Dynamic behavior of multi-layer heterogeneous composite magneto-elastic structures for surface wave scattering

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Abstract

The paper is focused on the surface wave field in functionally graded multi-layer transversely isotropic heterogeneous magneto-elastic reinforced media. The Geometry of the problem is formulated by considering the (n1) finite layer composite structure over a semi-infinite substance, occupying the domain: <x,y<, hi1zhi, i=1,2,n1, h0=0 and hn1z. Mechanical properties of magneto-elastic heterogeneous reinforced media in wave scattering are an essential part of this study. A generalized Haskell’s [1] technique has been applied to obtain the wave scattering relation in multi-layer heterogeneous magneto-elastic media using suitable boundary conditions. Estimated wave scattering relation is in affirmation with the general Love-type surface wave relation in case of a single layered medium over a semi-infinite substance as well as multi-layered media over the semi-infinite substance. A finite difference technique is derived to obtain the group and phase velocities with shear deformation in the magneto-elastic heterogeneous reinforced media. To study the group and phase velocity in a square grid, stability conditions for introducing finite difference techniques have been derived. Using graphical representation, it has been examined that phase velocity, group velocity, and wave scattering in the layered media are affected by heterogeneity, reinforced, magneto-elastic coupling parameters, and stability ratio.

Introduction

The surface wave scattering in functionally graded multi-layer composite magneto-elastic reinforced media play a significant role in several arenas of modern engineering including geophysics. Mechanical properties of magneto-elastic heterogeneous material are considered as an important area of study for research and development in defense, aerospace, and automotive industries. The wave scattering in layered media is characterized as the phase velocity of a harmonic wave, which is experienced as a frequency dependent function. If the amplitude of frequency dependent function decay with exponential coefficient in a layered medium, then it is identified that wave is attenuated in layered media. The subject field is interesting to read the behavior of wave scattering through the layered media under high speed impacts. Likewise, this serves to understand material behavior for vehicle impact, projectile impact or explosion resistance applications.

Large quantities of articles have been brought out the scattering of wave in functionally graded multi-layered media by demonstrating the analytical methods using the superposition principle. Nonetheless, the analytical result is rather complicated and sometimes does not generate exact results (Liu, et al. [2]). On the other hand, the finite difference method is a welfare numerical method to look into the vibrational attitude of solid structure. In view of this, we have considered the Haskell method and finite difference approach to find the dispersion relation and component velocity model including stability conditions of FDT in this paper. The matrix procedure used in the paper was first introduced by Thompson [3] and later Haskell [1] developed the procedure for obtain the phase velocity dispersion relations for Rayleigh and Love waves in multi-layered isotropic media. The idea of the Love wave propagation in multi-layer isotropic structure using a generalization of Haskell’s technique was introduced by Anderson [4]. Later, Chattopadhyay et al. [5] have used the Anderson’s methodology to examine the share wave dispersion in isotropic magneto-elastic media. They have considered (n1) finite layers over a half-space and established the accurate finite-difference technique to derive the general dispersion equation. In our paper, we have extended the methodology in the above cited papers to the composite magneto-elastic functionally graded heterogeneous reinforced solid media. The magneto-elastic effect is the change in magnetic flux when a force (mechanical stress) applied on ferromagnetic material. The mechanical characteristic of the reinforced substance is concrete member, including all the elements, which behave in concert as a single composite unit as long as they persist in an elastic state. It indicates that the two elements are confident together in order that there are no comparative displacements among them. The heterogeneity of the layered medium is considered as exponentially changing with elastic constants of the reinforced substance, which are performed toward the z-direction.

In the Last few years, some researchers start to pay attention to the dynamic behavior of functionally graded materials (FGMs) due to the vibration in the medium. A study of vibration analysis of the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model was studied by Kaddari et al. [6]. Karami et al. [7] have introduced the Galerkin’s approach for buckling analysis of functionally graded anisotropic different nanoplates boundary conditions. On the other hand, Bednarik et al. [8] formulated one-dimensional propagation of longitudinal elastic waves through functionally graded materials. A numerical approach of two-dimensional wave propagation in functionally graded materials was introduced by Berezovski et al. [9]. Also, an analytical study of wave propagation in functionally graded layered materials was discussed by Aksoy and Senocak [10]. Recently, Rabhi et al. [11] has introduced a new technique for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Also, Rahmani et al. [12] have studied the influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four unknown refined integral plate theory. Theoretical explanations and numerical results introduced in this paper are completely distinct than earlier published work and give valuable information on surface wave scattering in functionally graded media.

A critique of the literature indicates that a large number of the papers on wave scattering theory are published in the domain of Love, Rayleigh and Stoneley waves. The scattering of Love-type wave has pulled significant attention of researchers in two thousand century. The wave scattering equation for generalized Love and Rayleigh waves in isotropic multi-layered media was introduced by Crampin [13]. Chattopadhyay and Chaudhury [14] discussed the model for reflection of elastro-magnetic surface wave propagation in a finite thickness reinforced elastic medium. With the continuation of their work, Chattopadhyay et al. [5] have also derived the dispersion of shear wave in magneto-elastic reinforced layered media. A generalization of multi-frequency driven waves imaging via chirp excitations was reported by Michaels et al. [15]. On other side, Chaudhary et al. [16] have discussed the reflex and transmission of shear waves by a completely carrying reinforced elastic media arbitrated among two vertically heterogeneous viscoelastic solid half-space.

Recently, the investigation of transverse wave propagation in a solid medium has served as an exact recurrence relation with high accuracy solutions were presented by Farmer et al. [17]. Linton and Thompson [18] have demonstrated the persistence of the limed elastic wave above a circular tubular cavity in a substance. Also, Zhao et al. [19] have derived micromechanics models to calculate the compressive and tensile moduli of elastic solid media, depending on frequency. The scattering of elastic waves in a layer interlude with a Winkler foundation was investigated by Kaplunov et al. [20]. They [21] also exposed the elastic bending wave on the edge of a semi-infinite plate reinforced with a slip plate. On the other hand Manna and his group [22], [23], [24] have discussed the Love wave and Rayleigh wave propagation in a composite layered media with corrugated or plane boundaries overlying heterogeneous half-space. Teymouri et al. [25] have studied wave motion in multi-layered transversely isotropic porous media by the method of potential uses. The molding of very low-frequency (VLF) electromagnetic (EM) beam propagation in the Earth-ionosphere waveguide (WGEI) were introduced by Rapoport et al. [26]. Gao and Zhang [27] have formulated a methodology for the propagation of waves in a set of finite layer structure of anisotropic piezoelectric material.

In this paper, we examine the surface wave scattering in multilayer functionally graded heterogeneous transversely isotropic magneto-elastic reinforced media over a semi-infinite substance. The explicit finite difference scheme has been applied to find the closed form solution of the displacement of surface wave. Suitable boundary conditions help to capture the exact dispersion relation. For numerical simulation two types of reinforced composite structures have been considered: (1) metal composite structure (cf. Aluminum, Magnesium, Steel) and (2) fibre composite structure (cf. Fe whisker, SiC whisker, Quartz whisker and Al2O3 whisker). It has been remarked that the computed scattering relation turns to the general Love-type wave equation to the case of isotropic homogeneous layer over a semi-infinite substance. It has also been observed that the group and phase velocities are affected by magneto-elastic reinforced and non-homogeneity parameters. Stability criterion on the group and phase velocities for applying the FDT has been inserted. From the computational representation it has been noted that the group and phase velocities are stabled by the fixed value of the dimensionless stability parameter (1).

Section snippets

Geometry and mathematical formulations

Consider a set of (n1) finite thickness layers of magneto-elastic reinforced transversely isotropic heterogeneous media overlying a semi-infinite substance. As shown in Fig. 1, the surface wave is scattering along the x direction and amplitudes are acting positively downward (along z direction). The semi-infinite substance of the model is numbered as n and it goes decreasing as it grow up (down-top fashion).

The governing stress-strain relations for the scattering of Love-type surface wave in

Case-I

In the case of n=2, the scattering Eq. (30) takes the formtanα2(1)h1=iA2(1)A1(1)+eηh1[(A2(1))2(A1(1))2A1(2)A2(2)]

(3.1.1) In case the reinforced magneto-elastic media consists of single medium lying over a semi-infinite substance, is isotropic and heterogeneous i.e. n=2, P1(1)=μT(1), P2(1)=0, P3(1)=μT(1), P4(1)=ημT(1), P5(1)=0, P1(2)=μT(2), P2(2)=0, P3(2)=μT(2), P4(2)=ημT(2), P5(2)=0,tanα2(1)h1=μT(1)α2(1)μT(1)a2+ημp(1)H02sin2ϕ+eηh1[(μT(1))2(α2(1))2+{ημp(1)H02sin2ϕμT(1)η2}2μT(2)η/2ημp(2)H02

Stability condition of FDT on phase and group velocities

Finite Difference Technique: To study the stability conditions on group and phase velocities, we assume a small rectangular part of a layered structure. The xz plane is partitioned into small grids with equipotent increments of Δx and Δz towards the x and z directions, respectively. The time lengths are divided by using unit step-length of Δt. The space-time grids are defined byXj=jΔxj=0,1,2,,J,Zk=kΔzk=0,1,2,,K,Tl=lΔtl=0,1,2,,L.Let us denote v(m)(Xj,Zk,Tl)=vj,kl, then the Taylor expansion of

Numerical discussion and results

In order to demonstrate the outcome of diverse values of heterogeneity parameter, reinforced parameters and magneto-elastic field on the scattering of Love-type surface wave in composite multi-layered functionally graded heterogeneous media over a semi-infinite substance, the numerical computations were performed. Keep in mind that, the model in highly application for the manufacturing of aerospace and automotive industries, we have considered all the material properties of the carbon laminate

Summary and conclusion

The importance of reinforced parameters in a composite material is essentially one of the developmental mechanical properties of the material. Several types of fibers have been conducted for the composition of different composite bodies and they will apply the different result, depending on the characters. In this paper, we considered the model manly consented to the industries application such as aerospace, automotive, construction and many others demanded lighter and lighter materials polymer

Acknowledgments

Author Santanu Manna acknowledges the support through TEQIP-3 Collaborative Research and Internship fund from IIT Indore.

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