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Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method

Zhijian Wu, Li Guo*, Jun Hong
Jiangsu Key Laboratory of Engineering Mechanics, Department of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing, 214135, China
* Corresponding Author: Li Guo. Email:
(This article belongs to this Special Issue: Numerical Methods in Engineering Analysis, Data Analysis and Artificial Intelligence)

Computer Modeling in Engineering & Sciences 2023, 134(1), 611-636.

Received 07 December 2021; Accepted 06 February 2022; Issue published 24 August 2022


The local arc-length method is employed to control the incremental loading procedure for phase-field brittle fracture modeling. An improved staggered algorithm with energy and damage iterative tolerance convergence criteria is developed based on the residuals of displacement and phase-field. The improved staggered solution scheme is implemented in the commercial software ABAQUS with user-defined element subroutines. The layered system of finite elements is utilized to solve the coupled elastic displacement and phase-field fracture problem. A one-element benchmark test compared with the analytical solution was conducted to validate the feasibility and accuracy of the developed method. Our study shows that the result calculated with the developed method does not depend on the selected size of loading increments. The results of several numerical experiments show that the improved staggered algorithm is efficient for solving the more complex brittle fracture problems.


Phase-field model; brittle fracture; crack propagation; ABAQUS subroutine; staggered algorithm

Cite This Article

Wu, Z., Guo, L., Hong, J. (2023). Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method. CMES-Computer Modeling in Engineering & Sciences, 134(1), 611–636.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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