In this section, we start from the periodic system, i.e., the normal propagation of the time-harmonic SH elastic waves in a nanoscale periodic multilayered phononic crystal is considered. This structure depicted in Figure 1A is composed of materials A and B layers, with the thicknesses l 1 and l 2 , respectively, and l = l 1 + l 2 is the thickness of one unit-cell. The local coordinates of the monolayers are also given in the figure. We assume that the layered composite consists of ℕ unit-cells. Each unit-cell includes, unless otherwise stated, two sub-layers which are denoted by the subscript k = 1,2 . For this problem, the displacement components in the x - and y -directions, i.e., u x and u y are zero, and the only non-zero displacement u k is along the z -direction, which is perpendicular to the x y -plane. Then Eq. 9 can be simplified into the following form for the k th sub-layer μ k ∂ 2 u k ∂ x k 2 = ρ ω 2 ( a 2 ∂ 2 ∂ x k 2 − 1 ) u k    ( k = 1,2 ) (11)

By introducing the dimensionless local coordinate ζ k = x k / l , 0 ≤ ζ k ≤ l ¯ k = l k / l , Eq. 11 can be rewritten into the following dimensionless form ∂ 2 u k ∂ ζ k 2 + ϖ k 2 1 − ϖ k 2 τ k 2 u k = 0 (12) where ϖ k = ω l / c k is the non-dimensional frequency with c k = μ k / ρ k being the velocity of the transverse elastic wave, and τ k = a k / l is the ratio of the internal characteristic length and the external thickness of the unit-cell.

Then, the general harmonic solution for the k th sub-layer can be obtained, which has the following form: u k ( ζ k ) = ( U k e i q k ζ k + V k e − i q k ζ k ) e − i ω t (13) where q k = ϖ k 2 / ( 1 − τ k 2 ϖ k 2 ) , U k and V k are the unknown coefficients to be determined. According to Eqs 13, 5, the nonlocal stresses can be obtained as ς k ( ζ k ) = ∫ 0 l ¯ k μ k 2 τ k e − | ζ k − ζ k ′ | / τ k ∂ u k ∂ ζ k ' d ζ k ′ = i q k μ k 2 ( e i q k ζ k − e − ζ k / τ k 1 + i q k τ k − e i q k l ¯ k e ( ζ k − l ¯ k ) / τ k − e i q k ζ k 1 − i q k τ k ) U k − i q k μ k 2 ( e − i q k ζ k − e − ζ k / τ k 1 − i q k τ k − e − i q k l ¯ k e ( ζ k − l ¯ k ) / τ k − e − i q k ζ k 1 + i q k τ k ) V k (14)

Then, the state vector W = { u ¯ , ς ¯ } T with the bar denoting the dimensionless parameters is chosen in order to obtain the transfer matrix of the unit-cell. Based on the continuity conditions on the left (subscript L ) and right (subscript R ) sides of the interface of two adjacent sub-layers in the β th unit cell, the state vectors can be defined as W k L ( β ) = { u ¯ k , ς ¯ k } ζ k = 0 T = T k L { U k , V k } T W k R ( β ) = { u ¯ k , ς ¯ k } ζ k = l ¯ k T = T k R { U k , V k } T (15) where the matrices T k L and T k R can be obtained from Eqs 13, 5, whose elements are given by T k L ( 1,1 ) = T k L ( 1,2 ) = 1 ,   T k L ( 2,1 ) = − i μ k q k 1 − e − i q k l k e − l k τ k 2 ( 1 + i q k τ k ) ,   T k L ( 2,2 ) = i μ k q k 1 − e i q k l k e − l k τ k 2 ( 1 − i q k τ k ) T k R ( 1,1 ) = e − i q k l k ,   T k R ( 1,2 ) = e i q k l k ,   T k R ( 2,1 ) = − i μ k q k e − i q k l k − e − l k τ k 2 ( 1 − i q k τ k ) ,   T k R ( 2,2 ) = i μ k q k e i q k l k − e − l k τ k 2 ( 1 + i q k τ k ) (16)

Obviously, the two state vectors in Eq. 15 have the following relation by eliminating the common vector, i.e., W k R ( β ) = T k R T k L − 1 W k L ( β ) ≜ T k W k L ( β )             ( k = 1,2 ) (17) where T k = T k R T k L − 1 is the transfer matrix of the k th sub-layer.

The displacements and nonlocal stresses are continuous at the interface of two adjacent sub-layers in the same unit-cell and between the β th and the ( β − 1 ) th unit-cells, that is, W 2 R ( β − 1 ) = W 1 L ( β ) ,   W 1 R ( β ) = W 2 L ( β ) (18)

From Eqs 17, 18, the following relation can be obtained W 2 R ( β ) = T 2 W 2 L ( β ) = T 2 W 1 R ( β ) = T 2 T 1 W 1 L ( β ) = T 2 T 1 W 2 R ( β − 1 ) ≜ T β W 2 R ( β − 1 ) (19) which shows the relationship between the state vectors of the β th and the ( β − 1 ) th unit-cells, where T β = T 2 T 1 = T 2 R T 2 L − 1 T 1 R T 1 L − 1 is the transfer matrix between the two consecutive unit-cells, i.e., the transfer matrix of the β th unit-cell. For perfect periodic two-component PCs, T β = T 2 T 1 for all β = 1,2,3 , ⋯ , ℕ are the same and denoted as T . It should be noticed that the above derivation is applicable for not only the ordered periodic PNCs but also the disordered, quasi-periodic and aperiodic ones. However the transfer matrices of the “unit-cells” of the disordered, quasi-periodic and APNCs are different from those of the perfect periodic ones. For quasi-periodic or aperiodic structures, T β = T 2 T 1 are not all the same and the Bloch theory is not applicable, for example, Using the above method, for the aperiodic structure composed of ℕ unit-cells, the total transfer matrix can be obtained, that is, T t o t a l = T ℕ R T ℕ L − 1 T ( ℕ − 1 ) R T ( ℕ − 1 ) L − 1 T ( ℕ − 2 ) R ⋯ T 3 L − 1 T 2 R T 2 L − 1 T 1 R T 1 L − 1 (20)

The detailed mathematical derivation is not given here for the sake of brevity. Interested readers may refer to many publications for more details.

In this paper, we use the well-defined localization factor to characterize the band structures and localization phenomenon of 1D nanoscale layered PCs. The localization factor is defined as the minimum positive Lyapunov exponent which describes the average exponential rate of growth or attenuation of the wave amplitude (Gastanier and Pierre, 1997). And it can be calculated by using the Wolf’s method (Wolf et al., 1985) once the transfer matrix is obtained. If the dimension of the transfer matrices is 2 ƛ × 2 ƛ , then the smallest positive Lyapunov exponent ℓ ƛ is the localization factor. The expression for the localization factor ℓ ƛ of the system with ℕ unit-cells is given as follows: ℓ ƛ = lim ℕ → ∞ 1 ℕ ∑ β = 1 ℕ ln ‖ W ^ 2 R , ƛ ( β ) ‖ (21) where the vector in Eq. 21 is given by W ^ 2 R , ƛ ( β ) = W 2 R , ƛ ( β ) − ( W 2 R , ƛ ( β ) , v ƛ − 1 ( β ) ) v ƛ − 1 ( β ) − ⋯ − ( W 2 R , ƛ ( β ) , v 1 ( β ) ) v 1 ( β ) (22) in which v ƛ ( β ) = W ^ 2 R , ƛ ( β ) ‖ W ^ 2 R , ƛ ( β ) ‖ are orthogonal unit vectors, ( ⋅ , ⋅ ) denotes the dot-product, ‖ ⋅ ‖ is the vector norm, and ℕ represents the number of the unit-cells. The ƛ th orthogonal unit state vector W ^ 2 R , ƛ ( β ) is obtained through the iteration using the Gram–Schmidt orthonormalization procedures (Kissel, 1991). If the localization factor is equal to zero, the corresponding frequency intervals are known as pass-bands. Otherwise if the localization factor is positive, the frequency intervals are known as stop-bands or band-gaps. In this paper, only the normal incidence of SH wave is considered, thus, the dimension of the transfer matrix is 2 × 2 and ℓ 1 is the localization factor, which is denoted as ℓ in the following analysis.

Firstly, In order to check the correctness of the present method, the 1D nanoscale periodic layered structures arranged alternately by HfO2 ( A ) and ZrO2 ( B ), as shown in Figure 1A, are studied. The same material constants as the Ref. (Yan et al., 2020) are selected, and the results are shown in Figure 2A. We find that the present results (black solid lines) are in good agreement with the results (red solid lines) of Figure 2C in Ref. (Yan et al., 2020), which verifies the correctness and effectiveness of the current method. In the following, In order to reveal the influence of random disorder of the sub-layers on the band structures by considering the internal characteristic lengths and the external thicknesses, the localization factors in the cases of τ 1 = 0 , l 1 = l 2 = 0.5 , τ 2 = 0.18 + δ V 2 ( − 2 ⁡ ln ( S ) / S ) (i.e., disorder is considered only for the internal characteristic length of the second sub-layer B ) and τ 1 = 0.18 + δ V 1 ( − 2 ⁡ ln ( S ) / S ) , τ 2 = 0.18 + δ V 2 ( − 2 ⁡ ln ( S ) / S ) , l 1 = 0.5 + δ V 1 ( − 2 ⁡ ln ( S ) / S ) , l 2 = 0.5 + δ V 2 ( − 2 ⁡ ln ( S ) / S ) (i.e., disorder is considered for both internal characteristic lengths and external thickness of sub-layers A and B ), are calculated and the results are illustrated in Figures 2B,C. The disorder degree denoted by δ is assumed to be δ = 0,0.02 and 0.04 , respectively. It can be seen that there is one peak point a in Figure 2B and two peak points b , c in Figure 2C whose positions are determined by the cut-off frequency of a specific material sub-layer. The second peak point c in Figure 2C is a cut-off frequency beyond which the localization factor becomes stable and positive, which means that the elastic waves cannot propagate through the structure over the cut-off frequency. In addition, the dense band zones (DBZs) as defined in Ref (Yan et al., 2020) appears, where the localization factors are very big with multiple, dense, flat and narrow band-gaps in the frequency range ( 7.28 , 7.35 ) in Figure 2B and ( 5.47 , 5.56 ) in Figure 2C, meaning a very strong wave localization phenomenon. The disorder degree has little effect on the DBZ, the cut-off frequency and the localization factor whose frequencies are larger than the peak points a and c . However, when the disorder caused by the sub-layer’s length or thickness is introduced to the periodic phononic crystals, the disorder degree still has some influences on the band structures in both pass-bands and band gaps, which are demonstrated by the enlarged sub-figure in Figures 2B,C. For example, in the pass-band ( 6.519 , 6.704 ) for δ = 0 shown in the partly enlarged regions in Figure 2B, the localization factors become positive with the increase of δ , indicating a wave localization phenomenon, and the localization degree increases with the increase of δ in the pass-bands while the localization degree decreases with the increase of δ in the band gaps. Next, the disorder of the internal characteristic length and the external thickness of the same sub-layer, the disorder of different sub-layers and the number of disorder are considered to show more detailed results in Figures 3A–C, respectively. The disorder degree is δ = 0.08 . It can be seen from Figure 3A that the influences of the disorder caused by the internal characteristic lengths are a little bigger than those caused by the external thicknesses. In addition, in Figures 3B,C the localization factors change a lot and have a strong dependence on the disorder of different sub-layers and the number of disorder. Moreover, with the increase of the number of disorder, the degree of wave localization in the pass-bands increases, and the localization phenomenon becomes more and more obvious. From Figure 3, we can conclude that the disorder degree and the localization factor depend on different types of disorders of each sub-layer.

Due to the lack of periodicity, a finite but sufficiently large number of unit-cell n is very important to calculate the localization factor. After trial calculation, n = 1024 is taken in the following computations. Next, the influences of the material combinations on the band structures are analyzed by considering changing the order of material components, only the single sub-layer material and two sub-layers are both changed, respectively. Specifically, the order of material components is selected as HfO 2 / ZrO 2 , ZrO 2 / HfO 2 and HfO 2 / Cu , Cu / HfO 2 . Only the change of the second sub-layer material is taken as HfO 2 / Cu , HfO 2 / ZrO 2 and HfO 2 / Al . The examples where the materials of two sub-layers are both changed are ZrO 2 / Cu , HfO 2 / ZrO 2 and Cu / Al . The band structures for the nanoscale Thue-Morse laminate with the change in order of material components are plotted in Figure 4. The ratio of internal characteristic length and the thickness of the unit-cell is τ 1 = τ 2 = 0.18 . It can be seen that with the exchange of material components, the distance between the two peak points becomes narrower, and the two peak points and the first distinct band gap move to the low frequency zone. Furthermore, Compared with the two DBZs (5.42, 5.56) and (6.53, 7.34) for HfO 2 / ZrO 2 , the first and the second DBZs move left to (4.1, 4.21) and (4.98, 5.56) for ZrO 2 / HfO 2 , respectively. While the two DBZs for Cu / HfO 2 move left to (4.61, 4.72) and (5.3, 5.56) compared with the corresponding DBZs (5.31, 5.56) and (5.9, 6.54) for HfO 2 / Cu . It is worth noted that compared with the first peak points for HfO 2 / ZrO 2 and HfO 2 / Cu , the second peak points, i.e., the cut-off frequencies, stay in the almost same position for ZrO 2 / HfO 2 and Cu / HfO 2 , that is to say, the cut-off frequencies for ZrO 2 / HfO 2 and Cu / HfO 2 are almost ϖ = 5.56 which are exactly the right edges of the corresponding DBZs, i.e., the cut-off frequency of the material HfO 2 . In addition, when the frequency is higher than the cut-off frequency, the value of the localization factor will become stable at approximately 3.23 (as illustrated by the dashed lines in Figure 4).

In the following, we only change the material of the second sub-layer and the localization factors for the nanoscale aperiodic Thue-Morse sequences with different material combinations are presented in Figure 5. Here, the material combinations is taken as HfO 2 / Cu , HfO 2 / ZrO 2 and HfO 2 / Al , respectively. It can be observed that the peak points, i.e., the DBZs with the frequency ranges of (5.3, 5.56), (6.48, 6.54), (7.03, 7.35) and (7, 7.32) correspond to materials HfO 2 , Cu , ZrO 2 and Al , respectively. The band structures between the two DBZs have almost the same distributions, however, when the frequency is lower than the first peak point, the localization factors change a lot. More specifically, the first distinct band gap becomes wider and moves to the low frequency zone. Compared with Figures 5B,C shows the two DBZs are located at almost the same position, because the transverse wave velocities of the two materials ZrO 2 and Al , have almost similar values. In addition, the examples where the materials of two sub-layers are both changed are chosen as ZrO 2 / Cu , HfO 2 / ZrO 2 and Cu / Al , respectively. It can be seen from Figure 6 that the position of the peak point shown by the dashed line corresponds to materials Cu and HfO 2 because the shear wave velocities of the two materials have the similar values. The larger the difference in transverse wave velocity, the farther the two DBZs are from each other.

Next, the band structures of the systems with different aperiodic arrangements are calculated. As shown in Figures 1B–D, three different aperiodic arrangements are chosen, i.e., the Thue-Morse sequence, the Rudin-Shaprio sequence and the Fibonacci sequence, respectively. Figure 7 shows the influences of aperiodic arrangements on localization factors of the SH wave propagating normally in the nanoscale aperiodic systems consisting of HfO 2 / ZrO 2 for τ 1 = τ 2 = 0.18 . It can be seen that the tendencies of all curves are coincident for the three aperiodic systems, which implies the aperiodic arrangements have little effect on the peak points, the DBZs and the cut-off frequency. For example, the localization factors oscillate quickly with big values between the two peak points. The cut-off frequencies are all around about ϖ = 7.34 . However, the band structures before the first peak point becomes different for the three aperiodic arrangements. i.e., the main band gaps of the Fibonacci structure change a lot with the gradual disappearing and narrowing of the band gaps. Specifically, in the dot rectangle, compared with the Fibonacci structure, the localization factor is more like a defect state for the Thue-Morse and Rudin-Shapiro structures.

Additionally, the influences of the ratio of the mass density ρ 1 / ρ 2 and the ratio of the transverse wave velocity c 1 / c 2 on the localization factors are examined. Here, the nanoscale Cu / Al Fibonacci aperiodic laminate is selected as an example. It can be observed from Figure 8 that when the ratio of the mass density ρ 1 / ρ 2 is not equal to 1 , the first distinct band gap emerges, while for ρ 1 / ρ 2 = 1 , it disappears (enlarged regions shown in Figure 8). The positions of the two DBZs, the two peak points, the cut-off frequency and the localization factors whose frequencies are larger than those of the first peak point have no changes. For example, after the first peak point, the tendencies of all curves are coincident, i.e., the localization factors oscillate quickly with big values after the first peak point, and then the second peak point appears. More detailed results are illustrated in Figure 9 for the localization factors varying with the normalized frequency and ρ 1 / ρ 2 . Compared with the results in Figure 8, the first distinct band gap disappears for ρ 1 / ρ 2 = 1 , when ρ 1 / ρ 2 is less than 1 or ρ 1 / ρ 2 is bigger than 1, the first distinct band gap appears and becomes wider with ρ 1 : ρ 2 decreasing or increasing. Combining Figures 8, 9, we can conclude that the first distinct band gap will emerge for ρ 1 / ρ 2 ≠ 1 . and the ratio of the mass density has no influences on the peak points, the cut-off frequency, the DBZs and the localization factors whose frequencies are larger than those of the first peak point. From Figure 10, we can see that there is only one peak point when the ratio of transverse wave velocity is c 1 / c 2 = 1 , i.e., the cut-off frequency, and when the velocity ratio changes, the position of the peak point remains unchanged. However, when the ratio is not equal to 1 , there are two peak points. Among them, when the ratio of velocity is less than 1 , one peak point tends to the high-frequency region, and when the ratio is larger than 1 , a peak point tends to the low-frequency region. In addition, the low-frequency band gap is more likely to emerge with the increase of the ratio. This shows that the ratio of transverse wave velocity has significant effect on the cut-off frequency, the DBZ, the peak point and the low-frequency band gap. Furthermore, the localization factors varying with normalized frequency and c 1 / c 2 are presented in Figure 11. There is only one peak point when the ratio of transverse wave velocity is c 1 / c 2 = 1 , i.e., the cut-off frequency of material Al, and when the velocity ratio changes, the position of the peak point remains unchanged since c 2 is a fixed value. On the contrary, the peak point other than the cut-off frequency moves the high-frequency region when c 1 / c 2 is less than 1, and moves the low-frequency zone when c 1 / c 2 is larger than 1. Combining Figures 10, 11, we can conclude that the position and the number of the peak point, i.e., the DBZ have strong dependence on c 1 / c 2 . There exists a wide pass-band when c 1 / c 2 is less than 1, and the pass-band becomes wider with the decrease of c 1 / c 2 .

Finally, the influences of the structural parameter, i.e., the ratio of external characteristic thickness l 1 / l 2 and the ratio of internal characteristic length τ 1 / τ 2 on the localization factors are investigated. The localization factors for the nanoscale Fibonacci laminates with different values of l 1 / l 2 are presented in Figure 12, From Figure 12, we can see that the pass-bands and band-gaps are affected by varying the ratio of external characteristic thickness, while the two peak points, the two DBZs and the cut-off frequency remain in the same position. Detailed results are shown in Figure 13, the position of the cut-off frequency remains unchanged, but the localization degree enhances with the increase of l 1 / l 2 . From Figure 14, we can see that the bigger the difference of internal characteristic length between the two materials, the farther the distance between the two peak points. The first distinct band gap disappears with the increase of τ 1 / τ 2 . More detailed numerical results are shown in Figure 15, It can be seen that the cut-off frequency does not appear when τ 1 / τ 2 ≤ 0.2 , However, when 0.3 ≤ τ 1 / τ 2 ≤ 1 , the cut-off frequency appears and decreases with the τ 1 / τ 2 increasing. When τ 1 / τ 2 > 1 , the cut-off frequency remains unchanged and the first peak point tends to the low-frequency zone with the τ 1 / τ 2 increasing. In addition, the localization degree of the cut-off frequency, the pass-bands and the band gaps varies with the τ 1 / τ 2 . And the localization degree whose frequencies are larger than the cut-off frequency decreases with τ 1 / τ 2 increasing.