e ect of hygro-thermal loading on the two-dimensional...

Scientia Iranica B (2020) 27(4), 1916{1932

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

E�ect of hygro-thermal loading on the two-dimensionalresponse of a functionally graded piezomagneticcylinder under asymmetric loads

M. Gharibavi� and J. Yi

Department of Mechanical Engineering, East China University of Science and Technology, Shanghai, China.

Received 16 May 2018; received in revised form 26 January 2019; accepted 13 July 2019

KEYWORDSComplex Fourierseries;Functionally gradedmaterial;Piezomagneticmaterial;Asymmetric loading;Winkler elasticfoundation.

Abstract. In this article, a semi-analytical solution is presented in order to analyzea Functionally Graded Piezomagnetic (FGP) cylinder resting on an elastic foundationexposed to hygro-thermal loading. All mechanical, hygro-thermal, and magnetic propertieswere considered to be varying according to the power-law function through the thickness.The steady-state heat conduction and moisture di�usion equations were employed to attainthe moisture concentration and temperature distributions in the FGP cylinder. Theconstitutive equations as well as magnetic and mechanical equilibrium equations werecombined in order to derive three second-order di�erential equations in terms of magneticpotential and mechanical displacements. Variables were separated and the complex Fourierseries method was utilized to solve the governing equations. Numerical results revealed thee�ects of hygro-thermal loading, elastic foundation, and non-homogeneity constants onhygro-thermo-magneto-elastic response of the FGP cylinder. It was observed that hygro-thermal loading had remarkable e�ects on the behavior of the cylinder, leading to increasein the absolute values of the radial magnetic induction and radial, circumferential, andshear stresses.

© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

Functionally Graded (FG) materials can be distin-guished from each other by the compositional gradi-ents of their components. They have high thermalresistance and their thermal and mechanical propertiescontinuously vary with respect to the location. Theycan be designed for speci�c functions and applica-tions in many industries, including aerospace, nuclearpower, military, medicine, electronics, and biomate-

*. Corresponding author.E-mail address: [email protected] (M. Gharibavi)

doi: 10.24200/sci.2019.51043.1975

rials. In addition, FG materials with piezoelectricand piezomagnetic properties can be used in ultrasonictransducers as well as many electronic and engineeringapplications [1{7].

Thick-wall cylindrical vessels are crucial equip-ment used in oil, chemical, petroleum, petrochemical,and nuclear �elds. They are mostly employed in,e.g., power generation by fossil and nuclear fuels,storing gasoline in service stations in the petrochemicalindustry, and the chemical industry [8]. The thick-wall cylinder subjected to multi-physical loadings andsurrounded by an elastic foundation is a prime exampleof soil-structure interaction problems. The static anddynamic analyses of such structures leads to a deeperunderstanding of the response of the elastic foundation

M. Gharibavi and J. Yi/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 1916{1932 1917

under di�erent external loads. The dynamic e�ectsof soil-structure interaction appear in the earthquakeresponse of large-scale structures such as nuclear powerplants structures. Furthermore, the study of fracturemechanics, load transfer, and stress concentrations ofthe mentioned structures is another �eld of investiga-tion in solid mechanics [9{12].

Many researchers have inquired the mechan-ical response of smart structures such as FGpiezoelectricnpiezomagnetic cylinders and spheres un-der di�erent conditions of loading [13{18]. Hosseini etal. [16] o�ered a strain gradient elasticity formulationfor capturing the size e�ect in micro-scale structuresto analyze the thermo-elastic response of an FG micro-rotating cylinder. Jabbari et al. [19] presented a generaltheoretical analysis of the three-dimensional mechani-cal and thermal stresses in a hollow FG cylinder. Theyassumed that temperature distribution was a functionof radial and circumferential directions and solvedthe di�erential equations by employing generalizedBessel function and Fourier series method. Thermo-piezo-magneto-elastic response of rotating FunctionallyGraded Piezomagnetic (FGP) disks exposed to thermaland mechanical loadings was studied by GhorbanpourArani et al. [20]. In this study, they explained the ef-fects of non-homogeneity constant on the distributionsof stresses, displacement, and magnetic potential. Ananalytical solution for displacement and the strain andstress �eld in a rotating thick-wall cylinder made of FGmaterial subjected to the uniform external magneticand thermal �elds was presented by Hosseini andDini [4]. The magneto-thermo-elastic response of anFG annular sandwich disk was also investigated byZenkour [21]. Using exp-exp strain energy for modelingthe hyperelastic materials, Almasi et al. [22] carried outan analytical and numerical thermo-mechanical studyof an FG hyperelastic thick-wall pressure vessel. Anelastic-plastic analysis of FG spherical pressure vesselsunder internal pressure based on strain gradient plas-ticity was performed in [23]. A feedback gain controlalgorithm was employed by Barati and Jabbari [24] inthe two-dimensional piezothermoelastic analysis of anFG hollow sphere with integrated piezoelectric layersas a sensor and actuator subject to non-axisymmetricloads. They analytically solved the governing equationsutilizing the Legendre polynomials and the system ofEuler di�erential equations.

The study of the e�ects of hygro-thermal loadingon the behavior of smart materials and structures hasattracted the attention of many researchers in recentyears [25{27]. A three-dimensional discrete-layer modelwas developed by Smittakorn and Heyliger [28] foranalyzing rectangular plates, in which the transient andhygro-thermo-piezoelectric responses of plates wereevaluated under the coupled e�ects of mechanical,electrical, thermal, and moisture �elds. In another

work, Keles and Tutuncu [29] carried out free andforced vibration analysis of FG hollow cylinders andspheres using analytical solutions in which the materialproperties were assumed to vary based on the powerlaw function. The distributions of the temperature,moisture, displacement, and stress in an FGP circu-lar rotating disk under a coupled hygro-thermal �eldwere studied by Dai et al. [26]. Akbarzadeh andPasini [30] used a multi-physics model to study thee�ects of moisture, temperature, magnetic, electric,and mechanical loadings on the response of multi-layerand FG cylinders. Vinyas and Kattimani [31] presenteda 3D Finite Element (FE) method in order to analyzea magneto-electro-elastic plate subjected to hygro-thermal loads. In their work, an FE formulation wasinferred by employing the principle of total potentialenergy and linear coupled constitutive equations.

To the best of the authors' knowledge, the studyof the e�ects of hygro-thermal loading and elasticfoundation on the stress and strain �elds in the FGPcylinders under asymmetric loading is missing in theliterature. Hence, using the complex Fourier seriesmethod, with the assumption of power-law distributionfor FG materials, a cylinder made of piezomagneticmaterials embedded in a Winkler elastic foundationunder asymmetric hygro-thermo-magneto-mechanicalloadings is analyzed in this study.

2. Preliminaries

In this section, stress-strain relations and equilibriumequations are expressed and the formulation of theproblem is discussed. As shown in Figure 1, a cylinderis considered with the inner and outer radii of aand b, which is radially magnetized and exposed toasymmetric hygro-thermal and internal pressure loads.The piezomagnetic cylinder is embedded in a Winkler-type elastic foundation with sti�ness kw. According tothe power-law distribution, the material properties for

Figure 1. Schematics of the Functionally GradedPiezomagnetic (FGP) cylinder subject to physicalloadings.

1918 M. Gharibavi and J. Yi/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 1916{1932

the piezomagnetic cylinder are de�ned as [32]:

Cij = C0ijr

�1 ; dij = d0ijr

�1 ; gij = �0ijr

�1 ;

ki = k0i r�3 ; !i = !0

i r�3 ; �i = �0

i r�1+�2 ;

�i=�0i r�1+�2 ; qi=q0

i r�1+�2 ; i= 0

i r�1+�2 ; (1)

in which ki and !i are the thermal conductivity andmoisture di�usivity coe�cients, respectively. �1, �2,and �3 are non-homogeneity constants. The strainequations for the cylinder are written as [33]:

"rr =@u@r; "�� =

ur

+1r@v@�;

"r� =12

�1r@u@�

+@v@r� vr

�; (2)

where u and v are displacements in the radial andcircumferential directions, respectively. Magnetic �eldis considered Bi = �r and the constitutive equationsare expressed as [20,32,34]:

�rr =C11"rr + C12"�� + d11 ;r � #1T (r; �)

� %1 �M(r; �);

�� =C12"rr + C22"�� + d21 ;r � #2T (r; �)

� %2 �M(r; �);

�r� = C31"r� + C12"�� + d311r ;�;

Br=d11"rr+d12"��g11 ;r+q1T (r; �)+ 1 �M(r; �);

B� = d31"r� � g221r ;� + q2T (r; �) + 2 �M(r; �); (3)

in which �ij , Bi, , T , and �M are the stress, mag-netic induction, magnetic potential, temperature, andmoisture concentration, respectively. Cij , dij , #i, %i,gij , qi, and i are the elastic, piezomagnetic, thermalstress, hygroscopic stress, magnetic permeability, pyro-magnetic, and hygromagnetic coe�cients, respectively.The thermal and hygroscopic stresses are associatedwith the elastic, thermal expansion �i, and moistureexpansion �i coe�cients as follows [35,36]:

#1 = C11�r + C12��; #2 = C12�r + C22��;

%1 = C11�r + C12��; %2 = C12�r + C22��: (4)

The mechanical equilibrium equation and Maxwell'selectromagnetic relation in the cylindrical coordinatesystem may be expressed as [34,37]:

@�rr@r

+1r@�r�@�

+�rr � ��

r= 0;

@�r�@r

+1r@��@�

+2r�r� = 0;

@Br@r

+1r@B�@�

+Brr

= 0: (5)

3. Governing equations

In this section, the governing equations for the cylinderare given. Submitting Eqs. (2) and (3) into Eq. (5)leads to the coupled di�erential equations [33,34]:

C011r

2 @2u@r2 + (�1 + 1)C0

11r@u@r

+��1C0

12 � C022�u

+ C031@2u@�2 +

��1C0

12 � C031 � C0

22� @v@�

+�C0

12 + C031�r@2v@r@�

+ d011r

2 @2 @r2

+ d031@2 @�2 +

�d0

11 (�1 + 1)� d021�r@ @r

+�#0

2 � #01 (�1 + �2 + 1)

�r�2+1T

�#01r�2+2 @T

@r+�%0

2�%01 (�1+�2+1)

�r�2+1 �M

� %01r�2+2 @ �M

@r= 0; (6)�

C031 (�1 + 1) + C0

22� @u@�

+ (�1 + 1)C031r

@v@r

� C031 (�1 + 1) v +

�C0

31 + C012�r@2u@r@�

+ C031r

2 @2v@r2 + C0

22@2v@�2 +

�d0

21 + d031�r@2 @r@�

+ (�1 + 1) d031@ @�� #0

2r�2+1 @T

@�

� %02r�2+1 @ �C

@�= 0; (7)

�d0

11 (�1 + 1) + d021�r@u@r

+ d011r

2 @2u@r2 + �1d0

21u

+��1d0

21 � d031� @v@�

+�d0

21 + d031�r@2v@r@�

+ d031@2u@�2 � g0

11r2 @2 @r2 � g0

22@2 @�2

� (�1 + 1) g011r

@ @r

+ (�1 + �2 + 1) q01r�2+1T


+ q01r�2+2 @T

@r+ q0

2r�2+1 @T

@�

+ (�1 + �2 + 1) 01r�2+1 �M + 0

1r�2+2 @ �M

@r

+ 02r�2+1 @ �M

@�= 0: (8)

4. Hygrothermoelasticity

The two-dimensional heat conduction and moisturedi�usion relations in the cylindrical coordinates can bewritten as [36,38,39]:

1r@@r

�rKr

@T@r

�+

1r2

@@�

�K�

@T@�

�= 0; (9)

1r@@r

�r!r

@ �M@r

�+

1r2

@@�

�!�@ �M@�

�= 0: (10)

Simplifying Eqs. (9) and (10) yields:

r2 @2T@r2 + r (�3 + 1)

@T@r

+@2T@�2 = 0; (11)

r2 @2 �M@r2 + r (�3 + 1)

@ �M@r

+@2 �M@�2 = 0: (12)

The complex Fourier series is de�ned as the solution toEqs. (11) and (12):

T (r; �) =+1X

n=�1Tn(r)ein�;

Tn(r) =1

2�

+�Z��

T (r; �)e�in�d�; (13)

�M(r; �) =+1X

n=�1�Mn(r)ein�;

�Mn(r) =1

2�

+�Z��

�M(r; �)e�in�d�; (14)

where Tn(r) and �Mn(r) are the coe�cient of complexFourier series. By substituting Eqs. (13) and (14) intoEqs. (11) and (12), the following equations are given(please consider revising the distorted text):

r2 @2Tn@r2 + r (�3 + 1)

@Tn@r� n2Tn = 0; (15)

r2 @2 �Mn

@r2 + r (�3 + 1)@ �Mn

@r� n2 �Mn = 0: (16)

Using Eqs. (15) and (16), the temperature and moisture

distributions in the piezomagnetic cylinder are writtenas:

T (r; �) =+1X

n=�1(An1rmn1 +An2rmn2) ein�; (17)

�M(r; �) =+1X

n=�1

�Gn1rSn1 +Gn2rSn2

�ein�; (18)

in which An1, An2, Gn1, and Gn2 are hygro-thermalconstants. Also:

mn1 =12

��3 +

q�2

3 + 4n2�;

mn2 =12

��3 �

q�2

3 + 4n2�;

Sn1 =12

��3 +

q�2

3 + 4n2�;

Sn2 =12

��3 �

q�2

3 + 4n2�: (19)

5. Solution procedure

In the complex Fourier series method, the displace-ments u(r; �) and v(r; �) and the magnetic potential (r; �) are expressed as [40]:

u(r; �) =+1X

n=�1un(r)ein�;

v(r; �) =+1X

n=�1vn(r)ein�;

(r; �) =+1X

n=�1 n(r)ein�; (20)

where:

un(r) =1

2�

+�Z��

u(r; �)e�in�d�;

vn(r) =1

2�

+�Z��

v(r; �)e�in�d�;

n(r) =1

2�

+�Z��

(r; �)e�in�d�: (21)

By submitting Eq. (20) into Eqs. (6){(8), three di�er-ential equations can be derived:


C011r

2u00n + (�1 + 1)C011ru

0n

+��1C0

12 � C022 � n2C0

31�un

+��1C0

12 � C031 � C0

22�

(in)vn

+�C0

12 + C031�

(in)rv0n + d011r

2 00n

+�d0

11(�1 + 1)� d021�r 0n � n2d0

31 n

=�#0

1 (�1+�2+mn1+1)�#02�An1rmn1+�2+1

+�#0

1 (�1+�2+mn2+1)�#02�An2rmn2+�2+1

+�%0

1 (�1+�2+Sn1+1)� %02�Gn1rSn1+�2+1

+�%0

1 (�1+�2+Sn2+1)�%02�Gn2rSn2+�2+1; (22)�

C031 + C0

12�

(in)ru0n +�C0

31(�1 + 1) + C022�

(in)un

+ C031r

2v00n + (�1 + 1)C031rv

0n

��n2C022+(�1+1)C0

31�vn+

�d0

31+d021�

(in)r 0n

+ (�1 + 1)(in)d031 n

=#02(in)

�An1rmn1+�2+1 +An2rmn2+�2+1�

+ %02(in)

�Gn1rSn1+�2+1 +Gn2rSn2+�2+1� ; (23)

d011r

2u00n +�(�1 + 1)d0

11 + d021�ru0n

+��1d0

21 � n2d031�un +

�d0

21 + d031�

(in)rv0n

+��1d0

21 � d031�

(in)vn � g011r

2 00n

� (�1 + 1) g011r

0n + g0

22n2 n

=��q01(�1+�2+mn1+1)+(in)q0

2�An1rmn1+�2+1

��q01(�1+�2+mn2+1)+(in)q0

2�An1rmn1+�2+1

��q01(�1+�2+mn2+1)+(in)q0

2�An2rmn2+�2+1

�� 01(�1+�2+Sn1+1)+(in) 0

2�Gn1rSn1+�2+1

�� 01(�1+�2+Sn2+1)+(in) 0

2�Gn2rSn2+�2+1:

(24)

The symbols \0" and \00" indicate the �rst and secondderivatives with respect to the variable r, respectively.Eqs. (22){(24) are ordinary di�erential equations towhich the general solution can be formulated as:

ugn(r) = Br� ; vgn(r) = Cr� ; gn(r) = Dr� ; (25)

in which B, C, and D are unknown constants deter-mined by boundary conditions. Submitting Eq. (25)into Eqs. (22){(24) yields:�

C011� (�1 + �) +

��1C0

12 � C022 � n2C0

31��B

+��1C0

12�C031�C0

22�(in)+

�C0

12+C031�(in)�

�C

+�d0

11� (�1 + �)� d021� � n2d0

31�D = 0;��

C031 + C0

12�

(in)� +�C0

31 (�1 + 1) + C022�

(in)�B

+�C0

31� (�1 + �)� �n2C022 + (�1 + 1)C0

31��C

+��d0

31 + d021�

(in)� + (�1 + 1)(in)d031�D = 0;�

d011�(�1 + �) + d0

21� +��1d0

21 � n2d031��B

+��d0

21 + d031�

(in)� +��1d0

21 � d031�

(in)�C

+��g0

11�(�1 + �) + g022n

2�D = 0: (26)

The determinant of the system in Eq. (27) should beequal to zero in order to attain a nontrivial solutionto Eq. (26). Therefore, six roots �nj (j = 1; 2; :::; 6) ofEq. (27) are achieved:

un(r) =6Xj=1

Bnjr�nj ;

vn(r) =6Xj=1

LnjBnjr�nj ;

n(r) =6Xj=1

PnjBnjr�nj ; (27)

where Lnj is the relation between constants Bnj andCnj , and Pnj is the relation between constants Bnj andDnj calculated by Eq. (26) as:

Lnj =b1 � b6 � b4 � b3b3 � b5 � b2 � b6 ; Pnj =

b8 � b1 � b7 � b2b9 � b2 � b3 � b8 ;

j = 1; 2; � � � ; 6; (28)

b1 = C011�nj (�1 + �nj) +

��1C0

12 � C022 � n2C0

31�;

b2 =��1C0

12�C031�C0

22�

(in)+�C0

12+C031�

(in)�nj ;

b3 = d011�nj (�1 + �nj)� d0

21�nj � n2d031;

b4 =�C0

31+C012�

(in)�nj+�C0

31(�1+1)+C022�

(in);

b5 = C031�nj (�1 + �nj)� �n2C0

22 + (�1 + 1)C031�;

b6 =�d0

31 + d021�

(in)�nj + (�1 + 1)(in)d031;

b7 =d011�nj(�1+�nj)+d0

21�nj+��1d0

21�n2d031�;


b8 =�d0

21 + d031�

(in)�nj +��1d0

21 � d031�

(in);

b9 = �g011�nj (�1 + �nj) + g0

22n2: (29)

The particular solutions upn(r), vpn(r), and pn(r) are asfollows:

upn(r) =Q1nrmn1+�2+1 +Q2

nrmn2+�2+1

+Q3nrSn1+�2+1 +Q4

nrSn2+�2+1;

vpn(r) =Q5nrmn1+�2+1 +Q6

nrmn2+�2+1

+Q7nrSn1+�2+1 +Q8

nrSn2+�2+1;

pn(r) =Q9nrmn1+�2+1 +Q10

n rmn2+�2+1

+Q11n r

Sn1+�2+1 +Q12n r

Sn2+�2+1: (30)

Qjn (j = 1; 2; � � � ; 12) are constants. By substitutingEq. (30) into Eqs. (22){(24) and equating the coe�-cients of the identical powers, rSn1+�2+1, rSn2+�2+1,rSn1+�2+1, and rSn2+�2+1, we have:

�X1Q1n + �X2Q5

n + �X3Q9n = �Y1;

�X4Q1n + �X5Q5

n + �X6Q9n = �Y2;

�X7Q1n + �X8F 5

n + �X9Q9n = �Y3; (31)

�X10Q2n + �X11Q6

n + �X12Q10n = �Y4;

�X13Q2n + �X14Q6

n + �X15Q10n = �Y5;

�X16Q2n + �X17Q6

n + �X18Q10n = �Y6; (32)

�X19Q3n + �X20Q7

n + �X21Q11n = �Y7;

�X22Q3n + �X23Q7

n + �X24Q11n = �Y8;

�X25Q3n + �X26Q7

n + �X27Q11n = �Y9; (33)

�X28Q4n + �X29Q8

n + �X30Q12n = �Y10;

�X31Q4n + �X32Q8

n + �X33Q12n = �Y11;

�X34Q4n + �X35Q8

n + �X36Q12n = �Y12: (34)

The constants �Xi (i = 1; 2; :::; 36) and �Yj (j =1; 2; :::; 12) are presented in the appendix. The con-stants Qjn (j = 1; 2; :::; 12) are evaluated by the solutionto Eqs. (31){(34) as a system of algebraic equations.Utilizing Eqs. (27) and (30), we have:

un(r) =

0@ 6Xj=1

Bnjr�nj

1A+Q1nrmn1+�2+1

+Q2nrmn2+�2+1+Q3

nrSn1+�2+1+Q4

nrSn2+�2+1;

vn(r) =

0@ 6Xj=1

LnjBnjr�nj

1A+Q5nrmn1+�2+1

+Q6nrmn2+�2+1+Q7

nrSn1+�2+1+Q8

nrSn2+�2+1;

n(r) =

0@ 6Xj=1

PnjBnjr�nj

1A+Q9nrmn1+�2+1

+Q10n r

mn2+�2+1+Q11n r

Sn1+�2+1+Q12n r

Sn2+�2+1:(35)

For n = 0, the coe�cients Lnj and Pnj are unde�ned.Substituting n = 0 into Eqs. (22){(24) and followingthe solution procedure, the complete solutions foru0(r), v0(r), and �0(r) are derived as:

u0(r) =

0@ 4Xj=1

B0jr�0j

1A+Q10rm01+�2+1

+Q20rm02+�2+1 +Q3

0rS01+�2+1 +Q4

0rS02+�2+1;

v0(r) =6Xj=5

B0jr�0j ;

0(r) =

0@ 4Xj=1

P0jB0jr�0j

1A+Q90rm01+�2+1

+Q100 r

m02+�2+1+Q110 r

S01+�2+1+Q120 r

S02+�2+1;(36)

where:

P0j = ��C0

11�0j (�0j + �1) + �1C012 � C0

22�

[d011�0j (�0j + �1)� d0

21�0j ]: (37)

�05 and �06 can be written as:

�05 =��1 +

p�2

1 + 4(�1 + 1)2

;

�06 =��1 �p�2

1 + 4(�1 + 1)2

: (38)

Therefore, the radial and circumferential displacementsand magnetic potential are approximated by the sumof Eqs. (35) and (36):


u(r; �) =1X

n=�1;n 6=0

8<:0@ 6Xj=1

Bnjr�nj

1A+Q1

nrmn1+�2+1 +Q2

nrmn2+�2+1 +Q3

nrSn1+�2+1

+Q4nrSn2+�2+1 ein� +

0@ 4Xj=1

B0jr�0j

1A+Q1

0rm01+�2+1 +Q2

0rm02+�2+1 +Q3

0rS01+�2+1

+Q40rS02+�2+1; (39)

v(r; �) =1X

n=�1;n 6=0

8<:0@ 6Xj=1

LnjBnjr�nj

1A+Q5

nrmn1+�2+1 +Q6

nrmn2+�2+1 +Q7

nrSn1+�2+1

+Q8nrSn2+�2+1

9=; ein� +

0@ 6Xj=5

B0jr�0j

1A ; (40)

(r; �) =1X

n=�1;n 6=0

8<:0@ 6Xj=1

PnjBnjr�nj

1A+Q9

nrmn1+�2+1+Q10

n rmn2+�2+1+Q11

n rSn1+�2+1

+Q12n r

Sn2+�2+1

9=; ein� +

0@ 4Xj=1

P0jB0jr�0j

1A+Q9

0rm01+�2+1 +Q10

0 rm02+�2+1 +Q11

0 rS01+�2+1

+Q120 r

S02+�2+1: (41)

The displacement potential �eld of an FG piezoelectricsolid has been addressed in [41]. Submitting Eqs. (39){(41), (17), and (18) into Eq. (3), the components ofstress and magnetic induction are attained as:

�rr(r; �) =1X

n=�1;n 6=0

8<:0@ 6Xj=1

Z1njBnjr

�nj+�1�1

1A+ �W 1

nrmn1+�1+�2 + �W 2

nrmn2+�1+�2

+ �W 3nr

Sn1+�1+�2 + �W 4nr

Sn2+�1+�2

9=; ein�

+

0@ 4Xj=1

Z10jB0jr�0j+�1�1

1A+ �W 10 r

m01+�1+�2

+ �W 20 r

m02+�1+�2 + �W 30 r

S01+�1+�2

+ �W 40 r

S02+�1+�2 ; (42)

��(r; �) =1X

n=�1;n6=0

8<:0@ 6Xj=1

Z2njBnjr

�nj+�1�1

1A+ �W 5

nrmn1+�1+�2 + �W 6

nrmn2+�1+�2

+ �W 7nr

Sn1+�1+�2 + �W 8nr

Sn2+�1+�2

9=; ein�

+

0@ 4Xj=1

Z20jB0jr�0j+�1�1

1A+ �W 50 r

m01+�1+�2

+ �W 60 r

m02+�1+�2 + �W 70 r

S01+�1+�2

+ �W 80 r

S02+�1+�2 ; (43)

�r�(r; �) =1X

n=�1;n 6=0

8<:0@ 6Xj=1

Z3njBnjr

�nj+�1�1

1A+ �W 9

nrmn1+�1+�2 + �W 10

n rmn2+�1+�2

+ �W 11n rSn1+�1+�2 + �W 12

n rSn2+�1+�2

9=; ein�

+

0@ 6Xj=5

Z30jB0jr�0j+�1�1

1A ; (44)

Br(r; �) =1X

n=�1;n6=0

8<:0@ 6Xj=1

Z4njBnjr

�nj+�1�1

1A+ �W 13

n rmn1+�1+�2 + �W 14n rmn2+�1+�2

+ �W 15n rSn1+�1+�2 + �W 16

n rSn2+�1+�2

9=; ein�

+

0@ 4Xj=1

Z40jB0jr�0j+�1�1

1A+ �W 90 r

m01+�1+�2

+ �W 100 rm02+�1+�2 + �W 11

0 rS01+�1+�2

+ �W 120 rS02+�1+�2 ; (45)


B�(r; �) =1X

n=�1;n6=0

8<:0@ 6Xj=1

Z5njBnjr

�nj+�1�1

1A+ �W 17

n rmn1+�1+�2 + �W 18n rmn2+�1+�2

+ �W 19n rSn1+�1+�2 + �W 20

n rSn2+�1+�2

9=; ein�

+

0@ 6Xj=5

Z50jB0jr�0j+�1�1

1A ;(46)

where constants Zi0j , Zinj (i = 1; 2; :::; 5), �W in (i =

1; 2; :::; 20), and �W i0 (i = 1; 2 � � � 12) are de�ned in the

Appendix. It is noteworthy that Eqs. (42){(46) consistof six unknown constants Bnj , (j = 1; 2; :::; 6) and sixboundary conditions are needed to approximate Bnj .

6. Numerical results and discussion

6.1. Veri�cation of the presented solutionTo verify the solution, the results are depicted inFigure 2 for the axisymmetric FGP cylinder, regard-less of moisture concentration e�ects (i.e., �M = 0).Given the assumption of axisymmetric loadings (i.e.,� = 0), three coupled di�erential equations, namely,Eqs. (22){(24), derived in this study are reduced totwo coupled di�erential equations, as stated in [20].For this purpose, the geometrical parameters used area = 0:2 m and b = 1 m and the non-homogeneityconstant is assumed to be � = 1. Also, the boundaryconditions are set to �rr (r = a) = �rr (r = b) =0. The thermal, magnetic, and mechanical propertiesare chosen following Ghorbanpour Arani et al. [20].

Figure 2. Distribution of the radial stress. Comparisonbetween the present study and Ref. [20].

Figure 2 shows the distribution of the radial stress forthe purpose of veri�cation. It can be observed thatthe results of the present study have a good agreementwith those given in [20].

6.2. Current resultsIn this section, numerical results of the analysis ofthe FGP cylinder resting on an elastic foundationare discussed. The distributions of stress, displace-ment, magnetic potential, temperature, and moistureare illustrated. The inner and outer radii of thepiezomagnetic cylinder are assumed a = 1 m andb = 1:2 m, respectively. The material properties ofBaTiO3/CoFe2O4 are listed in Table 1 [36,37,42]. Theboundary conditions on the inner and outer surfaces ofthe piezomagnetic cylinder are expressed as follows:

T (a; �) = 60 cos 2�; T (b; �) = 100 cos 2� (K);

�M(a; �) = cos 2�; �M(b; �) = 3 cos 2� (kg/m3);

�rr(a; �)=10 cos 2�; �rr(b; �)=�kwu(b; �) (MPa);

�r�(a; �) = 0; �r�(b; �) = 0 (MPa);

(a; �)=104 cos 2�; (b; �) = 0 (W/A):

Figures 3 and 4 reveal the distributions of moistureconcentration and temperature, respectively. Asshown, the boundary conditions are satis�ed at theinner and outer surfaces of the FGP cylinder. Also,the maximum values of moisture concentration andtemperature occur at the angles of � = 0;��.

Table 1. Material properties of BaTiO3/CoFe2O4

[36,37,42].

C011 (GPa) 269.5

C012 (GPa) 170.5

C022 (GPa) 286

C031 (GPa) 170.5

d011 (N/Am) 699.7d0

21 (N/Am) 580.3d0

31 (N/Am) 550g0

11 (10�4 Ns2/C2) 1.57g0

22 (10�4 Ns2/C2) {5.9�r (10�6 1/K) 10�� (10�6 1/K) 10�r (10�4 m3/kg) 0.8�� (10�4 m3/kg) 1.1q01 (10�3 N/AmK) 6q02 (10�3 N/AmK) 6 0

1 (10�5 Nm2/Akg) 0 0

2 (10�5 Nm2/Akg) 0


Figure 3. Distribution of moisture concentration in theFunctionally Graded Piezomagnetic (FGP) cylinder.

Figure 4. Distribution of temperature in the FunctionallyGraded Piezomagnetic (FGP) cylinder.

Figure 5. Distribution of radial displacement in theFunctionally Graded Piezomagnetic (FGP) cylinder.

Figures 5{9 illustrate the distributions of theradial and circumferential displacement and radial,circumferential, and shear stresses, respectively. It isobserved in Figure 5 that the minimum values of theradial displacement occur at the angles of � = 0;��,whereas the critical values are achieved at the anglesof � = ��2 .

Figure 6. Distribution of circumferential displacement inthe Functionally Graded Piezomagnetic (FGP) cylinder.

Figure 7. Distribution of radial stress in the FunctionallyGraded Piezomagnetic (FGP) cylinder.

Figure 8. Distribution of circumferential stress in theFunctionally Graded Piezomagnetic (FGP) cylinder.

It is evident in Figures 7 and 9 that the boundaryconditions of radial and shear stresses are satis�edat the inner and outer surfaces of the cylinder. Asdepicted, the maximum values of radial stress areachieved at the angles of � = 0;��, and the valuesof shear stress are equal to zero at the angles of � =0;��. It can be seen in Figure 8 that the behavior of


Figure 9. Distribution of shear stress in the FunctionallyGraded Piezomagnetic (FGP) cylinder.

Figure 10. Distribution of magnetic potential in theFunctionally Graded Piezomagnetic (FGP) cylinder.

Figure 11. Distribution of radial magnetic induction inthe Functionally Graded Piezomagnetic (FGP) cylinder.

the circumferential stress is reversed at the radius ofr=a > 1:1. The distributions of the magnetic potentialand radial and circumferential magnetic inductions aredepicted in Figures 10{12. As shown, the maximumvalues occur at the angles of � = 0;��, whereas theminimum values arise at � = ��2 .

Figure 12. Distribution of circumferential magneticinduction in the Functionally Graded Piezomagnetic(FGP) cylinder.

Figure 13. Distribution of radial stress with di�erenthygrothermal loadings (� = �=3, � = 0:5, kw = 109 N/m3).

The e�ects of hygro-thermal loading on the re-sponse of the piezomagnetic cylinder at � = �=3are presented in Figures 13{16, wherein the non-homogeneity constants and foundation sti�ness areconsidered as �1 = �2 = �3 = � = 0:5 and kw =109 (N/m3) respectively. Figures 13{16 are depicted onthe basis of the temperature di�erence �T = To � Tiand moisture di�erence � �M = �Mo � �Mi. Figure 13indicates that increase in the hygro-thermal loadingenhances the absolute values of radial stress. As shownin Figure 14, increase in the hygro-thermal loadingraises the absolute values of circumferential stress. Itcan be seen in Figure 15 that the absolute valuesof shear stress increase by raising the hygro-thermalloading. As shown in Figure 16, raising the hygro-thermal loading leads to an increase in the absolutevalues of radial magnetic induction. Therefore, it canbe concluded that hygro-thermal loading has negativee�ects on the response of a thick-wall structure.


Figure 14. Distribution of circumferential stress withdi�erent hygrothermal loadings (� = �=3, � = 0:5,kw = 109 N/m3).

Figure 15. Distribution of shear stress with di�erenthygrothermal loadings (� = �=3, � = 0:5, kw = 109 N/m3).

7. Conclusion

In the current study, complex Fourier series methodwas employed to assess the hygro-thermo-magneto-elastic response of the piezomagnetic cylinders madeof Functionally Graded (FG) materials resting onan elastic foundation. The material properties wereconsidered based on the power-law function. Thecoupled di�erential equations in terms of mechanicaldisplacements and magnetic potential were solved usingthe separation of variables and complex Fourier seriesmethod. The main advantage of this method isthat any type of mechanical and magnetic boundaryconditions can be de�ned in it without any limita-tions. Numerical results obtained in this study wereevaluated to investigate the e�ects of hygro-thermalloading. It should be pointed out that in the uncoupled

Figure 16. Distribution of magnetic induction withdi�erent hygrothermal loadings (� = �=3, � = 0:5,kw = 109 N/m3).

hygrothemal problems, the temperature and moistureconcentration have not only the same manner, but alsosimilar e�ects on the behavior of the cylinder. Asobserved in the results, all the components of stress,displacement, magnetic potential, and magnetic induc-tion followed a harmonic pattern in the cross sectionof the piezomagnetic cylinder. Furthermore, hygro-thermal loading had considerable e�ects on the stressesand magnetic induction distributions. Moreover, theabsolute values of the radial, circumferential, and shearstresses and radial magnetic induction increased byraising the hygrothermal loading.

Nomenclature

�ij Stress components (Pa)"ij Strain componentsBi Magnetic induction (N/Am)u Radial displacement (m)v Circumferential displacement (m)T Temperature (K)

M Moisture concentration (kg/m3) Magnetic potential (Nm/C)Cij Elastic coe�cient (Pa)dij Piezomagnetic coe�cient (N/Am)

#i Thermal stress (N/m2K)%i Hygroscopic stress (Nm/kg)

gij Magnetic permeability (Ns2/C2)�i Thermal expansion coe�cient (1/K)

�i Moisture expansion coe�cient (m3/kg)qi Pyromagnetic coe�cient (N/AmK)


i Hygromagnetic coe�cient (Nm2/Akg)!i Moisture di�usivity (kg/ms�M)ki Thermal conductivity (W/mK)�i Non-homogeneity constantkw Elastic foundation sti�ness

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Appendix

�X1 =C011 (mn1 + �1 + �2 + 1) (mn1 + �2 + 1)

+��1C0

12 � n2C031 � C0

22�;

�X2 =��1C0

12 � C031 � C0

22�

(in)

+�C0

12 + C031�

(in) (mn1 + �2 + 1) ;

�X3 =d011 (mn1 + �1 + �2 + 1) (mn1 + �2 + 1)

� d021 (mn1 + �2 + 1)� n2d0

31;

�X4 =�C0

31 + C012�

(in) (mn1 + �2 + 1)

+�C0

31 (�1 + 1) + C022�

(in);

�X5 =C031(mn1 + �1 + �2 + 1) (mn1 + �2 + 1)

� �n2C022 + (�1 + 1)C0

31�;

�X6 =�d0

31 + d021�

(in) (mn1 + �2 + 1)

+ (�1 + 1) d031(in);

�X7 =d011 (mn1 + �2 + 1) (mn1 + �1 + �2 + 1)

+ d021 (mn1 + �1 + �2 + 1)� n2d0

31;

�X8 =�d0

21 + d031�

(in) (mn1 + �2 + 1)

+��1d0

21 � d031�

(in);

�X9 =�g011(mn1+�2+1) (mn1+�1+�2+1)+g0

22n2;

�X10 =C011(mn2 + �1 + �2 + 1) (mn2 + �2 + 1)

+��1C0

12 � C022 � n2C0

31�;

�X11 =��1C0

12 � C031 � C0

22�

(in)

+�C0

12 + C031�

(in) (mn2 + �2 + 1) ;


�X12 =d011 (mn2 + �2 + 1) (mn2 + �1 + �2 + 1)

� d021 (mn2 + �2 + 1)� n2d0

31;

�X13 =�C0

31 + C012�

(in) (mn2 + �2 + 1)

+�C0

31 (�1 + 1) + C022�

(in);

�X14 =C031(mn2 + �1 + �2 + 1) (mn2 + �2 + 1)

� n2C022 � (�1 + 1)C0

31;

�X15 =�d0

31 + d021�

(in) (mn2 + �2 + 1)

+ (�1 + 1) d031(in);

�X16 =d011 (mn2 + �2 + 1) (mn2 + �1 + �2 + 1)

+ d021 (mn2 + �1 + �2 + 1)� n2d0

31;

�X17 =�d0

21 + e031�

(in) (mn2 + �2 + 1)

+��1d0

21 � d031�

(in);

�X18 =� g011 (mn1 + �2 + 1) (mn2 + �1 + �2 + 1)

+ n2g022;

�X19 =C011(Sn1 + �1 + �2 + 1) (Sn1 + �2 + 1)

+��1C0

12 � n2C031 � C0

22�;

�X20 =��1C0

12 � C031 � C0

22�

(in)

+�C0

12 + C031�

(in) (Sn1 + �2 + 1) ;

�X21 =d011 (Sn1 + �1 + �2 + 1) (Sn1 + �2 + 1)

� d021 (Sn1 + �2 + 1)� n2d0

31;

�X22 =�C0

31 + C012�

(in) (Sn1 + �2 + 1)

+�C0

31 (�1 + 1) + C022�

(in);

�X23 =C031(Sn1 + �1 + �2 + 1) (Sn1 + �2 + 1)

� n2C022 � (�1 + 1)C0

31;

�X24 =�d0

31 + d021�

(in) (Sn1 + �2 + 1)

+ (�1 + 1) d031(in);

�X25 =d011 (Sn1 + �2 + 1) (Sn1 + �1 + �2 + 1)

+ d021 (Sn1 + �1 + �2 + 1)� n2d0

31;

�X26 =�d0

21+d031�

( in) (Sn1+�2+1)

+��1d0

21�d031�

(in);

�X27 =� g011 (Sn1+�2+1) (Sn1+�1+�2+1)+n2g0

22;

�X28 =C011(Sn2 + �1 + �2 + 1) (Sn2 + �2 + 1)

+��1C0

12 � n2C031 � C0

22�;

�X29 =��1C0

12 � C031 � C0

22�

(in)

+�C0

12 + C031�

(in) (Sn2 + �2 + 1) ;

�X30 =d011 (Sn2 + �1 + �2 + 1) (Sn2 + �2 + 1)

� d022 (Sn2 + �2 + 1)� n2d0

31;

�X31 =�C0

31 + C012�

(in) (Sn2 + �2 + 1)

+�C0

31 (�1 + 1) + C022�

(in);

�X32 =C031(Sn2 + �1 + �2 + 1) (Sn2 + �2 + 1)

� n2C022 � (�1 + 1)C0

31;

�X33 =�d0

31 + d021�

(in) (Sn2 + �2 + 1)

+ (�1 + 1) d031(in);

�X34 =d011 (Sn2 + �2 + 1) (Sn2 + �1 + �2 + 1)

+ d021 (Sn2 + �1 + �2 + 1)� n2d0

31;

�X35 =�d0

21 + d031�

(in) (Sn2 + �2 + 1)

+��1d0

21 � d031�

(in);

�X36 =�g011 (Sn2+�2+1) (Sn2+�1+�2+1)+n2g0

22;

�Y1 =�#0

1 (�1 + �2 +mn1 + 1)� #02�An1;

�Y2 = #02(in)An1;

�Y3 = � �q01 (�1 + �2 +mn1 + 1) + inq0

2�An1;

�Y4 =�#0

1 (�1 + �2 +mn2 + 1)� #02�An2;

�Y5 = #02(in)An2;

�Y6 = � �q01 (�1 + �2 +mn2 + 1) + (in)q0

2�An2;

�Y7 =�%0

1 (�1 + �2 + Sn1 + 1)� %02�Gn1;

�Y8 = %02(in)Gn1;

�Y9 = � � 01 (�1 + �2 + Sn1 + 1) + in 0

2�Gn1;

�Y10 =�%0

1 (�1 + �2 + Sn2 + 1)� %02�Gn2;

�Y11 = %02(in)Gn2;


�Y12 = � � 01 (�1 + �2 + Sn1 + 1) + in 0

2�Gn2;

�X01 =C0

11(m01 + �1 + �2 + 1) (m01 + �2 + 1)

+��1C0

12 � C022�;

�X02 =d0

11 (m01 + �1 + �2 + 1) (m01 + �2 + 1)

� d021 (m01 + �2 + 1) ;

�X03 =d0

11 (m01 + �2 + 1) (m01 + �1 + �2 + 1)

+ d021 (m01 + �1 + �2 + 1) ;

�X04 = �g0

11 (m01 + �2 + 1) (�1 + �2 +m01 + 1) ;

�X05 =C0

11(m02 + �1 + �2 + 1) (m02 + �2 + 1)

+��1C0

12 � C022�;

�X06 =d0

11 (m02 + �2 + 1) (m02 + �1 + �2 + 1)

� d021 (m02 + �2 + 1) ;

�X07 =d0

11 (m02 + �2 + 1) (m02 + �1 + �2 + 1)

+ d021 (m02 + �1 + �2 + 1) ;

�X08 = �g0

11 (m01 + �2 + 1) (m02 + �1 + �2 + 1) ;

�X09 =C0

11(S01 + �1 + �2 + 1) (S01 + �2 + 1)

+��1C0

12 � C022�;

�X010 =d0

11 (S01 + �1 + �2 + 1) (S01 + �2 + 1)

� d021 (S01 + �2 + 1) ;

�X011 =d0

11 (S01 + �2 + 1) (S01 + �1 + �2 + 1)

+ d021 (S01 + �1 + �2 + 1) ;

�X012 = �g0

11 (S01 + �2 + 1) (S01 + �1 + �2 + 1);

�X013 =C0

11(S02 + �1 + �2 + 1) (S02 + �2 + 1)

+��1C0

12 � C022�;

�X014 =d0

11 (S02 + �1 + �2 + 1) (S02 + �2 + 1)

� d022 (S02 + �2 + 1) ;

�X015 =d0

11 (S02 + �2 + 1) (S02 + �1 + �2 + 1)

+ d021 (S02 + �1 + �2 + 1) ;

�X016 = �g0

11 (S02 + �2 + 1) (S02 + �1 + �2 + 1) ;

�Y 01 =

�#0

1 (�1 + �2 +m01 + 1)� �02�A01;

�Y 02 = � �q0

1 (�1 + �2 +m01 + 1)�A01;

�Y 03 =

�#0

1 (�1 + �2 +m02 + 1)� #02�A02;

�Y 04 = � �q0

1 (�1 + �2 +m02 + 1)�A02;

�Y 05 =

�%0

1 (�1 + �2 + S01 + 1)� %02�G01;

�Y 06 = � � 0

1 (�1 + �2 + S01 + 1)�G01;

�Y 07 =

�%0

1 (�1 + �2 + S02 + 1)� %02�G02;

�Y 08 = � � 0

1 (�1 + �2 + S01 + 1)�G02;

Z1nj = C0

11�nj + C012 (inLnj + 1) + d0

11Pnj�nj ;

Z2nj = C0

12�nj + C022 (inLnj + 1) + d0

21Pnj�nj ;

Z3nj = inC0

31 + C031Lnj (�nj � 1) + ind0

31Pnj ;

Z4nj = d0

11�nj + d021 (inLnj + 1)� g0

11Pnj�nj ;

Z5nj = ind0

31 + d031Lnj (�nj � 1)� ing0

22Pnj ;

Z10j = C0

11�0j + C012 + d0

11P0j�0j ;

Z20j = C0

12�0j + C022 + d0

21P0j�0j ;

Z3nj = C0

31 (�0j � 1) ;

Z40j = d0

11�0j + d021 � g0

11P0j�0j ;

Z50j = d0

31 (�0j � 1) ;

�W 1n =C0

11 (mn1 + �2 + 1)Q1n + C0

12�Q1n + inQ5

n�

+ d011 (mn1 + �2 + 1)Q9

n � #01An1;

�W 2n =C0

11 (mn2 + �2 + 1)Q2n + C0

12�Q2n + inQ6

n�

+ d011 (mn2 + �2 + 1)Q10

n � #01An2;

�W 3n =C0

11 (Sn1 + �2 + 1)Q3n + C0

12�Q3n + inQ7

n�

+ d011 (Sn1 + �2 + 1)Q11

n � %01Gn1;

�W 4n =C0

11 (Sn2 + �2 + 1)Q4n + C0

12�Q4n + inQ8

n�

+ d011 (Sn2 + �2 + 1)Q12

n � %01Gn2;

�W 5n =C0

12 (mn1 + �2 + 1)Q1n + C0

22�Q1n + inQ5

n�

+ d021 (mn1 + �2 + 1)Q9

n � #02An1;

�W 6n =C0

12 (mn2 + �2 + 1)Q2n + C0

22�Q2n + inQ6

n�

+ d021 (mn2 + �2 + 1)Q10

n � #02An2;


�W 7n =C0

12 (Sn1 + �2 + 1)Q3n + C0

22�Q3n + inQ7

n�

+ d021 (Sn1 + �2 + 1)Q11

n � %02Gn1;

�W 8n =C0

12 (Sn2 + �2 + 1)Q4n + C0

22�Q4n + inQ8

n�

+ d021 (Sn2 + �2 + 1)Q12

n � %02Gn2;

�W 9n = inC0

31Q1n + C0

31 (mn1 + �2)Q5n + ind0

31Q9n;

�W 10n = inC0

31Q2n + C0

31 (mn2 + �2)Q6n + ind0

31Q10n ;

�W 11n = inC0

31Q3n + C0

31 (Sn1 + �2)Q7n + ind0

31Q11n ;

�W 12n = inC0

31Q4n + C0

31 (Sn2 + �2)Q8n + ind0

31Q12n ;

�W 13n =d0

11 (mn1 + �2 + 1)Q1n + d0

21�Q1n + inQ5

n�

� g011 (mn1 + �2 + 1)Q9

n + q01An1;

�W 14n =d0

11 (mn2 + �2 + 1)Q2n + d0

21�Q2n + inQ6

n�

� g011 (mn2 + �2 + 1)Q10

n + q01An2;

�W 15n =d0

11 (Sn1 + �2 + 1)Q3n + d0

21�Q3n + inQ7

n�

� g011 (Sn1 + �2 + 1)Q11

n + 01Gn1;

�W 16n =d0

11 (Sn2 + �2 + 1)Q4n + d0

21�Q4n + inQ8

n�

� g011 (Sn2 + �2 + 1)Q12

n + 01Gn2;

�W 17n =ind0

31Q1n + d0

31 (mn1 + �2)Q5n � ing0

22Q9n

+ q02An1;

�W 18n =ind0

31Q2n + d0

31 (mn2 + �2)Q6n � ing0

22Q10n

+ q02An2;

�W 19n =ind0

31Q3n + d0

31 (Sn1 + �2)Q7n � ing0

22F11n

+ 02Gn1;

�W 20n =ind0

31Q4n + d0

31 (Sn2 + �2)Q8n � ing0

22Q12n

+ 02Gn2;

�W 10 =C0

11 (m01 + �2 + 1)Q10 + C0

12Q10

+ d011 (m01 + �2 + 1)Q9

0 � #01A01;

�W 20 =C0

11 (m02 + �2 + 1)Q20 + C0

12Q20

+ d011 (m01 + �2 + 1)Q10

0 � #01A02;

�W 30 =C0

11 (S01 + �2 + 1)Q30 + C0

12Q30

+ d011 (S01 + �2 + 1)Q11

0 � %01G01;

�W 40 =C0

11 (S02 + �2 + 1)Q40 + C0

12Q40

+ d011 (S02 + �2 + 1)Q12

0 � %01G02;

�W 50 =C0

12 (m01 + �2 + 1)Q10 + C0

22Q10

+ d021 (m01 + �2 + 1)Q9

0 � #02A01;

�W 60 =C0

12 (m02 + �2 + 1)Q20 + C0

22Q20

+ d021 (m02 + �2 + 1)Q10

0 � #02A02;

�W 70 =C0

12 (S01 + �2 + 1)Q30 + C0

22Q30

+ d021 (S01 + �2 + 1)Q11

0 � %02G01;

�W 80 =C0

12 (S02 + �2 + 1)Q40 + C0

22Q40

+ d021 (S02 + �2 + 1)Q12

0 � %02G02;

�W 90 =d0

11 (m01 + �2 + 1)Q10 + d0

21F10

� g011 (m01 + �2 + 1)Q9

0 + q01A01;

�W 100 =d0

11 (m02 + �2 + 1)Q20 + d0

21Q20

� g011 (m02 + �2 + 1)Q10

0 + q01A02;

�W 110 =d0

11 (S01 + �2 + 1)Q30 + d0

21Q30

� g011 (S01 + �2 + 1)Q11

0 + 01G01;

�W 120 =d0

11 (S02 + �2 + 1)Q40 + d0

21Q40

� g011 (S02 + �2 + 1)Q12

0 + 01G02;

Biographies

Mehdi Gharibavi is a PhD candidate at East ChinaUniversity of Science and Technology, Shanghai, P.R.China. He received his MSc degree in MechanicalEngineering from Payame Noor University (PNU) inTehran, Iran, in 2015 and BSc degree from IslamicAzad University, Qazvin, Iran, in 2011. His currentresearch interests include control and stability of me-chanical systems, intelligent control and its industrialapplications, and medical device.

Jianjun Yi is a Professor at East China Universityof Science and Technology, Shanghai, P.R. China.Professor YI received his PhD in 1999, MSc in 1996,and BSc in 1991 in Mechanical Manufacturing and


Automation from Dalian University of Technology(DLUT), P.R. China. He worked as a post-doctoralfellow in the School of Mechanical Science and En-gineering, Huazhong University of Science and Tech-nology (HUST), from March 1999 to March 2001and in the Institute of Arti�cial Intelligence (AI),Zhejiang University (ZJU) from January 2005 to March2007. Also, he worked as a guest Professor at theTechnical University of Munich (TUM), Germany,

from September 2005 to October 2007. ProfessorYi is a Member of the IEEE and a senior memberof the China Mechanical Engineering Society. Hiscurrent research interests include digital control ofmechatronic systems, intelligent control and its indus-trial applications, distributed real-time control systemsand embedded systems, RFID and internet of things,automotive electronics, and ontology for knowledgemanagement.

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