Foam rheology at large deformation

Large deformations are prone to cause irreversible changes in materials structure, generally leading to either material hardening or softening. Aqueous foam is a metastable disordered structure of densely packed gas bubbles. We report on the mechanical response of a foam layer subjected to quasistatic periodic shear at large amplitude. We observe that, upon increasing shear, the shear stress follows a universal curve that is nearly exponential and tends to an asymptotic stress value interpreted as the critical yield stress at which the foam structure is completely remodeled. Relevant trends of the foam mechanical response to cycling are mathematically reproduced through a simple law accounting for the amount of plastic deformation upon increasing stress. This view provides a natural interpretation to stress hardening in foams, demonstrating that plastic effects are present in this material even for minute deformation.

Figure 1

(a) Sketch of the experimental setup: A thin layer of foam is deposited onto the horizontal bottom plate of the rheometer. The upper plate of the rheometer is periodically displaced along the $x$ axis by means of a motorized translation stage. Strain gauge provides the shear force. Layer thickness is fixed to $8\mathrm{mm}$. (b) Typical shear stress $\sigma$ vs shear strain $u$ for cycles of maximum strain, $u_m=0.375$. Loading starts at point A. Upon cycling, the stress tends to a limiting cycle between points B and C where the shear direction is reversed. Arrows show the cycling direction (velocity of the upper plate: grey, 0.3 mm/s; black, 0.5 mm/s).

Figure 2

(a) Shear cycles for increasing amplitude $u_m$. Starting at $\sigma =0$ and $u=0$, loading curves (A to B in Fig. 1) follow a universal nearly exponential law, underlined by the thick, light gray line. This line is the fit to the data using Eq. (1) with ${\sigma }_{\infty }=37$ Pa and $G=310$ Pa, the line thickness indicating fit errors (black: $u_m=0.0625$; dark gray: $u_m=0.375$; gray: $u_m=0.625$). (b) Evolution of shear stress at point C (Fig. 1) for the limit cycle as function of $u_m$. The solid line is the interpolation to Eq. (3) with ${\sigma }_{\infty }=39$ Pa and $G=225$ Pa. (c) Evolution of the stress at point C with number of cycles for $u_m=0.0625$. The dashed line represents the exponential dependence as predicted by Eq. (2) without adjustable parameter. The values ${\sigma }_{\infty }=39$ Pa and $G=225$ Pa are taken from the previous fit.