材料疲劳断裂的分数阶建模及动力学分析

分数阶微积分是刻画历史依赖、长距离相互作用以及遗传特性的有效工具。现已知道，分数阶微积分在粘弹性力学、流变学等方面有着重要的应用价值。近期发现，分数阶阿达马导数方程能够很好地描述材料疲劳断裂现象。虽然分数阶阿达马导数方程有众多的研究课题，但本项目重点研究分数阶阿达马导数在材料疲劳断裂分析中的应用以及研究分数阶阿达马导数方程的动力学行为。具体内容如下：(1) 建立准确反映材料疲劳断裂的分数阶阿达马导数方程，研究非线性分数阶阿达马导数方程的线性化问题，给出平衡点的稳定性判据。(2) 定义非线性分数阶阿达马导数方程的李雅普诺夫指数，给出李雅普诺夫指数界的估计。(3) 利用有限部分积分的思想，建立分数阶阿达马导数方程的数值仿真格式，并计算混沌系统的混沌吸引子。通过本项目的研究，推动分数阶动力学理论的深入发展，为现实应用奠定坚实的理论基础和提供强有力的技术支持。

Fractional calculus plays an efficient role in characterizing history dependence, long-range interactions, and heredity. It is known that fractional calculus has important applications in viscoelastic mechanics, rheology, etc. Recently, it has been found that fractional Hadamard derivatives can describe fatigue fracture of materials very well. Although there are lots of research topics in studying fractional Hadamard derivative equations (FHDEs), the presnt program mainly focuses on the applications of FHDEs in the fatigue crack of materials and the dynamical behaviors of FHDEs. In details, the contents of this program are planned as follows. (1) The FHDE model for fatigue fracture of materials will be established. The linearized theorem of nonlinear FHDE will be built up and the stability criteria of the equilibrium to FHDE will be given. (2) The definition of the Lyapunov exponents for FHDE will be proposed and their bounds will be determined. (3) The numerical scheme for FHDE will be established by using the idea of finite part integral. The chaotic attractor of chaotic FHDE will be computed and graphically presented. Through these studies, the applicants wish to deeply develop the theory of fractional dynamics in order to provide solid theoretical fundaments and strongly technical support for real applications.