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Influence of surface energy on the nanoindentation response of elastically-layered viscoelastic materials

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Abstract

In the context of integrated nonlinear viscoelastic contact mechanics, a nonlinear finite element model is developed to predict and analyze the quasistatic response of nanoindentation problems of an elastically-layered viscoelastic materials considering the surface elasticity effects. Effects of surface energy are accounted for by employing the Gurtin–Murdoch continuum model for surface elasticity. The linear viscoelastic response is modeled by the Schapery’s creep model with a Prony’s series to express the transient component in the creep compliance. The viscoelastic constitutive equations are cast into a recursive form that needs only the previous time increment rather than the entire strain history. To satisfy the contact constraints exactly, the Lagrange multiplier method is adopted to enforce the contact conditions into the system. The equilibrium indentation configuration is obtained through the Newton–Raphson iterative procedure. The developed model is verified then applied to investigate the quasistatic nanoindentation response of two different indentation problems with different geometry and loading conditions. Results show the significant effects of surface energy and viscoelasticity on the quasistatic nanoindentation response.

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References

  • Beegan, D., Chowdhury, S., Laugier, M.T.: Comparison between nanoindentation and scratch test hardness (scratch hardness) values of copper thin films on oxidised silicon substrates. Surf. Coat. Technol. 201, 5804–5808 (2007)

    Article  Google Scholar 

  • Cammarata, R.C.: Surface and interface stress effects in thin films. Prog. Surf. Sci. 46, 1–38 (1994)

    Article  Google Scholar 

  • Cammarata, R.C.: Surface and interface stress effects on interfacial and nanostructured materials. Mater. Sci. Eng. A 237, 180–184 (1997)

    Article  Google Scholar 

  • Chen, W.Q., Zhang, C.: Anti-plane shear Green’s functions for an isotropic elastic half-space with a material surface. Int. J. Solids Struct. 47(11), 1641–1650 (2010)

    Article  MATH  Google Scholar 

  • Chen, Z., Diebels, S.: Nanoindentation of soft polymers: modeling, experiments and parameter identification. Tech. Mech. 34(3–4), 166–189 (2014)

    Google Scholar 

  • Dingreville, R., Qu, J., Cherkaoui, M.: Surface free energy and its effect on the elastic behavior of nanosized particles, wires and films. J. Mech. Phys. Solids 53, 1827–1854 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Eshelby formalism for nano-inhomogeneities. Proc. R. Soc. A 461, 3335–3353 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Fischer, F.D., Waitz, T., Vollath, D., Simha, N.K.: On the role of surface energy and surface stress in phase-transforming nanoparticles. Prog. Mater. Sci. 53, 481–527 (2008)

    Article  Google Scholar 

  • Gad, A.I., Mahmoud, F.F., Alshorbagy, A.E., Ali-Eldin, S.S.: Finite element modeling for elastic nano-indentation problems incorporating surface energy effect. Int. J. Mech. Sci. 84, 158–170 (2014)

    Article  Google Scholar 

  • Gao, W., Yu, S., Huang, G.: Finite element characterization of the size dependent mechanical behaviour in nanosystems. Nanotechnology 17, 1118–1122 (2006)

    Article  Google Scholar 

  • Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  MATH  Google Scholar 

  • Gurtin, M.E., Weissmüller, J., Larché, F.: A general theory of curved deformable interfaces in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)

    Article  Google Scholar 

  • Hainsworth, S.V., Page, T.F.: Nanoindentation studies of the chemomechanical effect in sapphire. J. Mater. Sci. 29, 5529–5540 (1994)

    Article  Google Scholar 

  • Haj-Ali, R.M., Muliana, H.A.: Numerical finite element formulation of the Schapery nonlinear viscoelastic material model. Int. J. Numer. Methods Eng. 59(1), 25–45 (2004)

    Article  MATH  Google Scholar 

  • He, L.H., Lim, C.W.: Surface green function for a soft elastic half-space: influence of surface stress. Int. J. Solids Struct. 43, 132–143 (2006)

    Article  MATH  Google Scholar 

  • He, L.H., Lim, C.W., Wu, B.S.: A continuum model for size dependent deformation of elastic films of nano-scale thickness. Int. J. Solids Struct. 41, 847–857 (2004)

    Article  MATH  Google Scholar 

  • Huang, D.W.: Size dependent response of ultra-thin films with surface effects. Int. J. Solids Struct. 45, 568–579 (2008)

    Article  MATH  Google Scholar 

  • Huang, G.Y., Yu, S.W.: Effect of surface elasticity on the interaction between steps. J. Appl. Mech. 74, 821–823 (2007)

    Article  Google Scholar 

  • Intarit, P., Senjuntichai, T., Rajapakse, R.K.N.D.: Dislocations and internal loading in a semi-infinite elastic medium with surface stresses. Eng. Fract. Mech. 77, 3592–3603 (2010)

    Article  Google Scholar 

  • Intarit, P., Senjuntichai, T., Rungamornrat, J., Rajapakse, R.K.N.D.: Surface elasticity and residual stress effect on the elastic field of a nanoscale elastic layer. Interact. Multiscale Mech. 4, 85–105 (2011)

    Article  Google Scholar 

  • Jing, G.Y., Duan, H.L., Sun, X.M., Zhang, Z.S., Xu, J., Li, Y.D., et al.: Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys. Rev. B 73, 235409 (2006)

    Article  Google Scholar 

  • Koguchi, H.: Surface Green function with surface stresses and surface elasticity using Stroh’s formalism. J. Appl. Mech. 75, 061014 (2008)

    Article  Google Scholar 

  • Lai, J., Bakker, A.: 3-D Schapery representation for nonlinear viscoelasticity and finite element implementation. Comput. Mech. 18, 182–191 (1996)

    Article  MATH  Google Scholar 

  • Liu, C., Rajapakse, R.K.N.D., Phani, A.S.: Finite element modeling of beams with surface energy effects. J. Appl. Mech. 78, 031014 (2011)

    Article  Google Scholar 

  • Long, J.M., Wang, G.F.: Effects of surface tension on axisymmetric Hertzian contact problem. Mech. Mater. 56, 65–70 (2013)

    Article  Google Scholar 

  • Long, J.M., Wang, G.F., Feng, X.Q., Yu, S.W.: Two dimensional Hertzian contact problem with surface tension. Int. J. Solids Struct. 49, 1588–1594 (2012)

    Article  Google Scholar 

  • Lu, P., He, L.H., Lee, H.P., Lu, C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43, 4631–4647 (2006)

    Article  MATH  Google Scholar 

  • Mahmoud, F.F., El-Shafei, A.G., Abdelrahman, A.A., Attia, M.A.: Modeling of nonlinear viscoelastic contact problems with large deformations. Appl. Math. Model. 37, 6730–6745 (2013)

    Article  MathSciNet  Google Scholar 

  • Miller, R.E., Shenoy, V.B.: Size dependent elastic properties of nanosized structural elements. Nanotechnology 11(3), 139–147 (2000)

    Article  Google Scholar 

  • Mogilevskaya, S.G., Crouch, S.L., Stolarski, H.K.: Multiple interacting circular nano-inhomogeneities with surface/interface effects. J. Mech. Phys. Solids 56(6), 2298–2327 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Nix, W.D., Gao, H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425 (1998)

    Article  MATH  Google Scholar 

  • Pinyochotiwong, Y., Rungamornrat, J., Senjuntichai, T.: Rigid frictionless indentation on elastic half space with influence of surface stresses. Int. J. Eng. Sci. 71, 15–35 (2013)

    Article  MathSciNet  Google Scholar 

  • Ru, C.Q.: Simple geometrical explanation of Gurtin–Murdoch model of surface elasticity with clarification of its related versions. J. Phys. Mech. Astron. 53, 536–544 (2010)

    Article  MathSciNet  Google Scholar 

  • Schapery, R.A.: On the characterization of nonlinear viscoelastic materials. Polym. Eng. Sci. 9(4), 295–310 (1969)

    Article  Google Scholar 

  • Schapery, R.A.: Correspondence principles and a generalized J integral for large deformation and fracture analysis of viscoelastic media. Int. J. Fract. 25, 195–223 (1984)

    Article  Google Scholar 

  • Shaat, M., Eltaher, M.A., Gad, A.I., Mahmoud, F.F.: Nonlinear size dependent finite element analysis of functionally graded tiny bodies. Int. J. Mech. Sci. 77, 356–364 (2013a)

    Article  Google Scholar 

  • Shaat, M., Mahmoud, F.F., Alieldin, Alshorbagy, A.E.: Finite element analysis of graded nanoscale films. Finite Elem. Anal. Des. 74, 41–52 (2013b)

    Article  MathSciNet  MATH  Google Scholar 

  • Sharma, P., Wheeler, L.T.: Size dependent elastic state of ellipsoidal nano-inclusions incorporating surface/interface tension. J. Appl. Mech. 74, 447–454 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003)

    Article  Google Scholar 

  • Shenoy, V.B.: Atomistic calculations of elastic properties of metallic FCC crystal surfaces. Phys. Rev. B 71, 094104 (2005)

    Article  Google Scholar 

  • Tang, G., Shen, Y.L., Singh, D.R.P., Chawla, N.: Analysis of indentation-derived effective elastic modulus of metal-ceramic multilayers. Int. J. Mech. Mater. Des. 4(4), 391–398 (2008)

    Article  Google Scholar 

  • Tian, L., Rajapakse, R.K.N.D.: Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity. Int. J. Solids Struct. 44, 7988–8005 (2007)

    Article  MATH  Google Scholar 

  • Wang, G.F., Feng, X.Q.: Effects of surface stresses on contact problems at nanoscale. J. Appl. Phys. 101, 013510 (2007)

    Article  Google Scholar 

  • Wang, G., Bian, J., Feng, J., Feng, X.: Compressive behavior of crystalline nanoparticles with atomic-scale surface steps. Mater. Res. Express 2(1), 015006 (2015)

    Article  Google Scholar 

  • Wang, Z.Q., Zhao, Y.P., Huang, Z.P.: The effects of surface tension on the elastic properties of nano structures. Int. J. Eng. Sci. 48(2), 140–150 (2010)

    Article  MathSciNet  Google Scholar 

  • Wriggers, P.: Computational Contact Mechanics, 2nd edn. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  • Yakobson, B.I.: Nanomechanics. In: Goddard, W.A., Brenner, D.W., Lyshevski, S.E., Iafrate, G.J. (eds.) Handbook of Nanoscience, Engineering and Technology, pp. 17.1–17.18. CRC Press, Florida (2003)

    Google Scholar 

  • Zhao, X.J., Rajapakse, R.K.N.D.: Analytical solutions for a surface-loaded isotropic elastic layer with surface energy effects. Int. J. Eng. Sci. 47, 1433–1444 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, X.J., Rajapakse, R.K.N.D.: Elastic field of a nano-film subjected to tangential surface load: asymmetric problem. Eur. J. Mech. A Solids. 39, 69–75 (2013)

    Article  MathSciNet  Google Scholar 

  • Zhao, X. J.: Surface loading and Rigid Indentation of an Elastic Layer with Surface Energy Effects. Master thesis, The University of British Columbia, Vancouver, Canada (2009)

  • Zhou, S., Gao, X.L.: Solutions of half-space and half-plane contact problems based on surface elasticity. ZAMP 64, 145–166 (2013)

    MathSciNet  MATH  Google Scholar 

  • Zhou, S.: New Solution of Half Space Contact Problems Using Potential Theory, Surface Elasticity and Strain Gradient Elasticity, PhD dissertation, Texas A&M University (2011)

  • Zocher, M.A., Groves, S.E., Allen, D.H.: A three dimensional finite element formulation for thermoviscoelastic orthotropic media. Int. J. Numer. Methods Eng. 40, 2267–2280 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Design and Production Engineering, College of Engineering, Zagazig University, Zagazig, 44511, Egypt

    Alaa A. Abdel Rahman, Ahmed G. El-Shafei & Fatin F. Mahmoud

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  1. Alaa A. Abdel Rahman
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  2. Ahmed G. El-Shafei
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  3. Fatin F. Mahmoud
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Correspondence to Alaa A. Abdel Rahman.

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Abdel Rahman, A.A., El-Shafei, A.G. & Mahmoud, F.F. Influence of surface energy on the nanoindentation response of elastically-layered viscoelastic materials. Int J Mech Mater Des 12, 193–209 (2016). https://doi.org/10.1007/s10999-015-9301-6

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