Free-boundary dynamics in elastoplastic amorphous solids: The circular hole problem

We develop an athermal shear-transformation-zone (STZ) theory of plastic deformation in spatially inhomogeneous, amorphous solids. Our ultimate goal is to describe the dynamics of the boundaries of voids or cracks in such systems when they are subjected to remote, time-dependent tractions. The theory is illustrated here for the case of a circular hole in an infinite two-dimensional plate, a highly symmetric situation that allows us to solve much of the problem analytically. In spite of its special symmetry, this example contains many general features of systems in which stress is concentrated near free boundaries and deforms them irreversibly. We depart from conventional treatments of such problems in two ways. First, the STZ analysis allows us to keep track of spatially heterogeneous, internal state variables such as the effective disorder temperature, which determines plastic response to subsequent loading. Second, we subject the system to stress pulses of finite duration, and therefore are able to observe elastoplastic response during both loading and unloading. We compute the final deformations and residual stresses produced by these stress pulses. Looking toward more general applications of these results, we examine the possibility of constructing a boundary-layer theory that might be useful in less symmetric situations.

Figure 1

(Color online) A time-dependent nonmonotonic loading scheme corresponding to Eq. (4.2) with $T=8000{\tau }_0$ and ${\sigma }_p=2s_y$.

Figure 2

(Color online) The $r$ dependence of the dynamical variables at different times in the constant load case. The radius of the circle is shown in the upper left panel without the initial elastic expansion. At longer times the radius saturates. Note the relative small increase in radius. The stress is shown in the upper right panel, and the fields $D^{\mathrm{pl}}(r,t)$ and $\Lambda =e^{-1∕\chi }$ in the two lower panels, respectively. The parameters chosen are ${ϵ}_0=1$, $c_0=1$, ${\chi }_0=0.13$, ${\chi }_0=0.1$, $\mu =50s_y$, and ${\sigma }_0=2s_y$. The solid line corresponds to $t=2000{\tau }_0$, the dashed line to $t=4000{\tau }_0$, the dotted line to $t=6000{\tau }_0$, and the dotted-dashed line to $t=8000{\tau }_0$.

Figure 3

(Color online) The time-dependent nonmonotonic load case. All the material parameters are the same as those for Fig. 2, while the load is given by Eq. (4.2) with $T=8000{\tau }_0$ and ${\sigma }_p=2s_y$. The solid line corresponds to $t=2000{\tau }_0$, the dashed line to $t=4000{\tau }_0$, the dotted line to $t=6000{\tau }_0$, and the dotted-dashed line to $t=8000{\tau }_0$.

Figure 4

(Color online) Same as the preceding figure but with ${\sigma }_p=4s_y$.

Figure 5

(Color online) The profiles of $s(r,t)$ and $D^{\mathrm{pl}}(r,t)$ for the same external loading as in Fig. 3. The solid line corresponds to $t=4000{\tau }_0$, the dashed line to $t=4500{\tau }_0$, the dotted line to $t=5000{\tau }_0$, and the dotted-dashed line to $t=5500{\tau }_0$.

Figure 6

(Color online) A comparison between the exact solution (solid line) and the prediction of the boundary layer theory (open circles) for ${\sigma }_p=2s_y$.

Figure 7

(Color online) A comparison between the exact solution (solid line) and the prediction of the boundary layer theory (open circles) for ${\sigma }_p=3s_y$.

Figure 8

(Color online) A comparison between the exact solution (solid line) and the prediction of the boundary layer theory (open circles) for ${\sigma }_p=4s_y$.

Figure 9

(Color online) Comparison between the full solution (solid line) for $s(r,t)$ and the prediction of the boundary layer approximation (dashed line) for ${\sigma }_p=2s_y$ and for various values of $t$.