DDD-XRD Matlab implementation and validation

DDD-XRD approach

DDD-XRD: Matlab implementation and validation

Nicolas Bertin and Wei Cai

January 2018

---

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 $R=\|{\vec {x'}}-{\vec {x}}\|$ 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.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector $R_{a}$ defined as:

$R_{a}={\sqrt {R^{2}+a^{2}}}={\sqrt {R_{i}R_{i}+a^{2}}}={\sqrt {(x'_{i}-x_{i})(x'_{i}-x_{i})+a^{2}}}$

where $a$ denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points ${\vec {x}}_{a}$ and ${\vec {x}}_{b}$ can be analytically expressed as:

${\begin{aligned}u_{m,l}({\vec {x}})=&-{\frac {1}{8\pi }}b_{m}\epsilon _{jlk}A_{jk}({\vec {x}})-{\frac {1}{8\pi }}b_{i}\epsilon _{mik}A_{lk}({\vec {x}})\\&-{\frac {1}{8\pi (1-\nu )}}b_{i}\epsilon _{ijk}B_{jklm}({\vec {x}})\end{aligned}}$

where line integrals $A_{jk}({\vec {x}})$ and $B_{jklm}({\vec {x}})$ are given by:

$A_{jk}({\vec {x}})=t_{k}\left[3a^{2}d_{j}J_{05}+2d_{j}J_{03}+3a^{2}t_{j}J_{15}+2t_{j}J_{13}\right]$

${\begin{aligned}B_{jklm}({\vec {x}})=&t_{k}\left[(\delta _{mj}d_{l}+\delta _{jl}d_{m}+\delta _{lm}d_{j})J_{03}+(\delta _{mj}t_{l}+\delta _{jl}t_{m}+\delta _{lm}t_{j})J_{13}-3(d_{m}d_{j}d_{l})J_{05}\right.\\&\left.-3(d_{m}d_{j}t_{l}+d_{m}t_{j}d_{l}+t_{m}d_{j}d_{l})J_{15}-3(d_{m}t_{j}t_{l}+t_{m}d_{j}t_{l}+t_{m}t_{j}d_{l})J_{25}-3(t_{m}t_{j}t_{l})J_{35}\right]\end{aligned}}$

where ${\vec {t}}$ denotes the unit dislocation line tangent, and ${\vec {d}}={\vec {x}}_{0}-{\vec {x}}$ is the vector linking field point ${\vec {x}}$ to its orthogonal projection ${\vec {x}}_{0}$ on the dislocation line.

When adopting the following segment parametric representation

${\vec {x'}}={\vec {x}}_{0}+s{\vec {t}},\;s\in (s_{1},s_{2})$

an analytical expression for the line integrals $J_{ij}$ is obtained as follows:

${\begin{aligned}&J_{03}=\int _{s_{1}}^{s_{2}}{\frac {1}{R_{a}^{3}}}ds=\left.{\frac {s}{({\vec {d}}\cdot {\vec {d}}+a^{2})R_{a}}}\right|_{s_{1}}^{s_{2}}\\&J_{13}=\int _{s_{1}}^{s_{2}}{\frac {s}{R_{a}^{3}}}ds=\left.-{\frac {1}{R_{a}}}\right|_{s_{1}}^{s_{2}}\\&J_{05}=\int _{s_{1}}^{s_{2}}{\frac {1}{R_{a}^{5}}}ds=\left.{\frac {2s^{3}}{3({\vec {d}}\cdot {\vec {d}}+a^{2})^{2}R_{a}^{3}}}\right|_{s_{1}}^{s_{2}}+\left.{\frac {s}{({\vec {d}}\cdot {\vec {d}}+a^{2})R_{a}^{3}}}\right|_{s_{1}}^{s_{2}}\\&J_{15}=\int _{s_{1}}^{s_{2}}{\frac {s}{R_{a}^{5}}}ds=\left.-{\frac {1}{3R_{a}^{3}}}\right|_{s_{1}}^{s_{2}}\\&J_{25}=\int _{s_{1}}^{s_{2}}{\frac {s^{2}}{R_{a}^{5}}}ds=\left.{\frac {s^{3}}{3({\vec {d}}\cdot {\vec {d}}+a^{2})R_{a}^{3}}}\right|_{s_{1}}^{s_{2}}\\&J_{35}=\int _{s_{1}}^{s_{2}}{\frac {s^{3}}{R_{a}^{5}}}ds=\left.-{\frac {2({\vec {d}}\cdot {\vec {d}}+a^{2})}{3R_{a}^{3}}}\right|_{s_{1}}^{s_{2}}-\left.{\frac {s^{2}}{R_{a}^{3}}}\right|_{s_{1}}^{s_{2}}\end{aligned}}$

Matlab implementation

Code

- based on DDLab format - functions descreption

Validation: tringular loop test case

- what the test case does