Development of fundamental approaches for controlling the microstructure of liquid crystal elastomers in 4D printing based on extended micropolar theory

制造功能材料，其内部微观结构对外界刺激作出反应，变得越来越重要。它们通过改变形状和/或属性（如颜色、密度、电导率或弹性模量），为多功能零件和机器开辟了新的机会。一种制造这种动态结构的新技术是4D打印。该项目旨在为此类材料的研究和数学建模开发基本方法。尽管增材制造的实验和技术发展取得了重大进展，但关于材料微观结构行为的理论基础尚未得到充分发展。例如，目前的实验只是基于启发式。为了有效地开发多功能部件，需要准确地预测这些部件在外界刺激下的反应。为了实现这一目标，有必要开发数学模型来描述具有微观结构的材料的行为，这些微观结构响应外部刺激。最终，由于对实验的依赖较小，这使得多功能部件的开发过程更加高效。因此，提出了一种新的热声连续介质力学方法来阐述和解决微观结构变化的弹性材料问题。引入了考虑物质旋转自由度（DOF）的附加状态参数。它们的演化是基于与源项的平衡关系，模拟物质的结构变化。该方法的有效性将在4D打印液晶弹性体（LCE）中得到验证。为了发展，定义了几个在机械载荷或热刺激下增加复杂性的LCE结构实验。它们被形式化为初始边值问题（IBVP）。实验得到了这些ibvp的参考解。为了确定IBVPs的正式初始条件，还通过实验确定了3D打印LCE链内的微观结构取向。此外，利用推导出的方程组得到了解析解和数值解。数值解将由使用Python库FEniCS实现的有限元（FE）程序计算。对于简单的IBVPs，用解析解对程序进行了验证。对任意复杂性的IBVPs的数值解和实验解进行比较，既可以确定材料参数，也可以验证推导出的连续介质模型。此外，该工作流允许在计算模型中包含额外的自由度，并使用复杂的本构关系。在项目框架内开发的用于解决lce结构转换问题的软件随后可用于解决本项目未设想的任务。

Making functional materials, whose internal microstructure reacts to external stimuli, becomes increasingly important. They open up new opportunities for multifunctional parts and machines by changing their shape and/or properties, such as color, density, conductivity, or elastic moduli. One new technology to make such dynamic structures is 4D printing. This project aims at developing fundamental approaches for the study and mathematical modeling of such materials. Despite the significant advances in the experimental and technological development of additive manufacturing, the theoretical foundation regarding the behavior of materials with a microstructure has not yet been sufficiently developed. For instance, current experiments are only based on heuristics. For the efficient development of multifunctional parts, it would be desirable to predict the precise reaction of such parts subjected to an external stimulation. In order to achieve that, it is necessary to develop mathematical models describing the behavior of materials with a microstructure, which responds to external stimuli. Ultimately, due to the smaller dependence on experiments, this leads to a more efficient development process of multifunctional parts. Therefore, a new thermodynamically sound continuum mechanic approach to the formulation and solution of problems related to elastic materials with a changing microstructure is presented. Additional state parameters that take rotational Degrees Of Freedom (DOF) of matter into account are introduced. Their evolution is based on balance relations with source terms that simulate structural changes in the material. The effectiveness of this approach will be demonstrated for 4D printed liquid crystal elastomers (LCE). For the development, several experiments of increasing complexity for LCE structures under mechanical load or exposed to thermal stimulation are defined. They are formalized into Initial Boundary Value Problems (IBVP). Reference solutions to these IBVPs are obtained experimentally. In order to define formal initial conditions for the IBVPs, the microstructural orientation within 3D printed LCE strands is also determined experimentally. In addition, analytical and numerical solutions are found using the derived set of equations. The numerical solutions will be computed by a Finite Element (FE) program implemented using the Python library FEniCS. Regarding simple IBVPs, the analytical solutions are used to verify the program. The comparison of numerical and experimental solutions to IBVPs of arbitrary complexity allows for both the determination of material parameters, and the validation of the derived continuum model. Moreover, this workflow allows to include additional DOFs in the computational model and use complex constitutive relations. The software developed within the framework of the project for solving problems of structural transformations in LCEs can subsequently be used to solve tasks not envisaged by this project.