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Wave Theory of Impact and Professor Yury Rossikhin Contribution in the Field (A Memorial Survey)

  • Marina V. ShitikovaEmail author
Article
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Abstract

This survey article overviews the impact response of solids and structures within the framework of the wave theory of impact. It is dedicated to the bright memory of Professor Yury A. Rossikhin who has contributed a lot in the development of the wave theory of impact based on the theory of discontinuities and ray expansions, resulting in analytical solutions of intricate problems of impact interaction of solids possessing different features.

Keywords

conditions of compatibility fractional calculus models Hertz’s contact law ray expansions theory of discontinuities viscoelastic impact wave theory of impact 

Notes

Acknowledgments

The different aspects of the research in the field of impact dynamics carried out by Professor Rossikhin and his team at the Research Center on Dynamics of Solids and Structures, Voronezh State Technical University, were supported by the International Science Foundation (1994–1999), the Russian Ministry of Science and Education (1994–1996, 2004–2018, including the current Project No. 9.994.2017/4.6), Russian Foundation for Basic Research (1993–2018, including the current project No. 17-01-00490).

References

  1. 1.
    Y.A. Rossikhin and M.V. Shitikova, Transient Response of Thin Bodies Subjected to Impact: Wave Approach, Shock Vibr. Digest, 2007, 39, p 273–309Google Scholar
  2. 2.
    W.E. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Arnold, London, 1960Google Scholar
  3. 3.
    S. Abrate, Modeling of Impacts on Composite Structures, Compos. Struct., 2001, 51(2), p 129–138Google Scholar
  4. 4.
    L.B. Greszczuk, Damage in Composite Materials due to Low Velocity Impact, Impact Dynamics, Z.A. Zukas et al., Ed., Wiley, New York, 1982, p 55–94Google Scholar
  5. 5.
    K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985Google Scholar
  6. 6.
    Y.A. Rossikhin and M.V. Shitikova, Ray Method for Solving Boundary Dynamic Problems Connected with Propagation of Wave Surfaces of Strong and Weak Discontinuities, Appl. Mech. Rev., 1995, 48(1), p 1–39Google Scholar
  7. 7.
    Y.A. Rossikhin and M.V. Shitikova, The Method of Ray Expansions for Investigating Transient Wave Processes in Thin Elastic Plates and Shells, Acta Mech., 2007, 189, p 87–121Google Scholar
  8. 8.
    Y.A. Rossikhin and M.V. Shitikova, Ray Expansion Theory, Encyclopedia of Thermal Stresses, Vol 8, R.B. Hetnarski, Ed., Springer, Berlin, 2014, p 4108–4131Google Scholar
  9. 9.
    C. Zener, The Intrinsic Inelasticity of Large Plates, Phys. Rev., 1941, 59(7), p 669–673Google Scholar
  10. 10.
    Y.A. Rossikhin and M.V. Shitikova, Ray Expansions in Impact Interaction Problems, Encyclopedia of Continuum Mechanics, H. Altenbach and A. Öchsner, Ed., Springer, Berlin, 2018,  https://doi.org/10.1007/978-3-662-53605-6_99-1 Google Scholar
  11. 11.
    Y.A. Rossikhin, Impact of a Rigid Sphere onto an Elastic Half Space, Sov. Appl. Mech., 1986, 22(5), p 403–409Google Scholar
  12. 12.
    Y.A. Rossikhin and M.V. Shitikova, Problem of the Impact Interaction of an Elastic Rod with a Uflyand–Mindlin Plate, Int. Appl. Mech., 1993, 29(2), p 118–125Google Scholar
  13. 13.
    Y.A. Rossikhin and M.V. Shitikova, The Impact of a Rigid Sphere with an Elastic Layer of Finite Thickness, Acta Mech., 1995, 112(1–4), p 83–93Google Scholar
  14. 14.
    Y.A. Rossikhin and M.V. Shitikova, The Impact of Elastic Bodies upon Beams and Plates with Consideration for the Transverse Deformations and Extension of a Middle Surface, ZAMM, 1996, 76(Suppl. 5), p 433–434Google Scholar
  15. 15.
    Y.A. Rossikhin and M.V. Shitikova, The Impact of a Sphere upon a Timoshenko Thin-Walled Beam of Open Section with Due Account for Middle Surface Extension, ASME J. Press. Vessel Tech., 1999, 121(4), p 375–383Google Scholar
  16. 16.
    Y.A. Rossikhin and M.V. Shitikova, The Method of Ray Expansions for Solving Boundary-Value Dynamic Problems for Spatially Curved Rods of Arbitrary Cross-Section, Acta Mech., 2008, 200(3–4), p 213–238Google Scholar
  17. 17.
    Y.A. Rossikhin, M.V. Shitikova, and V.V. Shamarin, Dynamic Response of Spherical Shells Impacted by Falling Objects, Int. J. Mech., 2011, 5(3), p 166–181Google Scholar
  18. 18.
    Y.A. Rossikhin and M.V. Shitikova, Dynamic Response of a Pre-stressed Transversely Isotropic Plate to Impact by an Elastic Rod, J. Vibr. Control, 2009, 15(1), p 25–51Google Scholar
  19. 19.
    Y.A. Rossikhin and M.V. Shitikova, Dynamic Response of Pre-stressed Spatially Curved Thin-walled Beams of Open Profile Impacted by a Falling Elastic Hemispherical-nosed Rod. In: H. Altenbach and V. Eremeev, (Eds), Shell-like Structures, Series Advanced Structured Materials. Springer, Berlin, 2011, vol. 15, Chapter 14, pp. 183–202Google Scholar
  20. 20.
    Y.A. Rossikhin and M.V. Shitikova, Dynamic Response of Pre-Stressed Spatially Curved Thin-Walled Beams of Open Profile, Springer Series Springer Briefs in Applied Sciences and Technology, 2011Google Scholar
  21. 21.
    Y.A. Rossikhin and M.V. Shitikova, Impact Response of Circular Pre-stressed Orthotropic and Transversely Isotropic Plates, Int. J. Mech., 2012, 6(1), p 68–87Google Scholar
  22. 22.
    Y.A. Rossikhin and M.V. Shitikova, About Shock Interaction of Elastic Bodies with Pseudo-Isotropic Uflyand–Mindlin Plates, Proceedings of the International Symposium on Impact Engineering, Vol 2, I. Maekawa, Ed., Sendai, Japan, Nov 2–4, 1992, pp. 623–628.Google Scholar
  23. 23.
    Y.A. Rossikhin and M.V. Shitikova, A New Approach for Studying the Transient Response of Thin-Walled Beams of Open Profile with Cosserat-Type Micro-structure, Compos. Struct., 2017, 169, p 153–166Google Scholar
  24. 24.
    Y.A. Rossikhin, Impact of a Thermoelastic Rod Against a Heated Obstacle, Sov. Appl. Mech., 1978, 14(9), p 962–967Google Scholar
  25. 25.
    Y.A. Rossikhin and M.V. Shitikova, The Impact of a Thermoelastic Rod Against a Rigid Heated Barrier, J. Eng. Math., 2002, 44(1), p 83–103Google Scholar
  26. 26.
    Y.A. Rossikhin and M.V. Shitikova, Analysis of a Hyperbolic System Modelling the Thermo-Elastic Impact of Two Rods, J. Therm. Stresses, 2007, 30(9–10), p 943–963Google Scholar
  27. 27.
    Y.A. Rossikhin and M.V. Shitikova, D’Alembert’s Solution in Thermoelasticity—Impact of a Rod Against a Heated Barrier: Part I. A Case of Uncoupled Strain and Temperature Fields, J. Therm. Stresses, 2009, 32(1–2), p 79–94Google Scholar
  28. 28.
    Y.A. Rossikhin and M.V. Shitikova, D’Alembert’s Solution in Thermo-elasticity—Impact of a Rod Against a Heated Barrier: Part II. A Case of Coupled Strain and Temperature Fields, J. Therm. Stresses, 2009, 32(3), p 244–268Google Scholar
  29. 29.
    Y.A. Rossikhin and M.V. Shitikova, Analysis of the Thermoelastic Rod Collision with a Heated Rigid Wall with Due Account for Temperature and Strain Weak coupling, Int. J. Mech., 2014, 8(1), p 62–72Google Scholar
  30. 30.
    Y.A. Rossikhin and M.V. Shitikova, D’Alembert method in dynamic problems of thermoelasticity, Encycl Therm Stresses, Vol 2, R.B. Hetnarski, Ed., Springer, Berlin, 2014, p 859–872Google Scholar
  31. 31.
    V.L. Gonsovskii, Y.A. Rossikhin, and M.V. Shitikova, The Impact of a Thermoelastic Rod Against a Heated Barrier with Account of Finite Speed of Heat Propagation, J. Thermal Stresses, 1993, 16(4), p 437–454Google Scholar
  32. 32.
    Y.A. Rossikhin and V.V. Shitikov, The Hyperbolic Model with a Small Parameter for Studying the Process of Impact of a Thermoelastic Rod Against a Heated Rigid Barrier, Appl. Math. Sci., 2016, 10(41–44), p 2037–2050Google Scholar
  33. 33.
    Y.A. Rossikhin, M.V. Shitikova, and V.V. Shitikov, Ray Expansion Theory in the Problem of Impact of a Thermoelastic Rod against a Heated Wall, J. Therm. Stresses, 2018,  https://doi.org/10.1080/01495739.2018.1527663
  34. 34.
    H.W. Lord and Y. Shulman, A Generalized Dynamic Theory of Thermoelasticity, J. Mech. Phys. Solids, 1967, 15(5), p 299–309Google Scholar
  35. 35.
    A.E. Green and P.M. Naghdy, Thermoelasticity Without Energy Dissipation, J. Elasticity, 1993, 31(3), p 189–208Google Scholar
  36. 36.
    V.L. Gonsovskii, S.I. Meshkov, and Y.A. Rossikhin, Impact of a Viscoelastic Rod onto a Rigid Target, Sov. Appl. Mech., 1972, 8(10), p 1109–1113Google Scholar
  37. 37.
    Y.A. Rossikhin and M.V. Shitikova, Fractional-Derivative Viscoelastic Model of the Shock Interaction of a Rigid Body with a Plate, J. Eng. Math., 2008, 60(1), p 101–113Google Scholar
  38. 38.
    Y.A. Rossikhin and M.V. Shitikova, The Analysis of the Impact Response of a Thin Plate Via Fractional Derivative Standard Linear Solid Model, J. Sound Vibr., 2011, 330(9), p 1985–2003Google Scholar
  39. 39.
    T.K. Chang, Y.A. Rossikhin, M.V. Shitikova, and C.K. Chao, Application of Fractional-Derivative Standard Linear Solid Model to Impact Response of Human Frontal Bone, Theor. Appl. Fract. Mech., 2011, 56(3), p 148–153Google Scholar
  40. 40.
    Y.A. Rossikhin and M.V. Shitikova, Analysis of Two Colliding Fractionally Damped Spherical Shells in Modelling Blust Human Head Impacts, Central Eur. J. Phys., 2013, 11(6), p 760–778Google Scholar
  41. 41.
    Y.A. Rossikhin and M.V. Shitikova, Two Approaches for Studying the Impact Response of Viscoelastic Engineering Systems: An Overview, Comput. Math. Appl., 2013, 66, p 755–773Google Scholar
  42. 42.
    T.K. Chang, Y.A. Rossikhin, M.V. Shitikova, and C.K. Chao, Impact Response of a Fractionally Damped Spherical Shell, J. Mech., 2015, 31(1), p 47–53Google Scholar
  43. 43.
    I.I. Popov, Y.A. Rossikhin, M.V. Shitikova, and T.P. Chang, Impact Response of a Viscoelastic Beam Considering the Changes of Its Microstructure in the Contact Domain, Mech. Time Depend. Mat., 2015, 19(4), p 455–481Google Scholar
  44. 44.
    Y.A. Rossikhin and M.V. Shitikova, Dynamic Response of a Viscoelastic Plate Impacted by an Elastic Rod, J. Vibr. Control, 2016, 22(8), p 2019–2031Google Scholar
  45. 45.
    Y.A. Rossikhin, M.V. Shitikova, and M.G. Meza Estrada, Impact Response of a Timoshenko-Type Viscoelastic Beam Considering the Extension of Its Middle Surface, SpringerPlus, 2016, 5(1), p 1–18Google Scholar
  46. 46.
    Y.A. Rossikhin, M.V. Shitikova, and D.T. Manh, Modelling of the Collision of Two Viscoelastic Spherical Shells, Mech. Time-Depend. Mat., 2016, 20(4), p 481–509Google Scholar
  47. 47.
    Y.A. Rossikhin, M.V. Shitikova, and M.G. Meza Estrada, Modeling of the Impact Response of a Beam in a Viscoelastic Medium, Appl. Math. Sci., 2016, 10(49), p 2471–2481Google Scholar
  48. 48.
    Y.A. Rossikhin, M.V. Shitikova, and D.T. Manh, Normal Impact of a Viscoelastic Spherical Shell Against a Rigid Plate, WSEAS Trans. Appl. Theor. Mech., 2016, 11, p 125–128Google Scholar
  49. 49.
    Y.A. Rossikhin, M.V. Shitikova, and P.T. Trung, Application of the Fractional Derivative Kelvin–Voigt Model for the Analysis of Impact Response of a Kirchhoff–Love Plate, WSEAS Trans. Math., 2016, 15, p 498–501Google Scholar
  50. 50.
    Y.A. Rossikhin, M.V. Shitikova, and P.T. Trung, Analysis of the Viscoelastic Sphere Impact against a Viscoelastic Uflyand-Mindlin Plate Considering the Extension of its Middle Surface, Shock Vibr., 2017, 2017, Paper ID 5652023Google Scholar
  51. 51.
    Y.A. Rossikhin, M.V. Shitikova, and D.T. Manh, Comparative Analysis of Two Problems of the Impact Interaction of Rigid and Viscoelastic Spherical Shells, Int. J. Mech., 2017, 11, p 6–11Google Scholar
  52. 52.
    Y.A. Rossikhin, M.V. Shitikova, and P.T. Trung, Impact of a Viscoelastic Sphere Against an Elastic Kirchhoff–Love Plate Embedded into a Fractional Derivative Kelvin–Voigt Medium, Int. J. Mech., 2017, 11, p 58–63Google Scholar
  53. 53.
    Y.A. Rossikhin, M.V. Shitikova, and P.T. Trung, Low-velocity Impact Response of a Pre-stressed Isotropic Uflyand-Mindlin Plate, ITM Web Conf., 2017, vol. 9, Article ID 03005Google Scholar
  54. 54.
    Y.A. Rossikhin, M.V. Shitikova, and I.I. Popov, Dynamic Response of a Viscoelastic Beam Impacted by a Viscoelastic Sphere, Comp. Math. Appl., 2017, 73(6), p 970–984Google Scholar
  55. 55.
    I.I. Popov, T.P. Chang, Y.A. Rossikhin, and M.V. Shitikova, Experimental Study of Concrete Aging Effect on the Contact Force and Contact Time During the Impact Interaction of an Elastic Rod with a Viscoelastic Beam, J. Mech., 2017, 33(3), p 317–322Google Scholar
  56. 56.
    Y.A. Rossikhin and M.V. Shitikova, Classical Beams and Plates in a Fractional Derivative Medium, Impact Response, Encyclopedia of Continuum Mechanics, H. Altenbach and A. Öchsner, Ed., Springer, Berlin, 2018,  https://doi.org/10.1007/978-3-662-53605-6_86-1 Google Scholar
  57. 57.
    Y.A. Rossikhin and M.V. Shitikova, Fractional Derivative Timoshenko Beams and Uflyand-Mindlin Plates, Encyclopedia of Continuum Mechanics, H. Altenbach and A. Öchsner, Ed., Springer, Berlin, 2018,  https://doi.org/10.1007/978-3-662-53605-6_87-1 Google Scholar
  58. 58.
    Y.A. Rossikhin and M.V. Shitikova, Collision of Two Spherical Shells, Fractional Operator Models, Encyclopedia of Continuum Mechanics, H. Altenbach and A. Öchsner, Ed., Springer, Berlin, 2018,  https://doi.org/10.1007/978-3-662-53605-6_88-1 Google Scholar
  59. 59.
    Y.A. Rossikhin and M.V. Shitikova, Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids, Appl. Mech. Rev., 1997, 50(1), p 15–67Google Scholar
  60. 60.
    Y.A. Rossikhin and M.V. Shitikova, Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results, Appl. Mech. Rev., 2010, 63(1), p 1–52Google Scholar
  61. 61.
    Y.A. Rossikhin, Reflections on Two Parallel Ways in Progress of Fractional Calculus in Mechanics of Solids, Appl. Mech. Rev., 2010, 63(1), p 1–12Google Scholar
  62. 62.
    H. Hertz, Über die Berührung fester elastischer Körper, Journal reine und angewandte Mathematik, 1882, vol. 92, pp. 156–171 (English translation can be found in Miscellaneous Papers by H. Hertz, Jones and Schott, Eds., Macmillan, London, 1896)Google Scholar
  63. 63.
    A.W. Crook, A Study of Some Impacts Between Metal Bodies by a Piezo-electric Method, Proc. Roy. Soc. Series A. Math. Phys. Sci., 1952, A212(1110), p 377–390Google Scholar
  64. 64.
    S.P. Timoshenko, Zur Frage nach der Wirkung eines Stosses auf einen Balken, ZAMP Zeitschrift fur Mathematik und Physik, 1913, 62(1–4), p 198–209Google Scholar
  65. 65.
    K. Karas, Platten Unter Seitlichem Stoss, Ing. Archiv., 1939, 10, p 237–250Google Scholar
  66. 66.
    Y.A. Rossikhin and M.V. Shitikova, The Ray Method for Solving Boundary Problems of Wave Dynamics for Bodies Having Curvilinear Anisotropy, Acta Mech., 1995, 109(1–4), p 49–64Google Scholar
  67. 67.
    T. Thomas, Plastic Flow and Fracture in Solids, Academic Press, New York, 1961Google Scholar
  68. 68.
    Y.A. Rossikhin and M.V. Shitikova, A Ray Method of Solving Problems Connected with a Shock Interaction, Acta Mech., 1994, 102, p 103–121Google Scholar
  69. 69.
    S.P. Timoshenko, On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bar, Philos. Mag., 1921, 41(245), p 744–746Google Scholar
  70. 70.
    S.P. Timoshenko, Vibration Problems in Engineering, Van Nostrand, New York, 1928Google Scholar
  71. 71.
    Y.S. Uflyand, Wave Propagation Under Transverse Vibrations of Rods and Plates (in Russian), Prikl. Mat. Mech., 1948, 12(3), p 287–300Google Scholar
  72. 72.
    R.D. Mindlin, Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates, J. Appl. Mech., 1951, 18(1), p 31–38Google Scholar
  73. 73.
    E. Reissner, On the Theory of Bending of Elastic Plates, J. Math. Phys., 1944, 23(4), p 184–191Google Scholar
  74. 74.
    S.A. Ambartsumyan, Theory of Anisotropic Plates (in Russian), Fizmatlit, Moscow, 1967Google Scholar
  75. 75.
    I. Elishakoff, Ju. Kaplunov, and E. Nolde, Celebrating the Centenary of Timoshenko’s Study of Effects of Shear Deformation and Rotary Inertia, Appl. Mech. Rev., 2015, 67, Paper ID 060802Google Scholar
  76. 76.
    P.A. Zhilin, Mechanics of Deformable Directed Surfaces, Int. J. Solids Struct., 1976, 12(9–10), p 635–648Google Scholar
  77. 77.
    T. Kaneko, On Timoshenko’s Correction for Shear in Vibrating Beams, J. Phys., 1975, D8, p 1927–1936Google Scholar
  78. 78.
    J.R. Hutchinson, Shear Coefficients for Timoshenko Beam Theory, J. Appl. Mech., 2001, 68(1), p 87–92Google Scholar
  79. 79.
    P.A. Zhilin, Applied Mechanics. Theory of Thin Elastic Rods (in Russian), St. Petersburg Polytechnic University, St. Petersburg, 2007Google Scholar
  80. 80.
    H. Altenbach, An Alternative Determination of Transverse Shear Stiffnesses for Sandwich and Laminated Plates, Int. J. Solids Struct., 2000, 37(25), p 3503–3520Google Scholar
  81. 81.
    J.N. Reddy, A Refined Nonlinear Theory of Plates with Transverse Shear Deformation, Int. J. Solids Struct., 1984, 20, p 881–896Google Scholar
  82. 82.
    Y.A. Rossikhin and M.V. Shitikova, The Method of Ray Expansions for Solving Boundary-Value Dynamic Problems for Spatially Curved Rods of Arbitrary Cross-Section, Acta Mech., 2008, 200, p 213–238Google Scholar
  83. 83.
    Y.A. Rossikhin and M.V. Shitikova, Dynamic Response of Spatially Curved Thermoelastic Thin-Walled Beams of Generic Open Profile Subjected to Thermal Shock, J. Therm. Stresses, 2012, 35, p 205–234Google Scholar
  84. 84.
    Y.A. Rossikhin and M.V. Shitikova, Thermal Shock upon Thin-Walled Beams of Open Profile, Encyclopedia of Thermal Stresses, Vol 9, R.B. Hetnarski, Ed., Springer, Berlin, 2014, p 5146–5167Google Scholar
  85. 85.
    Y.A. Rossikhin and M.V. Shitikova, Transient Wave Velocities in Pre-stressed Thin-Walled Beams of Open Profile with Cosserat-Type Micro-structure, Compos. Part B: Eng., 2015, 83, p 323–332Google Scholar
  86. 86.
    N. Fantuzzi, L. Leonetti, P. Trovalusci, and F. Tornabene, Some Novel Numerical Applications of Cosserat Continua, Int. J. Comput. Methods, 2017, 15, p 1–38Google Scholar
  87. 87.
    E. Pasternak and A.V. Dyskin, On a Possibility of Reconstruction of Cosserat Moduli in Particulate Materials Using Long Waves, Acta Mech., 2014, 225(8), p 2409–2422Google Scholar
  88. 88.
    D. Bigoni and W.J. Drugan, Analytical Derivation of Cosserat Moduli Via Homogenization of Heterogeneous Elastic Materials, ASME J. Appl. Mech., 2007, 74, p 741–753Google Scholar
  89. 89.
    W. Nowacki, Theory of Asymmetric Elasticity, Pergamon Press, Oxford, 1986Google Scholar
  90. 90.
    Y.A. Rossikhin, M.V. Shitikova, and J.M. Muhammed Saleh Khalid, Impact-Induced Internal Resonance Phenomena in Nonlinear Doubly Curved Shallow Shells with Rectangular Base, Analysis and Modelling of Advanced Structures and Smart Systems, Advanced Structured Materials, Vol 81, H. Altenbach, E. Carrera, and G. Kulikov, Ed., Springer, Singapore, 2018, p 149–189Google Scholar
  91. 91.
    H.D. Conway and H.C. Lee, Impact of an Indenter on a Large Plate, ASME J. Appl. Mech., 1970, 37(1), p 234–235Google Scholar
  92. 92.
    Y.A. Rossikhin and M.V. Shitikova, Dynamic Stability of a Circular Pre-stressed Elastic Orthotropic Plate Subjected to Shock Excitation, Shock Vibr., 2006, 13(3), p 197–214Google Scholar

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© ASM International 2019

Authors and Affiliations

  1. 1.Research Center on Dynamics of Solids and StructuresVoronezh State Technical UniversityVoronezhRussian Federation

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