Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates

1 Introduction

Recent technological proliferation in engineering has resulted in an increased demand for multi-layered structures for numerous purposes. Of particular target in this study, however, is the symmetric three three-layered structure. Such structures, also known as sandwich structures, have been manufactured and utilised since long ago. These structures are popular for their lightweight combined with relatively large flexural stiffness, just to mention a few advantages. Such structures are in great demand and have found particular usage within modern aerospace, automotive, and civil engineering applications three-layered [1,2].

In view of the wider range of potential applications of layered structures, a number of researchers have extended the initial studies on linear isotropic media to include the influence of, for instance, anisotropy [3], and pre-stress [4] to mention a few. In this regard, some of the latest investigations include the analysis of various layered media with fixed-face boundary conditions and the study of long-wave behaviour posed by the dispersion relation of a pre-stressed incompressible elastic plate in both the low and high wave regimes [5–7].

Moreover, early studies of wave propagation problems were generally performed for the plane strain case. In comparison, the anti-plane shear case allows a more explicit analysis of the dispersion relation. Several studies in the plane strain case focus on asymptotic approaches for layered structures. For example, the development of long-wave theories for pre-stressed compressible elastic plates has been established by Rogerson and Prikazchikova [8]. In contrast, certain asymptotic procedures underlying multi-layered plate theories and devoted to the homogenisation of high-contrast periodic media have been introduced in different research works [9,10]. Asymptotic approximations for elastic plates were intensively considered first by several researchers [11–14]. More recently, multi-layered engineering structures, in particular, the four-ply symmetric laminates with contrast in the stiffness and density of outer and core layers, have been studied by various authors, see for example, Ryazantseva and Antonov [15]; and Kaplunov et al. [16], among others.

Besides, the importance of anti-plane shear deformations within solid mechanics has resulted in a number of studies [17–19]. The model features relative analytical simplicity in comparison with the plane strain problems. Anti-plane shear motion equation is a single second-order linear or quasi-linear partial differential equation rather than a higher-order or coupled system of partial differential equations. More so, for the anti-plane problem of a three-layered symmetric plate, one may read the research study by Prikazchikova et al. [20] for the asymptotic investigation of dispersion relation under low-frequency approximation. In contrast, we cite the recent works by Nuruddeen et al. [21] and Kaplunov et al. [22] for approximating the dispersion relation of a strongly inhomogeneous three-layered asymmetric laminate. Furthermore, the asymptotic analysis for a light core and stiff thin skin layers of different thicknesses has been studied by Kaplunov et al. [23] for a three-layered panel, while the examination of four contrasting material setups for a nonhomogeneous elastic five-layered laminate has been presented by Nuruddeen et al. [24].

However, the aim of the present work is to add to our understanding of layered and composite media by investigating the dynamic response of a symmetric three-layered pre-stressed compressible elastic structure with respect to the anti-plane shear motion. A couple of methods, including analytical, numerical simulation, and asymptotic, would all be employed to examine the situation. Thus, the study is organised as follows: the formulation of the problem is given in Section 2, while Section 3 analytically determines two cases of the dispersion relations for the governing structure with free and fixed face conditions, respectively. A numerical investigation of the obtained exact dispersion relations in Section 3 is carried out in Section 4, while Section 5 presents the corresponding asymptotic results for within the long-wave and low-frequency interval, and further establishes a comparative analysis. Finally, Section 6 provides a general conclusion.

2 Problem statement

This study primarily examines anti-plane shear waves in a symmetric three-layered laminate. The layers of the laminate are composed of pre-stressed compressible elastic material. The outer layers are identical and of thickness h , while the inner core layer is of thickness 2 h . The region occupied by the structure is shown in Figure 1.

Figure 1

A symmetric three-layered pre-stressed compressible elastic laminate.

The general equations of motion for a small time-dependent motion superimposed on a large homogeneous deformation are given by Ogden [25]. However, for our case of anti-plane shear motion with u 3 ( x 1 , x 2 , t ) as the only non-zero displacement, the single governing equation takes the form as follows:

where u 3 ( l ) = u 3 ( l ) ( x 1 , x 2 , t ) are the out-of-plane displacements in the spatial and temporal variables, x 1 x 2 , and t , respectively. Additionally, ρ is the density, l = r corresponds to the inner core layer, while l = s represents the two skin layers, respectively. In addition, Γ 13 ( l ) and Γ 23 ( l ) are material constants that can equally be represented in terms of components of the fourth-order elasticity tensor through Γ 13 ( l ) = C 1313 l and Γ 23 ( l ) = C 2323 l .

Thus, we seek solutions of Eq. (1) in the following form:

where U is an arbitrary constant, k denotes the wavenumber, υ is the phase wave speed, and q l is to be determined. It is now readily established that

from which we deduce the two solutions q l , l = r , s as follows:

Finally, we are able to obtain the displacements as a linear combinations of the two constants, associated with the two independent solution of q l indicated in (4), so

Then, on using the above solutions, we obtain the associated surface-traction components as follows:

where A i and B i , i = 1 , 2 , 3 are independent constants at each surface of the structure.

3 Derivation of the dispersion relations

As the laminate under consideration is symmetrical, this section makes use of the obtained displacements and traction components in Section 2 to comprehensively examine the dispersion relation associated with the structure when endowed with the free-faces and fixed-faces boundary conditions, respectively.

4 Numerical analysis

In this section, illustrative numerical solutions of the obtained dispersion relations are discussed. In doing so, we have made use of the scaled wavenumber K and the scaled frequency Ω ¯ = υ ¯ K for a given state of the pre-stress with parameters outlined in Table 1. In particular, we remark here that the anti-symmetric dispersion relation corresponding to both cases of free and fixed-faces given in Eqs. (20) and (24), respectively, provided implicit relationships.

Using the Mooney–Rivlin (MR) material parameters given in Table 1, we numerically portray in Figure 2(a) and (b) the dispersion curves for the respective dispersion relations associated with free-faces for the symmetric and anti-symmetric modes, respectively, while Figure 3(a) and (b) show the respective corresponding curves with fixed-faces, sequentially. In addition, these figures give the relationships between the scaled frequencies against the scaled wavenumbers.

Figure 2

Harmonic curves for the (a) symmetric (16) and (b) anti-symmetric (20) dispersion relations for the symmetric three-layered pre-stressed compressible elastic laminate with free-faces.

Figure 3

Harmonic curves for the (a) symmetric (23) and (b) anti-symmetric (24) dispersion relations for the symmetric three-layered pre-stressed compressible elastic laminate with fixed-faces.

Precisely, the dispersion curves for the symmetric dispersion relation shown in Figure 2(a) with free-faces provide a number of harmonic waves, with the first harmonic being the fundamental mode at zero; whereas all harmonic limits have a non-zero limit in the case of the anti-symmetric dispersion relation with free-faces shown in Figure 2(b). Furthermore, the dispersion curves for both the symmetric and anti-symmetric dispersion relations corresponding to fixed-faces conditions as shown in Figure 3(a) and (b) introduce harmonic waves with no zero limits. Finally, we remarked here that the fundamental mode is only observed in the case of symmetric dispersion relation with fixed-faces condition.

5 Asymptotic analysis

This section utilises the asymptotic analysis method to further examine or rather affirm the exactness of the obtained exact and numerical dispersion relations in the preceding sections asymptotically.

5.1 Free-faces case

As mentioned previously, we make consideration to both the symmetric and anti-symmetric dispersion relations with free-faces boundary conditions, independently.

5.1.1 Symmetric case

In this case, the exact symmetric dispersion relation determined in Eq. (16) will be analysed within the long-wave low-frequency region, that is, υ ¯ remains finite as K → 0 . On the other hand, in the long wave high-frequency region, υ ¯ → ∞ as K → 0 and the solutions q r 2 and q s 2 in (4) are negative, then, it is convenient to introduce q n = i q ˆ n . Thus, the scaled frequency Ω ¯ = υ ¯ K will be introduced in the following form:

(25) Ω ¯ 2 = Ω 0 + Ω 2 K 2 + O ( K 4 ) .

So, by expanding all the trigonometric functions in the exact symmetric dispersion relation given in Eq. (16) using the Taylor series, we derive the approximation in the low-frequency region as follows:

(26) Ω ¯ 2 = ( 1 ∕ 2 ) ( Γ 13 ( r ) + Γ 13 ( s ) ) K 2 + O ( K 4 ) .

More so, the exact symmetric dispersion relation given in Eq. (16) is expressed in the high-frequency region as follows:

(27) Γ ( r ) q r ˆ tan K q r ˆ + Γ ( s ) q s ˆ tan K q s ˆ = 0 .

Now, by considering the following expansion:

(28) K q ˆ l = Ω ¯ ∕ Γ 23 ( l ) 1 − Γ 13 ( l ) K 2 2 Ω ¯ 2 + … ,

together with the approximation given in Eq. (25), the dispersion relation expressed in Eq. (27) may be used to show that the frequency is indeed a solution of the following expression:

(29) Γ ( s ) tan ( Ω 0 ( n ) ∕ Γ 23 ( s ) ) + Γ 23 ( r ) tan ( Ω 0 ( n ) ∕ Γ 23 ( r ) ) = 0 ,

where Ω 0 ( n ) is the solution of Eq. (29) that equally defines the cut-off frequencies. The next order term will be Ω 2 = ψ 2 ( Ω 0 ) ∕ ψ 1 ( Ω 0 ) with

(30) ψ 1 ( Ω 0 ) = Γ 23 ( s ) F 1 ( Ω 0 ) + Γ 23 ( r ) F 2 ( Ω 0 ) + Ω 0 ( F 1 2 ( Ω 0 ) + 1 ) + Ω 0 ( F 2 2 ( Ω 0 ) + 1 ) , ψ ˜ 2 ( Ω 0 ) = − 1 2 ( Γ 13 ( s ) ( 1 + F 1 2 ( Ω 0 ) ) Ω 0 + Γ 13 ( r ) ( 1 + F 2 2 ( Ω 0 ) ) Ω 0 + Γ 23 ( s ) F 1 ( Ω 0 ) Γ 13 ( s ) + Γ 23 ( r ) F 2 ( Ω 0 ) Γ 13 ( r ) ) ,

where,

F i ( Ω 0 ) = tan Ω 0 Γ 23 ( i ) , i = 1 , 2 , 3 .

Finally, the scaled frequency given in Eq. (25) may therefore be written in the following form:

(31) Ω ¯ ( n ) 2 = Ω 0 ( n ) + [ ψ 2 ( Ω 0 ) ∕ ψ 1 ( Ω 0 ) ] K 2 + O ( K 4 ) .

5.1.2 Anti-symmetric case

For the anti-symmetric case, the exact anti-symmetric dispersion relation determined in Eq. (20) will be written as follows:

(32) Γ 23 ( s ) q s tan ( K q s ) tan ( K q r ) − Γ 23 ( r ) q r = 0 .

So, upon using the expansion given in Eq. (25), the leading order term may be expressed explicitly as follows

(33) Γ 23 ( s ) F 1 ( Ω 0 ( n ) ) F 2 ( Ω 0 ( n ) ) − Γ 23 ( r ) = 0 ,

where Ω 0 can be found by solving Eq. (33) numerically. The second term in this case can be introduced as

(34) Ω 2 = Φ 2 ( Ω 0 ) ∕ Φ 1 ( Ω 0 ) ,

with

(35) Φ 1 ( Ω 0 ) = Γ 23 ( r ) Γ 23 ( s ) F 1 ( Ω 0 ) F 2 ( Ω 0 ) Γ 13 ( s ) − Γ 13 ( r ) Γ 23 ( r ) + ( Γ 23 ( r ) F 1 ( Ω 0 ) F 2 2 ( Ω 0 ) C 1313 ( s ) + Γ 23 ( s ) F 1 2 ( Ω 0 ) × F 2 ( Ω 0 ) Γ 13 ( r ) + Γ 23 ( r ) F 1 ( Ω 0 ) Γ 13 ( s ) + Γ 23 ( s ) F 2 ( Ω 0 ) Γ 13 ( r ) ) Ω 0 ,

(36) Φ 2 ( Ω 0 ) = − Γ 23 ( r ) + ( Γ 23 ( r ) F 1 ( Ω 0 ) F 2 2 ( Ω 0 ) + Γ 23 ( s ) F 1 2 ( Ω 0 ) F 2 ( Ω 0 ) + Γ 23 ( r ) F 1 ( Ω 0 ) Γ 13 ( s ) + Γ 23 ( s ) F 2 ( Ω 0 ) ) Ω 0 + Γ 23 ( r ) Γ 23 ( s ) F 1 ( Ω 0 ) F 2 ( Ω 0 ) .

Thus, we can now express Ω ¯ 2 in the following form:

(37) Ω ¯ ( n ) 2 = Ω 0 ( n ) + [ Φ 1 ( Ω 0 ) ∕ Φ 2 ( Ω 0 ) ] K 2 + O ( K 4 ) .

Here we equally make use of the MR material parameters given in Table 1 to graphically illustrate the obtained asymptotic results in comparison with the exact dispersion relations earlier obtained for the three-layered pre-stressed compressible laminate with free-faces conditions. To elucidate further, Figure 4(a) and (b) compares the harmonic curves of exact and asymptotic dispersion relations in the cases of the symmetric and anti-symmetric vibration modes, respectively, where Ω ¯ is shown as a function of K . In particular, the graphical depictions of the exact symmetric dispersion relation found in Eq. (16) and both the leading and second-order approximations of the asymptotic expansion given in Eq. (31) are shown to be in good agreement in Figure 4(a); in fact, both the three harmonics and the associated fundamental mode preserve high-level of exactness between the two solutions. More so, in Figure 4(b), the asymptotic approximation of the anti-symmetric dispersion expression reported in Eq. (37) has been illustrated to be in perfect conformity with the exact anti-symmetric dispersion relation previously determined in Eq. (20).

Figure 4

Comparison between the exact and asymptotic dispersion relations for the (a) symmetric (16) and (31), and (b) anti-symmetric (20) and (37) cases associated with the symmetric three-layered pre-stressed compressible elastic laminate with free-faces.

5.2 Fixed-faces case

In the same manner, we asymptotically consider here both the symmetric and anti-symmetric dispersion relations with fixed-faces boundary conditions, separately.

5.2.1 Symmetric case

The dispersion relation given in Eq. (23) can be expressed in the form as follows:

(38) Γ 23 ( r ) q r ˆ tan K q s ˆ + Γ 23 ( s ) q s ˆ tan K q r ˆ = 0 .

Thus, a similar analysis just carried out in respect of the free-faces case can be performed for the fixed-faces here. The leading order term from Eq. (38) takes the following form:

(39) Γ 23 ( s ) tan ( Ω 0 ( n ) ∕ Γ 23 ( s ) ) + Γ 23 ( r ) tan ( Ω 0 ( n ) ∕ Γ 23 ( r ) ) = 0 .

The next order term of Eq. (38) gives

(40) Ω 2 = Φ ˜ 2 ( Ω 0 ) ∕ Φ ˜ 1 ( Ω 0 ) ,

where

(41) Φ ˜ 1 ( Ω 0 ) = Ω 0 [ Γ 23 ( r ) F 1 2 ( Ω 0 ) Γ 13 ( s ) + Γ 23 ( s ) F 2 2 ( Ω 0 ) Γ 13 ( r ) + Γ 23 ( s ) Γ 13 ( r ) + Γ 23 ( r ) Γ 13 ( s ) ] − ( Γ 13 ( r ) F 2 ( Ω 0 ) + Γ 13 ( s ) F 1 ( Ω 0 ) ) Γ 23 ( r ) Γ 23 ( s ) ,

(42) Φ ˜ 2 ( Ω 0 ) = Ω 0 [ Γ 23 ( s ) Γ 23 ( s ) F 2 2 ( Ω 0 ) F 1 ( Ω 0 ) + Γ 23 ( s ) Γ 23 ( 2 ) ] − ( Γ 23 ( r ) Γ 23 ( s ) F 1 ( Ω 0 ) + Γ 23 ( s ) Γ 23 ( r ) F 2 ( Ω 0 ) ) .

Then, we express the scaled frequency in the following form:

(43) Ω ¯ ( n ) 2 = Ω 0 ( n ) + [ Φ ˜ 2 ( Ω 0 ) ∕ Φ ˜ 1 ( Ω 0 ) ] K 2 + O ( K 4 ) .

5.2.2 Anti-symmetric case

For the anti-symmetric case, we may consider the exact anti-symmetric dispersion relation (24) in the following form:

(44) Γ 23 ( r ) q r tan ( K q r ) tan ( K q s ) − Γ 23 ( s ) q s = 0 .

The leading order term can be expressed explicitly from the numerical solution of the following relation:

(45) Γ 23 ( s ) F 1 ( Ω 0 ( n ) ) F 2 ( Ω 0 ( n ) ) − Γ 23 ( r ) = 0 ,

and the second term can be introduced as follows:

(46) Ω 2 = ψ ˜ 2 ( Ω 0 ) ∕ ψ ˜ 1 ( Ω 0 ) ,

with

(47) ψ ˜ 1 ( Ω 0 ) = Γ 23 ( r ) Γ 23 ( s ) F 1 ( Ω 0 ( n ) ) F 2 ( Ω 0 ( n ) ) Γ 13 ( s ) − Γ 13 ( r ) Γ 23 ( r ) + Ω 0 ( Γ 23 ( s ) F 2 ( Ω 0 ) Γ 13 ( r ) + Γ 23 ( s ) F 1 2 ( Ω 0 ) F 2 ( Ω 0 ) Γ 13 ( r ) + Γ 23 ( r ) F 1 ( Ω 0 ) Γ 13 ( s ) + Γ 23 ( r ) F 1 ( Ω 0 ) F 2 2 ( Ω 0 ) Γ 13 ( s ) ) ,

(48) ψ ˜ 2 ( Ω 0 ) = − Γ 23 ( r ) + Ω 0 ( Γ 23 ( r ) F 1 ( Ω 0 ) F 2 2 ( Ω 0 ) + Γ 23 ( s ) F 1 2 ( Ω 0 ) F 2 ( Ω 0 ) + Γ 23 ( r ) F 1 ( Ω 0 ) Γ 13 ( s ) + Γ 23 ( s ) F 2 ( Ω 0 ) ) + Γ 23 ( r ) Γ 23 ( s ) F 1 ( Ω 0 ) F 2 ( Ω 0 ) .

Finally, the following expansion is introduced:

(49) Ω ¯ ( n ) 2 = [ ψ ˜ 2 ( Ω 0 ) ∕ ψ ˜ 1 ( Ω 0 ) ] K 2 + O ( K 4 ) .

Similarly, we graphically examine the asymptotic results compared to the exact dispersion relations for the laminate under consideration with fixed-faces conditions. Figure 5(a) and (b) show comparative illustrations by comparing the harmonic curves of exact and asymptotic dispersion relations for both the symmetric and anti-symmetric dispersion relations. Without further delay, as may be seen in Figure 5(a), a comparison between exact and asymptotic dispersion relations for the symmetric modes reveals a perfect match; while the exact dispersion curves alongside the asymptotic ones are shown for the first three harmonics for the anti-symmetric counterpart. In both cases, a complete agreement between the two results has been observed.

Figure 5

Comparison between the exact and asymptotic dispersion relations for the (a) symmetric (23) and (43), and (b) anti-symmetric (24) and (49) cases associated with the symmetric three-layered pre-stressed compressible elastic laminate with fixed-faces.

6 Conclusion

This study presents a symmetric three-layered pre-stressed compressible elastic laminate in the context of anti-plane shear motion. The MR hyperelastic strain energy density model is used to describe the dynamic response of the pre-stressed materials. The dispersion curves, involving both the fundamental mode and harmonics, for the symmetric cases associated with free-faces conditions, analysed numerically and asymptotically. The presence of a fundamental mode in a symmetric free-faces case, whereas no fundamental mode was observed in the remaining cases comprising of the anti-symmetric case with free-faces and both the symmetric and anti-symmetric scenarios with fixed-faces. It was concluded that a low-frequency mode only exists in respect of symmetric motion with free-faces. A comparison between the asymptotic and exact results is also presented and a good agreement is observed for the first three harmonics.

Overall, the problem described in the modelling and formulation sections is simplified and the results are highly recommended that the same methods could be extended to analyse several layered and composite media in the context of anti-plane shear problems. Moreover, the study of generalised multi-layered structures with external forces is of particular interest. The obtained results may be extended to similar anti-plane problems for asymmetrical multi-layered structures, as well as to more complicated plane and 3D problems.