Adaptive multi-time-domain subcycling for crystal plasticity FE modeling of discrete twin evolution

Abstract

Crystal plasticity finite element (CPFE) models that accounts for discrete micro-twin nucleation-propagation have been recently developed for studying complex deformation behavior of hexagonal close-packed (HCP) materials (Cheng and Ghosh in Int J Plast 67:148–170, 2015, J Mech Phys Solids 99:512–538, 2016). A major difficulty with conducting high fidelity, image-based CPFE simulations of polycrystalline microstructures with explicit twin formation is the prohibitively high demands on computing time. High strain localization within fast propagating twin bands requires very fine simulation time steps and leads to enormous computational cost. To mitigate this shortcoming and improve the simulation efficiency, this paper proposes a multi-time-domain subcycling algorithm. It is based on adaptive partitioning of the evolving computational domain into twinned and untwinned domains. Based on the local deformation-rate, the algorithm accelerates simulations by adopting different time steps for each sub-domain. The sub-domains are coupled back after coarse time increments using a predictor-corrector algorithm at the interface. The subcycling-augmented CPFEM is validated with a comprehensive set of numerical tests. Significant speed-up is observed with this novel algorithm without any loss of accuracy that is advantageous for predicting twinning in polycrystalline microstructures.

Appendices

Appendix A: Evolution of slip system resistance

The athermal and thermal shear resistances in Eq. (8) and their evolutions are derived from dislocation-based mechanisms. Two types of dislocations are considered in the evolution of both the athermal ($s_{a}^\alpha$) and thermal ($s_{*}^\alpha$) shear resistances. They are: (i) the statistically stored dislocations (SSDs) and (ii) the geometrically necessary dislocations (GNDs) [50, 51]. SSDs are associated with the homogeneous components of plastic flow and are characterized by a vanishing net Burgers vector in the microstructure. GNDs correspond to stored polarized dislocation densities. They are related to the curl of plastic deformation gradient in the elastically unloaded configuration. GND accumulation is necessary for accommodating crystal lattice curvatures in single crystal bending or near grain boundaries of polycrystalline aggregates. The resulting athermal and thermal hardening rates due to the evolution of SSDs are given as:

$$\begin{aligned} {\dot{s}}^{\alpha }_{a,SSD}=h^{\alpha \beta }_a~~{|\dot{\gamma }}^{\beta }{sin(\varvec{n}_0}^{\alpha },{\varvec{t}_0}^{\beta })| \end{aligned}$$

(28a)

$$\begin{aligned} {\dot{s}}^{\alpha }_{*,SSD}=h^{\alpha \beta }_*~~{|\dot{\gamma }}^{\beta }{cos(\varvec{n}_0}^{\alpha },{\varvec{t}_0}^{\beta })| \end{aligned}$$

(28b)

where the coefficient matrices $h_a^{\alpha \beta }$ and $h_*^{\alpha \beta }$ represent the hardening of athermal and thermal shear resistances on the slip system $\alpha$ due to activity on slip system $\beta$. These matrices are derived to be:

$$\begin{aligned} h^{\alpha \beta }_a=q^{\alpha \beta }h^{\beta }_{a,ref}{\left| 1-\frac{s^{\beta }_{a,SSD}}{s^{\beta }_{a,sat}}\right| }^r sign\left( 1-\frac{s^{\beta }_{a,SSD}}{s^{\beta }_{a,sat}}\right) ~~~( \text{ no } \text{ sum } \text{ on }~\beta ) \end{aligned}$$

(29a)

$$\begin{aligned} h^{\alpha \beta }_*=q^{\alpha \beta }h^{\beta }_{*,ref}{\left| 1-\frac{s^{\beta }_{*,SSD}}{s^{\beta }_{*,sat}}\right| }^r sign\left( 1-\frac{s^{\beta }_{*,SSD}}{s^{\beta }_{*,sat}}\right) ~~~( \text{ no } \text{ sum } \text{ on }~\beta ) \end{aligned}$$

(29b)

where $s^{\alpha }_{a,sat}$ and $s^{\alpha }_{*,sat}$ are the athermal and thermal saturation stresses for hardening caused by the SSD population. The exponent r is a material constant. $h_{a,ref}$ and $h_{*,ref}$ are respectively the reference hardening rates for athermal and thermal slip resistances and $q^{\alpha \beta }$ is a matrix describing latent hardening.

The contribution of GNDs to the slip system hardening is from two sources, viz. (i) dislocation components $\rho _{GND,P}^\alpha$ parallel to the slip plane $\alpha$, which causes hardening due to the athermal shear resistance $s_{a}^\alpha$, and (ii) forest dislocation components $\rho _{GND,F}^\alpha$, which contributes to the hardening due to thermal shear resistance $s_{*}^\alpha$. These are given as:

$$\begin{aligned} s^{\alpha }_{a,GND}=c_1Gb\sqrt{{\rho }^{\alpha }_{P,GND}}\end{aligned}$$

(30a)

$$\begin{aligned} s^{\alpha }_{*,GND}=\frac{Q_{slip}}{c_2c_3b^2}\sqrt{{\rho }^{\alpha }_{F,GND}} \end{aligned}$$

(30b)

where G is the shear modulus, $Q_{slip}^{\alpha }$ is the effective activation energy for dislocation slip and $c_1,~ c_2, ~ c_3$ are constants representing the passing stress, jump-width, and obstacle-width, respectively. Accumulation of GNDs leads to a closure failure of the Burgers circuit in the intermediate configuration crystal lattice. This can be measured by the curl of the plastic deformation gradient per unit area in the reference configuration, corresponding to the Nye’s dislocation density tensor $\varvec{\varLambda } = -{(\mathbf {\nabla }_{X} \times {{\mathbf {F}}^{{p^T}}})^T}$ [52]. The relation between the Nye’s tensor and GND density components on each slip system may be expressed as [1]:

$$\begin{aligned} \varvec{\varLambda }= & {} \sum _{\alpha =1}^{nslip}\rho _{GND,s}^{\alpha }{\mathbf{{b}}_0^{\alpha } \otimes {\mathbf{{m}}_0^{\alpha }}}+\rho _{GND,et}^{\alpha }{\mathbf{{b}}_0^{\alpha } \otimes {\mathbf{{t}}_0^{\alpha }}}\nonumber \\&+\,\rho _{GND,en}^{\alpha }{\mathbf{{b}}_0^{\alpha } \otimes {\mathbf{{n}}_0^{\alpha }}} \end{aligned}$$

(31)

where $\varvec{\rho }_{GND,s}$, $\varvec{\rho }_{GND,et}$ and $\varvec{\rho }_{GND,en}$ are the GND density components with screw, in-slip-plane edge and normal-to-slip-plane edge characteristics. $\mathbf {t}^{\alpha }$ is the tangent vector of dislocation line expressed as $\mathbf {t}_0^{\alpha }=\mathbf {m}_0^{\alpha }\times \mathbf {n}_0^{\alpha }$, and $\mathbf b_0^\alpha$ is the Burgers vector for a slip system $\alpha$ in the reference configuration. For hcp crystals, there are more slip systems than the number of components in $\varvec{\varLambda }$. The solutions of $\rho _{GND,s}$, $\rho _{GND,et}$ and $\rho _{GND,en}$ are obtained by solving a constrained minimization problem of minimizing the $L_2$ norm of the GND densities subject to the constraint Eq. (31). The minimization problem may be expressed as:

$$\begin{aligned} \mathbf {\rho }_{GND}= & {} Arg\bigg [ Min\left\{ \{\mathbf {\rho }_{{GND}}\}^{T}\{\mathbf {\rho }_{{GND}}\}\right. \nonumber \\&\left. +\left\{ \varvec{\lambda }\right\} ^{T}(\left[ \mathbf{A}\right] \{\mathbf {\rho }_{{GND}}\}-\{\hat{\varvec{\varLambda }}\})\right\} \bigg ] \end{aligned}$$

(32)

where $\{ {\varvec{\rho }_{{GND}}} \}$ is a $36 \times 1$ vector column of GND components, $\left\{ \varvec{\lambda }\right\}$ is a $9 \times 1$ column vector containing components of the Lagrange multipliers, $\{ {\varvec{\hat{\varLambda }}} \}$ is $9 \times 1$ vector form of the Nye’s dislocation density tensor $\varvec{\varLambda }$, and $\left[ \mathbf{A} \right]$ is a $9 \times 36$ linear operator matrix containing the basis tensors ${{\mathbf {b}_0}^\alpha } \otimes {{\mathbf {m}_0}^\alpha }$, ${{\mathbf {b}_0}^\alpha } \otimes {{\mathbf {t}_0}^\alpha }$ and ${{\mathbf {b}_0}^\alpha } \otimes {{\mathbf {n}_0}^\alpha }$. The screw and edge GND components $\rho _{GND,s}$, $\rho _{GND,et}$ and $\rho _{GND,en}$ on each slip system respectively contribute to the parallel and forest GNDs $\rho _{GND,P}^\alpha$ and $\rho _{GND,F}^\alpha$ according to the following relation:

$$\begin{aligned} \rho _{GND,P}^{\alpha } =&\sum \limits _{\beta = 1}^{N_{slip}} {\chi ^{\alpha \beta }}\left[ {} \left| \rho _{GNDs}^\beta {\sin ({\mathbf {n}^\alpha _{0} },{\mathbf {m}^\beta _{0} })} \right| + \left| \rho _{GNDet}^\beta {\sin ({\mathbf {n}^\alpha _{0}},{\mathbf {t}^\beta _{0}})} \right| \right. \nonumber \\&\left. +\left| \rho _{GNDen}^\beta {\sin ({\mathbf {n}^\alpha _{0}},{\mathbf {n}^\beta _{0}})} \right| {} \right] \end{aligned}$$

(33a)

$$\begin{aligned} \rho _{GND,F}^\alpha =&\sum \limits _{\beta = 1}^{N_{slip}}{\chi ^{\alpha \beta }} \left[ {} \left| \rho _{GNDs}^\beta {\cos ({\mathbf {n}^\alpha _{0}},{\mathbf {m}^\beta _{0}})} \right| + \left| \rho _{GNDet}^\beta {\cos ({\mathbf {n}^\alpha _{0}},{\mathbf {t}^\beta _{0})}} \right| \right. \nonumber \\&\left. + \left| \rho _{GNDen}^\beta {\cos ({\mathbf {n}^\alpha _{0}},{\mathbf {n}^\beta _{0}})} \right| {} \right] \end{aligned}$$

(33b)

where $\chi _{\alpha \beta }$ is a matrix of coefficients that describes the interaction strength between different slip systems.

The total athermal and thermal resistances are expressed as the sum of parts related to the dislocation microstructure and those independent of the dislocation as:

$$\begin{aligned} s_a^{\alpha }=s_{a,0}^{\alpha }+\sqrt{(s^{\alpha }_{a,SSD})^2+(s^{\alpha }_{a,GND})^2} \end{aligned}$$

(34a)

$$\begin{aligned} s_*^{\alpha }={s}^{\alpha }_{*,0}+\sqrt{(s^{\alpha }_{*,SSD})^2+(s^{\alpha }_{*,GND})^2} \end{aligned}$$

(34b)

where $s_{a,0}$ and ${s}_{*,0}$ are initial resistances that do not depend on the dislocation microstructure.

The grain-size influences the shear resistances through two sources. First, the non-local GND model introduces a length scale dependence into the athermal and thermal hardening rates. Second, the initial yield stress at the onset of plastic flow is sensitive to the grain size [53, 54]. A Hall–Petch type relation has been proposed in [9, 53] to augment the initial thermal shear resistance $\hat{s}^{\alpha }_{*,0}$. The grain size dependent initial shear resistance is thus given as:

$$\begin{aligned} s^{\alpha }_{*,0}=\hat{s}^{\alpha }_{*,0}+\frac{K^\alpha }{\sqrt{D_g}} \end{aligned}$$

(35)

where $\hat{s}^{\alpha }_{*,0}$ is the initial resistance from defects that not related to grain-size, e.g., Periels resistance and the impurities. $D_g$ is the equivalent grain diameter and the parameter $K^\alpha =\sqrt{\frac{(2-\nu ) \pi \tau ^* G b^{\alpha }}{2(1-\nu )}}$, in which $\tau ^*$ is the barrier strength. In this paper, it is taken as $\tau ^*=0.01G$.

The twin boundaries act as barriers to glissile dislocations and cause hardening of slip systems. Considering the $\left\{ 10\bar{1}2\right\}$ extension twins have low energy coherent boundaries, the twin induced hardening contributes to the thermal slip resistance for the slip systems in the twinned region, expressed as:

$$\begin{aligned} \dot{s}^\alpha _{*,slip-twin}=\sum \limits _{\beta = 1}^{N_{twin}} {h^{\alpha \beta } } \left| {\dot{\gamma }^{\beta }_{tw} } \right| \end{aligned}$$

(36)

The hardening coefficient matrix $h^{\alpha \beta }$ defines the hardening on $\alpha$th—the slip system due to twin system $\beta$ activity. It is expressed as:

$$\begin{aligned} h^{\alpha \beta }=q^{\alpha \beta } h_{ref}^{\beta }\left| 1-\frac{s^{\beta }}{s_{sat}^{\beta }} \right| ^{r}\;sign\left( 1-\frac{s^{\beta }}{s_{sat}^{\beta }} \right) ~~~( \text{ no } \text{ sum } \text{ on }~\beta ) \end{aligned}$$

(37)

where $s^{\beta }=\sqrt{(s_{a}^{\beta })^2+(s_{*}^{\beta })^2}$, $q^{\alpha \beta }$ is a matrix describing latent hardening, and $h_{ref}^{\beta }$ is the reference hardening rate on twin system $\beta$.

Appendix B: Evolution of twin system resistance

The twin system shear resistance changes due to the interaction between twin boundaries and mobile dislocations, expressed as:

$$\begin{aligned} \dot{s}^\alpha _{tw}=\sum \limits _{\beta = 1}^{N_{slip}} {h^{\alpha \beta } } \left| {\dot{\gamma }^{\beta } } \right| \end{aligned}$$

(38)

where the hardening matrix $h^{\alpha \beta }$ defines the hardening of $\alpha$ twin system from the dislocation slip on $\beta$ system, which is written as:

$$\begin{aligned} h^{\alpha \beta }=q^{\alpha \beta }h^{\beta }_{ref}{\left| 1-\frac{s^{\beta }_a+s^{\beta }_*}{s^{\beta }_{sat}}\right| }^r sign\left( 1-\frac{s^{\beta }_a+s^{\beta }_*}{s^{\beta }_{sat}}\right) ~~~( \text{ no } \text{ sum } \text{ on }~\beta )\nonumber \\ \end{aligned}$$

(39)