Research Article

Static Bending, Vibration, and Buckling Responses of a Sandwich Beam Composed of Five Layers Considering Honeycomb Core and CNTRC with SMA Particles and Temperature-Dependent Material Properties Using SSDT

Mohsen Asgari , Mehdi Mohammadimehr ^* , Mohammad Arabzadeh-Ziari , Erfan Arabzadeh-Ziari

Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-53153, Iran

1.      Introduction

The sandwich structures consist of two thin face sheet layers with high strength and a thick, low-density core layer to achieve excellent properties. In the military and aerospace industries, it is very important to design a structure resistant to blast and impact loads [1–13]. Ghorbanpour Arani et al. [6] studied the free and forced vibrations of double viscoelastic piezoelectric nanobeam systems (DVPNBSs) using nonlocal viscoelasticity theory and the Euler–Bernoulli beam model.

Also, some technical expressions on the application of a sandwich beam are extensively used in aerospace, automotive, civil, marine, drone, and other high-performance structures because of their high specific stiffness and strength, excellent fatigue resistance, long durability, and many other superior properties compared to the conventional metallic materials [14–17].

1.1.  Structures and Applications of Carbon Nanotube

Carbon nanotubes have high strength and good thermal and electrical properties[18–21]. Currently, composites with carbon nanotube particles are used to construct aircraft bodies, transmission pipes, and Transistors[22, 23]. Also, functionally graded carbon nanotube (FG-CNT) is used in medical science as a platform for drug delivery [24–29]. Garg et al. [30] studied the vibration and bending analyses of FG-CNT reinforced composite (FG-CNTRC) for a sandwich beam. Ghorbanpour Arani et al. [31] investigated the vibration and instability of visco-elastically coupled CNTRC microtubes conveying fluid. Li et al. [32] investigated the effect of adding carbon nanotubes on the behavior of the structure by enhancing the volume fraction (V) of CNTs, and the strength of the composite increases. Excessive V-CNT has led to a decrease in strength. Zhang et al. [33] added carbon nanotubes to polymer materials to improve the mechanical properties of composites. Their experimental results have shown that epoxy composites reinforced by CNTs with a length of 5 to 20 micrometers have greater strength and toughness than shorter CNTs of 0.5 to 2 micrometers. Arshadi et al. [34] studied the vibration and bending of a sandwich microbeam with a functionally graded porous core (FG) and two polymer plates reinforced with graphene nanoplatelets (GPLs). Their results showed that the VA distribution type has the highest deviation, and the AV type has the lowest static bending value. Ghorbanpour Arani et al. [35] analyzed the stress of a long piezoelectric polymer hollow cylinder reinforced with carbon nanotubes under magnetic and mechanical loadings. The results showed that increasing V-CNTs increases the strength of the nanocomposite cylinder. Also, carbon nanotubes increase bending resistance and fracture toughness [36, 37]. Ghorbanpour Arani et al. [38] investigated the nonlinear dynamic analysis of an embedded functionally graded sandwich nanobeam (FGSNB) with integrated magnetically restrained layers. They concluded that the magnetic layers significantly affect the dynamic behavior of FGMSNB. Arefi et al. [39–41] investigated the free vibration analysis of a three-layer microbeam consisting of an elastic microcore and two piezomagnetic plates, which are located on the Pasternak base. Their results showed that the increase of three microscale parameters significantly increased three natural frequencies. Arefi et al. [42] studied the buckling and free vibration analyses of a sandwich beam comprised of softcore and functionally graded graphene platelets (GPLs) reinforced composite face sheets.

1.2.  Smart Structures and Applications of SMA

Shape-memory alloys are metals used for smart structures, such as KTS-SM alloy, one of the most reliable Ni-Ti alloys. KTS-SM alloy is used in the construction of hot water shut-off valves for water purification, evaporative emission control systems, dental implants, and fire dampers [43, 44]. Asadi et al. [45] analyzed the post-buckling behavior of multilayer beams with memory alloy under uniform heating. The fibers of this alloy are very effective at delaying the heat flux in the beam. The results have shown that increased fiber volume fraction or fiber prestressing increases the strain temperature. Alambeigi et al. [46] investigated forced vibration for a sandwich beam with FG porous core and composite layers embedded with SMA. Their numerical results have shown that with the increase in the volume fraction of memory alloys, the dimensionless natural frequency increases. Vishal et al. [47] investigated active control of vibration based on SMA as an actuator. Garafolo and McHugh [48] illustrated the effect of SMA in reducing vibration produced by aeroelastic flutter. Their results show that activating the embedded SMA wire leads to enhanced stiffness and a decrease in the vibration amplitude of the flutter. Rocha et al. [49] studied the post-tensioning mechanical feasibility of annealed glass beams by activating Fe-SMA reinforcement with external bonding. Zheng et al. [50] researched the reverse effect of shape memory alloys. They concluded that the shape memory alloy has a maximum reverse effect of 1.63%. Viet et al. [51] researched porous composite beams, including the FGP SMA layer. Their results showed that increasing the thickness of the SMA layer increases energy absorption. Lu et al. [52] investigated the method of improving the dynamic stiffness of structures at high temperatures with SMA actuators. Samadpour et al. [53] studied the vibration of thermally buckled SMA hybrid composite sandwich plates. Park et al. [54] research was done on the vibration of thermally post-buckled composite plates embedded with SMA. Their numerical results showed that using SMA increases the critical temperature. Kuang et al. [55] used optical fiber sensors to actively control smart composites with shape memory alloy sheets. Khalili et al. [56] studied the bending properties of composites reinforced with SMA wires. Their results showed that the pre-strained wires caused a 13% increase in the maximum bending load. Nejati et al. [57] analyzed the thermal vibration of composite reinforced with SMA wires. Their results showed that SMA wires increase the thermal frequency.

This work's main contributions and novelty are to consider simultaneously static bending, free vibration, and buckling responses of a sandwich beam composed of a five-layer beam using sinusoidal shear deformation theory (SSDT). In this work, a sandwich beam composed of five layers including a honeycomb core, carbon nanotubes reinforced composite (CNTRC) at the top and bottom of the core, and also shape memory alloy (SMA) in the form of nanoscale particles with matrix on top and bottom of CNTRC is obtained. Moreover, it must be explained that employing two types of nanoparticles: SMA is known for its smart material properties, and CNT is used as reinforcement in composite structures. This study derives the governing equations of equilibrium using the principle of minimum potential energy for deflection and buckling analyses, while Hamilton's principle is employed to obtain the governing equations of motion and then the natural frequencies of a sandwich beam. The obtained results investigate the influence of the core thickness, volume fraction of carbon nanotube particles, and SMA on the deflection, frequency, and buckling of a sandwich beam.

2.      Mathematical Modeling

Figure 1 shows the sandwich beam consists of a honeycomb core and two nanocomposite face sheets reinforced by CNT/Epoxy/fiber and SMA. To derive a mathematical model of the sandwich beam, the used assumptions and modifications are presented as follows:

• The sandwich structure is assumed to be in the elastic region. Thus, Hook’s law becomes correct.
• The top and bottom layers of cores have the same properties.
• It is assumed that the core and face sheets are fully bonded. Therefore, there is no slippage at the interfaces between the core and face sheets.
• For every point on a cross-section, the displacement fields are the same.
• In this research, the two-end boundary conditions are assumed to be simply supported, and the governing equations of motion are solved using Navier's type method.

Fig. 1. A schematic view of a sandwich beam  composed of five layers

Young’s modulus is calculated using the rule of mixture and Halpin–Tsai equations as follows [58]:

where  and  denote the Young's modulus and volume fraction of carbon nanotube (CNT) particles mixed with resin. ,  and  are the length, thickness, and width of the carbon nanotube.  and  are the volume fraction of CNT and fiber, respectively, and  and  denote Young's modulus of resin and fiber, respectively. Also, and  are the coefficients of the equipment [59].

where  is the Young's modulus of carbon nanotube (CNT) particles mixed with resin that is defined as follows:

where  is the shear modulus of carbon nanotube particles mixed with resin, and also,  is the shear modulus of fiber and the coefficients ,  and  are calculated as follows:

Mass density and Poisson’s ratio for CNT/Epoxy/fiber are obtained as:

 and  are defined as the volume fraction of nanocomposite fibers and epoxy resin and are related to each other by the following formula:

Table 1 shows the mechanical properties of the honeycomb core, resin, and reinforcements, including SMA and CNT.

Table 2 demonstrates the dependent material properties of CNT [60–62].

Table 1. The mechanical properties of the honeycomb core, resin, and reinforcements, including SMA and CNT in the present work

Table 2. Temperature-dependent material properties of CNT

3.      Governing Equations

3.1.  Displacement Fields

The displacement fields of sinusoidal shear deformation theory (SSDT) are considered as follows [63–68]:

where , , and  are the axial and transverse displacements and slope of the beam. Using equations (2-10), the following strain-displacement relations for a sandwich beam are obtained as follows:

where ,  , are the normal strains and , ,  denote the shear strains.

3.2.  Strain Energy

The strain energy variation for the composite beam is calculated as follows[69]:

where and  are the normal and shear stress for a sandwich beam, respectively.

Substituting Eq. (13) into Eq. (14), the following relation is obtained:

The resultant forces and moments are defined as:

where ,  are the axial and bending resultant forces, respectively. Also,  and  denote the higher shear and bending resultant forces, respectively.

Substituting Eq. (16) into Eq. (15), the following result is obtained:

3.3.  Kinetic Energy

The variation of kinetic energy for the CNT/Epoxy/fiber sandwich beam is calculated as follows:

3.4.  External Work

The variation of the total external work, including transverse load  buckling load  and axial load  can be formulated as:

where  and  are extensive loads applied to the composite beam in the x and z directions, respectively. Also, and  are the mechanical and thermal buckling loads, respectively.

3.5.  Hamilton’s Principle

The governing equations of motion are obtained based on Hamilton's principle, which is defined as follows[67]:

Using equation (24), the three governing equations of the five layers nanocomposite beam considering honeycomb core and CNTRC with SMA particles are obtained as follows:

In Eq. (25), the coefficients are defined as follows:

4.      Solution Method

Substituting the response of displacements into Eqs. (25) yields the stiffness, mass, and geometric stiffness of matrices as follows:

where:

Therefore, the mass matrix , bending stiffness matrix  and geometric stiffness matrix  are obtained as follows:

where:

The natural frequency of a sandwich beam is written as:

The following matrix form is used to calculate the deflection due to the application of transverse force (F):

To obtain the critical buckling load, we have:

5.      Thermal Flux

In this section, the one-dimensional heat flux of the composite beam is investigated based on Fig. 2 as follows:

where  is an empirical heat transfer coefficient is the surface area of the beam ,  is the perimeter of the beam    is the ambient temperature  and  is the thermal conductivity of the solid ,  is the cross-sectional area of the beam (m²) For constant , ,  and , Eq. (36) yields:

where  is the ambient temperature.

By solving Eq. (37), the following result is obtained:

where  is equal to the square root of the multiply  (an empirical heat transfer coefficient) by  (the perimeter of the beam) divided to the multiply  (the thermal conductivity of the solid) by  (the cross-sectional area of the beam) in which Eq. (38), A and B are defined as follows:

Fig. 2. A schematic view of steady heat conduction in a beam.

6.      Validation

To validate the results, they are compared with existing literature by Thai [70] for EBBT, TBT, and RBT in Tables 3 and 4. It is shown that the results of this article are in good agreement with other studies.

Table 3. Nondimensional fundamental frequency .

Table 4. Nondimensional critical buckling load .

7.      Results and Discussions

In this study, static bending, vibration, and buckling responses of a five layers nanocomposite sandwich beam considering honeycomb core, carbon nanotubes reinforced composite (Matrix and Resin) (CNTRC) in top and bottom of core, and also, shape memory alloy as well as matrix in top and bottom of CNTRC are investigated. Also, the influences of different parameters such as thickness ratio of core, V-CNT, V-SMA, and fiber orientation on deflection, natural frequency, and critical buckling force are studied.

Table 5. The efficiency coefficient for CNTs, mechanical and geometrical properties of carbon nanotubes, epoxy resin, fibers, and shape memory alloy

Figure 3 shows the effect of volume fraction for carbon nanotube particles and the aspect ratio (L/H) of the sandwich beam on deflection, frequency, and buckling load. Increasing the volume fraction of carbon nanotube particles leads to an increase in the stiffness of the nanocomposite beam. Then, it increases the critical buckling load and natural frequency and reduces the deflection.

(a)

(b)

(c)

Fig. 3. Effect of volume fraction of carbon nanotube particles on: (a) deflection, (b) natural frequency, (c) critical buckling load

Figure 4 shows the effect of volume fraction on the SMA-reinforced matrix of a sandwich beam. Although increasing V-SMA has a small effect on the beam's behavior, it reduces the deflection and increases the critical buckling load.

(a)

(b)

Fig. 4. Effect of volume fraction of SMA on: (a) deflection, (b) critical buckling load

Figure 5 shows the influence of fiber placement angles on composite beam behavior. The lowest deflection and highest natural frequency occur in mode and the highest deflection and lowest natural frequency occurs in mode . The highest and lowest critical buckling loads happen in  respectively in case  and . If  , the maximum and minimum critical buckling load occurs in mode  and , respectively. Also, if  the maximum and minimum buckling load occurs in mode  and , respectively.

(a)

(b)

(c)

Fig. 5. Effect of fiber angles on: (a) deflection, (b) natural frequency, (c) critical buckling load

Figure 6 shows the effect of the core on the total thickness ratio. Because the Young's modulus of the core is lower than the Young's modulus of carbon nanotube particles and shape memory alloy particles, by increasing  the equivalent stiffness decreases, according to Eqs. (29), (31), and (32), with a decrease in the stiffness matrix, the critical buckling load and natural frequency of the beam decrease, and the deflection of the beam increases.

(a)

(b)

(c)

Fig. 6.  Effect of core thickness on: (a) Deflection, (b) Natural frequency, (c) critical buckling load

The distribution of temperatures along the length of the sandwich beam is shown in Fig. 7. It is shown that with an increase across the beam length, temperature based on boundary conditions decreases. Figure 8 shows the heat flux across the beam length for various thickness ratios. It is concluded that with an increase in thickness ratio, the thickness of the core increases, while the thickness of the face sheets decreases because the conductivity coefficient for CNT is higher than the core; thus, the heat flux across the beam length decreases. Figure 9 shows that with increasing temperature changes, the structure becomes softer, and the critical buckling load decreases.

Fig. 7. Temperature variation along the length of the beam

Fig. 8. The heat flux variation along the length of the beam for various thickness ratio

Fig. 9. The effect of temperature change on critical buckling load

8.     Conclusions

This article investigated the vibration frequency, critical buckling load, and static bending of the five-layer composite beam of honeycomb core and carbon nanotubes reinforced with memory particles using the sinusoidal shear deformation theory. The governing equations of static equilibrium or motion were obtained using the principle of minimum potential energy and Hamilton's principle, respectively. The results of this research are stated as:

• Increasing the volume fraction of carbon nanotubes increases the vibration frequency and critical buckling load while decreasing beam deflection because the stiffness of a sandwich beam is enhanced.
• Increasing the volume fraction of SMA has very little effect on the deflection, vibration frequency, and critical buckling load of the beam.
• The lowest deflection and highest natural frequency occur in mode and the highest deflection and lowest natural frequency occur in mode  the highest and lowest critical buckling loads happen in , respectively, in case  and  If  
  the maximum and minimum critical buckling load occurs in mode  and   Also, if  the maximum and minimum buckling load occurs in mode  and  respectively.
• By increasing the ratio of the core thickness to the total thickness, the beam's vibration frequency and critical buckling load decrease, and the beam deflection increases because the sandwich beam becomes softer.
• The distribution of temperatures versus the length of the sandwich beam shows that with an increase across the beam length, temperature based on boundary conditions decreases.
• It is concluded that with an enhancement in thickness ratio, the heat flux decreases, it is due to increasing the thickness of the core; while the thickness of face sheets decreases, because the conductivity coefficient for CNT is higher than the core.
• It is shown that with increasing temperature changes, the structure becomes softer, and then the critical buckling load decreases.

Acknowledgments

The authors would like to thank the referees for their valuable comments and for increasing the quality of the present work. Also, the authors are thankful to the Iranian Nanotechnology Development Committee for supporting this research and the University of Kashan under Grant No. 1223097/5.

Funding Statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this article.

Nomenclature

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