Sets of physical constants are tabulated for three structural models of fibrous composites with fibers of four types: Thornel-300 carbon microfibers, graphite whiskers, carbon zigzag nanotubes, and carbon chiral nanotubes. The matrix for all the types of composites is always ÉPON-828 epoxy rosin (in some cases with polystyrene or pyrex additive). The values of the physical constants are commented on and used to study the distinctions in the evolution of three types of waves (plane longitudinal, plane transverse, and cylindrical) propagating in materials with soft and hard nonlinearities Keywords: hyperelastic materials, Murnaghan potential, complete sets of elastic constants, plane and cylindrical waves, hard and soft nonlinearitiesIntroduction. The present study is based on the publication [8] devoted to the relationship of the approaches and methods in all the historically established divisions of the mechanics of materials: macro-, meso-, micro-, and nanomechanics. As indicated in [8], the main common feature of these four divisions is that they account for the microstructure of materials and their main difference is the range of typical length scales for inhomogeneities in the microstructure of a material.Note that the first results illustrating the fundamental conclusions of the study [8] were presented in the paper [9], which was the first to propose to consider four types of fibrous composite materials: two with microlevel fibers (Thornel 300 carbon microfibers and graphite whiskers) and two with nanolevel fibers (zigzag and chiral graphite nanotubes). What these types of materials have in common is the materials of the fibers (carbon) and the matrix (ÉPON 828 epoxy resin). The mechanical constants of the matrix and fibers (density, Young's modulus, shear modulus, and Poisson's ratio) and the determinate microstructure of composites needed henceforth were described in [9]. Of the variety of structural models of composites, we have chosen the following two: the model based on effective elastic moduli (model 1), which pertains to macromechanics and accounts for the microstructure of a material by means of effective moduli, and the model of two-phase elastic mixture (model 2ñ), which pertains to micromechanics and nanomechanics and accounts for the microstructure in a more complex way, using the concepts of two interpenetrating and interacting continua. The choice fell on these models because they account for the microstructure of composites in so different ways that the complete sets of mechanical constants used by the models do not overlap and the mechanical phenomena described by the models are equally different and are characteristic of different levels: macro, micro, and nano. The fibrous materials under consideration are supposed to possess standard symmetry-transverse isotropy. The analytic expressions for the set of constants are derived from the theory of effective moduli [1, 2] for the former model and from the theory of elastic mixture [5,7,8] for the latter model. These expressions are presented i...

The perturbation (small-parameter) method is used to obtain the first three approximations for the problem of a harmonic longitudinal plane wave propagating in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The subsequent approximations are discussed. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that the third approximation corrects the prediction of the evolution of the initial wave profile. Which of the harmonics dominates depends on the distance traveled by the wave: the second harmonic is generated first, then it transforms into the fourth harmonic, and finally, as the distance increases, the eighth harmonic shows Keywords: harmonic longitudinal plane wave, quadratic nonlinear wave equation, perturbation method, generation, fourth harmonic, eighth harmonic, subsequent harmonicsIntroduction. This paper deals with the well-known physical phenomenon of second-or third-harmonic generation by a harmonic wave propagating in a nonlinear medium of arbitrary nature (water, gas, thermomechanical, optical, or acoustic medium, etc.). The essence of the phenomenon is often and very effectively demonstrated in a second-harmonic generation experiment where an optical wave propagating in an ammonium dihydrogen phosphate crystal changes from ruby light (red) to ultraviolet [26]. More abstractly, a harmonic wave enters a quadratic or cubic nonlinear medium, interacts with itself while propagating, and leaves the medium as a second or third harmonic, respectively. Thus, experiments confirm that a harmonic wave does generate a second harmonic in a quadratic nonlinear medium [1-3, 8, 26]. This phenomenon is theoretically described as a property of the solution of the corresponding nonlinear wave equation. The generation of the second harmonic by a longitudinal harmonic plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model was described in [5, 17] by a method traditional for nonlinear optics (Van der Pol's method of slowly varying amplitudes). Let us outline this approach.The following nonlinear wave equation is known [2, 3, 5, 10, 17] to describe the motion of a quadratic nonlinear longitudinal plane wave:

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