Isotropic version of the Peirce, Asaro, & Needleman (1983) phenomenological crystal plasticity constitutive law. The constitutive response is based on the second invariant of either the full, $\tnsr S$, or deviatoric, $\tnsr S^* = \operatorname{dev} (\tnsr S) = \tnsr S - \frac{1}{3} \operatorname{tr} (\tnsr S)$, second Piola––Kirchhoff stress tensor. The stress magnitude (von Mises stress in case of J2) 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 $g$




In accordance with the Peirce, Asaro, & Needleman (1983) law, the (average) shear rate is formulated as a power-law kinetic equation \begin{align} \label{eq: shear rate J2} \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 \\\label{eq: shear rate I2} \dot\gamma &= \dot\gamma_0 \left( \frac{\sqrt{3 I_2}}{M\,g} \right)^n = \dot\gamma_0 \left( \sqrt{\frac{3}{2}}\,\frac{\| \tnsr S \|}{M\,g} \right)^n \end{align} with $\dot\gamma_0$ a reference shear rate, $n$ the stress exponent (at constant structure), and $M$ an orientation (Taylor) factor.


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 J2} and \eqref{eq: shear rate I2}, 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 J2} and \eqref{eq: shear rate I2}. 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: \begin{equation} 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} \end{equation}



Plastic velocity gradient

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

\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} \\\tnsr L_\text p = \frac{\dot\gamma}{M}\, \frac{\tnsr S}{\| \tnsr S\|} &= \overbrace{\frac{\dot\gamma_0}{M} \left( \sqrt{\frac{3}{2}}\frac{1}{M\, g}\right)^n}{} \, \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,_{\scriptscriptstyle\tnsr S} & = \tnsr L_\text p,_{\scriptscriptstyle\tnsr S^*} \lcontract \,\tnsr S^*,_{\scriptscriptstyle\tnsr S} \\ & = \tnsr L_\text p,_{\scriptscriptstyle\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],_{\scriptscriptstyle\tnsr S^*} \lcontract \,\tnsrfour Q \\ & = k \left[ \tnsr S^* \odot \left[{{\| \tnsr S^* \|}^{n-1}} \right],_{\scriptscriptstyle\tnsr S^*} + \| \tnsr S^* \|^{n-1} \, \tnsr S^*,_{\scriptscriptstyle\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}} = \nonumber \\ & \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] = \nonumber \\ & \vctr g^i \otimes \vctr g^j \otimes \vctr g_k \otimes \vctr g_l \ k \, \| \tnsr S^* \|^{n-1} \left[ \delta_{ij}\,\delta^{kl} - 1/3\; \delta_i^l\,\delta_j^k + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}^{\phantom i l}_{i\cdot}\, {S^*}^{\phantom j k}_{j\cdot} \right] = \nonumber \\ & \vctr g^i \otimes \vctr g^j \otimes \vctr g_k \otimes \vctr g_l \ \frac{\dot\gamma}{M\,\| \tnsr S^* \|} \left[ \delta_{ij}\,\delta^{kl} - 1/3\; \delta_i^l\,\delta_j^k + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}^{\phantom i l}_{i\cdot}\, {S^*}^{\phantom j k}_{j\cdot} \right] \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


Logicals in material configuration

Logical Purpose
dilatation | I2 use I2 invariant instead of J2



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*}



D. Peirce, R. Asaro, & A. Needleman
Material rate dependence and localized deformation in crystalline solids
Acta Metall. 31 (1983) 1951–1976
Topic revision: r5 - 01 Feb 2017, PhilipEisenlohr

  • News
14 Sep 2020
CMCn2020 & DAMASK user meeting to be hosted at Max-Planck-Institut für Eisenforschung (cancelled)
22 Aug 2020
Release of first preview version of DAMASK 3
19 Feb 2020
DAMASK made it to the Advanved Engineering Materials Hall of Fame
26 Mar 2019
DREAM.3D 6.5.119
(released 2019/03/22) comes with a DAMASK export filter
25 Mar 2019
Release of version v2.0.3
21 Jan 2019
DAMASK overview paper finally published with full citation information available
01 Dec 2018
DAMASK overview paper now online
17 Sep 2018
CMCn2018 & DAMASK user meeting to be hosted at Max-Planck-Institut für Eisenforschung
22 May 2018
Release of version v2.0.2
01 Sep 2016
CMCn2016 & DAMASK user meeting to be hosted at Max-Planck-Institut für Eisenforschung
25 Jul 2016
Release of version v2.0.1
08 Mar 2016
Release of version v2.0.0
22 Feb 2016
New webserver up and running
09 Feb 2016
Migrated code repository from Subversion to GitLab
17 Dec 2014
Release of revision 3813
14 May 2014
Release of revision 3108
02 Apr 2014
Release of revision 3062
16 Oct 2013
Release of revision 2689
15 Jul 2013
Release of revision 2555
15 Feb 2013
Release of revision 2174
13 Feb 2013
Doxygen documentation
16 Dec 2012
Powered by MathJax rendering
23 Nov 2012
Release of revision 1955
15 Nov 2012
Release of revision 1924
01 Nov 2012
Updated sidebar
30 Oct 2012
Significant website updates and content extensions

This site is powered by FoswikiCopyright by the contributing authors. All material on this collaboration platform is the property of the contributing authors.
Ideas, requests, problems regarding DAMASK? Send feedback
§ Imprint § Data Protection