Modeling and simulation of strain-induced phase transformations under compression and torsion in a rotational diamond anvil cell

Strain-induced phase transformations (PTs) under compression and torsion in rotational diamond anvils are simulated using a finite-element approach. Results are obtained for three ratios of yield strengths of low-pressure and high-pressure phases and are compared with those for the compression without torsion from Levitas and Zarechnyy [Phys. Rev. B 82, 174123 (2010)]. Various experimental effects are reproduced, including a pressure self-multiplication effect, plateau at pressure distribution at the diffuse interface, simultaneous occurrence of direct and reverse PTs, and irregular stress distribution for PT to a weaker phase. The obtained results change the fundamental understanding of strain-induced PT in terms of interpretation of experimental measurements and the extracting of information on material processes from sample behavior. Intense radial plastic flow moves the high-pressure phase to the low-pressure region, which may lead to misinterpretation of measurements. Various interpretations based on a simplified equilibrium equation (for example, about zero yield strength of phase mixture and hydrostatic conditions during PT) appears to be wrong because of inapplicability of this equation for cases with large gradients of phase concentration and yield strength. The approach developed represents a tool for designing experiments for different purposes and for controlling PTs, and it opens unexpected ways to extract material information by combining simulation and experiment.

Figure 1

Geometry of quarter of a sample with boundary conditions.

Figure 2

(Color online) (a) Distributions of concentration of high-pressure phase $c$, (b) accumulated plastic strain $q$, and (c) pressure $p$ in a quarter of a cross section of the sample for the case with ${\sigma }_{2y}={\sigma }_{1y}$ for dimensionless applied axial force $F=4.44$ and different values of angle of rotation $\phi$. Pressure range $p_{\epsilon }^r<p<p_{\epsilon }^d$ is shown in magenta. The contours of this region are duplicated in Fig. 2a, for concentration, similar to all other figures with concentration fields in a sample. Maximum values of accumulated plastic strain $q$ for each of six compression stages are shown near color legend. The same is shown in all other figures with pressure and accumulated plastic strain fields in a sample. 1: $\phi =0.04$, 2: $\phi =0.14$, 3: $\phi =0.34$, 4: $\phi =0.78$, 5: $\phi =1.16$, and 6: $\phi =1.58$.

Figure 3

(Color online) (a) Distributions of pressure $p$, (b) concentration of high-pressure phase $c$, (c) shear stress ${\tau }_{zr}$, and (d) shear stress ${\tau }_{r\phi }$ along the radius of the contact surface of a sample $r$ for the case with ${\sigma }_{2y}={\sigma }_{1y}$ for dimensionless applied axial force $F=4.44$ and different values of angle of rotation $\phi$. 1: $\phi =0.04$, 2: $\phi =0.14$, 3: $\phi =0.34$, 4: $\phi =0.78$, 5: $\phi =1.16$, and 6: $\phi =1.58$. Diamonds ◇ designate two points of straight line according to Eq. (2).

Figure 4

(Color online) (a) Distributions of concentration of high-pressure phase $c$, (b) accumulated plastic strain $q$, and (c) pressure $p$ for the case with ${\sigma }_{2y}=5{\sigma }_{1y}$ for dimensionless applied axial force $F=4.44$ and different values of angle of rotation $\phi$. 1: $\phi =0.09$, 2: $\phi =0.38$, 3: $\phi =0.61$, 4: $\phi =0.94$, 5: $\phi =1.10$, and 6: $\phi =1.30$.

Figure 5

(Color online) (a) Distributions of pressure $p$, (b) concentration of high-pressure phase $c$, (c) shear stress ${\tau }_{zr}$, and (d) shear stress ${\tau }_{r\phi }$ along the radius of the contact surface of a sample $r$ for the case with ${\sigma }_{2y}=5{\sigma }_{1y}$ for dimensionless applied axial force $F=4.44$ and different values of angle of rotation $\phi$ in a quarter of cross section of the sample. 1: $\phi =0.09$, 2: $\phi =0.38$, 3: $\phi =0.61$, 4: $\phi =0.94$, 5: $\phi =1.10$, and 6: $\phi =1.30$. Diamonds ◇ designate two points of straight lines for ${\sigma }_y={\sigma }_{y1}$ and ${\sigma }_y={\sigma }_{y2}$ according to Eq. (2).

Figure 6

(Color online) (a) Distributions of concentration of high-pressure phase $c$, (b) accumulated plastic strain $q$, and (c) pressure $p$ for the case with ${\sigma }_{2y}=0.2{\sigma }_{1y}$ for dimensionless applied axial force $F=4.44$ (except 1) and different values of angle of rotation $\phi$. 1: $F=4.30$, $\phi =0.0$, 2: $\phi =0.09$, 3: $\phi =0.20$, 4: $\phi =0.64$, 5: $\phi =0.84$, and 6: $\phi =1.40$.

Figure 7

(Color online) (a) Distributions of pressure $p$, (b) concentration of high-pressure phase $c$, (c) shear stress ${\tau }_{zr}$, and (d) shear stress ${\tau }_{r\phi }$ along the radius of the contact surface of a sample $r$ for the case with ${\sigma }_{2y}=0.2{\sigma }_{1y}$ for dimensionless applied axial force $F=4.44$ (except 1) and different values of angle of rotation $\phi$. 1: $F=4.30$, $\phi =0.0$, 2: $\phi =0.09$, 3: $\phi =0.20$, 4: $\phi =0.64$, 5: $\phi =0.84$, and 6: $\phi =1.40$.

Figure 8

(Color online) (a) Comparison of distributions of pressure $p$, (b) concentration of high-pressure phase $c$, (c) accumulated plastic strain $q$ for compression (1, 3, and 5) and torsion (2, 4, and 6) for ${\sigma }_{y2}={\sigma }_{y1}$. The pressure at the contact surface at $r=0$ after pure compression and torsion is the same and equal to 8.40 (1 and 2), 9.40 (3 and 4), and 10.2 (5 and 6). The dimensionless force $F$ is 4.46 (1), 5.04 (3), 5.43 (5), and 4.44 (2, 4, and 6). Rotation angle $\phi$ is 0.04 (2), 0.60 (4), and 1.57 (6).

Figure 9

(Color online) (a) Comparison of distributions of pressure $p$ and (b) concentration of high-pressure phase $c$ for compression (1, 3, and 5) and torsion (2, 4, and 6) for ${\sigma }_{y2}={\sigma }_{y1}$. The pressure at the contact surface at $r=0$ after pure compression and torsion is the same and equal to 8.40 (1 and 2), 9.40 (3 and 4), and 10.2 (5 and 6). Dimensionless applied axial force $F$ is 4.46 (1), 5.04 (3), 5.43 (5), and 4.44 (2, 4, 6). Rotation angle $\phi$ is 0.04 (2), 0.60 (4), and 1.57 (6).

Figure 10

(Color online) (a) Comparison of distributions of pressure $p$, (b) concentration of high-pressure phase $c$, (c) accumulated plastic strain $q$ for compression (1, 3, and 5) and torsion (2, 4, and 6) for ${\sigma }_{y2}=5{\sigma }_{y1}$. The pressure at the contact surface at $r=0$ after pure compression and torsion is the same and equal to 12.4 (1 and 2), 16.2 (3 and 4), and 20.1 (5 and 6). Dimensionless force $F$ is 5.47 (1), 6.26 (3), 7.21 (5), and 4.44 (2, 4, and 6). Rotation angle $\phi$ is 0.20 (2), 0.52 (4), and 1.11 (6).

Figure 11

(Color online) (a) Comparison of distributions of pressure $p$ and (b) concentration of high-pressure phase $c$ for compression (1, 3, and 5) and torsion (2, 4, and 6) for ${\sigma }_{y2}=5{\sigma }_{y1}$. The pressure at the contact surface at $r=0$ after pure compression and torsion is the same and equal to 12.4 (1 and 2), 16.2 (3 and 4), and 20.1 (5 and 6). Dimensionless applied axial force $F$ is equal to 5.47 (1), 6.26 (3), 7.21 (5), and 4.44 (2, 4, and 6). Rotation angle $\phi$ is equal to 0.20 (2), 0.52 (4), and 1.11 (6).

Figure 12

(Color online) (a) Comparison of distributions of pressure $p$, (b) concentration of high-pressure phase $c$, (c) accumulated plastic strain $q$ for compression (1, 3, and 5) and torsion (2, 4, and 6) for ${\sigma }_{y2}=0.2{\sigma }_{y1}$. The pressure at the contact surface at $r=0$ after pure compression and torsion is the same and equal to 7.95 (1 and 2), 8.10 (3 and 4), and 8.80 (5 and 6). Dimensionless force $F$ is 5.07 (1), 5.42 (3), 6.11 (5), and 4.44 (2, 4, and 6). Rotation angle $\phi$ is 0.20 (2), 0.32 (4), and 0.64 (6).

Figure 13

(Color online) (a) Comparison of distributions of pressure $p$ and (b) concentration of high-pressure phase $c$ for compression (1, 3, and 5) and torsion (2, 4, and 6) for ${\sigma }_{y2}=0.2{\sigma }_{y1}$. The pressure at the contact surface at $r=0$ after pure compression and torsion is the same and equal to 7.95 (1 and 2), 8.10 (3 and 4), and 8.80 (5 and 6). Dimensionless applied axial force $F$ is equal to 5.07 (1), 5.42 (3), 6.11 (5), and 4.44 (2, 4, and 6). Rotation angle $\phi$ is equal to 0.20 (2), 0.32 (4), and 0.64 (6).

Figure 14

(Color online) Change in relative thickness $h/h_o$ of the sample vs angle of rotation $\phi$ for (1) ${\sigma }_{y2}={\sigma }_{y1}$, (2) ${\sigma }_{y2}=5{\sigma }_{y1}$, and (3) ${\sigma }_{y2}=0.2{\sigma }_{y1}$, and (0) according to Eq. (4).