  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.
