Meso mechanics and peridynamic modelling of fibrous through thickness reinforcement at high deformation rate

Abstract

Numerically modelling Z-pinned laminates is an important step in develop- ing damage tolerant structures using through thickness reinforcement such as Z-pins. This work is concerned with developing a reduced order model (ROM) capable of predicting the Z-pin bridging performance under mixed mode loading and at high deformation rates. The ROM is derived using the peridynamic formulation of the solid me- chanics equations of motion. For this purpose, rod and Timoshenko beam elements in peridynamic formulation were derived from first principles and validated against well established analytical results. The developed Tim- oshenko beam element is a non-linear element that accounts for moderate beam rotations. The rod peridynamic element is used as the basis of the Z-pin ROM model for the case of pure mode I loading. Two models were considered; the first model includes all key stages of the Z-pin response under mode I loading, i.e. bonded, debonding and frictional pull-out stages. The second model ignores the bonded and debonding stages and only considers the fric- tional pull-out stage. Both models yielded excellent prediction capabilities at different loading rates especially in terms of the energy absorbed by the Z-pin. This matched the experimental observation that most of the energy is absorbed during the frictional pull-out stage of the Z-pin response. The non-linear peridynamic beam element is used as the basis of the Z-pin ROM model for the general case of mixed mode loading. Three key external forces were considered under mixed mode loading; a lateral Win- kler foundation type force and an axial sliding and snubbing frictional forces. Ruina’s state based friction model was adopted to describe the rate depen- 2 dent behaviour of the sliding and snubbing frictional forces. The B-K failure criterion has been utilised to predict the fracture failure of the Z-pin. For this purpose, the energy release rate in peridynamic formulation has been developed. The resulting model had a number of parameters that needed to be defined. The developed model was calibrated against a subset of experimental data obtained from the literature in order to define unknown model param- eters. The calibrated model has been validated against a different subset of the experimental data. The model was able to predict the Z-pin bridging performance relatively accurately at different loading rates and mode mix- ities. More importantly, the model is very computationally efficient with a run time of the order of minutes using a non-optimised Matlab code that run on a desktop computer. This opens up the possibility of modelling Z-pinned laminates using a multi-scale approach where the ROM model is coupled with finite element packages to predict the performance of Z-pinned laminates.