DDD-XRD Matlab implementation and validation - Revision history

Nbertin: /* Analytical non-singular displacement gradient formulation */

2018-05-24T18:25:12Z

Analytical non-singular displacement gradient formulation

← Older revision		Revision as of 18:25, 24 May 2018
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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):		It can be shown that the displacement 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"		{\|border="0" align="center"
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	</math>		</math>
	\|}		\|}


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

Nbertin: /* Code */

2018-01-31T19:38:25Z

Code

← Older revision		Revision as of 19:38, 31 January 2018
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	The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:		The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

	[[~~File~~:ddd_xrd_matlab.tar \| ddd_xrd_matlab.tar]]		[[Media:ddd_xrd_matlab.tar \| ddd_xrd_matlab.tar]]

	Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment using the expressions presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.		Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment using the expressions presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

Nbertin: /* Code */

2018-01-20T03:09:21Z

Code

← Older revision		Revision as of 03:09, 20 January 2018
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	[[File:ddd_xrd_matlab.tar \| ddd_xrd_matlab.tar]]		[[File:ddd_xrd_matlab.tar \| ddd_xrd_matlab.tar]]

	Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.		Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment using the expressions presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

	=== Validation: triangular loop test case ===		=== Validation: triangular loop test case ===

Nbertin: /* Code */

2018-01-20T03:02:58Z

Code

← Older revision		Revision as of 03:02, 20 January 2018
Line 96:		Line 96:
	The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:		The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

	[[File:ddd_xrd_matlab.tar]]		[[File:ddd_xrd_matlab.tar \| ddd_xrd_matlab.tar]]

	Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.		Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

Nbertin: /* Code */

2018-01-20T03:02:30Z

Code

← Older revision		Revision as of 03:02, 20 January 2018
Line 96:		Line 96:
	The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:		The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

	[[ddd_xrd_matlab.tar]]		[[File:ddd_xrd_matlab.tar]]

	Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.		Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

Nbertin at 02:59, 20 January 2018

2018-01-20T02:59:56Z

← Older revision		Revision as of 02:59, 20 January 2018
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	=== Validation: triangular loop test case ===		=== Validation: triangular loop test case ===

	The code includes a test case (<tt>test_triangular_loop.m</tt>) to (i) ~~provide~~ an example of how to use the calculation functions and (ii) ~~perform~~ calculations that validate the non-singular displacement gradient formulation.		The code includes a test case (<tt>test_triangular_loop.m</tt>) that (i) provides an example of how to use the calculation functions, and (ii) performs calculations that validate the non-singular displacement gradient formulation.

	This test case does the following:		This test case does the following:

Nbertin at 02:59, 20 January 2018

2018-01-20T02:59:15Z

← Older revision		Revision as of 02:59, 20 January 2018
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	=== Code ===		=== Code ===

			The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:
	- based on DDLab format
	- functions descreption

			[[ddd_xrd_matlab.tar]]
⚫	=== Validation: ~~tringular~~ loop test case ===

			Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.
	- what the test case does

		⚫	=== Validation: triangular loop test case ===

			The code includes a test case (<tt>test_triangular_loop.m</tt>) to (i) provide an example of how to use the calculation functions and (ii) perform calculations that validate the non-singular displacement gradient formulation.

			This test case does the following:
			* generates a random triangular dislocation loop
			* computes the displacement gradient field along a line using the non-singular expression provided in Bertin and Cai, CMS, 2018.
			* compares it with the displacement gradient field obtained by numerically differentiating the displacement field
			* compares the stress obtained from the displacement gradient field with the non-singular stress expression provided in Cai et al., JMPS, 2006

Nbertin at 01:29, 20 January 2018

2018-01-20T01:29:24Z

← Older revision		Revision as of 01:29, 20 January 2018
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	\|}		\|}

	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.		where <math>R = \\|\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.

			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 <math>R_a</math> defined as:

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

			where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

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

			where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

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

			where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

			When adopting the following segment parametric representation

			{\|border="0" align="center"
			\|<math>
			\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
			</math>
			\|}

			an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

			{\|border="0" align="center"
			\|<math>
			\begin{align}
			&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}{3 R_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)}{3 R_a^3} \right\|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right\|_{s_1}^{s_2}
			\end{align}
			</math>
			\|}

Nbertin at 00:53, 20 January 2018

2018-01-20T00:53:06Z

@@ Line 4: / Line 4: @@
 <DIV>
 <P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
 </DIV>
@@ Line 12: / Line 15: @@
 == Analytical non-singular displacement gradient formulation ==
 == Matlab implementation ==

Nbertin: Created page with "
DDD-XRD approach

DDD-XRD: Matlab implementation and validation

2018-01-20T00:28:13Z

Created page with "DDD-XRD approach DDD-XRD: Matlab implementation and validation <DIV> DDD-XRD approach

DDD-XRD: Matlab implementation and validation
<DIV>
Nicolas Bertin and Wei Cai
</DIV>

<HR>

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 ==

== Matlab implementation ==