Vibration characteristics and damping properties of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures under hygrothermal conditions

Abstract

The present article deals with the vibration and damping characteristics of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structure in different hygrothermal conditions. Carbon nanotube reinforced polymer as a matrix phase and carbon fibre as a reinforcing phase are used, and carbon fibre is graded with uniform distribution along the thickness direction for the shell panel according to the power law distribution. The Mori–Tanaka scheme and strength of materials are used to determine the mechanical properties of such functionally graded carbon nanotubes reinforced hybrid composite materials. Finite element modelling has been done by considering an eight-noded shell element with the transverse shear effect according to Mindlin’s hypothesis, and an oblique coordinate system is used for the functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures. Damping is incorporated into such carbon nanotube–based hybrid skewed shell structure based on the Rayleigh damping model. A MATLAB-based in-house computer code has been developed for the proposed formulation and verified with published research work before using for the present dynamic analysis of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structure under hygrothermal conditions. The effect of the carbon nanotube, carbon fibre, material distribution as per power law index and hygrothermal conditions on the damping behaviour of such functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures have been studied. Furthermore, parametric studies are carried out for the first resonant frequency, absolute amplitude, settling time and carbon nanotube impact on the vibrational behaviour of different functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures under different hygrothermal conditions.

Keywords

Carbon nanotubes functionally graded hybrid nanocomposite material skewed shell structures hygrothermal conditions time and frequency responses

1. Introduction

The carbon nanotubes (CNTs), discovered by Iijima (1991), have attracted many researchers over the years, and extensive work has been carried out for the determination of elastic properties of CNT-reinforced composites which was pioneered by Thostenson et al. (2001) and Dai (2002). Elastic properties of aligned as well as randomly orientated CNT-based composites were obtained by Dong-Li et al. (2004). Wuite and Adali (2005) presented multi-scale analysis to investigate the stress and deflection of different CNT-reinforced nanocomposite beams. Conventional carbon fibre–reinforced polymer (CFRP)–based laminated composites are widely used in different engineering applications, and the study of crack and crack propagation in such CFRP composites has been carried out by Chandra et al. (1999). Koratkar et al. (2002, 2003, 2005) presented the result that a nanocomposite beam reinforced by 2% weight fraction of CNT fillers may exhibit 1000% increase in the loss modulus without increasing its weight. The interfacial interaction between CNT and polymer has been studied by Zhou et al. (2004) to characterise different structural damping of nanocomposites. CNTs improve the damping ratio of nanocomposites which is experimentally observed and was presented by Rajoria and Jalili (2005). Lin and Lu (2010) developed a mathematical formulation to determine the energy dissipation based on the interfacial friction between the CNT and polymer. The damping ratio of a hybrid composite (HC) can be enhanced with the help of CNTs which was observed by Khan et al. (2011) and Patnaik et al. (2020a, 2020b). The structural natural frequency increases as the CNT volume fraction increases which was investigated by Latibari et al. (2013) based on the stick–slip mechanism for interfacial interaction between the CNT and polymer. The mechanical responses of functionally graded carbon nanotubes reinforced composites (FG-CNTRC)–based nanocomposite structures have been extensively studied by many researchers. In recent years, the researchers have studied the free and forced vibration of different composite plate and shell structures. The dynamic response of FG-CNTRC–based plate structure was studied using a mesh free method by Lei et al. (2015). Nonlinear dynamics of the FG-CNTRC plate under lateral pressure was studied by Wang and Shen (2012).

The dynamic study of FG-CNTRC plates subjected to single moving mass was investigated by Malekzadeh et al. (2015). The effect of frequency- and temperature-dependent homogenised CNT–CFRP material property on the damping of different plate/shell structures was studied by Swain and Roy (2018). The damping characteristics of different FG-CNTRC–based shell structures was studied and discussed by Thomas and Roy (2017), whereas the effect of CNT on damping of such FG-CNTRC–based skew shell structures was presented by Kiani et al. (2018). The dynamic responses of different shell structures for predefined arbitrary velocity and path of moving load was studied by Kiani (2017a, 2017b). Mahapatra et al. (2016) presented the hygro-thermo-elastic constitutive relationship–based mathematical formulation for the nonlinear free vibration study of laminated composite shell structure. The literature studies on free and forced vibration of different functionally graded (FG) material-based skew shell structures are limited (e.g. Kandasamy and Singh (2006) and Shojaee et al. (2017)). The FG-based CNT-reinforced plate and shell structures were studied using the modified first-order shear deformation theory (FSDT)–based considering four-noded finite element (FE) for the thermal buckling analysis by Trabelsi et al. (2018, 2019) Frikha et al. (2018a, 2018b) and Zghal et al. (2018a, 2018b, 2018c).

Free vibration and damping analysis of FG-based multilayer shell structure with viscoelastic core materials was studied based on the FSDT by Biswal and Mohanty (2019). The FG-based face layer with the viscoelastic core was considered for the sandwich shell structures in vibration and damping analysis by Sahu et al. (2020). A refined theory was formulated to study the buckling of agglomerated hybrid nanocomposite beam structures under thermal environment based on the Halpin–Tsai homogenisation model, and the stability of multi-scale hybrid nanocomposite plates was also addressed by Dabbagh et al. (2019, 2020a, 2020b) and Ebrahimi et al. (2019, 2020). The FE-based higher order beam theory was developed to study the vibrational characteristics of multi-scale hybrid nanocomposite beam structures under thermal environment by Ebrahimi and Dabbagh (2018, 2019a, 2019b, 2019c). Vibration analysis was performed for the multi-scale hybrid nanocomposite-based plate/shell structures by considering the agglomeration effects of nanofillers by Ebrahimi and Dabbagh (2019a, 2019b, 2019c, 2020).

It is also found that extensive research work has been carried out for static and buckling analysis of FG-based CNT-reinforced shell structures in recent times, but any possible FG hybrid materials have not been considered for several applications under hygrothermal conditions.

Some recent research work is available on free vibration study of laminated shell and FG skewed shell structures. The free vibration of laminated shell structure was studied under hygrothermal conditions without considering any possible hybrid FG material with CNT reinforcement. Because of this limitation, the present study proposes the functionally graded carbon nanotubes reinforced hybrid composite (FG-CNTRHC) material model for skewed shell structure for better vibration and damping outcomes under different hygrothermal conditions. An arbitrary coordinate system is considered for skewing the structure, and the FSDT as per Mindlin’s hypothesis is used for the vibration analysis. In addition, the temperature- and moisture-dependent material properties are considered to determine the effective hygrothermal stiffness of such FG-CNTRHC shell structures for the modal analysis. An effect of CNTs on such FG-CNTRHC skewed shell structure under the hygrothermal environment is also observed by the parametric vibration study (viz. absolute amplitude, settling time, etc.).

2. Basic mathematical formulation

The FG-CNTRHC skewed shell structure is made of polymer, CNTs and carbon fibre with the different graded orientation per lamina through the thickness ranging from 0^o to 90^o. The CNTs are randomly orientated in the polymer and well dispersed, and it is assumed that the carbon fibre matrix is perfectly bonded. The different constituents of FG-CNTRHC are shown in Figure 1(a). Mathematical formulations for effective material properties and shell FE are presented in the following subsections.

2.1. Material properties based on strength of materials

The material properties for such FG-CNTRHC are determined using the Mori–Tanaka scheme and strength of materials method (Kundalwal and Ray, 2011). The effective material properties of the HC containing CNT-based polymer matrix and carbon fibre as a reinforcing phase are assumed to be transversely isotropic.

2.1.1. Effective material properties of nanocomposites

The elastic properties of CNT-reinforced polymer nanocomposites (i.e. modified matrix) are obtained considering completely random orientation of the CNT in the polymer (Yas and Heshmati, 2012). Assuming that the nanocomposite is isotropic, its bulk modulus and shear modulus are derived as follows

K_{N C} = K_{m} + \frac{V_{cnt} (δ_{r} - 3 K_{m} α_{r})}{3 (V_{m} + V_{cnt} α_{r})}, and G_{N C} = G_{m} + \frac{V_{cnt} (η_{r} - 3 G_{m} β_{r})}{3 (V_{m} + V_{cnt} β_{r})}

(1)

where

K_{m}

and

G_{m}

are the bulk modulus and shear modulus of the matrix, respectively. The parameters used in the above equation are available in the literature (Yas and Heshmati, 2012). Young’s modulus of the CNT-based nanocomposite material can be obtained as follows

E_{N C} = \frac{9 K_{N C} G_{N C}}{3 K_{N C} + G_{N C}}

(2)

where

K_{N C}, G_{N C}

and

E_{N C}

are the bulk modulus, the shear modulus and Young’s modulus of the nanocomposite, respectively.

2.1.2. Material properties of FG-CNTRHC based on strength of materials

The strength of materials method is used to reinforce the carbon fibre into the modified matrix with hexagonal representative volume element as shown in Figure 1(b). Rule of mixture and iso-field conditions are considered for induced stresses and strains in the HC with perfect bonding between the modified matrix and reinforcement (Thomas and Roy, 2017). The constitutive relations of such HC material are

{σ^{r}} = [C^{r}] {ε^{r}}; r = f, nanocomposite (NC)

(3)

v_{f} + v_{N C} = 1

(4)

Carbon fibre is graded using the power law for each lamina; volume fraction can be obtained as follows

V_{f} = V_{f min} + {(\frac{z}{h} + \frac{1}{2})}^{k} (V_{f max} - V_{f min})

(5)

The neutral axis ‘z’ $(- h / 2 \leq z \leq h / 2)$ and power law index ‘k’ are considered in the thickness direction. The orientation of carbon fibre in each lamina can be managed from the power law, which is given by

θ_{f} = θ_{f min} + {(\frac{z}{h} + \frac{1}{2})}^{k} (θ_{f max} - θ_{f min})

(6)

The transverse stresses induced in the modified matrix and reinforcement will be equal to the transverse stresses induced in the HC from iso-field conditions, whereas the longitudinal strains in the modified matrix, reinforcement and HC are equal from rule of mixture, and these can be written as

\begin{array}{l} {\begin{matrix} ε_{1}^{f} & σ_{2}^{f} & σ_{3}^{f} & σ_{23}^{f} & σ_{13}^{f} & σ_{12}^{f} \end{matrix}}^{T} = {\begin{matrix} ε_{1}^{N C} & σ_{2}^{N C} & σ_{3}^{N C} & σ_{23}^{N C} & σ_{13}^{N C} & σ_{12}^{N C} \end{matrix}}^{T} \\ = {\begin{matrix} ε_{1}^{H C} & σ_{2}^{H C} & σ_{3}^{H C} & σ_{23}^{H C} & σ_{13}^{H C} & σ_{12}^{H C} \end{matrix}}^{T} \end{array}

(7)

\begin{array}{l} v_{f} {\begin{matrix} σ_{1}^{f} & ε_{2}^{f} & ε_{3}^{f} & ε_{23}^{f} & ε_{13}^{f} & ε_{12}^{f} \end{matrix}}^{T} + v_{N C} {\begin{matrix} σ_{1}^{N C} & ε_{2}^{N C} & ε_{3}^{N C} & ε_{23}^{N C} & ε_{13}^{N C} & ε_{12}^{N C} \end{matrix}}^{T} \\ = {\begin{matrix} σ_{1}^{H C} & ε_{2}^{H C} & ε_{3}^{H C} & ε_{23}^{H C} & ε_{13}^{H C} & ε_{12}^{H C} \end{matrix}}^{T} \end{array}

(8)

The expression of stress and strain for HC lamina can be written as

{σ^{H C}} = [C^{H C}] {ε^{H C}}

(9)

[C^{H C}] = [C_{1}] {[V_{3}]}^{- 1} + [C_{2}] {[V_{4}]}^{- 1}

(10)

Other terms in the above equation are mentioned in Appendix 1.

The elastic properties of the FG structure after transformation can be obtained by

{[C^{H C}]}_{T} = {[T]}^{- 1} [C^{H C}] {[T]}^{- 1}

(11)

The transformation matrix [T] is also mentioned in Appendix 1. The transversely isotropic FG material constituents can be determined as follows

C_{11} = {[C_{11}^{H C}]}_{T}, C_{12} = {[C_{12}^{H C}]}_{T}, C_{23} = {[C_{23}^{H C}]}_{T}, C_{22} = {[C_{22}^{H C}]}_{T}, C_{55} = {[C_{55}^{H C}]}_{T}

(12)

2.2. FE modelling and analysis of FG-CNTRHC skewed shell structure under hygrothermal environment

An eight-noded C⁰ continuity, isoparametric quadrilateral element with five degrees of freedom per node is considered in the present formulation. The stress resultant-type Koiter’s shell theory has been considered to formulate the present FE formulation. Because Koiter’s shell theory is based on the Love–Kirchhoff assumptions, the effect of shear deformation was not considered by Koiter’s shell theory. This effect has been considered in the formulation of shell according to Mindlin’s hypothesis. An oblique coordinate system is considered for defining the skew shell structure geometry as shown in Figure 2(a). The stress resultant-type Koiter’s shell theory is used in the FE formulation by using the shear deformation effect according to Mindlin’s hypothesis (Roy et al., 2010).

2.2.1. Geometry of FG-CNTRHC skewed shell structure

The mid-surface of the shell surface can be described in terms of the position vector as

r (α_{1}, α_{2}) = X (α_{1}, α_{2}) \hat{i} + Y (α_{1}, α_{2}) \hat{j} + Z (α_{1}, α_{2}) \hat{k}

(13)

where

\hat{i}

\hat{j}

and

\hat{k}

are the unit vectors along the X, Y and Z axes, respectively. The normal Cartesian coordinate system (X, Y) can be written in the form of the oblique coordinate system (X^′, Y^′) as follows (Kiani et al., 2018)

\begin{array}{l} X^{'} = X - Y \tan (α) \\ Y^{'} = Y \sec (α) \\ Z^{'} = Z \end{array}

(14)

The tangent to the isoparametric curves (S_α1 and S_α2) is

r_{1} = \frac{\partial r}{\partial α_{1}}, and r_{2} = \frac{\partial r}{\partial α_{2}}

(15)

The vector joining two points on the middle surface (α₁, α₂) and (α₁+d α₁, α₂+d α₂) is given as

d s = (r_{1} d α_{1} + r_{2} d α_{2})

(16)

The scalar product of ds is

d s \cdot d s = (r_{1} \cdot r_{1}) d α_{1}^{2} + (r_{2} \cdot r_{2}) d α_{2}^{2}

(17)

Defining

A_{1} = \sqrt{r_{1} \cdot r_{1}} a n d A_{2} = \sqrt{r_{2} \cdot r_{2}}

(18)

As Lame’s parameters or measure numbers that are fundamentals for the understanding of curvilinear coordinates

d s^{2} = A_{1}^{2} d α_{1}^{2} + A_{2}^{2} d α_{2}^{2}

(19)

Because α₁ and α₂ are the independent coordinates

d s^{2} = d s_{α 1}^{2} + d s_{α 2}^{2}

(20)

\begin{array}{l} d s_{α 1} = A_{1} d α_{1} \\ d s_{α 2} = A_{2} d α_{2} \end{array}

(21)

The unit tangent vectors to the isoparametric curve S_α1 and S_α2 can be expressed, respectively, as

{\hat{e}}_{1} = \frac{r_{1}}{A_{1}} and {\hat{e}}_{2} = \frac{r_{2}}{A_{2}}

(22)

Subscripts 1 and 2 in r₁ and r₂ indicate the derivative with respect to α₁ and α₂, respectively. The unit normal vector to the tangent plane at any point on the reference surface can be expressed as

{\hat{e}}_{n} = \frac{r_{1} \times r_{2}}{| r_{1} \times r_{2} |}

(23)

The normal curvatures of the shell mid-surface can be expressed as

\frac{1}{R_{1}} = - \frac{{\hat{e}}_{n} \cdot r_{11}}{A_{1}^{2}} and \frac{1}{R_{2}} = - \frac{{\hat{e}}_{n} \cdot r_{22}}{A_{2}^{2}}

(24)

And the twist curvatures of the shell mid-surface can be expressed as

\frac{1}{R_{12}} = - \frac{{\hat{e}}_{n} \cdot r_{12}}{A_{1} \cdot A_{2}}

(25)

2.2.2. Isoparametric mapping

The shell mid-surface in the rectangular Cartesian coordinate system has been mapped into the parametric space (α₁, α₂), and the mid-surface in the parametric space has been divided into the required number of quadrilateral elements or sub-domains. The reference coordinates (ξ, η) map the quadrilateral elements in the curvilinear coordinates (α₁, α₂), that is a square as shown in Figure 2(b). Any point within an element in the parametric space has been approximated by the isoparametric mapping.

Hence, the curvilinear coordinates (α₁, α₂) of any point within an element may be expressed as

\begin{matrix} α_{1} = \sum_{i = 1}^{n d} N_{i} α_{1 i} \\ α_{2} = \sum_{i = 1}^{n d} N_{i} α_{2 i} \end{matrix}

(26)

The displacement components on the shell mid-surface at any point within an element is determined as

{\begin{array}{l} u_{0} \\ v_{0} \\ w \\ θ_{1} \\ θ_{2} \end{array}} = \sum_{i = 1}^{n d} N_{i} {\begin{array}{l} u_{0 i} \\ v_{0 i} \\ w_{i} \\ θ_{1 i} \\ θ_{2 i} \end{array}}

(27)

where nd is the number of nodes in an element and (α₁, α₂) is the coordinate of mid-surface at ith node in the curvilinear coordinate system.

u_{0 i}, v_{0 i} and w_{0 i}

are the deflection of mid-surface at ith node in α₁, α₂ and z directions. θ_1i is the rotation of normal at ith node about α₂ axis, and θ_2i is the rotation of normal at ith node about α₁ axis. N_i is the shape function corresponding to the ith node and shape functions of eight-noded serendipity element are given in Appendix 1.

2.2.3. Strain–displacement relationship

The five strain components of a curved shell structure may be expressed as

{[\begin{matrix} ε_{X^{'} X^{'}} & ε_{Y^{'} Y^{'}} & γ_{X^{'} Y^{'}} \begin{matrix} γ_{Y^{'} Z^{'}} & γ_{X^{'} Z^{'}} \end{matrix} \end{matrix}]}^{T} = {[\begin{matrix} \begin{matrix} ε_{X^{'} X^{'}}^{0} & ε_{Y^{'} Y^{'}}^{0} & γ_{X^{'} Y^{'}}^{0} \end{matrix} & \begin{matrix} γ_{Y^{'} Z^{'}}^{0} & γ_{X^{'} Z^{'}}^{0} \end{matrix} \end{matrix}]}^{T} + z {[\begin{matrix} \begin{matrix} k_{X^{'} X^{'}} & k_{Y^{'} Y^{'}} & k_{X^{'} Y^{'}} \end{matrix} & \begin{matrix} 0 & 0 & 0 \end{matrix} \end{matrix}]}^{T}

(28)

where

ε_{X^{'} X^{'}}^{0}, ε_{Y^{'} Y^{'}}^{0} and γ_{X^{'} Y^{'}}^{0}

are the in-plane strains of the mid-surface in the Cartesian coordinate system and

k_{X^{'} X^{'}}, k_{Y^{'} Y^{'}} and k_{X^{'} Y^{'}}

are the bending strains (curvatures) of the mid-surface in the Cartesian coordinate system. After incorporating the effect of transverse strain in Koiter’s shell theory, in-plane and transverse strain–displacement relations may be presented as described in the following subsections.

2.2.4. In-plane strain–displacement relation

The strain components on the mid-surface of the shell element are

{ε} = {[\begin{matrix} \begin{matrix} ε_{X^{'} X^{'}}^{0} & ε_{Y^{'} Y^{'}}^{0} & γ_{X^{'} Y^{'}}^{0} \end{matrix} & \begin{matrix} k_{X^{'} X^{'}} & k_{Y^{'} Y^{'}} & k_{X^{'} Y^{'}} \end{matrix} \end{matrix}]}^{T}

(29)

By using the 8-noded isoparametric shell element, the displacement component on the shell mid-surface at any point within an element can be expressed as

{\begin{matrix} u_{0} & v_{0} & w & θ_{1} & θ_{2} \end{matrix}}^{T} = [N] {d^{e}}

(30)

According to Koiter’s shell theory, the mid-surface strains and curvatures may be expressed in terms of field variables as

ε_{X^{'} X^{'}}^{0} = \frac{1}{A_{1}} \frac{\partial u}{\partial α_{1}} + \frac{v}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} + \frac{w}{R_{1}}

(31)

ε_{Y^{'} Y^{'}}^{0} = \frac{1}{A_{2}} \frac{\partial v}{\partial α_{2}} + \frac{u}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} + \frac{w}{R_{2}}

(32)

γ_{X^{'} Y^{'}}^{0} = \frac{1}{A_{1}} \frac{\partial v}{\partial α_{1}} + \frac{1}{A_{2}} \frac{\partial u}{\partial α_{2}} - \frac{u}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} - \frac{v}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} + \frac{2 w}{R_{12}}

(33)

k_{X^{'} X^{'}} = \frac{1}{A_{1}} \frac{\partial θ_{1}}{\partial α_{1}} + \frac{θ_{2}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} + \frac{1}{2 R_{12}} (\frac{1}{A_{1}} \frac{\partial v}{\partial α_{1}} - \frac{1}{A_{2}} \frac{\partial u}{\partial α_{2}} - \frac{u}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} + \frac{v}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}})

(34)

k_{Y^{'} Y^{'}} = \frac{1}{A_{2}} \frac{\partial θ_{2}}{\partial α_{2}} + \frac{θ_{1}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} - \frac{1}{2 R_{12}} (\frac{1}{A_{1}} \frac{\partial v}{\partial α_{1}} - \frac{1}{A_{2}} \frac{\partial u}{\partial α_{2}} - \frac{u}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} + \frac{v}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}})

(35)

k_{X^{'} Y^{'}} = [\begin{matrix} \frac{1}{A_{1}} \frac{\partial θ_{2}}{\partial α_{1}} + \frac{1}{A_{2}} \frac{\partial θ_{1}}{\partial α_{2}} - \frac{θ_{1}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} - \frac{θ_{2}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} \\ - \frac{1}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}}) (\frac{1}{A_{1}} \frac{\partial v}{\partial α_{1}} - \frac{1}{A_{2}} \frac{\partial u}{\partial α_{2}} - \frac{u}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} + \frac{v}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}}) \end{matrix}]

(36)

By using 8-noded isoparametric shape functions, the strain components at any point on the shell mid-surface can be expressed as

{ε} = \sum_{i = 1}^{8} [\begin{array}{l} \frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} & \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} & \frac{N_{i}}{R_{1}} & 0 & 0 \\ \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} & \frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} & \frac{N_{i}}{R_{2}} & 0 & 0 \\ \frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} - \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} & \frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} - \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} & \frac{2 N_{i}}{R_{12}} & 0 & 0 \\ - \frac{1}{2} \frac{1}{R_{12}} (\frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} + \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}}) & \frac{1}{2} \frac{1}{R_{12}} (\frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} + \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}}) & 0 & \frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} & \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} \\ \frac{1}{2} \frac{1}{R_{12}} (\frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} + \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}}) & - \frac{1}{2} \frac{1}{R_{12}} (\frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} + \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}}) & 0 & \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} & \frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} \\ C_{0} (\frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} + \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}}) & - C_{0} (\frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} + \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}}) & 0 & \frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} - \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{1}}{\partial α_{2}} & \frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} - \frac{N_{i}}{A_{1} A_{2}} \frac{\partial A_{2}}{\partial α_{1}} \end{array}] {\begin{array}{l} u_{0 i} \\ v_{0 i} \\ w_{i} \\ θ_{1 i} \\ θ_{2 i} \end{array}}

(37)

where

C_{0} = 1 / 2 ((1 / R_{1}) - (1 / R_{2}))

and the above expression can be simplified as follows

{ε} = [B_{b}^{e}] {d^{e}}

(38)

2.2.5. Transverse shear strain displacement relation

According to the First order shear deformation theory (FOST), the transverse shear strain vector of a doubly curved shell element is determined as

{\begin{matrix} γ_{Y^{'} Z^{'}} \\ γ_{X^{'} Z^{'}} \end{matrix}} = {\begin{matrix} θ_{2} + \frac{1}{A_{2}} \frac{\partial w}{\partial α_{2}} - \frac{u}{R_{12}} - \frac{v}{R_{2}} \\ θ_{1} + \frac{1}{A_{1}} \frac{\partial w}{\partial α_{1}} - \frac{u}{R_{1}} - \frac{v}{R_{12}} \end{matrix} {

(39)

and hence, the transverse shear strain at any point on the shell mid-surface can be expressed as

{\begin{matrix} γ_{Y^{'} Z^{'}} \\ γ_{X^{'} Z^{'}} \end{matrix}} = \sum_{k = 1}^{n d} [\begin{array}{l} - \frac{N_{i}}{R_{12}} & - \frac{N_{i}}{R_{2}} & \frac{1}{A_{2}} \frac{\partial N_{i}}{\partial α_{2}} & 0 & N_{i} \\ - \frac{N_{i}}{R_{1}} & - \frac{N_{i}}{R_{12}} & \frac{1}{A_{1}} \frac{\partial N_{i}}{\partial α_{1}} & N_{i} & 0 \end{array}] {\begin{array}{l} u_{0 i} \\ v_{0 i} \\ w_{i} \\ θ_{1 i} \\ θ_{2 i} \end{array}}

(40)

{\begin{matrix} γ_{Y^{'} Z^{'}} \\ γ_{X^{'} Z^{'}} \end{matrix}} = [B_{s}^{e}] {d^{e}}

(41)

The ABD can be written in following integral forms

(A_{i j} B_{i j} D_{i j}) = [\int_{- h / 2}^{h / 2} (\begin{matrix} 1 & z & z^{2} \end{matrix}) \bar{Q_{i j}} d z], (A_{i j}^{s}) = [\int_{- h / 2}^{h / 2} k_{s} \bar{Q_{i j}} d z]

(42)

The shear correction factor is taken 5/6 times in the present analysis, and the shell structure is divided into ‘n’ number of layers as shown in Figure 3. The elements of the ABD matrix are mentioned below

A_{i j} = \sum_{k = 1}^{n} {(\bar{Q_{i j}})}_{k} (z_{k} - z_{k - 1})

(43)

B_{i j} = \frac{1}{2} \sum_{k = 1}^{n} {(\bar{Q_{i j}})}_{k} (z_{k}^{2} - z_{k - 1}^{2}), Here, i = 1,3 and j = 1,3

(44)

D_{i j} = \frac{1}{3} \sum_{k = 1}^{n} {(\bar{Q_{i j}})}_{k} (z_{k}^{3} - z_{k - 1}^{3})

(45)

The transformed stiffness matrix is denoted by ${(\bar{Q_{i j}})}_{k}$ of each layer and the elemental in plane/bending stiffness $([K_{b b}^{e}])$ can be determined as follows

[K_{b b}^{e}] = \int {({[B_{b}^{e}]}^{T} [D_{b}] [B_{b}^{e}] d V)}_{v}

(46)

[K_{b b}^{e}] = \int {({[B_{b}^{e}]}^{T} [\begin{matrix} [A] & [B] \\ [B] & [D] \end{matrix}] [B_{b}^{e}] d Ω)}_{Ω}

(47)

The elemental transverse shear stiffness matrix $([K_{s s}^{e}])$ can be obtained as follows

[K_{s s}^{e}] = \int {({[B_{s}^{e}]}^{T} [D_{s}] [B_{s}^{e}] d V)}_{v}

(48)

[K_{s s}^{e}] = \int {({[B_{s}^{e}]}^{T} [\bar{D_{s}}] [B_{s}^{e}] d Ω)}_{Ω}

(49)

where

[D_{b}] and [D_{s}]

are the bending and shear rigidity matrices, respectively. The Gaussian quadrature scheme has been used to numerically determine the above stiffness matrices.

The external force vector $({F^{e}})$ can be expressed as follows

{F^{e}} = \int {({[N]}^{T} {f_{s}^{e} (x, y)} d A)}_{A}

(50)

2.2.6. Determination of stiffness under hygrothermal condition of FG-CNTRHC skewed shell structure

The in-plane strains due to the hygrothermal condition (Ram and Sinha, 1992) can be determined as follows

{ε}^{H T} = {[\begin{matrix} \begin{matrix} ε_{X^{'} X^{'}}^{H T 0} & ε_{Y^{'} Y^{'}}^{H T 0} & γ_{X^{'} Y^{'}}^{H T 0} \end{matrix} & \begin{matrix} k_{X^{'} X^{'}}^{H T} & k_{Y^{'} Y^{'}}^{H T} & k_{X^{'} Y^{'}}^{H T} \end{matrix} \end{matrix}]}^{T}

(51)

{(\begin{matrix} A_{i j} & B_{i j} & D_{i j} \end{matrix})}^{H T} = {[\int_{- h / 2}^{h / 2} (\begin{matrix} 1 & z & z^{2} \end{matrix}) \bar{Q_{i j}} d z]}^{H T}

(52)

where,

{\bar{(Q_{i j})}}_{k}^{H T} = {[\bar{Q_{i j}} e_{k}]}^{H T} and e_{k} = [T^{0}] {β_{1}, β_{2}}_{k}^{T} (C - C_{0}) + [T^{0}] {α_{1}, α_{2}}_{k}^{T} (T - T_{0})

(53)

where

[T^{0}] = [\begin{matrix} {cos}^{2} θ \\ {sin}^{2} θ \\ sin 2 θ \end{matrix} \begin{matrix} {sin}^{2} θ \\ {cos}^{2} θ \\ - sin 2 θ \end{matrix}]

A_{i j}^{H T} = \sum_{k = 1}^{n} {\bar{(Q_{i j})}}_{k}^{H T} (z_{k} - z_{k - 1})

(54)

B_{i j}^{H T} = \frac{1}{2} \sum_{k = 1}^{n} {\bar{(Q_{i j})}}_{k}^{H T} (z_{k}^{2} - z_{k - 1}^{2}), Here, i = 1,3 and j = 1,3

(55)

D_{i j}^{H T} = \frac{1}{3} \sum_{k = 1}^{n} {\bar{(Q_{i j})}}_{k}^{H T} (z_{k}^{3} - z_{k - 1}^{3})

(56)

The transformed hygrothermal stiffness matrix is denoted by ${\bar{(Q_{i j})}}_{k}^{H T}$ of each layer and the elemental in plane/bending stiffness is ${[K_{b b}^{e}]}^{H T}$

{[K_{b b}^{e}]}^{H T} = \int {({({[B_{b}^{e}]}^{T})}^{H T} {[D_{b}]}^{H T} {[B_{b}^{e}]}^{H T} d V)}_{v}

(57)

{[K_{b b}^{e}]}^{H T} = \int {({({[B_{b}^{e}]}^{T})}^{H T} {[\begin{matrix} [A] & [B] \\ [B] & [D] \end{matrix}]}^{H T} {[B_{b}^{e}]}^{H T} d Ω)}_{Ω}

(58)

2.2.7. Effective element stiffness matrix of the FG-CNTRHC skewed shell structure

The effective stiffness matrix of the element under the hydrothermal conditions can be expressed as

[K_{u u}^{e}] = ([K_{b b}^{e}] + [K_{s s}^{e}]) - {[K_{b b}^{e}]}^{H T}

(59)

The elemental dynamic equation can be written as

[M_{u u}^{e}] {{\ddot{d}}^{e}} + [K_{u u}^{e}] {d^{e}} = {F^{e}}

(60)

The elemental mass matrix $[M_{e}]$ is given as

[M_{e}] = \int_{- 1}^{1} \int_{- 1}^{1} {[N]}^{T} [ρ] [N] | J | d ξ d η

(61)

where

[ρ] = \sum_{k = 1}^{n} \int_{z_{k}}^{z_{k + 1}} [ρ^{e}]

(62)

$[ρ^{e}]$ is mentioned in Appendix 1

[N] = [\begin{matrix} N_{1} & 0 & 0 & 0 & 0 \\ 0 & N_{1} & 0 & 0 & 0 \\ 0 & 0 & N_{1} & 0 & 0 \\ 0 & 0 & 0 & N_{1} & 0 \\ 0 & 0 & 0 & 0 & N_{1} \end{matrix} \dots \begin{matrix} N_{n} & 0 & 0 & 0 & 0 \\ 0 & N_{n} & 0 & 0 & 0 \\ 0 & 0 & N_{n} & 0 & 0 \\ 0 & 0 & 0 & N_{n} & 0 \\ 0 & 0 & 0 & 0 & N_{n} \end{matrix}]

(63)

where

ρ^{k}

is the volumetric density at the midpoint of the kth layer which is a function of the power law index, and the mass matrix can be defined as

[M_{u u}^{e}] = \int_{- 1}^{1} \int_{- 1}^{1} \int_{- h / 2}^{h / 2} ρ {[N]}^{T} [N] d z | J | d ξ d η

(64)

The Rayleigh damping model is applied for skew FG-CNTRHC structures to calculate the damping matrix for higher degrees of freedom (Chowdhury and Dasgupta, 2003). The damping matrix [C_d] can be calculated as

{X}^{T} [C_{d}] {X} = [\begin{matrix} α + β ω_{1}^{2} & 0 & ... & 0 \\ 0 & α + β ω_{2}^{2} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & ... & 0 & α + β ω_{N}^{2} \end{matrix}] = [\begin{matrix} 2 ξ_{1} ω_{1} & 0 & ... & 0 \\ 0 & 2 ξ_{1} ω_{2} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & ... & 0 & 2 ξ_{N} ω_{N} \end{matrix}]

(65)

The eigenvector of the system is denoted by ${X}$ , whereas $α$ and $β$ are the damping coefficients to be determined for ‘N’ number of equations

ξ_{i} = \frac{α}{2 ω_{i}} + \frac{β ω_{i}}{2}

(66)

Frequency response analysis

The global equation of motion can be written as

[M] {\ddot{d}} + [C] {\dot{d}} + [K] {d} = {F}

(67)

The frequency-dependent dynamic response can be obtained as follows

{| d |} = | {[- [M] ω^{2} + i [C] ω + [K]]}^{- 1} {F_{0}} |

(68)

3. Results and discussions

Based on the formulation discussed in the previous section, a computer code has been developed which is capable of analysing the dynamics of the FG-CNTRHC shell structures under consideration. The developed computer code has been validated with already published results (shown in Table 1) before using the same for modelling and analysis of FG-CNTRHC shell structures. It is observed from Table 1 that the present FE code gives an excellent agreement with already published results (SK and Sinha, 2005).

Table 1.

Comparison of non-dimensional fundamental frequency parameters of laminated (0/90 ply orientation) spherical shell panel (a = b) with Simply-supported boundary condition for various R/a and a/h geometries.

Source	Element type/node	Mesh size	a/h	R/a
Source	Element type/node	Mesh size	a/h	5	10	20	50	100	Plate
ANSYS (commercial software)	Shell-181/4-node	10 × 10	10	10.662	10.273	9.822	9.721	9.687	9.671
ANSYS (commercial software)	Shell-181/4-node	20 × 20		10.616	10.228	9.779	9.677	9.644	9.630
ANSYS (commercial software)	Shell-181/4-node	30 × 30		10.608	10.220	9.772	9.670	9.637	9.622
ANSYS (commercial software)	Shell-281/8-node	10 × 10		9.844	9.455	9.122	8.966	8.936	8.919
Present	Serendipity/8-node	10 × 10		9.755	9.644	9.619	9.613	9.613	9.613
Refs. (Thomas and Roy, 2017)	Serendipity/8-node	10 × 10		9.204	8.977	8.919	8.903	8.901	8.900
Refs. (SK and Sinha, 2005)	Serendipity/8-node	—		9.230	8.984	8.921	8.904	8.901	8.900
ANSYS (commercial software)	Shell-181/4-node	10 × 10	100	30.264	17.168	12.733	11.738	11.586	11.572
ANSYS (commercial software)	Shell-181/4-node	20 × 20		30.073	17.059	12.652	11.663	11.511	11.497
ANSYS (commercial software)	Shell-181/4-node	30 × 30		30.037	17.039	12.637	11.649	11.497	11.483
ANSYS (commercial software)	Shell-281/8-node	10 × 10		29.506	16.783	12.451	11.465	11.310	11.292
Present	Serendipity/8-node	10 × 10		28.844	16.875	12.395	10.821	10.581	10.504
Refs. (Thomas and Roy, 2017)	Serendipity/8-node	10 × 10		28.841	16.704	11.840	10.063	9.7825	9.6873
Refs. (SK and Sinha, 2005)	Serendipity/8-node	—		28.826	16.706	11.841	10.063	9.7825	9.6873

3.1. Material properties of NCs and FG-CNTRHC materials

Strength of materials formulation has been used for determination of material properties of the FG-CNTRHC. The carbon fibre volume fraction and the fibre orientation are graded in the thickness direction of the shell using the power law index as shown in Figure 4(a) and (b). The basic constituents of such FG-CNTRHC material system are mentioned in Table 2. It is observed that the higher CNT volume fraction gradually increases the longitudinal material property of the FG-CNTRHC shell as shown in Figure 5(a) which is reinforced by graded carbon fibre 30%–60% in the thickness direction. The elastic property C₁₂ increases gradually up to its maximum value then decreases slightly, may be because of the poison ratio, whereas C₂₃ increases throughout along the thickness direction of the skewed shell which are shown in Figure 5(b) and (c). Value of C₂₂ increases as the volume fraction of carbon fibre increases but C₅₅ slightly increases then decreases along the thickness direction of the skewed shell as shown in Figure 5(d) and (e). It is clear from Figure 5(a)–(e) that the CNT volume fraction may enhance the elastic properties significantly. The smooth increment in the elastic properties of such FG-CNTRHC skewed shell is expected to avoid delamination of the fibres because of the mismatch in the elastic properties.

Table 2.

Basic constituents of functionally graded carbon nanotubes reinforced hybrid composite material system (Kundalwal and Ray, 2011).

Constituent	E₁₁ (GPa)	E₂₂ (GPa)	G₁₂ (GPa)	G₂₃ (GPa)	υ ₁₂	Density (kg/m³)
Carbon fibre	230.07	20.03	25.00	7.00	0.2986	1800
Carbon nanotube	1060.00	6.63	442.00	17.00	0.16	1440
Polymer (epoxy)	2.50	2.50	0.933	0.933	0.34	1150

Based on the above obtained material properties, FE modelling and analysis of FG-CNTRHC skewed shell structure is made by considering 10 × 10 elements (which is decided after convergence studies of displacement and natural frequency). To obtain time and frequency responses, a 16 layered skewed shell structure with ply orientation (0^°–90^°) having 0.25 mm thickness of each layer is considered for analysis of such FG structures under different hygrothermal conditions. The parametric study has also been carried out on such structure with varying CNT volume fraction into polymer, carbon fibre volume fraction in FG-CNTRHC, power law index, R/a and a/h. Based on the above considerations, various results have been obtained which are presented in the following subsections.

3.2. Maximum displacement and settling time of FG-CNTRHC–based skewed shell structure

The dynamic analysis is carried out for the FG-CNTRHC–based skewed shell structure under uniformly distributed load with the following simply-supported (SSSS) boundary conditions

X^{'} = 0, a : u^{0} = v^{0} = w^{0} = θ_{Y^{'}} = θ_{Z^{'}} = 0

Y^{'} = 0, a : u^{0} = v^{0} = w^{0} = θ_{X^{'}} = θ_{Z} = 0

The effect of the power law index (such as k = 0.2, 1 and 2) and CNT volume fractions (such as 0, 4 and 8) have been observed. In the present study, the skewing shell panel has been subjected to an impulse load of 10 N at the centre for a duration of $10 τ$ seconds (where $τ$ is the time period corresponding to the first natural frequency of the system), and impulse responses of this panel have been obtained with a time step of $τ$ seconds. Then, the transient response due to the impulse load is obtained by the DUHAMEL integration technique. These pieces of information are incorporated in the revised article. The time taken to decay from maximum amplitude of each response to 5% of its maximum amplitude is taken as settling time in the current study. The maximum displacement and settling time have been determined for the different skewed shell geometries (a/h = 10 and R/a = 1) with the different skewing angles (α = 30^°, 45^° and 60^°) as shown in Figure 6(a)–(c). Higher skewing angle increases the stiffness of the structure which leads to decrease in the displacements, and it can be seen in the presented results. Parametric studies for the different FG-CNTRHC–based skewed shell structures have been provided for their natural frequencies. The non-dimensional frequency parameter is defined as $\bar{ω} = ω a^{2} / h \sqrt{ρ_{m} / E_{m}}$ . Three skew angles are considered along with three CNT volume fractions. Higher natural frequencies are observed as the skew angle of such shell structure increases. FG-CNTRHC–based skewed shell structures with different CNT volume fractions $(v_{cnt} = 0 %, 4 % a n d 8 %)$ are used. The value of $v_{cnt} = 8 %$ significantly improves the damping of such structure than the lower CNT volume fractions. The free vibration analysis has been carried out for different skewed shell structures which are presented in Table 3.

Table 3.

First three non-dimensional natural frequency parameters $(\bar{ω} = ω a^{2} / h \sqrt{ρ_{m} / E_{m}})$ for different skewed shells with simply-supported boundary conditions.

a/h = 10	Skewed shell type		Cylindrical			Spherical
R/a	$α$	$v_{cnt} (%)$	${\bar{ω}}_{1}$	${\bar{ω}}_{2}$	${\bar{ω}}_{3}$	${\bar{ω}}_{1}$	${\bar{ω}}_{2}$	${\bar{ω}}_{2}$
1	30^o	0	4.5049	7.2336	7.8516	6.7542	10.6894	16.2013
		4	8.8444	15.5574	17.591	16.0447	23.9687	39.8484
		8	9.2429	16.1943	18.3221	16.3770	26.3864	40.1631
	45^o	0	5.3157	8.5357	11.8349	7.2635	11.4323	17.7977
		4	10.4364	18.3577	20.7574	17.24	25.124	41.7691
		8	10.9067	19.1093	21.6201	17.6118	27.901	42.4685
	60^o	0	5.7303	9.2015	12.7581	7.5749	12.1862	18.7743
		4	11.2504	19.7896	22.3765	18.1847	26.5912	43.0001
		8	11.7574	20.5998	23.3065	18.6228	29.4294	45.3734
10	30^o	0	2.8088	6.0067	7.5189	5.7762	9.3184	13.8605
		4	6.6737	14.5655	16.7694	13.8663	22.0635	33.7256
		8	6.9341	15.0584	17.4305	14.2933	23.9736	36.0921
	45^o	0	3.2891	7.0338	8.8046	6.0396	9.7955	14.4593
		4	7.8149	17.05621	19.6370	14.3572	23.1381	35.5704
		8	8.1199	17.6334	20.4111	15.351	25.5247	37.6187
	60^o	0	3.5151	7.5171	9.4095	6.2811	9.7638	14.7739
		4	8.3518	18.2279	20.9861	14.6803	23.7073	36.5822
		8	8.6777	18.8448	21.8134	14.9479	25.3977	37.7414
100	30^o	0	2.7705	6.0121	7.4850	3.6177	5.9731	9.0217
		4	6.6172	14.5202	16.7372	8.6845	14.1426	21.5981
		8	6.8710	15.0039	17.3946	9.0356	14.6383	22.3191
	45^o	0	3.2491	6.1696	9.1351	3.8202	6.2268	9.4611
		4	7.5295	14.6121	20.3965	9.1708	14.7434	22.6498
		8	7.8243	15.1175	21.1842	9.5416	15.2601	23.4060
	60^o	0	3.4634	6.2753	9.7379	3.9234	6.5681	9.9512
		4	8.0263	15.5762	21.7423	9.7714	15.6944	24.6306
		8	8.3406	16.1149	22.5819	9.7847	15.9601	26.5651

The skewing angle increases the stiffness of the shell structure because of the reduced surface area which can be seen in Figure 7(a) and (b). The skewing angle has dominant impact on the structural stiffness in comparison with the damping, where skewing of the structure needs to be done with a better skewing angle value without distorting the structure, and it is clear from the results that $α = 45 °$ may be the best skewing angle of the considered skewing angles for such shell structures. In Figure 7(c) and (d), the effect of the CNT volume fraction is shown for the skewing angle $α = 45 °$ and power law index 2. The settling time decreases as the CNT volume fraction increases, so clearly, it is evident from the obtained figures that CNTs are improving the damping of such FG-CNTHRC–based skewed shell structures.

3.3. Frequency response analysis of FG-CNTRHC skewed shell structure under hygrothermal conditions

The frequency response analysis has been done under three hygrothermal conditions, and resonant frequencies for the different FG-CNTRHC–based skewed shell structures with the SSSS boundary condition are obtained. The three combinations of moisture

(Δ C)

and temperature

(Δ T)

conditions are 95% and 25^o C, 95% and 50^o C, and 95% and 75^oC are considered as presented in Table 4. The parametric studies showed significant hygrothermal effect on such structures. It is found that frequencies decrease with the increased moisture concentration at different temperatures. The parametric study is also considered for three skewing angles which are presented in Table 5. The combination of skewing angle and inclusion of CNT exhibits better resonant frequencies and damping of FG-CNTRHC skewed shell structures under the hygrothermal environment.

Table 4.

Different hygrothermal conditions considered for functionally graded carbon nanotubes reinforced hybrid composite–based skewed shell structure.

Hygrothermal case	$Δ C and Δ T$
HT case 1	95% and 25^oC
HT case 2	95% and 50^oC
HT case 3	95% and 75^oC

Table 5.

Non-dimensional resonant frequency parameter $(\bar{ω} = ω a^{2} / h \sqrt{ρ_{m} / E_{m}})$ of functionally graded carbon nanotubes reinforced hybrid composite–based skewed shell structure with simply-supported boundary conditions under different hygrothermal environment (with $β_{1} = 0 and β_{2} = 0.44$ and $α_{1} = - 0.3 \times 10^{- 6} / °C$ and $α_{2} = 28.1 \times 10^{- 6} / ° C$ ) (Parhi et al., 2000; Zhou et al., 2019).

a/h = 10	Skewed shell type		Cylindrical			Spherical
R/a	$α$	$v_{cnt} (%)$	$Δ C (%) and Δ T (° C)$			$Δ C (%) and Δ T (° C)$
R/a	$α$	$v_{cnt} (%)$	95 and 25	95 and 50	95 and 75	95 and 25	95 and 50	95 and 75
1	30^o	0	3.4944	3.2524	3.1457	3.6178	3.4296	3.3577
		4	4.2973	4.2773	4.0190	4.5181	4.3616	4.2397
		8	4.7094	4.6797	4.5793	4.7937	4.7535	4.7132
	45^o	0	3.6604	3.4069	3.3146	3.6938	3.5297	3.5773
		4	4.6162	4.5947	4.4395	5.3828	5.2721	4.7739
		8	5.0118	4.9802	4.9797	5.7285	5.4712	5.3171
	60^o	0	3.932	3.6597	3.5647	4.5518	4.3675	4.1333
		4	5.323	5.2895	5.2092	6.8320	6.1262	6.0391
		8	5.7764	5.3492	5.3087	6.9658	6.6421	6.3448
10	30^o	0	2.7153	2.7045	2.6231	3.4364	2.5268	2.4191
		4	3.5234	3.3584	3.1596	3.9083	3.459	3.3616
		8	4.0231	3.9512	3.8768	4.0413	3.6759	3.6059
	45^o	0	2.8745	2.6412	2.2014	3.5062	2.6468	2.3435
		4	3.9638	3.8353	3.5501	3.8241	3.6119	3.4119
		8	4.8076	4.5464	4.4126	4.3008	3.7157	3.6111
	60^o	0	3.3562	3.4441	3.3984	3.5679	3.4432	3.1709
		4	4.5465	4.4566	4.1234	4.6195	4.1548	4.0182
		8	5.8461	5.5194	5.0415	4.7764	4.2017	4.0548
100	30^o	0	2.4703	2.3561	2.2142	2.5715	2.2521	2.0123
		4	3.4818	3.1539	3.0472	3.2966	3.2687	3.0571
		8	3.9837	3.9106	3.8302	3.8506	3.6621	3.5133
	45^o	0	2.5222	2.5023	2.0726	2.6937	2.3591	2.1363
		4	3.9170	3.6017	3.4998	3.5412	3.4971	3.1135
		8	4.7606	4.5011	4.2461	3.8627	3.5013	3.2517
	60^o	0	3.2939	3.4004	3.3412	2.8936	2.5342	2.3394
		4	4.4928	4.1852	4.0459	4.5274	4.1392	3.8214
		8	5.7888	5.4643	5.0087	4.5786	4.1859	3.9709

To study the hygrothermal effects on transverse displacements at the midpoint of the skewed shell and its resonant frequencies, the frequency response analysis of the simply supported cylindrical and spherical skewed shell structure is carried out for three different hygrothermal cases by considering 8% CNT volume fraction.

A point load of 10 N is applied in the transverse direction at the mid-node of the structure to perform the frequency analysis under different hygrothermal cases for different CNT volume fractions. A certain type of geometry (thick and deep shell i.e. R/a = 1 and a/h = 10) has been analysed, and the frequency response is obtained as shown in Figure 8(a) and (b). The effect of different hygrothermal cases is seen with the clear variation in the curves depicting the decrease in resonant frequency of such structure in hygrothermal case 3 because of high temperature and moisture concentration. Clearly, because of the hygrothermal environment, the induced stresses and strains are affecting the structural strength which results in increase of absolute displacement and decrease in the resonant frequencies. Different geometries (such as R/a = 1, 10 and 100 with a/h =10 and 100) are considered, and the frequency analysis is performed under all three hygrothermal cases so as to check the impact of different CNT volume fractions on such FG-CNTRHC skewed shell structures. The comparative study for the resonant frequency parameter is also carried out for different FG-CNTRHC skewed cylindrical and spherical shell structures as shown in Table 5. The frequency parameter decreases for the higher R/a and a/h geometries as the shell structure becomes shallow and thin. The skewing angle (α = 45°) significantly increases the structural stiffness of the structure which results in higher resonant frequencies.

3.4. Impulse-induced transient response analysis of FG-CNTRHC skewed shell structure under hygrothermal conditions

The transient response analysis is carried out for the simply supported cylindrical (R₂ > 10³⁰) and spherical (R₁ = R₂) skewed shell on square base (a = b) under impulse load. The settling time is obtained for the comparative study of the vibration damping of such shell structures. A thick and deep (R/a = 1 and a/h = 10) type of geometry is considered for skewed shell structures to obtain the transient response which is shown in Figure 9(a) and (b). The effect of different volume fractions of the CNT at 45^o skewing angle for Hygrothermal (HT) case 1 is performed thereafter from the obtained transient responses. It is clear from the obtained responses in the figures that higher volume fraction of CNTs improves the damping of such structures by lowering settling time. The comparative study also encompasses different geometries (viz. R/a = 1, 10 and 100 with a/h = 10 and 100) for the transient response analysis under all three hygrothermal cases as shown in Table 6. The higher R/a and a/h values make the structure shallow and thin hence exhibit higher settling time. It is clear from the results that CNT improves the damping of the FG-CNTRHC skewed cylindrical and spherical shell structures.

Table 6.

Settling time (in milliseconds) variation for different skewed shells with simply-supported boundary conditions with $k = 2$ and skewing angle $α = 45 °$ under different hygrothermal environment (with $β_{1} = 0 and β_{2} = 0.44$ and $α_{1} = - 0.3 \times 10^{- 6} / °C$ and $α_{2} = 28.1 \times 10^{- 6} / ° C$ ) (Parhi et al., 2000; Zhou et al., 2019).

a/h	R/a	$v_{cnt} (%)$	Cylindrical			Spherical
			$Δ C (%) and Δ T (° C)$			$Δ C (%) and Δ T (° C)$
			95 and 25	95 and 50	95 and 75	95 and 25	95 and 50	95 and 75
10	1	0	0.5098	0.5223	0.5336	0.5001	0.5176	0.5363
		4	0.5093	0.5259	0.5405	0.4962	0.5162	0.5358
		8	0.4434	0.4648	0.4885	0.4094	0.4311	0.4636
	10	0	0.5245	0.54	0.5543	0.5211	0.5357	0.5555
		4	0.5239	0.5385	0.5561	0.5109	0.5296	0.5511
		8	0.4565	0.478	0.4984	0.4174	0.4485	0.4839
	100	0	0.9413	0.9567	0.9682	0.9206	0.9401	0.9599
		4	0.9322	0.9477	0.9601	0.9156	0.9371	0.9529
		8	0.7824	0.8026	0.8216	0.7547	0.7861	0.8076
100	1	0	48.2	51.74	55.25	46.72	50.67	54.65
		4	40.2	44.78	49.19	37.51	42.43	47.41
		8	12.7	18.04	23.25	10.2	15.16	21.1
	10	0	49.6	52.55	55.47	47.21	50.26	53.4
		4	41.3	45.14	48.95	38.54	42.1	46.72
		8	13.1	17.62	21.88	11.4	15.92	20.06
	100	0	82.2	85.21	88.2	79.1	82.34	85.39
		4	67.5	71.56	75.57	62.14	66.15	70.66
		8	19.7	24.82	29.86	19.02	23.98	29.12

4. Conclusions

The vibration and damping analysis has been carried out numerically for different FG-CNTRHC skewed shell structures under hygrothermal conditions. The Mori–Tanaka scheme is used along with strength of materials method to obtain the effective elastic properties of FG-CNTRHC composite materials. The CNT is incorporated into such HC to avoid the delamination of the structure under the complex loading conditions. Furthermore, the FG distribution of carbon fibre volume fraction and its orientation in the thickness direction is varied by using the power law to strengthen the structure. The dynamic analysis is carried out by using such elastic properties of FG-CNTRHC composite materials for different skewed shell structures under hygrothermal conditions. The geometry of shell is defined by using an oblique coordinate system with a tangent to the edge for skewing the structure and said skewed shell structure is discretised by considering an eight-noded shell element for the FE analysis. Increasing the skewing angle causes a reduction in the surface area of the shell structure which leads to an increase in the material concentration, thereby resulting in higher natural frequencies. The stresses induced because of hygrothermal conditions are incorporated into dynamic analysis by calculating the effective stiffness of such skewed shell structures. The transient responses indicate that the CNT is more influential than carbon fibres in the shell structure because the CNT possess better damping properties under different hygrothermal conditions. Significant decrease in the settling time is observed by addition of CNTs in the hybrid FG skewed shell structure. The effect of thermal and moisture concentration results in increase in the absolute amplitudes of such structures. The frequency responses of such structures depict that CNTs have significant impact on the absolute amplitude in comparison with the first resonant frequency. So, it is evident that the inclusion of the CNT may improve the damping property of the FG-CNTRHC skewed shell structure under the hygrothermal conditions. It can be concluded that for hygrothermal condition–based applications, the CNT can play a vital role to achieve better dynamic responses by mitigating vibrations quickly for FG-CNTHRC skewed shell structures.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors kindly acknowledged the IMPRINT cell of the Ministry of Human Resource Development (MHRD) and Department of Science and Technology (DST), Government of India, for a project grant (No. 6292) under which the research work was carried out.

ORCID iD

Salur Srikant Patnaik

Appendix 1

(69)

[T] = [\begin{matrix} {cos}^{2} θ & {sin}^{2} θ & 0 & 0 & 0 & cos θ sin θ \\ {sin}^{2} θ & {cos}^{2} θ & 0 & 0 & 0 & - cos θ sin θ \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & cos θ & - sin θ & 0 \\ 0 & 0 & 0 & sin θ & cos θ & 0 \\ - 2 cos θ sin θ & 2 cos θ sin θ & 0 & 0 & 0 & {cos}^{2} θ - {sin}^{2} θ \end{matrix}]

(70)

[C_{1}] = v_{f} [\begin{matrix} C_{11}^{f} & C_{12}^{f} & C_{12}^{f} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(71)

[C_{2}] = [\begin{matrix} v_{N C} C_{11}^{N C} & v_{N C} C_{12}^{N C} & v_{N C} C_{12}^{N C} & 0 & 0 & 0 \\ C_{12}^{N C} & C_{11}^{N C} & C_{12}^{N C} & 0 & 0 & 0 \\ C_{12}^{N C} & C_{12}^{N C} & C_{11}^{N C} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44}^{N C} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44}^{N C} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44}^{N C} \end{matrix}]

(72)

[C_{3}] = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ C_{12}^{f} & C_{22}^{f} & C_{23}^{f} & 0 & 0 & 0 \\ C_{13}^{f} & C_{23}^{f} & C_{22}^{f} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44}^{f} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55}^{f} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{55}^{f} \end{matrix}]

(73)

[C_{4}] = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ C_{12}^{N C} & C_{11}^{N C} & C_{12}^{N C} & 0 & 0 & 0 \\ C_{12}^{N C} & C_{12}^{N C} & C_{11}^{N C} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44}^{N C} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44}^{N C} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44}^{N C} \end{matrix}]

(74)

[V_{1}] = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & v_{f} & 0 & 0 & 0 & 0 \\ 0 & 0 & v_{f} & 0 & 0 & 0 \\ 0 & 0 & 0 & v_{f} & 0 & 0 \\ 0 & 0 & 0 & 0 & v_{f} & 0 \\ 0 & 0 & 0 & 0 & 0 & v_{f} \end{matrix}]

(75)

[V_{2}] = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & v_{N C} & 0 & 0 & 0 & 0 \\ 0 & 0 & v_{N C} & 0 & 0 & 0 \\ 0 & 0 & 0 & v_{N C} & 0 & 0 \\ 0 & 0 & 0 & 0 & v_{N C} & 0 \\ 0 & 0 & 0 & 0 & 0 & v_{N C} \end{matrix}]

(76)

[V_{3}] = [V_{1}] + [V_{2}] {[C_{4}]}^{- 1} [C_{3}]

(77)

[V_{4}] = [V_{2}] + [V_{1}] {[C_{3}]}^{- 1} [C_{4}]

(78)

\begin{array}{l} N_{1} = \frac{1}{4} (1 - ξ) (1 - η) (- 1 - ξ - η) \\ N_{2} = \frac{1}{2} (1 - ξ^{2}) (1 - η) \\ N_{3} = \frac{1}{4} (1 + ξ) (1 - η) (- 1 + ξ - η) \\ N_{4} = \frac{1}{2} (1 + ξ) (1 - η^{2}) \\ N_{5} = \frac{1}{4} (1 + ξ) (1 + η) (- 1 + ξ + η) \\ N_{6} = \frac{1}{2} (1 - ξ^{2}) (1 + η) \\ N_{7} = \frac{1}{4} (1 - ξ) (1 + η) (- 1 - ξ + η) \\ N_{8} = \frac{1}{2} (1 - ξ) (1 - η^{2}) \end{array}

The Jacobian is expressed as (79)

| J | = \sqrt{a} \frac{\partial (α_{1}, α_{2})}{\partial (ξ, η)} = \sqrt{a} | J^{*} |

where (80)

\sqrt{a} = \sqrt{(r, 1 \cdot r, 1) (r, 2 \cdot r, 2) - {(r, 1 \cdot r, 2)}^{2}}

(81)

| J^{*} | = det [\begin{matrix} α_{1, ξ} & α_{2, ξ} \\ α_{1, η} & α_{2, η} \end{matrix}]

(82)

[ρ^{e}] = [\begin{matrix} ρ^{k} & 0 & 0 & ρ^{k} z & 0 \\ 0 & ρ^{k} & 0 & 0 & ρ^{k} z \\ 0 & 0 & ρ^{k} & 0 & 0 \\ ρ^{k} z & 0 & 0 & ρ^{k} z^{2} & 0 \\ 0 & ρ^{k} z & 0 & 0 & ρ^{k} z^{2} \end{matrix}]

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