Data-driven approach to gradient elasticity

Abstract

The structure of this thesis is organized as follows. Chapter 1 presents an introduction to Data-Driven Computational Mechanics. Herein, we introduce the motivation of this thesis, which is related to an extension of datadriven computing to gradient elasticity. In this chapter, we also briefly describe the problems in both one-dimensional and two-dimensional settings. Moreover, a review of published articles following the data-driven approach is presented in this chapter. In chapter 2, we summarize the governing equations. Fundamental equations required when the local theory is applied. This chapter also presents relationships between variable fields when we activate the long-range interactions in non-local theory. The important equations of the finite element method are shown in this chapter. The data-driven solver as well as its proof are presented in this chapter, which including this paradigm for local and non-local problems. Results from local problem are illustrated in the next chapter. The exact solutions for the local problem in two-dimensional setting is shown in this chapter. Also, the results from the finite element method are indicated in this chapter through steps. First, we discretize the domain into two resolutions and assign values to some parameters. Using these meshes as well as parameters, we obtain the stress values employing traditional finite element method and unconventional data-driven paradigm. We compare the outcomes from different methods to the exact solutions and evaluate the relative errors to study the convergence of the algorithms. In chapter 4, we conduct solutions for the non-local problem in both one-dimensional and two-dimensional settings. The one-dimensional problem is studied by evaluating the stress, strain and error and based on these results, we study the convergence of the paradigm. The solution and the accuracy of the two-dimensional problem are presented in this chapter.