Fully discrete approximation schemes for rate-independent crack evolution.

Area of Science:

• Computational Mechanics
• Materials Science
• Applied Mathematics

Background:

• Modeling crack propagation in materials requires handling rate-independent evolutionary systems with non-convex energies and time discontinuities.
• Existing solution concepts struggle to accurately capture these phenomena, particularly in the context of fracture mechanics.

Purpose of the Study:

• To develop and analyze two fully discrete approximation schemes for simulating the rate-independent evolution of a single crack in a 2D linear elastic material.
• To investigate the convergence properties of these schemes, ensuring discrete solutions approximate continuous models.

Main Methods:

• The study proposes two time-discrete schemes: one based on local minimization and a second regularized version.
• Both schemes feature adaptive time-stepping, refining time steps where crack tip discontinuities may occur.
• Discretization parameters include mesh size, crack increment, locality, and regularization parameters.

Main Results:

• Sufficient conditions are derived for the convergence of discrete interpolants to parametrized balanced viscosity solutions.
• The interplay between various discretization parameters and its effect on convergence is explored.
• Numerical simulations demonstrate the performance and effectiveness of the proposed approximation schemes.

Conclusions:

• The developed discrete approximation schemes provide a robust framework for modeling crack propagation.
• Adaptive time-stepping is crucial for accurately capturing discontinuities in crack tip evolution.
• The theoretical analysis and numerical results confirm the convergence and reliability of the methods for fracture mechanics applications.