Evolution of the theory of nonlinear waves in Murnaghan and Signorini materials

Abstract

The development of the classical Signorini nonlinear elastic model is continued. Recent year's published results on a recently developed area of the theory of nonlinear waves are outlined from a unified position and analyzed. The paper consists of six sections: explanation of the features of the class of waves under consideration; detailed description of the Signorini model; general formulation of wave-propagation problems based on the Signorini potential; analysis of results on plane and cylindrical waves obtained with the Signorini model; comparison of these results with those obtained with the Murnaghan model; discussion of the possible lines for the development of the research.Introduction. After many decades of neglect, the Signorini model is addressed here to clarify its capabilities in analyzing plane waves. Antonio Signorini (1888Signorini ( -1963, an outstanding Italian scientist, is by no means forgotten in modern mechanics. He is mentioned and quoted repeatedly in a number of classical books on continuum mechanics. For example, Chapter 9 of the book [87] discusses and elaborates upon Signorini's idea of iterative solution of nonlinear boundary-value problems in elasticity. Trusdell starts from Signorini's publication [80]. Of recent popularity is the Signorini problem with so-called Signorini boundary conditions, which is analyzed by mathematicians who represent many countries and schools of thought [20,33,36,37,39,42,43,45,48,77,78,90].The paper is structured as follows. Its goal is to unite recent results on waves in materials described by the Signorini elastic potential. A feature of the associated class of nonlinear problems is that they originally dealt with materials described by the Murnaghan potential. A general view of the achievements in solving wave problems with this model is presented in the reviews [17,18,60,[68][69][70] and, partially, in the recent monographs [9,19,30,50]. Therefore, results produced by the Murnaghan model are included here only for the sake of coherence and consistency. The present paper is mainly based on the author's publications [15,[25][26][27][28][29][30][31][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76] and consists of seven sections. The structure of the paper is described, features of the class of nonlinear elastic waves in materials described by elastic potentials are listed, and the main concepts are interpreted in Introduction. Section 1 describes the Signorini potential and gives some new comments on the Signorini constant, the Kelvin and Poynting effects, and Volterra distortions. Section 2 provides a general problem formulation for waves in nonlinear elasticity based on the Signorini potential and the transformation from the Eulerian to Lagrangian description of deformation, resulting in nonlinear wave equations in Lagrange coordinates. Section 3 preliminarily compares the nonlinear wave equations in the Murnaghan and Signorini models, including the solutions for longitudinal and transverse plane waves obtained with these models. Section...