Summary
Problems connected with a shock interaction of a thin elastic cylindrical bar and an elastic sphere with an Uflyand-Mindlin plate of a finite size are considered. In enumerated problems, an impact process is accompanied by material local deformations and propagation of wave surfaces of a strong or weak discontinuity in elastic bodies coming in contact. The local deformations are taken into account in terms of the Hertz's contact theory, but the dynamic deformations behind the fronts of incident and reflected waves are determined by means of truncated power series with variable coefficients (what is known as ray series). These two types of deformations are mated on the contact region boundary. As the truncated ray series, as a rule, are not uniformly applicable in the whole region of the wave solution existence, then for their improvement the method of “forward-area” regularization is used which is based on a combination of the recurrent equations of the ray method with the method of multiple scales. The method of power series (ray method) allows one to find the main characteristics of the impact theory in an analytical form.
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References
Timoshenko, S. P.: Vibration problems in engineering, 3rd ed. New York: Van Nostrand, 1928.
Uflyand, Ya. S.: Waves propagation under transverse vibrations of bars and plates. Prikl. Matem. Mekh.12, 287–300 (1948).
Mindlin, R. D.: High frequency vibrations of crystal plates. Quart. Appl. Math.19, 51–61 (1961).
Goldsmith, W.: Impact. London: Arnold 1960.
Mochihara, M., Tanaka, Y.: Behaviour of plates in the elastic range under transverse impact. Res. Repts. Kagoshima Techn. Coll.23, 35–44 (1989).
Ballmann, J., Staat, M.: Computation of impacts on elastic solids by method of bicharacteristics. In: Comp. Mech '88. Theory and Appl., Proc. Int. Conf. Comput. Eng. Sci.2, pp. 60.i.1.–60.i.4. Atlanta 1988.
Laturelle F. G.: Finite element analysis of wave propagation in an elastic half-space under step loading. Comput. Struct.32, 721–735 (1989).
Schonberg, W. P., Keer, L. M., Woo, T. K.: Low velocity impact of transversely isotropic beams and plates. Int. J. Solids Struct.23, 871–896 (1987).
Stein, E., Wriggers, P.: “Computational Mechanics” bei Festkörpern und Ingenieurstrukturen unter Finite-Element-Methoden. GAMM-Mitt.1, 3–39 (1989).
Achenbach, J. D., Reddy, D. P.: Note on wave propagation in linearly viscoelastic media. ZAMP18, 141–144 (1967).
Rossikhin, Yu. A.: Impact of a rigid sphere onto an elastic half-space. Sov. Appl. Mech.22, 403–409 (1986).
Rossikhin, Yu. A.: On non-stationary vibrations of plates on an elastic foundation. J. Appl. Math. Mech.42, 347–353 (1978).
Thomas, T. Y.: Plastic flow and fracture in solids. New York: Academic Press 1961.
Nayfeh, A. H.: Perturbation methods. New York: Wiley 1973.
Kamke, E.: Differentialgleichungen, Lösungsmethoden und Lösungen. Leipzig. Akademische Verlagsgesellschaft 1943.
Timoshenko, S. P.: Theory of elasticity. New York: McGraw-Hill 1934.
Mittal, R. K.: A simplified analysis of the effect of transverse shear on the response of elastic plates to impact loading. Int. J. Solids Struct.23, 1191–1203 (1987).
Zukas, J. A., Nicholas, T., Swift, H. F., Greszczuk, L. B., Curran, D. R.: Impact dynamics. New York: Wiley 1982.
Qian, Y., Swanson, S. R.: Experimental measurement of impact response in carbon/epoxy plates. AIAA J.28, 1069–1074 (1990).
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Rossikhin, Y.A., Shitikova, M.V. A ray method of solving problems connected with a shock interaction. Acta Mechanica 102, 103–121 (1994). https://doi.org/10.1007/BF01178521
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DOI: https://doi.org/10.1007/BF01178521