In the present paper, for the first time the vibration behavior of hybrid composite conical shells reinforced with shape memory alloy (SMA) fibers is investigated. The temperature-dependent properties of SMA fibers and composite are accurately considered. Using the one-dimensional constitutive law of Brinson, thermo-mechanical properties such as recovery stresses, elasticity modulus and shear modulus with uniform temperature change are calculated for SMA fibers. Love’s first approximation classical shell theory with von-Kármán type of geometrical nonlinearity is used in conjunction with Hamilton’s principle for deriving the equations of motion. A semi-analytical solution is presented so that trigonometric functions are applied in the circumferential direction, and the generalized differential quadrature method is used to discretize the equations of motion along the longitudinal direction. Finally, parametric studies are done to investigate the effects of volume fraction, pre-strain, location of SMA fibers, boundary conditions, semi-vertex angle of the cone, and temperature on the vibration characteristics of the SMA hybrid composite conical shells. It is shown that a proper utilization of SMA fibers significantly increases the fundamental frequency and vibration control of the SMA hybrid composite conical shells.
Hinweise
The original version of this article was revised due to a retrospective Open Access cancellation.
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1 Introduction
Laminated composite materials, because of their high stiffness-to-weight and high strength-to-weight ratios, have attracted noticeable attention in recent decades. Laminated conical shells are now widely used in various industries such as aerospace, marine, and automotive. Temperature of body and cape of high-speed flight vehicles due to aerodynamic resistance significantly rise. The increase in temperature in these structures causes thermal buckling and dynamic instability. In order to improve the vibration characteristics and buckling response, SMA fibers are widely used in the laminated structures. Therefore, it is necessary to study the dynamic behavior of laminated conical shells with embedded SMA fibers under thermal environment.
SMAs are a special category of smart materials which have two crystallographic phases involving the high-temperature phase called austenite and the low-temperature phase called martensite. SMAs have unique thermo-mechanical properties such as shape memory effect and pseudo-elasticity. The shape memory effect is the ability to recover large residual strains (to the primary configuration) by increasing the temperature under a solid–solid reversible phase transformation. The pseudo-elasticity is the ability to deform largely during the loading cycle which is fully recovered in a hysteresis loop upon unloading at the above austenite finish temperature. This property of SMA leads to vibration absorption and energy dissipation by the hysteresis loop. The shape memory effect generates a tensile recovery force which is used to modify buckling and control vibration of composite shells.
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An important factor in determining the properties of SMA is the volume fraction of martensite, which itself is a function of stress and operating temperature. Lagoudas [1] presented a comprehensive review of SMA models. In addition, thermo-mechanical characterization and constitutive modeling of SMAs are outlined. Rogers [2] presented experimental results of the active vibration and structural acoustic control utilizing SMA (Nitinol) fibers in hybrid composites (graphite/epoxy) with embedded SMA fibers by active strain energy tuning (ASET). Baz et al. [3] showed that the natural frequencies of flexible fiberglass composite beams are controlled by activating optimal sets of SMA (Nitinol) wires which are located along the axial direction. Lau [4] performed experimental and numerical studies on the vibration characteristics of laminated beams reinforced with SMA fibers. According to this study, the natural frequencies of laminated composite beams with embedded non-pre-strained SMA wires changed slightly. Also, it was concluded that the natural frequencies of laminated beams with embedded pre-strained SMA fibers increase in both the theoretical prediction and experimental results.
Park et al. [5] investigated the vibration behavior of thermally pre- and post-buckling composite plates embedded with SMA fibers. They extracted the temperature-dependent recovery stress and Young’s modulus of the SMA fibers using the experimental model by Cross et al. [6]. The numerical results of their study showed that the critical temperature is increased, and the thermal large deflection is decreased by using SMA fibers. In addition, the natural frequencies of the plate with SMA fibers may increase in the pre-buckled situation whereas in the post-buckled situation they are lower than those of the plate without SMA. Yongshenga and Shuangshuang [7] studied free and forced vibration of large deformation of a laminated composite plate embedded with SMA fibers. They employed the one-dimensional constitutive model of Brinson for the evaluation of the properties of SMA fibers. Results showed that the effects of temperature on the forced response behavior during phase transformation from martensite to austenite are significant.
Li et al. [8] examined the free vibration of thermally pre- and post-buckled aluminum circular thin plates embedded with Nitinol SMA fibers and used the one-dimensional Brinson’s model to predict the thermo-mechanical properties of SMA fibers. The results revealed that the embedding of SMA fibers in a circular plate increases the critical buckling temperature, decreases the post-buckling deflection, increases the natural frequencies in the pre-buckling situation for the clamped plates, and decrease the natural frequencies in the post-buckling situation for the simply supported plate. Shiau et al. [9] studied the effect of SMA fibers on the free vibration behavior of buckled cross-ply and angle-ply hybrid composite plats, using the finite-element method. They employed the Cross experimental data for extracting the temperature-dependent recovery stress and Young’s modulus of the SMA fibers.
Asadi et al. [10‐13] presented nonlinear vibration, free vibrations of thermally pre-/post-buckled, nonlinear thermal stability, and nonlinear dynamics of hybrid laminated composite beams reinforced with SMA fibers using the Euler–Bernoulli beam theory and the first-order shear deformation theory of Timoshenko. They considered the one-dimensional Brinson constitutive model for modeling the thermo-mechanical properties of Nitinol SMA fibers, including recovery stress and elasticity modulus. In their studies, the effect of volume fraction, pre-strain, location and direction of SMA fibers, temperature rise, and geometric parameters on static and dynamic characteristics of the hybrid laminated composite beams reinforced with SMA fibers in the thermally pre-/post-buckling situations are investigated. Malekzadeh et al. [14] and Karimi Mahabadi et al. [15] examined the influence of some geometrical and material properties on the free vibration response of the hybrid laminated composite plate embedded with SMA fibers.
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Forouzesh and Jafari [16] presented the analysis of the free and forced vibrations of a pseudo-elastic SMA cylindrical shell under time-dependent internal pressure with simple boundary conditions. They simulated the behavior of Nitinol SMA using the 3D Boyd–Lagoudas constitutive model and also extracted the equations of motion using Donnell-type classical shell theory and Hamilton’s principle. They used the differential quadrature method (DQM) and Newark method as the solution. Asadi et al. [17] developed semi-analytical solutions to investigate the efficiency and reliability of SMA fibers for improving the thermal buckling of cross-ply laminated composite cylindrical shells. Parhi and Singh [18] performed a nonlinear free vibration analysis of spherical and cylindrical shallow shell panels embedded with SMA fibers using the finite-element method. They extracted the governing equations by applying the principle of virtual work based on the higher-order shear deformation plate theory and von-Kármán’s nonlinear strain–displacement relations. In addition, the temperature-dependent recovery stress and Young’s modulus of the SMA fibers were determined using the experimental data reported by Cross. In their research, the effect of volume fraction of SMA fibers, pre-strain of SMA fibers, direction of SMA fibers, ratio of amplitude to thickness, ratio of radius to length, and boundary condition on the linear and nonlinear dimensionless frequency parameter of the panel are examined.
Salim et al. [19] indicated the effects of SMA volume fraction, temperature-dependence of material properties, layup orientation, and pre-strain of SMA wires on the natural frequency and buckling of SMA hybrid composite cylindrical shells. In addition, an optimization problem was offered to find the best layup orientation for layers with embedded SMAs in order to have the maximum fundamental frequency at a certain temperature.
A review of researches that were previously published shows that, although conical shells are widely used in vibration systems, there is no study on the vibration analysis of hybrid composite conical shells reinforced with SMA fibers. Hence, the novelty of this study is to develop the application of SMA fibers for reinforcing the shells in order to control the dynamic and vibration responses for the first time. Also, the influence of SMA fibers on the vibration behavior of SMA hybrid composite conical shells is investigated. The behavior of SMA fibers is evaluated by means of the one-dimensional constitutive law of Brinson. The equations of motion are obtained by using Love’s first approximation classical shell theory with the von-Kármán type of geometrical nonlinearity and Hamilton’s principle. To obtain the linearized dynamic stability equations, the concept of the adjacent equilibrium criterion is used. The pre-buckling forces are extracted based on a linear membrane pre-buckling analysis. A semi-analytical solution using the trigonometric functions and the generalized differential quadrature (GDQ) method is used to determine the natural frequencies of the SMA hybrid composite conical shell. Finally, the effect of material properties, geometry parameters, and boundary conditions on the dynamic response of an SMA hybrid composite conical shell is investigated in detail.
2 Constitutive equations of shape memory alloy fibers
In recent decades, many studies were done on the mathematical modeling of the behavior of shape memory alloys. Due to the great complexity of material modeling, one-dimensional models are commonly used in engineering applications.
According to the principles of thermodynamics, such as energy balance and Clausius–Duhem inequality, a one-dimensional constitutive relation was offered by Tanaka et al. [20]. Liang and Rogers [21] developed the Tanaka structural relations by considering an experimentally based cosine model for the martensite fraction as a function of stress and temperature during transformation. Brinson [22] improved Liang’s model with separating the martensite fraction into the stress-induced and temperature-induced components. According to Brinson’s model, the martensite fraction \(\xi \) is
where \(\xi _{\mathrm{s}} \) signifies the martensite fraction of the material induced by stress variation (detwinned martensite), and \(\xi _{\mathrm{T}} \) indicates the martensite fraction that is transformed by temperature (twinned martensite). Hence, the martensite fraction for various temperature is expressed as [22]
(I) The phase transformation conversion to detwinned martensite:
for \({\textit{T}}>M_{\mathrm{s}}\) and \(\sigma _{\mathrm{s}}^{\mathrm{cr}} +C_{\mathrm{M}} \left( {T-M_{\mathrm{s}} } \right)<\sigma <\sigma _{\mathrm{f}}^{\mathrm{cr}} +C_{\mathrm{M}} \left( {T-M_{\mathrm{s}} } \right) :\)
In relations (2) and (3), \(M_{\mathrm{f}} \), \(M_{\mathrm{s}} \), \(A_{\mathrm{s}} \), \(A_{\mathrm{f}} \), T, and \(\sigma \) are martensite finish temperature, martensite start temperature, austenite start temperature, austenite finish temperature, temperature and recovery stress, respectively. Also, subscript “0” denotes the initial condition of the parameter. In addition, \(C_{\mathrm{M}} \) and \(C_{\mathrm{A}} \) are ratios between the critical stress to temperature at the bound of conversion to martensite and austenite. \(\sigma _{\mathrm{s}}^{\mathrm{cr}} \) and \(\sigma _{\mathrm{f}}^{\mathrm{cr}} \) refer to critical stresses at the start and finish of the conversion of the martensitic variants (Fig. 1).
so that \(E_{\mathrm{A}} \) and \(E_{\mathrm{M}} \) denote the elastic modulus of SMA fibers in fully austenite and martensite phases, respectively. For the special case of uniaxial state of stress (which is used in this study), the Reuss scheme in comparison with the other methods has a good agreement with experimental results [23].
The constitutive equation for a shape memory alloy with non-constant material functions according to Brinson’s model is [22]:
where \(\varepsilon \), \(\varepsilon _L \), \(\Omega \) and \(\Theta \) represent strain, maximum strain, phase transformation coefficient, and thermo-elastic term. Considering that the initial conditions consist of stress zero (\(\sigma _0 =0)\) and \(\xi _{\mathrm{s}0} ={\varepsilon _0 }/{\varepsilon _L }\), Brinson’s equation (5) is simplified as [24]:
In this study, the reference temperature and initial temperature-induced martensite fraction \(\xi _{\mathrm{T}0} \) are assumed to be \(20\,^{\circ }\)C and zero, respectively, for Nitinol SMA. In addition, the material properties of Nitinol SMA fibers are given in Table 1.
Table 1
Material properties of the shape memory alloy fibers (Nitinol) [22]
The constitutive equation (6) which is coupled with the transformation equations (2) and (3) is used to calculate the stress–strain curves. Figure 2a shows stress–strain curves for two temperature cases, \(-\,10\,^{\circ }\)C to show the shape memory effect and the pseudo-elastic effect at \(50\,^{\circ }\)C in comparison with experimental results reported by Liang [25]. According to this Figure the selected material model in this study has a good accuracy and agreement to represent both the shape memory and pseudo-elastic effects. Figure 2b demonstrates stress–strain curves for the degradation of loading–unloading cycles at 45 \(^{\circ }\)C.
Fig. 2
Stress–strain curves of SMA with loading–unloading cycles
×
For determining the recovery stress in different pre-strains, Eq. (6) must be solved simultaneously with Eq. (3). Figure 3 compares recovery stress–temperature curves for different pre-strains in the present study and results reported by Asadi et al. [17] and Roh et al. [26]. This Figure verifies the correctness of modeling of material properties in the present study.
Fig. 3
Recovery stresses versus temperature for restrained SMA
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3 Micromechanical modeling of the hybrid laminated composite
In the considered hybrid laminated composite, it is assumed that SMA fibers are located in the innermost and outermost layers of the shell parallel with the fibers of the composite in these layers. In order to determine the effective thermo-mechanical material properties of a hybrid laminated composite, the multi-cell micromechanics approach [27] is applied. According to this approach, the resultant equations for the thermo-elastic properties such as \(E_{11} \), \(E_{22} \), \(G_{12} \), \(\upsilon _{12} \), \(\alpha _1 \), \(\alpha _2 \), and \(\rho \) of any SMA/graphite/epoxy layer are given in Appendix A. Local coordinates 1, 2, and 3 are directions aligned, perpendicular, and out-of-plane perpendicular with fibers, respectively, that are shown in Fig. 4b.
Fig. 4
Geometry of the hybrid laminated composite conical shell reinforced with SMA fibers
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4 Governing equations of motion
A hybrid laminated composite conical shell reinforced with SMA fibers with total thickness h and length L, referred to the coordinate system \(x, \theta , z\), is shown in Fig. 4. \(R_1\) and \(R_2 \) denote the radii of the cone at its small and large edges, and \(\alpha \) is a semi-vertex angle of the cone.
The displacement field based on the classical shell theory (Kirchhoff hypothesis) for the conical shell is defined in terms of the mid-plane displacement \((u_0 ,v_0 ,w_0 )\) as [28]:
Based on the classical theory of shells in Love’s first approximation theory incorporated with the von-Kármán geometrical nonlinearity, the components of strain on a generic point of the shell are expressed as [29]:
In the above relations, \(()_{,x} \) and \(()_{,\theta } \) denote the derivatives with respect to the axial and circumferential coordinates, respectively. Furthermore, \(R(x)=R_1 +x \sin \alpha \) stands for the radius of the conical shell at each cross section.
The elastic constitutive relations for the \(k{\mathrm{th}}\) layer of the hybrid laminate composite reinforced with SMA fibers in the global coordinate system subjected to thermal loading in a plane stress are given by:
In Eq. (9), \(\alpha \) is the thermal expansion coefficient, \(\beta \) delineates the angle of fibers with the x-axis, and \(\sigma ^{r}\) is the recovery stress generated by temperature induced reverse phase transformation of the pre-strained SMA fibers. Also, transformed stiffness \(\bar{{Q}}_{ij} \) are given in Appendix B.
Based on classical shell theory, the stress resultants for a thin shell are defined as:
In the above equations, the constant coefficients \(A_{ij} , \quad B_{ij} \) and \(D_{ij} \) indicate the stretching, coupling bending-stretching, and bending stiffness parameters, respectively, which are:
so that \(\Delta T=T-T_{\mathrm{ref}} \) is the difference of the reference temperature \(T_{\mathrm{ref}} \) to an arbitrary temperature T. Moreover, \(N^{r}\) and \(M^{r}\) refer to the force and bending moment resultants induced by the SMA fibers, which are expressed as:
In addition, \(\delta V\) presents the variation of potential energy of the external loads, and \(\delta K\) is the variation of the kinetic energy, that is:
A composite conical shell reinforced with SMA fibers under uniform heating and immovable edge supports is considered. Thermal loading of the shell in the pre-buckling state is axisymmetric; therefore, pre-buckling deformation of the shell is also axisymmetric. Generally, three approaches consisting of nonlinear bending, linear bending, and linear membrane approaches are used to determine the pre-buckling loads [31]. The linear membrane approach is used for moderately long shells, where the effect of the edge area function near the edges of the shell is not dominant. In the present study, the pre-buckling forces are obtained based on the linear membrane approach. According to this approach, the von-Kármán senses such as bending moments and curvatures are excluded from the axisymmetric and stability form of the equations of motion (18) and strain displacement relations (8). Therefore, the pre-buckling deformations of the conical shell are calculated using the solution of the following equations:
Here, superscript “0” denotes the pre-buckling parameters. According to the second equation of equilibrium (20), the circumferential stress resultant in the pre-buckling state is obtained as:
With substitution of the second strain–displacement relation (21) in the above equation, the pre-buckling lateral deflection of the conical shell is derived as
Upon replacing Eq. (22) in the first equilibrium equation (20), and solving the result differential equation, the axial stress resultant in the pre-buckling state is obtained as:
Substituting Eq. (21) in the above equation in conjunction with Eq. (23) leads to a first-order differential equation for the variable \(u_0 \). Finally, the axial stress resultant in the pre-buckling state is obtained from the exact solution of this equation with the boundary conditions \(u_0 =0\) at \(x=0\) and \(x=L\), which is expressed as follows:
To obtain the linearized dynamic stability equations, the concept of the adjacent equilibrium criterion is used [32]. According to this criterion, the components of displacements on the primary equilibrium position are defined with components \((u_0^0 ,v_0^0 ,w_0^0 )\). Considering an infinitesimal perturbation for the components of the displacement field \((u_0^1 ,v_0^1 ,w_0^1 )\) in the vicinity of the equilibrium position, a new adjacent equilibrium position is described. Based on the adjacent equilibrium criterion, the equations of motion (18) are simplified to the stability equations of motion as:
Herein, the linear partial differential operators \(L_{ij} \) for cross-ply conical shells are used [33]. In the present study, two types of boundary conditions that consist of simply supported (S) and clamped (C) are considered for each end of the cone conditions which are:
In this research, the generalized differential quadrature (GDQ) method (which was developed by Shu [34]) is used to discrete the equations of motion and boundary conditions to obtain the natural frequency of an SMA hybrid composite conical shell. According to the GDQ method, the SMA hybrid composite conical shell is discretized into K points in the x direction. The nth-order derivative of a function f(x) at any discrete point is approximated as:
$$\begin{aligned} \frac{\mathrm{d}^{(n)}f(x_i )}{\mathrm{d}x^{(n)}}\approx \sum _{j=1}^K {c_{ij}^{(n)} f(x_j )}{\text { for }}i=1,2,\ldots ,K{\text { and }}n=1,2,\ldots ,K-1 \end{aligned}$$
(29)
where \(f(x_j )\) is the function value at point \(x_j \), and \(c_{ij}^{(n)} \) are the weighting coefficients for the \(n{\mathrm {th}}\) order derivative of the function in the x direction. The weighting coefficients for the first-order derivative are defined in Appendix C. The accuracy of most numerical methods is highly sensitive to the distribution of nodal points. Shu et al. [35] investigated the effect of the distribution of points on the accuracy of the solutions in the beam and plate problems. They concluded that dragging points near the boundaries would significantly improve answers. In this regard, discrete points are selected coincident with the zeros of the Chebyshev polynomial in the following form:
Considering the stability equations of motion (27), trigonometric functions are used for the approximation of the displacement field components in the circumferential direction so that:
In the above relation, n is the full wave number in the circumferential direction, and \(\omega _n \) is the natural frequency corresponding to the \(n{\mathrm {th}}\) mode. With substituting relations (31) into Eq. (27) and using GDQ method, the stability equations of motion are discretized as:
where \(S_{ijk} \) are defined in Appendix D. Discretized simply supported boundary conditions at small and large edges are presented in the following equations, respectively:
Finally, by solving the above eigenvalue problem, the natural frequencies of the SMA hybrid composite conical shell are determined.
8 Numerical results and discussion
8.1 Comparative studies
Two comparative studies are presented in Tables 2 and 3 in order to validate the natural frequency and critical buckling temperature for cross-ply laminated conical shells.
Table 2
Comparison of fundamental frequency parameters for a cross-ply laminated conical shell with the simply supported boundary condition (\(\hbox {SS}_{3}\) according to Ref. [36])
In Table 2, the fundamental frequency parameters for cross-ply laminated conical shells with simply supported boundary conditions for two layers and infinite number of layers are compared with those reported by Shu [33], Tong [36], and Wu et al. [37]. Based on the classical shell theory, the free vibration problem in [33, 36, 37] is solved based on GDQ method, power series, and DQ method, respectively. It is seen from Table 2 that the presented numerical results are in good agreement with those available in the literature as the maximum relative difference between the results is 0.18%. It must be noted that the extension bending coupling term for cross-ply composite shell with two layers is maximum and with infinite number of layers approaches zero.
In Table 3, the critical buckling temperature parameters for cross-ply laminated conical shells with both edges simply supported and clamped–clamped are compared with those reported by Patel et al. [38] and Singh and Babu [39]. The linear eigenvalue problem for thermal buckling analysis of a laminated conical shell in [38] derived based on the first-order shear deformation theory is solved using a semi-analytical finite element method. In addition, the thermal buckling analysis of a laminated conical shell in [39] is performed based on the higher-order shear deformation theory, and the finite-element method is utilized. As the Table shows, there is not a significant difference between the results of this study and the results in the references as the average relative difference to [38, 39] is about 1.8% and 2.7%, respectively.
Table 3
Comparison of the critical buckling temperature parameter for a cross-ply laminated conical shell
Fundamental frequency of an SMA hybrid composite conical shells with SS boundary conditions (\(T=60\,^{\circ }\)C, \({R_1 }/h=500\), \(L/{R_1 }=1)\)
\(\alpha \)
Layup
Pre-strain
Fundamental frequency (Hz)
Without SMA
\(V_{\mathrm{s}} =5\% \)
\(V_{\mathrm{s}} =10\% \)
\(V_{\mathrm{s}} =15\% \)
\(30^\circ \)
\(\left[ {0_{\mathrm{SMA}} ,90} \right] \)
\(\varepsilon _0 =1\% \)
129.645 (12)
139.843 (12)
148.266 (12)
155.406 (12)
RI
7.9%
14.4%
19.9%
\(\varepsilon _0 =2\% \)
129.645 (12)
140.995 (12)
150.243 (12)
158.003 (12)
RI
8.7%
15.9%
21.9%
\(\left[ {0_{\mathrm{SMA}} ,90,0,90} \right] \)
\(\varepsilon _0 =1\% \)
149.936 (11)
153.947 (11)
157.735 (11)
161.307 (11)
RI
2.7%
5.2%
7.6%
\(\varepsilon _0 =2\% \)
149.936 (11)
154.474 (11)
158.706 (11)
162.655 (11)
RI
3%
5.8%
8.5%
\(45^\circ \)
\(\left[ {0_{\mathrm{SMA}} ,90} \right] \)
\(\varepsilon _0 =1\% \)
104.750 (12)
118.358 (12)
129.018 (12)
137.762 (12)
RI
13%
23.2%
31.5%
\(\varepsilon _0 =2\% \)
104.750 (12)
119.737 (12)
131.323 (12)
140.746 (12)
RI
14.3%
25.4%
34.4%
\(\left[ {0_{\mathrm{SMA}} ,90,0,90} \right] \)
\(\varepsilon _0 =1\% \)
122.479 (11)
128.095 (11)
133.221 (11)
137.924 (11)
RI
4.6%
8.8%
12.6%
\(\varepsilon _0 =2\% \)
122.479 (11)
128.740 (11)
134.396 (11)
139.543 (11)
RI
5.1%
9.7%
13.9%
\(60^\circ \)
\(\left[ {0_{\mathrm{SMA}} ,90} \right] \)
\(\varepsilon _0 =1\% \)
73.238 (11)
93.245 (11)
107.351 (11)
118.351 (11)
RI
27.3%
46.6%
61.6%
\(\varepsilon _0 =2\% \)
73.238 (11)
95.010 (11)
110.149 (11)
121.874 (11)
RI
29.7%
50.4%
66.4%
\(\left[ {0_{\mathrm{SMA}} ,90,0,90} \right] \)
\(\varepsilon _0 =1\% \)
87.983 (10)
96.620 (10)
104.074 (10)
110.635 (10)
RI
9.8%
18.3%
25.7%
\(\varepsilon _0 =2\% \)
87.983 (10)
97.487 (10)
105.605 (10)
112.697 (10)
RI
10.8%
20%
28.1%
Numbers in bracket mean the circumferential wave number
8.2 Convergence study
The convergence of GDQ results for the fundamental frequency of SMA hybrid composite conical shells with respect to the number of grid points is investigated in Table 4. According to this Table, it is clear that results converge for 21 grid points, and increasing the number of grid points does not affect the result accuracy.
Table 7
Fundamental frequency of SMA hybrid composite conical shells at the constant activation temperature (\(T=80\,^{\circ }\)C, \(V_{\mathrm{s}} =10\% \), \({R_1 }/h=500\), \(L/{R_1 }=1\), \(n=11\))
Fundamental frequency of SMA hybrid composite conical shells at the constant activation temperature (\(T=80\,^{\circ }\)C, \(\varepsilon _0 =1\% \), \({R_1 }/h=500\), \(L/{R_1 }=1\), \(n=11)\)
In this Section, the influence of SMA fibers volume fracture, pre-strain of SMA fibers, boundary condition, layup, and geometrical parameters on the natural frequencies of the SMA hybrid composite conical shells are investigated. In the present study, a hybrid cross-ply NiTi/graphite/epoxy laminated conical shell with geometric parameters \({R_1 }/h=500\), \(L/{R_1 }=1\), total thickness \(h=1\) mm, and same thickness for each layer is considered where the material properties are shown in Tables 1 and 5.
Fig. 5
The influence of SMA fibers pre-strain on the fundamental frequency of an SMA hybrid composite conical shell with SS boundary conditions
Fig. 6
Effect of temperature and SMA fibers volume fraction on the fundamental frequency of an SMA hybrid composite conical shell
×
×
The effects of SMA fibers pre-strain and volume fraction on the fundamental frequency of SMA hybrid composite conical shells for three different semi-vertex angle and two layup schemes at \(T=60\,^{\circ }\)C are presented in Table 6. As the Table shows, increasing the volume fraction and pre-strain of SMA fibers raises the fundamental frequency. This is due to the generation of higher tensile recovery force during phase transformation that raises the flexural rigidity. Investigation of the RI the parameter (rate of increasing the fundamental frequency in presence of SMA fibers with respect to the absence of SMA fibers in a laminated conical shell) indicates that adding pre-strained SMA fibers in higher semi-vertex angle is more efficient in increasing the fundamental frequency.
Fig. 7
Influence edge constraint on the fundamental frequency of SMA hybrid composite conical shells
Fig. 8
Variation of fundamental frequency of SMA hybrid composite conical shells versus semi-vertex angle
×
×
In Table 7, the fundamental frequencies of SMA hybrid composite conical shells for different pre-strains, semi-vertex angles, and various boundary conditions at \(T=80\,^{\circ }\)C are tabulated. As shown, an increase in pre-strain value in constant volume fraction of SMA fibers raises the fundamental frequency. According to RI parameters, increasing the fundamental frequency at \(80\,^{\circ }\)C is more apparent for the pre-strain value of SMA fibers from 0.2 to 1% in comparison with the range of 1–3%. Furthermore, embedding pre-strained SMA fibers to the first and eighth layer in a composite conical shell with SS boundary conditions is more efficient than for CC boundary conditions in increasing the fundamental frequency, whereas for the case of one reinforced layer it is the opposite.
Results in Table 8 show that for a constant activation temperature along the phase transformation region, an increase in the volume fraction of pre-strained SMA fibers is more efficient for an increase in the fundamental frequency of SMA hybrid composite conical shells.
The effect of SMA fibers pre-strain on the fundamental frequency of SMA hybrid composite conical shells at the uniform rise of temperature with simply supported boundary conditions is demonstrated in Fig. 5. From this Figure, it is obvious that the increase in SMA fibers pre-strain parameter leads to a rise of the fundamental frequency and the critical buckling temperature. It must be noted that the thermal buckling occurs when the natural frequency curve tends to zero. Also, as shown in this Figure, embedding SMA fibers in laminated conical shells at the temperature about austenite start temperature leads to a small decrease in the fundamental frequency by maximum 3%. This is because SMA fibers are denser than a graphite/epoxy composite so that they increase the weight of the structure. Also, rising the temperature after austenite start temperature results in generating tensile recovery force by phase transformation (between austenite start and austenite finish temperature) in SMA fibers that increases the fundamental frequency and the critical buckling temperature significantly.
In Fig. 6, the variation of the fundamental frequency versus the SMA fibers volume fraction of SMA hybrid composite conical shells with uniform temperature rise is depicted for SS, SC, CS, and CC boundary conditions, respectively. These diagrams show that embedding pre-strained SMA fibers in a laminated composite at temperature activation above the austenite start temperature raises the fundamental frequency. Also, the increase in SMA fibers volume fraction increases the fundamental frequency and the critical buckling temperature.
The influence of the boundary conditions on the fundamental frequency of SMA hybrid composite conical shells is demonstrated in Fig. 7. As expected, the SMA hybrid composite conical shell with CC boundary condition due to the edge constraints has the highest fundamental frequency and critical buckling temperature.
The variation of the fundamental frequency with respect to the semi-vertex angle of a conical shell is depicted in Fig. 8. Based on the Figure, adding the SMA fibers at any angle results in a noticeable increase in fundamental frequency and critical buckling temperature. Furthermore, it is visible that by increasing the semi-vertex angle fundamental frequency and critical buckling temperature of an SMA hybrid composite conical shell decrease.
Finally, the effect of SMA fibers position on the fundamental frequency of eight-layer cross-ply SMA hybrid composite conical shells is illustrated in Fig. 9. It is concluded that embedding SMA fibers in layers near to the inner surface of the conical shell is more effective on increasing of the fundamental frequency.
Fig. 9
Effect of SMA fibers position on the fundamental frequency of SMA hybrid composite conical shells
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9 Conclusions
In this research, the vibration behavior of hybrid composite conical shells reinforced with Nitinol SMA fibers was investigated. It was assumed that the material properties are temperature-dependent. The one-dimensional constitutive law of Brinson for modeling the thermo-mechanical properties was used. A variational approach and classical shell theory were used for deriving the equations of motion. The resulting stability equations of motion were solved using the GDQ method. According to the numerical study and the parametric study, it is concluded that:
Embedding pre-strained SMA fibers in a laminated composite at temperature activation above the austenite start temperature severely increases the fundamental frequency of the conical shell. This is due to the generation of a tensile recovery force during phase transformation that increases the stiffness of the SMA hybrid conical shell.
Adding pre-strained SMA fibers in a laminated composite at temperature activation below the austenite start temperature reduces the fundamental frequency of the conical shell. This is due to the lack of a tensile recovery force and increase of weight of the SMA hybrid conical shell.
Inserting pre-strained SMA fibers in a laminated composite shell severely increases the critical buckling temperature. In addition, an increase in volume fraction or pre-strain of SMA fibers is more efficient for increasing the critical buckling temperature of SMA hybrid composite conical shells. This is due to generation of a tensile recovery force during phase transformation.
At the constant activation temperature along the phase transformation region, an increase in volume fraction of pre-strained SMA fibers is more efficient than an increase in pre-strain parameter for increasing the fundamental frequency of SMA hybrid composite conical shells.
At constant activation temperature above the phase transformation region, an increase in the pre-strain parameter of SMA fibers is more efficient than an increase in the volume fraction parameter for increasing the fundamental frequency of SMA hybrid composite conical shells.
Adding pre-strained SMA fibers in a higher semi-vertex angle is more efficient in increasing the fundamental frequency.
Embedding pre-strained SMA fibers to the first and eighth layer in a composite conical shell with SS boundary conditions is more efficient than for CC boundary conditions in increasing the fundamental frequency, whereas for the case of one reinforced layer it is the opposite.
Placing pre-strained SMA fibers in the layers near the inner surface of the conical shell is more effective on increasing the fundamental frequency.
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where subscripts “m” and “s” stand for the composite matrix and SMA fibers, respectively. Also, the material parameters E, G, \(\upsilon \), \(\alpha \), and \(V_{\mathrm{s}} \) refer to Young’s modulus, shear modulus, Poisson’s ratio, thermal expansion coefficient, and volume fraction of SMA fibers, respectively.
Appendix B
The transformed stiffness components \(\bar{{Q}}_{ij} \) are
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