Evidence That Strain-Rate Softening Is Not Necessary for Material Instability Patterns

Strain-rate softening has been associated with a wide variety of material instabilities, from the Portevin–Le Chatelier effect in metal alloys to stick-slip motion in crust faults. Dynamic instability patterns have been recently discovered in brittle porous media: diffused, oscillatory, and erratic compaction. Using model simulations inspired by experiments with puffed rice, we question the link between these dynamic patterns and strain-rate sensitivity in such media. An important feature of our model is that it can recover strain-rate softening as an emergent phenomenon, without imposing it a priori at its microstructural scale. More importantly, the model also demonstrates that the full range of dynamic patterns can develop without presenting macroscopic strain-rate softening. Based on this counterexample model, we therefore argue that strain-rate softening should not be taken as a necessary condition for the emergence of instability patterns. Our findings in brittle porous media have implications on models that require strain-rate softening to explain earthquake and metal alloy instabilities.

Figure 1

Constant velocity experiments. (a) Experimental device and puffed rice. (b) Stress-strain curves for various compaction velocities. Each curve is offset 50 kPa from the previous one for clarity. Left ordinate: Stress scaled by the typical constrained modulus of the puffed rice, 15 MPa.

Figure 2

Variable velocity experiments. (a) Stress-strain curves for variable velocity tests (full lines) and a constant velocity test (dotted line). (b) Stress averaged over three different strain bins for variable velocity tests and constant velocity test at highest velocity ($V=300\mathrm{mm}·{\mathrm{s}}^{-1}$). Filled triangle, filled red circle, and open green square are variable velocity tests; the first two are shown in (a). Open blue circle is a constant high velocity test, also shown in (a). Corresponding scaled stresses in (a) and (b) are also shown, where stress values are scaled by the typical constrained modulus of the material. (c) Velocity field for the variable velocity test that corresponds to the black line in (a). (d) Velocity field for the constant velocity test that corresponds to the blue line in (a).

Figure 3

Spring-lattice model. (a) Lattice configuration (top) and spring force-displacement law (bottom) used for the simulations. (b) Stress scaled by the theoretical initial modulus ($(1+\sqrt{2}/l_0)k_0$) as a function of strain, for different constant velocity tests with $V=\{5\times {10}^{-6},2\times {10}^{-5},4\times {10}^{-5},8\times {10}^{-5},1.6\times {10}^{-4}\}$ using $k_0=0.004$, $F_{\mathrm{br}}=4\times {10}^{-6}$, $\eta =0.002$, and $a=0.01$. (c) Spatiotemporal vertical velocity fields, with symbols I–V that correspond to lines in (b) and data points in Fig. 4.

Figure 4

Scaled stress as a function of the strain rate, measured in strain range $\epsilon \in [0.01,0.05]$. All simulations have $k_0=0.004$ and $F_{\mathrm{br}}=4\times {10}^{-6}$. The different dotted lines correspond to different viscosities, from bottom to top $\eta =\{1.25\times {10}^{-4},2.5\times {10}^{-4},5\times {10}^{-4},0.001,0.002,0.004\}$. Data points denoted I–V correspond to simulations shown in Fig. 3.