# An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

## TL;DR

This paper introduces an adaptive space-time phase field approach for simulating dynamic fracture in brittle shells, integrating advanced isogeometric analysis and adaptive mesh refinement to accurately capture crack evolution.

## Contribution

It develops a novel phase field formulation for brittle shell fracture using LR NURBS and adaptive techniques, enabling detailed dynamic crack simulations on evolving surfaces.

## Key findings

- Successfully models dynamic crack propagation and branching.
- Demonstrates the effectiveness of adaptive LR NURBS in fracture simulation.
- Provides a comprehensive framework for brittle shell fracture analysis.

## Abstract

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff-Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell's mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith's theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires $C^1$-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-$\alpha$ method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton-Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.