Project Details
Projekt
A variational scale-dependent transition scheme: From Cauchy elasticity to the relaxed micromorphic continuum
Applicants
Professor Dr. Patrizio Neff; Professorin Dr.-Ing. Lisa Scheunemann; Professor Dr.-Ing. Jörg Schröder
Subject Area
Mathematics
Mechanics
Mechanics
Term
since 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 440935806
The overall goal of the project is to develop a physically motivated multiscale simulation environment without separation of length scales. In doing so, a relaxed micromorphic model (RMM) is defined on the macroscale and a Cauchy continuum on the microscale. This multiscale model will be developed especially for the simulation of metamaterials, starting from discrete microstructures. Based on the scientific findings of the first funding period, we aim to close the gap of the scale transition between both continua in the second funding period. For this purpose, a consistent homogenization procedure shall be developed. The finite element implementation is extended to a larger space than H(Curl) to achieve better flexibility in modeling. A central goal is to identify the effective moduli in the RMM. In a consistent two-scale approach, a suitable description of the effective (macroscopic) variables based on tensorial quantities of the microscale is necessary. Moreover, for the localization step, the identification of meaningful boundary conditions at the micro-scale is essential for the transfer of information between scales. The energetic equivalence between scales (Hill-Mandel condition) plays an important role in identifying these procedures. For all working points, we will address the following open questions in the field of higher order homogenization: 1. Is there a contribution to the strain energy from the fluctuations associated with higher order modes? In other words, do these fluctuations have to be periodic or not? If yes, then the second gradients of the deformation field cannot be controlled. 2. Does a unique RVE exist in the case of higher order theories? The analysis is often performed on a cluster of unit cells to remove boundary effects and account for convergent behavior in the central nit cell? This approach is questionable from an engineering perspective. 3. Does the higher order homogenization scheme provide convergent behavior for extreme cases, e.g., in a material with extremely hard and soft inclusions? This is not considered by asymptotic expansion homogenization as well as other theories. 4. Do the gradient effects disappear for homogeneous RVEs (which is a necessity)? 5. Does the classical first-order homogenization scheme arise when the separation of the scales is given?
DFG Programme
Priority Programmes
