ARO Project : W 911 NF - 04 - 1 - 0283 ( David M . Stepp , ARO Program Manager

The objective of this project is the development of a multi-scale simulation tool for designing thermo-mechanical processes to achieve control of microstructure-sensitive properties in materials used in applications of interest to the U.S. Army. As a part of continuing research effort, developments in the year 2004-05 are presented. An efficient large strain homogenization tool has been developed for evaluating thermo-mechanical response of 2D and 3D polycrystalline microstructures. Using microstructure images as input, the tool can be used by manufacturers to identify processing stages that would fine-tune the properties of microstructures. This decision making process has been automated using newly developed homogenization sensitivity analysis scheme for optimizing microstructuresensitive properties. Since material behavior is determined by its microstructure, a problem of interest is to evaluate the best microstructure for a particular application. Microstructure response is influenced by a combination of stereological and orientational features such as grain boundary mis-orientation distributions and orientation correlation apart from texture. Advanced statistical learning tools are being developed for exploring the feature space and to mine the best set of features that provides a desired property. Other developments in this year include a tool for modeling elasto-viscoplastic response of HCP materials. This complements our existing capabilities to model FCC and BCC metals using continuum ODF representation in Rodriguez space. Impact of these techniques in the overall goals will be discussed in this report. 1.1 Development of a tool for evaluating thermo-mechanical response of polycrystals Design for material performance requirements entails a multi-scale approach where effect of processes on the microstructure is incorporated in continuum analysis. Many engineering materials are polycrystalline in nature and the presence of crystallographic characteristics like texture and misorientations affects several important physical properties. Deformation process design for control of microstructure-sensitive properties involves development of a multi-scale tool where it would be possible to design required process sequence and macroscopic process parameters (die and preform shapes, forging velocities, etc.). Using inputs from the theory of homogenization, we have developed a microstructure material point simulator that can be directly used in multi-scale process simulations. The technique can be used to test real microstructures using meshing capabilities of the NIST software OOF-2 and parallel finite element processing tools. The key goals of this tool is to provide engineers with a userfriendly yet powerful environment for conducting virtual tests with microstructural images with the goal of designing microstructures with desired strength. Multi-scale homogenization techniques integrate microstructure modeling techniques with continuum approaches by providing a set of macro-micro linking assumptions and microstructure property averaging rules. The features of homogenization schemes include the absence of constitutive assumptions on the macro level, large deformations and rotations on the micro and macro level and independence of solution technique used on the micro scale. The technique is also referred to as FE owing to the use of finite element technique at two different scales. Microstructures are virtually loaded and tested in thermo-mechanical loading conditions using multi-scale boundary conditions arising from the theory of homogenization as applicable in the large strain context. Mechanical properties are obtained through associated volume averaging laws. Simulation of texture evolution in poly-crystals has been well studied in the past. Many of the related works apply the Taylor-type micro-macro transitions which assume a purely kinematical constraint such that all grains are subjected to the same deformation. This assumption satisfies compatibility but fails to account for equilibrium across grain boundaries. The effects of stereology and formation of disoriented regions within crystals due to nonuniform deformation are not taken into account. The present work employs a displacement-based Lagrangian FE formulation and utilizes alternative multi-scale transitions using the theory of homogenization based on averaging theorems for linking scales. In these models, Taylor assumption arises naturally as a linking assumption and new assumptions that also satisfy work averaging theorems are identified. Here, boundary conditions are chosen such that the variation of the internal work performed on arbitrary virtual displacements of the microstructure is equal to the work performed by the external loads on the microstructure. (a) (b) Figure. 1 The microstructure homogenization technique: (a) Each integration point is associated with an underlying microstructure. The microstructure reference configuration and the mapping to the present configuration are shown in contrast with the homogenized macro medium (b) Depiction of microstructure meshes employed in the analysis at each integration point of the macro component. The general assumption behind homogenization theory is that the deformation at the micro-scale consists of a homogeneous part seen at the macro-scale and an inhomogeneous part referred to as the fluctuation field due to the presence of the microstructure. The macroscopic stress is defined according to the virtual work consideration (HillMandel condition) through a volume average over the microstructure. An equilibrium state of the micro-structure at a certain stage of the deformation process is then assumed. Apart from the definitions above, in macro-problems with temperature effects, the temperature linking condition equates the macro and micro temperatures and the micro and macro mechanical dissipation. A Lagrangian Finite element scheme is employed here to solve the microstructure deformation problem. The weak form is solved in an incremental-iterative manner as a result of material non-linearities. A generalized microstructure homogenization procedure is implemented in an objectoriented and parallel environment in C++ and PetSc parallel toolbox and made applicable to both 2D and 3D microstructures, building from our earlier work on large deformation process modeling and design in reference [1]. The microstructure material point problem has been parallelized by efficiently partitioning microstructure elements to every processor. The multi-scale problems in the application examples will be parallelized in a different way, with individual microstructure problems solved in serial in each processor and the macro-grid elements being fully parallelized. 1.2 Homogenization examples The results of plane strain compression of a 99.98% pure FCC aluminum aggregate using homogenization are compared with Taylor models based on stress-strain curves and texture evolution. The process conditions used for the simulations are temperature of 300 K with strain rate of 6.667E-4 /s. The microstructures are simulated as a collection of 400 grains with each grain modeled with a single finite element. The corresponding initial ODF is plotted in Fig. 2(d). The ODF is obtained by assuming that each orientation acts as a Gaussian point source within the fundamental region. This converts the discrete set of 400 orientations to a continuous distribution of orientations in the fundamental region. Two sets of Taylor models are used. One is based on the ODF representation of crystals with a conservation law being followed based on our recent work [2]. This law conserves the crystal volume fractions over the polycrystal when grains reorient during deformation. The other is based on a discrete aggregate of grains. The constitutive law was calibrated with experimental results in [3] to suit Taylor based computations. In order to match the stress-strain curves obtained on deformation with experimental results when employing the homogenization model, the model parameters need to be re-calibrated. However, at present, the Taylor model is compared with homogenization, thus same constitutive law parameters are used for both cases. From Fig. 2(b) it can be seen that homogenization techniques provide a softer response than the Taylor model which theoretically provides the upper bound of the stress-strain curve for the given microstructure. Further, results of such simulations provide information of stresses within the microstructure allowing identification of areas where recrystallization might occur in subsequent annealing process. Homogenization has been extended towards 3D polycrystals also, and Fig. 3 provides comparisons of a 3D polycrystals stress-strain prediction with experimental results. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2