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Shape Evolution of Unstable, Flexural Cracks in Brittle Materials


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

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)


  1. 1.

    E.H. Yoffe, LXXV. The Moving Griffith Crack, Lond. Edinb. Dublin Philos. Mag. J. Sci., 1951, 42(330), p 739–750

  2. 2.

    J. Eshelby, Fracture meCHANICS, Sci. Prog., 1971, 59, p 161–179

  3. 3.

    L. Freund, Crack Propagation in an Elastic Solid Subjected to General Loading—I. Constant Rate of Extension, J. Mech. Phys. Solids, 1972, 20(3), p 129–140

  4. 4.

    E. Sharon and J. Fineberg, Confirming the Continuum Theory of Dynamic Brittle Fracture for Fast Cracks, Nature, 1999, 397, p 333

  5. 5.

    E. Sharon, S.P. Gross, and J. Fineberg, Local Crack Branching as a Mechanism for Instability in Dynamic Fracture, Phys. Rev. Lett., 1995, 74(25), p 5096–5099

  6. 6.

    E. Sharon and J. Fineberg, Microbranching Instability and the Dynamic Fracture of Brittle Materials, Phys. Rev. B, 1996, 54(10), p 7128–7139

  7. 7.

    T. Cramer, A. Wanner, and P. Gumbsch, Energy Dissipation and Path Instabilities in Dynamic Fracture of Silicon Single Crystals, Phys. Rev. Lett., 2000, 85(4), p 788

  8. 8.

    H. Bergkvist, An Investigation of Axisymmetric Crack Propagation, in ASTM STP 627 Fast Fracture and Crack Arrest (1977), pp. 312–335.

  9. 9.

    M.J. Buehler and H. Gao, Dynamical Fracture Instabilities Due to Local Hyperelasticity at Crack Tips, Nature, 2006, 439, p 307

  10. 10.

    F. Kerkhof, General Lecture Wave Fractographic Investigations of Brittle Fracture Dynamics, in Proceedings of an International Conference on Dynamic Crack Propagation, ed. by G.C. Sih (Springer, Dordrecht, 1973), pp. 3–35.

  11. 11.

    K. Ravi-Chandar and W. Knauss, An Experimental Investigation into Dynamic Fracture: I. Crack Initiation and Arrest, Int. J. Fract., 1984, 25(4), p 247–262

  12. 12.

    H. Gao, A Theory of Local Limiting Speed in Dynamic Fracture, J. Mech. Phys. Solids, 1996, 44(9), p 1453–1474

  13. 13.

    R. Dugnani and L. Ma, Energy Release Rate of Moving Circular-Cracks, Eng. Fract. Mech., 2019, 213, p 118–130

  14. 14.

    L.B. Freund and R.J. Clifton, On the Uniqueness of Plane Elastodynamic Solutions for Running Cracks, J. Elast., 1974, 4(4), p 293–299

  15. 15.

    N.F. Mott, Fracture of Metals: Theoretical Considerations, Engineering, 1948, 165, p 16–18

  16. 16.

    D.K. Roberts and A.A. Wells, The Velocity of Brittle Fracture, Engineering, 1954, 171, p 820–821

  17. 17.

    G.K. Bansal, On Fracture Mirror Formation in Glass and Polycrystalline Ceramics, Phil. Mag., 1977, 35(4), p 935–944

  18. 18.

    A.I.A. Abdel-Latif, R.C. Bradt, and R.E. Tressler, Dynamics of Fracture Mirror Boundary Formation in Glass, Int. J. Fract., 1977, 13(3), p 349–359

  19. 19.

    Y.M. Tsai, Exact Stress Distribution, Crack Shape and Energy for a Running Penny-Shaped Crack in an Infinite Elastic Solid, Int. J. Fract., 1973, 9(2), p 157–169

  20. 20.

    J. Craggs, The Growth of a Disk-Shaped Crack, Int. J. Eng. Sci., 1966, 4(2), p 113–124

  21. 21.

    B. Kostrov, The Axisymmetric Problem of Propagation of a Tension Crack, J. Appl. Math. Mech., 1964, 28(4), p 793–803

  22. 22.

    F. Erdogan, Crack Propagation Theories (Lehigh University, Bethlehem, 1967)

  23. 23.

    L.B. Freund, Dynamic Fracture Mechanics, Vol. 9 (Cambridge University Press, Cambridge, 1998), pp. 1689–1699.

  24. 24.

    R. Burridge and J.R. Willis, The Self-similar Problem of the Expanding Elliptical Crack in an Anisotropic Solid, Math. Proc. Camb. Philos. Soc., 2008, 66(2), p 443–468

  25. 25.

    L. Zhao, D. Bardel, A. Maynadier, and D. Nelias, Crack Initiation Behavior in Single Crystalline Silicon, Scr. Mater., 2017, 130, p 83–86

  26. 26.

    D. Sherman and I. Be’ery, Shape and Energies of a Dynamically Propagating Crack Under Bending, J. Mater. Res., 2003, 18(10), p 2379–2386

  27. 27.

    C.J. Newman, I.S. Raju, Stress Intensity Factor Equations for Cracks in Three-Dimensional Finite Bodies Subjected to Tension and Bending Loads (NASA Technical Memorandum, 1983)

  28. 28.

    R. Gol’dshtein and A. Mosolov, Fractal Cracks, J. Appl. Math. Mech., 1992, 56(4), p 563–571

  29. 29.

    R. Smith and J. Mecholsky, Jr., Application of Atomic Force Microscopy in Determining the Fractal Dimension of the Mirror, Mist, and Hackle Region of Silica Glass, Mater. Charact., 2011, 62(5), p 457–462

  30. 30.

    R. Dugnani, Z. Pan, M. Wu, Degradation of Consumer Products Glasses after Extended Use. 2009 MS&T Proc, vol. 1 (2009), pp. 500–509.

  31. 31.

    P. Dwivedi and D.J. Green, Indentation Crack-Shape Evolution during Subcritical Crack Growth, J. Am. Ceram. Soc., 1995, 78(5), p 1240–1246

  32. 32.

    A. Rabinovitch, V. Frid, D. Bahat, Wallner Lines Revisited, ed: AIP (2006)

  33. 33.

    D. Sherman, Fractography of Dynamic Crack Propagation in Silicon Crystal, Key Eng. Mater., 2009, 409(November), p 55–64

  34. 34.

    G.D. Quinn, NIST Recommended Practice Guide: Fractography of Ceramics and Glasses. NIST SP - 960-16 (2016)

  35. 35.

    T.A. Schwartz, The Fractography of Inorganic Glass (Massachusetts Institute of Technology, 1977)

  36. 36.

    C.L. Rountree, R.K. Kalia, E. Lidorikis, A. Nakano, L.V. Brutzel, and P. Vashishta, Atomistic Aspects of Crack Propagation in Brittle Materials: Multimillion Atom Molecular Dynamics Simulations, Annu. Rev. Mater. Res., 2002, 12(32), p 377–400

  37. 37.

    K. Sauthoff, M. Wenderoth, A.J. Heinrich, M.A. Rosentreter, K.J. Engel, T.C.G. Reusch, R.G. Ulbrich. Nonlinear Dynamic Instability in Brittle Fracture of GaAs. Phys. Rev. B: Condens. Matter Mater. Phys. 60(7), 4789–4795 (1999)

  38. 38.

    C.P. Chen, Fracture Mechanics Evaluation of GaAs (Final Report, 1984)

  39. 39.

    V.M. Bright, Y. Kim, and W.D. Hunt, Study of Surface Acoustic Waves on the 110 Plane of Gallium Arsenide, J. Appl. Phys., 1992, 71(2), p 597–605

  40. 40.

    A. Makishima and J.D. Mackenzie, Calculation of Bulk Modulus, Shear Modulus and Poisson’s Ratio of Glass, J. Noncryst. Solids, 1975, 17(2), p 147–157

  41. 41.

    J. Kim, D.-I.D. Cho, R.S. Muller, Why is (111) Silicon a Better Mechanical Material for MEMS? in Transducers’ 01 Eurosensors XV (Springer, 2001), pp. 662–665.

  42. 42.

    S. Adachi, Properties of Aluminium Gallium Arsenide (no. 7) (IET, 1993)

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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).

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