Papers by Keyword: Nye Tensor

Abstract: The Ginzburg-Landau (G-L) model possesses the thermodynamic foundation of energy minimization and is available for many dynamic formalisms, thus holds great potential for investigating the complex materials behaviors. The common ingredient in energy spawns the real-time control of diffusion potential and chemical mobility by integrating G-L model with CALPHAD technique. The coupling between martensitic transformation and dislocation evolution is achieved by mean of continuous mechanism. The updated G-L model is then validated against the martensitic transformation coupled with composition redistribution in Fe-C binary system. The modeling allows some deeper insights into the mechanisms of coupling effects behind the observed phenomena. It has been proven that the partitioning of carbon in steels is an ordinary diffusion governed by instantaneous diffusion potential and chemical mobility. The rough twin boundaries and retained austenite within the martensite should be attributed to the effect of dislocations. Although the developed model in this chapter has deficiencies, it sheds some lights on the integration of multi-physics models for a complex phase transformation.

Abstract: We develop a phase field model that describes the elastic distortion of a ferroelastic material with cubic anisotropy due to an arbitrary dislocation network and a uniform external load. The dislocation network is characterized using the Nye tensor and enters the formulation via a set of incompatibility constraints for the internal strain field. The long-range elastic response of the material is obtained by minimization of the free energy that accounts for higher order terms of the order parameters and symmetry-adapted strain gradients. To demonstrate the performance of the model, a minimal version of continuum dislocation dynamics is used to investigate the simultaneous evolution of the network of geometrically necessary dislocations and the internal strain field.

Abstract: We introduce a mesoscopic framework that is capable of simulating the evolution of dislocation networks and, at the same time, spatial variations of the stress, strain and displacement fields throughout the body. Within this model, dislocations are viewed as sources of incompatibility of strains. The free energy of a deformed solid is represented by the elastic strain energy that can be augmented by gradient terms to reproduce dispersive nature of acoustic phonons and thus set the length scale of the problem. The elastic strain field that is due to a known dislocation network is obtained by minimizing the strain energy subject to the corresponding field of incompatibility constraints. These stresses impose Peach-Koehler forces on all dislocations and thus drive the evolution of the dislocation network.

Abstract: Misorientation can be calculated over large datasets and a theme of this paper is the usefulness of examining the results statistically. Comparing the statistics of misorientations calculated from neighbouring pixels (or grains) with those calculated from pairs of pixels (or grains) selected at random helps to indicate deformation and recrystallisation mechanisms. Taking boundary length into account provides a link to grain boundary energy, and boundary length versus misorientation data should be used to examine how boundaries with different misorientations evolve through time. Time lapse misorientation maps indicate how orientation changes through time at particular points in a microstructure during in situ experiments. The size of areas which have changed orientation by particular amounts can be linked to boundary length and boundary migration velocities. When dealing with different phases, the statistics of angular relationships, akin to intraphase misorientation analysis, can indicate orientation relationships in the absence of prior knowledge, which is advantageous in investigating the plethora of minerals that make up the Earth.

Abstract: The Weighted Burgers Vector (WBV) is defined as the sum, over all types of dislocations, of [(density of intersections of dislocation lines with a map) x (Burgers vector)]. It can be calculated, for any crystal system, solely from orientation gradients in a map view, unlike the full dislocation density tensor, which requires gradients in the third dimension. No assumption is made about gradients in the third dimension and they may be non-zero. The only assumption involved is that elastic strains are small so the lattice distortion is entirely due to dislocations. Orientation gradients can be estimated from gridded orientation measurements obtained by EBSD mapping, so the WBV can be calculated as a vector field on an EBSD map. The magnitude of the WBV gives a lower bound on the magnitude of the dislocation density tensor when that magnitude is defined in a coordinate invariant way. The direction of the WBV can constrain the types of Burgers vectors of geometrically necessary dislocations present in the microstructure, most clearly when it is broken down in terms of lattice vectors. The WBV has five advantages over other measures of local lattice distortion. 1. It is a vector and hence carries more information than any scalar measure of local misorientation. 2. It has an explicit mathematical link to the individual Burgers vectors of dislocations. 3. Since it is derived via tensor calculus, it is not dependent on the map coordinate system, in contrast to existing measures of local misorientation which are not only scalar but dependent on the coordinate system used. 4. Calculation involves no assumptions about energy minimisation. 5. The numerical differentiation involved in calculating the WBV may introduce errors, but there is a direct mathematical link to a contour integral. The net Burgers vector content of dislocations intersecting an area of a map can be simply calculated by an integration round the edge of that area, a method which is fast and complements point-by-point WBV calculations. Errors in orientation measurement will have a much smaller effect here, and dislocations can be detected which are otherwise lost in the noise of any local calculation.

Abstract: Significant advances are reported in the application of HR-EBSD to the imaging of the dislocation structure of polycrystalline materials. The central assumption of the method is the compatibility of the total displacement field, which relates the (Nye) dislocation tensor to the (partially measurable) curl of the elastic displacement field. Two key challenges must be addressed, including: a) the fundamental limitation imposed by the electron-opacity of typical materials, which limits the measurement of gradients in the displacement field in the direction normal to the sample surface; and b) the inability of HR-EBSD to recover the spherical (elastic) distortions of the lattice. This second challenge can be overcome if a traction free boundary condition is applied. It is illustrated that consideration of the familiar stress equilibrium relations gives additional information, which may enable estimates of the missing components of the Nye tensor. An example of application of HR-EBSD to a Mg-Ce sample is presented.

Abstract: An extension to a previously published, novel stereological method is reported which infers experimentally inaccessible components of the Nye GND tensor. Limitations imposed by electron-opacity of metals prevent direct measurement of four components of the Nye tensor, but it is possible to use additional experimentally-obtainable information in connection with underlying field equilibrium equations to estimate these additional components. This approach uses derivatives to the infinitesimal elastic distortion tensor to reduce error imposed by pattern center inaccuracy.