arXiv subjects

Krishna Garikipati

Publications and source records attributed to Krishna Garikipati.

64 records · Page 4Linked to original sources

arXiv2016

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 field's evolution is localized to a mathematically sharp interface, and (b) those in which the configurational field's 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. 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 (Arch. Rat. Mech. Anal. 11, 1962, 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. Numerical examples are presented to demonstrate interface motion as well as evolving microstructures of crystal structures.

cond-mat.soft↗

arXiv2016

Multi-physics simulations of lithiation-induced stress in Li$_{\rm 1+x}$Ti$_2$O$_4$ electrode particles

Cubic spinel Li$_{\rm 1+x}$Ti$_2$O$_4$ is a promising electrode material as it exhibits a high lithium diffusivity and undergoes minimal changes in lattice parameters during lithiation and delithiation, thereby ensuring favorable cycleability. The present work is a multi-physics and multi-scale study of Li$_{\rm 1+x}$Ti$_2$O$_4$ that combines first principles computations of thermodynamic and kinetic properties with continuum scale modeling of lithiation-delithiation kinetics. Density functional theory calculations and statistical mechanics methods are used to calculate lattice parameters, elastic coefficients, thermodynamic potentials, migration barriers and Li diffusion coefficients. These quantities then inform a phase field framework to model the coupled chemo-mechanical evolution of electrode particles. Several case studies accounting for either homogeneous or heterogeneous nucleation are considered to explore the temporal evolution of maximum principle stress values, which serve to indicate stress localization and the potential for crack initiation, during lithiation and delithiation.

cond-mat.mtrl-sci↗

arXiv2016

Mechano-chemical 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 mechano-chemical 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 mechano-chemical 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 mechano-chemical spinodal decomposition should occur in metal hydrides such as ZrH$_{2-2c}$. 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.

physics.chem-ph↗

arXiv2016

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

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.

physics.comp-ph↗

arXiv2016

A three dimensional field formulation, and isogeometric solutions to point and line defects using Toupin's theory of gradient elasticity at finite strains

We present a field formulation for defects that draws from the classical representation of the cores as force dipoles. We write these dipoles as singular distributions. Exploiting the key insight that the variational setting is the only appropriate one for the theory of distributions, we arrive at universally applicable weak forms for defects in nonlinear elasticity. Remarkably, the standard, Galerkin finite element method yields numerical solutions for the elastic fields of defects that, when parameterized suitably, match very well with classical, linearized elasticity solutions. The true potential of our approach, however, lies in its easy extension to generate solutions to elastic fields of defects in the regime of nonlinear elasticity, and even more notably for Toupin's theory of gradient elasticity at finite strains(Arch. Rat. Mech. Anal., 11, 385, 1962). In computing these solutions we adopt recent numerical work on an isogeometric analytic framework that enabled the first three-dimensional solutions to general boundary value problems of Toupin's theory (Rudraraju et al. Comp. Meth. App. Mech. Engr., 278, 705, 2014). We first present exhaustive solutions to point defects, edge and screw dislocations, and a study on the energetics of interacting dislocations. Then, to demonstrate the generality and potential of our treatment, we apply it to other complex dislocation configurations, including loops and low-angle grain boundaries.

cond-mat.mtrl-sci↗

arXiv2015

Three-dimensional iso-geometric solutions to general boundary value problems of Toupin's gradient elasticity theory at finite strains

We present, to the best of our knowledge, the first complete three-dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity. We have chosen for our work, Toupin's theory [Arch. Rat. Mech. Anal., 11(1), 385-414, 1962]-one of the more general formulations of strain gradient elasticity. Our framework has three crucial ingredients: The first is iso-geometric analysis [Hughes et al., Comp. Meth. App. Mech. Engrg., 194(39-41), 4135-4195, 2005], which we have adopted for its straightforward and robust representation of C1-continuity. The second is a weak treatment of the higher-order Dirichlet boundary conditions in the formulation, which control the development of strain gradients in the solution. The third ingredient is algorithmic (automatic) differentiation, which eliminates the need for linearization "by hand" of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains. We present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions. We discuss size effects, the role of higher-order boundary conditions, and perhaps most importantly, the relevance of the framework to problems with elastic free energy density functions that are non-convex in strain space.

math-ph↗

arXiv2014

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 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.

q-bio.TO↗

arXiv2014

The mechano-chemistry of cytoskeletal force generation

In this communication, we propose a model to study the non-equilibrium process by which actin stress fibers develop force in contractile cells. The emphasis here is on the non-equilibrium thermodynamics, which is necessary to address the mechanics as well as the chemistry of dynamic cell contractility. In this setting we are able to develop a framework that relates (a) the dynamics of force generation within the cell and (b) the cell response to external stimuli to the chemical processes occurring within the cell, as well as to the mechanics of linkage between the stress fibers, focal adhesions and extra-cellular matrix.

q-bio.SC↗

arXiv2014

A computational study of stress fiber-focal adhesion dynamics governing cell contractility

We apply a recently developed model of cytoskeletal force generation to study a cell intrinsic contractility, as well as its response to external loading. The model is based on a non-equilibrium thermodynamic treatment of the mechano-chemistry governing force in the stress fiber-focal adhesion system. Our computational study suggests that the mechanical coupling between the stress fibers and focal adhesions leads to a complex, dynamic, mechano-chemical response. We collect the results in response maps whose regimes are distinguished by the initial geometry of the stress fiber-focal adhesion system, and by the external load on the cell. The results from our model connect qualitatively with recent studies on the force response of smooth muscle cells on arrays of polymeric microposts (Mann et al., Lab. on a Chip, 12, 731-740, 2012).

q-bio.SC↗

arXiv2006

Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

This work is based upon a coupled, lattice-based continuum formulation that was previously applied to problems involving strong coupling between mechanics and mass transport; e.g. diffusional creep and electromigration. Here we discuss an enhancement of this formulation to account for migrating grain boundaries. The level set method is used to model grain-boundary migration in an Eulerian framework where a grain boundary is represented as the zero level set of an evolving higher-dimensional function. This approach can easily be generalized to model other problems involving migrating interfaces; e.g. void evolution and free-surface morphology evolution. The level-set equation is recast in a remarkably simple form which obviates the need for spatial stabilization techniques. This simplified level-set formulation makes use of velocity extension and field re-initialization techniques. In addition, a least-squares smoothing technique is used to compute the local curvature of a grain boundary directly from the level-set field without resorting to higher-order interpolation. A notable feature is that the coupling between mass transport, mechanics and grain-boundary migration is fully accounted for. The complexities associated with this coupling are highlighted and the operator-split algorithm used to solve the coupled equations is described.

physics.comp-ph↗