Plastic instability of annular crystalline membrane in circular confinement

Understanding the mechanical instabilities of two-dimensional membranes has strong connection to the subjects of structure instabilities, morphology control, and materials failures. In this work, we investigate the plastic mechanism developed in the annular crystalline membrane system for adapting to the shrinking space, which is caused by the controllable gradual expansion of the inner boundary. In the process of plastic deformation, we find the continuous generation of dislocations at the inner boundary and their collective migration to the outer boundary; this neat dynamic scenario of dislocation current captures the complicated reorganization process of the particles. We also reveal the characteristic vortex structure arising from the interplay of topological defects and the displacement field. These results may find applications in the precise control of structural instabilities in packings of particulate matter and covalently bonded systems.

Figure 1

Schematic plot of the 2D annular crystalline membrane. The model consists of Lennard-Jones particles in triangular lattice confined in the annulus. The 2D deformation of the annular membrane is driven by the gradual expansion of the inner boundary, whose radius is denoted as ${\rho }_1$ and ${\rho }_1^{\prime }$ in the initial stress-free state and the deformed state, respectively. To avoid the introduction of topological defects at the outer boundary, the anchored particles within the outer annulus (in orange) are in triangular lattice that is commensurate with the lattice of the annular membrane.

Figure 2

Cumulative particle distributions over the deformed annular membrane at varying initial radius of the inner boundary. $N\left({\rho }^{\prime }\right)$ is the total number of particles within the circle of radius ${\rho }^{\prime }$ in the deformed lattice; ${\rho }^{\prime }\in [{\rho }_1+h,{\rho }_2]$. The expansion rate $h=0.8$. The simulation data (dots) could be well fitted by the theoretical results (red curves). The radius of the outer boundary ${\rho }_2=29.4$.

Figure 3

The pattern of dislocation current is revealed in the 2D plastic deformation of the annular crystalline membrane upon continuous expansion of the inner boundary. The displacement vectors of the particles as the radius of the inner boundary is increased from ${\rho }_i$ to ${\rho }_f$ are represented by the arrows; the values of ${\rho }_i$ and ${\rho }_f$ are given in the square brackets below each figure. (a) The red and blue arrows indicate the shearing deformation, leading to the emergence of the dislocation and the vortices (as indicated by the blue disks). The dislocation is composed of a pair of five- and seven-fold disclinations as represented by the red and blue dots. The boxed region in panel (a) is enlarged in panel (b) for showing the reorganization of the crystal lattice by the dislocation. With the expansion of the inner boundary, we observe the collective migration of the dislocations from the inner to the outer boundary, as shown in panels (e) and (f). In panel (f), the previous locations of the dislocations are indicated by red circles for visual convenience. The system is stress-free in the initial state, where the value of the radius of the inner boundary ${\rho }_1=5$ and ${\rho }_2=24.4$. The annular L-J lattice consists of 1574 particles.

Figure 4

Statistics of the emergent disclinations in the plastic deformation of the annular membrane driven by the expansion of the inner boundary. (a) Plot of the number of $\alpha$-fold disclinations $N_{\alpha }$ accumulated near the outer boundary vs the inner radius ${\rho }_1^{\prime }$ of the deformed membrane. The defects within the thin annular region of 1.5 lattice spacings away from the outer boundary are counted. (b) Plot of the relative number of $\alpha$-fold disclinations within the annular membrane vs the inner radius ${\rho }_1^{\prime }$ of the deformed membrane. The thin rims (1.5 lattice spacings) near both inner and outer boundaries are excluded. The four-, five-, seven-, and eight-fold disclinations are indicated by orange, red, blue, and yellow dots, respectively. In the initial stress-free state, the radius of the inner boundary ${\rho }_1=5, {\rho }_2=24.4$, and $N=1574$.