DFG SPP 2256 Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials
Nonlinear Fracture Dynamics: Modeling, Analysis, Approximation, and Applications
The correct modeling and efficient approximation of rapid deformations in nonlinear elastic and inelastic materials is a challenging task relevant for many engineering applications. Here we aim to develop efficient and reliable methods for the spatio-temporal approximation of dynamic models in solid mechanics at large strains. Our key interest lies in the investigation of criteria for the initiation and propagation of dynamic fracture in a variational setting in space and time: On the one hand, material discontinuities may arise as a property of the (weak) notion of solutions in finite strain elastodynamics with non-convex energy functionals. On the other hand, it has proved beneficial both from the analytical and computational point of view to regularize sharp material discontinuities using internal variables in terms of damage or phase field fracture models. It is our goal to establish relations between such different concepts and to systematically investigate in these models the interplay of dynamic wave propagation and purely dissipative effects such as phase field fracture and viscous damping both from the analytical and from the numerical point of view. In this context it is of importance to identify relations and quantify differences between models for finite strain elasticity and models for small strain elasticity. As a long term goal we aim to extend our methods to general finite strain models which also capture the evolution of plasticity and temperature.
Starting point for the proposed research are two of our recent results: the analysis and simulation of quasi-static phase field fracture models at finite strains based on modified invariants and a viscous evolution of the phase field variable, and the development of efficient methods for the approximation of elastodynamics at small strains based on the formulation as first-order hyperbolic system and using higher-order discontinuous Galerkin schemes. In this context, the discontinuous Galerkin method has already been successfully used to reformulate and analyse the convergence of a previously investigated fracture model with viscous damping.
For the first funding period, our main objectives are 1) the development of efficient and reliable methods for the approximation in space and time of finite strain models with and without phase field, 2) the investigation of propagation criteria for dynamic fracture and their corresponding formulation as phase field model, 3) a detailed study of the interplay of dynamic wave propagation and purely dissipative effects such as viscous damping and phase field fracture, and 4) the identification and quantification of differences between finite strain and small strain models.