Curved-fiber pull-out model for nanocomposites. Part 1: Bonded stage formulation

A bridging law and its application to the analysis of toughness of carbon nanotube-reinforced composites and pull-out of fibres grafted with nanotubes

“…The deformation of a CNT relative to the matrix under tensile loading is generally regarded to undergo three major stages [1,6,16,17]: (1) bonded stage when the CNT is fully bonded to the matrix during loading, and the CNT is generally assumed to deform elastically with the matrix; (2) debonding stage when a part of the CNT has debonded from the matrix and deforms against the frictional force exerted by the matrix, while the rest of the CNT is still bonded to the matrix; and (3) pull-out stage when the debonding propagates to the entire embedded length of the CNT, but due to friction it is progressively pulled out of the matrix. Although a direct observation of the whole deformation process is difficult, molecular dynamic simulations confirm these three stages of deformation [43][44][45].…”

“…In this regard, by assuming that the interface between the CNT and the matrix is described by the nonlinear shear cohesive model of Xu and Needleman [12,13], and neglecting the deformation of the CNT but considering the deformation of the matrix, Seshadri and Saigal [11] developed a nonlinear bridging law. Chen et al [16,17] developed the Fig. 1 A tri-linear bridging law (OABC) for pull-out of CNTs and conventional short fibres, where p denotes the pull-out force and δ denotes the pull-out displacement.…”

“…(1) most comprehensive pull-out model to date that gives a complete pull-out bridging law connecting the three stages of the deformation of the CNT. This model is based on the classical Lawrence shear-lag model [10]for a straight short fibre, but Chen et al [16,17] made several improvements by taking into account the curvature of the CNT, a nonvanishing force at the embedded end of the CNT that may be caused by entanglement, and a Coulomb friction law in the debonded section. Therefore, the bridging law of Chen at al.…”

“…For the curvature effect, they used a simplified analytical tri-linear bridging law that was developed based on the nonlinear bridging law for curved CNTs established by Chen et al [16,17]. The original bridging law of Chen et al [16,17] depicts the complete three-stage deformation of a curved CNT in a matrix, namely the elastic deformation in the bonded stage, the interfacial debonding stage, and the sliding pull-out stage. For the latter, it is assumed that the strength of the CNTs is described by a two-parameter Weibull distribution, where the axial stress is calculated by the formula of Chen et al [17] for a curved CNT.…”

“…Pavia and Curtin [18] predicted the strength and work of fracture of a nanofibre-reinforced ceramic composite characterized by a single crack bridged by wavy short nanofibres. A bridging law was developed using a shear-lag model of a unit cell, similar to the work of Chen et al [16,17], following the Lawrence shear-lag model [10]. However, the concept of a virtual pulley of Li et al [19] was used to simulate the frictional resistance at the exit point of the fibre from the matrix, so the contribution of the friction due to the curvature of the fibre was depicted by an exponential function.…”