Microstructural Evolution in Orthotropic Elastic Media

Abstract

We consider the problem of microstructural evolution in binary alloys in two dimensions. The microstructure consists of arbitrarily shaped precipitates embedded in a matrix. Both the precipitates and the matrix are taken to be elastically anisotropic, with different elastic constants. The interfacial energy at the precipitatematrix interfaces is also taken to be anisotropic. This is an extension of the inhomogeneous isotrpic problem considered by H.-J. Jou et al. (1997, J. Comput. Phys. 131, 109). Evolution occurs via diffusion among the precipitates such that the total (elastic plus interfacial) energy decreases; this is accounted for by a modified Gibbs-Thomson boundary condition at the interfaces. The coupled diffusion and elasticity equations are reformulated using boundary integrals. An efficient preconditioner for the elasticity problem is developed based on a small scale analysis of the equations. The solution to the coupled elasticity-diffusion problem is implemented in parallel. Precipitate evolution is tracked by special non-stiff time stepping algorithms that guarantee agreement between physical and numerical equilibria. Results show that small elastic inhomogeneities in cubic systems can have a strong effect on precipitate evolution. For example, in systems where the elastic constants of the precipitates are smaller than those of the matrix, the particles move toward each other, where the rate of approach depends on the degree of inhomogeneity. Anisotropic surface energy can either enhance or reduce this effect, depending on the relative orientations of the anisotropies. Simulations of the evolution of multiple precipitates indicate that the elastic constants and surface energy control precipitate morphology and strongly influence nearest neighbor interactions. However, for the parameter ranges considered, the overall evolution of systems with large numbers of precipitates is primarily driven by the overall reduction in surface energy. Finally,

A variational r‐adaption and shape‐optimization method for finite‐deformation elasticity

“…Our aim here is merely to illustrate how the variational adaption method can be applied to problems of shape optimization. A comprehensive study of the mechanics of symmetry-breaking transitions or the behavior of specific materials is beyond the scope of this work and may be found elsewhere (see, e. g., Voorhees [1985]; Voorhees et al [1988]; Voorhees [1992]; ; Jou et al [1997]; Leo et al [2000Leo et al [ , 2001, and references therein). Alternative approaches to shape optimization may also be found elsewhere (e. g., Ramm [1995, 1997]; Bendsoe and Kikuchi [1988]; Leo et al [1998]; Maute et al [1999]; Schleupen et al [2000]; Jog et al [2000]; Hou et al [2001]; Schwarz et al [2001]).…”

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

“…[36][37][38][39][40][41] In all the above studies, the central focus of investigation has been the study of equilibrium shapes of precipitates where the anisotropy exists only in the elastic energy. The coupled influence of both the interfacial energy and elastic anisotropies on the equilibrium morphologies is performed using the boundary integral method by Leo et al [42] In a study by Greenwood et al, [43] the authors have developed a phase-field model of microstructural evolution, where they study the morphological evolution of solid-state dendrites as function of anisotropies in both surface as well as elastic energy. While the study is not particularly aimed at the computation of equilibrium shapes, the authors illustrate transitions in the growth directions of solid-state dendrites from those dominated by the surface energy anisotropy to those along elastically soft directions, stimulated by a change in the relative strengths of the anisotropies.…”

Compact integration factor methods in high spatial dimensions

“…However, the exact treatment of the differential operator requires evaluating exponentials of the approximation matrix for the linear differential operator. For periodic systems, this calculation is cheap both in CPU and storage because the approximation matrix can be diagonalized in the Fourier space [1][2][3][4][5][6][7]. For non-periodic systems, in which the approximation matrices are not diagonal, storage and calculation of exponentials of the matrices are significantly more expensive.…”

“…Different approximation of the integral involving nonlinear term ℱ(u) results in either the integration factor (IF) method or the exponential time differencing (ETD) method [8]. For example, the second order integration factor Adams-Bashforth method (IFAB2 [9]) has the form (3) and the second order ETD method [2,9] has a form: (4) In (3) and (4), is a matrix of size n × n for a system with only one diffusive species, and u k is the approximate solution at the kth time step. The computational cost for updating u k+1 at one time step is of order of n 2 due to the three vector-matrix multiplications associated with and in (3).…”

“…For example, the second order integration factor Adams-Bashforth method (IFAB2 [9]) has the form (3) and the second order ETD method [2,9] has a form: (4) In (3) and (4), is a matrix of size n × n for a system with only one diffusive species, and u k is the approximate solution at the kth time step. The computational cost for updating u k+1 at one time step is of order of n 2 due to the three vector-matrix multiplications associated with and in (3). During the temporal updating, these two matrices remain the same for a fixed Δt, and they only need to be evaluated once initially from and be stored.…”