Dispersion of elastic waves in the three-layered inhomogeneous sandwich plate embedded in the Winkler foundations

Abstract

This research article investigates the behavior of elastic waves in an inhomogeneous isotropic three-layered sandwich plate with soft-core and stiff-skin layers embedded in Winkler foundations, using anti-plane shear motion. The study establishes the exact antisymmetric dispersion relation, low-frequency spectrum, and overall cut-off frequency of the wave propagation. A shortened polynomial dispersion relation is developed for the long-wave low-frequency regime by considering the contrasting material setup and compared with the exact dispersion relation. This article also provides exact and asymptotic formulas for stresses and displacements in each layer of the plate, as well as approximate one-dimensional equations of motion. The results suggest that the approximate equations of motion for the three-layered sandwich plate are valid throughout the entire low-frequency range, despite the presence of Winkler foundations on both sides of the plate. This research is significant as it provides insights into the behavior of waves in composite materials and can be used to improve the design of sandwich structures used in various engineering applications. Additionally, the findings that the approximate equations of motion for the three-layered sandwich plate are valid throughout the low-frequency range, despite the presence of Winkler foundations, can help simplify calculations and make the design process more efficient.

Introduction

Elastic waves dispersion in layered media is a complex and fascinating research topic with many potential applications. Advances in materials science are continually producing new materials with unique properties, and investigating the dispersion behavior of elastic waves in these materials could lead to optimized use in various applications. Understanding the dispersion behavior of elastic waves in layered media can also report to the development of new imaging techniques for more accurate and detailed imaging of internal structures. Additionally, studying the dispersion behavior of seismic waves in the Earth’s crust and mantle can provide valuable insights into the Earth's interior structure and processes. Also, understanding the dispersion behavior of elastic waves in layered materials can also inform the development of improved acoustic insulation materials. Overall, research in elastic waves dispersion in layered media has significant potential to impact a range of disciplines and applications from improving material design to earthquake detection to acoustic insulation, see for instance Refs.^1–5

Classical and nonclassical theories are used to study the dispersion relations of elastic waves in layered media. Classical theory assumes a homogeneous and isotropic medium and neglects microstructure effects, while nonclassical theory considers the microstructure and interaction between the constituent materials. The wavelet-based approach analyzes wavelets, while multiscale modelling accounts for the interaction between different length scales. The nonlocal elasticity theory considers nonlocal interactions between material points. Both classical and nonclassical theories are used to study dispersion relations, with each providing different levels of detail and accuracy depending on the properties of the medium and desired level of analysis. Layered structures, exclusively three layers sandwich plates are increasingly being employed as structural components in contemporary engineering applications. The waveguides, elastic plates, beams, laminates, panels, rods, as well as cylindrical shells, are some of the structures that may be found in multiply-bond layers, to highlight a few, see Refs.^6–25 In terms of layered structure design and structural analysis, which frequently has contrasting material and geometrical parameters, such as size, volume density, and stiffness. Layered structures offer advantages such as high resistance, lightweight, and strength; additionally, they have static and dynamic excitations. Utilizing efficient, creative, long-lasting, and sustainable structural systems is critical in modern engineering. Various analytical, numerical, and asymptotic techniques have also been used to investigate wave dispersion as well as propagation in multilayered and composite structures. Even though layered media mechanics has been a long-studied area, only a few publications have specifically addressed high contrast problems for highly inhomogeneous multilayered setups. High contrast in the mechanical characteristics of skins and cores in such structures demands the use of innovative techniques to estimate effective composite material properties. The application of asymptotic techniques^22–24 enhances a better understanding of the relationship between the contrast in parameters of its components and the lowest eigenfrequencies of the structure. The authors^{7,11–13,17–23} used asymptotic analysis to investigate the elastic waves dispersion in highly inhomogeneous multilayered structures having three and five layers, exclusively sandwich structures with contrasting material parameter setups, and analyzed the cut-off frequency properties. In addition, regarding the influence of external forces like a hygro-thermal response, thermal stress, rotation, magnetic fields, and viscoelasticity.^26–31 Likewise, for the significance of cracks and void pores on the propagation of waves in layered mediums, see Refs.^32,33 and the references therein. In the dynamical and static dispersion analysis of plates and beams interacting with elastic foundations, most researchers used innovative foundation models.^34–45 However, some models are so complex that they have just a few practical uses. As a result, researchers and engineers continue to choose basic but adequately realistic models, such as the Winkler model, ⁴⁶ the linear pressure model, the two-parameter model as well as the elastic continuum model. ⁴⁴ Authors^{7,35–39,43,47} have used the Winkler foundation model to investigate vibrational behavior or the effects of foundations. However, the paper aims to examine the elastic waves dispersion in an inhomogeneous isotropic sandwich plate having three layers with soft-core and stiff-skin layers, interacting with two-sided Winkler foundations due to their vibrant uses in current technology. We are particularly interested in establishing the structural parameter conditions that support the long-wave low-frequency regime. The three-layered plate symmetric about mid-plane interacting with the two-sided foundations is regarded to be symmetrical, and the symmetry simplifies the analysis by allowing the splitting of all quantities of the interest to be divided into even as well as odd components concerning the mid-plane.^1,35 By symmetry, one selects whether to examine the antisymmetric or symmetric modes. ¹³ However, the antisymmetric vibration mode is thought to have fulfilled the global low-frequency regime.^22,28

The underlying research article investigates the behavior of elastic waves in an inhomogeneous isotropic three-layered sandwich plate with soft-core and stiff-skin layers embedded in Winkler foundations, using anti-plane shear motion. The study establishes the exact antisymmetric dispersion relation, low-frequency spectrum, and overall cut-off frequency of the wave propagation. The article also provides exact and asymptotic formulas for stresses and displacements in each layer of the plate, as well as approximate one-dimensional equations of motion. The novelty of the work is that it develops a shortened polynomial dispersion relation for the long-wave low-frequency regime by considering the contrasting material setup and comparing it with the exact dispersion relation. The study finds that the approximate equations of motion for the three-layered sandwich plate are valid throughout the entire low-frequency range, despite the presence of Winkler foundations on both sides of the plate. The research provides insights into the behavior of waves in composite materials and can be used to improve the design of sandwich structures used in various engineering applications. The findings that the approximate equations of motion for the three-layered sandwich plate are valid throughout the low-frequency range, despite the presence of Winkler foundations, can help simplify calculations and make the design process more efficient. The results of this research can be applied in fields such as civil engineering, aerospace engineering, and mechanical engineering, where sandwich structures are commonly used.

The article is structured as follows: Section “Mathematical modelling” presents the mathematical modelling of the problem and outlines the contrasting material parameter setup to be analyzed. In Section “Solution of the problem,” a solution to the proposed problem is found, and exact relations for the stresses and displacements of each layer of the plate are determined. Section “Dispersion relations” determines the exact antisymmetric dispersion relation, cut-off frequency, polynomial dispersion relation, and shortened polynomial dispersion relation, subject to the contrasting setup, and compares these dispersion relations. Section “Asymptotic formulas for displacement and stresses” provides asymptotic formulas for the exact displacements and stresses. Section “Approximate equations of motion” compares the dispersion relations and provides an approximation of the equations of motion. Finally, Section “Conclusion” offers concluding remarks.

Mathematical modelling

We have taken an anti-plane shear motion of three-layered strongly inhomogeneous isotropic plate having core layer $-h_c\le x_2\le h_c$ , lower skin layer in $-(h_c+h_s)\le x_2\le -h_c,$ and $h_c\le x_2\le (h_c+h_s)$ upper skin layer, illustrated in Figure 1 below, with symmetry about the mid-plane. The constituents of the skin layers are the same but different from the core layer.

Figure 1.

Multi-material isotropic three-layered sandwich plate with mid-plane symmetric embedded in the Winkler foundations.

To formulate this problem, several assumptions are made. These include that the sandwich plate is thin and that the deformation of the plate is small compared to its thickness. Additionally, the face sheets are assumed to be isotropic and linearly elastic, while the core material is assumed to be elastic, homogeneous, and isotropic. The face sheets and core are bonded perfectly, and the plate is subjected to in-plane loading. The Winkler foundation is modeled as an elastic foundation that exerts a vertical reaction force on the plate proportional to its deflection. These assumptions simplify the analysis of the sandwich plate, and classical plate theory can be used to model its behavior. The equations of motion describing the vibration of each layer of the plate in $(x_1,x_2)$ coordinates take the following form:

$${\displaystyle \frac{\mathrm{\partial }{\sigma }_{13}^a}{\mathrm{\partial }{x}_1}+{\displaystyle \frac{\mathrm{\partial }{\sigma }_{23}^a}{\mathrm{\partial }{x}_2}={\rho }_a{\displaystyle \frac{{\mathrm{\partial }}^2{u}^a}{\mathrm{\partial }{t}^2},a=c,s,}}}$$

(1)

with $x_n(n=1,2)$ as the spatial variables, t as the temporal variable, $u^a(x_1,x_2,t)$ are the out of plane displacements, and $a=\mathrm{c}$ and $a=\mathrm{s}$ are representing the core and skin layers, respectively. The stresses ${\sigma }_{k3}^a,$ for $k=1,2$ are presented by

$${\sigma }_{k3}^a={\mu }_a{\displaystyle \frac{\mathrm{\partial }{u}^a}{\mathrm{\partial }{x}_k},k=1,2,}$$

(2)

where ${\mu }_a$ are the stiffnesses of the layers. We further prescribe the following interfacial conditions:

$$\begin{array}{rl}& u^{\mathrm{c}}(x_1,x_2,t)=u^{\mathrm{s}}(x_1,x_2,t)\mathrm{at}{x}_2=\pm h_c,\\ & {\sigma }_{23}^{\mathrm{c}}(x_1,x_2,t)={\sigma }_{23}^{\mathrm{s}}(x_1,x_2,t)\mathrm{at}{x}_2=\pm h_c.\end{array}$$

(3)

Additionally, considering the Winkler foundation or elastically restraint boundaries on both sides of the plate,^3,35 the upper and lower faces of the skin layers admit the following conditions:

$$\begin{array}{c}{\sigma }_{23}^{\mathrm{s}}(x_1,x_2,t)=\mp \theta u^{\mathrm{s}}(x_1,x_2,t),\mathrm{at}{x}_2=\pm (h_{\mathrm{c}}+h_{\mathrm{s}}),\end{array}$$

(4)

where $\theta$ is Winkler foundation stiffness. Also, when sandwich plate is not interacting with the Winker foundations (by allowing θ = 0), the current problem reduces to Prikazchikova et al. (2020).

In the current research, however, we are looking at the dispersion of elastic waves in sandwich plate driven by anti-plane shear, with symmetry about mid-plane along with conditions at the interface given in Eq. (3) as well as boundary conditions presented in Eq. (4) applied to the outer surfaces. In reference to material parameter contrasting setup corresponding to stiff-skin layers and a soft-core layer, investigated by Prikazchikova et al. (2020), within the zero cut-off frequency estimates, the exact dispersion relation will be studied asymptotically, with asymptotic relationship given in Eq. (5):

$$h\sim 1,\mu \ll 1,\rho \sim \mu .$$

(5)

Solution of the problem

In this section, we determined the exact solution to the posed problem. In this connection, Eqs. (1)–(4) will be used to calculate the relation of displacements along with stresses corresponding to each layer of the plate. Thus, we have the following classical wave equation through Eqs. (1) and (2):

$${\displaystyle \frac{{\mathrm{\partial }}^2{u}^a}{\mathrm{\partial }{x}_1^2}+{\displaystyle \frac{{\mathrm{\partial }}^2{u}^a}{\mathrm{\partial }{x}_2^2}={\displaystyle \frac{1}{c_a^2}{\displaystyle \frac{{\mathrm{\partial }}^2{u}^a}{\mathrm{\partial }{t}^2},a=\mathrm{c},\mathrm{s},}}}}$$

(6)

with $c_a=\sqrt{{\mu }_a/{\rho }_a},a=\mathrm{c},\mathrm{s},$

where $c_a$ are the transverse wave speeds. We also the above equation to admit the harmonic solution with the form

$$u^a(x_1,x_2,t)=v^a(x_2)\mathrm{Exp}(i(k{x}_1-\omega t)).$$

(7)

Now, since the three-layered plate interacting with a two-sided foundation is representing the symmetric framework about $x_2=0$ , we, therefore, analyze the anti-symmetric solution of the governing problem and obtain the solutions corresponding to the core and skin layers as follows:

$$v^{\mathrm{c}}(x_2)=A_1\sinh (\vartheta x_2),0\le x_2\le h_{\mathrm{c}},v^{\mathrm{s}}(x_2)=A_2\cosh (\vartheta x_2)+B_2\sinh (\vartheta x_2),h_{\mathrm{c}}\le x_2\le (h_{\mathrm{c}}+h_{\mathrm{s}})$$

(8)

Where $\vartheta =\sqrt{k^2-({\omega }^2/c_c^2)x_2}$ $i^2=-1,$ $\omega$ is frequency, and k is the wave number. Also, $A_1$ , $A_2$ , and $B_2$ are constants to be determined. After removing exponential components, the posed problem with solutions in Eq. (8) is linked to the interfacial and boundary conditions provided in Eqs. (3) and (4) give the dimensionless displacements as well as stresses in the following manner.

The core layer

$$\begin{array}{rl}& u^{\mathrm{c}}={\displaystyle \frac{h_c}{\alpha }(\sinh (\alpha {\eta }_{2_c})),}\\ & {\sigma }_{13}^{\mathrm{c}}=iK{\displaystyle \frac{{\mu }_c}{\alpha }(\sinh (\alpha {\eta }_{2_c})),}\\ & {\sigma }_{23}^{\mathrm{c}}={\mu }_c(\cosh (\alpha {\eta }_{2_c})).\end{array}$$

(9)

The skin layer

$$\begin{array}{rl}& u^{\mathrm{s}}={\displaystyle \frac{h_c}{\alpha }{\mathrm{\Psi }}_1(h\beta \cosh (h\beta ({\eta }_{2_s}-1))-G_2\sinh (h\beta ({\eta }_{2_s}-1))),}\\ & {\sigma }_{13}^{\mathrm{s}}=iK{\mathrm{\Psi }}_1{\displaystyle \frac{{\mu }_s}{\alpha }(h\beta \cosh (h\beta ({\eta }_{2_s}-1))-G_2\sinh (h\beta ({\eta }_{2_s}-1))),}\\ & {\sigma }_{23}^{\mathrm{s}}={\mathrm{\Psi }}_1{\mu }_s{\displaystyle \frac{\beta }{\alpha }(h\beta \sinh (h\beta ({\eta }_{2_s}-1))-G_2\cosh (h\beta ({\eta }_{2_s}-1))).}\end{array}$$

(10)

Where ${\mathrm{\Psi }}_1$ in Eq. (10) is given by

$${\mathrm{\Psi }}_1={\displaystyle \frac{\sinh (\alpha )}{h\beta \cosh (h\beta )+G_2\sinh (h\beta )}.}$$

(11)

The new scaling of $x_2$ variable, for Eqs. (9) and (10), is

$$\begin{array}{rl}& {\eta }_{2_c}={\displaystyle \frac{x_2}{h_c},0\le x_2\le h_{\mathrm{c}},}\\ & {\eta }_{2_s}={\displaystyle \frac{x_2-h_c}{h_s},h_{\mathrm{c}}\le x_2\le (h_{\mathrm{c}}+h_{\mathrm{s}}).}\end{array}$$

(12)

More,

$$\alpha =\sqrt{K^2-{\mathrm{\Omega }}^2},\beta =\sqrt{K^2-{\mathrm{\Omega }}^2{\displaystyle \frac{\mu }{\rho }}},$$

(13)

with

$$\mathrm{\Omega }={\displaystyle \frac{\omega h_c}{c_c},K=k{h}_c.}$$

(14)

Here, $\mathrm{\Omega }$ and K are the dimensionless frequency and wave number, correspondingly. Additionally, other dimensionless parameters are mentioned below

$$\mu ={\displaystyle \frac{{\mu }_c}{{\mu }_s},h={\displaystyle \frac{h_s}{h_c},\rho ={\displaystyle \frac{{\rho }_c}{{\rho }_s}.}}}$$

(15)

Furthermore, $G_1$ and $G_2$ are the Winkler foundation's dimensionless stiffnesses, originating in the skin and core layers of the plate, and are given by

$$G_1={\displaystyle \frac{\theta h_c}{{\mu }_c},G_2={\displaystyle \frac{\theta h_s}{{\mu }_s}.}}$$

(16)

Dispersion relations

The exact antisymmetric dispersion relation and cut-off frequency of the posed problem are provided in this section, and the resulting dispersion relation is then expressed in terms of its respective polynomial dispersion relation, as well as a shortened polynomial dispersion relation.

Exact antisymmetric dispersion relation (ASDR)

The analytical solution of the problem, provided in Eq. (8), is subject to the interfacial along with boundary conditions given in Eqs. (3) and (4), yielded a $3\times 3$ dispersion matrix expressed as

$$\left(\begin{array}{ccc}-\sinh (\alpha )& \cosh (\beta )& \sinh (\beta )\\ {\displaystyle \frac{\alpha {\mu }_c}{h_c}\cosh (\alpha )}& -{\displaystyle \frac{\beta {\mu }_s}{h_c}\sinh (\beta )}& -{\displaystyle \frac{\beta {\mu }_s}{h_c}\cosh (\beta )}\\ 0& N_1& N_2\end{array}\right),$$

(17)

where $N_1$ and $N_2$ are given by

$$\begin{array}{rl}& N_1=\theta \cosh (\beta (h+1))+{\displaystyle \frac{\beta {\mu }_s}{h_c}\sinh (\beta (h+1)),}\\ & N_2=\theta \sinh (\beta (h+1))+{\displaystyle \frac{\beta {\mu }_s}{h_c}\cosh (\beta (h+1)).}\end{array}$$

(18)

After setting the determinant of the aforementioned matrix to zero, the following exact ASDR is obtained.

$$\begin{array}{r}h{\beta }^2{G}_1\sinh (\alpha )\sinh (\beta h)+G_2\alpha \beta \cosh (\alpha )\cosh (\beta h)+G_1{G}_2(\beta \cosh (\beta h)\sinh (\alpha )\\ +\alpha \mu \cosh (\alpha )\sinh (\beta h))=0.\end{array}$$

(19)

Special case: However, without Winkler foundation, the ASDR provided in Eq. (19) match with result of Prikazchikova et al. (2020). By allowing $\theta =0$ , the current exact dispersion relation becomes Prikazchikova et al. (2020).

$$\begin{array}{c}\sinh (\alpha )\sinh (\beta h)\beta +\cosh (\alpha )\cosh (\beta h)\alpha \mu =0.\end{array}$$

Eq. (19) gives you the cut-off frequency at $K=0$ as

$$\sqrt{\rho }{G}_2(\mathrm{\Omega }+G_1\tan (\mathrm{\Omega }))+\sqrt{\mu }{G}_1(\rho G_2-h\mathrm{\Omega }\tan (\mathrm{\Omega }))\tan \left(h\sqrt{{\displaystyle \frac{\mu }{\rho }}}\mathrm{\Omega }\right)=0.$$

(20)

After omitting the negative frequencies, the single predicted cut-off frequency is as follows:

$$\mathrm{\Omega }\approx {\displaystyle \frac{1}{h}\sqrt{{\displaystyle \frac{\rho G_2(1+(1+h\mu )G_1)}{G_1\mu }}}\ll 1,}$$

(21)

The material parameters supporting the global low-frequency regime ²⁰ must be estimated in conjunction with the following global low-frequency inequalities

$$\begin{array}{rl}& {\mathrm{\Omega }}_1={\displaystyle \frac{1}{h}\sqrt{{\displaystyle \frac{\rho G_2(1+(1+h\mu )G_1)}{G_1\mu }}}\ll 1,}\\ & {\mathrm{\Omega }}_2=\sqrt{{\displaystyle \frac{G_2(1+(1+h\mu )G_1)}{G_1}}}\ll 1.\end{array}$$

(22)

The first two modes from the exact ASDR given in Eq. (19) are displayed in Figure 2 for the non-estimated range and Figure 3 for the estimated range of the zero cut-off frequencies, that is, satisfying the global frequency inequalities given in Eq. (22). For the long-wave low-frequency condition $(K\ll 1\mathrm{and}\mathrm{\Omega }\ll 1)$ Kaplunov et al. (1998), not all the dispersion curves lie in $0\le \mathrm{\Omega }\le 1$ in Figure 2 owing to the selection of parameters outside Eq. (22); while both curves lie within $0\le \mathrm{\Omega }\le 1$ in Figure 3.

Figure 2.

Dispersion curves from ASDR provided in Eq. (19) for non-estimated range with $h=1,\mu =3.5,\rho =3.5,G_1=0.6,G_2=1$ .

Figure 3.

Dispersion curve from ASDR provided in Eq. (19) for estimated range with $h=1,\mu =0.23,\rho =0.23,G_1=0.001,G_2=0.0003$ .

Polynomial dispersion relation

Using Taylor's series expansion, the polynomial dispersion relation has been obtained from the exact ASDR, given in Eq. (19) as follows:

$${\tau }_1+{\tau }_2{K}^2+{\tau }_3{K}^4+{\tau }_4{K}^2{\mathrm{\Omega }}^2+{\tau }_5{\mathrm{\Omega }}^2+{\tau }_6{\mathrm{\Omega }}^4+\dots =0,$$

(23)

With

$$G_2=G_{1}h\mu ,$$

(24)

and

$$\begin{array}{l}{\tau }_1=144h\mu G_1+144h\mu G_1^2+144h^2{\mu }^2{G}_1^2,\\ {\tau }_2=144h^2{G}_1+72h\mu G_1+72h^3\mu G_1+24h\mu G_1^2+72h^3\mu G_1^2+72h^2{\mu }^2{G}_1^2+24h^4{\mu }^2{G}_1^2,\\ {\tau }_3=24h^2{G}_1+24h^4{G}_1+6h\mu G_1+36h^3\mu G_1+6h^5\mu G_1+12h^3\mu G_1^2+6h^5\mu G_1^2+6h^2{\mu }^2{G}_1^2+12h^4{\mu }^2{G}_1^2,\\ {\tau }_4=24h^2{G}_1+24h^4{G}_1+6h\mu G_1+36h^3\mu G_1+6h^5\mu G_1+12h^3\mu G_1^2+6h^5\mu G_1^2+6h^2{\mu }^2{G}_1^2+12h^4{\mu }^2{G}_1^2,-12h^2{\mu }^2{G}_1^2-12h^4{\mu }^2{G}_1^2-{\displaystyle \frac{12h^3{\mu }^2{G}_1^2}{\rho }-{\displaystyle \frac{12h^5{\mu }^2{G}_1^2}{\rho }-{\displaystyle \frac{12h^4{\mu }^3{G}_1^2}{\rho },}}}\\ {\tau }_5=-72h\mu G_1-{\displaystyle \frac{144h^2\mu G_1}{\rho }-{\displaystyle \frac{72h^3{\mu }^2{G}_1}{\rho }-24h\mu G_1^2-72h^2{\mu }^2{G}_1^2-{\displaystyle \frac{72h^3{\mu }^2{G}_1^2}{\rho }-{\displaystyle \frac{24h^4{\mu }^3{G}_1^2}{\rho },}}}}\\ {\tau }_6=6h\mu G_1+{\displaystyle \frac{24h^4{\mu }^2{G}_1}{{\rho }^2}+{\displaystyle \frac{6h^5{\mu }^3{G}_1}{{\rho }^2}+{\displaystyle \frac{24h^2\mu G_1}{\rho }+{\displaystyle \frac{36h^3{\mu }^2{G}_1}{\rho }+6h^2{\mu }^2{G}_1^2+{\displaystyle \frac{6h^5{\mu }^3{G}_1^2}{{\rho }^2}+{\displaystyle \frac{12h^3{\mu }^2{G}_1^2}{\rho }+{\displaystyle \frac{12h^4{\mu }^3{G}_1^2}{\rho },}}}}}}}\end{array}$$

(25)

Shortened polynomial dispersion relation (SPDR)

For the material contrasting setup provided in Eq. (5), we explore the ASDR asymptotically within the predicted global low-frequency inequalities. Recall also from Eq. (16) that the dimensionless stiffnesses $G_1$ and $G_2$ of the Winkler foundation emerging in core and skin layers of sandwich plate are mentioned correspondingly by

$$G_1={\displaystyle \frac{\theta h_c}{{\mu }_c},G_2={\displaystyle \frac{\theta h_s}{{\mu }_s}.}}$$

Thus, from the above equation, and the dimensionless quantities mentioned in Eqs. (15) and (24), we have following relation that relates $G_1$ and $G_2$ as follows

$$G_2\ll G_1.$$

(26)

Now, assuming that the elastic foundation is made of a relatively soft material, ³⁵ we obtain the following asymptotic relation from immediate expression

$$G_2\ll G_1\ll 1.$$

(27)

Furthermore, using the relation given in Eq. (27) and coefficients of the polynomial provided in Eq. (25) at the leading orders, the corresponding asymptotic behavior is obtained

$${\tau }_1\sim \mu G_1,\mathrm{and}{\tau }_2\sim {\tau }_3\sim {\tau }_4\sim {\tau }_5\sim G_1,$$

(28)

with the following new coefficients

$$\begin{array}{r}\begin{array}{l}{\tau }_1=6\mu G_1,\\ {\tau }_2=6h{G}_1,\\ {\tau }_3=h{G}_1+24h^3{G}_1,\\ {\tau }_4=-h{G}_1-h{\mu }_{\rho }{G}_1-2h^3{\mu }_{\rho }{G}_1,\\ {\tau }_5=-6h{\mu }_{\rho }{G}_1,\\ {\tau }_6=h^3{\mu }_{\rho }^2{G}_1+h{\mu }_{\rho }{G}_1,\end{array}\end{array}$$

(29)

where ${\mu }_{\rho }=\mu /\rho$ . Therefore, shortened polynomial dispersion relation as under:

$$6\mu {\rho }^2{G}_1+h\rho G_1(6+K^2-{\mathrm{\Omega }}^2)(K^2\rho -\mu {\mathrm{\Omega }}^2)+h^3{G}_1(K^2\rho -\mu {\mathrm{\Omega }}^2{)}^2=0.$$

(30)

Thus, the comparison for the exact ASDR given in Eq. (19) and the SPDR provided in Eq. (30) has been made in Figure 4. Additionally, we also make notice of that a relatively soft foundation is considered by choosing $G_1$ and $G_2$ according to relationship established in Eq. (27). An exceptionally strong agreement can be observed in Figure 4 between the two approximations.

Figure 4.

Dispersion branch corresponding to exact ASDR Eq. (19) and SPDR given in Eq. (30) with $h=1,\mu =0.23,\rho =0.23,G_1=0.001,G_2=0.0003$ .

Asymptotic formulas for displacement and stresses

This section gives the asymptotic formulas corresponding to exact displacements and stresses earlier determined in Eqs. (9) and (10) for each layer of the plate. To achieve this objective, we make use of the following rescaling of wave number and frequency, respectively, of the form:

$$K^2=\mu {\xi }^2,{\mathrm{\Omega }}^2=\mu {\chi }^2.$$

(31)

Hence, substituting the above rescaling into Eqs. (9) and (10) alongside employing the material contrast relation given in Eq. (5), we obtain the following for leading order asymptotic displacements and stresses.

The core layer

$$\begin{array}{rl}& u^{\mathrm{c}}=h_c{\eta }_{2_c},\\ & {\sigma }_{13}^{\mathrm{c}}=i{\mu }_c\sqrt{\mu }\xi {\eta }_{2_c},\\ & {\sigma }_{23}^{\mathrm{c}}={\mu }_c.\end{array}$$

(32)

The skin layers

$$\begin{array}{rl}& u^{\mathrm{s}}={\displaystyle \frac{h_c}{1+G_2}(1-G_2({\eta }_{2_s}-1)),}\\ & {\sigma }_{13}^{\mathrm{s}}=i\sqrt{\mu }\xi {\displaystyle \frac{{\mu }_s}{1+G_2}(1-G_2({\eta }_{2_s}-1)),}\\ & {\sigma }_{23}^{\mathrm{s}}={\displaystyle \frac{{\mu }_s}{1+G_2}(-G_2+\mu ({\xi }^2-{\chi }^2)({\eta }_{2_s}-1)).}\end{array}$$

(33)

Special case. The resulting asymptotic formulas for displacements and stresses for antisymmetric modes are provided in Eqs. (32) and (33), reduces to the corresponding antisymmetric asymptotic formulas for displacements and stresses, ¹¹ when the elastic foundation is absent.

It is worth noting here that the following conclusion may be drawn via the asymptotic relation given in Eq. (5)

$${\displaystyle \frac{{\sigma }_{13}^{\mathrm{i}}}{{\mu }_i\sqrt{\mu }}\sim {\displaystyle \frac{{\sigma }_{23}^{\mathrm{i}}}{{\mu }_i}\sim {\displaystyle \frac{u^{\mathrm{i}}}{h_c},i=s,c,}}}$$

(34)

where s and c reflect the skin and core layers, respectively.

Approximate equations of motion

In this part, we will ascertain the approximate equations of motions by incorporating the contrast relation given in Eq. (5). First, we scale the time and longitudinal coordinates using

$$t={\displaystyle \frac{h_c}{c_{2_c}\sqrt{\mu }}\tau \mathrm{and}{x}_1={\displaystyle \frac{h_c}{\sqrt{\mu }}{\eta }_1,}}$$

(35)

Along with definitions of $x_2$ provided in Eq. (12). Furthermore, as a result of Eqs. (32) and (33), the normalized displacements and stresses are as follow

$$u_i=h_c{w}_i,{\sigma }_{13}^i={\mu }_i\sqrt{\mu }{Q}_{13}^i,{\sigma }_{23}^i={\mu }_c{Q}_{23}^i,i=c,s.$$

(36)

Further, substituting Eqs. (12), (35) and (36) into Eqs. (1)–(4), via Eq. (5) offer the following for the core and skin layers, respectively

$$\begin{array}{rl}& \mu {\displaystyle \frac{\mathrm{\partial }{Q}_{13}^c}{\mathrm{\partial }{\eta }_1}+{\displaystyle \frac{\mathrm{\partial }{Q}_{23}^c}{\mathrm{\partial }{\eta }_{2_c}}=\mu {\displaystyle \frac{{\mathrm{\partial }}^2{w}_c}{\mathrm{\partial }{\tau }^2},}}}\\ & Q_{13}^c={\displaystyle \frac{\mathrm{\partial }{w}_c}{\mathrm{\partial }{\eta }_1},Q_{23}^c={\displaystyle \frac{\mathrm{\partial }{w}_c}{\mathrm{\partial }{\eta }_{2_c}},}}\end{array}$$

(37)

$$\begin{array}{rl}& {\displaystyle \frac{\mathrm{\partial }{Q}_{13}^s}{\mathrm{\partial }{\eta }_1}+{\displaystyle \frac{1}{h}{\displaystyle \frac{\mathrm{\partial }{Q}_{23}^s}{\mathrm{\partial }{\eta }_{2_s}}={\mu }_{\rho }{\displaystyle \frac{{\mathrm{\partial }}^2{w}_s}{\mathrm{\partial }{\tau }^2},}}}}\\ & Q_{13}^s={\displaystyle \frac{\mathrm{\partial }{w}_s}{\mathrm{\partial }{\eta }_1},\mu h{Q}_{23}^s={\displaystyle \frac{\mathrm{\partial }{w}_s}{\mathrm{\partial }{\eta }_{2_s}},}}\end{array}$$

(38)

for $h\sim 1\sim {\mu }_{\rho }=\mu /\rho$ .

Additionally, the corresponding continuity and boundary conditions are established as well,

$$\begin{array}{rl}& w_c{|}_{{\eta }_{2_c=1}}=w_s{|}_{{\eta }_{2_s=0}},\\ & Q_{23}^c{|}_{{\eta }_{2_c=1}}=Q_{23}^s{|}_{{\eta }_{2_s=0}},\end{array}$$

(39)

$$\begin{array}{c}{Q}_{23}^s{|}_{{\eta }_{2_s=1}}={\displaystyle \frac{G_2}{h\mu }{w}_s{|}_{{\eta }_{2_s=1}}.}\end{array}$$

(40)

The asymptotic series expansion is then used in displacements as well as stresses, as shown below.

$$\begin{array}{rl}& w_i=w_{i,0}+\mu w_{i,1}+,\dots ,\\ & Q_{j3}^i=Q_{j3,0}^i+\mu Q_{j3,1}^i+,\dots ,i=c,s,j=1,2.\end{array}$$

(41)

Therefore, we have obtained the following leading order by substituting Eq. (41) into Eqs. (37)–(40),

$$Q_{13,0}^c={\displaystyle \frac{\mathrm{\partial }{w}_{c,0}}{\mathrm{\partial }{\eta }_1},{\displaystyle \frac{\mathrm{\partial }{Q}_{23,0}^c}{\mathrm{\partial }{\eta }_{2_c}}=0,Q_{23}^c={\displaystyle \frac{\mathrm{\partial }{w}_{c,0}}{\mathrm{\partial }{\eta }_{2_c}},}}}$$

(42)

$$\begin{array}{rl}& {\displaystyle \frac{\mathrm{\partial }{Q}_{13,0}^s}{\mathrm{\partial }{\eta }_1}+{\displaystyle \frac{1}{h}{\displaystyle \frac{\mathrm{\partial }{Q}_{23,0}^s}{\mathrm{\partial }{\eta }_{2_s}}={\mu }_{\rho }{\displaystyle \frac{{\mathrm{\partial }}^2{w}_{s,0}}{\mathrm{\partial }{\tau }^2},}}}}\\ & Q_{13}^s={\displaystyle \frac{\mathrm{\partial }{w}_{s,0}}{\mathrm{\partial }{\eta }_1},{\displaystyle \frac{\mathrm{\partial }{w}_{s,0}}{\mathrm{\partial }{\eta }_{2_s}}=0,}}\end{array}$$

(43)

with following continuity conditions

$$\begin{array}{rl}& w_{c,0}{|}_{{\eta }_{2_c=1}}=w_{s,0}{|}_{{\eta }_{2_s=0}},\\ & Q_{23,0}^c{|}_{{\eta }_{2_c=1}}=Q_{23,0}^s{|}_{{\eta }_{2_s=0}},\end{array}$$

(44)

and boundary condition

$$\begin{array}{c}{Q}_{23,0}^s{|}_{{\eta }_{2_s=1}}={\displaystyle \frac{G_2}{h\mu }{w}_{s,0}{|}_{{\eta }_{2_s=1}}.}\end{array}$$

(45)

It is straightforward to obtain the following from Eq. (43)

$$\begin{array}{c}{w}_{c,0}=p({\eta }_1,\tau ).\end{array}$$

(46)

The remaining quantities in Eqs. (42)–(45) are expressed in the terms of $p({\eta }_1,\tau )$ as follows

$$\begin{array}{c}{Q}_{13,0}^c={\eta }_{2_c}{\displaystyle \frac{\mathrm{\partial }p({\eta }_1,\tau )}{\mathrm{\partial }{\eta }_1},Q_{23,0}^c=p({\eta }_1,\tau ),w_{c,0}={\eta }_{2_c}p({\eta }_1,\tau ),}\end{array}$$

(47)

$$\begin{array}{c}{Q}_{13,0}^s={\displaystyle \frac{\mathrm{\partial }p({\eta }_1,\tau )}{\mathrm{\partial }{\eta }_1},Q_{23,0}^s=h\left({\mu }_{\rho }{\displaystyle \frac{{\mathrm{\partial }}^{2}p({\eta }_1,\tau )}{\mathrm{\partial }{\tau }^2}-{\displaystyle \frac{{\mathrm{\partial }}^{2}p({\eta }_1,\tau )}{\mathrm{\partial }{\eta }_1^2}}}\right)({\eta }_{2_s}-1)+{\displaystyle \frac{G_2}{\mu h}p({\eta }_1,\tau ).}}\end{array}$$

(48)

By using Eq. (42)

$$\begin{array}{c}h\left({\mu }_{\rho }{\displaystyle \frac{{\mathrm{\partial }}^{2}p({\eta }_1,\tau )}{\mathrm{\partial }{\tau }^2}-{\displaystyle \frac{{\mathrm{\partial }}^{2}p({\eta }_1,\tau )}{\mathrm{\partial }{\eta }_1^2}}}\right)=p({\eta }_1,\tau )\left({\displaystyle \frac{G_2}{h\mu }-1}\right),\end{array}$$

(49)

then

$$\begin{array}{c}{Q}_{23,0}^s=p({\eta }_1,\tau ){\eta }_{2_s}\left({\displaystyle \frac{G_2}{h\mu }-1}\right).\end{array}$$

(50)

Eq. (43) takes the form

$$\begin{array}{c}{\displaystyle \frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{\eta }_1^2}-{\displaystyle \frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{\tau }^2}-{\displaystyle \frac{1}{h}\left(1-{\displaystyle \frac{G_2}{h\mu }}\right)p({\eta }_1,\tau )=0,}}}\end{array}$$

(51)

which can be expressed in original variables, where $u_s(x_1,t)\approx p({\eta }_1,\tau )$ , we obtain

$$\begin{array}{c}{\displaystyle \frac{{\mathrm{\partial }}^2{u}_s}{\mathrm{\partial }{x}_1^2}-{\displaystyle \frac{{\mathrm{\partial }}^2{u}_s}{\mathrm{\partial }{t}^2}-{\displaystyle \frac{1}{h}\left(1-{\displaystyle \frac{G_2}{h\mu }}\right)u_s=0.}}}\end{array}$$

(52)

Now, utilizing the expression $u_s=\mathrm{Exp}(-i(-k{x}_1+\omega t))$ , we obtain the following dispersion relation

$$\begin{array}{c}{k}^2+{\omega }^2-{\displaystyle \frac{1}{h}\left(1-{\displaystyle \frac{G_2}{h\mu }}\right)=0.}\end{array}$$

(53)

As highlighted in Figure 5, which coincides with the ASDP determined in Eq. (19) and SPDR provided in Eq. (30), for the first mode. It is also consistent with the determined exact ASDP and the SPDR in Section 4. In addition, ${\mu }_c$ and ${\rho }_c$ are stiffness and density of core layer of sandwich plate, respectively. The core is considered to be made of aluminum material ³⁰ in the depiction below with $h=1,{\mu }_c=246\times {10}^{10}\mathrm{Pa},{\rho }_c=2.66\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3},\mu =0.023,\rho =0.008,G_1=0.008,G_2=\mathrm{0.0001.}$

Figure 5.

Comparison of the exact ASDP Eq. (19), SPDR given in Eq. (30) and dispersion relation Eq. (53) obtained through Eq. (52).

Conclusion

In conclusion, the dispersion of elastic waves in an inhomogeneous three-layered sandwich plate with a soft-core and stiff-skin layers interacting with a two-sided Winkler foundations, driven by anti-plane shear has been investigated by adopting an asymptotic technique. The governing equations have been reduced to ordinary differential equations with the help of specified boundary and interfacial conditions, the exact displacements along with stresses in corresponding layers, and exact ASDR have been computed analytically. Due to multiple industrial applications of soft-stiff layer combinations,^{7–11,19,23,25} the exact ASDR and its related polynomial dispersion relation, as well as the SPDR, subject to the long-wave low-frequency region, have been investigated. Additionally, asymptotic formulas for the exact stresses and displacement have also been derived for the respective layer of the plate. It is remarkable that in the absence of the Winkler foundation by allowing $\theta =0$ , the results of this study are validated and shown to be consistent with the recently investigated. ¹¹ Furthermore, using the estimated equations of motion given in Eq. (52), we derived a dispersion relation presented in Eq. (53) that coincides with the exact ASDR given in Eq. (19) and the SPDR given in Eq. (30). Additionally, the Mathematica software has been deployed for all computational simulations and graphical representations in this study.

Finally, it is suggested that similar research might be conducted to ascertain the symmetric anti-plane shear vibration of a multilayered plate embedded in the Winkler foundation or other innovative foundation models. Furthermore, this study might be extended to other layered plates with varying compositions and material contrasts interacting with different foundation models.