Abstract
Based on a two-dimensional atomistic model, we study the bifurcation associated with crack propagation and kinking. In the study, the varying parameter is the tensile and shear strain applied at the boundary. Both mode-I and mixed mode-I/II loading are considered.
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References
J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity (Dover Publications Inc., New York, 1994)
A.A. Griffith, Philos. Trans. R. Soc. Lond. A 221, 163 (1921)
G.R. Irwin, J. Appl. Mech. 24, 361 (1957)
A. Azhdari, S. Nemat-Nasser, J. Mech. Phys. Solids 44, 929 (1996)
B. Cotterell, J.R. Rice, Int. J. Fract. 16, 155 (1980)
M.E. Gurtin, P. Podio-Guidugli, J. Mech. Phys. Solids 46, 1343 (1998)
K. Hayashi, S. Nemat-Nasser, Int. J. Solids Struct. 17, 107 (1981)
B.L. Karihaloo, Mech. Mater. 1, 189 (1982)
W. Yang, Z. Suo, C.F. Shih, Proc. R. Soc. Lond. A 433, 679 (1991)
H. Gao, C.-H. Chiu, Int. J. Solids Struct. 29, 947 (1992)
H.-Y. Yeh, C.H. Kim, Eng. Fract. Mech. 48, 595 (1994)
Y. Zhao, Eng. Fract. Mech. 34, 891 (1989)
J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of vector fields (Springer-Verlag, Berlin, 1983)
R. Thompson, C. Hsieh, V. Rana, J. Appl. Phys. 42, 3154 (1971)
C. Hsieh, J. Appl. Phys. 44, 2051 (1973)
E.R. Fuller, R.M. Thomson, Lattice theories of fracture, in Fracture Mechanics of Ceramics; Proceedings of the International Symposium, University Park, PA., 1978
D. Holland, M. Marder, Adv. Phys. 11, 793 (1999)
M. Mardar, X. Liu, Phys. Rev. Lett. 71, 2417 (1993)
M. Marder, S.P. Gross, J. Mech. Phys. Solids 43, 1 (1995)
L.I. Slepyan, Sov. Phys. Doklady 26, 538 (1981)
R.S. Elliott, J.A. Shaw, N. Triantafyllidis, J. Mech. Phys. Solids 50, 2463 (2002)
I. Plans, A. Carpio, L.L. Bonilla, Europhys. Lett. 81, 36001 (2008)
I. Plans, A. Carpio, L.L. Bonilla, Chaos Sol. Fract. 42, 1623 (2009)
V. Shastry, D. Farkas, Molecular statics simulation of crack propagation in alpha-Fe using EAM potentials, Materials Research Society Symposium Proceedings, Boston, 1995 (1996), pp. 75–80
P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62, 251 (1990)
T. Zhu, J. Li, S. Yip, Proc. R. Soc. Lond. A 462, 1741 (2006)
B. Lawn, Fracture of Brittle Solids (Cambridge University Press, Cambridge, 1993)
W.C. Rheinboldt, SIAM J. Numer. Anal. 17, 221 (1980)
W.C. Rheinboldt, J.V. Burkardt, ACM Trans. Math. Software 9, 215 (1983)
R. Seydel, Practical Bifurcation and Stability Analysis (Springer-Verlag, New York, 2010)
I.S. Duff, J.K. Reid, ACM Trans. Math. Software 22, 187 (1995)
M. Kubíček, A. Klič, Appl. Math. Comp. 13, 125 (1983)
D.C. Liu, J. Nocedal, Math. Prog. B 45, 503 (1989)
K.S. Cheung, S. Yip, Model. Simul. Mater. Sci. Eng. 2, 865 (1994)
J. Carr, Applications of Centre Manifold Theory (Springer-Verlag, New York, 1981)
T. Ma, S. Wang, Bifurcation Theory and Applications (World Scientific, Singapore, 2005)
Dynamic Bifurcations, edited by E. Benoît (Springer-Verlag, Berlin, 1990)
N.R. Lebovitz, A.I. Pesci, SIAM J. Appl. Math. 55, 1117 (1995)
J. Su, The IMA Volumes in Mathematics and its Applications 122, 203 (2001)
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Li, X. A bifurcation study of crack initiation and kinking. Eur. Phys. J. B 86, 258 (2013). https://doi.org/10.1140/epjb/e2013-40145-9
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DOI: https://doi.org/10.1140/epjb/e2013-40145-9