Cubically nonlinear elastic waves: Wave equations and methods of analysis

• Published in: International applied mechanics Vol. 39; no. 10; pp. 1115 - 1145
• Main Authors: CATTANI, C, RUSHCHITSKII, Ya. Ya
• Format: Journal Article
• Language: English
• Published: London Consultants Bureau 01.10.2003; New York, NY
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; Structural and continuum mechanics; Theory and numerical methods; Wave equation; Wavelet transformation; Potential well; Plane wave; Mixture theory; Polarized wave; Mindlin model; Elastic wave; Elastodynamics; Hyperelasticity; Cosserat material; Non linear wave
• ISSN: 1063-7095; 1573-8582
• DOI: 10.1023/B:INAM.0000010366.48158.48

Studies on cubically nonlinear elastic waves are reviewed: the state of affairs in research on cubically and quadratically nonlinear waves in physics is briefly reported; a general view on nonlinear elastic waves is formulated; and mainly hyperelastic materials are considered. A description is given to well-known elastic potentials and various modifications of the Murnaghan potential such as the classical modification (in five forms), Guz's modification, Mindlin-Eringen modification, modification for a Cosserat pseudocontinuum, modification for Le Roux gradient theory, and two modifications for the theory of elastic mixtures. A procedure of transition from the potential to wave equations is described; and the corresponding wave equations for plane polarized waves are written for all the modifications mentioned above. Three basic methods for solving wave equations are considered and commented on: the method of successive approximations, the method of slowly varying amplitudes, and a wavelet-based method. The last method is discussed in more detail and exemplified.