Competition between slicing and buckling underlies the erratic nature of paper cuts (2024)

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  Figure 1

  The physics of paper cuts. (a) Paper cuts are a common injury that can occur if skin contacts a sheet of paper (thickness $t$, area density $\rho$). It causes significant pain and discomfort and is often associated with a slicing motion (arrow). The underlying physical processes remain poorly understood. (b) However, it is well established that cuts frequently occur in the thickness range of $t=0.05\text{−}0.1\mathrm{mm}$ (this range includes magazines and office paper) [5], whereas thinner and thicker paper is relatively safe. We propose that a competition between slicing and buckling determines which paper types can cut. If the sheet is too thin, it buckles and loses structural integrity before initiating a fracture. In contrast, thicker sheets smoothly indent the surface and distribute the load over a greater area. The slicing motion enhances the likelihood of cutting, which peaks at the most hazardous thickness $t_h\approx 65\mu \text{m}$ (see Fig.2).

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  Figure 2

  Experimental setup and data classification scheme. (a) Schematic of the experiment used to quantify contact processes between a soft solid and a sheet of paper in relative motion. The standardized paper and gelatin samples were held by 3D-printed clamps. The vertical indentation depth, $d$, the speed (arrow), and the slicing angle $\varphi$ of the paper sheet were controlled using a micromanipulator. A video of each experiment was recorded and the stresses [normal (${\sigma }_n$) and tangential (${\sigma }_t$) to the gelatin surface] were measured using two load cells. (b–d) Representative data illustrating three regimes. (b) Thin paper ($t=30\mu \mathrm{m}$, $\varphi ={15}^{\circ })$ buckles because the normal load exceeds the buckling threshold ${\sigma }_b$ before reaching peak applied stress, ${\sigma }_a>{\sigma }_b$. (c) Intermediate paper ($t=65\mu \mathrm{m}$, $\varphi ={15}^{\circ })$ cuts because the cutting threshold ${\sigma }_{n,c}$ is exceeded before reaching ${\sigma }_b$ or ${\sigma }_a$. (d) Finally, thick paper ($t=220\mu \mathrm{m}$, $\varphi ={15}^{\circ })$ indents the surface because the dispersed normal force is insufficient to breach the surface or buckle the paper. (See also Supplemental Video S1 [14] and additional details in the text.)

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  Figure 3

  A competition between slicing and buckling governs paper cuts. The phase diagram shows the outcome of each experiment (dots) as a function of thickness $t$ and slicing angle $\varphi$. The outcome of a cutting attempt depends on how the thresholds relate to each other. If $\varphi <{\varphi }_h$, then there exists a range of thicknesses for which ${\sigma }_{n,c}$ is lower than both ${\sigma }_a$ and ${\sigma }_b$ where cutting is observed. When the slicing angle $\varphi$ is sufficiently small, nearly all types of paper cuts (red shaded domain, label: cutting). However, the probability peaks at the most hazardous thickness $t_h\approx 65\mu \text{m}$ (between printed magazines and office paper) and angle ${\varphi }_h\approx {20}^{\circ }$. Outside this zone, the peak applied stress either exceeds the buckling limit (label: buckling) or simply causes an indentation (label: indentation) (top right). The mechanical model [Eqs.(1, 2, 3, 4)] is consistent with observations (solid lines mark model transitions between domains). Error bars: $t\pm 5\mu \mathrm{m}$ and $\varphi \pm 2^{\circ }$. See additional details in the text.

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  Figure 4

  The Papermachete uses discarded traction sectionsof dot-matrix paper as a blade. (a) Technical drawing and (b) photograph of the recyclable paper-knife. The single-use paper blade is fixed in the clip by magnets while the handle facilitates convenient use. (c) The Papermachete can cut into a variety of plant- and animal-based products. The cuts were performed by hand at the slicing angle of $\varphi \approx {10}^{\circ }$ at speeds of approximately 1cm/s in the direction of the arrow.

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  Figure 5

  Definitions of surface characteristics after the experiment. The horizontal line is the edge and is included to demonstrate the cut into the sample.

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  Figure 6

  Measured maximum applied stress ${\sigma }_a$ plotted as a function of relative indentation depth $d/t$ (dots). An unweighted least-squares to Eq.(2) yields the estimate $E_s=3.13\pm 0.08$ kPa (solid line).

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  Figure 7

  Experimentally measured buckling force compared to Euler's beam equation, which stipulates $F_b\sim E_p{t}^{3}w/h^2$ . Here, $E_p$ is the paper's elastic modulus, $t$ and $w$ its thickness and width, and $h$ the free height of the sheet (Fig.1 in the text). Data compare reasonably well with the simple model.

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  Figure 8

  The normal stress ${\sigma }_{n,c}$ required for the onset of cutting diminishes with slicing angle $\varphi$. Our experimental data (blue) as well as data from Reyssat etal. [11] are not inconsistent with a linear fit according to Eq.(7): (solid line) derived from the numerical model proposed [11]. The fitted slope is the material parameter ${\sigma }_{t,c}=0.84\pm 0.06$ MPa.

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  Figure 9

  Phase diagram constructed for paper height $h=28\mathrm{mm}$ (i.e., twice the height used to construct the phase diagram in Fig.3). Peaks at the most hazardous thickness $t_h\approx 105\mu \text{m}\mathrm{m}$ (between printed magazines and office paper) and angle ${\varphi }_h\approx {12}^{\circ }$.

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