Flutter and divergence instability of nanocomposite sandwich plate with magnetostrictive face sheets

Journal of Sound and Vibration 457 (2019) 240e260

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Flutter and divergence instability of nanocomposite sandwich plate with magnetostrictive face sheets Zahra Khoddami Maraghi Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 October 2018 Received in revised form 15 May 2019 Accepted 3 June 2019 Available online 8 June 2019 Handling Editor: P. Joseph

This research points out the positive results of structural deformations and follower forces exerted on a sandwich composite plate. This phenomenon that is named “flutter” is a dynamic instability and defines the ductility of the structure in fluid flow. Herein, the effective elastic properties of the reinforced composite by carbon nanotubes are acquired by the rule of mixture. Pasternak foundation has developed normal and shear modulus. The upper and lower layers of the sandwich are selected from the smart magnetostrictive sheets. A feedback control system pursues the magnetization effects of Terfenol-D films on the flutter vibration characteristics of sandwich plate. Five coupled equations of motion are derived using Hamilton's principle. These equations are solved by differential quadrature methods. Results of this study detected the critical values of follower forces corresponding to the start of flutter and divergence instabilities considering the effect of thickness ratios of the sandwich plate and velocity feedback gain, volume fraction of fibers and temperature gradient. The results of this research are used in the diagnosis of new structures used in aerospace, military and facilities industries. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Flutter Divergence Nanocomposite sandwich plate Feedback control system Magnetostrictive face sheets

1. Introduction Elhami and Zeinali [1] noted that the instability in the structures is static and dynamic, in static instability the frequency is zero and in dynamic instability the two frequencies of the structure are closer to each other under the influence of the follower forces, Specifically the system is highly unstable and The exerted forces on the structures can be divided into two groups, conservative and non-conservative. The hydrodynamic and aerodynamic loads, forces exerted to the parts and connections of a mechanical system, forces because of magnetic fields and electrical and fluid flow are most non-conservative types of the follower forces. One of the well-known examples is the inside - page forces, when they're changing the shape of the structure, they preserve their direction. The follower forces (Fig. 1) are among the categories of non-conservative forces, so that their direction is proportional to the changing shape of the structure and because they follow the geometry of the structure we call them follower forces. The buckling phenomenon can easily be studied using a dead load on the structure, while the simulations of follower forces are difficult to be assimilated in the lab. According to the study by Kim and Kim [2] and Jayaraman and Struthers [3], the instability of a plate is detected under a non-conservative force:

E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.jsv.2019.06.002 0022-460X/© 2019 Elsevier Ltd. All rights reserved.

Z. Khoddami Maraghi / Journal of Sound and Vibration 457 (2019) 240e260

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Fig. 1. Rectangular plate under follower forces.

1. The divergence of natural frequency at zero 2. Flutter caused by coupling two vibration modes If the transverse displacement of the plates is considered, there will be an increasing term. If the value of the transverse displacement be considered in this case the transverse displacement of the plate will grow exponentially over time and flutter instability happening. Yang et al. [4]studied the critical dynamic pressure for panel flutter of composite plate with viscoelastic mid-layer in supersonic airflow and used damping model, Hamilton's principle and Galerkin method to derive and solve the partial differential equations and found out if the viscoelastic damping gets increased further the flutter resistance of the composite plate will be enhanced. Kuo [5] investigated the supersonic flutter of rectangular composite plates while considering the effects of variable fiber spacing. This paper will show the sequence of the natural mode may be altered and the two natural frequencies may be close to each other because of the fiber distribution that may change the distributed stiffness and mass of the plate. Song and Li [6] have analyzed the active Aeroelastic flutter analysis and vibration control at the flutter bounds of the supersonic composite laminated plates with the piezoelectric patches that bonded on the top and bottom surfaces and designed a controller by the velocity feedback and proportionate feedback control algorithm to obtain the active damping and stiffness. Results showed that the Aeroelastic flutter characteristics can be improved and the Aeroelastic vibration response amplitudes can be reduced. Kim and Kim [2] have performed the dynamic stability analysis of an isotropic, orthotropic and a symmetrically laminated composite plate under a pure follower force. Kirchhoff-Love and Mindlin plate theory was put upon theoretical analysis and its results were compared with finite element method. Jayaraman and Struthers [3] have analyzed the effects of divergence and flutter instabilities of a rectangular, orthotropic plate subject upon follower forces and discussed the effects of the tangential follower parameter, aspect ratio, boundary condition, and the magnitude of the critical load on the stability of plate. Guo et al. [7] studied the effects of the dynamic characteristics and stability of rectangular plate subjected upon uniformly distributed tangential follower forces and presented the differential equations of motion considering thermo-elastic coupling for rectangular plate under tangential follower forces and calculated the dimensionless complex frequencies of the moving rectangular plate for different boundary conditions, also found out that composites that consists two or more kinds of materials will produce desirable properties when they're together, that can't be achieved with any of the constituents alone and mentioned “fibres with high strength and high modulus are used in a matrix material of fibre-reinforced composite”. Composite materials and sandwich structures have been interesting for researchers for a long time but in recent decades smart materials have created a breakthrough in these investigations. Also Lei et al. [8] has presented free vibration analysis of Functionally graded (FG) Nano-composite plates reinforced by single-walled carbon nanotubes (CNTs) using the element-free kp-Ritz method and inferred that “The composite was assumed to be graded through the thickness direction according to several linear distributions of the volume fraction of CNTs” and also presented in several examples the effects of CNT volume fraction, width-to-thickness ratio, aspect ratio, temperature change on natural frequencies and mode shapes of FG-CNT reinforced composite (CNTRC) plates. Natarajan et al. [9] have developed bending and free flexural vibration behaviour of sandwich plates with CNT reinforced face sheets based on higherorder structural theory and considered the in-plane and rotary inertia terms in the formulation. Herein, the governing equations were solved for a sandwich plate with homogeneous core and CNT reinforced face sheets also they examined the influence of volume fraction of the CNT, core-to-face sheet thickness and thickness ratio on the global/local response of different sandwich plates. Malekzadeh et al. [10] has investigated the vibration of laminated plates resting on Pasternak foundation by considering the effect of non-ideal boundary conditions and initial stresses because of in-plane loads and used the LindstedtePoincare perturbation technique to solve the problem and obtained the frequencies and mode shapes of the plate also compared their results with finite element simulation using ANSYS software and showed the effects of various parameters like the stiffness of foundation, boundary conditions and in-plane stresses on the vibration of the plate. Lee et al. [11]studied the transient response of laminated composite plates with embedded smart material layers using a unified plate theory that includes the classical, first-order and third-order plate theories, noted that Terfenol-D layers were used to control the vibration suppression and their findings showed the effect of material properties, smart layer position and smart layer thickness on the vibration suppression of plate.

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HYPERLINK "https://www.sciencedirect.com/science/article/pii/S1270963818305868” yl "!” yo ” et al. [12] was the first presenter of experimental application of the Parametric Flutter Margin method for identification of aeroelastic instabilities also while They were in a stable system demonstrating and excellent agreement with the reference instability conditions had positively identified the flutter and divergence conditions of the original test model. Chen et al. [13] by establishing a nonlinear aeroelastic model had carried out the numerical simulations on the post-flutter response of a flexible cantilever plate also they noticed that with the increase of inflow velocity the plate undergoes period-1, period-3 and non-periodic motions, considering that both geometric and aerodynamic nonlinearities are prevailing the post-flutter behaviour. Li et al. [14] with Hertzian contact and in an axial flow have developed a complete dynamical model of a plate and they also studied flutter instability, limit cycles flutter, lock-in motions, quasi-periodic motions, and quasi-periodic divergence and by using the Galerkin method they have discretized the governing nonlinear partial differential equation of the system in space and time domains. Ultimately, results are showing that after instability, the plate loses its stability by flutter and then undergoes limit cycle motions because of the contact nonlinearity. Hong [15] studied the transient response of thermal stresses and centre displacement in laminated magnetostrictive plates under thermal vibration and used the velocity feedback control to vibration suppression in a three-layer laminated magnetostrictive plate with four simply supported edges and His results showed an efficient method to compute the results including shear deformation effect with a few grid points. Instability of nanocomposite sandwich plate made of two smart magnetostrictive face sheets is the new topic which can't be found in the literature. Magnetostrictive materials (MsM) are some of the most affordable options in control systems because of their reciprocal nature, also adduced that magneto-mechanical coupling in MSM can be used in the stability of systems. In this research, the magnetostrictive face sheets are used in a feedback control system and the role of follower force is studied (as a single potential function that satisfies the conservation principle) when flutter and divergence instabilities occur. The results of this study have investigated that the flutter and divergence instability on the vibration frequency of the sandwich plate can be useful in many industries. 2. Structural definition A schematic diagram of a sandwich plate is illustrated in Fig. 2 which has indicated geometrical parameters of length a, width b and thickness 2hm þ hc (index “c” refers to core and index “m” refer to the magnetostrictive face sheet). As shown in Fig. 2 the sandwich plate is composed of three layers: a. Central composite core reinforced by CNT, b. Two face sheets made of MsM. The strain-displacement relations are separately written for each layer afterward forces and moments are obtained ultimately, the total energy including the energy of CNTRC core and magnetostrictive face sheet is obtained and the equations of motion are derived using Hamilton's principle. 3. Third order shear deformation theory (TSDT) TSDT extends the first order shear deformation theory (FSDT) by assuming that: - Shear strain and consequently shear stress are not constant through the plate thickness where for a moderately thick plate, TSDT leads to better results. - The strain equations do not need a shear correction factor. - The displacement field accommodates a quadratic variation of the transverse shear through the thickness and the vanishing of transverse shear stresses on the top and bottom surfaces of the plate as mentioned in Wang's paper [17] and Reddy's book [18]. ~ V ~ and W ~ of TSDT for plates: Reddy [18] has given the displacement field U,

 3
~ y; z; tÞ ¼ u ðx; y; tÞ þ zq ðx; y; tÞ  4 z q ðx; y; tÞ þ v w ðx; y; tÞ Uðx; 0 1 1 0 2 3h vx  3 ~ y; z; tÞ ¼ v ðx; y; tÞ þ zq ðx; y; tÞ  4 z q ðx; y; tÞ þ v w ðx; y; tÞ Vðx; 0 2 3 h2 2 vy 0

(1)

~ Wðx; y; z; tÞ ¼ w0 ðx; y; tÞ where u0 ðx; y; tÞ; v0 ðx; y; tÞ; w0 ðx; y; tÞ are displacement function along ðx; y; zÞ directions. q1 ðx; y; tÞ; q2 ðx; y; tÞ are rotations about x and y axis. t is time and h is the thickness of the plate that can be different for each layer. Sandwich plate is considered

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as a monolithic structure where the displacement of each layer is assumed to be the same, mentioned in Eq. (1). The linear strain field for TSDT is obtained using Hooke's law. The linear strain field for TSDT is obtained using Hooke's law (Appendix A). 4. Constitutive equations Brockmann [19] presented the Stress-strain relation for orthotropic and isotropic materials following equation:

2

2

3

6 sxx 7 6 Q 11 6 syy 7 6 Q 6 7 6 21 6 txy 7 ¼ 6 0 6 7 6 6 txz 7 6 0 4 5 4 tyz 0

Q 12 Q 22 0 0 0

0 0 Q 44 0 0

32 3 0 76 εxx 7 6 7 0 7 76 εyy 7 6 gxy 7 0 7 76 7 6 7 0 7 54 gxz 5 gyz Q 66

0 0 0 Q 55 0

(2)

where sxx ; syy are normal stresses, txy ; txz ; tyz are shear stresses, εxx ; εyy are normal strains and gxy ; gxz ; gyz are shear strains. Also Q ij the constant terms of engineering are in this way:

For orthotropic materials : Q 11 ¼ Q 22 ¼

E11 ; ð1  y12 :y21 Þ

Q 12 ¼

y12

Q 44 ¼ Q 55 ¼ Q 66 ¼ G12 ;

E11

¼

E12 y12 E21 y21 ;Q ¼ ð1  y12 :y21 Þ 21 ð1  y12 :y21 Þ

y21 E22 (3)

For isotropic materials : E11 ¼ E22 ¼ E12 ¼ E; E ; Q 11 ¼ Q 22 ¼  1  y2

G12 ¼

E 2ð1 þ yÞ

Q 12 ¼ Q 21 ¼ 

Ey 1y

2

; Q 44 ¼ Q 55 ¼ Q 66 ¼

E 2ð1 þ yÞ

in which E and y are Young's modulus and Poisson's ratio of isotropic materials is given respectively. For orthotropic materials, E11 ; E22 are normal Young's modulus, E12 is transverse Young's modulus and G12 is shear modulus. Poisson's ratios y12 ; y21 are defined as y12 ¼ εyy =εxx ; y12 ¼ εyy =εxx and are obtained from standard tests, such as tensile test or pure shear tests. Above equations are valid for CNTRC core as orthotropic material and magnetostrictive face sheets as isotropic material (C ij instead Q ij ). Herein, the following section is presented to obtain the elastic properties of the central core that reinforced by CNTs. 4.1. CNTRC core Consider a CNTRC plate with thickness hc and dimensions ða bÞ .This CNTRC plate is made of two different isotropic materials that called fibres (SWCNT) and polymer matrix (PmPV). The reinforcement fibres are uniformly distributed in length of the plate, According to the extended rule of mixture, the material properties of CNTRC plates can be expressed as [8]: m E11 ¼ h1 VCNT ECNT 11 þ Vm E

h2 E22

h3 G12

¼ ¼

VCNT ECNT 22 VCNT GCNT 12

þ

Vm Em

þ

Vm Gm

(4)

where superscript “m” defines matrix and “CNT” defines CNT fibers. Also, VCNT and Vm are the volume fractions of CNT and matrix. Shen [20] in Eq. (4) has introduced the accounting way of the load transfer between the fibers and matrix such as the surface effect, strain gradient effect, and intermolecular coupled effect on the equivalent material properties of CNTRCs, CNT efficiency parameters hj (j ¼ 1,2, 3). Poisson's ratio for CNTRC core is also introduced as follows [8]: m y12 ¼ VCNT yCNT 12 þ Vm y m where yCNT 12 and y are Poisson's ratios of fibers and matrix, respectively.

(5)

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Z. Khoddami Maraghi / Journal of Sound and Vibration 457 (2019) 240e260 Table 1 Material properties of SWCNTs (10,10) with h ¼ 0:34 nm [8]. Han and Elliott [17]

ECNT 11

ECNT 22

GCNT 12

SWCNTs (10,10)

600 GPa

10 GPa

17.2 GPa

Table 2 CNT efficiency parameter for VCNT ¼ 0:11; 0:14; 0:17 [23]. VCNT

Rule of Mixture

h1

h2

VCNT ¼ 0:11 VCNT ¼ 0:14 VCNT ¼ 0:17

0.149 0.150 0.149

0.934 0.941 1.381

Lei et al. [8] stated that the effective thickness of fibers plays an important role to estimate the material properties of CNTs. Han and Elliott [21] have obtained a new value for SWCNTs. (10, 10) with h ¼ 0:34 nm that has been presented in Table 1. Also, Lei et al. [8] and Wang and Shen [22] were reported typical values of effective material properties of SWCNTs (10, 10). Comparing the results of the extended rule of the mixture and molecular dynamic (MD) simulations, the CNT efficiency parameter is estimated such as estimations that have been listed in Table 2 [23]. 4.2. Magnetostrictive face sheets Using the articles provided by Hong [24], Murty and Anjanappa [25], Daneshmehr and Rajabpoor [26], the following magneto-mechanical coupling can be used to consider the effect of magnetic field on MsM:

2

3

2

32

3

sms 7 6 0 0 e31 76 0 7 6 xx 6 6 sms 7 76 7 yy 5 ¼  4 0 0 e32 54 0 5 4 ms Hz 0 0 e34 txy

(6)

Superscript “ms” refers to magnetic stress that summarized by Eq. (6) (C ij for magnetostrictive face sheets instead Q ij for composite core). Hz and e31 ; e32 ; e34 are the magnetic field intensity and the magnetostrictive coupling modules that can be expressed as it follows [24e26]:

Hz ¼ Kc Iðx; y; tÞ ¼ Kc CðtÞ

vw0 ðx; y; tÞ vw ðx; y; tÞ ¼ Kvfc 0 vt vt

(7)

where Kc , IðtÞ and CðtÞ are the coil constant, coil current and the control gain in which Kc CðtÞ is introduced as velocity feedback gain. Fig. 3 displays a simple control loop that has been used in this sandwich for magnetostrictive layers. The magnetic field is applied on the face sheets at z-direction, so magnetostrictive coupling modules e31 ; e32 ; e34 are determined as it follows [25]:

e31 ¼ e31 cos2 q þ e32 sin2 q e32 ¼ e31 sin2 q þ e32 cos2 q e34 ¼ ðe31  e32 Þ sin q sin q

(8)

where the q represents the direction of the magnetic anisotropy. The values of e31 ; e32 ; e34 are reported in Table 4. By applying stresses and strain obtained from three layers in the next step, with the help of the energy method, equations of motion are derived. 5. Energy method in sandwich plate The energy method in applied mechanics is the relationship between stress, strain, displacement, material properties, and environmental effects in the form of energy or work carried out by internal and external forces. The energy method is a suitable way to formulate the governing equations on solid mechanics. This method in complex systems gives approximate solutions using a set of partial differential equations. In this section, the energy method is used to obtain the governing equations. Strain energy ðUÞ of the rectangular sandwich plate is calculated as [16]:

Z. Khoddami Maraghi / Journal of Sound and Vibration 457 (2019) 240e260

245

Fig. 2. Rectangular plate under follower forces.

Fig. 3. A simple schematic of the control loop.

U¼

1 2

Z





sxx εxx þ syy εyy þ txy gxy þ txz gxz þ tyz gyz dx dy dz

V Top

Usandwich ¼ UCore þ U Bottom Face sheet þ U Face sheet

¼

1 2

hc Z2 Zb Za hc  2

0

hc Z2



1 þ 2

hm 

1 þ 2

hc 2

ðsxx εxx þsyy εyy þ txy gxy þ txz gxz þ tyz gyz

Core

dx dy dz

0

(9) Zb Za 0

0



m m sm xx εxx þsyy εyy þ txy gxy

 lowerSheet

dx dy dz

0

hc þ hm 2Z Zb Za hc 2





m m sm xx εxx þsyy εyy þ txy gxy

 upperSheet

dx dy dz

0

The kinetic energy of sandwich plate can be stated as [16]:

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Z. Khoddami Maraghi / Journal of Sound and Vibration 457 (2019) 240e260

1 ¼ 2

KSandwich ¼ KCore þ 2KFace sheet

h Z c =2

hc =2

þ

hc =2hm

þ

hc =2þh Z m

1 2

hc =2

A

" 2  2  # ! ~ ~ ~ 2 vU vV vW rc þ þ dAdz vt vt vt

" 2  2  # ! Z ~ ~ ~ 2 vU vV vW rm þ þ dAdz vt vt vt

h Z c =2

1 2

Z

(10)

A

Z

rm

A

" 2  2  # ! ~ ~ ~ 2 vU vV vW þ þ dAdz vt vt vt

where rc and rm are the density of core and sheets respectively. According to the above equation, the next step is to calculate the stresses and strains in three layers. For this purpose, Reddy's theory for moderately thick plate is defined. 6. Follower force Follower forces are one of the common forces considered in applied mechanics and have a single potential function, such as loads of satisfying the conservation principle and their works are independent of the path during any admissible movement of the body. The simplest and most common loads are the dead loads, which remain constant both in size and magnitude during the deformation and motion of the body. On the other hand, non-conservative forces are very popular in nature. An example of these forces is the force that gets a result from the mechanical interaction of a continuous environment with its own perimeter that is known as cross - effect force. As it had been mentioned earlier, the in-plane forces preserve their direction while changing the shape of the structure, as the follower forces remain proportional to the structural deformation. The first variation of work done by follower force ðP0 Þ for use in the energy method as given by Refs. [1e3].



dWFollower force ¼  ∬ P0

v2 w 0 vx2

 ðx  aÞ dw0 dx dy

(11)

where dw0 is the variation form of displacement in z-direction. 7. Pasternak foundation Pasternak foundation model is capable of considering the transverse shear and normal loads. Ding et al. described the Pasternak model in detail in their book [27]. Herein, the bottom surface of the sandwich plate is continuously supported by an elastic foundation. Elastic medium applies the force ðFÞ on the surface of plate ða  bÞ:

 F ¼ Kw w0  gx

v2 w 0 vx

2

þ

v2 w 0

 (12)

2

vy

in which kw , gx are Winkler foundation parameter, shear foundation parameter. Therefore, the external work ðSÞ because of the elastic foundation is calculated.

S¼

1 2

Zb Za F w0 dxdy : 0

(13)

0

8. Hamilton's principle Z

t2

As Reddy said in his book [18], d t1

L dt ¼ 0; represents Hamilton's principle for the conservative motion of a particle. t1

and t2 are the arbitrary instants of time that the particle was moved by conservative forces. In mathematical terms the line integral over the Lagrangian function ðLÞ is an extremum for the path motion. Hamilton's principle is used for dynamics system and presents a variational form in the determination of motion equations. The principle can be stated in an analytical form where the first variation form of equations must be zero, as it follows [18]:

Z. Khoddami Maraghi / Journal of Sound and Vibration 457 (2019) 240e260

d

Zt2 h

i
USandwich  KSandwich þ S þ WFollower force dt ¼ 0

247

(14)

t1

where dUSandwich , dKSandwich are the variation of strain energy and kinetic energy, and d S; dWFollower force are the variation of external work due to elastic foundation and follower force. for TSDT and afterward using dimensionless parameters which had been introduced in Eq. (15):

8 > > < Core : hc
u v w 
x y Q ij a 0 0 0 ; ; ; For ðz; hÞ ¼ ; ; ðU; V; WÞ ¼ ; g ¼ ; Qij ¼ > a b b a b h Em > Face sheet : hm :     C ij Ec hc hm hm hc hc Ii ; ; ðac ; bc Þ ¼ ; ; Ibi ¼ iþ1 ði ¼ 2; 4; 6Þ ;2 ¼ ;d ¼ ; ðam ; bm Þ ¼ Em Em hm a b a b hm sffiffiffiffiffiffi eij CðtÞKc t Em kw hm gx P Gij ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; t ¼ ; gxa ¼ ;q ¼ 0 ;K ¼ a rm bw Em a:Em b Em a Em rm Cij ¼

(15)

The equations of motions are obtained by setting the coefficient dU ; dV; dW; dq1 ; dq2 equal to zero and are given in Z hc =2 Z ðhc =2þhm Þ zi dz þ zi dz; ði ¼ 2; 4; 6Þ. Appendix A, where Ii ¼ ðhc =2þhm Þ

hc =2

9. Solution procedure using differential quadrature method (DQM) First of all, Eqs. (A2)-(A6) has to be used to separate the spatial and temporal:

Uðz; h; tÞ ¼ Uðz; hÞ eumn t Vðz; h; tÞ ¼ Vðz; hÞ eumn t Wðz; h; tÞ ¼ Wðz; hÞ eumn t q1 ðz; h; tÞ ¼ q1 ðz; hÞ eumn t q2 ðz; h; tÞ ¼ q2 ðz; hÞ eumn t

(16)

qffiffiffiffiffi where umn ¼ Ua Ermm is the dimensionless frequency (U is the dimension frequency). In DQM, the differential equations get changed into the first algebraic equations. Herein, the partial derivatives of a function (F) are approximated by a specific variable at discontinuous points by a set of weighting series. It supposed that F be a function representing U; V; W; q1 and q2 while considering the variables x and h (0 < x < 10 < h < 1) when Nx  Nh be the grid points along these variables with following derivative as mentioned in Shu's book [28].

  dn F xi ; hj dx

n

¼

dm F xi ; hj

 ¼

dhm

Nh X

ðmÞ

Bjl Fðxi ; hl Þ m ¼ 1; :::; Nh  1

(17)

l¼1

  dnþm F xi ; hj m

dx dh ðnÞ

 ðnÞ  Aik F xk ; hj n ¼ 1; :::; Nx  1; i ¼ 1; :::; Nx ; j ¼ 1; :::; Nh

k¼1



n

Nx X

¼

Nx X Nh X

ðnÞ ðmÞ

Aik Bjl Fðxk ; hl Þ

k¼1 l¼1 ðmÞ

where Aik and Bjl

are the weighting coefficients that are using Chebyshev polynomials for the positions of the grid points

whose recursive formulae can be found in Ref. [25]. If DQM exerts using Eq. (17) into governing equations that presented in Eqs. (A2)-(A6), the standard form of the equation of motion will be obtained. Considering simply supported boundary conditions of an eigenvalue problem is derived in the eigenvalues of a state-space matrix which are introduced as the dimensionless frequency. pffiffiffiffiffiffiffiffiffiffiffiffiffi To verify the accuracy of the results of this numerical method, the linear frequency umn ¼ Ua2 r:h=D; D ¼ EI=ð1 y2 Þ for an isotropic square plate has been compared in Table 3 for three modes. A good agreement between the results of the present work and other published papers as Wang and Shen [22], Bardell [29], Wang et al. [30] and Leissa [31] shows that the correct solution is used.

248

Z. Khoddami Maraghi / Journal of Sound and Vibration 457 (2019) 240e260 Table 3 Comparison of linear frequency for an isotropic square plate ða =b ¼ 1; a=h ¼ 300; y ¼ 0:3Þ.

Leissa [30] Bardell [28] Wang et al. [29] Wang and Shen [18] (DQM)Present work

u11

u12

u13

19.7392 19.7392 19.7392 19.7362 19.4250

49.3480 49.3480 49.3453 49.3431 49.3392

98.6960 98.7162 98.6268 98.6765 98.0423

Mode shapes of vibration for the isotropic square plate are plotted in Fig. 4 for combination of simply supported (S) and clamped (C) boundary conditions on the four edges of the plate such as SSSS-CCCC-CSCS boundary conditions.

Fig. 4. Mode shapes of vibration for isotropic square plate.

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10. Numerical results and discussion Herein, divergence and flutter instability of sandwich plate were analyzed when the plate resting on Pasternak foundation, the sandwich plate was contained an elastic composite core that reinforced by CNT fibers and the fibers were uniformly distributed and their material properties were obtained by the rule of Mixture. Two smart face sheets have surrounded the CNTRC core where the three layers vibrated as an integrated structure while A feedback loop was used to control the vibration of the sandwich plate in presence of magnetic field. In order to investigate the stability of sandwich, Pasternak model was also used, afterward the instability of sandwich plate was investigated by velocity gain feedback, aspect ratio, thickness ratio, volume fraction of fibers, temperature and elastic medium. The face sheets have been made of Terfenol-D and its properties have been listed in Table 4. The presented solution method by comparing the results of Natarajan et al.’s article. This paper [9] has studied the sandwich plate with titanium alloy core ðTi  6Al  4V : EH ¼ 122:56ð1  4:586 104 TÞ GPa; yH ¼ 0:29; rH ¼ 4429 Kg=m3 Þ and CNT reinforced composites in a uniform arrangement. Natarajan et al. have obtained the vibration frequency of sandwich plate for six finite element models with the help of high order theories. the results shows that the DQM has a good agreement in comparison with this reference because of the extent of the equations of motion (Eq. (17)) in a sandwich plate and the limitation of controlling multiple terms, the comparison was only possible for two value of a =ð2hf þhH Þ and hH =hf ¼ 2; u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6a2 =h rH =EH ; VCNT ¼ 0:17 for uniformly distributed as given in Table 5 in which hf is face sheet thickness and hH ; rH ; EH is thickness, density and Young's modulus of core. The diverging instability which is called the buckling instability occurs when the imaginary part of the frequency tends to zero. Fig. 5a shows the variation of the image part of the frequency with increasing the following force for various thickness ratios of the composite core as shown in Fig. 5a, it is evident that divergence occurs in this case as the frequency tends to reduce with relatively high gradient and tends towards zero and close values to it. The values of the force in which the frequency is close to zero are the critical values of the system, and the system remains its stability until it reaches a critical value. As soon as the following force reaches a critical value, the unstable conditions will be provided and the system is prone to buckling. Herein, it is important to find and identify critical values. The imaginary part of the dimensionless frequency in Fig. 5b shows that the dimensionless frequency is zero and then it starts to grow suddenly. In other words, sustainability areas contain values of frequency, in which the real part is much larger than the imaginary part ðReðuÞ ¼ 0Þ and in the unstable areas the imaginary part is larger than the real part ðImðuÞ ¼ 0Þ. Another important point is that by shifting the thickness ratio of the core, the critical values of the following forces will increase as the thickness ratio increases. Fig. 6a and b are showing the dimensionless frequency variation in terms of the following for different value of the core to face sheet thickness ratio ðg ¼ 1; ac ¼ 0:2; Kvfc ¼ 103 ; h ¼ 0:34 nm; VCNT ¼ 0:17Þ. Both figures demonstrate that critical values of following force will increase by increasing the critical values of following force and the stability of those areas will grow larger. Fig. 7a and b are showing the dimensionless frequency variations in terms of the follower force for different values of the volume fraction of carbon nanotube fibers at Kvfc ¼ 103 ; h ¼ 0:34 nm g ¼ 1; d ¼ 10; ac ¼ 0:2. Fig. 7a and b are showing that by strengthening the composite core while using carbon nanotube fiber, critical values of the following force will be highly influenced and the stability of the system increases significantly. When VCNT ¼ 0:17 the critical values is more than four times greater in VCNT ¼ 0. the results show that increasing the composite strength using carbon fiber increases their resistance to external forces and increases the stability of the structures.

Table 4 Material properties of face sheet (Terfnol_D) and polymer matrix (PmPV) [8,15]. Properties

E

y

r

e31 ¼ e32

Terfenol-D PmPV

30e9 Pa ð3:51  0:0047TÞ GPa

0:25 0:34

9:25  103 kg=m3 1190 kg =m3

442:55 N =ðm:AÞ -

Table 5 Comparison of results with Natarajan et al.’s paper [9]. a =ð2hf þ hH Þ Theory HSDT13 HSDT11A HSDT11B HSDT9 TSDT7 FSDT Present work

5 3.8203 3.8108 4.1736 4.1677 4.3199 4.2504 4.2867

Error %12.21 %12.5 %2.71 %2.86 %0.77 %0.85

10 4.4397 4.4373 4.6095 4.6078 4.6655 4.6426 4.6481

Error %4.69 %4.75 %0.84 %0.87 %0.37 %0.12

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Fig. 5. a) variation of image part of frequency versus follower force in different ac . (b) variation of real part of frequency versus follower force in different ac .

Fig. 8a and b are showing the dimensionless frequency variation in terms of follower force in different elastic medium for

g ¼ 1; d ¼ 10; ac ¼ 0:2; Kvfc ¼ 103 ; h ¼ 0:34 nm; VCNT ¼ 0:17. The following two figures confirm that previous results of other papers and indicate that the presence of the elastic medium increases the stiffness of system and increases its stability. It is evident from the figure that the Winkler and Pasternak models increase the critical values of the following forces under the circumstances such as appropriate elastic medium can be used to delay the instability of structures.

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Fig. 6. a) variation of image part of frequency versus follower force in different d. (b) variation of real part of frequency versus follower force in different d.

10.1. Flutter instability Elastic structures getting influenced by non-conservative forces can show flutter phenomena. Flutter means the stability drop in the form of oscillations with incremental oscillatory slopes. One of the most important issues in engineering is the problem of follower forces that are important because of the asymmetric nature of the matrices derived from this type of loading. the results presented in this paper are related to the identification of flutter in a sandwich structure influenced by the following forces, which can be extended by further research on other types of non-conservative forces. In critical loads, which are known as flutter loads, the structure suffers from vibrations because of initial disorders, at forces lower than the critical load, the vibrations are damped, the follower force acts as a damper, while on the above, one of the modes will become

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Fig. 7. a) variation of image part of frequency versus follower force in different VCNT . (b) variation of real part of frequency versus follower force in different VCNT .

negative and the oscillations grow exponentially unless there is a way of limiting motion that is not discussed here. The term "soft flutter” refers to the state in which the damping ratio changes slowly with the change in load. When, by increasing the load in flutter conditions, a sudden drop in the damping value occurs, hard flutter occurs. In flutter topics, the second type of flutter is more important. In this case, a stable system may become unstable with a slight increase in load. Flutter analysis in the structure of this paper begins with a set of ordinary differential equations. These equations are transformed into a onedimensional value problem and the stability characteristics are in the form of eigenvalue problem and for VCNT ¼ 0:11; Kvfc ¼ 103 ; g ¼ 1; d ¼ 5; ac ¼ 0:2 and "SCSC" discussed.

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Fig. 8. a) variation of image part of frequency versus follower force in different elastic mediums. (b) variation of real part of frequency versus follower force in different elastic mediums.

Comparison of the theoretical results in the recent decades with experimental data suggests that there is a correlation in the answers, i.e., the damping of the modes at the lower loads of a flutter load is considered zero which is not true. the main reason for these shortcomings is that the equations are assumed to be linear and two dimensional. In Fig. 9a, the imaginary part is considered. When qb ¼ 0 the first and second dimensionless frequencies are spaced apart, As the frequencies increase, they start to get close together and the analysis indicates the coupling between torsional and flexural modes. Flutter occurs when the frequencies are merged, where roots are doubled in two pairs. Flutter loads in this state are ðqf Þ and flutter

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Fig. 9. a) variation of image part of frequency versus follower force for two vibration mods. (b) variation of real part of frequency versus follower force for two vibration mods.

frequency is ðuf Þ. This phenomenon can occur in higher modes as well. The real part of the frequency remains zero in Fig. 9b until a flutter occurs. After that, the real part of one of the roots will be positive and the other real part will be negative. Fig. 10 shows a flutter load and flutter frequency in the three wavenumbers m ¼ 1; 2; 3 . by increasing the wavenumbers, flutter load and flutter frequency will be significantly increased and in other words, flutter is delayed. The distance between natural frequencies has a large impact on flutter bar. Generally, the closer the structural frequencies converge, the critical interaction between the modes will occur faster and a flutter load will decrease. Designers always try to increase the distance between frequencies by altering the mass distribution or increase of stiffness, to increase the flutter charge. Fig. 11 shows the effect of reinforcing fibers on flutter load. Strengthening the composite core using carbon Nanofibers while increasing its strength can flutter load and delay its occurrence, but flutter frequency changes are insignificant compared to flutter load.

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Fig. 10. Variation of image part of frequency versus follower force for three wave numbers.

Fig. 11. Variation of image part of frequency versus follower force for different VCNT .

The results of the previous sections show that increasing the thickness of the composite core leads to a reduction of the dimensionless frequency of sandwich plate. According to the Fig. 12, the increase in this parameter also reduces both flutter and flutter frequency because of its effect on the stiffness of the structure. Fig. 13 shows the effect of velocity feedback control parameter on flutter load and flutter frequency. by increasing the velocity control parameter in the given range in Fig. 13, flutter frequency has decreased, but the flutter load does not change significantly. 11. Conclusion Instability analysis of a sandwich plate with composite core and magnetostrictive face sheet is a novel topic that has been studied in this research for the first time. To consider the magnetization effect of face sheet, a control feedback system was

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Fig. 12. Variation of image part of frequency versus follower force for different d.

Fig. 13. Variation of image part of frequency versus follower force for different Kvfc .

used and velocity feedback gain as controlling parameter introduced. Pasternak foundation was developed to evaluate normal and shear modulus. Reddy's plate as a third order shear deformation plate theory was used to derive the equations of motion. Set of equations were solved by two-dimensional DQM and following results were concluded: ❖Follower force has damping effect on vibration of sandwich plate so that with increasing the follower force; the sandwich plate behaves divergent instability in the first-order mode. Increasing the follower forces leads to the divergence and flutter instability in a sandwich plate. In divergence instability with increasing the amount of the following force, the frequency reduces with a relatively high gradient, it tends to reduce to zero and close values to it and the sandwich is on

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the verge of buckling. Flutter occurs when two vibration modes are integrated into each other. In this case, vibrations are created with high fluctuations and will make the sandwich unstable. ❖Normal and shear modulus significantly increase the dimensionless frequency of sandwich plate. ❖Increasing the volume fraction of fibers also increases the critical load because of carbon fiber reinforces composite core will increase the strength and hardness of the sandwich plate. ❖Increasing temperature reduces the stiffness of the sandwich plate and decreases the frequency. Therefore, increasing the temperature can act as system instability. ❖Both face sheets can be utilized to vibration suppression of sandwich plate using control feedback system where velocity feedback gain as a control parameter reduces the frequency of sandwich plate and delays the occurrence of flutter. Such an intelligent system with the postponement of instability that can increase the useful life of the structure and its efficiency. Sandwich composites can be used in aerospace industry because of the good performance of composites, strength, ductility, and weight. Such studies in flutter instability can be useful in detecting destructive seismic loads in aircraft wings. Using this type of sandwiches in giant turbine blades, in addition to increasing strength, can adjust the vibration frequency of the system and prevent flutter.

Acknowledgements This work was financially supported by Mahallat institute of higher education. The author also thanks Professor J.N. Reddy for the comprehensive book on the principles of the energy method and Professor A. Ghorbanpour Arani for guidance and new ideas. Appendix A

  v v 4 z3 v v2 u0 ðx; y; tÞ þ z q1 ðx; y; tÞ  q ðx; y; tÞ þ w ðx; y; tÞ ; 0 vx vx 3 h2 vx 1 vx2   v v 4 z3 v v2 ¼ v0 ðx; y; tÞ þ z q2 ðx; y; tÞ  q ðx; y; tÞ þ w ðx; y; tÞ ; 0 vy vy 3 h2 vy 2 vy2

εxx ¼ εyy

εzz ¼ 0

  1 v 1 v 2 z3 v v2 v0 ðx; y; tÞ þ z q2 ðx; y; tÞ  w q ðx; y; tÞ þ ðx; y; tÞ ; 2 0 2 vx 2 vx 3 h2 vx vyvx   1 v 1 v 2 z3 v v2 ðx; ðx; u0 ðx; y; tÞ þ z q1 ðx; y; tÞ  w y; tÞ þ y; tÞ ; q þ 1 0 2 vy 2 vy 3 h2 vy vyvx  1 v 1 z2
v w0 ðx; y; tÞ þ q1 ðx; y; tÞ  2 2 q1 ðx; y; tÞ þ w0 ðx; y; tÞ ; εxz ¼ 2 vx 2 vx h  2 1 v 1 z v w0 ðx; y; tÞ þ q2 ðx; y; tÞ  2 2 q2 ðx; y; tÞ þ w0 ðx; y; tÞ : εyz ¼ 2 vy 2 vy h εxy ¼

(A1)

dU ¼ dUcore þ dUSheet ¼  1=2 C44 b g þ

 2Q11 am

d2 U d2 U d2 V d2 V d2 V d2 U  C11 a 2  1=2 C44 a  1=2 C21 a  1=2 C12 a þ aε 2 2 dhdz dhdz dhdz dh dt dz

d2 U dz

2

 Q44 g bm

! Core

d2 U d2 W d2 V d2 V d2 V d2 U þ G31 a2m  Q44 am  Q12 am  Q21 am þ 2 am 2 2 dtdz dhdz dhdz dhdz dh dt

dV ¼ dVCore þ dVSheet ¼ þ

d2 V

d2 V d2 U d2 U d2 U a d2 V  1=2 g C44 a 2  C22 b 2  1=2 C44 b  1=2 C21 b  1=2 C12 b þ ε dhdz dhdz dhdz g dt2 dh dz  2Q22 bm

d2 V dh2



Q44 am

d2 V

g

dz2

þ G32 am bm

d2 W dtdh

 Q44 bm

d2 U dhdz

 Q12 bm

d2 U dhdz

 Q21 bm

d2 U dhdz

(A2)

! ; Sheet

!

þ2

Core

d2 V

am g dt2

(A3)

! ; Sheet

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dW ¼ dWCore þ dWSheet ¼ 

4 d2 W 4 d2 W 2 d3 q2 2 d3 q2 C a2 C12 a2 b C21 a2 b  C66 b2 2   2 2 15 55 dz2 15 315 315 dh dhz dhz



4 d3 q 2 d3 q 4 d3 q 4 d3 q 4 d3 q2 C44 b2 a 2 1  C21 b2 a 2 1  C22 b3 32  C11 a3 31  C44 a2 b 2 315 315 315 dh dz 315 dh dz 315 dh dz dhz



3 4 4 dq 2 dq 7 dq 1 2 d q 4d W C a 1 C b a 2 1  1=2C66 b 2 þ C66 a 2 þ C22 b 15 55 dz 315 12 30 252 d h d h dh4 dh dz

1 d4 W 1 d4 W 1 d4 W 1 d4 W C44 a2 b2 C11 a4 4 þ C12 a2 b2 C21 a2 b2 þ þ 2 2 2 2 2 126 252 252 252 dh dz dz dh dz dh2 dz ! 2 4 3 d3 q1 1 4 d4 W 4 2 d3 q2 1 2 2 d4 W 2 d W þ a ε 2  a ε þ a bε 2  a b ε 2 2 þa ε 2 2 315 315 dt dh dt dt dz 252 dt dh 252 dt2 dz þ

Core

4 2 32 32 d4 W d2 W 4d W 2d W Q22 Ib6bm 4 þ Q11 Ib6a4m 4 þ 8Q55 Ib2a2m 2  16Q66 Ib4bm 2 þ 9 9 dh dh dz dz 2

þ8Q66 Ib2bm

d2 W 2

dh

 16Q55 Ib4a2m

d2 W 2

dz



4 32 d3 q 2 d W Ib6a2m bm 2 2  8=3Q11 Ib4a3m 31 9 dt dh dz

3

þ

3 32 d q dq dq 3d q Q11 Ib6a3m 31  16Q55 Ib4am 1 þ 8Q55 Ib2am 1  8=3Q22 Ib4bm 32 9 dz dz dh dz

þ

3 32 d3 q 32 d3 q dq 3d q Q Ib6bm 32 þ 8=3Ib4a2m bm 2 2  Ib6a2m bm 2 2  16Q66 Ib4bm 2 9 22 9 dh dh dt dh dt dh

þ8Q66 Ib2bm þ8=3Ib4a3m Q66 bm þ

dq2 d2 W d2 U d2 V d2 W dW þ 2a2m 2 þ G31 am þ G32 am  Q55 a2m 2 þ cbd am dtdz dtdh dt dh dt dz

d3 q1 2

dt dz



32 d4 W d2 W 32 d3 q Ib6a4m Ib6a3m 2 1  Q66 b2m 2  2 9 9 dt dz dh dt2 dz

dq2 dq 16 d3 q d3 q 2 2  Q55 am 1 þ Q21 Ib6bm am 2 1  4=3Q12 Ib4bm am 2 1 9 dh dz dh dz dh dz

32 d3 q d3 q 16 d3 q Q Ib6b2m am 2 1  4=3Q21 Ib4b2m am 2 1 þ Q12 Ib6b2m am 2 1 9 44 9 dh dz dh dz dh dz 2

8=3Q44 Ib4bm am 4=3Q12 Ib4a2m bm

d3 q1 2

dh dz

þ

32 d3 q2 d3 q2 Q44 Ib6a2m bm  4=3Q21 Ib4a2m bm 2 9 dhdz dhdz2

d3 q2

d3 q2

dhdz

dhdz

 8=3Q44 Ib4a2m bm 2

16 d3 q2 Q21 Ib6a2m bm 2 9 dhdz

16 d q2 64 d4 W 32 d4 W Q Ib6a2m bm þ Q44 Ib6a2m b2m þ Q12 Ib6a2m b2m 2 2 2 9 12 9 9 dh dz dh2 dz2 dhdz ! 32 d4 W v2 W þ Q21 Ib6a2m b2m w þ q ðz  1Þ; 0 b 2 2 9 dh2 dz vz þ

3

þ 2

Sheet

(A4)

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259

dq1 ¼ dq1Core þ dq1Sheet ¼ 23 2 d3 W 4 d3 W 2 d3 W C dq þ C db2 a 2 þ C44 db2 a 2 þ C21 db2 a 2 30 55 1 315 12 315 315 dh dz dh dz dh dz 17 d2 q2 17 d2 q2 4 d3 W 17 d2 q2 4 dW C12 dba C44 dba C11 da3 3  C21 dba   þ þ C55 da 630 630 15 dz dhz 630 dhz 315 d hz dz ! 2 17 17 d2 q 17 d2 q 4 d3 W 32 d3 W 2 d q1 C44 db C11 da2 21 þ   da2 ε 21  da3 ε 2 þ Q55 q1  Q11 Ib6a3m 3 2 630 315 315 315 9 dt dz dh dt dz dz Core

dW dW d2 q 16 d2 q d2 q 8Q55 Ib2am þ 16Q55 Ib4am þ 8=3Q44 Ib4b2m 21  Q44 Ib6b2m 21 þ 16=3Q11 Ib4a2m 21 dz dz 9 dh dh dz 32 d2 q d2 q d2 q d3 W d3 W  Q11 Ib6a2m 21  Q44 Ib2b2m 21  2Q11 Ib2b2m 21 þ 8=3Q11 Ib4a3m 3 þ 4=3Q21 Ib4b2m am 2 9 dh dh dz dz dz dz þ4=3Q12 Ib4b2m am 2

þ8=3Q44 Ib4bm am

d3 W 2

dh dz d3 W dh2 dz



(A5)

32 d3 W 16 d3 W 16 d3 W Q44 Ib6b2m am 2  Q21 Ib6b2m am 2  Q12 Ib6b2m am 2 9 9 9 dh dz dh dz dh dz

 Q21 Ib2bm am

d2 q2 d2 q2 16 d2 q2 þ 8=3Q21 Ib4bm am  Q12 Ib6bm am 9 dhdz dhdz dhz

16 d2 q2 16 d2 q2 d2 q2 d2 q2  Q44 Ib6bm am  Q21 Ib6bm am þ 8=3Q44 Ib4bm am þ 8=3Q12 Ib4bm am 9 9 dhdz dhdz dhdz dhdz d2 q2 d2 q2 32 d2 q d2 q 32 d3 W Ib6a2m 21 þ 2Ib2a2m 21 þ Ib6a3m 2  Q44 Ib2bm am þ 9 9 dhdz dhdz dt dt dt dz ! 2 3 d q d W dW 8Q55 Ib2q1  16=3Ib4a2m 21 þ 16Q55 Ib4q1  8=3Ib4a3m 2 þ Q55 am ; dz dt dz dt Q12 Ib2bm am

Sheet

dq2 ¼ dq2Core þ dq2Sheet ¼

4 2 d3 W 2 d3 W 17 d2 q1 C dq þ C da2 b C21 da2 b C12 dab þ  2 2 15 66 2 315 12 315 630 dhz dhz dhz

3 4 17 d2 q1 4 d3 W 17 d2 q1 4 dW 3d W C22 db C44 dab C44 da2 b C21 dab þ  þ  þ C db 3 2 315 630 630 dh dhz 315 dhz 15 66 dh dhz ! 2 17 d2 q 17 17 d2 q 4 d3 W dW 2 d q2 C da2 22  C22 db  þ da2 ε 22  da3 ε 2 þ Q66 q2 þ Q66 bm 2 630 44 315 315 315 dh dt dz dh dt dz Core

2 2 d q2

d3 W

3 3 32 2 d W 2 d W Q þ16Q66 Ib4q2 þ 2Ib2am 2 þ 4=3Q12 Ib4bm a2m  Ib6 b a þ 4=3Q Ib4 b a 44 21 m m m m 2 2 2 9 dt dhdz dhdz dhdz

16 d3 W 16 d3 W d3 W 16 d2 q1  Q12 Ib6bm a2m  Q21 Ib6bm a2m þ 8=3Q44 Ib4bm a2m  Q44 Ib6bm am 2 2 2 9 9 9 dhdz dhdz dhdz dhdz 16 d2 q1 d2 q1 d2 q1 d2 q1  Q21 Ib6bm am  Q21 Ib2bm am  Q12 Ib2bm am þ 8=3Q12 I4bbm am 9 dhdz dhdz dhdz dhdz Q44 Ib2bm am 8Q66 Ib2bm

d2 q1 d2 q1 d2 q1 16 d2 q1 þ 8=3Q44 Ib4bm am þ 8=3Q21 Ib4bm am  Q12 Ib6bm am 9 dhdz dhdz dhdz dhdz

dW dW d3 W 32 d3 W d3 W þ 16Q66 Ib4bm þ 8=3Q22 Ib4b3m 3  Q22 Ib6b3m 3  8=3Ib4bm a2m 2 dh dh 9 dh dh dt dh

2 2 32 d3 W 32 16 d2 q d2 q 2d q 2d q þ Ib6bm a2m 2  Q22 Ib6bm 22  Q44 Ib6a2m 22 þ 8=3Q44 Ib4a2m 22 þ 16=3Q22 Ib4bm 22 9 9 9 dt dh dh dh dz dz ! d2 q d2 q 32 d2 q d2 q 2 Ib6a2m 22  8Q66 Ib2q2  16=3Ib4a2m 22 2Ib2bm Q22 22  Q44 Ib2a2m 22 þ 9 dh d t dt dz Sheet

(A6)

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