Abstract
This paper proposes an analytical model for the Boussinesq problem between a tilted rigid punch and an elastic half space to enable the analysis of elastic deformations inside and outside a contact area. Inside the contact area, two types of pressure distributions are applied: one generates a flat elastic deformation, and the other produces a tilted elastic deformation. The projection of this elastic deformation varies depending on the observed horizontal direction because the elastic deformation is non-axisymmetric. To calculate integrals for the non-axisymmetric elastic deformation, we use the polar coordinate system with two angular coordinates, which can enable the calculation of an integral at any arbitrary point. The proposed model can obtain the relationship between the pressure distribution and the elastic deformations inside and outside a contact area from any arbitrary direction. In addition, the normal load and torque applied inside the contact area are obtained, and these parameters are normalized using the contact radius and the elastic modulus. At the zero-pressure point around the contact edge, the elastic deformation is smooth.
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References
Hertz, H.: On the contact of elastic solids. J. Reine Angew. Math. 92, 156–171 (1881). https://doi.org/10.1515/crll.1882.92.156
Johnson, K.L., Kendall, K., Roberts, A.D.: Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. Math. Phys. Eng. Sci. 324, 301–313 (1971). https://doi.org/10.1098/rspa.1971.0141
Liechti, K.M., Schnapp, S.T., Swadener, J.G.: Contact angle and contact mechanics of a glass/epoxy interface. Int. J. Fract. 86, 361–374 (1997). https://doi.org/10.1023/A:1007472628431
Myshkin, N.K., Petrokovets, M.I., Kovalev, A.V.: Tribology of polymers: adhesion, friction, wear, and mass-transfer. Tribol. Int. 38, 910–921 (2005). https://doi.org/10.1016/j.triboint.2005.07.016
Momozono, S., Takeuchi, H., Iguchi, Y., Nakamura, K., Kyogoku, K.: Dissipation characteristics of adhesive kinetic friction on amorphous polymer surfaces. Tribol. Int. 48, 122–127 (2012). https://doi.org/10.1016/j.triboint.2011.11.016
Fujiwara, R., Hemthavy, P., Takahashi, K., Saito, S.: The effect of surface conductivity and adhesivity on the electrostatic manipulation condition for dielectric microparticles using a single probe. J. Micromech. Microeng. 26, 055010 (2016). https://doi.org/10.1088/0960-1317/26/5/055010
Baek, D., Saito, S., Takahashi, K.: Estimating work of adhesion using spherical contact between a glass lens and a PDMS block. J. Adhes. Sci. Technol. 32, 158–172 (2018). https://doi.org/10.1080/01694243.2017.1343519
Greenwood, J.A., Williamson, J.B.P., Bowden, F.P.: Contact of nominally flat surfaces. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 295, 300–319 (1966). https://doi.org/10.1098/rspa.1966.0242
Majumdar, A., Bhushan, B.: Fractal model of elastic–plastic contact between rough surfaces. J. Tribol. 113, 1–11 (1991). https://doi.org/10.1115/1.2920588
Persson, B.N.J.: Contact mechanics for randomly rough surfaces. Surf. Sci. Rep. 61, 201–227 (2006). https://doi.org/10.1016/j.surfrep.2006.04.001
Sridhar, I., Johnson, K.L., Fleck, N.A.: Adhesion mechanics of the surface force apparatus. J. Phys. Appl. Phys. 30, 1710–1719 (1997). https://doi.org/10.1088/0022-3727/30/12/004
Taljat, B., Zacharia, T., Kosel, F.: New analytical procedure to determine stress–strain curve from spherical indentation data. Int. J. Solids Struct. 35, 4411–4426 (1998). https://doi.org/10.1016/S0020-7683(97)00249-7
Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A., Suresh, S.: Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Mater. 49, 3899–3918 (2001). https://doi.org/10.1016/S1359-6454(01)00295-6
Pelletier, H.: Predictive model to estimate the stress–strain curves of bulk metals using nanoindentation. Tribol. Int. 39, 593–606 (2006). https://doi.org/10.1016/j.triboint.2005.03.019
Beghini, M., Bertini, L., Fontanari, V.: Evaluation of the stress–strain curve of metallic materials by spherical indentation. Int. J. Solids Struct. 43, 2441–2459 (2006). https://doi.org/10.1016/j.ijsolstr.2005.06.068
Saito, S., Ochiai, T., Yoshizawa, F., Dao, M.: Rolling behavior of a micro-cylinder in adhesional contact. Sci. Rep. 6, 34063 (2016). https://doi.org/10.1038/srep34063
Green, A.E.: On Boussinesq’s problem and penny-shaped cracks. Math. Proc. Camb. Philos. Soc. 45, 251–257 (1949). https://doi.org/10.1017/S0305004100024804
Sneddon, I.N.: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57 (1965). https://doi.org/10.1016/0020-7225(65)90019-4
Barber, J.R.: Some polynomial solutions for the non-axisymmetric Boussinesq problem. J. Elast. 14, 217–221 (1984). https://doi.org/10.1007/BF00041668
Menga, N., Carbone, G.: The surface displacements of an elastic half-space subjected to uniform tangential tractions applied on a circular area. Eur. J. Mech. A Solids 73, 137–143 (2019). https://doi.org/10.1016/j.euromechsol.2018.07.011
Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1969)
Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (2013)
Takahashi, K., Mizuno, R., Onzawa, T.: Influence of the stiffness of the measurement system on the elastic adhesional contact. J. Adhes. Sci. Technol. 9, 1451–1464 (1995). https://doi.org/10.1163/156856195X00121
Sneddon, I.N.: Boussinesq’s problem for a flat-ended cylinder. Math. Proc. Camb. Philos. Soc. 42, 29 (1946). https://doi.org/10.1017/S0305004100022702
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Iguchi, Y., Hemthavy, P., Saito, S. et al. Analytical solution of elastic deformations inside and outside circular contact area between tilted rigid punch and elastic half space. Acta Mech 230, 4311–4320 (2019). https://doi.org/10.1007/s00707-019-02505-9
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DOI: https://doi.org/10.1007/s00707-019-02505-9