Slicing Softly with Shear

A soft solid is more easily sliced using a combination of normal and shearing deformations rather than diced by squeezing down on it normally with the same knife. To explain why this is so, we experimentally probe the slicing and dicing of a soft agar gel with a wire, and complement this with theory and numerical simulations of cutting of a highly deformable solid. We find that purely normal deformations lead to global deformations of the soft solid, so that the blade has to penetrate deeply into the sample, well beyond the linear regime, to reach the relatively large critical stress to nucleate fracture. In contrast, a slicing motion leads to fracture nucleation with minimal deformation of the bulk and thus a much lower barrier. This transition between global and local deformations in soft solids as a function of the angle of shear explains the mechanics of the paper cut and design of guillotine blades.

Figure 1

(a) Schematic of cutting experiment. A wire of radius $R\approx 150\mu \mathrm{m}$ is used to cut through a piece of soft gel ($\mu \approx 35\mathrm{kPa}$) with a velocity $V$. The thickness of the sample is $h\approx 5\mathrm{mm}$. Shear is controlled via the kinematic angle $\theta$ between the direction of penetration by the wire and the normal to the surface of the gel: the normal velocity is kept constant ($V_n=0.2\mathrm{mm}/\mathrm{s}$), while the tangential component $V_t$ varies as $\theta$ changes. (b) The gel surface deforms strongly before the initiation of penetration, and then rebounds partially; these phases correspond to the nucleation and growth phases of the fracture for purely normal indentation (top) and increased shearing motion (bottom). We see that shearing reduces the maximum deformation of the gel before fracture is nucleated. (c) The normal force $F_n$ as a function of penetration depth for various angles of cutting $\theta$ characterizing the relative rate of shearing. (d) The tangential shear force $F_t$ as a function of penetration depth for various angles of cutting $\theta$. (e) The peak normal force $F_n^m$ as a function of the shear angle $\theta$ for a fixed normal velocity drops by a factor of about 5 as the shear angle increases from 0° to 75°. (f) The peak shear force $F_t^m$ as a function of the shear angle $\theta$ for the same normal displacement rate shows a shallow maximum but at a value that is nearly an order of magnitude smaller than the normal force.

Figure 2

Numerical simulations of periodic slabs with shear modulus $\mu$ that are squeezed or sheared by a wire of radius $R$. In (a) the slab is squeezed normally by a force $f_n=15\mu R$ per unit length. In (b) $f_n=7\mu R$ and the slab is, in addition, sheared tangentially by $f_t=f_n/2$. Color indicates the magnitude of the largest principal (tensile) stress ${\sigma }_1$. The insets show tensile stress at the surface of the slab, for various $f_n$, as a function of distance $r$ from the middle line in the undeformed configuration.

Figure 3

Simulated slicing of a thick ($h=30R$) three-dimensional slab. (a) A snapshot with slicing angle $\theta ={tan}^{-1}(V_t/V_n)=45°$. Colors indicate tensile stress at the surface ($\mathrm{\text{red}}=\mathrm{\text{high}}$). (b),(c) $F_n$ and $F_t$ as a function of indentation depth $d$ of the wire for $\theta =0°$, 60°, and 75°. The plots extend to the point where the extreme tensile stress at the surface reaches ${\sigma }^{*}=2\mu$. (d),(e) The required $F_n$ and $F_t$ to reach ${\sigma }^{*}$ as a function of $\theta$ (boxes). Scaled experimental data from Fig. 1 is shown for comparison (dots).