Analytical and numerical studies on local and nonlocal elastic bars in tension and neutralizer-based iterative methods


Student thesis: Doctoral Thesis

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  • Xiaowu ZHU

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Awarding Institution
Award date3 Oct 2012


In this thesis, we study some problems associated with both local and nonlocal elasticity and, present the so-called neutralizer-based iterative methods for integral equations of the second kind. The details are as follows. The local problem is on the strain softening of materials. Strain-softening, i.e., the decrease of stress with the increase of strain, which is a common post-peak phenomenon that has been recorded for a variety of materials. Snap-back due to strain softening may be one of the most interesting and the most common structural instability phenomena observed in experiments. There have been many efforts in the past decades to investigate strain-softening with localization experimentally, numerically, and analytically. However, there is not any analytical study with general nonlinear constitutive relations in the open literature which explores the role played by the convexity of the constitutive curve of the softening part and the coupling effect between this convexity and the size. Also, both snap-back and snap-through were observed in some experiments, but no analytical results are available for explaining the transition from snap-back to snap-through. Nonlocal elasticity is a growing direction of continuum mechanics nowadays. There are many works contributed to this area. Due to the essential difficulty of the integral equation, analytical approximate solutions are usually prohibited. Thus, many existing literature devote to apply the approximate differential models suggested by Eringen et al. However, one weak point of the approximations is that the possible boundary effects, which are present in the integral formulations, are neglected. Thus, it would be desirable to have proper differential formulations which take into account such effects. For this purpose, a first step is to know both qualitative and quantitative behavior near the ends (e.g., the influences of the parameters). Thus, some analytical solutions are needed to provide convincing results. In this thesis, firstly for the local problem, we modify an existing model and set up the stress-strain equations for the structure in the post-peak region, which are nonlinear as compared with the bilinear case in the literature. After some analysis, we derive the mathematical conditions for the occurrence of several important curves as frequently observed in experiments, including the snap-through (which cannot be captured by the bilinear assumptions). Two examples are also given to illustrate these cases, and the post-peak curves are consistent with our theoretical predictions. Secondly, for a static tension problem in nonlocal elasticity (the uniform case), we apply an existing iterative method that are efficient for a special kind of kernel to handle the resultant integral equation. By explicitly evaluating the integral in the second iterative solution, we are able to get a good approximate analytical solution for this problem. Some features of the nonlocal theory can then be closely examined, especially the boundary effects. It seems that the analytical results obtained here would give some insights into nonlocal theory, particularly for its applications in nanomaterials. Moreover, in view of the boundary effects, we also present a new model for nonuniform nonlocal bar-a varying volume fraction in the nonlocal phase. The numerical results of different shapes of materials show that, as compared with a uniform bar and that in local elasticity, the model herein shows more features, such as stress concentration. Thirdly, we present neutralizer-based iterative methods for integral equations of the second kind. As is known to all, there is an abundant of numerical techniques for solving integral equations, such as Neumann series, multi-grid method, GMRES and so on. Here, we introduce the concept of neutralizer, which is sometimes used in dealing with integrals (e.g., the asymptotic expansions of integrals), to obtain the neutralizer-based iterative method. Some features of such iterative method is numerically explored. Several meaningful examples are given, showing that the methods perform well as compared with some related methods.

    Research areas

  • Integral equations, Elasticity, Numerical solutions, Iterative methods (Mathematics)