Mixed mode fracture in fibre reinforced polymer composites

It has been widely reported that the delamination toughness of adhesive joints and polymer matrix composites can vary considerably depending on the mode of loading. For this reason, extensive research has been aimed at devising fracture test methods based on beam-like geometries, which enable the testing of these joint systems under variable modes of loading. However, central to the analysis of these mixed mode test methods, is the definition of a consistent parameter for the characterisation of the mode mixity in the fracture process zone for a given geometry and loading arrangement.This requirement has led to the development of a number of contrasting analytical partitioning theories in the literature which aim to address this problem. The most notable of these are the initial global analysis by Williams and a subsequent local analysis by Hutchinson and Suo. However, significant differences exist between the local and global approaches in the case of fracture in asymmetric geometries, and experimental evidence has been put forward on various occasions supporting each. Other semi-analytical and analytical partitioning theories have also been put forward by Davidson et al. and, more recently, Wang \& Harvey. However, much confusion still surrounds the area of mixed mode partitioning in beam-like geometries, and considerable disagreement remains within the fracture mechanics community as to which partitioning theory, if any, should be used in practice.In this thesis, a new semi-analytical partitioning scheme is proposed based on the findings from a finite element analysis of fracture in beam-like geometries. The fracture process zone is modelled using a cohesive zone model and the energy going into both opening and shearing, and hence the mode mixity, is obtained using a mode decomposed J-integral approach. A parametric study is carried out to examine the effect of various substrate and cohesive properties on the mode mixity. In an initial study, it is found that if no damage is modelled (purely elastic case), the numerical partitioning closely follows the local partitioning of Hutchinson and Suo. Once damage is introduced using the cohesive zone model, it is found that the numerical partition deviates away from the local solution and in all cases tends towards the global solution of Williams.The parametric study reveals that the numerical mode partition moves towards the global solution when the substrate stiffness $(E)$ or cohesive toughness $(G)$ is increased, or the cohesive strength $(t)$ is decreased. Each of these parameters that affect the mode mixity are linked through the developed cohesive zone length $(l_{cz}=f(EG/t^{2}))$. For each individual test geometry, it is found that the mode partition is uniquely dependent on the numerical cohesive zone length. At this point, it is hypothesised that the size of the cohesive zone relative to the size of the $K$ dominant region is the controlling factor in the mode mixity, where the upper and lower bounds of the partitioning solution are given by the local and global analytical solutions respectively (region of $K$ dominance scales with smallest characteristic dimensions). To test this hypothesis, the normalised mode mixity is plotted against the normalised cohesive zone length for all cases over a range of geometries, loadings and cohesive properties. A unique dependency is observed, thus supporting the proposed hypothesis. This hypothesis is further tested using a different cohesive zone formulation and it is still found to hold true. A best fit exponential function is fitted to the observed unique dependency and this forms the basis for the proposed semi-analytical cohesive analysis (SACA).SACA works on the basis that if the cohesive zone length can be estimated from known cohesive properties, then the mode mixity can be estimated from the unique dependency curve. Analytical expressions exist, which were presented by Suo (mode I) and Cox \& Yang (mode II), that can accurately estimate cohesive zone lengths in beam like geometries under mode I and mode II loadings. It is shown in this work that these expressions can also be used to accurately estimate the cohesive zone length under mixed mode loadings by using the maximum of the mode I and mode II cohesive lengths, where the mode I and mode II cohesive zone lengths are obtained by using the energy dissipated in mode I and mode II respectively. As the mode partition is not known initially for each case, the final estimated length, and hence mode mixity, is obtained using an iterative procedure. The final solution is shown to be independent of the initial guess of mode mixity. This procedure is coded into an excel macro and is available for download on the UCD Centre of Adhesion and Adhesives \href{http://adhesion.ucd.ie/caa/CAA_MixedMode.html}{webpage}.Finally, the SACA approach is assessed using an experimental program. In this series of experiments, the true delamination failure locus of a carbon fibre-epoxy composite is obtained using a range of symmetric double cantilever beam specimens loaded with uneven bending moments. A number of asymmetric fracture tests are then carried out and partitioned using a range of analytical partitioning theories including the new SACA approach. It is found that the SACA partitioning of the asymmetric specimens produces the best fit to the symmetrically measured failure locus. This experimental result supports the numerical findings that the true partitioning in the general case is neither local nor global and can only be accurately estimated by accounting for the effect of damage at the crack tip, as in the SACA approach. The SACA partitioning is also applied to previously contrasting experimental data available in the literature, and good agreement is obtained in all cases, suggesting that the proposed semi-analytical cohesive analysis provides an efficient and accurate method for predicting mixed mode partitions.