Second order shear lag theory

The standard shear lag theory for elastic aligned short-fibre composites is extended to allow for a gradient of overall strain. The result is a one-dimensional strain gradient theory of the Toupin–Mindlin type. All parameters are the same as in the standard theory, and in the limits of weak strain gradients, large fibre aspect ratios or low elastic modulus ratios, the standard theory is recovered. The gradient effect is illustrated by a simple one-dimensional boundary value problem: a vertical composite rod fixed at both ends and loaded by gravity. The fibre length significantly affects the solution when the fibres are rigid and their length is near the rod length; but otherwise the effect is weak.