Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Axisymmetric and Other States

A rigorous approach of nonlinear continuum mechanics is used to derive nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. Nonlinearity is introduced by means of metric coefficients, the Cauchy—Green strain tensor, and the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. Quadratically nonlinear wave equations are derived for three states (configurations): (i) axisymmetric configuration dependent on the radial and axial coordinates and independent of the angular coordinate, (ii) configuration dependent on the angular coordinate, and (iii) axisymmetric configuration dependent on the radial coordinate. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. Six different systems of wave equations are written __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 72–84, June 2005.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

The propagation and interaction of hyperelastic cylindrical waves are studied. Nonlinearity is introduced by means of the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. To analyze wave propagation, an asymptotic representation of the Hankel function of the first order and first kind is used. The second-order analytical solution of the nonlinear wave equation is similar to that for a plane longitudinal wave and is the sum of the first and second harmonics, with the difference that the amplitudes of cylindrical harmonics decrease with the distance traveled by the wave. A primary computer analysis of the distortion of the initial wave profile is carried out for six classical hyperelastic materials. The transformation of the first harmonic of a cylindrical wave into the second one is demonstrated numerically. Three ways of allowing for nonlinearities are compared __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 73–82, July 2005.     

Modeling cylindrical waves in nonlinear elastic composites

A procedure of deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves in composite materials modeled by a mixture with two elastic constituents is outlined. Nonlinearity is introduced by metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential. It is the quadratic nonlinearity of all governing relations. For a configuration (state) dependent on the radial coordinate and independent of the angular and axial coordinates, quadratically nonlinear wave equations for stresses are derived and a relationship between the components of the stress tensor and partial strain gradient is established. Four combinations of physical and geometrical nonlinearities in systems of wave equations are examined. Nonlinear wave equations are explicitly written for three of the combinations __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 63–72, June 2007.     

Cubically Nonlinear Versus Quadratically Nonlinear Elastic Waves: Main Wave Effects

This paper is a review of studies on quadratically and cubically nonlinear elastic waves in elastic materials. The main methods for analysis of the wave equations are demonstrated. The main wave phenomena are described. The disproportion between the achievements in the analyses of quadratically and cubically nonlinear waves is pointed out—cubically nonlinear waves have been studied much less     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

Analytical Solutions of Nonlinear Static Problems for Threads of Finite Stiffness Under Active Loading

The behavior of elastoplastic threads of finite stiffness under lateral bending is analyzed. Geometrical and physical nonlinearities are taken into account. The material is assumed to be elastoplastic. The nonlinear equations describing the stress—strain state of threads are derived using the virtual-displacement principle. Numerical results are discussed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 121–129, June 2005.     

Physical constants for one type of nonlinearly elastic fibrous micro-and nanocomposites with hard and soft nonlinearities

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 __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 47–60, December 2005.