Influence of Weak Shock Wave on the Dynamic Stress State of Foam Materials

Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


This paper proposes the technique for the analysis of weak shock wave influence on the dynamic stress state of foam materials with negative and positive Poisson’s ratio. The investigation of the dynamic behavior of foam materials under the action of weak shock waves is performed in the framework of couple stress elasticity, where one can account for the influence of shear rotation deformation in structurally inhomogeneous media. For the solution of the non-stationary problem, Fourier transforms are used. The calculation of transforms of dynamic stresses in the foam medium is performed by using the boundary integral equation method and the theory of complex variable functions in the framework of couple stress elasticity. The numerical implementation of the developed algorithm is based on the method of mechanical quadrature and collocation technique. For calculation of originals of dynamic stresses discrete, Fourier transform is used. The distribution of dynamic hoop stresses in a positive and negative Poisson’s ratio foam medium with tunnel cavities under the action of a weak shock wave is investigated. The algorithm is effective in the analysis of the dynamic behavior of the foam media with tunnel defects of various cross-sections.


Auxetic Couple stress elasticity Time domain problem 


  1. 1.
    Lakes, R.S.: Experimental micro mechanics methods for conventional and negative poisson’s ratio cellular solids as cosserat continua. J. Eng. Mater. Tech. 113, 148–155 (1991)CrossRefGoogle Scholar
  2. 2.
    Grima, J., Attard, D., Gatt, R., Cassar, R.: A novel process for the manufacture of auxetic foams and for their re-conversion to conventional form. Adv. Eng. Mater. 11(7), 533–535 (2009)CrossRefGoogle Scholar
  3. 3.
    Lakes, R.S.: Physical meaning of elastic constants in cosserat, void, and microstretch elasticity. J. Mech. Mater. Struct. 11(3), 217–229 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Li, D., Dong, L., Lakes, R.: A unit cell structure with tunable poisson’s ratio from positive to negative. Mater. Lett. 164, 456–459 (2016)CrossRefGoogle Scholar
  5. 5.
    Rueger, Z., Lakes, R.S.: Cosserat elasticity of negative poisson’s ratio foam: experiment. Smart Mater. Struct. 25, 1–8 (2016)CrossRefGoogle Scholar
  6. 6.
    Altenbach, H., Eremeyev, V.: Cosserat media. In: Generalized Continua from the Theory to Engineering Applications, vol. 541, pp. 65–130 (2013)Google Scholar
  7. 7.
    Hassanpour, S., Heppler, G.: Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Math. Mech. Solids 12, 1–19 (2015)zbMATHGoogle Scholar
  8. 8.
    Lim, T-Ch.: Auxeticity of concentric auxetic-conventional foam rods with high modulus interface adhesive. Materials 11(2), 223–235 (2018)CrossRefGoogle Scholar
  9. 9.
    Hou, X., Deng, Z., Zhou, J.: Simplistic analysis for the wave propagation properties of conventional and auxetic cellular structures. Int. J. Numer. Anal. Model. 2(4), 298–314 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Adler, L., Warmuth, F., Lodes, M., Osmanlic, F., Körner, C.: The effect of a negative poisson’s ratio on thermal stresses in cellular metallic structures. Smart Mater. Struct. 25, 1–9 (2016)CrossRefGoogle Scholar
  11. 11.
    Strek, T., Jopek, H., Idczak, E., Wojciechowski, K.: Computational modelling of structures with non-intuitive behaviour. Materials 10, 1386–1402 (2017)CrossRefGoogle Scholar
  12. 12.
    Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solid Struct. 48(18), 2496–2510 (2011)CrossRefGoogle Scholar
  13. 13.
    Savin, G.N., Shulga, N.A.: Dynamic plane problem of the moment theory of elasticity. Appl. Mech. 3(6), 216–221 (1967)Google Scholar
  14. 14.
    Hadjesfandiari, A.R., Dargush, G.F.: Fundamental solutions for isotropic size-dependent couple stress elasticity. Int. J. Solid Struct. 50(9), 1253–1265 (2013)CrossRefGoogle Scholar
  15. 15.
    Mikulich, O., Shvabyuk, V., Pasternak, Ia., Andriichuk, O.: Modification of boundary integral equation method for investigation of dynamic stresses for couple stress elasticity. Mech. Res. Commun. 91, 107–111 (2018)Google Scholar
  16. 16.
    Tabak, E.G., Rosales, R.R.: Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection. Phys. Fluids 6(5), 1874–1892 (1994)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wang, Y., Gioia, G., Cuitiño, A.: The deformation habits of compressed open-cell solid foams. J. Eng. Mater. Tech. 122, 376–378 (2000)CrossRefGoogle Scholar
  18. 18.
    Nowacki, W.: The Linear Theory of Micropolar Elasticity. Springer, New York (1974)CrossRefGoogle Scholar
  19. 19.
    Mikulich, O., Shvabyuk, V., Sulym, H.: Dynamic stress concentration at the boundary of an incision at the plate under the action of weak shock waves. Acta Mechanica et Automatica 11(3), 217–221 (2017)CrossRefGoogle Scholar

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© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Lutsk National Technical UniversityLutskUkraine

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