Elastoplastic behavior of yield stress fluids

By means of the incremental elastic modulus for small deformations superimposed on creep and subsequent recovery tests, we follow the structural state of various soft-jammed systems (yield stress fluids) in their solid regime. We demonstrate that the solid state of the material is associated with a persistent elastic network of constant elastic modulus up to yielding (solid-liquid transition), while progressively more additional elastoplastic elements are involved. The main features of these elastoplastic elements, i.e., increase of both the plastic and the elastic deformation components with the square of the shear stress, reveal the fundamental characteristics of a simple generic model (independent of material structure) describing the different components of the mechanical behavior of such systems in their solid regime.

Figure 1

Elastic modulus of the emulsion measured from oscillations under low stress amplitude as a function of time during creep tests under various shear stress values (see legend). The red squares correspond to the solid regime ($\tau <{\tau }_c$); the blue circles correspond to the liquid regime ($\tau >{\tau }_c$). The inset shows the corresponding deformation undergone by the fluid in time (same symbols).

Figure 2

Elastic modulus for low oscillations during creep tests as a function of the imposed shear stress value rescaled by the yield stress, for different materials: emulsion (orange squares), Carbopol gel (blue circles), laponite suspension (red stars), bentonite suspension (black diamonds). The legend gives the material yield stress values. The horizontal dotted lines indicate the (mean) value ($G_{\infty }$) retained for the elastic modulus in the solid regime for each material. The inset shows the loss modulus measured during the same creep tests by means of small oscillations (same symbols). The dashed lines in the inset correspond to the calculated $G^{\prime \prime }$ values at first order according to theory (see text). Note that for laponite and bentonite the moduli values are taken 20 s after the test beginning, i.e., before the significant evolutions observed later during the flow (see Appendix pp1).

Figure 3

Shear stress vs induced deformation during recovery tests (stress release) for the emulsion: elastic (reversible) component (crossed squares); plastic (irreversible) component (open squares); total deformation (filled squares). The dotted line is the equation $\tau =G_{\infty }\gamma$, with $G_{\infty }$ taken as the constant $G^{\prime }$ (490 Pa for the emulsion) in small-oscillations tests. The inset shows a typical recovery test: deformation induced during creep test, then deformation recovery for stress release, allowing to distinguish the two (elastic and plastic) components.

Figure 4

Additional elastic deformation (filled symbols) with regard to the deformation of the constant elastic network, and plastic deformation (open symbols), as a function of the stress applied for different materials (same type of data in the inset). The dotted lines (of slope 2) correspond to the model fitted to data (see text) with $\alpha =0.8$ (emulsion); $\alpha =1$ (gel); $\alpha =0.5$ [colloid (laponite)]; $\alpha =1.2$ [clay (bentonite)].

Figure 5

(a) Representation of the simplest model reproducing the constant elastic network and some additional elastoplastic component. (b) Resulting elastic, plastic, and total deformation components when submitted to stress, then recovery, for different stress levels.

Figure 6

Representation of the complete model (see text) in discrete form.

Figure 7

Gel: (a) Deformation vs time during creep tests under different stress levels; (b) elastic modulus under low stress oscillations during creep tests; (c) shear stress vs induced deformation during recovery tests (stress release): elastic (reversible) component, plastic (irreversible) component, and total deformation. The dotted line is the equation $\tau =G_{\infty }\gamma$, with $G_{\infty }$ taken as the constant $G^{\prime }$ (183 Pa here) in small-oscillations tests. The inset shows a zoom of these data on low stress values to better appreciate the deviation of elastic deformation from linearity in that range.

Figure 8

Bentonite suspension: (a) Deformation vs time during creep tests under different stress levels; (b) elastic modulus under low stress oscillations during creep tests; (c) shear stress vs induced deformation during recovery tests (stress release): elastic (reversible) component, plastic (irreversible) component, and total deformation. The dotted line is the equation $\tau =G_{\infty }\gamma$, with $G_{\infty }$ taken as the constant $G^{\prime }$ (680 Pa here) in small-oscillations tests. Note that for (a) the initial time is the time at which the stress is imposed, while for (b) it is the time just after preshear.

Figure 9

Laponite suspension: (a) deformation vs time during creep tests under different stress levels; the horizontal dashed line is a guide for the eye; (b) elastic modulus under low stress oscillations during creep tests; (c) shear stress vs induced deformation during recovery tests (stress release): elastic (reversible) component, plastic (irreversible) component, and total deformation. The dotted line is the equation $\tau =G_{\infty }\gamma$, with $G_{\infty }$ taken as the constant $G^{\prime }$ (135 Pa here) in small-oscillations tests. Note that for (a) the initial time is the time at which the stress is imposed, while for (b) it is the time just after preshear.

Figure 10

(a) Deformation vs time observed for creep tests on bentonite suspension at different stress levels. The horizontal dotted lines show the level of the plateau deformation retained as the measure of the elastoplastic deformation. (b) Deformation vs time observed with Carbopol gel during recovery tests, after creep tests under different stress levels. The horizontal dotted lines show the level of the plateau deformation retained as the measure of the residual (plastic) deformation.

Figure 11

Flow curve data for the different materials as deduced from steady-state flows during creep tests. The continuous lines are the Herschel-Bulkley models fitted to the different material data: emulsion ($k=7\mathrm{Pa}{\mathrm{s}}^n$, $n=0.38$), gel ($k=12\mathrm{Pa}{\mathrm{s}}^n$, $n=0.4$), bentonite ($k=0.027\mathrm{Pa}{\mathrm{s}}^n$, $n=1$), and laponite ($k=0.02\mathrm{Pa}{\mathrm{s}}^n$, $n=1$).