Features

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.


parent_gray

Microstructure parameterization

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


parent_gray

Kinetics

Deformation

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.

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 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}


parent_gray

Kinematics

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}


parent_gray

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


parent_gray

Logicals in material configuration

Logical Purpose
dilatation | I2 use I2 invariant instead of J2


parent_gray

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*}


parent_gray

References

[1]
D. Peirce, R. Asaro, & A. Needleman
Material rate dependence and localized deformation in crystalline solids
Acta Metall. 31 (1983) 1951–1976

This topic: Documentation > Background > Plasticity > Isotropic
Topic revision: 01 Feb 2017, PhilipEisenlohr
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