crystal spin. Polycrystal simulations of crystallographic texture development using this model are compared to the results of similar calculations using the Taylor model, to finite element simulations of a model polycrystal, and to experimental data. The model incorporating neighbor interactions is shown to result in improved texture predictions, in terms of both the intensity levels and the locations of certain texture components.

Elastoplastic finite element analyses of metal deformations using polycrystal constitutive models

A finite element formulation using an elastoplastic polycrystal constitutive theory is presented. The crystal constitutive equations are expressed in an intermediate configuration obtained by unloading the deformed crystal without rotation from the current configuration to a stress-free state through small elastic strains. These constitutive relations and an associated integration procedure are implemented in a pressure-velocity (mixed) finite element formulation. Both small scale (finite element dimensions comparable to crystal dimensions) and large scale (finite element dimensions much larger than the crystal size) problems are solved to illustrate the formulations application to metal deformations, including the determination of the R-value in a tensile specimen and the prediction of residual stresses in a thick ring deformed in an upsetting operation.