Net Burgers Density Vector Fields in Crystal Plasticity

This PhD thesis consists of five complementary chapters. Chapters 2 through 4 constitute the basis of research papers to be published subsequently. These three chapters summarize the state of a single crystal undergoing elastoplastic deformation. The studies presented in this thesis primarily deal with experimental and computational concepts that enable the calculation, measurement and extraction of the spatially resolved net Burgers density vector and the geometrically necessary dislocation densities (GNDs), which reveal the small scale continuum characteristics of a single crystal in the elastoplastic state. The calculation methodology of a new validation parameter, β, which is the orientation of the net Burgers density vector, is given in chapter 2. This new parameter, β, enables us to validate the elastic-plastic constitutive relations. Since the existing methods used for validation cannot give direct information about the state of the material, the β-variable is introduced for elastic- plastic constitutive models. β-fields, which are essentially contour maps of β-variables on two dimensional spatial coordinates, are used to monitor the activity regions of effective slip systems. Chapters 2 through 4 present a comprehensive analysis of the spatially resolved net Burgers density vector, along with the length scale characterization of dislocation structures and validation of constitutive relations. The studies presented in this work are the outcome of experimental and computational research. The experimental work consists of the indentation of a nickel single crystal deformed through a quasi-statically applied line load parallel to the [110] crystallographic orientation. The line load was applied onto (001) surface of the single crystal by a tungsten carbide wedge indenter with a 90◦ included angle. A two-dimensional deformation field on an indented single crystal, in which the only non-zero lattice rotation occurs in the plane of deformation and only three effective in-plane slip systems are activated, was investigated. The mid-section of the deformed single crystal was exposed by EDM and polished electrochemically. The in-plane lattice rotations were measured by high-resolution electron backscattered diffraction (HR-EBSD). The Nye's dislocation density components, lattice curvatures, GNDs and net Burgers density vectors were calculated. Therefore, the β- variable and the β-fields are calculated both experimentally, analytically and numerically in Chapter 2. A qualitative comparison of the three methods showed that the β-field obtained from experimental measurements agrees with those obtained from analytical and numerical methods. The directions of the net Burgers density vector, which are used to determine the boundaries of the slip activity regions, are also given in Chapter 2. Chapter 3 mainly deals with the hardening parameters associated with strain hardening rules utilized in finite element simulations, and investigates the sensitivity of the β-variable to parameters such as latent hardening ratio, initial hardening modulus and saturation strength. The study revealed that a change in the saturation strength has a significant effect on both magnitude of the β-variable and the boundary of the slip activity regions. Chapter 4 presents a length scale analysis associated with dislocation structures such as cell size and cell wall width. The methods presented in this chapter employ the SEM- based continuum method and Fourier Analysis. As-measured GNDs are extracted along the local crystallographic traces, and a quasi-periodic arrangement of dislocation structures is obtained. The extracted GND functions are truncated, interpolated, and filtered. Finally, Fourier Transform is applied to obtain a relationship between cell size and cell wall width of the dislocation structures. The results are compared with those obtained by TEM micrographs. Whereas TEM micrographs characterize the dislocation structures in small scale, the method that is presented in this chapter provides multi scale characterization, which is an order of magnitude larger. Concluding remarks and recommendations for future studies are given in Chapter 5.