J. Patrick Wilber

The Convexity Properties of a Class of Constitutive Models for Biological Soft Issues

During the last three decades, the theory of nonlinear elasticity has been used extensively to model biological soft issues. Although this research has generated many different models to describe the constitutive response of these issues, surprisingly little work has been done to analyze the mathematical features of the various models proposed. Here we carefully examine the convexity properties, specifically the Legendre-Hadamard and the strong ellipticity conditions, for a class of soft-issue models related to the classical Fung model. For various versions of models within this class, we discover necessary and sufficient conditions that must be satisfied for a model to have one or both of these convexity properties. We interpret our results mechanically and discuss the related implications for constitutive modeling. Also, we explore some more general points on how convexity can be studied and on the relation among different convexity properties.

The asymptotic problem for the springlike motion of a heavy piston in a viscous gas

This paper treats the classical problem for the longitudinal motion of a piston separating two viscous gases in a closed cylinder of finite length. The motion of the gases is governed by singular initial-boundary-value problems for parabolic-hyperbolic partial differential equations depending on a small positive parameter e, which characterizes the ratios of the masses of the gases to that of the piston. (The equation of state giving the pressure as a function of the specific volume need not be monotone and the viscosity may depend on the specific volume.) These equations are subject to a transmission condition, which is the equation of motion of the piston. The specific volumes of the gases are shown to have a positive lower bound at any finite time. This bound leads to the theorem asserting that (under mild smoothness restrictions) the initial-boundary-value problem has a unique classical solution defined for all time. The main emphasis of this paper is the treatment of the asymptotic behavior of solutions as e \ 0. It is shown that this solution admits a rigorous asymptotic expansion in e consisting of a regular expansion and an initial-layer expansion. The reduced problem, for the leading term of the regular expansion (which is obtained by setting e = 0), is typically governed by an equation with memory, rather than by an ordinary differential equation of the sort governing the motion of a mass on a massless spring. The reduced problem nevertheless has a 2-dimensional attractor on which the dynamics is governed precisely by such an ordinary differential equation.

Invariant manifolds describing the dynamics of a hyperbolic–parabolic equation from nonlinear viscoelasticity

The governing equations for a collection of dynamical problems for heavy rigid attachments carried by light, deformable, nonlinearly viscoelastic bodies are studied. These equations are a discretization of a nonlinear hyperbolic–parabolic partial differential equation coupled to a dynamical boundary condition. A small parameter measuring the ratio of the mass of the deformable body to the mass of the rigid attachment is introduced, and geometric singular perturbation theory is applied to reduce the dynamics to the dynamics of the slow system. Fenichel theory is then applied to the regular perturbation of the slow system to prove the existence of a low-dimensional invariant manifold within the dynamics of the high-dimensional discretization.

Discrete-to-continuum modeling of weakly interacting incommensurate chains

In this paper we use a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls.

Buckling Instabilities in Coupled Nanoscale Structures

We consider the bending of two nanotubes coupled together with van der Waal forces acting transverse to the axis, and subject to axial loads. The nanotubes are modeled as elastica while the interaction forces are derived from a Lennard-Jones 12-6 potential. The elastica are assumed to be a fixed distance apart at their ends, not necessarily equal to the equilibrium distance as identified from the Lennard-Jones potential. Therefore, the equilibrium configuration is not necessarily straight. As the compressive axial force increases, the beams can undergo buckling instability and the critical load depends not only on the material properties of the structure, but the geometry of the system as well. The continuum model is subjected to a Galerkin reduction to develop a reduced set of equations that can be used to calculate the equilibrium configuration of the system as well as the stability of these configurations. We show that the buckling instability in this model is significantly affected by the presence of the interaction force as well as the separation of the nanotubes at their ends.© 2005 ASME

Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

In this paper, we derive a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The starting point is a discrete atomistic model for the two lattices which are assumed to have slightly different lattice parameters and, possibly, a small relative rotation. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We use a discrete-to-continuum procedure to obtain the continuum model which recovers both qualitatively and quantitatively the behaviour observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching and bending deformation that accommodates the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters, which can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the details of the deformation associated with the domain walls.

Euler elastica as a Γ-limit of discrete bending energies of one-dimensional chains of atoms

In the 1920s, Hencky proposed a discrete elastica model describing a chain of identical rigid bars connected by torsional springs. Hencky observed that this discrete elastica model converges to Euler’s elastica as the number of bars increases while their lengths decrease, and Hencky’s bar-chain model has been used primarily as an approximation of Euler’s elastica. A Hencky-type bar-chain model can also be incorporated into a Frenkel–Kontorova-type discrete atomistic model, where the joints and bars represent the atoms and interatomic bonds, respectively, while the entire chain of atoms interacts with either a substrate or other chains. The energy of a continuum system corresponding to this Frenkel–Kontorova-type model can then be recovered by taking an appropriate discrete-to-continuum limit. Developing a correct limiting procedure for the discrete elastica establishes the bending component of this continuum energy. In this paper we use Γ-convergence to rigorously show that as the bar length in the discrete elastica model we consider goes to 0, the bending energies of the chain Γ-converge to the continuum bending energy associated with Euler’s elastica.