Probabilistic crack bridge model reflecting random bond properties and elastic matrix deformation

A semi-analytical probabilistic model of an isolated composite crack bridge is presented in this paper. With the assumptions of heterogeneous fibrous reinforcement embedded in an elastic matrix the model is capable of evaluating the stress and strain fields in both fibers and matrix. In order to be applicable as a representative unit in models at higher scales, the micromechanical response of the composite crack bridge is homogenized by using a probabilistic approach. Specifically, the mean response of a crack bridge is obtained as the integral of the response of a single fiber over the domain of random variables weighted by their joint probability density function. This approach has been used by the authors in a recent publication describing a single crack bridge with rigid matrix. The main extension of the present crack bridge model is the incorporation of elastic matrix deformations and of boundary conditions restricting fiber debonding at the crack bridge boundaries. The latter extension is needed to reflect the effects of interactions with neighboring cracks within a tensile specimen with multiple cracks. The model is verified against three limiting cases with known analytical solutions (fiber bundle model, crack bridge with rigid matrix, mono-filament in elastic matrix) and is shown to be in exact conformity with all of these limiting cases.