×

On an invariance principle for unilateral contact at a bimaterial elastic interface. (English) Zbl 1211.74192

Summary: This paper examines the axisymmetric elastostatic problem related to the unilateral receding contact at a pre-compressed smooth bimaterial elastic interface. The separation at the interface is caused by the action of axisymmetric stress fields of unequal magnitude, which are applied at any location of the separate halfspace regions. The analysis of the problem focuses on the determination of the zone of separation as a function of the pre-compression, the magnitudes and locations of the axisymmetric stress fields inducing the separation, and the elasticity characteristics of the halfspace regions. It is found that the radius of the separation zone can be evaluated in explicit form. In the particular instance when the loadings applied at the surface of the halfspace regions are equal in magnitude and distribution, the analysis reveals that the radius of the separation zone is independent of the elasticity properties of the halfspace regions and can be evaluated in exact closed form.

MSC:

74R10 Brittle fracture

Software:

CONTACT
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] (Aliabadi, M. H.; Brebbia, C. A., Computational Methods in Contact Mechanics (1993), Computational Mechanics Publications, Elsevier Applied Science: Computational Mechanics Publications, Elsevier Applied Science Amsterdam, The Netherlands) · Zbl 0790.73004
[2] Barenblatt, G. I., The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially symmetric cracks, J. Appl. Math. Mech. (PMM), 23, 622-636 (1959) · Zbl 0095.39202
[3] Barenblatt, G. I., The mathematical theory of equilibrium cracks in brittle fracture, (Dryden, H. L.; von Karman, Th., Advances in Applied Mechanics, vol. 7 (1962)), 55-129
[4] D.M. Barnett, L.M. Keer, J.W. Rudnicki, T.C.T. Ting (Eds.), Special Topics in the Theory of Elasticity: A Volume in Honor of Professor John Dundurs. Int. J. Solids Struct. 32 (1995) 269-567; D.M. Barnett, L.M. Keer, J.W. Rudnicki, T.C.T. Ting (Eds.), Special Topics in the Theory of Elasticity: A Volume in Honor of Professor John Dundurs. Int. J. Solids Struct. 32 (1995) 269-567 · Zbl 0863.00026
[5] Boussinesq, J., Applications des potentials a l’etude de l’equilibre et du mouvement des solides elastique (1885), Gauthier-Villars: Gauthier-Villars Paris · JFM 18.0932.01
[6] P.G. Ciarlet, Mathematical Elasticity, Three-Dimensional Elasticity, vol. 1, North Holland, Amsterdam, 1988; P.G. Ciarlet, Mathematical Elasticity, Three-Dimensional Elasticity, vol. 1, North Holland, Amsterdam, 1988 · Zbl 0648.73014
[7] A. Curnier (Ed.), Proceedings of the International Symposium on Contact Mechanics, Presses Polytech. et Univ. Romandes, Lausanne, 1992; A. Curnier (Ed.), Proceedings of the International Symposium on Contact Mechanics, Presses Polytech. et Univ. Romandes, Lausanne, 1992 · Zbl 0796.73001
[8] (de Pater, A. D.; Kalker, J. J., Mechanics of Contact Between Deformable Media, Proceedings of the IUTAM Symposium, Enschede (1975), Delft University Press: Delft University Press Delft) · Zbl 0308.00012
[9] Dundurs, J.; Stippes, M., The role of elastic constants on certain contact problems, J.Appl. Mech., 37, 965-970 (1970)
[10] Dundurs, J.; Comninou, M., An old elasticity problem in a unilateral setting, J. Elasticity, 12, 231-238 (1982) · Zbl 0487.73125
[11] Dundurs, J.; Tsai, K. C.; Keer, L. M., Contact between elastic bodies with wavy surfaces, J. Elasticity, 3, 109-115 (1973)
[12] Duvaut, G.; Lions, J. L., Inequalities in Mechanics and Physics (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0331.35002
[13] G. Fichera, The Signorini elastostatic problems with ambiguous boundary conditions, Proceedings of the International Conference on the Application of the Theory of Functions in Continuum Mechanics, Tiblisi, vol. 1, 1963; G. Fichera, The Signorini elastostatic problems with ambiguous boundary conditions, Proceedings of the International Conference on the Application of the Theory of Functions in Continuum Mechanics, Tiblisi, vol. 1, 1963
[14] Fichera, G., Boundary value problems of elasticity with unilateral constraints, (Truesdell, C., Handbuch der Physik, Mechanics of Solids II, vol. VIa/2 (1972), Springer-Verlag: Springer-Verlag Berlin), 391-424
[15] Fremond, M., Contact with adhesion, (Moreau, J. J.; Panagiotopoulos, P. D.; Strang, G., Topics in Non-smooth Mechanics (1988), Birkhauser Verlag: Birkhauser Verlag Basel), 157-185 · Zbl 0669.73079
[16] L.A. Galin, Contact Problems in the Classical Theory of Elasticity, in: I.N. Sneddon (Ed.), Engl. Trans., Tech. Rep. G16447, North Carolina State College, Raleigh, NC, 1961; L.A. Galin, Contact Problems in the Classical Theory of Elasticity, in: I.N. Sneddon (Ed.), Engl. Trans., Tech. Rep. G16447, North Carolina State College, Raleigh, NC, 1961
[17] Gladwell, G. M.L., Contact Problems in the Classical Theory of Elasticity (1980), Sijthoff and Noordhoff: Sijthoff and Noordhoff Alphen Aan den Rijn, The Netherlands · Zbl 0431.73094
[18] Gladwell, G. M.L.; Hara, T., The contact problem for a rigid obstacle pressed between two dissimilar halfspaces, Quart. J. Mech. Appl. Math., 34, 251-263 (1981) · Zbl 0462.73093
[19] Gladwell, G. M.L., On contact problems for a medium with rigid flat inclusions of arbitrary shape, Int. J. Solids Struct., 32, 383-389 (1995) · Zbl 0865.73052
[20] Goodman, L. E., Developments of the three-dimensional theory of elasticity, (Herrmann, G., R.D. Mindlin and Applied Mechanics (1972), Pergamon Press: Pergamon Press Oxford), 25-64
[21] Green, A. E.; Zerna, W., Theoretical Elasticity (1968), Clarendon Press: Clarendon Press Oxford · Zbl 0155.51801
[22] Gurtin, M. E., Linear theory of elasticity, (Truesdell, C., Handbuch der Physik, Mechanics of Solids II, vol. VIa/2 (1972), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0123.40803
[23] Haslinger, J.; Janovsky, V., Contact problem with friction, (Brilla, J., Trends in Applications of Pure Mathematics to Mechanics, vol. IV (1983), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston), 74-100 · Zbl 0615.73120
[24] Hertz, H., Uber die Beruhrung fester elastischer Korper, J. fur die reine und angew. Mathematik, 49, 156-171 (1882) · JFM 14.0807.01
[25] Hertz, H., Gesammelte Werke, Band 1 (1895), Johann Ambrosius Barth: Johann Ambrosius Barth Leipzig
[26] Johnson, K. L., Contact Mechanics (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0599.73108
[27] Johnson, K. L.; Greenwood, J. A.; Higginson, J. G., The contact of elastic regular wavy surfaces, Int. J. Mech. Sci., 27, 383-396 (1985) · Zbl 0576.73105
[28] Kalker, J. J., Three-Dimensional Elastic Bodies in Rolling Contact (1990), Kluwer Academic Publisher: Kluwer Academic Publisher Dordrecht, The Netherlands · Zbl 0709.73068
[29] Kikuchi, N.; Oden, J. T., Contact Problems in Elasticity: A study of Variational Inequalities and Finite Element Methods (1988), SIAM: SIAM Philadelphia · Zbl 0685.73002
[30] Kinderlehrer, D.; Stampaccia, G., An Introduction to Variational Inequalities, and their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[31] Klabring, A., Mathematical programming in contact problems (Chapter 7), (Aliabadi, M. H.; Brebbia, C. A., Computational Methods in Contact Mechanics (1993), Computational Mechanics Publ., Elsevier Applied Science: Computational Mechanics Publ., Elsevier Applied Science Amsterdam), 233-263 · Zbl 0790.73004
[32] Love, A. E.H., A Treatise on the Mathematical Theory of Elasticity (1927), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0063.03651
[33] Love, A. E.H., Boussinesq’s problem for a rigid cone, Quart. J. Math., 10, 161-175 (1939) · Zbl 0022.27304
[34] Lur’e, A. I., Three-Dimensional Problems in the Theory of Elasticity (1965), Wiley-Interscience: Wiley-Interscience New York · Zbl 0202.56204
[35] Mindlin, R. D., A force at a point in the interior of a semi-infinite solid, Physics, 7, 195-202 (1936) · JFM 62.1531.03
[36] Mindlin, R. D.; Deresiewicz, H., Elastic spheres in contact under varying oblique forces, J. Appl. Mech., 75, 327-344 (1953) · Zbl 0051.41202
[37] (Moreau, J. J.; Panagiotopoulos, P. D.; Strang, G., Topics in Nonsmooth Mechanics (1988), Birkhauser Verlag: Birkhauser Verlag Basel) · Zbl 0646.00014
[38] Mura, T., Micromechanics of Defects in Solids (1987), Martinus Nijhoff Publ: Martinus Nijhoff Publ Dordrecht, The Netherlands · Zbl 0652.73010
[39] Panagiotopoulos, P. D., Inequality Problems in Mechanics and Applications (1985), Birkhauser Verlag: Birkhauser Verlag Basel · Zbl 0579.73014
[40] W. Prager, Unilateral constraints in mechanics of continua. Atti del Convegno Lagrangiano, Acc. Sci. Torino, 1963, pp. 181-191; W. Prager, Unilateral constraints in mechanics of continua. Atti del Convegno Lagrangiano, Acc. Sci. Torino, 1963, pp. 181-191 · Zbl 0125.41301
[41] Selvadurai, A. P.S., Elastic Analysis of Soil-Foundation Interaction, Developments in Geotechnical Engineering, vol. 17 (1979), Elsevier Science Publishers: Elsevier Science Publishers Amsterdam · Zbl 0406.73016
[42] Selvadurai, A. P.S., Separation at a pre-fractured bi-material geological interface, Mech. Res. Comm., 21, 83-88 (1994) · Zbl 0795.73064
[43] Selvadurai, A. P.S., A unilateral contact problem for a rigid disc inclusion embedded between two dissimilar elastic halfspaces, Quart. J. Mech. Appl. Math., 47, 493-510 (1994) · Zbl 0808.73066
[44] Selvadurai, A. P.S., (Partial Differential Equations in Mechanics, The Biharmonic Equation and Poisson’s Equation, vol. 2 (2000), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0967.35002
[45] Signorini, A., Sopra alcune questioni di elastostatica, Atti Soc. Ital. Per il Progresso delle Scienze, 11, 143-148 (1933) · JFM 59.1413.02
[46] Sneddon, I. N., Fourier Transforms (1951), McGraw-Hill: McGraw-Hill New York · Zbl 0099.28401
[47] Sneddon, I. N.; Lowengrub, M., Crack Problems in the Classical Theory of Elasticity (1969), John Wiley: John Wiley New York · Zbl 0201.26702
[48] Truesdell, C., Invariant and complete stress functions for general continua, Arch. Rational Mech. Anal., 4, 1-29 (1960) · Zbl 0089.40803
[49] Ia.S. Ufliand, Survey of Applications of Integral Transforms in the Theory of Elasticity, in: I.N. Sneddon (Ed.), Trans., Tech. Rep. 65-1556, North Carolina State College, Raleigh, NC, 1965; Ia.S. Ufliand, Survey of Applications of Integral Transforms in the Theory of Elasticity, in: I.N. Sneddon (Ed.), Trans., Tech. Rep. 65-1556, North Carolina State College, Raleigh, NC, 1965
[50] Villaggio, P., A unilateral contact problem in elasticity, J. Elasticity, 10, 113-119 (1980) · Zbl 0425.73097
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.