A constrained LArge Time INcrement method for a gradient-enhanced damage model

Abstract

An important aspect when modelling quasi-brittle materials is a robust and efficient solution algorithm which can cope with highly non-linear behaviour such as snap-backs and bifurcations. An alternative to the widely used incremental-iterative Newton-Raphson algorithm is the non-incremental LArge Time INcrement (LATIN) method. This algorithm differs significantly from conventional incremental methods since the whole loading process is iteratively calculated in a single time increment using two solution stages. In the local solution stage, stresses and strains verify the non-linear constitutive laws as well as a local search equation which searches for a new solution using the fields of the previous iteration. In the global solution stage, stress-strain couples are calculated which satisfy both structural equilibrium and a global search equation which searches for an improved solution using the stresses and strains of the local solution stage. In other words, the local and non-linear behaviour is separated from global and linear behaviour. In the literature, LATIN algorithms are usually applied to elastoplastic hardening problems. In the few papers that investigate strain-softening problems using LATIN algorithms, the softening behaviour is attributed to discontinuous and therefore local failure. In this contribution, the constrained LATIN algorithm developed in will be applied to the continuous modelling of quasi-brittle failure. The problem is regularised using the implicit gradient-enhanced damage model in which a modified Helmholtz equation is coupled to the standard equilibrium equation. Although coupled problems have been successfully modelled using the LATIN method, the implementation of a gradient-enhanced damage model in a LATIN framework is not straightforward since the non-linear constitutive behaviour can no longer be separated from the non-local (or global) behaviour. Special attention will therefore be devoted to implementation aspects of the algorithm. The performance of the proposed methodology is demonstrated by means of several numerical examples.