Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture

Chih-Hung Chen, Tristan Cambonie, Veronique Lazarus, Matteo Nicoli, Antonio J. Pons, and Alain Karma

Phys. Rev. Lett. 115, 265503 – Published 30 December 2015

Abstract

A planar crack generically segments into an array of “daughter cracks” shaped as tilted facets when loaded with both a tensile stress normal to the crack plane (mode I) and a shear stress parallel to the crack front (mode III). We investigate facet propagation and coarsening using in situ microscopy observations of fracture surfaces at different stages of quasistatic mixed-mode crack propagation and phase-field simulations. The results demonstrate that the bifurcation from propagating a planar to segmented crack front is strongly subcritical, reconciling previous theoretical predictions of linear stability analysis with experimental observations. They further show that facet coarsening is a self-similar process driven by a spatial period-doubling instability of facet arrays.

Received 17 September 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.265503

Physics Subject Headings (PhySH)

Fracture

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Chih-Hung Chen^1,*, Tristan Cambonie², Veronique Lazarus², Matteo Nicoli¹, Antonio J. Pons³, and Alain Karma^1,†

¹Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA
²Laboratoire FAST, Univ Paris Sud, CNRS, Université Paris-Saclay, F-91405 Orsay, France
³Department of Physics, Polytechnic University of Catalonia, Terrassa, Barcelona 08222, Spain

^*ch.chen@neu.edu
^†a.karma@neu.edu

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Vol. 115, Iss. 26 — 31 December 2015

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Images

Figure 1
In situ microscope images (a)–(g) of fatigue cracks in Plexiglas at different stages of crack advance in mixed-mode $I + III$ loading depicted schematically in (h) and corresponding example of crack front segmentation in phase-field simulation (i). $K_{III} / K_{I} \approx 0.3$ in (a)–(e) and $\approx 0.5$ in (f) and (g); (a), (b), and (f) are experimental views from a direction approximately perpendicular to the plane of the parent crack with facets propagating downwards, while views (c), (d), (e), and (g) are views with the crack propagation direction out of the page. Views (c), (d), and (e) correspond to different stages of crack advance increasing from (c) to (e). Broken (pristine) regions of the samples appear in black (light blue) or darker (lighter) gray depending on the viewing direction. The bar scale is 1 mm in all images. The red dashed lines in (a) highlight the curved fronts of two facets as guides to the eye; curved tips are clearly visible in (f). (i) Snapshots of phase-field fracture surfaces ( $ϕ = 1 / 2$ surfaces) at different stages of crack advance increasing from top to bottom, showing that energetically favored $A$ facets [18] propagate ahead of $B$ facets, eventually outgrowing them completely. Simulation parameters are $G / G_{c} = 1.5$ , $K_{III} / K_{I} = 0.5$ , and box dimensions $D_{x} = 307 ξ$ , $D_{y} = 100 ξ$ , and $D_{z} = 200 ξ$ .
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Figure 2
Snapshots of phase-field simulations illustrating the destabilization of planar crack growth for $K_{III} / K_{I} = 0.4$ . The crack propagation length $a$ increases from (a) to (d), and both the crack front (blue lines) and its in-plane and out-of-plane projections (red lines) are shown. (e) Plot of linear instability threshold ${(K_{III} / K_{I})}_{c}$ versus $D_{y} / Λ$ , where $Λ$ represents the mean facet spacing. Planar growth is unstable (stable) above (below) the filled circles where error bars are defined in Ref. [30]. In all simulations, $G = 1.5 G_{c}$ , $D_{x} = 230 ξ$ , and $D_{z} = Λ = 60 ξ$ .
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Figure 3
(a) Comparison of facet tilt angles obtained from experiments and simulations, where red and blue arrows indicate the instability thresholds of planar crack propagation for $D_{y} / Λ = 1$ and $D_{y} / Λ = 2$ , respectively [see Fig. 2], and theoretically predicted assuming shear-free facets (dashed line) [16, 24]. (b) Snapshots of a phase-field simulation for $D_{y} / Λ = 1$ demonstrating the subcritical nature of the bifurcation from a planar to segmented crack propagation. A segmented front solution for $K_{III} / K_{I} = 0.5$ ( $θ = 31 °$ ) was used as an initial condition in a simulation for $K_{III} / K_{I} = 0.07$ , causing the facet angle to relax to a lower steady-state value ( $θ = 11.2 °$ ) (see Movie 2 of [30]). (c) Out-of-plane and in-plane (inset) crack front projections. In all simulations, $D_{x} = 154 ξ$ , $D_{y} = D_{z} = 60 ξ$ , $Λ = 60 ξ$ , and $G = 1.5 G_{c}$ . (d) Schematic diagram of subcritical bifurcation recapitulating the experimental and simulation results with solid (dashed) lines representing stable (unstable) solutions.
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Figure 4
(a) Illustration of spatial period-doubling instability in a phase-field simulation for $K_{III} / K_{I} = 0.5$ ; out-of-plane and in-plane projections of crack fronts are plotted in the top panel and the bottom panel, respectively (see Movie 3 of [30]). (b) Semilog plot of difference of tip positions along the propagation $x$ axis between leading and lagging facets versus scaled time for different $K_{III} / K_{I}$ . Inset: Coarsening rate $β$ versus $K_{III} / K_{I}$ obtained from experiments and phase-field simulations. In all simulations, $D_{x} = 307 ξ$ , $D_{y} = 60 ξ$ , $D_{z} = 120 ξ$ , $Λ = 60 ξ$ , and $G = 1.5 G_{c}$ .
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