Comparison of shear-lag theory and continuum fracture mechanics for modeling fiber and matrix stresses in an elastic cracked composite lamina

This study analyzes fiber tensile and matrix shear stresses near the crack tip ina transversely cracked, unidirectional, fiber-reinforced lamina under a remote tensile stress applied in the fiber direction. The two-dimensional lamina consists of parallel, equally-spaced elastic fibers with elastic matrix in-between, and contains a row of up to a few hundred contiguous fiber breaks aligned transverse to the fiber direction forming a central transverse crack. Using the break-influence superposition (BIS) technique, a recently developed method for analyzing a shear-lag model first introduced by Hedgepeth, we calculate the tensile and shear stress concentrations in the fibers and matrix, respectively. These are compared to tensile and shear stresses calculated using Linear Elastic Fracture Mechanics (LEFM) and the complete elasticity solution both for the continuum limit of a homogeneous, orthotropic elastic material with a transverse central crack loaded in Mode I. For the shear-lag model a critical scaling parameter for examining the stress behavior away from the crack tip along the fiber direction is root E*/G*, where E* and G* are composite in-plane stiffness constants along the fiber direction and in shear, respectively. In addition to these parameters, the LEFM and complete elasticity solutions also involve the effective transverse stiffness and Poisson's ratio. For a sizable crack (consisting of 100 or more fiber breaks), the fiber tensile stresses ahead of the crack tip along the crack plane calculated from the BIS approach achieve excellent agreement with the LEFM solution down to the scale of one fiber diameter and even better agreement with the complete solution both in the near crack tip field and far field, regardless of the composite stiffness constants. The profiles of the fiber tensile and matrix shear stresses along the fiber direction show generally good agreement, with the agreement improving as the composite stiffness transverse to the fiber direction grows.