Mechanics of Fibre Reinforced Cement Systems
The mechanical behaviour of fibre reinforced cements is a result of the complex interaction between the all the constituents of the composite constitutes such as the macroscopic behaviour between the strength and stiffness of both fibre and cementitious matrix, matrix toughness, mechanical interaction between the fibres and matrix and supplementary effects of polymeric additives or aggregates [66, 68].
The stress vs. strain (a vs. 6) curve generally defines the fibre bridging property across the matrix crack. That is the average tensile strength (a) transmitted across the uniform crack opening (6) in a uniaxial tensile specimen. The link between the composite (fibre, matrix and interface) and composite tensile ductility is gained from the a vs. 6 curve (Fig. 3.4).
Monolithic and fibre reinforcement a vs. 6 curves for different failure behaviour following loading under bending or tension conditions (Fig. 3.5a). The material strength or stress maximum before failure depends on two aspects: (i) strain to failure and (ii) the area below the curve or energy necessary to yield material failure. These properties will increase in the order of failure modes (i.e. brittle < tension softening
Fig. 3.4 Schematic representation of relation between composite material properties
Fig. 3.5 Fracture of cement and composites. (a) Typical stress-strain curves of different composites.  © by RILEM reprint with permission from springer Science + Business Media B.V. (b) Complementary energy C originated from a vs. 6 curves (stress (a) against crack opening displacement (6)) . (c) Schematic representation of cracks in brittle and quasi-brittle materials (Griffith crack, top) and strain hardening composites (steady state, bottom)  © by JCI - reprinted with permission
or quasi-brittle materials < strain hardening or strictly pseudo strain hardening; high ductile materials). High amounts of energy are absorbed during fracture of fibre reinforced cement based composites when compared with pure brittle failure, which leads to two or more fractures. Whereas for materials that undergo quasi-brittle composites, fracture is associated with a peak load and exhibits a tension softening behaviour, during this process the fibre bridging across the single crack opening delay the fracture via dissipation of the fracture energy - however the composite no longer demonstrates an ability to bear substantial load after the peak load reached.
Fibre reinforced composites with pseudo strain-hardening behaviour (elastic- plastic behaviour) demonstrates the highest ductility. For such composites even at peak load, the strain to failure can be several percent due to network of multiple cracks that develop and therefore, substantially more energy is dissipated during the fracture process [67, 68]. Energy dissipation happens at different sites and mechanisms during fracture. Some of the important sites are the frontal process zone (FPZ) where the fibres influence the fracture process before the crack tip, behind the crack and adjacent to the crack plane [62, 69, 70]. These behaviours of composite failure are based on different mechanisms. When tensile load is applied on a brittle solid, the stress intensity reaches a critical value where catastrophic failure occurs due to fast crack propagation from the initial crack (monolithic CPC) . Whereas fibre reinforced composites that exhibit quasi-brittle behaviour, the linear elastic region ends with first crack of the composite and softening of the material is observed as crack propagates [67, 71]. The strength of the composite was determined by the behaviour of FPZ.
In pseudo strain-hardening composites, the composite can withstand substantial load after initial loss of stiffness in the elastic region. With increasing tensile loading multiple microcracks are formed, which could be visualised microscopically due to composite ductility . The multiple cracking of fibre reinforced composites is based on strength of criterion and energy criterion. It is explained by the complementary energy on the о vs. 6 curve, which shows stress against crack opening displacement (Fig. 3.5b). Fibre pull-out occurs at a low peak strength when the interface is too weak, whereas fibre rupture takes place when the interface is strong with a small value of critical opening (6p). In both cases, the area between rising part of о vs. 6 curve and the о-axis is very small (complementary energy). When complementary energy is large, the crack will remain flat as it propagates so that the steady state crack opening will be < 6p, and the composite maintain tensile load carrying capacity post-propagation (Fig. 3.5c). The shape of the о vs. 6 curve plays an important role in determining the uniaxial tensile load (strain-hardens or tension- softens) of the composite . In composite based cements that undergo strain hardening, the main crack from initial notch is delayed and further loading can take place before real crack growth can begin . High bending capacity of fibre- reinforced cements is a consequence of its high ductility. Also, ductility and fracture energy dissipation are higher in composites that undergo strain hardening when compared with composites that experience tension softening. Optimisations of these two variables are essential when attempting to improve the mechanical properties of fibre reinforced cements.