Shiva Rudraraju

Mechanochemical spinodal decomposition: A phenomenological theory of phase transformations in multi-component, crystalline solids

We present a phenomenological treatment of diffusion-driven martensitic phase transformations in multi-component crystalline solids that arise from non-convex free energies in mechanical and chemical variables. The treatment describes diffusional phase transformations that are accompanied by symmetry-breaking structural changes of the crystal unit cell and reveals the importance of a mechanochemical spinodal, defined as the region in strain–composition space, where the free-energy density function is non-convex. The approach is relevant to phase transformations wherein the structural order parameters can be expressed as linear combinations of strains relative to a high-symmetry reference crystal. The governing equations describing mechanochemical spinodal decomposition are variationally derived from a free-energy density function that accounts for interfacial energy via gradients of the rapidly varying strain and composition fields. A robust computational framework for treating the coupled, higher-order diffusion and nonlinear strain gradient elasticity problems is presented. Because the local strains in an inhomogeneous, transforming microstructure can be finite, the elasticity problem must account for geometric nonlinearity. An evaluation of available experimental phase diagrams and first-principles free energies suggests that mechanochemical spinodal decomposition should occur in metal hydrides such as ZrH2−2c. The rich physics that ensues is explored in several numerical examples in two and three dimensions, and the relevance of the mechanism is discussed in the context of important electrode materials for Li-ion batteries and high-temperature ceramics. A treatment to describe structural phase transformations in materials has been developed by researchers in the USA. Under favorable energetic conditions, some multicomponent solids such as lithium-ion battery electrodes can decompose from a uniform phase into two co-existing phases having different compositions. This spinodal decomposition affects the properties of a range of structural and electronic materials. Now, Krishna Garikipati at the University of Michigan and colleagues have modeled spinodal decomposition triggered by instabilities and variations in either strain or composition throughout a material. The model requires solving a system of complex nonlinear interactions through a computational framework, and represents a new phenomenological description of spinodal decomposition that could potentially be extended to other phase-transformation phenomena. The model may play a role in the study and development of practically relevant materials such as battery electrodes or nanoelectronics.

Elastic Free Energy Drives the Shape of Prevascular Solid Tumors

It is well established that the mechanical environment influences cell functions in health and disease. Here, we address how the mechanical environment influences tumor growth, in particular, the shape of solid tumors. In an in vitro tumor model, which isolates mechanical interactions between cancer tumor cells and a hydrogel, we find that tumors grow as ellipsoids, resembling the same, oft-reported observation of in vivo tumors. Specifically, an oblate ellipsoidal tumor shape robustly occurs when the tumors grow in hydrogels that are stiffer than the tumors, but when they grow in more compliant hydrogels they remain closer to spherical in shape. Using large scale, nonlinear elasticity computations we show that the oblate ellipsoidal shape minimizes the elastic free energy of the tumor-hydrogel system. Having eliminated a number of other candidate explanations, we hypothesize that minimization of the elastic free energy is the reason for predominance of the experimentally observed ellipsoidal shape. This result may hold significance for explaining the shape progression of early solid tumors in vivo and is an important step in understanding the processes underlying solid tumor growth.

Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition

Abstract We consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn–Hilliard model with Toupin’s theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.

Multiphysics Modeling of Reactions, Mass Transport and Mechanics of Tumor Growth

The biochemical dynamics involved in tumor growth can be broadly classified into three distinct spatial scales: the tumor scale, the cell-ECM interactions and the sub-cellular processes. This work presents the tumor scale investigations, which are expected to eventually lead to a system-level understanding of the progression of cancer. Many of the macroscopic phenomena of physiological relevance, such as tumor size and shape, formation of necrotic core and vascularization, proliferation and metastasis of cell populations, external mechanical interactions, etc., can be treated within a continuum framework by modeling the evolution of various species involved by transport equations coupled with mechanics. This framework is an extension of earlier work (Garikipati et al. in J. Mech. Phys. Solids 52:1595–1625, 2004; Narayanan et al. in Biomech. Model. Mechanobiol. 8:167–181, 2009, J. Phys. Condens. Matter. 22:194122, 2010) based on the continuum theory of mixtures for modeling biological growth. Specifically, the focus is on demonstrating the effectiveness of mechano-transport coupling in simulating tumor growth dynamics and explaining some in vitro observations like mechanics-induced ellipsoidal tumor shapes. Additionally, this work also seeks to demonstrate the effectiveness of tools like adaptive mesh refinement and automatic differentiation in handling highly nonlinear, coupled multiphysics systems.

A variational treatment of material configurations with application to interface motion and microstructural evolution

We present a unified variational treatment of evolving configurations in crystalline solids with microstructure. The crux of our treatment lies in the introduction of a vector configurational field. This field lies in the material, or configurational, manifold, in contrast with the traditional displacement field, which we regard as lying in the spatial manifold. We identify two distinct cases which describe (a) problems in which the configurational fields evolution is localized to a mathematically sharp interface, and (b) those in which the configurational fields evolution can extend throughout the volume. The first case is suitable for describing incoherent phase interfaces in polycrystalline solids, and the latter is useful for describing smooth changes in crystal structure and naturally incorporates coherent (diffuse) phase interfaces. These descriptions also lead to parameterizations of the free energies for the two cases, from which variational treatments can be developed and equilibrium conditions obtained. For sharp interfaces that are out-of-equilibrium, the second law of thermodynamics furnishes restrictions on the kinetic law for the interface velocity. The class of problems in which the material undergoes configurational changes between distinct, stable crystal structures are characterized by free energy density functions that are non-convex with respect to configurational strain. For physically meaningful solutions and mathematical well-posedness, it becomes necessary to incorporate interfacial energy. This we have done by introducing a configurational strain gradient dependence in the free energy density function following ideas laid out by Toupin (1962, Elastic materials with couple-stresses. Arch. Ration. Mech. Anal., 11, 385–414). The variational treatment leads to a system of partial differential equations governing the configuration that is coupled with the traditional equations of nonlinear elasticity. The coupled system of equations governs the configurational change in crystal structure, and elastic deformation driven by elastic, Eshelbian, and configurational stresses. Numerical examples are presented to demonstrate interface motion as well as evolving microstructures of crystal structures.