Test Case: Frank-Read Source - Revision history

Jieyin: /* Conclusion */

2008-02-13T17:24:36Z

Conclusion

← Older revision		Revision as of 17:24, 13 February 2008
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	2. Numerical results are closer to theoretical ones at small '''a''' and large '''L'''.		2. Numerical results are closer to theoretical ones at small '''a''' and large '''L'''.

	3. Numerical critical stress for '''a''' well fits theoretical results for '''a/5'''		3. Numerical critical stress for '''a/5''' well fits theoretical results for '''a'''

	Some of the plots above will be updated when more accurate data are available for a=1 (or less) and L>6000.		Some of the plots above will be updated when more accurate data are available for a=1 (or less) and L>6000.

Jieyin: /* F-R Test */

2008-02-13T17:24:03Z

F-R Test

← Older revision		Revision as of 17:24, 13 February 2008
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	Longer simulation time can help eliminate the intersections at large l.		Longer simulation time can help eliminate the intersections at large l.

	A interesting phenomenon is that numerical result for a/5 well fits analytical result for a. Relative error goes to 0.~~001~~ when l goes to 10,000.		A interesting phenomenon is that numerical result for a/5 well fits analytical result for a. Relative error goes to 0.01 when l goes to 10,000.

	[[Image:Frstr_L_a10a50.jpg]]		[[Image:Frstr_L_a10a50.jpg]]

Jieyin: /* Conclusion */

2008-02-13T17:22:02Z

Conclusion

← Older revision		Revision as of 17:22, 13 February 2008
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	==Conclusion==		==Conclusion==

	After conducting several test cases of Frank-Read Source using		After conducting several test cases of Frank-Read Source using DDLab, we have shown:

			1, The critical stress is linearly dependent on ln(L/a)/L;

			2. Numerical results are closer to theoretical ones at small '''a''' and large '''L'''.

			3. Numerical critical stress for '''a''' well fits theoretical results for '''a/5'''

			Some of the plots above will be updated when more accurate data are available for a=1 (or less) and L>6000.

Jieyin: /* F-R Test */

2008-02-13T17:14:56Z

F-R Test

← Older revision		Revision as of 17:14, 13 February 2008
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	4.3. lmax influence		4.3. lmax influence

			Above results are based on the setting that lmax=l, lmin=l/10 other than the default lmax=1000, limin=200. Though the dislocation line is less smoother, numerical results are the same for the case l=5000. For computation simplicity, we can set lmax and lmin proportional to l and get reasonable results.

			==Conclusion==

			After conducting several test cases of Frank-Read Source using
	Currently there is some deviation from the theoretical value of critical applied stress; however, we can clearly see the linear relationship. Following figure shows the critical applied stress is linearly dependent on lnN/N.



	By varying a, which is the core radius, we can derive the critical stresses and find its linear dependence on ln a, as long as L is fixed.

Jieyin: /* F-R Test */

2008-02-13T17:06:52Z

F-R Test

← Older revision		Revision as of 17:06, 13 February 2008
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	When a varies from 1 to 50 and l is fixed at 5000, numerical results are given, shown in following figures		When a varies from 1 to 50 and l is fixed at 5000, numerical results are given, shown in following figures

	[[Image:Frstr_a_l5000.jpg]]		[[Image:Frstr_a_l5000.jpg]]

	Same as the formula, the critical stress has a linear dependence on ln(L/a)/L		Same as the formula, the critical stress has a linear dependence on ln(L/a)/L

	[[Image:Frstr_lnlal_l5000.jpg ]]		[[Image:Frstr_lnlal_l5000.jpg ]]

	We can see the decrease in relative error.		We can see the decrease in relative error.

	[[Image:Re_a_l5000.jpg ]]		[[Image:Re_a_l5000.jpg ]]

	When the length of initial segment increases, relative error seems to decrease when a approaches 0.		When the length of initial segment increases, relative error seems to decrease when a approaches 0.

	[[Image:RE_a_l16.jpg]]		[[Image:RE_a_l16.jpg]]

Jieyin: /* F-R Test */

2008-02-13T17:06:02Z

F-R Test

← Older revision		Revision as of 17:06, 13 February 2008
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	When a varies from 1 to 50 and l is fixed at 5000, numerical results are given, shown in following figures		When a varies from 1 to 50 and l is fixed at 5000, numerical results are given, shown in following figures

	[[Image:Frstr_a_l5000.jpg]]		[[Image:Frstr_a_l5000.jpg]]

	Same as the formula, the critical stress has a linear dependence on ln(L/a)/L		Same as the formula, the critical stress has a linear dependence on ln(L/a)/L

	[[Image:Frstr_lnlal_l5000.jpg ]]		[[Image:Frstr_lnlal_l5000.jpg ]]

	We can see the decrease in relative error.		We can see the decrease in relative error.
	[[Image:Re_a_l5000.jpg ]]		[[Image:Re_a_l5000.jpg ]]
			When the length of initial segment increases, relative error seems to decrease when a approaches 0.

			[[Image:RE_a_l16.jpg]]





	4.3. lmax influence		4.3. lmax influence

Jieyin: /* F-R Test */

2008-02-13T16:37:18Z

F-R Test

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	4.2. a dependence		4.2. a dependence

			When a varies from 1 to 50 and l is fixed at 5000, numerical results are given, shown in following figures

			[[Image:Frstr_a_l5000.jpg]]

			Same as the formula, the critical stress has a linear dependence on ln(L/a)/L

			[[Image:Frstr_lnlal_l5000.jpg ]]

			We can see the decrease in relative error.
			[[Image:Re_a_l5000.jpg ]]

Jieyin: /* F-R Test */

2008-02-13T16:08:27Z

F-R Test

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	[[Image:Frstr_L_a10a50.jpg]]		[[Image:Frstr_L_a10a50.jpg]]

			It shows that analytical result will be more accurate at smaller a/L ratio.

		⚫	4.2. a dependence






⚫	4.2. a dependence

	4.3. lmax influence		4.3. lmax influence

Jieyin: /* F-R Test */

2008-02-13T16:06:16Z

F-R Test

← Older revision		Revision as of 16:06, 13 February 2008
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	[[Image:Frstr_L_a50.jpg]]		[[Image:Frstr_L_a50.jpg]]

			According to the formula, there is a linear relation between the critical stress and segment length L, which was also proven by numerical results.

			[[Image:Frstr_lnLL_a50.jpg]]

			The relative error decreases from 0.26 to 0.20 when the initial segment coordinate parameter l goes from 1000 to 10,000; however, the error can't be negligible. Better convergence should occur at a smaller a value. Numerical tests shows that, when a=1, relative error decreases from 0.20 to 0.18 when l goes from 1000 to 6000. Lower error is expected for longer segments.

			[[Image:Frstr_L_a1.jpg]]

			Longer simulation time can help eliminate the intersections at large l.

			A interesting phenomenon is that numerical result for a/5 well fits analytical result for a. Relative error goes to 0.001 when l goes to 10,000.

			[[Image:Frstr_L_a10a50.jpg]]

Jieyin: /* F-R Test */

2008-02-13T15:34:49Z

F-R Test

← Older revision		Revision as of 15:34, 13 February 2008
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	Based on the maximum distance criteria, we can easily determine the critical applied stress for different initial straight line lengths. Here is a graph showing the relationship between <math>\sigma_c</math> and L at a=50.		Based on the maximum distance criteria, we can easily determine the critical applied stress for different initial straight line lengths. Here is a graph showing the relationship between <math>\sigma_c</math> and L at a=50.

		⚫	[[Image:Frstr_L_a50.jpg]]



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	4.3. lmax influence		4.3. lmax influence

	Based on the maximum distance criteria, we can easily determine the critical applied stress for different initial straight line lengths. Here is a graph showing the relationship between <math>\sigma_c</math> and L.

⚫	[[Image:~~Newfr2_40pt1~~.jpg]]

	Currently there is some deviation from the theoretical value of critical applied stress; however, we can clearly see the linear relationship. Following figure shows the critical applied stress is linearly dependent on lnN/N.		Currently there is some deviation from the theoretical value of critical applied stress; however, we can clearly see the linear relationship. Following figure shows the critical applied stress is linearly dependent on lnN/N.


	[[Image:Newfr2_40pt2.jpg]]

	By varying a, which is the core radius, we can derive the critical stresses and find its linear dependence on ln a, as long as L is fixed.		By varying a, which is the core radius, we can derive the critical stresses and find its linear dependence on ln a, as long as L is fixed.