Abstract
A surface slip band due to contact by a rectangular rigid flat-ended indenter is investigated. An inclined Zener–Stroh crack model is proposed to simulate the slip band. By using the fundamental solution of a single dislocation in a half plane as Green’s function, the Zener–Stroh crack is modeled with continuously distributed dislocations. It leads to a singular integral equation of the first kind, which is solved with the Gauss–Chebyshev numerical quadrature, and then stress intensity factors (SIFs) at the crack tips are evaluated. It is demonstrated that the Zener–Stroh crack model can efficiently capture micro deformation behavior of the surface slip band due to contact. With this model, the corresponding relations of the applied load, the slip band length, the relative sliding displacement of slip band and SIFs are obtained. Compared with the experimental results, it is shown that the surface Zener–Stroh crack model to contact slip band can well address such kind of contact damage problems.
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Venkataraman, G., Chung, Y.W., Nakasone, Y., Mura, T.: Free-energy formulation of fatigue crack initiation along persistent slip bands-calculation of S–N curves and crack depths. Acta Metall. Mater. 38(1), 31–40 (1990)
Tanaka, K.: A theory of fatigue crack initiation at inclusions. Metall. Trans. Phys. Metall. Mater. Sci. 13(1), 117–123 (1982)
Tanaka, K., Mura, T.: A dislocation model for fatigue crack initiation. J. Appl. Mech. Trans. ASME 48(1), 97–103 (1981)
Mura, T., Nakasone, Y.: A theory of fatigue crack initiation in solids. J. Appl. Mech. Trans. ASME 57(1), 1–6 (1990)
Lin, T.H., Ito, Y.M.: Mechanics of a fatigue crack nucleation mechanism. J. Mech. Phys. Solids 17(6), 511–523 (1969)
Lin, M.R., Fine, M.E., Mura, T.: Fatigue crack initiation on slip bands—theory and experiment. Acta Metall. 34(4), 619–628 (1986)
Fan, H., Keer, L.M., Mura, T.: Near surface crack initiation under contact fatigue. Tribol. Trans. 35(1), 121–127 (1992)
Fan, H., Keer, L.M., Mura, T.: The effect of plastic-deformation on crack initiation in fatigue. Int. J. Solids Struct. 28(9), 1095–1104 (1991)
Cheng, W., Cheng, H.S., Keer, L.M.: Experimental investigation on rolling sliding contact fatigue-crack initiation with artificial defects. Tribol. Trans. 37(1), 1–12 (1994)
Suresh, S., Nieh, T.G., Choi, B.W.: Nano-indentation of copper thin films on silicon substrates. Scr. Mater. 41(9), 951–957 (1999)
Schuh, C.A., Mason, J.K., Lund, A.C.: Quantitative insight into dislocation nucleation from high-temperature nanoindentation experiments. Nat. Mater. 4(8), 617–621 (2005)
Kiely, J.D., Hwang, R.Q., Houston, J.E.: Effect of surface steps on the plastic threshold in nanoindentation. Phys. Rev. Lett. 81(20), 4424–4427 (1998)
Kiely, J.D., Houston, J.E.: Nanomechanical properties of Au (111), (001), and (110) surfaces. Phys. Rev. B 57(19), 12588–12594 (1998)
Mason, J.K., Lund, A.C., Schuh, C.A.: Determining the activation energy and volume for the onset of plasticity during nanoindentation. Phys. Rev. B 73(5), 054102 (2006)
Gerberich, W.W., Nelson, J.C., Lilleodden, E.T., Anderson, P., Wyrobek, J.T.: Indentation induced dislocation nucleation: the initial yield point. Acta Mater. 44(9), 3585–3598 (1996)
Fivel, M.C., Robertson, C.F., Canova, G.R., Boulanger, L.: Three-dimensional modeling of indent-induced plastic zone at a meso-scale. Acta Mater. 46(17), 6183–6194 (1998)
Corcoran, S.G., Colton, R.J., Lilleodden, E.T., Gerberich, W.W.: Anomalous plastic deformation at surfaces: nanoindentation of gold single crystals. Phys. Rev. B 55(24), 16057–16060 (1997)
Chiu, Y.L., Ngan, A.H.W.: Time-dependent characteristics of incipient plasticity in nanoindentation of a Ni\(_{3}\)Al single crystal. Acta Mater. 50(6), 1599–1611 (2002)
Ma, L.F., Korsunsky, A.M., Wiercigroch, M.: Dislocation model of localized plastic deformation initiated with a flat punch. Int. J. Solids Struct. 47(7–8), 1082–1089 (2010)
Rice, J.R., Thomson, R.: Ductile versus brittle behavior of crystals. Philos. Mag. 29(1), 73–97 (1974)
Ma, L.F., Korsunsky, A.M.: Surface dislocation nucleation from frictional sliding contacts. Int. J. Solids Struct. 45(22–23), 5936–5945 (2008)
Yan, B., Zhao, J.: Dislocation nucleation near a sharp indenter in contact problems. Int. J. Fract. 155(2), 119–125 (2009)
Qiu, Y.K., Zhang, P., Ma, L.F.: Collinear micro-shear-bands model for grain-size and precipitate-size effects on the yield strength. Theor. Appl. Mech. Lett. 8(4), 245–251 (2018)
Ma, L.F., Yari, N., Wiercigroch, M.: Shear stress triggering brittle shear fracturing of rock-like materials. Int. J. Mech. Sci. 146, 295–302 (2018)
Zener, C.: The micro-mechanism of fracture. In: Jonassen, F., Roop, W.P., Bayless, R.T. (eds.) Fracturing of Metals, pp. 3–31. American Society for Metals, Cleveland (1948)
Stroh, A.N.: The formation of cracks as a result of plastic flow. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 223(1154), 404–414 (1954)
Weertman, J.: Zener–Stroh crack, Zener–Hollomon parameter, and other topics. J. Appl. Phys. 60(6), 1877–1887 (1986)
Fan, H.: Interfacial Zener–Stroh crack. J Appl Mech Trans ASME 61(4), 829–834 (1994)
Chen, Y.Z.: Multiple Zener–Stroh crack problem in an infinite plate. Acta Mech. 170(1–2), 11–23 (2004)
Xiao, Z.M., Chen, B.J., Fan, H.: A Zener–Stroh crack in a fiber-reinforced composite material. Mech. Mater. 32(10), 593–606 (2000)
Zhao, J.F., Xiao, Z.M.: An inclined Zener–Stroh crack near a free surface. Int. J. Solids Struct. 41(20), 5663–5675 (2004)
Xiao, Z.M., Chen, B.J.: Stress analysis for a Zener–Stroh crack interacting with a coated inclusion. Int. J. Solids Struct. 38(28–29), 5007–5018 (2001)
Fan, M., Yi, D.K., Xiao, Z.M.: Fracture behavior investigation on an arbitrarily oriented sub-interface Zener–Stroh crack. Acta Mech. 226(5), 1591–1603 (2014)
Gao, Y.F., Bower, A.F., Kim, K.S., Lev, L., Cheng, Y.T.: The behavior of an elastic-perfectly plastic sinusoidal surface under contact loading. Wear 261(2), 145–154 (2006)
Gong, Z.Q., Komvopoulos, K.: Effect of surface patterning on contact deformation of elastic–plastic layered media. J. Tribol. Trans. ASME 125(1), 16–24 (2003)
Majumdar, A., Bhushan, B.: Fractal model of elastic–plastic contact between rough surfaces. J. Tribol. 113(1), 1–11 (1991)
Pei, L., Hyun, S., Molinari, J.F., Robbins, M.O.: Finite element modeling of elasto-plastic contact between rough surfaces. J. Mech. Phys. Solids 53(11), 2385–2409 (2005)
Persson, B.: Elastoplastic contact between randomly rough surfaces. Phys. Rev. Lett. 87(11), 116101 (2005)
Giannakopoulos, A.E., Lindley, T.C., Suresh, S.: Overview no. 129—aspects of equivalence between contact mechanics and fracture mechanics: theoretical connections and a life-prediction methodology for fretting-fatigue. Acta Mater. 46(9), 2955–2968 (1998)
Langer, S., Weatherley, D., Olsen-Kettle, L., Finzi, Y.: Stress heterogeneities in earthquake rupture experiments with material contrasts. J. Mech. Phys. Solids 61(3), 742–761 (2013)
Lengliné, O., Elkhoury, J.E., Daniel, G., Schmittbuhl, J., Toussaint, R., Ampuero, J.P., Bouchon, M.: Interplay of seismic and aseismic deformations during earthquake swarms: an experimental approach. Earth Planet. Sci. Lett. 331–332(2), 215–223 (2012)
Rubinstein, S.M., Cohen, G., Fineberg, J.: Detachment fronts and the onset of dynamic friction. Nature 430(7003), 1005–1009 (2004)
Svetlizky, I., Fineberg, J.: Classical shear cracks drive the onset of dry frictional motion. Nature 509(7499), 205–208 (2014)
Sneddon, I.N.: Boussinesq’s problem for a flat-ended cylinder. Math. Proc. Camb. Philos. Soc. 42(1), 29–39 (1946)
Yu, H.H., Shrotriya, P., Gao, Y.F., Kim, K.S.: Micro-plasticity of surface steps under adhesive contact: part I—surface yielding controlled by single-dislocation nucleation. J. Mech. Phys. Solids 55(3), 489–516 (2007)
Hills, D.A., Kelly, P.A., Dai, D.N., Korsunsky, A.M.: Solution of Crack Problems: The Distributed Dislocation Technique. Springer, Berlin (1996)
Nowell, D., Hills, D.A.: Open cracks at or near free edges. J. Strain Anal. 22, 177–185 (1987)
Erdogan, F., Gupta, G.D.: On the numerical solution of singular integral equations. Q. Appl. Math. 29(4), 525–534 (1972)
Weertman, J.: Dislocation Based Fracture Mechanics. World Scientific, Singapore (1996)
Krenk, S.: On the use of the interpolation polynomial for solutions of singular integral equations. Q. Appl. Math. 32(4), 479–484 (1975)
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This work is partially supported by National Natural Science Foundation of China (Grant No. 41630634).
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Appendices
Appendix A: The influence function and transformation relationship in two coordinate systems
The influence function due to a single dislocation in a half plane was given by Hills et al. [46]
where
Note that in global and local coordinate systems we have
Equation (A.2) can be expressed in local coordinate system as
The transformation matrix M is given by
Appendix B
The regular term \({K}'\left( {t;s} \right) \) of singular integral Eq. (19) is
where
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Zhang, Y., Ma, L. Surface Zener–Stroh crack model to slip band due to contact. Arch Appl Mech 90, 221–234 (2020). https://doi.org/10.1007/s00419-019-01606-0
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DOI: https://doi.org/10.1007/s00419-019-01606-0