Abstract
The integral representation of the relevant boundary problems together with the difference approximation schemes of the time solution is used to simulate the nonlinear wave dynamics of threedimensional heterogeneous deformable media. The time solution of the integral equations is based on the Kelvin—Somigliana fundamental solution, the Newmark -scheme, the collocation approximation, and the predictor-corrector of the method of the theory of elastoplastic media flow with anisotropic hardening. The parametrization of the domain surface occupied by the medium, i.e., its internal boundaries, is made using the quadratic boundary elements. Similar volume elements are used to calculate the integrals with the bulk (inertial and viscous) forces and determine the plastic deformation zones. The complex histories of the combined shock loading of the composite piecewise homogeneous media slowly changing in time in the presence of the local zones of singular perturbation of the solution are considered. The discrete domain method was developed, based on which the practice-relevant nonlinear problems of stress wave propagation in heterogeneous media with the concentrators were solved.
Similar content being viewed by others
References
Wrobel, L.C. and Aliabadi, M.H., The Boundary Element Method, New York: Wiley, 2007.
Liu, Y.J., Mukherjee, S., Nishimura, N., Schanz, M., et al., Recent advances and emerging applications of the boundary element method, Appl. Mech. Rev., 2012, pp. 64, p. 38. doi 10.1115/1.4005491
Costabel, M., Time-dependent problems with the boundary integral equation method, in Encyclopedia of Computational Mechanics, New York: Wiley, 2004, pp. 703–721.
Bazhenov, V.G. and Igumnov, L.A., Metody granichnykh integral’nykh uravnenii i granichnykh elementov v reshenii zadach trekhmernoi dinamicheskoi teorii uprugosti s sopryazhennymi polyami (Methods of Boundary Integral Equations and Boundary Elements in Solving Problems of the Three-Dimensional Dynamic Theory of Elasticity with Conjugate Fields), Moscow: Fizmatlit, 2008.
Romanov, V.G. and Kabanikhin, S.I., Inverse problems of wave propagation, in Matematicheskoe modelirovanie v geofizike (Mathematical Modeling in Geophysics), Novosibirsk: Vychisl. Tsentr SO AN SSSR, 1987, pp. 151–167.
Vatul’yan, A.O., Obratnye zadachi v mekhanike deformiruemogo tverdogo tela (Inverse Problems in the Mechanics of a Deformable Solid), Moscow: Fizmatlit, 2007.
Hatzigeorgiou, G.D. and Beskos, D.E., Dynamic inelastic structural analysis by the BEM: a review, Eng. Anal. Boundary Elem., 2011, vol. 35, no. 2, pp. 159–169.
Hsiao, G.C. and Wendland, W.L., Boundary Integral Equations, Berlin: Springer, 2008.
Hatzigeorgiou, G.D., Dynamic inelastic analysis with BEM: results and needs, in Recent Advances in Boundary Element Methods, Manolis, G.D. and Polyzos, D., Eds., Springer, Berlin, 2009, pp. 193–208.
Petushkov, V.A., Numerical realization of the method of boundary integral equations applied to nonlinear problems of the mechanics of deformation and destruction of bulk bodies, in Sb. nauchnykh trudov ITPM SO AN SSSR Modelirovanie v mekhanike (Modelling in Mechanics, Collection of Scientific Articles of Khristianovich Inst. Theor. Appl. Mech., Sib. Branch of Acad. Sci. USSR), Novosibirsk: ITPM SO AN SSSR, 1989, Vol. 3 (20), No. 1, pp. 133–156.
Petushkov, V.A. and Zysin, V.I., Application package MEGRE-3D for numerical modeling of nonlinear processes of deformation and destruction of bulk bodies. Algorithms and implementation in the EU OS, in Pakety prikladnykh programm: Programmnoe obespechenie matematicheskogo modelirovaniya (Applied Software Packages: Mathematical Modeling Software), Moscow: Nauka, 1992, pp. 111–126.
Petushkov, V.A., Simulation of nonlinear deformation and fracture of heterogeneous media based on the generalized method of integral representations, Mat. Model., 2015, vol. 27, no. 1, pp. 113–130.
Petushkov, V.A. and Potapov, A.I., Numerical solutions of three-dimensional dynamic problems in the theory of elasticity, in Sb. dokladov Sed’mogo Vsesoyuznogo s’ezda po teoreticheskoi i prikladnoi mekhanike (Proceedings of the 7th All-Union Symposium on Theoretical and Applied Mechanics), Moscow, 1991, pp. 286–287.
Petushkov, V.A. and Frolov, K.V., Dynamics of hydroelastic systems under pulsed excitation, in Dinamika konstruktsii gidroaerouprugikh sistem (Dynamics of Constructions of Hydroaeroelastic Systems), Moscow: Nauka, 2002, pp. 162–202.
Guz’, A.N., Uprugie volny v telakh s nachal’nymi (ostatochnymi) napryazheniyami (Elastic Waves in Bodies with Initial (Residual) Stresses), Kiev: A.S.K., 2004.
Petushkov, V.A and Shneiderovich, R.M, Thermoetastic plastic deformation of comrgated shells of revolution at finite displacements, Strength Mater., 1979, vol. 11, no. 6, pp. 578–585.
Wheeler, L.T. and Sternberg, E, Some theorems in classical elastodynamics (uniqueness theorems and elastodynamic equations homogeneous and isotropic elastic media), Arch. Ration. Mech. Anal., 1968, vol. 31, no. 1, pp. 51–90.
Kupradze, V.D. and Burchuladze, T.V., Dynamic problems in the theory of elasticity and thermoelasticity, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., 1975, vol. 7, pp. 163–294.
Costabel, M., Boundary integral operators on lipschitz domains: elementary results, SIAM J. Math. Anal., 1988, vol. 19, no. 3, pp. 613–626.
Nintcheu, Fata S., Treatment of domain integrals in boundary element methods, Appl. Numer. Math., 2012, vol. 62, no. 6, pp. 720–735.
Strang, G. and Fix, G., An Analysis of the Finite Element Method, 2nd ed., Cambridge, MA: Wellesley-Cambridge Press, 2008.
Rjasanow, S. and Steinbach, O., The Fast Solution of Boundary Integral Equations, Heidelberg: Springer, 2007.
Hsiao, G.C., Steinbach, O., and Wendland, W.L., Domain decomposition methods via boundary integral equations, J. Comput. Appl. Math., 2000, vol. 125, pp. 521–537.
Petushkov, V.A., Boundary integral equation method in the modeling of nonlinear deformation and failure of the 3D inhomogeneous media, Vestn. Samar. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2014, no. 2 (35), pp. 96–114.
Soares, D., Dynamic analysis of elastoplastic models considering combined formulations of the time-domain boundary element method, Eng. Anal. Boundary Elem., 2015, vol. 55, pp. 28–39.
Kramer, S.L., Geotechnical Earthquake Engineering, Englewood Cliffs, New York: Prentice-Hall, 1996.
Makhutov, N.A., Petushkov, V.A., and Zysin, V.I., Numerical solution by the method of integral representations of the problem of a curvilinear fracture of a normal detachment in a three-dimensional elastoplastic body, Probl. Prochn., 1988, no. 7, pp. 83–91.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.A. Petushkov, 2018, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2018, No. 5.
About this article
Cite this article
Petushkov, V.A. Nonlinear Deformation of Three-Dimensional Piecewise Homogeneous Media in Stress Waves. J. Mach. Manuf. Reliab. 47, 451–463 (2018). https://doi.org/10.3103/S1052618818050096
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1052618818050096