B09 - Materials with discontinuities on many scales
Head(s): Dr. Martin Heida (WIAS), Prof. Dr. Marita Thomas (FU Berlin)
Participating institution(s): FU Berlin, WIAS
Research area: Mathematics
The evolution of macroscopically visible material defects, such as geological faults in the lithosphere or fractures in engineering structures, originates and is influenced by material discontinuities on much smaller scales: Interfaces where components of different elastic properties or rheology meet, e.g., grain boundaries, provide a natural location for failure and crack propagation. Often plastic deformation precedes the onset of fracture. The resulting heterogeneities and discontinuities appear to be random in their spatial distribution or geometry. The goal of this project is to develop novel mathematical multiscale models for the onset and propagation of fracture by combining analytical methods for dissipative solids , delamination , and phase-field fracture models [9, 10] with stochastic and fractal homogenization [2–5]. For
this, different upscaling scenarios will be investigated also paying attention to the variational structure of the small-scale models [TH21] by means of the framework of GENERIC (General Equation of Non-Equilibrium Reversible Irreversible Coupling).
In previous phases of CRC 1114 in  the concept of fractal homogenization was developed for elliptic operators coupled to an iteratively constructed set of interfaces. The method considers a hierarchy of models, each originating from the interruption of the self-similar iterative construction after K steps and accounts for the modeling error resulting from this interruption. The first scenario aims at extending this concept to constrained volume damage models with an additional parameter h accounting for the typical thickness of damageable layers. Here the convergence properties of approximating solutions and their relation as h ! 0 to those of models with sharp cracks across all scales have to be investigated (Figure 1).
The second scenario starts from a prescribed ergodic random network of sharp interfaces along which delamination may develop in terms of adhesive contact or brittle fracture , governed by a coupled system of nonlinear, nonsmooth partial differential equations. By combining methods from  with stochastic homogenization [2, 5] novel multiscale phase-field
fracture models for crack nucleation shall be deduced. In this scenario we have two small parameters: # relates to the grain size that results from the microscopically prescribed network and h relates to the average thickness of cracks in the upscaled phase field model (Figure 2 a)).
Finally, in a third scenario, the coarse graining techniques to be developed for above scenarios shall be further generalized to the mutual coupling of materials with different plastic and viscoelastic rheologies and to the interaction of the bulk material with damage and friction processes along sharp and diffuse interfaces. The results shall be incorporated into models of B01 and C09 and help to build a bridge between small-scale (B08) and large scale models (B01).
Stochastic homogenization may also play a role in collaborations with B05 and C02.