Doubly periodic arrays of bridged cracks and short fibre-reinforced cementitious composites

Abstract

This paper presents a superposition approach for studying the influence of bridging forces upon the opening of multiple cracks in elastic solids under unidirectional tensile loading. The bridging forces may be purely elastic and proportional to the crack-opening displacements, but an elasto-plastic bridging law is more likely to represent reality in a short fibre-reinforced solid. The fibres debond from the elastic matrix at a certain critical crack-opening and thereafter provide a residual bridging force due to frictional pull-out. From a mathematical point of view, the elasto-plastic bridging law introduces an additional (logarithmic) singularity at the point of discontinuity in the bridging force, besides the square root singularity at the crack tips. These singularities have been analytically isolated, so that only regular functions are subjected to numerical integration. The double infinite summations necessary for the solution of multiple cracks have been found to be divergent in earlier studies. The superposition procedure developed in this paper eliminates double infinite series and thus the problem of divergence. The mathematical solutions are used to study the influence of varying amounts of short fibre reinforcement upon the complete tensile macroscopic response (including strain hardening and tension softening) of two fibre-reinforced cementitious composites: a conventional fibre-reinforced cement and a high-performance fibre-reinforced (DSP) cement. It is shown that the model of multiple bridged cracks accurately predicts the prolonged nonlinear strain-hardening and the initial tension-softening response of both these cementitious composites.