Mathematical modelling of mechanics of nonsingular defects in elastic and electro-elastic materials at small scales using gradient-enhanced continuum field theories

智能、先进或新颖材料等现代材料的研究在材料与工程科学、应用数学、固体物理和纳米力学中具有重要意义，由于它们在新兴技术中的大量应用而不断受到关注。这类材料的主要特点是其复杂的微观结构、电弹性耦合效应以及缺陷的存在。对于在这些材料中发生的各种现象的物理现实描述，需要高阶广义连续介质场论，因为它们能够捕获在小尺度上的现象，而经典连续介质理论无效，导致非物理奇点，因此它们是重要的。另一方面，由于高阶项要求使用严格的数学方法，增加了数学建模的复杂性。该项目的目的是在小尺度上提供具有缺陷，微观结构和电弹性耦合效应的材料力学的有效数学模型，系统地发展，扩展和评估，如果必要的话，梯度增强连续体场理论。本项目的重点在于梯度增强弹性、电弹性和塑性连续体理论，其内部特征长度对于描述经典理论中没有出现的柔性电和纳米压电等效应非常重要。该项目将为材料在小尺度上的力学建模提供独特、创新和重要的贡献，其中缺陷、潜在的微观结构和电弹性耦合效应起着至关重要的作用，使用和发展梯度增强连续体场理论，有效到Ångström-scale，在纳米技术中有很好的应用。

The study of modern materials like smart, advanced or novel materials is of great importance in materials and engineering sciences, applied mathematics, solid state physics and nanomechanics receiving continuously growing interest due to their numerous applications in new and emerging technologies. To the main characteristics of such materials belong their complex microstructure, electro-elastic coupling effects as well as the presence of defects. For the physically realistic description of various phenomena occurring in these materials, generalized continuum field theories of higher-order are required, since they are able to capture phenomena at small scales where classical continuum theories are not valid leading to unphysical singularities and for this reason they are significant. On the other hand, the complexity of the mathematical modelling is increased due to the higher-order terms demanding the use of rigorous mathematical methods. The aim of the project is providing an efficient mathematical modelling of mechanics of materials with defects, microstructure and electro-elastic coupling effects at small scales developing systematically, extending and assessing, if necessary, gradient-enhanced continuum field theories. The focus of this project lies on gradient-enhanced elasticity, electroelasticity and plasticity continuum theories with internal characteristic lengths important for the description of effects such as flexoelectricity and nanopiezoelectricity, which do not appear in classical theories. This project will provide a unique, innovative and important contribution to the modelling of mechanics of materials at small scales where defects, the underlying microstructure and electro-elastic coupling effects play an essential role, using and developing gradient-enhanced continuum field theories valid down to the Ångström-scale with promising applications in nanotechnology.