Internal Stress in a Model Elastoplastic Fluid

Plastic materials can carry memory of past mechanical treatment in the form of internal stress. We introduce a natural definition of the vorticity of internal stress in a simple two-dimensional model of elastoplastic fluids, which generates the internal stress. We demonstrate how the internal stress is induced under external loading, and how the presence of the internal stress modifies the plastic behavior.

Figure 1

(a) System’s geometry and the initial transient loading on the boundary (thick arrows) at $x=0$. The fluid velocity $\partial u(x,y,t)/\partial t$ lies along the $z$ axis. (b) Time protocol of loading, $s(t)$ (see the text).

Figure 2

(a) The flux of the vorticity of the internal stress (VIS), $\mathit{J}$ in units of ${\sigma }_{\mathrm{Y}}/\tau$, at $t=100\tau$. (b) The spatially distributed stress components $({\sigma }_{xz},{\sigma }_{yz})$ at $t=500\tau$ shown in the $({\sigma }_{xz},{\sigma }_{yz})$ plane. The points on the curves give the stress components realized on the rectangular grids in the $(x,y)$ plane, spaced by $15\Delta x$ and $16\Delta y$, respectively. The dotted circle represents the yielding threshold, $|\mathbf{\sigma }|={\sigma }_{\mathrm{Y}}$. (c) The same data as (b) shown on the physical $(x,y)$ plane (in units of ${\sigma }_{\mathrm{Y}}$). (d) The VIS, $\omega$, for the same data as (b), shown by gray scales and contours. [Only the range of $0\le x\le 60\xi$ is shown in (a),(c),(d), with the axes labeled in units of $\xi$.] The thick dots (●) in (c),(d) at $x=17.0\xi$, $26.8\xi$ ($y>0$), $26.8\xi$ ($y<0$), and $46.4\xi$ indicate the locations where the subsequent yield starts under the specific uniform external shear stresses: $({\sigma }_{xz}^{\mathrm{ext}},{\sigma }_{yz}^{\mathrm{ext}})={\sigma }^{\mathrm{ext}}(\cos \theta ,\sin \theta )$ with $\theta =\pi /2$, 0, $\pi$, $-\pi /2$, respectively (see the text).

Figure 3

The log-log plot of the two-dimensional spatial Fourier spectral intensity of $\omega$, integrated over the angle, vs the amplitude of the wave numbers [in units of $q_0\equiv (2\xi {)}^{-1}$, the wave number beyond which the yielded fluid has no propagative modes]. The exponent of the hydrodynamic regime ($q<q_0$) is read out to be about $-1/3$.