Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Coupling resonance mechanism of interfacial stratification of sandwich plate structures excited by SH waves:

Coupling resonance mechanism of interfacial stratification of sandwich plate structures excited... For interfacial stratification mechanism of sandwich plate structures, the forced propagation solution of interface shear stress excited by SH waves is derived by global matrix methods and integral transformation methods. The necessary condition of interfacial shear delamination excited by alternating stress is analyzed with the interface fatigue failure theory. The exact value of forced propagation solution is calculated by adaptive Gauss–Kronrod quadrature numerical integration methods, which is verified via finite element methods. The coupling resonance mechanism of interface shear stratification is revealed by the forced vibration solution and the mass-spring model. The effects of excitation frequency, structural parameters, accretion, and matrix materials on the interfacial shear delamination are analyzed and discussed by practical cases of vibration de-accretion. For interface shear stratification of sandwich structures, the optimal exci- tation frequency as well as substrate thickness and accretion thickness is the value at coupling resonance, around which the interfacial shear stratification interval is formed by the interface fatigue failure criteria. In the stratification or/and antistratification design excited by vibrations, the excitation source and structure could be optimized by the method. Therefore, the results have important theoretical value for the extension and application of vibration stratification or/ and antistratification technology. Keywords Sandwich plate, SH waves, interfacial stratification, coupling resonance mechanism, interface shear stress, forced prop- agation solution Introduction Sandwich plates are often encountered in practice, but the interface stratification has long puzzled researchers in 1–9 vibration de-accretion, delamination dismantling, and even anti-layering design. Therefore, it is of great the- oretical value to study the interface stratification of sandwich structures loaded by stress waves. The problem of interfacial stratification of layered medium excited by elastic waves has been emphasized in 10,11 vibration deicing/defrosting. It was observed in experiments that ice layers accreted on the metal plate are 12,13 stratified instantaneously when the ultrasonic transducer is turned on. The finite element analysis confirms 14,15 that the excited interface shear stress is greater than the adhesive shear strength. A lot of research has been 16–26 done, and some of it has even been applied in related fields. However, due to the lack of theoretical analysis methods, the interface stratification mechanism is still unclear. The dynamic analysis of sandwich structures has always been one of the research hotspots in solid mechanics, 27–30 especially the derivation and solution of wave equations. Wave equations in sandwich plates are established School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, People’s Republic of China State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, People’s Republic of China Corresponding author: Jiu H Wu, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi 710049, People’s Republic of China. Email: ejhwu@mail.xjtu.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1167 0(0) by differential theory, variational theory, higher-order shear deformation theory, etc., which are usually partial 31–39 differential equations. Using variational iteration methods, homotopy perturbation methods, integral trans- formation methods, Green’s function methods, separation of variables methods, and so on, wave equations are 40–53 solved. The analytical model of wave motion in laminates is established with the boundary conditions by 54–59 stiffness-matrix methods, transfer-matrix methods, and global-matrix methods. By means of classical differential theory, the equilibrium equation is obtained by force analysis of micro elements in solid. On the basis of the equilibrium equation, the constitutive equation, and the geometric equation, the Navier’s wave equation is established. Using the variational theory based on minimum potential energy principle and/or minimum residual energy principle, the wave equation is established through solving the func- 61–65 tional standing value problem, which does not need strong local differentiability of each variable. The vibra- tion governing equation in sandwich plates is derived by higher-order shear deformable theory in which the shear correction factor is not required, the free vibration behavior influenced by structural parameters is investigated, 66–72 and it is found that the model has high precision. The free propagation solution of SH waves in plates with unknown coefficients was obtained via separation of 73,74 variable methods. Using variational iteration methods, the solution of wave equation first is approximated with the possible unknowns, and then a correction functional is constructed by a general Lagrange multiplier, 75,76 which can be identified optimally through the variational theory. Variational iteration methods can be used to solve homogenous and inhomogeneous partial differential equations in bounded and unbounded domains flex- ibly. By homotopy perturbation methods, a homotopy with an imbedding parameter p 2 [0,1] is constructed by 77–79 the homotopy technique, and the imbedding parameter is considered as a “small parameter.” Therefore, homotopy perturbation methods take full advantage of traditional perturbation methods and homotopy techni- ques, which can be used to solve a system of coupled linear as well as nonlinear differential equations. By integral transformation methods and Green’s function methods, the forced vibration solution of Love waves in the clad- 80–82 ding structure excited by point sources or line sources is derived which contains complex definite integrals. Complex integrals are solved numerically by Trapezoidal methods, Simpson methods, and Newton–Cotes methods, where the adaptive Gauss–Kronrod quadrature method in Simpson methods is particularly effective 83–85 for the integral of oscillation functions. For sandwich structures, a lot of research has been done. However, up to now, there is a lack of the forced propagation solution of SH waves partly because of the complexity of theoretical analysis and partly because of the difficulty of numerical integrations. Therefore, the interface stratification mechanism of sandwich structures has not been deeply analyzed and understood. In the paper, the forced vibration solution of sandwich structures excited by SH waves is derived. The exact solution is obtained by numerical integration methods. The necessary condition of interfacial stratification excited by dynamic shear stress is discussed. On the basis of the analytical solution, the coupling resonance mechanism of interface shear stratification is proposed. Finally, the influence of structural parameters, accretion, and matrix materials on the interfacial shear stress and interfacial stratification intervals is analyzed in accretion–substrate– accretion sandwich structures. Forced propagation solutions and necessary conditions of interfacial shear delamination of sandwich structures excited by SH waves Wave equations of sandwich structures excited by SH waves are established which are solved by integral trans- formation methods and global matrix methods. Further, forced propagation solutions of interface shear stress are obtained. The critical condition for interfacial shear delamination excited by the symmetrical alternating shear stress is obtained. Forced propagation solutions of interfacial shear stress of sandwich structures excited by SH waves For an isotropic solid plate, using u for the displacement of the particle, if the first Lame constant, the second Lame constant, density, and thickness are represented, respectively, by k, l, q, and h, the tensor notation of 31,60 classical Navier’s equation of elastic waves under the external force F is as follows @ u ðk þ lÞu þ lu þ F ¼ q ði; j ¼ 1; 2; 3Þ (1) j;ij i;jj @t 1168 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)3 Here, the lower right corner represents the axes. The wave travels in the x direction. The thickness direction of the plate is x . According to the plane strain assumption, only u is non-zero for SH waves, that is, u ¼ u ¼ 0. Furthermore, 3 1 2 u is independent of x 3 3 �ixt u ðx ; x ; tÞ¼ U ðx ; x Þe (2) 3 1 2 3 1 2 Substituting displacement components into equation (1), then it reduces to @ u lr u þ F ¼ q (3) @t where 2 2 @ @ r ¼ þ (4) 2 2 @x @x 1 2 The sandwich plate consists of three layers of medium as shown in Figure 1. The upper right corner of parameters represents the sequence number of layers. In engineering practice, such as ultrasonic defrosting, the middle substrate plate is often loaded with an excitation source F as shown in Figure 1. SH waves in sandwich structures excited by plane wave sources propagate along the direction of x . Moreover, the interfaces are all rigidly connected; in other words, the displacements and stresses are continuous at the interfaces, and the surfaces are free of tractions. For sandwich structures, as shown in Figure 1, the particle displacement of SH waves in each layer medium satisfies the following wave equation, respectively ðnÞ @ u ðnÞ ðnÞ 2 ðnÞ 3 l r u ¼ q ; n ¼ 1; 3 (5) @t ð2Þ @ u ð2Þ ð2Þ 2 ð2Þ 3 l r u þ F ¼ q (6) @t Here ðnÞ ðnÞ �ixt u ¼ U e ; n ¼ 1; 2; 3 (7) 3 3 �ixt F ¼ f d x e (8) 0 ðÞ 1 1; x ¼ 0 d x ¼ (9) ðÞ 1 0; x 6¼ 0 x ¼ 2pf (10) where f is the excitation amplitude and f is the excitation frequency. Figure 1. Sandwich structures. 4 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1169 0(0) Using the following stress–displacement relation @u r ¼ l (11) @x the boundary conditions are expressed as follows ð1Þ ð1Þ r ðx ¼�h Þ¼ 0 (12) ð3Þ ð2Þ ð3Þ r ðx ¼ h þ h Þ¼ 0 (13) ð1Þ ð2Þ r ðx ¼ 0Þ¼ r ðx ¼ 0Þ (14) 2 2 23 23 ð1Þ ð2Þ u ðx ¼ 0Þ¼ u ðx ¼ 0Þ (15) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ r ðx ¼ h Þ¼ r ðx ¼ h Þ (16) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ u ðx ¼ h Þ¼ u ðx ¼ h Þ (17) 2 2 23 23 Thus, the problem is reduced to solving a set of second-order nonhomogeneous partial differential equations with boundary conditions, which could be solved by many ways, such as variational iteration methods, homotopy perturbation methods, integral transformation methods, and so on. The integral transformation is a classical, convenient, and intuitive method for solving partial differential equations, in which partial differential equations are eliminated into ordinary differential equations by the integral transformation technique. In the paper, we use integral transformation methods to solve the problem. We assume that U ðs; x Þ is Fourier transform of U ðx ; x Þ, and the following integral transform formulas are 3 2 3 1 2 used in the paper. �isx U ðs; x Þ¼ U ðx ; x Þe dx (18) 3 2 3 1 2 1 2p �1 isx ^ 1 U ðx ; x Þ¼ U ðs; x Þe ds (19) 3 1 2 3 2 �1 �ixt For the sake of analysis, the time harmonic factor e is dropped from both sides of wave equations (5)–(6). Then, we take the Fourier transform on x to decouple wave equations, thus reading equations (20) and (21) ðnÞ d U ðnÞ 3 ðnÞ2 þ q U ¼ 0; n ¼ 1; 3 (20) dx ð2Þ d U ð2Þ f 3 ð2Þ2 þ q U ¼� (21) ð2Þ dx l where ðnÞ ðnÞ2 2 2 q ¼ k � s ; n ¼ 1; 2; 3 (22) ðnÞ k ¼ ; n ¼ 1; 2; 3 (23) ðnÞ sffiffiffiffiffiffiffiffi ðnÞ ðnÞ c ¼ ; n ¼ 1; 2; 3 (24) ðnÞ q 1170 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)5 Here, s is the integration factor, which satisfies the following condition according to equation (22). ðnÞ ðnÞ �k < s < k ; n ¼ 1; 2; 3 (25) T T And likewise, we take the Fourier transform of boundary conditions (12) to (17) on x ð1Þ ð1Þ r ^ ðx ¼�h Þ¼ 0 (26) ð3Þ ð2Þ ð3Þ r ^ ðx ¼ h þ h Þ¼ 0 (27) ð1Þ ð2Þ r ^ ðx ¼ 0Þ¼ r ^ ðx ¼ 0Þ (28) 2 2 23 23 ð1Þ ð2Þ u^ ðx ¼ 0Þ¼ u^ ðx ¼ 0Þ (29) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ r ^ ðx ¼ h Þ¼ r ^ ðx ¼ h Þ (30) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ u^ ðx ¼ h Þ¼ u^ ðx ¼ h Þ (31) 2 2 23 23 In addition, we also take the Fourier transform of equation (11) on x @u^ r ^ ¼ l (32) @x Thus, the problem is transformed into a solvable system of ordinary differential equations with boundary conditions. Therefore, we could obtain the single matrix as controls the whole system by global matrix methods. The solutions of equations (20) and (21) are assumed to be ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ U ¼ A cosðq x Þþ B sinðq x Þ; n ¼ 1; 3 (33) 2 2 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ U ¼ A cosðq x Þþ B sinðq x Þþ C (34) 2 2 (n) (2) (n) (2) Here, A , A , B , and B are the undetermined coefficients �f ð2Þ C ¼ (35) ð2Þ ð2Þ 2 2 l ðk � s Þ After equations (33) and (34) are substituted into equations (26)–(31) by equation (32), we are able to obtain (1) (2) (3) (1) (2) (3) the following system of inhomogeneous equations with respect to unknowns A , A , A , B , B , and B , which is written in the matrix form as follows 2 32 3 2 3 ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ sinðq h Þ 0 0 cosðq h Þ 00 A 0 6 76 7 6 7 6 ð3Þ ð2Þ ð3Þ ð3Þ ð2Þ ð3Þ 76 ð2Þ 7 6 7 00 �sin½q ðh þ h Þ� 0 0 cos½q ðh þ h Þ� A 0 6 6 6 7 7 7 6 76 7 6 7 6 ð1Þ ð1Þ ð2Þ ð2Þ 76 ð3Þ 7 6 7 00 0 l q �l q 0 A 0 6 76 7 6 7 6 76 7 ¼ 6 7 ð2Þ ð1Þ 6 76 7 6 7 1 �1 0 0 0 0 B �C 6 76 7 6 7 6 76 7 6 7 ð2Þ ð2Þ ð2Þ ð2Þ ð3Þ ð3Þ ð3Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð3Þ ð3Þ ð3Þ ð2Þ ð2Þ 6 76 7 6 7 0 �l q sinðq h Þ l q sinðq h Þ 0 l q cosðq h Þ�l q cosðq h Þ B 4 54 5 4 5 ð2Þ ð2Þ ð2Þ ð3Þ ð2Þ ð2Þ ð2Þ ð3Þ ð2Þ ð3Þ 0 cosðq h Þ�cosðq h Þ 0 sinðq h Þ�sinðq h Þ B (36) 6 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1171 0(0) Thus, the whole system is controlled by a single matrix (i.e. the global matrix), which consists of six equations with six unknowns. Solving equation (36), we get ð2Þ ð1Þ A ¼ (37) ð3Þ D Z ð1Þ ð2Þ ð1Þ B ¼� (38) D Z ð2Þ ð3Þ Here ð2Þ Z ¼ðZ M þ M � Z ÞC (39) ð2Þ ð1Þ ð2Þ ð1Þ ð1Þ D ðM � Z M Þ ð1Þ ð2Þ ð1Þ ð1Þ Z ¼ �ðZ M þ M Þ� (40) ð3Þ ð1Þ ð2Þ ð1Þ D X ð2Þ Table 1 Material parameters. Young’s Poisson’s Density Shear Shear wave Material modulus (GPa) ratio (kg/m ) modulus (GPa) velocity (m/s) Frost 7.759 0.343 800 2.661 1900 Rime ice 8.759 0.337 900 3.276 1907 Glaze ice 9.759 0.331 950 3.666 1964 Magnesium 43.4 0.297 1740 16.7 3100 Aluminum 69.4 0.337 2700 23.4 3140 Titanium 102 0.35 4600 37.778 2866 Steel 205.0 0.28 7850 80 3194 Molybdenum 297.3 0.297 10200 114.3 3350 Figure 2. Analytical and simulation results of normalized interface shear stress amplitude and interfacial shear delamination interval in excitation frequency domain as the accretion frost layer thickness is 0.1 mm, the substrate aluminum plate thickness is 1 mm, and ð1Þ ½r ~ � ¼ 10. �1 1172 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)7 E N YE ð1Þ ð2Þ ð2Þ Z ¼ � N (41) ð1Þ ð1Þ E E N þ E N ð2Þ ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð1Þ D ¼ sinðq h Þ (42) ð1Þ ð1Þ ð1Þ D ¼ cosðq h Þ (43) ð2Þ Figure 3. Analytical and simulation results of normalized interface shear stress amplitude and interfacial shear delamination interval in accretion frost layer thickness domain as the excitation frequency is 20 kHz, and the substrate aluminum plate thickness is 1 mm and ð1Þ ½r ~ � ¼ 10. �1 Figure 4. Analytical and simulation results of normalized interface shear stress amplitude and interfacial shear delamination interval in substrate aluminum plate thickness domain as the excitation frequency is 20 kHz, and the accretion frost layer thickness is 0.25 mm ð1Þ and ½r ~ � ¼ 10. �1 8 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1173 0(0) ð3Þ ð2Þ ð3Þ E ¼ sin½q ðh þ h Þ� (44) ð1Þ ð3Þ ð2Þ ð3Þ E ¼ cos½q ðh þ h Þ� (45) ð2Þ ð2Þ ð2Þ M ¼ sinðq h Þ (46) ð1Þ ð2Þ ð2Þ M ¼ cosðq h Þ (47) ð2Þ ð3Þ ð2Þ N ¼ sinðq h Þ (48) ð1Þ ð3Þ ð2Þ N ¼ cosðq h Þ (49) ð2Þ ð2Þ ð2Þ l q X ¼ (50) ð1Þ ð1Þ l q ð3Þ ð3Þ l q Y ¼ (51) ð2Þ ð2Þ l q (2) (3) (2) (3) The other coefficients A , A , B , and B are also determined in turn similarly by the global matrix (36). Figure 5. Normalized interface shear stress amplitude in excitation frequency and accretion frost layer thickness domains as the substrate aluminum plate thickness is 1 (a), 1.5 (b), and 2 mm (c). 1174 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)9 Finally, taking the inverse Fourier transform of equations (33) and (34) to x , we obtain the closed solution of displacement component ðjÞ ðnÞ ðnÞ �ixt isx u ðx ; x Þ¼ e U ðs; x Þe ds; n ¼ 1; 2; 3 (52) 1 2 2 3 3 ðjÞ �k ðjÞ ð1Þ ð2Þ ð3Þ k ¼ minðk ; k ; k Þ (53) T T T T Here, the upper and lower limits of integral (52) must satisfy the condition (25). Further, the closed solution of horizontal shear stress is also obtained by equations (52) and (11) ðjÞ ðnÞ ðnÞ �ixt ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ isx r ¼ l e q ½�A sinðq x Þþ B cosðq x Þ�e ds (54) 2 2 ðjÞ �k At last, forced propagation solutions of interface shear stress of sandwich structures could be written as follows ðjÞ ð1Þ ð1Þ �ixt ð1Þ ð1Þ isx r ðx ¼ 0Þ¼ l e B q e ds (55) ðjÞ �k ð1Þ Figure 6. Interfacial shear delamination intervals in excitation frequency and accretion frost layer thickness domains as ½r ~ � ¼ 10, �1 and the substrate aluminum plate thickness is 1 (a), 1.5 (b) and 2 mm (c). 10 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1175 0(0) ðjÞ ð2Þ ð2Þ ð2Þ �ixt ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ isx r ðx ¼ h Þ¼ l e q ½B cosðq h Þ� A sinðq h Þ�e ds (56) ðjÞ �k Obviously, the interfacial shear stress (55) and (56) are symmetric alternating stress of time harmonic, which of the amplitude depends on the amplitude of excitation source. It is worth noting that the upper and lower limits of integral (55) and (56) contain the integrating factors s, which could be calculated by numerical integration methods. In addition, the method could be extended to laminates with any number of layers. Necessary conditions of interface stratification excited by the symmetrical alternating shear stress According to the maximum stress principle of interface fatigue failure, when the maximum value of alternating stress acting on the interface is greater than the allowable fatigue stress, the interfacial debonding occurs. Obviously, the peak of a symmetric alternating stress is equal to its amplitude. Therefore, the necessary conditions for interface shear stratification are expressed as follows � � � � ð1Þ ð1Þ �r ðx ¼ 0Þ� > ½r � (57) 2 �1 � � � � ð2Þ ð2Þ ð2Þ �r ðx ¼ h Þ� > ½r � (58) 2 �1 Figure 7. Normalized interface shear stress amplitude in excitation frequency and substrate aluminum plate thickness domains as the accretion frost layer thickness is 0.1 (a), 0.2 (b), and 0.3 mm (c). 1176 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 11 (1) (2) Here, [r ] and [r ] are the allowable shear fatigue stress of lower interface of the first and second layers, –1 –1 separately, which are both constants that relate to the interface material and environmental conditions. For equations (55) and (56), f is only the coefficient of interface shear stress. Therefore, dividing both sides of equations (57) and (58) by the excitation amplitude f , we obtain ð1Þ  ð1Þ r ~ ðx ¼ 0Þ > ½r ~ � (59) 2 �1 ð2Þ ð2Þ ð2Þ r ~ ðx ¼ h Þ > ½r ~ � (60) 2 �1 Here ð1Þ r ðx ¼ 0Þ ð1Þ 2 r ~ ðx ¼ 0Þ ¼   (61) ð2Þ ð2Þ r ðx ¼ h Þ ð2Þ ð2Þ 23 r ~ ðx ¼ h Þ ¼   (62) Figure 8. Interfacial shear delamination intervals in excitation frequency and substrate aluminum plate thickness domains as ð1Þ ½r ~ � ¼ 10, and the accretion frost layer thickness is 0.1 (a), 0.2, (b) and 0.3 mm (c). �1 12 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1177 0(0) ð1Þ ½r � �1 ð1Þ ½r ~ � ¼ (63) �1 ð2Þ ½r � ð2Þ �1 ½r ~ � ¼ (64) �1 Equations (59) and (60) are the necessary condition for shear stratification of lower interface of the first and second layers, separately, which of the left is the normalized interface shear stress amplitude, and the right is the normalized allowable shear fatigue stress. In the application, sandwich plates with the same accretion layer on both surfaces are often encountered such as metal plates exposed to cold air that frost and/or ice evenly on both surfaces. Because of the symmetry of structure and excitation source, the necessary condition of shear delamination for the upper and lower interface is the same. Therefore, we take the symmetrical sandwich structure as an example to explore the law of interface shear stratification in the following sections. Coupling resonance mechanism of interfacial shear stratification and finite element verification In this section, analytic results of forced propagation solution of normalized interface shear stress amplitude are calculated in the excitation frequency, accretion layer thickness, and matrix thickness domains by adaptive Figure 9. Normalized interface shear stress amplitude in accretion frost layer thickness and substrate aluminum plate thickness domains as the excitation frequency is 20 (a), 22.5 (b), and 25 kHz (c). 1178 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 13 Gauss–Kronrod quadrature numerical integration methods (see Figures 2 to 4), which are verified by the finite element simulation method. The coupling resonance effect of interface shear stress amplitude is revealed. Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is studied by equation (59). Material parameters are �1 shown in Table 1. As shown in Figure 2, the normalized interface shear stress amplitude first increases with the increase of excitation frequency, then resonates as the excitation frequency reaches 32 kHz, and finally becomes unstable in high frequency region. Obviously, the coupled resonance frequency is 32 kHz. Because of the coupling reso- nance between interface shear stress amplitude and excitation frequency, the interfacial shear delamination inter- val of 19–35 kHz is formed around the coupling resonance frequency of 32 kHz as shown in the gray region in Figure 2. As shown in Figure 3, the normalized interface shear stress amplitude first increases with the increase of accretion frost layer thickness, then resonates as the accretion frost layer thickness reaches 0.355 mm, and finally becomes unstable in large frost layer thickness region. Obviously, the frost layer thickness of coupled resonance is 0.355 mm. Because of the coupling resonance between interface shear stress amplitude and frost layer thickness, the interfacial shear delamination interval of 0.08–0.38 mm is formed around 0.355 mm as shown in the gray region in Figure 3. As shown in Figure 4, the normalized interface shear stress amplitude first increases with the increase of substrate aluminum plate thickness, then resonates as the substrate aluminum plate thickness reaches 2.75 mm, and finally becomes unstable in large substrate aluminum plate thickness region. Obviously, the substrate alu- minum plate thickness of coupled resonance is 2.75 mm. Because of the coupling resonance between interface Figure 10. Interfacial shear delamination intervals in accretion frost layer thickness and substrate aluminum plate thickness domains ð1Þ as ½r ~ � ¼ 10 and the excitation frequency is 20 (a), 22.5 (b), and 25 kHz (c). �1 14 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1179 0(0) shear stress amplitude and substrate aluminum plate thickness, the interfacial shear delamination interval of 0.25– 2.8 mm is formed around 2.75 mm as shown in the gray region in Figure 4. As shown in Figures 2 to 4, the analysis results are in good agreement with the simulation results. The influence of excitation frequency, accretion frost layer thickness, and substrate aluminum plate thickness on the normalized interface shear stress amplitude is similar. In other words, the normalized interface shear stress amplitude has a coupling resonance effect with each of excitation frequency, accretion frost layer thickness, and substrate alumi- num plate thickness. The interfacial shear delamination interval is formed around the location of coupling res- onance peak. The coupling of structure intrinsic property and excitation frequency results into the resonance of interface shear stress amplitude, which is determined by structural parameters and material parameters. The resonance frequency of interface shear stress amplitude, that is, the natural frequency of interfacial shear delamination of sandwich structures, is determined by structure intrinsic property. Therefore, when the excitation frequency is close to or equal to the natural frequency, the excited interface shear stress amplitude resonates, and the interface shear delamination interval is formed. Obviously, the frequency as well as accretion frost layer thickness and substrate aluminum plate thickness at coupling resonance is the optimal for interface shear stratification. Therefore, the method could be used to optimize the excitation source and structure design of interface shear stratification. Figure 11. Normalized interface shear stress amplitude in excitation frequency and accretion thickness domains as magnesium substrate plate thickness is 1 mm, and the accretion is frost (a), rime ice (b), and glaze ice (c). 1180 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 15 The influence of structural parameters on normalized interface shear stress amplitude and interface shear stratification interval The influence of substrate thickness, accretion layer thickness, and excitation frequency on the normalized interface shear stress amplitude and interfacial shear delamination intervals is analyzed and discussed in the section. The influence of substrate thickness in excitation frequency and accretion layer thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and accretion frost layer thickness domains when the substrate aluminum plate thickness is 1, 1.5, and 2 mm, separately, see Figure 5. ð1Þ Taking ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 6). �1 As shown in Figure 5, the distributions of normalized interface shear stress amplitude in excitation frequency and frost layer thickness domains are like the shape of a volcano. But the location of crater from the origin is reduced with the increase of substrate thickness. Therefore, the excitation frequency as well as frost layer thickness at coupling resonance decreases with the increase of substrate thickness. As shown in Figure 6, the distributions of interfacial shear delamination intervals in excitation frequency and frost layer thickness domains are like floating clouds. But the location of distribution from the origin is reduced with the increase of substrate thickness. In addition, the excitation frequency decreases with the increase of frost layer thickness in distributions. Figure 12. Interfacial shear delamination intervals in excitation frequency and accretion thickness domains as magnesium substrate ð1Þ plate thickness is 1 mm, ½r ~ � ¼ 10, and the accretion is frost (a), rime ice (b), and glaze ice (c). �1 16 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1181 0(0) The influence of accretion layer thickness in excitation frequency and substrate thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and substrate aluminum plate thickness domains when the accretion frost layer thickness is 0.1, 0.2, and 0.3 mm, separately (see Figure 7). ð1Þ Taking ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 8). �1 As shown in Figure 7, the distributions of normalized interface shear stress amplitude in excitation frequency and aluminum plate thickness domains are like the shape of a volcano. But the location of crater from the origin is reduced with the increase of frost layer thickness. Therefore, the excitation frequency as well as aluminum plate thickness at coupling resonance decreases with the increase of frost layer thickness. As shown in Figure 8, the distributions of interfacial shear delamination intervals in excitation frequency and aluminum plate thickness domains are like floating clouds. But the location of distribution from the origin is reduced with the increase of frost layer thickness. In addition, the excitation frequency decreases with the increase of aluminum plate thickness in distributions. The influence of excitation frequency in accretion layer thickness and substrate thickness domains The normalized interface shear stress amplitude is calculated in accretion frost layer thickness and substrate aluminum plate thickness domains when the excitation frequency is 20, 22.5, and 25 kHz, separately ð1Þ (see Figure 9). Taking ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) �1 (see Figure 10). Figure 13. Normalized interface shear stress amplitude in excitation frequency and substrate thickness domains as accretion layer thickness is 0.3 mm, and the accretion is frost (a), rime ice (b), and glaze ice (c). 1182 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 17 As shown in Figure 9, the distributions of normalized interface shear stress amplitude in frost layer thickness and aluminum plate thickness domains are like the shape of a volcano. But the location of crater from the origin is reduced with the increase of excitation frequency. Therefore, the frost layer thickness as well as aluminum plate thickness at coupling resonance decreases with the increase of excitation frequency. As shown in Figure 10, the distributions of interfacial shear delamination intervals in frost layer thickness and aluminum plate thickness domains are like fish scales. But the location of distribution from the origin is reduced with the increase of excitation frequency. In addition, the frost layer thickness decreases with the increase of aluminum plate thickness in distributions. As shown in Figures 5 to 10, in the excitation frequency, matrix thickness, and accretion layer thickness, the distribution of normalized interface shear stress amplitude in any two of them domains is like a volcano in which the two of them at coupling resonance decrease with the increase of the other one, and the distribution of interfacial shear delamination interval in any two of them domains is like the floating clouds or fish scales which of the location from the origin is reduced with the increase of the other one. It is well known that coupling resonance is an inherent property of structures. For the accretion–substrate– accretion symmetrical structure with fixed material properties, the coupling resonance effect of interfacial shear delamination among excitation frequency, matrix thickness, and accretion layer thickness could be explained by a single degree of freedom system consisting of a mass block and a spring which of the natural frequency x is calculated as x ¼ k=m, where m is the mass of block and k is the elasticity modulus of spring. Undoubtedly, the (1) (2) (2) (1) total mass m(h ,h ) increases with the increase of matrix thickness h as well as accretion layer thickness h . (1) (2) 2 ð1Þ ð2Þ Thus, the natural frequency could be written as x ¼ k=mðh ; h Þ. When k is a constant, for x , h , and h , as 0 0 Figure 14. Interfacial shear delamination intervals in excitation frequency and substrate thickness domains as accretion layer ð1Þ thickness is 0.3 mm, ½r ~ � ¼ 10, and the accretion is frost (a), rime ice (b), and glaze ice (c). �1 18 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1183 0(0) long as two of them are determined, the other one can be obtained, which is also the value at coupling resonance. Moreover, it is easy to deduce that, for excitation frequency, substrate thickness, and accretion layer thickness, when any two of them increase, the other one decreases. In the structural optimization design of interface shear stratification, we could adjust excitation frequency, substrate thickness, and accretion layer thickness at any time by using the above conclusions. For example, in ultrasonic defrosting, the optimal excitation frequency is determined by substrate thickness and accretion layer thickness, or the optimal substrate thickness is determined by excitation frequency and accretion layer thickness, or the optimal accretion layer thickness is determined by substrate thickness and excitation frequency. The influence of accretion medium on normalized interface shear stress amplitude and interfacial shear stratification interval The influence of accretion medium on the normalized interface shear stress amplitude and interfacial shear stratification intervals in excitation frequency, accretion thickness, and substrate thickness domains is analyzed and discussed in the section. In excitation frequency and accretion thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and accretion thickness ð1Þ domains when the accretion is frost, rime ice, and glaze ice, separately (see Figure 11). Taking ½r ~ � ¼ 10, �1 the interfacial shear delamination interval is investigated by equation (55) (see Figure 12). Figure 15. Normalized interface shear stress amplitude in accretion layer thickness and substrate thickness domains as the exci- tation frequency is 20 kHz, and the accretion is frost (a), rime ice (b), and glaze ice (c). 1184 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 19 As shown in Figure 11, the distributions of normalized interface shear stress amplitude in excitation frequency and accretion thickness domains are like the shape of a mountain. But the peak is reduced when the accretion is frost, rime ice, and glaze ice in order. Therefore, the excitation frequency as well as accretion thickness at coupling resonance increases when the accretion is frost, rime ice, and glaze ice in order. As shown in Figure 12, the distributions of interfacial shear delamination intervals in excitation frequency and accretion thickness domains are like flow lines. But the location of distribution from the origin increases when the accretion is frost, rime ice, and glaze ice in order. In addition, the excitation frequency decreases with the increase of accretion thickness in distributions. In excitation frequency and substrate thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and substrate thickness ð1Þ domains when the accretion is frost, rime ice, and glaze ice, separately (see Figure 13). Taking ½r ~ � ¼ 10, �1 the interfacial shear delamination interval is investigated by equation (55) (see Figure 14). As shown in Figure 13, the distributions of normalized interface shear stress amplitude in excitation frequency and substrate thickness domains are like the shape of a volcano. But the location of crater from the origin increases when the accretion is frost, rime ice, and glaze ice in order. Therefore, the excitation frequency as well as substrate thickness at coupling resonance increases when the accretion is frost, rime ice, and glaze ice in order. Figure 16. Interfacial shear delamination intervals in accretion layer thickness and substrate thickness domains as the excitation ð1Þ frequency is 20 kHz, ½r ~ � ¼ 10, and the accretion is frost (a), rime ice (b), and glaze ice (c). �1 20 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1185 0(0) As shown in Figure 14, the distributions of interfacial shear delamination intervals in excitation frequency and substrate thickness domains are like floating clouds. But the location of distribution from the origin increases when the accretion is frost, rime ice, and glaze ice in order. In addition, the excitation frequency decreases with the increase of substrate thickness in distributions. In accretion layer and substrate thickness domains The normalized interface shear stress amplitude is calculated in accretion layer and substrate thickness domains ð1Þ when the accretion is frost, rime ice, and glaze ice, separately (see Figure 15). Taking ½r ~ � ¼ 10, the interfacial �1 shear delamination interval is investigated by equation (55) (see Figure 16). As shown in Figure 15, the distributions of normalized interface shear stress amplitude in accretion layer thickness and substrate thickness domains are like the shape of a mountain. But the peak is reduced when the accretion is frost, rime ice, and glaze ice in order. Therefore, the accretion layer thickness as well as substrate thickness at coupling resonance increases when the accretion is frost, rime ice, and glaze ice in order. As shown in Figure 16, the distributions of interfacial shear delamination intervals in accretion layer thickness and substrate thickness domains are like fish scales. But the location of distribution from the origin increases when the accretion is frost, rime ice, and glaze ice in order. In addition, the accretion layer thickness decreases with the increase of substrate thickness in distributions. As shown in Figures 11 to 16, among the excitation frequency, matrix thickness, and accretion layer thickness, the distribution of normalized interface shear stress amplitude in any two of them domains is like a volcano/ mountain in which the two of them at coupling resonance increase when the accretion is frost, rime ice, and glaze Figure 17. Normalized interface shear stress amplitude in excitation frequency and accretion thickness domains as the substrate thickness is 1 mm, and the substrate is titanium (a), steel (b), and molybdenum (c). 1186 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 21 ice in order, and the distribution of interfacial shear delamination interval in any two of them domains is like the flow lines or floating clouds or fish scales which, of the location from the origin, increases when the accretion is frost, rime ice, and glaze ice in order. When the accretion is frost, rime ice, and glaze ice in order, both the shear modulus and the density increase, but the shear modulus grows faster than the density, which could be seen from the formula c ¼ l=q because of the increase of shear wave velocity. The material properties of accretion layer affect the overall material properties of sandwich plate. Therefore, the natural frequency of interfacial shear stratification of sandwich structure increases when the accretion is frost, rime ice, and glaze ice in order because the natural frequency x increases in the formula x ¼ k=m when the increase rate of stiffness k is greater than the increase rate of mass m. When the excitation source and substrate plate are determined, the accretion layer material with smaller shear modulus is more favorable at the same thickness for the interface shear stratification. The accretion material sometimes varies with environmental conditions. For example, frost, rime ice, and glaze ice could be formed on metal plates due to differences in ambient temperature and humidity. Therefore, for ultrasonic de-accretion, the frost layer could be layered and removed more easily under the same conditions. The influence of substrate medium on normalized interface shear stress amplitude and interfacial shear stratification interval The influence of substrate medium on the normalized interface shear stress amplitude and interfacial shear stratification intervals in excitation frequency, accretion thickness, and substrate thickness domains are analyzed and discussed in the following section. Figure 18. Interfacial shear delamination intervals in excitation frequency and accretion layer thickness domains as the substrate ð1Þ thickness is 1 mm, ½r ~ � ¼ 10, and the substrate is titanium (a), steel (b), and molybdenum (c). �1 22 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1187 0(0) In excitation frequency and accretion thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and accretion thickness domains when the substrate is titanium, steel, and molybdenum, separately (see Figure 17). Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 18). �1 As shown in Figure 17, the distributions of normalized interface shear stress amplitude in excitation frequency and accretion thickness domains are like the shape of a mountain. But the peak is reduced when the substrate is titanium, steel, and molybdenum in order. Therefore, the excitation frequency as well as accretion thickness at coupling resonance increases when the substrate is titanium, steel, and molybdenum in order. As shown in Figure 18, the distributions of interfacial shear delamination intervals in excitation frequency and accretion layer thickness domains are like flow lines. But the location of distribution from the origin increases when the substrate is titanium, steel, and molybdenum in order. In addition, the excitation frequency decreases with the increase of accretion thickness in distributions. In excitation frequency and substrate thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and substrate thickness domains when the substrate is titanium, steel, and molybdenum, separately, see Figure 19. Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 20). �1 As shown in Figure 19, the distributions of normalized interface shear stress amplitude in excitation frequency and substrate thickness domains are like the shape of a mountain. But the peak is reduced when the substrate is Figure 19. Normalized interface shear stress amplitude in excitation frequency and substrate thickness domains as the accretion frost layer thickness is 0.3 mm, and the substrate is titanium (a), steel (b), and molybdenum (c). 1188 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 23 titanium, steel, and molybdenum in order. Therefore, the excitation frequency as well as substrate thickness at coupling resonance increases when the substrate is titanium, steel, and molybdenum in order. As shown in Figure 20, the distributions of interfacial shear delamination intervals in excitation frequency and substrate thickness domains are like floating clouds. But the location of distribution from the origin increases when the substrate is titanium, steel, and molybdenum in order. In addition, the excitation frequency decreases with the increase of substrate thickness in distributions. In accretion thickness and substrate thickness domains The normalized interface shear stress amplitude is calculated in accretion thickness and substrate thickness domains when the substrate is titanium, steel, and molybdenum, separately, see Figure 21. Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 22). �1 As shown in Figure 21, the distributions of normalized interface shear stress amplitude in accretion thickness and substrate thickness domains are like the shape of a mountain. But the peak is reduced when the substrate is titanium, steel, and molybdenum in order. Therefore, the accretion thickness as well as the substrate thickness at coupling resonance increases when the substrate is titanium, steel, and molybdenum in order. As shown in Figure 22, the distributions of interfacial shear delamination intervals in accretion thickness and substrate thickness domains are like fish scales. But the location of distribution from the origin increases when the Figure 20. Interfacial shear delamination intervals in excitation frequency and substrate thickness domains as the accretion frost ð1Þ layer thickness is 0.3 mm, ½r ~ � ¼ 10, and the substrate is titanium (a), steel (b), and molybdenum (c). �1 24 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1189 0(0) Figure 21. Normalized interface shear stress amplitude in accretion thickness and substrate thickness domains as the excitation frequency is 20 kHz, and the substrate is titanium (a), steel (b), and molybdenum (c). substrate is titanium, steel, and molybdenum in order. In addition, the accretion thickness decreases with the increase of substrate thickness in distributions. As shown in Figures 17 to 22, among the excitation frequency, matrix thickness, and accretion layer thickness, the distribution of normalized interface shear stress amplitude in any two of them domains is like a mountain in which the two of them at coupling resonance increase when the substrate is titanium, steel, and molybdenum in order, and the distribution of interfacial shear delamination interval in any two of them domains is like flow lines or floating clouds or fish scales which of the location from the origin increases when the substrate is titanium, steel, and molybdenum in order. When the substrate is titanium, steel, and molybdenum in turn, both the shear modulus and the density increase, but the shear modulus grows faster than the density, which could be seen from the formula c ¼ l=q because of the increase of shear wave velocity. The material properties of substrate affect the overall material properties of sandwich plate. The natural frequency of interfacial shear stratification of sandwich structure increases because in the formula x ¼ k=m, when the increase rate of stiffness k is greater than the increase rate of mass m, the natural frequency x increases. In the structural design of interfacial shear delamination, it is very important to choose the right base material. The results show that titanium is more suitable as the matrix material among titanium, steel, and molybdenum for ultrasonic defrosting. In other words, the method could be used to determine the optimal matrix material for interfacial shear delamination. 1190 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 25 Figure 22. Interfacial shear delamination intervals in accretion thickness and substrate thickness domains as the excitation frequency ð1Þ is 20 kHz, ½r ~ � ¼ 10, and the substrate is titanium (a), steel (b), and molybdenum (c). �1 Conclusions The interface shear stratification mechanism of sandwich structures is investigated in the paper. The forced propagation solution of sandwich constructions excited by SH waves is derived by global matrix methods and integral transformation methods. Combined with the interface fatigue failure theory, the necessary condition of interface shear delamination excited by symmetrical alternating stresses is analyzed and discussed. The analytical results of normalized interface shear stress amplitude are calculated by adaptive Gauss–Kronrod quadrature numerical integration methods. The effects of structural parameters, accretion, and matrix materials on the normalized interfacial shear stress amplitude and interfacial shear stratification intervals are analyzed and discussed. It is found that the interface shear stress amplitude first increases and then resonates with the increase of excitation frequency as well as accretion thickness and matrix thickness. The coupling resonance frequency of interfacial shear delamination is an inherent property of sandwich structure, which is determined by structure sizes and material properties. In other words, when the excitation frequency is equal to the coupling resonance fre- quency, the interface shear stress resonates, and interface shear delamination must occur. The coupling resonance effect of interfacial shear delamination could be explained by using a single degree of freedom system. In addition, in excitation frequency, matrix thickness, and accretion thickness, the interface shear stress amplitude in any two of the domains increases with the increase of the other one. For interfacial shear delamination, frost is the best accretion among frost, rime ice, and glaze ice, and titanium is the best substrate among titanium, steel, and molybdenum. 26 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1191 0(0) The results could provide a theoretical basis for the structure as well as excitation source optimization of interface shear stratification in ultrasonic de-accretion technology and preventing laminates layering. But it is worth noting that the results only apply to the case of infinite plate. The problem of finite plate will be further studied in the future. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding This work was supported by the National Natural Science Foundation of China (grant number 51675401). ORCID iD Feng Guo https://orcid.org/0000-0003-2816-7817 References 1. Gupta A and Ghosh A. Isogeometric static and dynamic analysis of laminated and sandwich composite plates using nonpolynomial shear deformation theory. Compos Part B-Eng 2019; 176: 107295. 2. Sahla M, Saidi H, Draiche K, et al. Free vibration analysis of angle-ply laminated composite and soft core sandwich plates. Steel Compos Struct 2019; 33: 663–679. 3. Gupta A and Ghosh A. Static and transient analysis of sandwich composite plates using isogeometric analysis. Mech Adv Mater Struct 2019; 26: 81–87. 4. Yang Y, Boom R, Irion B, et al. Recycling of composite materials. Chem Eng Process 2012; 51: 53–68. 5. Pickering SJ. Recycling technologies for thermoset composite materials-current status. Compos Part A Appl Sci Manuf 2006; 37: 1206–1215. 6. Tao C, Ji H, Qiu J, et al. Characterization of fatigue damages in composite laminates using Lamb wave velocity and prediction of residual life. Compos Struct 2017; 166: 219–228. 7. Higuchi R, Okabe T and Nagashima T. Numerical simulation of progressive damage and failure in composite laminates using XFEM/CZM coupled approach. Compos Part A Appl Sci Manuf 2017; 95: 197–207. 8. Palacios J, Smith E, Rose J, et al. Ultrasonic de-icing of wind-tunnel impact icing. J Aircr 2011; 48: 1020–1027. 9. Tan H, Xu G, Tao T, et al. Experimental investigation on the defrosting performance of a finned-tube evaporator using intermittent ultrasonic vibration. Appl Energ 2015; 158: 220–232. 10. Adachi K, Saiki K, Sato H, et al. Ultrasonic frost suppression. Jpn J Appl Phys 2003; 42: 682–685. 11. Wang Z. Recent progress on ultrasonic de-icing technique used for wind power generation, high-voltage transmission line and aircraft. Energ Build 2017; 140: 42–49. 12. Palacios J, Smith E, Rose J, et al. Instantaneous de-icing of freezer ice via ultrasonic actuation. AIAA J 2011; 49: 1158–1167. 13. Li D and Chen Z. Experimental study on instantaneously shedding frozen water droplets from cold vertical surface by ultrasonic vibration. Exp Therm Fluid Sci 2014; 53: 17–25. 14. Tan H, Tao T, Xu G, et al. Experimental study on defrosting mechanism of intermittent ultrasonic resonance for a finned- tube evaporator. Exp Therm Fluid Sci 2014; 52: 308–317. 15. Kalkowski MK, Waters TP and Rustighi E. Delamination of surface accretions with structural waves: piezo-actuation and power requirements. J Intel Mater Syst Struct 2017; 28: 1454–1471. 16. Overmeyer A, Palacios J and Smith E. Ultrasonic de-icing bondline design and rotor ice testing. AIAA J 2013; 51: 2965–2976. 17. Budinger M, Pommier-Budinger V, Napias G, et al. Ultrasonic ice protection systems: analytical and numerical models for architecture tradeoff. J Aircr 2016; 53: 680–690. 18. Venna SV and Lin YJ. Mechatronic development of self-actuating in-flight deicing structures. IEEE/ASME Trans Mechatron 2006; 11: 585–592. 19. Venna SV, Lin YJ and Botura G. Piezoelectric transducer actuated leading edge de-icing with simultaneous shear and impulse forces. J Aircr 2007; 44: 509–515. 20. Zeng J and Song B. Research on experiment and numerical simulation of ultrasonic de-icing for wind turbine blades. Renew Energ 2017; 113: 706–712. 21. Wang Z, Xu Y, Su F, et al. A light lithium niobate transducer for the ultrasonic de-icing of wind turbine blades. Renew Energ 2016; 99: 1299–1305. 22. Wang Y, Xu Y and Lei Y. An effect assessment and prediction method of ultrasonic de-icing for composite wind turbine blades. Renew Energ 2018; 118: 1015–1023. 1192 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 27 23. Habibi H, Cheng L, Zheng H, et al. A dual de-icing system for wind turbine blades combining high-power ultrasonic guided waves and low-frequency forced vibrations. Renew Energ 2015; 83: 859–870. 24. Wang Y, Xu Y and Huang Q. Progress on ultrasonic guided waves de-icing techniques in improving aviation energy efficiency. Renew Sust Energ Rev 2017; 79: 638–645. 25. Wang D, Tao T, Xu G, et al. Experimental study on frosting suppression for a finned-tube evaporator using ultrasonic vibration. Exp Therm Fluid Sci 2012; 36: 1–11. 26. Tan H, Zhang X, Zhang L, et al. Ultrasonic influence mechanism of a cold surface frosting process and an optimised defrosting technique. Appl Therm Eng 2019; 153: 113–127. 27. Bednarik M, Cervenka M, Lotton P, et al. Analytical solutions for elastic SH-waves propagating through an isotropic inhomogeneous layer. Compos Struct 2019; 220: 875–887. 28. Golub MV, Fomenko SI, Bui TQ, et al. Transmission and band gaps of elastic SH waves in functionally graded periodic laminates. Int J Solids Struct 2012; 49: 344–354. 29. Mardanshahi A, Shokrieh MM and Kazemirad S. Identification of matrix cracking in cross-ply laminated composites using Lamb wave propagation. Compos Struct 2020; 235: 111790 30. Zhu F, Pan E, Qian Z, et al. Waves in a generally anisotropic viscoelastic composite laminated bilayer: impact of the imperfect interface from perfect to complete delamination. Int J Solids Struct 2020; 202: 262–277. 31. Pollard HF. Sound waves in solids. London: Pion, 1977. 32. He JH. Generalized variational principles for buckling analysis of circular cylinders. Acta Mech 2020; 231: 899–906. 33. He JH. Generalized hellinger-reissner principle. J Appl Mech 2000; 67: 326–331. 34. He JH. Coupled variational principles of piezoelectricity. Int J Eng Sci 2001; 39: 323–341. 35. He JH. An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams. Appl Math Lett 2016; 52: 1–3. 36. He JH. Hamilton’s principle for dynamical elasticity. Appl Math Lett 2017; 72: 65–69. 37. He JH. Inverse problems of determining the unknown shape of oscillating airfoils in compressible 2D unsteady flow via variational technique. Aircr Eng Aerosp Tech 2000; 72: 18–24. 38. Al-Furjan MSH, Habibi M, Jung DW, et al. A computational framework for propagated waves in a sandwich doubly curved nanocomposite panel. Eng Comput-Germany. Epub ahead of print 3 August 2020: 1–18. https://doi.org/10.1007/ s00366-020-01130-8 39. Al-Furjan MSH, Oyarhossein MA, Habibi M, et al. On the wave propagation of the multi-scale hybrid nanocomposite doubly curved viscoelastic panel. Compos Struct. Epub ahead of print 15 September 2020; 255: 112947. DOI: 10.1016/j. compstruct.2020.112947. 40. He JH. Variational iteration method – a kind of non-linear analytical technique: some examples. Int J Nonlin Mech 1999; 34: 699–708. 41. He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 2000; 114: 115–123. 42. He JH. Homotopy perturbation technique. Comput Method Appl Mech Eng 1999; 178: 257–262. 43. He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003; 135: 73–79. 44. He JH and El-Dib YO. Homotopy perturbation method for Fangzhu oscillator. J Math Chem 2020; 58: 2245-2253. 45. Anjum N and He JH. Homotopy perturbation method for N/MEMS oscillators. Math Method Appl Sci Epub ahead of print 24 June 2020: 1–15. DOI: 10.1002/mma.6583. 46. Debnath L and Bhatta D. Integral transforms and their applications. Boca Raton, FL: CRC Press, 2014. 47. Tuan HS and Ponamgi SR. Surface excitation of a Love wave. IEEE Trans Sonics Ultrason 1972; 19: 442–443. 48. Williams AR, Feibelman PJ and Lang ND. Green’s-function methods for electronic-structure calculations. Phys Rev B 1982; 26: 5433. 49. Sakurai T. Green’s function methods for potential magnetic fields. Sol Phys 1982; 76: 301–321. 50. Rungger I, Droghetti A and Stamenova M. Non-equilibrium Green’s function methods for spin transport and dynamics. In: Handbook of materials modeling: methods: theory and modeling. Berlin, Germany: Springer, 2020, pp.957–983. 51. Alekseev AS and Mikhailenko BG. The solution of dynamic problems of elastic wave propagation in inhomogeneous media by a combination of partial separation of variables and finite-difference methods. J Geophys 1980; 48: 161–172. 52. Polyanin AD and Zhurov AI. Separation of variables in PDEs using nonlinear transformations: applications to reaction– diffusion type equations. Appl Math Lett 2020; 100: 106055. 53. Ali EYAA. The transform of partial differential equations to ordinary differential equations by using integral transforms and separation of variables methods. Master’s Dissertation, University of Gezira, Sudan, 2017. 54. Yilmaz C, Topal S, Ali HQ, et al. Non-destructive determination of the stiffness matrix of a laminated composite structure with lamb wave. Compos Struct 2020; 237: 111956. 55. Ding L, Zhu HP and Wu L. Analysis of mechanical properties of laminated rubber bearings based on transfer matrix method. Compos Struct 2017; 159: 390–396. 56. Knopoff L. A matrix method for elastic wave problems. Bull Seismol Soc Am 1964; 54: 431–438. 57. Gao J, Lyu Y, Zheng M, et al. Modeling guided wave propagation in multi-layered anisotropic composite laminates by state-vector formalism and the Legendre polynomials. Compos Struct 2019; 228: 111319. 28 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1193 0(0) 58. Ogilvy JA. A model for the ultrasonic inspection of composite plates. Ultrasonics 1995; 33: 85–93. 59. Li X, Wang H and Li G. Reanalysis assisted metaheuristic optimization for free vibration problems of composite laminates. Compos Struct 2018; 206: 380–391. 60. Kolsky H. Stress waves in solids. Chelmsford, MA: Courier Corporation, 1963. 61. He JH. Generalized equilibrium equations for shell derived from a generalized variational principle. Appl Math Lett 2017; 64: 94–100. 62. He JH. Hybrid problems of determining unknown shape of bladings in compressible S2-flow in mixed-flow turbomachin- ery via variational technique. Aircr Eng Aerosp Technol 1999; 71: 154–159. 63. He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006; 20: 1141–1199. 64. He JH. A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718. 65. He JH. Fractal calculus and its geometrical explanation. Results Phys 2018; 10: 272–276. 66. Allam O, Draiche K, Bousahla AA, et al. A generalized 4-unknown refined theory for bending and free vibration analysis of laminated composite and sandwich plates and shells. Comput Concr 2020; 26: 185–201. 67. Menasria A, Kaci A, Bousahla AA, et al. A four-unknown refined plate theory for dynamic analysis of FG-sandwich plates under various boundary conditions. Steel Compos Struct 2020; 36: 355–367. 68. Rabhi M, Benrahou KH, Kaci A, et al. A new innovative 3-unknowns HSDT for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Geomech Eng 2020; 22: 119. 69. Chikr SC, Kaci A, Bousahla AA, et al. A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin’s approach. Geomech Eng 2020; 21: 471–487. 70. Refrafi S, Bousahla AA, Bouhadra A, et al. Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations. Comput Concr 2020; 25: 311–325. 71. Rahmani MC, Kaci A, Bousahla AA, et al. Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory. Comput Concr 2020; 25: 225–244. 72. Boussoula A, Boucham B, Bourada M, et al. A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates. Smart Struct Syst 2020; 25: 197–218. 73. Polyanin AD. Functional separation of variables in nonlinear PDEs: general approach, new solutions of diffusion-type equations. Math Basel 2020; 8: 90. 74. Polyanin AD and Zhurov AI. Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations. Commun Nonlinear Sci 2016; 31: 11–20. 75. He JH and Wu XH. Variational iteration method: new development and applications. Comput Math Appl 2007; 54: 881–894. 76. He JH. Variational iteration method – some recent results and new interpretations. J Comput Appl Math 2007; 207: 3–17. 77. He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004; 151: 287–292. 78. He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004; 156: 527–539. 79. He JH. Asymptotology by homotopy perturbation method. Appl Math Comput 2004; 156: 591–596. 80. Tuan HS and Ponamgi SR. Excitation of Love waves in a thin film layer by a line source. IEEE Trans Sonics Ultrason 1972; 19: 9–14. 81. Hashimoto K and Yamaguchi H. Excitation and propagation of shear-horizontal-type surface and bulk acoustic waves. IEEE Trans Ultrason Ferroelectr Freq Control 2001; 48: 1181–1188. 82. Singh AK, Singh S, Kumari R, et al. Impact of point source and mass loading sensitivity on the propagation of an SH wave in an imperfectly bonded FGPPM layered structure. Acta Mech. Epub ahead of print June 2020; 231(6): 2603–2627. DOI: 10.1007/s00707-020-02659-x. 83. Gander W and Gautschi W. Adaptive quadrature – revisited. BIT Numer Math 2000; 40: 84–101. 84. Calvetti D, Golub G, Gragg W, et al. Computation of Gauss-Kronrod quadrature rules. Math Comp 2000; 69: 1035–1052. 85. Johnson RW. Algorithm 988: AMGKQ: an efficient implementation of adaptive multivariate Gauss-Kronrod quadrature for simultaneous integrands in Octave/MATLAB. ACM Trans Math Softw 2018; 44: 1–19. 86. Rahman A, Huang H, Ai C, et al. Fatigue performance of interface bonding between asphalt pavement layers using four- point shear test set-up. Int J Fatig 2019; 121: 181–190. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

Coupling resonance mechanism of interfacial stratification of sandwich plate structures excited by SH waves:

Download PDF
28 pages

Loading next page...
 
/lp/sage/coupling-resonance-mechanism-of-interfacial-stratification-of-sandwich-Sx20xK70qV

References (86)

Publisher
SAGE
Copyright
Copyright © 2022 by SAGE Publications Ltd unless otherwise noted. Manuscript content on this site is licensed under Creative Commons Licenses
ISSN
0263-0923
eISSN
2048-4046
DOI
10.1177/1461348420979468
Publisher site
See Article on Publisher Site

Abstract

For interfacial stratification mechanism of sandwich plate structures, the forced propagation solution of interface shear stress excited by SH waves is derived by global matrix methods and integral transformation methods. The necessary condition of interfacial shear delamination excited by alternating stress is analyzed with the interface fatigue failure theory. The exact value of forced propagation solution is calculated by adaptive Gauss–Kronrod quadrature numerical integration methods, which is verified via finite element methods. The coupling resonance mechanism of interface shear stratification is revealed by the forced vibration solution and the mass-spring model. The effects of excitation frequency, structural parameters, accretion, and matrix materials on the interfacial shear delamination are analyzed and discussed by practical cases of vibration de-accretion. For interface shear stratification of sandwich structures, the optimal exci- tation frequency as well as substrate thickness and accretion thickness is the value at coupling resonance, around which the interfacial shear stratification interval is formed by the interface fatigue failure criteria. In the stratification or/and antistratification design excited by vibrations, the excitation source and structure could be optimized by the method. Therefore, the results have important theoretical value for the extension and application of vibration stratification or/ and antistratification technology. Keywords Sandwich plate, SH waves, interfacial stratification, coupling resonance mechanism, interface shear stress, forced prop- agation solution Introduction Sandwich plates are often encountered in practice, but the interface stratification has long puzzled researchers in 1–9 vibration de-accretion, delamination dismantling, and even anti-layering design. Therefore, it is of great the- oretical value to study the interface stratification of sandwich structures loaded by stress waves. The problem of interfacial stratification of layered medium excited by elastic waves has been emphasized in 10,11 vibration deicing/defrosting. It was observed in experiments that ice layers accreted on the metal plate are 12,13 stratified instantaneously when the ultrasonic transducer is turned on. The finite element analysis confirms 14,15 that the excited interface shear stress is greater than the adhesive shear strength. A lot of research has been 16–26 done, and some of it has even been applied in related fields. However, due to the lack of theoretical analysis methods, the interface stratification mechanism is still unclear. The dynamic analysis of sandwich structures has always been one of the research hotspots in solid mechanics, 27–30 especially the derivation and solution of wave equations. Wave equations in sandwich plates are established School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, People’s Republic of China State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, People’s Republic of China Corresponding author: Jiu H Wu, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi 710049, People’s Republic of China. Email: ejhwu@mail.xjtu.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1167 0(0) by differential theory, variational theory, higher-order shear deformation theory, etc., which are usually partial 31–39 differential equations. Using variational iteration methods, homotopy perturbation methods, integral trans- formation methods, Green’s function methods, separation of variables methods, and so on, wave equations are 40–53 solved. The analytical model of wave motion in laminates is established with the boundary conditions by 54–59 stiffness-matrix methods, transfer-matrix methods, and global-matrix methods. By means of classical differential theory, the equilibrium equation is obtained by force analysis of micro elements in solid. On the basis of the equilibrium equation, the constitutive equation, and the geometric equation, the Navier’s wave equation is established. Using the variational theory based on minimum potential energy principle and/or minimum residual energy principle, the wave equation is established through solving the func- 61–65 tional standing value problem, which does not need strong local differentiability of each variable. The vibra- tion governing equation in sandwich plates is derived by higher-order shear deformable theory in which the shear correction factor is not required, the free vibration behavior influenced by structural parameters is investigated, 66–72 and it is found that the model has high precision. The free propagation solution of SH waves in plates with unknown coefficients was obtained via separation of 73,74 variable methods. Using variational iteration methods, the solution of wave equation first is approximated with the possible unknowns, and then a correction functional is constructed by a general Lagrange multiplier, 75,76 which can be identified optimally through the variational theory. Variational iteration methods can be used to solve homogenous and inhomogeneous partial differential equations in bounded and unbounded domains flex- ibly. By homotopy perturbation methods, a homotopy with an imbedding parameter p 2 [0,1] is constructed by 77–79 the homotopy technique, and the imbedding parameter is considered as a “small parameter.” Therefore, homotopy perturbation methods take full advantage of traditional perturbation methods and homotopy techni- ques, which can be used to solve a system of coupled linear as well as nonlinear differential equations. By integral transformation methods and Green’s function methods, the forced vibration solution of Love waves in the clad- 80–82 ding structure excited by point sources or line sources is derived which contains complex definite integrals. Complex integrals are solved numerically by Trapezoidal methods, Simpson methods, and Newton–Cotes methods, where the adaptive Gauss–Kronrod quadrature method in Simpson methods is particularly effective 83–85 for the integral of oscillation functions. For sandwich structures, a lot of research has been done. However, up to now, there is a lack of the forced propagation solution of SH waves partly because of the complexity of theoretical analysis and partly because of the difficulty of numerical integrations. Therefore, the interface stratification mechanism of sandwich structures has not been deeply analyzed and understood. In the paper, the forced vibration solution of sandwich structures excited by SH waves is derived. The exact solution is obtained by numerical integration methods. The necessary condition of interfacial stratification excited by dynamic shear stress is discussed. On the basis of the analytical solution, the coupling resonance mechanism of interface shear stratification is proposed. Finally, the influence of structural parameters, accretion, and matrix materials on the interfacial shear stress and interfacial stratification intervals is analyzed in accretion–substrate– accretion sandwich structures. Forced propagation solutions and necessary conditions of interfacial shear delamination of sandwich structures excited by SH waves Wave equations of sandwich structures excited by SH waves are established which are solved by integral trans- formation methods and global matrix methods. Further, forced propagation solutions of interface shear stress are obtained. The critical condition for interfacial shear delamination excited by the symmetrical alternating shear stress is obtained. Forced propagation solutions of interfacial shear stress of sandwich structures excited by SH waves For an isotropic solid plate, using u for the displacement of the particle, if the first Lame constant, the second Lame constant, density, and thickness are represented, respectively, by k, l, q, and h, the tensor notation of 31,60 classical Navier’s equation of elastic waves under the external force F is as follows @ u ðk þ lÞu þ lu þ F ¼ q ði; j ¼ 1; 2; 3Þ (1) j;ij i;jj @t 1168 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)3 Here, the lower right corner represents the axes. The wave travels in the x direction. The thickness direction of the plate is x . According to the plane strain assumption, only u is non-zero for SH waves, that is, u ¼ u ¼ 0. Furthermore, 3 1 2 u is independent of x 3 3 �ixt u ðx ; x ; tÞ¼ U ðx ; x Þe (2) 3 1 2 3 1 2 Substituting displacement components into equation (1), then it reduces to @ u lr u þ F ¼ q (3) @t where 2 2 @ @ r ¼ þ (4) 2 2 @x @x 1 2 The sandwich plate consists of three layers of medium as shown in Figure 1. The upper right corner of parameters represents the sequence number of layers. In engineering practice, such as ultrasonic defrosting, the middle substrate plate is often loaded with an excitation source F as shown in Figure 1. SH waves in sandwich structures excited by plane wave sources propagate along the direction of x . Moreover, the interfaces are all rigidly connected; in other words, the displacements and stresses are continuous at the interfaces, and the surfaces are free of tractions. For sandwich structures, as shown in Figure 1, the particle displacement of SH waves in each layer medium satisfies the following wave equation, respectively ðnÞ @ u ðnÞ ðnÞ 2 ðnÞ 3 l r u ¼ q ; n ¼ 1; 3 (5) @t ð2Þ @ u ð2Þ ð2Þ 2 ð2Þ 3 l r u þ F ¼ q (6) @t Here ðnÞ ðnÞ �ixt u ¼ U e ; n ¼ 1; 2; 3 (7) 3 3 �ixt F ¼ f d x e (8) 0 ðÞ 1 1; x ¼ 0 d x ¼ (9) ðÞ 1 0; x 6¼ 0 x ¼ 2pf (10) where f is the excitation amplitude and f is the excitation frequency. Figure 1. Sandwich structures. 4 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1169 0(0) Using the following stress–displacement relation @u r ¼ l (11) @x the boundary conditions are expressed as follows ð1Þ ð1Þ r ðx ¼�h Þ¼ 0 (12) ð3Þ ð2Þ ð3Þ r ðx ¼ h þ h Þ¼ 0 (13) ð1Þ ð2Þ r ðx ¼ 0Þ¼ r ðx ¼ 0Þ (14) 2 2 23 23 ð1Þ ð2Þ u ðx ¼ 0Þ¼ u ðx ¼ 0Þ (15) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ r ðx ¼ h Þ¼ r ðx ¼ h Þ (16) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ u ðx ¼ h Þ¼ u ðx ¼ h Þ (17) 2 2 23 23 Thus, the problem is reduced to solving a set of second-order nonhomogeneous partial differential equations with boundary conditions, which could be solved by many ways, such as variational iteration methods, homotopy perturbation methods, integral transformation methods, and so on. The integral transformation is a classical, convenient, and intuitive method for solving partial differential equations, in which partial differential equations are eliminated into ordinary differential equations by the integral transformation technique. In the paper, we use integral transformation methods to solve the problem. We assume that U ðs; x Þ is Fourier transform of U ðx ; x Þ, and the following integral transform formulas are 3 2 3 1 2 used in the paper. �isx U ðs; x Þ¼ U ðx ; x Þe dx (18) 3 2 3 1 2 1 2p �1 isx ^ 1 U ðx ; x Þ¼ U ðs; x Þe ds (19) 3 1 2 3 2 �1 �ixt For the sake of analysis, the time harmonic factor e is dropped from both sides of wave equations (5)–(6). Then, we take the Fourier transform on x to decouple wave equations, thus reading equations (20) and (21) ðnÞ d U ðnÞ 3 ðnÞ2 þ q U ¼ 0; n ¼ 1; 3 (20) dx ð2Þ d U ð2Þ f 3 ð2Þ2 þ q U ¼� (21) ð2Þ dx l where ðnÞ ðnÞ2 2 2 q ¼ k � s ; n ¼ 1; 2; 3 (22) ðnÞ k ¼ ; n ¼ 1; 2; 3 (23) ðnÞ sffiffiffiffiffiffiffiffi ðnÞ ðnÞ c ¼ ; n ¼ 1; 2; 3 (24) ðnÞ q 1170 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)5 Here, s is the integration factor, which satisfies the following condition according to equation (22). ðnÞ ðnÞ �k < s < k ; n ¼ 1; 2; 3 (25) T T And likewise, we take the Fourier transform of boundary conditions (12) to (17) on x ð1Þ ð1Þ r ^ ðx ¼�h Þ¼ 0 (26) ð3Þ ð2Þ ð3Þ r ^ ðx ¼ h þ h Þ¼ 0 (27) ð1Þ ð2Þ r ^ ðx ¼ 0Þ¼ r ^ ðx ¼ 0Þ (28) 2 2 23 23 ð1Þ ð2Þ u^ ðx ¼ 0Þ¼ u^ ðx ¼ 0Þ (29) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ r ^ ðx ¼ h Þ¼ r ^ ðx ¼ h Þ (30) 2 2 23 23 ð2Þ ð3Þ ð2Þ ð2Þ u^ ðx ¼ h Þ¼ u^ ðx ¼ h Þ (31) 2 2 23 23 In addition, we also take the Fourier transform of equation (11) on x @u^ r ^ ¼ l (32) @x Thus, the problem is transformed into a solvable system of ordinary differential equations with boundary conditions. Therefore, we could obtain the single matrix as controls the whole system by global matrix methods. The solutions of equations (20) and (21) are assumed to be ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ U ¼ A cosðq x Þþ B sinðq x Þ; n ¼ 1; 3 (33) 2 2 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ U ¼ A cosðq x Þþ B sinðq x Þþ C (34) 2 2 (n) (2) (n) (2) Here, A , A , B , and B are the undetermined coefficients �f ð2Þ C ¼ (35) ð2Þ ð2Þ 2 2 l ðk � s Þ After equations (33) and (34) are substituted into equations (26)–(31) by equation (32), we are able to obtain (1) (2) (3) (1) (2) (3) the following system of inhomogeneous equations with respect to unknowns A , A , A , B , B , and B , which is written in the matrix form as follows 2 32 3 2 3 ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ sinðq h Þ 0 0 cosðq h Þ 00 A 0 6 76 7 6 7 6 ð3Þ ð2Þ ð3Þ ð3Þ ð2Þ ð3Þ 76 ð2Þ 7 6 7 00 �sin½q ðh þ h Þ� 0 0 cos½q ðh þ h Þ� A 0 6 6 6 7 7 7 6 76 7 6 7 6 ð1Þ ð1Þ ð2Þ ð2Þ 76 ð3Þ 7 6 7 00 0 l q �l q 0 A 0 6 76 7 6 7 6 76 7 ¼ 6 7 ð2Þ ð1Þ 6 76 7 6 7 1 �1 0 0 0 0 B �C 6 76 7 6 7 6 76 7 6 7 ð2Þ ð2Þ ð2Þ ð2Þ ð3Þ ð3Þ ð3Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð3Þ ð3Þ ð3Þ ð2Þ ð2Þ 6 76 7 6 7 0 �l q sinðq h Þ l q sinðq h Þ 0 l q cosðq h Þ�l q cosðq h Þ B 4 54 5 4 5 ð2Þ ð2Þ ð2Þ ð3Þ ð2Þ ð2Þ ð2Þ ð3Þ ð2Þ ð3Þ 0 cosðq h Þ�cosðq h Þ 0 sinðq h Þ�sinðq h Þ B (36) 6 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1171 0(0) Thus, the whole system is controlled by a single matrix (i.e. the global matrix), which consists of six equations with six unknowns. Solving equation (36), we get ð2Þ ð1Þ A ¼ (37) ð3Þ D Z ð1Þ ð2Þ ð1Þ B ¼� (38) D Z ð2Þ ð3Þ Here ð2Þ Z ¼ðZ M þ M � Z ÞC (39) ð2Þ ð1Þ ð2Þ ð1Þ ð1Þ D ðM � Z M Þ ð1Þ ð2Þ ð1Þ ð1Þ Z ¼ �ðZ M þ M Þ� (40) ð3Þ ð1Þ ð2Þ ð1Þ D X ð2Þ Table 1 Material parameters. Young’s Poisson’s Density Shear Shear wave Material modulus (GPa) ratio (kg/m ) modulus (GPa) velocity (m/s) Frost 7.759 0.343 800 2.661 1900 Rime ice 8.759 0.337 900 3.276 1907 Glaze ice 9.759 0.331 950 3.666 1964 Magnesium 43.4 0.297 1740 16.7 3100 Aluminum 69.4 0.337 2700 23.4 3140 Titanium 102 0.35 4600 37.778 2866 Steel 205.0 0.28 7850 80 3194 Molybdenum 297.3 0.297 10200 114.3 3350 Figure 2. Analytical and simulation results of normalized interface shear stress amplitude and interfacial shear delamination interval in excitation frequency domain as the accretion frost layer thickness is 0.1 mm, the substrate aluminum plate thickness is 1 mm, and ð1Þ ½r ~ � ¼ 10. �1 1172 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)7 E N YE ð1Þ ð2Þ ð2Þ Z ¼ � N (41) ð1Þ ð1Þ E E N þ E N ð2Þ ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð1Þ D ¼ sinðq h Þ (42) ð1Þ ð1Þ ð1Þ D ¼ cosðq h Þ (43) ð2Þ Figure 3. Analytical and simulation results of normalized interface shear stress amplitude and interfacial shear delamination interval in accretion frost layer thickness domain as the excitation frequency is 20 kHz, and the substrate aluminum plate thickness is 1 mm and ð1Þ ½r ~ � ¼ 10. �1 Figure 4. Analytical and simulation results of normalized interface shear stress amplitude and interfacial shear delamination interval in substrate aluminum plate thickness domain as the excitation frequency is 20 kHz, and the accretion frost layer thickness is 0.25 mm ð1Þ and ½r ~ � ¼ 10. �1 8 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1173 0(0) ð3Þ ð2Þ ð3Þ E ¼ sin½q ðh þ h Þ� (44) ð1Þ ð3Þ ð2Þ ð3Þ E ¼ cos½q ðh þ h Þ� (45) ð2Þ ð2Þ ð2Þ M ¼ sinðq h Þ (46) ð1Þ ð2Þ ð2Þ M ¼ cosðq h Þ (47) ð2Þ ð3Þ ð2Þ N ¼ sinðq h Þ (48) ð1Þ ð3Þ ð2Þ N ¼ cosðq h Þ (49) ð2Þ ð2Þ ð2Þ l q X ¼ (50) ð1Þ ð1Þ l q ð3Þ ð3Þ l q Y ¼ (51) ð2Þ ð2Þ l q (2) (3) (2) (3) The other coefficients A , A , B , and B are also determined in turn similarly by the global matrix (36). Figure 5. Normalized interface shear stress amplitude in excitation frequency and accretion frost layer thickness domains as the substrate aluminum plate thickness is 1 (a), 1.5 (b), and 2 mm (c). 1174 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3)9 Finally, taking the inverse Fourier transform of equations (33) and (34) to x , we obtain the closed solution of displacement component ðjÞ ðnÞ ðnÞ �ixt isx u ðx ; x Þ¼ e U ðs; x Þe ds; n ¼ 1; 2; 3 (52) 1 2 2 3 3 ðjÞ �k ðjÞ ð1Þ ð2Þ ð3Þ k ¼ minðk ; k ; k Þ (53) T T T T Here, the upper and lower limits of integral (52) must satisfy the condition (25). Further, the closed solution of horizontal shear stress is also obtained by equations (52) and (11) ðjÞ ðnÞ ðnÞ �ixt ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ isx r ¼ l e q ½�A sinðq x Þþ B cosðq x Þ�e ds (54) 2 2 ðjÞ �k At last, forced propagation solutions of interface shear stress of sandwich structures could be written as follows ðjÞ ð1Þ ð1Þ �ixt ð1Þ ð1Þ isx r ðx ¼ 0Þ¼ l e B q e ds (55) ðjÞ �k ð1Þ Figure 6. Interfacial shear delamination intervals in excitation frequency and accretion frost layer thickness domains as ½r ~ � ¼ 10, �1 and the substrate aluminum plate thickness is 1 (a), 1.5 (b) and 2 mm (c). 10 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1175 0(0) ðjÞ ð2Þ ð2Þ ð2Þ �ixt ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ isx r ðx ¼ h Þ¼ l e q ½B cosðq h Þ� A sinðq h Þ�e ds (56) ðjÞ �k Obviously, the interfacial shear stress (55) and (56) are symmetric alternating stress of time harmonic, which of the amplitude depends on the amplitude of excitation source. It is worth noting that the upper and lower limits of integral (55) and (56) contain the integrating factors s, which could be calculated by numerical integration methods. In addition, the method could be extended to laminates with any number of layers. Necessary conditions of interface stratification excited by the symmetrical alternating shear stress According to the maximum stress principle of interface fatigue failure, when the maximum value of alternating stress acting on the interface is greater than the allowable fatigue stress, the interfacial debonding occurs. Obviously, the peak of a symmetric alternating stress is equal to its amplitude. Therefore, the necessary conditions for interface shear stratification are expressed as follows � � � � ð1Þ ð1Þ �r ðx ¼ 0Þ� > ½r � (57) 2 �1 � � � � ð2Þ ð2Þ ð2Þ �r ðx ¼ h Þ� > ½r � (58) 2 �1 Figure 7. Normalized interface shear stress amplitude in excitation frequency and substrate aluminum plate thickness domains as the accretion frost layer thickness is 0.1 (a), 0.2 (b), and 0.3 mm (c). 1176 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 11 (1) (2) Here, [r ] and [r ] are the allowable shear fatigue stress of lower interface of the first and second layers, –1 –1 separately, which are both constants that relate to the interface material and environmental conditions. For equations (55) and (56), f is only the coefficient of interface shear stress. Therefore, dividing both sides of equations (57) and (58) by the excitation amplitude f , we obtain ð1Þ  ð1Þ r ~ ðx ¼ 0Þ > ½r ~ � (59) 2 �1 ð2Þ ð2Þ ð2Þ r ~ ðx ¼ h Þ > ½r ~ � (60) 2 �1 Here ð1Þ r ðx ¼ 0Þ ð1Þ 2 r ~ ðx ¼ 0Þ ¼   (61) ð2Þ ð2Þ r ðx ¼ h Þ ð2Þ ð2Þ 23 r ~ ðx ¼ h Þ ¼   (62) Figure 8. Interfacial shear delamination intervals in excitation frequency and substrate aluminum plate thickness domains as ð1Þ ½r ~ � ¼ 10, and the accretion frost layer thickness is 0.1 (a), 0.2, (b) and 0.3 mm (c). �1 12 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1177 0(0) ð1Þ ½r � �1 ð1Þ ½r ~ � ¼ (63) �1 ð2Þ ½r � ð2Þ �1 ½r ~ � ¼ (64) �1 Equations (59) and (60) are the necessary condition for shear stratification of lower interface of the first and second layers, separately, which of the left is the normalized interface shear stress amplitude, and the right is the normalized allowable shear fatigue stress. In the application, sandwich plates with the same accretion layer on both surfaces are often encountered such as metal plates exposed to cold air that frost and/or ice evenly on both surfaces. Because of the symmetry of structure and excitation source, the necessary condition of shear delamination for the upper and lower interface is the same. Therefore, we take the symmetrical sandwich structure as an example to explore the law of interface shear stratification in the following sections. Coupling resonance mechanism of interfacial shear stratification and finite element verification In this section, analytic results of forced propagation solution of normalized interface shear stress amplitude are calculated in the excitation frequency, accretion layer thickness, and matrix thickness domains by adaptive Figure 9. Normalized interface shear stress amplitude in accretion frost layer thickness and substrate aluminum plate thickness domains as the excitation frequency is 20 (a), 22.5 (b), and 25 kHz (c). 1178 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 13 Gauss–Kronrod quadrature numerical integration methods (see Figures 2 to 4), which are verified by the finite element simulation method. The coupling resonance effect of interface shear stress amplitude is revealed. Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is studied by equation (59). Material parameters are �1 shown in Table 1. As shown in Figure 2, the normalized interface shear stress amplitude first increases with the increase of excitation frequency, then resonates as the excitation frequency reaches 32 kHz, and finally becomes unstable in high frequency region. Obviously, the coupled resonance frequency is 32 kHz. Because of the coupling reso- nance between interface shear stress amplitude and excitation frequency, the interfacial shear delamination inter- val of 19–35 kHz is formed around the coupling resonance frequency of 32 kHz as shown in the gray region in Figure 2. As shown in Figure 3, the normalized interface shear stress amplitude first increases with the increase of accretion frost layer thickness, then resonates as the accretion frost layer thickness reaches 0.355 mm, and finally becomes unstable in large frost layer thickness region. Obviously, the frost layer thickness of coupled resonance is 0.355 mm. Because of the coupling resonance between interface shear stress amplitude and frost layer thickness, the interfacial shear delamination interval of 0.08–0.38 mm is formed around 0.355 mm as shown in the gray region in Figure 3. As shown in Figure 4, the normalized interface shear stress amplitude first increases with the increase of substrate aluminum plate thickness, then resonates as the substrate aluminum plate thickness reaches 2.75 mm, and finally becomes unstable in large substrate aluminum plate thickness region. Obviously, the substrate alu- minum plate thickness of coupled resonance is 2.75 mm. Because of the coupling resonance between interface Figure 10. Interfacial shear delamination intervals in accretion frost layer thickness and substrate aluminum plate thickness domains ð1Þ as ½r ~ � ¼ 10 and the excitation frequency is 20 (a), 22.5 (b), and 25 kHz (c). �1 14 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1179 0(0) shear stress amplitude and substrate aluminum plate thickness, the interfacial shear delamination interval of 0.25– 2.8 mm is formed around 2.75 mm as shown in the gray region in Figure 4. As shown in Figures 2 to 4, the analysis results are in good agreement with the simulation results. The influence of excitation frequency, accretion frost layer thickness, and substrate aluminum plate thickness on the normalized interface shear stress amplitude is similar. In other words, the normalized interface shear stress amplitude has a coupling resonance effect with each of excitation frequency, accretion frost layer thickness, and substrate alumi- num plate thickness. The interfacial shear delamination interval is formed around the location of coupling res- onance peak. The coupling of structure intrinsic property and excitation frequency results into the resonance of interface shear stress amplitude, which is determined by structural parameters and material parameters. The resonance frequency of interface shear stress amplitude, that is, the natural frequency of interfacial shear delamination of sandwich structures, is determined by structure intrinsic property. Therefore, when the excitation frequency is close to or equal to the natural frequency, the excited interface shear stress amplitude resonates, and the interface shear delamination interval is formed. Obviously, the frequency as well as accretion frost layer thickness and substrate aluminum plate thickness at coupling resonance is the optimal for interface shear stratification. Therefore, the method could be used to optimize the excitation source and structure design of interface shear stratification. Figure 11. Normalized interface shear stress amplitude in excitation frequency and accretion thickness domains as magnesium substrate plate thickness is 1 mm, and the accretion is frost (a), rime ice (b), and glaze ice (c). 1180 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 15 The influence of structural parameters on normalized interface shear stress amplitude and interface shear stratification interval The influence of substrate thickness, accretion layer thickness, and excitation frequency on the normalized interface shear stress amplitude and interfacial shear delamination intervals is analyzed and discussed in the section. The influence of substrate thickness in excitation frequency and accretion layer thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and accretion frost layer thickness domains when the substrate aluminum plate thickness is 1, 1.5, and 2 mm, separately, see Figure 5. ð1Þ Taking ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 6). �1 As shown in Figure 5, the distributions of normalized interface shear stress amplitude in excitation frequency and frost layer thickness domains are like the shape of a volcano. But the location of crater from the origin is reduced with the increase of substrate thickness. Therefore, the excitation frequency as well as frost layer thickness at coupling resonance decreases with the increase of substrate thickness. As shown in Figure 6, the distributions of interfacial shear delamination intervals in excitation frequency and frost layer thickness domains are like floating clouds. But the location of distribution from the origin is reduced with the increase of substrate thickness. In addition, the excitation frequency decreases with the increase of frost layer thickness in distributions. Figure 12. Interfacial shear delamination intervals in excitation frequency and accretion thickness domains as magnesium substrate ð1Þ plate thickness is 1 mm, ½r ~ � ¼ 10, and the accretion is frost (a), rime ice (b), and glaze ice (c). �1 16 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1181 0(0) The influence of accretion layer thickness in excitation frequency and substrate thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and substrate aluminum plate thickness domains when the accretion frost layer thickness is 0.1, 0.2, and 0.3 mm, separately (see Figure 7). ð1Þ Taking ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 8). �1 As shown in Figure 7, the distributions of normalized interface shear stress amplitude in excitation frequency and aluminum plate thickness domains are like the shape of a volcano. But the location of crater from the origin is reduced with the increase of frost layer thickness. Therefore, the excitation frequency as well as aluminum plate thickness at coupling resonance decreases with the increase of frost layer thickness. As shown in Figure 8, the distributions of interfacial shear delamination intervals in excitation frequency and aluminum plate thickness domains are like floating clouds. But the location of distribution from the origin is reduced with the increase of frost layer thickness. In addition, the excitation frequency decreases with the increase of aluminum plate thickness in distributions. The influence of excitation frequency in accretion layer thickness and substrate thickness domains The normalized interface shear stress amplitude is calculated in accretion frost layer thickness and substrate aluminum plate thickness domains when the excitation frequency is 20, 22.5, and 25 kHz, separately ð1Þ (see Figure 9). Taking ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) �1 (see Figure 10). Figure 13. Normalized interface shear stress amplitude in excitation frequency and substrate thickness domains as accretion layer thickness is 0.3 mm, and the accretion is frost (a), rime ice (b), and glaze ice (c). 1182 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 17 As shown in Figure 9, the distributions of normalized interface shear stress amplitude in frost layer thickness and aluminum plate thickness domains are like the shape of a volcano. But the location of crater from the origin is reduced with the increase of excitation frequency. Therefore, the frost layer thickness as well as aluminum plate thickness at coupling resonance decreases with the increase of excitation frequency. As shown in Figure 10, the distributions of interfacial shear delamination intervals in frost layer thickness and aluminum plate thickness domains are like fish scales. But the location of distribution from the origin is reduced with the increase of excitation frequency. In addition, the frost layer thickness decreases with the increase of aluminum plate thickness in distributions. As shown in Figures 5 to 10, in the excitation frequency, matrix thickness, and accretion layer thickness, the distribution of normalized interface shear stress amplitude in any two of them domains is like a volcano in which the two of them at coupling resonance decrease with the increase of the other one, and the distribution of interfacial shear delamination interval in any two of them domains is like the floating clouds or fish scales which of the location from the origin is reduced with the increase of the other one. It is well known that coupling resonance is an inherent property of structures. For the accretion–substrate– accretion symmetrical structure with fixed material properties, the coupling resonance effect of interfacial shear delamination among excitation frequency, matrix thickness, and accretion layer thickness could be explained by a single degree of freedom system consisting of a mass block and a spring which of the natural frequency x is calculated as x ¼ k=m, where m is the mass of block and k is the elasticity modulus of spring. Undoubtedly, the (1) (2) (2) (1) total mass m(h ,h ) increases with the increase of matrix thickness h as well as accretion layer thickness h . (1) (2) 2 ð1Þ ð2Þ Thus, the natural frequency could be written as x ¼ k=mðh ; h Þ. When k is a constant, for x , h , and h , as 0 0 Figure 14. Interfacial shear delamination intervals in excitation frequency and substrate thickness domains as accretion layer ð1Þ thickness is 0.3 mm, ½r ~ � ¼ 10, and the accretion is frost (a), rime ice (b), and glaze ice (c). �1 18 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1183 0(0) long as two of them are determined, the other one can be obtained, which is also the value at coupling resonance. Moreover, it is easy to deduce that, for excitation frequency, substrate thickness, and accretion layer thickness, when any two of them increase, the other one decreases. In the structural optimization design of interface shear stratification, we could adjust excitation frequency, substrate thickness, and accretion layer thickness at any time by using the above conclusions. For example, in ultrasonic defrosting, the optimal excitation frequency is determined by substrate thickness and accretion layer thickness, or the optimal substrate thickness is determined by excitation frequency and accretion layer thickness, or the optimal accretion layer thickness is determined by substrate thickness and excitation frequency. The influence of accretion medium on normalized interface shear stress amplitude and interfacial shear stratification interval The influence of accretion medium on the normalized interface shear stress amplitude and interfacial shear stratification intervals in excitation frequency, accretion thickness, and substrate thickness domains is analyzed and discussed in the section. In excitation frequency and accretion thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and accretion thickness ð1Þ domains when the accretion is frost, rime ice, and glaze ice, separately (see Figure 11). Taking ½r ~ � ¼ 10, �1 the interfacial shear delamination interval is investigated by equation (55) (see Figure 12). Figure 15. Normalized interface shear stress amplitude in accretion layer thickness and substrate thickness domains as the exci- tation frequency is 20 kHz, and the accretion is frost (a), rime ice (b), and glaze ice (c). 1184 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 19 As shown in Figure 11, the distributions of normalized interface shear stress amplitude in excitation frequency and accretion thickness domains are like the shape of a mountain. But the peak is reduced when the accretion is frost, rime ice, and glaze ice in order. Therefore, the excitation frequency as well as accretion thickness at coupling resonance increases when the accretion is frost, rime ice, and glaze ice in order. As shown in Figure 12, the distributions of interfacial shear delamination intervals in excitation frequency and accretion thickness domains are like flow lines. But the location of distribution from the origin increases when the accretion is frost, rime ice, and glaze ice in order. In addition, the excitation frequency decreases with the increase of accretion thickness in distributions. In excitation frequency and substrate thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and substrate thickness ð1Þ domains when the accretion is frost, rime ice, and glaze ice, separately (see Figure 13). Taking ½r ~ � ¼ 10, �1 the interfacial shear delamination interval is investigated by equation (55) (see Figure 14). As shown in Figure 13, the distributions of normalized interface shear stress amplitude in excitation frequency and substrate thickness domains are like the shape of a volcano. But the location of crater from the origin increases when the accretion is frost, rime ice, and glaze ice in order. Therefore, the excitation frequency as well as substrate thickness at coupling resonance increases when the accretion is frost, rime ice, and glaze ice in order. Figure 16. Interfacial shear delamination intervals in accretion layer thickness and substrate thickness domains as the excitation ð1Þ frequency is 20 kHz, ½r ~ � ¼ 10, and the accretion is frost (a), rime ice (b), and glaze ice (c). �1 20 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1185 0(0) As shown in Figure 14, the distributions of interfacial shear delamination intervals in excitation frequency and substrate thickness domains are like floating clouds. But the location of distribution from the origin increases when the accretion is frost, rime ice, and glaze ice in order. In addition, the excitation frequency decreases with the increase of substrate thickness in distributions. In accretion layer and substrate thickness domains The normalized interface shear stress amplitude is calculated in accretion layer and substrate thickness domains ð1Þ when the accretion is frost, rime ice, and glaze ice, separately (see Figure 15). Taking ½r ~ � ¼ 10, the interfacial �1 shear delamination interval is investigated by equation (55) (see Figure 16). As shown in Figure 15, the distributions of normalized interface shear stress amplitude in accretion layer thickness and substrate thickness domains are like the shape of a mountain. But the peak is reduced when the accretion is frost, rime ice, and glaze ice in order. Therefore, the accretion layer thickness as well as substrate thickness at coupling resonance increases when the accretion is frost, rime ice, and glaze ice in order. As shown in Figure 16, the distributions of interfacial shear delamination intervals in accretion layer thickness and substrate thickness domains are like fish scales. But the location of distribution from the origin increases when the accretion is frost, rime ice, and glaze ice in order. In addition, the accretion layer thickness decreases with the increase of substrate thickness in distributions. As shown in Figures 11 to 16, among the excitation frequency, matrix thickness, and accretion layer thickness, the distribution of normalized interface shear stress amplitude in any two of them domains is like a volcano/ mountain in which the two of them at coupling resonance increase when the accretion is frost, rime ice, and glaze Figure 17. Normalized interface shear stress amplitude in excitation frequency and accretion thickness domains as the substrate thickness is 1 mm, and the substrate is titanium (a), steel (b), and molybdenum (c). 1186 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 21 ice in order, and the distribution of interfacial shear delamination interval in any two of them domains is like the flow lines or floating clouds or fish scales which, of the location from the origin, increases when the accretion is frost, rime ice, and glaze ice in order. When the accretion is frost, rime ice, and glaze ice in order, both the shear modulus and the density increase, but the shear modulus grows faster than the density, which could be seen from the formula c ¼ l=q because of the increase of shear wave velocity. The material properties of accretion layer affect the overall material properties of sandwich plate. Therefore, the natural frequency of interfacial shear stratification of sandwich structure increases when the accretion is frost, rime ice, and glaze ice in order because the natural frequency x increases in the formula x ¼ k=m when the increase rate of stiffness k is greater than the increase rate of mass m. When the excitation source and substrate plate are determined, the accretion layer material with smaller shear modulus is more favorable at the same thickness for the interface shear stratification. The accretion material sometimes varies with environmental conditions. For example, frost, rime ice, and glaze ice could be formed on metal plates due to differences in ambient temperature and humidity. Therefore, for ultrasonic de-accretion, the frost layer could be layered and removed more easily under the same conditions. The influence of substrate medium on normalized interface shear stress amplitude and interfacial shear stratification interval The influence of substrate medium on the normalized interface shear stress amplitude and interfacial shear stratification intervals in excitation frequency, accretion thickness, and substrate thickness domains are analyzed and discussed in the following section. Figure 18. Interfacial shear delamination intervals in excitation frequency and accretion layer thickness domains as the substrate ð1Þ thickness is 1 mm, ½r ~ � ¼ 10, and the substrate is titanium (a), steel (b), and molybdenum (c). �1 22 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1187 0(0) In excitation frequency and accretion thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and accretion thickness domains when the substrate is titanium, steel, and molybdenum, separately (see Figure 17). Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 18). �1 As shown in Figure 17, the distributions of normalized interface shear stress amplitude in excitation frequency and accretion thickness domains are like the shape of a mountain. But the peak is reduced when the substrate is titanium, steel, and molybdenum in order. Therefore, the excitation frequency as well as accretion thickness at coupling resonance increases when the substrate is titanium, steel, and molybdenum in order. As shown in Figure 18, the distributions of interfacial shear delamination intervals in excitation frequency and accretion layer thickness domains are like flow lines. But the location of distribution from the origin increases when the substrate is titanium, steel, and molybdenum in order. In addition, the excitation frequency decreases with the increase of accretion thickness in distributions. In excitation frequency and substrate thickness domains The normalized interface shear stress amplitude is calculated in excitation frequency and substrate thickness domains when the substrate is titanium, steel, and molybdenum, separately, see Figure 19. Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 20). �1 As shown in Figure 19, the distributions of normalized interface shear stress amplitude in excitation frequency and substrate thickness domains are like the shape of a mountain. But the peak is reduced when the substrate is Figure 19. Normalized interface shear stress amplitude in excitation frequency and substrate thickness domains as the accretion frost layer thickness is 0.3 mm, and the substrate is titanium (a), steel (b), and molybdenum (c). 1188 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 23 titanium, steel, and molybdenum in order. Therefore, the excitation frequency as well as substrate thickness at coupling resonance increases when the substrate is titanium, steel, and molybdenum in order. As shown in Figure 20, the distributions of interfacial shear delamination intervals in excitation frequency and substrate thickness domains are like floating clouds. But the location of distribution from the origin increases when the substrate is titanium, steel, and molybdenum in order. In addition, the excitation frequency decreases with the increase of substrate thickness in distributions. In accretion thickness and substrate thickness domains The normalized interface shear stress amplitude is calculated in accretion thickness and substrate thickness domains when the substrate is titanium, steel, and molybdenum, separately, see Figure 21. Taking ð1Þ ½r ~ � ¼ 10, the interfacial shear delamination interval is investigated by equation (55) (see Figure 22). �1 As shown in Figure 21, the distributions of normalized interface shear stress amplitude in accretion thickness and substrate thickness domains are like the shape of a mountain. But the peak is reduced when the substrate is titanium, steel, and molybdenum in order. Therefore, the accretion thickness as well as the substrate thickness at coupling resonance increases when the substrate is titanium, steel, and molybdenum in order. As shown in Figure 22, the distributions of interfacial shear delamination intervals in accretion thickness and substrate thickness domains are like fish scales. But the location of distribution from the origin increases when the Figure 20. Interfacial shear delamination intervals in excitation frequency and substrate thickness domains as the accretion frost ð1Þ layer thickness is 0.3 mm, ½r ~ � ¼ 10, and the substrate is titanium (a), steel (b), and molybdenum (c). �1 24 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1189 0(0) Figure 21. Normalized interface shear stress amplitude in accretion thickness and substrate thickness domains as the excitation frequency is 20 kHz, and the substrate is titanium (a), steel (b), and molybdenum (c). substrate is titanium, steel, and molybdenum in order. In addition, the accretion thickness decreases with the increase of substrate thickness in distributions. As shown in Figures 17 to 22, among the excitation frequency, matrix thickness, and accretion layer thickness, the distribution of normalized interface shear stress amplitude in any two of them domains is like a mountain in which the two of them at coupling resonance increase when the substrate is titanium, steel, and molybdenum in order, and the distribution of interfacial shear delamination interval in any two of them domains is like flow lines or floating clouds or fish scales which of the location from the origin increases when the substrate is titanium, steel, and molybdenum in order. When the substrate is titanium, steel, and molybdenum in turn, both the shear modulus and the density increase, but the shear modulus grows faster than the density, which could be seen from the formula c ¼ l=q because of the increase of shear wave velocity. The material properties of substrate affect the overall material properties of sandwich plate. The natural frequency of interfacial shear stratification of sandwich structure increases because in the formula x ¼ k=m, when the increase rate of stiffness k is greater than the increase rate of mass m, the natural frequency x increases. In the structural design of interfacial shear delamination, it is very important to choose the right base material. The results show that titanium is more suitable as the matrix material among titanium, steel, and molybdenum for ultrasonic defrosting. In other words, the method could be used to determine the optimal matrix material for interfacial shear delamination. 1190 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 25 Figure 22. Interfacial shear delamination intervals in accretion thickness and substrate thickness domains as the excitation frequency ð1Þ is 20 kHz, ½r ~ � ¼ 10, and the substrate is titanium (a), steel (b), and molybdenum (c). �1 Conclusions The interface shear stratification mechanism of sandwich structures is investigated in the paper. The forced propagation solution of sandwich constructions excited by SH waves is derived by global matrix methods and integral transformation methods. Combined with the interface fatigue failure theory, the necessary condition of interface shear delamination excited by symmetrical alternating stresses is analyzed and discussed. The analytical results of normalized interface shear stress amplitude are calculated by adaptive Gauss–Kronrod quadrature numerical integration methods. The effects of structural parameters, accretion, and matrix materials on the normalized interfacial shear stress amplitude and interfacial shear stratification intervals are analyzed and discussed. It is found that the interface shear stress amplitude first increases and then resonates with the increase of excitation frequency as well as accretion thickness and matrix thickness. The coupling resonance frequency of interfacial shear delamination is an inherent property of sandwich structure, which is determined by structure sizes and material properties. In other words, when the excitation frequency is equal to the coupling resonance fre- quency, the interface shear stress resonates, and interface shear delamination must occur. The coupling resonance effect of interfacial shear delamination could be explained by using a single degree of freedom system. In addition, in excitation frequency, matrix thickness, and accretion thickness, the interface shear stress amplitude in any two of the domains increases with the increase of the other one. For interfacial shear delamination, frost is the best accretion among frost, rime ice, and glaze ice, and titanium is the best substrate among titanium, steel, and molybdenum. 26 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1191 0(0) The results could provide a theoretical basis for the structure as well as excitation source optimization of interface shear stratification in ultrasonic de-accretion technology and preventing laminates layering. But it is worth noting that the results only apply to the case of infinite plate. The problem of finite plate will be further studied in the future. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding This work was supported by the National Natural Science Foundation of China (grant number 51675401). ORCID iD Feng Guo https://orcid.org/0000-0003-2816-7817 References 1. Gupta A and Ghosh A. Isogeometric static and dynamic analysis of laminated and sandwich composite plates using nonpolynomial shear deformation theory. Compos Part B-Eng 2019; 176: 107295. 2. Sahla M, Saidi H, Draiche K, et al. Free vibration analysis of angle-ply laminated composite and soft core sandwich plates. Steel Compos Struct 2019; 33: 663–679. 3. Gupta A and Ghosh A. Static and transient analysis of sandwich composite plates using isogeometric analysis. Mech Adv Mater Struct 2019; 26: 81–87. 4. Yang Y, Boom R, Irion B, et al. Recycling of composite materials. Chem Eng Process 2012; 51: 53–68. 5. Pickering SJ. Recycling technologies for thermoset composite materials-current status. Compos Part A Appl Sci Manuf 2006; 37: 1206–1215. 6. Tao C, Ji H, Qiu J, et al. Characterization of fatigue damages in composite laminates using Lamb wave velocity and prediction of residual life. Compos Struct 2017; 166: 219–228. 7. Higuchi R, Okabe T and Nagashima T. Numerical simulation of progressive damage and failure in composite laminates using XFEM/CZM coupled approach. Compos Part A Appl Sci Manuf 2017; 95: 197–207. 8. Palacios J, Smith E, Rose J, et al. Ultrasonic de-icing of wind-tunnel impact icing. J Aircr 2011; 48: 1020–1027. 9. Tan H, Xu G, Tao T, et al. Experimental investigation on the defrosting performance of a finned-tube evaporator using intermittent ultrasonic vibration. Appl Energ 2015; 158: 220–232. 10. Adachi K, Saiki K, Sato H, et al. Ultrasonic frost suppression. Jpn J Appl Phys 2003; 42: 682–685. 11. Wang Z. Recent progress on ultrasonic de-icing technique used for wind power generation, high-voltage transmission line and aircraft. Energ Build 2017; 140: 42–49. 12. Palacios J, Smith E, Rose J, et al. Instantaneous de-icing of freezer ice via ultrasonic actuation. AIAA J 2011; 49: 1158–1167. 13. Li D and Chen Z. Experimental study on instantaneously shedding frozen water droplets from cold vertical surface by ultrasonic vibration. Exp Therm Fluid Sci 2014; 53: 17–25. 14. Tan H, Tao T, Xu G, et al. Experimental study on defrosting mechanism of intermittent ultrasonic resonance for a finned- tube evaporator. Exp Therm Fluid Sci 2014; 52: 308–317. 15. Kalkowski MK, Waters TP and Rustighi E. Delamination of surface accretions with structural waves: piezo-actuation and power requirements. J Intel Mater Syst Struct 2017; 28: 1454–1471. 16. Overmeyer A, Palacios J and Smith E. Ultrasonic de-icing bondline design and rotor ice testing. AIAA J 2013; 51: 2965–2976. 17. Budinger M, Pommier-Budinger V, Napias G, et al. Ultrasonic ice protection systems: analytical and numerical models for architecture tradeoff. J Aircr 2016; 53: 680–690. 18. Venna SV and Lin YJ. Mechatronic development of self-actuating in-flight deicing structures. IEEE/ASME Trans Mechatron 2006; 11: 585–592. 19. Venna SV, Lin YJ and Botura G. Piezoelectric transducer actuated leading edge de-icing with simultaneous shear and impulse forces. J Aircr 2007; 44: 509–515. 20. Zeng J and Song B. Research on experiment and numerical simulation of ultrasonic de-icing for wind turbine blades. Renew Energ 2017; 113: 706–712. 21. Wang Z, Xu Y, Su F, et al. A light lithium niobate transducer for the ultrasonic de-icing of wind turbine blades. Renew Energ 2016; 99: 1299–1305. 22. Wang Y, Xu Y and Lei Y. An effect assessment and prediction method of ultrasonic de-icing for composite wind turbine blades. Renew Energ 2018; 118: 1015–1023. 1192 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control 40(3) 27 23. Habibi H, Cheng L, Zheng H, et al. A dual de-icing system for wind turbine blades combining high-power ultrasonic guided waves and low-frequency forced vibrations. Renew Energ 2015; 83: 859–870. 24. Wang Y, Xu Y and Huang Q. Progress on ultrasonic guided waves de-icing techniques in improving aviation energy efficiency. Renew Sust Energ Rev 2017; 79: 638–645. 25. Wang D, Tao T, Xu G, et al. Experimental study on frosting suppression for a finned-tube evaporator using ultrasonic vibration. Exp Therm Fluid Sci 2012; 36: 1–11. 26. Tan H, Zhang X, Zhang L, et al. Ultrasonic influence mechanism of a cold surface frosting process and an optimised defrosting technique. Appl Therm Eng 2019; 153: 113–127. 27. Bednarik M, Cervenka M, Lotton P, et al. Analytical solutions for elastic SH-waves propagating through an isotropic inhomogeneous layer. Compos Struct 2019; 220: 875–887. 28. Golub MV, Fomenko SI, Bui TQ, et al. Transmission and band gaps of elastic SH waves in functionally graded periodic laminates. Int J Solids Struct 2012; 49: 344–354. 29. Mardanshahi A, Shokrieh MM and Kazemirad S. Identification of matrix cracking in cross-ply laminated composites using Lamb wave propagation. Compos Struct 2020; 235: 111790 30. Zhu F, Pan E, Qian Z, et al. Waves in a generally anisotropic viscoelastic composite laminated bilayer: impact of the imperfect interface from perfect to complete delamination. Int J Solids Struct 2020; 202: 262–277. 31. Pollard HF. Sound waves in solids. London: Pion, 1977. 32. He JH. Generalized variational principles for buckling analysis of circular cylinders. Acta Mech 2020; 231: 899–906. 33. He JH. Generalized hellinger-reissner principle. J Appl Mech 2000; 67: 326–331. 34. He JH. Coupled variational principles of piezoelectricity. Int J Eng Sci 2001; 39: 323–341. 35. He JH. An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams. Appl Math Lett 2016; 52: 1–3. 36. He JH. Hamilton’s principle for dynamical elasticity. Appl Math Lett 2017; 72: 65–69. 37. He JH. Inverse problems of determining the unknown shape of oscillating airfoils in compressible 2D unsteady flow via variational technique. Aircr Eng Aerosp Tech 2000; 72: 18–24. 38. Al-Furjan MSH, Habibi M, Jung DW, et al. A computational framework for propagated waves in a sandwich doubly curved nanocomposite panel. Eng Comput-Germany. Epub ahead of print 3 August 2020: 1–18. https://doi.org/10.1007/ s00366-020-01130-8 39. Al-Furjan MSH, Oyarhossein MA, Habibi M, et al. On the wave propagation of the multi-scale hybrid nanocomposite doubly curved viscoelastic panel. Compos Struct. Epub ahead of print 15 September 2020; 255: 112947. DOI: 10.1016/j. compstruct.2020.112947. 40. He JH. Variational iteration method – a kind of non-linear analytical technique: some examples. Int J Nonlin Mech 1999; 34: 699–708. 41. He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 2000; 114: 115–123. 42. He JH. Homotopy perturbation technique. Comput Method Appl Mech Eng 1999; 178: 257–262. 43. He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003; 135: 73–79. 44. He JH and El-Dib YO. Homotopy perturbation method for Fangzhu oscillator. J Math Chem 2020; 58: 2245-2253. 45. Anjum N and He JH. Homotopy perturbation method for N/MEMS oscillators. Math Method Appl Sci Epub ahead of print 24 June 2020: 1–15. DOI: 10.1002/mma.6583. 46. Debnath L and Bhatta D. Integral transforms and their applications. Boca Raton, FL: CRC Press, 2014. 47. Tuan HS and Ponamgi SR. Surface excitation of a Love wave. IEEE Trans Sonics Ultrason 1972; 19: 442–443. 48. Williams AR, Feibelman PJ and Lang ND. Green’s-function methods for electronic-structure calculations. Phys Rev B 1982; 26: 5433. 49. Sakurai T. Green’s function methods for potential magnetic fields. Sol Phys 1982; 76: 301–321. 50. Rungger I, Droghetti A and Stamenova M. Non-equilibrium Green’s function methods for spin transport and dynamics. In: Handbook of materials modeling: methods: theory and modeling. Berlin, Germany: Springer, 2020, pp.957–983. 51. Alekseev AS and Mikhailenko BG. The solution of dynamic problems of elastic wave propagation in inhomogeneous media by a combination of partial separation of variables and finite-difference methods. J Geophys 1980; 48: 161–172. 52. Polyanin AD and Zhurov AI. Separation of variables in PDEs using nonlinear transformations: applications to reaction– diffusion type equations. Appl Math Lett 2020; 100: 106055. 53. Ali EYAA. The transform of partial differential equations to ordinary differential equations by using integral transforms and separation of variables methods. Master’s Dissertation, University of Gezira, Sudan, 2017. 54. Yilmaz C, Topal S, Ali HQ, et al. Non-destructive determination of the stiffness matrix of a laminated composite structure with lamb wave. Compos Struct 2020; 237: 111956. 55. Ding L, Zhu HP and Wu L. Analysis of mechanical properties of laminated rubber bearings based on transfer matrix method. Compos Struct 2017; 159: 390–396. 56. Knopoff L. A matrix method for elastic wave problems. Bull Seismol Soc Am 1964; 54: 431–438. 57. Gao J, Lyu Y, Zheng M, et al. Modeling guided wave propagation in multi-layered anisotropic composite laminates by state-vector formalism and the Legendre polynomials. Compos Struct 2019; 228: 111319. 28 Guo and Wu Journal of Low Frequency Noise, Vibration and Active Control1193 0(0) 58. Ogilvy JA. A model for the ultrasonic inspection of composite plates. Ultrasonics 1995; 33: 85–93. 59. Li X, Wang H and Li G. Reanalysis assisted metaheuristic optimization for free vibration problems of composite laminates. Compos Struct 2018; 206: 380–391. 60. Kolsky H. Stress waves in solids. Chelmsford, MA: Courier Corporation, 1963. 61. He JH. Generalized equilibrium equations for shell derived from a generalized variational principle. Appl Math Lett 2017; 64: 94–100. 62. He JH. Hybrid problems of determining unknown shape of bladings in compressible S2-flow in mixed-flow turbomachin- ery via variational technique. Aircr Eng Aerosp Technol 1999; 71: 154–159. 63. He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006; 20: 1141–1199. 64. He JH. A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718. 65. He JH. Fractal calculus and its geometrical explanation. Results Phys 2018; 10: 272–276. 66. Allam O, Draiche K, Bousahla AA, et al. A generalized 4-unknown refined theory for bending and free vibration analysis of laminated composite and sandwich plates and shells. Comput Concr 2020; 26: 185–201. 67. Menasria A, Kaci A, Bousahla AA, et al. A four-unknown refined plate theory for dynamic analysis of FG-sandwich plates under various boundary conditions. Steel Compos Struct 2020; 36: 355–367. 68. Rabhi M, Benrahou KH, Kaci A, et al. A new innovative 3-unknowns HSDT for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Geomech Eng 2020; 22: 119. 69. Chikr SC, Kaci A, Bousahla AA, et al. A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin’s approach. Geomech Eng 2020; 21: 471–487. 70. Refrafi S, Bousahla AA, Bouhadra A, et al. Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations. Comput Concr 2020; 25: 311–325. 71. Rahmani MC, Kaci A, Bousahla AA, et al. Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory. Comput Concr 2020; 25: 225–244. 72. Boussoula A, Boucham B, Bourada M, et al. A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates. Smart Struct Syst 2020; 25: 197–218. 73. Polyanin AD. Functional separation of variables in nonlinear PDEs: general approach, new solutions of diffusion-type equations. Math Basel 2020; 8: 90. 74. Polyanin AD and Zhurov AI. Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations. Commun Nonlinear Sci 2016; 31: 11–20. 75. He JH and Wu XH. Variational iteration method: new development and applications. Comput Math Appl 2007; 54: 881–894. 76. He JH. Variational iteration method – some recent results and new interpretations. J Comput Appl Math 2007; 207: 3–17. 77. He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004; 151: 287–292. 78. He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004; 156: 527–539. 79. He JH. Asymptotology by homotopy perturbation method. Appl Math Comput 2004; 156: 591–596. 80. Tuan HS and Ponamgi SR. Excitation of Love waves in a thin film layer by a line source. IEEE Trans Sonics Ultrason 1972; 19: 9–14. 81. Hashimoto K and Yamaguchi H. Excitation and propagation of shear-horizontal-type surface and bulk acoustic waves. IEEE Trans Ultrason Ferroelectr Freq Control 2001; 48: 1181–1188. 82. Singh AK, Singh S, Kumari R, et al. Impact of point source and mass loading sensitivity on the propagation of an SH wave in an imperfectly bonded FGPPM layered structure. Acta Mech. Epub ahead of print June 2020; 231(6): 2603–2627. DOI: 10.1007/s00707-020-02659-x. 83. Gander W and Gautschi W. Adaptive quadrature – revisited. BIT Numer Math 2000; 40: 84–101. 84. Calvetti D, Golub G, Gragg W, et al. Computation of Gauss-Kronrod quadrature rules. Math Comp 2000; 69: 1035–1052. 85. Johnson RW. Algorithm 988: AMGKQ: an efficient implementation of adaptive multivariate Gauss-Kronrod quadrature for simultaneous integrands in Octave/MATLAB. ACM Trans Math Softw 2018; 44: 1–19. 86. Rahman A, Huang H, Ai C, et al. Fatigue performance of interface bonding between asphalt pavement layers using four- point shear test set-up. Int J Fatig 2019; 121: 181–190.

Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Dec 13, 2020

Keywords: Sandwich plate; SH waves; interfacial stratification; coupling resonance mechanism; interface shear stress; forced propagation solution

There are no references for this article.