Mathematical analysis and modeling of the evolution of magnetoelastic materials

几十年来，磁弹性材料因其迷人的性质和在航空、生物医学、能源技术等领域作为驱动器和传感器的广泛应用而引起了人们的技术和学术兴趣。人们已经发现和研究了一些新型材料，如超磁致伸缩材料、磁弹性薄膜、铁磁形状记忆合金、磁弹性超材料和泡沫或磁性流体。对这类材料的彻底数学理解需要先进的分析工具以及与工程师和物理学家的跨学科合作。在这个项目中，我们专注于考虑大变形和微磁性的磁粘弹性材料的创新的随时间变化的数学模型。虽然有相当多的文献关于磁弹性材料的静态模型，但关于时间相关系统的出版物在很大程度上仅限于弹性或磁效应。对于弹性效应、磁效应和时间演化的耦合，我们采用了一种新的方法，该方法由Pi、她的团队前成员和刘春共同发起和发展。对于均匀材料，关于解的存在唯一性的几个结果已经被证明，这也是正在进行的项目的一部分。这一建议的目的是借助锐界面模型和弥散界面模型来推进非均质磁粘弹性材料的建模。在非均质材料的界面上，力学和/或磁性发生了巨大的变化。这产生了一种较弱的正则性，这在分析研究相应的偏微分方程组的适定性时引起了特殊的数学挑战。此外，我们将加强关于磁粘弹性材料数学模型的跨学科讨论，这是对这种材料有充分基础的理解所必需的，从而也将促进知识从数学研究向材料科学的转移。

Magnetoelastic materials have been of technological and academic interest for decades due to their fascinating properties and various applications, e.g., as actuators and sensors in aeronautics, biomedicine, energy technology etc. Several novel materials have been found and studied like giant magnetostrictive materials, magnetoelastic membranes, ferromagnetic shape-memory alloys, magnetoelastic metamaterials and foams or magnetic fluids. A thorough mathematical understanding of such materials requires advanced analytical tools and interdisciplinary cooperations with engineers and physicists. In this project we focus on innovative time-dependent mathematical models for magnetoviscoelastic materials allowing for large deformations and micromagnetism. While there is quite some literature on static models for magnetoelastic materials, the publications on time-dependent systems are largely limited to either elastic or magnetic effects. For the coupling of elastic effects, magnetic effects and temporal evolution, we apply a novel approach which was initiated and developed by the PI, former members of her team and Chun Liu. For homogeneous materials, several results on the existence and uniqueness of solutions have been proved yet, also as part of the ongoing project. The objective of this proposal is to advance the modelling of heterogeneous magnetoviscoelastic materials with the help of sharp as well as diffuse interface models. At interfaces of heterogeneous materials, mechanical and/or magnetic properties change drastically. This yields a weaker regularity that causes special mathematical challenges in the analytical investigation of the well-posedness of the corresponding systems of partial differential equations. In addition, we will intensify the interdisciplinary discussion on the mathematical modeling of magnetoviscoelastic materials required for a well-founded understanding of such materials and will thus also promote the transfer of knowledge from mathematical research to materials science.