DFG SPP 2256 Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials

Nonlinear Fracture Dynamics: Modeling, Analysis, Approximation, and Applications

This project is financially supported by the German Research Foundation (DFG) within the priority programme "DFG SPP 2256 - Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials). The first funding period is from 2020 until 2023.

Project description

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.

Project-related events

• Closing Conference (23rd -- 25th February 2021) "Structures in Evolution: Theory and Applications" within the Thematic Einstein Semester, winter term 2020/21
• TES-Seminar on Energy-based Mathematical Methods and Thermodynamics within the Thematic Einstein Semester, winter term 2020/21
• "MA4M: Mathematical Analysis for Mechanics" (23 -- 25 November 2020) within the Thematic Einstein Semester, winter term 2020/21
• "Kick-off Conference" (26 -- 30 October 2020) within the Thematic Einstein Semester, winter term 2020/21
• "Student Compact Course" (12 -- 23 October 2020) within the Thematic Einstein Semester, winter term 2020/21

Project-related publications

• K. Friebertshäuser, M. Thomas, S. Tornquist, K. Weinberg and C. Wieners: Dynamic Phase-Field Fracture in Viscoelastic Materials using a First-Order Formulation, to appear in PAMM, 2022. (PDF)
• K. Friebertshäuser, M. Thomas, S. Tornquist, K. Weinberg and C. Wieners: Dynamic fracture with a continuum-kinematics-based peridynamic and a phase-field approach, to appear in PAMM, 2022. (PDF)
• M. Thomas, S. Tornquist and C. Wieners: Approximating Dynamic Phase-Field Fracture in Viscoelastic Materials with a First-Order Formulation for Velocity and Stress, 2022. (PDF)
• M. Heida and M. Thomas: GENERIC for dissipative solids with bulk-interface interaction. WIAS-Preprint 2872, to appear in Springer AWM series volume "Research in the Mathematics of Materials Science", 2022.

Talks and posters

• S. Tornquist: Dynamic Phase-Field Fracture in Viscoelastic Materials using a First-Order Formulation. 92nd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), August 15-19, 2022, RWTH Aachen University, Germany
• M. Thomas: First-order formulation for dynamic phase-field fracture in visco-elastic materials. Workshop PHAse-field MEthods in applied sciences -- PHAME2022, May 23-27, 2022, INDAM, Rome, Italy.
• M. Thomas: First-order formulation for dynamic phase-field fracture in visco-elastic materials. Workshop Beyond Elasticity: Advances and Research Challenges, May 16-20, 2022, CIRM Luminy, France.
• M. Thomas: Convergence analysis for fully discretized damage and phase-field fracture models. FBP 2021 International Conference on Free Boundary Problems, September 13-17, 2021 (online).
• S. Tornquist: Analysis of dynamic phase-field fracture. 9th BMS Student Conference, March 03-05, 2021 (online).
• S. Tornquist: Temporal regularity of solutions to a dynamic phase-field fracture model in visco-elastic materials. MA4M: Mathematical Analysis for Mechanics, November 23-25, 2020 within the Thematic Einstein Semester, winter term 2020/21 (online).
• S. Tornquist: Dynamic phase-field fracture in visco-elastic materials. Student Compact Course - Variational Methods for Fluids and Solids, October 12-23, 2020 within the Thematic Einstein Semester, winter term 2020/21 (online).