摘 要
随着材料科学的不断进步，对粘弹性材料性能的理解和研究愈发深入。粘弹性材料因其独特的应力-应变关系，即在受到外力作用时既表现出弹性又表现出粘性的特性，而被广泛应用于各种工程领域。在粘弹性材料的建模过程中，分数阶微分方程因其独特的性质而展现出显著的优势。本文旨在探讨分数阶微分方程在粘弹性材料建模中的应用，分析其优势并讨论相关研究进展。在粘弹性材料的建模中，传统的整数阶微分方程往往难以准确描述材料的应力-应变关系。而分数阶微分方程则通过引入分数阶导数，提供了一种更为灵活和准确的建模工具。分数阶导数的引入，使得模型能够更好地捕捉粘弹性材料的记忆效应和松弛特性，从而更准确地描述材料的力学行为。在分数阶微分方程的应用中，研究人员通过建立分数阶粘弹性本构关系，实现了对粘弹性材料力学行为的精确描述。这些模型不仅能够准确模拟材料的静态力学行为，还能够有效地模拟材料的动态力学行为，如蠕变、应力松弛等。此外，分数阶微分方程还可用于研究粘弹性材料的频率响应和阻尼特性，为材料的设计和优化提供了有力的支持。在分数阶微分方程的研究中，一些重要的进展值得关注。
关键词：分数阶微分方程 粘弹性材料 建模
Abstract
With the continuous progress of materials science, the understanding and research of viscoelastic material properties are getting deeper and deeper. Viscoelastic materials are widely used in various engineering fields because of their unique stress-strain relationship, that is, they exhibit both elasticity and viscosity when subjected to external forces. In the modeling process of viscoelastic materials, fractional differential equations show significant advantages because of their unique properties. This paper aims to discuss the application of fractional differential equation in viscoelastic material modeling, analyze its advantages and discuss the related research progress. In the modeling of viscoelastic materials, it is often difficult to accurately describe the stress-strain relationship of the materials with the traditional integer order differential equations. Fractional differential equations provide a more flexible and accurate modeling tool by introducing fractional derivatives. With the introduction of fractional derivatives, the model can capture the memory effect and relaxation characteristics of viscoelastic materials better, so as to describe the mechanical behavior of materials more accurately. In the application of fractional-order differential equations, researchers have achieved an accurate desc ription of the mechanical behavior of viscoelastic materials by establishing fractional-order viscoelastic constitutive relations. These models can not only accurately simulate the static mechanical behavior of materials, but also effectively simulate the dynamic mechanical behavior of materials, such as creep and stress relaxation. In addition, fractional differential equations can also be used to study the frequency response and damping characteristics of viscoelastic materials, which provides strong support for the design and optimization of materials. In the study of fractional differential equations, some important advances are worth paying attention to.
Keywords: Fractional differential equation Viscoelastic material modeling
目 录
1 引言 1
2 分数阶微分方程基础 1
2.1 分数阶微分方程的基本概念 1
2.2分数阶微分方程的数学属性 2
3 分数阶模型在粘弹性材料中的识别与响应分析 2
3.1 粘弹性材料的分数阶本构关系 2
3.2 分数阶模型的建立与参数识别 3
3.3 分数阶模型在动态分析中的应用 3
4 分数阶模型在具体粘弹性材料中的应用 4
4.1 聚合物材料的分数阶模型应用 4
4.2 复合材料的分数阶模型应用 4
4.3 生物材料的分数阶模型应用 4
5 结论 5
致 谢 6
参考文献 7
