Analysis of horizontally polarized shear waves on a highly inhomogeneous loaded bi-material plate

1.   Introduction

Propagation of surface waves has been broadly studied in the past and recent literature sequel to its enormous significance in diverse technological formations, comprising both the plane and corrugated media. This significance is notable in a variety of cutting-edge research in the fields of biomechanics, material science, fluid mechanics, coated and composite media, seismology, and vibration control, to mention a few, read ^{[1,2,3,4,5,6,7,8]}. Moreover, several other areas of modern engineering get pleasure from noteworthy consideration by theorists with regards to the vibration and propagation phenomena in different structural shapes and bodies. We mention here the propagation of waves in a functionally graded beam via approximation approach ^[9], and the nature of the flexural waves in a piezoelectric layer in the presence of an elastic foundation ^[10]. We also recall the diverse propagation considerations in composite and multi-layered structures – involving various external effects, like inhomogeneity, thermal and magnetic effects, rotation, initial stress, and material contrasts among others, see ^{[11,12,13,14,15,16,17,18,19,20,21,22]} and the references therein. In addition, as the propagation of waves has been tackled in a variety of media; see the famous book of propagation of waves in solid media by Achenbach ^[23], there has been a number of reported scenarios of some motivating consideration in the classical literature, like the works of Dutta ^[24] who analyzed the propagation of longitudinal waves in an elastic medium with material inhomogeneity. We also recall the findings of Bhattacharyya and Bera ^[25] with regard to the propagation of waves in elastic structure amidst random and linear material properties, respectively, via the application of the decomposition approach. More so, a case of a wave vibrating in an inhomogenous rod was equally examined by Ahmad and Zaman ^[26] both analytically and asymptotically; see also the recent examination of the influence of temporal deviation on a vibrating elastic substrate by Alzaidi et al. ^[27].

Furthermore, to review some of the important results/literature with regard to the propagation of waves in elastic multi-layered and coated media, we start off by recalling the work of Kaplunov et al. ^[28] on the dispersion of plane elastic waves on an inhomogeneous three layers panel. There, the propagation of waves was analyzed within the low frequency and long-wave conditions, for the four different material contrasts of typical sandwich panels. We equally recall the work of Dai et al. ^[29] on the propagation of plane surface waves on a coated half-space. Both examinations given in ^[28,29] were carried out with the help of the asymptotic analysis method; one would equally read the recent works reported in ^{[30,31,32,33,34]} for more consideration, through the help of an asymptotic analysis method. Additionally, Vinh and Linh ^[35] and Vinh et al. ^[36] studied the propagation of Rayleigh waves on an orthotropic and isotropic coated media, respectively; for more on the applications and other deliberations on coated media, interested readers could read ^[37,38] and the references therewith. Besides, we equally recall the new study about the propagation of Love-type surface waves on a coated structure of infinite extents, by Manna et al. ^[39,40]; cases of coated orthotropic and coated porous anisotropic extended layers were respectively examined in the references. Lastly, Selim and Althobaiti ^[41] examined the longitudinal wave propagation on a nano-tube over elastic support, using a wave-based approach; see also ^[42,43] for some relevant studies in the area.

However, as most of the papers so far reviewed in the literature examined the situation of either a single- or inhomogeneous multi-layered structure with constant material properties as in ^{[11,15,16,17,20,22,28,31,32,33]} via the asymptotic approximation approach, with the exception of a few consideration as in the recent work of Mubaraki et al. ^[34]; it will be more pertinent to explore the situation of inhomogeneous multi-layered structures with variable-dependent material properties - thereby compounding the inhomogeneity property of the composite media. In this regard, the current manuscript examines the propagation of horizontally polarized shear waves on the dispersion of a highly inhomogeneous thin bi-material plate with a loaded end surface boundary on one end, due to the Winkler elastic foundation, while a stress-free condition is assumed on the other end. An analytical procedure of solution will be used to get hold of exact vibrational displacements and dispersion relation; in addition to the utilization of the asymptotic approximation approach for the derivation of the approximate dispersion relation. Additionally, the influence of the material inhomogeneity and that of the Winkler elastic foundation will be examined on the propagation of horizontally polarized shear waves on the governing structure. More so, some inference will be deduced in the end, to pave a way for future considerations. Lastly, the organization of the current paper takes the following format: Section 2 presents the formulation of the aiming model. Section 3 derives the exact analytical solution of the model, while Section 4 derives the consequential exact dispersion relation. Section 5 acquires the related cut-off frequency and static equations; while the approximate dispersion relation is determined in Section 6. Moreover, Sections 7 and 8 give the discussion of the obtained results, and the concluding remarks, respectively.

2.   Problem statement

Consider a highly inhomogeneous bi-material plate of constant thicknesses $h_{1}$ and $h_{2},$ occupying the regions $-h_1\le x_{2}\le0$ and $0\le x_{2}\le h_2$, respectively. More so, the individual layers of the plate $-$ and $+$ are perfectly joined down the domain $-\infty < x_{1} < \infty$ with each other, see Figure 1.

Additionally, we have considered the equation of horizontally polarized shear waves ^[20,44,45] to describe the propagation of waves in the respective layers of the plate. Horizontally polarized shear motion, or alternatively the anti-plane shear motion has been described by Horgan (1995) ^[45] as "an interesting two-dimensional mathematical model arising in solid mechanics involving a single second-order linear or quasi-linear partial differential equation. This model has the virtue of relative mathematical simplicity without loss of essential physical relevance. Anti-plane shear deformations are one of the simplest classes of deformations that solids can undergo".

Therefore, the equations of horizontally polarized shear motions, modeling the governing bi-material plate in the displacement ${\bf{w}} = (0, 0, w_3)$ are given in the respective layers as follows ^[16,20,34]

where $-$ denotes the upper layer and $+$ denotes the lower layer, $w_3^{\mp} = w_3^{\mp}(x_{1}, x_{2}, t)$ are the out-of-plane displacements, $\rho_{*}^{\mp}(x_{2})$ are space-dependent densities; while $\tau_{j3}^{\mp} = \tau_{j3}^{\mp}(x_{1}, x_{2}, t)$ are the respective stresses for $j = 1, 2,$ and defined by

where $\mu_{*}^{\mp}(x_{2})$ are space-dependent Lame's elastic parameters. More so, "pure anti-plane shear motions governed by Eq (2.1) are usually called the horizontally polarized shear motion", ^[23]. Thus, as the plate is considered to be of a highly inhomogeneous nature, the material constituents comprising the densities $\rho_{*}^{\mp}(x_{2})$ and the Lame's elastic parameters $\mu_{*}^{\mp}(x_{2})$ in the respective layers of the bi-material plate are further presumed to be inhomogeneous, that is, they are $x_{2}$-dependents. More specifically, they are represented as follows ^[27]

where $\rho^{\mp}$ and $\mu^{\mp}$ are the respective constant densities and Lame's elastic parameters, respectively, in the upper and lower plates; while $\alpha$ is the dimensional material inhomogeneity. Moreover, as asserted by Alzaidi et al. ^[27], the material parameters involving the Lame's constants (shear modulus') $\mu_{*}^{\mp}(x_{2})$ and the material densities $\rho_{*}^{\mp}(x_{2})$ in both layers are considered to have an exponentially decaying variation along their respective thicknesses. Such scenario can be seen as the case of the distribution of material inhomogeneity along the thickness of the structure; take the case of corrosion distribution in an elastic bar ^[26], or the rate of penetration of an external agent over/across a sheet of metal. In addition, for various considerations of inhomogeneous structures where an exponentially growing variation along the depth of the different elastic media was examined, the recent works by Yigit and Sahin ^[9] on elastic beams, Mandi et al. ^[12] on the double-layered structure beneath an inhomogeneous half-plane, and Mubarali et al. ^[34] on the doubly coated elastic half-plane could be an eye-opener in this regard. More so, some additional material non-homogeneities such as linearly varying material parameters, quadratically material parameters, and exponentially varying material parameters can equally be seen in the work of Shekhar and Parvez. ^[46]; see also ^{[25,26,27,34]} and the references therewith.

Furthermore, in this regard, there are quite a few other computational methods that compute the wave propagation in highly inhomogeneous materials, such as finite elements, boundary elements, finite difference, and local interaction simulation approach; for such methods, we suggest the reader(s) to read the interesting review work by Shen and Cesnik ^[47]. In the same vein, it is quite significant to mention the role being played by the concept of dissipation of wave in various media; more on such studies with regard to structure-preserving circumstances can be found in ^{[48,49,50,51,52]} and the references therein. Indeed, the dissipation of the wave is an important characteristic of the horizontally polarized shear wave. In the end, however, some work interrelated to the wave vibration of the shells can be found in the recent examinations ^[53,54,55].

Furthermore, we prescribe classical conditions that are associated with a typical multi-structure comprising interfacial and boundary conditions as follows

where $Q$ is a tangential load that is considered to be due to an elastic Winkler foundation, which is expressed as ^[11,34]

where $a$ is the dimensional stiffness of the Winkler's foundation, while $w^{-}$ is the vibrational displacement of the upper layer, where the load is being applied on.

3.   Exact analytical solution

In order to solve the governing model, we start off by substituting the constitutive equation expressed in Eq $(2.2)$ in to the equation of motion given in Eq $(2.1)$ to thus obtain the following wave-like equation

where $\alpha$ is an inhomogeneity parameter, and $c_{\mp}$ are the transverse speeds in the upper and lower regions, respectively, and given by

More so, with the consideration of the harmonic wave solution, we further assume the model to admit the travelling wave solution of the following format ^[20]

where $\xi$ and $\theta$ represent the dimensional wavenumber, and frequency, respectively; with the wave propagating along $x_1$ direction, and $x_2$ direction represents the thickness direction. Thus, with this development, Eq $(3.1)$ reduces to the following ordinary differential equation

where

Further, the two equations in Eq $(3.4)$ for the lower and the upper layers pose the following characteristic equations, respectively,

So, from the above equation, the following roots are obtained

Therefore, the exact analytical solution in the upper and lower layers takes the following expression

where $A^{\mp}$ and $B^{\mp}$ are constants to be found right-away after utilising the outlined perfect interfacial and surface boundary conditions.

Therefore, we proceed further by non-dimensionalizing the roots of the characteristic equation expressed in Eq $(3.7)$ as follows

with

where $r$ is the scaled inhomogeneity parameter, and $\Theta$ and $K$ are the dimensionless wavenumber and frequency, respectively, given by

together with the dimensionless Lame's elastic parameter $\mu,$ density $\rho,$ and thickness $h$ given sequentially as

Moreover, a new dimensionless parameter $b$ is equally discovered denoting the stiffness of the Winkler foundation expressed as

Therefore, the respective dimensionless displacements (solutions) and stresses in the two layers of the bi-material plate are determined after ignoring the exponential factor $e^{i(\xi x_1-\theta t)}$ as follows

displacements $w_3^{\mp}$:

shear stresses $\tau_{13}^{\mp}$:

shear stresses $\tau_{23}^{\mp}$:

where

while $\zeta^{-}_2$ and $\zeta^{+}_2$ are scaled variables valid over $-h_1 \le x_2\le 0$ and $0\le x_2\le h_2,$ respectively. Additionally, we graphically illustrate in Figure 2 the effects of the (a) inhomogeneity parameter $r$ (with fixed $b = 0.25$), and that of the (b) Winkler foundation parameter $b$ (with fixed $r = 0.25$) on the propagation of horizontally polarized shear waves on the structure under consideration. From these sub-figures, the vibrational displacement in the upper layer ($'-': -1\le \zeta_2\le 0$) is observed to decrease steadily down to the prescribed perfect interfacial conditions, which are well satisfied on the interface (at $\zeta_2 = 0$), then followed by an almost linear behavior in the lower layer ($'+': 0\le \zeta_2\le 1$) of the plate. Again, one will note from Figure 2(a) that an increase in the inhomogeneity parameter causes an increment in the vibrational displacement; while an increase in the Winkler foundation parameter decreases the vibration, as observed from Figure 2(b).

4.   Exact dispersion relation

The derivation of the exact dispersion relation is very important in such a structural arrangement comprising of multiply laminated segments. Thus, this section aims to analytically determine the consequential exact dispersion relation with the help of the outlined interfacial and surface conditions.

Now, we determine the consequential exact dispersion relation in this scenario by considering the load $Q$ prescribed in Eq (2.5) on one side of the plate, together with perfect interfacial conditions given in Eq (2.4). Then, the form of interfacial and boundary conditions (including the load condition) yields the following exact dispersion matrix

More so, the vanishing point of the determinant of the above dispersion matrix $(\det(A) = 0)$ gives the following consequential dispersion relation

or more explicitly after using the expressions for $m_{jj}$ and $n_{jj}$ given in the Eq (3.9) as follows

where

Besides, the absence of the material inhomogeneity in the two layers of the plate, that is, when $r = 0,$ causes the obtained consequential dispersion relation in Eq (4.3) to further reduce to the following

where $\gamma$ and $\lambda$ are again given in the Eq (3.10).

Obviously, in the case of no load, that is, when $b = 0,$ the dispersion relation expressed in Eq (4.3) reduces to

Moreover, in the absence of material inhomogeneity in the two layers of the plate, that is, when $r = 0,$ the obtained consequential dispersion relation above for the highly inhomogeneous bi-material plate with stress-free boundaries further reduces to the following

Thus, the reduced consequential dispersion relation determined above corresponds to an unloaded inhomogeneous bi-material plate with stress-free faces; of course, the inhomogeneity here is due to the lamination of two homogeneous plates to make a solid bi-material structure.

5.   Cut-off frequency and static equation

This section specifically refers to the above-obtained consequential exact dispersion relations by deeply scrutinizing them with regard to their respective cut-off frequencies and static equations. In this regard, the propagation of horizontally polarized shear waves on a highly inhomogeneous bi-material plate under the assumption of low-frequency and long-wave is achieved when ^[28]

where $\Theta$ and $K$ denote the dimensionless frequency and wave number, respectively; see Eq (3.11) for their explicit expressions. Additionally, interested readers can equally read ^[17,28] and the references therewith for more related studies concerning the significance of material contrasts.

Hence, in what follows, we determine the cut-off frequency and static equation associated with the governing structure.

5.1.   Cut-off frequency

The presence of the scaled inhomogeneity parameter $r$ in the consequential dispersion relation needs to vanish in order to get to the desired target, that is, $r = 0.$ Thus, in this case, the reduced dispersion relation given in the Eq (4.5) would be used. Then, the determination of the cut-off frequency requires the dimensionless wavenumber to vanish, that is, $K = 0.$ With this, the cut-off frequencies are obtained as follows

which visibly reveals cut-off at $\Theta = 0$ and additional ($ad$) one over the transcendental function as

What's more, in the case of no load, that is when $b = 0,$ the cut-off frequency determined in Eq (5.2) reduces to the following

which obviously reveals cut-off frequencies over the low-frequency range of $\Theta \ll 1,$ that is at $\Theta = 0$; one would equally find additional roots associated with the transcendental function, possibly numerically!

5.2.   Static equation

Again, we determine the static equation for $K$ by setting the dimensionless frequency $\Theta = 0$ in the reduced dispersion relation to obtain

which visibly admits a root at $K = 0,$ and an additional and additional ($ad$) one over the transcendental function as

Equally, in the case of no load, that is when $b = 0,$ the static equations determined in Eq (5.5) reduces to the following

which gives an apparent root at $K = 0$, and extra ones at the zeros of the sine hyperbolic function.

Lastly, the analytically obtained cut-off frequency and static equation as rightly determined in Eqs $(5.2)$ and $(5.5)$ are depicted in Figures 3 and 4, sequentially. More so, one would see the variational effect of the Winkler foundation parameter $b$ on these equations.

6.   Approximate dispersion relation

The present section makes use of the asymptotic approximation approach, through the application of the effective boundary conditions to derivate the approximate dispersion relation of the earlier determined consequential exact dispersion relation. In doing so, the governing SH equations of motions would be approximated, and subsequently, get hold of the approximate dispersion relation for the loaded highly inhomogeneous bi-material plate via the stated procedure.

6.1.   Treating the upper layer (–)

In this section, the effect of the upper layer $-$ will be analyzed in the long-wave region, that is,

where $h_1$ is the thickness of the upper layer, and $L$ is the wave length. Thus, we will approximately solve the equation of motion given in Eq $(2.1)$ subject to the boundary conditions expressed in Eq $(2.4)$.

Now, it is appropriate to set the following boundary conditions at the interface ($x_{2} = 0$) as follows

where the function $v^{-} = v^{-}\left(x_{1}, t\right)$ is assumed to be known.

Below, we adopt the asymptotic methodology similar to that for thin elastic structures that was equally used by Mubaraki et al. ^[32] to approximately analyze an elastic half-space coated by an inhomogeneous layer. With that, let us introduce the following scaling variables

and

where $\mu^{-}$ and $\rho^{-}$ are constants, $c_{-} = \sqrt{\frac{\mu^{-}}{\rho^{-}}}$ and all quantities with the asterisk are assumed to have the same asymptotic order.

Therefore, with this development, the equation of motion given in Eq $(2.1)$ via the constitutive relations expressed in Eq $(2.2)$ can then be explicitly re-written as

where

Next, the governing boundary conditions become

More so, we expand the displacements and stresses involved asymptotically as follows

Consequently, Eq (6.4) at leading order yields

subject to the following boundary conditions from Eq (6.6) as follows

Then, the solutions of the above system is found as

Finally, in terms of the original dimensional form, the expressions for the stresses at the interface ($x_{2} = 0$) may be obtained as follows

More so, as a special case, substituting the exponential inhomogeneity coefficient in Eq $(2.3)$ into Eq $(6.11)$, we arrive at the boundary condition at the interface ($x_{2} = 0$), given as

6.2.   Asymptotic dispersion relation

Now, having effectively suppressed the effect of the entire upper layer through the boundary condition determined in Eq (6.12), we further make use of the same procedure to re-represent the interfacial and boundary (stress-free) conditions associated with the lower layer earlier prescribed in Eq (2.4) as follows

Thus, without loss of generality, the aiming approximate consequential dispersion relation is determined as follows

where $n_{11}$ and $n_{22}$ are defined earlier in Eq (3.9) as $n_{11} = \frac{h}{2}\left(r-\sqrt{r^{2}-4\lambda^{2}}\right)$ and $n_{22} = \frac{h}{2}\left(r+\sqrt{r^{2}-4\lambda^{2}}\right)$. In addition, we graphically compare the exact and approximate dispersion relation in Figure 5 with regard to the posed fundamental mode. The exactitude between the two curves is perfectly observed.

Hence, having perfectly approximated the obtained consequential exact dispersion relation given in Eq $(4.3)$ by the approximate one as determined in Eq $(6.14),$ we further attempt to graphically show the effect of the inhomogeneity parameter $r$ in Figure 6; while Figure 7 shows the effect of the Winkler foundation parameter $b$ on the propagation of horizontally polarized shear waves via the obtained approximate dispersion relation in Eq $(6.14).$

7.   Discussion of results

The present section attempts to shed more light on the obtained results thereby discussing the effects of the involved parameters on the propagation and dispersion of horizontally polarized shear waves on the bi-material inhomogeneous plate; and on the other hand, discuss the exactitude of the obtained exact and approximate consequential dispersion relations. Recall that the case of a loaded highly inhomogeneous multilayered elastic solid structure is considered, where the upper layer is loaded with a Winkler foundation; moreover, the absence of this foundation (when $b = 0$) rendered the structure to have stress-free faces. Besides, an analytical procedure was primarily utilized for the determination of the analytical solutions, as well as the consequential exact dispersion relations, before the subsequent deployment of the asymptotic procedure by utilizing the effective boundary conditions to determine the equivalent approximate dispersion relations. In addition, such a scenario of varying material properties along the depth of the bi-material plate can be seen as the case of the distribution of material inhomogeneity; take the case of corrosion distribution/spread in an elastic bar or the rate of penetration of an external agent over/across a sheet of metal.

Now, let us begin by discussing the effects of the inhomogeneity parameter $r$ and that of the Winkler foundation parameter $b,$ on the obtained dimensionless vibrational displacements $w^{\mp}$ (see Eq (3.14)) in the respective regions of the plate. This scenario is thus depicted in Figure 2, with 2(a) illustrating the effects of $r,$ while 2(b) illustrates the effects of $b,$ on the upper $(-1 \le \zeta_2 \le 0)$ and lower $(0 \le \zeta_2 \le 1)$ layers of the plate. From this figure, it is noted from Figure 2(a) that an increase in the inhomogeneity parameter causes an increment in the vibrational displacement; while an increase in the Winkler foundation parameter decreases the vibration, as observed from Figure 2(b).

In addition, the cut-off frequency and static equation are analytically acquired in Eqs $(5.2)$ and $(5.5),$ sequentially. More so, as the requirement for the existence of these equations, the inhomogeneity parameter is bound to vanish, thereby allowing only the Winkler foundation parameter to remain. Thus, one would see the variational effects of the Winkler foundation parameter $b$ on these equations as rightly captured in Figures 3 and 4. In addition, as the cut-off frequencies are the roots of the periodic transcendental equations (Eq $(5.2)$), an oscillatory behavior has been observed upon which the effect of the Winkler foundation parameter continuously sinusoids; while the curves of the static expression increase with an increase in the Winkler foundation parameter.

Next, we discuss Figure 5, which graphically compares the obtained consequential exact and approximate dispersion relation as given in Eqs $(4.3)$ and Eq $(6.14),$ respectively. Recall also that the approximate dispersion relation was determined asymptotically by getting hold of the related effective boundary conditions, and thereafter, derived the approximate equation of motion, which later revealed the approximate dispersion relation in Eq $(6.14).$ Thus, Figure 5 compares the two results by comparing the respective exact and approximate fundamental modes; amazingly, a perfect agreement is realized. Moreover, the selection of the values of the involving parameters ensures a low-frequency harmonic, $\Theta \ll 1,$ as the validity for the derived asymptotic equation extends to almost the entire low-frequency range.

Therefore, with the recorded level of exactness between the exact and approximate dispersion curves as shown in Figure 5, we further analyze the effects of the inhomogeneity parameter $r$ in Figure 6, and that of the Winkler elastic foundation parameter $b$ in Figure 7, on the propagation of horizontally polarized shear waves on the governing structure from the derived approximate relation. Hence, from Figure 6, the influence of the inhomogeneity parameter is quite obvious on the vibration – through the fundamental model; take a look at the case when $r = 0,$ where the mode propagates at zero. Then, the propagation is followed by a deviation away from the low-frequency region, as the inhomogeneity parameter increases. However, in Figure 7, the effect of the Winkler elastic foundation on the propagation of waves on the governing highly inhomogeneous structure has been noted to be insignificant in the fundamental mode (see the first curve, where all the curves matched on it), as for all the values of $b$, the propagation remains the same. Nevertheless, the variation is significantly visible on the first harmonic. More so, an increase in the Winkler elastic foundation parameter pushes away the first harmonic from the low-frequency interval.

8.   Conclusions

In conclusion, the current manuscript examined the propagation of horizontally polarized shear waves on the dispersion of a highly inhomogeneous thin perfectly bonded bi-material plate with a load due to the Winkler elastic foundation. An analytical procedure of solution was deployed to determine the exact expressions of the consequential vibrational displacements in the two regions of the plate and the respective dispersion relations; in addition to the exploitation of effective boundary conditions for the asymptotic validation of the obtained exact dispersion relations. The vibrational displacements in both layers have been observed to be enhanced by an increase in the inhomogeneity parameter; at the same time lessened with an increment in the foundation parameter. Additionally, a perfect approximation between the exact and approximate dispersion relation has been realized, with the validity for the relation extending to almost the entire low-frequency range. The influence of the material inhomogeneity parameter on the fundamental mode is noted to be obvious, as the more it is increased, the more the deviation away from the low-frequency region. On the other hand, the presence of the Winkler elastic foundation has been observed to affect the first harmonic mode, in comparison with the inhomogeneity parameter. More so, an increase in the foundation parameter narrowed the chances of low-frequency propagation.

Conflict of interest

The authors declare no conflict of interest.