How Supertough Gels Break

Itamar Kolvin, John M. Kolinski, Jian Ping Gong, and Jay Fineberg

Phys. Rev. Lett. 121, 135501 – Published 26 September 2018

Abstract

Fracture of highly stretched materials challenges our view of how things break. We directly visualize rupture of tough double-network gels at >50% strain. During fracture, crack tip shapes obey a $x \sim y^{1.6}$ power law, in contrast to the parabolic profile observed in low-strain cracks. A new length scale $ℓ$ emerges from the power law; we show that $ℓ$ scales directly with the stored elastic energy and diverges when the crack velocity approaches the shear wave speed. Our results show that double-network gels undergo brittle fracture and provide a testing ground for large-strain fracture mechanics.

Received 9 May 2018
Revised 24 July 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.135501

Physics Subject Headings (PhySH)

Continuum mechanics Elastic deformation Elastic modulus Fracture

Elastomers Front propagation Polymer gels

Polymers & Soft MatterNonlinear DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Itamar Kolvin^1,2, John M. Kolinski^1,3, Jian Ping Gong⁴, and Jay Fineberg¹

¹The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
²UC Santa Barbara, Santa Barbara, California, 93106, USA
³École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
⁴Faculty of Advanced Life Science and Soft Matter GI-CoRE, Hokkaido University, Sapporo, 001-0021, Japan

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Vol. 121, Iss. 13 — 28 September 2018

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Images

Figure 1
(a) A $T = 1.3 − mm$ -thick rectangular sheet of DN gel is uniformly loaded by two rigid grips. A load cell monitors the tension $F$ in the sheet. (b) The nominal tensile stress versus the stretch along the $y$ axis for uniformly loaded samples prior to fracture. Colors represent the values of $L$ used. Inset: A schematic stress-stretch loop depicting the loading and unloading cycle. Shaded areas define the work dissipated in a cycle $U_{hys}$ and the stored elastic energy $U_{el}$ following the loading step. (c) The work density due to stretching $W_{stretch}$ as a function of the maximum stretch $λ_{\max}$ . $W_{stretch}$ was obtained by integrating the area under the average of the curves in (b). These measurements (red line) compare well with previous studies [9, 10]. Red shading denotes standard deviations. (d) The stored elastic energy $U_{el} = W_{stretch} - U_{hys}$ computed from (c) using published values of $U_{hys} / W$ [10].
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Figure 2
(a) Virgin gel samples are first uniformly stretched. A crack is then inserted at the middle of the sample’s $y$ edge. Propagating cracks are imaged via a fast camera as shown. (b) Image sequence of an accelerating crack and (c) the CTODs extracted from each image and translated so that the crack tip is at the origin of axes. Inset: The velocity trace, where the colored symbols denote the frames (i)–(iii) in (b). (d) CTODs extracted from each image in (b) follow a $\sim 1.6$ power law.
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Figure 3
The prefactor $a$ extracted from power-law fits $x = a y^{1.64}$ over the range $| y | < 3 mm$ to CTODs for steadily propagating cracks (blue points are averages over >10 images at a given $v$ , error bars show standard deviations), accelerating cracks (green circles, blue squares, orange triangles—each set of points represents a separate experiment; each point corresponds to a single image). A linear regression over all $a$ values intersects the $v$ axis at $v ≃ 15 m / s$ (red dashed line). Inset: Averages and standard deviations (error bars) of the exponent $b$ when fitting CTODs of steady propagating cracks to $x = a y^{b}$ . Each point corresponds to a single experiment. The dashed red line lies at the average $b = 1.64 \pm 0.08$ over all experiments.
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Figure 4
In situ determination of the longitudinal wave velocity along the $x$ axis $c_{p}^{x}$ . (a) At the initiation of fracture an elastic compressive wave propagates along the $x$ direction. DIC analysis of the region highlighted in red (top) is presented in three consecutive snapshots which yield a pulselike $E_{x x}$ disturbance traveling at a constant velocity $c_{p}^{x}$ (bottom panels) (the time interval between frames is 0.125 msec). (b) The variation of measured $c_{p}^{x}$ with the background stretch for different sample geometries (symbols).
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Figure 5
Dissipation of energy in the dynamic fracture of DN gels. (a) The energetic measure, $G_{ℓ} = \frac{1}{2} ρ c_{p}^{2} ℓ$ , approximates the stored elastic energy per unit area $U_{el} L$ , which is an upper bound for the energy dissipated in the fracture process. $U_{el} L$ is estimated from the quasistatic loading data in Fig. 1. (b) Growth of elastic energy dissipated in fracture with crack velocity.
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