# Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading

- 335 Downloads
- 10 Citations

## Abstract

There have been a number of recent papers by various authors addressing static fracture in the setting of the linearized theory of elasticity in the bulk augmented by a model for surface mechanics on fracture surfaces with the goal of developing a fracture theory in which stresses and strains remain bounded at crack-tips without recourse to the introduction of a crack-tip cohesive-zone or process-zone. In this context, surface mechanics refers to viewing interfaces separating distinct material phases as dividing surfaces, in the sense of Gibbs, endowed with excess physical properties such as internal energy, entropy and stress. One model for the mechanics of fracture surfaces that has received much recent attention is based upon the Gurtin-Murdoch surface elasticity model. However, it has been shown recently that while this model removes the strong (square-root) crack-tip stress/strain singularity, it replaces it with a weak (logarithmic) one. A simpler model for surface stress assumes that the surface stress tensor is Eulerian, consisting only of surface tension. If surface tension is assumed to be a material constant and the classical fracture boundary condition is replaced by the jump momentum balance relations on crack surfaces, it has been shown that the classical strong (square-root) crack-tip stress/strain singularity is removed and replaced by a weak, logarithmic singularity. If, in addition, surface tension is assumed to have a (linearized) dependence upon the crack-surface mean-curvature, it has been shown for pure mode I (opening mode), the logarithmic stress/strain singularity is removed leaving bounded crack-tip stresses and strains. However, it has been shown that curvature-dependent surface tension is insufficient for removing the logarithmic singularity for mixed mode (mode I, mode II) cracks. The purpose of this note is to demonstrate that a simple modification of the curvature-dependent surface tension model leads to bounded crack-tip stresses and strains under mixed mode I and mode II loading.

## Keywords

Fracture Interfacial mechanics Singular integro-differential equations## Mathematics Subject Classification

45 74## Notes

### Acknowledgement

The author gratefully acknowledges support for this research from the Air Force Office of Scientific Research through the task number 12RX13COR and the Air Force Materials and Manufacturing Directorate under contract # FA8650-07-D-5800.

## References

- 1.Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. In: Argon, A.S. (ed.) Advances in Applied Mechanics, vol. VII, pp. 55–129. Academic Press, San Diego (1962) Google Scholar
- 2.Broberg, K.B.: Cracks and Fracture. Academic Press, San Diego (1999) Google Scholar
- 3.Cherepanov, G.P.: Mechanics of Brittle Fracture. McGraw-Hill, New York (1979) MATHGoogle Scholar
- 4.Cherepanov, G.P.: Fracture. Krieger, Melbourne (1998) MATHGoogle Scholar
- 5.Gibbs, J.W.: Collected Works vol. 1. Longmans, New York (1928) MATHGoogle Scholar
- 6.Kaninnen, M.F., Popelar, C.H.: Advanced Fracture Mechanics. Oxford University Press, London (1985) Google Scholar
- 7.Kim, C.I., Schiavone, P., Ru, C.-Q.: Analysis of a mode iii crack in the presence of surface elasticity and a prescribed non-uniform surface traction. Z. Angew. Math. Phys.
**61**, 555–564 (2010) CrossRefMATHMathSciNetGoogle Scholar - 8.Kim, C.I., Schiavone, P., Ru, C.-Q.: The effects of surface elasticity on an elastic solid with mode iii crack: complete solution. J. Appl. Mech.
**77**, 021011 (2010) ADSCrossRefGoogle Scholar - 9.Kim, C.I., Schiavone, P., Ru, C.-Q.: Analysis of plane-strain crack problems (mode i and mode ii) in the presence of surface elasticity. J. Elast.
**104**, 397–420 (2011) CrossRefMATHMathSciNetGoogle Scholar - 10.Kim, C.I., Schiavone, P., Ru, C.-Q.: The effect of surface elasticity on a mode-iii interface crack. Arch. Mech.
**63**, 267–286 (2011) MATHMathSciNetGoogle Scholar - 11.Kim, C.I., Schiavone, P., Ru, C.-Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc., Math. Phys. Eng. Sci.
**467**, 3530–3549 (2011) ADSCrossRefMATHMathSciNetGoogle Scholar - 12.Kim, C.I., Ru, C.-Q., Schiavone, P.: A clarification of the role of crack-tip conditions in linear elasticity with surface effects. Math. Mech. Solids (2012). doi: 10.1177/1081286511435227 Google Scholar
- 13.Morini, L., Piccolroaz, A., Mishuris, G., Radi, E.: Integral identities for a semi-infinite interfacial crack in anisotropic elastic bimaterials. arXiv:1205.1321 (2012)
- 14.Piccolroaz, A., Mishuris, G.: Integral identities for a semi-infinite interfacial crack in 2d and 3d elasticity. J. Elast.
**110**(2), 117–140 (2013) CrossRefMATHMathSciNetGoogle Scholar - 15.Rajagopal, K.R., Walton, J.R.: Modeling fracture in the context of a strain-limiting theory of elasticity: a single anti-plane shear crack. Int. J. Fract.
**169**, 39–48 (2011) CrossRefGoogle Scholar - 16.Rice, J.R.: Mathematical analysis in the mechanics of fracture. In: Liebowitz, H. (ed.) Fracture—An Advanced Treatise, vol. II, pp. 191–311. Pergamon, Elmsford (1968) Google Scholar
- 17.Sendova, T., Walton, J.R.: A new approach to the modeling and analysis of fracture through extension of continuum mechanics to the nanoscale. Math. Mech. Solids
**15**, 368–413 (2010) CrossRefMATHMathSciNetGoogle Scholar - 18.Slattery, J.C., Sagis, L., Oh, E.-S.: Interfacial Transport Phenomena, 2nd edn. Springer, Berlin (2007) MATHGoogle Scholar
- 19.Walton, J.R.: A note on fracture models incorporating surface elasticity. J. Elast.
**109**, 95–102 (2012). doi: 10.1007/s10659-011-9569-7 CrossRefMATHMathSciNetGoogle Scholar - 20.Zemlyanova, A.Y., Walton, J.R.: Modeling of a curvilinear planar crack with a curvature-dependent surface tension. SIAM J. Appl. Math.
**72**(5), 1474–1492 (2012) CrossRefMATHMathSciNetGoogle Scholar