Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires, using a hyperbolic plate theory

Accepted Manuscript

Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires, using a hyperbolic plate theory M. Shariyat , A. Mozaffari , M.H. Pachenari PII: DOI: Reference:

S0307-904X(16)30633-3 10.1016/j.apm.2016.11.028 APM 11454

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

18 February 2016 27 October 2016 30 November 2016

Please cite this article as: M. Shariyat , A. Mozaffari , M.H. Pachenari , Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires, using a hyperbolic plate theory, Applied Mathematical Modelling (2016), doi: 10.1016/j.apm.2016.11.028

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Highlights  Impact responses of visco-composite plates with visco-SMA wires are investigated.  Effects of viscoelastic and phase-transformation-based dissipations are compared.  A hyperbolic plate theory is proposed and employed.  A refined constitutive model is proposed for the hierarchical SMA wire.  Zero-shear traction condition is satisfied for the viscoelastic orthotropic plate.

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Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires, using a hyperbolic plate theory M. Shariyat*a, A. Mozaffari†b, M.H. Pachenari‡b a

Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran 19991-43344, Iran. Faculty of Aerospace Engineering, K.N. Toosi University of Technology, Tehran 19991-43344, Iran.

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Abstract In the present paper, a comprehensive study is made on effects of the viscoelastic and the phasetransformation-based dissipations and their interactions on impact responses of viscoelastic composite plates with damping treated (structural hierarchy) shape memory alloy (SMA) wires, for the first time. In contrast to almost all of the available researches, a high-order hyperbolic plate theory that includes not only odd but also even functions of the transverse coordinate, is proposed and employed here. While a hierarchical viscoelastic constitutive law is employed for both the orthotropic and SMA materials, Brinson’s constitutive law is refined to include the loading fluctuations and structural hierarchy of the SMA wire, simultaneously. The traditional Hertz and Yang-Sun contact laws are modified accordingly. The resulting highly nonlinear piecewise-defined integro-differential finite element governing equations are solved by an iterative algorithm within each time step. The presented discussions show that in contrast to the common belief, the zero-shear traction condition on the top and bottom surfaces of the viscoelastic orthotropic plate cannot be satisfied by the available plate theories, even for the symmetric lamination schemes. Results show that the viscoelasticity and phasetransformation effects on the resulting dynamic responses are more pronounced for the low and high energy impacts, respectively.

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Keywords: Hierarchical shape memory alloys; Viscoelastic composites plates; Refined phasetransformation model; Hyperbolic plate theory; Finite element method.

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1 Introduction Dissipation of the induced undesired impact energies, due to falling or thrown rigid objects, may enhance strengths of the structures. The viscoelastic and phase-transformation-based dissipations are among the main structural dissipative sources wherein, the first type induces an additional structural damping [1] whereas the second type dissipates a considerable portion of the stored strain energy through a hysteretic behavior [2]. While both the matrices and fibers of the composites exhibit viscoelastic behaviors whose intensity vary with humidity and temperature [3], the shape memory alloys (SMAs) may dissipate the impact energies and consequently, reduce the resulting stresses and displacements [4-6] through successive phase transformations and reorientations. Some researchers have studied effects of the viscoelastic layers on the dynamic responses of the reinforced composite plates. Shariyat [7,8] and Alipour and Shariyat [9] have investigated vibration and dynamic buckling behaviors of the viscoelastic composite plates. Cupial and Niziol [10] studied vibration of a three-layered composite plate with a viscoelastic mid-layer. Vangipuram and Ganesan [11] investigated buckling and vibration of rectangular composite viscoelastic sandwich plates under *

Corresponding author, Professor, E-mail addresses: [email protected] and [email protected] Tel.: +98 9122727199; Fax: +98 21 88674748, zip code: 19991-43344. Web page: http://wp.kntu.ac.ir/shariyat/publications.html. † Assistant Professor, E-mail address: [email protected] ‡ Ph.D. candidate, E-mail address: [email protected].

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thermal loads. Zheng and Deng [12] studied free vibration of the multi-layer viscoelastic composite plates. Kim et al. [13] investigated dynamic behavior of multi-layer viscoelastic composite plates undergoing large deformations. A numerical algorithm to analyze dynamic response of composite plates with anisotropic viscoelastic materials was developed by Yi et al. [14]. Meunier and Shanoi [15] investigated dynamic response of sandwich plates with viscoelastic PVC foam cores. Ferreira et al. [16] presented a layerwise finite element model for vibration analysis of sandwich laminated plates with frequency-dependent viscoelastic cores and laminated anisotropic face layers, using Carrera’s Unified Formulation (CUF). Rossikhin and Shitikova [17] discussed two approaches for dynamic response determination of viscoelastic Bernoulli-Euler beams transversely impacted by elastic spheres. Shariyat and Farzan Nasab [18] investigated low velocity impact of the single-layer functionally graded viscoelastic plates. Recently, Shariyat and Hosseini [19] accomplished a lowvelocity impact analysis on the viscoelastic laminated/sandwich composite plates, using a global-local plate theory. The superelastic dissipative nature of the SMAs stems from a reversible solid-state phase transformation between an austenite phase comprised of a high symmetry crystal structure and a lower symmetry martensite phase and vice versa [20,21]. When no slip occurs between the composite plate and the embedded SMA wires, considering the time-varying non-uniform distribution of the martensite volume fraction along the individual SMA wires is a crucial issue. Influences of the pseudoelastic nature of the SMAs on low-velocity impact responses of the composite plates were investigated by Wu et al. [22] using Brinson’s model and the finite element method. Shariyat and Moradi [23] and Shariyat and Hosseini [24] investigated eccentric impact responses of pre-stressed composite plates with embedded SMA wires, based on various plate theories. Apart from the mathematical constitutive description of the SMA, the employed plate theory may remarkably affect accuracy of the predicted responses of the impacted plate with embedded SMA wires. A wide variety of the equivalent single-layer plate theories has been proposed so far for estimation of the transverse variations of the in-plane displacement components. The third-order theory Reddy [25], the m-th order theory of Matsunaga [26], the sinusoidal [27] or trigonometric [28,29], hyperbolic [30], inverse hyperbolic [31], and exponential theories [32] are among the aforementioned theories. Neves et al. [33] presented a quasi-3D hybrid polynomial and trigonometric shear deformation theory for the functionally graded plates. Wang and Shi [34] developed a thirdorder shear plate theory accounting for the transverse shear stress continuity at the interlaminar interfaces of laminated plates. Almost all of the available high-order equivalent single-layer theories have been presented in terms of odd functions of the transverse coordinate, to satisfy the zero shear stress condition on the top and bottom layers and choosing the mid-surface as the reference plane. In the present paper, the dissipative effects and interactions of the viscoelastic nature of the composite and SMA materials and the phase-transformation of the SMA wires on impact responses of composite plates with shape memory wires are studied. A new odd-even hyperbolic plate theory is also proposed and employed in development of the formulation. While a hierarchical constitutive law is employed for the composite plate, Brinson’s constitutive equation of the SMA is refined to include the reverse loading and the structural hierarchy (viscoelasticity) effects, simultaneously. The contact laws are reconstructed based on the mentioned constitutive equations. The resulting highly nonlinear piecewise-defined integro-differential finite element governing system of equations is solved by an incremental and iterative solution algorithm to enable tracing the localized and instantaneous variations of the martensite and austenite volume fractions for each point of the SMA wires. Different and non-linear contact laws are utilized for the loading and unloading events. 2 The governing equations 2.1 The proposed odd-even hyperbolic plate theory The considered rectangular laminated composite plate with embedded SMA wires is shown in Fig. 1. Length, width, and thickness of the plate are denoted by a, b, and h, respectively. The plate is subjected to an impact by a rigid spherical indenter whose radius and initial velocity are denoted by R and V, respectively. The employed coordinate system is shown in Fig. 1. The z-coordinate of the plate may be measured from an arbitrary but fixed layer and is positive upward. Since the ideal plate theory must contain but odd and even function as well as infinite powers of the transverse coordinates (in the sense of MacLaurin series), the following description of the

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ACCEPTED MANUSCRIPT displacement field that is an enhanced form of the theory introduced recently by Shariyat and Hosseini [24], may be proposed: u( x, y ,z, t ) u0 ( x, y , t ) z x ( x, y , t ) x ( x , y , t ) f1 ( z ) x ( x , y , t ) f 2 ( z ),

v( x, y ,z, t )

v0 ( x, y , t )

z

y

( x, y , t )

y

( x , y , t ) f1 ( z )

y

( x, y , t ) f 2 ( z ),

(1)

w( x, y , z, t ) w0 ( x, y , z, t ) where u0 , v0 , x , and y are displacements and rotations of the reference plane: u0

u( x, y,0, t ), u, z

x

z 0

v0

,

v ( x, y ,0, t ) v, z

y

(2) z 0

z 2 sinh

z , h

z 2 cosh

f2 ( z)

z h

(3)

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f1 ( z )

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and f1 ( z ) and f 2 ( z ) are odd and even functions respectively:

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Fig. 1 The geometric parameters and coordinate system of the considered hybrid composite plate and the indenter.

traction conditions: XT , xz z z yz

z zT

YT ,

xz z z B

XB,

yz z z B

YB .

(4)

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T

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Indeed, according to MacLaurin series expansion, the first terms of the in-plane displacements appeared in Eq. (1) contain the zeroth- and first-order powers of the z coordinate whilst the remaining terms contain the next powers, according to Eq. (3). The two-dimensional in-plane functions ( x, x, y , y ) may be determined in terms of ( u0 , x , v0 , y , w0 ), through imposing the following

CE

where, the subscripts T and B stand for the top and bottom surfaces of the plate, respectively. For the present problem: X T ,YT , X B ,YB 0 (5)

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2.2 The constitutive and contact laws of the viscoelastic composite plate and the viscoelastic SMA wires According to Voigt and Reuss micromechanical homogenization models respectively, the effective material properties of the fiber-matrix composite may be determined for directions parallel and perpendicular to the fibers (denoted by 1 and 2, respectively), as follows: E 2f Em2 G12f G12 m 1 1 1 2 12 Ec E f V f EmVm , Ec , Gc 2 2 12 EmV f E f Vm Gm V f G12f Vm (6) 12 c

Vf

12 f

Vm

12 m

,

c

Vf

f

Vm

m

where V is the volume fraction and the subscripts f, m and c denote the fiber, matrix, and composite, respectively. Eq. (6) is valid for all time instants. Since it is assumed that the SMA wires are parallel to the composite fibers, similar equations may be written for the effective material properties of the SMA-composite combination where the composite plate plays the role of the matrix. Therefore, Eq. (6) may be extended for a composite lamina with embedded SMA wires as:

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1 Eeff

E1f V f

12 Geff

13 Geff

12 eff

13 eff

Em1 Vm

E 2f Em2 Es

Eeff2

EsVs ,

Em2 EsV f

G12f Gm12Gs 12 m

12 f

G GsV f Vf

12 f

12 f

G GsVm 12 m

Vm

Vs

12 s

12 m s

G G V ,

E 2f EsVm

E 2f Em2Vs

,

(7)

,

Vf

Vm

f

m

Vs

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where, the subscript s stands for the SMA.

Fig. 2 The employed standard solid viscoelastic model for a representative volume of the three-phase mixture of materials of the composite plate with embedded SMA wires.

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The standard hierarchical solid viscoelastic [18,35] model shown in Fig. 2, is considered for the three-phase mixture of the composite lamina with SMA wires. According to this model, time variations of the viscoelastic modulus of the lamina may be expressed as: Ev t E1 E2 e t (8)

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where the superscript v stands for viscoelastic,

is the relaxation parameter, and E1 is the elastic

modulus associated with the steady-state behavior. Therefore: Ev 0 E1 E2

(9)

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Eq.(8) may be rewritten as: Evi t E1 i E2 i e i t (i 1,2) Furthermore, the shear modulus of the plate may be interpreted as: Gv12 t G112 G212e 3t

(10)

CE

(11)

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Therefore, based on Eqs. (7) and (9): E1 1 E1f ( )V f Em1 ( )Vm Es1 ( )Vs , E1 2 G1

12

E 21 E2

E 2f ( ) Em2 ( ) Es ( )

Em2 ( ) Es ( )V f G1

E 2f ( ) Es ( )Vm

Gm12 ( )Gs ( )V f Em1 (0)Vm

G12f ( )Gs ( )Vm

2 m

E (0) Es (0)V f

G12f ( )Gm12 ( )Vs

Es1 (0)Vs E1 1 ,

E 2f (0) Em2 (0) Es (0)

2

G 212 G 213

,

G12f ( )Gm12 ( )Gs ( )

13

E1f (0)V f

E 2f ( ) Em2 ( )Vs

2 f

E (0) Es (0)Vm 12 f

2 f

2 m

E (0) E (0)Vs

E1 2 ,

12 m

G (0)G (0)Gs (0) 12 m

G (0)Gs (0)V f

G12f (0)Gs (0)Vm

G12f (0)Gm12 (0)Vs

5

G112

,

(12)

ACCEPTED MANUSCRIPT For a hierarchical viscoelastic composite lamina, the constitutive equation may be written as [2]: dC ijkl (t ) (13) kl ( ) d 0 0 d d Because the stress and strain tensors are respectively, contravariant and covariant tensors of the rank two. The non-zero components of the contravariant elasticity coefficients matrix are: t

(t )

C ijkl (t

C1111

Evs1 (t )

C

1

1313

)

,

G (t ),

where G

kl

( )

C ijkl (0)

d

C1122

C 2211

2323

23 vs

12 21 13 vs

23 vs

d

C

12

t

kl (t )

Evs2 (t )

1

Evs2 (t )

C 2222

,

1

12 21

,

C1212

Gvs12 (t ),

(13)

12 21

G (t )

can be determined based on Reuss rule as well.

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ij

Brinson’s constitutive equation [36] for the pseudoelastic SMA wires has the following form:

(14) E ( )( 0) s 0 s0 where, , s , and are volume fractions of the martensite and stress-induced detwinned martensite phases and the transformation function, respectively. The subscript 0 denotes the initial quantities. The modulus of elasticity appeared in Eq. (14) may be determined based on a proper micromechanical rule in terms of volume fractions and Young’s moduli of the austenite and martensite phases ( E A and EM , respectively); so that, based on Voigt’s rule of mixtures, one may write: (15) E ( ) EA ( EM EA ) , L E( )

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where, L is the maximum recoverable strain. Since in the reverse loading, the diagram has to be an anti-symmetric one, signs of the second and third terms of the right-hand side of Eq. (14) has to be changed; so that, we can modify Brinson’s model as follows: (16) E ( )( sgn( ) ( ) s ( 0 ) s0 0 0)

s0

cos

AC

1

CE

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However, this model can be used for the pseudoelastic materials only. For super-viscoelastic SMAs, one can extend the constitutive law (16) as follows: t dE (t ) (t ) E (0) (t ) sgn( ) E ( 0 ) L s 0 { ( ) [ 0 sgn( ) L s ]}d (17) 0 0 d If the strain value at the end of each time event is denoted by 0 , the term sgn( ) L of Eqs. (16) and (17) may be omitted and vice versa, i.e., these terms are alternates. For temperatures above the austenite start ( T As ), two phase transformation situations may be imagined. The relevant martensite volume fractions may be determined as [36]: i) Conversion from the austenite to the detwinned martensite phase ( cr cr C ( T M ) C ( T M ) s M s f M s ): s

2

cr s

cr f

cr f

CM (T

1

Ms )

s0

(18)

2

ii) Conversion from detwinned martensite to austenite phase ( CA (T 0

2

cos

Af

As

T

In Eqs. (18) and (19),

As cr s

,

CA cr f

1 ,

s0 s

s0

(

0

)

As )

CA (T

As ) )

(19)

0

, As , A f , and M s are the critical stresses associated with the start and

finish of the phase transformation process, the start and finish temperatures of the austenite phase transformation, and temperature of the martensite transformation start, respectively. CM and CM are slopes of the martensite and austenite transformation curves in the stress-temperature plane. To develop the finite element formulations, it is appropriate to use the following vector convention for the stress and strain components:

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x

y

xy

xz

yz

(20)

T

,

x

y

xy

xz

yz

For the general case where the kth layer contains embedded SMA wires, Eq. (17) may be modified as: Vs( k ) [ σ (0k )

C( k ) (0)

(k )

t

(t ) 0

Vs( k ) sgn(

(k ) s

dC( k ) (t d

) E(

(k ) 0

)

)

ε(k ) ( )

sgn( 0 0 0 0

(k ) s

)

(k ) L

] d

(21)

(k ) L s0

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σ ( k ) (t )

(k ) 0

0 0 0 0

*

( , t) and σ*s k ( , t) terms contain integral operators that have to be updated with time:

CE

*k

and the

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where, Vs , and Es are the volume fraction and modulus of elasticity of the SMA material, respectively, and in the geometric coordinates: t dC( k ) (t ) (k ) (k ) *( k ) (22) σ ( k ) (t ) C( k ) (0) ( k ) (t ) ( ) d S * ( , t) ( k ) σ*s( k ) ( , t) 0 d where Vs( k ) [ 0( k ) sgn( (sk ) ) L ( k ) ]cos 2 Vs( k ) [ 0( k ) sgn( (sk ) ) L ( k ) ]sin 2 Vs( k ) [ 0( k ) sgn( (sk ) ) L ( k ) ]sin cos 0 0 (23) Vs( k ) sgn( (sk ) )[ E ( 0( k ) ) L ( k ) ]cos 2 Vs( k ) sgn( (sk ) )[ E ( 0( k ) ) L ( k ) ]sin 2 S * Vs( k ) sgn( (sk ) )[ E ( 0( k ) ) L ( k ) ]sin cos 0 0

t

(...)

C 0 (...)

0

dC( k ) t d

(...) d

(23) dC( k ) (t ) * σ d S 0 d and C is the stiffness matrix in the geometric coordinates: (24) C(t ) Τ 1CT where T is the rotation matrix. Eq. (22) may be computed based on a proper integration scheme, such as the trapezoidal, Simpson, or the most accurate Gauss-Legendre numerical integration method: t

AC

* s

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6

Ci0j ( k )

(t )

t (k ) j

dCij

n 1

(t )

m

j 1

d

j 1 m 0

dC

6 n j 1

m t j

m t i

t (n m) t

(25)

(k )

0 ij

t j

d

t i

(S * )i( k )

t

where m is the m-th integration weighting coefficient and t is the integration time step. Turner [37] proposed relating Hertz contact force ( Fc ) to the indentation value ( ) as follows:

8 2Gxy / 3 (1

xy

)

1

1 2 xz

Ex / E y 1

2 xy

1

3 2

R

3 2

kc

,

2

( E x / 2Gxz )

,

2

xz

1

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Fc

xy

2 xy

1

(26)

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where, k c is the contact stiffness. For viscoelastic layers with embedded SMA wires, the elastic moduli are functions of the time and martensite volume fraction: Ex Ex ( , t ), E y E y ( , t ), Gxy Gxy ( , t ) (27) For the unloading phase, Yang and Sun [38] proposed using the following equation: 5/2

Fc

0

Fmax max

0

where Fmax is the maximum contact force reached during the loading phase, maximum indentation, and 0 is the permanent indentation, if any.

(28) max

is the relevant

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2.3 The finite element form of the integro-differential governing equations of motion The hyperbolic description of the displacement field proposed in Eq. (1) may be expressed in the following matrix form: ( z) ( x, y, t )

where u v w 1 z

z 2 sinh 0

0 0

0

AC T

u0

x

z h

z 2 cosh

CE

0 0

z h

PT

T

x

(29)

ED

( x, y, z, t )

x

v0

0 0

0

1 z

0

0 0

y

x

y

0 z 2 sinh 0

0 z h

z 2 cosh 0

0 z h

0

(30)

1

w0

Therefore, the strain matrix may be determined from: u, x 0 0 x x v, y 0 0 y y u, y v, x 0 xy y x w, x u, z 0 xz z x 0 w v yz z y ,y ,z

8

(31)

ACCEPTED MANUSCRIPT where the comma and symbols denote partial differentiation operations. In contrast to the strain components, the stress components expressions are layer-dependent; so that, according to Eq. (25) ( k σ0 0 ):

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k *k *k *k *k (32) σ ( , t) σ s ( , t) ( , t) σ s ( , t) To provide a finite element representation, the plate is discretized by means of the quadratic rectangular elements [39] shown in Fig. 1. Therefore, the in-plane variations of the displacement components may be traced through the following equation: x, y (t ) (33) where, is the shape functions matrix and is the nodal values vector. Therefore: (z) x, y (t ) (z,x, y) (t ) (34) By imposing the four boundary conditions (4), the higher-order terms of the displacement field v0 w0 ). Since description ( x x x y ) may be interpreted in terms of ( u0 x y

interpreted in terms of

u0

v0

x

y

u0 ( x, y , t ) (t )

y

( x, y , z , t )

( z, , t )

( z, , 1 , *k

2

,

3

instead of

:

(35)

( z, , t ) ( x, y ) ( t )

, t ) ( x , y ) (t )

PT

σ

k

T

ED

w0

w0

M

x

v0

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based on Eq. (32), the stress components of the viscoelastic plate contain integrals of time, in contrast to the elastic plates, explicit expressions cannot be presented for the higher-order terms and the boundary conditions may be incorporated through a numerical procedure. The relevant complicated symbolic mathematical manipulations have been carried out, using MAPLE software. The same procedure may be used to refine the available plate theories. But the researchers have assumed that theories of the elastic plate may be used identically for the viscoelastic plate; an assumption that cannot be justified. Therefore, the zero-shear stress condition on the top and bottom surfaces of the viscoelastic plate cannot be satisfied by the available theories, even for the symmetric lamination schemes. Based on the above discussion, the displacement, strain, and stress matrices may be

( , t) ( x, y , z , , 1 ,

2

,

3

( x, y , z, , 1 ,

, t ) (t ) σ

*k s

2

,

3

, t ) (t )

( , t)

T

(U 0

CE

where and are the new shape functions matrix and nodal values vector of , respectively. The governing equations of the plate may be derived using Hamilton’s principle: W )dt

(36)

0

AC

where increments of the strain energy ( U ) and work of the external loads (including the inertia forces and moments) ( W ) may be determined from: n

U

k 1 n

hk

1

T

σ ( k ) dz d

hk hk

(37) 1

)T (

(

)T

σ*s( k ) dz d

*( k )

hk

k 1 n

hk

W k 1

hk

n

hk

k 1

1

(

(k ) T

)

(k )

dzd

Fc ( w x

a /2, y b /2

)

(38) 1

(

T

) (

T

) (

) dzd

k

hk

9

3/2

(w x

a / 2, y b / 2

)

ACCEPTED MANUSCRIPT where and are the mass density and area of the reference plane, respectively. In addition to the nodal displacement parameters, the indentation ( ) is an unknown parameter and should be added to degrees of freedom of the whole plate. Therefore, the nodal vector of the displacement parameters of the whole plate has to be augmented to include this degree of freedom: (39) where is vector of the nodal displacement parameters of the whole plate. Therefore, the U and W energies of the entire plate may be found by summing up those of the individual elements: e

(e)

n

hk

U e 1 e

(

T

) (

σ

*( k )

)

*( k ) s

dz d

(e)

n

hk

W e 1

T

hk

k 1

n

1

1

T

(

T

) (

) (

) dzd

)T k

(

hk

k 1

(40)

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n

3/2

)T

(

x a /2, y b /2

YT

(41)

where n ( e ) is number of the elements and the bar symbol indicates relevance to the entire degrees of T freedom of the plate. Sine Eq. (36) has to be valid for any arbitrary time interval and any , based on Eqs. (36), (40) and (41), one may deduce that:

e 1

hk

n

e 1

T

(

) (

)dzd

hk

k 1 N

1

N

1

T

(

*( k )

) (

)dz d

hk

n

e 1

(e)

hk

k 1

(42)

(e)

hk

1

) σ ( ) T

(

* s

hk

k 1

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(e)

n

(k )

dz d

M

N

(

)T

YT

x a / 2, y b / 2

(e)

n

e 1

hk

n

k

hk

)T

(

n

hk

AC

N

e 1

k 1

n

1

hk

k 1

3/ 2

) (

(

)

dC 0 d

x a / 2, y b / 2

1

(

C

0( k )

m

dC d

(

m t

m t

)

t (n m) t

(e)

(k )

(S )

t

dz d

(43)

n 1

m 0

* t

n

(e)

dC0( k ) d

YT

)T

hk

t

t

)dzd

T

CE

e 1

(

T

hk

k 1

N

1

PT

N

ED

In other words, the resulting system of equations is a non-linear piecewise-defined integro-differential one. According to Eq. (25), Eq. (42) may be rewritten as:

dz d

or in a compact form ( ) ( ) ( ) The governing equation of motion of the indenter is: mi w0i Fc or

10

(44) (45)

ACCEPTED MANUSCRIPT

mi (

wx

a /2, y b / 2

)

kc

3/2

0

(46)

where mi , Fc and w0i are the mass of the indenter, contact force, and displacement of the indenter, respectively. It is worth mentioning that: w 0 0 1 (47) w0i

wx

(48)

a /2 , y b /2

Eq. (45) has to be assembled with Eq. (43) to impose the load boundary condition of the plate.

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3 Solution procedure of the resulting augmented nonlinear system of equations The edge conditions of the plate must be incorporated before solving the finite element governing equations; by either eliminating the rows and columns associated with the zero quantities or using the penalty method [40]. The time-dependency of the governing equations may be solved through using Newmark’s numerical time integration method. However, since the governing equations are integrodifferential ones, the accumulated integrations terms of Eq. (43) have to be updated for the successive time instants, based on the following initial conditions:

0 t 0

(49)

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Due to the nonlinear and piecewise-defined nature of the constitutive equation of the SMA wires and the viscoelastic nature of the materials, all the mass, stiffness, and force matrices of the hybrid composite plate are dependent on the nodal displacement parameters. Therefore, an incremental solution regarding the displacement components may be accomplished within each time step. Since the constitutive law of the SMA is a piecewise-defined function and different for the martensite to austenite and vice versa phase transformations, a proper transformation checking algorithm has to be employed. The algorithm proposed by Shariyat and Hosseini [24] is utilized here. During solving the governing system of equations in each iteration, the displacement parameters and consequently, the stress components (by using a trapezoidal numerical integration) and the martensite volume fraction of all the nodal points are determined. Therefore, these quantities may be used to update the mass, stiffness, and force matrices of the plate and the process continues till a convergence criterion is met. After then, the hierarchical integrals have to be updated to proceed to the next time step.

4 Results and discussions

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4.1 The base specifications employed in derivation of the results The integration time step is adopted according to a convergence study whose results are not presented here due to restriction of the paper length. To accurately trace the time history of the displacement parameters, an integration time step in the order of the 10 6 sec that is much less than the fundamental period time of the structure and especially much less than the response time of the structure is adopted. Dimensions of the chosen [0/90/0/90/0]s Graphite/Epoxy plate are assumed to be 100mm×100mm×5mm. The plate is simply supported and centrally impacted by a spherical steel indenter. The plate is composed of 10 layers with identical thicknesses. The following general specifications are adopted for the viscoelastic composite plate and the indenter: Plate E1 129GPa, E2 ( ) 7.5GPa, G12 ( ) G13 ( ) 3.5GPa, 12 0.33, 13 1540kgm 3 , E1 / E2

0.4,

0.2

Indenter R 6mm, E 200GPa, 0.3, 7850kgm 3 , V 30m / s It is evident that for a composite plate without viscoelastic characteristics: E2 / E1 G2 / G1

0.

The viscoelastic to elastic moduli ratio is chosen as a representative one, according to some researchers (e.g., [42]) and can be used for comparative studies. The Ni-Ti SMA wires are embedded in the top and bottom layers and have the following material properties [36]

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EA

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26.3GPa, 49C , M s cr f

A

M

6448.1kgm 3 ,

0.3,

18.4C , M f

9C , CM

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13.8MPa / K ,

170 MPa.

Volume fraction of the SMA wires in the mentioned layers is assumed to be 50%.

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4.2 The adequate mesh size Choice of the adequate number of elements has been accomplished based on a convergence analysis. Table 1 gives values of the key parameters of the impact (i.e., the maximum lateral deflection, indentation, and contact force of the laminated composite plate) for various mesh sizes. As may be noted from results of Table 1, the 20×20 mesh size may be considered to lead to convergent results. For this reason, the next results are extracted based on the mentioned mesh size. Table 1 Effects of the mesh size on the key parameters of the impact responses of the composite plate. Maximum lateral deflection (mm)

Maximum indention (mm)

Maximum contact force (kN)

6×6 8×8 10×10 12×12 14×14 16×16 18×18 20×20

0. 40999 0.410683 0.404077 0.432766 0.429209 0.424013 0.424662 0.424658

0.51650 0.50375 0.495212 0.490285 0.487168 0.484997 0.482384 0.482133

9.279 8.937 8.711 8.581 8.499 8.400 8.368 8.368

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4.3 Verification of the results Since present problem has not been considered by other researchers before, verification of present results is accomplished through comparing present results with those of Wang and Tsai [43] for forced vibration of an isotropic homogeneous viscoelastic plate. Wang and Tsai has presented a finite element formulation based on the bending version of the first-order shear-deformation plate theory.

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Fig. 3 The viscoelasticity model employed by Wang and Tsai [43].

Wang and Tsai [43] considered the viscoelasticity model shown in Fig. 3 for a simply supported square plate with a=10m and h=1m dimensions and the following material properties: E0 9.8 107 N / m2 , E1 2.45 107 N / m2 , 2200kg / m3 , 0.35, 1 2.744 108 Ns / m2 Therefore, the Prony series form of the relaxation modulus becomes: E (t ) 1.96 107 7.84 107 e t /2.24 N / m2 (50) 2 The plate is subjected to a uniformly distributed transvers step load (in time) with q=10N/m intensity. Present transient results for lateral deflection of the central point of the elastic and viscoelastic plates are compared in Fig. 4 with those of Wang and Tsai [43]. The two types of results show a good concordance. The main source of discrepancy is that the in-plane deformations of the mid-plane was neglected by Wang and Tsai [43], i.e., they used a pure bending theory. From phenomenological point of view, the

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reason, lateral deflection of the plate grows with time. However, by increasing between the elastic and viscoelastic responses becomes smaller.

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Fig. 4 A comparison between present results for the elastic and viscoelastic plates and results of Wang and Tsai [43].

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As a next verification example, present results are compared with the experimental results. Probably, the only published experimental results on impact responses of composite plates with SMA wires are those of Shariyat and Niknami [44]. In this stage of verification, present results are compared with the experimental results of Shariyat and Niknami [44] and results obtained using the displacement-based (and not mixed) layerwise theory of reference [45], in the framework of Carrera's Unified Formulation (CUF) [45] for sandwich plates with transversely-flexible layers (LD1). Therefore, in the considered layerwise theory on the base of CUF, in contrast to the present formulation, transverse variations of the transverse displacement component are assumed to be linear, within each layer. Details of the experimental work accomplished by the first author and his co-author may be found in Ref. [44]. The five-layer 2mm thickness and 100 mm edge length thick [0/90/0/90/0] composite plate was manufactured by a layup technique from a woven IRGWP200 E-glass fibers, CY219 epoxy resin, and HY5161 hardener. The volume fraction of the glass fibers was about 59%. Thicknesses of the layers were identical. Niti Superelastic Spool REF SESP020 SMA wires of 0.51mm diameter were embedded in the bottom layer of the plate and constituted 10% of the volume fraction of the mentioned layer. Mechanical properties of the composite plate and the SMA wires are: 1540kgm 3 E-glass-epoxy laminate: E1 E2 23GPa, G12 9GPa, 12 0.3, SMA wires:

EA

70GPa, EM

Ms

9.4C , M f

26.3GPa, 0C , C A

A

M

0.3,

13.8 MPa / C , CM

13

6450kgm 3 , As 6.527 MPa / C ,

l

18C , Af 0.067

32C ,

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Radius of the hemispherical nose of the indenter was 8 mm. An indenter mass of 2.685 kg and a falling height of 0.2m were used to extract the results. The resulting initial velocity of the indenter was 1.981 m/sec. An integration time step in the order of (sec) that is much less than the fundamental period time of the structure and especially much less than the response time of the structure was adopted to accurately trace the time history of the displacement parameters. Present results for the time history of the contact force are compared in Fig. 5 with the experimental results and results of the layerwise transversely-flexible CUF model [45]. Generally, all the three types of results have shown a good concordance. The considered plate is composed of relatively rigid and not soft layers and on the other hand, our hyperbolic plate theory estimates transverse variations of the in-plane displacement components with higher accuracies (from the mathematical point of view) in comparison to the polynomial-based theories. Furthermore, in contrast to majority of the available plate theories, present theory it is inherently matched to the viscoelastic nature of the materials. Reddy [25] has claimed that since the three-dimensional, layerwise, and the more complicated theories include more degrees of freedom, their system of equations may be a compliant, float, and more sensitive system from the numerical point of view; so that, results of such theories may sometimes be less accurate than those of the lower-order theories [25], especially for thinner plates. Fig. 5 reveals that employing the layerwise CUF model for the present case of a composite plate with relatively transversely rigid layers (present manuscript is concerned only with this type of plates), has led to a more compliant plate and consequently, lower contact forces and a larger contact time. For this reason, results of the present formulation are closer to the experimental results.

Fig. 5 A comparison among present, layerwise CUF, and experimental results for time history of the contact force for a composite plate with embedded SMA wires.

4.4 Results of the base composite plate (without the sources of dissipations) To enable a comparison among the viscoelastic and phase-transformation-based dissipative effects on the impact responses of the plate, determination of the impact responses of the original laminated composite plate is a vital task. Specifications of the composite plate are given in Sec. 4.1. Results are derived for three distinct initial velocities of the indenter: V=10, 20, and 30m/s. Time histories of the contact force, lateral deflection of the central point of the plate, and indentation are plotted in Figs. 6 to 8, respectively.

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Fig. 6 Time history of the contact force of the base composite plate, for various initial velocities of the indenter.

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Since mass of the indenter is small, the response time of the plate (contact time) is very short. It is evident that as the indenter becomes more heavy, the repulsion time becomes longer. Moreover, the contact time is smaller for higher initial velocities of the indenter (at the limit of quasi-static impact, the contact approaches infinity). Asymmetry of the responses curves reveals that different contact laws have been used in the loading and unloading phases. Figs. 6 and 7 imply that since mass of the indenter is small, the resulting contact force and lateral deflections are approximately proportional to the initial velocity of the indenter. Comparing Figs. 6 and 8 with 7 reveals that the maximum lateral deflection occurs in times after occurrence of the maximum indentation (in contrast to predictions of the discrete models).

Fig. 7 Time history of the lateral deflection of the central point of the base composite plate, for various initial velocities of the indenter.

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Fig. 8 Time history of the indentation of the base composite plate, for various initial velocities of the indenter.

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4.5 Influence of the viscoelastic nature of the composite and SMA materials In the present section, effects of the viscoelastic nature of the material properties on the impact responses of the composite plate are investigated. In this regard, the SMA wires are assumed to act inactively, i.e., they exhibit a viscoelastic behavior without phase transformation, like the composite materials. In Figs. 9 to 11, time histories of the contact force, lateral deflection, and indentation of the elastic and viscoelastic composite plates are compared, respectively. It may be deduced from these figures that viscoelasticity leads to higher rigidities; so that, the contact force increases and the contact time (as well as the time associated with the maximum contact force) decreases. It is evident that when the indenter collides with a stiffer plate (with less movability), the resulting contact force will be higher. On the other hand, due to the higher stiffness of the materials, the lateral deflection and indentation of the viscoelastic plate are smaller, in spite of the higher contact force. Indeed, in the solid viscoelastic model, the elastic and viscoelastic elements are parallel; so that, their forces are summed. Fig. 9 shows that for identical initial velocities of the indenter, the areas beneath the contact force-time curves of the elastic and viscoelastic plates are almost equal. The obtained maximum contact force may also be checked approximately. Since each half of the contact force is almost a parabola, one may write the following approximate equation for the loading phase, for the first region of the contact force: t 2 Fc dt m V ( Fc ) max t( Fc )max V (51) 0 3m Since the resulting bending moments are proportional to the second derivatives of the lateral deflection, it may be deduced that the resulting bending moments and stresses are smaller in the viscoelastic plate. Figs. 9 and 11 show that loading and unloading regions of the time histories of the contact force and the indentation are closer to the symmetric situation in comparison to the elastic plate; so that slopes of the curves of the viscoelastic plate are larger than those of the elastic plate, in the unloading region.

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Fig. 9 A comparison between time histories of the contact force of the elastic and viscoelastic composite plates, for various initial velocities of the indenter.

Fig. 10 A comparison between time histories of the lateral deflection of the central points of the elastic and viscoelastic composite plates, for various initial velocities of the indenter.

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Fig. 11 A comparison between time histories of the indentation of the elastic and viscoelastic composite plates, for various initial velocities of the indenter.

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The E1/E2 ratio plays an important role on the dynamic responses of the viscoelastic structures. Effects of this ratio on time histories of the contact force, lateral deflection, and indentation of the viscoelastic composite plate are illustrated in Figs. 12 to 14, respectively. Figs. 12 to 14 show that as the E1/E2 ratio increases, the plate becomes less rigid; so that, the contact time and the lateral deflection of the plate increase while the contact force decreases. The resulting decrease in the rigidity of the plate has in turn increased the indentation of the viscoelastic plate.

Fig. 12 Influence of the E1/E2 ratio on time history of the contact force of the viscoelastic composite plate.

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Fig. 13 Influence of the E1/E2 ratio on time history of the lateral deflection of the central point of the viscoelastic composite plate.

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Fig. 14 Influence of the E1/E2 ratio on time history of the indentation of the viscoelastic composite plate. 4.6 Effects of the phase transformation of the SMA wires on behavior enhancement of a viscoelastic composite plate with viscoelastic SMA wires (light indenter) The foregoing sections have demonstrated the significant effects of the viscoelastic nature of the composite and SMA materials on the impact responses. In the present and next sections, effects of the phase-transformation-based dissipation on the impact responses of the viscoelastic plate are studied. To this end, results of two types of plates are compared: (i) a viscoelastic composite plate with inactive (without phase transformation) viscoelastic SMA wires and (ii) a viscoelastic composite plate with embedded viscoelastic SMA wires that undergo phase transformation as well. The study is divided into two stages wherein in the first stage, these effects are compared for a small indenter mass (low impact energy) whereas in the next stage, a larger indenter mass is adopted. The first stage is accomplished in the present section and the second stage is left to the next section. Results of the small indenter mass (i.e., the 7.1gr indenter mass, according to the specifications mentioned in Sec. 4.1) are shown in Figs. 15 to 17. As may be noted from these figures, when both

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the viscoelastic and superelasticity natures of the material properties are considered, the plate becomes slightly stiffer due to the pseudoelastic behavior of the SMA wires, even though the elastic modulus of the SMA wires is lower than that of the composite plate. At the same time, the response (contact) time has become slightly larger due to the martensite softening of the SMA material. Fig. 16 reveals that these effects are more pronounced on the lateral deflection of the plate. Comparing Figs. 15 to 17 with Figs. 9 to 11, respectively show that in cases of small indenter mass, the superelastic effects are almost ignorable in comparison to the viscoelastic effects. Indeed, in this case, the volume fraction of the martensite phase is very small and local (almost restricted to the impact region).

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Fig. 15 Effects of the phase transformation of the SMA wires on enhancement of time history of the contact force of the viscoelastic composite plate.

Fig. 16 Effects of the phase transformation of the SMA wires on enhancement of time history of the lateral deflection of the viscoelastic composite plate.

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Fig. 17 Effects of the phase transformation of the SMA wires on enhancement of time history of the indentation of the viscoelastic composite plate.

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4.7 Effects of the phase transformation of the SMA wires on behavior enhancement of a viscoelastic composite plate with viscoelastic SMA wires (heavy indenter) To assess effects of the higher indenter masses, a viscoelastic Graphite/Epoxy composite plate with the material and geometric information mentioned in Sec. 1, but with a [0/90/0/90/0] lamination scheme is considered. The SMA wires, whose diameters are now larger, are embedded in the top and bottom layers of the plate. Two indenter masses: 2 and 4kg are adopted but diameter of nose of the indenter is chosen as the previous case, i.e., R= 6mm. Modulus of elasticity of the indenter is E=26.3GPa. Moreover, to prevent the SMA wires from saturation, it is assumed that the ambient temperature is about 60C. Therefore, the forward and transverse transformation stresses are about 400 MPa higher than those of Sec. 4.6. Based on these specifications, volume fraction of the martensite phase is much higher in the present case, especially for the V=20m/s initial velocity of the indenter. Furthermore, in this case, in contrast to the previous case, majority of the plate regions experience martensite phase transformation. The resulting time histories of the contact force and lateral deflection are compared in Figs. 18 and 19 respectively, for viscoelastic plates with and without SMA wires and various masses and initial velocities of the indenter. As may be readily noted from Figs. 18 and 19, the phase transformation phenomenon has considerably affected responses of the plate. Furthermore, due the resulting stiffening, the lateral deflection and contact time of the plate have reduced. Since the response time is larger than that of the preceding section due to the higher indenter mass, responses of the higher vibration modes of the plate have superimposed on the fundamental one. Effects of the viscoelastic behavior of the composite plate and the SMA wires on the stress-strain diagram of the SMA wires at the impacted point of the plate may be observed in Fig. 20, for various magnitudes of the indenter mass and velocity. It may be readily noted from Fig. 19 that the viscoelastic behavior of the materials, leads to a stiffer plate; so that, load bearing of the SMA wires decreases and consequently, smaller stresses will be induced in these wires. This phenomenon, eventually leads to smaller hysteresis loops and reduces contribution of the SMA wires in dissipation of the undesired strain energies, as Fig. 20 confirms. Fig. 20 shows that the SMA wires are not saturated yet; since the strains are smaller than the maximum recoverable strain, i.e., the strain at the end of the full martensite transformation.

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Fig. 18 Influence of the initial velocity and phase transformation on contact forces of viscoelastic composite plates with embedded SMA wires (heavy indenter).

Fig. 19 Influence of the initial velocity and phase transformation on lateral deflections of viscoelastic composite plates with embedded SMA wires (heavy indenter).

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Fig. 20 Effects of the viscoelasticity of the composite plate and SMA wires on the stress-strain diagram of the impacted point of the plate, for different indenter masses and velocities (T=60C).

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5 Conclusions In the present paper, a quantitative comparison is made between effects of the viscoelasticity and the superelasticity phenomena on the impact responses of composite plates with embedded SMA wires, using a new high-order hyperbolic plate theory. A hierarchical constitutive law is employed for both the viscoelastic composite plate and SMA wires, Brinson’s constitutive equation is refined to include the reverse loading and the viscoelasticity effects of the SMA wire, and Hertz contact law is modified accordingly. Some of the extracted practical conclusions are: - Due to establishing hysteretic loops and dissipating a significant portion of the stored strain energy, the SMA wires increase the apparent stiffness of the plate and strength of the plate against the impact. - When the indenter mass is small, effects of the viscoelasticity on the resulting dynamic responses are more pronounced in comparison to those of the phase-transformation phenomenon. - Higher indenter masses lead to greater martensite volume fractions and consequently, smaller lateral deflections and larger contact forces; so that effects of the phase transformation phenomenon become more pronounced. - The growth in the contact force is not necessarily accompanied by a reduction in the contact time. Sometimes, the contact force grows due to hysteretic nature of the material during the phase transformation but at the same time, the contact time decreases due to transformation to the softer martensite phase. - The viscoelastic behavior of the plate leads to smaller hysteresis loops and reduces contribution of the SMA wires in dissipation of the undesired strain energies of the plate. References [1] Marques SPC, Creus GJ. Computational viscoelasticity. Heidelberg: Springer; 2012. [2] Lecce L, Concilo A. Shape memory alloy engineering for aerospace, structural and biomedical applications. Oxford: Butterworth-Heinemann, Elsevier; 2015. [3] Muliana AH, Sawant S. Responses of viscoelastic polymer composites with temperature and time dependent constituents. Acta Mech 2009; 204: 155–173. [4] Khalili SMR, Botshekanan Dehkordi M, Carrera E, Shariyat M. Non-linear dynamic analysis of a sandwich beam with pseudoelastic SMA hybrid composite faces based on higher order finite element theory. Compos Struct 2013; 96: 243-255.

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