Wave propagation characteristics of the electrically GNP-reinforced nanocomposite cylindrical shell
Abstract
In this article, wave propagation characteristics of a size-dependent graphene nanoplatelet (GNP) reinforced composite cylindrical nanoshell coupled with piezoelectric actuator (PIAC) and surrounded with viscoelastic foundation is presented. The effects of small scale are analyzed based on nonlocal strain gradient theory (NSGT) which is an accurate theory employing exact length scale parameter and nonlocal constant. The governing equations of the GNP composite cylindrical nanoshell coupled with PIAC have been evolved using Hamilton’s principle and solved with assistance of the analytical method. For the first time in the current study, wave propagation electrical behavior of a GNP composite cylindrical nanoshell coupled with PIAC based on NSGT is examined. The results show that, by decreasing the PIAC thickness, extremum values of phase velocity occur in the lower values of the wave number. Another important result is that, by increasing GPL%, the effects of PIAC thickness on the phase velocity decrease. Finally, influence of PIAC thickness, wave number, applied voltage, and different GPL distribution patterns on phase velocity is investigated using mentioned continuum mechanics theory. Useful suggestion of this research is that for designing of a nanostructure coupled with PIAC attention should be given to PIAC thickness and applied voltage, simultaneously. The outputs of the current study can be used in the structural health monitoring and ultrasonic inspection techniques.
Keywords
Wave propagation Maxwell equation Graphene nanoplatelet Piezoelectric actuator NSGT Applied voltage1 Introduction
Researches proved that subjoining even a very meager amount of graphene into primary polymer matrix can desperately improve its mechanical, thermal and electrical properties [5, 6, 7, 8, 9]. It is worse to mention that nanostructures reinforced with GNP are more applicable in engineering design, so focus on dynamic modeling of the nanostructure with GNP reinforcement is useful and important. Furthermore, polymer matrix reinforced by various types of nanofillers is one of the most efficient and easily extruded nanocomposite materials with a wide range of applications such as field effect transistors, electromechanical actuators, biosensors and chemical sensors, solar cells, photoconductor and superconductor devices. For this reason, the investigation of their mechanical characteristics is a great interest for engineering design and manufacturing. In the field reinforcement structures, Habibi et al. [10, 11, 12, 13, 14] with the aid of some methods improved the mechanical property of macro-structures. Dong et al. [15] presented an analytical study on linear and nonlinear vibration characteristics and dynamic responses of spinning FG graphene-reinforced thin cylindrical shells with various boundary conditions and subjected to a static axial load. Dong et al. [16] investigated the buckling behavior of FG graphene-reinforced porous nanocomposite cylindrical shells with spinning motion and subjected to a combined action of external axial compressive force and radial pressure. Dong et al. [17] concerned with free vibration characteristics of the functionally graded graphene-reinforced porous nanocomposite cylindrical shell with spinning motion. In their result section, detailed parametric studies on natural frequencies and critical spinning speeds of the GPL-reinforced porous nanocomposite cylindrical shell are carried out, especially, effect of initial hoop tension on vibration characteristics of the spinning cylindrical shell is numerically discussed. Jang et al. [18] presented the postbuckling and buckling behaviors of FG multilayer nanocomposite beams reinforced with graphene platelets (GNPs). They investigated that GNPs have a remarkable reinforcing effect on the buckling and postbuckling of nanocomposite beams. In another work, Feg et al. [19] found out nonlinear bending behavior of a novel class of multilayer polymer composite beams reinforced with graphene platelets (GNPs). They studied that beam with a higher value fraction of GNPs and symmetric distribution in such a way is less sensitive to the nonlinear deformation.
None of the above researches have taken size effects into account in wave propagation analysis of structures. Continuum mechanics theories including classical theory [20, 21] and size-dependent theories are used to model micro/nanostructures. As classical theory does not consider submicron discontinuities of structure, so it cannot capture size-dependent effects when scale turns to micro or nano. Size-dependent theories including nonlocal [22, 23, 24, 25, 26, 27, 28, 29, 30], strain gradient [31, 32] and couple stress [33, 34, 35, 36, 37, 38] theories are better choices and present more accurate outputs in these cases. It should be noted that mentioned theories consist of size-dependent parameters which their exact values must be determined by experimental data or numerical simulations [39, 40, 41]. For simple structures such as graphene sheets or carbon nanotubes, production of material for experiment or simulation is a straightforward process, but for composite structures, the process becomes complicated and encourages the researchers to approximate the models through mathematics and theories. Gul and Aydogdu [42] employed some length scale-dependent theories for investigation of wave propagation in double-walled carbon nanotubes. As a comparative research, they compared doublet mechanics results with classical elasticity, strain gradient theory, nonlocal theory and lattice dynamics. Experimental wave frequencies and doublet mechanics theory of graphite have good agreement with each other, so they showed that doublet mechanics theory has higher accuracy.
Ard and Aydogdu [43] studied torsional wave propagation for a multiwalled carbon nanotubes in the framework of Eringen’s nonlocal elasticity theory. They considered effect of van der Waals interaction and reported that the interaction has an important role in torsional wave propagation. In another work, Islam et al. [44] represented size effects on torsional wave propagation of circular nanostructure, such as nanoshafts, nanorods and nanotubes. They demonstrated the significance of considering the integral nonlocal model and nanoscale relations in dispersion characteristics of circular nanostructures. Aydogdu [45] employed nonlocal elasticity theory for studying longitudinal wave propagation in multiwalled carbon with including van der Waals force effect in the axial direction. In the result of this paper, the effects of various parameters on wave propagation were examined in detail. In recent years, incorporating the local and nonlocal curvatures in constitutive relations, NSGT has emerged. Based on this theory, the stress of submicron-scale structures appears in both nonlocal stress and pure strain gradient stress fields. Lim et al. [46] used thermodynamic framework to derive NSGT equations, so that the higher-order nonlocal parameters and the nonlocal gradient length coefficients were considered.
Applications of NSGT in vibration analysis of nanostructures [47, 48, 49] have attracted many researcher’s attentions. Zeighampour et al. [50] investigated wave propagation in double-walled carbon nanotube surrounded by Winkler foundation using the nonlocal strain gradient theory. In their results, nonlocal strain gradient theory and classical theory were compared in terms of the influences of nonlocal and material length scale parameters, wave number, fluid velocity and stiffness of elastic foundation on phase velocity. In another work [51], they modeled a composite cylindrical micro/nanoshells and studied the variation of phase velocity versus material length scale parameters and nonlocal constant. According to that paper, an increase in material length scale increases the phase velocity while nonlocal parameter acts vice versa. Zeighampour et al. [52] conducted a wave propagation study on a thin cylindrical nanoshell surrounded by visco-Pasternak foundation based on nonlocal strain gradient theory. The viscoelastic properties were modeled by Kelvin–Voigt theory. They indicated that the structure has a better stability condition using strain gradient theory in comparison with classical theory. In the field of the wave propagation behavior of a structure coupled with piezoelectric actuators, Lat. Am et al. [53] studied wave propagation of the two-layer piezoelectric composite structure. As a parametric study, they in this work showed the effects of volume fraction, thickness and elastic constant on the wave dispersion of the structure. Arani et al. [54] investigated wave dispersion of the FG carbon nanotube-reinforced piezoelectric composite. They in this work included the visco-Pasternak in their mathematical modeling. The structure was subjected to magnetic and electric fields. Their results showed that external voltage has a significant effect on the wave desperation behavior. Zhou [55] modeled the piezoelectric cylindrical shells and investigated surface effect on wave dispersion of the nanostructure. Their results presented that at the higher mode surface effect has a significant effect on the wave propagation of the structure. Bishe et al. [56] presented wave propagation in smart laminated composite cylindrical shells reinforced with carbon nanotubes in hygrothermal environments. They investigated the effects of temperature/moisture variation, CNT volume fraction and orientation, piezoelectricity, shell geometry, stacking sequence and material properties of the host substrate laminated composite shell at different axial and circumferential wave numbers and the results of their work showed that the temperature/moisture variation influences moderately on the dispersion solutions of smart laminated CNT-reinforced composite shells. Bishe et al. [57] focused on the wave propagation in piezoelectric cylindrical composite shells reinforced with angled and randomly oriented CNTs. Bishe et al. [58] studied and analyzed wave propagation in a piezoelectric cylindrical composite shell reinforced with CNTs by using the Mori–Tanaka micromechanical model and considering the transverse shear effects and rotary inertia via the first-order shear deformation shell theory. Bishe et al. [59] investigated wave behavior in a piezoelectric coupled laminated fiber-reinforced composite cylindrical shell by considering the transverse shear effects and rotary inertia. In the results of their work presented a comparison of dispersion solutions from different shell theories with different axial and circumferential wave numbers and piezoelectric layer thickness is provided to illustrate the transverse shear and rotary inertia effects on wave behavior of a laminated fiber-reinforced composite shell. Guo et al. [60] analyzed the effects of FG interlayers on the wave propagation in covered piezoelectric/piezomagnetic cylinders. They in this work showed that high-order modes are more sensitive to the gradient interlayers, while the low-order modes are more sensitive to the electromagnetic surface conditions. The wave propagation of the porous FG plates with the aid of some shear deformation theories was analyzed by Yahia et al. [61]. As an application, they showed that their results are useful for ultrasonic inspection. Also, they presented the effect of porosity on the wave propagation behavior of the structure. The present study investigates the wave propagation piezoelectric behavior of a GPLRC cylindrical nanoshell coupled with PIAC based on NSGT with considering the calibrated values of nonlocal constant and material length scale parameter for the first time. In this regard, influence of wave number, critical applied voltage and GNP distribution pattern on phase velocity are investigated using mentioned continuum mechanics theory.
2 Mathematical modeling
2.1 Nonlocal strain gradient theory
2.2 Constitutive equations for nanocomposite core and piezoelectric layers
2.3 Piezoelectric layers
2.4 Displacement field of the cylindrical shell
2.5 Derivation of governing equations and boundary conditions
2.6 External work
2.7 Kinetic energy
2.8 Strain energy
3 Solution procedure
In Eq. (38), \(c\) and \(m\) are called phase velocity and wave number of a laminated nanocomposite cylindrical shell. These parameters are propagation speeds of the particles in a laminated nanocomposite cylindrical shell. With considering \(\mu = 0\), the phase velocity of classical continuum theory is computed.
3.1 Parametric study
Material properties of the epoxy and GPL [66]
Material properties | Epoxy | GPL |
---|---|---|
Young’s modulus (GPa) | 3 | 1010 |
Density (kg m^{−3}) | 1200 | 1062.5 |
Poisson’s ratio | 0.34 | 0.186 |
Material properties of piezoelectric layer which is composed of BiTiO_{3}–CoFeO_{4} [40]
Material constants | BiTiO_{3}–CoFeO_{4} |
---|---|
Elastic (GPa) | \(c_{11} = 226,\,\,c_{12} = 125,\,\,c_{13} = 124,\,\,c_{33} = 216,\,\,c_{44} = 44.2 ,\,c_{55} = 44.2,\,\,c_{66} = 50.5\) |
Piezoelectric (C m^{−2}) | \(e_{31} = - 2.2,\,e_{33} = 9.3,\,\,e_{15} = 5.8\) |
Dielectric (10^{−9} C V m^{−1}) | \(s_{11} = 5.64,\,s_{22} = 5.64,\,s_{33} = 6.35\) |
Piezomagnetic (N A m^{−1}) | \(q_{15} = 275,\,q_{31} = 290.1,\,q_{33} = 349.9\) |
Magnetoelectric (\(10^{ - 12}\) N V \({\text{C}}^{ - 1}\)) | \(d_{11} = 5.367,\,\,d_{33} = 2737.5\) |
Magnetic (\(10^{ - 6}\) N\({\text{s}}^{2}\) \({\text{C}}^{ - 2}\)) | \(r_{11} = - 297,\,\,r_{33} = 83.5\) |
Thermal moduli (\(10^{5}\) N km^{−2}) | \(\beta_{1} = 4.74,\,\,\,\beta_{3} = 4.53\) |
Pyroelectric (\(10^{ - 6}\) C \({\text{N}}^{ - 1}\)) | \(P_{3} = 25\) |
Pyromagnetic (\(10^{ - 6}\) N A m \({\text{K}}^{ - 1}\)) | \(\lambda_{3} = 5.19\) |
Mass density (\(10^{3}\) kg m^{−3}) | \(\rho = 5.55\) |
The effect of different PIAC thickness, pattern of GNP and mode numbers on phase velocity (km/s) of GNPRC nanoshell with s = 1(1/nm), g_{GPL} = 1%, R = 1 nm, h = R/10
n = 1 | n = 2 | n = 3 | ||||
---|---|---|---|---|---|---|
h_{p}= h/20 | h_{p}= h/10 | h_{p}= h/20 | h_{p}= h/10 | h_{p}= h/20 | h_{p}= h/10 | |
Pure epoxy | ||||||
Φ (mVolt) | ||||||
0 | 0.950952 | 1.135213 | 0.571641 | 0.671976 | 0.750838 | 0.852311 |
1 | 0.897843 | 1.098203 | 0.477231 | 0.607062 | 0.682519 | 0.802453 |
2 | 0.841338 | 1.059881 | 0.359352 | 0.534309 | 0.606549 | 0.749282 |
3 | 0.780703 | 1.020101 | 0.174627 | 0.449928 | 0.519583 | 0.692037 |
Pattern 1 | ||||||
Φ (mVolt) | ||||||
0 | 1.353548 | 1.438770 | 0.846692 | 0.883556 | 1.207535 | 1.217339 |
1 | 1.316756 | 1.409735 | 0.782695 | 0.835218 | 1.166244 | 1.182940 |
2 | 1.278890 | 1.380079 | 0.720790 | 0.783900 | 1.123434 | 1.147509 |
3 | 1.239851 | 1.349763 | 0.648726 | 0.728973 | 1.078926 | 1.110948 |
Pattern 2 | ||||||
Φ (mVolt) | ||||||
0 | 1.352978 | 1.438318 | 0.825098 | 0.866104 | 1.120738 | 1.144654 |
1 | 1.316169 | 1.409273 | 0.762958 | 0.816729 | 1.076119 | 1.107998 |
2 | 1.278284 | 1.379607 | 0.695278 | 0.764165 | 1.029565 | 1.070087 |
3 | 1.239225 | 1.349279 | 0.620248 | 0.707703 | 0.980804 | 1.030781 |
Pattern 3 | ||||||
Φ (mVolt) | ||||||
0 | 1.354507 | 1.439530 | 0.867739 | 0.900698 | 1.287439 | 1.285275 |
1 | 1.317744 | 1.410512 | 0.808897 | 0.853336 | 1.248794 | 1.252744 |
2 | 1.279909 | 1.380875 | 0.745417 | 0.803181 | 1.208915 | 1.219345 |
3 | 1.240904 | 1.350578 | 0.675993 | 0.749674 | 1.167672 | 1.185005 |
Pattern 4 | ||||||
Φ (mVolt) | ||||||
0 | 1.354502 | 1.439649 | 0.835099 | 0.875785 | 1.168384 | 1.191728 |
1 | 1.317731 | 1.410579 | 0.773634 | 0.826913 | 1.125595 | 1.156530 |
2 | 1.279835 | 1.380888 | 0.706837 | 0.774960 | 1.081113 | 1.120225 |
3 | 1.240731 | 1.350535 | 0.633021 | 0.719259 | 1.034719 | 1.082703 |
After the minimum value, phase velocity is improved with increasing the wave number. Figures 8 and 9 demonstrate that, by decreasing the PIAC thickness, the minimum and maximum values of the phase velocity shift to left. For a better comprehensive, by decreasing the PIAC thickness, extremum values of phase velocity are seen in the lower values of the wave number. As an important result, by decreasing the PIAC thickness, the phase velocity decreases in the all range of the wave number. In addition, by increasing the applied voltage, the phase velocity decreases in the all range of the wave number. As a comparison report, the PIAC thickness and applied voltage have direct and inverse effects on the phase velocity of the nanostructure, respectively. Furthermore, as a new report of literature, comparison of the effects of PIAC thickness and applied voltage on the phase velocity can be found from these figures which show that by decreasing the effect of PIAC thickness the effect of applied voltage on the phase velocity become more significant. In another word, these figures show that, by increasing the PIAC thickness, the graphs of Fig. 9 are closely than the graphs of Fig. 10. As a useful suggestion of this research is that for designing of a nanostructure coupled with PIAC should be attention to the PIAC thickness and applied voltage, simultaneously. It is worth to mention that increasing the PIAC thickness and applied voltage has negative and positive effects on the stability of the nanostructure, respectively.
4 Conclusion
- 1.
The results show that, by decreasing the PIAC thickness, extremum values of the phase velocity occur in the lower values of the wave number.
- 2.
It is observed that, by increasing GPL%, the effects of PIAC thickness on the phase velocity decrease.
- 3.
The PIAC thickness and applied voltage have direct and inverse effects on the phase velocity of the nanostructure, respectively.
- 4.
The results demonstrate that, by adding GNP in the pure epoxy matrix, the phase velocity of the nanostructure improves.
- 5.
Another important result is that, by increasing GPL%, the effects of PIAC thickness on the phase velocity decrease.
Notes
References
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