Dislocation Density Based Hardening

A dislocation density based polycrystal plasticity model hardening law (see Estrin et al. and Toth et al.).

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]\displaystyle{ \rho_{t} = f\rho_{w} + (1-f)\rho_{c} }[/math]

Where ρ_t is the total dislocation density, [math]\displaystyle{ f }[/math] is the volume fraction of the cell walls, ρ_c and ρ_w 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]\displaystyle{ 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]\displaystyle{ f }[/math], is associated with the resolved shear strain rate, [math]\displaystyle{ \gamma^r }[/math], and the saturation value of [math]\displaystyle{ f }[/math] at large strains and initial volume fraction, [math]\displaystyle{ f_{\infty} }[/math] and [math]\displaystyle{ f_o }[/math] respectively, which are constants. [math]\displaystyle{ \tilde\gamma^{r} }[/math], is the rate of variation of [math]\displaystyle{ f }[/math] with resolved shear strain rate, [math]\displaystyle{ \gamma^r }[/math].

      [math]\displaystyle{ 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]\displaystyle{ \tau^{r}_{c} = \alpha G b \sqrt{\rho_{c}}\bigg(\frac{\dot{\gamma^{r}_{c}}}{\dot{\gamma_{0}}}\bigg ) ^\frac{1}{m} }[/math]

      [math]\displaystyle{ \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]\displaystyle{ \dot{\gamma}_0 }[/math] is a reference shear strain rate, [math]\displaystyle{ \dot{\gamma}^r_c }[/math] is the shear strain rate of the cell interior, [math]\displaystyle{ \dot{\gamma^{r}_{w}} }[/math] is the shear strain rate of the cell wall, [math]\displaystyle{ 1\over m }[/math] is the strain rate sensitivity parameter, where m is inversely proportional to the absolute temperature:

      [math]\displaystyle{ 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]\displaystyle{ \tau^{r} = f \tau^{r}_{w} + (1-f) \tau^{r}_{c} }[/math]

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

      [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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*,k_o are constants, and similar to m, n is the strain rate sensitivity parameter, where n is inversely proportional to the absolute temperature:

      [math]\displaystyle{ 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]\displaystyle{ \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 law can set the following properties:

Property	Description	Units	Default
tayM	Taylor Factor, [math]\displaystyle{ M }[/math]	none	3.2
rhoW	Initial dislocation density in cell wall, [math]\displaystyle{ \rho_w }[/math]		1e13
rhoC	Initial dislocation density in cell interior, [math]\displaystyle{ \rho_c }[/math]		1e14
fo	Initial volume fraction, [math]\displaystyle{ f_o }[/math]	none	0.25
flim	Saturation value of f at large strain, [math]\displaystyle{ f_{\infty} }[/math]	none	0.06
fsto	The rate of variation of f, with resolved shear strain rate, [math]\displaystyle{ \tilde{\gamma}^{r} }[/math]	[math]\displaystyle{ s^{-1} }[/math]	3.2
sto	Reference shear strain rate, [math]\displaystyle{ \dot{\gamma}_0 }[/math]	[math]\displaystyle{ s^{-1} }[/math]	1e6
m	Strain rate sensitivity exponent, [math]\displaystyle{ m }[/math]		50
n	Strain rate sensitivity exponent, [math]\displaystyle{ n }[/math]		10
alp	material constant, [math]\displaystyle{ \alpha }[/math]		0.25
burg	Burgers Vector, [math]\displaystyle{ b }[/math]	m	2.56e-10
K1	Proportionality constant between total dislocation density and grain size, [math]\displaystyle{ K }[/math]		10
alpstar	Material constant, [math]\displaystyle{ \alpha * }[/math]		0.120
betastar	Material constant, [math]\displaystyle{ \beta * }[/math]		0.006
Atd	Temperature proportionality constant for m, [math]\displaystyle{ A }[/math]	Kelvin	30000
Btd	Temperature proportionality constant for n, [math]\displaystyle{ 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]\displaystyle{ 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 stress free temperature and not set using hardening law properties.
• Default material parameters are for copper, see Lemiale et al.

History Data

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

      [math]\displaystyle{ \alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p|| }[/math]

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

The current yield stress (σ_y) 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.