Nonlinear Waves In Solids Research Articles

Nonlinear waves in solids with slow dynamics: an internal-variable model.

In heterogeneous solids such as rocks and concrete, the speed of sound diminishes with the strain amplitude of a dynamic loading (softening). This decrease, known as 'slow dynamics', occurs at time scales larger than the period of the forcing. Also, hysteresis is observed in the steady-state response. The phenomenological model by Vakhnenko et al. (2004 Phys. Rev. E 70, 015602. (doi:10.1103/PhysRevE.70.015602)) is based on a variable that describes the softening of the material. However, this model is one dimensional and it is not thermodynamically admissible. In the present article, a three-dimensional model is derived in the framework of the finite-strain theory. An internal variable that describes the softening of the material is introduced, as well as an expression of the specific internal energy. A mechanical constitutive law is deduced from the Clausius-Duhem inequality. Moreover, a family of evolution equations for the internal variable is proposed. Here, an evolution equation with one relaxation time is chosen. By construction, this new model of the continuum is thermodynamically admissible and dissipative (inelastic). In the case of small uniaxial deformations, it is shown analytically that the model reproduces qualitatively the main features of real experiments.

Nonlinear damage detection and localization via an innovative metamaterial-based sensor

In recent years, acoustic metamaterials have attracted increasing scientific interest for very diverse technological applications ranging from sound abatement to ultrasonic imaging, mainly due to their ability to act as band-stop filters. At the same time, the concept of chaotic cavities has been recently proposed as an efficient tool to enhance the quality of nonlinear signal analysis, particularly in the ultrasonic/acoustic case. The goal of the present work is to merge the two concepts to propose a metamaterial-based device that can be used as a natural and selective linear filter for the detection of signals resulting from the propagation of elastic waves in nonlinear solids, e.g., in the presence of damage, and as a detector for the damage itself in time reversal experiments. Numerical simulations and experimental measurements based on scanning laser Doppler vibrometer demonstrate the feasibility of the approach and the potential of the device in providing improved signal to noise ratios and enhanced focusing on scatterer locations.

Low-frequency wave propagation in post-buckled structures

Nonlinear wave propagation in solids and material structures provides a physical basis to derive nonlinear canonical equations which govern disparate phenomena such as vortex filaments, plasma waves, and traveling loops. Nonlinear waves in solids however remain a challenging proposition since nonlinearity is often associated with irreversible processes, such as plastic deformations. Finite deformations, also a source of nonlinearity, may be reversible as for hyperelastic materials. In this work, we consider geometric bucking as a source of reversible nonlinear behavior. Namely, we investigate wave propagation in initially compressed and post-buckled structures with linear-elastic material behavior. Such structures present both intrinsic dispersion, due to buckling wavelengths, and nonlinear behavior. We find that dispersion is strongly dependent on pre-compression and we compute waves with a dispersive front or tail. In the case of post-buckled structures with large initial pre-compression, we find that wave propagation is well described by the KdV equation. We employ finite-element, difference-differential, and analytical models to support our conclusions.

Proper formulation of viscous dissipation for nonlinear waves in solids

To model nonlinear viscous dissipative motions in solids, acoustical physicists usually add terms linear in Ė, the material time derivative of the Lagrangian strain tensor E, to the elastic stress tensor σ derived from the expansion to the third (sometimes fourth) order of the strain energy density E=E(tr E,tr E(2),tr E(3)). Here it is shown that this practice, which has been widely used in the past three decades or so, is physically wrong for at least two reasons and that it should be corrected. One reason is that the elastic stress tensor σ is not symmetric while Ė is symmetric, so that motions for which σ+σ(T)≠0 will give rise to elastic stresses that have no viscous pendant. Another reason is that Ė is frame-invariant, while σ is not, so that an observer transformation would alter the elastic part of the total stress differently than it would alter the dissipative part, thereby violating the fundamental principle of material frame indifference. These problems can have serious consequences for nonlinear shear wave propagation in soft solids as seen here with an example of a kink in almost incompressible soft solids.

Hierarchical structures in complex solids with microscales

In nonclassical mechanics it is natural to deal with the problem of the propagation of nonlinear waves in solids with different internal structural scales (Engelbrecht, J., Pastrone, F., Braun, M., and Berezovski, A. Hierarchy of waves in nonclassical materials. In Universality of Nonclassical Nonlinearity (Delsanto, P. P., ed.). Springer, 2007, 29–48). The choice of suitable microstrain functions of the microdisplacements, of their time derivatives as strain velocities, allows us to obtain the field equations via a variational principle (see Pastrone, F., Cermelli, P., and Porubov, A. V. Nonlinear waves in 1-D solids with microsctructure. Mater. Phys. Mech., 2004, 7, 9–16; Casasso, A. and Pastrone, F. Wave propagation in solids with vectorial microstructures. Wave Motion, doi: 10.1016/j.wavemoti.2009.12.006; Porubov, A. V., Pastrone, F., and Maugin, G. A. Selection of two-dimensional nonlinear strain waves in micro-structured media. C. R. Acad. Sci. Paris, 2004, Ser. I 337, 513–518) in three different cases: one-dimensional solids with two different microscales, two-dimensional solids with microstructures, and plane granular media. In all cases the hierarchical structure of equations due to the scales in materials is evident.

Theory of nonlinear waves in solids undergoing strong rearrangements of their crystalline structure

The nonlinear theory of propagation of nonlinear localized waves (like kinks and solitons), connected with the movement of defects in crystals, is developed. It is assumed that crystals possess a complicated lattice consisting of two sub-lattices. The arbitrary large displacements of sub-lattices, , are considered. An additional element of translational symmetry is employed in the theory. The element is typical for complicated lattices, however, it has not been introduced in physics of solid state before. It is evident that the relative displacement of sub-lattices for one period (or for an integer of periods) to a superposition of the sub-lattice with itself reproduces the structure of a complicated lattice. This means that the complicated lattice energy should be a periodic function of the relative rigid displacement of sub-lattices, , invariant to such a translation. The variational equations of macro- and micro-displacements turned out to be a nonlinear generalization of the linear equations of acoustic and optical modes obtained by Karman, Born and Huang. Some exact solutions are obtained for the one-dimensional case, and their specific features are revealed.

On the generalized ultraspherical or Gegenbauer functions of fractional orders

Fractional calculus, i.e. the theory of derivatives and integrals of non-integer orders, enjoys growing interest not only among mathematicians, but also among physicists and engineers (see [J.L. Wenger, F.R. Norwood (Eds.), IUTAM Symposium––Nonlinear Waves in Solids, ASME/AMR, Fairfield, NJ, 1995; J. Alloys Compd. 211/212 (1994) 534; An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons Inc., New York, 1993; On Two Definitions of Fractional Calculus, Slovak Academy of Sciences, Institute of Experimental Physics, UEF-96 ISBN 80-7099-252-2, 1996; B. Ross (Ed.), Fractional Calculus and its Applications, Lecture Notes in Mathematics, vol. 457, Springer-Verlag, Berlin, 1975, p. 1; Phys. Scripta 43 (1991) 174; IEEE Trans. Dielect. Electr. Insulation 1(5) (1994) 826]). In this paper we generalize the Gegenbauer or ultraspherical polynomials C n λ ( x) for n=1,2,…, λ>−(1/2) to functions C n, α λ ( x) where α is any positive number. We prove that this definition generalizes and interpolates the properties of C n λ ( x). The generalized Legendre and Chebyshev polynomials of fractional orders will be studied as special cases. The hypergeometric and R-functions representation will be given.

Generation of Spin-Wave Envelope Dark Solitons

Experimental study of nonlinear waves in different physical systems is an attractive area of research, not only from the fundamental point of view, but also for future applications of nonlinear properties of solids. So far the most impressive results have been demonstrated in nonlinear guided-wave optics (see, e.g., Refs. [1,2]). In a sharp contrast, the experimental study of nonlinear waves in solidstate systems has demonstrated much slower progress, in particular due to dissipative losses that make the observation of large nonlinear effects in real systems difficult. Nevertheless, spin-wave bright envelope solitons have been observed in magnetic films, for different orientations of the magnetic field and propagation direction [3,4]. Similar results have been reported for other types of nonlinear waves in solids, e.g., acoustic envelope solitons generated in a quartz crystal [5]. The first observation of microwave magnetic-envelope dark solitons was reported by Chen et al. [6], who generated spin waves propagating perpendicularly to the direction of a bias magnetic field in a tangentially magnetized single-crystal yttrium iron garnet (YIG) film. For such a geometry, the dispersion and nonlinear coefficients have the same sign [7] and, therefore, dark solitons can be generated. The experimental results reported in Ref. [6] revealed unusual features of the soliton generation when the number of generated solitons was changing with the input power from even to odd. The purpose of this Letter is twofold. First, we demonstrate that in the case when an input pulse without any phase modulation enters a nonlinear dispersive medium at a certain point, the generated localized wave acquires an induced spatial phase shift accumulated during its generation, the phase shift being inversely proportional to the wave group velocity. Such a phase shift is negligible for large group velocities, e.g., for optical solitons in fibers. However, for wave propagation in solids, the induced phase is no longer small, and its effect becomes important, as in the case of spin waves. Second, based on this general concept, we shed light on the experimental results reported in Ref. [6]. We show that an arbitrary small phase shift across the initial pulse can change the character of the soliton generation, and both an odd and even number of dark solitons can emerge. For the same shape and duration of the input pulse, this effect is determined only by the pulse amplitude as observed in [6]. First, we discuss the phenomenon of the phase shift accumulated during the pulse generation. We consider the evolution of a slowly varying wave envelope Asz, td described by the nonlinear Schrodinger (NLS) equation, i µ ›A

Special Issue on IUTAM Symposium: Nonlinear Waves in Solids

Special Issue on IUTAM Symposium: Nonlinear Waves in Solids

Euromech colloquium 212 non-linear waves in solids : Toledo, Spain July 1–4 1986

Euromech colloquium 212 non-linear waves in solids : Toledo, Spain July 1–4 1986

Behavior of Finite-Amplitude Sound Waves in Nonlinear Solids

This paper reviews nonlinear elasticity theory, with special emphasis on waves whose amplitude is just large enough that nonlinearity must be taken into account. Methods for obtaining traveling-plane-wave solutions and the mathematical and physical properties of these solutions are discussed. Particular attention is given to that part of the theory that relates experimental measures of distortion to the third-order elastic constants. Finally, the relevance of these constants to other areas of physics is mentioned. [Research sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corporation and by a National Science Foundation grant to the Georgia Institute of Technology.]