几类非线性固体结构系统的整体动力行为研究

发布时间：2018-04-23 22:39

本文选题：无穷维动力系统 + 长时间动力行为　；参考：《太原理工大学》2016年博士论文

【摘要】：梁、板、壳结构是工程领域中基本且至关重要的承重构件,在长期复杂服役环境下,其稳定性直接影响整个构件的使用寿命。因此,研究这些构件的长时间动力行为及其动力稳定性具有重要的理论价值和实际意义。大多数工程中的弹性构件实际上接近力学系统的连续体,在讨论有势力场作用下的力学系统稳定性时,连续系统的稳定性涉及到数学上的非线性偏微分方程、无限自由度动力系统的定性研究以及无限维空间的几何理论等,因此,基于动力学观点对非线性弹性系统稳定性开展系统研究成为关注的热点和焦点。近年来,关于弹性梁、板方程(组)解的存在性、唯一性、渐近性等动力行为的研究取得了许多可喜的成果。然而,对于解的长时间动力行为研究的结果相对较少。由于吸引子是描述时间趋于无穷大时系统的长时间动力行为的重要指标,而分形维数是刻划吸引子的几何特征量,所以吸引子的存在性及其维数估计成为无穷维动力系统研究的重要课题,也是近年来比较活跃的前沿问题。在本文中,我们针对几类具有强阻尼、结构阻尼或外阻尼的固体结构系统作了以下工作。首先,研究了一类满足Dirichlet边界条件的具有强阻尼和外阻尼Kirchhoff型非自治弹性梁系统解的长时间动力行为。利用算子半群理论证明了系统连续解的存在唯一性;把自治系统的半群理论推广到非自治系统的过程理论,通过引入等价范数,在一定条件下,利用能量一致先验估计得到系统所生成的过程的有界吸收集;通过过程分解技术,构造恰当的能量泛函,将过程分解成两部分,使得一部分满足紧致性,而另一部分满足压缩性质,成功地证明了所对应过程的紧的核截面的存在性,从而得到系统所生成的过程的一致吸引子的存在性。其次,研究了一类具有强阻尼和结构阻尼Kirchhoff型热弹梁耦合系统解的长时间动力行为。在系数的一定范围内,利用算子半群理论证明了系统存在唯一的mild解;以半群理论为依据,构造合适的泛函,获得等价的泛函系统,利用能量一致先验估计得到半群的有界吸收集,进而证明了系统所生成的解半群的整体吸引子的存在性。最后,研究了一类具有强阻尼的热弹板耦合系统解的长时间动力行为。利用算子半群理论证明了系统存在唯一的连续解;通过引入等价范数,能量方法和一系列精细的先验估计得到半群的有界吸收集,进而证明了系统所生成的解半群存在整体吸引子;通过变分方法与能量一致先验估计得到吸引子的Hausdorff维数估计。
[Abstract]:Beam, plate and shell structure are basic and important load-bearing components in engineering field. Under the long-term complex service environment, their stability directly affects the service life of the whole member. Therefore, it is of great theoretical and practical significance to study the long-term dynamic behavior and its dynamic stability of these components. Most elastic members in engineering are actually close to the continuum of the mechanical system. When discussing the stability of the mechanical system under the action of the force field, the stability of the continuous system is related to the mathematical nonlinear partial differential equation. The qualitative study of infinite degree of freedom dynamical system and the geometric theory of infinite dimensional space, etc., therefore, the systematic research on the stability of nonlinear elastic system based on the viewpoint of dynamics has become a hot spot and focus. In recent years, many gratifying results have been obtained in the study of the existence, uniqueness and asymptotic behavior of the solutions of elastic beam and plate equations. However, there are relatively few results on the long-term dynamic behavior of solutions. Because the attractor is an important index to describe the long-time dynamic behavior of the system when the time tends to infinity, the fractal dimension is the geometric characteristic quantity of the attractor. Therefore, the existence of attractor and its dimension estimation have become an important subject in the study of infinite dimensional dynamical systems, and are also active frontier problems in recent years. In this paper, we have done the following work for several solid structural systems with strong damping, structural damping or external damping. Firstly, the long-time dynamic behavior of a class of Kirchhoff type nonautonomous elastic beam systems with strong damping and external damping satisfying the Dirichlet boundary condition is studied. The existence and uniqueness of continuous solution are proved by using operator semigroup theory, the semigroup theory of autonomous system is extended to the process theory of nonautonomous system, and the equivalent norm is introduced, under certain conditions, The bounded absorption set of the process generated by the system is obtained by using the energy consistent prior estimation, and the proper energy functional is constructed by the process decomposition technique, and the process is decomposed into two parts, so that one part satisfies the compactness. The other part satisfies the squeezing property, and proves the existence of the compact kernel cross section of the corresponding process successfully, and thus obtains the existence of the uniform attractor of the process generated by the system. Secondly, the long-time dynamic behavior of a Kirchhoff type thermoelastic beam coupling system with strong damping and structural damping is studied. In a certain range of coefficients, the existence of a unique mild solution is proved by using the operator semigroup theory, and an equivalent functional system is obtained by constructing a proper functional based on the semigroup theory. The bounded absorption set of Semigroups is obtained by energy uniform prior estimation, and the existence of global attractors of solution Semigroups generated by the system is proved. Finally, the long time dynamic behavior of a coupled thermoelastic plate system with strong damping is studied. The existence of unique continuous solution is proved by using operator semigroup theory, and the bounded absorption set of semigroup is obtained by introducing equivalent norm, energy method and a series of fine prior estimates. Furthermore, it is proved that there exists a global attractor in the solution semigroup generated by the system, and the Hausdorff dimension estimation of the attractor is obtained by using the variational method and the energy consistent prior estimation.
【学位授予单位】：太原理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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