Local Stress Relaxation and Shear Banding in a Dry Foam under Shear

We have developed a realistic simulation of 2D dry foams under quasistatic shear. After a short transient, a shear-banding instability is observed. These results are compared with measurements obtained on real 2D (confined) foams. The numerical model allows us to exhibit the mechanical response of the material to a single plastication event. From the analysis of this elastic propagator, we propose a scenario for the onset and stability of the flow localization process in foams, which should remain valid for most athermal amorphous systems under creep flow.

Figure 1

(a) Snapshot of a simulated foam with $16\times 48$ bubbles. The polydispersity is 6%. The foam has periodic boundary conditions along the $x$ direction. Films laying at upper and lower edges are fixed; shearing is obtained by moving the lower edge along the $x$ direction. (b) Example of a topological change ($T1$ process) occurring inside the foam upon shearing.

Figure 2

Localization process in a simulated foam. (a) The $y$ position of the $T1$ events as a function of the wall displacement $D$ expressed in bubble diameter. (b) Distribution of the $y$ positions of the $T1$ events for $D<15$ (transient regime) and (c) $D>20$ (localized regime). In the latter, the dotted line shows the gradient $\frac{\partial \overline{v}}{\partial y}(y)$ of the associated plastic flow profile $\overline{v}(y)$.

Figure 3

Comparison of time averaged measurements obtained from (a) experiments on 2D foams in a Couette cell and (b) simulated foams. $\overline{v}$ is the tangential velocity rescaled by the wall velocity $v_0$. $\sqrt{\overline{v_{\perp }^2}}$ is the normal mean square displacement, associated with a time lapse $\tau =0.25d/v_0$, where $d$ is the average bubble diameter. The rapid drop of $\sqrt{\overline{v_{\perp }^2}}$ in (b) far from the moving wall is due to the relatively small width of the simulated foam ($W=16$), and hence the presence of the other confining wall.

Figure 4

Displacement and variational shear stress field associated with a single $T1$ event. Each of these results has been obtained by averaging over $\sim 100$ different $T1$ events located at the same distance $y_{T1}$ from the shearing wall. (a) and (b): line averaged displacement profiles (expressed in bubble diameter $d$), for a $T1$ event located at $y_{T1}=3$ and $8$, respectively. $\delta {ϵ}_{T1}$ represents the mean strain released by the $T1$ event. (c) and (d): Corresponding shear stress variation fields. Red color indicates an increase of the stress (relative to the imposed shear stress), blue color indicates a stress relaxation. The arrows show approximately the motion of the rearranging bubbles during the $T1$ event.

Figure 5

Profile of shear stress variance $⟨[\sigma (x,y)-\overline{\sigma }{]}^2⟩$ in real (left) and simulated (right) foams. Solid lines correspond to foams in the fully localized regime. Dashed lines correspond to freshly prepared samples (no shearing). All data have been rescaled with the long time limit shear stress. Inset: shear stress probability distributions at lines $y=1$ (solid) and $y=10$ (dashed), respectively, in the localized regime.