Interface Balance Laws, Growth Conditions and Explicit Interface Modeling Using Algebraic Level Sets for Multiphase Solids with Inhomogeneous Surface Stress

Pavan Kumar Vaitheeswaran, Purdue University

Abstract

Many natural and engineering phenomena including cancer cell growth, solidification, and crack propagation may be classified as phase evolution problems. These phenomena may have significant physiological or engineering impact, but are difficult to analyze due to the complexity of their interface geometry evolution and the complexity of their governing equations. The goal of this dissertation is to derive the thermodynamic conditions governing the evolution of the interfaces and an efficient computational procedure for modeling the interface evolution. The first contribution towards the goal is to derive the thermodynamic conditions at a moving interface in a body subject to large deformation with multiple diffusing species and arbitrary surface stress. A pillbox procedure is used to form balance laws at an interface, analogous to conventional balance laws in the bulk. The thermodynamic conditions that result from interface free-energy inequality lead to the analytical form of the configurational force for bodies subject to mechanical loads, heat and multiple diffusing species. The derived second law condition naturally extends the Eshelby energy-momentum tensor to include species diffusion terms. The above second law restriction is then used to derive the condition for the growth of new phases in a body undergoing finite deformation subject to inhomogeneous, anisotropic surface stress. Next, a general, finite-deformation, arbitrary surface stress form of phase nucleation condition is derived by considering uncertainty in growth of a small nucleus. The probability of nucleation is shown to naturally depend on a theoretical estimate of critical volumetric energy density, which is directly related to the surface stress. The classical nucleation theory is shown to result as a simplified special case. Towards the goal of computationally simulating evolving interfaces, an interface tracking approach called enriched isogeometric analysis (EIGA) is adopted in this thesis. The phase boundary is represented explicitly using parametric splines, with the physical fields isogeometrically defined on the interface geometry, and immersed in a non-conforming underlying domain. The behavioral field solution at a point is given by a convex blending of the underlying solution and the interface solution. Signed algebraic level sets are used as a measure of distance to model the weakening influence of a phase interface with distance. These level sets are generated from the implicitization of geometries using resultants, and are smooth, monotonic with distance, and exact on the boundary. Furthermore, the sign of these level sets enable classifying points as lying inside or outside a given closed geometry. The generation of these level sets is found to fail often for even simple three-dimensional surfaces, where the Dixon resultant used for implicitization is either identically zero or unsigned. A maximal-rank submatrix approach is adopted in this thesis to recover the implicitization for surfaces with identically zero resultants. Also, a polynomial square root procedure is developed to extract sign from unsigned resultants. The proposed approach is demonstrated on three-dimensional electrostatic and electromigration problems, and is used to simulate electromigration experiments conducted on Copper-TiN line structures. Since EIGA is an explicit interface method, topological changes that are common in phase evolution pose geometric challenges, such as computing intersection between boundaries of coalescent phases. This is overcome in this thesis by using Boolean compositions on algebraic level sets. These compositions can be done algebraically using R-functions, and provide level sets for merged phases.

Degree

Ph.D.

Advisors

Subbarayan, Purdue University.

Subject Area

Mathematics|Mechanics

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