Vibration Characteristics of Shear Thickening Fluid-Based Sandwich Structures

The vibration attenuation mechanism of shear thickening ﬂ uid- (STF-) ﬁ lled sandwich structures was investigated in this study. Structural equivalent damping, sti ﬀ ness, and mass increased simultaneously with the increase in the volume fraction of shear thickening ﬂ uid. However, the damping ratio decreased and natural frequency increased with the increase in structural mass. Thus, the damping ratio was not a monotonically increasing function of the volume fraction of STF. A modi ﬁ ed shear strain model of the damping layer was developed based on the following conditions: (1) under the condition of small strain, shear thickening ﬂ uid was regarded as linear viscoelastic material, and (2) the warpage of the sandwich beam was considered during deformation and the in ﬂ uence of STF on the shear strain of sandwich beam. According to the modi ﬁ ed shear strain model of the damping layer, the shear thickening occurred at 1Hz to 20Hz during vibration. Therefore, the resonance point of the structure shifted to the left. The predictions were in excellent agreement with the experimental results. The results demonstrated that shear thickening ﬂ uid improved the vibration damping performance of the sandwich structure, while the thickening ability was not the higher, the better.


Introduction
Viscoelastic damping sandwich structures have not only high stiffness, light weight, and high deformation resistance but also high damping properties. It is a more widely used damping technology in vibration and noise reduction at present [1]. Although the passive control method has the advantages of low cost and design simplicity, it is difficult to give an effective response to external changes. These weaknesses were resolved by the development of smart materials, such as shape memory polymers (SMPs) [2], magnetorheological (MR) [3], and electrorheological (ER) fluids [4]. Sandwich structures with these smart materials need only an external power supply to be activated and have the advantage of better adaptability to external changes [5]. However, long-term use of external power supply damages other devices due to electric breakdown. One approach to avoiding the requirement of an external power supply is to use a material whose characteristics are changed with the loading conditions [6].
Till 2006, the application of shear thickening fluid (STF) in sandwich structures has grown in importance [7]. STF is composed of particulate phase and carrier fluid, demonstrating a significant increase in viscosity when the shear rate exceeded the critical shear rate. The thickening process is quick and reversible [8]. On account of the remarkable properties, STF is applied to body armors [9,10], dampers [11], polishing operations [12], and machining processes [13]. After integrating STF with sandwich structures, the stiffness [7] and damping properties [14] of STF-filled sandwich structures can be designed by STF with the different rheological properties. Viscoelastic damping sandwich structures consist of a damping layer (viscoelastic damping materials) and a structural layer [15]. The mismatch of stiffness between the structural layer and damping layer led to large shear deformation of the damping layer and then resulted in increased energy dissipation [16]. Therefore, the dynamic properties of sandwich structures change with the shear rheological property of STF. The shear thickening properties of STF have been extensively studied [17]. However, there is little research on the impact of the thickening ability of STF on the vibration performance of sandwich structures [18].
Fischer et al. first explored the effect of STF on the dynamic properties of sandwich structures. The results suggested that the resonant frequency and damping raised as the viscosity of STF increased. The stiffness of sandwich structures increased under better interfacial stress transfer [19]. In a separate study published later, the influence of particle morphology on structural dynamic characteristics was investigated. The results reflected that these parameters can directly affect shear moduli of STF, and therefore, the damping properties of the sandwich structure changed [20]. Based on magnetorheological fluid sandwich beam theory, the influence of thickness ratio of the surface layer to the middle layer, excitation frequency, amplitude, and location on the natural frequency of the sandwich structures integrating STF was analyzed by Wei et al. [21]. Gürgen and Sofuoğlu investigated the effects of STF on vibration performance of tubes under different boundary conditions with vibration tests [22]. Later, dynamic properties of PEG-400 and STFfilled sandwich structures were researched for contrast. The experimental results were consistent with Fischer's research conclusions [23]. The starting point for previous studies was strain sensitivity of STF [24,25]. Additionally, the shear strain of the STF interfacial layer was calculated based on classical beam theory [26]. The effects of STF on the dynamic properties of sandwich structures were deduced by the occurrence of shear thickening behavior. The impact of factors influencing STF rheology, vibration conditions, and structure design on sandwich structures was also considered in other studies. Generally, the vibration attenuation mechanism of shear thickening fluid-(STF-) filled sandwich structures was not deeply studied. The shear strain model of Fischer ignored the warpage of the sandwich beam during deformation and the influence of STF on the shear strain of the sandwich beam [27,28]. Besides, Newtonian fluids and shear-thinning fluids were neglected in these studies, making it difficult to determine whether the thickening characteristics or the viscous properties of fluid played a role.
In this study, the vibration performance of PEG-400, 18 vol.% STF, 49 vol.% STF, and 61 vol.% STF-filled sandwich structures was studied to demonstrate the vibration attenuation mechanism of STF-filled sandwich structures. The shear rheological measurements were performed to reveal shear rheological behavior. Then, the vibration experiment was conducted to research the influence of PEG and STF on the dynamic properties of sandwich structures. The modified shear strain model was developed to analyze the vibration attenuation mechanism of shear thickening fluid on the vibration performance of sandwich structures.

Modeling Shear Strain of Damping Layer
The change in the dynamic properties of sandwich structures was correlated with damping layer rheological response under shear deformation. STF was a shear strainsensitive material, with its damp and rigid performance increasing when the critical shear strain was exceeded. To analyze the effect of STF on the dynamic properties of sandwich structures, the modified shear strain model based on Fischer's calculation [19] was developed. Previous models ignored the warpage of the sandwich beam during deformation and the influence of STF on the shear strain of the sandwich beam. These conditions were considered when the modified model was established. Shear thickening fluid was regarded as linear viscoelastic material under small strain to simplify the calculation [21]. Figure 1(a) is the schematic diagram of the cantilever sandwich beam filled with STF under external excitation. The sandwich beam consisted of three layers. The lower and upper layers were aluminum alloy with thicknesses of h 1 and h 3 , and the middle layer was STF with a thickness of h 2 . The length and width of the sandwich beam were a and b, respectively. Based on the classic sandwich plate theory, some assumptions were made about the model. Specifically, (1) the bond of each layer has no slippage during deformation; (2) the transverse displacement w of each layer was equal; (3) the tensile stress along the length direction of the STF layer was ignored; and (4) the section of each layer remained flat after being deformed.
Based on the above assumptions, the x-x section in the STF-filled sandwich beam was taken for analysis. The deformation relationship of the x-x section is illustrated in Figure 1(b) (considering that the STF layer had large shear deformation, the x-x section warped after deformation, and the plane was no longer kept). The displacements of the sections of the lower and upper layers in the x -direction after deformation were u 1 and u 3 , respectively; the transverse displacement of each layer in the y -direction was w. Assuming that the displacement of the STF layer in the x -direction was u 2 , the shear strain γ of the sandwich beam was expressed as where According to Equation (1), the shear stress was where the complex shear modulus was G * = G ′ + iG ″ ; G ′ and G ″ denoted the storage modulus and the loss modulus of STF, respectively. As illustrated in Figure 1(c), the total shear force after the deformation of the x-x section was The shear force of the section was related to the transverse distributed load q, obtained from Figure 1(d): From Equation (5), q was the partial derivative of shear force S with respect to x, namely, Given the inertial force of the sandwich beam during deformation, the transverse distributed load q was given as where m denoted the mass per unit length of the sandwich beam and f ðx, tÞ represented the periodic excitation force, described as where F and ω indicated the amplitude and frequency of the excitation force, respectively; x 0 denoted the position coordinate of the excitation force; δ represented the Dirac delta function. Substituting Equations (3) and (4) into Equation (6) yielded where EI = E 1 I 1 + E 3 I 3 .
By taking the partial derivative of γ in Equation (1) with respect to x and substituting the partial derivative into Equation (9) to eliminate u 1 and u 3 , it was obtained that

Advances in Polymer Technology
The transverse displacement of the STF-filled sandwich beam in the time domain could be represented as [21] whereω n represented then -order natural frequency of the sandwich beam. Substituting Equations (7), (8), and (11) into Equation (10) yielded where The boundary conditions of the cantilever sandwich beam satisfied the following conditions: (1) The shear strain at the fixed end was 0 (2) The shear rate at the fixed end was 0 By integrating Equation (12) through the boundary conditions (14) and (15), the shear strain of the STF layer was obtained as

Preparation of STFs and STF-Filled Sandwich Structures.
STFs were prepared by dispersing silica particles (from Evonik Degussa Co., Ltd.) with a mean particle size of 650 nm into polyethylene glycol medium with M w values of 400 (PEG-400, from Sinopharm Chemical Reagent Co., Ltd.). Figure 2(a) exhibits the transmission electron microscope (TEM) images of silica particles. The silica particles were monodisperse and spherical. By obtaining a homogeneous suspension, the sample was placed in a vacuum drying oven for 24 hours at 25°C to remove the bubbles. In the study, STFs at the conditions of different concentrations (18 vol.%, 49 vol.%, and 61 vol.% for SiO 2 ) were produced. STF-filled sandwich structures were prepared by one layer of 3 mm thick STF and two layers of 1 mm thick aluminum alloy (5052 type aluminum alloy). Silicone rubber with a width and a thickness of 3 mm was applied for sealing around the sandwich beams. A bakelite with a length of 30 mm, a width of 10 mm, and a thickness of 3 mm was placed at the clamping end. Figure 2(b) displays STF-filled sandwich structures, with the dimension of 260 × 10 × 5 mm 3 .

Method
3.2.1. Rheology Tests. The rheological behavior of PEG and STF was explored by an Anton-Paar Physica MCR 301 rheometer. The steady-shear experiments were performed with 25 mm diameter parallel plates. The frequency sweep and strain amplitude sweep tests were conducted on a cone plate with a diameter of 50 mm and an angle of 1°.

Vibration
Tests. The vibration experiment was conducted by the vibration beam testing (VBT) set-up to investigate the influence of PEG and STF on the dynamic properties of sandwich structures, as illustrated in Figure 3. The boundary conditions of all samples were single-fixed. One end of the sample was clamped, and the other end was excited to generate vibration from a shaker (MS-100). The force signal was collected by a piezoelectric pressure sensor (CL-YD-303). The sandwich beam acceleration signal was measured by the acceleration sensor (KISTLER 8640A50). The power spectrum of the vibration signal was obtained by CRAS modal analysis software. The experimental design is presented in Table 1. The neat sandwich structure is sealed with silicone rubber to ignore the effect of silicone rubber on the damping.  Figure 4. PEG-400 was the dispersion medium of shear thickening fluid. As a Newtonian fluid, the viscosity of PEG-400 did not change with the shear rate. With the addition of silica particles, rheological properties and initial viscosity of PEG-400 solution presented various kinds. Terminal hydroxyl and abundant oxygen atoms in PEG-400 formed a considerable number of hydrogen bonds with the silanol groups of silica particles [29]. Hence, the initial viscosity of STF was greater than that of PEG-400. 18 vol.% STF presented as non-Newtonian shearthinning fluids. In the initial stages, electrostatic force and Brownian force acted on silica particles. The particles 4 Advances in Polymer Technology repelled one another and were arranged orderly. With the increasing shear rate, the original aggregates of particles were dispersed. According to the hydrodynamic cluster theory [30], the collisions between particles were neglected when the number of silica particles was small. Therefore, 18 vol.% STF possessed shear-thinning property. Besides, 49 vol.% STF exhibited continuous shear thickening (CST) behavior, and its viscosity slowly increased at the thickening stage. Additionally, 61 vol.% STF had a discontinuous shear thickening (DST) phenomenon, and its viscosity increased rapidly at the transition point. Chen et al. [31] investigated microstructural features of STF. The force between silica particles changed from hydrodynamic forces to contact friction force, resulting in the transition from CST to DST. A stable friction contact network was formed along with the DST phenomenon occurring. The silica particles were subject to compression during the DST stage [25,32].

Dynamic Rheological.
The changes in structure's damping and stiffness were induced by the STF's dynamic viscoelastic characteristics [33]. Considering that the frequency and strain varied in vibration, the effects of frequency and strain amplitude on the viscoelasticity of the damping layer were explored. The viscoelastic moduli of PEG-400 and STF versus angular frequency at the strain of 100% are presented in Figure 5  5 Advances in Polymer Technology mainly exhibited elasticity characteristics because its storage modulus was higher than the loss modulus. After the addition of silica particles, the viscous properties of mixtures were significant since the particle chain breakage resulted in permanent deformation when the angular frequency was high. The modulus of 61 vol.% STF changed suddenly in which significant nonlinear viscoelastic behavior appeared.
Regarding most polymer solutions, the strain amplitude in the linear region was generally 0.01-0.1. The nonlinear region appeared with the increase in the strain [34]. The large amplitude oscillatory shear (LAOS) tests were performed to characterize the nonlinear viscoelastic properties of STF. Figure 5 These results implied that shear thickening occurred more easily with the increase in the strain and angular frequency. Fourier rheology was a general quantitative method for LAOS tests. In the linear region, the stress response contained none but the first harmonic, while the multiharmonic response was observed in the nonlinear region [35,36]. The output moduli measured by the method of LAOS tests generally were the first-harmonic moduli. Figures 5(a)-5(d) suggest that it was difficult to characterize the nonlinear properties of materials because the contribution of higher order was ignored. G L ′ , G M ′ , η L ′ , and η M ′ were defined to study the instantaneous nonlinear characteristics. The local tangent modulus at γ = 0 of the elastic Lissajous-Bowditch curves was G M ′ (minimum-strain modulus), and local secant modulus at γ/γ 0 = x = 1 of the elastic Lissajous-Bowditch curves was G L ′ (large-strain modulus) [37]; similarly, η M ′ and η L ′ represented the modulus at the lowest and highest shear rates, respectively. According to the graphics demon-stration, the parameters of quantitative nonlinear characteristics were also defined. The strain-stiffening ratio (S) and the shear thickening ratio (T) can be obtained by where S = 0 demonstrated linear response, S > 0 suggested strain stiffening with one cycle, and S < 0 indicated strain softening with one cycle. T = 0 represented linear response, T > 0 signified shear thickening with one cycle, and T < 0 amounted to shear thinning with one cycle. The minimum-strain modulus G M ′ equated to the zero-strain elasticity, whereas the large-strain modulus G L ′ captured the large strain elasticity. Furthermore, the strain-stiffening ratio S correctly captured the true relative magnitude of the intracycle strain stiffening. A similar approach defined shear thickening ratio T. η M ′ captured the viscous material coefficient at zero shear rate, whereas η L ′ represented the viscous material coefficient at maximum strain rate. The sign of S and T was determined by e 3 and v 3 , respectively. The degree of nonlinear contribution was measured by e 3 /e 1 and v 3 /v 1 . Among them, e 3 and v 3 were dominant components in nonlinear response, and e 1 and v 1 were linear components. Therefore, the changes of e 3 /e 1 and v 3 /v 1 values should be analyzed at different frequencies and strains [34].
The measured e 3 /e 1 and v 3 /v 1 values from LAOS tests of 49 vol.% STF and 61 vol.% STF with the angular frequency at 1.5 rad/s are presented in Figures 6(a) and 6(b). The results revealed that the elastic nonlinearity of 61 vol.% STF was strain softening (e 3 < 0) under a small strain; after that, the strain hardening appeared (e 3 > 0). Additionally, 49 vol.% STF first exhibited elastic linearity (e 3 = 0). Gradually, the elastic properties changed from linearity to strain hardening. With the increase in STF concentration, the e 3 /e 1 values were added. The reason was that the more the silica particles of STF, the longer the length of particle chains. At the oscillatory shear strain, more particle chains twisted and rotated. Thus, the performance of elasticity of 61 vol.% STF became better. As reflected in Figure 6(b), the viscous nonlinearity of 49 vol.% STF and 61 vol.% STF improved little by little with the increasing strain. The variation trend of the ratio of viscous coefficients was similar to the elastic coefficients' ratios. More particle chains increased resistance to particle motion. Therefore, the viscous properties of 61 vol.% STF were enhanced. Comparative analysis implied that the e 3 /e 1 values increased and the viscous deformation occurred at small strains with the increase of the volume fraction of STF.  Advances in Polymer Technology critical point, the greater the angular frequency, the higher the viscoelastic moduli. Thus, elastic and viscous coefficients ratios increased [38].

Dynamic
Response of the Sandwich Structure. Figure 7(a) provides a frequency response curve of NSS. The frequency corresponding to the peak value of vibration acceleration was the natural frequency ω n . According to the halfpower bandwidth method [39], the damping ratio ζ was calculated by  where f 1 and f 2 denoted the frequency corresponding to 0.707 time peak value. Owing to different damping layers, the natural frequency and damping ratios of the structure were changed in variety, as presented in Figure 7(b). The natural frequencies of each specimen are illustrated in Figure 7(c). With the increasing content of silica particles, the natural frequency increased and became lower than that of the neat sandwich structure. The natural frequency of the sandwich structure was affected by the mass matrix and stiffness matrix. Bending stiffness was related to the moment of inertia and elastic modulus. It can be obtained that where E and I Z represented elastic modulus and moment of inertia, respectively. The moment of inertia of rectangular section was where B and H denoted the width and height of the rectangular, respectively. Generally, the moment of inertia of the rectangular hollow section was smaller than that of the rectangular solid section. According to Equation (21), the moment of inertia of the hollow section was that of a solid section of the same width and height minus that of a solid section, which was the same size as the middle hollow 9 Advances in Polymer Technology section [40]. Thus, the moment of inertia of the neat sandwich structure was the smallest. The elastic moduli of the lower and upper aluminum alloy layers were 70 GPa. However, the maximum modulus of elasticity in shear of the viscous liquid was about 10 5 Pa. The farther away from the neutral layer, the larger the normal stress was. Under the same stress, the strain of the neat sandwich structure was slightly lower than that of the viscous fluid-filled sandwich structure. Therefore, the bending stiffness of the neat sandwich structure was a little higher. The viscoelastic moduli of mixtures were enhanced with the increase in silica particles' volume fraction. Moreover, the natural frequency rose when the modulus was improved. The variation trend of the natural frequency with damping layers was consistent with these inferences. The influence of stiffness on natural frequency was stronger than that of mass in these samples. Consequently, the effect of mass on natural frequency was not discussed in this part. Figure 7(d) exhibits the damping ratio of each sandwich structure. The damping ratio rose first and then decreased with increasing silica particles. The damping ratio could be computed as where c eq and k eq denoted equivalent damping and equivalent stiffness, respectively. The damping ratio of sandwich structure was proportional to equivalent damping and inversely proportional to equivalent stiffness and mass of sandwich structure. The equivalent damping was decided by the viscous properties of STF; the equivalent stiffness was determined by the elastic moduli of the lower and upper aluminum and STF. Obviously, with the increasing of volume fraction of SiO 2 , equivalent damping, equivalent stiffness, and mass of sandwich structure increased. When the volume fraction of SiO 2 was 18%, the increase in equivalent damping played a major role. With the further increase of volume fraction of SiO 2 , the increase in equivalent stiffness and mass determined the damping ratio. Therefore, at higher silica loadings, damping got lower. The constrained damping layer was mainly subjected to shear during vibration. The dynamic properties of sandwich structures were primarily connected with the shear properties of the damping layer [41]. PEG-400 belonged to Newtonian fluid, and its viscosity did not vary with the shear strain. Nonetheless, the damping properties of STF were distinct under different shear strain conditions. The action of the damping layer on structural dynamic performance was analyzed by the previous shear strain model. In this study, x was set to be a, and the maximum strain was obtained by simplifying Equation (16): The vibration test results suggested that the exciting force of each sample was about 10 -2 N. The values of a and h were constant, and the cos ðωtÞ values were from 0 to 1. The maximum strain was affected by the complex shear modulus alone. The γ max of 61 vol.% STF and 49 vol.% STF was estimated to be 10 -3 % and 1%, respectively, following the Cox-Merz rule [42]: The shear thickening occurred at 1-20 Hz under dynamic conditions. Therefore, the resonance point of the structure shifted to the left after adding STF. The vibration attenuation mechanism of STF-filled sandwich structures is illustrated in Figure 8. Equation (16) indicated that the shear strain of STF was different. As demonstrated by Figure 8(a), the shear strain depended on where STF was. The shear thickening behavior occurred when the shear strain was greater than the critical shear strain, as indicated in Figure 8 bonds might be formed between one end group hydroxyl and surface silanol group. Hydrogen bonds may also be formed between PEG-400 molecules and silica particles. Compared with the one end group hydroxyl, the silanol groups were more acidic. PEG-400 was more likely to bond with silica particles than to form hydrogen bonds between PEG-400 molecules [43]. The formation of particle chain increased E and c eq . Additionally, the natural frequency of the neat sandwich structure was larger than that of the viscous fluid-filled sandwich structure since the maximum E value was smaller than 70 GPa. ζ was proportional to c eq . Nevertheless, k eq and m remarkably affected the structural damping ratio. Therefore, ζ was not a monotonically increasing function of silica particles.

Conclusions
The vibration characteristics of fluids with different rheological properties filled with sandwich structures were investigated in this study. The results did not demonstrate that the larger the thickening ability, the better the vibration damping performance of the sandwich structure. STF improved the stiffness and damping properties of sandwich structures as a result of its high viscosity. With the increasing content of silica particles, the natural frequency and equivalent damping of STF-filled sandwich structures increased. According to the modified shear strain model of the damping layer, shear thickening behavior occurred under fixed strain amplitudes. Besides, the energy dissipation and stress transfer increased owing to the shear thickening effect. However, STF significantly influenced the mass of the structure, which increased by 33.6% in max. With the increasing mass, the damping ratio decreased, and natural frequency increased. Therefore, the vibrational properties were not monotonically increasing functions of the content of STF. The results demonstrated that after integrating STF with sandwich structures, sandwich structures filled with STF give effective response on external changes without external power supply. Sandwich structures filled with STF can be used for aeronautics, transportation, and buildings. In all these applications, adaptive control of structural vibration is realized by shear rate sensitivity of STF.

Data Availability
The data used to support the findings of this study are available from the corresponding authors upon request.

Conflicts of Interest
There are no conflicts to declare.