One-dimensional compression of a saturated elastoviscoplastic medium

A theoretical and experimental study is presented of the one-dimensional compression of a networked suspension. Particular attention is given to relatively rapid compression where we extend previous works by including an elastoviscoplastic constitutive relation. Solutions of a one-dimensional model are presented, and asymptotic limits explored, for compressions controlling either displacement or load. The results are compared to complementary laboratory experiments using cellulose fiber suspensions, with the material functions appearing in the model calibrated by independent experiments. Measurements of load and local solid velocity as a function of displacement during compression and unloading gauge the importance of elastic effects. The comparison between experiment and theory is satisfying, demonstrating a dramatic improvement over existing inelastic constitutive models in reproducing the observed differential spatial compaction.

Figure 1

Sketches showing the constitution of each material element in our model elastoviscoplastic medium (left), and the four simplest types of quasistatic compression dynamics (right; (a)–(d). On the right, the light blue shading indicates when the material has yielded, the black (solid or dashed) lines show $P_{{}_Y}\left(\varphi \right)$, and the red (solid or dashed) lines indicate $\widehat{F}\left(\varphi \right)$. Quasistatic compression follows either $P_{{}_Y}\left(\varphi \right)$ or $\widehat{F}\left(\varphi \right)$ depending on their relative magnitudes; the sections of these curves that are followed are drawn by solid lines (and are dashed otherwise). The gray-shaded rectangles illustrate the response when compression is interrupted by a cycle of limited unloading and reloading. If this response is elastic, but begins from a point on the compressive yield stress curve, the unloading-reloading follows a translated elastic curve $\widehat{F}\left(\varphi \right)+\text{const}$ indicated by the red dotted lines.

Figure 2

Compression curves interrupted by limited unloading-reloading cycles for a hydrogel suspension. The short red lines indicate local fits to each unloading-reloading section. The insets show the original piston position vs load data, and the values for $\mathcal{E}\left(\varphi \right)$ extracted from the loading-reloading fits.

Figure 3

Compression curves interrupted by limited unloading-reloading cycles for a suspension of cellulose fibers. Two tests are shown, conducted at compression rates of $1\mu \mathrm{m}/\mathrm{s}$; in the first test, the unloading rate is the same, but in the second it is increased to $10\mu \mathrm{m}/\mathrm{s}$. The red line shows the pure compression test calibrating $P_{{}_Y}\left(\varphi \right)$. The inset shows a detail of one of the cycles, together with the fit giving $\mathcal{E}\left(\varphi \right)$.

Figure 4

Compression curves interrupted by limited unloading-reloading cycles for the model Eq. (1) with $(\mathcal{E},\widehat{\mathrm{\Lambda }},P_{{}_Y})=({\mathcal{E}}_{*},{\eta }_{*},p_{*}){\varphi }^2$. In (a) scaled stress $\mathcal{P}=\widehat{\mathcal{P}}/p_{*}$ (blue curve) is plotted for $\epsilon /\gamma \equiv {\eta }_{*}U/\left(h_0{p}_{*}\right)=10$ and $\epsilon /\left(\gamma \lambda \right)\equiv {\mathcal{E}}_{*}/p_{*}=4$, where $U/h_0$ is the compression rate (see Sec. 3c) The black dashed and dotted lines display ${\mathrm{\Pi }}_{{}_Y}=P_{{}_Y}/p_{*}$ and the translated $F=\widehat{F}/p_{*}$ curves. For (b) we show solutions in which the compression rate is increased so that $\epsilon /\gamma \equiv {\eta }_{*}U/\left(h_0{p}_{*}\right)={10}^{(j-2)/2}$ with $j=0$, 1, $...$, 4 (from blue to red). The insets show details of one of the cycles [with the different solutions successively offset horizontally by 0.04 in (b) for clarity].

Figure 5

A sketch of the two-phase model geometry.

Figure 6

Interrupted compression curves for differential compaction with $\epsilon ={10}^{-2}, \gamma \lambda /\epsilon =0.316, a=3, n=m=2$ and the values of $\gamma$ indicated. Panel (a) shows the prescribed piston position $h\left(t\right)$, along with a density plot of ${log}_{10}\varphi$ for $\gamma =0.1$; red contours indicate the yield surfaces where $\mathcal{P}={\mathrm{\Pi }}_{{}_Y}\left(\varphi \right)$. Two compression curves are shown in (b) (offset from one another); the black dotted lines show the corresponding quasistatic solution. Details of the loops arising during one unloading-reloading cycle are shown in (c), with $\gamma$ ranging from 0.1 (red) to 10 (blue), offset successively (by factors of 1.2 and 1.1 in the horizontal and vertical, respectively) from the low-$\gamma$ (red) loop for clarity.

Figure 7

Differential compaction for $\epsilon =\gamma ={10}^{-2}, a=3, n=m=2$ using a constant piston speed up to $t=\frac{1}{2}$. Shown are density plots of velocity $u(z,t)$ and solid fraction $\varphi (z,t)$ for (a) $\lambda =0.01$ and (b) $\lambda =0.316$. The red lines are the yield surfaces, and the black dashed lines show the contours $u=2^j\times {10}^{-3}$ for $j=1,...,4$ to illustrate finer details not visible on this color scale. Below we plot time series of (c) the mean, top, and bottom solid fractions and (d) piston load, for solutions with $\lambda ={10}^{(j-6)/2}$ for $j=0,...,5$ (from blue to red).

Figure 8

A similar plot to Fig. 7, but with a parabolic piston motion for $t<1$.

Figure 9

Parabolic compression solutions with $\epsilon =1, \gamma =0.01$ and $a=n=m=2$. Time series of the bottom and top solid fractions, $\varphi (0,t)$ and $\varphi (h,t)$, are plotted in (a) for $\lambda =0.01$, 0.1, 1, 4, 12.5, and 25 (blue to red). Also shown are the mean solid fraction $\overline{\varphi }\left(t\right)$ (dashed) and the prediction in Eq. (36) (red dots). For the solution with $\lambda =25$, we present (b) density plots of $u(z,t)$ and $\varphi (z,t)$, and the profiles of (c) $\varphi (z,t)$ and (d) $\mathcal{P}(z,t)$ at the times indicated by the dotted lines in (b). The (red) dots in (c) show the prediction from Eq. (33), based on the value of ${\varphi }_T$ from the numerical solution, and the (red) dashed lines in (d) show the translated elastic stress function $F=\epsilon ({\varphi }^m-{\varphi }_0^m)/\left(m\lambda \gamma \right)+{\varphi }_0^n$.

Figure 10

Fixed-load compression ($\gamma =1$) with $a=n=m=2$ and $R=25{\mathrm{\Pi }}_{{}_Y}\left({\varphi }_0\right)$. In (a)–(c) ${\epsilon }^{-1}\lambda =\frac{1}{4}$ and $\epsilon ={10}^j$ for $j=-4, -3, ...$, 2 (blue to red). Shown are (a) $h\left(t\right)$, (b) spatial profiles of $\varphi (z,t)$ when $h\approx 0.69$, and (c) ${\varphi }_T\left(t\right)=\varphi (h,t)$ and $\varphi (0,t)$. The horizontal lines in (a) and (c) show $({\varphi }_E,h_E)$ (dotted) and $(h_{\infty },{\varphi }_{\infty })$ (dot-dashed); the corresponding curves show Eq. (37) for the elastic and plastic limits. Also shown in (b) (shifted to the right for clarity) are predictions from Eq. (33), based on the values of ${\varphi }_T$ from the numerical solutions. Panel (d) shows a similar plot to (c), but for solutions with $\epsilon =0.01$ and $4{\epsilon }^{-1}\lambda =1, \frac{1}{3}, {10}^{-1}, {10}^{-2}$, and ${10}^{-3}$ (from red to blue). The inset shows $h\left(t\right)$.

Figure 11

Fitted constitutive functions for the pulp suspension, showing (a) $P_{{}_Y}\left(\varphi \right)$ and $\mathcal{E}\left(\varphi \right)$ and (b) permeability. The symbols show the actual experimental measurements; the solid red lines are the fits. The dashed red line in (a) shows the fit for $P_{{}_Y}$ multiplied by a factor of 20. The lighter blue symbols denote results obtained from the low-load tests reported by [18].

Figure 12

Compressions for cellulose fibers performed at varying initial velocities ($U=5,15,...,55$ mm/s) with the same initial and final heights. Panel (a) shows experimental results. Each test is conducted four times; the points show the average and the error bars band a standard deviation. Panel (b) shows the corresponding results for the theoretical model with (solid) and without (dotted) elasticity. The vertical dashed line indicates when the piston stops, and the black line shows the load assuming that compression arises quasistatically along the $P_{{}_Y}$ curve.

Figure 13

Solid velocities, $u(z,t)=\widehat{u}/U$, as densities on the ($t,z)\mathrm{plane}$ for $U=1.5$, 10, 30, and 55 mm/s (from left to right). For each column, particle tracking measurements are shown on the top, the predictions of the elastoviscoplastic theory in the middle, and those of the viscoplastic model on the bottom. The vertical dashed lines show where the piston stopped, and the red contours in the theoretical plots show the yield surfaces. The PTV data include the velocities computed for the piston.