A new unified arc-length method for damage mechanics problems

TL;DR

This paper introduces a novel unified arc-length method for damage mechanics problems that improves convergence, robustness, and efficiency in solving continuum damage mechanics issues, especially during material softening stages.

Contribution

The paper develops a new unified arc-length method that treats external force as an independent variable, enhancing solver robustness and efficiency for damage mechanics problems.

Findings

Significantly faster than existing solvers by 1-2 orders of magnitude.

Demonstrates superior ability to handle snap-backs and critical increments.

Effective for both local and non-local damage problems.

Abstract

The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, in the presented examples, the proposed UAL method is 1-2 orders of magnitude faster than force-controlled arc-length and monolithic Newton-Raphson solvers.