Boundary-Value Problems for New Classes of Nonlinearly Elastic Materials

项目申请：DMS-0202834PI: Cornelius O. HorganInstitution: University of virginia标题：新型非线性弹性材料的边值问题摘要：研究连续介质力学中描述类橡胶材料在大应变作用下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，材料模型在大应变下表现出应变硬化。在弹性分子理论中，要研究的模型被称为非高斯模型，因为它们涉及组成材料的聚合物链的端到端距离的分布函数是非高斯分布。从现象学的角度来看，模型可分为两类：具有极限链可拓性的模型和幂律模型。前者具有在极限应变处出现应力奇点的特点，从而对数学提出了新的挑战。提出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表述为非线性常微分方程和偏微分方程的边值问题，后者涉及二阶拟线性椭圆偏微分方程和这些方程的耦合系统。由于具有极限链可拓的非线性弹性材料模型的应力响应与目前科学和工程实践中使用的经典模型有很大不同，因此本文的研究具有重要的实际意义。这里提出的工作与橡胶和橡胶样材料的机械行为（对汽车，航空航天和国防工业至关重要）以及生物软组织的行为有关，包括天然和组织衍生的工程生物材料。在工业实践中，基于有限元的大型商用计算机代码主要基于经典材料模型，它们对大应变下类橡胶材料的预测值得重新评估。特别是，橡胶断裂导致汽车和飞机轮胎退化是当前重要的技术问题，所提出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。为了保证在现代技术中安全可靠地利用类橡胶材料，这里提出的这类基础数学研究是必要的。

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.