Numerical modeling of diffusion-induced deformation
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INTRODUCTION
THE recent trend toward decreasing size scale in electromechanical devices has led to a renewed interest in deformation during diffusion. Deformation induced by the Kirkendall effect in such devices can lead to unexpected behavior or product failures. In a recent article, Boettinger et al.[1] modeled this phenomenon. Following the work of Stephenson[2] and Daruka et al.,[3] they introduced a linear viscoelastic constitutive relation for the deformation. Boettinger et al. then considered several test problems in binary diffusion couples, where the partial molar volumes were assumed to be equal and constant, and the interdiffusion and intrinsic diffusivities were also assumed to be constant. The diffusion field for these cases was onedimensional (1-D), and the deformation was either 1-D or two-dimensional (2-D), depending on the surface traction boundary conditions. Because of the relatively simple geometry and constant material properties, Boettinger et al. were able to find analytical solutions for the composition and displacement. However, this restricts the application of the method and suggests the need for numerical implementations that can readily handle more complex geometries and nonconstant material properties. In this work, we describe a formulation that is suitable for numerical computation, enabling the examination of more complex problems. The paper is organized as follows. After introducing some basic thermodynamic relations needed for the derivations that follow, we develop the governing equations for a deformable solid undergoing isothermal diffusion. The diffusion equations are written in two forms: one referenced to the velocity field vM associated with inert markers, and J.A. DANTZIG, W. Grafton and Lillian B. Wilkins Professor of Mechanical Engineering, is with the Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801. Contact e-mail: [email protected] W.J. BOETTINGER, NIST Fellow, Metallurgy Division, J.A. WARREN, Leader, Thermodynamics and Kinetics Group, Metallurgy Division, G.B. McFADDEN, Mathematical and Computation Sciences Division, Information Technology Laboratory, and S.R. CORIELL, Guest Scientist, are with the National Institute of Standards and Technology, Gaithersburg, MD 20899. R.F. SEKERKA, University Professor of Physics, Mathematics and Materials Science, is with Carnegie Mellon University, Pittsburgh, PA 15213. Manuscript submitted February 16, 2006. METALLURGICAL AND MATERIALS TRANSACTIONS A
the other referenced to the velocity field vV of the local center of volume. A linear viscoelastic constitutive model for the deformation is introduced to complete the set of governing equations. We show that when the computations are performed using the volumetric formulation, one can take advantage of well-known techniques from computational fluid dynamics to solve the diffusion/deformation problem, and then later convert to the marker formulation to compute the deformation. The method is verified using the example problem
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