تجزیه و تحلیل حساسیت در مقیاس چندطولی برای کنترل خواص وابسته به بافت در پردازش تغییر شکل

Material property evolution during processing is governed by the evolution of the underlying microstructure. We present an efficient technique for tailoring texture development and thus, optimizing properties in forming processes involving polycrystalline materials. The deformation process simulator allows simulation of texture formation using a continuum representation of the orientation distribution function. An efficient multi-scale sensitivity analysis technique is then introduced that allows computation of the sensitivity of microstructure field variables such as slip resistances and texture with respect to perturbations in macro-scale forming parameters such as forging rates, die shapes and preform shapes. These sensitivities are used within a gradient-based optimization framework for computational design of material property distribution during metal forming processes. Effectiveness of the developed computational scheme is demonstrated through computationally intensive examples that address control of properties such as Young’s modulus, strength and magnetic hysteresis loss in finished products.

Realization of optimal material properties is important to address the critical performance needs of hardware components in aerospace, naval and automotive applications. Newly emerging property design strategies for metallic materials are aimed towards tailoring microstructural subsystems by controlling processes that govern their evolution (Olson, 1997). An example is in composite design, where techniques that enable tailoring of microstructure topology have allowed identification of structures with interesting extremal properties such as negative thermal expansion (Sigmund and Torquato, 1996) and negative Poisson’s ratio (Lakes, 2000). One such technique for optimizing properties of metallic materials, comprised of a polycrystalline microstructure, involves tailoring of preferred orientation of crystals manifested as the crystallographic texture. During forging and extrusion processes, mechanisms such as crystallographic slip and lattice rotation drive formation of texture and variability in property distributions in polycrystalline materials. A possible method for designing property distribution in such materials is to control the deformation so that textures with desired properties are obtained. Several applications exist where certain textures are desirable to improve properties of materials. For example, a Goss texture is desirable in transformer cores to reduce power losses during magnetization (Rollett et al., 2001). In deep drawing, a high value of texture-dependent R parameter ( Hosford, 1993) and low planar anisotropy is necessary to prevent earing and to increase drawability of the sheet. Recent developments in microstructure-sensitive design have addressed problems such as computing optimal textures that lead to desired properties from the space of all possible textures (Adams et al., 2001 and Kalidindi et al., 2004). The problem of identification of processing paths that lead to such optimal textures is being addressed using novel means such as representation of processing paths in microstructure spaces using spectral (Li et al., 2005) or reduced order representations (Sundararaghavan and Zabaras, 2007) and using gradient optimization techniques (Acharjee and Zabaras, 2003 and Sundararaghavan and Zabaras, 2006). However, the success of such process design techniques has only been demonstrated at the microstructural length scale. The novelty in this work is that process design is performed using two different length scales. The macro-scale is associated with the component being modelled (10-310-3–101101 m) and the meso-scale is characterized by the underlying polycrystalline microstructure (10-610-6–10-310-3 m). We address the design problem of computation of macro-scale parameters such as forging velocity, die and preform shapes such that microstructure evolution is tailored towards achieving desired properties. The optimization problem involves minimizing an objective function that is an error measure between the desired property and the numerically calculated properties for a given set of macro-scale parameters. A sensitivity analysis scheme is used for calculating the gradient of the objective function and to drive the optimization procedure (Srikanth and Zabaras, 2001 and Zabaras et al., 2003). Posed in a multi-scale sense, the approach is used to compute sensitivities of microstructural fields such as slip resistance, crystal orientations due to perturbations in macro-scale parameters. These sensitivities are exactly defined using a set of field equations developed by directly differentiating the governing equations with respect to small perturbations in the macro-scale process parameters. An averaging principle is then developed to compute sensitivity of stress and various material properties at the macroscopic level from microstructural sensitivity fields. Evolution of the micro-scale during forming is modelled using continuum representation of texture (Kumar and Dawson, 1996, Kumar and Dawson, 1997 and Ganapathysubramanian and Zabaras, 2005) and incorporates crystal elasto-plasticity through the constitutive equations of Anand and Kothari (1996). Effectiveness of the developed computational scheme is demonstrated through examples involving control of properties such as Young’s modulus, strength and magnetic hysteresis loss in finished products. The paper is arranged as follows. In Section 2, the direct deformation model is briefly defined. This includes details about the microstructure representation, the constitutive model for polycrystalline materials and the kinematic problem in the context of a multi-length scale analysis. Section 3 develops the multi-length scale kinematic sensitivity problem. Section 4 considers a set of examples to demonstrate the accuracy, performance and applicability of the proposed algorithms.

Selection of macro-scale process parameters such as die shape, preform shape and forging velocity to control properties such as strength and stiffness is a challenging multi-scale problem due to the need to relate such macro-scale parameters with microstructure evolution. In this work, we presented a multi-scale optimization strategy for designing thermo-mechanical processes so that desired microstructure-sensitive properties are realized. Specifically, a two-scale continuum sensitivity formulation was developed that allows efficient computation of sensitivities of microstructure field variables such as slip resistances and texture with respect to perturbations in macro-scale forming parameters such as forging rates, die shapes and preform shapes. These sensitivities have been successfully employed in a gradient optimization framework for controlling properties such as hysteresis losses, the yield stress distribution and Young’s modulus in complex deformation processes by optimally altering die and preform shapes. The algorithm is computationally efficient and converges to the desired response within a few iterations. The simulator can also be easily extended towards computational design of other orientation-dependent properties such as thermal conductivity or the thermal expansion coefficient (Kocks et al., 2000) as well as in the design of devices with desired optical properties (Bunge, 1983). Other areas of applicability of this approach include calibration of material models or contact parameters through minimization of error between known experimental measurements and those predicted by the multi-scale model. The extension of the technique to address materials design in multiple stage forming processes is needed but the modifications involved are rather straightforward (Zabaras et al., 2003). Effort is currently on to improve the micro-scale model by incorporating higher-order features such as grain size and orientation correlations, in addition to crystallographic texture, using finite element homogenization schemes (Sundararaghavan and Zabaras, 2006).