A new unified arc-length method for damage mechanics problems

TL;DR

This study presents the unified arc-length (UAL) method for continuum damage mechanics to address convergence challenges during material softening and to improve computational efficiency. By treating the entire external force vector ${\boldsymbol f}^{ext}$ as an independent variable and coupling it with the displacement and non-local variables, UAL traces the equilibrium path with larger, adaptive arc-length steps while maintaining accuracy, outperforming traditional Newton-Raphson and force-controlled arc-length approaches. The authors derive the analytical tangent matrices for both local and non-local gradient damage models, develop two solution schemes (Partitioned Consistent and Partitioned Non-Consistent), and validate the method on 1D and 2D benchmark problems exhibiting sharp snap-backs, using the Mazars damage model. Results show that UAL achieves 1–2 orders of magnitude faster performance than FAL and NR, with robust convergence across highly localized damage scenarios and mesh sensitivities. The approach holds promise for efficient, reliable simulation of damage evolution in engineering materials and structures under complex loading.

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.

Figures (32)

• Figure 1: Schematic representation of a 2D domain with local (left) and non-local (right) damage idealization.
• Figure 2: Schematic of the Newton-Raphson scheme where $\boldsymbol{x}$ represents the independent variables and $\lambda$ represents the loadfactor. The left superscript is the increment number and the left subscript represents the iteration number. In every increment the system is advanced by $\Delta \lambda$ and an iterative process is required to achieve convergence.
• Figure 3: (a) Schematic representation of the force-controlled arc-length method where $\boldsymbol{x}$ represents the independent variables and $\lambda$ represents the loadfactor. Unlike the Newton-Raphson method, both $\boldsymbol{x}$ and $\lambda$ are updated simultaneously in each iteration. $\Delta l$ refers to the arc-length used to define the search radius within which $\Delta \boldsymbol{x}$ and $\Delta \lambda$ are calculated; (b) Force-controlled arc-length method with representative arc-length contours at different points on the equilibrium path; In FAL, $\boldsymbol{f}^{ext}$ = $\lambda$$\boldsymbol{q}$ and $\lambda$ is an unknown.
• Figure 4: Schematic representation of the unified arc-length method, where $\delta \bar{m}$ is the fraction of the total applied displacement load $\boldsymbol{u}^p$ acting on the system at a given increment.
• Figure 5: Schematic representation of the influence of different arc-length values on the UAL performance.