Abstract
A finite-form solution for the problem of an elastic body weakened by a straight nano-crack and disturbed by a screw dislocation in the mode-III deformation is proposed. The boundary equations incorporating the surface effects are established and reduced to single one via the integration over the crack boundary in the physical region. The cracked physical region is then mapped onto an image region with circular hole. By using Muskhelishvili (Some basic problems of the mathematical theory of elasticity, Noordhoff, Groningen, 1953) method and Cauchy’s integral formula, the reduced boundary equation is further converted into a first-order differential equation. Consequently, the solution of the differential equation is mapped back to the physical region and the complex potential representing the displacement is formulated in finite form. The results show that the stresses at the crack tip are finite and the stress singularity is eliminated completely by the incorporation of the surface effects. The non-singularity of the stresses at the crack tip derived from the present article radically differs from the square-root singularity obtained from the classic theory of the elastic fracture and, however, is reasonable from the physical essence. The numerical results shows that when the surface effects are incorporated, the magnitudes of the stresses and the image forces near the crack tip are reduced, which are strongly dependent on the crack size. The corresponding results approach those from the classic theory as the crack length increases. The results by the present solution are also compared with those from other solutions.
Similar content being viewed by others
References
Abraham, F.F., Broughton, J.Q., Bernstein, N., Kaxiras, E.: Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhys. Lett. 44, 783–787 (1998)
Buehler, M.J., Gao, H.J.: Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nat. Lond. 439, 307–310 (2006)
Dewapriya, M.A.N., Meguid, S.A.: Atomistic modeling of out-of-plane deformation of a propagating Griffith crack in graphene. Acta Mech. 228, 3063 (2017). https://doi.org/10.1007/s00707-017-1883-7
Hille, E.: Ordinary Differential Equations in the Complex Domain. Dover Publications, New York (1997). ISBN:978-0486696201
England, A.H.: Complex Variable Methods in Elasticity. Wiley, London (1971)
Fang, Q.H., Liu, Y.W.: Size-dependent interaction between an edge dislocation and a nanoscale inhomogeneity with interface effects. Acta Mater. 54(16), 4213–4220 (2006)
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)
Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)
He, L.H., Li, Z.R.: Impact of surface stress on stress concentration. Int. J. Solids Struct. 43(20), 6208–6219 (2006)
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(2), 293–298 (2010)
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)
Lee, S.: The image force on the screw dislocation around a crack of finite size. Eng. Fract. Mech. 27(5), 539–545 (1987)
Luo, J., Xiao, Z.M.: Analysis of a screw dislocation interacting with an elliptical nano inhomogeneity. Int. J. Eng. Sci. 47(9), 883–893 (2009)
Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1953)
Ou, Z.Y., Pang, S.D.: A screw dislocation interacting with a coated nano-inhomogeneity incorporating interface stress. Mater. Sci. Eng. A 528(6), 2762–2775 (2011)
Rabczuk, T., Belytschko, T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61(13), 2316–2343 (2004)
Rabczuk, T., Zi, G., Bordas, S., Nguyen-Xuanet, H.: A simple and robust three-dimensional cracking-particle method without enrichment. Comput. Methods Appl. Mech. Eng. 199(37–40), 2437–2455 (2010)
Sharma, P., Ganti, S.: Size-dependent Eshelby’s tensor for embedded nano-lnclusions incorporating surface/interface energies. J. Appl. Mech. 72(4), 663–671 (2005)
Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82(4), 535–537 (2003)
Shodja, H.M., Ahmadzadeh-Bakhshayesh, H., Gutkin, M.Y.: Size-dependent interaction of an edge dislocation with an elliptical nano-inhomogeneity incorporating interface effects. Int. J. Solids Struct. 49(5), 759–770 (2012)
Sun, C.T., Jin, Z.H.: Fracture Mechanics. Academic Press, Oxford (2012)
Wang, X., Fan, H.: Interaction between a nanocrack with surface elasticity and a screw dislocation. Math. Mech. Solids 22(2), 1–13 (2015)
Wong, E.W., Sheehan, P.E., Lieber, C.M.: Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277(26), 1971–1975 (1997)
Walton, J.R.: A note on fracture models incorporating surface elasticity. J. Elast. 109, 95–102 (2012)
Acknowledgements
This work was supported by the National Key Research and Development Plan of China (Grant No. 2016YFC0303700), and the Key Laboratory for Damage Diagnosis of Engineering Structures of Hunan Province.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, Z., Xiao, W., Xi, J. et al. Finite-form solution for anti-plane problem of nanoscale crack. Arch Appl Mech 90, 385–396 (2020). https://doi.org/10.1007/s00419-019-01615-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-019-01615-z