Wave propagation in a semi-infinite heteromodular elastic bar subjected to a harmonic loading

Abstract

In this paper we deal with one-dimensional wave propagation in a material that reacts differently to compression and tension. A possible approach to describe such materials is the heteromodular (or bimodular) elastic theory: a piece-wise linear theory with different elastic moduli depending on the stress state. We consider a one-dimensional problem concerning non-stationary wave propagation in a semi-infinite heteromodular elastic body subjected to a suddenly applied harmonic loading. For a medium where the difference of elastic moduli for tension and compression is a small quantity, we obtain an approximate analytical solution of the problem using an asymptotic technique. Then we compare the asymptotic solutions obtained with numerical results and demonstrate a good agreement between them. The spectral characteristics of the constructed solution can be compared with experimental data obtained from dynamical experiments with materials displaying pronounced heteromodular properties.

Introduction

Rocks, soils, and oil and tar sands are complex materials containing pores, cracks, and other defects. If this is the case, their constitutive behavior can be nonlinear and stress-dependent, which implies that loading changes the properties of the material. If materials react differently to compression and tension, this can have a strong influence on the wave propagation. A possible approach to describe such materials is heteromodular (or bimodular) elastic theory: a piece-wise linear theory with different elastic moduli depending on the stress state. It is known [1], [2] that many natural and artificial materials possess this property, see, for instance, concrete, or cracked and defective rocks. The formulation of such a theory for the three-dimensional case is a difficult task. The equations of heteromodular elasticity are always strongly nonlinear and there is as yet no three-dimensional heteromodular theory in canonical form [2]. Different approaches to construct the theory are suggested by Ambartsumyan and Khachatryan [2], [1], Lomakin and Rabotnov [3], Oleinikov [4].

Most of the studies concerning heteromodular media only consider the statical properties, not wave propagation. This is due to the complexity of the corresponding dynamical problems. Even for the one-dimensional case, there are only a few simplest dynamical problems investigated [2], [5], [6], [7], [8], [9], [10], [11], [12]. Maslov and Mosolov [5], [6] investigated discontinuous wave fronts propagation in a one-dimensional heteromodular elastic medium, gave the classification for types of local solutions for the corresponding partial differential equation (3), proved several theorems concerning the existence and uniqueness of the solution, and considered several model problems. Dudko et al. [11] demonstrated the formation of “a rigid domain” during a compression–tension cycle of the loading (see Section 3 of this paper) and the formation of a shock wave during a tension–compression cycle (see Section 4). Khachatryan [13] (see also [2]) considered oscillation of a finite bar and obtained the solution the form of the Fourier series. Wave propagation in a no-tension medium was considered by Maslov and Mosolov [5], Maslov and Antsiferova [7] and by Maslov et al. [10]. In the above referenced studies the solutions are constructed based on the general form of the D'Alembert solution for the wave equation.

Kulikovskii and Pekurovskaya [9] investigated the possible transitions of propagating wave fronts. Yang and Wang [14] used FEM modeling to investigate the longitudinal vibration of a heteromodular bar. Abeyaratne and Knowles [8] considered the Riemann problem for the corresponding partial differential equation and investigated self-similar solutions. Lucchesi and Pagni [15] considered the Riemann problem and proved that a unique solution always exists except for the special case of no-tension materials. Recently, Kharenko et al. [16] considered free longitudinal vibration of both infinite and finite heteromodular bar using high-resolution methods based on the finite-element approach.

We consider a one-dimensional problem concerning non-stationary wave propagation in a semi-infinite heteromodular elastic body subjected to a suddenly applied harmonic loading. For a medium where the difference of elastic moduli for tension and compression is a small quantity, we obtain an approximate analytical solution of the problem using an asymptotic technique. The use of the asymptotic technique appears to be very efficient and allows us to investigate the problem in detail. Then we compare the obtained asymptotic solutions with our numerical results and demonstrate a good agreement between them. The asymptotic technique we use is quite similar to the one applied before in the context of wave propagation in an elastic body subjected to a moving load [17] and in elastic bar composed of a material, which is capable of undergoing phase transitions [18]. The results of our investigation give us a general idea of the character of acoustic wave propagation we may expect in such media, and of the technique to apply in more complex cases.