Julian Marr

• The selection of suitable materials represents a demanding task in the context of the increasing electrification of vehicles. This is mainly due to the fact that not only mechanical but also multiphysical requirements for the material properties have to be taken into account simultaneously. An example of this is the requirement for an electrically insulating yet good heat-conducting material with low thermal expansion. Polymer-based composites offer the possibility of specifically influencing the physical material properties by the use of suitable fillers. The macroscopic material properties result from the cause-effect principle: manufacturing process \(\rightarrow\) structure \(\rightarrow\) properties depending on the complex microstructure of the polymer matrix composite. The design of materials requires the consideration of this process chain in the reverse order and can be regarded as an inverse problem. Consequently, the design of such functional materials is diverse and sophisticated. The present work aims to answer the question of how the microstructure of composite materials must be tailored in order to exhibit the desired effective thermomechanical material behavior. A purely experimental development of such materials is generally a time-consuming and cost-intensive process, as many prototypes have to be produced and tested. Computer-aided multiscale methods enable a virtual characterisation of composites by a direct modelling of the heterogeneity at the microstructure level. These methods offer the potential to replace expensive experiments and to shorten the product development time. In the work at hand, a simulation-based optimization methodology for the design of functional materials is presented. For that purpose, a metamodel-based approach is proposed, which relies on the approximation of the structure-property relationship (SPR) in dependence of physical parameters of the microstructure. This enables an efficient solution of multiphysical material optimization problems in high-dimensional parameter spaces. The developed methodology is subsequently applied to the optimization of thermomechanical properties of particle reinforced and short fiber reinforced polymers. Chapter 2 covers some basic aspects of continuum mechanics for the modeling of thermoelastic materials and heat conduction. The introduced notations and relations are applied in the following sections to compute effective material properties of composites by numerical simulation at the microscopic level. In Chapter 3, the metamodel-based optimization methodology for the design of functional composites is presented. For this purpose, a structural optimization problem is formulated for the material design. A description of the macroscopic material properties in terms of tensor quantities enables a consideration of direction-dependent (anisotropic) requirements imposed on materials. As design variables, physical parameters of the microstructure are selected, since these can be directly affected during the manufacturing process. These comprise parameters of the geometrical microstructure (e.g., the filler volume fraction) and the material properties (e.g., the Young's modulus) of the individual constituents. Several aspects such as the large computational effort and the influence of stochastic effects caused by the heterogeneous microstructure, motivate the development of the surrogate-based optimization methodology. This is based on an approximation of the SPR within the entire design space. A major advantage of such metamodels is the efficient use of global optimization methods compared to the direct application of the micromechanical simulation model, especially for high-dimensional problems. To create the metamodels, in a first step certain parameter combinations (experimental designs) are specified on the basis of an experimental design scheme. For each of these designs, a representative volume element is generated that represents the particular configuration of the microstructure. The corresponding effective material properties are computed by numerical homogenization under specification of periodic boundary conditions. As a result of these steps, a material database is provided that describes the SPR for a discrete set of parameterized microstructures. In the next step, the effective properties are approximated within the entire parameter space on the basis of this material data. Polynomial response surface models and Kriging interpolations are introduced for that purpose. The use of (global) sensitivity analysis methods to quantify the relative importance of the different design variables is discussed. Moreover, a simple concept for the evaluation of the robustness of the designs and for robust optimization is presented. Chapter 4 provides a compact overview of particle reinforced and short fiber reinforced polymers. In addition to a description of the matrix and filler materials relevant for technical applications and their influence on the macroscopic properties, the manufacturing and application areas of these materials are briefly discussed. In Chapter 5, the metamodel-based optimization method is demonstrated by means of specific material examples. For that purpose, in Section 5.1 a three-phase particle reinforced material with large contrast in the thermal conductivities of the individual constituents is considered, where a certain thermal conductivity is to be adjusted. The filler volume fraction and phase fractions as well as geometrical parameters of the filler particles are considered as design variables. It turns out that with a relatively small number of samples a reasonably good approximation of the macroscopic thermal conductivity can be obtained. The filler volume fraction and the phase fractions prove to be significant influencing variables, in addition to the aspect ratio of the platelet-shaped particles. As optimization method, differential evolution is employed to determine different optimal microstructures. A validation of the optimization results by an evaluation of the micromechanical model shows only slight deviations. Subsequently, a gradient-based optimization method and a sequential metamodel-based method are utilized as alternative optimization methods and are compared with the presented approach. A validation of the homogenization method by experimental measurements shows a satisfactory agreement. In Section 5.2, the methodology is applied to optimize the viscoelastic properties of short fiber reinforced polymers. In addition to the fiber volume fraction, the fiber orientation distribution and the linear elastic material properties of the fiber material are considered as design variables. Within the concept of the second-order Advani-Tucker fiber orientation tensor, the fiber orientation distribution can be described by only two parameters lying within the fiber orientation triangle. A generalized Maxwell model is used to model the viscoelastic behavior of the polymer matrix and the corresponding material parameters are determined based on experimentally determined creep curves. A viscoelastic material database is established from the results of creep simulations. This is used to create Kriging interpolations of the orthotropic components of the creep compliance tensor. Differential evolution is employed to solve the material optimization problem considering the constrained design space imposed by the fiber orientation triangle. A validation of the optimization results using the micromechanical model shows only minor deviations here as well. Finally, the proposed method is applied to solve multiphysical optimization problems in Section 5.3. To consider thermal and mechanical requirements of the three-phase material simultaneously, a multi-objective optimization problem is formulated. A relatively large number of ten design variables are considered, which include the thermomechanical properties of the fillers in addition to the filler volume fraction and the phase fractions. Based on the material databases generated separately for thermal and mechanical properties, Kriging models are created to approximate the effective thermomechanical properties. An genetic algorithm is applied to the metamodels for optimization. The application of the method is demonstrated by two examples. The results show that by changing the phase fractions, the thermal expansion can be optimized at the expense of the thermal conductivity. Compared to the unrestricted problem, compliance with a maximum macroscopic Young's modulus in the form of a constraint can be achieved by reducing the Young's modulus of the platelet-shaped particles. The main findings of this work can be summarized as follows: The large number of function evaluations required when using global optimization methods is associated with a large numerical effort caused by the generation of representative volume elements and the simulation of different load cases. For this purpose, a surrogate-based optimization methodology is proposed in that work, which partly enables an economical solution of thermomechanical structural optimization problems at all. Further advantages of this method are the efficient identification of different design alternatives, the consideration of varying target properties without the necessity of renewed numerical simulations, as well as the evaluation of the robustness of different designs and the sensitivity of individual parameters. Existing models can also be extended to include other physical properties and can be used for multiphysical optimization. In contrast to existing works, this thesis presents a holistic process chain for the optimization of physical parameters of the microstructure of particle reinforced and short fiber reinforced polymers. A validation of the optimization results obtained with this methodology by comparison with an evaluation of the numerical model shows only minor deviations for the presented examples. The approach proves to be quite general and can also be applied to other types of composite materials. In combination with advanced manufacturing techniques and the selection of suitable fillers, the proposed method offers the potential to accelerate the product development process and to reduce the number of time-consuming experiments.