Two kinds of contact problems in three-dimensional icosahedral quasicrystals

doi:10.1007/s10483-015-2006-6

Abstract

Abstract: Two kinds of contact problems, i.e., the frictional contact problem and the adhesive contact problem, in three-dimensional(3D) icosahedral quasicrystals are discussed by a complex variable function method. For the frictional contact problem, the contact stress exhibits power singularities at the edge of the contact zone. For the adhesive contact problem, the contact stress exhibits oscillatory singularities at the edge of the contact zone. The numerical examples show that for the two kinds of contact problems, the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone. The phonon-phason coupling constant has a significant effect on the contact stress intensity, while has little impact on the contact stress distribution regulation. The results are consistent with those of the classical elastic materials when the phonon-phason coupling constant is 0. For the adhesive contact problem, the indentation force has positive correlation with the contact displacement, but the phonon-phason coupling constant impact is barely perceptible. The validity of the conclusions is verified.

Key words: complex variable function method, Riemann-Hilbert problem, contact problem, three-dimensional(3D) icosahedral quasicrystal, singularity

2010 MSC Number:

O346.1

Xuefen ZHAO, Xing LI, Shenghu DING. Two kinds of contact problems in three-dimensional icosahedral quasicrystals. Applied Mathematics and Mechanics (English Edition), 2015, 36(12): 1569-1580.

TrendMD

References

[1] Hertz, H. On the contact of elastic solids(in German). Journal für die Reine und Angewandte Mathematik, 92, 156-171(1881)
[2] Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, London(1927)
[3] Galin, L. A. Contact Problems in the Classical Theory of Elasticity, Gostekhizdat, Gostekhizdat, Moscow/Leningrad(1953)
[4] Gladwell, G. M. L. Contact Problems in the Classical Elasticity Theory, Beijing Institute of Technology Press, Beijing(1991)
[5] Bagault, C., Nelias, D., and Baietto, M. C. Contact analyses for anisotropic halfspace:effect of the anisotropy on the pressure distribution and contact area. Journal of Tribology, 134, 1-8(2012)
[6] Bagault, C., Nelias, D., Baietto, M. C., and Ovaert, T. C. Contact analyses for an isotropic halfspace coated with an anisotropic layer:effect of the anisotropy on the pressure distribution and contact area. International Journal of Solids and Structures, 50, 43-54(2013)
[7] Mehmet, A. G. Closed-form solution of the two-dimensional sliding frictional contact problem for an orthotropic medium. International Journal of Mechnaical Sciences, 87, 72-88(2014)
[8] Alinia, Y., Guler, M. A., and Adibnazari, S. On the contact mechanics of a rolling cylinder on a graded coating, part 1:analytical formulation. Mechanics of Materials, 68, 207-216(2014)
[9] Chidlow, S. J., Chong, W. W. F., and Teodorescu, M. On the two-dimensional solution of both adhesive and non-adhesive contact problems involving functionally graded materials. European Journal of Mechanics, A:Solids, 39, 86-103(2013)
[10] Peric, D. and Owen, D. R. J. Computational model for 3D contact problems with friction based on the penalty method. International Journal for Numerical Methods in Engineering, 35, 1289-1309(1992)
[11] Shao, S. B., Gang, X. Y., Li, S., and Chai, S. A reduced quadratic programming method for elastic contact problems(in Chinese). Mechanics and Practice, 36, 604-610(2014)
[12] Yan, X. M., Guo, X. P., Sun, T. S., and Xu, K. J. The linear spread loci of SMS4(in Chinese). Journal of Qingdao University(Natural Science Edition), 27, 1-8(2014)
[13] Migorski, S. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete and Continuous Dynamical Systems, Series B, 6, 1339-1356(2006)
[14] Liu, Z. and Migorski, S. Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete and Continuous Dynamical Systems, Series B, 9, 129-143(2008)
[15] Migorski, S., Ochal, A., and Sofonea, M. Weak solvability of a piezoelectric contact problem. European Journal of Applied Mathematics, 20, 145-167(2009)
[16] Migorski, S., Ochal, A., and Sofonea, M. A dynamic frictional contact problem for piezoelectric materials. Journal of Mathematical Analysis and Applications, 361, 161-176(2010)
[17] Mikae, B. and Mirceal, S. Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation. Applied Mathematics and Computation, 215, 2978-2991(2009)
[18] Matei, A. and Ciurcea, R. Weak solvability for a class of contact problems. Annals of the Academy of Romanian Scientists Series on Mathematics and Its Applications, 2, 25-44(2010)
[19] Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53, 1951-1953(1984)
[20] Fan, T. Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications, Science Press, Beijing(2010)
[21] Peng, Y. Z. and Fan, T. Y. Crack and indentation problems for one-dimensional hexagonal quasicrystals. The European Physical Journal B, 21, 39-44(2001)
[22] Guo, J. H., Jing, Y., and Riguleng, S. A semi-inverse method of a Griffith crack in one-dimensional hexagonal quasicrystals. Applied Mathematics and Computation, 219, 7445-7449(2013)
[23] Yang, L. Z., Gao, Y., Ernian, P., and Natalie, W. An exact solution for a multilayered twodimensional decagonal quasicrystal plate. International Journal of Solids and Structures, 51, 1737-1749(2014)
[24] Fan, T. Y. and Guo, L. H. The final governing equation and fundamental solution of plane elasticity of icosahedral quasicrystals. Physics Letters A, 341, 235-239(2005)
[25] Zhou, W. M. and Song, Y. H. Moving screw dislocation in cubic quasicrystal. Applied Mathematics and Mechanics(English Edition), 26(12), 1611-1614(2005) DOI 10.1007/BF03246270
[26] Li, L. H. and Fan, T. Y. Final governing equation of plane elasticity of icosahedral quasicrystals and general solution based on stress potential function. Chinese Physics Letters, 9, 2519-2521(2006)
[27] Li, L. H. and Fan, T. Y. Complex variable method for plane elasticity of icosahedral quasicrystals and elliptic notch problem. Science in China, Series G:Physics, Mechanics and Astronomy, 51, 773-780(2008)
[28] Azhazha, V. M., Borisova, S. S., Dub, S. N., Malykhin, S. V., Pugachov, A. T., Merisov, B. A., and Khadzhay, G. Y. Mechanical behavior of Ti-Zr-Ni quasicrystals during nanoindentayion. Physics of the Solid State, 47, 2262-2267(2005)
[29] Mukhopadhyay, N. K., Belger, A., and Paufler, R. Nanomechanical characterization of Al-Co-Ni decagonal quasicrystals. Philosophical Magazine, 86, 999-1005(2006)
[30] Peng, Y. Z. and Fan, T. Y. Crack and indentation problems for one-dimensional hexagonal quasicrystals. The European Physical Journal B, 21, 39-44(2001)
[31] Zhou, W. M. Contact problem in decagonal two-dimensional quasicrystal. Journal of Beijing Institute of Technology, 10, 51-55(2001)
[32] Yin, S. Y., Zhou, W. M., and Fan, T. Y. Contact problem in octagonal two-dimensional quasierystalline material(in Chinese). Chinese Quarterly of Mechanics, 23, 255-259(2002)
[33] Wu, Y. F. Indentation Analysis of Piezoelectric Materials and Quasicrystals(in Chinese), Ph. D. dissertation, Zhejiang University, Hangzhou, 97-148(2012)
[34] Gao, Y. and Ricoeur, A. Three-dimensional Green's function for two-dimensional quasicrystal bimaterials. Proceedings of the Royal Society of London, Series A:Mathematical and Physical, 467, 2622-2642(2011)
[35] Wang, X., Zhang, J. Q., and Guo, X. M. Two kinds of contact problems in decagonal quasicrystalline materials of point group 10 mm(in Chinese). ACTA Mechanica Sinica, 37, 169-174(2005)
[36] Zhou, W. M., Fan, T. Y., and Yin, S. Y. Axisymmetric contact problem of cubic quasicrystalline materials(in Chinese). Acta Mechanica Solidarity Sinica, 15, 68-74(2002)