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Shape Evolution of Unstable, Flexural Cracks in Brittle Materials

Abstract

In this manuscript, the crack-shape evolution of unstable, flexural cracks was modeled using a continuum mechanics approach applied to dynamic, brittle, and elliptical cracks. The crack-shape evolution was first computed analytically for both flat and fractal cracks and subsequently compared to experimental values inferred from fractographic features present on silicate glass, gallium arsenide, and silicon single crystals fractured in bending. The trends predicted by the analytical model were consistent with the fractographic observations prior to the onset of crack-tip instabilities. It was determined that the effect of the surface roughness was to slow down the crack and to decrease the local crack-front radius of curvature which, in flexural cracks, led to crack shapes elongated along the free surface direction. This work represents the first attempt to predict the crack-shape evolution in rough, brittle samples fractured in bending and to understand the role played by the surface roughness during fast, unstable propagation.

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Abbreviations

\(a \left( {a_{0} , a_{m} } \right)\) :

Crack depth/radius (m)

\(A\) :

Crack extension area (\({\text{m}}^{2}\))

\(A_{\text{m}}\) :

Mirror constant (\({\text{MPa}}\sqrt {\text{m}}\))

\(A\left( v \right)\) :

Modified function of crack speed (\(/\))

\(c \left( {c_{0} } \right)\) :

Crack length (along the maximum tensile surface for circular/elliptic crack) (m)

\(c_{1}\) :

Extension wave speed (m/s)

\(c_{2}\) :

Shear wave speed (m/s)

\(c_{\text{R}}\) :

Rayleigh wave speed (m/s)

\(D\) :

Acoustic waves’ influence domain (\({\text{m}}^{2}\))

\(D^{*}\) :

Fractal increment (/)

\(E\) :

Young’s modulus (MPa)

\(f^{{\prime }}\) :

Slope of step-hackle lines (/)

\(G_{0}^{c} ,G_{\text{d}}^{c}\) :

Static/dynamic energy release rate (MPa m)

\(g\) :

Universal function (/)

\(H\) :

Plate thickness (m)

\(k_{\text{p}} , k_{\text{c}}\) :

Displacement field integration constants (/)

\(K_{\text{I}} \left( {K_{\text{Ic}} } \right)\) :

(Critical) stress intensity factor (\({\text{MPa}}\sqrt {\text{m}}\))

\(\partial \ell\) :

Crack’s tip infinitesimal circular width (m)

\(u \left( {u_{x} , u_{y} } \right)\) :

Crack opening displacement (m)

\(v \left( v_{\text{T}} , \tilde{v}_{\text{T}} , v_{\max} \right)\) :

Crack-tip velocity (terminal/maximum) (m/s)

\(w \left( {w_{\text{c}} } \right)\) :

Normalized cracked tip velocity (\(w \equiv v/c_{\text{R}}\), m/s)

\(r \left( {r_{0} } \right)\) :

Crack length in radial direction (m)

\(R_{\text{m}}\) :

Mirror radius (m)

\(S\) :

Surface energy (J)

\(U_{K}\) :

Kinetic energy (J)

\(U_{0}\) :

Strain energy (J)

\(Y \left( {Y_{\text{S}} , Y_{\text{C}} , Y^{{\prime }} } \right)\) :

Shape factor (for straight/circular crack, modified shape factor) (/)

\(\alpha , \beta\) :

Geometry angles (rad)

\(\sigma_{\text{f}}\) :

Fracture strength (MPa)

\(\varepsilon \left( {\varepsilon_{y} } \right)\) :

Crack opening strain (/)

\(\lambda\) :

Density (\({\text{kg/m}}^{3}\))

\(\rho \left( {\rho_{0} } \right)\) :

Local radius of curvature (m)

\(\gamma\) :

Surface energy per unit area (J/m2)

\(\upsilon\) :

Poisson’s ratio (/)

\(\varphi\), \(\phi\) :

Projected angle (rad)

\(\varOmega\) :

Local crack extension area (m2)

\(\varPi\) :

Total potential energy (J)

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Correspondence to Roberto Dugnani.

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Ma, L., Sun, H. & Dugnani, R. Shape Evolution of Unstable, Flexural Cracks in Brittle Materials. J. of Materi Eng and Perform (2020). https://doi.org/10.1007/s11665-020-04657-5

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Keywords

  • brittle fracture
  • crack evolution
  • flexural crack
  • fractal surface
  • gallium arsenide
  • silicate glass
  • silicon