Hyperelastic Shear Lag Model

On the basis of the analysis of deformation of a representative volume element, a micromechanical model is derived to describe the elastic modulus of a unidirectional short-fiber composite under tension in the reinforcement direction. The analysis includes an exact solution to the hyperelastic equations for the deformed matrix and an approximate solution to the equations for the fiber material. The solution is provided for a neo-Hookean material. Formulas are derived to relate the elastic strain energy to the macroscopic longitudinal strain of the composite and to describe the longitudinal and radial deformation of the matrix and fiber material. The main result is a formula that relates the initial tangential elastic modulus of the composite (an analog of Young's modulus in linear elasticity) to the mechanical characteristics of the composite constituents (namely, the ratio of the elastic modulus of the matrix material to the elastic modulus of the fiber), as well as to the geometric characteristic (fiber length-to-diameter ratio) and volume fraction of fibers in the composite. The derived results are compared with other analytical models, as well as with the known results of finite element and boundary element modeling. The results generalize the well-known shear lag (SL) model to hyperelastic materials and are obtained via a more rigorous analysis than the original model.