Boundary-Value Problems for New Classes of Nonlinearly Elastic Materials

建议：DMS-0202834PI：Cornelius O.Hgan研究所：弗吉尼亚大学题目：新型非线性弹性材料的边值问题研究人员将研究连续介质力学中描述大应变橡胶类材料力学行为的新型非线性弹性本构模型的数学边值问题。与文献中使用的经典模型相比，材料模型在大应变时表现出应变硬化。在分子弹性理论中，要研究的模型被称为非高斯模型，因为它们涉及组成非高斯材料的聚合物链的端到端距离的分布函数。从现象学的角度来看，这些模型可以分为两类：具有极限链延伸性的模型和幂定律模型。前者的特点是在极限应变时出现应力奇异性，从而提出了新的数学挑战。提出了用非线性连续介质力学理论来分析这些材料的一些基本力学问题。这些具有挑战性的问题将被描述为非线性常微分方程组和偏微分方程组的边值问题，后者涉及二阶拟线性椭圆型偏微分方程组和这类方程组的耦合组。由于具有极限链伸长性的非线性弹性材料模型的应力响应与目前在科学和工程实践中使用的经典模型的应力响应有很大的不同，因此本文的研究具有重要的实际意义。本文的工作与橡胶和橡胶类材料(对汽车、航空航天和国防工业至关重要)的力学行为以及包括天然和组织衍生生物工程材料在内的生物软组织的行为有关。在工业实践中，基于有限元的大型商业计算机程序主要基于经典材料模型，它们对橡胶类材料在大应变下的预测需要重新评估。特别是，橡胶断裂导致汽车和飞机轮胎退化是当前的一个重要技术问题，拟议的工作将有助于解决这一问题。生物材料的应用包括动脉壁力学的研究，重点是应变硬化在心血管疾病发展中的作用。这项工作是跨学科的，涉及工程力学、应用数学和物理学的现代方法。为了确保橡胶类材料在现代技术中的安全可靠使用，必须进行这种类型的基础数学研究。

Proposal: DMS-0202834PI: Cornelius O. HorganInstitution: University of VirginiaTitle: Boundary-value problems for new classes of nonlinearly elastic materials ABSTRACTThe 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.