Toughness of a composite in which sliding between fibers and matrix is rate-sensitive

It has been common to use brittle constituents to form tough composites. For example, ceramic fibers and a ceramic matrix are brittle, but their composite can be tough, provided the matrix can slide relative to the fibers. Here we study the effect of rate-dependent sliding on toughness. Consider a crack through the matrix, with the fibers being intact and bridging the crack. The composite is subject to a tensile load normal to the crack. Both the fibers and the matrix are elastic, and the sliding stress between them is linear in their relative velocity. Far away from the crack, the matrix does not slide relative to the fibers, and the deformation is elastic. Near the crack, the matrix slides relative to the fibers, and the deformation is inelastic. When the rate of the applied load is low, the sliding stress is low, so that tension in each fiber is distributed over a long length. Breaking the fiber dissipates elastic energy over a long length of the fiber. This de-concentration of stress leads to high toughness. When the rate of the applied load is high, the sliding stress is also high, so that tension in the fiber is concentrated in a short length near the crack plane. This concentration of stress leads to low toughness. We model this rate-sensitive toughness using a shear lag model. The strain in the fiber satisfies a diffusion equation. When the composite is subjected to load at a constant strain rate, before the fiber breaks, the sliding zone increases with time. We discuss stress de-concentration in various materials.