"Hierarchical fiber bundle strength statistics"

Multi-scale modeling is currently one of the most active research topics in a wide range of disciplines. In this thesis we develop innovative hierarchical multi-scale models to analyze the probabilistic strength of fiber bundle structures. The Fiber Bundle Model (FBM) was developed initially by Daniels (1945), and then expanded, modified and generalized by many authors. Daniels considered a bundle of N fibers with identical elastic properties under uniform tensile stress. When a fiber breaks, the load from the broken fiber is distributed equally over all the remaining fibers (global load sharing). The strength of fibers is assigned randomly most often according to the Weibull probability distribution. In chapter 2, we develop for the first time an ad hoc hierarchical theory designed to tackle hierarchical architectures, thus allowing the determination of the strength of macroscopic hierarchical materials from the properties of their constituents at the nanoscale. The results show that the mean strength of the fiber bundle is reduced when scaling up from a fiber bundle to bundles of bundles. The hierarchical model developed in this study enables the prediction of strength values in good agreement with existing experimental results. This new ad hoc extension of the fiber bundle model is used for evaluating the role of hierarchy on structural strength. Different hierarchical architectures of fiber bundles have been investigated through analytical multiscale calculations based on a fiber bundle model at each hierarchical level. In general, we find that an increase in the number of hierarchical levels leads to a decrease in the strength of material. On a more abstract level, the hierarchical fiber bundle model (HFBM), an extension of the fiber bundle model (FBM) presented in this thesis, can be applied to any hierarchical system. FBMs are an established method helpful to understand hierarchical strength. Another extension of Daniels‘ theory for bimodal statistical strength has been implemented to model flaws in carbon nanotube fibers such as joints between carbon nanotubes, where careful analysis is necessary to assess the true mean strength. This model provides a more realistic description of the microscopic structure constituted by a nanotube-nanotube joint than a simple fiber bundle model. We demonstrate that the disorder distribution and the relative importance of the two failure modes have a substantial effect on mean strength of the structure. As mentioned, the fiber bundle model describes a collection of elastic fibers under load. The fibers fail successively and for each failure, the load is redistributed among the surviving fibers. In the fiber bundle model, the survival probability is defined as a ratio between number of surviving fibers and the total number of fibers in the bundle. We find that this classical relation is no longer suitable for a bundle with a small number of fibers, so that it is necessary to implement a modification into the probability function. It is possible to predict snap-back instabilities by inserting this modification in the theoretical expression of the load-strain (F-ε) relationship for the bundle, as discussed in chapter 4. Scrutiny into the composition of natural, or biological materials convincingly reveals that high material and structural efficiency can be attained, even with moderate-quality constituents, by hierarchical topologies, i.e., successively organized material levels. This is shown in chapter 5, where a composite bundle with two different types of fibers is considered, and an improvement in the mean strength is obtained for some specific hierarchical architectures, indicating that both hierarchy and material ―mixing‖ are necessary ingredients to obtain improved mechanical properties. In Chapter 6, we consider a novel modeling approach, namely we introduce self healing in a fiber bundle model. Here, we further assume that failed fibers are replaced by new unstressed fibers. This process has been characterized by introducing a self healing parameter which has been implemented into the survival probability function of the fiber. General conclusions of the research efforts presented in this thesis are given in chapter 7. This is followed by suggestions for further research and a brief outlook.