$\def\rcontract{{\scriptscriptstyle\circ\bullet}} \def\lcontract{{\scriptscriptstyle\bullet\circ}}$

## Features

Isotropic version of the Peirce, Asaro, & Needleman (1983) phenomenological crystal plasticity constitutive law. The von Mises stress related to the second invariant $J_2 = \frac{1}{2} \tnsr S^* : {\tnsr S^*}^\text T$ of the stress deviator $\tnsr S^* = \operatorname{dev} (\tnsr S) = \tnsr S - \frac{1}{3} \operatorname{tr} (\tnsr S)$ is considered as driving dislocation motion in lieu of slip system-resolved shear stresses.

## Microstructure parameterization

The microstructure is parameterized by a single internal state variable that captures the resistance to deformation:
• flowstress $g$

## Kinetics

### Deformation

In accordance with the Peirce, Asaro, & Needleman (1983) law, the (average) shear rate is formulated as a power-law kinetic equation $$\label{eq: shear rate} \dot\gamma = \dot\gamma_0 \left( \frac{\sqrt{3 J_2}}{M\,g} \right)^n = \dot\gamma_0 \left( \sqrt{\frac{3}{2}}\,\frac{\| \tnsr S^* \|}{M\,g} \right)^n,$$ with $\dot\gamma_0$ a reference shear rate, $n$ the stress exponent (at constant structure), and $M$ an orientation (Taylor) factor.

### Structure

Again following the hardening behavior suggested in Peirce, Asaro, & Needleman (1983) the flow stress $g$ evolves in time due to deformation from its initial value $g_0$ towards a saturation value $g_\infty$ according to \begin{align} \label{eq: hardening} \dot g &= \dot\gamma \left( h_0 + h_\text s \ln{\dot\gamma} \right) \left| 1 - g/g_\infty \right|^a \operatorname{sign} \left( 1 - g/g_\infty \right), \end{align} with free parameters $h_0$ and $a$. The parameter $h_\text s = \mathrm d h_0/\mathrm d \ln \dot\gamma$ introduces a strain rate sensitivity of the hardening slope.

To capture more than the power-law rate dependency of the saturation stress inherent in \eqref{eq: shear rate}, we make use of the empirical relation \begin{align} \dot\gamma &= A\left( \sinh \left( B\, g_\infty^\ast\right)^C \right)^D \nonumber \\ &= A\left( \sinh \left( B\, g_\infty\left(\dot\gamma/\dot\gamma_0\right)^{1/n}\right)^C \right)^D, \end{align} where the factor $\left(\dot\gamma/\dot\gamma_0\right)^{1/n}$ corrects the (experimentally observed) saturation stress $g_\infty^\ast$ for the rate sensitivity introduced by the deformation kinetics \eqref{eq: shear rate}. Parameters $A$, $B$, $C$, and $D$ allow for fitting.

The value of $A$ is used to switch between constant saturation stress and rate-sensitive saturation behavior: $$g_\infty = \begin{cases} \tau_\text{sat} & \text{if } A = 0 \\ \tau_\text{sat} + \left.\left( \operatorname{asinh} \left( \left( \dot\gamma/A \right)^{1/D} \right) \right)^{1/C} \middle/ \left(B \left( \dot\gamma/\dot\gamma_0 \right )^{1/n}\right)\right. & \text{otherwise} \end{cases}$$

## Kinematics

Instead of summing the different slip system contributions ($\dot\gamma^\alpha\, \vctr b^\alpha \otimes \vctr n^\alpha$) to the plastic velocity gradient as is done in Peirce, Asaro, & Needleman (1983), in the present model the "direction" of $\tnsr L_\text p$ is set equivalent to that of $\tnsr S^*$:

\begin{align} \tnsr L_\text p = \frac{\dot\gamma}{M}\, \frac{\tnsr S^*}{\| \tnsr S^*\|} &= \underbrace{\frac{\dot\gamma_0}{M} \left( \sqrt{\frac{3}{2}}\frac{1}{M\, g}\right)^n}_{k} \, \tnsr S^*\, \| \tnsr S^*\|^{n-1} \end{align} where $k$ abbreviates the constant prefactor in preparation of the next section.

### Tangent $\partial\tnsr L_\text p / \partial \tnsr S$

The sensitivity of the plastic velocity gradient with respect to the second Piola–Kirchhoff stress is a required output of all plastic constitutive laws and used in the implicit stress calculation. In the following, we mostly adopt the elegant notation scheme introduced by Olaf Kintzel.

\begin{align*} \tnsr L_\text p,_{\scriptstyle\tnsr S} & = \tnsr L_\text p,_{\scriptstyle\tnsr S^*} \lcontract \,\tnsr S^*,_{\scriptstyle\tnsr S} \\ & = \tnsr L_\text p,_{\scriptstyle\tnsr S^*} \lcontract \underbrace{\left[ \tnsr I \otimes \tnsr I - \frac{1}{3} \tnsr I \odot \tnsr I \right]}_{\tnsrfour Q} \\ & = k \left[\tnsr S^* \|\tnsr S^*\|^{n-1} \right],_{\scriptstyle\tnsr S^*} \lcontract \,\tnsrfour Q \\ & = k \left[ \tnsr S^* \odot \left[{{\| \tnsr S^* \|}^{n-1}} \right],_{\scriptstyle\tnsr S^*} + \| \tnsr S^* \|^{n-1} \, \tnsr S^*,_{\scriptstyle\tnsr S^*} \right] \lcontract \, \tnsrfour Q \\ & = k \left[ \tnsr S^* \odot ( n-1 ) \|\tnsr S^*\|^{n-2} \, \frac{\tnsr S^*}{\| \tnsr S^* \|} + \| \tnsr S^*\|^{n-1} \, \tnsr I \otimes \tnsr I \right] \lcontract \, \tnsrfour Q \\ & = k \left[ \tnsr S^* \odot \tnsr S^* ( n-1 ) \|\tnsr S^*\|^{n-3} + \| \tnsr S^* \|^{n-1} \, \tnsr I \otimes \tnsr I \right] \lcontract \, \tnsrfour Q \\ & = k \left[ \tnsr S^* \odot \tnsr S^* ( n-1 ) \|\tnsr S^*\|^{n-3} \right] \lcontract \tnsrfour Q + k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q \\ & = k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q + k ( n-1 ) \, \|\tnsr S^*\|^{n-3} \, \left[ \tnsr S^* \odot \tnsr S^* \lcontract \, \tnsr I \otimes \tnsr I - \frac{1}{3}\, \tnsr S^* \odot \tnsr S^* \lcontract \, \tnsr I \odot \tnsr I \right] \\ & = k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q + k \left( n-1 \right) \, \|\tnsr S^*\|^{n-3} \, \left[ \tnsr S^* \odot \tnsr S^* - \frac{1}{3} \underbrace{(\tnsr S^* : \tnsr I)}_{=\,\operatorname{tr}(\tnsr S^*)\,=\,\tnsr 0} \tnsr S^* \odot \tnsr I \right] \\ & = k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q + k \left( n-1 \right) \, \|\tnsr S^*\|^{n-3} \, \tnsr S^* \odot \tnsr S^* \\ & = k \, \| \tnsr S^* \|^{n-1} \left[ \tnsrfour Q + ( n-1 ) \, \frac{\tnsr S^*}{\| \tnsr S^* \|} \odot \frac{\tnsr S^*}{\| \tnsr S^* \|} \right] \end{align*} It is useful to rewrite this equation in index notation. \begin{align*} & \vctr g^i \otimes \vctr g_k \otimes \vctr g_l \otimes \vctr g^j \ \frac{\partial \{L_\text p\}_{ij}}{\partial S_{kl}} = \\ & \vctr g^i \otimes \vctr g_k \otimes \vctr g_l \otimes \vctr g^j \ k \, \| \tnsr S^* \|^{n-1} \left[ \delta_i^k\,\delta^l_j - 1/3\; \delta_{ij}\,\delta^{kl} + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}_{ij}\, {S^*}^{kl} \right] \end{align*}

A change of tensor basis is performed to transform $\left [\tnsr L_\text p,_{\scriptstyle\tnsr S}\right]^\text R = \tnsr L_\text p;_{\scriptstyle\tnsr S}$. In index notation, this corresponds to: \begin{align*} \Big[\vctr g^i \otimes \vctr g_k \otimes \vctr g_l \otimes \vctr g^j & \ k \, \| \tnsr S^* \|^{n-1} \left[ \delta_i^k\,\delta^l_j - 1/3\; \delta_{ij}\,\delta^{kl} + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}_{ij}\, {S^*}^{kl} \right]\Big]^\text R = \\ \phantom{\Big[}\vctr g^i \otimes \vctr g^j \otimes \vctr g_k \otimes \vctr g_l & \ k \, \| \tnsr S^* \|^{n-1} \left[ \delta_i^k\,\delta^l_j - 1/3\; \delta_{ij}\,\delta^{kl} + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}_{ij}\, {S^*}^{kl} \right] = \tnsr L_\text p;_{\scriptstyle\tnsr S} \end{align*}

## Parameters in material configuration

To set the above parameters use the following (case-insensitive) naming scheme in a material.config file:

Parameter Name
$\dot\gamma_0$ gdot0
$M$ Taylorfactor
$n$ n
$h_0$ h0
$h_\text s$ h0_slope, slopeLnRate
$g_0$ tau0
$g_\infty$ tausat
$A$ tausat_sinhFitA
$B$ tausat_sinhFitB
$C$ tausat_sinhFitC
$D$ tausat_sinhFitD
$a$ a, w0

## Appendix

In the following, $\tnsr A$ denotes an arbitrary second-order tensor and $\tnsr I$ the second-order identity tensor. Notation of operators is explained in detail here.

### Norm of a tensor

$\|\tnsr A\| := \left( \tnsr A \cdot \tnsr A\right)^{1/2}$

### Deviatoric part of a tensor

$\tnsr A^* := \operatorname{dev}\tnsr A = \tnsr A - \tnsr I \operatorname{tr} (\tnsr A)/3$

### Derivative of the norm of a tensor with respect to that tensor

\begin{align*} \| \tnsr A \|_{,\tnsr A} & = \left[ \left( \tnsr A \cdot \tnsr A \right)^{1/2} \right]_{,\tnsr A} \\ & = \frac{1}{2} \left( \tnsr A \cdot \tnsr A \right)^{-1/2} \left( \tnsr A \cdot \tnsr A \right)_{,\tnsr A} \\ & = \frac{1}{2} \frac{1}{\| \tnsr A \|} \left[ \tnsr A_{,\tnsr A} \rcontract \tnsr A + \tnsr A \lcontract \tnsr A_{,\tnsr A} \right] \\ & = \frac{1}{2} \frac{1}{\| \tnsr A \|} \left[ \tnsr I \otimes \tnsr I \rcontract \tnsr A + \tnsr A \lcontract \tnsr I \otimes \tnsr I \right] \\ & = \frac{\tnsr A}{\|\tnsr A\|} \end{align*}

### Derivative of the deviator of a tensor with respect to that tensor

\begin{align*} \tnsr A^*_{,\tnsr A} & = \left[\tnsr A - \tnsr I \operatorname{tr} (\tnsr A)/3 \right]_{,\tnsr A} \\ & = \tnsr A_{,\tnsr A} - \frac{1}{3} \left[ \tnsr I \odot \operatorname{tr}(\tnsr A)_{, \tnsr A} \right] \\ & = \tnsr I \otimes \tnsr I - \frac{1}{3} \tnsr I \odot \tnsr I \end{align*}

## References

[1]
D. Peirce, R. Asaro, & A. Needleman
Material rate dependence and localized deformation in crystalline solids
Acta Metall. 31 (1983) 19511976
Topic revision: r47 - 07 Mar 2015, PhilipEisenlohr

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