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Transient nonlinear responses of an auxetic honeycomb sandwich plate under impact loads
International Journal of Impact Engineering 134 (2019) 103383
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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Transient nonlinear responses of an auxetic honeycomb sandwich plate under impact loads
T
⁎
Junhua Zhanga, , Xiufang Zhua, Xiaodong Yangb, Wei Zhangb a b
College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing, 100192, PR China College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Honeycomb sandwich plate Auxetic Nonlinear transient response Dynamic loads
In this paper, we study the nonlinear transient responses of an auxetic (negative Poisson's ratio) honeycomb sandwich plate under impact dynamic loads. The partial differential equations of the honeycomb sandwich plate based on Reddy's higher shear deformation theory and Hamilton's principle are obtained. The nonlinear ordinary differential equations are then derived by Galerkin truncation method. The dynamic responses of the honeycomb sandwich plate subjected to different loads, such as step loading, air-blast loading, sinusoidal loading, triangular loading and incremental loading, respectively, have been investigated. The contributions of total thickness, core thickness ratio, Poisson's ratio, cell inclination angle and blast types to transient responses of the plate are discussed in detail by numerical simulations. The effect of geometrical parameters on the dynamics of the plate and the effect of different blast loads on the plate are obtained. It is found that the honeycomb sandwich plate with negative Poisson's ratio would be a better choice compared with the positive one for some structures under dynamic loads.
1. Introduction Driven by design of lightweight materials with vibration and energy absorption characteristics for impact applications, the advanced composite materials are widely used in space station structures, aircraft and spacecraft structures, architectures and bridge structures. Especially multi-cell material sandwich structures are good at energy absorption and impact resistance. It is known that most natural materials are characterized by positive Poisson's ratio. However, these traditional materials can form structures that exhibit negative Poisson's ratio, i.e. auxetic, which have lower yielding strength and higher stiffness compared with that of positive Poisson's ratio. For example, the traditional honeycomb structures realize the metamaterial properties of negative Poisson's ratio by changing the combination of cells. Consequently, many researches related to the structures with negative Poisson's ratio have been reported in literatures [1–3]. Wan et al. [4] proposed a theoretical approach to calculate the Poisson's ratios of honeycombs according to the large deflection theory. The deformation curves of the tilting part of honeycombs, strains and Poisson's ratios in two orthogonal directions are obtained. Zhang et al. [5] investigated the effect of auxetic honeycomb cellular microstructure on the in-plane crushing behaviors and energy absorptions by using finite element method. The most common sandwich structure at present is made of two ⁎
rigid metallic thin face sheets and a low-density soft core. Many research works focused on this kind of structures. Huang et al. [6] presented a new design about auxetic honeycombs which consisted of a reentrant hexagonal component and a thin plate. The authors studied the in-plane mechanical properties of the auxetic honeycombs. Two kinds of multi-functional layered auxetic honeycombs have been proposed by Sun et al. [7], one is anisotropic re-entrant honeycomb and the other is isotropic chiral honeycomb. The formula of Young's moduli is also given. Brischetto et al. [8] used 3D printing technique to study mechanical behaviors of honeycomb sandwich structure. The progresses of mechanical properties, calculation and application of auxetic materials have been reviewed in [9]. With the emergence of the advanced auxetic sandwich structures and their wide applications in the structures of aircraft and spacecraft, the nonlinear dynamics the sandwich plates under the dynamic loads, such as explosion loads, and acoustic explosion loads produced by fuel explosion has been paid more and more attentions [10–12]. Duc et al. [13] studied vibrations and nonlinear dynamics of sandwich composite cylindrical panels with auxetic honeycomb cores on elastic foundations subjected to mechanical, blasting and damping loads. Imbalzano et al. [14] analyzed and compared the resistance performances of auxetic and conventional honeycomb cores and metal facets against impulsive loadings. Damanpack et al. [15] investigated active control of
Corresponding author. E-mail address: [email protected] (J. Zhang).
https://doi.org/10.1016/j.ijimpeng.2019.103383 Received 4 May 2018; Received in revised form 27 July 2019; Accepted 20 August 2019 Available online 22 August 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Impact Engineering 134 (2019) 103383
J. Zhang, et al.
geometric nonlinear dynamic responses of sandwich beams under impact of integrated piezoelectric sensor. The effect of geometric nonlinearity, the flexibility of the core layer and the dynamic response of piezoelectric patch arrangement, frequency and loss factor of the beams was analyzed in detail. Zhang et al. [16,17] investigated tensile mechanical properties of auxetic hexagonal honeycomb structures. Karagiozova et al. [18] studied energy absorptions of miura-ori patterned open cell metamaterials under in-plane impact dynamic compressions. The performance of different sandwich panels under impact loading was evaluated by parametric studies. Wang et al. [19] studied the dynamic response including deformation and damage modes of honeycomb sandwich plates under the blast loading by the experimental and theoretical studies. Mu et al. [20] analyzed the displacements, deformation, frequencies and stress of a composite sandwich plate with transversely compressible core under time-dependent impact. Rekatsinasa et al. [21] gave the transient impact responses of thick sandwich composite plates under low velocity impacts. The nonlinear transient responses of laminated basalt composite plate under dynamic loads is studied by Basturk et al. [22], it is shown that the laminated basalt composites are good at dynamic loads resistance. Kazanci [23] summarized the responses of various types of explosion pulse models loaded on laminated composite plates. Feli and Namdari Pour [24] proposed the strain and energy of honeycomb sandwich composite plate under high-velocity impact. Jagtap and Lai [25] researched nonlinear dynamic responses of functionally graded laminated plates under stochastic loads. Salami [26, 27] investigated responses of sandwich beams with soft cores subjected low velocity impact by using Ritz method. As key structures in aircraft, space station and marine structures, honeycomb sandwich structures may be subjected to time-dependent pulses, such as step pulses and triangular loading, which may cause serious damage to structures. Auxetic honeycomb sandwich structures are good at energy absorption and impact resistance. For example, Qiao and Chen [28] explored impact resistance of double arrowhead auxetic honeycombs and gave relations between impact velocity and dynamic plateau stress. Qi et al. [29] investigated experimentally and numerically responses of auxetic honeycomb-cored sandwich panels as protective system under impact and close-in blast. Wang et al. [30] gave parametric optimization for double-V auxetic sandwich panels under air blast load. Gao et al. [31] showed multi-objective optimization of an auxetic cylindrical crash box on energy absorption. Energy absorption of 3D printed auxetic honeycomb structures was given in [32, 33]. Therefore, it is of importance to study the dynamic response of these composite honeycomb structures subjected blast loads. In particular, auxetic honeycomb sandwich plate, which has metamaterial property, may be better in vibration absorptions than the traditional honeycombs. It is proved in this investigation. Nonlinear dynamic responses of an auxetic honeycomb sandwich plate are analyzed. The honeycomb sandwich plate is subjected to step loading, air-blast loading, triangular loading, sine loading and incremental loading. The influence of total thickness, the thickness ratio, cell inclination angle, Poisson's ratio and blast types to transient responses of the plate are discussed in detail by the numerical simulations.
Fig. 1. Honeycomb sandwich plate with concave hexagon core.
According to higher order shear deformation theory, the displacements of u, v and w can be expressed as
u (x , y, z , t ) = u 0 (x , y, t ) + zϕx (x , y, t ) −
4 3⎛ ∂w0 ⎞ z ϕ + 3h2 ⎝ x ∂x ⎠
(1a)
v (x , y, z , t ) = v0 (x , y, t ) + zϕy (x , y, t ) −
4 3⎛ ∂w0 ⎞ z ϕ + 3h2 ⎝ y ∂y ⎠
(1b)
⎜
⎟
w (x , y, z , t ) = w0 (x , y, t )
(1c)
where u0, v0 and w0denote the displacements at the middle surface, and ϕx, ϕy are the transversal normal rotations of middle surface. Using of the von Karman large geometric deformation theory on the plate, the strain-displacement relations and curvature-displacement relationship can be obtained as (0) (1) (3) ⎧ εxx ⎫ ⎧ εxx ⎫ ⎧ εxx ⎫ ε ⎪ (1) ⎪ ⎪ (3) ⎪ ⎧ xx ⎫ ⎪ (0) ⎪ 3 εyy = ε yy + z ε yy + z ε yy ⎨ ⎬ ⎨ εxy ⎬ ⎨ ⎬ ⎨ ⎬ (0) ⎪ (1) ⎪ (3) ⎪ ⎩ ⎭ ⎪ εxy ⎪ εxy ⎪ εxy ⎩ ⎭ ⎩ ⎭ ⎩ ⎭
(2a)
γ (0) γ (2) ⎧ γyz ⎫ = ⎧ yz ⎫ + z 2 ⎧ yz ⎫ γ ⎨ γ (2) ⎬ ⎨ γ (0) ⎬ ⎭ ⎨ ⎩ xz ⎬ ⎩ xz ⎭ ⎩ xz ⎭
(2b)
where ∂u0 1 ⎧ + 2 (0) ∂x ⎧ εxx ⎫ ⎪ ⎪ (0) ⎪ ⎪ ∂v 0 1 ε yy = + 2 ∂y ⎨ ⎬ ⎨ (0) ⎪ εxy ⎪ ⎪ ∂u0 ∂v ⎩ ⎭ ⎪ + ∂x0 + ⎩ ∂y
∂w 0 2 ∂x
( )
⎫ ⎪ ⎪ ∂w 0 ∂y ⎬ ∂w 0 ∂w 0 ⎪ + ∂x ∂y ⎪ ⎭
( )
∂ϕ
⎧ x (1) ⎧ εxx ⎫ ⎪ ∂x ⎪ (1) ⎪ ⎪ ∂ϕy ε yy = ⎨ ⎬ ⎨ ∂y (1) ⎪ γxy ⎪ ⎪ ∂ϕx ⎩ ⎭ ⎪ ∂y + ⎩
2. Equations of motion We consider a honeycomb plate with thickness of h and core thickness of hc. The coordinate system is located on the middle surface of the plate, and u, v and w represent the displacement components in x, y and z directions, respectively. The lengths in x and y directions are a and b respectively, as shown in Fig. 1. The sandwich plate consists of one core layer and two layers of skins covering it. The surface on one side of z > 0 is called as upper skin, and the surface on the other side of z < 0 is called as lower skin and the skin is made of isotropic materials. The materials of the honeycomb panel is aluminum, whose modulus is Es, density is ρs, Poisson's ratio is νs, and shear modulus is Gs.
∂ϕ
⎫ ⎪ γ (0) ⎪ ⎧ yz ⎫ ⎧ ϕy + , = ⎬ ⎨ γ (0) ⎬ ⎨ ϕ + ∂ϕ y ⎪ ⎩ xz ⎭ ⎩ x ∂x ⎪ ⎭
⎧ x + (3) ⎧ εxx ⎫ ⎪ ∂x ⎪ ∂ϕy ⎪ (3) ⎪ ε yy = −c1 + ⎨ ⎬ ⎨ ∂y (3) ⎪ ∂ϕx ⎪ γxy ⎪ ⎩ ⎭ ⎪ ∂y + ⎩ 4 3h2
2
∂2w 0 ∂x 2 ∂2w 0 ∂y 2 ∂ϕ y ∂x
(3a)
∂w 0 ⎫ ∂y ∂w 0 ⎬ ∂x ⎭
⎫ ⎪ γ (2) ⎧ ϕy + ⎪ ⎧ yz ⎫ = −c2 , (2) ⎨ϕ + ⎬ ⎨γ ⎬ xz ⎭ ⎩ x ∂2w ⎪ ⎩ + 2 ∂x ∂y0 ⎪ ⎭
(3b)
∂w 0 ⎫ ∂y ∂w 0 ⎬ ∂x ⎭
(3c)
and c2 = 3c1. and c1 = The honeycomb sandwich plate with negative Poisson ratio is orthotropic and the constitutive equation can be written as 2
International Journal of Impact Engineering 134 (2019) 103383
J. Zhang, et al.
(k ) (k ) Q11 Q12 0 0 0 ⎫ ε σxx (k ) ⎧ ⎪ ⎧ xx ⎫ ⎪ (k ) (k ) ⎧σ ⎫ 0 0 0 ⎪ ⎪ εyy ⎪ ⎪Q21 Q22 ⎪ yy ⎪ ⎪γ ⎪ (k ) σyz yz = 0 0 Q44 0 0 ⎬⎨γ ⎬ ⎨ σxz ⎬ ⎨ xz k ( ) ⎪ ⎪ ⎪ 0 0 0 Q55 0 ⎪⎪ ⎪ ⎪ ⎪ γxy ⎪ ⎪ ⎩ σxy ⎭ ⎭ (k ) ⎩ 0 0 0 0 Q 66 ⎭ ⎩
∂Q¯ x ∂x
+
+
∂Q¯ y
+
∂y
(N
∂ ∂x
∂w 0 xx ∂x
+ Nxy
∂2Pxy
∂2P c1 ⎛ 2xx ∂x
+ 2 ∂x ∂y +
⎝
= I0 w¨ 0 − c12 I6 (4)
(
∂2w¨ 0 ∂x 2
+
∂2Pyy ⎞ ∂y 2
∂2w¨ 0 ∂y 2
⎠
∂w 0 ∂y
) + (N ∂ ∂y
∂w 0 xy ∂x
+ Nyy
∂w 0 ∂y
) + J ⎛⎝
∂ϕ¨x
)
+ p (t )
) + c ⎡⎢⎣I ( 1
3
∂u¨ 0 ∂x
∂v¨0 ∂y
+
4
∂x
+
∂ϕ¨y ∂y
⎞⎤ ⎠⎥ ⎦ (10c)
where (k ) Q11 =
E1(k )
(k ) , Q22 = (k )
(k ) ν21 1 − ν12
¯ xy ¯ xx ∂M ∂w¨ ∂M − Q¯ x = J1 u¨ 0 + K2 ϕ¨x − c1 J4 0 + ∂y ∂x ∂x
E2(k ) (k ) (k ) ν21 1 − ν12
Q66 = G12, Q44 = G23, Q55 = G13, Q21 = Q12
(5)
¯ xy ∂M
(k ) (k ) (k ) (k ) (k ) , G13 , G23 , ν12 and ν21 denote the skin and core where E1(k ) , E2(k ) , G12 layer elastic modulus, shear modulus and Poisson's ratio, respectively. The superscripts of k = 1, 3 represent the upper and lower skins respectively, and k = 2 indicates the core layer. The governing equations are established by using the Hamilton principle, and the expression of Hamilton principle is as follows
∫t
t2
(δU − δK ) dt +
1
∫t
t2
δWdt = 0
∂x
N
Ii = ∑k = 1 ∫z
δϕ˙x − c1 (I3 u˙ 0 + I4 ϕ˙x − c1 I6 φ˙ x ) δφ˙ x − c1 (I3 v˙ 0 + I4 ϕ˙y − c1 I6 φ˙ y ) δφ˙ y
ρ(k ) (z )idz , (i = 0, 1, 2, …, 6)
∂w0 ∂w0 φ˙ x = ϕ˙x + , φ˙ y = ϕ˙y + ∂x ∂y and
(12a)
(1) (3) ⎧ε ⎫ ⎧ε ⎫ ⎧ Mxx ⎫ ⎧ D11 D12 0 ⎫ ⎪ xx ⎪ ⎧ F11 F12 0 ⎫ ⎪ xx ⎪ (1) (3) Myy = D21 D22 0 ε yy + F21 F22 0 ε yy ⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ ⎬⎨ ⎬ (1) D F 0 0 0 0 M 66 66 xy ⎩ ⎭ ⎩ ⎭ ⎪ ⎪ ⎪ γ γ (3) ⎪ ⎩ ⎭ ⎩ xy ⎭ ⎩ xy ⎭
(12b)
(1) (3) ⎧ε ⎫ ⎧ εxx ⎫ ⎧ Pxx ⎫ ⎧ F11 F12 0 ⎫ ⎪ xx ⎧ H11 H12 0 ⎫ ⎪ (3) ⎪ (1) ⎪ Pyy = F21 F22 0 ε yy + H21 H22 0 ε yy ⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ ⎬⎨ ⎬ (1) 0 H66 ⎭ ⎪ γ (3) ⎪ ⎩ Pxy ⎭ ⎩ 0 0 F66 ⎭ ⎪ γxy ⎪ ⎩ 0 ⎩ ⎭ ⎩ xy ⎭
(12c)
(0) (1) (3) (0) (1) (3) = ∫Ω [Nxx δεxx + Mxx δεxx − c1 Pxx δεxx + Nyy δε yy + Myy δε yy − c1 Pyy δε yy 0
(0)
(0) − c2 Ry δγyz
(8b)
(Aij , Dij , Fij , Hij ) =
where Ω0 represents the middle surface of the laminate, and p(t) is an uniformly distributed load acting on the plate.
∫− h 2
⎧Qα ⎫ = R ⎨ ⎭ ⎩ α⎬
h 2
∫− h 2
1 σαz ⎧ 2 ⎫ dz z ⎨ ⎭ ⎩ ⎬
(Aij , Dij , Fij ) =
2
+
∂Nyy ∂y
∂w¨ = I0 v¨0 + J1 ϕ¨y − c1 I3 0 ∂y
(13a)
h /2
∫−h/2 Qij (1, z2, z 4) dz, (i, j = 4, 5)
(13b)
¯ xx = 0 x = 0 and x = a, v0 = w0 = ϕy = M
(9a)
¯ yy = 0 y = 0 and y = b, u 0 = w0 = ϕx = M (9b)
Nyy
in which subscripts α and β denote x and y. Substituting δU and δV from Eqs. (7) and (8) into Eq. (6), one can obtain the following governing equations
∂Nxy ∂w¨ ∂Nxx = I0 u¨ 0 + J1 ϕ¨x − c1 I3 0 + ∂y ∂x ∂x
h /2
∫−h/2 Qij (1, z2, z 4, z 6) dz, (i, j = 1, 2, 6)
The boundary conditions of simply supported rectangular plates are
h 2
∫− h ρ0 (z )idz, (i = 0, 1, 2, ···,6)
(12e)
For the laminated composite plate made of orthotropic layers, the stiffness coefficients Aij,Dij,Fij and Hij are expressed as
(8a)
⎧1 ⎫ σαβ z dz , ⎨ z3⎬ ⎩ ⎭
(12d)
(2)
γ γ ⎧ Ry ⎫ = ⎧ D44 0 ⎫ ⎧ yz ⎫ + ⎧ F44 0 ⎫ ⎧ yz ⎫ (0) (2) 0 D 0 F R ⎬ ⎨ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ 55 55 ⎭ ⎩ γxz ⎬ ⎭ ⎩ γxz ⎭ ⎩ ⎩ x⎭ ⎩ ⎭
(0) (1) (3) (0) (0) (0) + Nxy δγxy + Mxy δγxy − c1 Pxy δγxy + Qx δγxz − c2 Rx δγxz + Q y δγyz
∫A p (t ) δwdA
(2)
γ γ ⎧Q y ⎫ = ⎧ A 44 0 ⎫ ⎧ yz ⎫ + ⎧ D44 0 ⎫ ⎧ yz ⎫ (0) 0 D 0 A ⎨ ⎨ ⎨Q x ⎬ 55 ⎬ ⎨ γ (2) ⎬ 55 ⎬ ⎨ γ ⎬ ⎭ ⎩ xz ⎭ ⎩ ⎭ ⎩ xz ⎭ ⎭ ⎩ ⎩
δU
h 2
(11)
⎧ε ⎫ ⎧ Nxx ⎫ ⎧ A11 A12 0 ⎫ ⎪ xx (0) ⎪ Nyy = A21 A22 0 ε yy ⎬⎨ ⎬ ⎨ ⎬ ⎨ 0 A66 ⎭ ⎪ γ (0) ⎪ ⎩ Nxy ⎭ ⎩ 0 ⎩ xy ⎭
(0)
⎧ Nαβ ⎫ Mαβ = ⎨ ⎬ P ⎩ αβ ⎭
4 , c2 = 3c1. 3h2
(7)
where
∂x
zk+1 k
(0)
+ (I0 v˙ 0 + I1 ϕ˙y − c1 I3 φ˙ y ) δv˙ 0 + (I1 v˙ 0 + I2 ϕ˙y − c1 I4 φ˙ y ) δϕ˙y
δW =
(10e)
The internal-strain relations are expressed as
= ∫Ω [(I0 u˙ 0 + I1 ϕ˙x − c1 I3 φ˙ x ) δu˙ 0 + (I1 u˙ 0 + I2 ϕ˙x − c1 I4 φ˙ x ) 0
∂Nxy
∂w¨ − Q¯ y = J1 v¨0 + K2 ϕ¨y − c1 J4 0 ∂y
Ji = Ii − c1 Ii + 2, K2 = I2 − 2c1 I4 + c12 I6, c1 =
δK
Ii =
∂y
¯ αβ = Mαβ − c1 Pαβ , (α, β = 1, 2, 6), Q¯ α = Qα − c2 R α , (α = 4, 5) M
where δU is the potential energy, δK is the kinetic energy, and δW is the virtual work done by external forces. The kinetic energy and potential energy can be casted in the following equations:
− I0 w˙ 0 δw˙ 0] dxdy
¯ yy ∂M
where
(6)
1
+
(10d)
y − 0, b
= 0,
∫0
b
Nxx
x − 0, a dy
(14)
To truncate the governing partial differential Eq. (10), the displacements u0, v0, w0, ϕx, ϕy can be expressed as:
(10a)
(10b) 3
u 0 = u1 cos
πy 3πy 3πx πx sin + u2 cos sin , a b a b
(15a)
v0 = v1 sin
πy 3πy 3πx πx cos + v2 sin cos , a b a b
(15b)
International Journal of Impact Engineering 134 (2019) 103383
J. Zhang, et al.
w0 = w1 sin
πy 3πy 3πx πx sin + w2 sin sin , a b a b
ϕx = ϕx1 cos
πy 3πy 3πx πx sin + ϕx 2 cos sin , a b a b
3.4. Triangular loading (15c) The function of the triangular loading can be given as [22] (15d)
p (t ) = πy 3πy 3πx πx ϕy = ϕy1 sin cos + ϕy2 sin cos , a b a b
(15e)
πy 3πy 3πx πx sin + p2 (t )sin sin , a b a b
(15f)
16p (t )
where pn (t ) = , (n = 1, 2) . 3π 2 Considering the lateral nonlinear vibrations of the honeycomb sandwich plate in w direction, the inertial terms of other directions u and v are neglected. Hence the truncated nonlinear system with two degrees of freedom for laminated plates is obtained by using the Galerkin method:
w¨ 1 + β11 w1 + β12 w1 w22 + β13 w12 w2 + β14 w13 + β15 w23 = β16 p1 (t )
(16a)
w¨ 2 + β21 w2 + β22 w12 w2 + β23 w1 w22 + β24 w23 + β25 w13 = β26 p2 (t )
(16b)
t
p (t ) =
The Newton iteration method is used to solve the transient responses of the honeycomb sandwich plate by using FORTRAN software. In order to verify the correctness of this method and procedure, the transient response results are compared with the results of ANSYS software by considering a simply supported auxetic honeycomb sandwich plate subjected to step load as shown in Fig. 3. The finite element model of the plate in Figs. 3 and 5 is modeled as shell element with shell 181 from commercial software ANSYS Mechanic APDL 14.2. The Physical parameters is Anisotropic, the honeycomb plate model is Rectangle. The dividing grid is Smart Size 1, the analysis type is Transient, the type of applied load is On Area. Choosing the geometric midpoint of honeycomb plate as the observation point. The boundary conditions are simply supported. The material of honeycomb plate is aluminum, and the Poisson's ratio of the material is νs = 0.3. The length and width of the plate are a = b = 0.2m , thickness is h = 0.01m , respectively. In addition, the step load of intensity is p0 = 106N / m2 . The parameters of theoretical model and finite element model are all same. The displacements of w direction are selected to compare with that from theoretical model. It indicates from Fig. 3 that the results obtained from the truncated system by FORTRAN in this paper are in good agreement with those obtained by ANSYS. Fig. 4 is the cell of the honeycomb cores layer with negative Poisson's ratio. l1 and l2 are lengths of straight and sloping walls of the cellular, t1 and t2 are their thickness, the cellular inclined angle is θ. The equivalent elastic parameters of the honeycomb core layer used in Eq. (5) with k = 2 were computed from the sizes of the cell, in which tension and compression of the honeycomb cores wall and bending deformation are considered. The formulas can be found from paper [34]. The following two tables represent changes of equivalent Poisson's ratios and equivalent Young's moduli in X-axis with different cell (2) inclination angles θ. Table 1 gives Poisson's ratios ν12 under different cell inclination angles θ. It can be seen from Table 1 that the symbol of θ determines the sign of the equivalent Poisson's ratio, and with increase of angle θ and the geometric size l2/l1, the absolute values of equivalent Poisson's ratios also increase. Table 2 gives values of equivalent moduli E1(2) under different cell inclination angles θ. As can be seen from Table 2, no matter whether the cells are traditional honeycombs or auxetic honeycombs, the equivalent moduli increases with
(17)
where r represents the pulse length which can change the time of the blast pulse impacting on the structure, as shown in Fig. 2(a). 3.2. Exponential air blast loading If the explosion source is far from the plate, it is assumed that the explosion load is uniform and can be given by the Friedlander exponential decay equation as [23] (18)
where p0 is the peak pressure, tp represents the positive phase blast duration, α is the waveform decay parameter and t is the total time, see Fig. 2(b) for details. 3.3. Sinusoidal loading The function of the sinusoidal pulse can be given
p (t ) =
⎧ p sin 0 ⎨0 ⎩
( ) πt tp
0 < t < rtp t < 0, t > rtp
(21)
3.6. Validation of the method
The function of the step pulse is represented as
−α t tp
⎧ p0 tp 0 < t < tp ⎨ 0 t < 0, t > tp ⎩
where p0 is the peak blasting pressure and p(t) is proportional to t, see Fig. 2(e) for details.
3.1. Step loading
p (t ) = p0 (1 − t / tp) e
(20)
The function of incremental pulse can be given
In this section, nonlinear transient responses of the honeycomb sandwich plate under several uniform time-dependent blast pulses are investigated by numerical simulations. The blast loads are step loading, air-blast loading, sinusoidal loading, triangular loading and incremental loading, respectively. The expressions of the blast loading are summarized as follows.
0 < t < rtp t < 0, t > rtp
0 < t < rtp t < 0, t > rtp
3.5. Incremental loading
3. Transient responses of the plate under blast loads
p0 p (t ) = ⎧ 0 ⎨ ⎩
t ) tp
where r represents the pulse length factor. The shape of the pulse is determined by the variation of r. When r = 1 the triangular pulse is a special form of air blast and when tp → ∞ it can be converted to a step pulse. For r = 2 the load function corresponds to the centrosymmetric form of tp, while r ≠ 2, the pulse turns into an asymmetric N-pulse. It is evident that when r > 1 the negative phase of the pulse is covered, as shown in Fig. 2(d).
where u1, u2, v1, v2, w1, w2, ϕx1, ϕx2, ϕy1, ϕy2 represent the generalized coordinates. The expression of the excitation force correspondingly becomes
p (t ) = p1 (t )sin
⎧ p0 (1 − ⎨ 0 ⎩
(19)
It can be seen that r may determine the form of the sine pulse, see Fig. 2(c) for details. 4
International Journal of Impact Engineering 134 (2019) 103383
J. Zhang, et al.
Fig. 2. (a). Step loading, (b). Exponential air blast loading, (c). Sinusoidal loading, (d). Triangular loading, (e). Incremental loading.
the increase of the cell angle θ. When the angle θ is same, the equivalent modulus of auxetic honeycombs is smaller than that of traditional honeycombs. In order to verify the superiority of the honeycomb structures with
negative Poisson ratio, the transient responses of metal plate and honeycomb sandwich plate both subjected to step load are compared in Fig. 5 by finite element model. The geometric parameters are chosen as follows, the lengths a = b = 0.2m , and the thickness of the plate and
Fig. 3. Comparison of transient responses of the honeycomb sandwich plate. 5
International Journal of Impact Engineering 134 (2019) 103383
J. Zhang, et al.
blast load, sinusoidal load, triangular load and incremental load, will be studied through numerical simulations.
3.7. Effect of geometrical parameters under step loads The length and width of the plate is a = b = 1m , and other parameters are same as that of Fig. 5. Nonlinear responses of the auxetic honeycomb plate under step load as shown in Fig. 6. The step load of intensity is p0 = 106N / m2 and time duration is tp = 1.5ms . The time step is dt = 10−8ms and the total time is t = 10ms . Fig. 6(a) represents the effect of honeycomb cell inclined angle θ on lateral displacements under step load with the width is h = 0.01m and hc / h = 0.9, and the pulse length factor is r = 1. It can be seen clearly from Fig. 6(a) that the larger of the cell inclined angle, the greater of the lateral displacement caused by the load. Therefore, when the cell inclined angle θ = 30∘, the honeycomb sandwich plate has the best ability to absorb energy in Fig. 6(a). Fig. 6(b) shows the time history of displacement for various total thickness (0.01 m, 0.05 m, 0.10 m) of the honeycomb sandwich plate. Other parameters of the system are same as that of Fig. 6(a). It is observed that the total thickness of the plate plays a great effective role on the lateral displacements of the system. The depicted results show that the decrease of the total thickness of the plate extremely increases the response amplitudes. While the vibration frequencies and lateral displacements have an inverse relationship, therefore, we can increase the capacity of the system to withstand the impact by increasing the total thickness of the plate. In Fig. 6(c), the transient response of the honeycomb sandwich plate is described by changing the core-thickness ratio (0.8, 0.85, 0.9, 0.95) and other parameters are chosen as same as aforementioned. One can see clearly that the greater of the core thickness, the greater of the amplitude of the lateral displacements under the same loads. Because of the honeycomb core is soft, it is easier to deform than the homogeneous metal material under blast loads. Fig. 6(d) depicts the time history of lateral displacement under step (2) load with different equivalent Poisson's ratio ν12 as −0.9612 and 1.6019 when the cellular inclined angle θ = 30∘ and θ = −30∘, respectively, as show in Table 1. The values of other corresponding parameters are chosen as same as that of Fig. 5. Fig. 6(d) indicates that the auxetic honeycomb sandwich plate has larger lateral displacements when it received the same intensity of the load, but after the sudden load, the lateral displacements is less than the plate with the positive Poisson's ratio. This shows that the auxetic honeycomb sandwich panel has better properties on vibration absorption than the honeycomb plate with positive Poisson's ratio. The effect of the pulse length factor r on lateral displacements under step load is investigated in Fig. 6(e). The length factor r represents the impact time of the step pulse on the plate. It is evident that as r increases, the maximal lateral displacements and vibration frequencies
Fig. 4. The unit cell of concave hexagonal honeycomb core.
Table 1 (2) Poisson's ratios ν12 with different cell angels θ. θ
-π/3 -π/4 -π/6 π/6 π/4 π/3
l2/l1 1.5
2
2.5
3
7.4442 2.9712 1.2899 −0.6450 −1.0674 −1.9947
8.8549 3.6097 1.6019 −0.9612 −1.7240 −3.5035
10.2156 4.3695 1.9100 −1.2733 −2.3681 −4.9590
11.5290 4.8507 2.2141 −1.5815 −3.0002 −6.3638
Table 2 The equivalent moduli E1(2) (×109GPa) with different cell angels θ. θ
-π/3 -π/4 -π/6 π/6 π/4 π/3
l2/l1 1.5
2
2.5
3
1.1982 0.4142 0.2076 0.1038 0.1488 0.3211
1.4253 0.5032 0.2578 0.1547 0.2403 0.5639
1.6443 0.5904 0.3074 0.2050 0.3301 0.7982
1.8557 0.6762 0.3564 0.2546 0.4182 1.0243
thickness of the core layer are h = 0.01m ,hc = 0.009m respectively, the cellular parameters as shown in Fig. 4 are l1 = 0.01m , l2 = 2l1, t1 = t2 0.001m , the cellular inclined angle is θ = 30∘. The honeycomb plate is made of aluminum, and it's modulus is Es = 69GPa, the density is ρs = 2700kg / m3 , Poisson's ratio is νs = 0.3, and shear modulus is Gs = 27GPa . It can be obtained from Fig. 5 that the deflections of auxetic honeycomb plate are larger than that of the metal plate which indicate that the honeycomb has a good energy absorption capacity under impact forces. Next, the nonlinear transient dynamic responses of the honeycomb sandwich plate under the mentioned loads, including step load, air-
Fig. 5. Comparison of transient responses of auxetic honeycomb sandwich plate with metal plate under step loads. 6
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Fig. 6. (a). The effect of cell inclined angle θ on lateral displacements under step load, (b). The effect of total thickness h on lateral displacements under step load, (c). The core-thickness ratio (hc/h) versus lateral displacements under step load, (d). The effect of Poisson's ratios on lateral displacements under step load, (e). The effect of the length factor r on lateral displacements under step load.
loading drop. Therefore, the loading form is the easiest to be absorbed by the honeycomb plate at α = 1in Fig. 7(a). Fig. 7(b) expresses the dynamic responses of the plate under airblast load for different duration time tp. The waveform decay parameter α = 1, other parameters are same as that of Fig. 7(a). It can be seen from the small diagram that the larger of the duration time tp resulting in the slower decay rate of the load. The lateral displacement of the plate decreases when tp decreases. When the duration time tp is equal to 2, the transverse displacement and the vibrated frequency of the plate are most appropriate. Fig. 7(c) depicts the time history of lateral displacements under airblast load with different equivalent Poisson's ratio. In Fig. 7(c), the lateral displacement of auxetic honeycomb sandwich plate is similar as that of the positive Poisson when subjected the same intensity of the load, but after the sudden load, the lateral displacements of the auxetic honeycomb plate is less. This shows that the auxetic honeycomb sandwich plate has better properties on vibration absorption than that of the honeycomb plate with positive Poisson's ratio.
also increase and vice versa. The pulse length factor r actually represents the loading time, so the longer the loading time is, the larger transverse displacement and frequency of the honeycomb plate are.
3.8. Effect of air-blast loads Nonlinear responses of the auxetic honeycomb plate under air-blast load are obtained in Fig. 7. The air-blast load of intensity is p0 = 106N / m2 , the time step is dt = 108ms , and the parameters of the plate are same as that of Fig. 5. Fig. 7(a) describes the effect of the waveform decay parameter α on the nonlinear transient response of the honeycomb plate, the duration time tp = 2 m s . It can be seen from the small diagram that the higher the decay parameter α means the lower the maximum blast pressure. The lateral displacements of the plate decrease when α increases and the vibration frequencies decreases while the α increases as expected. When the waveform decay parameter α = 1, the lateral displacements and the vibrated frequencies of the plate are most appropriate. When t < 2 m s the residual displacement is smaller under the pulse loading with a faster decaying rate. But after t = 2 m s , the residual displacement is independent with the speed of 7
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Fig. 7. (a). The effect of different waveform parameter α on lateral displacements under air-blast load, (b). The effect of duration time tp on lateral displacements under air-blast load, (c). The effect of Poisson's ratios on lateral displacements under air-blast load.
3.9. Effect of sinusoidal load Fig. 8 shows the effect of the length factor r on the transient responses of the auxetic honeycomb sandwich plate under sinusoidal loads. The intensity of sinusoidal load is p0 = 106N / m2 , the time step is dt = 10−8ms . It can be seen from the small diagram that the length factor r determines the shape of the sinusoidal pulse. The maximum impact force of the three sinusoidal loads is same, and the larger of the length factor r, the longer of the time of the load on the honeycomb plate. As is illustrated from the results that when r = 1 the lateral displacements and vibration frequencies are reached maximize.
Fig. 9. The effect of shock pulse length factor r on lateral displacements under triangular load.
3.10. Effect of triangular loads Fig. 9 indicates nonlinear dynamics of the plate under triangular load with the length factor r change. The triangular load of intensity is p0 = 106N / m2 , the time step is dt = 10−8ms . As is illustrated from the diagram that the larger of the length factor r, the larger of the lateral displacements.
3.11. Effect of incremental loads Fig. 10 represents the nonlinear dynamics of the auxetic honeycomb plate under the incremental load given by Eq. (21). Fig. 10(a) is the effect of duration time tp on the nonlinear responses of the plate. It can be seen from the small diagram that the larger of the duration time tp means the slower the increase rate of the load in the same time. The lateral displacement of the plate increases when the tp decreases. When the duration time tp = 2 , the transverse displacement and the vibrated frequency of the plate are most appropriate in Fig. 10(a). Fig. 10(b) depicts the time history of lateral displacements under incremental load with different equivalent Poisson's ratios. It can be seen from that the lateral displacement of auxetic honeycomb sandwich plate is smaller than that of the positive one. In terms of shock absorption capacity, the honeycomb sandwich plate with negative Poisson's ratio is superior to that with positive Poisson's ratio under any loads. Fig. 11 compares the effects of several dynamic loads on transient response of the auxetic honeycomb sandwich plate. As shown in Fig. 11,
Fig. 8. The effect of the length factor r on lateral displacements under sinusoidal load. 8
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Fig. 10. (a). The effect of time duration tp on lateral displacements under incremental load, (b). The Poisson's ratios versus lateral displacements under incremental load.
the lateral displacements of the step load are slightly higher than other loads, but after the step load, the lateral displacement and vibration frequency are much smaller and slower than other loads. Therefore, it can be concluded that the step load is the most easily absorbed by the honeycomb model among these loads, and the sin load is the hardest to be absorbed in these five kinds of loads. 4. Transient responses with other direction of the unit cells In order to check the effect of the unit cells on crashworthiness, the auxetic honeycomb cores arranged as in Fig. 1 is rotated 90∘ around the X-axis, as shown in Fig. 12. The absorption of vibrations is compared between the auxetic honeycombs and conventional honeycombs in Fig. 13. The geometric parameters are chosen as follows, the lengths a = b = 1m , and the thickness of the plate and thickness of the core layer are h = 0.021m ,hc = 0.02m respectively. Other parameters are same as that of Fig. 5. The step load of intensity is p0 = 2 × 106N / m2 and time duration is tp = 2ms . The time step is dt = 10−8ms and the total time is t = 20ms . It can be seen clearly from Fig. 13 that the auxetic honeycomb sandwich plate has larger lateral displacements when it received the same intensity of the load, but after the sudden load, the lateral displacement is less than the plate with the positive Poisson's ratio. The conclusion obtained here is consistent with that of in Fig. 6(d) and the lateral displacement of the auxetic honeycomb plate is much less than the honeycomb plate with positive Poisson's ratio after the impact load.
Fig. 12. Honeycomb sandwich plate of varying the direction of the unit cells.
5. Conclusions In this paper, the equations of motion for honeycomb sandwich plate subjected to several dynamic loads are obtained according to vonKarman deflection, theory Hamilton's principle and the Galerkin method and FORTRAN software. The results obtained by our methods
Fig. 13. The effect of Poisson's ratios on lateral displacements under step load.
Fig. 11. Comparison of different dynamic loads on central lateral displacements. 9
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are compared with ANSYS software to verify the validation of the methods. The transient responses of the honeycomb sandwich plate with negative and positive Poisson's ratio subjected explosive loads are compared through same geometric parameters of the plate. The results show that the lateral displacements of the auxetic honeycomb sandwich plate is relatively lower compared to that of the honeycomb plate with positive Poisson's ratio. This shows that the auxetic honeycomb sandwich plate have better properties on vibration absorption than the honeycomb plate with positive Poisson's ratio. Numerical analysis about the transient response of the honeycomb sandwich plate under several dynamic loads is obtained, considering the influence of total thickness, thickness ratio, cell inclination angle, Poisson's ratio, shock pulse length factor and duration time, etc. The effect of geometrical parameters on the dynamics of the plate is obtained under the same blast load. The numerical results show that the larger of the cell inclined angle, the greater of the amplitudes of the plate. The increase of the total thickness results in the decrease of the amplitudes and larger of the core-thickness ratio, the larger of the amplitude of the plate is. As to different kinds of load, it can be seen from the numerical simulations that the amplitude of the lateral displacement of the step load is slightly higher than other loads for the same pulse length factor r, but after the sudden load, the lateral displacement and vibration frequency are much smaller and slower than other loads. From a large number of detailed numerical simulations, it can be concluded that the honeycomb sandwich structures are good at energy absorption and impact resistance. The auxetic honeycomb sandwich plates may be better or at least provide a preferable alternative for the honeycomb sandwich plates with positive Poisson's ratio.
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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Transient nonlinear responses of an auxetic honeycomb sandwich plate under impact loads
T
⁎
Junhua Zhanga, , Xiufang Zhua, Xiaodong Yangb, Wei Zhangb a b
College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing, 100192, PR China College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Honeycomb sandwich plate Auxetic Nonlinear transient response Dynamic loads
In this paper, we study the nonlinear transient responses of an auxetic (negative Poisson's ratio) honeycomb sandwich plate under impact dynamic loads. The partial differential equations of the honeycomb sandwich plate based on Reddy's higher shear deformation theory and Hamilton's principle are obtained. The nonlinear ordinary differential equations are then derived by Galerkin truncation method. The dynamic responses of the honeycomb sandwich plate subjected to different loads, such as step loading, air-blast loading, sinusoidal loading, triangular loading and incremental loading, respectively, have been investigated. The contributions of total thickness, core thickness ratio, Poisson's ratio, cell inclination angle and blast types to transient responses of the plate are discussed in detail by numerical simulations. The effect of geometrical parameters on the dynamics of the plate and the effect of different blast loads on the plate are obtained. It is found that the honeycomb sandwich plate with negative Poisson's ratio would be a better choice compared with the positive one for some structures under dynamic loads.
1. Introduction Driven by design of lightweight materials with vibration and energy absorption characteristics for impact applications, the advanced composite materials are widely used in space station structures, aircraft and spacecraft structures, architectures and bridge structures. Especially multi-cell material sandwich structures are good at energy absorption and impact resistance. It is known that most natural materials are characterized by positive Poisson's ratio. However, these traditional materials can form structures that exhibit negative Poisson's ratio, i.e. auxetic, which have lower yielding strength and higher stiffness compared with that of positive Poisson's ratio. For example, the traditional honeycomb structures realize the metamaterial properties of negative Poisson's ratio by changing the combination of cells. Consequently, many researches related to the structures with negative Poisson's ratio have been reported in literatures [1–3]. Wan et al. [4] proposed a theoretical approach to calculate the Poisson's ratios of honeycombs according to the large deflection theory. The deformation curves of the tilting part of honeycombs, strains and Poisson's ratios in two orthogonal directions are obtained. Zhang et al. [5] investigated the effect of auxetic honeycomb cellular microstructure on the in-plane crushing behaviors and energy absorptions by using finite element method. The most common sandwich structure at present is made of two ⁎
rigid metallic thin face sheets and a low-density soft core. Many research works focused on this kind of structures. Huang et al. [6] presented a new design about auxetic honeycombs which consisted of a reentrant hexagonal component and a thin plate. The authors studied the in-plane mechanical properties of the auxetic honeycombs. Two kinds of multi-functional layered auxetic honeycombs have been proposed by Sun et al. [7], one is anisotropic re-entrant honeycomb and the other is isotropic chiral honeycomb. The formula of Young's moduli is also given. Brischetto et al. [8] used 3D printing technique to study mechanical behaviors of honeycomb sandwich structure. The progresses of mechanical properties, calculation and application of auxetic materials have been reviewed in [9]. With the emergence of the advanced auxetic sandwich structures and their wide applications in the structures of aircraft and spacecraft, the nonlinear dynamics the sandwich plates under the dynamic loads, such as explosion loads, and acoustic explosion loads produced by fuel explosion has been paid more and more attentions [10–12]. Duc et al. [13] studied vibrations and nonlinear dynamics of sandwich composite cylindrical panels with auxetic honeycomb cores on elastic foundations subjected to mechanical, blasting and damping loads. Imbalzano et al. [14] analyzed and compared the resistance performances of auxetic and conventional honeycomb cores and metal facets against impulsive loadings. Damanpack et al. [15] investigated active control of
Corresponding author. E-mail address: [email protected] (J. Zhang).
https://doi.org/10.1016/j.ijimpeng.2019.103383 Received 4 May 2018; Received in revised form 27 July 2019; Accepted 20 August 2019 Available online 22 August 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Impact Engineering 134 (2019) 103383
J. Zhang, et al.
geometric nonlinear dynamic responses of sandwich beams under impact of integrated piezoelectric sensor. The effect of geometric nonlinearity, the flexibility of the core layer and the dynamic response of piezoelectric patch arrangement, frequency and loss factor of the beams was analyzed in detail. Zhang et al. [16,17] investigated tensile mechanical properties of auxetic hexagonal honeycomb structures. Karagiozova et al. [18] studied energy absorptions of miura-ori patterned open cell metamaterials under in-plane impact dynamic compressions. The performance of different sandwich panels under impact loading was evaluated by parametric studies. Wang et al. [19] studied the dynamic response including deformation and damage modes of honeycomb sandwich plates under the blast loading by the experimental and theoretical studies. Mu et al. [20] analyzed the displacements, deformation, frequencies and stress of a composite sandwich plate with transversely compressible core under time-dependent impact. Rekatsinasa et al. [21] gave the transient impact responses of thick sandwich composite plates under low velocity impacts. The nonlinear transient responses of laminated basalt composite plate under dynamic loads is studied by Basturk et al. [22], it is shown that the laminated basalt composites are good at dynamic loads resistance. Kazanci [23] summarized the responses of various types of explosion pulse models loaded on laminated composite plates. Feli and Namdari Pour [24] proposed the strain and energy of honeycomb sandwich composite plate under high-velocity impact. Jagtap and Lai [25] researched nonlinear dynamic responses of functionally graded laminated plates under stochastic loads. Salami [26, 27] investigated responses of sandwich beams with soft cores subjected low velocity impact by using Ritz method. As key structures in aircraft, space station and marine structures, honeycomb sandwich structures may be subjected to time-dependent pulses, such as step pulses and triangular loading, which may cause serious damage to structures. Auxetic honeycomb sandwich structures are good at energy absorption and impact resistance. For example, Qiao and Chen [28] explored impact resistance of double arrowhead auxetic honeycombs and gave relations between impact velocity and dynamic plateau stress. Qi et al. [29] investigated experimentally and numerically responses of auxetic honeycomb-cored sandwich panels as protective system under impact and close-in blast. Wang et al. [30] gave parametric optimization for double-V auxetic sandwich panels under air blast load. Gao et al. [31] showed multi-objective optimization of an auxetic cylindrical crash box on energy absorption. Energy absorption of 3D printed auxetic honeycomb structures was given in [32, 33]. Therefore, it is of importance to study the dynamic response of these composite honeycomb structures subjected blast loads. In particular, auxetic honeycomb sandwich plate, which has metamaterial property, may be better in vibration absorptions than the traditional honeycombs. It is proved in this investigation. Nonlinear dynamic responses of an auxetic honeycomb sandwich plate are analyzed. The honeycomb sandwich plate is subjected to step loading, air-blast loading, triangular loading, sine loading and incremental loading. The influence of total thickness, the thickness ratio, cell inclination angle, Poisson's ratio and blast types to transient responses of the plate are discussed in detail by the numerical simulations.
Fig. 1. Honeycomb sandwich plate with concave hexagon core.
According to higher order shear deformation theory, the displacements of u, v and w can be expressed as
u (x , y, z , t ) = u 0 (x , y, t ) + zϕx (x , y, t ) −
4 3⎛ ∂w0 ⎞ z ϕ + 3h2 ⎝ x ∂x ⎠
(1a)
v (x , y, z , t ) = v0 (x , y, t ) + zϕy (x , y, t ) −
4 3⎛ ∂w0 ⎞ z ϕ + 3h2 ⎝ y ∂y ⎠
(1b)
⎜
⎟
w (x , y, z , t ) = w0 (x , y, t )
(1c)
where u0, v0 and w0denote the displacements at the middle surface, and ϕx, ϕy are the transversal normal rotations of middle surface. Using of the von Karman large geometric deformation theory on the plate, the strain-displacement relations and curvature-displacement relationship can be obtained as (0) (1) (3) ⎧ εxx ⎫ ⎧ εxx ⎫ ⎧ εxx ⎫ ε ⎪ (1) ⎪ ⎪ (3) ⎪ ⎧ xx ⎫ ⎪ (0) ⎪ 3 εyy = ε yy + z ε yy + z ε yy ⎨ ⎬ ⎨ εxy ⎬ ⎨ ⎬ ⎨ ⎬ (0) ⎪ (1) ⎪ (3) ⎪ ⎩ ⎭ ⎪ εxy ⎪ εxy ⎪ εxy ⎩ ⎭ ⎩ ⎭ ⎩ ⎭
(2a)
γ (0) γ (2) ⎧ γyz ⎫ = ⎧ yz ⎫ + z 2 ⎧ yz ⎫ γ ⎨ γ (2) ⎬ ⎨ γ (0) ⎬ ⎭ ⎨ ⎩ xz ⎬ ⎩ xz ⎭ ⎩ xz ⎭
(2b)
where ∂u0 1 ⎧ + 2 (0) ∂x ⎧ εxx ⎫ ⎪ ⎪ (0) ⎪ ⎪ ∂v 0 1 ε yy = + 2 ∂y ⎨ ⎬ ⎨ (0) ⎪ εxy ⎪ ⎪ ∂u0 ∂v ⎩ ⎭ ⎪ + ∂x0 + ⎩ ∂y
∂w 0 2 ∂x
( )
⎫ ⎪ ⎪ ∂w 0 ∂y ⎬ ∂w 0 ∂w 0 ⎪ + ∂x ∂y ⎪ ⎭
( )
∂ϕ
⎧ x (1) ⎧ εxx ⎫ ⎪ ∂x ⎪ (1) ⎪ ⎪ ∂ϕy ε yy = ⎨ ⎬ ⎨ ∂y (1) ⎪ γxy ⎪ ⎪ ∂ϕx ⎩ ⎭ ⎪ ∂y + ⎩
2. Equations of motion We consider a honeycomb plate with thickness of h and core thickness of hc. The coordinate system is located on the middle surface of the plate, and u, v and w represent the displacement components in x, y and z directions, respectively. The lengths in x and y directions are a and b respectively, as shown in Fig. 1. The sandwich plate consists of one core layer and two layers of skins covering it. The surface on one side of z > 0 is called as upper skin, and the surface on the other side of z < 0 is called as lower skin and the skin is made of isotropic materials. The materials of the honeycomb panel is aluminum, whose modulus is Es, density is ρs, Poisson's ratio is νs, and shear modulus is Gs.
∂ϕ
⎫ ⎪ γ (0) ⎪ ⎧ yz ⎫ ⎧ ϕy + , = ⎬ ⎨ γ (0) ⎬ ⎨ ϕ + ∂ϕ y ⎪ ⎩ xz ⎭ ⎩ x ∂x ⎪ ⎭
⎧ x + (3) ⎧ εxx ⎫ ⎪ ∂x ⎪ ∂ϕy ⎪ (3) ⎪ ε yy = −c1 + ⎨ ⎬ ⎨ ∂y (3) ⎪ ∂ϕx ⎪ γxy ⎪ ⎩ ⎭ ⎪ ∂y + ⎩ 4 3h2
2
∂2w 0 ∂x 2 ∂2w 0 ∂y 2 ∂ϕ y ∂x
(3a)
∂w 0 ⎫ ∂y ∂w 0 ⎬ ∂x ⎭
⎫ ⎪ γ (2) ⎧ ϕy + ⎪ ⎧ yz ⎫ = −c2 , (2) ⎨ϕ + ⎬ ⎨γ ⎬ xz ⎭ ⎩ x ∂2w ⎪ ⎩ + 2 ∂x ∂y0 ⎪ ⎭
(3b)
∂w 0 ⎫ ∂y ∂w 0 ⎬ ∂x ⎭
(3c)
and c2 = 3c1. and c1 = The honeycomb sandwich plate with negative Poisson ratio is orthotropic and the constitutive equation can be written as 2
International Journal of Impact Engineering 134 (2019) 103383
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(k ) (k ) Q11 Q12 0 0 0 ⎫ ε σxx (k ) ⎧ ⎪ ⎧ xx ⎫ ⎪ (k ) (k ) ⎧σ ⎫ 0 0 0 ⎪ ⎪ εyy ⎪ ⎪Q21 Q22 ⎪ yy ⎪ ⎪γ ⎪ (k ) σyz yz = 0 0 Q44 0 0 ⎬⎨γ ⎬ ⎨ σxz ⎬ ⎨ xz k ( ) ⎪ ⎪ ⎪ 0 0 0 Q55 0 ⎪⎪ ⎪ ⎪ ⎪ γxy ⎪ ⎪ ⎩ σxy ⎭ ⎭ (k ) ⎩ 0 0 0 0 Q 66 ⎭ ⎩
∂Q¯ x ∂x
+
+
∂Q¯ y
+
∂y
(N
∂ ∂x
∂w 0 xx ∂x
+ Nxy
∂2Pxy
∂2P c1 ⎛ 2xx ∂x
+ 2 ∂x ∂y +
⎝
= I0 w¨ 0 − c12 I6 (4)
(
∂2w¨ 0 ∂x 2
+
∂2Pyy ⎞ ∂y 2
∂2w¨ 0 ∂y 2
⎠
∂w 0 ∂y
) + (N ∂ ∂y
∂w 0 xy ∂x
+ Nyy
∂w 0 ∂y
) + J ⎛⎝
∂ϕ¨x
)
+ p (t )
) + c ⎡⎢⎣I ( 1
3
∂u¨ 0 ∂x
∂v¨0 ∂y
+
4
∂x
+
∂ϕ¨y ∂y
⎞⎤ ⎠⎥ ⎦ (10c)
where (k ) Q11 =
E1(k )
(k ) , Q22 = (k )
(k ) ν21 1 − ν12
¯ xy ¯ xx ∂M ∂w¨ ∂M − Q¯ x = J1 u¨ 0 + K2 ϕ¨x − c1 J4 0 + ∂y ∂x ∂x
E2(k ) (k ) (k ) ν21 1 − ν12
Q66 = G12, Q44 = G23, Q55 = G13, Q21 = Q12
(5)
¯ xy ∂M
(k ) (k ) (k ) (k ) (k ) , G13 , G23 , ν12 and ν21 denote the skin and core where E1(k ) , E2(k ) , G12 layer elastic modulus, shear modulus and Poisson's ratio, respectively. The superscripts of k = 1, 3 represent the upper and lower skins respectively, and k = 2 indicates the core layer. The governing equations are established by using the Hamilton principle, and the expression of Hamilton principle is as follows
∫t
t2
(δU − δK ) dt +
1
∫t
t2
δWdt = 0
∂x
N
Ii = ∑k = 1 ∫z
δϕ˙x − c1 (I3 u˙ 0 + I4 ϕ˙x − c1 I6 φ˙ x ) δφ˙ x − c1 (I3 v˙ 0 + I4 ϕ˙y − c1 I6 φ˙ y ) δφ˙ y
ρ(k ) (z )idz , (i = 0, 1, 2, …, 6)
∂w0 ∂w0 φ˙ x = ϕ˙x + , φ˙ y = ϕ˙y + ∂x ∂y and
(12a)
(1) (3) ⎧ε ⎫ ⎧ε ⎫ ⎧ Mxx ⎫ ⎧ D11 D12 0 ⎫ ⎪ xx ⎪ ⎧ F11 F12 0 ⎫ ⎪ xx ⎪ (1) (3) Myy = D21 D22 0 ε yy + F21 F22 0 ε yy ⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ ⎬⎨ ⎬ (1) D F 0 0 0 0 M 66 66 xy ⎩ ⎭ ⎩ ⎭ ⎪ ⎪ ⎪ γ γ (3) ⎪ ⎩ ⎭ ⎩ xy ⎭ ⎩ xy ⎭
(12b)
(1) (3) ⎧ε ⎫ ⎧ εxx ⎫ ⎧ Pxx ⎫ ⎧ F11 F12 0 ⎫ ⎪ xx ⎧ H11 H12 0 ⎫ ⎪ (3) ⎪ (1) ⎪ Pyy = F21 F22 0 ε yy + H21 H22 0 ε yy ⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ ⎬⎨ ⎬ (1) 0 H66 ⎭ ⎪ γ (3) ⎪ ⎩ Pxy ⎭ ⎩ 0 0 F66 ⎭ ⎪ γxy ⎪ ⎩ 0 ⎩ ⎭ ⎩ xy ⎭
(12c)
(0) (1) (3) (0) (1) (3) = ∫Ω [Nxx δεxx + Mxx δεxx − c1 Pxx δεxx + Nyy δε yy + Myy δε yy − c1 Pyy δε yy 0
(0)
(0) − c2 Ry δγyz
(8b)
(Aij , Dij , Fij , Hij ) =
where Ω0 represents the middle surface of the laminate, and p(t) is an uniformly distributed load acting on the plate.
∫− h 2
⎧Qα ⎫ = R ⎨ ⎭ ⎩ α⎬
h 2
∫− h 2
1 σαz ⎧ 2 ⎫ dz z ⎨ ⎭ ⎩ ⎬
(Aij , Dij , Fij ) =
2
+
∂Nyy ∂y
∂w¨ = I0 v¨0 + J1 ϕ¨y − c1 I3 0 ∂y
(13a)
h /2
∫−h/2 Qij (1, z2, z 4) dz, (i, j = 4, 5)
(13b)
¯ xx = 0 x = 0 and x = a, v0 = w0 = ϕy = M
(9a)
¯ yy = 0 y = 0 and y = b, u 0 = w0 = ϕx = M (9b)
Nyy
in which subscripts α and β denote x and y. Substituting δU and δV from Eqs. (7) and (8) into Eq. (6), one can obtain the following governing equations
∂Nxy ∂w¨ ∂Nxx = I0 u¨ 0 + J1 ϕ¨x − c1 I3 0 + ∂y ∂x ∂x
h /2
∫−h/2 Qij (1, z2, z 4, z 6) dz, (i, j = 1, 2, 6)
The boundary conditions of simply supported rectangular plates are
h 2
∫− h ρ0 (z )idz, (i = 0, 1, 2, ···,6)
(12e)
For the laminated composite plate made of orthotropic layers, the stiffness coefficients Aij,Dij,Fij and Hij are expressed as
(8a)
⎧1 ⎫ σαβ z dz , ⎨ z3⎬ ⎩ ⎭
(12d)
(2)
γ γ ⎧ Ry ⎫ = ⎧ D44 0 ⎫ ⎧ yz ⎫ + ⎧ F44 0 ⎫ ⎧ yz ⎫ (0) (2) 0 D 0 F R ⎬ ⎨ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ 55 55 ⎭ ⎩ γxz ⎬ ⎭ ⎩ γxz ⎭ ⎩ ⎩ x⎭ ⎩ ⎭
(0) (1) (3) (0) (0) (0) + Nxy δγxy + Mxy δγxy − c1 Pxy δγxy + Qx δγxz − c2 Rx δγxz + Q y δγyz
∫A p (t ) δwdA
(2)
γ γ ⎧Q y ⎫ = ⎧ A 44 0 ⎫ ⎧ yz ⎫ + ⎧ D44 0 ⎫ ⎧ yz ⎫ (0) 0 D 0 A ⎨ ⎨ ⎨Q x ⎬ 55 ⎬ ⎨ γ (2) ⎬ 55 ⎬ ⎨ γ ⎬ ⎭ ⎩ xz ⎭ ⎩ ⎭ ⎩ xz ⎭ ⎭ ⎩ ⎩
δU
h 2
(11)
⎧ε ⎫ ⎧ Nxx ⎫ ⎧ A11 A12 0 ⎫ ⎪ xx (0) ⎪ Nyy = A21 A22 0 ε yy ⎬⎨ ⎬ ⎨ ⎬ ⎨ 0 A66 ⎭ ⎪ γ (0) ⎪ ⎩ Nxy ⎭ ⎩ 0 ⎩ xy ⎭
(0)
⎧ Nαβ ⎫ Mαβ = ⎨ ⎬ P ⎩ αβ ⎭
4 , c2 = 3c1. 3h2
(7)
where
∂x
zk+1 k
(0)
+ (I0 v˙ 0 + I1 ϕ˙y − c1 I3 φ˙ y ) δv˙ 0 + (I1 v˙ 0 + I2 ϕ˙y − c1 I4 φ˙ y ) δϕ˙y
δW =
(10e)
The internal-strain relations are expressed as
= ∫Ω [(I0 u˙ 0 + I1 ϕ˙x − c1 I3 φ˙ x ) δu˙ 0 + (I1 u˙ 0 + I2 ϕ˙x − c1 I4 φ˙ x ) 0
∂Nxy
∂w¨ − Q¯ y = J1 v¨0 + K2 ϕ¨y − c1 J4 0 ∂y
Ji = Ii − c1 Ii + 2, K2 = I2 − 2c1 I4 + c12 I6, c1 =
δK
Ii =
∂y
¯ αβ = Mαβ − c1 Pαβ , (α, β = 1, 2, 6), Q¯ α = Qα − c2 R α , (α = 4, 5) M
where δU is the potential energy, δK is the kinetic energy, and δW is the virtual work done by external forces. The kinetic energy and potential energy can be casted in the following equations:
− I0 w˙ 0 δw˙ 0] dxdy
¯ yy ∂M
where
(6)
1
+
(10d)
y − 0, b
= 0,
∫0
b
Nxx
x − 0, a dy
(14)
To truncate the governing partial differential Eq. (10), the displacements u0, v0, w0, ϕx, ϕy can be expressed as:
(10a)
(10b) 3
u 0 = u1 cos
πy 3πy 3πx πx sin + u2 cos sin , a b a b
(15a)
v0 = v1 sin
πy 3πy 3πx πx cos + v2 sin cos , a b a b
(15b)
International Journal of Impact Engineering 134 (2019) 103383
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w0 = w1 sin
πy 3πy 3πx πx sin + w2 sin sin , a b a b
ϕx = ϕx1 cos
πy 3πy 3πx πx sin + ϕx 2 cos sin , a b a b
3.4. Triangular loading (15c) The function of the triangular loading can be given as [22] (15d)
p (t ) = πy 3πy 3πx πx ϕy = ϕy1 sin cos + ϕy2 sin cos , a b a b
(15e)
πy 3πy 3πx πx sin + p2 (t )sin sin , a b a b
(15f)
16p (t )
where pn (t ) = , (n = 1, 2) . 3π 2 Considering the lateral nonlinear vibrations of the honeycomb sandwich plate in w direction, the inertial terms of other directions u and v are neglected. Hence the truncated nonlinear system with two degrees of freedom for laminated plates is obtained by using the Galerkin method:
w¨ 1 + β11 w1 + β12 w1 w22 + β13 w12 w2 + β14 w13 + β15 w23 = β16 p1 (t )
(16a)
w¨ 2 + β21 w2 + β22 w12 w2 + β23 w1 w22 + β24 w23 + β25 w13 = β26 p2 (t )
(16b)
t
p (t ) =
The Newton iteration method is used to solve the transient responses of the honeycomb sandwich plate by using FORTRAN software. In order to verify the correctness of this method and procedure, the transient response results are compared with the results of ANSYS software by considering a simply supported auxetic honeycomb sandwich plate subjected to step load as shown in Fig. 3. The finite element model of the plate in Figs. 3 and 5 is modeled as shell element with shell 181 from commercial software ANSYS Mechanic APDL 14.2. The Physical parameters is Anisotropic, the honeycomb plate model is Rectangle. The dividing grid is Smart Size 1, the analysis type is Transient, the type of applied load is On Area. Choosing the geometric midpoint of honeycomb plate as the observation point. The boundary conditions are simply supported. The material of honeycomb plate is aluminum, and the Poisson's ratio of the material is νs = 0.3. The length and width of the plate are a = b = 0.2m , thickness is h = 0.01m , respectively. In addition, the step load of intensity is p0 = 106N / m2 . The parameters of theoretical model and finite element model are all same. The displacements of w direction are selected to compare with that from theoretical model. It indicates from Fig. 3 that the results obtained from the truncated system by FORTRAN in this paper are in good agreement with those obtained by ANSYS. Fig. 4 is the cell of the honeycomb cores layer with negative Poisson's ratio. l1 and l2 are lengths of straight and sloping walls of the cellular, t1 and t2 are their thickness, the cellular inclined angle is θ. The equivalent elastic parameters of the honeycomb core layer used in Eq. (5) with k = 2 were computed from the sizes of the cell, in which tension and compression of the honeycomb cores wall and bending deformation are considered. The formulas can be found from paper [34]. The following two tables represent changes of equivalent Poisson's ratios and equivalent Young's moduli in X-axis with different cell (2) inclination angles θ. Table 1 gives Poisson's ratios ν12 under different cell inclination angles θ. It can be seen from Table 1 that the symbol of θ determines the sign of the equivalent Poisson's ratio, and with increase of angle θ and the geometric size l2/l1, the absolute values of equivalent Poisson's ratios also increase. Table 2 gives values of equivalent moduli E1(2) under different cell inclination angles θ. As can be seen from Table 2, no matter whether the cells are traditional honeycombs or auxetic honeycombs, the equivalent moduli increases with
(17)
where r represents the pulse length which can change the time of the blast pulse impacting on the structure, as shown in Fig. 2(a). 3.2. Exponential air blast loading If the explosion source is far from the plate, it is assumed that the explosion load is uniform and can be given by the Friedlander exponential decay equation as [23] (18)
where p0 is the peak pressure, tp represents the positive phase blast duration, α is the waveform decay parameter and t is the total time, see Fig. 2(b) for details. 3.3. Sinusoidal loading The function of the sinusoidal pulse can be given
p (t ) =
⎧ p sin 0 ⎨0 ⎩
( ) πt tp
0 < t < rtp t < 0, t > rtp
(21)
3.6. Validation of the method
The function of the step pulse is represented as
−α t tp
⎧ p0 tp 0 < t < tp ⎨ 0 t < 0, t > tp ⎩
where p0 is the peak blasting pressure and p(t) is proportional to t, see Fig. 2(e) for details.
3.1. Step loading
p (t ) = p0 (1 − t / tp) e
(20)
The function of incremental pulse can be given
In this section, nonlinear transient responses of the honeycomb sandwich plate under several uniform time-dependent blast pulses are investigated by numerical simulations. The blast loads are step loading, air-blast loading, sinusoidal loading, triangular loading and incremental loading, respectively. The expressions of the blast loading are summarized as follows.
0 < t < rtp t < 0, t > rtp
0 < t < rtp t < 0, t > rtp
3.5. Incremental loading
3. Transient responses of the plate under blast loads
p0 p (t ) = ⎧ 0 ⎨ ⎩
t ) tp
where r represents the pulse length factor. The shape of the pulse is determined by the variation of r. When r = 1 the triangular pulse is a special form of air blast and when tp → ∞ it can be converted to a step pulse. For r = 2 the load function corresponds to the centrosymmetric form of tp, while r ≠ 2, the pulse turns into an asymmetric N-pulse. It is evident that when r > 1 the negative phase of the pulse is covered, as shown in Fig. 2(d).
where u1, u2, v1, v2, w1, w2, ϕx1, ϕx2, ϕy1, ϕy2 represent the generalized coordinates. The expression of the excitation force correspondingly becomes
p (t ) = p1 (t )sin
⎧ p0 (1 − ⎨ 0 ⎩
(19)
It can be seen that r may determine the form of the sine pulse, see Fig. 2(c) for details. 4
International Journal of Impact Engineering 134 (2019) 103383
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Fig. 2. (a). Step loading, (b). Exponential air blast loading, (c). Sinusoidal loading, (d). Triangular loading, (e). Incremental loading.
the increase of the cell angle θ. When the angle θ is same, the equivalent modulus of auxetic honeycombs is smaller than that of traditional honeycombs. In order to verify the superiority of the honeycomb structures with
negative Poisson ratio, the transient responses of metal plate and honeycomb sandwich plate both subjected to step load are compared in Fig. 5 by finite element model. The geometric parameters are chosen as follows, the lengths a = b = 0.2m , and the thickness of the plate and
Fig. 3. Comparison of transient responses of the honeycomb sandwich plate. 5
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blast load, sinusoidal load, triangular load and incremental load, will be studied through numerical simulations.
3.7. Effect of geometrical parameters under step loads The length and width of the plate is a = b = 1m , and other parameters are same as that of Fig. 5. Nonlinear responses of the auxetic honeycomb plate under step load as shown in Fig. 6. The step load of intensity is p0 = 106N / m2 and time duration is tp = 1.5ms . The time step is dt = 10−8ms and the total time is t = 10ms . Fig. 6(a) represents the effect of honeycomb cell inclined angle θ on lateral displacements under step load with the width is h = 0.01m and hc / h = 0.9, and the pulse length factor is r = 1. It can be seen clearly from Fig. 6(a) that the larger of the cell inclined angle, the greater of the lateral displacement caused by the load. Therefore, when the cell inclined angle θ = 30∘, the honeycomb sandwich plate has the best ability to absorb energy in Fig. 6(a). Fig. 6(b) shows the time history of displacement for various total thickness (0.01 m, 0.05 m, 0.10 m) of the honeycomb sandwich plate. Other parameters of the system are same as that of Fig. 6(a). It is observed that the total thickness of the plate plays a great effective role on the lateral displacements of the system. The depicted results show that the decrease of the total thickness of the plate extremely increases the response amplitudes. While the vibration frequencies and lateral displacements have an inverse relationship, therefore, we can increase the capacity of the system to withstand the impact by increasing the total thickness of the plate. In Fig. 6(c), the transient response of the honeycomb sandwich plate is described by changing the core-thickness ratio (0.8, 0.85, 0.9, 0.95) and other parameters are chosen as same as aforementioned. One can see clearly that the greater of the core thickness, the greater of the amplitude of the lateral displacements under the same loads. Because of the honeycomb core is soft, it is easier to deform than the homogeneous metal material under blast loads. Fig. 6(d) depicts the time history of lateral displacement under step (2) load with different equivalent Poisson's ratio ν12 as −0.9612 and 1.6019 when the cellular inclined angle θ = 30∘ and θ = −30∘, respectively, as show in Table 1. The values of other corresponding parameters are chosen as same as that of Fig. 5. Fig. 6(d) indicates that the auxetic honeycomb sandwich plate has larger lateral displacements when it received the same intensity of the load, but after the sudden load, the lateral displacements is less than the plate with the positive Poisson's ratio. This shows that the auxetic honeycomb sandwich panel has better properties on vibration absorption than the honeycomb plate with positive Poisson's ratio. The effect of the pulse length factor r on lateral displacements under step load is investigated in Fig. 6(e). The length factor r represents the impact time of the step pulse on the plate. It is evident that as r increases, the maximal lateral displacements and vibration frequencies
Fig. 4. The unit cell of concave hexagonal honeycomb core.
Table 1 (2) Poisson's ratios ν12 with different cell angels θ. θ
-π/3 -π/4 -π/6 π/6 π/4 π/3
l2/l1 1.5
2
2.5
3
7.4442 2.9712 1.2899 −0.6450 −1.0674 −1.9947
8.8549 3.6097 1.6019 −0.9612 −1.7240 −3.5035
10.2156 4.3695 1.9100 −1.2733 −2.3681 −4.9590
11.5290 4.8507 2.2141 −1.5815 −3.0002 −6.3638
Table 2 The equivalent moduli E1(2) (×109GPa) with different cell angels θ. θ
-π/3 -π/4 -π/6 π/6 π/4 π/3
l2/l1 1.5
2
2.5
3
1.1982 0.4142 0.2076 0.1038 0.1488 0.3211
1.4253 0.5032 0.2578 0.1547 0.2403 0.5639
1.6443 0.5904 0.3074 0.2050 0.3301 0.7982
1.8557 0.6762 0.3564 0.2546 0.4182 1.0243
thickness of the core layer are h = 0.01m ,hc = 0.009m respectively, the cellular parameters as shown in Fig. 4 are l1 = 0.01m , l2 = 2l1, t1 = t2 0.001m , the cellular inclined angle is θ = 30∘. The honeycomb plate is made of aluminum, and it's modulus is Es = 69GPa, the density is ρs = 2700kg / m3 , Poisson's ratio is νs = 0.3, and shear modulus is Gs = 27GPa . It can be obtained from Fig. 5 that the deflections of auxetic honeycomb plate are larger than that of the metal plate which indicate that the honeycomb has a good energy absorption capacity under impact forces. Next, the nonlinear transient dynamic responses of the honeycomb sandwich plate under the mentioned loads, including step load, air-
Fig. 5. Comparison of transient responses of auxetic honeycomb sandwich plate with metal plate under step loads. 6
International Journal of Impact Engineering 134 (2019) 103383
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Fig. 6. (a). The effect of cell inclined angle θ on lateral displacements under step load, (b). The effect of total thickness h on lateral displacements under step load, (c). The core-thickness ratio (hc/h) versus lateral displacements under step load, (d). The effect of Poisson's ratios on lateral displacements under step load, (e). The effect of the length factor r on lateral displacements under step load.
loading drop. Therefore, the loading form is the easiest to be absorbed by the honeycomb plate at α = 1in Fig. 7(a). Fig. 7(b) expresses the dynamic responses of the plate under airblast load for different duration time tp. The waveform decay parameter α = 1, other parameters are same as that of Fig. 7(a). It can be seen from the small diagram that the larger of the duration time tp resulting in the slower decay rate of the load. The lateral displacement of the plate decreases when tp decreases. When the duration time tp is equal to 2, the transverse displacement and the vibrated frequency of the plate are most appropriate. Fig. 7(c) depicts the time history of lateral displacements under airblast load with different equivalent Poisson's ratio. In Fig. 7(c), the lateral displacement of auxetic honeycomb sandwich plate is similar as that of the positive Poisson when subjected the same intensity of the load, but after the sudden load, the lateral displacements of the auxetic honeycomb plate is less. This shows that the auxetic honeycomb sandwich plate has better properties on vibration absorption than that of the honeycomb plate with positive Poisson's ratio.
also increase and vice versa. The pulse length factor r actually represents the loading time, so the longer the loading time is, the larger transverse displacement and frequency of the honeycomb plate are.
3.8. Effect of air-blast loads Nonlinear responses of the auxetic honeycomb plate under air-blast load are obtained in Fig. 7. The air-blast load of intensity is p0 = 106N / m2 , the time step is dt = 108ms , and the parameters of the plate are same as that of Fig. 5. Fig. 7(a) describes the effect of the waveform decay parameter α on the nonlinear transient response of the honeycomb plate, the duration time tp = 2 m s . It can be seen from the small diagram that the higher the decay parameter α means the lower the maximum blast pressure. The lateral displacements of the plate decrease when α increases and the vibration frequencies decreases while the α increases as expected. When the waveform decay parameter α = 1, the lateral displacements and the vibrated frequencies of the plate are most appropriate. When t < 2 m s the residual displacement is smaller under the pulse loading with a faster decaying rate. But after t = 2 m s , the residual displacement is independent with the speed of 7
International Journal of Impact Engineering 134 (2019) 103383
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Fig. 7. (a). The effect of different waveform parameter α on lateral displacements under air-blast load, (b). The effect of duration time tp on lateral displacements under air-blast load, (c). The effect of Poisson's ratios on lateral displacements under air-blast load.
3.9. Effect of sinusoidal load Fig. 8 shows the effect of the length factor r on the transient responses of the auxetic honeycomb sandwich plate under sinusoidal loads. The intensity of sinusoidal load is p0 = 106N / m2 , the time step is dt = 10−8ms . It can be seen from the small diagram that the length factor r determines the shape of the sinusoidal pulse. The maximum impact force of the three sinusoidal loads is same, and the larger of the length factor r, the longer of the time of the load on the honeycomb plate. As is illustrated from the results that when r = 1 the lateral displacements and vibration frequencies are reached maximize.
Fig. 9. The effect of shock pulse length factor r on lateral displacements under triangular load.
3.10. Effect of triangular loads Fig. 9 indicates nonlinear dynamics of the plate under triangular load with the length factor r change. The triangular load of intensity is p0 = 106N / m2 , the time step is dt = 10−8ms . As is illustrated from the diagram that the larger of the length factor r, the larger of the lateral displacements.
3.11. Effect of incremental loads Fig. 10 represents the nonlinear dynamics of the auxetic honeycomb plate under the incremental load given by Eq. (21). Fig. 10(a) is the effect of duration time tp on the nonlinear responses of the plate. It can be seen from the small diagram that the larger of the duration time tp means the slower the increase rate of the load in the same time. The lateral displacement of the plate increases when the tp decreases. When the duration time tp = 2 , the transverse displacement and the vibrated frequency of the plate are most appropriate in Fig. 10(a). Fig. 10(b) depicts the time history of lateral displacements under incremental load with different equivalent Poisson's ratios. It can be seen from that the lateral displacement of auxetic honeycomb sandwich plate is smaller than that of the positive one. In terms of shock absorption capacity, the honeycomb sandwich plate with negative Poisson's ratio is superior to that with positive Poisson's ratio under any loads. Fig. 11 compares the effects of several dynamic loads on transient response of the auxetic honeycomb sandwich plate. As shown in Fig. 11,
Fig. 8. The effect of the length factor r on lateral displacements under sinusoidal load. 8
International Journal of Impact Engineering 134 (2019) 103383
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Fig. 10. (a). The effect of time duration tp on lateral displacements under incremental load, (b). The Poisson's ratios versus lateral displacements under incremental load.
the lateral displacements of the step load are slightly higher than other loads, but after the step load, the lateral displacement and vibration frequency are much smaller and slower than other loads. Therefore, it can be concluded that the step load is the most easily absorbed by the honeycomb model among these loads, and the sin load is the hardest to be absorbed in these five kinds of loads. 4. Transient responses with other direction of the unit cells In order to check the effect of the unit cells on crashworthiness, the auxetic honeycomb cores arranged as in Fig. 1 is rotated 90∘ around the X-axis, as shown in Fig. 12. The absorption of vibrations is compared between the auxetic honeycombs and conventional honeycombs in Fig. 13. The geometric parameters are chosen as follows, the lengths a = b = 1m , and the thickness of the plate and thickness of the core layer are h = 0.021m ,hc = 0.02m respectively. Other parameters are same as that of Fig. 5. The step load of intensity is p0 = 2 × 106N / m2 and time duration is tp = 2ms . The time step is dt = 10−8ms and the total time is t = 20ms . It can be seen clearly from Fig. 13 that the auxetic honeycomb sandwich plate has larger lateral displacements when it received the same intensity of the load, but after the sudden load, the lateral displacement is less than the plate with the positive Poisson's ratio. The conclusion obtained here is consistent with that of in Fig. 6(d) and the lateral displacement of the auxetic honeycomb plate is much less than the honeycomb plate with positive Poisson's ratio after the impact load.
Fig. 12. Honeycomb sandwich plate of varying the direction of the unit cells.
5. Conclusions In this paper, the equations of motion for honeycomb sandwich plate subjected to several dynamic loads are obtained according to vonKarman deflection, theory Hamilton's principle and the Galerkin method and FORTRAN software. The results obtained by our methods
Fig. 13. The effect of Poisson's ratios on lateral displacements under step load.
Fig. 11. Comparison of different dynamic loads on central lateral displacements. 9
International Journal of Impact Engineering 134 (2019) 103383
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are compared with ANSYS software to verify the validation of the methods. The transient responses of the honeycomb sandwich plate with negative and positive Poisson's ratio subjected explosive loads are compared through same geometric parameters of the plate. The results show that the lateral displacements of the auxetic honeycomb sandwich plate is relatively lower compared to that of the honeycomb plate with positive Poisson's ratio. This shows that the auxetic honeycomb sandwich plate have better properties on vibration absorption than the honeycomb plate with positive Poisson's ratio. Numerical analysis about the transient response of the honeycomb sandwich plate under several dynamic loads is obtained, considering the influence of total thickness, thickness ratio, cell inclination angle, Poisson's ratio, shock pulse length factor and duration time, etc. The effect of geometrical parameters on the dynamics of the plate is obtained under the same blast load. The numerical results show that the larger of the cell inclined angle, the greater of the amplitudes of the plate. The increase of the total thickness results in the decrease of the amplitudes and larger of the core-thickness ratio, the larger of the amplitude of the plate is. As to different kinds of load, it can be seen from the numerical simulations that the amplitude of the lateral displacement of the step load is slightly higher than other loads for the same pulse length factor r, but after the sudden load, the lateral displacement and vibration frequency are much smaller and slower than other loads. From a large number of detailed numerical simulations, it can be concluded that the honeycomb sandwich structures are good at energy absorption and impact resistance. The auxetic honeycomb sandwich plates may be better or at least provide a preferable alternative for the honeycomb sandwich plates with positive Poisson's ratio.
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