Hysteresis in modeling of poroelastic systems: Quasistatic equilibrium

The behavior of hysteretic, coupled elastic and fluid systems is modeled. The emphasis is on quasistatic equilibrium in response to prescribed chemical potential ($\mu$) protocols and prescribed stress ($\sigma$) protocols. Hysteresis arises in these models either from the presence of hysterons or from the presence of self-trapping internal fields. This latter mechanism is modeled in finite element calculations which serve to illustrate the creation of hysteresis in a range of circumstances that go from conventionally hysteretic systems, a sandstone, to systems like a wood fiber. An essential ingredient in the behavior of these systems, the interaction between the mechanical variables and the fluid variables, is accorded special attention. The proper venue for the exploration of these systems is $(\mu ,\sigma )$ space and appropriate $\mu$ protocols, $\sigma$ protocols, and combined $\mu -\sigma$ protocols.

Figure 1

System. A set of parallel pores of length $c$, areal density $n_A$ (on area $a\times b$) that have radii normally distributed around $\langle R\rangle =0.1\hspace{0.5em}\mu$m. A pore is in the empty state, $E$, when there is at most a thin fluid film on the surface. It is in the filled state, $F$, after capillary condensation. The pores are coupled by the strain on the absorbent that is caused by forces due to the fluid configurations.

Figure 2

Preisach space. The chemical potential pairs $({\mu }_F,{\mu }_E)$ are plotted as open circles. [The values of $({\mu }_F,{\mu }_E)$ are made dimensionless by scaling by ${\gamma }_{\mathit{LV}}/(n_L\langle R\rangle )$.] The upper black line is the diagonal ${\mu }_E={\mu }_F$ and the lower black line is at ${\mu }_E=2{\mu }_F$. For a pore of radius $R$ the chemical potential for $F\to E$ is at $-2{\gamma }_{\mathit{LV}}/(n_{L}R)\to -2\langle R\rangle /R$. The $({\mu }_F,{\mu }_E)$ pairs are above the line ${\mu }_E=2{\mu }_F$ because the instability that leads to the $E\to F$ transition is at finite film thickness, that is, at ${\mu }_F<-{\gamma }_{\mathit{LV}}/(n_{L}R)={\mu }_E/2$. For pores of large radius ${\mu }_F\to {\mu }_E/2$ and ${\mu }_F\to 0^{-}$. The inset in the lower right is the ${\mu }^{*}$ protocol, where the horizontal axis is “time.”

Figure 3

Moisture content as a function of relative humidity. For no interaction between pores, $\Lambda =0$ in Eq. (5), the isotherm (open circles) is controlled entirely by the $({\mu }_F,{\mu }_E)$ pairs. For $\Lambda =-0.025$ the interaction distorts the isotherm (solid circles), particularly at $u>0.8$ where the interaction is very strong. The approximately linear behavior of $u$ near $R_H\to 1$ is due to the change in pressure that occurs when all pores are full of fluid and $r\to +\infty$, where $r$ is the radius of curvature of liquid at the pore end. The curve without symbols is for the case that $({\mu }_F,{\mu }_E)$ shift with the strain as in Eq. (5) but the pore radii are taken as unchanged in the calculation of $u$.

Figure 4

Compressive strain. The moisture dependent term in the compressive strain, Eq. (5), is plotted as a function of ${\mu }^{*}$ for the two isotherms in Fig. 3, $\Lambda =0$ (open square at $0$) and $\Lambda =-0.025$ (solid circles for $\mu$ increase and open circles for $\mu$ decrease). The arrows show the direction of the chemical potential change along the curves. There is reversible change in strain with change in chemical potential along the double-tipped arrows, that is, no change in state. As is often the case the strain stays in. The inset is a schematic representation of the strain-$\mu$ result of Amberg and McIntosh [15, 16].

Figure 5

Elastic system. (a) The triangular elastic elements are elastically identical having elastic constants $(E,\nu )=(1.0,0.3)$. (b) Each elastic element carries a set of three internal forces that are along the line from triangle centroid to vertex and can at most change sign, $\eta =\pm 1$. (c) The system is tethered at $x=0$ on the left and in $y=0.5$ at the left center. It is driven from the right-hand side by a force protocol. The $x$ strain is defined as the change in the average $x$ displacement of the right-hand side. The $y$ strain is defined as the change in the average $y$ separation of the top and bottom.

Figure 6

$({\sigma }_o,{\sigma }_c)$ Preisach space. (Top) Preisach space with off-diagonal elements, ${\sigma }_c⩽{\sigma }_o$, used in the calculations leading to the results in Fig. 8 (left panels). (Bottom) Preisach space with diagonal elements only, ${\sigma }_c={\sigma }_o$, used in the calculations leading to the results in Fig. 8 (middle, right panels). (Inset) Schematic of $\mu$ protocol used in all calculations.

Figure 7

Forces on system. Each node experiences a net internal force which is the sum of the forces from the elements that touch it. (Left) The net internal forces are shown as lines pointing from the nodes (dots) at applied stress ${\sigma }_{\mathit{xx}}=-0.31$ (a compressive applied stress with most elastic elements in the $\eta =+1$ state). (Right) The net internal forces are shown as lines pointing from the nodes (dots) at applied stress ${\sigma }_{\mathit{xx}}=+0.30$ (a tensile applied stress with most elastic elements in the $\eta =-1$ state). Note, the internal forces on a node are in random directions and typically balanced (small) in the interior of the system in both cases. The forces are unbalanced primarily on the system surface where they point mostly inward (outward) on the left (right).

Figure 8

Strain vs applied stress. For all panel sets (top) is the $x$ strain as a function of stress and (bottom) is the $y$ strain as a function of stress. (Left two panels) The strength of the internal forces is nonzero and the Preisach space is off-diagonal. (Center two panels) The strength of the internal forces is zero and the Preisach space is diagonal. (Right two panels) The strength of the internal forces is nonzero and Preisach space is diagonal. The stress protocol in all cases is that shown in the inset in Fig. 6 (bottom) and in all cases $\Lambda =0.035$.

Figure 9

Effect of internal forces. The $x$ strain as a function of stress for three values of the internal force, $\Lambda =0$ (asterisks), $\Lambda =0.1$ (open squares), and $\Lambda =0.2$ (solid circles). The Preisach space is diagonal. The elastic elements respond to the internal stress. The hysteresis loops are sharp because of the strength of the internal forces (compare to Fig. 8). If this circumstance obtained in physical realizations the sharp step, a mechanical avalanche, would be associated with a burst of acoustic emission.

Figure 10

End point memory. The $x$ strain as a function of stress. For this case the interaction was taken to be very strong so that the transition of the internal forces from closed (compressed) to open (in tension) is very abrupt (compare to Figs. 8 and 9).

Figure 11

$({\mu }_F,{\mu }_E)$ Preisach space. The pairs $({\mu }_F,{\mu }_E={\mu }_F)$ used for the studies in Figs. 12 to 17. (Inset) Schematic of the $\mu$ protocol used in Figs. 12 to 16.

Figure 12

Moisture content vs chemical potential. The moisture content, $u$, as a function $\mu$ for the chemical potential protocol in the inset of Fig. 11 [$\mu$ increase (solid circles) and $\mu$ decrease (open circles)]. Compare to the strain for the same $\mu$ protocol in Fig. 13. The moisture content stays in.

Figure 13

Strain vs chemical potential. The $x$ strain as a function $\mu$ for the chemical potential protocol in the inset of Fig. 11 [$\mu$ increase (solid circles) and $\mu$ decrease (open circles)]. Compare to the moisture content for the same $\mu$ protocol in Fig. 12.

Figure 14

$u$-$\epsilon$ hysteresis. The moisture content as a function of the strain for the chemical potential protocol in the inset of Fig. 11, $\mu$ increase (solid circles) and $\mu$ decrease (open circles). See Figs. 12 and 13.

Figure 15

Congruence test. The moisture content as a function of $\mu$ for a modification of the chemical potential protocol in the inset of Fig. 11 to test for congruence. The modification involves two interior loops, $\mu =0\to -{0.12}^{+}\to 0$ on $\mu$ increase and $\mu =-{0.12}^{+}\to 0\to -{0.12}^{+}$ on $\mu$ decrease. The part of the $u-\mu$ curve not shown overlays that in Fig. 12. There is qualitatively similar lack of congruence in the corresponding strain-$\mu$ curve.

Figure 16

Applied stress. The moisture content as a function of $\mu$ for three repetitions of the chemical potential protocol in the inset of Fig. 11 at successively larger applied stress, ${\sigma }_{\mathit{xx}}=-0.0025,0.0,+0.0025$ (solid circles, open circles, and asterisks, respectively).

Figure 17

End point memory $(\sigma ,\mu )$ protocol. (Top) The chemical potential as a function of time. (Bottom) The applied stress as a function of time. The chemical potential-stress protocol from $t=50$ to $t=200$ is a closed loop in $(\mu ,\sigma )$ space. The strain response and moisture response to this protocol are shown in Fig. 17.

Figure 18

End point memory. The $x$ strain (left) and moisture content (right) as a function of $\mu$ for the $(\mu ,\sigma )$ protocol in Fig. 18. A closed loop in $(\mu ,{\sigma }_{\mathit{xx}})$ begins at the feathers of the right arrow (the end of the arrow opposite the tip) and returns to these feathers ($50\to 200$ in Fig. 18).