OSUPDOCS - User contributions [en]

Particle-Based Boundary Conditions

2014-03-31T15:39:43Z

Fagant:

__TOC__
== Introduction ==

Particle-based boundary conditions are used to apply conditions directly to particles for loads, tractions, heat fluxes, and concentration fluxes. Particle-based boundary conditions are typically applied to particles on the boundary of the object. Besides particle conditions, simulations can set [[Grid-Based Boundary Conditions]].

== Particle-Based Boundary Condition in Scripted Files ==

All particle-based boundary conditions are created by a series of [[Particle BC Shape Commands|shape commands]] that select particles in the simulation. The [[Particle BC Shape Commands|shape commands]] define shapes (lines or arcs) and all particles withing those shapes are selected.. In scripted 2D simulations, the commands are

* [[Particle BC Shape Commands#Line (2D or Axisymmetric)|LoadLine]] - select particles along a line
* [[Particle BC Shape Commands#Arc (2D or Axisymmetric)|LoadArc]] - select particles along an arc
* [[Particle BC Shape Commands#Rectangle (2D or Axisymmetric)|LoadRect]] - select particles within a rectangle

In scripted 3D simulations, the commands is:

* [[Particle BC Shape Commands#Box or Cylinder (3D)|LoadBox]] - select particles within a box or a cylinder

The commands within these [[Particle BC Shape Commands|shape commands]] are used to set [[Setting Forces and Fluxes|load, traction, heat flux, and/or concentration flux conditions]].

== Particle-Based Boundary Condition in XML Files ==

All particle-based boundary conditions must be set up within a single <ParticleBCs> element. The format is

<ParticleBCs>
(one or more particle BC shape commands)
...
<LoadBCs>
(one or more explicit boundary conditions)
</LoadBCs >
</ParticleBCs>

There are two ways to specify particle boundary conditions. The most common approach is to generate boundary conditions using one or more [[Particle BC Shape Commands|shape commands]] to select particles and assign [[Setting Forces and Fluxes|specified load, traction, heat flux, and or concentration flux conditions]] to those particles. The other way is to [[Explicit Particle Boundary Conditions|explicitly list each particle condition]]. The explicit method is shown in the <tt><LoadBCs></tt> section above; it is limited to load conditions, and is usually generated with other software. You can use both [[Particle BC Shape Commands|shape commands]] and a [[Explicit Particle Boundary Conditions|<tt><LoadBCs></tt> section]] in the same input file.

Dislocation Density Based Hardening

2014-01-22T07:05:38Z

Fagant:

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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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 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>.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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), <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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description !! Units !! Default
|-
| 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
|-
| 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
|-
| 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
|-
| sto || Reference shear strain rate, <math>\dot{\gamma}_0</math> || <math>s^{-1}</math> || 1e6
|-
| m || Strain rate sensitivity exponent, <math>m</math> || none || 50
|-
| n || Strain rate sensitivity exponent, <math>n</math> || none || 10
|-
| alp || material constant, <math>\alpha</math> || || 0.25
|-
| 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> || none || 10
|-
| alpstar || Material constant, <math>\alpha *</math> || || 0.120
|-
| betastar || Material constant, <math>\beta *</math> || || 0.006
|-
| Atd || Temperature proportionality constant for m, <math>A</math> || <math>Kelvin</math> || 30000
|-
| 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
|-
| MMG || Shear Modulus, <math>G</math> || <math>MPa</math> || 48.93e3
|}

* If using temperature dependence, temperature must be absolute set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.
* Default material parameters are for copper, see Lemiale et al<ref name="Lemiale">Lemiale et. al., "Grain refinement under high strain rate impact: A numerical approach," ''Comp. Mater. Sci.'', '''48''', 124-132 (2010)</ref>.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

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

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

== References ==

<references/>

Dislocation Density Based Hardening

2014-01-20T03:45:39Z

Fagant:

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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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 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>.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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), <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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description !! Units !! Default
|-
| tayM || Taylor Factor, <math>M</math> || none || 3.2
|-
| rhoW || Initial dislocation density in cell wall, <math>\rho_w</math> || || 1e13
|-
| rhoC || Initial dislocation density in cell interior, <math>\rho_c</math> || || 1e14
|-
| fo || Initial volume fraction, <math>f_o</math> || none || 0.25
|-
| 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
|-
| sto || Reference shear strain rate, <math>\dot{\gamma}_0</math> || <math>s^{-1}</math> || 1e6
|-
| m || Strain rate sensitivity exponent, <math>m</math> || || 50
|-
| n || Strain rate sensitivity exponent, <math>n</math> || || 10
|-
| alp || material constant, <math>\alpha</math> || || 0.25
|-
| burg || Burgers Vector, <math>b</math> || m || 2.56e-10
|-
| K1 || Proportionality constant between total dislocation density and grain size, <math>K</math> || || 10
|-
| alpstar || Material constant, <math>\alpha *</math> || || 0.120
|-
| betastar || Material constant, <math>\beta *</math> || || 0.006
|-
| Atd || Temperature proportionality constant for m, <math>A</math> || Kelvin || 30000
|-
| 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 || none || 0
|-
| MMG || Shear Modulus, <math>G</math> || MPa || 48.93e3
|}

* If using temperature dependence, temperature must be absolute set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.
* Default material parameters are for copper, see Lemiale et al<ref name="Lemiale">Lemiale et. al., "Grain refinement under high strain rate impact: A numerical approach," ''Comp. Mater. Sci.'', '''48''', 124-132 (2010)</ref>.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

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

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

== References ==

<references/>

Dislocation Density Based Hardening

2014-01-20T03:25:41Z

Fagant:

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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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>.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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), <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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description !! Units !! Default
|-
| tayM || Taylor Factor, <math>M</math> || none || 3.2
|-
| rhoW || Initial dislocation density in cell wall, <math>\rho_w</math> || || 1e13
|-
| rhoC || Initial dislocation density in cell interior, <math>\rho_c</math> || || 1e14
|-
| fo || Initial volume fraction, <math>f_o</math> || none || 0.25
|-
| 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
|-
| sto || Reference strain rate, <math>\dot{\gamma}_0</math> || <math>s^{-1}</math> || 1e6
|-
| m || Strain rate sensitivity exponent, <math>m</math> || || 50
|-
| n || Strain rate sensitivity exponent, <math>n</math> || || 10
|-
| alp || material constant, <math>\alpha</math> || || 0.25
|-
| burg || Burgers Vector, <math>b</math> || || 2.56e-10
|-
| K1 || Proportionality constant between total dislocation density and grain size, <math>K</math> || || 10
|-
| alpstar || Material constant, <math>\alpha *</math> || || 0.120
|-
| betastar || Material constant, <math>\beta *</math> || || 0.006
|-
| Atd || Temperature proportionality constant for m, <math>A</math> || Kelvin || 30000
|-
| 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 || none || 0
|-
| MMG || Shear Modulus, <math>G</math> || MPa || 48.93e3
|}

* If using temperature dependence, temperature must be absolute set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.
* Default material parameters are for copper, see Lemiale et al<ref name="Lemiale">Lemiale et. al., "Grain refinement under high strain rate impact: A numerical approach," ''Comp. Mater. Sci.'', '''48''', 124-132 (2010)</ref>.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

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

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

== References ==

<references/>

Dislocation Density Based Hardening

2014-01-18T13:07:50Z

Fagant:

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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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>.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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), <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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description
|-
| tayM || Taylor Factor
|-
| rhoW || Initial dislocation density in cell wall, ρw
|-
| rhoC || Initial dislocation density in cell interior, ρt
|-
| fo || Initial volume fraction, fo. Unitless.
|-
| flim || Saturation value of f at large strain, flim. Unitless.
|-
| fsto || The rate of variation of f, with resolved shear strain rate.
|-
| sto || Reference strain rate.
|-
| m || Strain rate sensitivity exponent.
|-
| n || Strain rate sensitivity exponent.
|-
| alp || material constant, α.
|-
| burg || Burgers Vector, b.
|-
| K1 || Proportionality constant between total dislocation density and grain size, K.
|-
| alpstar || Material constant, α*.
|-
| betastar || Material constant, β*.
|-
| Atd || Temperature proportionality constant for m. 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. * see note below table
|-
| MMG || Shear Modulus. Enter in units of MPa.
|}

* If using temperature dependence, temperature must be absolute set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.
* Default material parameters are for copper, see Lemiale et al<ref name="Lemiale">Lemiale et. al., "Grain refinement under high strain rate impact: A numerical approach," ''Comp. Mater. Sci.'', '''48''', 124-132 (2010)</ref>.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

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

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

== References ==

<references/>

Dislocation Density Based Hardening

2014-01-18T12:57:22Z

Fagant:

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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, <math>f</math> is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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>.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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), <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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<math>\tau^{r} = f \tau^{r}_{w} + (1-f) \tau^{r}_{c}</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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description
|-
| tayM || Taylor Factor
|-
| rhoW || Initial dislocation density in cell wall, ρw
|-
| rhoC || Initial dislocation density in cell interior, ρt
|-
| fo || Initial volume fraction, fo. Unitless.
|-
| flim || Saturation value of f at large strain, flim. Unitless.
|-
| fsto || The rate of variation of f, with resolved shear strain rate.
|-
| sto || Reference strain rate.
|-
| m || Strain rate sensitivity exponent.
|-
| n || Strain rate sensitivity exponent.
|-
| alp || material constant, α.
|-
| burg || Burgers Vector, b.
|-
| K1 || Proportionality constant between total dislocation density and grain size, K.
|-
| alpstar || Material constant, α*.
|-
| betastar || Material constant, β*.
|-
| Atd || Temperature proportionality constant for m. 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. * see note below table
|-
| MMG || Shear Modulus. Enter in units of MPa.
|}

* If using temperature dependence, temperature must be absolute set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.
* Default material parameters are for copper, see Lemiale et al<ref name="Lemiale">Lemiale et. al., "Grain refinement under high strain rate impact: A numerical approach," ''Comp. Mater. Sci.'', '''48''', 124-132 (2010)</ref>.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

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

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

== References ==

<references/>

Dislocation Density Based Hardening

2014-01-18T03:10:03Z

Fagant:

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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, f is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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, γr, and the saturation value of f at large strains and initial volume fraction, finf and fo respectively, which are constants. \~γ^{r}, is the rate of variation of f with resolved shear strain rate, $ \tilde{\gamma} $.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<math>\tau^{r} = f \tau^{r}_{w} + (1-f) \tau^{r}_{c}</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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description
|-
| tayM || Taylor Factor
|-
| rhoW || Initial dislocation density in cell wall, ρw
|-
| rhoC || Initial dislocation density in cell interior, ρt
|-
| fo || Initial volume fraction, fo. Unitless.
|-
| flim || Saturation value of f at large strain, flim. Unitless.
|-
| fsto || The rate of variation of f, with resolved shear strain rate.
|-
| sto || Reference strain rate.
|-
| m || Strain rate sensitivity exponent.
|-
| n || Strain rate sensitivity exponent.
|-
| alp || material constant, α.
|-
| burg || Burgers Vector, b.
|-
| K1 || Proportionality constant between total dislocation density and grain size, K.
|-
| alpstar || Material constant, α*.
|-
| betastar || Material constant, β*.
|-
| Atd || Temperature proportionality constant for m. 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. *
|-
| MMG || Shear Modulus. Enter in units of MPa.
|}

* If using temperature dependence, temperature must be absolute set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.
* Default material parameters are for copper, see Lemiale et al<ref name="Lemiale">Lemiale et. al., "Grain refinement under high strain rate impact: A numerical approach," ''Comp. Mater. Sci.'', '''48''', 124-132 (2010)</ref>.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

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

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

== References ==

<references/>

Dislocation Density Based Hardening

2014-01-17T06:17:16Z

Fagant:

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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, f is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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, γr, and the saturation value of f at large strains and initial volume fraction, finf and fo respectively, which are constants. \~γ^{r}, is the rate of variation of f with resolved shear strain rate, $ \tilde{\gamma} $.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<math>\tau^{r} = f \tau^{r}_{w} + (1-f) \tau^{r}_{c}</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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description
|-
| tayM || Taylor Factor
|-
| rhoW || Initial dislocation density in cell wall, ρw
|-
| rhoC || Initial dislocation density in cell interior, ρt
|-
| fo || Initial volume fraction, fo. Unitless.
|-
| flim || Saturation value of f at large strain, flim. Unitless.
|-
| fsto || The rate of variation of f, with resolved shear strain rate.
|-
| sto || Reference strain rate.
|-
| m || Strain rate sensitivity exponent.
|-
| n || Strain rate sensitivity exponent.
|-
| alp || material constant, α.
|-
| burg || Burgers Vector, b.
|-
| K1 || Proportionality constant between total dislocation density and grain size, K.
|-
| alpstar || Material constant, α*.
|-
| betastar || Material constant, β*.
|-
| Atd || Temperature proportionality constant for m. Units K^{-1}.
|-
| Btd || Temperature proportionality constant for n. Units K^{-1}.
|-
| 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. *
|-
| MMG || Shear Modulus. Enter in units of MPa.
|}

* If using temperature dependence, temperature must be absolute set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.
* Default material parameters are for copper, see Lemiale et al<ref name="Lemiale">Lemiale et. al., "Grain refinement under high strain rate impact: A numerical approach," ''Comp. Mater. Sci.'', '''48''', 124-132 (2010)</ref>.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

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

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

== References ==

<references/>

Dislocation Density Based Hardening

2014-01-17T05:48:02Z

Fagant:

A dislocation density based polycrystal plasticity model [[Hardening Laws|hardening law]] (see Toth at 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 total dislocation density is made up of these two variables added together via a rule of mixtures:

     
<math>\rho_{t} = f\rho_{w} + (1-f)\rho_{c}</math>

Where ρt is the total dislocation density, f is the volume fraction of the cell walls, ρc and ρw 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:

     
<math>d = \frac{K}{\sqrt{\rho_{t}}}</math>

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, γr, and the saturation value of f at large strains and initial volume fraction, finf and fo respectively, which are constants. \~γ^{r}, is the rate of variation of f with resolved shear strain rate, $ \tilde{\gamma} $.

     
<math>f = f_{\infty} + (f_{o} - f_{\infty}) e^{\frac{-\gamma^{r}}{\tilde{\gamma}^{r}}}</math>

For the kinetic equations, the resolved shear stress is related to the resolved plastic shear strain rate. The two different dislocation densities give rise to two scalar stresses in the cell wall and cell interiors.

     
<math>\tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\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:

     
<math>m = \frac{A}{T}</math>

Where A is a constant. The overall behavior of the composite structure, described with two dislocation densities, is defined by the scalar quantity obtained using the rule of mixtures below.

     
<math>\tau^{r} = f \tau^{r}_{w} + (1-f) \tau^{r}_{c}</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.

     
<math>
\dot{\rho}_{c} = \alpha^{\ast}\frac{1}{\sqrt{3}}\frac{\sqrt{\rho_{w}}}{b}\dot{\gamma_{w}} - \beta^{\ast}\frac{6 \dot{\gamma_{c}}}{b d (1-f) ^\frac{1}{3}} - k_{o} \bigg ( \frac{\dot{\gamma_{c}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{c}} \rho_{c}
</math>

     
<math>
\dot{\rho}_{w} = \frac{6 \beta^{\ast} \dot{\gamma}_{c} (1-f)^{\frac{2}{3}}}{bdf} + \frac{\sqrt{3}\beta^{\ast} \dot{\gamma}_{c} (1-f) \sqrt{\rho_{w}}}{fb} - k_{o} \bigg ( \frac{\dot{\gamma_{w}}}{\dot{\gamma_{0}}} \bigg ) ^{- \frac{1}{n}} \dot{\gamma_{w}} \rho_{w}
</math>

Where β*,α*,ko are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

     
<math>n = \frac{B}{T}</math>

As the interface between the cell interior and cell wall must satisfy strain compatibility, the resolved shear strain rate is equal within each phase of the composite structure.

     
<math>\dot{\gamma}^{r}_{c} = \dot{\gamma}^{r}_{w} = \dot{\gamma}^{r}</math>

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.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description
|-
| tayM || Taylor Factor
|-
| rhoW || Initial dislocation density in cell wall, ρw
|-
| rhoC || Initial dislocation density in cell interior, ρt
|-
| fo || Initial volume fraction, fo. Unitless.
|-
| flim || Saturation value of f at large strain, flim. Unitless.
|-
| fsto ||
|-
| sto || Reference strain rate.
|-
| m || Strain rate sensitivity exponent.
|-
| n || Strain rate sensitivity exponent.
|-
| alp || material constant, α.
|-
| burg || Burgers Vector, b.
|-
| K1 || Proportionality constant between total dislocation density and grain size, K.
|-
| alpstar || Material constant, α*.
|-
| betastar || Material constant, β*.
|-
| Atd || Temperature proportionality constant for m. Units K^{-1}.
|-
| Btd || Temperature proportionality constant for n. Units K^{-1}.
|-
| 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. *
|-
| MMG || Shear Modulus. Enter in units of MPa.
|}

* If using temperature dependence, temperature must be set in Kelvin. The reference temperature, T0, is set using the simulations [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.

== History Data ==

This [[Hardening Laws|hardening law]] defines six history variables, which are the cumulative equivalent plastic strain (absolute) defined as:

     
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where dεp is the incremental plastic strain tensor in one time step.

The current yield stress (σy) 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 fourth history variable is the average grain size (d) in metres.

The fifth and sixth history variables are the dislocation densities in the cell interior and wall respectively.

== References ==

<references/>