DDD-XRD Matlab implementation and validation: Difference between revisions

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<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
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<P ALIGN="CENTER">January 2018</P>
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== Analytical non-singular displacement gradient formulation ==
== Analytical non-singular displacement gradient formulation ==

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

{|border="0" align="center"
|<math>
\begin{align}
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{align}
</math>
|}

where <math>R = \sqrt{\vec{x'}-\vec{x}}</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.



== Matlab implementation ==
== Matlab implementation ==

=== Code ===

- based on DDLab format
- functions descreption

=== Validation: tringular loop test case ===

- what the test case does

Revision as of 00:53, 20 January 2018

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 produced by a dislocation loop Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} with Burgers vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{b}} can be written as (see Bertin and Cai, CMS, 2018):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = \sqrt{\vec{x'}-\vec{x}}} is the norm of the distance vector linking the field point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}} and the coordinate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x'}} spanning the dislocation line, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk}} is the permutation tensor, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} is Poisson's ratio of the medium. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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