Study on CNT reinforced functionally graded sandwich conical shell subjected to low-velocity impact under thermal environment

Abstract

In the present work, low-velocity impact behavior of pre-twisted carbon nanotubes (CNTs) reinforced functionally graded (FG) sandwich conical shell panels is studied using finite element method. The top and bottom face sheets are reinforced with single-walled CNTs reinforced FG materials with various patterns. The temperature-dependent material properties of the CNTs reinforced functionally graded facings are evaluated using micromechanical models. The dynamic equilibrium equation of the impacted sandwich panel is formulated using Lagrange’s equation. A modified Hertzian contact law is used to evaluate contact force. The solutions of the resulting equations are obtained using Newmark’s time integration method. After validation study of the present method, the effects of the CNTs grading pattern, velocity of the impactor, size of the impactor, CNTs volume fraction, operating temperature, angle of twist, and core-to-facing thickness ratio on the low-velocity impact response of the CNTs reinforced functionally graded sandwich conical shell panel are studied in detail. Numerical results show that increasing the volume fraction of CNTs increases the contact force while a lower contact force is predicted at elevated temperatures.

Keywords

Sandwich structures conical shell carbon nanotubes low-velocity impact finite element method

Introduction

The impact-induced vibration of a structural part is dependent on the contact force generated due to impact. The low-velocity impact of a structural part has a significant interest in many situations in engineering applications. Sandwich techniques are used to make robust and lightweight structures for potential applications in the energy, aviation, automobile, and shipping industries. Two functionally graded face layers are used with an intermediate core (isotropic homogeneous) layer to get a functionally graded sandwich structure. The facing layers support the main bending strength and the core bears the shear strength of the sandwich structures. Composite material exhibits superior strength and stiffness. However, composite structures are vulnerable to fiber rupture, matrix cracking, delamination, and facing/core separation due to low-velocity impact with foreign bodies during fabrication or operation¹ resulting in structural instability. Impact resistance of the composites must be enhanced to have a better structural design. In 1991, Iijima first reported that Carbon nanotubes (CNTs) have exhibit astonishing mechanical properties (Young’s modulus is over 1 Tpa; tensile strength is 200 GPa).² These properties are ideal for reinforcing it to the conventional composite and functionally graded composites. Thus, the use of CNTs with functionally graded material (FGM) provides improved mechanical, electrical, and thermal properties. Carbon nanotubes reinforced functionally graded material has excellent energy-absorbing capacity in addition to other mechanical properties over conventional composite fibers.² The reinforcement of functionally graded CNTs into polymeric facings enhances the impact resistance significantly. Therefore, CNTs reinforced functionally graded facings could be used in sandwich panels under impact loads. Hence, a detailed study on low velocity impact characteristics of CNT reinforced functionally graded sandwich panels is highly required for safety during operation. Effective elastic moduli of CNTs-reinforced functionally graded material can be determined by molecular dynamic (MD) simulations, classical shell theory (CST), and finite element method (FEM).³ Addition of CNTs up to a certain limit can enhance the elastic moduli of the CNTs reinforced FGM.⁴ An advanced composite named as functionally graded reinforcing optimal amount of CNTs was first proposed by Shen.⁵ The concept, modeling, and mechanical behavior of FG-CNT structures are explained by Liew et al.⁶ Kwon et al.⁷ successfully fabricated this advanced material. Bending, buckling, and vibration behavior of the graded distribution of CNTs are more efficient compared to uniform distribution.^8–10 Kiani¹¹ used the first-order shear deformation theory in combination with the Ritz method to investigate post-buckling characteristics of CNTs reinforced functionally graded sandwich plate in a thermal environment. Meher and Panda¹² analyzed the free vibration of FG-CNTRC sandwich curved panels in uniform temperature. Qi et al.¹³ performed the thermoelastic analysis of stiffened sandwich doubly curved plate with FGM core under low-velocity impact. The impact behavior of conventional laminated fiber-reinforced composites, functionally graded structures, and sandwich structures was investigated by numerous researchers.^14–21 Recently, Fan et al.²² investigated the low-velocity impact response of an auxetic nanocomposite laminate beam. In contrast, the research works concerning low-velocity impact analysis of CNTs reinforced functionally graded sandwich structures are very few. Wang et al.²³ investigated analytically the impact characteristics of CNTRC single-layer and multi-layer sandwich plates in a uniform thermal environment based on modified Hertzian contact law. In another study, Feli and Rashidi²⁴ included the spring–mass model in the analytical model to predict the influence of CNTs grading pattern on low-velocity impact response of CNTRC sandwich plates. The dynamic behavior of laminated conical shells subjected to impact load was not much explored, although conical shells have many practical applications in modern engineering. An analytical approach was presented by Azizi²⁵ to study the low-velocity impact behavior of agglomerated SWCNTs-reinforced conical sandwich frusta using spring–mass–damper (SMD) model. Das et al.²⁶ presented the time-dependent response of impact-induced functionally graded conical shell considering porosity.

From the past literature it is observed that very few works on low-velocity impact response of functionally graded conical shell are available. Reinforcing the CNTs into functionally graded conical shell with sandwich structures and analyzing the response in a thermal environment are the novelty of this study. In this work, a finite element based method is considered to predict the impact response of a cantilever pre-twisted CNTs reinforced functionally graded sandwich conical shell panel with top and bottom facings and an isotropic middle core. The material properties of both facing and core layers of the sandwich panels are assumed to be temperature-dependent, including various patterns of CNTs distribution in the facings. The effective thermo-mechanical properties of CNT reinforced functionally graded facings are evaluated based on micromechanical models. The dynamic equilibrium equation of the CNT reinforced functionally graded sandwich panel is formulated using Lagrange’s equation of motion, wherein modified Hertzian contact law is used to predict the contact force. The resulting equations of motion are solved using Newmark’s time integration method. After confirming the accuracy and constancy of the present formulation, the effects of various parameters on the impact response of pre-twisted CNT reinforced functionally graded sandwich conical shell panel are analyzed and presented.

Theoretical formulation

Model of conical shell

A shallow conical shell made of CNTs reinforced functionally graded sandwich material is considered in this analysis. The shell is made of two CNTs reinforced FGM face-sheets and an isotropic core as shown in Figure 1. The parameter of a thin shallow conical shell panel denoted by L, b, s, β₀, α₀, h, h_fgm, h_core, φ_v, and φ₀ represents the blade length, blade base width, cone length, major and minor radii at the fixed end of the conical shell, total thickness, top and bottom CNTs reinforced FGM face-sheets thickness, isotropic core thickness, vertex angle, and base subtended angle, respectively. A curve (curvature in x- and y-direction) shallow conical shell with twist (radius of twist =r_xy) can be characterized through its middle surface by²⁷

z = - \frac{1}{2} [\frac{x^{2}}{r_{x}} + 2 \frac{x y}{r_{x y}} + \frac{y^{2}}{r_{y}}]

(1)

where r_x and r_y represent the radii of curvature in the x- and y-directions while r_xy denotes the radius of twist. Length (L) of the shell and twist angle (ψ) are related as

\tan ψ = - \frac{L}{r_{x y}}

(2)

Figure 1.

Geometry of the untwisted shallow sandwich conical shell panel.

Figure 2 represents a cantilever conical shell made of functionally graded sandwich material idealized as turbo machinery blade. The curvature radius in x-direction is infinity. The neutral plane of the sandwich conical shell can be expressed as

z = \frac{y^{2}}{2 r_{y}} - \frac{x y}{r_{x y}}

(3)

Figure 2.

Geometry of twisted CNTs reinforced functionally graded sandwich conical shell idealized as turbomachinery blade conical shell.

The equation of the ellipse at any cross-section of the conical shell defined as

{(\frac{y}{α})}^{2} + {(\frac{z + β}{α})}^{2} = 1

(4)

The varying radius of curvature $(R_{y})$ of the conical shell is expressed as

\begin{array}{l} R_{y} = \frac{{[1 + (\frac{d^{2} z}{d^{2} y})]}^{\frac{3}{2}}}{(\frac{d^{2} z}{d^{2} y})} \\ = s {(\frac{α}{s})}^{2} {(\frac{β}{s})}^{2} {[{(\frac{s}{β})}^{2} + {(\frac{y}{b_{0}})}^{2} {(\frac{b_{0}}{s})}^{2} {(\frac{s}{α})}^{2} {{(\frac{α}{s})}^{2} - {(\frac{s}{β})}^{2}}]}^{\frac{3}{2}} \end{array}

(5)

These parameters are related as

\frac{β}{s} = \tan (\frac{θ_{v}}{2}) [1 - \frac{x}{s}]

(6)

\frac{b_{0}}{s} = 2 \sin (\frac{θ_{v}}{2}) \sqrt{\frac{{(\tan (\frac{θ_{0}}{2}))}^{2}}{{(\cos (\frac{θ_{v}}{2}))}^{2} + {(\tan (\frac{θ_{0}}{2}))}^{2}}}

(7)

\frac{b}{s} = \frac{b_{0}}{s} [1 - \frac{x}{s}]

(8)

(\frac{α}{s}) = \frac{(\frac{b}{s}) (\frac{β}{s}) \tan (\frac{θ_{0}}{2})}{\sqrt{4 {(\frac{β}{s})}^{2} {(\tan (\frac{θ_{0}}{2}))}^{2} - {(\frac{b}{s})}^{2}}}

(9)

CNTs-reinforced FGM facings

Armchair single-walled (10, 10) carbon nanotubes (SWCNTs) are reinforced longitudinally in the functionally graded top and bottom facings of the sandwich conical shell panel. The reinforcement patterns in the thickness direction can be uniformly distributed (UD) or functionally graded (FG). The commonly found CNTs grading patterns are UD, FG-V, FG-O, and FG-X.²⁸ Considering these patterns in the present study, five different varieties of CNT reinforced functionally graded (FG) sandwich conical shell panels are considered as described in Figure 3.

FG-U: Both facings are made of uniformly distributed CNTs FGM. The volume fraction V_CNT is expressed as

V_{C N T} = V_{C N T}^{e} b o t h f o r t o p a n d b o t t o m f a c e i n g

(10)

FG-VΛ: Top and bottom facings are made of FG-V and FG-Λ CNTs FGM, respectively. The volume fraction V_CNT is expressed as

V_{C N T} = 2 V_{C N T}^{e} (\frac{2 z - h_{c o r e}}{h_{f g m}}) f o r t o p f a c e i n g

(11)

V_{C N T} = - 2 V_{C N T}^{e} (\frac{2 z + h_{c o r e}}{h_{f g m}}) f o r b o t t o m f a c e i n g

(12)

FG-ΛV: Top and bottom facings are made of FG-Λ and FG-V CNTs FGM, respectively. The volume fraction V_CNT is expressed as

V_{C N T} = 2 V_{C N T}^{e} (\frac{h_{c o r e} + h_{f g m} - 2 z}{h_{f g m}}) f o r t o p f a c e i n g

(13)

V_{C N T} = 2 V_{C N T}^{e} (\frac{h_{c o r e} + h_{f g m} + 2 z}{h_{f g m}}) f o r b o t t o m f a c e i n g

(14)

FG-XX: Both facings are made of FG-X FGM, respectively. The volume fraction V_CNT is expressed as

V_{C N T} = 2 V_{C N T}^{e} (| \frac{4 z - 2 h_{c o r e} - h_{f g m}}{h_{f g m}} |) f o r t o p f a c e i n g

(15)

V_{C N T} = 2 V_{C N T}^{e} (| \frac{4 z + 2 h_{c o r e} + h_{f g m}}{h_{f g m}} |) f o r b o t t o m f a c e i n g

(16)

FG-OO: Both facings are made of FG-O CNTs FGM, respectively. The volume fraction V_CNT is expressed as

V_{C N T} = 2 V_{C N T}^{e} (1 - | \frac{4 z - 2 h_{c o r e} - h_{f g m}}{h_{f g m}} |) f o r t o p f a c e i n g

(17)

V_{C N T} = 2 V_{C N T}^{e} (1 - | \frac{4 z + 2 h_{c o r e} + h_{f g m}}{h_{f g m}} |) f o r b o t t o m f a c e i n g

(18)

Figure 3.

Grading patterns of CNTs in sandwich conical shell panel.

Overall volume fraction of CNTs ( $V_{C N T}^{e})$ as per equations (10)–(18) is defined as

V_{C N T}^{e} = \frac{w_{C N T}}{w_{C N T} + \frac{ρ^{C N T}}{ρ^{f g m}} - \frac{ρ^{C N T}}{ρ^{f g m}} w_{C N T}}

(19)

where $w_{C N T}$ indicates the mass fraction of CNTs; $ρ^{C N T}$ and $ρ^{f g m}$ denote the mass densities of CNTs and FGM matrix, respectively.

The effective material property of the conical shell is obtained by the extended rule of mixture expressed as

E_{11} = η_{1} V_{C N T} E_{11}^{C N T} + V_{f g m} E_{f g m} + V_{m} E_{m}

(20)

\frac{η_{2}}{E_{22}} = \frac{V_{C N T}}{E_{22}^{C N T}} + \frac{V_{f g m}}{E_{f g m}} + \frac{V_{m}}{E_{m}}

(21)

\frac{η_{3}}{G_{12}} = \frac{V_{C N T}}{G_{12}^{C N T}} + \frac{V_{f g m}}{G_{f g m}} + \frac{V_{m}}{E_{m}}

(22)

where $E_{11}^{C N T}$ , $E_{22}^{C N T}$ , and $G_{12}^{C N T}$ are the longitudinal modulus, transverse modulus, and shear modulus of elasticity and $η_{i}$ (i = 1,2,3) is the efficiency parameter of CNTs. In the entire analysis, it is assumed that $G_{12} = G_{13} = G_{23}$ . $E_{m}$ and $G_{m}$ are Young’s modulus and shear modulus of the isotropic core. $E_{f g m}$ and $G_{f g m}$ are the effective Young’s modulus and shear modulus of the FGM. The apparent elasticity modulus in longitudinal (E₁₁) and transverse direction ((E₁₂) of CNTs reinforced functionally graded sandwich conical shell may be obtained as per equations (20) and (21). The volume fraction of isotropic core, FGM matrix ( $V_{f g m}$ ), and that of CNTs ( $V_{C N T}$ ) of top and bottom facings is related as

V_{C N T} + V_{f g m} + V_{c o r e} = 1

(23)

It is to be noted that for the facing, the volume percentage of core is zero and for the core the volume percentage of CNTs and FGM is zero. The effective mass density and Poisson’s ratio of the conical shell can be computed as

ρ = V_{C N T} ρ_{C N T} + V_{f g m} ρ_{f g m} + V_{m} ρ_{m}

(24)

υ = V_{C N T} υ^{C N T} + V_{f g m} ρ_{f g m} + V_{m} υ_{m}

(25)

where $ρ_{C N T}$ , $ρ_{m}$ , and $ρ_{f g m}$ are the densities of the CNT, isotropic core, and the FGM matrix, respectively. $υ^{C N T}$ , $υ_{f g m}$ , and $υ_{m}$ are the Poisson’s ratio of the CNT isotropic core and the FGM matrix, respectively

α_{11} = V_{C N T} α_{11}^{C N T} + V_{f g m} α_{f g m} + V_{m} α_{m}

(26)

\begin{array}{l} α_{22} = (1 + υ_{12}^{C N T}) V_{C N T} α_{22}^{C N T} + (1 + υ_{m}) V_{f g m} α_{f g m} \\ + (1 + υ_{m}) V_{m} α_{m} - υ_{12} α_{11} \end{array}

(27)

where $α_{11}^{C N T}$ and $α_{22}^{C N T}$ are the longitudinal and transverse thermal expansion co-efficient of CNT, respectively. $α_{m}$ is the thermal expansion coefficient of the matrix.

Finite element formulation

An isoparametric shell element with eight node having five degrees of freedom each node (three translations (u, v, w) and two rotations $(θ_{x}, θ_{y})$ is considered to model the conical shell. The shape functions of the shell element corresponding to the eight noded shells²⁹ are given by

\begin{array}{l} Q_{i} = 0.25 (1 + ζ ζ_{i}) (1 + η η_{i}) \\ (ζ ζ_{i} + η η_{i} - 1) for i = 1, 2, 3, 4 \end{array}

(28)

Q_{i} = 0.5 (1 + ζ ζ_{i}) (1 - η^{2}) for i = 5, 7

(29)

Q_{i} = 0.5 (1 + η η_{i}) (1 - ζ^{2}) for i = 6, 8

(30)

(u, v, w) are the field displacement and at any point along the coordinate system of the conical shell are written as

u (x, y, z) = u^{*} (x, y) + z θ_{x} (x, y)

(31)

v (x, y, z) = v^{*} (x, y) + z θ_{y} (x, y)

(32)

w (x, y, z) = w^{*} (x, y)

(33)

$u^{*}$ and $v^{*}$ represent the in-plane displacements and $w^{*}$ represents the transverse displacement of the mid-plane of the shell. $θ_{x}$ and $θ_{y}$ are the rotations about X- and Y-axis, respectively. For the conical shell, linear strain–displacement relation of material coordinate scheme is written as

\begin{array}{l} {ε} = {ε_{x x}^{*} ε_{y y}^{*} ε_{x y}^{*} ε_{x z}^{*} ε_{y z}^{*}}^{T} \\ + Z {k_{x x} k_{y y} k_{x y} k_{x z} k_{y z}}^{T} \end{array}

(34)

Where

[\begin{array}{l} ε_{x x}^{*} \\ ε_{y y}^{*} \\ ε_{x y}^{*} \\ ε_{x z}^{*} \\ ε_{y z}^{*} \end{array}] = [\begin{matrix} \partial / \partial x & 0 & 0 & 0 & 0 \\ 0 & \partial / \partial y & 1 / R_{y} & 0 & 0 \\ \partial / \partial y & \partial / \partial x & 2 / R_{x y} & 0 & 0 \\ 0 & 0 & \partial / \partial x & 1 & 0 \\ 0 & 0 & \partial / \partial y & 0 & 1 \end{matrix}] {\begin{cases} u^{*} \\ v^{*} \\ w^{*} \\ θ_{x} \\ θ_{y} \end{cases}}

(35)

And

[\begin{array}{l} k_{x x} \\ k_{y y} \\ k_{x y} \\ k_{x z} \\ k_{y z} \end{array}] = [\begin{matrix} \partial / \partial x & 0 \\ 0 & \partial / \partial y \\ \partial / \partial y & \partial / \partial x \\ 0 & 0 \\ 0 & 0 \end{matrix}] {\begin{cases} θ_{x} \\ θ_{y} \end{cases}}

(36)

$ε_{x x}^{*}$ and $ε_{y y}^{*}$ are mid plane normal strain along X- and Y-direction, respectively, $ε_{x y}^{*}$ mid plane shear strain, and $ε_{x z}^{*}$ and $ε_{y z}^{*}$ are the transverse shear strain. ${k}$ is the curvature of the mid plane of the shell. Stress-vector under a thermal environment of the conical shell is expressed as

{σ} = [S] {ε - α Δ T}

(37)

where ${ϵ} = {ε_{x x} ε_{y y} ε_{x y} ε_{x z} ε_{y z}}^{T}$ and ${σ} = {[σ_{x x} σ_{y y} σ_{x y} τ_{x z} τ_{y z}]}^{T}$ , and ${α} = {α_{11} α_{22} 000}^{T}$ is the co-efficient of thermal expansion and $Δ T$ ∆T is the uniform temperature gradient along the thickness direction. The elasticity constant matrix of the shell (S) is expressed as

[S] = [\begin{matrix} S_{11} & S_{12} & 0 & 0 & 0 \\ S_{12} & S_{22} & 0 & 0 & 0 \\ 0 & 0 & S_{66} & 0 & 0 \\ 0 & 0 & 0 & S_{44} & 0 \\ 0 & 0 & 0 & 0 & S_{55} \end{matrix}]

(38)

The linear elastic strain energy of the shell element can be written as

U_{L} = \frac{1}{2} {\int_{v} {\bar{ε}}}^{Τ} {\bar{σ}} d v

(39)

where stress-resultant vector ${\bar{σ}} = [D] {\bar{ε}}$ and the strain vector ${\bar{ε}} = [B] {δ_{e}}$ . (B) represent the matrix of strain displacement. Replacing these values U_total can be written as

U_{L} = \frac{1}{2} {δ_{e}}^{T} [K_{e l e}] {δ_{e}}

(40)

Element elastic stiffness matrix is denoted by $[K_{e l e}]$ and it could be expressed as

[k_{e l e}] = \int_{- 1}^{1} \int_{- 1}^{1} {[B]}^{T} [D] [B] | J | d ζ d η

(41)

where $| J |$ refers to the determinant of the Jacobian matrix. At moderate speed, the kinetic energy of the rotating shell element is written as

K_{1} = \frac{1}{2} \int_{V} ρ {\overset{\cdot}{δ}}^{T} {\overset{\cdot}{δ}} d v + {{\overset{\cdot}{δ}}_{e}}^{T} {F_{c e}}

(42)

K_{1} = \frac{1}{2} \int_{A} {{\overset{\cdot}{δ}}_{m}}^{T} [M_{c e}] {{\overset{\cdot}{δ}}_{m}} d A + {{\overset{\cdot}{δ}}_{e}}^{T} {F_{c e}}

(43)

At any point the displacement vector can be expressed as

{δ} = {u, v, w}^{T} = [T] {δ_{m}}^{T}, {δ_{m}} = {u^{*}, v^{*}, w^{*}, θ_{x}, θ_{y}}^{T}

(44)

and element inertia $[M_{c e}] [M_{c e}]$ and transformation matrix [T] of the conical shell are expressed as

[M_{c e}] = \int_{- h / 2}^{h / 2} {[T]}^{T} ρ [T] d z

(45)

[T] = [\begin{matrix} 1 & 0 & 0 & z & 0 \\ 0 & 1 & 0 & 0 & z \\ 0 & 0 & 1 & 0 & 0 \end{matrix}]

(46)

Now, equation (43) can be written as

K_{1} = \frac{1}{2} {{\overset{\cdot}{δ}}_{e}}^{T} [M_{e}] {{\overset{\cdot}{δ}}_{e}} + {{\overset{\cdot}{δ}}_{e}}^{T} {F_{c e}}

(47)

where ${δ_{m}} = [N] {{\overset{\cdot}{δ}}_{e}}$ and $[M_{e}]$ is the elemental mass matrix of the conical shell expressed as

[M_{e}] = \int_{- 1}^{1} \int_{- 1}^{1} {[W]}^{T} [M_{c e}] [W] | J | d ζ d η

(48)

The element centrifugal force vector ${F_{ce}}$ due to rotation is given as

{F_{c e}} = ρ \int_{v} {[W]}^{T} [A] {\begin{matrix} h_{x} + x \\ h_{y} + y \\ h_{z} + z \end{matrix}} d v

(49)

where [W] is the shape function, $ρ$ is the mass density, and ${h_{x} h_{y} h_{z}}$ represents fixed translational offset with respect to the shell coordinate system. Angular velocity components matrix $[A]$ causing acceleration vector is written as

[A] = [\begin{matrix} (Ω_{y}^{2} + Ω_{z}^{2}) & - Ω_{x} Ω_{y} & - Ω_{x} Ω_{z} \\ - Ω_{x} Ω_{y} & (Ω_{x}^{2} + Ω_{z}^{2}) & - Ω_{y} Ω_{z} \\ - Ω_{x} Ω_{z} & - Ω_{y} Ω_{z} & (Ω_{x}^{2} + Ω_{y}^{2}) \end{matrix}]

(50)

The strain energy contributed by initial stresses generated due to temperature and rotation is expressed as

U_{t o t a l} = \int_{v} {ε^{'}} {σ_{0 T}} d v + \int_{v} {ε^{'}} {σ_{0 R}} d v

(51)

where ${ε^{'}}$ represents the vector of non-linear stress components and ${σ_{0 T}}$ and ${σ_{0 R}}$ are the initial stress vector that arises due to temperature and rotation, respectively. For the conical shell, the non-linear strain components are expressed as

{ε^{'}} = \frac{1}{2} [\begin{matrix} ξ_{x}^{T} & 0 & 0 \\ 0 & ξ_{y}^{T} & 0 \\ ξ_{y}^{T} & ξ_{x}^{T} & 0 \\ ξ_{z}^{T} & 0 & ψ_{x}^{T} \\ 0 & ξ_{z}^{T} & ψ_{y}^{T} \end{matrix}] {\begin{cases} ξ_{x} \\ ξ_{y} \\ ξ_{z} \end{cases}} = \frac{1}{2} [A_{x}] {ξ}

(52)

Where

\begin{array}{l} ξ_{x}^{T} = [\frac{\partial u}{\partial x}, \frac{\partial v}{\partial x}, \frac{\partial w}{\partial x}] \\ ξ_{x}^{T} = [\frac{\partial u}{\partial y}, \frac{\partial v}{\partial x} + \frac{w}{R_{y}}, \frac{\partial w}{\partial y} - \frac{v}{R_{y}}] \\ ξ_{z}^{T} = [\frac{\partial u}{\partial z}, \frac{\partial v}{\partial z}, \frac{\partial w}{\partial z}] \end{array}}

(53)

{ξ} = [G] {δ_{e}}

(54)

where $[G]$ is the matrix consists of derivatives of shape functions. Substituting equations (35) and (37) in equation (34), we have

U_{t o t a l} = \frac{1}{2} \int_{v} {δ_{e}}^{T} {[G]}^{T} {[A_{x}]}^{T} {σ_{0 T}} d v + \frac{1}{2} \int_{v} {δ_{e}}^{T} {[G]}^{T} {[A_{x}]}^{T} {σ_{0 R}} d v

(55)

Again

\begin{array}{l} {[A_{x}]}^{T} {σ_{0 T}} = [P_{σ T}] {ξ} = [P_{σ T}] [G] {δ_{e}} a n d \\ {[A_{x}]}^{T} {σ_{0 R}} = [P_{σ R}] {ξ} = [P_{σ R}] [G] {δ_{e}} \end{array}

(56)

where $[P_{σ T}]$ and $[P_{σ R}]$ are the matrices of initial in-plane stress resultants that arise due to thermal load and centrifugal force. So, the additional strain energy stored in the shell element is given by

U_{t o t a l} = \frac{1}{2} {δ_{e}}^{T} ([K_{t h e}] + [K_{r e}]) {δ_{e}}

(57)

where $[K_{t h e}]$ and $[K_{r e}]$ are the elemental geometric stiffness matrices due to temperature and rotation.

Dynamic equilibrium equation in global form is derived from Lagrange’s equation of motion in absence of the Coriolis effect ²⁸

[M] {\overset{\cdot\cdot}{δ}} + ([K] + [K_{t h}] + [K_{r}]) {δ} = F_{c}

(58)

where $[M]$ , $[K]$ , $[K_{t h}]$ , and $[K_{r}]$ are the global mass, elastic stiffness, geometric stiffness matrices due to thermal load, and geometric stiffness matrices due to rotation, respectively. $[K_{r}]$ depends on initial stress distribution due to the rotation. Global displacement vector is ${δ}$ and nodal external force is $F_{c}$ .

Governing contact law

A rigid spherical steel ball impacts the top surface of the conical shell at low velocity. The impactor mass is not considered to the analysis system as impactor bounces off the shell instantaneously after impacting on the surface. Goldsmith³⁰ has formulated the governing equations of low-velocity, low-energy impact equations between a sphere and a plate considering the pressure distribution from an impact incident. After the impact, the target shell is deformed locally (at impact location) as well as globally. The present material used in this analysis is anisotropic in nature and indentation due to impact causes permanent deformation. Therefore, the coefficient of restitution is not considered in the impact model. Conway³¹ established a contact force model considering Hertzian-type contact that is appropriate for transversely isotropic materials. This model has been used to state the contact between the impactor and FGM materials.³² For the Hertzian contact law, contact force (F_C) can be calculated during loading unloading cycle as³²

F_{c} = K_{m} α^{\frac{3}{2}} 0 \leq α \leq α_{m a x}

(59)

While {F_C} is the global contact force vector resulting from impact and is given by

{F_{C}} = {000 \dots \cdot\cdot F_{C i} \cdot\cdot \dots 0 0 0}^{T}

(60)

where F_Ci is the contact forces at the node“i”. α is the relative change in distance between the impactor’s center and the middle surface of target shell and $α_{m a x}$ is the maximum local indentation. The modified contact stiffness ( $K_{m})$ based on modified Hertzian contact law is given by³³

K_{m} = \frac{16 R_{r}^{\frac{1}{2}}}{\frac{66}{7} (K_{t a r g} + K_{i m p t}) {(ε)}^{\frac{3}{2}}}

(61)

where $ε$ depends on shape of the impactor and target surface. For the low-velocity impact case, the value of $ε$ depends on contact behavior. The value of $ε$ is taken as 2 for the present the low-velocity impact case. $R_{r}$ is a constant which depends on curvatures of impactor as well as target

\frac{1}{R_{r}} = \frac{1}{R_{x}^{i m p t}} + \frac{1}{R_{y}^{i m p t}} + \frac{1}{R_{x}^{t a r g}} + \frac{1}{R_{y}^{t a r g}}

(62)

For spherical impactor, the principal radii $R_{x}^{i m p t}$ and $R_{y}^{i m p t}$ are equal. The stiffness $K_{t a r g}$ of equation (61) can be written as

K_{t a r g} = \frac{\sqrt{A_{22} [{(G_{12} + A_{11} A_{22})}^{2} - {(G_{12} + A_{12})}^{2}]}}{\sqrt{G_{x} 4 π^{2}} . (A_{11} A_{22} - A_{12}^{2})}

(63)

where $G_{12}$ is defined in equation (22) and the other parameters are defined as follows

A_{11} = \frac{E_{12} (1 - υ_{12})}{(1 - υ_{12} - 2 υ_{12}^{2} \frac{E_{11}}{E_{12}})}

(64)

A_{22} = \frac{E_{11} (1 - υ_{11}^{2} \frac{E_{11}}{E_{12}})}{(1 + υ_{11}) (1 - υ_{11} - 2 υ_{11}^{2} \frac{E_{11}}{E_{12}})}

(65)

A_{12} = \frac{E_{11} υ_{11}}{(1 - υ_{11} - 2 υ_{11}^{2} \frac{E_{11}}{E_{12}})}

(66)

$K_{i m p t}$ as per equation (61) can also be determined in a similar way of $K_{t a r g}$ in equation (62). It can be noted that for the rigid impactor as in the present case, $K_{i m p t} = 0$ . The equation of motion of the rigid impactor is governed by

m_{i} {\overset{\cdot\cdot}{w}}_{i} + F_{c i} = 0

(67)

m_i is the mass of the impactor and ${\overset{\cdot\cdot}{w}}_{i}$ is the acceleration of the impactor. Neglecting the contribution of plate displacements along global x- and y-directions, the indentation α can be written as

α (t) = w_{i} (t) - w_{p} (x_{c}, y_{c}, t) \cos ψ

(68)

where w_i and w_p are displacement of impactor and target plate along global z-direction at the impact point (L/2,b/2), respectively, while Ψ is the angle of twist. The components of force at impact point in global directions are given by

F_{i x} = 0, F_{i y} = F_{C} \sin ψ a n d F_{i z} = F_{C} \cos ψ

(69)

The solution for the equation of motion given by equations (58) and (67) is solved by Newmark’s integration algorithm.³⁴

Convergence study and validation

The numerical results obtained from the in-house finite element code are evaluated to study the impact behavior of the cantilever pre-twisted sandwich conical shell panel having CNTs-reinforced nanocomposite facings and homogenous core. Convergence and validation studies are performed to examine the stability and correctness of the present methods. Thereafter, a detailed parametric study is carried out to see the effect of different parameters on the low-velocity impact behavior. Temperature-dependent material properties of FGM constituents (Ti6Al4V-ZrO₂) for the facing are considered as per reference.³⁵ The simple power law exponent with a linear variation of the FGM constituents along the thickness direction is considered. The outer surface (top-most and bottom-most) is considered ceramic (ZrO₂) rich. The (10, 10) single-walled CNTs are considered as a reinforcement and its properties along with efficiency parameters are taken from reference.¹²

In order to ensure the convergence of the FE mesh and steadiness of the time integration technique, the time histories of contact force and target displacement for FG-VΛ CNT reinforced composite sandwich conical shell panel are determined. Conical shells (r_x = ∞) having rectangular plan-form (L/b) =5.59, curvature ratio (b/r_y) =0.5, aspect ratio (L/s) =0.7 and twist angle (Ψ) =15°, core-to-facing thickness ratio (h_core/h_fgm) = 2, $V_{C N T}^{e}$ =0.12, impactor diameter (D_imp) = 12.7 mm, and impactor velocity = 5 m/s at reference temperature (T =300 K) are considered. The obtained results are illustrated in Figure 4, considering a variety of mesh sizes and time steps. The error among the results is found to decrease when the mesh becomes finer and the time step becomes smaller. Hence, an 8×8 mesh size with a time step of 1 μs is reasonably considered for the subsequent analyses.

Figure 4.

Convergence of (a) contact force and (b) target displacement for pre-twisted CNTs reinforced functionally graded sandwich conical panel: CNTs volume fraction 0.12, core-to-facing thickness ratio 0.5, FG VΛ pattern and twist angle (Ψ) =15°, VOI= 5 m/s, and reference temperature = 300 K.

After the convergence study, two different examples are considered for comparison. First, work of Wang et al.²³ to validate the present method with respect to impact response of nanocomposite sandwich plate. In this example, an SSSS FG-VΛ CNTRC sandwich plate is considered to be impacted centrally with a spherical impactor. The matrix for this structure was considered as PMMA where CNTs are reinforced. Due to unavailability of the FGM matrix in the open literature, the example with PMMA material property²⁸ is considered for validation. The time histories of the contact force and plate deflection along with those of Wang et al.²³ are illustrated in Figure 5. For second validation, an isotropic simply supported FGM beam (153.5 mm length, 10 mm width, and 15 mm thick) having rectangular planner form impacted centrally by a steel sphere of 12.7 mm radius with an initial velocity of impactor (VOI) 2.0 m/sec is considered based on Kiani et al.³⁶ work. This provides an integral equation for a modified Hertzian contact law expressed as in equation (59). Figure 6 depicts the comparisons of time histories of contact force and indentation obtained from present FEM and that of Kiani et al.³⁶. This shows a good match between the present FEM analysis and the reference results.

Figure 5.

Time histories of (a) contact force and (b) plate displacement of an SSSS FG-VΛ CNTRC sandwich plate under low-velocity impact at three different temperatures T = 300 K, 500 K, and 700 K (L/ b = 1, b/h =10, ℎ_f =1 mm, ℎ_core =8 mm, $(\frac{6^{"}}{16})$ VeCNT = 0.12, D_imp=12.7 mm, and VOI=3 m/s).

Figure 6.

(a) Contact force and (b) indentation histories of an FGM beam clamped at both ends. 153.5 L= 10 mm b= 15 mm, D_imp=12.7 mm, and VOI=2 m/s.

Results and discussion

After convergence and validation of the present model, the effects of various parameters on the low-velocity impact response of pre-twisted CNTs reinforced functionally graded sandwich conical panel with CNTs reinforced functionally graded facings and titanium alloy core are investigated. In the present study, the dimensions of length (L) and width (b) are taken as 0.8 m and 0.143 m, respectively. The other parameters of the conical shells (r_x=∞) are considered as (L/b) =5.59, curvature ratio (b/r_y) =0.5, and aspect ratio (L/s) =0.7. A steel impactor having material properties E =205 GPa, v =0.3, and p = 7800 kg/m³ is assumed to impact at the central point (L/2, b/2) of the target shell.

Effect of CNTs grading pattern

The effect of CNTs grading pattern on the impact response of pre-twisted CNTs reinforced functionally graded sandwich conical panel at temperature 300 K is determined for five different CNT grading patterns, namely, FG-U, FG-VΛ, FG-ΛV, FG-OO, and FG-XX. The CNTs with volume fraction 0.12, core-to-facing thickness ratio 0.5, VOI 5 m/s, and twist angle (Ψ) =15° are used. The time histories of contact force, target displacement, impactor velocity, and impactor displacement are shown in Figures 7(a)–(d), respectively. From Figure 7(a), it is observed that FG-VΛ grading pattern predicts the highest contact force with the shortest contact duration, while the case of FG-ΛV possesses the minimum value of contact force with the longest contact duration among the five grading patterns. This is due to the highest transverse Young’s moduli of the surface with densely populated CNTs in the case of the FG-VΛ pattern as evident from CNTs distribution pattern in Figure 3. It is worth mentioning that the material properties of the impact surfaces can affect the contact force. However, the contact force for sandwich conical panel having FG-VΛ grading profile is slightly larger than FG-XX, although their contact surfaces have the same material properties. It can be inferred that the contact force is affected by the material properties of the contact surface as well as the non-contact surface. For the same reason, FG-VΛ has the lowest shell displacement, while FG-ΛV has the highest central shell displacement, as depicted in Figure 7(b). FG-VΛ pattern predicts the highest impactor displacement while FG-ΛV shows a lower value. Other variants are within the range of these two (Figure 7(c)). Impactor bounces back with higher velocities for FG-VΛ pattern and FG-ΛV pattern shows it bounces back with lower velocities. Other variants are within the range of these two (Figure 7(d)).

Figure 7.

Variation of CNTs grading pattern on impact response of CNTs reinforced functionally graded sandwich conical panel at reference temperature 300 K: CNTs volume fraction 0.12, core-to-facing thickness ratio 0.5, VOI=5 m/s, and twist angle (Ψ) =15°: (a) contact force, (b) target displacement, (c) impactor displacement, and (d) impactor velocity.

Effect of initial velocity of the impactor

Effect of initial velocity on the impact behavior of CNTs reinforced functionally graded sandwich conical panel having core-to-facing thickness ratio 2 and twist angle (Ψ) =15° at elevated temperature 500 K is shown in Figure 8. An FG-VΛ grading pattern with CNTs volume fraction 0.12 in the sandwich panel is considered. Three different initial velocities of the spherical steel impactor, that is, 1 m/s, 3 m/s, and 5 m/s, are assumed to obtain the contact force, target displacement, impactor displacement, and impactor velocity. The time histories of contact force, target displacement, impactor displacement, and impactor velocity are displayed in Figure 8(a)–(d), respectively. As observed in the figures, an increase in impactor velocity would lead to a higher peak of contact force, higher shell displacement, and lower contact duration due to increase in the initial kinetic energy of the impactor. The time reaching the peak of contact force becomes shorter, while the peak time of the shell displacement is unaltered with the increase of initial velocity. It is also found that the contact force is much more affected than the contact duration time due to changes in the initial velocity of the impactor. Higher impactor displacements are observed for higher impactor velocity. The slope of time history curves for the velocity of impactor is enhanced with higher impactor velocity. It is also to be noted that the velocity of the impactor comes down to zero value subsequently it falls down to more negative value for higher impactor velocity.

Figure 8.

Effect of initial velocity of the impactor on impact response of CNTs reinforced functionally graded sandwich conical panel at reference temperature 300 K: CNTs volume fraction 0.12, core-to-facing thickness ratio 0.5, FG VΛ pattern, and twist angle (Ψ) =150: (a) contact force, (b) target displacement, (c) impactor displacement, and (d) impactor velocity.

Effect of size of the impactor

Figures 9(a)–(d) illustrate the effect of impactor size on impact response of CNTs reinforced functionally graded sandwich conical panel. In this case, an FG-VΛ sandwich conical panel having CNTs volume fraction 0.12, core-to-facing thickness ratio of 0.5, and pre-angle (Ψ) =15° at reference temperature 300 K is taken into consideration. Three different spherical steel impactors are selected with diameters of 9.525 mm $(\frac{8^{"}}{16})$ , 12.7 mm $(\frac{10^{"}}{16})$ , and 15.875 mm $V_{C N T}^{e}$ . As observed in Figure 9(a), an increase in impactor diameter would lead to higher peak contact force, longer contact duration, and larger target displacement (Figure 9(b)) within the considered time span. This may be attributed to the higher value of contact stiffness of the system and the larger mass and energy of the impactor with its bigger diameter. Further, the peak time of contact force becomes longer for the impactor of bigger diameter, whereas the peak time of shell displacement is nearly independent of the impactor diameter. Impactor displacement corresponding to impact force is higher for a larger size of the impactor as shown in Figure 9(c). Impactor velocity for larger size impactors is higher as depicted in Figure 9(d).

Figure 9.

Effect of size of impactor on impact response of CNTs reinforced functionally graded sandwich conical panel at reference temperature 300 K: CNTs volume fraction 0.12, core-to-facing thickness ratio 0.5, VOI=5 m/s, FG VΛ pattern, and twist angle (Ψ) =150: (a) contact force, (b) target displacement, (c) impactor displacement, and (d) impactor velocity.

Effect of CNTs volume fraction

The influence of CNTs volume fraction on the impact response of CNTs reinforced functionally graded sandwich conical panel considering FG-VΛ grading profile at the reference temperature (T = 300 K) is demonstrated in Figure 10. The impact responses are determined considering three different values of CNTs volume fraction $V_{C N T}^{e}$ = 0.12, 0.17, and 0.28 with core-to-facing thickness ratio 0.5 and twist angle (Ψ) =15°. Figure 10(a) shows that an increase in CNTs volume fraction results in higher contact force and shorter contact duration. This is due to the fact that an increase in the CNTs volume fraction results in an increase in the contact stiffness of the sandwich panel with FG-VΛ grading pattern at the impact point. This increase in the contact stiffness is also responsible for the reduction in the target displacement with an increment of $V_{C N T}^{e}$ as shown in Figure 10(b). It is further to note that the time for attaining the peak contact force and shell displacement becomes shorter as $V_{C N T}^{e}$ increases. Impactor displacement is higher as $V_{C N T}^{e}$ increases, though the displacement shows a lower trend as it returns as depicted in Figure 10(c). Impactor velocity (Figure 10(d)) shows a higher trend as $(V_{C N T}^{e})$ increases.

Figure 10.

Effect of CNTs volume fraction on impact response of CNTs reinforced functionally graded sandwich conical panel at reference temperature 300 K: VOI=5 m/s, core-to-facing thickness ratio 0.5, FG VΛ pattern, and twist angle (Ψ) =150: (a) contact force, (b) target displacement, (c) impactor displacement, and (d) impactor velocity.

Effect of thermal environment

The influence of the thermal environment on the impact response of pre-twisted CNTs reinforced functionally graded sandwich conical panel is depicted in Figure 11. In this figure, the results are obtained for three different temperatures: T = 300 K, 500 K, and 700 K considering two grading patterns, namely, FG-VΛ and FG-ΛV with CNTs volume fraction $(V_{C N T}^{e})$ = 0.12, core-to-facing thickness ratio 0.5, and pre-twist angle = 15°. As evident in Figure 11(a), a rise in temperature of the environment results in longer contact duration and a higher peak value of contact force irrespective of the CNTs grading pattern. The effect of thermal environment on target displacement is similar to contact duration, that is, central shell displacement increases with the rise in temperature, as shown in Figure 11(b). The elastic properties of both facing and core layers decrease with an increase in the operating temperature. In addition, there is a decrease in the contact stiffness which is, in turn, dependent on the transverse Young’s modulus of the facing on which the impact occurs. At elevated temperatures, the sandwich construction becomes more flexible, which results in lower contact force and higher contact duration. Figure 11(c) shows that the impactor displacement reduces at elevated temperatures. On the other hand, impactor bounces back with a lower velocity at elevated temperature as depicted in Figure 11(d).

Figure 11.

Variation of operating temperature on impact response of CNTs reinforced functionally graded sandwich conical panel for FG VΛ and FG ΛV pattern: VOI=5 m/s, core-to-facing thickness ratio 0.5, and twist angle (Ψ) =150: (a) contact force, (b) target displacement, (c) impactor displacement, and (d) impactor velocity.

Effect of pre-twist angle

Figure 12 illustrates the effect of pre-twist angle (ψ = 0°, 15°, 30°, and 45°) on the impact behavior of FG-VΛ CNTs reinforced functionally graded sandwich conical panel with CNTs volume fraction ( $V_{C N T}^{e})$ = 0.12 and core-to-facing thickness ratio 0.5 at temperature 300 K. It is clear from Figure 12(a) that peak value of contact force increases with an increase in pre-twist angle. The pre-twisted structures are stiffer than the untwisted structures since the structural stiffness decreases with increasing pre-twist angle. However, an increase in pre-twist angle results in higher local indentation at the impact point, which in turn increases the resulting contact force. For the same reason, the peak value of shell displacement decreases with an increase in pre-twist angle, as shown in Figure 12(b). However, during the time span the peak time of contact force and the peak time of target displacement are found to be almost independent of the pre-twist angle. From Figure 12(c) and (d), a higher impactor displacement and higher impactor velocities are observed as the twist angles increased.

Figure 12.

Variation of twist angle (Ψ) on impact response of CNTs reinforced functionally graded sandwich conical panel at reference temperature 300 K: VOI=5 m/s, CNTs volume fraction 0.12, core-to-facing thickness ratio 0.5, and FG VA pattern:(a) contact force, (b) target displacement, (c) impactor displacement, and (d) impactor velocity.

Effect of core-to-facing thickness ratio

The effect of core-to-facing thickness ratio on the impact behavior of pre-twisted CNTs reinforced functionally graded sandwich conical panel with FG-VΛ grading profile, with CNTs volume fraction ( $V_{C N T}^{e} = 0.12$ = 0.12, and pre-twist angle =15°, is presented in Figure 13. In this regard, the time histories of contact force and target displacement for three different values of core-to-facing thickness ratio =0.5, 1, and 2 are obtained and presented in Figure 13(a) and ( b), respectively. As seen from these figures, an increase in the core-to-facing thickness ratio leads to lower contact force and higher shell displacement. In view of the fact that the overall thickness of the sandwich panel is constant, therefore, the increasing ratio involves a thicker core and thinner facings. Obviously, the lower thickness of the facing (comprising CNTs reinforced Ti6Al4V-ZrO₂ FGM with ZrO₂ in outer surface) causes lower bending stiffness as CNTs reinforced FGM is stiffer than the Titanium alloy core, which in turn leads to lower contact force. Accordingly, the peak value of target displacement decreases when the ratio is increased. It is further noted that the contact duration and peak time of contact force remain unaltered, whereas the peak time of shell displacement becomes longer with an increase in the ratio. Impactor displacement enhances as core-to-facing thickness ratio increases as depicted in Figure 13(c). It is observed from Figure 13(d) that impactor bounces back with lower velocity as core-to-facing thickness ratio enhances.

Figure 13.

Variation of core-to-facing thickness ratio on impact response of CNTs reinforced functionally graded sandwich conical panel at reference temperature 300 K: VOI=5 m/s, CNTs volume fraction 0.12, FG VA pattern, and twist angle (Ψ) =150: (a) contact force, (b) target displacement, (c) impactor displacement, and (d) impactor velocity.

Conclusion

Low-velocity impact responses of pre-twisted CNTs reinforced functionally graded sandwich conical panel with isotropic core are presented using finite element technique. The effective thermo-mechanical properties of the CNTs reinforced functionally graded sandwich facings are evaluated using micromechanical models. Lagrange’s equation is employed to formulate the dynamic equilibrium equation of the system wherein modified Hertzian contact law is included to evaluate the contact force. The resulting dynamic equilibrium equations are solved by implementing Newmark’s time integration method. The following conclusion can be obtained based on the present numerical analysis.

(a) Among the considered five different CNTs grading profiles in the CNTs reinforced functionally graded sandwich conical panel, the FG-VΛ type exhibits the largest contact force, shortest contact duration, and smallest shell displacement. In contrast, the smallest contact force, longest contact duration, and largest target displacement are evident in the case of FG-ΛV type.

(b) Higher impactor displacement is found for FG-VΛ pattern while the impactor bounces back with higher velocities for this pattern. On the other hand, FG-ΛV pattern shows the reverse trend.

(c) The contact force and target displacement are found to increase when the initial velocity and size of the impactor are increased.

(d) The impact responses are highly affected by CNTs volume fraction. For instance, the higher value of CNTs volume fraction results in higher contact force with shorter contact duration and lower target displacement.

(e) With the rise in temperature of the environment, the contact force decreases, whereas contact duration and target displacement increase.

(f) The higher value of pre-twist angle leads to a higher value of contact force and a lower value of target displacement.

(g) With the increase of core-to-facing thickness ratio, the contact force increases and target displacement decreases.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Rashtriya Uchchatar Shiksha Abhiyan.

ORCID iD

Apurba Das

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