Mathematical models of inhomogeneous nonlinear viscoelastic solids and associated applications.

为了准确地模拟材料在外加载荷下的变形方式，本构模型是必不可少的。在力学方面，它们通常将应力(单位面积的力)与应变或应变率联系起来。“线性”材料遵循简单的线性关系：应力与应变成正比(胡肯固体)或应力与应变率成正比(牛顿流体)。线性粘弹性材料的行为介于这两种理想介质之间。这些模型通常适用于小变形或小应变率的材料。然而，它们没有充分描述广泛的材料，包括橡胶和其他感兴趣的弹性体、泡沫和软组织，主要是因为它们可以经历大变形，但也因为它们的材料响应不是线性的。当所讨论的介质是不均匀的，例如，它可能被夹杂物或纳米填料填充时，这种行为在模型中甚至更难捕捉。加入这种填料的原因通常与改善某些类型的材料响应(硬度、导电性等)有关。本项目将专注于发展与粘弹性固体相关的非线性粘弹性的基本数学理论，特别是那些不均匀的固体。自20世纪60年代以来，已经提出了一小部分理论来模拟特定类型变形下的特定非线性粘弹性材料。它们的复杂程度各不相同，从简单的数学表达式到复杂的公式。因此，第一个目标将涉及确定这些模式之间的联系：它们在哪里重叠，在哪里不重叠，以及它们如何在某些限度内相互融合。目前还不了解这些联系，明确的理解将对应用数学和材料科学的社区非常有益。学生还将研究某些类型的非线性粘弹性行为(例如，应变相关松弛)如何被某些模型而不是其他模型所适应。特别是，填充的(不均匀的)弹性体通常表现出在均匀材料中不存在的强烈的非线性行为。为了适应这种强烈的非线性行为，将发展新的本构模型。这些模型的起点将是准线性粘弹性，在能够准确表示某些类型的行为和能够在计算模型中实现之间的平衡方面，这被视为一个有用的起点。主要是在这些模型中研究应变相关松弛的概念，以及它们对一系列非线性材料变形模式的实验数据的拟合能力。特别令人感兴趣的是与句法泡沫相关的现有实验数据，这种材料广泛用于从航空航天、海洋、运动服装和无损评估等一系列应用中。本构模型将被整合到开源有限元软件中。

Constitutive models are essential in order to accurately model the way that materials deform under applied load. In terms of mechanics they typically relate stress (force per unit area) to strain or rate of strain. "Linear" materials are governed by simple linear relations: stress is proportional to strain (Hookean solids) or stress is proportional to rate of strain (Newtonian fluids). Linear viscoelastic materials behave somewhere between these two idealized media. These models typically apply to materials that undergo small deformations or small rates of strain. However they do not adequately describe a broad range of materials that include rubber and other elastomers of interest, foams and soft tissues predominantly because they can undergo large deformation but also because their material reponse is not linear. This behavior is even more difficult to capture in models when the medium in question is inhomogeneous, e.g. it may be filled with inclusions or nano-fillers. Reasons to include such fillers are usually associated with a desire to improve certain types of material response (stiffness, conductivity, etc.)This project will focus on developing the fundamental mathematical theory of nonlinear viscoelasticity associated with viscoelastic solids, and particularly those that are inhomogeneous. Since the 1960s a small number of theories have been proposed that model specific nonlinear viscoelastic materials under certain types of deformation. These vary in complexity, ranging from simple mathematical expressions to complex formulations. A first objective will therefore involve determining the links between these models: where they overlap and where they do not and how they may converge to each other in certain limits. These linkages are currently not understood and a clear understanding would be extremely beneficial to communities in applied mathematics and materials science. The student will also investigate how certain types of nonlinear viscoelastic behavior (e.g. strain dependent relaxation) can be accommodated by certain models but not others. In particular filled (inhomogeneous) elastomers often exhibit strong nonlinear behavior that is not present in homogeneous materials. In order to accommodate this strongly nonlinear behavior, new constitutive models will be developed. The starting point of the models will be quasi-linear viscoelasticity, which is viewed as a useful starting point in terms of a balance between being able to accurately represent certain types of behaviour and also being able to be implemented in computational models. Chiefly, the notion of strain-dependent relaxation will be investigated in these models and their ability to fit experimental data on a range of deformation modes of nonlinear materials. Of specific interest is existing experimental data associated with Syntactic foams, materials that are widely used in an array of applications ranging from aerospace, marine, sportswear and non-destructive evaluation. The constitutive models will be incorporated into open-source finite element software.