Characterization of the Functionally Graded Shear Modulus of a Half-Space - Publication - MOST Wiedzy


Characterization of the Functionally Graded Shear Modulus of a Half-Space


In this article, a method is proposed for determining parameters of the exponentialy varying shear modulus of a functionally graded half-space. The method is based on the analytical solution of the problem of pure shear of an elastic functionally graded half-space by a strip punch. The half-space has the depth-wise exponential variation of its shear modulus, whose parameters are to be determined. The problem is reduced to an integral equation that is then solved by asymptotic methods. The analytical relations for contact stress under the punch, displacement of the free surface outside the contact area and other characteristics of the problem are studied with respect to the shear modulus parameters. The parameters of the functionally graded half-space shear modulus are determined (a) from the coincidence of theoretical and experimental values of contact stresses under the punch and from the coincidence of forces acting on the punch, or (b) from the coincidence of theoretical and experimental values of displacement of the free surface of the half-space outside the contact and coincidence of forces acting on the punch, or (c) from other conditions. The transcendental equations for determination of the shear modulus parameters in cases (a) and (b) are given. By adjusting the parameters of the shear modulus variation, the regions of “approximate-homogeneous” state in the functionally graded half-space are developed.


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artykuły w czasopismach
Published in:
Mathematics no. 8, pages 1 - 18,
ISSN: 2227-7390
Publication year:
Bibliographic description:
Zelentsov V., Lapina P., Mitrin B., Eremeev V.: Characterization of the Functionally Graded Shear Modulus of a Half-Space// Mathematics -Vol. 8,iss. 4 (2020), s.1-18
Digital Object Identifier (open in new tab) 10.3390/math8040640
Bibliography: test
  1. Shiraki, Y.; Usami, N. (Eds.) Silicon-Germanium (SiGe) Nanostructures: Production, Properties and Applications in Electronics; Woodhead Publishing: Cambridge, UK, 2011. open in new tab
  2. Miyamoto, Y.; Kaysser, W.A.; Rabin, B.H.; Kawasaki, A.; Ford, R.G. (Eds.) Functionally Graded Materials: Design, Processing and Applications; Springer: New York, NY, USA, 1999. open in new tab
  3. Gibson, R.E. Some results concerning displacements and stresses in a non-homogeneous elastic half-space. Geotechnique 1967, 17, 58-67. doi:10.1680/geot.1967.17.1.58. open in new tab
  4. Selvadurai, A.P.S.; Singh, B.M.; Vrbik, J. A Reissner-Sagoci problem for a non-homogeneous elastic solid. J. Elast. 1986, 16, 383-391. doi:10.1007/BF00041763. open in new tab
  5. Giannakopoulos, A.E.; Suresh, S. Indentation of solids with gradients in elastic properties: Part I. Point force. Int. J. Solids Struct. 1997, 34, 2357-2392. doi:10.1016/S0020-7683(96)00171-0. open in new tab
  6. Giannakopoulos, A.E.; Suresh, S. Indentation of solids with gradients in elastic properties: Part II. Axisymmetric indentors. Int. J. Solids Struct. 1997, 34, 2393-2428. doi:10.1016/S0020-7683(96)00172-2. open in new tab
  7. Guler, M.A.; Erdogan, F. Contact mechanics of graded coatings. Int. J. Solids Struct. 2004, 41, 3865-3889. doi:10.1016/j.ijsolstr.2004.02.025. open in new tab
  8. Selvadurai, A.P.S.; Katebi, A. Mindlin's problem for an incompressible elastic half-space with an exponential variation in the linear elastic shear modulus. Int. J. Eng. Sci. 2013, 65, 9-21. doi:10.1016/j.ijengsci.2013.01.002. open in new tab
  9. Tokovyy, Y.; Ma, C.-C. An analytical solution to the three-dimensional problem on elastic equilibrium of an exponentially-inhomogeneous layer. J. Mech. 2015, 31, 545-555. doi:10.1017/jmech.2015.17. open in new tab
  10. Selvadurai, A.P.S.; Katebi, A. The Boussinesq-Mindlin problem for a non-homogeneous elastic halfspace. Z. Angew. Math. Phys. 2016, 67, 68. doi:10.1007/s00033-016-0661-z. open in new tab
  11. Altenbach, H.; Eremeyev, V.A. Eigen-vibrations of plates made of functionally graded material. Comput. Mater. Con. 2009, 9, 153-177.
  12. Kulchytsky-Zhyhailo, R.; Bajkowski, A. Analytical and numerical methods of solution of three-dimensional problem of elasticity for functionally graded coated half-space. Int. J. Mech. Sci. 2012, 54, 105-112. doi:10.1016/j.ijmecsci.2011.10.001. open in new tab
  13. Awojobi, A.O. On the hyperbolic variation of elastic modulus in a non-homogeneous stratum. Int. J. Solids Struct. 1976, 12, 739-748. doi:10.1016/0020-7683(76)90039-1. open in new tab
  14. Chen, P.; Chen, S. Contact behaviors of a rigid punch and a homogeneous half-space coated with a graded layer. Acta Mech. 2012, 223, 563-577. doi:10.1007/s00707-011-0581-0. open in new tab
  15. Aizikovich, S.M.; Vasil'ev, A.S.; Volkov, S.S. The axisymmetric contact problem of the indentation of a conical punch into a half-space with a coating inhomogeneous in depth. J. Appl. Math. Mech. 2015, 79, 500- 505. doi:10.1016/j.jappmathmech.2016.03.011. open in new tab
  16. Çömez, İ. Contact problem for a functionally graded layer indented by a moving punch. Int. J. Mech. Sci. 2015, 100, 339-344. doi:10.1016/j.ijmecsci.2015.07.006. open in new tab
  17. Su, J.; Ke, L.-L.; Wang, Y.-S. Axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space under a conducting punch. Int. J. Solids Struct. 2016, 90, 45-59. doi:10.1016/j.ijsolstr.2016.04.011. open in new tab
  18. Vasiliev, A.S. Penetration of a spherical conductive punch into a piezoelectric half-space with a functionally graded coating. Int. J. Eng. Sci. 2019, 142, 230-241. doi:10.1016/j.ijengsci.2019.06.006. open in new tab
  19. Volkov, S.S.; Vasiliev, A.S.; Aizikovich, S.M.; Mitrin, B.I. Axisymmetric indentation of an electroelastic piezoelectric half-space with functionally graded piezoelectric coating by a circular punch. Acta Mech. 2019, 230, 1289-1302. doi:10.1007/s00707-017-2026-x. open in new tab
  20. Kudish, I.I.; Pashkovski, E.; Volkov, S.S.; Vasiliev, A.S.; Aizikovich, S.M. Heavily loaded line EHL contacts with thin adsorbed soft layers. Math. Mech. Solids 2020, 25, 1011-1037. doi:10.1177/1081286519898878. open in new tab
  21. Liu, T.J.; Ke, L.L.; Wang, Y.S.; Xing, Y.M. Stress analysis for an elastic semispace with surface and graded layer coatings under induced torsion. Mech. Based Des. Struct. Mach. 2015, 43, 74-94. doi:10.1080/15397734.2014.928222. open in new tab
  22. Nakamura, T.; Wang, T.; Sampath, S. Determination of properties of graded materials by inverse analysis and instrumented indentation. Acta Mater. 2000, 48, 4293-4306. doi:10.1016/S1359-6454(00)00217-2. open in new tab
  23. Gu, Y.; Nakamura, T.; Prchlik, L.; Sampath, S.; Wallace, J. Micro-indentation and inverse analysis to characterize elastic-plastic graded materials. Mater. Sci. Eng. A 2003, 345, 223-233. doi:10.1016/S0921-5093(02)00462-8. open in new tab
  24. Bocciarelli, M.; Bolzon, G.; Maier, G. A constitutive model of metal-ceramic functionally graded material behavior: Formulation and parameter identification. Comput. Mater. Sci. 2008, 43, 16-26. doi:10.1016/j.commatsci.2007.07.047. open in new tab
  25. Huang, L.; Yang, M.; Zhou, X.; Yao, Q.; Wang, L. Material parameter identification in functionally graded structures using isoparametric graded finite element model. Sci. Eng. Compos. Mater. 2016, 23, 685-698. doi:10.1515/secm-2014-0289. open in new tab
  26. Chen, B.; Chen, W.; Wei, X. Characterization of elastic parameters for functionally graded material by a meshfree method combined with the NMS approach. Inverse Probl. Sci. Eng. 2018, 26, 601-617. doi:10.1080/17415977.2017.1336554. open in new tab
  27. Aizikovich, S.; Vasiliev, A.; Seleznev, N. Inverse analysis for evaluation of the shear modulus of inhomogeneous media by torsion experiments. Int. J. Eng. Sci. 2010, 48, 936-942. doi:10.1016/j.ijengsci.2010.05.013. open in new tab
  28. Aizikovich, S.M. Die-caused shear of an inhomogeneous elastic half-space of special form. Mech. Solids 1978, 13, 74-80.
  29. Horgan, C.O.; Miller, K.L. Antiplane shear deformations for homogeneous and inhomogeneous anisotropic linearly elastic solids. J. Appl. Mech. 1994, 61, 23-29. doi:10.1115/1.2901416. open in new tab
  30. Achenbach, J.D.; Balogun, O. Anti-plane surface waves on a half-space with depth-dependent properties. Wave Motion 2010, 47, 59-65. doi:10.1016/j.wavemoti.2009.08.002. open in new tab
  31. Ting, T.C.T. Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity. Wave Motion 2011, 48, 335-344. doi:10.1016/j.wavemoti.2010.12.001. open in new tab
  32. Cao, X.; Jin, F.; Kishimoto, K. Transverse shear surface wave in a functionally graded material infinite half space. Philos. Mag. Lett. 2012, 92, 245-253. doi:10.1016/j.wavemoti.2010.12.001. open in new tab
  33. Sarychev, A.; Shuvalov, A.; Spadini, M. Surface shear waves in a half-plane with depth-variant structure. J. Optim. Theory Appl. 2019, 184, 21-42. doi:10.1007/s10957-019-01501-2. open in new tab
  34. Pritz, T. Five-parameter fractional derivative model for polymeric damping materials. J. Sound Vib. 2003, 265, 935-952. doi:10.1016/S0022-460X(02)01530-4. open in new tab
  35. Alaimo, A.; Orlando, C.; Valvano, S. Analytical frequency response solution for composite plates embedding viscoleastic layers. Aerospace Sci. Tech. 2019, 92, 429-445. doi:10.1016/j.ast.2019.06.021. open in new tab
  36. Zelentsov, V.B.; Lapina, P.A.; Mitrin, B.I.; Kudish, I.I. An antiplane deformation of a functionally graded half-space. Contin. Mech. Thermodyn. 2019. doi:10.1007/s00161-019-00783-1. open in new tab
  37. Zelentsov, V.B.; Lapina, P.A.; Eremeyev, V.A. Identification of shear modulus parameters of half-space inhomogeneous by depth. AIP Conf. Proc. 2019, 2188, 040018. doi:10.1063/1.5138427. open in new tab
  38. Kim, J.I.; Kim, W.-J.; Choi, D.J.; Park, J.Y.; Ryu, W.-S. Design of a C/SiC functionally graded coating for the oxidation protection of C/C composites. Carbon 2005, 43, 1749-1757. doi:10.1016/j.carbon.2005.02.025. open in new tab
  39. Timoshenko, S.P.; Goodier, J.N. Theory of Elasticity; McGraw Hill: New York, NY, USA, 1970. open in new tab
  40. Vorovich, I.I.; Aleksandrov, V.M.; Babeshko, V.A. Neklassicheskiye Smeshannyye Zadachi Teorii Uprugosti [Non-Classical Mixed Problems of the Theory of Elasticity]; Nauka Publishers: Moscow, Russia, 1974. (In Russian) open in new tab
  41. Gradshteĭn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products; Academic Press: New York, NY, USA, 1965.
  42. Brychkov, Y.A.; Prudnikov, A.P. Integral Transformations of Generalized Functions; Gordon & Breach Science Publishers, CRC-Press: New York, NY, USA; London, UK, 1989. open in new tab
  43. Noble, B. Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations;
  44. Bateman, H.; Erdelyi, A. Tables of Integral Transforms;
  45. McGraw-Hill: New York, NY, USA, 1954. open in new tab
  46. Aleksandrov, V.M.; Belokon', A.V. Asymptotic solution of a class of integral equations and its application to contact problems for cylindrical elastic bodies. J. Appl. Math. Mech. 1967, 31, 718-724. doi:10.1016/0021-8928(67)90011-1. open in new tab
  47. Dong, Z.; Sun, X.; Liu, W.; Yang, H. Measurement of free-form curved surfaces using laser triangulation. Sensors 2018, 18, 3527. doi:10.3390/s18103527. open in new tab
  48. Lu, Q.; Pan, D.; Bai, J.; Wang, K. Optical acceleration measurement method with large non-ambiguity range and high resolution via synthetic wavelength and single wavelength superheterodyne interferometry. Sensors 2018, 18, 3418. doi:10.3390/s18103417. open in new tab
  49. Zelentsov, V.B.; Sadyrin, E.V.; Sukiyazov, A.G.; Shubchinskaya, N.Y. On a method for determination of Poisson's ratio and Young modulus of a material. MATEC Web Conf. 2018, 226, 03027. doi:10.1051/matecconf/201822603027. open in new tab
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