Nonlinear analytical study of thin laminated composite plate reinforced by nanoparticles under high-velocity impact

Thin-Walled Structures 127 (2018) 446–458

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Nonlinear analytical study of thin laminated composite plate reinforced by nanoparticles under high-velocity impact R. Mohamadipoora, M.H. Polb, E. Zamania, a b

T

⁎

Department of Mechanical Engineering, Shahrekord University, Shahrekord, Iran Department of Mechanical Engineering, Tafresh University, Tafresh, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Nanocomposite Nonlinear strain High velocity impact Energy absorption Delamination

An analytical modeling is developed in a range of elastic waves propagation to investigate the nonlinear behavior of thin laminated composite plate. These plates reinforced by nanoparticles under high-velocity impact. Four crucial regions are considered: fracture region, nonlinear deformation region, local movement region and delamination region. In addition, based on a new deformation function, new equations are developed for assessing the normal and shear strains and computing the energy absorbed in different regions. Results of the present study show a good agreement with the available experimental data of other researchers at velocities beyond the ballistic limit velocity (i.e. V > VB.L≈ 120 m/s).

1. Introduction

epoxy. They observed that V50 can be increased by adding nanoparticles. Pol and Liaghat [10,11] conducted an experimental study on the ballistic behavior of glass/epoxy hybrid nanocomposites. Their results show that the energy absorption capability and mechanical properties of the composite are significantly enhanced by adding nanoparticles. Various studies have been carried out to model the process of penetration of projectiles into targets made of thin composite and nanocomposite plates. These studies have introduced various energy absorption mechanisms, including tensile fiber failure, secondary fiber elastic deformation, kinetic energy of cones formed on the back face of the target, delamination, matrix cracking, shear plugging and friction between projectile and target during the penetration process [12–18]. Among analysis models, Mines et al. [12] and Morye et al. [13] primarily presented those simple models that were based on empirical results. These models, in order to assess ballistic behavior, needed ballistic testing. It was a defect in predicting ballistic behavior and the design of composite structures under ballistic impact. Therefore, Naik et al. [14] presented a completely analytical prediction of the ballistic behavior of woven composite plates by dividing of the ballistic impact time to several intervals. Balaganesan et al. [15] computed the ballistic limit velocity in nanocomposites reinforced with nano-clay particles by Naik et al. [14] model relations. It should be noted that fiber width is substituted with projectile diameter in the strain calculation of Naik et al. model [14]. In addition, Sanchez et al. [16,17] developed an analytical and non-dimensional model based on Morye et al. [13] and Naik et al. [14] model. They used this developed model to study the

In recent years, the nanocomposites have been used widely in commercial applications, e.g. space science, and military and aerospace technologies. This is because of the enhanced mechanical properties such as specific strength and stiffness, fracture toughness, impact energy absorption, vibration damping. This improved performance is generally due to the distinctive strengthening of matrix phase and the modification of interfacial properties; e.g. chemical bonds and dispersing of nanoparticles in the phase matrix [1]. One of the critical requirements of the nanocomposite structures during their service life is their protection against high velocity impacts. Therefore, high velocity ballistic impact and its effect on the armour materials have attracted a lot of attention in recent years [2–6]. Among certain studies [7–11], Experimental results show that some nanoparticles can improve the mechanical properties and ballistic resistance of the nanocomposites. Mohagheghian et al. [7] studied the nanocomposite plates with nanoclay, which have been subjected to quasi-static penetration, as well as dynamic impact testing. They demonstrated decreased energy absorption capability of nanocomposite plates in quasi-static testing. Avila et al. [8] studied the effect of nano-clay and nano-graphite on highvelocity impact properties in glass/epoxy. They demonstrated that the addition of nano-clay and nano-graphite plates to laminated glass/ epoxy composites increase their resistance to high-velocity impacts. They also exerted a remarkable effect on the fracture mechanism. Naik et al. [9] investigated the effect of adding of nanoparticle on the ballistic impact behavior of unidirectional E-glass/epoxy laminates and

⁎

Correspondence to: P.O.B. 115, Shahrekord, Iran. E-mail address: [email protected] (E. Zamani).

https://doi.org/10.1016/j.tws.2018.02.010 Received 25 August 2017; Received in revised form 20 January 2018; Accepted 8 February 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 127 (2018) 446–458

R. Mohamadipoor et al.

Nomenclature A Aij b C. C Cl Ct C1,2,3 d D Dij exx , e yy exy ef efxi , efyi emxi , emyi

emxyi

ė0 ė E0 Eij ET ELi Efxi, fyi Efi

Eeli

Edi

Ebi Emi

EKE0 EKEi ELMi

Cross-sectional area of fracture region Extensional stiffness Stress wave transmission factor Corrective coefficient of delamination region Longitudinal wave velocity Transverse wave velocity strain-rate constants Projectile diameter Average flexural stiffness Flexural stiffness Normal strain component Shear strain component Fracture strain of the target Strain variations in the fracture region Maximum normal strain at the impact point along x,y directions within ith time interval Maximum shear strain at the impact point within ith time interval Quasi static strain rate Dynamic strain rate Quasi-static modulus vector Young's modulus Effective modulus vector Total strain energy Total energy absorbed by different mechanisms at the end of ith time interval Energy absorbed by fracture region along x,y directions during ith time interval Total Energy absorbed by fracture region during ith time interval Energy absorbed by nonlinear deformation region during ith time interval Energy absorbed by delamination region during ith time interval Bending energy during ith time interval Membrane energy during ith time interval

Fi Fd GII Gij h ILSSG / E L MCi mp Mx , y . xy N Nl Nfi Nx , y . xy Qij r, θ rd rli rti SR vi (r ) Vi Vs W ∆Zi Wi (r ) x, y z ∆Zi ∆t ρ σx , y τxy

Initial kinetic energy of projectile Kinetic energy of the projectile during ith time interval Kinetic energy of the local moving part within ith time interval Contact force during ith time interval Delamination critical load Strain energy release rate in mode II Shear modulus Plate thickness Interlaminar shear strength glass/epoxy Length of plate Mass of the local movement during ith time interval Mass of the projectile The moment resultant Number of time intervals Number of layers Number of failed layers during ith time interval The force resultant Stiffness matrix In-plane polar coordinate Delamination radius Longitudinal wave radius Transverse wave radius Effective strengths vector The field velocity at the end of ith time interval The projectile velocity at the end of ith time interval Initial velocity of projectile The work of external load Central deflection of plate at the end of ith time interval The local deformation function during ith time interval In-plane rectangular coordinates Out of plane rectangular coordinates Central deflection of plate at the end of ith time interval Time interval Density of the plate Normal stress component along x,y directions Shear stress component

2. Mathematical model

ballistic behavior of multi-layered composite plates made of woven glass fibers. They investigated the effect of two dimensionless ratios (geometry and density ratios) on ballistic limit velocity, contact time, and the energy absorption mechanisms. Among the limitations of the previous models [14–17], it is possible to name the uncertainty of strain variations along the thickness and the assumption that all points of the conical region move with one velocity. Then, Pol et al. [18] yielded a more accurate behavior for ballistic impact in 2D woven composite and nanocomposite targets. They determined the relationship for strain variations between different layers of nanocomposite target and investigated the possibility of layer-by-layer fracture during the process of penetration. As it can be understood from the review of the literature, most analytical models have been considered the ballistic behavior of nanocomposite materials in a one-dimensional form. Due to the limited measurement and no sufficient information, simplistic assumptions are used in this theories. Moreover, in order to calculate the initiation and boundary of the delamination region, a damage threshold, less than the strain of the failure, has been used. In this study, a function is considered that is similar to circular plates under concentrated load by adding the effects of the transverse wave's propagation. Then, with the help of the von Kármán relations in different directions, a simultaneous study of the variations of nonlinear strains is conducted in different regions and thickness direction. Moreover, the variation of velocity, the absorbed energy in different regions, kinetic energy, and delamination are estimated.

2.1. Description of the model assumptions A number of analytical models have been developed to investigate the behavior of composite targets upon high velocity impacts. However, most of these analytical models are one-dimensional. In the present study, a two-dimensional time-dependent analytical model is developed to predict nonlinear deformations and strains, energy absorption of various fracture modes, amount of delamination damage region of a nanocomposite laminate and residual velocity of the projectile. The following presuppositions have been taken into account in developing this model: – The stress-strain relationship of the nanocomposite laminates is linear during the impact [18,19]. This assumption is based on empirical observations of stress-strain curve changes during tensile tests of the samples made with different percentages. – The projectile is rigid and remains undeformed during the ballistic impact [13–17]. This assumption is also taken from the previous observations, which showed that the projectile is not distorted after the high velocity impact [14,18]. – In-plane deformations are little and negligible as compared to the large transverse deformations of the nanocomposite laminate [19,20]. 447

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– The energy absorbed by the fracture region in each layer remains constant after fracture [18]. As time progresses, the amount of strain increases in the first layer and the velocity of the projectile decreases. Hence, along with strain variations up to the failure strain in this layer, the energy absorbed by the fracture region is calculated. After a while the first layer is broken suddenly because the strain of the first layer has reached the failure strain. Therefore, the first layer is removed and the projectile reaches the second layer. The energy absorbed by the first layer stays constant and does not change. In the second layer, the total energy absorbed by the fracture region is obtained from the total energy absorbed by the fracture region in the first and second layers. In the third layer, the total energy absorbed by the fracture region is obtained from the total energy absorbed by the fracture region in the third layer and the first and second layers. This process will continue, and the energy absorbed by the fracture region will be fixed at each eliminated layer. When all layers are broken, the laminated nanocomposite plate is perforated. The total energy absorbed by the fracture region of the laminated nanocomposite plate is obtained from the sum of energies absorbed by the fracture regions in different layers. – The strain of the nonlinear deformation region is independent of that of the fracture region [14,18]. In previous studies, it has been shown that the strain in the fracture region is greater than the strain in the nonlinear deformation region, because the fracture region is located directly below the projectile. Therefore, this region undergoes extreme stress at high strain rates. However, the strains change nonlinearly between the fracture region and the boundary that the transverse wave traveled. – The energy absorbed by the delamination mechanism remain constant in each fractured layer. As time progresses, the amount of contact force changes in the first layer and the projectile's velocity decreases. By comparing the delamination critical load and contact force, the initiation, expansion and the absorbed energy of the delamination are estimated in the remaining layers. When the layer is fractured, the delamination area of layer is remained constant. – The absorbed energy through the matrix cracking mechanism is negligible relative to other absorbed energies [13,14,18]. If the thickness of laminated nanocomposite plate is few, the most part of the thickness will fail under compression-shear mode. The remaining small thickness will fail under tension-shear mode. Hence, matrix cracking will be insignificant in the thin laminated nanocomposite target [21]. – The heat generated by the frictional resistance between the projectile and target can be ignored [13–15]. The frictional resistance is proportional to the normal force, and friction coefficient. The region of contact between the two materials is very limited and insignificant, especially for thinner samples [22]. Hence, due to the low thickness of the plate in this study, it is neglected.

1 ⎛ dσ ⎞ ρ ⎝ dε ⎠

(1)

Ct = Cl ( ε (1+ε ) − ε )

(2)

Cl =

Where ε and ρ are the strain created in the laminated nanocomposite plate and density, respectively. Hence, the distances traveled by the transverse and longitudinal waves within the ith time interval are given as follows [18]:

rti =

rli =

d + 2

d + 2

N =i

∑ ctN ∆t

(3)

N =1

N =i

∑ clN ∆t

(4)

N =1

Where d, ∆t and N represent the projectile diameter, the time interval and the number of time interval in during the ballistic impact event, respectively.

2.2. Determining nonlinear deflection, strains and stresses First, it is assumed that in-plane deformations are extremely diminutive and hence negligible in comparison to transverse deformations [19,20,25]. Afterwards, the deformation function of an isotropic circular plate subjected to concentrated load is considered. The following transverse deformation function is developed for ballistic impacts on thin laminated nanocomposite plates by introducing some slight variation. This function satisfies the geometric boundary conditions including a zero deformation and slope in the boundary of longitudinal wave. In addition to being dependent on r coordinates, the value of this function varies with the propagation of the transverse wave: 2

2

r r ⎡ r ⎤ wi (r ) = ∆Zi ⎢2 ⎛ ⎞ ln ⎛ ⎞ + ⎜⎛1−⎛ ⎞ ⎟⎞ ⎥ rti ⎠ ⎝ rti ⎠ ⎝ ⎝ rti ⎠ ⎠ ⎝ ⎣ ⎦ ⎜

⎟

⎜

⎟

⎜

⎟

(5)

Where ∆Zi represents plate displacement at the end of the ith time interval, r stands for polar coordinates, and r 2 = x 2 + y 2 . In woven laminated nanocomposites, as the longitudinal wave propagates in the yarns direction, the stress wave attenuation would take place. In result, the stress is reduced from the point of impact up to the point where the longitudinal stress wave has reached. Therefore, similarly, strain varies between maximum value at the impact point and zero value at the point which the longitudinal stress wave has just reached. According to von Kármán, the large deformation strains in polar coordinate are as follows [26]:

Furthermore, in this study, the following four different regions are taken into consideration after the ballistic impact on the nanocomposite plate (Fig. 1): 1. The fracture region, which is located right under the projectile. 2. The distance between the fracture region and the transverse wave traveled. This region is named ‘the nonlinear deformation region’. 3. The distance between the transverse wave traveled and the end of the plate. This region is named ‘the undeformed region’. 4. The delamination region where delamination damage occurs. Upon a ballistic impact, the longitudinal and transverse stress waves propagate in the radial direction of nanocomposite target. The velocities of propagation of longitudinal and transverse waves are as follows [23,24]: Fig. 1. Plate regions during the ballistic impact process.

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sin(θ) ∂ ⎛ ∂w ⎞ ∂w ∂w (r ) ∂2w ∂ ∂w = cos(θ) , = cos(θ) ⎛ ⎞ − , exx ∂x ∂r ∂x 2 ∂r ⎝ ∂x ⎠ r ∂θ ⎝ ∂x ⎠ 1 ∂w 2 ∂ 2w = ⎛ ⎞ −z 2 2 ⎝ ∂x ⎠ ∂x

⎡ Nx ⎤ ⎡ A11 ⎢ Ny ⎥ ⎢ A12 ⎢N ⎥ ⎢ ⎢ xy ⎥ = ⎢ A16 ⎢ Mx ⎥ ⎢ B11 ⎢ M ⎥ ⎢B ⎢ y ⎥ ⎢ 12 B ⎢ ⎦ ⎣ 61 ⎣ Mxy ⎥

(6)

cos (θ) ∂ ⎛ ∂w ⎞ ∂w ∂w (r ) ∂2w ∂ ∂w = sin (θ) = sin (θ) ⎛ ⎞ + , , e yy ∂y ∂r ∂y 2 ∂r ⎝ ∂y ⎠ r ∂θ ⎝ ∂y ⎠ ⎜

⎟

⎜

⎟

A12 A22 A26 B12 B22 B62

A16 A26 A66 B16 B26 B66

B11 B12 B16 D11 D12 D16

(0) ⎡ exx ⎤ B16 ⎤ ⎢ (0) ⎥ e B26 ⎥ ⎢ yy ⎥ ⎢ (0) ⎥ B66 ⎥ ⎢ 2exy ⎥ ⎥ (1) ⎥ D16 ⎥ ⎢ exx D26 ⎥ ⎢ (1) ⎥ ⎢ e yy ⎥ D66 ⎥ ⎦ ⎢ (1) ⎥ 2e ⎢ ⎣ xy ⎥ ⎦

(16)

hk2 − hk2 hk3 − hk3 ⎤ , ⎥ 2 3 ⎦

(17)

B12 B22 B16 D12 D22 D26

2

∂ 2w 1 ∂w = ⎛ ⎞ −z 2 ∂y 2 ⎝ ∂y ⎠ ⎜

n

⎟

(7)

[Aij , Bij , Dij] =

Where A, B, and D represent extensional stiffness, bending-extensioncoupling stiffness, and bending stiffness, respectively. For the symmetric laminated nanocomposite plates, Bij is zero and for the orthotropic laminated nanocomposite plates, the terms A16, A26, D16 and D26 are Zero. Therefore,

(8) By substituting the deflection function from Eq. (5) into Eqs. (6)–(8) (nonlinear strain equations), the following equations are developed for strain variation in terms of radius and thickness (Fig. 2).

8cos (θ)2

∆Zi2 r 2ln

exx =

() r rti

4z ∆Zi −

rti4 ∆Zi2 r 2ln

8sin (θ)2 e yy =

r 2 rti

()

rti4

(

cos (θ)2

+ ln

( )) r rti

rti2

4z ∆Zi +

(

cos (θ)2

(9)

− ln

( )−1) r rti

rti2

⎣

k=1

cos(θ) ∂ ⎛ ∂w ⎞ ∂ 2w ∂ ∂w 1 ∂w ∂w ∂ 2w = sin(θ) ⎛ ⎞ + , exy = −z ∂x ∂y ∂r ⎝ ∂x ⎠ r ∂θ ⎝ ∂x ⎠ 2 ∂x ∂y ∂x ∂y

2

∑ Qij(k) ⎡⎢hk − hk −1,

[Q ]k = [T ]−1 [Q]k [T ]−T

(18)

⎡Q11 Q12 0 ⎤ [Q]k = ⎢Q12 Q22 0 ⎥ ⎢ 0 0 Q66 ⎥ ⎣ ⎦

(19)

Q11 =

(10)

E11 , 1−ν12 ν21

Q12 =

ν12 E22 , 1−ν12 ν21

Q22 =

E22 , 1−ν12 ν21

Q33 = G12 (20)

8cos (θ) sin (θ) ∆Zi2 r 2ln exy =

r 2 rti

()

4z ∆Zicos (θ) sin (θ) rti2

−

rti4

Where, E11, E22 are respectively the modulus of elasticity along and perpendicular to the fiber direction, ν12 is the Poisson's ratio and G12 is the shear modulus.

(11)

Maximum strains at the impact point are determined by setting r = d/2, θ = 0, 90, 45 in Eqs. (6 and 7). Hence, the following simplified equations are computed in terms of thickness:

2 ∆Zi2 d 2ln emxi =

d 2 2rti

( )

−

rti4 4z ∆Ziln

=−

rti2

4z ∆Ziln emxi = − emxyi = 0 r =

rti2 d , 2

d 2 2rti

( ),

d 2rti

2

d ,θ=0 2 2

emyi =

The strain energy in a laminated nanocomposite plate is computed by accounting von Kármán nonlinear strains and ignoring in-plane deformation [25]. In the plate nonlinear deformation region, the strain energy is expressed in simple form through the following equation [26]:

emyi

rti2

emxyi = 0 r =

( ), d 2rti

( ( )) ,

4z ∆Zi 1+ln

d ⎞ 2 ∆ Zi2 d2ln ⎛ ⎝ 2rti ⎠ rti4

UT = Ub + Um =

(12)

⎜

emxi =

d 2 2rti

( )

rti4

d 2 2rti

( )

−

−

(

1 2

+ ln rti2

4z ∆Zi

rti4 ∆Zi2 d 2ln

=

−

rti4 ∆Zi2 d 2ln

=

( )

4z ∆Zi

(

1 2

+ ln rti2

( )) ,

emyi

( )) ,

emxyi

d 2rti

d 2rti

2z ∆Zi d π r= ,θ= 2 4 rti2

⎜

2

2

2

⎩

⎟

⎝

⎠

4

2

⎜

⎟

⎝

⎠

2

2

⎟

⎜

4

+

θ = 90

d 2 2rti

2

2

∬ ⎧⎨D11 ⎛ ∂∂xw2 ⎞ +2D12 ∂∂xw2 ∂∂yw2 +4D66 ⎛ ∂∂x ∂wy ⎞

1 1 ∂w ∂w ∂w + D22 ⎛ 2 ⎞ + A11 ⎛ ⎞ + A12 ⎛ ⎞ ⎛ ⎞ 4 2 ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠

⎟

(13)

∆Zi2 d 2ln

1 2

2 ∂ 2w

d ⎞⎞ 4z ∆ Ziln ⎛1 + ln ⎛ ⎝ 2rti ⎠ ⎠ ⎝ , rti2 ⎜

−

2.3. Energy absorption in the nonlinear deformation region

2

1 ∂w ∂w 2 ∂w A22 ⎛ ⎞ +A66 ⎛ ⎞ ⎛ ⎞ ⎫ dxdy 4 ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎬ ⎝ ∂y ⎠ ⎭ ⎜

⎟

⎟

⎜

⎟

(21)

For the thin orthotropic symmetric composite plate, the bending strain energy absorbed in the transverse wave propagation region rti , which leads to the deformation of the elastic plate, is given in the

(14)

By assuming in plane stress state, σzz = σ3 = 0 . Therefore, the stressstrain relation in the kth layer is as follows [26,27]: k

σ k ⎡Q11 Q12 Q16 ⎤ ⎡ exx ⎤ ⎡ σxx ⎤ e ⎢ yy ⎥ = ⎢Q21 Q22 Q26 ⎥ ⎢ yy ⎥ ⎢ ⎥ ⎢ 2exy ⎥ σ ⎥ ⎢ xy ⎣ ⎦ ⎦ ⎣Q61 Q62 Q66 ⎦ ⎣

(15)

The force and moment resultant are defined as follows: (it is worth noting that shear forces do not appear in the classical thin plate theory) [26]:

Fig. 2. (a) Schematic of the plate prior to ballistic impact, (b) Schematic of local deformation of the plate after ballistic impact.

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R. Mohamadipoor et al.

following exponential equation [14]:

following equation:

Ubi = 4×

1 2

π 2

rti

∂ 2w ∂ 2w ∂ 2w + D22 ⎛ 2 ⎞ +2D12 2 2 ∂x ∂y ⎠ ⎝ ∂y ⎠

⎜

0

⎜

⎟

⎝

⎩

d 2

efxi, fyi = emxi, myi b x , y / d

2

2

2

∫ ∫ ⎧⎨D11 ⎛ ∂∂xw2 ⎞

⎞ ⎫ rdrdθ +4D66 ⎛ ⎝ ∂x ∂y ⎠ ⎬ ⎭ ⎜

Where efxi, fyi represents strain in the fracture region along x and y directions, emxi, myi stands for maximum strain (i.e. strain at the impact point, along x and y directions), x and y are the distances from the impact point, b is the stress wave propagation factor, and d is the projectile diameter. It is assumed that width of fracture region is equal with the projectile diameter. The stress wave propagation is function of the geometry of laminated nanocomposites, the mechanical and physical properties of the materials. This parameter is computed separately for each nanocomposite plate. The strain values of different layers of the laminated nanocomposite target subjected to ballistic impact is computed by Eqs. (12–14), and then the fracture region is investigated by comparing these values with the criterion of maximum strain around the impact point in each layer during each time interval [28]:

2

∂ 2w

⎟

(22)

By integrating Eq. (22) through Eq. (5), the following simplified equation is developed for the bending energy versus the radius of the propagated transverse wave rti at each point: 2

π ∆Zi2 ⎧ ⎛ 2 3 2 d d −2d 2ln ⎛ ⎞ ⎟⎞ D11 ⎜3rti − 4 2 rti4 ⎨ ⎝ ⎝ rti ⎠ ⎠ ⎩

Ubi =

⎜

⎟

2

2

d d 1 3 + ⎜⎛2rti2 − d 2−4d 2ln ⎛ ⎞ ⎟⎞ D12 + ⎜⎛3rti2 − d 2−2d 2ln ⎛ ⎞ ⎟⎞ D22 r 2 2 4 2 ti ⎝ ⎠⎠ ⎝ rti ⎠ ⎠ ⎝ ⎝ ⎜

⎟

⎜

⎫ + (4rti2 − d 2) D66 = K1i ∆Zi2 ⎬ ⎭

⎟

< 0 failuredoesnotoccur emxi, myi − ef ⎧ ≥ ⎨ ⎩ 0 failureoccurs

(23)

Considering Eq. (21) for the thin orthotropic symmetric composite plate, the membrane energy absorbed in the regions of propagation of the transverse wave rti is given the following equation:

1 Umi = 4× 8

π 2

rti

∫ ∫ ⎧⎨A11 ⎛⎝ ∂∂wx ⎞⎠ 0

⎩

d 2

4

4

⎜

⎟

Efx , fy =

K2i = KA1i + KA2i + KA3i + KA4i

⎜

3

⎟

⎜

Efxi, fyi = A

⎟

2

27 6 ⎛ d ⎞ 9 6 ⎛ d ⎞⎫ + d ln d ln r 32 2 32 ti ⎝ ⎠ ⎝ 2rti ⎠ ⎬ ⎭ ⎜

⎟

⎜

(26) 3

2πA12 ⎧ 6 2 6 54 6 ⎛ d ⎞ 36 6 ⎛ d ⎞ + d − d ln d ln 2rti − 64 32 32 81rti8 ⎨ ⎝ 2rti ⎠ ⎝ 2rti ⎠ ⎩ ⎜

⎟

⎜

⎜

⎟

⎜

Nf

(27)

+ 3

⎟

⎜

⎜

⎟

e

r

e

(

) dx

) dx

(35)

2.5. Energy absorption due to the local movement of the plate

3

⎜

⎟

During the ballistic impact event, local movement occurs on the rear surface of the target that absorbs some of the projectile energy. In this study, the local velocity field is considered similar to Eq. (5). This variable function is expressed in terms of the transverse wave propagation and radius coordinates by the following equation at ith time interval:

2

36 6 ⎛ d ⎞ 12 6 ⎛ d ⎞ ⎫ + d ln d ln 32 32 ⎝ 2rti ⎠ ⎝ 2rti ⎠ ⎬ ⎭ ⎜

(

(28) 4

⎟

r

∑ 4×A ∫0 li ∫0 fxi σ (ε ) dε

⎟

⎜

⎜

(34)

Where Nf is the number of fractured layers and Efi is total Energy absorbed by fracture region during ith time interval.

2πA66 ⎧ 6 4 6 108 6 ⎛ d ⎞ 72 6 ⎛ d ⎞ + d − d ln d ln 4rti − r 64 32 2 32 81rti8 ⎨ ti ⎝ ⎠ ⎝ 2rti ⎠ ⎩ −

)

σ (ε ) dε dx

⎟

2

KA4i =

(33)

∑ 4×A ∫0 li ∫0 fyi σ (ε ) dε i=1

27 6 ⎛ d ⎞ 9 6 ⎛ d ⎞⎫ − + d ln d ln 32 32 ⎝ 2rti ⎠ ⎝ 2rti ⎠ ⎬ ⎭ ⎟

efxi, fyi

Nf

2πA22 ⎧ 6 3 6 81 6 ⎛ d ⎞ 54 6 ⎛ d ⎞ = + d − d ln d ln 3rti − r 64 32 2 32 81rti8 ⎨ ti ⎝ ⎠ ⎝ 2rti ⎠ ⎩ ⎜

σ (e ) dε

i=1

4

KA3i

r li

∫0 (∫0

Efi = Efxi + Efyi =

18 6 ⎛ d ⎞ 6 6 ⎛ d ⎞⎫ − + d ln d ln r 32 2 32 ⎝ ti ⎠ ⎝ 2rti ⎠ ⎬ ⎭ ⎟

efx , fy

⎟

2

⎜

∫0

Since the fracture region in question is ¼ of the plate, the foregoing expression is multiplied by 4 in each time interval. The total energy absorbed in the fracture region by laminates is the sum of the fracture energy of each layer along x and y-direction:

⎟

4

KA2i =

Adx

h

2πA11 ⎧ 6 3 6 81 6 ⎛ d ⎞ 54 6 ⎛ d ⎞ + d − d ln d ln 3rti − r 64 32 2 32 81rti8 ⎨ ⎝ ti ⎠ ⎝ 2rti ⎠ ⎩ −

x

region along x and y directions, A = N d (where h, N, d are total plate thickness, the number of nanocomposite plate layers and the projectile diameter, respectively). Therefore, by considering the longitudinal wave expansion, the energy absorbed by the fracture of one layer equals the following equation:

(25)

4

KA1i =

∫0

Where Efx , fy represents the fracture energy of each layer in the fracture (24)

By integrating Eq. (24) through Eq. (5), the following simplified variable equation is developed:

,

(32)

Where ef represents fracture strain of the target. The fracture energy of each layer is calculated by strain energy. Therefore,

∂w ∂w 2 ∂w 2 + A22 ⎛ ⎞ +2A12 ⎛ ⎞ ⎛ ⎞ ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠

∂w 2 ∂w 2 +4A66 ⎛ ⎞ ⎛ ⎞ ⎫ rdrdθ ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎬ ⎭

Umi = K2i ∆Zi4

(31)

⎟

⎟

(29)

Finally, the energy absorbed by the nonlinear deformation region is obtained in the ith time interval as follows:

Eeli = Ubi + Umi

(30)

2

2

r r ⎡ r ⎤ vi (r ) = Vi ⎢2 ⎛ ⎞ ln ⎛ ⎞ + ⎜⎛1−⎛ ⎞ ⎟⎞ ⎥ r r r ⎣ ⎝ ti ⎠ ⎝ ti ⎠ ⎝ ⎝ ti ⎠ ⎠ ⎦ ⎜

2.4. Energy absorption in the fracture region

⎟

⎜

⎟

⎜

⎟

(36)

Where Vi represents the velocity of the plate center at the end of the ith time interval. Hence, the kinetic energy of the local movement of the plate can be expressed by the following integral in the ith time interval:

The strain in each layer is maximal at the impact point and decreases with getting away from center. This can be represented by the 450

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ELMi =

1 2

2π

∫0 ∫0

rti

⎡ d = π ⎢ 2 ρhVi 2rdr +4 0 ⎣ = MCi Vi2

∫

2 ⎡ Fdel ⎛ 1 − 1 ⎞−2G ⎤ πr δr = 0 II ⎢ π ⎝ 16D ⎥ d d 32D ⎠ ⎣ ⎦

ρhvi 2 (r ) rdrdθ

∫d 2

rti

2

2

r r ⎡ r ρhVi ⎢2 ⎛ ⎞ ln ⎛ ⎞ + ⎜⎛1−⎛ ⎞ rti ⎠ ⎝ rti ⎠ ⎝ ⎝ rti ⎠ ⎝ ⎣ ⎜

⎟

⎜

⎟

⎜

⎟

⎞ ⎤ rdr ⎤ ⎟⎥ ⎥ ⎠⎦ ⎦

1

Fd = 8π (GII D) 2

⎧ ⎪ ⎪1 ⎨8 ⎪ ⎪ ⎩

rd = 2

⎡ 72 d6ln ⎛ d ⎞ − 96 d6ln ⎛ d ⎞ + 108 d4ln ⎛ d ⎞ r 2 + 34 d6 − 81 d4r 2 + 54 d2r 4 ⎤ ⎫ ti ti ti ⎥ ⎪ ⎢ 64 4 16 64 64 16 ⎝ 2rti ⎠ ⎝ 2rti ⎠ ⎝ 2rti ⎠ ⎦⎪ + ⎣ ⎬ 27rti4 ⎪ ⎪ ⎭

In this section, it is assumed that the projectile and the local moving part on the back face of the target are traveling at an identical velocity equal to the velocity at the end of the ith time interval. 2.6. Energy absorption due to matrix cracking and delamination During the impact occurrence, prior to the fracture of layers, matrix cracking and delamination take place around the impact point that absorb some of the primary energy of the projectile. The energy absorbed through these mechanisms is considerably less than the one absorbed by other energy absorption mechanisms, such as nonlinear deformation, tensile failure, and kinetic energy of movement of the nanocomposite plate. It is such that Morye et al. and Pol et al. [13,18] ignored them. Debonding between adjacent layers depend on the stresses applied on that interface: the normal stress σz and two interlaminar shear stress τyz and τxz [28,29]. By using the classical theory of thin composite plates, the normal stress in thichness direction is zero. Therefore, the critical load is used to investigate delamination phenomenon. The delamination is estimated by comparing the contact force and the critical load of delamination. Hence, first the delamination critical load is computed. Consider a circular composite plate at radius rd under concentrated force Fd (Fig. 3). In this case, transversal displacement is given by the following equation:

Ed = (C . C )(πrd2) GII

EKE =

ELi = Efi + Eeli + Edi

(40)

a

F2 r 2 Fd w (r ) rdrdθ = d d 2 64πD πa

(41)

The delamination energy Ed is obtained by multiplying the mode II critical energy release rate (GII ) by the circular surface, as follows:

Ed = πrd2 GII

(47)

(48)

Where Efi, Eeli, and Edi are the energy absorbed in the fracture region, the energy absorbed through the nonlinear deformation, and the delamination energy at the end of the ith time interval. Considering the results of previous research, the matrix cracking energy has been ignored in view of its considerably lower share in comparison to that of other energy absorption mechanisms [14,18]. Once the initial kinetic energy of the projectile and the energy absorption during each time interval is determined, projectile velocity can be determined for the next time interval. After determining projectile velocity, various

The work of the external concentrated load is obtained by the following integral: 2π

1 mp Vs2 2

Where mp and Vs are the projectile mass and the initial velocity, respectively. To analyzes the ballistic impact event, the contact time can be divided into equal time intervals at length δti . The total energy absorbed by the target at the end of the ith time interval is given by the following equation:

Where D represents mean bending stiffness. By substituting Eq. (39) into the strain energy equation and integration in polar coordinates, the strain energy is given by the following equation:

∫0 ∫0

(46)

The total of ballistic impact time is divided into shorter subdivisions. Afterwards, by considering the multi-stage penetration process, the amount of kinetic and absorbed energies in each subdivision are determined. The residual velocity at the end of ith time interval is computed by considering the energy balance of the projectile. At the onset, in the first impact time interval, the whole energy is in the form of the kinetic energy of the projectile, which is given by the following equation:

(39)

W=

(45)

2.7. Projectile energy balance

Fd ⎡ 2 r 1 w (r ) = 2r ln + (rd2 − r 2) ⎤ D = (3D11+2D12 +4D33 +3D22) ⎥ 16πD ⎢ rd 8 ⎣ ⎦

rd2 Fd2 16πD

Fd 2πh (ILSS )

Delamination propagation in laminated nanocomposite plates is not identical in all directions [6,12,14,18]. In the front side of the plate, a small almost-circular delamination region exists. Besides, the shape of the delamination region in the rear side resembles a butterfly; therefore, this delamination region is not fully circular in the lower layers. In addition, the delamination radius in the lower layers is greater than the one in the upper layers. This is because of the delamination radius remains constant in each fractured layer. Hence, the correction coefficient (C.C) of the delamination region is taken into account in relation to the circular fracture surface. Finally, the delamination energy is given by the following equation:

(38)

2ET =

(44)

If the contact force at each moment is greater than the delamination critical load, the delamination damage occurs. The delamination radius rd can be determined by dividing delamination critical load by interlaminar shear strength (ILSS)G/E and the thickness of the nanocomposite plate as follows:

(37)

MCi

= πρhd2

(43)

(42)

The parameter which changes virtually is rd. The critical load of delamination is obtained by setting the virtual variation of the external load work equal to the sum of the virtual variation of strain and delamination energy. therefore,

Fig. 3. Circular nanocomposite plate subjected to concentrated load.

451

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parameters such as projectile displacement, strain, contact force, and energy absorbed through different mechanisms for the various intervals are determined. The energy balance of the projectile at the end of the ith time interval is given by the following equation:

Table 1 Specifications of the projectile and the target [18]. Projectile characteristics

(49)

EKE 0 = EKEi + ELi + ELMi

d mp Lp h N L b ε0̇ C1 C2 C3 GII Δt

Target characteristics

After substitution of the equations of absorbed energy into Eqs. (48, 49) and simplification, the following equation is obtained for the velocity at the end of the ith time interval:

mp Vs2−2ELi

Vi 2 =

(2MCi + mp)

(50)

The contact force or the force resistant to the projectile motion can be computed by the following equation:

Fi =

mp (Vi −1 − Vi )

Table 2 Mechanical properties of the nanocomposites (0%, 3%, 5%) in quasi-static loading [18].

(51)

∆t

The distance traveled by the projectile (ΔZi) within the ith time interval is given by the following equation: n = i −1

∆Zi =

∑ n=1

(Vn −2 + Vn −1) ∆t 2

(52)

3. Formulation of dependency of mechanical properties on strain rate

(53)

ė {Er } = {E0} ⎛⎜1+C2,3 Ln ⎛ ⎞ ⎞⎟ ⎝ e0̇ ⎠ ⎠ ⎝

(54)

⎜

⎟

⎜

⎟

The samples are tested at strain ė0 , and the strength matrix {S0} as well as the elasticity modulus matrix {E0} is determined. Afterwards, the strength matrix {Sr } and the elasticity modulus matrix {Er } are determined at each strain rate ė in question. Hence, C1 is the strain rate coefficient for the strength matrix and C2 and C3 are strain rate coefficients for the longitudinal and shear elasticity moduli. The strain rate ė by Eqs. (53, 54) in each time interval is given as follows:

ė =

∆emxi e − emxi −1 = mxi ∆t ∆t

Wt (%)

ρ (kg/m3)

E11 , E22 (GPa)

ef (%)

G12 (GPa)

ILSS (MPa)

0 3 5

1907 1925 1937

8.2 9.2 7.8

1.8 2 1.9

2.5 3 2.5

20.0 18.5 19.8

program based on the analytical model. Comparison and validation are carried out using the Pol et al.’s results [18] in two velocity ranges, i.e. near the ballistic limit velocity and beyond the ballistic limit velocity. The samples are made of 2D woven-glass/epoxy composites reinforced by nano-clay particles through a hand lay-up procedure. Tables 1, 2 present the projectile specifications and the mechanical properties of the nanocomposite plates. It should be noted that due to the lack of information in reference [10]. As mentioned in Section 3, the normal failure strain values at different strains rate are determined after multiplication of certain coefficients by the maximum strain. These coefficients are determined by matching experimental results with the theories (Table 3). Table 4 presents the results of projectile residual velocity for different samples in two different impact velocities, i.e. near the ballistic limit velocity and beyond the ballistic limit velocity. The results show that the present model can predict the residual velocity of thin laminated nanocomposite plate reinforced by nanoparticle with an acceptable accuracy. As presented in Table 4, there is a high consistency between the theoretical and the experimental results of projectile residual velocity at impact velocities beyond the ballistic limit. However, in velocities near the ballistic limit, the difference from the experimental results is 28% on average. Reasons for the differences between the predicted and measured values in velocities near to the ballistic limit are explained as follows:

In most cases, mechanical properties are a function of the loading rate. Various research has been conducted on the dependency of mechanical properties of laminated nanocomposite plates on strain rate. The results have demonstrated that strength, longitudinal and transverse modulus in high strain rate loading are more than static loading. [29–32]. Therefore, in the ballistic loading case, nanocomposites undergo higher stiffness due to strain rate sensitivity. In the present paper, the effect of strain rate on the strength and modulus of the thin laminated nanocomposite plate is given by the following equations [29,33]:

ė {Sr } = {S0} ⎜⎛1+C1 Ln ⎛ ⎞ ⎞⎟ ⎝ e0̇ ⎠ ⎠ ⎝

10 mm 8.9 gr 30 mm 2.6 mm 12 150 mm 0.95 1.3*10− 4 0.08 0.05 0.05 1000 j/m2 1 × 10−7 s

• One reason appears to be the inaccurate measurement of residual velocity by the measurement instrument in experiment test. • Using simplified assumptions especially assumption 1 in Section 2,

(55)

Moreover, there is a considerable spread with respect to fracture strain of nanocomposite at high strain rates. Previous studies have presented various coefficients approximately between 0.6 and 2.2 for nanocomposites reinforced with plain woven fabric glass [18,34]. In the present study, the fracture strain values at different strain rates will be determined after applying the suitable coefficients for two ranges velocity i.e. near or beyond the ballistic limit velocity.

which probably is important in initial velocity near to ballistic limit.

As demonstrated, the results of the function with large deformation enjoy high consistency with the experimental results. Adding of nanoparticles improves the mechanical properties of the composite plate and leads to increased energy absorption and decreased residual velocity. Table 3 The failure strain of the nanocomposites after applying strain rate effect (0%, 3%, 5%).

4. Results and discussion 4.1. Comparison of analytical and experimental results First, the parameters of projectile residual velocity and the delamination region radius are determined using a mathematical software 452

Wt (%)

0

3

5

Near ballistic limit velocity Above ballistic limit velocity

2.95 2.25

3.14 2.50

3.05 2.25

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Table 4 Comparisons of experimental and analytical results of projectile residual velocity. Range of velocity

Near ballistic limit velocity Above ballistic limit velocity

Impact velocity (m/s)

Wt (%)

134

0 3 5 0 3 5

169

Residual velocity (m/s)

Error %

Experimental result [18]

Analytical result

(42–51) (30–41) (60–68) (116–122) (113–116) (128–131)

62.39 62.08 75.18 119.65 115.42 129.58

22.33 51.41 10.56 1.92 0.5 1.08

Fig. 4. Delamination radius expansion versus time of nanocomposite plates (0%, 3%, 5%) near the ballistic limit velocity.

An investigation of the results presented in Table 4 reveals that the 3% sample has the highest energy absorption and the lowest residual velocity in both initial velocity regions due to possessing of the best mechanical properties compared to other samples; this expectation is well predicted by the analytical model. In terms of analyzing high velocity impact, large contact forces result from target's resistance to impact. Therefore, these large vertical forces cause very large flexural deformations and energy absorption. As time progresses, the flexural deformation and contact force increases. If the contact force is greater than the delamination critical load, the delamination damage can occur. Although the energy absorbed by the delamination mechanism is low as compared to other energy absorption, it is effective in transferring stress and energy absorption to other layers. Table 5 presents the radius of the delamination region in various proportions of nano-clays at velocities near and beyond the ballistic limit velocity. As observed, the difference between analytical and experimental delamination radius in velocities near the ballistic limit and beyond the ballistic limit are 12.6% and 1.46%, respectively. Hence, similar to the results of projectile residual velocity, there is a higher accuracy between the analytical and experimental results at velocities beyond the ballistic limit. Furthermore, the results of the analytical model show that delamination region radius is decreased by increasing the impact velocity. This is highly consistent with the experimental results, too. As the projectile penetrates into the target, the contact force increases and as a result the radius of the delamination region expands. According to the results of Table 5, it can be observed that in the experimental results, the radius of the 3% and 5% sample delamination region increased initially and then decreased slightly. Referring to the analytic relations of the calculation of the delamination radius, Eq. (45), it is determined that the radius of the delamination region is inversely related to the interlaminar shear strength of the layers. Therefore, by reducing the interlaminar shear strength between layers of the 3% sample, the radius of the delamination increases. Also, by increasing the interlaminar shear strength between the layers of the 5% sample, the delamination radius decreases. These results are consistent with experimental results in Fig. 5 in such a way that the final

Fig. 5. Delamination radius expansion versus time of nanocomposite plates (0%, 3%, 5%) beyond the ballistic limit velocity.

delamination radius expansion of the 5% sample is less than 0% sample. The delamination radius in velocities near the ballistic limit is higher than beyond the ballistic limit velocity (Figs. 4, 5). That is because, in impact velocities beyond the ballistic limit, there is not sufficient time for propagation through the laminated nanocomposites; therefore, all layers fracture and the penetration process ends. As illustrated in Figs. 4 and 5, in both velocity ranges, at first the delamination radius reaches a constant value and then increases gradually until complete penetration or projectile stop. As stated in Section 2.6, with the aid of Eq. (44), an imposed damage threshold of force is obtained in order to keep the layers together and prevent them from separating. Therefore, if the contact force exceeds this damage threshold, it initiates a separation between the layers. Figs. 6 and 7 illustrate the variation of delamination critical force of nanocomposite plates with different proportions of nano-clays versus time in velocities near and beyond the ballistic limit. Figs. 6 and 7 show the changes of the critical force of the delamination over time with oscillation, because the damage force in Eq. (44) depends on the mechanical properties of the target. As shown in Figs. 6 and 7, increased impact velocity leads to increased delamination critical load; hence, the initiation of delamination occurs later. In contrast, the addition of nano-clay particles leads to decreased in delamination critical load in comparison to that of the unadulterated sample at velocities near and beyond the ballistic limit.

Table 5 Comparison of experimental and analytical results of the delamination region. Range of velocity

Impact velocity (m/s)

Near ballistic limit velocity

134

Above ballistic limit velocity

169

Wt (%)

0 3 5 0 3 5

Contact Force (KN)

16.46 16.63 16.70 20.76 20.98 20.86

453

Delamination radius (mm)

Error %

Experimental result [18]

Analytical result

20.86 21.93 20.42 14.81 16.39 14.41

22.78 24.76 23.64 14.57 16.19 14.41

9.2 12.9 15.7 1.62 1.22 1.55

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Delamination Critical Load (N)

6000

3%

5%

5000 4000 3000 2000 1000 0 0

50

100

150

200

250

time (×10-7 s) Fig. 6. Delamination critical force variation versus time of nanocomposite plates (0%, 3%, 5%) at velocities near the ballistic limit.

Delamination Critical Load (N)

10000

0%

3%

5%

9000 8000 7000

6000 5000 4000 3000 2000

1000 0 0

20

40

60

80

100

120

time (×10- s)

Fig. 7. Delamination critical force variation versus time of nanocomposite plates (0%, 3%, 5%) at velocities beyond the ballistic limit.

4.2. Variation of the nonlinear strains of nanocomposite plate and projectile velocity At the beginning of this section, the nonlinear normal and shear strains variation in the first and last layers of nanocomposite plates were investigated near the impact location versus the time in both velocity ranges. Then, by determining the maximum strain on each layer, the variation of the number of fractured layers versus time at velocities near and beyond the ballistic limit are presented. At the end of this section, the projectile velocity variation accompanied by maximum strain variation in the first and last layers of nanocomposite plates are determined with time in both initial velocity ranges. Figs. 8 and 9 illustrate the variation of nonlinear normal and shear strains near the impact point in the first and last layers of nanocomposite laminates during impact event with various percentages of nano-clays in both velocity ranges. According to Eqs. 9 and 11, it is obvious that the nonlinear normal and shear strain is maximized at θ = 0, 45° respectively. Hence, by substituting r = d/2 and θ = 0, 45° into Eqs. 12 and 14, respectively, the nonlinear normal and shear strains near the impact location is calculated in each layer. After analyzing the nonlinear strains of specified points, the highest nonlinear normal strain in each layer is at r = d/2 and θ = 0 and the highest nonlinear shear strain in each layer is at r = d/2 and θ = 45. As shown in Figs. 8 and 9, the nonlinear normal strains are larger than the nonlinear shear strains in the first and last layers. Therefore, for comparing with the failure criterion (maximum strain) in each layer, the nonlinear normal strains are used. Also, the normal strain and nonlinear shear strain of the first layer begin to increase like a straight line with much gradient after the projectile impact. They reach to their maximum in less than a microsecond and then break up. It should be noted that in the present model, when the projectile impacts, the strain of each layer is calculated simultaneously. Therefore, by adding the thickness effect in subsequent layers, the strains in the subsequent layers decrease as compared to the first layer. The curve of the strain variation increases with less gradient. The present model is also based on the multi-stage solution method, so that the next stage variables depend on the parameters of the previous step. Therefore, the absorbed

Fig. 8. Nonlinear normal and shear strain variation of first through last layers of nanocomposite plates at velocities near the ballistic limit a) 0%, b) 3%, c) 5%.

energy will change very much with the propagation of transverse waves, projectile displacement and velocity during impact event. These changes will be more intense in the end layers, especially the last layer, which will be close to the end of the penetration process, and will be accompanied by more oscillations. If the amplitude of these oscillations increases, they can lead to instability or divergence of the solving process. Figs. 10 and 11 illustrate the variation of the number of failed layers of nanocomposite plates during impact event with various percentages of nano-clay in both velocity ranges using the criterion of maximum strain. As illustrated, the variation of the number of failed layers with the velocity above the ballistic limit is greater than that the velocity near the ballistic limit. In both velocity limits, elementary layers are failed in turn and quickly. Due to the reduction of energy and velocity of projectile, the last layers exhibit the greatest resistance to fracturing. According to Figs. 10 and 11, it can be seen that in each of the velocity ranges, the failure of 5% sample is faster than the 0% sample. This could have the following reasons: 1. By looking at mechanical properties in Table 2, it is found that the elastic modulus E11, E22 and the interlaminar shear strength of the 454

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Number of failed layers

14

3%

5%

Fracture line

12 10 8 6 4 2 0 0

20

40

60

80

100

120

time (×10-7 s)

Fig. 11. Variation of the number of failed layers of nanocomposite plates (0%, 3%, 5%) at velocities beyond the ballistic limit.

Fig. 9. Nonlinear normal and shear strain variation of first through last layers of nanocomposite plates at velocities beyond the ballistic limit a) 0%, b) 3%, c) 5%.

0%

Number of failed layers

14

3%

5%

Fracture line

12 10 8 6 4 2 0 0

50

100

150

200

250

time (×10-7 s) Fig. 10. Variation of the number of failed layers of nanocomposite plates (0%, 3%, 5%) at velocities near the ballistic limit. Fig. 12. Velocity variation accompanied by maximum strain variation of first through last layers of nanocomposite plates at velocities near the ballistic limit a) 0%, b) 3%, c) 5%.

layer of the 5% sample have been reduced to 0% sample. The failure strain of the 5% sample is slightly more than the 0% sample while the final strength and the fracture toughness of 5% sample is lower than the 0% sample. 2. Among the important parameters in the present analytical model is the expansion of the delamination radius and increasing the energy absorption of the delamination. According to Fig. 5, the

delamination radius of the 5% sample is lower than the 0% sample. As a result, the delamination energy of the 5% sample is lower than the 0% sample in the penetration process. 3. Comparison of Figs. 9-a to 9-c show that the slope of the nonlinear 455

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Fig. 14. Variation of energy absorption by the nanocomposite plate (0%, 3%, 5%) at velocities near the ballistic limit.

Fig. 15. Variation of energy absorption by the nanocomposite plate (0%, 3%, 5%) at velocities beyond the ballistic limit.

is penetrated. As shown in Figs. 12 and 13, due to changes in the strain of the last layer and the velocity of projectile, it is understood that the slope of the last layer strain curve and the projectile velocity versus time are interrelated. Especially, at velocities above the ballistic velocity, the slope of the velocity curve decreases with increasing gradient of the strain curve. Regarding Fig. 13, the slope of the last layer strain curve in the 5% sample is more than the other samples. As a result, the highest residual velocity is expected for this sample, which is also consistent with experimental results.

Fig. 13. Velocity variation accompanied by maximum strain variation of first through last layers of nanocomposite plates at velocities beyond the ballistic limit a) 0%, b) 3%, c) 5%.

shear strain curves of the 5% sample is higher than the 0% sample. Thus, this matter is in good agreement with reducing the stiffness and strength of the 5% sample in comparison to the 0% sample. It is also an indication of a sooner failure of the 5% sample than the 0% sample.

4.3. Variation of energy absorption by nanocomposite laminates As mentioned in the description of the analytical model, the four significant regions of energy absorption, i.e. the fracture region, the nonlinear deformation region, the plate local movement region and the delamination region, play a significant role in the absorption of the kinetic energy of the projectile. Figs. 14 and 15 illustrate the variation of energy absorption by the nanocomposite plate with different proportions of nano-clays in both velocity ranges. The greatest energy absorption by the nanocomposite plate takes place in the nonlinear deformation region. At first, the kinetic energy of the local moving part is zero. As time passes, the mass of the local moving part increases, the projectile velocity decreases, and the kinetic energy of the local moving part increases. Near the end of the complete penetration process, due to the significant decrease in projectile velocity, the kinetic energy of the local moving part

Figs. 12 and 13 demonstrate the velocity variation and maximum nonlinear strain variation in the first and last layers of nanocomposite plates during impact event in both velocity ranges. As demonstrated by Eqs. (12–14), nonlinear strain variation in each layer is dependent on the central deflection of the plate and the propagation of the transverse wave. With the passage of time, strain increases in the layer and projectile velocity decreases. Therefore, after some time, as the projectile and the transverse wave progress, the first layer is suddenly fractured because the strain of the first layer has reached the fracture strain. This process continues to the point where all layers reach fracture strain and the laminated nanocomposite target 456

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b)

450

Transverse Wave velocity (m/s)

Transverse Wave Velocity (m/s)

a)

400 350 300

0%

250

3%

5%

200 150 100 50

0 0

50

100

150 (×10-7 s)

time

200

600 500 400 300

0%

3%

5%

200 100 0

250

0

20

40

60

80

time (×10-7 s)

100

120

Fig. 16. Transverse wave propagation velocity variation versus time in nanocomposite plates (0%, 3%, 5%) at velocities (a) near the ballistic limit and (b) beyond the ballistic limit.

b)

0.018

0.016

0%

3%

Transverse Wave radius (m)

Transverse Wave radius (m)

a)

5%

0.014 0.012 0.01 0.008 0.006 0.004

0.002

0.012

0%

0.01

3%

5%

0.008 0.006 0.004 0.002 0

0 0

50

100

time

150 (×10-7 s)

200

0

250

20

40

time

60

(×10-7

80

100

120

s)

Fig. 17. Transverse wave propagation radius variation versus time in nanocomposite plates (0%, 3%, 5%) at velocities (a) near the ballistic limit and (b) beyond the ballistic limit.

propagation and the 3% sample has the highest velocity and radius of transverse wave propagation in both velocity.

decreases. The energy absorption in the nonlinear deformation region is greater than that in the fracture region, because the volume of the nonlinear deformation region is considerably greater than the volume of the fracture region. Other mechanisms such as the local movable region and the delamination absorb little energy. According to Figs. 14 and 15, the maximum and minimum values of the energy absorption estimation in both velocity ranges are for 3% and 5% samples, respectively. They are also in good agreement with experimental results. As shown in Table 2 and Figs. 10–13, the 3% sample has the highest resistance to projectile penetration and the lowest residual velocity of the projectile. The 5% sample has the minimum resistance to projectile penetration and the highest residual velocity of projectile, especially in the velocity beyond the ballistic limit, which is compatible with the energy absorption estimation of Figs. 14 and 15.

5. Conclusions In the present paper, a two-dimensional analytical model is proposed for investigating the ballistic impact behavior of nanocomposite laminates based on the energy balance of the projectile and the calculation of the energy absorbed during various time interval. The results are as follows:

• Generally, the predictions of analytical model shows good agreement with the experimental results. • Variation of nonlinear strains in each layer depend on the central deflection of the plate as well as the transverse wave propagation. • The highest nonlinear normal strain in each layer is at r = d/2 and

4.4. Investigation of velocity and radius of propagation of stress waves Considering Eqs. 1 and 3, variation of velocity and radius of the transverse wave propagation in nanocomposite plates with various proportions of nano-clays are investigated in both velocity ranges. Figs. 16 and 17 respectively illustrate the variations of velocity and radius of the transverse wave propagation in nanocomposite plates with various proportions of nano-clay in both velocity ranges. As illustrated in Figs. 17.a and 17.b, the radius of propagation of the transverse wave in both velocity ranges extends respectively by approximately 15 mm and 11 mm in the nanocomposite plate. This makes the effect of wave reflection of the support unnecessary to investigate. As shown in Section 2 in describing the analytical model of ballistic impact, the velocity and radius of transverse wave propagation are important parameters in determining the strain, velocity, and energy absorption of target. Therefore, according to Figs. 16 and 17, it is clear that the 5% sample has the lowest velocity and radius of transverse wave

• •

• 457

θ = 0 and the highest nonlinear shear strain in each layer is at r = d/2 and θ = 45. The nonlinear normal strains are larger than the nonlinear shear strains in the first and last layers. Therefore, for comparing the failure criterion (maximum strain) in each layer, the nonlinear normal strains are used. Addition of nanoparticles improves the mechanical properties of composite plates, helps to increase the amount of energy absorption, and decreases the residual velocity. Hence, thanks to its enjoyment of the most favorable properties of all samples, the 3% sample enjoys the highest energy absorption and lowest residual velocity in both velocity ranges. This expectation is clearly predicted by the analytical model. The energy absorbed by the nonlinear deformation region of the nanocomposite plate is more than that absorbed by the fracture region.

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• The delamination radius at the impact velocity near the ballistic limit is greater than that beyond the ballistic limit. • Addition of nano-clay particles reduces the delamination critical

[16] S.K. García-Castillo, S. Sánchez-Sáez, E. Barbero, Impact behaviour of preloaded glass/polyester woven plates, Compos. Sci. Technol. 69 (6) (2009) 711–717. [17] S.K. García-Castillo, S. Sánchez-Sáez, E. Barbero, Nondimensional analysis of ballistic impact on thin woven laminate plates, Int. J. Impact Eng. 39 (1) (2012) 8–15. [18] M.H. Pol, G.H.H. Liaghat, E. Zamani, A. Ordys, Investigation of the ballistic impact behavior of 2D woven glass/epoxy/Nano clay nanocomposites, Compos. Mater. 49 (12) (2015) 1449–1460. [19] S.L. Phoenix, P.K. Porwal, A new membrane model for the ballistic impact response and V50 performance of multi-ply fibrous systems, Solids Struct. 40 (2003) 6723–6765. [20] M.S. Hoo Fatt, D. Sirivolu, A wave propagation model for the high velocity impact response of a composite sandwich panel, Int. J. Impact Eng. 37 (2) (2010) 117–130. [21] B.A. Gama, Gillespie Jr. J.W., A quasi-static penetration model of ballistic penetration of thick-section composites. In: Proceedings of the 52, 20th Annual Technical Conference, American Society for Composites, Philadelphia, PA; September 7–9, 2005. [22] R.H. Zee, C.Y. Hsieh, Energy loss partitioning during ballistic impact of polymer composites, Polym. Compos. 14 (3) (1993) 265–271. [23] J.C. Smith, F.L. McCrackin, H.F. Scniefer, Stress–strain relationships in yarns subjected to rapid impact loading. Part V. Wave propagation in long textile yarns impacted transversely, Res. Natl. Bur. Stand. 60 (1958) 517–534. [24] B. Gu, Analytical modeling for the ballistic perforation of planar plain-woven fabric target by projectile, Compos. Part B: Eng. 34 (4) (2003) 361–371. [25] S. Zhu, G.B. Chai, Low-velocity impact response of fiber-metal laminates – a theoretical approach, Proc. Inst. Mech. Eng. Part L: J. Mater.: Des. Appl. 228 (4) (2016) 301–311. [26] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells Theory and Analysis, CRC, London, 1945. [27] M. Latifi, M. Kharazi, H.R. Ovesy, Nonlinear dynamic response of symmetric laminated composite beams under combined in-plane and lateral loadings using full layerwise theory, Thin-Walled Struct. 104 (2016) 62–70. [28] S. Abrate, Impact Engineering of Composite Structures, Springer, NewYork, 2011. [29] L.J. Deka, S.D. Bartus, U.K. Vaidya, Damage evolution and energy absorption of Eglass/polypropylene laminates subjected to ballistic impact, Mater. Sci. 43 (2008) 4399–4410. [30] M. Shokrieh, M.J. Omidi, Investigating the transverse behavior of Glass–Epoxy composites under intermediate strain rates, Compos. Struct. 93 (2) (2011) 690–696. [31] N.K. Naik, P. Yernamma, N.M. Thoram, R. Gadipatri, V.R. Kavala, High strain rate tensile behavior of woven fabric E-glass/epoxy composite, Polym. Test. 29 (1) (2010) 14–22. [32] K. Naresh, K. Shankar, R. Velmurugan, N.K. Gupta, Statistical analysis of the tensile strength of GFRP, CFRP and hybrid composites, Thin-Walled Struct. (2016), http:// dx.doi.org/10.1016/j.tws.2016.12.021. [33] C.F. Yen, A ballistic material model for continuous-fiber reinforced composites, Int. J. Impact Eng. 46 (2012) 11–22. [34] N.K. Naik, V.R. Kavala, High strain rate behavior of woven fabric composites under compressive loading, Mater. Sci. Eng. A 474 (1–2) (2008) 301–311.

force and increases the delamination radius in comparison to the neat sample at velocities near and beyond the ballistic limit.

References [1] L. Sun, R.F. Gibson, F. Gordaninejad, J. Suhr, Energy absorption capability of nanocomposites: a review, Compos. Sci. Technol. 69 (14) (2009) 2392–2409. [2] U. Farooq, P. Myler, Finite element simulation of carbon fibre-reinforced composite laminates subjected to low velocity impact using damage induced static load-deflection methodology, Thin-Walled Struct. 97 (2015) 63–73. [3] S. Patela, S. Ahmad, P. Mahajanb, Safety assessment of composite beam under ballistic impact, Thin-Walled Struct. (2017), http://dx.doi.org/10.1016/j.tws.2017. 05.027. [4] K.S. Pandya, Ch.V.S. Kumar, N.S. Nair, P.S. Patil, N.K. Naik, Analytical and experimental studies on ballistic impact behavior of 2D woven fabric composites, Damage Mech. 24 (4) (2015) 471–511. [5] P.R.S. Reddy, T.S. Reddy, V. Madhua, A.K. Gogia, V.R. Kavala, Behavior of E-glass composite laminates under ballistic impact, Mater. Des. 84 (2015) 79–86. [6] K.S. Pandya, J.R. Pothnis, G. Ravikumar, N.K. Naik, Ballistic impact behavior of hybrid composites, Mater. Des. 44 (2013) 128–135. [7] I. Mohagheghian, G.J. Mcshane, W.J. Stronge, Impact response of polyethylene nanocomposites, Eng. Procedia 10 (2011) 704–709. [8] A.F. Avila, A.S. Netob, H.N. Juniorb, Hybrid nanocomposites for mid-range ballistic protection, Int. J. Impact Eng. 38 (8–9) (2011) 669–676. [9] K.S. Pandya, K. Akella, M. Joshi, N.K. Naik, Ballistic impact behavior of carbon nanotube and nanosilica dispersed resin and composites, Appl. Phys. 112 (2012) (113522-113522). [10] M.H. Pol, G.H. Liaghat, Investigation of the high velocity impact behavior of nanocomposites, Polym. Compos. 37 (4) (2016) 1173–1179. [11] M.H. Pol, G.H. Liaghat, F. Hajiaarazi, Effect of nanoclay on ballistic behavior of woven fabric composites: experimental investigation, Compos. Mater. 47 (13) (2013) 1563–1573. [12] R.A.W. Mines, A.M. Roach, N. Jones, High velocity perforation behavior of polymer composite laminates, Int. J. Impact Eng. 22 (6) (1999) 561–588. [13] S.S. Morye, P.J. Hine, R.A. Duckett, D.J. Carr, I.M. Ward, Modelling of the energy absorption by polymer composites upon ballistic impact, Compos. Sci. Technol. 60 (14) (2000) 2631–2642. [14] N.K. Naik, P. Shrirao, B.C.K. Reddy, Ballistic impact behavior of woven fabric composites: formulation, Int. J. Impact Eng. 32 (10) (2006) 1521–1552. [15] G. Balaganesan, R. Velmurugan, M. Srinivasan, N.K. Gupta, K. Kanny, Energy absorption and ballistic limit of nanocomposite laminates subjected to impact loading, Int. J. Impact Eng. 74 (2014) 57–66.

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