Abstract
In this chapter, some regularities in the propagation of plane waves in hypoelastic materials are expounded. The statement is divided into three parts. In the first part, the main facts from the theory of hypoelastic materials and the basic notions are introduced and discussed, and the necessary information on elastic plane waves is given. Part 2 is devoted to the transition from the general nonlinear case to a linearized model of hypoelastic material analysis, which includes linearized constitutive equations. The key point is the possibility of analyzing the presence of initial stresses and initial velocities. Part 3 presents an example plane waves exploration in the presence of initial stresses and initial velocities. Here, the influence of the initial state on the types and number of plane waves is studied along with a general approach and the simplest case of initially isotropic material. The wave effects characteristic for hypoelastic materials are described. In particular, the effect of blocking the initiation of certain types of plane waves by means of initial stresses is predicted.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)
Cattani, C., Rushchitsky, J.J.: Generation of the third harmonic by the plane waves in materials with Murnaghan potential. Int. Appl. Mech. 38(12), 1482–1487 (2002)
Cattani, C., Rushchitsky, J.J.: Cubically nonlinear elastic waves: wave equations and methods of analysis. Int. Appl. Mech. 39(10), 1115–1145 (2003)
Cattani, C., Rushchitsky, J.J.: Cubically nonlinear elastic waves versus quadratically ones. Int. Appl. Mech. 39(12), 1361–1399 (2003)
Cattani, C., Rushchitsky, J.J.: Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure. World Scientific, Singapore/London (2007)
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)
Drumheller, D.S.: Introduction to Wave Propagation in Nonlinear Fluids and Solids. Cambridge University Press, Cambridge (1998)
Erofeev, V.I.: Wave Processes in Solids with Microstructure. World Scientific, Singapore/London (2003)
Fedorov, F.I.: Theory of Elastic Waves in Crystals. Plenum, New York, NY (1968)
Germain, P.: Cours de mechanique des milieux continus. Tome 1. Theorie generale, (Course of Continuum Mechanics. vol. 1, General Theory), Masson et Cie, Editeurs, Paris (1973)
Green, A.E., Adkins, J.E.: Large Elastic Deformations and Nonlinear Continuum Mechanics. Oxford University Press/Clarendon Press, London (1960)
Guz, A.N.: Uprugie volny v tielakh s nachalnymi (ostatochnymi) napriazheniiami (Elastic Waves in Bodies with Initial (Residual) Stresses. А.С.К, Kiev (2004)
Guz, A.N.: Uprugie volny v tielakh s nachalnymi napriazheniiami (Elastic Waves in Bodies with Initial Stresses). Naukova dumka, Kiev (1987)
Guz, A.N.: Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies. Springer, Berlin (1999)
Guz, A.N., Makhort, F.G., Gushcha, O.I., Lebedev, V.K.: Osnovy ultra-zvukovogo nierazru-shaiushchego metoda opredelenia napriazhenii v tvierdykh tielakh (Foundations of Ultra-Sound Method of Determination of Stresses in Solid Bodies). Naukova Dumka, Kiev (1984)
Guz, A.N., Rushchitsky, J.J.: Nanomaterials. On mechanics of nanomaterials. Int. Appl. Mech. 39(11), 1271–1293 (2003)
Guz, A.N., Rushchitsky, J.J.: Short Introduction to Mechanics of Nanocomposites. Scientific and Academic Publishing, Rosemead (2012)
Guz, A.N., Rushchitsky, J.J., Guz, I.A.: Introduction to Mechanics of Nanocomposites. Aka-demperiodika, Kiev (2010)
Guz, A.N., Zhuk, A.P., Makhort, F.G.: Volny v sloie s nachalnymi napriazheniiami. (Waves in a Layer with Initial Stresses). Naukova Dumka, Kiev (1986)
Kauderer, H.: Nichtlineare Mechanik (Nonlinear Mechanics). Springer, Berlin (1958)
Lur’e, A.I.: Nonlinear Theory of Elasticity. North-Holland, Amsterdam (1990)
Lur’e, A.I.: Theory of Elasticity. Springer, Berlin (1999)
Prager, W.: Einfūhrung in die Kontinuumsmechanik. Birkhäuser, Basel (1961). Introduction to Mechanics of Continua. Ginn, Boston (1961)
Rushchitsky, J.J.: Interaction of waves in solid mixtures. Appl. Mech. Rev. 52(2), 35–74 (1999)
Rushchitsky, J.J.: Development of microstructural theory of two-phase mixtures as applied to composite materials. Int. Appl. Mech. 36(5), 315–335 (2000)
Rushchitsky, J.J.: On types and number of plane waves in hypoelastic materials. Int. Appl. Mech. 41(11), 1288–1298 (2005)
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)
Rushchitsky, J.J., Tsurpal, S.I.: Khvyli v materialakh z mikrostrukturoiu (Waves in Materials with the Microstructure). SP Timoshenko Institute of Mechanics, Kiev (1998)
Truesdell, C.: A First Course in Rational Continuum Mechanics. John Hopkins University, Baltimore, MD (1972)
Author information
Authors and Affiliations
Exercises
Exercises
-
1.
The division of elastic materials into hyperelastic, elastic, and hypoelastic ones is well known in mechanics. Where else in mechanics of materials are the prefixes “hyper” and “hypo” used (for example, perhaps someone proposed a classification scheme—hyperplastic, plastic, and hypoplastic materials).
-
2.
Provide other definitions for the stress change rate that are different from the Jaumann’s definition.
-
3.
Formulate an H-condition.
-
4.
Compare the Christoffel equation for hypoelastic materials with initial stresses (8.15) with the Christoffel equation for hyperelastic materials with initial stresses (see, for example, [19]).
-
5.
Try to comment more in depth the presence of initial velocity in the Christoffel equation (8.18).
-
6.
Discuss, from the point of view of mechanics, the impossibility of exciting a plane wave in hypoelastic materials.
-
7.
Discuss, from point of view of mechanics, the filtering of plane waves in hypoelastic materials.
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Rushchitsky, J.J. (2014). Nonlinear Plane Waves in Hypoelastic Materials. In: Nonlinear Elastic Waves in Materials. Foundations of Engineering Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-00464-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-00464-8_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00463-1
Online ISBN: 978-3-319-00464-8
eBook Packages: EngineeringEngineering (R0)