Modelling and computation of microstructured materials by generalised continua
Zusammenfassung der Projektergebnisse
The behavior of most materials (e.g. composites, polycrystals, granular media, soft tissues) is influenced by inhomogeneously distributed microscale properties. A reliable and efficient use of microstructured materials in engineering, nanotechnology, biomedical application and earth sciences requires and justifies high efforts in their mathematical modeling and numerical treatment. Since the classical continuum theories are not capable of capturing internal microstructures and size effects, generalized continuum theories have to be introduced. The principle focus of this research project was the numerical and theoretical treatment of generalized continua. Therefore, the focus has been on the following three different problems. First, the treatment and application of: micromorphic, microstructured continua, second, an extensive investigation and improvement of C1-continuous discretization methods for the treatment of the constrained micromorphic or gradient elastic theory and third, a comprehensive revision and characterization of generalized plasticity formulations. As mentioned above, the first part dealt with the micromorphic continuum theory. In an initial step, the material and spatial motion problem was formulated. A straightforward approach was used for the finite element implementation of the spatial and material motion problem of micromophic hyperelasticity. It was shown that the material force method is well suitable for the treatment of defect mechanics problems for materials with microstructure. Additionally, a micromorphic continuum approach was incorporated in a multiscale approach to capture the mesostructure of material layers. Subsequently, to deal with the task of gradient elasticity, a new computational methodology, the C1-natural element method (C1-NEM) has successfully been applied.The C1-NEM combines the easy imposition of essential boundary conditions and simple achievement of the C1-continuity requirement. Additionally, the C1-NEM was compared to different numerical methods such as several C1-continuous finite elements, closing a gap in the current state-of-the-art. Two additional aspects arose while using the C1-continuous finite elements. First, the algorithm presented and used for the application the gradient elasticity showed to be ineffcient. Therefore, a new mesh optimization tool has been proposed. Additionally, the usage of subparametric elements has shown significant errors, when the conventional procedure of linear interpolation of the boundary is used´. We could show that much better results can be achieved by a least square fit of the boundary. Additional work is done in the setup of a method to create geometries with irregular triangular meshes to further improve the performance of the elements.
