Mechanics of Composite Materials

, Volume 55, Issue 3, pp 297–314 | Cite as

Modeling the Dynamic Behavior of Rigid-Plastic Thin Reinforced Curvilinear Plates with a Hole on a Viscous Foundation

  • T. P. RomanovaEmail author

A mathematical model is developed for the dynamic behavior of rigid-plastic thin reinforced layered curvilinear, hinge-supported or clamped, plates with an arbitrary free hole. The plates are on a viscous foundation and are subjected to the action of a dynamic load of explosive type uniformly distributed on its surface. The plates are hybrid, multilayered, and fibrous, with their layers distributed symmetrically with respect to the middle surface. In each layer, the reinforcing fibers are located parallel or normal to the external contour of plate. The structural model of a reinforced layer is used. Depending on intensity of the load, various dynamic deformation modes of the plates are possible. From the principle of virtual power, with account of d’Alembert’s principle, the equations of dynamic behavior of the plates are obtained and the conditions for their implementation are determined for each of the modes. Analytical expressions for estimation of their limit loads are obtained. The variant of quasi-isotropic reinforcement is considered. Numerical examples for a reinforced elliptic plate with a circular hole are given.


curvilinear reinforced plate rigid-plastic model free hole viscous foundation explosive load limit load 


  1. 1.
    C. W. Lim, S. Kitipornchai, and K. M. Liew, “Free vibration analysis of doubly connected super elliptical laminated composite plates,” Compos. Sci. Technol., 58, 435-445 (1998). (97) 00167-XGoogle Scholar
  2. 2.
    E. Altunsaray, “Static deflections of symmetrically laminated quasi-isotropic super-elliptical thin plates,” Ocean Engineering, 141, 337-350 (2017). CrossRefGoogle Scholar
  3. 3.
    A. P. Yankovskii, “Viscoplastic dynamics of metallic composite shells of layered-fibrous structure under the action of loads of explosive type. I. Statement of the problem and method for solution,” J. Mathematical Sci., 192, No. 6, 623-633 (2013). CrossRefGoogle Scholar
  4. 4.
    A. P. Yankovskii, “Employing the time-explicit method of central differences for numerically modeling the dynamic behavior elastoplastic flexible reinforced plates,” Wychislit. Mekh. Splosh. Sred, 9, No. 3, 279-297 (2016). Scholar
  5. 5.
    O. N. Popov, A. P. Malinovskii, M. O. Moiseenko, and T. A. Тreputneva, “On the problem of calculation of inhomogeneous structural beyond the limit of elasticity,” Vest. TGASU, Вест. ТГАСУ, No. 4, 127-142 (2013).Google Scholar
  6. 6.
    N. A. Abrosimov, A.V. Elesin, and N. A. Novosel’tsev, “Numerical analysis of the effect of reinforcement structure on the dynamic behavior and ultimate deformability of composite shells of revolution,” Mech. Compos. Mater., 50, No. 2, 313-326 (2014). CrossRefGoogle Scholar
  7. 7.
    F. D. Morinière, R. C. Alderliesten, and R. Benedictus, “Modelling of impact damage and dynamics in fibre-metal laminates. A review,” Int. J. Impact Eng., 67, 27-38 (2014). CrossRefGoogle Scholar
  8. 8.
    H. Arora, P. Del Linz, and J. P. Dear, “Damage and deformation in composite sandwich panels exposed to multiple and single explosive blasts,” Int. J. Impact Eng., 104, 95-106 (2017).CrossRefGoogle Scholar
  9. 9.
    N. Jones, “Some recent developments in the dynamic inelastic behavior of structures,” Ships and Offshore Structures, 1, No. 1, 37-44 (2006).CrossRefGoogle Scholar
  10. 10.
    Z. Wang, G. Lu, F. Zhu, and L. Zhao, “Load-carrying capacity of circular sandwich plates at large deflection,” J. Eng. Mech., 143, No. 9, 04017057-1-12 (2017).Google Scholar
  11. 11.
    Yu. V. Nemirovskii and T. P. Romanova, “Dinamic Resistance of Plane Plastic Barriers [in Russian], Novosibirsk: Izd. GEO, (2009).Google Scholar
  12. 12.
    Yu. V. Nemirovskii, “On a condition of plasticity (strength) for a reinforced layer,” Prikl. Mekh. Tekhn. Fiz., 10, No. 5, 81-88 (1969).Google Scholar
  13. 13.
    Yu. V. Nemirovskii and B. S. Resnikoff, “On limit equilibrium of reinforced slabs and effectiveness of their reinforcement,” Archiwum Inzynierii Ladowej, XXI, No. 1, 57-67 (1975).Google Scholar
  14. 14.
    A. R. Rzhanitsyn, Limit Equilibrium of Plates and Shells [in Russian], M.: Nauka (1983).Google Scholar
  15. 15.
    N. A. Abrosimov and V. G. Bazhenov, “Nonlinear Problems of the Dynamics of Composite Structures [in Russian], N. Novgorod, Izd. NNGU (2002).Google Scholar
  16. 16.
    Yu. V. Nemirovskii and N. A. Fedorova, Mathematical Modeling of Plane Structures of Reinforced Fibrous Materials [in Russian], Krasnoyarsk, Izd. SFU (2010).Google Scholar
  17. 17.
    S. A. Аmbartsumyan, Theory of Anisotropic Plates: Strength, Stability, and Vibrations [in Russian], M., Nauka (1987).Google Scholar
  18. 18.
    S. G. Lekhnitskii, Anizotropic Plates [in Russian], M., GITTL (1957).Google Scholar
  19. 19.
    Yu. V. Nemirovskii and B. S. Reznikov, Stregth of Structural Members of Composite Materials [in Russian], Novosibirsk, Nauka, Sib. Otdelenie (1986).Google Scholar
  20. 20.
    V. N. Mazalow and Yu. V. Nemirovsky, “Dynamic bending of rigid-plastic annular plates,” Int. J. Non-Linear Mechanics, 11, No. 1, 25-40 (1976).CrossRefGoogle Scholar
  21. 21.
    N. Jones, “Finite deflections of rigid-viscoplastic strain-hardening annular plate loaded impulsively,” Trans. ASME, J. Appl. Mech., 35, No. 2, 349-356 (1968).CrossRefGoogle Scholar
  22. 22.
    N. Jones, “Finite deflections of a simply supported rigid-plastic annular plate loaded dynamically,” Int. J. Solids and Struct., 5, No. 6, 593-603 (1968).CrossRefGoogle Scholar
  23. 23.
    J. Lellep and K. Torn, “Dynamic plastic behavior of annular plates with transverse shear effects,” Int. J. Impact Eng., 34, No. 6, 1061-1080 (2007).CrossRefGoogle Scholar
  24. 24.
    H. R. Aggarwal and C. M. Ablow, “Plastic bending of an annular plate by uniform impulse,” Int. J. Non-Linear Mechanics, 6, No. 1, 69-80 (1971).CrossRefGoogle Scholar
  25. 25.
    Yu. V. Nemirovskii and T. P. Romanova, “Dynamics of a rigid-plastic regular polygonal plate with a hole under the action of explosive loadings,” Boundary-value problems and mathematical modeling: Sb. St. 9 Vseros. Nauch. Konf., November 28-29, 2008, Novokuznetsk. In 3 vol. 1. / NFI GOU <KemGU>; under ed. V. O. Kaledin, Novokuznetsk, 93-97 (2008).Google Scholar
  26. 26.
    Yu. V. Nemirovskii and T. P. Romanova, “Mechanics of the dynamic behavior of rigid-plastic curvilinear plate with an arbitrary free hole,” Teor. Prikl. Mekh., Mezhdunar. Nauch. Tekhn. Sb., Minsk, BNTU, No. 23, 26-34 (2007).Google Scholar
  27. 27.
    T. P. Romanova and Yu. V. Nemirovsky, “Dynamic rigid-plastic deformation of arbitrarily shaped plates,” J. Mechanics of Materials and Structures, 3, No. 2, 313-334 (2008). CrossRefGoogle Scholar
  28. 28.
    Yu. V. Nemirovskii and T. P. Romanova, “Dynamics of rigid-platic curvilinear plate of variable thickness with an arbitrary internal hole,” Prikl. Mekh., 46, No. 3, 70-82 (2010). Scholar
  29. 29.
    Yu. V. Nemirovskii and T. P. Romanova, “Dynamics of a round rigid-plastic plate with an arbitrary free internal hole,” Nauka. Promyshl. Oborona (NPO-2008) Trudy 9 Vseros. Nauch. Tekhn. Konf., Novosibirsk, April, 23-25, 2008. Novosibirsk: NGTU, 262-274 (2008).Google Scholar
  30. 30.
    T. P. Romanova, “Modeling the dynamic bending rigid-plastic reinforced layered round plates with an arbitrary hole on a viscous foundation at explosive loadings,” Probl. Prochn. Plastichn., Iss. 79, No. 3, 267-284 (2017).Google Scholar
  31. 31.
    T. P. Romanova, “Modeling of dynamic bending of rigid-plastic reinforced layered curvilinear plate with supported circular hole under explosive loads,” PNRPU Mechanics Bulletin, No. 3, 167-187 (2017).
  32. 32.
    T. P. Romanova, “Modeling the dynamic bending of rigid-plastic hybrid composite elliptical plates with a rigid insert,” Mech. Compos. Mater., 53, No. 5, 809-828 (2017). CrossRefGoogle Scholar
  33. 33.
    A. A. Savelov, Plane Curves [in Russian], M., Gos. izd. fiz.-mat. liter. (1960)Google Scholar
  34. 34.
    M. I. Erkhov, Theory of Ideally Plastic Bodies and Structures [in Russian], M., Nauka (1978).Google Scholar
  35. 35.
    A. H. Keil, Problems of plasticity in naval structures: explosive and impact loadings,” Mekhanika (collection of translations), No. 2, 197-223 (1961).Google Scholar
  36. 36.
    L.V. Kantorovich and G. P. Аkilov, Functional Analysis [in Russian], M., Nauka (1984).Google Scholar
  37. 37.
    T. P. Romanova, “Carrying capacity and optimization of three-layer reinforced concrete annular plate, supported on the internal contour,” PNRPU Mechanics Bulletin, No. 3, 114-132 (2015).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.S. A. Hristianovich Institute of Theoretical and Applied Mechanics of Siberian Branch of the Russian Academy of SciencesNovosibirskRussia

Personalised recommendations