【作者】 彭小春；

【导师】 陈文艺； 戴晖辉；

【作者基本信息】 武汉大学 ， 基础数学， 2010， 博士

【摘要】 本文研究两类由带预应力的可压缩超弹性细棒在轴向拉伸作用下产生的局部化问题。其一为关于一般性材料的小局部化问题,另一为关于具体材料的大局部化问题。局部化通常表现为局部应变的集中,它反映了材料在局部大变形下性质的退化,故而其在材料结构安全评估中非常重要。目前已有不少关于这方面的研究。但是,在三维背景下,就我们所知,关于小局部化的分析研究还非常少,关于大局部化目前为止还没有此类研究。本文的主要目的是建立一个三维模型并求其分析解,进而用所得结果描述或预测一些关键的实验现象。本文的模型建立在小变形作用于有限均匀变形的理论之上,在数学中相当于对动力系统的平凡(常数)解作小的扰动。最初的结构方程是非常复杂的混合型偏微分方程组,我们通过耦合渐进展开法将其近似地转化为一个渐进模型方程。进一步将其转化成一阶非线性动力系统并利用相平面分析,我们得到在自然边界条件下的分析解。这些解是以积分形式表示的。对两类问题我们都讨论了细棒的几何形态(半径与长度之比)对非平凡解的影响。对小局部化问题,本文的结果能捕捉一些关键的实验现象,比如应力应变曲线的回溯现象,对细棒的不同几何形态应力应变曲线的峰值后行为的不唯一性,等等。Blatz-Ko材料是一种具体的用来模拟大变形的各向同性超弹性材料。为研究由其构成的细棒在外力作用下的大局部化问题,我们通过引进一种新方法,将耦合渐进展开法这一通常用来处理小应变问题的方法推广成能处理大应变问题。这是本文的主要贡献。我们的分析结果能对Blatz-Ko材料在大变形情形下的行为给出一些预测。更多还原

【Abstract】 In this thesis, we study two localization problems in pre-stressed slender cylin-ders composed of compressible hyperelastic materials subjected to axial forces. one is the small localization for materials with a general form, the other is the large localization for a specific material.Localization, represented as local strain concentration, is a manifestation of the degradation of material properties with localized large deformations. Due to its importance in structural safety assessment, much research has been, conducted to resolve experimental, theoretical and computational issues associated with localization problems. However, in a three-dimensional setting, the analytical studies on the small localization are quite few, and as far as we know, there is not any analytical result for the large localization available in literature.The main purpose of this thesis is to construct a three-dimensional.model and use it to present some analytical solutions to capture or predict some key experimental features.Our model is constructed on the basis of the theory of small elastic defor-mations superimposed on a finite elastic deformation,.which is represented as perturbations of a trivial (constant) solution of a dynamics system in mathe-matics. The coupled series-asymptotic expansion method is used to derive the normal form equation from the original complicated system of nonlinear PDEs. By writing the normal form equation into a first-order dynamical system and with a phase-plane analysis, we manage to solve the natural boundary value problems analytically. The asymptotic solutions in terms of integrals are obtained. And the influences of the geometry (radius-length ratio) of the cylinders on the properties of the non-trivial solutions are also discussed for the both two problems. The analytical results obtained for small strains can capture some key features in experiments by others, such as the snap-back phenomenon of the stress-strain response, the nonuniqueness of the post-peak behavior for different radius-length ratios, and so on.For the Blatz-Ko material, a specific isotropic hyperelastic material used to model large deformations, we develop the coupled series-asymptotic method, a localized method to deal with small strains, to a global one by introducing a novel methodology and use it to deal with large strains. This is the main contribution of this thesis. Our analytical results give some predictions for the behavior of Blatz-Ko material undergoing large deformations.更多还原