Stochastic regularisation of lattice modelling for the failure of quasi-brittle materials

A regular periodic triangular lattice model is developed in this paper to examine the isotropic damage of cementitious materials. The mesh size dependency of current lattice models, which results in a non-unique mechanical response, is addressed through consideration of statistical distributions of lattice beam strengths. The theory of ‘stochastic regularisation’ is presented for a simple 1D parallel bar model, and applied directly to a 2D plane stress lattice discretisation of uniaxial tensile experiments on notched specimens. The main issues to be considered when applying the theory to two dimensions are: (i) stress concentrations at crack tips; (ii) the existence of multiple cracking paths, and; (iii) the increase in ‘effective length’ of beams as damage progresses. These issues are addressed, in part, by the implementation of a two-part strength distribution, which can significantly improve objectivity, whilst maintaining realistic crack patterns without the need for explicit modelling of the mesostructure.