Abstract
Composite shells, which are being widely used in engineering applications, are often under thermal loads. Thermal loads usually bring thermal stresses in the structure which can significantly affect its static and dynamic behaviors. One of the possible solutions for this matter is embedding Shape Memory Alloy (SMA) wires into the structure. In the present study, thermal buckling and free vibration of laminated composite cylindrical shells reinforced by SMA wires are analyzed. Brinson model is implemented to predict the thermomechanical behavior of SMA wires. The natural frequencies and buckling temperatures of the structure are obtained by employing Generalized Differential Quadrature (GDQ) method. GDQ is a powerful numerical approach which can solve partial differential equations. A comparative study is carried out to show the accuracy and efficiency of the applied numerical method for both free vibration and buckling analysis of composite shells in thermal environment. A parametric study is also provided to indicate the effects of like SMA volume fraction, dependency of material properties on temperature, layup orientation, and prestrain of SMA wires on the natural frequency and buckling of Shape Memory Alloy Hybrid Composite (SMAHC) cylindrical shells. Results represent the fact that SMAs can play a significant role in thermal vibration of composite shells. The second goal of present work is optimization of SMAHC cylindrical shells in order to maximize the fundamental frequency parameter at a certain temperature. To this end, an eightlayer composite shell with four SMAreinforced layers is considered for optimization. The primary optimization variables are the values of SMA angles in the four layers. Since the optimization process is complicated and time consuming, Genetic Algorithm (GA) is performed to obtain the orientations of SMA layers to maximize the first natural frequency of structure. The optimization results show that using an optimum stacking sequence for SMAHC shells can increase the fundamental frequency of the structure by a considerable amount.
Keywords
Shape memory alloys; Composite shells; Free vibration; Thermal buckling; Optimization; Genetic algorithm
1. INTRODUCTION
SMAs are a class of new smart materials that have been receiving increased attention due to their two unique thermomechanical characteristics, Shape Memory Effect (SME) and PseudoElasticity (PE). SME refers to the material ability to recover coldforged permanent large strains (up to 10%) upon a mild increase in material temperature. These permanent strains can also be recovered by loading in opposite direction of inelastic strain which is called ferroelasticity. At high temperatures, SMAs behave pseudoelasticity and can recover large strains during mechanical loadingunloading patterns. This property of SMAs produces a hysteretic loop that is responsible of energydissipation. The unique properties of SMAs lie in the phase transition between martensite and austenite. Setting SMA components in the form of wires into composite laminates can control the static and dynamic structural response. That is why some researchers have been motivated to analyze thermomechanical behavior of smart structures with SMA components, experimentally or/and numerically. Birman (1997) Birman, V., (1997). Stability of functionally graded shape memory alloy sandwich panels. Journal of Smart Material Structure 6: 278–286. studied the stability loss phenomenon in a simplysupported rectangular plates embedded with SMA fibers subjected to uniaxial loading. As concluded, nonuniform distribution of fibers through the width is more influential than the conventional uniform fiber dispersion. To examine the influence of SMA fibers on the stability characteristics of SMAHC plates, a series of experimental studies were carried out by Thompson and Loughlan (2001) Thompson, S.P., Loughlan, J., (2001). Enhancing the postbuckling response of composite panel structure utilizing shape memory alloy actuators  a smart structural concept. . Journal of Composite Structures 51: 21–36. and Loughlan et al. (2002) Loughlan, J., Thompson, S.P., Smith, H., (2002). Buckling control using embedded shape memory actuators and the utilization of smart technology in future aerospace platforms. Journal of Composite Structures 58: 319–347. . Results of these studies accept the high influence of SMA fibers on the buckling delay and alleviation of postbuckling deflections of SMAHC plates. Roh et al. (2004) Roh J.H., Oh I.K., Yang S.M., Han J.H., Lee I., (2004). Thermal postbuckling analysis of shape memory alloy hybrid composite shell panels. Smart materials and structures 13: 13371344. analyzed thermal postbuckling of SMA composite shell panels by using the finite element method formulated on the basis of the layerwise theory and Brinson model. They showed that embedding SMA wires in the composite shell panel could prevent the snapping phenomenon that is one of the unstable postbuckling behaviors. Using Galerkin approximate, harmonic balance method and Brinson model, Yongsheng and Shuangshuang (2007) Yongsheng, R., Shuangshuang, S., (2007). Large amplitude flexural vibration of the orthotropic composite plate embedded with shape memory alloy fibers. Chinese Journal of Aeronautics 20:415424. examined large amplitude flexural vibration of the orthotropic composite plate with embedded SMA wires. Kuo et al. (2009) Kuo, S.Y., Shiau, L.C., Chen, K.H., (2009). Buckling analysis of shape memory alloy reinforced composite laminates. Journal of Composite Structures 90:188195. employed experimental data from the SMA curves and investigated buckling of SMA reinforced composite laminates using finite element method. Li et al (2010) Li, S.R., Yu, W.S., Batra, R.C., (2010). Free vibration of thermally pre/postbuckled circular thin plates embedded with shape memory alloy fibers. Journal of Thermal Stresses 33:79–96. analyzed free vibration of thermally per/postbuckled circular thin plates with embedded SMA fibers based on Brinson model and by using shooting method. Mirzaeifar et al (2011) Mirzaeifar, R., Shakeri, M., DesRoches, R., Yavari, A., (2011). A semianalytic analysis of shape memory alloy thickwalled cylinders under internal pressure. Journal of Arch Appl Mech 81: 1093–1116. presented a semi analytical analysis of thickwalled SMA cylinder under internal pressure. Free vibration analysis of buckled SMA reinforced crossply and angleply plates was performed by Shiau et al. (2011) Shiau, L.C., Kuo, S.Y., Chang, S.Y., (2011). Free vibration of buckled SMA reinforced composite laminates. . Journal of Composite Structures 93: 2678–2684. . They investigated the effects of SMAs on the vibrational behavior of structure by varying the SMA fiber spacing. Using FEM and Brinson model, Khalili et al. (2013) Khalili, S.M.R., Botshekanan Dehkordi, M., Carrera, E., Shariya, M., (2013). Nonlinear dynamic analysis of a sandwich beam with pseudoelastic SMA hybrid composite faces based on higher order finite element theory. . Journal of Composite Structures 96:243–255. analyzed geometrically nonlinear dynamic response of flexible sandwich beams with pseudoelastic SMAHC face sheets. Asadi et al. (2013a) Asadi, H., Bodaghi, M., Shakeri, M., Aghdam, M.M., (2013a). On the free vibration of thermally pre/postbuckled shear deformable SMA hybrid composite beams. . Journal of Aerospace Science and Technology 31:73–86. examined the nonlinear free vibration of SMA composite beams in thermally pre/postbuckled domains based on firstorder shear deformation theory and Brinson model. Also Asadi et al. (2013b) Asadi, H., Bodaghi, M., Shakeri, M., Aghdam, M.M., (2013b). An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams. European Journal of Mechanics A/Solids 42:454468. presented an exact closedform solution for buckling temperature, postbuckling deformation and temperaturedeformation equilibrium path of symmetric and asymmetric simply supported SMAHC beams under uniform temperature rise. Recently, Thermal bifurcation behavior of crossply laminated composite cylindrical shells embedded with SMA fibers was analyzed by Asadi et al. (2015) Asadi H., Kiani Y., Aghdam M.M., Shakeri M., (2015). Enhanced thermal buckling of laminated composite cylindrical shells with shape memory alloy. Journal of composite materials 50(2): 114. . Properties of the constituents were assumed to be temperaturedependent. Donnell’s kinematic assumptions accompanied with the vonKarman type of geometrical nonlinearity were used to derive the governing equations of the shell. Also, onedimensional constitutive law of Brinson was implemented to predict the behavior of SMA fibers through the heating process. Forouzesh and Jafari (2015) Forouzesh, F., Jafari, A. A., (2015). Radial vibration analysis of pseudoelastic shape memory alloy thin cylindrical shells by the differential quadrature method. Journal of ThinWalled Structures, 93:158–168. investigated the radial vibrations of SMAHC cylindrical shells under harmonic internal pressure based on Donnelltype classical shell theory. They used Boyd–Lagoudas model to simulate the nonlinear thermomechanical behavior of SMA fibers and implemented GDQ method and Newmark approach for the analysis. Nonlinear free vibration of thermally buckled SMAHC sandwich plate was examined by Samadpour et al. (2015) Samadpour, M., Sadighi, M., Shakeri, M., Zamani, H., (2015). Vibration analysis of thermally buckled SMA hybrid composite sandwich plate. Journal of Composite Structures 119:251–263. based on shear deformation plate theories. They showed that SMA fibers can significantly affect the natural frequency and postbuckling deflection of sandwich plates. Using the mixed LW (layerwise)/ESL (equivalent single layer) models, Botshekanan Botshekanan Dehkordi et al. (2016) Botshekanan Dehkordi, M., Khalili, S.M.R., Carrera E., (2016). Nonlinear transient dynamic analysis of sandwich plate with composite facesheets embedded with shape memory alloy wires and flexible core based on the mixed LW (layerwise)/ESL (equivalent single layer) models. Composites Part B 87: 5974. studied nonlinear dynamic analysis of sandwich plate with flexible core and SMAHC face sheets. Brinson model was implemented to predict the thermomechanical behavior of SMA fibers. The effect of volume fraction and location of SMAs, the thickness of face sheets, plate aspect ratio, and boundary conditions on dynamic of structure were examined. Recently, Parhi and Singh (2016) Parhi, A., Singh, B.N., (2016). Nonlinear Free Vibration Analysis of Shape Memory Alloy Embedded Laminated Composite Shell Panel. Journal of Mechanics of Advanced Materials and Structures 24: 713724. presented nonlinear free vibration analysis of SMAHC spherical and cylindrical composite shell panels. The governing equations were derived based on higherorder shear deformation plate theory and using nonlinear von Karman strain displacement relations, and were solved by applying ninenodded isoperimetric element. The influence of prestrain of SMAs, volume fraction of SMAs, temperature, and curvature on the linear and nonlinear frequency of structure were discussed in details.
As can be found from the literature survey, although worthwhile researches have been dedicated to analyze composite structures reinforced by SMA wires, a few works have been devoted to study optimization of SMAHC structures. Optimization of SMAHC plates with crossply layup subjected to lowvelocity impact was performed by Birman et al. (1996) Birman, V., Chandrashekhara, K., Sain, S., (1996). An approach to optimization of shape memory alloy hybrid composite plates subjected to lowvelocity impact. Journal of Composites Part B 439446. . The variations of volume fractions of SMA fibers in each direction subject to a constraint on the total volume fraction of the SMAs were considered for the optimization problem. It was shown that using optimum SMA in the plate significantly reduces deflections and stresses. Recently, Kamarian and Shakeri (2017) Kamarian, S., Shakeri, M., (2017). Thermal buckling analysis and stacking sequence optimization of rectangular and skew shape memory alloy hybrid composite plates. Composites Part B : Engineering, 116: 137152. optimized SMAHC skew plates with respect to thermal buckling. They used a meta heuristic algorithm called Firefly Algorithm (FA) for stacking sequence optimization of the plate in order to maximize the critical buckling temperature. It was found that optimization of orientation of SMA wires can improve the buckling temperature of composite structures by a considerable amount.
To the best of the authors' knowledge, there is no published work on the optimization of cylindrical shells with SMA wires. Thus, in the present work, analysis and optimization of SMAHC cylindrical shells are presented. In the first part of results, free vibration and buckling analysis of cylindrical shells under thermal environments are studied. The effect of some parameters like volume fraction of SMAs, prestrain of SMAs, and temperaturedependency of materials on the behavior of structure are examined. GDQ method, which is an efficient numerical method for problems with partial differential equations, is employed to discretize the governing equations and solve the Eigen value problems in order to obtain the natural frequencies and critical buckling temperatures. Then, in the second part of present work, stacking sequence optimization of cylindrical shells reinforced by SMA wires is presented in order to have the maximum natural frequency of structure at a certain temperature. To this end, since the problem cannot be computed analytically and the process takes too much time, GA is employed to predict the best solutions. GA is one of the most approved heuristic methods for optimization problems and has been successfully applied in for various objective functions in composite structures, such as buckling loads, weight, fundamental frequencies, deflection, etc. ( Wu et al. (2012) Wu, Z., Weaver, P.M., Raju, G., Kim, B.C., (2012). Buckling analysis and optimisation of variable angle tow composite plates. Journal of ThinWalled Structures 60:163–172. , Sliseris and Rocens (2013) Sliseris, J., Rocens, K., (2013). Optimal design of composite plates with discrete variable stiffness. Journal of Composite Structures 98:15–23. , LeManh, and Lee (2014) LeManh, T., Lee, J., (2014). Stacking sequence optimization for maximum strengths of laminated composite plates using genetic algorithm and isogeometric analysis. Journal of Composite Structures 116:357–363. , An et al (2015) An, H., Chen, S., Huang, H., (2015). Laminate stacking sequence optimization with strength constraints using twolevel approximations and adaptive genetic algorithm. Journal of Struct Multidisc Optim 51:903–918. , Xu et al. (2015) Xu, C., Lin, S., Yang, Y., (2015). Optimal design of viscoelastic damping structures using layerwise finite element analysis and multiobjective genetic algorithm. Journal of Computers and Structures 157:1–8. , Vosoughi et al. (2016) Vosoughi, A.R., Dehghani Forkhorji, H., Roohbakhsh, H., (2016). Maximum fundamental frequency of thick laminated composite plates by a hybrid optimization method. Journal of Composites Part B 86:254260. , Mashrouteh et al. (2017) Mashrouteh, H., Rahnamayan, S., Esmailzadeh, E., (2017). Optimal Vibration Control and Innovization for Rectangular Plate. Journal of IEEE 549556. , Ou and Mak (2017) Ou, D., Mak, C.M., (2017). Optimization of natural frequencies of a plate structure by modifying boundary conditions. Journal of the Acoustical Society of America 142: 5662. ).
2. Equilibrium equations
Here, the recovery stress of SMA wires is calculated based on the simplified form of the Brinson model ( Brinson and Huang, 1996 Brinson, L. C., Huang, M. S., (1996). Simplifications and Comparisons of Shape Memory Alloy Constitutive Models. Journal of Intelligent Material Systems and structures 7: 108114. ) in which the martensite volume fraction ξ is separated into the stressinduced ξ_{s} and the temperatureinduced components ξ_{T} as equations (1 and 2 )).
where ξ denotes the maximum residual strain and the Young’s modulus is expressed based on the Reuss model as ( Auricchio and Sacco, 1997 Auricchio, F., Sacco, E., (1997). A onedimensional model for superelastic shape memory alloys with different elastic properties between austenite and martensite. International Journal of Non Linear Mechanics 32:1101I 114. )
In equation (3) , E_{A} and E_{M} represents Young’s modulus of the SMA in the pure austenite and the pure martensite phases, respectively. According to ( Brinson, 1993 Brinson, L.C., (1993). Onedimensional constitutive behavior of shape memory alloys: thermomechanical derivation with nonconstant material functions and redefined martensite internal variable. Journal of Intelligent Material Systems and Structures 4: 229242. ), the martensite fractions during heating stage when T>A_{s} and C_{A} (TA_{f})<σ<C_{A}(TA_{s}) can be calculated as equation (4)
in which a subscript ‘0’ indicates the initial state of a parameter and the constant C_{A} is the slope of the curve of the critical stress for reverse phase transformation. Since the martensite fraction depends on the stress and temperature, transformation kinetics must be coupled with equation (4) to formulate a complete governing equation for SMAs. The elastic properties of an SMA/graphite/epoxy layer are found in Appendix.
Consider a composite cylindrical shell as shown in Figure 1 . The strains in terms of the midsurface displacement components (u, ν, w) are defined as equation (5)
Where e_{x}, e_{Ɵ,} e_{xƟ} are the components of axial, circumferential and shear strain, respectively. Considering
where ó_{x}, ó_{Ɵ} and ó_{xƟ} are the components of axial, circumferential and shear, stress, respectively. In equation (6) ,
in which, the stress resultants are defined as equations (8)
Where N^{T}, M^{T}, N^{r} and M^{r} represent thermal force resultant, thermal moment resultant, inplane force and bending moment resultants induced by the SMA fibers respectively defined in Appendix. Furthermore, Q_{x} and Q_{Ɵ} denote shear forces in x and Ɵ direction respectively and defined as equation (9)
3. GDQ Method
GDQ approach is used to solve the governing equation of the SMAHC shell. In the purposed method, the nth order of a continuous function f(x, z) with respect to x at a given point x_{i} can be approximated as a linear sum of weighting values at all of the discrete points in the domain of x , i.e. ( Shu, 2000 Shu, C., (2000). Differential quadrature and its application in engineering, Berlin: Springer. ):
In equation (10)N is the number of sampling points, and c_{ij}^{n} is the x_{i} dependent weight coefficients. In order to determine the weighting coefficients, the Lagrange interpolation basic functions are used as test functions, and explicit formulation for computing these weighting coefficients can be obtained as equations (11 and 12 )) ( Shu, 2000 Shu, C., (2000). Differential quadrature and its application in engineering, Berlin: Springer. ):
where
for the firstorder derivative (n=1), and for higherorder derivative, one can use the equations (13 and 14 )) iteratively ( Shu, 2000 Shu, C., (2000). Differential quadrature and its application in engineering, Berlin: Springer. ):
A simple and natural choice of the grid distribution is the uniform grid spacing rule. However, it was found that nonuniform grid spacing yields results with better accuracy. Hence, in this work, the ChebyshevGaussLabatto quadrature points are used ( equations (15) ), that is ( Shu, 2000 Shu, C., (2000). Differential quadrature and its application in engineering, Berlin: Springer. ),
More details about GDQ method can be found in Shu, (2000) Shu, C., (2000). Differential quadrature and its application in engineering, Berlin: Springer. , Shu and Richards (1992) Shu, C., Richards, B.E., (1992). Application of generalized differential quadrature to solve twodimensional incompressible NavierStockes equations. International Journal For Numerical Methods In Fluids 15:791798. .
4. Genetic algorithm
GA is a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such an inheritance, mutation, selection, and cross over (also called recombination). The basic genetic algorithm is as follows ( Sivanandam and Deepa, 2008 Sivanandam, S.N., Deepa, S.N., (2008). Introduction to Genetic Algorithms, Springer Berlin Heidelberg New York. ):
[start] Genetic random population of n chromosomes (suitable solutions for the problem)
[Fitness] Evaluate the fitness f(x) of each chromosome x in the population.
[New population] Create a new population by repeating following steps until the new population is complete.
[selection] select two parent chromosomes from a population according to their fitness. The better fitness, the bigger chance to get selected.
[crossover] with a crossover probability, cross over the parents to form new offspring (children). If no crossover was performed, offspring is the exact copy of parents.
[Mutation] with a mutation probability, mutate new offspring at each locus (position in chromosome)
[Accepting] Place new offspring in the new population.
[Replace] Use new generated population for a further sum of the algorithm.
[Test] if the end condition is satisfied, stops, and returns the best solution in current population.
[Loop] Go to step2 for fitness evaluation.
The flowchart of the proposed algorithm is shown in Figure 2 . More details about the algorithms are found in Haftka and Gurdal (1992) Haftka, R.T., Gurdal, Z., (1992) Elements of structural optimization, Springer Science+Business Media BV (Dordrecht). , Gurdal et al. (1999) Gurdal, Z., Haftka, R.T., Hajela, P., (1999). Design and optimization of laminated composite materials, John Wiley Sons (New York). , etc.
5. RESULTS
5.1. Verification
To verify the proficiency of GDQ method, two numerical examples are carried out for comparison. As the first example, accuracy of the method is investigated in evaluating fundamental natural frequency parameter of the composite shell (without SMA) for clampedsimply supported boundary conditions. The material parameters of each layer are given as:
First natural frequency parameter of the composite shell (without SMA) for clampedsimply supported boundary conditions
5.2. Free vibration
In this section, numerical results on free vibration of the SMAHC shell are presented. It is assumed that the cylindrical shell is made of NiTi / graphite/epoxy. Material properties of NiTi fibers and graphiteepoxy are considered to be temperaturedependent presented in Tables 3 and 4 , respectively. Brinson model is employed to estimate thermomechanical behavior of SMAs. The influence of temperature dependency of materials, volume fraction and prestrain of SMAs, and stacking sequence of layers on vibrational behavior of composite shells in prebuckling region are examined.
The material properties of graphiteepoxy ( Asadi et al., 2013a Asadi, H., Bodaghi, M., Shakeri, M., Aghdam, M.M., (2013a). On the free vibration of thermally pre/postbuckled shear deformable SMA hybrid composite beams. . Journal of Aerospace Science and Technology 31:73–86. , Rasid et al., 2011 Rasid, Z.A., Zahari, R., Ayob, A., Majid, D.L., Rafie, A.S.M., (2011). Thermal postbuckling of shape memory alloy composite plates under nonuniform temperature distribution. International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering 5:16491654. )
5.2.1. Recover stress
Figure 3 illustrates the effect of temperature on the recovery stress of SMA fibers (which is constrained to maintain the deformation). It is seen from this figure that large internal stresses are produced when the transformation to austenite occurs. It is worth noting that SMA recovery stress is tensile while the temperature raise leads to the compressive thermal stress in the structure as it can be found from Equation.(8) . Therefore, SMA recovery stress can reduce the thermal stress and improve the performance of the structure. The influences of temperature on the axial and circumferential forces produced in the structure are shown in Figure 4 . This figure shows that the tensile SMA recovery stresses induced in the structure can contribute to delay the critical buckling temperature.
Variations of axial and circumferential forces generated by SMA fibers in the structure versus temperature
5.2.2. Temperature dependency of materials
Here, the effect of temperature dependency of material properties on free vibration of SMAHC cylindrical shells is demonstrated. Variations of the fundamental frequency with respect to temperature in the prebuckled domains are depicted in Figure. 5 . In this figure, TD denotes that the material properties are temperature dependent and TID represents the assumption of constant material properties. As one can observe from Figure 5 , in the absence of SMA fibers, temperature dependency of material properties does not have any significant effects on the free vibration of composite shells while the difference between the natural frequencies becomes more significant for shells with embedded SMA fibers at high temperatures. It means the influence of temperature dependency is considerable in evaluating free vibration of SMAHC structures. It can be also found that TD material properties contribute to higher natural frequency and critical buckling temperature. Therefore, to attain more accurate results, only TD case is addressed in the following.
Influences of temperature and volume fraction of SMA fibers on the fundamental frequency of composite shells layup
5.2.3. Volume fraction and prestrain of SMA fibers
Variations of fundamental frequency parameter of composite shells with respect to SMA volume fraction and prestrain are depicted in Figures 5 and 6 . The numerical results provided in Figure 5 describe that an increase in volume fraction of SMA fibers results in an increase in critical buckling temperature of the structure. It should be mentioned that the temperature at which the natural frequencies reach to zero is buckling temperature. However, the results state a different scenario regarding the vibrational behavior of structure. It is seen that the fundamental frequency of structure is not necessarily increased with the increase of SMA fiber volume fraction. This is because of the fact that though more SMA fiber volume fraction leads to more stiffness in the structure, the weight of structure is also increased. Therefore, at low temperatures, SMA fibers have a destructive effect on the free vibration behavior of structure.
Variations of the fundamental frequency of simply supported
Now, influence of prestrain of SMA fibers on the natural frequency of composite shell is illustrated. Various values of prestrain of SMA fibers are given and free vibrations are demonstrated in Figure 6 . This figure represents that different values of prestrain do not have any significant effects on the vibrational behavior of SMAHC shell when T<A_{f} but for temperatures more than A_{f,} with the increase of prestrain value, both natural frequencies and critical buckling temperature increase. This is due to the fact that the weight of structure is not changed but the stiffness is increased when T>A_{f}. Numerical comparison specifies that by increasing the prestrain of SMAs from 0.1% to 1%, the buckling temperature can improve up to 140% (from 102 to 245).
5.2.4. Stacking sequence of layers
The effect of stacking sequence of layers on the variations of fundamental frequency of SMAHC laminate shell is investigated in Figure 7 by considering various layup configurations. It is obviously found that the stacking sequence of layers play an important role in vibrational behavior of the SMAHC structures. As can be observed from Figure 7 , an unsuitable layup may lead to destructive influence of SMA fibers on free vibration of structures. As an example, for
Effect of layup orientation on the free vibration and thermal buckling behavior of simply supported shell
5.3. Optimization of SMAHC cylindrical shell
The main objective of optimization in the present work is to find the best layup orientation for SMAreinforced layers so that to maximize the fundamental frequency parameter in a certain temperature for a constant amount of SMA volume fraction. As an example, an eightlayer shell is considered with the layup orientation
It is also assumed that fiber orientations of SMA layers take integer values (e.g. 0 ^{0} C, 1^{0}, …, 90^{0}). If the analytical solution is applied for the optimization problem, the process becomes so complicated and time consuming. In other words, the formed discrete space contains more than
CONCLUSIONS
In this paper, two goals were followed. First, thermal vibration of SMAHC cylindrical shells in prebuckling region was analyzed. The numerical results were presented based on classical shell theory which is valid only for thin shells. To obtain the natural frequencies, GDQ method was implemented to solve the governing equations. The influences of SMA volume fraction, layup orientation, prestrain of SMA fibers, and temperaturedependency of material on the natural frequency and buckling of structure were examined. The numerical results showed that at lowtemperature region, SMAs have a destructive role for natural behavior of structure, but they can significantly improve the vibrational behavior and thermal buckling when the temperatures are high. It was also concluded that, at low temperatures, the variations of prestrain of SMAs cannot considerably change the vibrational characteristics of the composite structures. Another important conclusion is that the orientation of SMAs in the layers plays an effective role in vibrations of SMAHC shells under thermal environments and can sharply vary the fundamental frequency of the structure. Therefore, an optimization problem was presented to find best stacking sequence for layers with embedded SMAs in order to have the maximum fundamental frequency at a certain temperature. To this end, GA as a heuristic algorithm was employed. It is worth noting that, like other meta heuristic algorithms, GA does not necessarily guarantee the best global optimal solution. However, it at least suggests one of the best answers for optimization problem in short time.
Appendix
where the subscripts ‘m’ and ‘s’ mean the composite matrix and SMA fiber, respectively. Also, parameters E, G, í, á, ñ and V_{s} are Young modulus, shear modulus, Poisson ratio, thermal expansion coefficient, material density and volume fraction of SMA fibers, respectively
References
 An, H., Chen, S., Huang, H., (2015). Laminate stacking sequence optimization with strength constraints using twolevel approximations and adaptive genetic algorithm. Journal of Struct Multidisc Optim 51:903–918.
 Asadi H., Kiani Y., Aghdam M.M., Shakeri M., (2015). Enhanced thermal buckling of laminated composite cylindrical shells with shape memory alloy. Journal of composite materials 50(2): 114.
 Asadi, H., Bodaghi, M., Shakeri, M., Aghdam, M.M., (2013a). On the free vibration of thermally pre/postbuckled shear deformable SMA hybrid composite beams. . Journal of Aerospace Science and Technology 31:73–86.
 Asadi, H., Bodaghi, M., Shakeri, M., Aghdam, M.M., (2013b). An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams. European Journal of Mechanics A/Solids 42:454468.
 Auricchio, F., Sacco, E., (1997). A onedimensional model for superelastic shape memory alloys with different elastic properties between austenite and martensite. International Journal of Non Linear Mechanics 32:1101I 114.
 Birman, V., (1997). Stability of functionally graded shape memory alloy sandwich panels. Journal of Smart Material Structure 6: 278–286.
 Birman, V., Chandrashekhara, K., Sain, S., (1996). An approach to optimization of shape memory alloy hybrid composite plates subjected to lowvelocity impact. Journal of Composites Part B 439446.
 Botshekanan Dehkordi, M., Khalili, S.M.R., Carrera E., (2016). Nonlinear transient dynamic analysis of sandwich plate with composite facesheets embedded with shape memory alloy wires and flexible core based on the mixed LW (layerwise)/ESL (equivalent single layer) models. Composites Part B 87: 5974.
 Brinson, L. C., Huang, M. S., (1996). Simplifications and Comparisons of Shape Memory Alloy Constitutive Models. Journal of Intelligent Material Systems and structures 7: 108114.
 Brinson, L.C., (1993). Onedimensional constitutive behavior of shape memory alloys: thermomechanical derivation with nonconstant material functions and redefined martensite internal variable. Journal of Intelligent Material Systems and Structures 4: 229242.
 Civalek, O., (2007). Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach. Journal of Computational and Applied Mathematics 205:251–271.
 Forouzesh, F., Jafari, A. A., (2015). Radial vibration analysis of pseudoelastic shape memory alloy thin cylindrical shells by the differential quadrature method. Journal of ThinWalled Structures, 93:158–168.
 Gurdal, Z., Haftka, R.T., Hajela, P., (1999). Design and optimization of laminated composite materials, John Wiley Sons (New York).
 Haftka, R.T., Gurdal, Z., (1992) Elements of structural optimization, Springer Science+Business Media BV (Dordrecht).
 Kamarian, S., Shakeri, M., (2017). Thermal buckling analysis and stacking sequence optimization of rectangular and skew shape memory alloy hybrid composite plates. Composites Part B : Engineering, 116: 137152.
 Khalili, S.M.R., Botshekanan Dehkordi, M., Carrera, E., Shariya, M., (2013). Nonlinear dynamic analysis of a sandwich beam with pseudoelastic SMA hybrid composite faces based on higher order finite element theory. . Journal of Composite Structures 96:243–255.
 Kuo, S.Y., Shiau, L.C., Chen, K.H., (2009). Buckling analysis of shape memory alloy reinforced composite laminates. Journal of Composite Structures 90:188195.
 LeManh, T., Lee, J., (2014). Stacking sequence optimization for maximum strengths of laminated composite plates using genetic algorithm and isogeometric analysis. Journal of Composite Structures 116:357–363.
 Li, S.R., Yu, W.S., Batra, R.C., (2010). Free vibration of thermally pre/postbuckled circular thin plates embedded with shape memory alloy fibers. Journal of Thermal Stresses 33:79–96.
 Loughlan, J., Thompson, S.P., Smith, H., (2002). Buckling control using embedded shape memory actuators and the utilization of smart technology in future aerospace platforms. Journal of Composite Structures 58: 319–347.
 Mashrouteh, H., Rahnamayan, S., Esmailzadeh, E., (2017). Optimal Vibration Control and Innovization for Rectangular Plate. Journal of IEEE 549556.
 Mirzaeifar, R., Shakeri, M., DesRoches, R., Yavari, A., (2011). A semianalytic analysis of shape memory alloy thickwalled cylinders under internal pressure. Journal of Arch Appl Mech 81: 1093–1116.
 Ou, D., Mak, C.M., (2017). Optimization of natural frequencies of a plate structure by modifying boundary conditions. Journal of the Acoustical Society of America 142: 5662.
 Parhi, A., Singh, B.N., (2016). Nonlinear Free Vibration Analysis of Shape Memory Alloy Embedded Laminated Composite Shell Panel. Journal of Mechanics of Advanced Materials and Structures 24: 713724.
 Rasid, Z.A., Zahari, R., Ayob, A., Majid, D.L., Rafie, A.S.M., (2011). Thermal postbuckling of shape memory alloy composite plates under nonuniform temperature distribution. International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering 5:16491654.
 Reddy, J.N., (2004). Mechanics of laminated composite plates and shells Theory and analysis, CRC Press LLC (Boca Raton).
 Roh J.H., Oh I.K., Yang S.M., Han J.H., Lee I., (2004). Thermal postbuckling analysis of shape memory alloy hybrid composite shell panels. Smart materials and structures 13: 13371344.
 Samadpour, M., Sadighi, M., Shakeri, M., Zamani, H., (2015). Vibration analysis of thermally buckled SMA hybrid composite sandwich plate. Journal of Composite Structures 119:251–263.
 Shen, H.S., (2008). Thermal Postbuckling Behavior of Anisotropic Laminated Cylindrical Shells with TemperatureDependent Properties. Journal of AIAA 46:185193.
 Sheng, G.G., Wang, X., (2007). Thermal Vibration, Buckling and Dynamic Stability of Functionally Graded Cylindrical Shells Embedded in an Elastic Medium. Journal of Reinforced Plastics and Composites 27:117134.
 Shiau, L.C., Kuo, S.Y., Chang, S.Y., (2011). Free vibration of buckled SMA reinforced composite laminates. . Journal of Composite Structures 93: 2678–2684.
 Shu, C., (2000). Differential quadrature and its application in engineering, Berlin: Springer.
 Shu, C., Du, H., (1997). Free vibration analysis of laminated composite cylindrical shells by DQM. Journal of Composites Part B 28B:267274.
 Shu, C., Richards, B.E., (1992). Application of generalized differential quadrature to solve twodimensional incompressible NavierStockes equations. International Journal For Numerical Methods In Fluids 15:791798.
 Sivanandam, S.N., Deepa, S.N., (2008). Introduction to Genetic Algorithms, Springer Berlin Heidelberg New York.
 Sliseris, J., Rocens, K., (2013). Optimal design of composite plates with discrete variable stiffness. Journal of Composite Structures 98:15–23.
 Thompson, S.P., Loughlan, J., (2001). Enhancing the postbuckling response of composite panel structure utilizing shape memory alloy actuators  a smart structural concept. . Journal of Composite Structures 51: 21–36.
 Vosoughi, A.R., Dehghani Forkhorji, H., Roohbakhsh, H., (2016). Maximum fundamental frequency of thick laminated composite plates by a hybrid optimization method. Journal of Composites Part B 86:254260.
 Wu, Z., Weaver, P.M., Raju, G., Kim, B.C., (2012). Buckling analysis and optimisation of variable angle tow composite plates. Journal of ThinWalled Structures 60:163–172.
 Xu, C., Lin, S., Yang, Y., (2015). Optimal design of viscoelastic damping structures using layerwise finite element analysis and multiobjective genetic algorithm. Journal of Computers and Structures 157:1–8.
 Yongsheng, R., Shuangshuang, S., (2007). Large amplitude flexural vibration of the orthotropic composite plate embedded with shape memory alloy fibers. Chinese Journal of Aeronautics 20:415424.
Publication Dates

Publication in this collection
2018
History

Received
07 May 2016 
Reviewed
05 Sept 2017 
Accepted
28 Sept 2017