Stochastic approaches for damage evolution in quasi-brittle materials

The damage process in quasi-brittle materials is a complex phenomenon in which heterogeneity plays an important role. This heterogeneity may imply that the exact failure mode can be highly dependent upon the precise spatial distribution of initial imperfections. To model this inhomogeneity stochastic material properties must be assumed in numerical simulations. However, the use of a stochastic approach does not resolve the issue of the change of character of the governing differential equations during progressive damage. As a consequence, the initial value problem becomes ill-posed at the onset of strain localisation and the finite element calculations suffer from a severe mesh dependence. A simulation technique that describes the true failure process must incorporate both a regularisation of the standard continuum during progressive damage and a stochastic description of the random continuum. This statement will be substantiated in this contribution. To do so, we will present finite element analyses of direct tension tests with a standard local damage model and with a nonlocal damage model. The randomness in the damage process will be introduced by considering the initial damage threshold of the continuum damage model as a random field, characterized by a relevant distribution and autocorrelation function. The response statistics calculated by the Monte Carlo technique will be presented for two different levels of discretisation. The non local and random field formulations rely both on the introduction of length parameter: the internal length of the nonlocal continuum and the correlation distance of the random field. The effect of the relative variation of the correlation distance and the internal length will also be investigated.