Elastobuoyant Heavy Spheres: A Unique Way to Study Nonlinear Elasticity

Large deformations in soft elastic materials are ubiquitous, yet systematic studies and methods to understand the mechanics of such huge strains are lacking. Here, we investigate this complex problem systematically with a simple experiment: by introducing a heavy bead of radius $a$ in an incompressible soft elastic medium. We find a scaling law for the penetration depth ($\delta$) of the bead inside the softest gels as $\delta \sim a^{3/2}$, which is vindicated by an original asymptotic analytic model developed in this article. This model demonstrates that the observed relationship is precisely at the demarcating boundary of what would be required for the field variables to either diverge or converge. This correspondence between a unique mathematical prediction and the experimental observation ushers in new insights into the behavior of the deformations of strongly nonlinear materials.

Popular Summary

In both laboratory experiments and nature, it is common for a soft material to deform when a stress is applied to it. In particular, extra-large deformations in soft elastic materials are ubiquitous—for example, in polymeric gels, adhesives, or biological tissues—yet systematic studies and methods to understand the mechanics of such huge strains are distinctly lacking. Here, we investigate this complex problem using a simple experiment.

We gently deposit a rigid sphere on the surface of a soft elastic medium (a polymer gel). The sphere reaches its equilibrium (i.e., elastobuoyant) position inside the gel, determined by the balance of its weight and the gel’s elastic forces, almost instantaneously. We study the elastobuoyant depths of steel spheres systematically by varying their sizes and placing them in gels of different elastic moduli. After each measurement, we remove the sphere from the gel using a magnet. By measuring the depth of submersion ($\delta$)—from the top surface of the gel to the bottom of the steel sphere—we experimentally determine a scaling relationship between this depth and the size of the sphere (radius $a$) in the large deformation limit: $\delta \sim a^{3/2}$. We find that the experimentally observed scaling law for the penetration depth is the supported asymptotic analytic model, first proposed here.

We expect that our findings will be useful in cases where large deformations of a material are observed, such as delicate surgeries in soft tissues.

Figure 1

Snapshots and schematics of the experiment. (Top panels) Side views of two transparent cells filled with a polyacrylamide gel with shear modulus 1160 Pa (a) and 13 Pa (b). Two identical steel beads (5 mm diameter) have been placed on the free air-gel interface. The vertical downshifts are, respectively, $\delta =0.03{\delta }_0$ and $\delta =320{\delta }_0$. (Bottom panels) Schematic representation of the deformation fields in the respective experiments.

Figure 2

Experimental evidence of reversibility of the gels. (a) A 10-mm-diameter steel sphere immersed in a 140-Pa gel, when slightly disturbed from its elastobuoyant position via an electromagnet, undergoes underdamped oscillations about its equilibrium depth ${\delta }_{\mathrm{eq}}$. (b) Depth of an elastobuoyant bead (2.8 mm diameter) in a 13-Pa gel is plotted as a function of the strength of an external vertical magnetic force. The depth at zero force indicates the equilibrium-elastobuoyant position of the sphere. The data (red, blue, and green) shown here are from three different experiments, where the closed symbols indicate the loading cycles and the open symbols indicate the unloading cycles. (c) The depth of submersion $\delta$ of a 5-mm-diameter steel sphere in a soft gel ($\mu \sim 13\mathrm{Pa}$) is shown, varying as a function of its temperature. The experiments in the cooling cycle were performed first, after which the gel was heated systematically to obtain the data for the heating cycle.

Figure 3

Dimensionless depth of spheres ($\delta /{\delta }_0$) plotted as a function of its dimensionless radius $a/{\delta }_0$, for various shear moduli of the gels from $\mu =13\mathrm{Pa}$ to 2930 Pa. The grey lines indicate power-law curves [$\delta /{\delta }_0\sim (a/{\delta }_0{)}^{\alpha }$] with $\alpha =2$ and $\alpha =1.5$ for the two asymptotic limits for the normalized data. (Upper inset) Depths ($\delta$) versus radii ($a$) for all the spheres. The plot area is divided into two domains, the boundary indicating $\delta =2a$. The data points above the boundary indicate that the spheres were entirely below the gel’s surface. (Lower inset) Best fit [$\delta /{\delta }_0=k(a/{\delta }_0{)}^{\alpha }$] for the softest gel (13 Pa) highlighted (see text).

Figure 4

Sketches corresponding to the steps of the calculation of the elastic energy. (a) Reference state (no deformation). (b) A point load applied at the free surface. The displacement at the application point is $\delta$. (c) A sphere of radius $a$ indents the free surface over the distance $\delta$. (d) Mapping from the reference state (solid lines, with the bead at the surface) to the deformed state. A point of the gel in the rest state is located with coordinates $(r,z)$. Note that $\tilde{r}$ is the distance from the initial contact point of the bead. In the deformed state, the point that was at ($r$, $z$) is located at $R(r,z)$, $Z(r,z)$.

Figure 5

Sketch of the (dimensionless) energy density profiles ${\mathcal{W}}_0$ (e.g., dotted line) and $\mathcal{W}$ (e.g., solid line) as a function of the distance $\tilde{r}$ to the initial contact point. The dashed horizontal double arrows highlight two domains, corresponding to (i) $\tilde{r}\gg a$, where the (dimensionless) elastic energy densities $\mathcal{W}$ and ${\mathcal{W}}_0$ are similar, and (ii) $\tilde{r}\ll \delta$, where they are different.

Figure 6

Strain energy density function given by the incompressible Gent material model $W=-\frac{1}{2}\mu J_m\log \{1-[(I_1-3)/J_m]\}$ [21], with $J_m=97$, plotted in log-log scales (black solid line). Note that $J_m+3$ is the limiting value of the first invariant $I_1$. The strain energy density function for the neo-Hookean material is plotted with a dashed line. The two gray straight lines indicate slopes 1 and 2, i.e., values of the local exponent ${\alpha }_1$ equal to 1 and 2. Here, $I_{10}^{*}$ is the value of the first invariant beyond which the local exponent is larger than $3/2$.

Figure 7

(a) Snapshots of the experiment of loading a needle (diameter $D$ and grafted with about 5-nm layer of polydimethylsiloxane chains) inside the gel with a thin layer of silicone oil (AR20, Sigma Aldrich) on its surface. The oil above the gel reduces friction by lubricating the contact between the needle and the gel during indentation. The red dye demarcates the interface between the gel and silicone oil. After unloading the needle from the gel, the dyed interface retracts to its original position, showing that there was no fracture in the gel. (b) For these experiments of loading a sharp needle (inset) inside the gel ($\mu =478\mathrm{Pa}$), the nondimensional force $P/\mu D^2$ varies almost as the square of the nondimensional displacement $\delta /D$. An uncertainty analysis yields the value of the power of $\delta /D$ as $1.99\pm 0.03$. (c) When the same needle is indented in the gel while undergoing vertical vibrations, a fine fracture is induced by the indentation. The needle undergoes square-wave oscillations (amplitude of vibration of about 0.64 mm) along its axial direction, which were generated by a waveform generator (Agilent, Model 33120A), connected to a mechanical oscillator (Pasco Scientific, Model No: SF-9324) via an amplifier (Sherwood, Model No: RX-4105).