# Shear horizontal waves in a nonlinear elastic layer overlying a rigid substratum

## Summary (2 min read)

### 1. Introduction

- Elastic waves propagating in an unbounded media are non-dispersive i.e. phase velocities of waves are constants.
- Below some of these works will be reviewed to relate the present work to them.
- The materials of the layer and the half space are both assumed to be homogeneous, isotropic and compressible hyper-elastic.
- In [17], Ahmetolan and Teymur studied the propagation of nonlinear SH waves and the formation of Love waves in a double layered plate each having nite thickness.
- An NLS equation is derived which governs the nonlinear self modulation of waves asymptotically and the e ect of nonlinearity on the propagation characteristics is discussed.

### 2. Formulation of the Problem

- Let (x1, x2, x3) and (X1, X2, X3) be, respectively, the spatial and material coordinates of a point referred to the same rectangular Cartesian system of axes.
- The summation convention on repeated indices is implied in (2.1) and in the sequel of this section, and Latin and Greek indices have respective ranges (1, 2, 3) and (1, 2).
- Let us now assume that the constituent material of the layer is nonlinear, homogeneous, isotropic and compressible hyper-elastic.
- For any material de ned by (2.5), if the rst two equations in (2.2) are satis ed by a given solution of the third equation in (2.2), then the motion (2.1) can exist in the medium in the absence of body forces.
- ∂I3 = 0, whenever the invariants are given by (2.7).

### 3. Asymptotic analysis of the nonlinear SH waves

- 0 is a small parameter which measures the weakness of the nonlinearity, {x0, t0, y} are fast variables describing the fast variations in the problem while {x1, x2, . . . , t1, t2, . . .} are slow variables describing the slow variations.
- Now, rst writing the equation of motion (2.17), the boundary conditions (2.18) and (2.19) in terms of the new independent variables (3.1) and then employing the asymptotic expansion (3.2) in the resulting expressions and collecting the terms of like powers of in ε , the authors obtain a hierarchy of problems from which it is possible to determine un, successively.
- Note that the dispersion relation (3.19) (or (3.21)) is the same of the dispersion relation for the antisymmetric motion of SH waves in an elastic isotropic plate with the thickness 2h occupying the region between the planes Y = h and Y = −h.
- Note that, the rst order solution given in (3.27) and the solution of the linear problem are of the same form(see [29]).
- The explicit form of the vector b (3) 3 is not given here, since it represents the third harmonic interactions and therefore in the sequel it will not be required.

### 4. Concluding remarks

- The variation of C, Vg, Γ, ∆, and Γ∆ with the non-dimensional wave number K = kh for the rst three branches of the dispersion relation (3.21) are calculated and they are plotted in Fig(1), Fig(2), and Fig(3) respectively.
- The NLS equation (3.67), as in this work, asymptotically describes the self modulation of the monochromatic plane waves in a nonlinear dispersive medium[30, 31].
- This solution is known as envelope soliton or bright soliton [30, 31, 32].
- Several investigator have shown that the shear stress is a nonlinear function of the strain in certain soils and nT < 0, that is the response of the soil is softening in shear (see e.g.[33] and references given there).
- An experimental result about the observation of solitons in soil mechanics was reported in [34] by Dimitriu.

### Acknowledgment

- The authors Would like to thank the referee for the invaluable comments leading to improvements to this paper.

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### Cites background from "Shear horizontal waves in a nonline..."

...Nonlinear SH wave (or surface SH wave) solutions in homogeneous media are also known in different waveguides, such as in a layer [17], a plate [16], a two-layered plate [15], or a layered half-space [27]....

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...On the other hand, in the homogeneous problem, the propagation of SH waves in the geometrically same layer is considered for material properties such as hyperelastic, isotropic, homogeneous, and generalized neoHookean (for a similar problem, see [17])....

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2 citations

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##### References

4,124 citations

### "Shear horizontal waves in a nonline..." refers background in this paper

...(see, e.g. Ewing et al. [1], Love [2], Achenbach [3], Graf [4], Farnell [5], Maugin [6])....

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...[1], Love [2], Achenbach [3], Graf [4], Farnell [5], Maugin [6])....

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3,582 citations

### "Shear horizontal waves in a nonline..." refers background in this paper

...A uniform layer of a nonlinear soil overlying a rigid bedrock is an example from soil dynamics (see for example[22])....

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1,707 citations

### "Shear horizontal waves in a nonline..." refers background in this paper

...5) φ(η) = φ0 tanh[(−∆/2Γ)φ0η], V0 = 2KΓ which represents the propagation of a phase jump [30, 31, 32]....

[...]

...67), as in this work, asymptotically describes the self modulation of the monochromatic plane waves in a nonlinear dispersive medium[30, 31]....

[...]

...This solution is known as envelope soliton or bright soliton [30, 31, 32]....

[...]

...On the other hand for disturbances that tend to a uniform state at in nity the envelope dark solitons exist for Γ∆ < 0 [30, 31]....

[...]

1,621 citations

### "Shear horizontal waves in a nonline..." refers background in this paper

...[1], Love [2], Achenbach [3], Graf [4], Farnell [5], Maugin [6])....

[...]

170 citations

### "Shear horizontal waves in a nonline..." refers result in this paper

...For an extensive review of most of these works we refer to Parker and Maugin [7], Maugin [8], Parker [9], Mayer [10], Norris [11], Porubov [12]....

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