An analytic traction-displacement model for a reinforcing ligament bridging a crack at an arbitrary angle, including elastic, frictional, snubbing, yielding, creep, and fatigue phenomena

Abstract

A micromechanical model is developed that generates analytic expressions for the crack displacement vector $u$ given an arbitrary far-field stress state ${\sigma }_a$ for a crack that is bridged by an array of ligaments oriented at an arbitrary angle with respect to the crack plane. The model is applicable to various materials, e.g., fibrous ceramic composites, or polymer composites reinforced by stitches or z-pins or woven tows, and deals with interfacial friction, enhanced friction due to increased contact pressure (“snubbing”), and the possibility of ligament deflection enabled by yield or damage. The model also conveniently incorporates ligament failure and rate dependent phenomena (fatigue or creep). Adaptability of the model is enabled by the definition of a standard Reference Model, which generates analytic expressions for the crack displacement for given possible yield, ligament deflection, and friction and snubbing effects and is invariant for all geometrical and material choices. The switching on or off and the strengths of all phenomena are governed by assigning values to a handful of material parameters. The material parameters will generally be calibrated against data in a top-down strategy, the model thereby mapping material selection onto engineering fracture via the predicted bridging relationship $u\left[{\sigma }_a\right]$. The relationship $u\left[{\sigma }_a\right]$ can depend strongly on bi-angular ligament orientation. Yield and deflection can change $u\left[{\sigma }_a\right]$ qualitatively, e.g., by creating fracture surface contact even when ${\sigma }_a$ includes substantial opening tension. Snubbing has significant effects, including possible stabilization of the pullout of a finite ligament. Since model output is computed via analytic expressions, its speed will support the model's use in large-scale material simulations or as constraining physical information in machine learning algorithms.