具有缺陷的超弹性材料球体中的空穴分岔
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摘要
由于材料内部空穴的生成、增长以及相邻空穴的贯通是材料损伤和破坏的重要机理，因此如何预测材料内部的空穴生成和增长引起了广泛关注。本文利用分岔理论和动力学理论中的基本观点、方法和结论，对具有缺陷的超弹性材料组成的球体内部的空穴生成和增长的拟静态问题以及空穴生成后随时间的运动规律等方面进行了系统的研究，得到了一些新的结论，同时给出了相应的数值分析。
本文的主要工作如下：
1．第三章中，研究了具有缺陷的不可压超弹性材料组成的球体在表面均布的拉伸死载荷作用下的空穴分岔问题。得到了描述球体内部的空穴生成和增长的空穴分岔方程。证明了空穴分岔方程的平凡解支上存在唯一的分岔点；给出了左分岔和右分岔的判别条件，并指出了向左分岔的空穴分岔解还会发生二次转向点分岔。利用奇点理论给出了空穴分岔方程在分岔点的等价正规形。将缺陷参数分成若干个区域，证明了当缺陷参数属于某些区域时，具有缺陷的超弹性材料球体发生空穴分岔时的临界载荷比无缺陷的小；并利用最小势能原理讨论了空穴分岔方程的解在各个参数区域内的稳定性，解释了空穴生成的突变现象。
2．第四章中，分别研究了两类具有缺陷的可压缩超弹性材料球体在表面均布的径向拉伸作用下的空穴分岔问题。得到了两类材料球体的径向变形函数的参数型解析解和参数型空穴分岔解，同时指出了本章中给出的第一种缺陷材料组成的球体与无缺陷材料的相比会提前发生空穴分岔。最后讨论了空穴分岔方程的解的稳定性。
3．第五章中，研究了具有缺陷的不可压超弹性材料组成的球体在表面均布的拉伸死载荷作用下的径向运动问题。得到了与时间相关的空穴运动方程。以一类具有缺陷的不可压超弹性材料组成的球体为例，研究了材料缺陷对空穴运动方程的非平凡解的存在性及其定性性质的影响。证明了当拉伸载荷超过其临界值时，球体内部就会有空穴生成，并且证明了空穴生成后随时间的运动是它的导数具有第一类间断点的周期运动。
4．第六章中，研究了具有缺陷的不可压超弹性材料球壳在内、外表面分别受到突加死载荷作用的径向运动问题。得到了描述球壳内表面的径向运动的二阶非线性常微分方程。对微分方程进行了动力学行为的分析，讨沦了各个参数对方程解的定性性质的影响。给出了球壳产生周期振动和最终被破坏的数值算例。讨论了材料的缺陷参数对球壳产生周期振动的影响。同时给出了相应的数值模拟。
5．第七章中，研究了周期阶梯加载时各向同性的neo-Hookean材料组成的球体
内部的空穴生成和运动.‘首先对给定的常值拉仲载荷作用的问题进行了动力学行为
分析，并且指出了不同初始条件下的(广义)周期解的存在性.然后对周期阶梯加载
函数，研究了空穴运动方程的解的定性性质，指出了在不同的加载形式和不同初始
条件下的(广义)周期解的存在性.同时给出了相应的数值模拟.
Generally, cavity formation, growth and run-through of the adjacent cavities in solid materials are considered as important mechanisms of failure and fracture. Thus, prediction of cavity formation and growth in materials has long attracted much attention. In this dissertation, by means of the basic viewpoints, methods and conclusions of bifurcation theory and dynamical theory, the quasi-static problems of cavity formation and growth, motion rules of the formed cavity along with time in the interior of the spheres that are composed of imperfect hyper-elastic materials are discussed systemically. Some new results are obtained and the corresponding numerical analyses are carried out. The main works are as follows:
1. In Chapter 3, problems of cavitated bifurcation for a solid sphere composed of imperfect incompressible hyper-elastic materials, subjected to a surface tensile dead load, are examined. The cavitated bifurcation equation that describes cavity formation and growth in the interior of the sphere is obtained. It is proved that there exists a unique bifurcation point on the trivial solution of the cavitated bifurcation equation. And the distinguished conditions of left-bifurcation and right-bifurcation are carried out. It is presented that secondary turning point can be occurred on the cavitated bifurcation solution which bifurcates locally to the left. Equivalent normal forms of the cavitated bifurcation equation at the bifurcation point were presented by using singularity theory. The imperfect parameters' plane is divided into several regions, and it is proved that the critical dead load for the sphere composed of imperfect hyper-elastic materials is less than that for perfect hyper-elastic materials as
the imperfect parameter belongs to some regions. Stability of solutions of the cavitated bifurcation equation is discussed in each region by using the minimal potential principle, and the catastrophic phenomenon of cavity formation is explained.
2. In Chapter 4, problems of cavitated bifurcation for two spheres, which are composed of two different kinds of imperfect compressible hyper-elastic materials, were examined, respectively. The parametric analytic solutions of the radially deformed function and the parametric cavitated bifurcation solutions are all obtained for the two spheres. It is pointed out that the critical stretch of the sphere composed of the first kind of materials given in this chapter is less than that for the perfect materials. Finally, stability of solutions of the cavitated bifurcation equation is discussed.
3. In Chapter 5, problem of radial motion for a solid sphere composed of an imperfect incompressible hyper-elastic material, subjected to a surface tensile dead load, is examined. And a cavity motion equation with respect to time is obtained. As an example that the sphere is composed of an imperfect incompressible hyper-elastic material, effects of material imperfection on the existence and the qualitative properties of the nontrivial solutions of the cavity motion equation were studied. It is proved that a cavity forms in the interior of the sphere as the surface tensile dead load exceeds certain critical value, and the motion of the formed cavity along
with time is periodic motion which the derivative of the solution has the first kind of discontinuous point.
4. In Chapter 6, problem of radial motion for a spherical shell composed of an imperfect incompressible hyper-elastic material under sudden dead loads at the inner- and outer-surface respectively, is examined. And a second-order nonlinear differential equation that describes radial motion of the inner-surface of the sphere is obtained. Dynamical behavior of the differential equation is analyzed, and effects of each parameter on the qualitative properties of the solutions of the equation are discussed in detail. After that, numerical examples of period vibration and destroy of the spherical shell are presented. The effects of material imperfection on period vibration of the spherical shell are discussed.
本文的主要工作如下：
1．第三章中，研究了具有缺陷的不可压超弹性材料组成的球体在表面均布的拉伸死载荷作用下的空穴分岔问题。得到了描述球体内部的空穴生成和增长的空穴分岔方程。证明了空穴分岔方程的平凡解支上存在唯一的分岔点；给出了左分岔和右分岔的判别条件，并指出了向左分岔的空穴分岔解还会发生二次转向点分岔。利用奇点理论给出了空穴分岔方程在分岔点的等价正规形。将缺陷参数分成若干个区域，证明了当缺陷参数属于某些区域时，具有缺陷的超弹性材料球体发生空穴分岔时的临界载荷比无缺陷的小；并利用最小势能原理讨论了空穴分岔方程的解在各个参数区域内的稳定性，解释了空穴生成的突变现象。
2．第四章中，分别研究了两类具有缺陷的可压缩超弹性材料球体在表面均布的径向拉伸作用下的空穴分岔问题。得到了两类材料球体的径向变形函数的参数型解析解和参数型空穴分岔解，同时指出了本章中给出的第一种缺陷材料组成的球体与无缺陷材料的相比会提前发生空穴分岔。最后讨论了空穴分岔方程的解的稳定性。
3．第五章中，研究了具有缺陷的不可压超弹性材料组成的球体在表面均布的拉伸死载荷作用下的径向运动问题。得到了与时间相关的空穴运动方程。以一类具有缺陷的不可压超弹性材料组成的球体为例，研究了材料缺陷对空穴运动方程的非平凡解的存在性及其定性性质的影响。证明了当拉伸载荷超过其临界值时，球体内部就会有空穴生成，并且证明了空穴生成后随时间的运动是它的导数具有第一类间断点的周期运动。
4．第六章中，研究了具有缺陷的不可压超弹性材料球壳在内、外表面分别受到突加死载荷作用的径向运动问题。得到了描述球壳内表面的径向运动的二阶非线性常微分方程。对微分方程进行了动力学行为的分析，讨沦了各个参数对方程解的定性性质的影响。给出了球壳产生周期振动和最终被破坏的数值算例。讨论了材料的缺陷参数对球壳产生周期振动的影响。同时给出了相应的数值模拟。
5．第七章中，研究了周期阶梯加载时各向同性的neo-Hookean材料组成的球体
内部的空穴生成和运动.‘首先对给定的常值拉仲载荷作用的问题进行了动力学行为
分析，并且指出了不同初始条件下的(广义)周期解的存在性.然后对周期阶梯加载
函数，研究了空穴运动方程的解的定性性质，指出了在不同的加载形式和不同初始
条件下的(广义)周期解的存在性.同时给出了相应的数值模拟.
Generally, cavity formation, growth and run-through of the adjacent cavities in solid materials are considered as important mechanisms of failure and fracture. Thus, prediction of cavity formation and growth in materials has long attracted much attention. In this dissertation, by means of the basic viewpoints, methods and conclusions of bifurcation theory and dynamical theory, the quasi-static problems of cavity formation and growth, motion rules of the formed cavity along with time in the interior of the spheres that are composed of imperfect hyper-elastic materials are discussed systemically. Some new results are obtained and the corresponding numerical analyses are carried out. The main works are as follows:
1. In Chapter 3, problems of cavitated bifurcation for a solid sphere composed of imperfect incompressible hyper-elastic materials, subjected to a surface tensile dead load, are examined. The cavitated bifurcation equation that describes cavity formation and growth in the interior of the sphere is obtained. It is proved that there exists a unique bifurcation point on the trivial solution of the cavitated bifurcation equation. And the distinguished conditions of left-bifurcation and right-bifurcation are carried out. It is presented that secondary turning point can be occurred on the cavitated bifurcation solution which bifurcates locally to the left. Equivalent normal forms of the cavitated bifurcation equation at the bifurcation point were presented by using singularity theory. The imperfect parameters' plane is divided into several regions, and it is proved that the critical dead load for the sphere composed of imperfect hyper-elastic materials is less than that for perfect hyper-elastic materials as
the imperfect parameter belongs to some regions. Stability of solutions of the cavitated bifurcation equation is discussed in each region by using the minimal potential principle, and the catastrophic phenomenon of cavity formation is explained.
2. In Chapter 4, problems of cavitated bifurcation for two spheres, which are composed of two different kinds of imperfect compressible hyper-elastic materials, were examined, respectively. The parametric analytic solutions of the radially deformed function and the parametric cavitated bifurcation solutions are all obtained for the two spheres. It is pointed out that the critical stretch of the sphere composed of the first kind of materials given in this chapter is less than that for the perfect materials. Finally, stability of solutions of the cavitated bifurcation equation is discussed.
3. In Chapter 5, problem of radial motion for a solid sphere composed of an imperfect incompressible hyper-elastic material, subjected to a surface tensile dead load, is examined. And a cavity motion equation with respect to time is obtained. As an example that the sphere is composed of an imperfect incompressible hyper-elastic material, effects of material imperfection on the existence and the qualitative properties of the nontrivial solutions of the cavity motion equation were studied. It is proved that a cavity forms in the interior of the sphere as the surface tensile dead load exceeds certain critical value, and the motion of the formed cavity along
with time is periodic motion which the derivative of the solution has the first kind of discontinuous point.
4. In Chapter 6, problem of radial motion for a spherical shell composed of an imperfect incompressible hyper-elastic material under sudden dead loads at the inner- and outer-surface respectively, is examined. And a second-order nonlinear differential equation that describes radial motion of the inner-surface of the sphere is obtained. Dynamical behavior of the differential equation is analyzed, and effects of each parameter on the qualitative properties of the solutions of the equation are discussed in detail. After that, numerical examples of period vibration and destroy of the spherical shell are presented. The effects of material imperfection on period vibration of the spherical shell are discussed.
引文
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