Non-linear load-deflection response of SMA composite plate resting on winkler-pasternak type elastic foundation under various mechanical and thermal loading

Thin-Walled Structures 129 (2018) 391–403

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Non-linear load-deflection response of SMA composite plate resting on winkler-pasternak type elastic foundation under various mechanical and thermal loading Behrang Tavousi Tehrani, Mohammad-Zaman Kabir

T

⁎

Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

A R T I C LE I N FO

A B S T R A C T

Keywords: Lateral pressure Shape memory alloy Brinson’s model Galerkin technique Preliminary imperfection

This study investigates the non-linear load-deflection behavior of a composite plate reinforced with Shape Memory Alloy (SMA) fibers. The plate is subjected to the uniform lateral pressure and thermo-mechanical loading while resting on a Winkler-Pasternak type elastic foundation. In SMA fibers, simple one-dimensional Brinson’s model is implemented to determine tensile recovery stresses due to the phase transformation. The nonlinear semi-analytical solution is formulated for the examination of the large deflections including the preliminary geometrical imperfection. The governing equations of equilibrium are derived in terms of displacement and stress function. The Galerkin technique is chosen to solve the nonlinear partial differential equations of motions. A detailed parametric study including different SMA material properties, SMA fiber pre-strain values, SMA fiber volume fraction, foundation stiffness, and activation temperature are examined.

1. Introduction In recent years, enormous progresses have been made to use smart materials for reinforcing structural systems. Shape memory alloys are greatly used in the engineering applications such as actuators, sensors, and damping devices. Researchers have focused on two main features of the SMA which are large transformation strain recovery and huge energy absorption capacity. The first one is known as shape memory effect and the second one is recognized as superelasticity characteristics. The shape memory effect (SME) can be used to generate the largely tensile recovery forces; these forces can reduce the deflection of structures under different loading conditions, enhance vibration response, and improve stability in reinforced composite structures. The other feature, the superelasticity effect, can be used in structural applications which require energy absorption. The shape memory alloys, through the SME feature, show temperature dependent properties that can be used to enhance the load-deflection behavior of composite structures. Composite plates are used in the airplane structural components, ships, aero vehicles, and cars. In such structures, the behavior of the plate under various thermo-mechanical loads is a major issue. Composite plates are employed in structural components, which are subjected to in-plane loading such as, wing skin structures, aircraft fuselage sections, and launch vehicle booster tankages. Also, Composite plates can be used in nuclear and petrochemical industries because of their inherent

⁎

highly specific stiffness and strength. In some structural plate components such as plate elements in the bottom of ships, lateral pressure significantly decreases the strength of the plate. This type of loading occurs due to the aerodynamic forces in aerospace vehicles, hydrostatic water pressure, and uniform blast load. Therefore, SMA reinforced plates can be used to enhance their load bearing capacity or stability. Composite plates containing embedded SMA fibers with simple configurations and desired features can be manufactured in a laboratory. Some literature survey about the static and dynamic response of SMA composite plates and shells is as follows: Thompson & Loughlan [1,2] conducted an experimental study on the post-buckling response of SMA composite plate. They inserted SMA wires through rubber sleeve tubes located in the plate’s midplane. In addition, they performed a numerical investigation on linear thermal buckling based on FEM using NASTRAN finite element package. The authors reported that in load levels nearly three times of critical buckling load, deflections were reduced considerably. The reduction happened even by incorporating a relatively low-volume fraction of SMA wires. Also, Thompson & Loughlan [3] carried out the control of post-buckling behavior in thin composite plates using the smart material. Hassanli and Samali [4] investigated the buckling of curved laminated composite panels reinforced with SMA fibers. The panels were studied under different geometric conditions. Ho et al. [5] performed

Corresponding author. E-mail addresses: [email protected] (B. Tavousi Tehrani), [email protected] (M.-Z. Kabir).

https://doi.org/10.1016/j.tws.2018.04.017 Received 11 May 2017; Received in revised form 15 April 2018; Accepted 23 April 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 129 (2018) 391–403

B. Tavousi Tehrani, M.-Z. Kabir

Nomenclature

x, y, z a, b, c ξ, ξ s, ξ T ξ 0, ξ s0, ξ T0

Es (ξ), EA , EM

Gs (ξ) A s , Af Ms , Mf

CA , CM f σ scr, σ cr σ ε,εL , ε 0 Θ T, ΔT, ΔT*

α s, Vs

E1, E1m E2 , E2m

G12 , G12m α1, α1m

α2, α2m

υ12 , υ12m , υs

u 0 , v0, w0 ε x , ε y , γxy ε0x , ε0y , γ0xy

Cartesian coordinates Length, width, and total thickness of the plate Total, stress-induced, and temperature induced martensite fraction Initial total, Initial stress-induced, and Initial temperature induced martensite fraction SMA’s Young modulus, SMA’s Young modulus in austenite phase, and SMA’s Young modulus in martensite phase Shear modulus of SMA Austenite start temperature, austenite finish temperature Martensite start temperature, martensite finish temperature Brinson’s model constant parameters Critical phase transformation stresses Stress in SMA Strain, maximum recoverable strain, and initial residual strain of SMA Thermo-elastic tensor in SMA Temperature, temperature change (activation temprature), reference buckling temperature SMA thermal expansion coefficient, SMA volume fraction Young modulus and matrix Young modulus in direction 1 Young modulus and matrix Young modulus in direction 2 Shear modulus and shear modulus of matrix Thermal expansion coefficient and matrix thermal expansion coefficient in direction 1 Thermal expansion coefficient and matrix thermal expansion coefficient in direction 2 Poisson’s ratio, matrix Poisson’s ratio, and SMA Poisson’s ratio

Displacement components of mid-surface Strain components Strain components of mid-surface

κ 0x , κ 0y , κ 0xy φx , φy γxz , γyz σx , σy , τxy Q ij (i, j = 1, 2, 6) α x, α y

Curvatures Rotation terms about the y -axis and x -axis Transverse shear strains Stress components in the laminate Reduced stiffness matrix Thermal expansion coefficient in direction x and y Reference thermal expansion coefficient α0 σ rx , σ rx , σ r SMA recovery stress in x and y direction, SMA recovery stress in its fiber direction In-plane force resultants of the plate Nx , Ny , Nxy Mx , My , Mxy Moment resultants of the plate Nrx , Nry Recovery force of the SMA fibers NTx , NTy Thermal force resultants of the plate Uniform edge compressive loads Nx0 , Ny0 Shear force resultants Q x , Qy Aij, Dij (i, j = 1, 2, 4, 5, 6) Total extensional (in-plane) and bending plate stiffness aij Inverse of total extensional stiffness t p, N Thickness of ply, total number of plies K Shear correction factor w *0 Initial geometric imperfection μ Imperfection amplitude Φx , Φy Functions of displacements in rotation function m, n Number of half-waves in x and y directions q ef , K1, K2 Foundation reaction, Winkler and Pasternak elastic foundation modulus k1, k2 Dimensionless forms of Winkler and Pasternak elastic foundation modulus Q Uniform lateral pressure F Airy stress function ∆x, ∆y Average end-shortening displacement E Young modulus of isotropic material λT, λ x Dimensionless forms of Thermal and inplane compressive load R Biaxial ratio, Ny /Nx

than the other locations. The response of an SMA composite plate under the effect of thermal and aerodynamic loads was examined by Ibrahim et al. [8]. Rasid et al. [9] and Park et al. [10] used finite element method to study thermal post-buckling of SMA composite plates. A different finite element method was used by Tawfik et al. [11] to examine the stability behavior of SMA composite panels. In addition, VonKarman nonlinear strains were considered in the formulations. A 3-D layer-wise displacement theory was applied by Kumar and Singh [12] to analyze the thermal post-buckling composite plate reinforced with SMA fibers. Turner [13] examined a finite element analysis on the thermoelastic response of SMA hybrid structures. Duan et al. [14] implemented the finite element method for predicting critical buckling temperature and post-buckling deflection of composite plates reinforced with the shape memory alloys. An incremental method consisting of small temperature increment was presented in the study. Another FEM method was employed to determine critical buckling temperature, post-buckling behavior, and vibration about the buckled equilibrium position of SMA composite plate by Guo and Lee [15]. The results demonstrated that it was feasible to suppress the thermal deformation by using a suitable percentage of SMA volume fraction. Peraza Hernandez et al. [16] studied the structurally stable

an experimental study to investigate the thermal post-buckling of isotropic and composite panels embedded with shape memory alloy. The experiment revealed that when the temperature increased above the critical buckling level, the passive plates with embedded SMA strips showed less deformation than that of plates without SMA. Promising results were obtained by reduction in buckling deflection of Aluminum/ SMA sandwich panels. Asadi et al. [6] studied the nonlinear thermal stability of imperfect laminated composite plates based on higher order shear deformation theory. They presented a closed-form formulation to determine the load-deflection path for a rectangular plate with simply supported edges. They also reported that SMA fibers could stabilize imperfect composite plates when they were embedded with relatively high SMA volume fraction. When the critical buckling temperature was lower than the austenite start temperature, SMA recovery forces stabilized the plate. Shiau et al. [7] studied the post-buckling reinforced laminated plates with SMA by utilizing nonlinear finite element method. They used experimental data for SMA recovery stress. The results revealed that an increase in SMA fiber volume fraction and prestrain led to generate higher tensile recovery force. The recovery force increased the plate stiffness in the post-buckling region. The concentration of SMA fibers in the center of the plate was more effective 392

Thin-Walled Structures 129 (2018) 391–403

B. Tavousi Tehrani, M.-Z. Kabir

design process of structures. Ganilova and Cartmell [39] presented a model for the vibration analysis of a composite plate embedded with shape memory alloy. The work offered an analytical solution using the hybrid WKB-Galerkin method. Some dynamic properties of a cantilever SMA composite beam with SMA wires in the straight and zig-zag configuration have been investigated by John and Hariri [40]. The governing equations solved analytically and numerically. The SMA wires caused a momentary shift in natural frequencies and reduction in the amplitude of vibration. Control of natural frequencies of a clamped supported composite SMA beam has been studied by Lau et al. [41]. Based on the Euler-Bernoulli, Timoshenko, and third order beam theory, analytical investigation of the vibration response of SMA beams have been performed by Barzegari et al. [42]. Samadpour et al. [43] presented a study on the non-linear free vibration of thermally buckled sandwich SMA plates to reduce the deflections. They reported that an increase in SMA pre-strain and volume fraction led to a decline in natural frequencies. Li et al. [44] presented an analytical investigation of the vibrations of thermally pre/post-buckled thin circular plates reinforced with SMA fibers. They concluded that the thermal buckling and the vibration response of the plate could be controlled by adjusting the SMA volume fraction, SMA pre-strain, and temperature rise. Soltanieh et al. [45] proposed an accurate algorithm for behavior investigations of super-elastic SMA composite plates under impulse loading. Based on the presented literature review, the following points are realized:

configurations in shape memory alloy actuated plates through an analytical approach. They used Rayleigh-Ritz method in their work and provided a closed-form expression for the deflection of the plates. Birman [17] presented a linear stability analysis of the FGM/SMA sandwich panels using Galerkin procedure. Asadi et al. [18] developed a third-order shear deformation theory to study non-linear thermal instability of moving SMA/FG sandwich. They used original Brinson’s model and Galerkin technique to model the plate behavior. In a recent study, Kabir and Tavousi Tehrani [19] have focused on thermo-mechanical buckling and post-buckling of SMA composite plates through mathematical analysis. Balapgola et al. [20], Cho and Rhee [21], Daghia et al. [22], and Lee and Lee [23] used finite element approach to investigate deflection response of SMA composite plate, deflection response of cantilever SMA composite shell due to the thermal loads, Shape and stiffness control of SMA composite plates, and post-buckling of SMA composite shells, respectively. Asadi et al. [24] proposed an exact solution for the non-linear thermal stability of hybrid laminated composite Timoshenko beams reinforced with SMA fibers. Asadi et al. [25] used Euler-Bernoulli beam theory to investigate vibration response and thermal stability of SMA composite beams through an analytical approach. Ghomshei et al. [26] reported the non-linear transient response of a thick composite beam with SMA layer using finite element model. Based on the variational asymptotic method, Ren and Sun [27] presented an analytical model for a study on a thin-walled laminated box beam embedded with shape memory alloy fibers. The results revealed that SMA fiber volume fraction and actuation temperature significantly influenced the non-linear displacement of the beam. Choi et al. [28] investigated experimentally and analytically the thermal buckling and post-buckling response of a composite beam which was actuated by the shape memory alloy wires. The results proved that SMA wire actuators could be utilized to study the thermal instability problem of a real composite beam under intense heating environments. Lee et al. [29] presented an analytical model of the SMA-Composite beam using the alleged cut and paste method to predict its thermal buckling and post-buckling behavior. Roh et al. [30] performed FEM to investigate the thermal post-buckling behavior of SMA hybrid composite shell panels based on the layer-wise theory. Abdollahi et al. [31] carried out an analytical research on the nonlinear thermal stability of SMA wire-embedded hybrid laminated composite beam resting on elastic foundation. The theory of Timoshenko beams with Von-Karman non-linear strains was used in their study. To discretize the equilibrium equation, they utilized the generalized differential quadrature method. The post-buckling curves were plotted for different values of the SMA volume fraction, the magnitude of SMA wire pre-strain, and non-linear elastic foundation parameters. The constitutive behavior of SMA wires was captured based on the Brinson’s model. Any increase, either in the SMA volume fraction or in the SMA pre-strain led to decrease in lateral deflection in the postbuckling curve. Marfia and Sacco [32] studied the effect of super-elastic shape memory alloy in SMA composite beams. Soltanieh et al. [33] investigated the snap buckling of the shallow SMA hybrid composite cylindrical shells. Many works have been carried out to investigate the impact and vibration response of SMA composite structures. Forouzesh and Jafari [34] presented the radial vibrations of simply supported super-elastic SMA cylindrical shells under internal pressure. In their study, the SMA behavior was simulated via the Boyd-Lagoudas model. Khalili et al. [35] studied low-velocity impact response of SMA composite plate by using an analytical approach. Niknami and Shariyat [36] studied the impact response of SMA composite plates using a refined contact law. Shariyat and Farzan Nasab [37] used a semi-analytical method to investigate low-velocity impact response of an FGM plate with partially supported edges and a Winkler-type elastic foundation. Shokuhfar et al. [38] presented an analytical and mathematical model to demonstrate the effect of low-velocity impact upon the SMA composite plate. They found that the SMA volume fraction had an important effect on the

1. According to the mentioned studies, the lack of a solution using Galerkin technique and Airy stress function to investigate the loaddeflection response of the SMA composite plate under simultaneous thermal and mechanical load is obvious. 2. The influence of only one kind of SMA material was considered until now. The effect of different kind of SMA materials still needs further investigations. 3. Some loading types were thermal and the others were mechanical alone; but in the real world, structures are subjected to the combination of loads. For instance, the composite panels are loaded with mechanical forces while simultaneously confronting thermal shocks or lateral pressures. Also, the present work has focused to consider mechanical lateral pressure while the plate is supported on all edges. 4. The authors have not received any reported work on the influence of both elastic foundation (specifically two-parameter elastic foundation or Winkler-Pasternak elastic foundation [46]) and SMA on the load-deflection behavior of a composite plate. 5. Many researchers have studied mainly the buckling and post-buckling (stability), vibration analysis, and impact response of SMA composite structures. Due to lack of any results on the load-deflection response (bending behavior) of SMA composite plates, the authors have been encouraged to consider the present practical issue. According to the existing literature, the lack of any study on the non-linear load-deflection response of initially imperfect symmetric SMA composite plates under different loading conditions, while resting on a two-parameter elastic foundation, has motivated the authors to study this subject. Plate resting on an elastic foundation is a common model for several types of practical engineering problems and real-life applications. For instance, plates on elastic support models are always used to analyze shell panels in ships and aircraft vehicles, buckling and bending of the face sheets of sandwich panels, and the foundation of building and column supports. The authors have been motivated to consider the influence of a proper elastic foundation model in the present work. 2. One-dimensional Brinson’s constitutive for SMA modeling The Brinson’s one 393

dimensional

model made

a

significant

Thin-Walled Structures 129 (2018) 391–403

B. Tavousi Tehrani, M.-Z. Kabir

Table 1 Thermo-mechanical properties of the two different SMA materials.

improvement to the Tanaka’s model and the Liang and Rogers’s model [47]; It shows a simple constitutive equation to simulate the characteristics of the SMA. In Brinson’s model, the total martensite fraction (ξ ) is sum of the stress-induced martensite fraction (ξ s ) and the temperature induced martensite fraction (ξ T ) as:

ξ = ξ s+ ξ T

(1)

The Young modulus of SMA is assumed to be based on Reuss definition [48] as:

(

1+

)ξ

EA −1 EM

SMA#1 [47].

SMA#2 [50].

(EA , EM) GPa (Ms , Mf ) °C (A s, Af ) ℃ (CA , CM) MPa/℃

(67, 23.6) (18.4, 9) (34.5, 49) (13.8, 8) (100, 170)

(31.5, 20) (48.4, 43.9) (68, 73.85) (6.32, 6.73) (25, 78)

f MPa (σ scr, σ cr ) Θ MPa/℃

α s ℃−1 (υs , εL , ξ T0)

EA

Es (ξ) =

Material Property

0.55

0.55

10.26×10−6 (0.33, 0.067, 0)

10.26×10−6 (0.33, 0.041, 0)

(2)

where EA and EM are the SMA’s Young modulus in the 100% austenite phase and the 100% martensite phase, respectively. The tensile recovery stress of SMA during phase transformation and under restrained conditions (embedded in the composite matrix), can be calculated by using the simple Brinson’s model [49] as:

σ = E(ξ)(ε−εL ξ s) + Θ∆T

(3)

where σ , ε , Θ, ∆T and εL refer to the stress, strain, thermo-elastic tensor, temperature change, and maximum recoverable strain, respectively. The reference temperature is assumed to be 20 ℃. Calculation of the phase transformation from martensite to austenite can be expressed [47] as:

for T>A s, CA (T−Af ) σ CA (T−A s) ξ=

ξ0 ⎧ ⎡ π σ ⎤ ⎫ cos (T−A s − ) +1 ⎥ ⎬ ⎢ 2⎨ A A C − f s A ⎦ ⎣ ⎭ ⎩

ξ s = ξ s0 − ξ T = ξ T0 −

ξ s0 ξ0

Fig. 2. SMA#1 temperature-dependent elastic modulus.

(ξ 0−ξ)

ξ T0 ξ0

(ξ 0−ξ)

(4)

where the subscript “0” denotes the initial condition of the parameter. The constant CA shows the relationship between temperature and critical phase transformation stress in Brinson’s model. A s , Af , and T stand for austenite start temperature, austenite finish temperature, and temperature, respectively. As the SMA fibers are restrained to freely recover their initial strains (SMA fibers are restrained in composite medium), large tensile recovery stress is generated during phase transformation. This recovery stress can be used to control the load-deflection behavior of a composite plate. Consider a rectangular laminated composite plate with plane dimensions of a × b and total thickness of t . Fig. 1 shows a diagram of SMA composite plate over a two-parameter elastic foundation and its coordinates system. The plate is on continuous spring elements. A shear layer interacts between the spring elements by connecting the ends of the springs to a plate which deforms only by transverse shear [46]. Table 1 provides two SMA material properties.

Fig. 3. SMA#2 temperature-dependent elastic modulus.

Figs. 2 and 3 show the temperature-dependent elastic modulus of an SMA fiber during the phase transformation designated by SMA#1 and SMA#2. The curves have been plotted using Eqs. (1)-(4) for different SMA pre-strain magnitudes. The SMA fiber is embedded in the composite matrix. Fig. 4 shows the stress-strain curves of the SMA and experimental data obtained from work done by Liang [51] and Sayyaadi et al. [50] for SMA#1 and SMA#2, respectively. There is a good agreement between the Brinson’s Model and the experimental work for engineering applications. Figs. 5 and 6 show the SMA recovery stress for different pre-strain values based on the simple Brinson’s model Eq. (3). Also, the results for pre-strains equal to 1.34%,1%, and 0.5% are presented as per by Lee et al. [52], Roh et al. [30] and Abdollahi et al. [31]. Minor differences in the magnitude of recovery stresses versus validating data are due to implementing of new simplified Brinson’s formulation (Eq. (3)) instead of original Brinson’s formulation [47] to determine the recovery stress.

Fig. 1. The SMA composite plate resting over a two-parameter elastic foundation and its coordinates system. 394

Thin-Walled Structures 129 (2018) 391–403

B. Tavousi Tehrani, M.-Z. Kabir

α1 =

Es (ξ)Vsα s + E1m (1−V)α s 1m Es (ξ)Vs + E1m (1−V) s

α2 =

⎡ α2m Vs − Vs (α2m − α s) ⎤ E2m ⎢ ⎥ α2m (1− Vs ) + E2m ⎥ E2 ⎢ − − 1 V 1 s ⎢ ⎥ Es (ξ) ⎣ ⎦

Gs (ξ) =

(9)

(

)

Es (ξ) 2(1+υs )

(10)

(11)

where the “m” and “s” subscripts stand for the composite matrix and SMA fibers, respectively. Also, material parameters E1, E2 , G12 , υ12 , α1, α2 , Gs (ξ) and Vs denote the Young modulus, shear modulus, Poisson’s ratio, thermal expansion coefficient in fiber direction, thermal expansion coefficient perpendicular to the fiber direction, SMA shear modulus and SMA volume fraction, respectively. In the SMA composite plate, SMA fibers are aligned with the direction of graphite fibers. In this study, the first-order shear deformation plate theory is used to analyze the load-deflection response of SMA composite plate. The nonlinear strains through the plate thickness at a distance z from the mid-plane are [54]:

Fig. 4. Stress-strain curves of SMA#1 and SMA#2.

0 0 ⎡ εx ⎤ ε ⎡ κx ⎤ ⎡ x⎤ ⎢ 0 ⎥ ⎥ ⎢ 0 ⎢ εy ⎥ = ⎢ εy ⎥ + z ⎢ κ y ⎥ ⎢ γxy ⎥ ⎢ 0 ⎥ ⎢ κ0 ⎥ γ ⎣ ⎦ ⎣ xy ⎦ ⎣ xy ⎦

∂w *0 ∂w0 ∂u 0 1 ∂w 2 ⎤ ⎡ + ⎛ 0⎞ + ⎥ ⎢ ∂x ∂x ∂x 2 ⎝ ∂x ⎠ 0 ⎥ ⎡ εx ⎤ ⎢ 2 ⎥ ∂w *0 ∂w0 ⎢ ε0 ⎥ ⎢ ∂v0 1 ⎛ ∂w0 ⎞ = + ⎜ + ⎟ ⎥ ⎢ y⎥ ⎢ ∂ ∂ ∂ ∂ y 2 y y y ⎝ ⎠ ⎥ ⎢ γ0 ⎥ ⎢ xy ⎣ ⎦ ⎢ ∂u 0 ∂w *0 ∂w0 ∂w *0 ∂w0 ⎥ ∂v0 ∂w0 ∂w0 ⎥ ⎢ + + + + ∂y ∂x ∂y ∂x ∂y ∂y ∂x ⎥ ⎢ ⎦ ⎣ ∂y ∂φx ⎡ ⎤ ⎢ ⎥ ∂x 0 ⎥ ⎡ κx ⎤ ⎢ ∂ φ y ⎥ ⎢ κ0 ⎥ ⎢ = ⎥ ⎢ y⎥ ⎢ ∂y ⎥ ⎢ κ0 ⎥ ⎢ xy ∂ φ ⎦ ⎢ ∂φx ⎣ y ⎥ + ⎢ ⎥ ∂x ⎦ ⎣ ∂y

Fig. 5. SMA#1 recovery stress.

⎡ φ + ∂w0 ⎤ x γ ∂x ⎥ xz ⎡ ⎤=⎢ ⎢ γyz ⎥ ⎢ ∂w0 ⎥ ⎣ ⎦ ⎢ φy + ⎥ ∂y ⎦ ⎣

where ε x and ε y are the normal strains and γxy is the shear strain. u 0 and v0 are plate displacements in the x and y directions and w0 is the lateral displacement. φx and φy represent rotation terms about the y and x axes. Stress-strain in a laminate reinforced with SMA fibers (in global coordinates), is [55]:

Fig. 6. SMA#2 recovery stress.

3. Constitutive equations

r σ Q Q 0 ⎤ ⎛ ⎡ εx ⎤ α ⎡ σx ⎤ ⎡ x ⎤ ⎡ 11 12 ⎡ x⎤ ⎞ r ε ⎢ σy ⎥ = ⎢Q12 Q22 0 ⎥ × ⎜ ⎢ y ⎥ − ⎢ α y ⎥ ∆T⎟ + Vs ⎢ σ y ⎥ ⎥ ⎜⎢ γ ⎥ ⎢ ⎥ ⎟ ⎢ τxy ⎥ ⎢ 0 0 Q66 ⎦ ⎝ ⎣ xy ⎦ ⎢ ⎣0⎥ ⎦ ⎠ ⎣ ⎦ ⎣ ⎣0⎦

In this study, the effective engineering parameters, E1, E2 , G12 , υ12 , α1 and α2 are based on the multi-cell micromechanics approach [53] as:

E1 = Es (ξ)Vs + E1m (1−V) s

(

)

⎤ ⎥ ⎥ ⎥ ⎦

⎡ Vs G12 = G12m ⎢ (1− Vs ) + G ⎢ 1− Vs 1− G12m ⎢ s (ξ) ⎣

(

υ12 = υs Vs + υ12m (1−V) s

)

⎤ ⎥ ⎥ ⎥ ⎦

(13)

where Q ij defines the reduced stiffness matrix [54] and σ r is the SMA recovery stress. The stress resultants are defined as [55]:

(5)

⎡ Vs E2 = E2m ⎢ (1− Vs ) + E2m ⎢ 1 − V s 1− E (ξ) ⎢ s ⎣

(12)

σ ⎡ Mx ⎤ ⎞ t ⎡ x⎤ ⎢ My ⎥ ⎟ = ∫ 2t ⎢ σy ⎥ (1, z)dz − ⎢ ⎥ 2 ⎢ τxy ⎥ M ⎟ ⎣ ⎦ ⎣ xy ⎦ ⎠

(6)

⎛ ⎡ Nx ⎤ ⎜ ⎢ Ny ⎥, ⎜ ⎢ Nxy ⎥ ⎦ ⎝⎣

(7)

⎡Q x ⎤ = ∫ 2 ⎡ τxz ⎤ dz − t ⎢ τyz ⎥ ⎢Qy ⎥ 2 ⎣ ⎣ ⎦ ⎦

t

(8) 395

⇒

⎡ φ + ∂w0 ⎤ x ∂x ⎥ ⎡Q x ⎤ = K ⎡ A 44 A 45 ⎤ ⎢ ⎢ ⎢Qy ⎥ ⎢ A 45 A55 ⎥ ∂w0 ⎥ ⎣ ⎦ ⎣ ⎦ ⎢ φy + ⎥ ∂y ⎦ ⎣

(14)

Thin-Walled Structures 129 (2018) 391–403

B. Tavousi Tehrani, M.-Z. Kabir

where Nx , Ny , Nxy , Mx , My , Mxy , Nrx and Nry are the force resultants, Moment resultants, and the recovery forces of the SMA, respectively; Also, each ply has the same thickness of tp . The shear correction factor K = 5/6. The recovery and thermal force resultants are [54]: r

⎡ Nx ⎤ ⎢ Nry ⎥ = ⎢ ⎥ ⎣0⎦

− w0,xx w *0,yy – w0,yy w *0,xx

r

N

⎡σ x ⎤

p=1

⎢ ⎥ ⎣0⎦

N

∑ p=1

a11F,yyyy + a22F,xxxx − (2a12 + a 66)F,xxyy = w0,xy 2 − w0,xx w0,yy + w0,xy w *0,xy

⎡ Q11α x + Q12 α y ⎤ ∫tp − 1 ⎢Q12 α x + Q22 α y ⎥ ΔTdz ⎢ ⎥ 0 ⎣ ⎦

− w0,xx w *0,yy – w0,yy w *0,xx

tp

r T ⎡ N3 × 1 ⎤ = ⎡ A3 × 3 0 ⎤ ⎡ ε3 × 3 ⎤ − ⎡ N3 × 3 ⎤ + ⎡ N3 × 3 ⎤ ⎢ M3 × 1⎥ ⎢ 0 D3 × 3 ⎥ ⎢ κ3 × 3 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎣ ⎦ ⎣ ⎦ ⎦⎣ ⎦ ⎣ ⎦ ⎣

(15)

4. Methodology In this study, two types of in-plane boundary conditions are considered:

• SS-1 case

The SS-1 denotes the boundary conditions that plate edges are free to move. The boundary conditions correspond to assuming roller supports along the edges. The uniform edge compressive loads are as:

(16)

Aij and Dij are the plate stiffnesses defined as: t

∫− 2t [Qij](1,z2)dz

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

2

Nx0 = Nx , Ny0 = Ny , R = Ny /Nx (17)

Where R is the biaxial ratio.

The SS-2 expresses the immovability at all edges for in-plane conditions. These conditions are identical when the average end-shortening displacements vanish.

w0 = Wsin

−1 b a ∂u 0 dxdy = 0, ∫ ∫ b 0 0 ∂x −1 b a ∂v0 ∆y = dxdy = 0 ∫ ∫ a 0 0 ∂y ∆x =

4.1. Thermal loading

(19)

Composite laminated structures are often subjected to thermal loading while supported by an elastic foundation. In this study, the main assumptions in the non-linear load-deflection response of SMA composite plate under thermal loading are as:

where q ef is the foundation reaction force per unit area, K1 and K2 are the Winkler and Pasternak elastic foundation modulus, respectively. The equilibrium equations of an imperfect composite plate resting on an elastic foundation are [6]:

• Plates are rectangular with simply supported edges SS-2 (Immovable

Nx,x + Nxy,y = 0 Nxy,x + Ny,y = 0 Mx,x + Mxy,y − Q x = 0 My,y + Mxy,x − Qy = 0

edges).

Table 2 Thermo-mechanical properties of the Composite matrix.

Q x,x + Qy,y +N x (w0,xx + w *0,xx) + 2Nxy (w0,xy + w *0,xy) + Ny (w0,yy + w *0,yy) + q −K1w0 + K2 (w0,xx + w0,yy) = 0

(20)

where q is lateral pressure loading. The Airy stress function F is defined to satisfy the first and the second equilibrium Eq. (20) [54]:

Nx = Fyy , Ny = Fxx , Nxy = −Fxy

(25)

where ∆x and ∆y are the average end-shortening displacement of the plate in the x and y directions, respectively. This boundary condition is equivalent to a pin support at the edges of the plate. The stress function can be determined using Eqs. (18), (20), (23) and the conditions in Eqs. (24) and (25). The load-deflection paths can be reached by determining the Airy stress function and applying Galerkin technique on the last equation in the equilibrium Eq. (20). These procedures are conducted through a MATLAB code. To achieve this goal, an appropriate admissible deflection function should be chosen to satisfy the boundary conditions.

(18)

where μ is the imperfection amplitude, m and n are the half-wave numbers. Φx and Φy are functions of W that can be fully determined. A temperature-dependent composite matrix material is used in the modeling of the SMA composite plate. The Graphite Epoxy material property is presented in Table 2. The load-deflection relation of the two-parameter elastic foundation is defined as [46]:

∂ 2w ∂2w0 ⎞ q ef = K1w0 − K2 ⎛⎜ 20 + ⎟ ∂y 2 ⎠ ⎝ ∂x

(24)

• SS-2 case

In the present study, simply supported at all edges as boundary conditions, is considered. The lateral deflection of the plate (w0 ), plate geometric imperfection (w *0 ), and the rotation functions (φx and φy ) are assumed to be:

nπy mπx sin b a nπy mπx w *0 = μtsin sin b a nπy mπx φx = Φx cos sin b a nπy mπx φy = Φy sin cos b a

(23)

Where a is the inverse of A, the extensional stiffness matrix.

where p and N represent the ply and total number of plies. By calculating the mathematical integrations above, the relations between stress resultants and strains can be expressed as:

(Aij, Dij) =

(22)

By substituting Eq. (16) into the compatibility Eq. (22), one can obtain the compatibility equation in terms of the Airy stress function and the lateral displacement component as follows:

t ∑ ∫tpp−1 ⎢σ ry ⎥ Vsdz

T

⎡ Nx ⎤ ⎢ NT ⎥ = ⎢ y⎥ ⎣0⎦

ε x,yy + ε y,xx − γxy,xy = w0,xy 2 − w0,xx w0,yy + w0,xy w *0,xy

(21)

The compatibility equation of the plate in terms of the strains and the lateral displacement component is as follows [54]: 396

Mechanical Parameter

Graphite/Epoxy [6].

E1m (GPa)

155(1 − 3.53×10−4∆T)

E2m (GPa)

8.07(1 − 4.27×10−4∆T)

G12m (GPa)

4.55(1 − 6.06×10−4∆T)

α1m (°C−1)

−0.07×10−6 (1 − 1.25×10−3∆T)

α2m (°C−1) υ12m

30.1×10−6 (1 + 0.41×10−4∆T) 0.22

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• All necessary information is clearly presented in the figures, such as plate dimensions, material properties, and foundation parameters. Plates are under uniform temperature rise (∆T ) and initial constant • uniform lateral pressure (q ). • Initial imperfection (μ ) is considered in all plates. • Plates are with or without elastic foundation (Winkler (K , 0)) or 1

Winkler-Pasternak (K1, K2 )).

4.2. In-plane mechanical loading In this study, the main considerations in the non-linear load-deflection of SMA strengthened composite plates under in-plane mechanical loading are as:

• Plates are rectangular with simply supported edges SS-1 (Movable edges). • Plates are under growing uniform in-plane edge compressive load • • • •

Fig. 7. Comparison of thermal load-deflection curves for an imperfect composite plate resting on a two-parameter elastic foundation.

(Nx ), initial constant uniform lateral pressure (q ) and initial thermal load (Activation temperature). Initial imperfection (μ ) is considered in all plates. Plates are with or without elastic foundation (Winkler (K1, 0) or Winkler-Pasternak (K1, K2 )). The first parametric analysis is conducted to show the influence of elastic foundation and lateral pressure. In the other parametric investigations, the results are presented for usage of SMA#1 and SMA#2, respectively.

the thermal load-deflection path of an SMA composite plate. The acronym “WOS” stands for “a plate Without SMA”. This figure is a plot of normalized thermal load versus the maximum dimensionless lateral displacement. The figure shows that the present method is also in good agreement with the work done by Asadi et al. [6]. 5.1.2. In-plane mechanical loading The load-deflection response of a composite plate under an increasing in-plane mechanical loading and constant lateral pressure while resting on an elastic foundation is verified through a comparison study. Fig. 9 shows the load-deflection curves of a cross-ply arrangement of composite plate compared to the work done by Shen [57]. The properties of the composite are stated in the figure. It simply shows that the comparison is well justified.

4.3. Lateral pressure loading Lateral pressure significantly decreases the strength of plates. This type of loading occurs as the aerodynamic forces in aerospace vehicles or hydrostatic water pressure. In this study, the main considerations in the non-linear load-deflection of SMA reinforced composite plates under this type of loading are as:

5.1.3. Lateral pressure loading As part of validation of the presented method, Fig. 10 illustrates the load-deflection curves of an isotropic plate under uniform lateral pressure load by neglecting the foundation interaction. The reference works are those done by Shen [58], Iyengar & Naqvi [59], Gorji [60] and Levy [61]. In addition, the effect of two types of in-plane boundary conditions is presented. Obviously, Fig. 10 shows good agreement between the current method and the reference works.

• Plates are rectangular with simply supported edges SS-1 and SS2 (Movable and immovable edges). • All necessary information is clearly presented in the figures, such as plate dimensions, material properties, and foundation parameters. • Plates are under increasing uniform lateral pressure and initial thermal load (Activation temperature). • Initial imperfection (μ ) is considered in all plates. • Plates are with or without elastic foundation (Winkler (K , 0) or 1

Winkler-Pasternak (K1, K2 )).

5.2. Parametric studies

5. Results and discussions

5.2.1. Thermal loading The plate central deflection induced by applied thermal loading are

The methodology established in Section 4 is applied here to the SMA strengthened composite plate. Primary, the validation study of the response of the composite plate under three types of loading is examined. Afterward, a series of parametric studies are conducted to show the influence of SMA fibers and elastic foundation on the plate load-deflection behavior. 5.1. Validation study 5.1.1. Thermal loading To evaluate the accuracy of the load-deflection curve of a thermally loaded composite plate, the dimensionless load-deflection response of the plate is compared to the work done by Shen [56], see Fig. 7. It is notable that the results of Shen’s study have been obtained using a mixed Galerkin-perturbation technique to determine the thermal loaddeflection curve. The results show that the proposed method is in good agreement with the published data. Another validation study is performed on two composite plates with and without SMA reinforcement. Fig. 8 depicts a comparative study on

Fig. 8. Comparison of thermal load-deflection of imperfect SMA composite plate. 397

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Fig. 9. In-plane mechanical load-deflection curves for a square composite plate under combined loadings while resting on an elastic foundation.

Fig. 12. The effect of SMA volume fraction on thermal load-deflection of the imperfect composite plate (SMA#1).

Fig. 10. Load-deflection of the square plate under uniform lateral pressure. Fig. 13. The effect of SMA volume fraction on thermal load-deflection of the imperfect composite plate (SMA#2).

plate. The plate is under the initial lateral load, without considering the elastic foundation. As can be seen, the reverse martensite phase transformation of SMA fibers occurs during the thermal loading and generates considerable tensile recovery force that enhances the plate behavior. All of the deflection curves are coincident prior to the beginning of phase transformation. By initiation of phase transformation, the plates with a higher SMA volume fraction show higher curves and lower deflection in a certain thermal load. Figs. 14 and 15 validate the carried out analysis. In addition, the influence of the elastic foundation has been considered. Similar results have been achieved and shown in the figures. Generation of higher recovery force due to the higher SMA volume fraction and temperature load significantly reduces the maximum central displacement. Obviously, in Figs. 13 and 15, SMA#2 material makes lower recovery force at high temperatures compared to the SMA#1 material. In order to assess the effect of SMA pre-strain on the thermal loaddeflection of SMA composite plates, general examples are presented in Figs. 16 and 17. These figures show that the phase transformation region is expanded when the SMA pre-strain value increases. Furthermore, higher SMA pre-strain leads to the higher tensile recovery force. This additional recovery force can be implemented to enhance the plate behavior. The plate corresponding to the curve number 5 (q = 0 ) in Fig. 16, behaves almost linearly in comparison with the other curves.

Fig. 11. The influence of elastic foundation parameters on thermal load-deflection of the imperfect composite plate.

plotted in Fig. 11. The plate is under initial constant lateral load and uniform temperature is growing while resting on an elastic foundation. The thermal path shows initial lateral displacement in the presence of lateral pressure. The load-deflection curves of the plates resting over the Pasternak or Winkler elastic foundation are placed higher than the ones without elastic foundations. Figs. 12 and 13 illustrate the influence of different SMA volume fractions on the thermal load-deflection response of the SMA composite 398

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Fig. 14. The effect of SMA volume fraction on thermal load-deflection of the imperfect composite plate resting on an elastic foundation (SMA#1).

Fig. 17. The effect of SMA pre-strain on thermal load-deflection of the imperfect composite plate resting on an elastic foundation (SMA#2).

Fig. 15. The effect of SMA volume fraction on thermal load-deflection of the imperfect composite plate resting on an elastic foundation (SMA#2).

Fig. 18. The influence of lateral pressure and elastic foundation parameters on in-plane Mechanical load-deflection of an imperfect composite plate.

Fig. 16. The effect of SMA pre-strain on thermal load-deflection of the imperfect composite plate resting on an elastic foundation (SMA#1).

Fig. 19. The influence of SMA volume fraction, lateral pressure and elastic foundation on the in-plane Mechanical load-deflection of an imperfect SMA composite plate (SMA#1).

5.2.2. In-plane mechanical loading In this subsection, the non-linear in-plane mechanical load-deflection response of SMA composite plate is studied. Fig. 18 shows the influence of elastic foundation and lateral pressure load on the load-deflection response of an imperfect WOS composite plate. The figure clearly illustrates the increase in deflection due to the lateral pressure and reduction in deflection due to the influence of two-parameter elastic foundation.

The next two figures (Figs. 19 and 20) illustrate the effect of different SMA volume fraction and SMA material’s type on the plate loaddeflection curves. The figures depict enhancement in the plate behavior under a combination of thermal, lateral, and in-plane loads but the effect of SMA#2 in enhancement is much lower compared to the SMA#1. This difference can be explained according to the fact that the SMA#2 produces lower tensile recovery force compared to the SMA#1 399

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Fig. 20. The influence of SMA volume fraction, lateral pressure and elastic foundation on the in-plane Mechanical load-deflection of an imperfect SMA composite plate (SMA#2).

Fig. 23. Load-deflection of the laterally loaded composite plate resting on an elastic foundation.

Fig. 24. Load-deflection of the laterally loaded SMA composite plate with different activation temperatures (SMA#1).

Fig. 21. The influence of activation temperature, lateral pressure and elastic foundation on the in-plane Mechanical load-deflection of an imperfect SMA composite plate (SMA#1).

on the effect of SMA wires or fibers on the post-buckling of a composite plate through a closed-form solution and non-linear finite element method. Figs. 21 and 22 show the influence of activation temperature, lateral pressure, and an elastic foundation on the in-plane mechanical loaddeflection response of an imperfect SMA composite plate. The figures show that as activation temperature increases, the lateral deflection decreases. As a result, the plate can withstand higher mechanical loads (see Fig. 21). Fig. 22 shows the same trend, this time using SMA#2. It is also interesting to note that the curves number 3 and 4 in Fig. 21 are almost coincident. The coincidence occurs because there is no considerable additional recovery force at ∆T = 100℃. 5.2.3. Lateral pressure loading In this part, a non-linear load-deflection analysis has been presented for a laminated SMA plate subjected to uniform lateral pressure while resting either on a Winkler or a Pasternak elastic foundation or without any elastic foundation. It should be pointed out that in the SS-2 boundary condition the activation temperature (∆T ) acts as a thermal load due to the restrained in-plane boundary conditions. Fig. 23 illustrates the effect of elastic foundation on the load-deflection behavior of a composite plate under lateral pressure. The Winkler and the two-parameter (Pasternak) elastic foundations are considered in the plate response. Figs. 24 and 25 show the effect of different SMA activation temperatures on the non-linear load-deflection behavior of SMA composite plate without any elastic foundation. All necessary information has

Fig. 22. The influence of activation temperature, lateral pressure and elastic foundation on the in-plane Mechanical load-deflection of an imperfect SMA composite plate (SMA#2).

material in the given activation temperature. In both figures, higher SMA volume fraction leads to the higher curves. Similar improvement in the response of SMA composite plate due to different SMA volume fractions (at a fixed activation temperature) has been reported in Kabir and Tavousi Tehrani [19], Shiau et al. [7] and Rasid et al. [62] work. It is worth noting that the above-mentioned reference works are focused 400

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Fig. 25. Load-deflection of the laterally loaded SMA composite plate with different activation temperatures (SMA#2).

Fig. 27. Load-deflection of laterally loaded SMA composite plate with different activation temperatures while resting on two-parameter elastic foundation (SMA#2).

Fig. 26. Load-deflection of laterally loaded SMA composite plate with different activation temperatures while resting on two-parameter elastic foundation (SMA#1).

Fig. 28. Load-deflection of laterally loaded SMA composite plate with different SMA volume fraction while resting on two-parameter elastic foundation (SMA#1).

been presented in the figures; and in all of them, the effect of two types of boundary conditions has been investigated as well. It can be noted that an increase in activation temperature results in a higher recovery tensile force in the SMA fibers (See Figs. 5 and 6). The lateral deflection curves for SMA composite plate greatly differ with those of a plate without any SMA fibers. At ∆T = 100℃, the SMA fibers generate no additional recovery force relative to the ∆T = 80℃; therefore, there is no considerable enhancement for the plate deflection at ∆T = 100℃ by using of SMA#1. It is seen, in Fig. 25, that using SMA#2 makes low difference in the results. The effect of activation temperature is diminished compared to apply SMA#1. This is according to different phase transformation temperatures in SMA#2 that lead to a delay in generation of the recovery force. The figure clearly shows that ∆T = 60℃ has the lowest improvement on the load-deflection of the SMA plate. From these results, it cannot conclude that a higher activation temperature always leads to a better performance; it is observed that the performance strongly depends on the SMA material property and the activation temperature range. Figs. 26 and 27 illustrate the effect of an elastic foundation and the benefits of the recovery force of SMA fiber on the load-deflection response of composite plate. The figures give the non-linear load–deflection curves of two SMA composite plates embedded with SMA#1 and SMA#2 fibers resting on an elastic foundation. It is seen that the effects of both elastic foundation stiffness and high SMA activation temperature result in response curves that are placed significantly lower than those of in the WOS plate. Note that in SMA#1, at

∆T = 100 ℃, the recovery tensile force is just slightly greater than the case at ∆T = 80℃. The influence of SMA volume fraction on the load-deflection behavior of SMA composite plates under lateral load is illustrated in Figs. 28 and 29. As expected, the results show that by an increase in SMA volume fraction, a decrease in central deflection occurs. Again, the effect of SMA#2 volume fraction on load-deflection behavior is quite different (Fig. 29). The load-deflection curves are nearly coincident in the case of SMA#2 and the recovery force has a small improvement on the plate load-deflection behavior. The SMA#1 and SMA#2 austenite start temperatures are 35.5 °C and 72.3 °C, respectively. At ∆T = 60 ℃ (T = 20 + 60 = 80 ℃), SMA#1 produces more than six times recovery force than that of SMA#2. At this temperature, SMA#2 makes relatively small recovery force. It can be concluded that SMA material properties, specifically phase transformation temperatures, has a great effect on the generation of tensile recovery force.

6. Conclusions A semi-analytical approach has been performed to investigate the non-linear load-deflection response of an SMA composite plate under thermal, mechanical in-plane and lateral mechanical loadings. The formulation is based on the first-order shear deformation theory. The following conclusions can be drawn from the present work: 401

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Fig. 29. Load-deflection of laterally loaded SMA composite plate with different SMA volume fraction while resting on two-parameter elastic foundation (SMA#2).

• The effect of the Winkler-Pasternak type elastic foundation on the • • • • • •

load-deflection responses of a rectangular SMA composite plate is considered in the present study, for the first time. In SMA material, specifically the phase transformation temperatures, and activation temperatures play a key role in the enhancement of the plate behavior. Generally, any increase in the SMA volume fraction, fiber pre-strain or activation temperature results in a reduction in the nonlinear deflection. During heating, when the temperature rises through the phase transformation region, considerable decrease can be noted in the plate deflection. This reduction in deflection only begins when the temperature increases above the austenite start temperature. According to the recovery stress diagrams, the SMA activation temperature should be above the austenite starting temperature in order to achieve a considerable improvement in the plate load-deflection behavior. Increase in SMA pre-strain leads to a development in the phase transformation region (higher austenite finish temperature in restrained condition). In the current study, it is seen that the SMA material properties and activation temperature both play important roles in plate enhancement against deflection.

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