Gabriella A. Pinter

A Probabilistic Multiscale Approach to Hysteresis in Shear Wave Propagation in Biotissue

Motivated by a problem involving wave propagation through viscoelastic biotissue, we present a theoretical framework for treating hysteresis as multiscale phenomena which must be averaged across distributions of internal variables. The resulting systems entail probability measure-dependent partial differential equations for which we establish well-posedness in a framework that leads readily to computationally useful approximations.

Modeling of Nonlinear Hysteresis in Elastomers under Uniaxial Tension

As a fundamental component of an overall program in modeling smart material damping devices, we consider inactive host material models for moderate to highly filled rubbers undergoing uniaxial tensile deformations. Beginning from a neo-Hookean strain energy function formulation for nonlinear extension, we develop general constitutive models for both quasi-static and dynamic deformations of a viscoelastic rod. The constitutive laws are nonlinear and contain hysteresis through a Boltzmann superposition integral term. The resulting integropartial differential equations models are shown to be equivalent to the usual Lagrangian dynamic distributed parameter models coupled with linear ordinary differential equations for internal variables (internal strains). Comprehensive well-posedness results (existence, uniqueness and continuous dependence) are summarized in a discussion of theoretical aspects of the systems. The models are validated with experiments designed and carried out explicitly for this study. In particular, quasi-static Instron experimental data are used in a least squares inverse problem formulation to estimate nonlinear elastic and nonlinear viscoelastic contributions to the general stress-strain constitutive laws proposed. It is shown that the models provide an excellent prediction of nested hysteresis loops manifested in the data. These models are then used as initial estimates in determining the nonlinear hysteretic constitutive laws for the dynamic experiments. It is shown that in cases of more highly filled rubbers, multiple internal variable models lead to best fits to the data.

Multiscale Considerations in Modeling of Nonlinear Elastomers

We present a survey of results from an extended project focused on the understanding of the dynamic behavior of elastomers or filled rubbers. This entailed experimental, modeling, computational and theoretical efforts. Of particular emphasis are the nonlinear and hysteretic aspects of dynamic deformations.

Compression of bonded rubber blocks

Abstract Many practical rubber component designs involve the compression of rubber blocks bonded to rigid metal plates. The rigorous analysis of such components is very difficult. However, a simple approximate method developed by Gent gives reasonably satisfactory predictions for stiffness and stress distribution provided deformations are kept small. Gents “pressure” method is briefly reviewed for the classical cases of a long strip and a cylindrical disk. The method is then applied to more complex cases of a rectangular block and an annulus. The usual “incompressibility” assumption is relaxed to “near incompressibility” to yield more accurate solutions. Predictions have been verified using linear finite element analysis.

A nonlinear beam equation

Abstract : Existence and uniqueness of weak solutions to a nonlinear beam equation is established under relaxed assumptions (locally Lipschitz plus affine domination) on the nonlinearity in the stiffness constitutive law. The results provide alternatives to previous theories requiring rather stringent monotonicity assumptions. The techniques and arguments are applicable to a large class of nonlinear second order (in time) partial differential equation systems.

Asymptotic behavior of an SI epidemicmodel with pulse removal

In this paper, we discuss an SI epidemic model with pulse removal from the infectiveclass at fixed time intervals with both exponential and logistic type underlying population dynamics. This model has significance when dealing with animal diseases with no recovery, or when we consider isolation in human diseases. We provide a rigorous analysis of the asymptotic behavior of the percentage of infected individuals, the total number of infected individuals, and the total population in our model. We show that periodic removal/isolation is a feasible strategy to control the spread of the disease.

Maxwell-systems with nonlinear polarization

We establish well-posedness results for a model describing the propagation of high-intensity electromagnetic waves in a nonlinear medium. The nonlinear material properties are represented by a nonlinear polarization in the form of a convolution. We also include some remarks on potential applications.

Dynamics of a virus-host model with an intrinsic quota

In this work we develop and analyze a mathematical model describing the dynamics of infection by a virus of a host population in a freshwater environment. Our model, which consists of a system of nonlinear ordinary differential equations, includes an intrinsic quota, that is, we use a nutrient (e.g., phosphorus) as a limiting element for the host and potentially for the virus. Motivation for such a model arises from studies that raise the possibility that on the one hand, viruses may be limited by phosphorus (Bratbak et al. [17]), and on the other, that they may have a role in stimulating the host to acquire the nutrient (Wilson [18]). We perform an in-depth mathematical analysis of the system including the existence and uniqueness of solutions, equilibria, asymptotic, and persistence analysis. We compare the model with experimental data, and find that biologically meaningful parameter values provide a good fit. We conclude that the mathematical model supports the hypothesized role of stored nutrient regulating the dynamics, and that the coexistence of virus and host is the natural state of the system.

Approximation Results for Parameter Estimation in Nonlinear Elastomers

In this paper we present an approximation framework and theoretical convergence results for a class of parameter estimation problems for general abstract nonlinear hyperbolic systems. These systems include as a special case those modeling a large class of nonlinear elastomers.

APPROXIMATION RESULTS FOR PARAMETER ESTIMATION IN A CLASS OF ABSTRACT NONLINEAR HYPERBOLIC SYSTEMS

Abstract In this paper, we present an approximation framework and theoretical convergence results for a class of parameter estimation problems for general abstract nonlinear hyperbolic systems. These systems include as a special case those modeling a large class of nonlinear elastomers.