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

制造内部微观结构对外部刺激做出反应的功能材料变得越来越重要。它们通过改变形状和/或特性（如颜色、密度、导电性或弹性模量）为多功能部件和机器开辟了新的机会。制造这种动态结构的一种新技术是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，但关于具有微结构的材料行为的理论基础尚未得到充分发展。例如，目前的实验仅基于化学计量学。为了有效地开发多功能部件，期望预测这些部件受到外部刺激的精确反应。为了实现这一目标，有必要开发描述具有微观结构的材料行为的数学模型，该微观结构对外部刺激做出响应。最终，由于对实验的依赖性较小，这导致多功能部件的开发过程更有效。因此，提出了一种新的声学连续介质力学方法来制定和解决与具有变化的微观结构的弹性材料相关的问题。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，源项模拟材料中的结构变化。这种方法的有效性将在4D打印液晶弹性体（LCE）中得到证明。对于发展，几个实验的复杂性增加的LCE结构下的机械载荷或暴露于热刺激的定义。它们被形式化为初始边值问题（IBVP）。通过实验获得这些IBVP的参比溶液。为了定义IBVP的正式初始条件，还通过实验确定了3D打印LCE股线内的微观结构取向。此外，分析和数值解被发现使用导出的方程组。将通过使用Python库FEniCS实现的有限元（FE）程序计算数值解。对于简单的IBVP，分析解决方案被用来验证程序。数值和实验解决方案的任意复杂的IBVP的比较允许的材料参数的确定，并验证所得到的连续模型。此外，该工作流程允许在计算模型中包括额外的自由度并使用复杂的本构关系。在该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该项目未设想的任务。

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.