Krishna Garikipati

Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity

Abstract A new finite element method for fourth-order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem. The method combines concepts from the continuous Galerkin (CG) method, the discontinuous Galerkin (DG) method and stabilization techniques. A brief review of the CG method, the DG method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth-order elliptic problems. A continuous/discontinuous Galerkin (C/DG) method is proposed which uses C 0 -continuous interpolation functions and is formulated in the primary variable only. The advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided. In the context of thin bending theory, the C/DG method leads to a formulation where displacements are the only degrees of freedom, and no rotational degrees of freedom need to be considered. The main feature of the C/DG method is the weak enforcement of continuity of first and higher-order derivatives through stabilizing terms on interior boundaries. Consistency, stability and convergence of the method are shown analytically. Numerical experiments verify the theoretical results, and applications are presented for Bernoulli–Euler beam bending, Poisson–Kirchhoff plate bending and a shear layer problem using Toupin–Mindlin strain gradient theory.

A continuum treatment of growth in biological tissue: The coupling of mass transport and mechanics

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.

Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network

A new class of micromechanically motivated chain network models for soft biological tissues is presented. On the microlevel, it is based on the statistics of long chain molecules. A wormlike chain model is applied to capture the behavior of the collagen microfibrils. On the macrolevel, the network of collagen chains is represented by a transversely isotropic eight chain unit cell introducing one characteristic material axis. Biomechanically induced remodeling is captured by allowing for a continuous reorientation of the predominant unit cell axis driven by a biomechanical stimulus. To this end, we adopt the gradual alignment of the unit cell axis with the direction of maximum principal strain. The evolution of the unit cell axis’ orientation is governed by a first-order rate equation. For the temporal discretization of the remodeling rate equation, we suggest an exponential update scheme of Euler-Rodrigues type. For the spatial discretization, a finite element strategy is applied which introduces the current individual cell orientation as an internal variable on the integration point level. Selected model problems are analyzed to illustrate the basic features of the new model. Finally, the presented approach is applied to the biomechanically relevant boundary value problem of an in vitro engineered functional tendon construct.

Characterization of contact electromechanics through capacitance-voltage measurements and simulations

Electrostatically actuated polysilicon beams fabricated in the multiuser MEMS process (MUMPs) are studied, with an emphasis on the behavior when the beam is in contact with an underlying silicon nitride dielectric layer. Detailed two-dimensional (2-D) electromechanical simulations, including the mechanical effects of stepups, stress-stiffening and contact, as well as the electrical effects of fringing fields and finite beam thickness, are performed. Comparisons are made to quasi-2-D and three-dimensional simulations. Pull-in voltage and capacitance-voltage measurements together with 2-D simulations are used to extract material properties. The electromechanical system is used to monitor charge buildup in the nitride which is modeled by a charge trapping model. Surface effects are included in the simulation using a compressible-contact-surface model. Monte Carlo simulations reveal the limits of simulation accuracy due to the limited resolution of input parameters.

A discontinuous Galerkin method for the Cahn-Hilliard equation

A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn-Hilliard equation. The Cahn-Hilliard equation is a fourth-order nonlinear parabolic partial differential equation, originally proposed to model phase segregation of binary alloys. The developed discontinuous Galerkin approach avoids the need for mixed finite element methods, coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth-order spatial derivatives. The variational formulation of the discontinuous Galerkin method, its implementation and numerical examples are presented. In this communication, it is also shown under what conditions the method is stable, and an error estimate in an energy-type norm is presented. The method is evaluated by comparison with a standard finite element treatment in which the Cahn-Hilliard equation is decomposed into two coupled partial differential equations.

The Role of Coherency Strains on Phase Stability in LixFePO4: Needle Crystallites Minimize Coherency Strain and Overpotential

We investigate the role of coherency strains on the thermodynamics of two-phase coexistence during Li (de)intercalation of LixFePO4. We explicitly account for the anisotropy of the elastic moduli and analytically derive coupled chemical and mechanical equilibrium criteria for two-phase morphologies observed experimentally. Coherent two-phase equilibrium leads to a variable voltage profile of individual crystallites within the two-phase region as the dimensions of the crystallite parallel to the interface depend on the phase fractions of the coexisting phases. With a model free energy for LixFePO4, we illustrate the effect of coherency strains on the compositions of the coexisting phases and on the voltage profile. We also show how coherency strains can stabilize intermediate solid solutions at low temperatures if phase separation is restricted to Li diffusion along the b-axis of olivine LixFePO4. A finite element analysis shows that long needlelike crystallites with the long axis parallel to the a lattice vector of LixFePO4 minimize coherency strain energy. Hence, needlelike crystallites of LiFePO4 reduce the overpotential needed for Li insertion and removal and minimize mechanical damage, such as dislocation nucleation and crack formation, resulting from large coherency strain energies.

A variational multiscale approach to strain localization – formulation for multidimensional problems

Abstract The multiscale approach to strain localization problems developed earlier, is generalized to higher dimensions. The key idea is the identification of the fine scale field with a component of the displacement that has a large gradient. The difference between the total displacement and the fine scale is the coarse scale field. The formulation is variationally based, with the weak form as the point of departure. With suitable assumptions on the space of fine scale fields, the weak form is separable into coarse and fine scale parts. The weak form of the fine scale problem is used to eliminate the fine scales from the formulation. This is achieved by way of a projection onto the fine scale space and allows a reformulation of the full problem in terms of coarse scale components alone. In this framework, fine scale interpolations are identified that ensure sharp resolution of localized displacement irrespective of the underlying finite element mesh. It is demonstrated that by correctly accounting for the “microstructural” fine scale behavior, the formulation results in solutions devoid of pathological mesh sensitivity. For problems wherein the detailed structure of the localized displacement is of interest, recovery of the fine scale components can be performed to reconstruct the entire displacement field. Several numerical examples are presented that demonstrate the robustness of the formulation.

A study of strain localization in a multiple scale framework—The one-dimensional problem

Abstract This work approaches strain localization by recognizing the multiple scales inherent in the problem. A component associated with the region of localized strain (typically, one with high gradient) is referred to as the fine scale. It represents the microstructure. The field obtained by removing the fine scale from the total solution is referred to as the coarse scale. The aim of the multiple scale method advanced here is to derive a model for the coarse scale field that accounts for the finite scale. This process eliminates the fine scale from the problem, yet retains its effect. In applying this framework to the nonlinear problems with which localized strains are associated, a crucial step is a first-order approximation of the relevant relations. The fully nonlinear problem is solved by an iterative scheme. By accounting directly for the microstructure the multiple scale model recovers the regularizing effects of various alternative formulations for softening strain localization. It thus presents itself as a unifying framework for such models. Numerical solutions are shown to be invariant with respect to the discretization. Furthermore, for cases in which the displacements assume a distinct profile within the localization band, the multiple scale model provides a resolution that can be made as accurate as desired even with the coarsest mesh possible. The model is applied to strain localization problems that arise in inviscid and viscoplastic solids. Numerical simulations are presented that demonstrate the efficacy of the approach.

The non-equilibrium thermodynamics and kinetics of focal adhesion dynamics.

Background We consider a focal adhesion to be made up of molecular complexes, each consisting of a ligand, an integrin molecule, and associated plaque proteins. Free energy changes drive the binding and unbinding of these complexes and thereby controls the focal adhesions dynamic modes of growth, treadmilling and resorption. Principal Findings We have identified a competition among four thermodynamic driving forces for focal adhesion dynamics: (i) the work done during the addition of a single molecular complex of a certain size, (ii) the chemical free energy change associated with the addition of a molecular complex, (iii) the elastic free energy change associated with deformation of focal adhesions and the cell membrane, and (iv) the work done on a molecular conformational change. We have developed a theoretical treatment of focal adhesion dynamics as a nonlinear rate process governed by a classical kinetic model. We also express the rates as being driven by out-of-equilibrium thermodynamic driving forces, and modulated by kinetics. The mechanisms governed by the above four effects allow focal adhesions to exhibit a rich variety of behavior without the need to introduce special constitutive assumptions for their response. For the reaction-limited case growth, treadmilling and resorption are all predicted by a very simple chemo-mechanical model. Treadmilling requires symmetry breaking between the ends of the focal adhesion, and is achieved by driving force (i) above. In contrast, depending on its numerical value (ii) causes symmetric growth, resorption or is neutral, (iii) causes symmetric resorption, and (iv) causes symmetric growth. These findings hold for a range of conditions: temporally-constant force or stress, and for spatially-uniform and non-uniform stress distribution over the FA. The symmetric growth mode dominates for temporally-constant stress, with a reduced treadmilling regime. Significance In addition to explaining focal adhesion dynamics, this treatment can be coupled with models of cytoskeleton dynamics and contribute to the understanding of cell motility.

The Kinematics of Biological Growth

The kinematic aspects of biological growth models are reviewed by paying attention to the handful of crucial ideas on which modern treatments rest. Both surface and volumetric growth are considered. A critical appraisal is presented of the geometric and physical features of the models. Links are made to the mathematical treatment of growth and evolving interface phenomena in other physical problems. Computational issues are pointed out wherever appropriate.