Boundary-Value Problems for New Classes of Nonlinearly Elastic Materials

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

Proposal: DMS-0202834 PI: Cornelius O. Horgan Institution: University of Virginia Title: Boundary-value problems for new classes of nonlinearly elastic materials ABSTRACT The investigator will study mathematical boundary-value problems in continuum mechanics for new classes of nonlinear elastic constitutive models describing the mechanical behavior of rubber-like materials undergoing large strains. The material models exhibit strain hardening at large strains, in contrast to the classic models used in the literature. In the molecular theory of elasticity, the models to be investigated are called non-Gaussian, because they involve a distribution function for the end-to-end distance of the polymer chain composing the material that is non-Gaussian. From the phenomenological point of view, the models can be divided into two classes: models with limiting chain extensibility and power-law models. The former have the feature that a stress singularity occurs at the limiting strain and thus offer new mathematical challenges. It is proposed to use theories of nonlinear continuum mechanics to analyze some fundamental mechanical problems for these materials. These challenging problems will be formulated as boundary-value problems for nonlinear ordinary and partial differential equations, the latter involving second-order quasilinear elliptic partial differential equations and coupled systems of such equations. Since the stress response of nonlinearly elastic material models with limiting chain extensibility is significantly different from that of classical models currently used in science and engineering practice, the proposed research should have important practical implications. The work proposed here is relevant to the mechanical behavior of rubber and rubber-like materials (of critical importance to the automotive, aerospace and defense industries) as well as to the behavior of biological soft tissues including both natural and tissue derived engineered biomaterials. In industrial practice, finite-element based large-scale commercial computer codes are based primarily on classical material models and their predictions for rubber-like materials at large strains warrant reassessment. In particular, fracture of rubber leading to automobile and aircraft tire degradation is an important current technological problem to which the proposed work will contribute. Applications to biological materials include the study of arterial wall mechanics with emphasis on the role of strain hardening in the development of cardiovascular disease. The work is interdisciplinary involving modern methods of engineering mechanics, applied mathematics and physics. Fundamental mathematical studies of the type proposed here are necessary to ensure the safe reliable utilization of rubber-like materials in modern technology.