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The Simplest Linear Waves in Elastic Materials

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Nonlinear Elastic Waves in Materials

Part of the book series: Foundations of Engineering Mechanics ((FOUNDATIONS))

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Abstract

The first part of this chapter presents essential information on linear elastic waves. It involves basic equations of wave propagation in the theory of elasticity, volume and shear waves, the classic wave equation and basic facts associated with it, features, and terminology. The classic examples of breaking the correctness in mathematical statements (Helmholtz and Taylor instabilities and as well as Hadamard example and John sentence) are proposed, which describe the generation of waves. Special attention is given to plane waves and their classical analysis. In the second part, a model of linear elastic mixture (a structural model of the second order) that is distinct from the model of averaged elastic moduli used in the first part is considered. Here the basic equations of wave propagation in elastic mixtures, volume and shear waves, and plane waves and a brief analysis of them are presented.

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References

  1. Achenbach, J.D., Hermann, G.: Wave motion in solids with lamellar structuring. In: Hermann, G. (ed.) Proceedings of Symposium on Dynamics of Structured Solids, New York, 23–46 (1968)

    Google Scholar 

  2. Atkin, R.J., Crain, R.E.: Continuum theory of mixtures: basic theory and historical developments. Quart. J. Mech. Appl. Math. 29(2), 209–244 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bedford, A., Drumheller, D.S.: Theories of immiscible and structured mixtures. Int. J. Eng. Sci. 21(8), 863–960 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bedford, A., Drumheller, D.S.: Introduction to Elastic Wave Propagation. Wiley, Chichester (1994)

    Google Scholar 

  5. Bedford, A., Drumheller, D.S., Sutherland, H.J.: On modelling the dynamics of composite materials. In: Nemat-Nasser, S. (ed.) Mechanics Today, vol. 3, pp. 1–54. Pergamon Press, New York (1976)

    Chapter  Google Scholar 

  6. Bedford, A., Stern, M.: Toward a diffusing continuum theory of composite materials. Trans. ASME J. Appl. Mech. 38(1), 8–14 (1971)

    Article  Google Scholar 

  7. Bedford, A., Sutherland, H.J.: A lattice model for stress wave propagation in composite materials. Trans. ASME J. Appl. Mech. 40(1), 157–164 (1973)

    Google Scholar 

  8. Ben-Amoz, M.: On wave propagation in laminated composites. I. Propagation parallel to the laminates. Int. J. Eng. Sci. 14(1), 43–56 (1975)

    Article  Google Scholar 

  9. Ben-Amoz, M.: On wave propagation in laminated composites. II. Propagation normal to the laminates. Int. J. Eng. Sci. 14(1), 57–67 (1975)

    Article  Google Scholar 

  10. Bowen, P.M.: Mixtures and EM field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. III, pp. 1–127. Academic, New York (1976)

    Google Scholar 

  11. Bolotin, V.V.: Ob izgibe plit sostoiashchikh iz bolshogo chisla sloev (On bending of plates consisting of a great number of layers). Proc. Acad. Sci. USSR Mech. Eng. 3, 65–72 (1963)

    Google Scholar 

  12. Bolotin, V.V.: Osnovnyie uravneniia teorii armirovannykh sred (Basic equations of the theory of reinforced media). Mechanika Polymerov (Soviet Compos Mater) 2, 26–37 (1965)

    Google Scholar 

  13. Bolotin, V.V.: Vibration of layered elastic plates. Proc. Vib. Probl. 4(4), 331–346 (1963)

    MathSciNet  Google Scholar 

  14. Bowen, P.M.: Toward a thermodynamics and mechanics of mixtures. Arch. Ration. Mech. Anal. 24(5), 370–403 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  15. Broutman, L.J., Krock, R.H. (eds.): Composite Materials. In 8 vols. Academic, New York (1974–1975)

    Google Scholar 

  16. Cattani, C., Rushchitsky, J.J.: Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure. World Scientific, Singapore (2007)

    MATH  Google Scholar 

  17. Dieulesaint, E., Royer, D.: Ondes elastiques dans les solides. Application au traitement du signal (Elastic waves in solids. Application to a signal processing). Masson et Cie, Paris (1974)

    Google Scholar 

  18. Drumheller, D.S., Bedford, A.: Wave propagation in elastic laminates using a second order microstructure theory. Int. J. Solids Struct. 10(10), 61–76 (1974)

    Article  MATH  Google Scholar 

  19. Eringen, A.C.: Theory of micromorphic materials with memory. Int. J. Eng. Sci. 10(7), 623–641 (1972)

    Article  MATH  Google Scholar 

  20. Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple microelastic solids. Int. J. Eng. Sci. 2(2), 189–203 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hegemier, G.A., Nayfeh, A.N.: A continuum theory for wave propagation in composites. Case 1: propagation normal to the laminate. Trans. ASME J. Appl. Mech. 40(2), 503–510 (1973)

    Article  Google Scholar 

  22. Hegemier, G.A., Bache, T.C.: A continuum theory for wave propagation in composites. Case 2: propagation parallel the laminates. J. Elas. 3(2), 125–140 (1973)

    Article  Google Scholar 

  23. Hegemier, G.A., Bache, T.C.: A general continuum theory with the microstructure for the wave propagation in elastic laminated composites. Trans. ASME J. Appl. Mech. 41(1), 101–105 (1974)

    Article  MATH  Google Scholar 

  24. Herrmann, G., Kaul, R.K., Delph, T.G.: On continuum modelling of the dynamic behaviour of layered composites. Arch. Mech. 28(3), 405–421 (1978)

    Google Scholar 

  25. Lempriere, B.: On practicability of analyzing waves in composites by the theory of mixtures, Lockheed Palo Alto Research Laboratory, Report No. LMSC-6-78-69-21, 76–90 (1969)

    Google Scholar 

  26. Maugin, G.A.: A continuum approach to magnon-phonon couplings—I, II. Int. J. Eng. Sci. 17(11), 1073–1091, 1093–1108 (1979)

    Google Scholar 

  27. Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. North Holland, Amsterdam (1988)

    MATH  Google Scholar 

  28. Maugin, G.A., Eringen, A.C.: On the equations of the electrodynamics of deforarmble bodies of finite extent. J. de Mechanique. 16(1), 101–147 (1977)

    MATH  MathSciNet  Google Scholar 

  29. McNiven, H.D., Mengi, Y.: A mathematical model for the linear dynamic behavior of two-phase periodic materials. Int. J. Solids Struct. 15(1), 271–280 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  30. McNiven, H.D., Mengi, Y.: A mixture theory for elastic laminated composites. Int. J. Solids Struct. 15(1), 281–302 (1979)

    Article  MathSciNet  Google Scholar 

  31. McNiven, H.D., Mengi, Y.: Propagation of transient waves in elastic laminated composites. Int. J. Solids Struct. 15(1), 303–318 (1979)

    Article  MathSciNet  Google Scholar 

  32. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nigmatulin, R.I.: Dinamika mnogofaznykh sred (Dynamics of Multi-phase Media), In 2 parts. Nauka, Moscow (1987)

    Google Scholar 

  34. Peck, J.S., Gurtman, G.A.: Dispersive pulse propagation parallel to the interfaces of a laminated composite. Trans. ASME J. Appl. Mech. 36(2), 479–484 (1969)

    Article  MATH  Google Scholar 

  35. Pobedria, B.E.: Mechanika kompozicionnykh materialov (Mechanics of Composite Materials). Moscow University Press, Moscow (1984)

    Google Scholar 

  36. Pouget, J.: Electro-acoustic echoes in piezoelectric powders. In: Vaugin, G.A. (ed.) Proceedings of the IUTAM-IUPAP Symposium. North Holland, Amsterdam, pp. 177–184 (1984)

    Google Scholar 

  37. Robinson, C.W., Leppelmeier, G.W.: Experimental verification of dispersion relations for layered composites. Trans. ASME J. Appl. Mech. 41(1), 89–91 (1974)

    Article  Google Scholar 

  38. Rushchitsky, J.J.: To the plane problem of theory of mixture of two solids. Soviet Appl. Mech. 10(2), 52–61 (1974)

    Google Scholar 

  39. Rushchitsky, J.J.: On certain case of wave propagation in the mixture of elastic materials. Soviet Appl. Mech. 14(1), 25–33 (1978)

    Google Scholar 

  40. Rushchitsky, J.J.: Determination of physical constants of the theory of mixture of elastic materials using the experimentally obtained dispersion curves. Int. Appl. Mech. 15(6), 26–32 (1979)

    Google Scholar 

  41. Rushchitsky, J.J.: Elementy teorii smesi (Elements of the Theory of Mixtures). Naukova Dumka, Kyiv (1991)

    Google Scholar 

  42. Rushchitsky, J.J.: Interaction of waves in solid mixtures. Appl. Mech. Rev. 52(2), 35–74 (1999)

    Article  Google Scholar 

  43. Rushchitsky, J.J.: Theory of Waves in Materials. Ventus Publishing ApS, Copenhagen (2011)

    Google Scholar 

  44. Rushchitsky, J.J.: Certain class of nonlinear hyperelastic waves: classical and novel models, wave equations, wave effects. Int. J. Appl. Math. Mech. 8(6), 400–443 (2012)

    Google Scholar 

  45. Rushchitsky, J.J., Tsurpal, S.I.: Khvyli v materialakh z mikrostrukturoiu (Waves in Materials with the Microstructure). SP Timoshenko Institute of Mechanics, Kiev (1998)

    Google Scholar 

  46. Sutherland, H.J.: Dispersion of acoustic waves by an alumina-epoxy mixture. J. Compos. Mater. 13(1), 35–47 (1979)

    Article  MathSciNet  Google Scholar 

  47. Sutherland, H.J.: On the separation of geometric and viscoelastic dispersion in composite materials. Int. J. Solids Struct. 11(3), 233–246 (1975)

    Article  Google Scholar 

  48. Tiersten, T.R., Jahanmir, M.: A theory of composites modeled as interpenetrating solid continua. Arch. Ration. Mech. Anal. 54(2), 153–163 (1977)

    Article  MathSciNet  Google Scholar 

  49. Voelker, L.E., Achenbach, J.D.: Stress waves in a laminated medium generated by transverse forces. J. Acoust. Soc. Am. 46(6), 1213–1216 (1969)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Rheology, S.P.Timoshenko Institute of Mechanics The National Academy of Sciences, Kyiv, Ukraine

    Jeremiah J. Rushchitsky

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  1. Jeremiah J. Rushchitsky
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Exercises

Exercises

  1. 1.

    Establish a connection between the appearance in elastic materials of only volume and shear waves and the presence in these materials of universal deformations (five kinds, only).

  2. 2.

    Identify the most frequently used variant of D’Alembert formula in theory of waves and evaluate its simplicity.

  3. 3.

    Which sense is in introducing the frequency and circular frequency to describe harmonic waves in materials?

  4. 4.

    Derive Rayleigh’s formula, which links group and phase velocities, and construct for some simple case of the dispersion law a plot of the dependence of group velocity on wave number based on the plot of the dependence of phase velocity on wave number. Compare the plots.

  5. 5.

    Analyze whether the existence of standing waves contradict the existence of running waves within the framework of the same approach.

  6. 6.

    It is well known that Christoffel’s equation for plane waves transforms the mechanical problem on the types and number of plane waves into a mathematical problem on finding eigenvalues and eigenvectors. Verify the supposed conditions in the mathematical problem on mechanical properties (on the tensor c iklm ) and comment on them from a mechanical point of view.

  7. 7.

    Explain the use of the term energy continualization in some structural models of the second order. Why just energy? Look for the answer in the procedure of transition from a piecewise homogeneous structure to a homogeneous one.

  8. 8.

    Show the difference between effective property and effective stiffness. Explain why the term stiffness is used.

  9. 9.

    Assess the originality of the lattice model to describe wave propagation in fibrous composites and explain the different applicabilities of the model to the propagation of waves along and across fibers.

  10. 10.

    Read more in depth about the theory of granular piezoelectric powders and assess an adequacy of description of wave propagation based on this theory (see the publications of Maugin in the reference list).

  11. 11.

    The factor (−1)α in formula (4.30) determines, which component transmits and which receives linear impulse in a mixture of two elastic components. As a rule, one of two components is stiffer. Which number (α = 1 or α = 2) should be taken for this component?

  12. 12.

    Try to explain the origin of dispersion of waves in the linear theory of mixtures.

  13. 13.

    Consider the cutoff frequency and compare the theoretical prediction of this phenomenon with experimental observations of wave propagation in composite materials (first consult the reference list).

  14. 14.

    Read more (though not too much) on in-phase and antiphase vibrations of particles and establish appearance of antiphase vibrations of mixture components with the possible debonding of the composites.

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Rushchitsky, J.J. (2014). The Simplest Linear Waves in Elastic Materials. In: Nonlinear Elastic Waves in Materials. Foundations of Engineering Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-00464-8_4

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  • DOI: https://doi.org/10.1007/978-3-319-00464-8_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-00463-1

  • Online ISBN: 978-3-319-00464-8

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