Effect of smart stiffening procedure on low-velocity impact response of smart structures

Journal of Materials Processing Technology 190 (2007) 142–152

Effect of smart stiffening procedure on low-velocity impact response of smart structures S.M.R. Khalili ∗ , A. Shokuhfar, F. Ashenai Ghasemi Department of Mechanical Engineering, K.N. Toosi University of Technology, Vafadar East Ave, 4th Tehran pars SQ., Tehran, Iran Received 2 January 2007; accepted 28 February 2007

Abstract The paper presents the response of smart hybrid composite plate subjected to low-velocity impact. The low-velocity impact response of the composite plate embedded with shape memory alloy (SMA) wires is investigated. The SMA wires are embedded within the layers of the composite laminate. The first-order shear deformation theory as well as the Fourier series method is utilized to solve the dynamic governing equations of the hybrid composite plate analytically. The interaction between the impactor and the plate is modeled with the help of two degrees-of-freedom system, consisting of springs-masses. The Choi’s linearized Hertzian contact model is used in the impact analysis of the laminated hybrid composite plate. The stiffness of the composite structures classified into two new groups. Interactive and non-interactive effects of these stiffnesses are studied too. In addition, a procedure named smart stiffening procedure (SSP) was used to improve the impact resistance of the composite structures. It was seen that by using of the SSP, the mechanical characteristics of the structure could be improved the most. © 2007 Elsevier B.V. All rights reserved. Keywords: Smart structure; Shape memory alloy (SMA); Composite; Stiffness; Impact

1. Introduction Advanced structural materials with shape memory alloy (SMA) wires have attracted nowadays, because of their potential uses: performance enhancement, shape control, damage tolerance, vibration suppression and self-repair. One of the ways to prevent the damage of the multilayered laminated composite structures is embedding the SMA wires inside the polymer composites, because the excited SMA wires can generate recovery tensile stresses inside the structure and hence reduce the deflections and the in-plane strains and stresses of the structure. Abrate [1–3] studied the impact behavior of composite laminates extensively. Olsson [4–6] classified low-velocity impacts in two categories, known as the small mass and the large-mass impact. This classification is based on the ratio of impactor mass to the target (here hybrid composite plate) mass. Small impactor masses cause a small mass response dominated by shear and

∗

Corresponding author. Tel.: +98 021 77343300; fax: +98 021 77334338. E-mail address: [email protected] (S.M.R. Khalili).

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.02.052

flexural waves in which the load, deflection, and flexural strains are out of phase. Impactor masses much larger than the laminate mass would cause a ‘quasi-static’ large-mass response, in which the peak load, deflection, and strains are more or less in phase. Khalili et al. [7–12] investigated the free damped vibrations as well as low-velocity impact response of laminated composite and sandwich plates using higher-order theories. Sun and Chen [13] examined the positive effect of tensile stresses to improve the impact resistance of the composite materials. Birman et al. [14] demonstrated that if the SMA wires were embedded inside the traditional polymer composites, tensile stresses could be generated inside the structures. Hence, the impact resistance of the structures would be increased. He also demonstrated that if some SMA wires were embedded inside different layers of the structure, the global deflection of the structure would be reduced in low-velocity impacts. He used constitutive relationship and associated micromechanics for a hybrid SMA composite material, which were considered in a number of studies [15,16]. In addition, some of the researchers [16–18] verified the above theories by experiments. Roh and Kim [19,20] embedded the SMA wires inside a traditional multilayered laminated composite structure and utilized the energy balance and the FEM method

S.M.R. Khalili et al. / Journal of Materials Processing Technology 190 (2007) 142–152

to study the effect of the same parameters worked in Birman’s article [14]. In the present research, a complete model was developed so that the effect of low-velocity impact upon the multilayered laminated smart composite plates demonstrated. In addition, the effect of stiffening of the structures was discussed introducing a procedure named smart stiffening procedure (SSP) to reach the most impact resistance of the composite structures. The effect of using SMA wires as well as some parameters such as the volume fraction and the orientation of the SMA wires on impact response of smart hybrid composite plate is studied in details as discussed shortly before in Ref. [21]. 2. Smart stiffening procedure The stiffness of the structures has a great effect on their overall mechanical behavior. The stiffness is an important factor in order to study the tensile or the bending strength, the natural frequency, the impact resistance and the fatigue or the fracture strength of structures. Therefore, it is not strange that the improving of this parameter is one of the aims of many researchers. Although the stiffness is an important factor during the study of the characteristics of the structures, it seems that there were only small attentions on it. The stiffness is usually called to the resistance of a material against the elastic deformation. In isotropic materials, stiffness is recognized by a single digit named as modulus of elasticity, which can be found by means of the slope of the stress–strain diagram of the tensile strength tests. A high stiff material, with a high modulus of elasticity maintains its size and shape even under a large elastic loading [22]. In a unidirectional composite lamina, the stiffness is not exactly known as the modulus of elasticity of the lamina. Modulus of elasticity of a lamina may be modeled as three single digits in three principle coordinates of the structure, but its stiffness can be described with the help of a matrix named stiffness matrix (not introduced with three single digits). Because of their more complicated nature, there is a more different case for the stiffness of a composite laminate. Here there is not the above three single digits for defining of the modulus of elasticity of the structure either. This means that the modulus of elasticity of a composite laminate has no meaning. However, we can present the elastic properties of a composite laminate with a more complicated matrix from what we have for a composite lamina. The matrix that can also be named the stiffness matrix is contained of three independent matrices, which are usually named as the extensional stiffness matrix, the coupling stiffness matrix, and the bending stiffness matrix. These matrices are defined completely in most of the composite reference books like in Ref. [23]. Therefore, the stiffness is defined for all of the materials (here we only talk about composite materials) while the modulus of elasticity is not. Because of this, we decided to talk on the stiffness more and to do this in a certain format we classified the total stiffness of a structure as:

143

Fig. 1. Classifying the stiffnesses of a structure and some of the affective parameters on them.

(1) essential stiffness, (2) acquired stiffness. Essential stiffness is the stiffness that a structure is reached after its manufacturing process and when it is ready to use. Some of the affective parameters on this stiffness are like the fibers and the matrix material, the fibers orientation, the stacking sequence, the interface type between the fibers and the matrix. Acquired stiffness is the stiffness that a structure gains after of its manufacturing process and when it is in use. Some of the effective parameters on this stiffness are like the thermal stresses, the prestressing, and the environmental effects, which they usually apply to the structure after the end of manufacturing process of the structure and when it is in use. Therefore, it seems that the total stiffness of a structure can be concluded of these two stiffnesses, which are shown in Fig. 1 briefly. Therefore, we can write these stiffnesses as follows: S T = SE ± S A

(1)

in which ST , SE , and SA are the total stiffness, the essential stiffness, and the acquired stiffness of the structure, respectively. The questions, which can be asked, are that which of the above stiffnesses are the more effective to improve the repose of the structure in different applications and what does the “±” symbol in Eq. (1) means. The answers are easy but difficult! Because each of the above mentioned groups of the stiffnesses can increase or decrease the positive effects of stiffening a structure in a certain application. For example, the essential stiffness of a composite structure will be increased by increasing of the volume fraction of its fibers. However, if we increase this parameter more and more, the load sharing between the fibers and the matrix becomes low, which is not a good effect for composite structures. In addition, acquired stiffness of a composite structure may be increased by prestressing of the structure. Nevertheless, if we increase the above positive prestressing more and more, this may results in the fibers go on failure. Therefore, as it can be seen, we need a procedure with the aim of the smart stiffening of the structures in a manner that this stiffening only results in positive effects in the structures. We call this, smart stiffening procedure. In this research, we try to show the different effects which origins from the changing of the above mentioned stiffnesses in smart structures. Here, SMA wires are used to improve the impact resistance of the structure. As the SMA wires show their

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unique properties (applying the positive tensile stresses in the structure) when they are activated (by heating), we can call this stiffening, acquired stiffening. Because the SMA wires are embedded in the structure, before the manufacturing process of the structure is finished, presence of these wires changes the stiffness of the structure lonely, which is an essential stiffening itself. By smart using of these stiffening, we can improve the stiffness of the structures more and more. To see these effects, we must first develop the required equations to analyze the impact response of these structures as follow. 3. Constitutive equations Constitutive equations of principal stress–strain relationship for a SMA hybrid laminated composite are as follows [14,27]: ⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎡ m ¯ ¯ ¯ 0 ⎪ Q ⎪ 11 ⎨ σ1 ⎪ ⎬ ⎪ ⎨ Q11 Q12 ⎬⎪ ⎨ ε1 ⎪ ⎬ ⎪ ⎨ σr ⎪ ⎬ ⎢ ¯ 12 Q ¯ 22 ¯m σ2 0 ε2 = Q + 0 ks − ⎣ Q 12 ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎩ ¯ 66 ⎭ ⎩ γ12 ⎭ ⎩ 0 ⎭ τ12 0 0 Q 0      ¯ 55 Q ¯ 54 τ13 Q γ13 = ¯ 54 Q ¯ 44 τ23 γ23 Q

Also in Eq. (3) [14]: ⎧ i ⎫ ⎧ r ⎫ T ⎪ ⎬ ⎪ ⎨ Nx − Nx ⎪ ⎨ Nx ⎪ ⎬ = Nyr − NyT N i = Nyi ⎪ ⎪ ⎭ ⎪ ⎩ ⎩ i ⎪ ⎭ Nxy 0 ⎧ ⎧ r ⎫ ⎫ ⎪ ⎪ ⎬ ⎨ ksx hx ⎪ ⎨ Nx ⎪ ⎬ Nyr = σr ksy hy T ⎪ ⎪ ⎪ ⎭ ⎩ ⎩ r ⎪ ⎭ 0 Nxy ⎧ T⎫ ⎡ ¯m c ⎫ ¯ m αct ) ⎤ ⎧ (Q11 αl + Q ⎪ Nx ⎬ ⎪ ⎪ kcx hx ⎪ 12 ⎨ ⎬ ⎨ ¯ m αc + Q ¯ m αct ) ⎥ NyT = ⎢ k h T ⎣ (Q ⎦ cy y 12 l 22 ⎪ ⎪ ⎪ ⎩ T ⎪ ⎭ ⎭ ⎩ 0 Nxy 0

¯m Q 12 ¯m Q 22

0

⎤⎧ c⎫ ⎪ ⎨ αl ⎪ ⎬ ⎥ 0 ⎦ αct kc T ⎪ ⎪ ¯m ⎩ 0 ⎭ Q 66

(5)

0

(2)

where {σ} represents the stresses in the principle directions. In addition, the matrix {ε} represents the strains in the principle where: ¯ ij and Q ¯ m , respectively, represent the reduced stiffdirections. Q ij ness matrices for the SMA hybrid composite and the composite kcx = 1 − ksx (6) medium (without the SMA wires). σ r represents the recovery kcy = 1 − ksy stress which can be determined analytically or by experiments [14]. αci (i = l, t) represents the thermal expansion coefficients of In the above equation, ksx and ksy are the volume fractions composite medium which is calculated based on the temperature of SMA wires in the plies oriented in the x and y directions, difference T between the current and the reference temperarespectively, and hx and hy are the total thicknesses of composite tures. ks and kc , respectively, represent the volume fractions of medium in the x and y directions. the SMA wires and the composite medium. Because of discontinuity function of stresses through the 4. Governing equations thickness in each layer, it is possible to determine the constitutive equation by considering the force-couple resultants in terms of The plate equations developed by Whitney and Pagano stresses, using integration of Eq. (2) through the plate thickness, [24] are used as they included the effect of transverse shear which yields:            N Aij Bij Ni Ni Nr − NT ε0 = + ; = ; i, j = 1, 2, 6 M Bij Dij (3) Mi Mi κ Mr − MT {S} = [ksh Aij ]{γ};

i, j = 4, 5

where N and S are vectors of forces and M is the vector of moments, respectively. Aij , Bij , and Dij are the components of extensional and shear, coupling, and bending stiffness matrices, respectively. Nr and Mr are, respectively, the vectors of the recovery stress resultants and the stress moments generated in the SMA wires. NT and MT are, respectively, the vectors of the thermal stress resultants and moments in the structure. Also ε0 and γ are the mid-plane and shear strains, respectively, and κ is the curvatures. In addition, ksh is the shear correction factor. In symmetrically laminated cross-ply plates: B = Mi = Mr = MT = 0

(4)

deformations. The assumed displacement field is: u = u0 (x, y, t) + zψx (x, y, t) v = v0 (x, y, t) + zψy (x, y, t) w=

(7)

w0 (x, y, t)

where u0 , v0 and w0 are the plate displacements in x, y and z directions at the plate mid-plane and ψx and ψy are the shear rotations in the x and y directions. Reducing the equation to specially orthotropic form (Bij = 0, A16 = A26 = D16 = D26 = 0), and adding the uniform in-plane initial stress resultants Nxi and

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discussed before by the author [21]: m2 z¨ 2 + F = 0 m1 z¨ 1 + Kbs z2 + Km z32 − F = 0

Fig. 2. Schematic view of the SMA hybrid composite plate impacted by a spherical mass.

Nyi (see Eq. (2)) as discussed by Sun and Chattopadhyay [25] results in:

(11)

where F is the contact force, m1 and m2 represents, respectively, the mass of the hybrid plate and the impactor, z1 and z2 are, respectively, the relative displacements of the hybrid plate and the impactor masses. K1 = Kbs is the bending-shear stiffness, Kb is the bending stiffness, Ks is the shear stiffness, and Km is the membrane stiffness of the hybrid plate. Since the membrane stiffness Km is low in polymer composite materials especially in low velocities and low deflections, ignoring this parameter from the current model in Fig. 3(a) would result in changing the model into a simplified one (Fig. 3(b)). Hence, using Choi and Lim’s linearized model [29] instead of nonlinear Hertzian contact law, the contact force can be obtained as: F (t) = Kl α = Kl (z2 − z1 ) 1/3

2/3

Kl = Fm Kc

¨x D11 ψx,xx + D66 ψx,yy + (D12 + D66 )ψy,xy − ksh A55 ψx − ksh A55 w,x = I ψ ¨y (D12 + D66 )ψx,xy + D66 ψy,xx + D22 ψy,yy − ksh A44 ψy − ksh A44 w,y = I ψ

(12)

(8)

¨ ksh A55 ψx,x + (ksh A55 + Nxi )w,xx + ksh A44 ψy,y + (ksh A44 + Nyi )w,yy + q = ρw ksh is the shear correction factor introduced by Mindlin [26], normally taken to be π2 /12. In addition, q is the dynamic normal load (transverse impact) over the plate and:  h/2 (Aij , Bij , Dij ) = Qkij (1, z, z2 ) dz −h/2 (9)  h/2 2 (ρ, I) = ρ0 (1, z ) dz

where K2 = Kl represents the linearized contact coefficient in Choi’s linearized contact law, Fm is the maximum predicted contact force, and Kc represents the contact stiffness in the improved Hertzian contact law, which can be calculated based

−h/2

where ρ0 represents the density of each layer and ρ is the total density of the plate. In addition, I is the moment of inertia and h is the thickness of the plate. Qij (i, j = 1–6) is the reduced inplane stiffness matrix and Qij (i, j = 4, 5) are the transverse shear stiffnesses that is defined in Ref. [24]. In this work, attention is focused upon a simply supported rectangular plate with the dimensions of a and b. Hence, the boundary conditions are as follows: w = ψx,x = ψy = 0; w = ψy,y = ψx = 0;

at x = 0, a at y = 0, b

(10)

Fig. 2 shows a SMA hybrid composite plate, which is in contact with an impactor mass. 5. Dynamic response of the plate 5.1. Contact force In the present analysis, two degrees-of-freedom springsmasses model [3,28] is utilized to determine the contact force of the impact (Fig. 3). The equation of motion is as follows, as

Fig. 3. A two-degrees-of-freedom springs-masses model: (a) with the membrane stiffness and (b) without the membrane stiffness.

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on the following equation [21]: Kc =

4 R1/2 2 3 ((1 − ν )/E) + (1/E22 )

(13)

where R is the curvature radius, ν the Poisson’s ratio and E is the elastic modulus of the isotropic impactor. Because the plate is not isotropic, it must be mentioned that the parameter E22 is the transverse elastic modulus of the top lamina of the structure. Considering the free body diagram in Fig. 3(b) and replacing the value of F in Eq. (11) with the value presented by Choi’s differential equations after rearrangements, it can be converted to the following form: m1 z¨ 1 = −K1 z1 − K2 (z1 − z2 ) m2 z¨ 2 = −K2 (z2 − z1 )

(14)

This system can be solved easily by Runge-Kutta method using ODE solver of the MATLAB software. 5.2. Deflection and stress–strain analysis The solution to the dynamic problem is based on expansions of the loads, displacement, and rotations in double Fourier series. Each expression is assumed to be separately consisting of a function of position and a function of time. Furthermore, by neglecting the effects of rotary inertia [30], the current problem could be converted to a system of ordinary differential equations of second-order in time for the Fourier coefficients of the transverse deflection [14]. The contact force obtained by the present procedure (see Eq. (12)) is used as an input for further analysis of the hybrid composite plate, like the deflection under the point of impact and the failure prediction of the plate, etc. The load function can be demonstrated as follows [31,32]:  mπ   nπ   q(x, y, t) = Pmn (t) sin x sin y (15) α b m n

In this work the rotary inertia effect are neglected following the Mindlin [26]. Hence, using Eqs. (8) and (17), the system of Eq. (8) can be reduced to the following system of ordinary decouple differential equations: ⎤ ⎧⎡ ⎤⎫ ⎡ L11 L12 L13 ⎪ ⎨ Amn (t) ⎪ ⎬ ⎥ ⎢ ⎥ ⎢ ⎣ L12 L22 L23 ⎦ ⎣ Bmn (t) ⎦ ⎪ ⎪ ⎩ ⎭ Wmn (t) L13 L23 L33 ⎧⎡ ⎤⎫ 0 ⎪ ⎪ ⎨ ⎬ ⎢ ⎥ 0 = ⎣ (18) ⎦ ⎪ ⎪ ⎩ ⎭ ¨ Pmn (t) − ρhWmn (t) where: L11 = D11

 mπ 2

+ D66

 nπ 2

+ ksh A55 a  bmπ   nπ  L12 = L21 = (D12 + D66 ) b  mπ  a L13 = L31 = ksh A55 a  mπ 2  nπ 2 + D22 + ksh A44 L22 = D66 a  nπ  b L23 = L32 = ksh A44 b  mπ 2  nπ 2 i + (ksh A44 + Nyi ) L33 = (ksh A55 + Nx ) a b  h/2 ρ= ρ0 dz −h/2

Following Christoforou procedure [32] and using the changes of variables as below: Amn (t) = KA Wmn (t); Bmn (t) = KB Wmn (t) L12 L23 − L13 L22 L12 L13 − L11 L23 KA = ; KB = L11 L22 − L212 L11 L22 − L212

where Pmn (t) are the terms of the Fourier series. For a concentrated load located at the point (xc , yc ):  mπ   nπ  4F (t) sin xc sin yc (16) Pmn (t) = ab a b

Eq. (18) simplifies as follows:

where F(t) is the impact load (see Eq. (12)) and a and b are, respectively, the plate length and width. The impact solution for a rectangular plate with simply supported boundary conditions is assumed to be in the following form [33,34]:

where:

ψx (x, y, t) = ψy (x, y, t) =

∞ 

Amn (t) cos

m,n=1 ∞ 

Bmn (t) sin

m,n=1

w(x, y, t) =

∞ 

m,n=1

Wmn (t) sin

 mπ  a

 mπ  a

 mπ  a

x sin

x cos

x sin

 nπ  b  nπ  b

 nπ  b

y y

(17)

y

where Amn (t), Bmn (t) and Wmn (t) are the time dependent coefficients.

Pmn (t) 2 ¨ mn (t) + ωmn W Wmn (t) = ρh

2 ωmn =

L13 KA + L23 KB + L33 ρh

(19)

(20)

2 is the square of the fundamental frequencies of the plate. ωmn If the value of m = n = 1 is inserted in the above expression, the value of Kbs in Eq. (11) could be calculated as follows: 2 Kbs = m1 ω11

(21)

For zero initial displacement and velocity, the solution of Eq. (19), namely the value of Wmn (t), would be easily calculated based on the Runge-Kutta method of 4th and 5th ranks and using a software like MATLAB and its ODE 45 solver. Substituting these values in Eq. (17), the values of w, ψx and ψy would be easily calculated.

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147

If the above quantities are determined, then the values of strains (εx , εy , γ xy , γ xz , and γ yz ) will be easily calculated by the following equations, as discussed earlier by the author [7]: ∞ 

εx = zψx,x = −z

Amn

 mπ  a

m,n=1

εy = zψy,y = −z

∞ 

sin

 mπ  a

x sin

 nπ  b

y (22)

Bmn

 nπ  b

m,n=1

sin

 mπ  a

x sin

 nπ  b

y (23)

γxy = z(ψx,y + ψy,x ) = −z

∞  

Amn

 nπ  b

m,n=1

× cos γyz = ψy +

γxz = ψx +

 mπ  a

x cos

+ Bmn

 nπ  b

 mπ  a

Fig. 4. Comparison of the contact force histories.

y

(24)

∞    nπ  ∂w = Wmn (t) + Bmn (t) ∂y b m,n=1  mπ   nπ  × sin x cos y a b ∞    mπ  ∂w = Wmn (t) + Amn (t) ∂x a m,n=1  mπ   nπ  × cos x sin y a b

6. Model verification

(25)

(26)

To ensure the accuracy of the present model, the contact force determined from the springs-masses model is compared with the contact-force determined from the direct solution method based on the exact impactor mass and velocity values [33], the inverse model prediction [34] and the experiment [35]. Fig. 4 shows a good agreement in the results. The effect of increasing the number of terms of the Fourier series included in the solution for the transverse deflection of the plate is illustrated in Fig. 5. Fig. 5 shows that a reasonable solution is obtained using as much as 9 terms, but the convergence is demonstrated with 100 terms.

where z is the thickness coordinate of the structure. The stress–strain relations of any lamina with respect to x, y and z-axes are given by: ⎡ ⎤ ⎡ ⎤⎡ ⎤      σxx Q11 Q12 Q16 εx σxz Q55 Q54 εxz ⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎣ σyy ⎦ = ⎣ Q12 Q22 Q26 ⎦ ⎣ εy ⎦ ; σyz Q54 Q44 εyz σxy Q16 Q26 Q66 εxy

(27)

where [Qij ]k is the stiffness matrix of kth layer and is defined as ¯ ij ] [T1 ]−T ; [Qij ]k = [T1 ]−1 [Q k −1 ¯ ij ] [T2 ]−T ; [Qij ] = [T2 ] [Q k

k

(i, j = 1, 2, 6) (i, j = 5, 4)

(28)

in which [T1 ]k and [T2 ]k are transformation matrices for the kth ¯ ij ] is the reduced stiffness matrix of the lamina in the layer and [Q principal material coordinates and are given in Ref. [30]. Using the elastic coefficients, the reduced stiffness matrix elements are related to the engineering properties as follows [30]: (ν12 E22 ) , (1 − ν12 ν21 )

¯ 11 = Q

E11 , (1 − ν12 ν21 )

¯ 12 = Q

¯ 22 = Q

E22 , (1 − ν12 ν21 )

¯ 66 = G12 , Q

Q44 = G23

Q55 = G13 , (29)

where E, G and ν denote, respectively, the Young’s moduli, shear moduli and the Poisson’s ratio of the material. The plate is a laminated composite with a symmetrical lay-up of plies.

Fig. 5. Effect of varying the number of terms in Fourier series on transverse deflection of the plate.

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Table 1 Geometrical and material properties of the hybrid composite plate and the impactor [14,27] Geometrical properties of the SMA hybrid composite Boundary conditions Simply supported Length = width 200 mm Lay-up [0 90 0 90 0]s Ply thickness 0.269 mm Material properties of composite medium (Glass-epoxy) E11 = 23.062 GPa; E22 = E33 = 10.789 GPa G13 = G12 = 11.92 GPa; G23 = 4.68 GPa ν12 = ν13 = ν23 = 0.344 ρ = 1796 kg/m3 Properties of the SMA wires (Ni–Ti based in austenite form) E = 70 GPa, ν = 0.33, ρ = 6500 kg/m3 σ r = 220 MPa at T = 39 ◦ C Properties of the steel wires E = 197 GPa, ν = 0.25, ρ = 7800 kg/m3 Properties of the impactor E = 207 GPa, ν = 0.3, ρ = 7800 kg/m3 Tip diameter = 12.7 mm Weight = 1.5 kg Velocity = 2.00 m/s

7. Results and discussions The hybrid composite plate used is symmetric and cross ply. Geometrical and material properties of the composite plate as well as the impactor are presented in Table 1. The laminate consists of 10 layers, which are numbered from top to bottom. 7.1. Interactive effect of the stiffnesses In this research, the SMA wires are embedded only parallel to the fibers of each lamina of the composite medium (the composite without SMA wires). If the orientations of the SMA wires do not be parallel to the fibers of the composite lamina medium, the mismatching of the fibers and the SMA wires causes some non-fiber spaces in which contain only brittle resin. This phenomenon causes brittle fracture and results in the reduction of structure properties [36]. The volume fraction of the SMA wires are considered to be ksx + ksy = 0.00 up to 1.00. The wires were placed along the x and y direction simultaneously and equally (in each lamina), which corresponds to the 0◦ and 90◦ fiber orientation in composite medium. Fig. 6 shows the effect of reduction of w/ h ratio (a non-dimensional transverse deflection, which is the ratio of composite plate deflection to its thickness) and increase of contact force with respect to increasing of the volume fraction of the SMA wires in the composite medium, which origins from increasing of the acquired stiffness of the structure. In addition, we have the essential stiffening effect, which origins of embedding the SMA wires during the manufacturing process of the structure. Therefore, in this case we see an interaction effect between these two stiffnesses. As it is visible, increasing of the SMA wires more than 0.40 (40%) of the whole structure has only

Fig. 6. Effect of increasing the volume fraction of the SMA wires on nondimensional deflection (w/ h) history and on the contact force history.

little effect on the reducing of w/ h ratio and so the improving of the impact resistance of the structures. This is maybe why Birman et al. [14] and Roh and Kim [20] choose their maximum volume fraction of SMA wires in their researches 0.40 and 0.30, respectively. Because of this, we also concentrate our efforts on volume fraction of the SMA wires from 0.20 up to 0.40. Here we see that positive effect of increasing of volume fraction of the SMA wires (essential stiffness) which results in improving of the prestressing of the structure (acquired stiffness) reduce from a special value (here 0.40). So increasing of the essential stiffness of a structure cannot be a positive factor at all. In the next step, we put the SMA wires only in two layers symmetrically with the volume fraction of the SMA wires equal to ksx = 0.00, 0.12, 0.16, 0.20. Fig. 7(a and b) shows the effect of embedding the SMA wires on the w/ h ratio and the contact force of the structure. Fig. 7(a) shows the maximum value of w/ h ratio decreases from 1.55 in the composite medium to 0.75 in the smart hybrid composite in which the SMA wires have the volume fraction of 0.20 (curve 4). Thus, 52% reduction is occurred. Another interesting result belongs to the maximum pick of these curves. As it is visible from Fig. 7(a), there is not only one pick point in each curve. Except the curve 4, in all other curves the first pick point is not the maximum one. Because the first pick point is happened nearly at the end of the contact time (CT) of the impact phenomena (Fig. 7(b)), it can be concluded that by choosing the appropriate volume fraction of the SMA wires, the decrease of the oscillation motion of the structure will be the most. Fig. 7(a) shows this movement of the maximum pick points of the curves 2–4 from the right side of Fig. 7(a) to the left side of it. As it is evident in Fig. 7(b), by increasing the volume fraction of SMA wires, the maximum contact force (MCF) increases from 803 N in the case of composite without SMA wires to 1028 N in the case that the volume fraction of the SMA wires is equal to 0.20. The MCF was increased by 28%, while the maximum contact force time (MCFT) tends to move to the right side of the diagram by the distance d in Fig. 7(b), and the CT

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149

Fig. 8. Effect of the SMA wires orientation in the different SMA wires volume fractions, on the w/ h ratio of the structure.

Fig. 7. Effect of volume fraction of the SMA wires on: (a) w/ h ratio and (b) contact force history.

was increased from 371 to 444 ␮s. Thus, the shocking effect of the impact force transfers to the plate decreases and a weaker impact inflicted upon the structure. Therefore, reducing the damage imparted to the structure, increasing the damage tolerance properties (Fig. 8). Here we see that embedding the SMA wires and so a positive prestress in the structure results in to improve the essential and the acquired stiffness of the structure and improving the impact resistance of the structure. However, it is visible that by increasing the volume fraction of the SMA wires and changing the essential stiffness of the structure, the acquired stiffness increases too and so the impact resistance of the structure increases. Here, we see the dependent of the essential stiffness and the acquired stiffness (an interactive effect). 7.2. Non-interactive effect of the stiffnesses This section dealt with the effect of the orientation and the through thickness location of the SMA wires inside the composite structure. Since the SMA wires with volume fraction of 0.20, reinforced the structure resistance more effectively, at this stage, the volume fraction is chosen 0.20 and they are embedded only in two layers symmetrically. The SMA wires are embedded

along the x-axis. Table 2 shows the effect of through thickness location of the SMA wires in the structure. Because the volume fraction of the SMA wires is constant, the positive prestress in the structure is the same for all cases. This means that the amount of increasing the acquired stiffness of the structure is the same for all cases but the essential stiffness of the structure changes by changing the location of the SMA wires through the thickness of the structure (non-interactive effect). As it is obvious, embedding the SMA wires along the x-axis and in layers 5 and 6 results in the most reduction effect of w/ h ratio of the structure (52%) in comparison with the embedding them in all other cases, specially in comparison with embedding them in all layers (37%). This means that at least for small volume fraction of the embedded SMA wires in the structure, the through thickness location, and the orientation of the SMA wires in the structure has an important role in improving the impact resistance of the structure. As it can be seen from Table 2, this reduction in case 3 is 0.75 and in case 6 is 0.97, which shows a 22% different. Therefore, by increasing the essential stiffness of the structure, we can increase the total stiffness of the structure without changing of its acquired stiffness. Table 2 The SMA wires are embedded in the structure [0 90 0 90 0]s 0.20 SMA wires

(1) Direction x, layers 1, 10 (2) Direction x, layers 3, 8 (3) Direction x, layers 5, 6 (4) Direction y, layers 2, 9 (5) Direction y, layers 4, 7 (6) Direction x and y, all layers

Data Max. w/ h

Max. F (N)

MCFT (␮s)

CT (␮s)

0.93

1980

112

226

0.79

1027

221

448

0.75

1026

216

444

0.90

1028

219

451

0.92

1027

217

445

0.97

1101

204

413

126 124 244 247 129 126 249 246 244 246 180 239 2345 2344 1221 1222 2346 2345 1223 1222 1222 1221 1644 1423 0.61 0.56 0.61 0.54 0.68 0.69 0.62 0.61 0.60 0.63 0.63 0.61 324 324 485 486 325 324 487 486 485 484 386 428 160 160 232 232 158 163 233 232 236 231 188 208 1630 1630 1126 1026 1630 1628 1127 1127 1126 1126 1383 1257 0.59 0.62 0.69 0.63 0.78 0.75 0.75 0.75 0.73 0.71 0.66 0.70 359 358 445 446 359 358 446 445 445 445 390 413 177 174 213 217 180 177 220 220 220 220 193 204 1261 1261 1026 1026 1261 1261 1026 1026 1026 1026 1159 1101 0.66 0.68 0.73 0.82 1.03 1.00 1.05 1.06 1.03 1.04 0.70 0.97 Direction x, layers 1, 3, 8, 10 Direction x, layers 1, 5, 6, 10 Direction x, layers 3, 8, 5, 6 Direction y, layers 2, 4, 7, 9 Direction x and y, layers 1,2,9,10 Direction x and y, layers 1, 4, 7, 10 Direction x and y, layers 2,3,8,9 Direction x and y, layers 2, 5, 6, 9 Direction x & y Layers 3, 4, 7, 8 Direction x and y, layers 4, 5, 6, 7 Direction x, layers 1, 3, 5, 6, 8, 10 Direction x and y, all layers

MCFT (␮s) Max. F (N) Max. w/ h Max. w/ h Max. F (N)

MCFT (␮s)

CT (␮s) Max. w/ h

Max. F (N)

MCFT (␮s)

CT (␮s)

Column three: 0.40 SMA Column two: 0.30 SMA Data SMA wires

In this part, we want to show the effect of using the SMA wires, which increases the essential and the acquired stiffnesses of the structure together (here in an interactive effect mode) with the case of using the steel wires, which only increases the essential stiffnesses of the structure (in a non-interactive mode). Table 4 shows the effect of embedding the SMA wires and the steel wires in both x and y direction in the structure. Both of the wires are embedded in replace of the glass fibers in the composite medium from a volume fraction of 0.00 up to 1.00. It can be seen from Table 4 that embedding 0.40 of the SMA wires results in 61% reduction in w/ h ratio of the structure. This reduction is origin from increasing of the essential and the acquired stiffness of the structure simultaneously (an interactive effect). It may also be seen that embedding 40% of the steel wires results in

Table 3 The SMA wires are embedded in the structure [0 90 0 90 0]s

7.3. A comparison between the interactive and the non-interactive effect of the stiffnesses

CT (␮s)

To study the non-interactive effect of the stiffnesses more, we put the SMA wires in more than two layers and in different effective volume fractions (it means from 0.20 up to 0.40 as described in the last section) as it can be seen in Table 3. Comparing Table 2 and column one of Table 3, which have the same acquired stiffness but the different essential stiffness, presents us some interesting results. Embedding the SMA layers in only x or only in y direction but in more than two layers seems to be more effective than embedding them in only two layers. This is not surprising because by embedding the SMA wires in more than two layers, the essential stiffness of the structure distributes more uniformly so the in-plane strains and stresses of the structure are distributed in a more uniform way and the impact resistance of the structure improves too. Here, we see again that embedding the SMA wires in x direction is more beneficial because of the greater negative effect of thermal stresses in x direction, which can be decreased by embedding the SMA wires in x direction more. This could be an important factor during the optimization procedures. Looking at the columns two and three of Table 3 leads us to more interesting results. The column two and three data show that the more positive effect of orientation of the SMA wires change from x direction to y. As it is visible in column three, embedding the SMA wires in the y direction can be a little better (65% decrease of w/ h of the structure from the case of no SMA wires in the structure) than embedding them in the x direction (59%) or embedding them in the all layers (61%). This means that on the contrary to the low volume fractions of the SMA wires (here 0.20), when the volume fraction of the SMA wires increase in the structure, the negative compression thermal effect of the composite medium decrease, so embedding the SMA wires in y direction improves the impact resistance of the structure more. This is not strange, because the volume fraction of the fibers of the composite medium in the y direction is less than the x direction, so embedding the SMA wires in the y direction may improve the impact resistance of the structure a little more. Again in a constant acquired stiffness and only by changing of the essential stiffness of the structure (non-interactive effect), we see that we can reach the most beneficial properties of the structure.

256 255 527 534 256 255 538 533 528 524 370 437

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Column one: 0.20 SMA

150

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151

Table 4 Effect of the SMA and the steel wires which are embedded in the glass-epoxy structure [0 90 0 90 0]s Volume fraction

0.00 0.20 0.40 0.60 0.80 1.00

Data SMA wires

Steel wires

Max. w/ h

Max. F (N)

MCFT (␮s)

CT (␮s)

Structure weight (kg)

Max. w/ h

Max. F (N)

MCFT (␮s)

CT (␮s)

Structure weight (kg)

1.55 0.97 0.61 0.56 0.51 0.43

803 1101 1423 1805 2323 3221

183 204 212 204 188 155

371 413 437 433 399 325

0.20 0.29 0.40 0.50 0.60 0.70

1.55 1.50 1.44 1.39 1.30 0.91

803 1168 1560 2047 2788 4634

183 206 212 200 179 125

371 417 429 412 362 253

0.20 0.32 0.45 0.58 0.71 0.84

only 7% reduction in w/ h ratio of the structure. This reduction is origin only from increasing of the essential stiffness of the structure (a non-interactive effect). It is noticeable that embedding the steel wires instead of the SMA wires will result in a 12.5% more increase of the structure weight, which is a negative effect in designing of the structures too. Again, we see that the stiffening of a structure must be done in a smart procedure. Because increasing the stiffness of a structure by embedding the steel wires in it (only increasing the essential stiffness) can be less effective than embedding the SMA wires in it. The stiffness of the SMA wires is less than 35% of the steel wires (Table 1) but they can decrease the w/ h ratio of the structure 54% more by increasing the total stiffness of the structure with exerting positive prestressing in it (increasing the essential and the acquired stiffness of the structure together). 8. Conclusions Response of low-velocity impact upon smart hybrid composite structures was modeled using the first-order shear deformation theory and Fourier series method to solve the system of governing differential equations of the plate analytically. The interaction between the impactor and the plate is also modeled with a system having two degrees-of-freedom, consisting of springs-masses. The model accuracy was verified with results of the published papers. The results of the above research also demonstrated that using of the SMA wires within the traditional hybrid composite plates improves the global behavior of the structure against the impact. The plate with the SMA wires damp more uniformly and rapidly than the plate without the SMA wires after the impact. The results indicated that some parameters like the volume fraction, the orientation and the through thickness location of the SMA wires are important factors affecting the impact process and the design of the structures. The volume fraction of the SMA wires could affect the MCF, the MCFT, and the CT. Hence, the shock effect of the impact imparted to the plate can be changed accordingly. It was also seen that the orientation and the through thickness location of the SMA wires are dependent of the volume fraction of the SMA wires. A new procedure named smart stiffening procedure introduced for smart stiffening of the smart structures in a best way.

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