Abstract
We study the effect of surface elasticity on an arc-shaped crack in a linearly elastic isotropic homogeneous material under antiplane shear deformation. The surface mechanics is incorporated by using a continuum-based surface/interface model of Gurtin and Murdoch. We obtain a complete solution by reducing the problem to two decoupled first-order Cauchy-type singular integro-differential equations. It is shown that different from the case of a straight crack, the stresses exhibit both the weak logarithmic and the strong square root singularities at the tips of the arc crack.
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Antipov Y.A., Schiavone P.: Integro-differential equation for a finite crack in a strip with surface effects. Q. J. Mech. Appl. Math. 64, 87–106 (2011)
Chen T., Dvorak G.J., Yu C.C.: Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mech. 188, 39–54 (2007)
Gurtin M.E., Murdoch A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
Gurtin M.E., Murdoch A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)
Gurtin M.E., Weissmuller J., Larche F.: A general theory of curved deformable interface in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)
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 18, 59–66 (2013)
Kim C.I., Schiavone P., Ru C.Q.: The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. ASME J. Appl. Mech. 77, 021011-1–021011-7 (2010a)
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 (2010b)
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 (2011a)
Kim C.I., Schiavone P., Ru C.Q.: The effects of surface elasticity on mode-III interface crack. Arch. Mech. 63, 267–286 (2011b)
Kim C.I., Schiavone P., Ru C.Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc. Lond. A 467, 3530–3549 (2011c)
Lengyel T.H., Schiavone P.: Displacement field in an elastic solid with mode-III crack and first-order surface effects. J. Mech. Mater. Struct. 7(8), 783–794 (2012)
Markenscoff X., Dundurs J.: Annular inhomogeneities with eigenstrain and interphase modeling. J. Mech. Phys. Solids 64, 468–482 (2014)
Ru C.Q.: Simple geometrical explanation of Gurtin–Murdoch model of surface elasticity with clarification of its related versions. Sci. China 53, 536–544 (2010)
Sigaeva, T., Schiavone, P.: The effect of surface stress on an interface crack in linearly elastic materials. Math. Mech. Solids (2014). doi:10.1177/1081286514534871
Steigmann D.J., Ogden R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond. A. 453, 853–877 (1997)
Walton J.R.: A note on fracture models incorporating surface elasticity. J. Elast. 109, 95–102 (2012)
Zemlyanova, A.Y.: Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary. arXiv preprint arXiv:1409.3501 (2014)
Zemlyanova A.Y., Walton J.R.: Modeling of a curvilinear planar crack with a curvature-dependent surface tension. SIAM J. Appl. Math. 72, 1474–1492 (2012)
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Wang, X. A mode III arc-shaped crack with surface elasticity. Z. Angew. Math. Phys. 66, 1987–2000 (2015). https://doi.org/10.1007/s00033-014-0482-x
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DOI: https://doi.org/10.1007/s00033-014-0482-x