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Fracture path in brittle thin sheets: a unifying review on tearing

Abstract

We review several studies of crack path in brittle thin sheets where large out of plane bending is involved. Fracture path are observed to be very reproducible. We present a unifying framework based on an energetic point of view. A simplified description, where the sheet is considered to behave as an inextensible fabric, captures important features of experiments: the fact that fracture path seems to obey geometry. We quantify the possible effects of additional bending and stretching terms, and estimate the validity of the model.

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Notes

  1. See Villermaux and Vandenberghe (2013) for a review of recent studies on dynamic tearing of brittle thin sheets.

  2. However we should keep in mind that \(G_c\) is not a well defined material property, and may depend on the details of the plastic flow in the process zone: fracture toughness usually depends on the nature of the loading (mode mixity).

  3. Because of inherent assumption in the kinematics of Kirchhoff equations the shear stresses diverge like \(r^{-3/2}\) instead of \(r^{-1/2}\), a feature which can be corrected within a Timoshenko–Reissner plate theory (Bui 1978; Zehnder and Viz 2005). But these effects are only relevant at a distance to the tip inferior to the thickness.

  4. This argument defined the shape of Eiffel’s tower (Eiffel 1900) for a maximum rigidity against distributed wind loading.

  5. We find again the elasto-capillary lengthscale (Bico et al. 2004; Roman and Bico 2010) over which surface and bending energies equilibrate.

  6. As the slenderness ratio \(e=t/L\) vanishes, 3D-elasticity converges towards the inextensible, infinitely bendable model if the normalized loading \(\eta =F/Etw\) follows \(\eta \sim e\). This falls in the conditions (2), which can be rewritten into \(1\gg \eta \gg e^2\).

  7. The experimental flap shapes are more complex because converging effects due to finite bending rigidity (Sect. 4.1) has to be considered, and may dominate over the diverging terms for weak substrate curvature, strong adhesion, or thick sheets (Kruglova et al. 2011).

  8. Things are more complex if there are several non-connected cuts.

  9. An argument based on the same principle was developed in Atkins (2007) for the case of a cylindrical rigid tool. With the additional assumption that the crack tip was assumed to be located on a circle centered on the tip but with larger radius. Because no elastic energy was attributed to the sheet, the ratio \(dz/ds\) was minimized to obtain the angular velocity of the crack, and therefore the crack path.

    Fig. 11
    figure 11

    (left) The crack path convex hull defines “flaps” (white area) which easily bend around the tool. If the tool were to advance by \(dz\) and cross the boundary of the convex hull (in orange), it would generate stretching in the inextensible film. Instead the crack tip will propagate (right) by a distance \(ds\) so that the pushed front (orange boundary) advances and the tool still belongs to the flap region (white). The direction of propagation for the crack that minimize \(ds/dz\) is perpendicular to the pushed front

  10. Interactive software written by B.Audoly, Institut Jean Le Rond d’Alembert, CNRS/UPMC, 2003.

  11. It can be interpreted as a size around the crack tip where strains become large.

  12. But condition (24) is more restrictive, the condition on the length of the flap depends on its width.

  13. An additional condition ensures that curvature in folds is localized on an area much smaller than the system size.

References

  • Adda-Bedia M, Amar M (1996) Stability of quasi-equilibrium cracks under uniaxial loading. Phys Rev Lett 76(1996):1497–1500

    Google Scholar 

  • Amestoy M, Leblond JB (1992) Crack paths in plane situationii, detailed form of the expansion of the stress intensity factors. Int J Solids Struct 29(4):465501

    Article  Google Scholar 

  • Argon A (1959) Surface cracks on glass. Proc R Soc A 250:472

    Article  Google Scholar 

  • Atkins A (1995) Opposite paths in the tearing of sheet materials. Endeavour 19(1):2–10

    Article  Google Scholar 

  • Atkins A (2007) Wiggly crack paths in the tearing of thin films. Eng Fract Mech 74:1018–1025. doi:10.1016/j.engfracmech.2006.12.006

    Article  Google Scholar 

  • Audoly B, Pomeau Y (2010) Elasticity and geometry: from hair curls to the non-linear response of shells. OUP, Oxford

    Google Scholar 

  • Audoly B, Reis PM, Roman B (2005) Cracks in thin sheets: when geometry rules the fracture path. Phys Rev Lett 95(025):502 doi:10.1103/PhysRevLett.95.025502

    Google Scholar 

  • Bayart E, Boudaoud A, Adda-Bedia M (2010) On the tearing of thin sheets. Eng Fract Mech 77:18491856. doi:10.1016/j.engfracmech.2010.03.006

    Article  Google Scholar 

  • Bayart E, Boudaoud A, Adda-Bedia M (2011) Finite-distance singularities in the tearing of thin sheets. Phys Rev Lett 106(194):301. doi:10.1103/PhysRevLett.106.194301

    Google Scholar 

  • Ben-Amar M, Pomeau Y (1997) Crumpled paper. Proc Math 453:729–755

    Google Scholar 

  • Bico J, Roman B, Moulin L, Boudaoud A (2004) Elastocapillary coalescence in wet hair. Nature 432(7018):690–690. doi:10.1038/432690a

    Google Scholar 

  • Bourdin B, Francfort G, Marigo JJ (2008) The variational approach to fracture. J Elast 91:5–148. doi:10.1007/s10659-007-9107-3

    Article  Google Scholar 

  • Bui H (1978) Mécanique de la rupture fragile. Masson, Paris

    Google Scholar 

  • Cerda E, Hamm L, Roman B, Romero V (2012) Film mince d’emballage amorce de dchirure. INPI (2953499)

  • Cerup-Simonsen B, Tornqvist R, Lutzen M (2009) A simplified grounding damage prediction method and its application in modern damage stability requirements. Marine Struct 22: 62–83

    Google Scholar 

  • Chambolle A, Francfort G, Marigo JJ (2009) When and how do cracks propagate? J Mech Phys Solids 57:16141622

    Article  Google Scholar 

  • Cohen Y, Procaccia I (2010) Dynamics of cracks in torn thin sheets. Phys Rev E 81:066103. doi:10.1103/PhysRevE.81.066103

  • Cotterell B, Rice J (1980) Slightly curved or kinked cracks. Int J Fract 16(2):155169

    Article  Google Scholar 

  • Davidovitch B, Schroll RD, Vella D, Adda-Bedia M, Cerda EA (2011) Prototypical model for tensional wrinkling in thin sheets. Proc Natl Acad Sci USA 108(45):18227–18232. doi:10.1073/pnas.1108553108

    Article  CAS  Google Scholar 

  • Dillard D, Hinkley J, Johnson W, St. Clair T (1994) Spiral tunneling cracks induced by environmental stress cracking in larc-tpi. J Adhesion 44: 1–2, 51–67. doi:10.1080/00218469408026616

  • Duplaix S (2008) Jacques Villéglé, la comdie urbaine. Centre Georges Pompidou Service Commercial

  • Eiffel G (1900) La Tour de Trois Cent Metres. Société des Imprimeries Le Mercier, Paris

    Google Scholar 

  • Ghatak A, Mahadevan L (2003) Crack street: the cycloidal wake of a cylinder tearing through a thin sheet. Phys Rev Lett 91:215507

    Google Scholar 

  • Gladden J, Belmonte A (2007) Motion of a viscoelastic micellar fluid around a cylinder: flow and fracture. Phys Rev Lett 98(223):501. doi:10.1103/PhysRevLett.98.224501

    Google Scholar 

  • Goldstein R, Salganik R (1974) Brittle fracture of solids with arbitrary cracks. Int J Fract 10(2):507523

    Google Scholar 

  • Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Sol 57(235501):342368

    Google Scholar 

  • Hamm E, Reis P, Leblanc M, Roman B, Cerda E (2008) Tearing as a test for mechanical characterization of thin adhesive films. Nat Mater 7:386–390. doi:10.1038/nmat2161

    Google Scholar 

  • Hui C-Y, Zehnder AT, Potdar YK (1998) Williams meets von Karman: Mode coupling and nonlinearity in the fracture of thin plates International Journal of Fracture 93:409–429

    Google Scholar 

  • Kendall K (1971) The adhesion and surface energy of elastic solids. J Phys D Appl Phys 4:1186–1195

    Article  Google Scholar 

  • Kendall K (1975) Thin-film peeling- the elastic term. J Phys D Appl Phys 8:115

    Article  Google Scholar 

  • Kruglova O, Brau F, Villers D, Damman P (2011) How geometry controls the tearing of adhesive thin films on curved surfaces. Phys Rev Lett 107(164):303. doi:10.1103/PhysRevLett.107.164303

    Google Scholar 

  • Landau L, Lifshitz E (1967) Theory of elasticity. Mir

  • Lawn B (1993) Fracture of brittle solids. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Lebental M (2007) Chaos quantique et micro-lasers organiques. Ph.D. Thesis, Univ Paris XI. http://tel.archives-ouvertes.fr/tel-00194350

  • Leblond J (2003) Mécanique de la rupture fragile et ductile. Hermes Science Publications

  • Leung KT, Jozsa L, Ravasz M, Nda Z (2001) Spiral cracks without twisting. Nature 410(6825):166. doi:10.1038/35065517

    Google Scholar 

  • Lobkovsky AE, Gentges S, Li H, Morse D, Witten TA (1995) Scaling properties of stretching ridges in a crumpled elastic sheet. Science 270:1482–1485

    Article  CAS  Google Scholar 

  • Love A (1944) Treatise on the mathematical theory of elasticity. Dover, New York

  • Mansfield EH (1989) The bending and stretching of plates. Cambridge university press, Cambridge

    Book  Google Scholar 

  • Marigo JJ, Meunier N (2006) Hierarchy of one-dimensional models in nonlinear elasticity. J Elast 83:1–28. doi:10.1007/s10659-005-9036-y

    Article  Google Scholar 

  • Meyer DC, Leisegang T, Levin A, Paufler P, Volinsky A (2004) Tensile crack patterns in mo/si multilayers on si substrates under high-temperature bending. Appl Phys A 78:303–305. doi:10.1007/s00339-003-2340-0

    Google Scholar 

  • Monsalve A, Gutierrez I (2000) Application of a modified rigid plastic model to the out-plane fracture of ‘easy open end cans’. Int J Fract 102:323–339. doi:10.1023/A:1007625512996

    Article  CAS  Google Scholar 

  • Néda Z, t Leung K, Józsa L, Ravasz M (2002) Spiral cracks in drying precipitates. Phys Rev Lett 88(9):095502. doi:10.1103/PhysRevLett.88.095502

    Google Scholar 

  • O’keefe R (1994) Modeling the tearing of paper. Am J Phys 62(4):299–305. doi:10.1119/1.17570

    Google Scholar 

  • Pogorelov A (1988) Bendings of surfaces and stability of shells. American Mathematical Society, Providence

  • Reis P, Kumar A, Shattuck MD, Roman B (2008) Unzip instabilities: straight to oscillatory transitions in the cutting of thin polymer sheets. Eur Phys Lett 82(64):002. doi:10.1209/0295-5075/82/64002

    Google Scholar 

  • Roman B, Bico J (2010) Elasto-capillarity: deforming an elastic structure with a liquid droplet. J Phys-Condens Mater 22(49). doi:10.1088/0953-8984/22/49/493101

  • Roman B, Gay C, Clanet C (2013) Pendulum, drops and rods: a physical analogy. Am J Phys (submitted)

  • Roman B, Reis PM, Audoly B, De Villiers S, Vigui V, Vallet D (2003) Oscillatory fracture paths in thin elastic sheets/oscillatory fracture paths in thin elastic sheets. C R Mecanique 331:811–816. doi:10.1016/j.crme.2003.10.002

    Google Scholar 

  • Romero V (2010) Spiraling cracks in thin sheets. Ph.D. Thesis, UPMC/USACH. http://pastel.archives-ouvertes.fr/pastel-00552045

  • Romero V, Hamm E, Cerda E (2013) Spiral tearing of thin films. Soft Matter (in press). doi:10.1039/c3sm50564b

  • Ronsin O, Heslot F, Perrin B (1995) Experimental study of quasistatic brittle crack propagation. Phys Rev Lett 75(12):2352–2355. doi:10.1103/PhysRevLett.75.2352

    Google Scholar 

  • Sendova M, Willis K (2003) Spiral and curved periodic crack patterns in sol-gel films. Appl Phys A Mater Sci Process 76:957–959. doi:10.1007/s00339-002-1757-1

  • Stein M, Hedgepeth J (1961) Analysis of partly wrinkled membranes. Tech. rep, NASA, Langley research center, Langley Field, VA

  • Struik D (1988) Lectures on classical differential geometry. Dover, New York

    Google Scholar 

  • Takei A, Roman B, Bico J, Hamm E, Melo F (2013) Forbidden directions for the fracture of thin anisotropic sheets: an analogy with the wulff plot. Phys Rev Lett 110(144):301. doi:10.1103/PhysRevLett.110.144301

    Google Scholar 

  • Tallinen T, Mahadevan L (2011) Forced tearing of ductile and brittle thin sheets. Phys Rev Lett 107(245):502. doi:10.1103/PhysRevLett.107.245502

    Google Scholar 

  • Timoshenko S, Woinowski-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New York

    Google Scholar 

  • Vella D, Wettlaufer JS (2007) Finger rafting: a generic instability of floating elastic sheets. Phys Rev Lett. doi:10.1103/PhysRevLett.98.088303

  • Vermorel R (2010) Elasticité et fragmentation solide. Ph.D. Thesis, Univ. Provence Aix-Marseille. PRL 104:175502. doi:10.1103/PhysRevLett.104.175502

  • Vermorel R, Vandenberghe N, Villermaux E (2009) Impacts on thin elastic sheets. Proc R Soc Lond A 465:823842

    Google Scholar 

  • Vermorel R, Vandenberghe N, Villermaux E (2010) Radial cracks in perforated thin sheets. Phys Rev Lett 104

  • Villermaux E, Vandenberghe N (2013) Geometry and fragmentation of soft brittle impacted bodies. Soft Matter (in press). doi:10.1039/C3SM50789K

  • Wan N, Xu J, Lin T, Xu L, Chen K (2009) Observation and model of highly ordered wavy cracks due to coupling of in-plane stress and interface debonding in silica thin films. Phys Rev B. doi:10.1103/PhysRevB.80.014121

  • Wierzbicki T, Trauth KA, Atkins AG (1998) On diverging concertina tearing. J Appl Mech 65:990

    Article  Google Scholar 

  • Williams M (1961) The bending stress distribution at the base of a stationary crack. J Appl Mech 28:7882

    Article  Google Scholar 

  • Witten TA (2007) Stress focusing in elastic sheets. Rev Mod Phys 79(2):643–675. doi:10.1103/RevModPhys.79.643

    Google Scholar 

  • Xiaa C, Hutchinson JW (2000) Crack patterns in thin films. J Mech Phys Solids 48:1107–1131

    Article  Google Scholar 

  • Yang B, Ravi-Chandar K (2001) Crack path instabilities in a quenched glass plate. J Mech Phys Solids 49(2001):91–130

    Google Scholar 

  • Yuse A, Sano M (1993) Transition between crack patterns in quenched glass plates. Nature 362:329

    Google Scholar 

  • Zehnder AT, Viz MJ (2005) Fracture mechanics of thin plates and shells under combined membrane, bending, and twisting loads. Appl Mech Rev 58(1):37

    Article  Google Scholar 

Download references

Acknowledgments

I thank K. Ravichandar for his suggestions and comments. I also thank José Bico and Basile Audoly for invaluable help.

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Appendix

Appendix

Peeling an elastica

We consider an inextensible rod, with a tangent having an angle \(\theta \) at the point of curvilinear abscissa \(s\), subject to a horizontal force \(F\) (see Fig. 23). The torque balance on an element with size \(ds\) reads \(dM/ds + F \sin \theta =0\), where the constitutive relation \(M=Bd\theta /ds\) can be used. Here the bending rigidity is \(B=Et^3/12(1-\nu ^{2})\), where \(t\) is the thickness of the sheet, \(E\) its Young modulus, and \(\nu \) Poisson’s ratio.

Fig. 23
figure 23

A portion of the shape of a peeled elastica with an angle \(\phi =\pi \) (left) leads to an angle \(\phi \) when rotated (right)

Finally, the elastica equation is Landau and Lifshitz (1967), Love (1944)

$$\begin{aligned} B \ddot{\theta }+\frac{F}{w} \sin \theta =0 \end{aligned}$$

with the boundary conditions \(\theta (0)=0\), and a force \(F\), but no torque applied at \(s=L\), \(\dot{\theta }(L)=0\). We can expect the flap to be curved only on a localized region near the clamped condition \(s=0\). What is the size of this region? Dimensional analysis directly shows that the only length-scale left in the problem is \(\sqrt{Bw/F},\) so that the flap shapes for different loading and rigidity will all be similar, up to a simple scaling factor, as long as they are long compared to this radius of curvature, \(L\gg \sqrt{Bw/F}.\) The elastic energy per unit width which only depends on \(B\) and \(F/w\) can only be written as \(E_{el}/w = a \sqrt{FB/w}\).

These results are also found by estimating the radius of curvature of the fold \(R\) from a torque balance. The torque \(Bw/R \sim FR\) is produced by force \(F\) with a lever arm of the order of \(R\). Because \(1/R \sim \sqrt{ F/Bw}\), we also find that \(E_{el} \sim Bw/R \sim \sqrt{BFw}.\) We also note that the bending energy density scales like \(F/w.\)

In fact these quick arguments can be made exactly because an explicit solution is available in the case where \(L=\infty \): we first normalize all distances by the typical length \(\sqrt{Bw/F}\) and find \(\ddot{\theta }+ \sin \theta =0\). Here we look for the solution where with the condition \(\theta (0)=0, \theta (\infty )=\pi , \dot{\theta }(\infty )=0.\) These solutions are the same as the 2D meniscus of a liquid under gravity and surface tension (Roman et al.).

A first integral of this equation gives \(\dot{\theta }^2 /2 = 1+\cos \theta \), using the boundary conditions at \(s=\infty \). If we keep \(\dot{\theta }>0\), this can be rewritten into \(\dot{\theta }= 2\cos \theta /2,\) which can be integrated into

$$\begin{aligned} \sin (\theta /2) = \tanh (s). \end{aligned}$$

This implicit solution with \(s \in [0,\infty ]\) corresponds to a peeling angle \(\phi =\pi \). But for a different peeling angle \(\phi \), the solution is simply a rotated portion of the same solution \(s \in [s_0, \infty ]\), where \(\tanh (s_0)= \sin (\pi /2-\phi /2)=\cos (\phi /2)\), as seen in Fig. 23.

We compute the nondimensional elastic energy using these solutions:

$$\begin{aligned} E_{el}/\sqrt{FBw}&= \int \limits _{s_0}^\infty \dot{\theta }^2/2 ds = 2 \int \limits _{s_0}^\infty \cos ^2(\theta /2)\\ ds&= 2 [\tanh (s)]_{s_0}^\infty = 2[1-\cos (\phi /2)]. \end{aligned}$$

Finally we obtain

$$\begin{aligned} E_{el} = 2 \sqrt{FBw}[1-\cos (\phi /2)] \end{aligned}$$
(25)

Another estimate gives in \(\int \dot{\theta }^2 /2 ds = \int (1+\cos \theta ) ds = l-\delta \), where \(l\) and \(\delta \) are the distances on Fig. 23. In dimensional terms, we find

$$\begin{aligned} E_{el} = F(l-\delta ), \end{aligned}$$
(26)

which shows that

$$\begin{aligned} l-\delta = 2\sqrt{Bw/F} [1-\cos (\phi /2)] \end{aligned}$$
(27)

and

$$\begin{aligned} E_{el}= \frac{4 Bw}{(l-\delta )}[1-\cos (\phi /2) ]^2 \end{aligned}$$
(28)

Yet another interesting quantity is based on direct integration, which shows that \( h= \int \sin \theta = -[\dot{\theta }]_{s_0}^\infty = 2\cos (\theta (0))=2\sin (\phi /2)\). In dimensional form, this means that

$$\begin{aligned} h = 2 \sqrt{Bw/F}\sin (\phi /2) = (l-\delta )\frac{\sin (\phi /2)}{1-\cos (\phi /2)} \end{aligned}$$
(29)

and to the elastic energy

$$\begin{aligned} E_{el} = \frac{ 4Bw}{h} [1-\cos (\phi /2)] \sin (\phi /2) \end{aligned}$$
(30)

Why does the crack loose memory (almost) instantaneously?

In the pulling configuration of pulling on an adhering sheet (Fig. 16), the past history of the crack only enters the problem through the shape of the flap. We consider that the flap continues to have a cylindrical shape invariant in the \(z\) direction. The elastic energy reads

$$\begin{aligned} E_{el}= \frac{B}{2}\int \limits _{0}^{\infty } w(u) \kappa ^2(u) du \end{aligned}$$

where \(u\) is the curvilinear abscissa along the fold, and the function \(\kappa (.)\) is the curvature of the fold, an universal function that depends on \((l-\delta )^{-1}\).  As the cracks propagate by \(\delta s\), this energy varies for two reasons: the profile \(w(l)\) is modified because the origin of the fold has advanced by \(\delta l\), and the curvature profile is modified (because \(l-\delta \) has changed).

$$\begin{aligned} \delta E_{el}&= \frac{B}{2}\int \limits _{0}^{\infty } [ w(u+\delta l)-w(u)]\kappa ^2(u) du \\&\quad + \frac{B}{2}\int \limits _{0}^{\infty } w(u) \delta [\kappa ^2(u)] du \end{aligned}$$

The key point is that the curvature profile is localized on a small region with size \(r\) comparable to \(l-\delta .\) If we assume that on this small lengthscale, \(w(u+\delta l)-w(u)\) can be replaced by \(\delta l (dw/du)_{u=0} \) and \(w(u)\sim w(0)\), we get

$$\begin{aligned} \delta E_{el}&= \frac{dw}{du}\Bigg )_{u=0} \!\! \!\! \delta l \frac{B}{2} \int \limits _{0}^{\infty } \kappa ^2(u) du\\&\quad + w(0) \frac{B}{2}\delta \left[ \int \limits _{0}^{\infty } \kappa ^2(u)du \right] \end{aligned}$$

In the first term we recognize the elastic energy of a slice of fold with unit width, multiplied by the the variation \(\delta w\). Because of the invariance of the fold in direction \(z\), this is exactly

$$\begin{aligned} d w \frac{\partial E_{el} }{\partial w } \Bigg )_{(l-\delta )} \end{aligned}$$

whereas the second term is in fact a derivative where the width \(w=w(0)\) is held constant:

$$\begin{aligned} \frac{\partial E_{el} }{\partial (l-\delta )} \Bigg )_{w} d (l-\delta ) \end{aligned}$$

so that we recover the equations of Sect. 4.1

$$\begin{aligned} dE_{el}= \frac{\partial E_{el} }{\partial (l-\delta )} \Bigg )_{w} d (l-\delta ) + d w \frac{\partial E_{el} }{\partial w } \Bigg )_{(l-\delta )}. \end{aligned}$$

When inserted in Griffith’s criterion, all the quantities depend on \(w\) and \(dw/ds\), so that finally the equation of evolution of the width can only be a first order equation of the type \(dw/ds = \mathcal{F} (w)\): the evolution of the inter-crack distance \(w\) only depends on its actual value, not on the past.

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Roman, B. Fracture path in brittle thin sheets: a unifying review on tearing. Int J Fract 182, 209–237 (2013). https://doi.org/10.1007/s10704-013-9869-5

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Keywords

  • Plate mechanics
  • Crack path
  • Thin sheets