The numerical solution of continuum damage mechanics (CDM) problems suffers from critical points during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. Displacement-controlled arc-length methods were developed to address these challenges, but are currently applicable only to geometrically non-linear problems. In this work, we present a novel displacement-controlled arc-length (DAL) method for CDM problems in both local damage and non-local gradient damage versions. The analytical tangent matrix is derived for the DAL solver in both of the local and the non-local models. In addition, several consistent and non-consistent implementation algorithms are proposed, implemented, and evaluated. Unlike existing force-controlled arc-length solvers that monolithically scale the external force vector, the proposed method treats the external force vector 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. Such a flexible approach renders the proposed solver to be substantially more efficient and versatile than existing solvers used in CDM problems. The considerable advantages of the proposed DAL algorithm are demonstrated against several benchmark 1D problems with sharp snap-backs and 2D examples with various boundary conditions and loading scenarios, where the proposed method drastically outperforms existing conventional approaches in terms of accuracy, computational efficiency, and the ability to predict the complete equilibrium path including all critical points.

翻译：连续介质损伤力学（CDM）问题的数值求解在材料软化阶段面临临界点难题，导致现有迭代求解器需要在计算开销与求解精度之间进行权衡。位移控制弧长法虽旨在应对这些挑战，但目前仅适用于几何非线性问题。本文针对局部损伤与非局部梯度损伤两种CDM模型，提出了一种新型位移控制弧长（DAL）方法。我们推导了局部与非局部模型下DAL求解器的解析切线矩阵，并设计、实现和评估了多种一致性与非一致性算法。与现有统一缩放外力向量的力控制弧长求解器不同，本文方法将外力向量视为独立变量，基于外力向量所有节点变化量确定系统在平衡路径上的位置。这种灵活策略使所提求解器在CDM问题中显著优于现有求解器的效率与通用性。通过多个具有急剧回弹特性的1D基准问题及包含不同边界条件与加载场景的2D算例，验证了所提DAL算法在精度、计算效率及完整预测包含所有临界点的平衡路径能力方面的显著优势，其性能大幅超越现有传统方法。