Test Case: Binary Junction - Revision history

Wcash: moved TOC

2008-07-26T00:34:02Z

moved TOC

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	[[William Cash]]* and [[Seokwoo Lee]]		[[William Cash]]* and [[Seokwoo Lee]]
	----		----
			__TOC__

	==Introduction==		==Introduction==
	This tutorial will:		This tutorial will:

Wcash: messing with titles

2008-07-26T00:32:34Z

messing with titles

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			=DDLab Binary Junction Tutorial=
	[[William Cash]]* and [[Seokwoo Lee]]		[[William Cash]]* and [[Seokwoo Lee]]
	----		----

Wcash: added mention of the problem with mobfcc0.m in current DDLab release

2008-03-20T05:50:06Z

added mention of the problem with mobfcc0.m in current DDLab release

← Older revision		Revision as of 05:50, 20 March 2008
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	[[William Cash]] and [[Seokwoo Lee]]		[[William Cash]]* and [[Seokwoo Lee]]
	----		----

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	[[image:LC_bowed.jpg\|right\|thumb\|225px\|Figure 14: Bowing of the binary junction should be avoided]]		[[image:LC_bowed.jpg\|right\|thumb\|225px\|Figure 14: Bowing of the binary junction should be avoided]]
	Physically, the dislocation formed by the junction between two intersecting dislocations on differing slip planes is sessile, because it is composed of a partial dislocation in both slip planes. If one of the partials were to glide in its respective slip plane, it would force the other to climb. However in DDLab, the junction is formed using its normal split and merge topological changes, which means the nodes of the junction are not sessile. These operations enforce Burgers vector conservation, but calculate the slip plane by taking the cross product of the resulting Burgers vector and line direction. In the case of this tutorial, <math>\mathbf{b}=\frac{1}{2}[\,1\,1\,0\,]</math> and <math>\mathbf{n}=[\,0\,0\,1]</math> for the junction dislocation. The nodes at the end of the junction are constrained to only move along the line of intersection, because they are connected to segments in both slip planes. On the other hand, the nodes on the interior of the junction do not have such a restraint and are free to move in the <math>[\,0\,0\,1]</math> plane. If an external shear stress is applied to the junction, a situation such as that in Figure 14 can result. This bowing of the junction is clearly not physical and should be avoided.		Physically, the dislocation formed by the junction between two intersecting dislocations on differing slip planes is sessile, because it is composed of a partial dislocation in both slip planes. If one of the partials were to glide in its respective slip plane, it would force the other to climb. However in DDLab, the junction is formed using its normal split and merge topological changes, which means the nodes of the junction are not sessile. These operations enforce Burgers vector conservation, but calculate the slip plane by taking the cross product of the resulting Burgers vector and line direction. In the case of this tutorial, <math>\mathbf{b}=\frac{1}{2}[\,1\,1\,0\,]</math> and <math>\mathbf{n}=[\,0\,0\,1]</math> for the junction dislocation. The nodes at the end of the junction are constrained to only move along the line of intersection, because they are connected to segments in both slip planes. On the other hand, the nodes on the interior of the junction do not have such a restraint and are free to move in the <math>(\,0\,0\,1)</math> plane. If an external shear stress is applied to the junction, a situation such as that in Figure 14 can result. This bowing of the junction is clearly not physical and should be avoided.
	==References==		==References==
	<references/>		<references/>
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	==Downloads==		==Downloads==
	[[Media:LC_Junction_DDLab.tar.gz\|LC_Junction_DDLab.tar.gz]] - DDLab tutorial files. (Note: best if extracted as a subdirectory of the DDLab folder)		[[Media:LC_Junction_DDLab.tar.gz\|LC_Junction_DDLab.tar.gz]] - DDLab tutorial files. (Note: best if extracted as a subdirectory of the DDLab Inputs folder)

			<font color="red">Note: there are currently some issues with the fcc mobility law in the latest release of DDLab (12/18/2007) that we hope to work out in the future. There is also a typo in <tt>mobfcc0.m</tt> involving <tt>BScrew</tt>, <tt>Bedge</tt>, etc. If you have any trouble resolving this issue please contact the author for a corrected version of the code.</font>


	[[Media:LC_lock.pdf\|LC_lock.pdf]]. - Past version of this tutorial. (Note: There are some errors associated with the junction length calculation and the codes have been significantly modified)		[[Media:LC_lock.pdf\|LC_lock.pdf]]. - Past version of this tutorial. (Note: There are some errors associated with the junction length calculation and the codes have been significantly modified)


			----
			<nowiki>*</nowiki> Please contact [[William Cash]] with any questions or concerns

Wcash: /* Description of Files */

2008-03-20T05:18:41Z

Description of Files

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	This section provides a brief description of all the files used in this tutorial and their hierarchy. The main bullets are the primary scripts, and the sub-bullets are the functions or scripts they call.		This section provides a brief description of all the files used in this tutorial and their hierarchy. The main bullets are the primary scripts, and the sub-bullets are the functions or scripts they call.

	*'''<tt>LC_Junction.m</tt>'''- This script calculates the length of the junction formed between the two dislocations in the absence of stress, as described in the Junction Length Calculation section. It generates the surface plot of normalized junction length as a function of orientation shown in Figure ~~_____~~. It stores the final dislocation data and junction length (<tt>rn</tt>, <tt>links</tt>, and <tt>junc_length</tt>) in <tt>LC_junction_data.mat</tt>.		*'''<tt>LC_Junction.m</tt>'''- This script calculates the length of the junction formed between the two dislocations in the absence of stress, as described in the Junction Length Calculation section. It generates the surface plot of normalized junction length as a function of orientation shown in Figure 9. It stores the final dislocation data and junction length (<tt>rn</tt>, <tt>links</tt>, and <tt>junc_length</tt>) in <tt>LC_junction_data.mat</tt>.
	**'''<tt>LC_make_dis.m</tt>'''- The script that generates the ~~coordinate~~ of one fixed node for all 21 initial orientations of both dislocations from <math>\,-\pi</math> to <math>\,\pi</math>. The midpoint of all the dislocations is the origin		**'''<tt>LC_make_dis.m</tt>'''- The script that generates the coordinates of one fixed node for all 21 initial orientations of both dislocations from <math>\,-\pi</math> to <math>\,\pi</math>. The midpoint of all the dislocations is the origin
	**'''<tt>LC_relax.m</tt>'''- Script with the input deck for DDLab with zero applied stress. It requires <tt>D1</tt> and <tt>D2</tt>, one fixed node position for dislocations 1 and 2 respectively; and <tt>dis_length</tt>, the initial length of the dislocations, as inputs.		**'''<tt>LC_relax.m</tt>'''- Script with the input deck for DDLab with zero applied stress. It requires <tt>D1</tt> and <tt>D2</tt>, one fixed node position for dislocations 1 and 2 respectively; and <tt>dis_length</tt>, the initial length of the dislocations, as inputs.
	**'''<tt>LC_junc_length.m</tt>'''- Function that calculates the length of the resulting binary junction. It takes the resulting <tt>rn</tt> and <tt>connectivity</tt> matrices from <tt>LC_relax.m</tt> as inputs and returns only junction length.		**'''<tt>LC_junc_length.m</tt>'''- Function that calculates the length of the resulting binary junction. It takes the resulting <tt>rn</tt> and <tt>connectivity</tt> matrices from <tt>LC_relax.m</tt> as inputs and returns only junction length.


	*'''<tt>LC_dissociate_orientation.m</tt>'''- This script determines the critical stress to dissociate a binary junction based on the orientation of the dislocations. The orientation angle of dislocation 1,<math>\,\phi_1</math> , equals <math>\frac{2\pi}{10}</math> and those of dislocation 2,<math>\,\phi_2</math> , are <math>\frac{\pi}{10}</math>, <math>\frac{2\pi}{10}</math>, and <math>\frac{3\pi}{10}</math>. The junction data is loaded from <tt>LC_junction_data.mat</tt>. The user inputs a guess at the critical shear stress in coordinate system 2 (see the Resolved Shear Stress Calculation section). The resulting critical shear stresses are plotted with <math>\phi_2</math> and junction length. It is best to start with a conservative guess to avoid having the dislocations dissociate and begin acting as Frank-Read sources, which is costly computationally.		*'''<tt>LC_dissociate_orientation.m</tt>'''- This script determines the critical stress to dissociate a binary junction based on the orientation of the dislocations. The orientation angle of dislocation 1,<math>\,\phi_1</math> , equals <math>\frac{2\pi}{10}</math> and those of dislocation 2,<math>\,\phi_2</math> , are <math>\frac{\pi}{10}</math>, <math>\frac{2\pi}{10}</math>, and <math>\frac{3\pi}{10}</math>. The junction data is loaded from <tt>LC_junction_data.mat</tt>. The user inputs a guess at the critical shear stress in coordinate system 2 (see the Resolved Shear Stress Calculation section). The resulting critical shear stresses are plotted with <math>\,\phi_2</math> and junction length. It is best to start with a conservative guess to avoid having the dislocations dissociate and begin acting as Frank-Read sources, which is costly computationally.
	**'''<tt>LC_stress.m</tt>'''- Script with the DDLab input deck for the user specified <math>\,\tau_{13}</math> in coordinate system 2. It performs the stress transformation described in the Resolved Shear Stress Calculation section. It also requires that <tt>dis_length</tt>, the initial length of the dislocations prior to junction formation, is specified.		**'''<tt>LC_stress.m</tt>'''- Script with the DDLab input deck for the user specified <math>\,\tau_{13}</math> in coordinate system 2. It performs the stress transformation described in the Resolved Shear Stress Calculation section. It also requires that <tt>dis_length</tt>, the initial length of the dislocations prior to junction formation, is specified.

Wcash: /* Resolved Shear Stress Calculation */

2008-03-20T05:11:49Z

Resolved Shear Stress Calculation

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	where <math>\mathbf{\sigma}</math> and <math>\mathbf{\sigma'}</math> are the stress tensor defined in the original and new coordinate systems, respectively. <math>\mathbf{Q}</math> is the transformation matrix, where its columns are the components of the basis <math>\mathbf{e'}</math> written in terms of the basis <math>\mathbf{e}</math>. If <math>\mathbf{e'}</math> and <math>\mathbf{e}</math> are chosen as orthonormal bases, then <math>\mathbf{Q}</math> is an orthogonal matrix consisting of directional cosines, and <math>\mathbf{Q^{-1}}</math> equals <math>\mathbf{Q^T}</math>.		where <math>\mathbf{\sigma}</math> and <math>\mathbf{\sigma'}</math> are the stress tensor defined in the original and new coordinate systems, respectively. <math>\mathbf{Q}</math> is the transformation matrix, where its columns are the components of the basis <math>\mathbf{e'}</math> written in terms of the basis <math>\mathbf{e}</math>. If <math>\mathbf{e'}</math> and <math>\mathbf{e}</math> are chosen as orthonormal bases, then <math>\mathbf{Q}</math> is an orthogonal matrix consisting of directional cosines, and <math>\mathbf{Q^{-1}}</math> equals <math>\mathbf{Q^T}</math>.

	The resolved shear stress, <math>\,\tau_{13}</math> in basis <math>\mathbf{e'}</math> (Figure 6) is applied to the formed binary junction. This stress is transformed using the following algorithm in <tt>LC_stress.m</tt>:		The resolved shear stress, <math>\,\tau_{13}</math> in basis <math>\mathbf{e'}</math> (Figure 7) is applied to the formed binary junction. This stress is transformed using the following algorithm in <tt>LC_stress.m</tt>:

	<blockquote style="background: white; border: 0; padding: rem; width: 525px;"><pre>		<blockquote style="background: white; border: 0; padding: rem; width: 525px;"><pre>
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	where <math>\,\mu</math> is the shear modulus and <math>\,b</math> is the magnitude of the Burgers vector. From the relation, it is obvious that dissociation can occur more easily with longer dislocation arms. If the initial length of the two dislocations increases, the resulting junction length ~~increase~~. Also, the length of dislocation arms become longer (not shown), resulting in lower critical resolved shear stresses.		where <math>\,\mu</math> is the shear modulus and <math>\,b</math> is the magnitude of the Burgers vector. From the relation, it is obvious that dissociation can occur more easily with longer dislocation arms. If the initial length of the two dislocations increases, the resulting junction length increases. Also, the length of dislocation arms become longer (not shown), resulting in lower critical resolved shear stresses.

	===Description of Files===		===Description of Files===

Wcash: /* Initial Orientation */

2008-03-20T04:50:05Z

Initial Orientation

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	===Initial Orientation===		===Initial Orientation===
	[[image:LC_formation.png\|right\|thumb\|250px\|Figure 1: Formation of a Lomer-Cottrell lock]]		[[image:LC_formation.png\|right\|thumb\|250px\|Figure 1: Formation of a Lomer-Cottrell lock]]
	These simulations are performed using the material constants for gold in order to conform with past computational and experimental results obtained by Madec et. al<ref name="Madec">''From Dislocation Junctions to Forest Hardening,'' Phys. Rev. Lett. 89 255508 (2002), R. Madec, B. Devincre, and L.P. Kubin</ref>, although any fcc metal could be used. In order to simplify the simulations, two dislocations with identical lengths, but on differing slip planes, are initially placed such that they intersect at their center. The first dislocation glides on the <math>(\,1\,1\,1\,)</math> plane - henceforth referred to as dislocation 1. On the other hand, dislocation 2 glides on the <math>(\,1\,1\,\bar{1}\,)</math> plane. Both of these slip planes are part of the <math>\{\,1\,1\,1\,\}</math> slip system in fcc crystals. The dislocations have Burgers vectors of <math>\frac{1}{2}[\,1\,0\,\bar{1}\,]</math> and <math>\frac{1}{2}[\,0\,1\,1\,]</math>, respectively, regardless of their line direction <math>\mathbf{\xi}</math>. A Lomer-Cottrell junction could form at the intersection of these two planes, i.e. along the <math>[\,\bar{1}\,1\,0\,]</math> direction, if the conditions are correct, as shown in Figure 1. Thus, it is desired to determine the junction behavior at all ~~possible~~ combinations of line directions.		These simulations are performed using the material constants for gold in order to conform with past computational and experimental results obtained by Madec et. al<ref name="Madec">''From Dislocation Junctions to Forest Hardening,'' Phys. Rev. Lett. 89 255508 (2002), R. Madec, B. Devincre, and L.P. Kubin</ref>, although any fcc metal could be used. In order to simplify the simulations, two dislocations with identical lengths, but on differing slip planes, are initially placed such that they intersect at their center. The first dislocation glides on the <math>(\,1\,1\,1\,)</math> plane - henceforth referred to as dislocation 1. On the other hand, dislocation 2 glides on the <math>(\,1\,1\,\bar{1}\,)</math> plane. Both of these slip planes are part of the <math>\{\,1\,1\,1\,\}</math> slip system in fcc crystals. The dislocations have Burgers vectors of <math>\frac{1}{2}[\,1\,0\,\bar{1}\,]</math> and <math>\frac{1}{2}[\,0\,1\,1\,]</math>, respectively, regardless of their line direction <math>\mathbf{\xi}</math>. A Lomer-Cottrell junction could form at the intersection of these two planes, i.e. along the <math>[\,\bar{1}\,1\,0\,]</math> direction, if the conditions are correct, as shown in Figure 1. Thus, it is desired to determine the junction behavior at all combinations of line directions.

	If we consider all possible orientations of a straight dislocation of constant length and we take the origin to be the initial point of contact of the dislocations, we can define an end point (i.e. fixed node), <math>\mathbf{P}</math>, of the dislocation by a parametric equation of <math>\,\theta</math>:		If we consider all possible orientations of a straight dislocation of constant length and we take the origin to be the initial point of contact of the dislocations, we can define an end point (i.e. fixed node), <math>\mathbf{P}</math>, of the dislocation by a parametric equation of <math>\,\theta</math>:
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	where <math>R</math> is half the length of the dislocation; <math>\mathbf{u}</math> is a unit vector in the direction of the line of intersection, <math>[\,\bar{1}\,1\,0\,]</math> ; and <math>\mathbf{n}</math> is the unit glide plane ~~normal~~. The vectors <math>\mathbf{P}</math> and <math>-\mathbf{P}</math> are then the end points of the dislocation. It is obvious from the equation that varying <math>\,\theta</math> from <math>[\,-\pi,\,\pi\,]</math> generates a circle on the glide plane. We have chosen to simulate 21 evenly distributed line orientations for both dislocations in this study. This results in a discretization for both dislocations shown in Figure 2, where the points represent one fixed node in each orientation and the initial line direction <math>\mathbf{\xi}</math> is taken as the vector from the origin to that point. <tt>LC_make_dis.m</tt> calculates the position of an end node of each dislocation in this manner using the following code:		where <math>\,R</math> is half the length of the dislocation; <math>\mathbf{u}</math> is a unit vector in the direction of the line of intersection, <math>[\,\bar{1}\,1\,0\,]</math> ; and <math>\mathbf{n}</math> is the unit normal of the glide plane. The vectors <math>\mathbf{P}</math> and <math>-\mathbf{P}</math> are then the end points of the dislocation. It is obvious from the equation that varying <math>\,\theta</math> from <math>[\,-\pi,\,\pi\,]</math> generates a circle on the glide plane. We have chosen to simulate 21 evenly distributed line orientations for both dislocations in this study. This results in a discretization for both dislocations shown in Figure 2, where the points represent one fixed node in each orientation and the initial line direction <math>\mathbf{\xi}</math> is taken as the vector from the origin to that point. <tt>LC_make_dis.m</tt> calculates the position of an end node of each dislocation in this manner using the following code:

	<blockquote style="background: white; border: 0; padding: 1em; width: 550px;"><pre>		<blockquote style="background: white; border: 0; padding: 1em; width: 550px;"><pre>

Wcash: /* Introduction */

2008-03-20T04:40:15Z

Introduction

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	==Introduction==		==Introduction==
	This tutorial will:		This tutorial will:
	*Study the formation and dissociation of Lomer-Cottrell (binary) junctions in fcc crystals using ~~both~~ DDLab		*Study the formation and dissociation of Lomer-Cottrell (binary) junctions in fcc crystals using DDLab
	*Introduce running a batch of simulations, and calculating junction length and critical resolved shear stress to cause dissociation of the junction		*Introduce running a batch of simulations, and calculating junction length and critical resolved shear stress to cause dissociation of the junction
	*Determine the dependence of junction length and critical stress at dissociation on the initial orientation of the intersecting dislocations		*Determine the dependence of junction length and critical stress at dissociation on the initial orientation of the intersecting dislocations
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	A copy of the input decks used in this tutorial can be downloaded from the link at the bottom of this page.		A copy of the input decks used in this tutorial can be downloaded from the link at the bottom of this page.

	A tutorial for ParaDiS will be created in the ~~near~~ future.		A tutorial for ParaDiS will be created in the future.

	==Background==		==Background==

Wcash: /* Junction Dissociation */

2008-01-14T21:57:05Z

Junction Dissociation

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	image:LC_length_tc_vs_length.jpg\|Figure 13: Critical stress at various dislocation lengths		image:LC_length_tc_vs_length.jpg\|Figure 13: Critical stress at various dislocation lengths
	</gallery>		</gallery>

			<font color="red">We can discuss in more detail which dislocation length is the controlling parameter for critical stress. Right now, the relaxed dislocation structure has five segments and their lengths are all related in this particular series of simulations. More simulations with different geometries are needed to determine which length is the controlling parameter.</font>

	===Junction Dislocations in DDLab===		===Junction Dislocations in DDLab===

Wcash: /* References */

2008-01-14T21:52:04Z

References

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	==References==		==References==
	<references/>		<references/>
			[1] From Dislocation Junctions to Forest Hardening, Phys. Rev. Lett. 89 255508 (2002), R. Madec, B. Devincre, and L.P. Kubin

			<font color="red">Currently, the references only show up automatically to logged in users. We are looking for a solution to this problem. For now we are just writing the references by hand.</font>

	==Downloads==		==Downloads==

Wcash: added section about the sessile junction

2008-01-13T09:47:25Z

added section about the sessile junction

← Older revision		Revision as of 09:47, 13 January 2008
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	*Determine the dependence of junction length and critical stress at dissociation on the initial orientation of the intersecting dislocations		*Determine the dependence of junction length and critical stress at dissociation on the initial orientation of the intersecting dislocations
	*Determine the dependence of junction length and critical stress at dissociation on the initial length of the dislocations		*Determine the dependence of junction length and critical stress at dissociation on the initial length of the dislocations
	*Discuss any limitations. Physically the junction should be sessile. But in the simulation, the inner nodes of the junction segment are able to move. The end nodes of the junction are confined to the intersection line as they should. ~~(work in progress)~~		*Discuss any limitations. Physically the junction should be sessile. But in the simulation, the inner nodes of the junction segment are able to move. The end nodes of the junction are confined to the intersection line as they should.

	This tutorial assumes that the reader is familiar with the basics of DDLAB. New users should read [[DDLab_Manual_01]] before proceeding.		This tutorial assumes that the reader is familiar with the basics of DDLAB. New users should read [[DDLab_Manual_01]] before proceeding.
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	</gallery>		</gallery>

			===Junction Dislocations in DDLab===

		⚫	[[image:LC_bowed.jpg\|right\|thumb\|225px\|Figure 14: Bowing of the binary junction should be avoided]]
			Physically, the dislocation formed by the junction between two intersecting dislocations on differing slip planes is sessile, because it is composed of a partial dislocation in both slip planes. If one of the partials were to glide in its respective slip plane, it would force the other to climb. However in DDLab, the junction is formed using its normal split and merge topological changes, which means the nodes of the junction are not sessile. These operations enforce Burgers vector conservation, but calculate the slip plane by taking the cross product of the resulting Burgers vector and line direction. In the case of this tutorial, <math>\mathbf{b}=\frac{1}{2}[\,1\,1\,0\,]</math> and <math>\mathbf{n}=[\,0\,0\,1]</math> for the junction dislocation. The nodes at the end of the junction are constrained to only move along the line of intersection, because they are connected to segments in both slip planes. On the other hand, the nodes on the interior of the junction do not have such a restraint and are free to move in the <math>[\,0\,0\,1]</math> plane. If an external shear stress is applied to the junction, a situation such as that in Figure 14 can result. This bowing of the junction is clearly not physical and should be avoided.
	==References==		==References==
	<references/>		<references/>
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	[[Media:LC_lock.pdf\|LC_lock.pdf]]. - Past version of this tutorial. (Note: There are some errors associated with the junction length calculation and the codes have been significantly modified)		[[Media:LC_lock.pdf\|LC_lock.pdf]]. - Past version of this tutorial. (Note: There are some errors associated with the junction length calculation and the codes have been significantly modified)
⚫	[[image:LC_bowed.jpg\|right\|thumb\|225px\|Figure X]]