Coordinate transformation methodology for simulating quasistatic elastoplastic solids

Molecular dynamics simulations frequently employ periodic boundary conditions where the positions of the periodic images are manipulated in order to apply deformation to the material sample. For example, Lees-Edwards conditions use moving periodic images to apply simple shear. Here, we examine the problem of precisely comparing this type of simulation to continuum solid mechanics. We employ a hypoelastoplastic mechanical model, and develop a projection method to enforce quasistatic equilibrium. We introduce a simulation framework that uses a fixed Cartesian computational grid on a reference domain, and which imposes deformation via a time-dependent coordinate transformation to the physical domain. As a test case for our method, we consider the evolution of shear bands in a bulk metallic glass using the shear transformation zone theory of amorphous plasticity. We examine the growth of shear bands in simple shear and pure shear conditions as a function of the initial preparation of the bulk metallic glass.

Figure 1

(a) Lees-Edwards boundary conditions in three dimensions where the $z$ coordinate points upward. The system of interest is shown in yellow and outlined in black dashed lines; other periodic copies are shown in brown. Periodic copies of the system above and below are set to move with a specific velocity, imposing a specific strain rate $\dot{\gamma }$ on the system. (b) A graphical depiction of a domain transformation $\mathbf{T}\left(t\right)$ that maps a fixed reference domain $\mathbf{X}$ to a sheared physical domain $\mathbf{x}$.

Figure 2

The initial configuration for the transformed to nontransformed comparison. Here, $a=0.3, \eta =1.2$, and ${\chi }_{bg}=550\text{K}$ in the opacity function.

Figure 3

Snapshots of the effective temperature field $\chi (\mathbf{x},t)$ for the (a), (c), (e) nontransformed and (b), (d), (f) transformed simulation. Simple shear deformation is imposed via a domain transformation. The initial condition in $\chi$ corresponds to a helix of elevated $\chi$ as described in Eq. (39) and depicted in Fig. 2. ${\chi }_{bg}=550\text{K}$ in the opacity function in all panels. (a), (b) $t=2.88\times {10}^5{t}_s, a=0.7, \eta =1.25$. (c), (d) $t=4.02\times {10}^5{t}_s, a=0.8, \eta =1.35$. (e), (f) $t=6\times {10}^5{t}_s, a=0.9, \eta =1.5$.

Figure 4

$L_2$ norm of the $\chi , \mathbf{v}$, and $\mathit{\sigma }$ simulation field differences between the transformed and nontransformed methods computed using Eq. (40) in a simple shear simulation. (Top left) A comparison of the three different field norms on a grid of size $256\times 256\times 128$. (Top right, bottom left, bottom right) The velocity, effective temperature, and stress norm differences, respectively, for varying levels of discretization $N=N_x=N_y=2N_z$.

Figure 5

The initial conditions for the cylindrical inclusion numerical experiments. ${\chi }_{bg}=600\text{K}, a=0.3$, and $\eta =1.2$ in the opacity function.

Figure 6

Snapshots of the effective temperature distribution $\chi (\mathbf{X},t)$. Simple shear deformation is imposed via a domain transformation. The initial condition in $\chi$ corresponds to a cylindrical inclusion as described in Sec. 5a1 and shown in Fig. 5. On the left, clamped boundary conditions in $Z$ are used, while on the right, Lees-Edwards boundary conditions are used. ${\chi }_{bg}=600\text{K}$ in the opacity function in all subfigures. (a), (b) $t=5\times {10}^5{t}_s$. $a=0.3$, and $\eta =1.2$. (c), (d) $t=1.25\times {10}^6{t}_s$. $a=0.45$ and $\eta =1.55$. (e), (f) $t=2\times {10}^6{t}_s$. $a=0.55$ and $\eta =1.6$.

Figure 7

Snapshots of the effective temperature field at $t=0t_s$. All simulations use nonperiodic boundary conditions in $Z$ and apply simple shear deformation. For all plots, values of $a=0.25$ and $\eta =1.3$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 8

Snapshots of the effective temperature field at $t=4\times {10}^5{t}_s$. All simulations use nonperiodic boundary conditions in $Z$ and apply simple shear deformation. For all plots, values of $a=0.45$ and $\eta =1.75$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 9

Snapshots of the effective temperature field at $t=6\times {10}^5{t}_s$. All simulations use nonperiodic boundary conditions in $Z$ and apply simple shear deformation. For all plots, values of $a=0.45$ and $\eta =1.75$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 10

Snapshots of the effective temperature field at $t={10}^6{t}_s$. All simulations use nonperiodic boundary conditions in $Z$ and apply simple shear deformation. For all plots, values of $a=0.75$ and $\eta =2$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 11

Snapshots of the effective temperature field at $t=0t_s$. All simulations use Lees-Edwards boundary conditions. For all plots, values of $a=0.25$ and $\eta =1.3$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 12

Snapshots of the effective temperature field at $t=4\times {10}^5{t}_s$. All simulations use Lees-Edwards boundary conditions. For all plots, values of $a=0.45$ and $\eta =1.75$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 13

Snapshots of the effective temperature field at $t=6\times {10}^5{t}_s$. All simulations use Lees-Edwards boundary conditions. For all plots, values of $a=0.45$ and $\eta =1.75$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 14

Snapshots of the effective temperature field at $t={10}^6{t}_s$. All simulations use Lees-Edwards boundary conditions. For all plots, values of $a=0.75$ and $\eta =2$ are used. ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$ in each pane. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575$, and $600\text{K}$, respectively.

Figure 15

Snapshots of the effective temperature distribution $\chi (\mathbf{X},t)$ for a quasistatic simulation. Pure shear deformation is imposed via a domain transformation with an initial condition corresponding to a sequence of blips of elevated $\chi$ lying roughly along the superdiagonal of the simulation domain. This simulation uses periodic boundary conditions in all three directions. ${\chi }_{bg}=550\text{K}$ in the opacity function for all panels. (a) $t=0t_s, a=0.75, \eta =1.2$. (b) $t=5\times {10}^4{t}_s, a=0.75, \eta =1.2$. (c) $t=8\times {10}^4{t}_s, a=0.75, \eta =1.25$. (d) $t={10}^5{t}_s, a=0.75, \eta =1.25$. (e) $t=2\times {10}^5{t}_s, a=0.4, \eta =1.6$. (e) $t=4\times {10}^5{t}_s, a=1.1, \eta =2.45$.

Figure 16

Snapshots of the effective temperature distribution $\chi (\mathbf{X},t)$ for a quasistatic simulation. Pure shear deformation is imposed via a domain transformation with an initial condition corresponding to a sequence of blips of elevated $\chi$ lying roughly along the superdiagonal of the simulation domain. This simulation uses nonperiodic boundary conditions in $Z$ and is periodic in the $X$ and $Y$ directions. ${\chi }_{bg}=550\text{K}$ in the opacity function in all panels. (a) $t=0t_s, a=0.75, \eta =1.2$. (b) $t=5\times {10}^3{t}_s, a=0.75, \eta =1.2$. (c) $t={10}^4{t}_s, a=0.75, \eta =1.45$. (d) $t=1.5\times {10}^4{t}_s, a=0.75, \eta =1.45$. (e) $t=4\times {10}^5{t}_s, a=1.75, \eta =1.75$.

Figure 17

Snapshots of the effective temperature field at $t=0t_s$ with pure shear transformation imposed on the domain. All simulations use periodic boundary conditions. For all plots, a value of $a=0.25$ and $\eta =1.3$ is used in the opacity function, and ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575,\text{and}600\text{K}$, respectively.

Figure 18

Snapshots of the effective temperature field at $t=3\times {10}^5{t}_s$ with a pure shear transformation imposed on the domain. All simulations use periodic boundary conditions. For all plots, a value of $a=0.55$ and $\eta =1.5$ is used in the opacity function, and ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575,\text{and}600\text{K}$, respectively.

Figure 19

Snapshots of the effective temperature field at $t=6\times {10}^5{t}_s$ with a pure shear transformation imposed on the domain. All simulations use periodic boundary conditions. For all plots, a value of $a=0.75$ and $\eta =1.6$ is used in the opacity function, and ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575,\text{and}600\text{K}$, respectively.

Figure 20

Snapshots of the effective temperature field at $t=1.5\times {10}^6{t}_s$ with a pure shear transformation imposed on the domain. All simulations use periodic boundary conditions. For all plots, a value of $a=1.35$ and $\eta =1.5$ is used in the opacity function, and ${\chi }_{bg}$ is set to ${\mu }_{\chi }-25\text{K}$. (a)–(f) have ${\mu }_{\chi }=450,500,525,550,575,\text{and}600\text{K}$, respectively.