Dislocation Density Based Hardening - Revision history

Nairnj: /* Hardening Law Properties */

2015-06-03T00:20:51Z

Hardening Law Properties

Nairnj at 22:57, 2 June 2015

2015-06-02T22:57:35Z

← Older revision		Revision as of 22:57, 2 June 2015
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	where dε<sub>p</sub> is the incremental plastic strain tensor in one time step.		where dε<sub>p</sub> is the incremental plastic strain tensor in one time step.

	The current yield stress (σ<sub>y</sub>) in [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]], and the current plastic strain rate (dα/dt in 1/~~sec~~). These variables are stored as history variables #1, #2, and #3.		The current yield stress (σ<sub>y</sub>) in [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]], and the current plastic strain rate (dα/dt in [[ConsistentUnits Command#Legacy and Consistent Units\|1/time units]] ). These variables are stored as history variables #1, #2, and #3.

	The #4 history variable is the average grain size (d) in metres.		The #4 history variable is the average grain size (d) in metres.

Nairnj at 22:04, 2 June 2015

2015-06-02T22:04:21Z

← Older revision		Revision as of 22:04, 2 June 2015
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	\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table \|\| none \|\| 0		\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table \|\| none \|\| 0
	\|-		\|-
	\| MMG \|\| Shear Modulus, <math>G</math> \|\| ~~MPa~~ \|\| 48.93e3		\| MMG \|\| Shear Modulus, <math>G</math> \|\| [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]] \|\| 48.93e3
	\|}		\|}

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	where dε<sub>p</sub> is the incremental plastic strain tensor in one time step.		where dε<sub>p</sub> is the incremental plastic strain tensor in one time step.

	The current yield stress (σ<sub>y</sub>) in ~~MPa~~, and the current plastic strain rate (dα/dt in 1/sec). These variables are stored as history variables #1, #2, and #3.		The current yield stress (σ<sub>y</sub>) in [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]], and the current plastic strain rate (dα/dt in 1/sec). These variables are stored as history variables #1, #2, and #3.

	The #4 history variable is the average grain size (d) in metres.		The #4 history variable is the average grain size (d) in metres.

Nairnj: /* Hardening Law Properties */

2015-04-14T04:16:14Z

Hardening Law Properties

← Older revision		Revision as of 04:16, 14 April 2015
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	\| n \|\| Strain rate sensitivity exponent, <math>n</math> \|\| none \|\| 10		\| n \|\| Strain rate sensitivity exponent, <math>n</math> \|\| none \|\| 10
	\|-		\|-
	\| alp \|\| material constant, <math>\alpha</math> \|\| \|\| 0.25		\| alp \|\| material constant, <math>\alpha</math> \|\| none \|\| 0.25
	\|-		\|-
	\| burg \|\| Burgers Vector, <math>b</math> \|\| <math>m</math> \|\| 2.56e-10		\| burg \|\| Burgers Vector, <math>b</math> \|\| <math>m</math> \|\| 2.56e-10
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	\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| none \|\| 10		\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| none \|\| 10
	\|-		\|-
	\| alpstar \|\| Material constant, <math>\alpha *</math> \|\| \|\| 0.120		\| alpstar \|\| Material constant, <math>\alpha *</math> \|\| none \|\| 0.120
	\|-		\|-
	\| betastar \|\| Material constant, <math>\beta *</math> \|\| \|\| 0.006		\| betastar \|\| Material constant, <math>\beta *</math> \|\| none \|\| 0.006
	\|-		\|-
	\| Atd \|\| Temperature proportionality constant for m, <math>A</math> \|\| K \|\| 30000		\| Atd \|\| Temperature proportionality constant for m, <math>A</math> \|\| K \|\| 30000

Nairnj at 18:20, 22 January 2014

2014-01-22T18:20:58Z

Fagant at 07:05, 22 January 2014

2014-01-22T07:05:38Z

Fagant at 03:45, 20 January 2014

2014-01-20T03:45:39Z

← Older revision		Revision as of 03:45, 20 January 2014
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	Where d is the average cell size and K is a proportionality constant.		Where d is the average cell size and K is a proportionality constant.
	The relation for volume fraction of the dislocation density in the cell walls, <math>f</math>, is associated with the shear strain rate, <math>\~~dot\gamma_r~~</math>, and the saturation value of <math>f</math> at large strains and initial volume fraction, <math>f_{\infty}</math> and <math>f_o</math> respectively, which are constants. <math>\tilde\gamma^{r}</math>, is the rate of variation of <math>f</math> with resolved shear strain rate, <math>~~\tilde{~~\gamma}</math>.		The relation for volume fraction of the dislocation density in the cell walls, <math>f</math>, is associated with the resolved shear strain rate, <math>\gamma^r</math>, and the saturation value of <math>f</math> at large strains and initial volume fraction, <math>f_{\infty}</math> and <math>f_o</math> respectively, which are constants. <math>\tilde\gamma^{r}</math>, is the rate of variation of <math>f</math> with resolved shear strain rate, <math>\gamma^r</math>.


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	\| fsto \|\| The rate of variation of f, with resolved shear strain rate, <math>\tilde{\gamma}^{r}</math> \|\| <math>s^{-1}</math> \|\| 3.2		\| fsto \|\| The rate of variation of f, with resolved shear strain rate, <math>\tilde{\gamma}^{r}</math> \|\| <math>s^{-1}</math> \|\| 3.2
	\|-		\|-
	\| sto \|\| Reference strain rate, <math>\dot{\gamma}_0</math> \|\| <math>s^{-1}</math> \|\| 1e6		\| sto \|\| Reference shear strain rate, <math>\dot{\gamma}_0</math> \|\| <math>s^{-1}</math> \|\| 1e6
	\|-		\|-
	\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| \|\| 50		\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| \|\| 50
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	\| alp \|\| material constant, <math>\alpha</math> \|\| \|\| 0.25		\| alp \|\| material constant, <math>\alpha</math> \|\| \|\| 0.25
	\|-		\|-
	\| burg \|\| Burgers Vector, <math>b</math> \|\| \|\| 2.56e-10		\| burg \|\| Burgers Vector, <math>b</math> \|\| m \|\| 2.56e-10
	\|-		\|-
	\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| \|\| 10		\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| \|\| 10

Fagant at 03:25, 20 January 2014

2014-01-20T03:25:41Z

← Older revision		Revision as of 03:25, 20 January 2014
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	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
	! Property !! Description		! Property !! Description !! Units !! Default
	\|-		\|-
	\| tayM \|\| Taylor Factor		\| tayM \|\| Taylor Factor, <math>M</math> \|\| none \|\| 3.2
	\|-		\|-
	\| rhoW \|\| Initial dislocation density in cell wall, ~~ρ~~<~~sub~~>w</~~sub~~>		\| rhoW \|\| Initial dislocation density in cell wall, <math>\rho_w</math> \|\| \|\| 1e13
	\|-		\|-
	\| rhoC \|\| Initial dislocation density in cell interior, ~~ρ~~<~~sub~~>t</~~sub~~>		\| rhoC \|\| Initial dislocation density in cell interior, <math>\rho_c</math> \|\| \|\| 1e14
	\|-		\|-
	\| fo \|\| Initial volume fraction, f<~~sub~~>o</~~sub~~>. ~~Unitless.~~		\| fo \|\| Initial volume fraction, <math>f_o</math> \|\| none \|\| 0.25
	\|-		\|-
	\| flim \|\| Saturation value of f at large strain, f<~~sub~~>~~lim~~</~~sub~~>. ~~Unitless.~~		\| flim \|\| Saturation value of f at large strain, <math>f_{\infty}</math> \|\| none \|\| 0.06
	\|-		\|-
	\| fsto \|\| The rate of variation of f, with resolved shear strain rate.		\| fsto \|\| The rate of variation of f, with resolved shear strain rate, <math>\tilde{\gamma}^{r}</math> \|\| <math>s^{-1}</math> \|\| 3.2
	\|-		\|-
	\| sto \|\| Reference strain rate.		\| sto \|\| Reference strain rate, <math>\dot{\gamma}_0</math> \|\| <math>s^{-1}</math> \|\| 1e6
	\|-		\|-
	\| m \|\| Strain rate sensitivity exponent.		\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| \|\| 50
	\|-		\|-
	\| n \|\| Strain rate sensitivity exponent.		\| n \|\| Strain rate sensitivity exponent, <math>n</math> \|\| \|\| 10
	\|-		\|-
	\| alp \|\| material constant, α.		\| alp \|\| material constant, <math>\alpha</math> \|\| \|\| 0.25
	\|-		\|-
	\| burg \|\| Burgers Vector, b.		\| burg \|\| Burgers Vector, <math>b</math> \|\| \|\| 2.56e-10
	\|-		\|-
	\| K1 \|\| Proportionality constant between total dislocation density and grain size, K.		\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| \|\| 10
	\|-		\|-
	\| alpstar \|\| Material constant, α*.		\| alpstar \|\| Material constant, <math>\alpha *</math> \|\| \|\| 0.120
	\|-		\|-
	\| betastar \|\| Material constant, β*.		\| betastar \|\| Material constant, <math>\beta *</math> \|\| \|\| 0.006
	\|-		\|-
	\| Atd \|\| Temperature proportionality constant for m~~. Units K.~~		\| Atd \|\| Temperature proportionality constant for m, <math>A</math> \|\| Kelvin \|\| 30000
	\|-		\|-
	\| Btd \|\| Temperature proportionality constant for n~~. Units K.~~		\| Btd \|\| Temperature proportionality constant for n, <math>B</math> \|\| Kelvin \|\| 14900
	\|-		\|-
	\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table		\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table \|\| none \|\| 0
	\|-		\|-
	\| MMG \|\| Shear Modulus~~. Enter in units of~~ MPa.		\| MMG \|\| Shear Modulus, <math>G</math> \|\| MPa \|\| 48.93e3
	\|}		\|}

Fagant at 13:07, 18 January 2014

2014-01-18T13:07:50Z

← Older revision		Revision as of 13:07, 18 January 2014
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	<math>\tau^{r} = f \tau^{r}_{w} + (1-f) \tau^{r}_{c}</math>		<math>\tau^{r} = f \tau^{r}_{w} + (1-f) \tau^{r}_{c}</math>

			Where <math>\tau^{r}</math> is the resolved shear stress in the material. The yield stress, <math>\sigma_y</math> is proportional to this term via the Taylor Factor, M.


			<math>\sigma_y = M \tau^{r}</math>

	The evolution equations for the dislocation density in the cell interior and cell wall are given below. Their evolution rate is a function of addition, subtraction and annihilation of dislocations. The growth of both can be attributed to Frank-Read sources at the cell wall or interior interface. The loss of cell interior dislocations can also be attributed to the movement from cell interior into the cell wall. The annihilation of cell dislocations is governed by cross-slip, whereby positive and negative dislocations, or “out of phase” dislocations cancel each other out, this is the last term in each equation.		The evolution equations for the dislocation density in the cell interior and cell wall are given below. Their evolution rate is a function of addition, subtraction and annihilation of dislocations. The growth of both can be attributed to Frank-Read sources at the cell wall or interior interface. The loss of cell interior dislocations can also be attributed to the movement from cell interior into the cell wall. The annihilation of cell dislocations is governed by cross-slip, whereby positive and negative dislocations, or “out of phase” dislocations cancel each other out, this is the last term in each equation.

Fagant at 12:57, 18 January 2014

2014-01-18T12:57:22Z

← Older revision		Revision as of 00:20, 3 June 2015
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	\| tayM \|\| Taylor Factor, <math>M</math> \|\| none \|\| 3.2		\| tayM \|\| Taylor Factor, <math>M</math> \|\| none \|\| 3.2
	\|-		\|-
	\| rhoW \|\| Initial dislocation density in cell wall, <math>\rho_w</math> \|\| m<sup>-2</sup> \|\| 1e13		\| rhoW \|\| Initial dislocation density in cell wall, <math>\rho_w</math> \|\| [[ConsistentUnits Command#Legacy and Consistent Units\|1/length<sup>2</sup> units]] \|\| 1e13
	\|-		\|-
	\| rhoC \|\| Initial dislocation density in cell interior, <math>\rho_c</math> \|\| m<sup>-2</sup> \|\| 1e14		\| rhoC \|\| Initial dislocation density in cell interior, <math>\rho_c</math> \|\| [[ConsistentUnits Command#Legacy and Consistent Units\|1/length<sup>2</sup> units]] \|\| 1e14
	\|-		\|-
	\| fo \|\| Initial volume fraction, <math>f_o</math> \|\| none \|\| 0.25		\| fo \|\| Initial volume fraction, <math>f_o</math> \|\| none \|\| 0.25
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	\| flim \|\| Saturation value of f at large strain, <math>f_{\infty}</math> \|\| none \|\| 0.06		\| flim \|\| Saturation value of f at large strain, <math>f_{\infty}</math> \|\| none \|\| 0.06
	\|-		\|-
	\| fsto \|\| The rate of variation of f, with resolved shear strain rate, <math>\tilde{\gamma}^{r}</math> \|\| ~~s<sup>-~~1</~~sup>~~ \|\| 3.2		\| fsto \|\| The rate of variation of f, with resolved shear strain rate, <math>\tilde{\gamma}^{r}</math> \|\| [[ConsistentUnits Command#Legacy and Consistent Units\|1/time units]] \|\| 3.2
	\|-		\|-
	\| sto \|\| Reference shear strain rate, <math>\dot{\gamma}_0</math> \|\| ~~s<sup>-~~1</~~sup>~~ \|\| 1e6		\| sto \|\| Reference shear strain rate, <math>\dot{\gamma}_0</math> \|\| [[ConsistentUnits Command#Legacy and Consistent Units\|1/time units]] \|\| 1e6
	\|-		\|-
	\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| none \|\| 50		\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| none \|\| 50
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	\| alp \|\| material constant, <math>\alpha</math> \|\| none \|\| 0.25		\| alp \|\| material constant, <math>\alpha</math> \|\| none \|\| 0.25
	\|-		\|-
	\| burg \|\| Burgers Vector, <math>b</math> \|\| ~~<math>m</math>~~ \|\| 2.56e-10		\| burg \|\| Burgers Vector, <math>b</math> \|\| [[ConsistentUnits Command#Legacy and Consistent Units\|length units]] \|\| 2.56e-10
	\|-		\|-
	\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| none \|\| 10		\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| none \|\| 10

← Older revision		Revision as of 18:20, 22 January 2014
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	A dislocation density based polycrystal plasticity model [[Hardening Laws\|hardening law]] (see Estrin et al.<ref name="Estrin">Estrin et al., "A dislocation-based model for all hardening stages in large strain deformation," ''Acta mater.'', '''46''', 5509-5522 (1998)</ref> and Toth et al.<ref name="Toth">Toth et al., "Strain hardening at large strains as predicted by dislocation based polycrystal plasticity model," ''J. Eng. Mat. and Techn.'', '''124''', 71-77 (2002)</ref>).		A dislocation density based polycrystal plasticity model [[Hardening Laws\|hardening law]] (see Estrin et al.<ref name="Estrin">Estrin et al., "A dislocation-based model for all hardening stages in large strain deformation," ''Acta mater.'', '''46''', 5509-5522 (1998)</ref> and Toth et al.<ref name="Toth">Toth et al., "Strain hardening at large strains as predicted by dislocation based polycrystal plasticity model," ''J. Eng. Mat. and Techn.'', '''124''', 71-77 (2002)</ref>).


	The model allows for the tracking of dislocation density and provides a response variable that describes the grain size (or dislocation cell size) based on the variation in dislocation density. It considers the cell or grain to be made of two phases; a cell wall and cell interior, each with its own dislocation density. These two distinct dislocation densities are the internal variables of the model.		The model allows for the tracking of dislocation density and provides a response variable that describes the grain size (or dislocation cell size) based on the variation in dislocation density. It considers the cell or grain to be made of two phases; a cell wall and cell interior, each with its own dislocation density. These two distinct dislocation densities are the internal variables of the model.
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	<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>		<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>


	Where ρ<sub>t</sub> is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρ<sub>c</sub> and ρ<sub>w</sub> are the dislocation densities in the cell interior and cell walls respectively.		Where ρ<sub>t</sub> is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρ<sub>c</sub> and ρ<sub>w</sub> are the dislocation densities in the cell interior and cell walls respectively.
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	The strength of this physically-based microstructure model is that it can be incorporated into a numerical model that operates at the continuum level, and through the material constitutive behavior provide information at a microscopic level (microstructure).		The strength of this physically-based microstructure model is that it can be incorporated into a numerical model that operates at the continuum level, and through the material constitutive behavior provide information at a microscopic level (microstructure).
	It was formulated to be applied to processes that incur severe plastic deformation, and makes assumptions based on this. The dislocation cell size is assumed to be the same size as the subgrains. It also assumes that these dislocation cells or subgrains will ultimately replace the larger grains. This assumption is adequate in processes where there is large grain refinement.		It was formulated to be applied to processes that incur severe plastic deformation, and makes assumptions based on this. The dislocation cell size is assumed to be the same size as the subgrains. It also assumes that these dislocation cells or subgrains will ultimately replace the larger grains. This assumption is adequate in processes where there is large grain refinement.


	== Hardening Law Properties ==		== Hardening Law Properties ==
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	\| tayM \|\| Taylor Factor, <math>M</math> \|\| none \|\| 3.2		\| tayM \|\| Taylor Factor, <math>M</math> \|\| none \|\| 3.2
	\|-		\|-
	\| rhoW \|\| Initial dislocation density in cell wall, <math>\rho_w</math> \|\| <~~math~~>~~m^{~~-2}</~~math~~> \|\| 1e13		\| rhoW \|\| Initial dislocation density in cell wall, <math>\rho_w</math> \|\| m<sup>-2</sup> \|\| 1e13
	\|-		\|-
	\| rhoC \|\| Initial dislocation density in cell interior, <math>\rho_c</math> \|\| <~~math~~>~~m^{~~-2}</~~math~~> \|\| 1e14		\| rhoC \|\| Initial dislocation density in cell interior, <math>\rho_c</math> \|\| m<sup>-2</sup> \|\| 1e14
	\|-		\|-
	\| fo \|\| Initial volume fraction, <math>f_o</math> \|\| none \|\| 0.25		\| fo \|\| Initial volume fraction, <math>f_o</math> \|\| none \|\| 0.25
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	\| flim \|\| Saturation value of f at large strain, <math>f_{\infty}</math> \|\| none \|\| 0.06		\| flim \|\| Saturation value of f at large strain, <math>f_{\infty}</math> \|\| none \|\| 0.06
	\|-		\|-
	\| fsto \|\| The rate of variation of f, with resolved shear strain rate, <math>\tilde{\gamma}^{r}</math> \|\| <~~math~~>~~s^{~~-1}</~~math~~> \|\| 3.2		\| fsto \|\| The rate of variation of f, with resolved shear strain rate, <math>\tilde{\gamma}^{r}</math> \|\| s<sup>-1</sup> \|\| 3.2
	\|-		\|-
	\| sto \|\| Reference shear strain rate, <math>\dot{\gamma}_0</math> \|\| <~~math~~>~~s^{~~-1}</~~math~~> \|\| 1e6		\| sto \|\| Reference shear strain rate, <math>\dot{\gamma}_0</math> \|\| s<sup>-1</sup> \|\| 1e6
	\|-		\|-
	\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| none \|\| 50		\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| none \|\| 50
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	\| betastar \|\| Material constant, <math>\beta *</math> \|\| \|\| 0.006		\| betastar \|\| Material constant, <math>\beta *</math> \|\| \|\| 0.006
	\|-		\|-
	\| Atd \|\| Temperature proportionality constant for m, <math>A</math> \|\| ~~<math>Kelvin</math>~~ \|\| 30000		\| Atd \|\| Temperature proportionality constant for m, <math>A</math> \|\| K \|\| 30000
	\|-		\|-
	\| Btd \|\| Temperature proportionality constant for n, <math>B</math> \|\| ~~<math>Kelvin</math>~~\|\| 14900		\| Btd \|\| Temperature proportionality constant for n, <math>B</math> \|\| K \|\| 14900
	\|-		\|-
	\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table \|\| none \|\| 0		\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table \|\| none \|\| 0
	\|-		\|-
	\| MMG \|\| Shear Modulus, <math>G</math> \|\| ~~<math>~~MPa~~</math>~~ \|\| 48.93e3		\| MMG \|\| Shear Modulus, <math>G</math> \|\| MPa \|\| 48.93e3
	\|}		\|}

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	The #5 and #6 history variables are the dislocation densities in the cell interior and wall respectively.		The #5 and #6 history variables are the dislocation densities in the cell interior and wall respectively.


	== References ==		== References ==

	<references/>		<references/>

← Older revision		Revision as of 07:05, 22 January 2014
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	\| tayM \|\| Taylor Factor, <math>M</math> \|\| none \|\| 3.2		\| tayM \|\| Taylor Factor, <math>M</math> \|\| none \|\| 3.2
	\|-		\|-
	\| rhoW \|\| Initial dislocation density in cell wall, <math>\rho_w</math> \|\| \|\| 1e13		\| rhoW \|\| Initial dislocation density in cell wall, <math>\rho_w</math> \|\| <math>m^{-2}</math> \|\| 1e13
	\|-		\|-
	\| rhoC \|\| Initial dislocation density in cell interior, <math>\rho_c</math> \|\| \|\| 1e14		\| rhoC \|\| Initial dislocation density in cell interior, <math>\rho_c</math> \|\| <math>m^{-2}</math> \|\| 1e14
	\|-		\|-
	\| fo \|\| Initial volume fraction, <math>f_o</math> \|\| none \|\| 0.25		\| fo \|\| Initial volume fraction, <math>f_o</math> \|\| none \|\| 0.25
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	\| sto \|\| Reference shear strain rate, <math>\dot{\gamma}_0</math> \|\| <math>s^{-1}</math> \|\| 1e6		\| sto \|\| Reference shear strain rate, <math>\dot{\gamma}_0</math> \|\| <math>s^{-1}</math> \|\| 1e6
	\|-		\|-
	\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| \|\| 50		\| m \|\| Strain rate sensitivity exponent, <math>m</math> \|\| none \|\| 50
	\|-		\|-
	\| n \|\| Strain rate sensitivity exponent, <math>n</math> \|\| \|\| 10		\| n \|\| Strain rate sensitivity exponent, <math>n</math> \|\| none \|\| 10
	\|-		\|-
	\| alp \|\| material constant, <math>\alpha</math> \|\| \|\| 0.25		\| alp \|\| material constant, <math>\alpha</math> \|\| \|\| 0.25
	\|-		\|-
	\| burg \|\| Burgers Vector, <math>b</math> \|\| m \|\| 2.56e-10		\| burg \|\| Burgers Vector, <math>b</math> \|\| <math>m</math> \|\| 2.56e-10
	\|-		\|-
	\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| \|\| 10		\| K1 \|\| Proportionality constant between total dislocation density and grain size, <math>K</math> \|\| none \|\| 10
	\|-		\|-
	\| alpstar \|\| Material constant, <math>\alpha *</math> \|\| \|\| 0.120		\| alpstar \|\| Material constant, <math>\alpha *</math> \|\| \|\| 0.120
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	\| betastar \|\| Material constant, <math>\beta *</math> \|\| \|\| 0.006		\| betastar \|\| Material constant, <math>\beta *</math> \|\| \|\| 0.006
	\|-		\|-
	\| Atd \|\| Temperature proportionality constant for m, <math>A</math> \|\| Kelvin \|\| 30000		\| Atd \|\| Temperature proportionality constant for m, <math>A</math> \|\| <math>Kelvin</math> \|\| 30000
	\|-		\|-
	\| Btd \|\| Temperature proportionality constant for n, <math>B</math> \|\| Kelvin \|\| 14900		\| Btd \|\| Temperature proportionality constant for n, <math>B</math> \|\| <math>Kelvin</math>\|\| 14900
	\|-		\|-
	\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table \|\| none \|\| 0		\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table \|\| none \|\| 0
	\|-		\|-
	\| MMG \|\| Shear Modulus, <math>G</math> \|\| MPa \|\| 48.93e3		\| MMG \|\| Shear Modulus, <math>G</math> \|\| <math>MPa</math> \|\| 48.93e3
	\|}		\|}

← Older revision		Revision as of 12:57, 18 January 2014
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	A dislocation density based polycrystal plasticity model [[Hardening Laws\|hardening law]] (see Estrin et al<ref name="Estrin">Estrin et. al., "A dislocation-based model for all hardening stages in large strain deformation," ''Acta mater.'', '''46''', 5509-5522 (1998)</ref> and Toth et al<ref name="Toth">Toth et. al., "Strain hardening at large strains as predicted by dislocation based polycrystal plasticity model," ''J. Eng. Mat. and Techn.'', '''124''', 71-77 (2002)</ref>).		A dislocation density based polycrystal plasticity model [[Hardening Laws\|hardening law]] (see Estrin et al.<ref name="Estrin">Estrin et al., "A dislocation-based model for all hardening stages in large strain deformation," ''Acta mater.'', '''46''', 5509-5522 (1998)</ref> and Toth et al.<ref name="Toth">Toth et al., "Strain hardening at large strains as predicted by dislocation based polycrystal plasticity model," ''J. Eng. Mat. and Techn.'', '''124''', 71-77 (2002)</ref>).


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	Where ρ<sub>t</sub> is the total dislocation density, f is the volume fraction of the cell walls, ρ<sub>c</sub> and ρ<sub>w</sub> are the dislocation densities in the cell interior and cell walls respectively.		Where ρ<sub>t</sub> is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρ<sub>c</sub> and ρ<sub>w</sub> are the dislocation densities in the cell interior and cell walls respectively.

	The grain or cell size is determined as proportional to the inverse of the square root of the total dislocation density:		The grain or cell size is determined as proportional to the inverse of the square root of the total dislocation density:
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	Where d is the average cell size and K is a proportionality constant.		Where d is the average cell size and K is a proportionality constant.
	The relation for volume fraction of the dislocation density in the cell walls, f, is associated with the shear strain rate, ~~γ~~<~~sub~~>r</~~sub~~>, and the saturation value of f at large strains and initial volume fraction, f<~~sub~~>~~inf~~</~~sub~~> and f<~~sub~~>o</~~sub~~> respectively, which are constants. \~γ^{r}, is the rate of variation of f with resolved shear strain rate, $ \tilde{\gamma} $.		The relation for volume fraction of the dislocation density in the cell walls, <math>f</math>, is associated with the shear strain rate, <math>\dot\gamma_r</math>, and the saturation value of <math>f</math> at large strains and initial volume fraction, <math>f_{\infty}</math> and <math>f_o</math> respectively, which are constants. <math>\tilde\gamma^{r}</math>, is the rate of variation of <math>f</math> with resolved shear strain rate, <math>\tilde{\gamma}</math>.


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	<math>\tau^{r}_{w} = \alpha G b \sqrt{\rho_{w}}\bigg(\frac{\dot{\gamma^{r}_{w}}}{\dot{\gamma_{0}}}\bigg ) ^\frac{1}{m}</math>		<math>\tau^{r}_{w} = \alpha G b \sqrt{\rho_{w}}\bigg(\frac{\dot{\gamma^{r}_{w}}}{\dot{\gamma_{0}}}\bigg ) ^\frac{1}{m}</math>

	Where α is a constant, G is the shear modulus, b is the Burgers vector (a constant dependent on the crystal structure of the metal, i.e. fcc, hpc, bcc), $ \dot{\~~gamma_{0~~}~~} $~~ is a reference shear strain rate, \dot{γ}^r_c is the shear strain rate of the cell interior, $ \dot{\gamma^{r}_{w}} $ is the shear strain rate of the cell wall, 1/m is the strain rate sensitivity parameter, where m is inversely proportional to the absolute temperature:		Where α is a constant, G is the shear modulus, b is the Burgers vector (a constant dependent on the crystal structure of the metal, i.e. fcc, hpc, bcc), <math>\dot{\gamma}_0</math> is a reference shear strain rate, <math>\dot{\gamma}^r_c</math> is the shear strain rate of the cell interior, <math>\dot{\gamma^{r}_{w}}</math> is the shear strain rate of the cell wall, <math>1\over m</math> is the strain rate sensitivity parameter, where m is inversely proportional to the absolute temperature:


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	\| Btd \|\| Temperature proportionality constant for n. Units K.		\| Btd \|\| Temperature proportionality constant for n. Units K.
	\|-		\|-
	\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. *		\| tempDepend \|\| Turn temperature dependence of m and n on or off. 0 is off, 1 is on. If 1, must have conduction on for effect. * see note below table
	\|-		\|-
	\| MMG \|\| Shear Modulus. Enter in units of MPa.		\| MMG \|\| Shear Modulus. Enter in units of MPa.