Dynamics and Properties of the Cohesive Zone in Rapid Fracture and Friction

Neri Berman, Gil Cohen, and Jay Fineberg

Phys. Rev. Lett. 125, 125503 – Published 17 September 2020

Abstract

The cohesive zone is the elusive region in which material fracture takes place. Here, the putatively singular stresses at a crack’s tip are regularized. We present experiments, performed on PMMA, in which we visualize the cohesive zone of frictional ruptures as they propagate. Identical to shear cracks, these ruptures range from slow velocities to nearly the limiting speeds of cracks. We reveal that the cohesive zone is a dynamic quantity; its spatial form undergoes a sharp transition between distinct phases at a critical velocity. The structure of these phases provides an important window into material properties under the extreme conditions that occur during fracture.

Received 18 March 2020
Revised 28 June 2020
Accepted 12 August 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.125503

Physics Subject Headings (PhySH)

Continuum mechanics Fracture Friction

Front propagation

Condensed Matter, Materials & Applied PhysicsNonlinear DynamicsPolymers & Soft Matter

Authors & Affiliations

Neri Berman, Gil Cohen, and Jay Fineberg^*

The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

^*jay@mail.huji.ac.il

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Issue

Vol. 125, Iss. 12 — 18 September 2020

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Images

Figure 1
Experimental system and data acquisition. (a) Schematic of the experimental system. Normal $F_{N}$ and shear $F_{S}$ forces are applied to PMMA blocks. Rosette-type strain gauges, mounted 3.5 mm above the interface, enable instantaneous strain measurements. Local material displacements on the interface are measured via grooves engraved on the upper block’s contacting surface. (b) Groove edge locations are imaged by total internal reflection [1] each $0.8 μ \sec$ [20]. (c) Separate light sources simultaneously image the real area of contact, $A (x, t)$ of the (i) groove edge displacements, $u_{x} (t)$ , and (ii) frictional crack dynamics over the entire 150 mm interface every $1.6 μ \sec$ . Each optical path is imaged by a separate fast camera producing simultaneous measurements of $u_{x} (t, y = 0)$ and $C_{f}$ . (d) Typical measurements of $u_{x} (t)$ when averaged across the $z$ axis. The bright (dark) borders trace $u_{x} (t)$ motion (blue lines) of each edge. (e) $A (x, t)$ , averaged in $z$ and normalized by $A_{0} = A (x, t_{0})$ , where $t_{0}$ is prior to crack initiation. Colors reflect $A (x, t) / A_{0}$ . $C_{f}$ are extracted from crack tip locations, $x_{tip}$ . Red box: grooved surface location.
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Figure 2
Displacement measurements, $u_{x}$ , on the interface for $C_{f} = 707 \frac{m}{s}$ (0.56 $C_{R}$ ). (a) $u_{x} (t)$ [Fig. 1(d)] transformed to $u_{x} (x - x_{tip})$ . $u_{x}^{(i)} (x - x_{tip})$ of the $i$ th groove edges vary widely in amplitude for the same rupture front, each corresponding to a fracture energy, $Γ_{i}$ . (b) Distribution of $Γ_{i}$ from (a). $Γ_{i}$ as determined by fitting $u_{x}^{(i)} (x - x_{tip})$ to Eq. (1). $Γ_{i}$ are distributed around the value (blue line) of $Γ$ measured by the strain gauge above the grooves. (c) The mean value of $u_{x}^{(i)} (x - x_{tip})$ (red) compares well to LEFM predictions (black) using $Γ$ measured by the strain gauge. Inset: fitting the strain components to LEFM (orange curve) yields $Γ$ at strain gage locations.
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Figure 3
Slip velocity dependence on $C_{f}$ . (a) Slip velocity, $v_{x} (x - x_{tip})$ , profiles for front velocities $0 < C_{f} < C_{R}$ . $v_{x}$ (see color map) increase with $C_{f}$ . Measurements are prior to the return of waves reflected by system boundaries. (b) Peak slip velocity, $v_{peak}$ , of $v_{x}$ versus $C_{f}$ . Marker shapes indicate different experiments, red markers denote the experiments presented in (a). Three representative error bars are displayed. Yellow line: $v_{peak}$ versus $C_{f}$ calculated from the cohesive zone model [14] in Eq. (2).
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Figure 4
Properties of the slip velocity form. (a) The two collapsed forms are qualitatively different. The high $C_{f}$ regime (bottom) monotonically increases before a slow decay in $v_{x}$ . In low $C_{f}$ regime (top) $v_{x} (x)$ has a sharp local maximum followed by an abrupt increase to $v_{peak}$ and a subsequent rapid decay in $x$ . (b) When normalizing the slip velocity by $v_{peak}$ , profiles collapse into two distinct forms for low and high $C_{f}$ . The critical front velocity separating the two regimes is $C_{c} = 0.8 \pm 0.06 C_{R}$ . (c) Comparison of the two regimes to the cohesive zone model used in Fig. 3. The model, using $x_{c} = 3.2 mm$ and $Γ = 3.5 {J m}^{- 2}$ , provides a reasonable description only behind the crack tip for $C_{f} > C_{c}$ . In the $C_{f} < C_{c}$ regime, this simple model fails for all $x$ . Note that the distance between the points in each measurement presented is less than $1 μ m$ .
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Figure 5
Shear stress changes with $C_{f}$ . (a) Comparisons of $A (x - x_{tip})$ (blue), $v_{x} (x - x_{tip})$ (green, top; red, bottom), and the LEFM solution (black) for $v_{x}$ . For both $C_{f} < C_{c}$ (top) and $C_{f} > C_{c}$ (bottom) slip ( $v_{x} > 0$ ) initiates upon the crack tip’s arrival at $x_{tip} = x_{nl}$ . The locations $x_{nl}$ (denoted by arrows) are, therefore, the last point where the LEFM solution (hence linear elasticity) is valid. Typical error bars, resulting from $v_{x}$ fluctuations [see Fig. 4(a)], are noted. (b) Green: the $C_{f}$ dependence of the shear stress value, $τ_{nl}$ , at the onset of the nonlinear region defined at $x - x_{tip} = x_{nl}$ , where the LEFM solution starts to break down. Red: $C_{f}$ dependence of the peak shear stress of the cohesive zone, $τ_{p}$ , calculated with the cohesive zone model, Eq. (2) [14] at the measured $v_{peak}$ . Both quantities exhibit similar behavior, with a clear transition at $C_{f} \approx C_{c}$ . Both $τ_{p}$ and $τ_{nl}$ vs $v_{peak}$ are presented in [20]. Inset: solid line indicates equality. While $τ_{p}$ is greater than $τ_{nl}$ for $C_{f} < C_{c}$ , beyond $C_{c}$ their values coincide.
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