Propagation and interaction of non-linear elastic plane waves in soft solids

Using the perturbation method of weakly non-linear asymptotics we analyze the propagation and interaction of elastic plane waves in a model of a soft solid proposed by Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44]. We derive the evolution equations for the wave amplitudes and find analytical formulas for all interaction coefficients of quadratically non-linear interacting waves. We show that in spite of the assumption of almost incompressibility used in Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44], the model behaves essentially like that of a compressible isotropic material. Both the structure of the equations and the interaction patterns are similar. The models differ, however, in the elastic constants that characterize them, and hence the values of the coefficients in the evolution equations and the values of the interaction coefficients differ.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Inhomogeneous plane waves in layered materials including fluid, solid and porous layers

The propagation of plane waves in layered media including fluid, solid and porous layers is modelled with transfer matrices, the three Biot waves being taken into account in the porous layers.     

Nonresonance parametric interactions of surface waves in isotropic solid bodies

A solution in parametric approximation is given of the nonlinear interaction problem of Rayleigh surface waves propagating in a solid body with given elastic fields. Shortened equations that govern the modulation effect of the surface waves, and also expressions for the modulation index in terms of the third-order elasticity constant, are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 163–172, July–August, 1973.     

Second harmonic generation in elastic surface waves on an isotropic solid

A procedure for solving the problem of non-linear propagation of elastic surface waves is given. An expression for the particle displacement of the second harmonic is obtained. It is shown that the amplitude of the harmonic increases linearly with distance and time and is proportional to the square of the amplitude of the fundamental frequency wave.     

Reflection and transmission of plane waves from fluid-piezothermoelastic solid interface

The reflection and transmission of plane waves from a fluid-piezothermoelastic solid interface are studied. The expressions for amplitude ratios and energy ratios corresponding to reflected waves and transmitted waves are derived analytically. The piezo-thermoelastic solid half-space is assumed to have 6mm type symmetry and assumed to be loaded with water. The effects of angle of the incidence, the frequency, the specific heat, the relaxation time, and the thermal conductivity on the reflected and transmitted energy ratios are studied numerically for a particular model of cadmium selenide (CdSe) and water. Some special cases are also studied.     

Propagation of plane elastic waves at a plane interface between two electro-microelastic solid half-spaces

This work is concerned with the wave propagation and their reflection and transmission from a plane interface between two different electro-microelastic solid half-spaces in perfect contact. It is found that there exist five basic waves in an infinite electro-microelastic solid, namely an independent longitudinal micro-rotational wave, two sets of coupled longitudinal waves influenced by the electric effect, and two sets of coupled transverse waves. The existence of the two sets of coupled longitudinal waves is new. In the absence of microstretch and electric effects, these two coupled longitudinal waves reduce to a longitudinal displacement wave of micropolar elasticity. Amplitude and energy ratios of various reflected and transmitted waves are presented when (i) a set of coupled longitudinal wave is made incident and (ii) a set of coupled transverse wave is made incident. Numerical computations have been performed for a particular model and the variations of amplitude and energy ratios are obtained against the angle of incidence. The results obtained are depicted graphically. It has been verified that the sum of energy ratios is equal to unity at the interface and the amplitude ratios of reflected and transmitted waves depend upon the angle of incidence, frequency and elastic properties of the media. Results of some earlier workers have also been reduced from the present formulation.     

On the types and number of plane waves in hypoelastic materials

General principles are formulated for modeling the elastic deformation of materials and analyzing plane waves in nonlinearly elastic materials such as hyperelastic, hypoelastic, and those governed by the general law of elasticity. The results of studying the propagation of plane waves in hypoelastic materials are further outlined. The influence of initial stresses and initial velocities on the types and number of plane waves is studied. Wave effects characteristic of hypoelastic materials are predicted theoretically. One of such effects is blocking of certain types of plane waves by initial stresses __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 96–107, November 2005.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Some remarks on the growth behavior of one-dimensional shock waves in non-linear materials

The growth behavior of both compressive and expansive one-dimensional shock waves which propagate into an unstrained region of a non-linear material exhibiting anelastic response, in the sense of Eckart, is analyzed. In each case, a differential equation governing the growth of the amplitude of the shock is derived and it is shown that a critical strain gradient may be defined. The growth behavior of the waves closely resembles the growth behavior of compressive and expansive shock waves propagating in sufficiently smooth non-linear materials with fading memory, i.e., in materials which can be approximated by linear viscoelastic materials for small relative strains.     

Reflection of plane waves in thermoelastic microstructured materials under the influence of gravitation

This paper presents an analysis of wave propagation in a microstretch elastic medium in the context of the Green–Naghdi (GN) theory. Moreover, the dissipation and the influence of gravity on reflected waves have also been investigated. In the present article, five reflected waves propagate into the medium for any incident wave. The problem is solved numerically, and the amplitude ratios are graphically represented allowing for a comparison between the simple GN theory and the case in which one considers the effect of gravity on waves.     

Evolutionary behavior of induced discontinuities behind one dimensional shock waves in non-linear elastic materials

The governing differential equation of induced discontinuities behind one dimensinal shock waves in non-linear elastic materials has been derived. This equation depends, in particular, on the shock amplitude itself. Therefore, its solution depends on the solution of the governing equation of the shock amplitudes which, in turn, depend on the induced discontinuities. It is shown in the special case pertaining to a first-order approximation that there exists a critical shock amplitude S _c such that the evolutionary behavior of the induced discontinuities depends on the relative magnitudes of the shock amplitudes and S _c. However, in the special case pertaining to a second-order approximation the evolutionary behavior of the induced discontinuities is monotone.     

Analysis of plane and cylindrical nonlinear hyperelastic waves in materials with internal structure

A comparative analysis of two types of hyperelastic waves—plane waves (with plane front) and cylindrical waves (with curved front)—is offered. The propagation of the waves is studied theoretically for quadratically nonlinear hyperelastic media and numerically for a class of unidirectional fibrous composite materials. Hyperelasticity is described using the classical Murnaghan potential and a structural model of the first order—the model of effective constants. The internal structure of materials is described by this model and is at the micro-or nanolevels in numerical analysis. Particular attention is given to the evolution of the wave profile. It is studied in three stages: (i) derivation of nonlinear wave equations, (ii) construction of solutions in the form of plane and cylindrical waves, and (iii) numerical analysis of the evolution of these waves in composites with microlevel (Thornel) or nanolevel (Z-CNT) fibers. The main similarities and differences between plane longitudinal and cylindrical waves are shown. The most unexpected result is the striking difference between the evolution patterns numerically observed for plane and cylindrical wave profiles __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 21–46, October 2006.     

Transverse wave at a plane interface between isotropic elastic and unsaturated porous elastic solid half-spaces

Based on the poroelasticity theory, this article investigates the reflection and transmission characteristics of an incident plane transverse wave at a plane interface between an isotropic elastic half-space and an unsaturated poroelastic solid half-space. For this purpose, the effect of the saturation degree and frequency on the properties of the four bulk waves in unsaturated porous medium, i.e., three longitudinal waves and one transverse wave, are discussed at first. Two general cases of mode conversion are considered: (i) The initial transverse wave is incident from an unsaturated poroelastic half-space to the interface, and (ii) the initial transverse wave is incident from an elastic solid half-space to the interface. The expressions for the partition of energy at the interface during transmission and reflection process of waves are presented in explicit forms. At last, numerical computations are performed for these two cases and the results obtained are depicted, respectively. The variation of the amplitude ratios and energy ratios with the saturation degree and incident angle is illustrated in detail. It is also verified that, at the interface, the sum of energy ratios is approximately equal to unity as expected.