DDD-XRD Matlab implementation and validation

DDD-XRD approach

DDD-XRD: Matlab implementation and validation

Nicolas Bertin and Wei Cai

January 2018

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This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

Analytical non-singular displacement gradient formulation

It can be shown that the deformation gradient $G_{ml}=u_{m,l}$ produced by a dislocation loop $C$ with Burgers vector ${\vec {b}}$ can be written as (see Bertin and Cai, CMS, 2018):

${\begin{aligned}u_{m,l}({\vec {x}})=&-{\frac {1}{8\pi }}\oint _{C}b_{m}\epsilon _{jlk}R_{,ppj}dx'_{k}-{\frac {1}{8\pi }}\oint _{C}b_{i}\epsilon _{mik}R_{,ppl}\,dx'_{k}\\&-{\frac {1}{8\pi (1-\nu )}}\oint _{C}b_{i}\epsilon _{ijk}R_{,mjl}\,dx'_{k}\end{aligned}}$

where $R={\sqrt {{\vec {x'}}-{\vec {x}}}}$ is the norm of the distance vector linking the field point ${\vec {x}}$ and the coordinate ${\vec {x'}}$ spanning the dislocation line, $\epsilon _{ijk}$ is the permutation tensor, and $\nu$ is Poisson's ratio of the medium. $R_{,ijk}=\partial ^{3}R/\partial x_{i}\partial x_{j}\partial x_{k}$ denotes the third derivative of the radius vector wrt. the field coordinate.

Matlab implementation

Code

- based on DDLab format - functions descreption

Validation: tringular loop test case

- what the test case does