Abstract
This paper presents a rigorous procedure, based on the concepts of nonlinear continuum mechanics, to derive nonlinear wave equations that describe the propagation and interaction of plane hyperelastic waves. The nonlinearity is introduced by the Signorini potential and represents the quadratic nonlinearity of all governing relations with respect to displacements. A configuration (state) of the elastic medium dependent on the abscissa is analyzed. Analytic transformations are used to go over from the Eulerian to the Lagrangian description of nonlinear deformation and from the invariants of the Almansi finite-strain tensor to the invariants of the Cauchy-Green finite-strain tensor. Nonlinear wave equations describing the propagation of plane longitudinal and transverse waves in Signorini’s materials are derived, and the strain and true-stress tensors are analytically expressed in terms of the deformation gradient. These wave equations are compared with those based on the Murnaghan model. Their similarities and differences are indicated. It is shown that the new Signorini constant can be identified from the Lamé and Murnaghan constants
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References
A. N. Guz, Elastic Waves in Bodies with Initial (Residual) Stresses [in Russian], A.S.K., Kyiv (2004).
A. N. Guz, Elastic Waves in Prestressed Bodies [in Russian], Vol. 1, Naukova Dumka, Kyiv (1987).
L. K. Zarembo and V. A. Krasil’nikov, An Introduction to Nonlinear Acoustics [in Russian], Nauka, Moscow (1966).
V. V. Krylov and V. A. Krasil’nikov, An Introduction to Physical Acoustics [in Russian], Nauka, Moscow (1986).
A. I. Lurie, Theory of Elasticity, Springer, Berlin (1999).
A. I. Lurie, Nonlinear Theory of Elasticity, North-Holland, Amsterdam (1990).
J. J. Rushchitsky and S. I. Tsurpal, Waves in Microstructural Materials [in Ukrainian], Inst. Mekh. im. S. P. Tymoshenka, Kyiv (1998).
L. I. Sedov, Mechanics of Continuous Media, World Scientific, Singapore (1997).
C. Cattani and J. J. Rushchitsky, “Cubically nonlinear elastic waves: Wave equations and methods of analysis,” Int. Appl. Mech., 39, No. 10, 1115–1145 (2003).
C. Cattani and J. J. Rushchitsky, “Cubically nonlinear versus quadratically nonlinear elastic waves: Main wave effects,” Int. Appl. Mech., 39, No. 12, 1361–1399 (2003).
C. Cattani and J. J. Rushchitsky, “Nonlinear cylindrical waves in Signorini’s hyperelastic material,” Int. Appl. Mech., 42, No. 7, 1361–1399 (2006).
A. N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Springer Verlag, Berlin (1999).
J. J. Rushchitsky, “Interaction of waves in solid mixtures,” Appl. Mech. Rev., 52, No. 2, 35–74 (1999).
J. J. Rushchitsky, “Extension of the microstructural theory of two-phase mixtures to composite materials,” Int. Appl. Mech., 36, No. 5, 586–614 (2000).
J. J. Rushchitsky, “Quadratically nonlinear cylindrical hyperelastic waves: Derivation of wave equations for plane-strain state,” Int. Appl. Mech., 41, No. 5, 496–505 (2005).
J. J. Rushchitsky, “Quadratically nonlinear cylindrical hyperelastic waves: Derivation of wave equations for axisymmetric and other states,” Int. Appl. Mech., 41, No. 6, 646–656 (2005).
J. J. Rushchitsky, “Quadratically nonlinear cylindrical hyperelastic waves: Primary analysis of evolution,” Int. Appl. Mech., 41, No. 7, 770–777 (2005).
C. Cattani, J. J. Rushchitsky, and S. V. Sinchilo, “Physical constants for one type of nonlinearly elastic fibrous micro-and nanocomposites with hard and soft nonlinearities,” Int. Appl. Mech., 41, No. 12, 1368–1377 (2005).
A. Signorini, “Transformazioni termoelastiche finite,” Annali di Matematica Pura ed Applicata, Serie IV, 22, 33–143 (1943).
A. Signorini, “Transformazioni termoelastiche finite,” Annali di Matematica Pura ed Applicata, Serie IV, 30, 1–72 (1949).
A. Signorini, “Transformazioni termoelastiche finite. Solidi Incomprimibili. A Mauro Picone nel suo 70ane compleano,” Ànnali di Matematica Pura ed Applicata, Serie IV, 39, 147–201 (1955).
A. Signorini, Questioni di elasticite non linearizzata, Edizioni Cremonese, Roma (1959).
A. Signorini, “Questioni di elasticite non linearizzata e semilinearizzata,” Rendiconti di Matematica, 18, No. 1–2, 95–139 (1959).
A. Signorini, “Transformazioni termoelastiche finite. Solidi Vincolati. A Giovanni Sansone nel suo 70ane compleano,” Annali di Matematica Pura ed Applicata, Serie IV, 51, 320–372 (1960).
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Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 58–67, August 2006.
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Cattani, C., Rushchitsky, J.J. Nonlinear plane waves in Signorini’s hyperelastic material. Int Appl Mech 42, 895–903 (2006). https://doi.org/10.1007/s10778-006-0157-1
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DOI: https://doi.org/10.1007/s10778-006-0157-1