Wave propagation in a semi-infinite heteromodular elastic bar subjected to a harmonic loading☆
Highlights
►► We study 1D waves in a material that reacts differently to compression and tension. ► We obtain an analytical solution of the problem using an asymptotic technique. ► The expressions for strain, particle velocity and displacements were obtained. ► Analytics has been compared with numerics to demonstrate a good agreement. ► Spectral characteristics can be used to compare with experimental data
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.
Section snippets
Mathematical formulation
We begin with some intuitive ideas. The schematic of the stress–strain curve for a heteromodular material is presented in Fig. 1.
Consider the longitudinal oscillation of a heteromodular bar. Let the Young modulus of the bar in compression be and the Young modulus of the bar in tension be , where . The governing equation for the longitudinal oscillation of the bar iswhere is the mass of the bar per unit length, u is the displacement, X is the
First auxiliary problem: formation of a rigid domain during a compression–tension cycle
Let the external force F(t) in Eq. (9) be as follows:The structure of the solution of this problem can be illustrated by Fig. 3. The upper part of Fig. 3 is the characteristic plane for the problem. At the lower part of Fig. 3, the plot of the strain at the free end of the bar is shown.
At t=0 and , forward and backward fronts of the perturbation in the bar are formed. They propagate at speeds and , respectively. It may be noted that the strain remains
Second auxiliary problem: formation of a shock wave during a tension–compression cycle
Let the external force F(t) be as follows:
Now look at Fig. 5, where the characteristic plane for the second problem is presented. Again at t=0 the forward frontof the perturbation in the bar is formed. It is possible to show that at time , when changes its value from positive to negative, a shock wave is formed. Note that for the shock wave under considerationIn what follows, we denote the position of this shock wave by .
The second
External harmonic excitation
Let the external force F(t) be given by formula (11). In this section we assume the time t to be an arbitrary quantity and the distance x to be such that . To construct the solution we use the following assumptions:
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Quasi-rigid domains are assumed to be rigid domains.
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We assume that left-running waves generated at do not influence the near-field of the wave-field.
Conclusions
The model of a heteromodular elastic body could be useful for geophysical applications in cases where a nonlinear response to an external harmonic excitation is experimentally observed.
The approximate analytical solution describing near-field of the wave pattern in a heteromodular bar subjected to a harmonic loading is obtained using an asymptotic technique. Although the governing equation contains a non-differentiable nonlinearity, the use of asymptotic techniques allows one to investigate the
Acknowledgments
The work was supported by Shell International E.&P. in frames of the project “Waves in complex media” (CRDF Grant RG0-1318(8)-ST-02). One of the authors (Serge Gavrilov) is grateful to the Russian Foundation for Basic Research for the financial support (Grant 11-01-00385-a) that allowed him to finalize this work.
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This work was started in 2004 in collaboration with Dr. Gérard C. Herman, who showed a great interest in the subject and suggested important ideas. Due to an unexpected decease of Dr. G.C. Herman on 24.08.2006, a tragic event for all those who were fortunate to know him and to work with him, the first author of the paper (Serge Gavrilov) had to continue and finish the work on his own, and he takes the responsibility for anything that merits criticism in this paper.