Scholarly article on topic 'Dimensional Two Steady State Thermal and Mechanical Stresses of a Poro-FGM Spherical Vessel'

Dimensional Two Steady State Thermal and Mechanical Stresses of a Poro-FGM Spherical Vessel Academic research paper on "Materials engineering"

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{FGM / Poroelastic / "Hollow Sphere" / Thermoporoelasticity}

Abstract of research paper on Materials engineering, author of scientific article — Sajad Karampour

Abstract In this study, an analytical method is developed to obtain mechanical and thermal stress and mechanical displacement in two dimensional steady (r,θ) state a functionally graded porous material hollow sphere (FGpm). It is assumed that properties of poro, and FGM material is changed through thickness according to power law functions. heat conduction equation is obtained for obtaining temperature distribution and navier equations analytically using legendre polynomials and Euler differential equations system for investigating displacements changes and stress and potential functions for different indices power indices.

Academic research paper on topic "Dimensional Two Steady State Thermal and Mechanical Stresses of a Poro-FGM Spherical Vessel"

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Procedia - Social and Behavioral Sciences 46 (2012) 4880 - 4885

WCES 2012

Dimensional two steady state thermal and mechanical stresses of a

Poro-FGM spherical vessel

Sajad Karampour a *

aDepartment of mechanical engineering, Omidiyeh Branch, Islamic Azad University, Omidiyeh, Iran

Abstract

In this study, an analytical method is developed to obtain mechanical and thermal stress and mechanical displacement in two dimensional steady (r,0) state a functionally graded porous material hollow sphere (FGpm). It is assumed that properties of poro, and FGM material is changed through thickness according to power law functions. heat conduction equation is obtained for obtaining temperature distribution and navier equations analytically using legendre polynomials and Euler differential equations system for investigating displacements changes and stress and potential functions for different indices power indices. © 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of Prof. Dr. Huseyin Uzunboylu Keywords: FGM; Poroelastic; Hollow Sphere; Thermoporoelasticity

l.Intruduction

Functionally graded materials (FGMs) are made of a mixture with arbitrary composition of two different materials,and the volume fraction of each material changes continuously and gradually. The FGMs concept is applicable to many industrial fields such as chemical plants, electronics, biomaterials and so on[1]. Thick hollow sphere analysis made of FGM under mechanical and thermal loads and in asymmetric and two-dimensional (r,0) state was conducted investigating navier equations and using legendre polynomials [2]. Ootao and Tanigawa derived the three-dimensional transient thermal stresses of a non-homogeneous hollow sphere with a rorating heat source [3]. Jabbari presented the analytical solution of one and two-dimensional steady state thermoelastic problems of the FGM cylinder [4].Two-dimensionalnon-axisymmetric transient mechanical and thermal stresses in a thick hollow cylinder is presented by Jabbari et al [5]. Porous spheres of nanometer to micrometer dimensions are being pursued with great interest because of several possible technical applications in catalysis, drug delivery systems, separation techniques, photonics, as well as piezoelectric and other dielectric devices [6-7]. The study of the thermomechanical response of fluid saturated porous materials is important for several branches of engineering [8-12]. Inspite of conducted studies on spherical and cylindrical vessel made of FGM to obtain mechanical displacements and mechanical and thermal stresses, it has not done any study on composition of poro and FGM materials. This study investigates the effect of mechanical and thermal stresses two dimensional steady (r,0) state a poro FGM hollow sphere that Fluid trapped in the porous medium is located in undrain conditions.

2. Analysis

Consider a thick spherical vessel of inside radius a and outside radius b made of poro FGM. It is assume that the mechanical and thermal loads and their associated boundary conditions are such that the stress field is a function of variables r and q . For the assumed condition, the strain-displacement relations are

u 1 U D 1/1 , x

£rr = u,r, ff* = - + - ■ve, - + - cot 0, = -(- u,0+ »r- -X (1)

r r w r r 2 r 2

* Corresponding author. Tel.: +98-916-737-3814; fax: +98-652-322-2533. E-mail address: Sajad.karampour2010@gmail.com

1877-0428 © 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of Prof. Dr. Huseyin Uzunboylu doi:10.1016/j.sbspro.2012.06.353

where u and v are the displacement components along the r- and 0 - directions, respectively.the linear constitutive relations for a functionally graded porous material hollow sphere (FGpm) can be express

°rr= C33^ + + C13£# -ypôrr-zrT(r^Gf), °ee = Cnem + C13Err + Ci2s# ~pee ~zeT(r,e) CT#=Cn£# +C13£rr+Ci2£m- ypS^-zT (r,ff), crg = 2CAA£re , =0 (2)

Where <7. and By are the stress and strain tensors, r(r, 0) is the temperature distribution determined from the heat conduction equation, zt and is the coefficients of thermal expansion in effective stress, Cy are the elastic constants, ^ and M are Biots coefficient of effective stress and Biots Moduls. there for

z = zf2m, Zr= C33a r+C^ag + ^,=^33+2^)« , ^=C1ft + CBar+C12«, = (C„ + Cn + CB)« (3)

Pressure equation in porous environment

p=M (£-Yë) (4)

For undrain condition

¿■ = 0, p = -№t=-№ (Srr+Bffl + (5)

Where Ç and e are variation of fluid content and volumetric strain. The equilibrium equations, disregarding the body forces and the inertia terms, are

crrrr+ -are „+- (2(Jrr—GQg— CT + a r9 œt 0) = 0,vre,r + -0ee,g+1 - )cot ^ + 3<*re ) = 0 (6)

r r r 2

Substituting Eqs.(5) in to Eqs.(2) lead to

^rr= C3>,r + C1*3%> + C1*3^r " zT(r,0),0gg = C*13£rr+ CnEee + Cu£& " zeT(r,ff)

= ^rr + C1*2^ + Cn*» - ZTtr^XCrO = 2C44£r6 (7)

Cn= C^-fM, C*2 = C22 - y2M, c; = C,3-r2M, C; = Cn - fM, C12 = Cl2- y2M (8)

With this hypothesis that

C.=C/m (9)

Employing a change of variable fi = cos 0 and using Eqs. (1)-(9), the equilibrium equations in terms of the displacement components (Navier

equations) are obtained as

, 2 C!(m +1) C1*1 + C* 1 C 1 C * C

(m+ c^a-^,^ ( Cf+14)

(-v^Si^+v co^),+_3_( C13Ct + 1)_ C4^(C»+ C12))^^ SinO + v cot^) = C- rm~ l(2(m +1)- f yr + rT, ), (10) r C * CC'

33 w33 ^33

12 , t 1 1\ 11 V. v1 h1 \U2n . C13 -

1 1 Cn ,v *((m + 2) +Ck + (—--ni^v- -—(1 + C

{Vrr+L {m + 2K+ + v cot*)" -2 (( m + 2) + p^ + Crr"1) ) (1 +

r 2 T=T ^ M ) r2 ' p 1_7 C r C

r r C44 ' C44 Z C44 C44

(m + ? + C11 +C12)u - , , ,

(m + ^ CBC±)uM= -B^rm^H-ST (11)

r CAA C A

f = ^ (12) zr

3. Temperature distribution

The heat conduction equation and the thermal boundary conditions, respectively, are

T„ + (- + -)Tr + (1-^)7^=0, a<r<b, -1<//<1, (13) k r r

where k is the coefficient of thermal conduction. It is assumed that the thermal conduction coefficient for the poro FGM sphere is k (r)= kmrm (14)

Here, km= where k(a) is the conduction coefficient of the inner radius of FG material at r = a and m is the material constant.

Substituting Eq. (14) into Eq. (13) yields the FGM heat conduction equation as

(m + —T + — (1-//-)7, -fj. TM=0 (15)

Solution of the conduction equation may be assumed in the form of Legendre series as T (r,p) = Y/n (r) pn (jt) (16)

where Tn (r) as the coefficient of Legendre series may be illustrated as 2n +1 f1 N , ^ , 2n +1 t"r,

Using Eq. (12), Eq. (11) may be written as

T (r) = ^Jl, T (r,ti) p„ (ji)dp = ^ J/ (r,0) p„ (cos0)sin(0)J0 (17)

00 1 i

XiTV) + - (m + 2)T'n (r)) pn W+J-qi-S) pi (M)-2fp'„ (n)Tn (r)} = 0 (18)

to r r2

Employing the change of variable given in Appendix B, results into the separation of independent variables in Eq. (18), which may be written as

n 4- n

T:(r) + (m + 2) - T (r)--— Tn (r) = 0 (19)

The above equation is the Euler equation. Thus, the solution may be written in the form

Tn (r)= A^ (20)

Substituting Eq. (20) into Eq. (19) yields

_ ~(m + 1)±V(m + 1) + 4n(n + 1) (2i)

Pn\,2~ 2 ( )

The general solution of Eq. (18) is

Tn (r)= An - rA- (22)

Thus, the temperature distribution becomes

T (r,fi)= £( Anl^Pn1+ An - r"n2) Pn (M) (23)

Which, here An1, An2 constants are obtained from thermal boundary conditions for internal and external radius as below. Cu T'{a,e) + C12 r(a,e) = fXo\ C21 T'(b,0)+ C22 r(b,0) = /2(0) (24) Where f^d) and /2(#) are inside and outside radius temperature shown boundary condition. 4. Stress distributions

The Navier equations (10) and (11) may be solved by the direct method of analysis employing the series solution introduced by Jabbari et al. [4, 5]. The solution of the Navier equations (10) and (11) is assumed in the form of Legendre series as

u(r,/f)=£Un (r)Pn (ji), v(r,/f)=£Vn (r)p'n (M)(1-ß2) (25)

n=0 n=0

where un(f) andvn(r) are functions of r. Substituting Eqs. (25) into Eqs. (10) and (11) and then using form of Legendre series to separate the independent variables r and ß lead to

< (r) + (m + 2)1 u'n (r ) + -1 (CBm-n(n +1) 2C13 C12) un (r ) +1 (n(n + iX^C^4 ))v' (r) +

r r C33 C33 ^^ r C^í

n(n + 1) ,Cn(m +1) (C4^+(C11+ C1

2 (r) = ,^((2((m + 1)-f) + PaVaf^Umn* 1)-f)A„2+ Ajjy^

* * * C33 C33 C33

- C * ^ n1r

^m+finl-1 Ï

44 ^44

1 1 C ' C ' 1 C * 1 C *+ C*

v'n (r) + -(m + 2)vn (r)--2(( m + 2) + n(n + + Vn (r) — (1 + =^K (r)- —(m + 2 + 12

^ ( A^r^-1 + An 2 r"**2"1)

)un(r) (27)

This is the system of Euler differential equations. Thus, the solution of homogeneous part of Eqs. (26) and (27) may be assumed in the form

ug (r) = Br", vg (r) = Cr* (28)

where B and C are constants to be found using the given boundary conditions. Substituting Eqs. (28) into the homogeneous parts of Eqs. (26) and (27) yields

r¡ +(m+2)7 + (2

■ n(n +1)

C44 . 2(C13 (C11 C12))

C + C n(n +1)( ^C^44

+ n(n + 1) (C1*3(^ + 1) - (C1*1 + C1*2 )) )

-(1 + Cr-ft-(m + 2 + ^C^2)

C44 C44

rj2+(m + 2)rç-((m + 2) + n(n + 1) C1

C44 C44

To obtain the non-trivial solution of the above equation, the determinant of coefficients of constants B and C must be vanished. This leads to the evaluation of the eigenvector 77 obtained as

(r¡2+(m + 2)^ + (2

C'3m-n(n +1) C4t+ 2(C'3 (Cl1+ C'2)) ))x({i12+(m + 2)rç-((m + 2) + n(n + 1) + S2)-

((«(« + 1)C1yr44^ + n(n + 1)(C13(^ +1) (B44+(C1^ C12)))^ (_(1 + _ (m + 2 + ^-1^))) = 0

^33 w33 w44 w44

Thus, the general solution, utilizing the linearity lemma, is a linear combination of all values of eigenvalues and are obtained as

(r ) = £ V. vn (r) = £ Nn.

Where, using Eq. (29), Nn may be founded as

N = (C^W2 + C^(m + 2)^ + 2C,*m- n(n +1^4+ 2^*3 - C + C^,))) j=1234 nn (n(n+1)(C1*3+C44^+n(n+1)(C;(^ 1)-(C4^ o C;))) , "'

The particular solution of Eqs. (26) and (27) are assumed As

< (r) = Dn2rm+Pn2+\ vp (r) = + (33)

Substituting Eqs. (33) into Eqs. (26)-(27) yields

d1Dn1r^n1+ m~ 1 + d2 Dn 2 r^"1 + d3 Dn 3r^n1~1+ d( Dn 4 r^"1 = d5rA1+m"1 + d6 r"«™-1

d7 Dp3 r^"1"1 + dDS^ + d9Dnlrmii^+ dDr^ = + dy^1

where coefficients d1 through d12are presented in Appendix A. Equating the coefficients of the identical powers yields

d1Dn1+ d3 Dn

d10Dn2+ d8 Dnd= ^ (36)

d5 •

d9 Dm+ d7 Dn3= d11

Here, Dni{i = 1,...,4) are obtained solving the two systems of algebraic equations. The complete solutions for displacements are the sum of Eqs. (31) and (33) and are

Un (r) = i Bn/^+Dn!^1 + AXn2+m+1, Vn (r) = ±NnJBn/"n Dn4 r^n2+1 (37)

For n = 0, the system of Navier equations (26)-(27) lead to the following single differential equation, as

<(r)+(m+2)- u>(r)+-(-

1 , 2 CÙ(m+1)-Cj'J-C1*2

)u>(r)= M2(m + 1)- /)+ Ai)i„i rm^~l+{(2{m+1)-/)+¡¡^ rm+^ (38)

^33 ^33

solving the homogeneous part of Eq. (38) provide the complete solution for ug (r) as

(r ) = X b0 r0

constants to be found using the given boundary conditions.

solving the non_homogeneous part of Eq. (38) provide the complete solution for up (r) as u0P(r) = D01rAl+m+1 + D02rA2+m+1, where D01, and D02 are constants to be found using the given boundary conditions.

Thus, the complete solution of radial and circumferential displacement components for all values of n, using Eqs. (37) and (40), is

where B0, and TJ0 are (40)

uM=t Bo r01 )+A>1

yfioi+mt-1 D yPo2+m+1 ^

^B^'+Dy^m+^ D,

fin2+ m+1

JjNB/" +D„3r^+m+1 + Dy-2+m* j=1

X1-//)1'

CP» (41)

"„^(Ltooi*C3^Doj(09oj 1) + CV°1+2m -C^

Using the strain-displacement relations given by Eqs. (1) and relations (2), the radial, circumferential and shear stresses are obtained as the strains are

- -C * -C '

J - 33 ^33

00 4 C * C *

(2,(2,(1 n;+ =3(2 + j(j + \)Nnj )Bn/"+m-1 + Dnl((Pnl+ m +1) + -=$(2 + n(n +1) NnJ + Dnl((Jinl+ m +1)

j= 1 J=1 C33 C33

+S(2+ "(" +DNjS) r^+DM S m+1H S(2+ "(" + 1)Nnj~~))r^n2+^m -■Cz*-( ))x Pa (/,)

ee\ = C^3(2^o] + CCC2)B0Jr*>1 m1 + Doi ((fi0J + m + 1) + C11C;C12)/»1+2m -C^ (Ao1r^2m +A2r^2i H +

J 1=1 C13 C13 C13 W

2(2(li + m1 + Dj ((/3„i+m + 1H(C1CCl))/l2^-CCrr(A„1^"^2^A„2 r*2+2m ))P, (

№1 1=1 C13 C13 C13 [1J

2N„iB„/"n+m-^ D„3 r^2m+D„4 r

P„2+2m

C±n(n + 1)P, (//>

v- i '

v- 1 '

C * - C * f 1

.+( )M p:

= C44(£

2B:j N (1,- Dnl((finl+ m) D3-!)^+Dnl((pnl+ m)

xp: ^)(^//2)1/2)

Appendix A

4 =0» + A1+1)0» + /U + ('» + 2)(m + /7„1+D + (2Cim n{n + 1)+J*(C'3 (C" + Cl2))), d2=(m + pn 2+X)(m +¡in 2) +

(m + 2Xm + A ^1H(2C1>-„(^1) C<C13 -(Cn + O , d,=( „(„ + 1)(C,1;+ C44))(^/?„^1) +

C33 C33

(„(„+1x0+1)-(C44+C1+C^ („(„+1)(C;+C44), 1), („(„+1x0+1)-(C44 +C^

(-C-^ d4=(—C5- 2+ (-c-

d, = (Cf)(m +/?.,+1)(m +P„1) + ((m + 2)pr 2e21 )(m + fl„1+1) -„(„ + 1)(Ce25), d^ (Cf)(m + f)„2+ 1)(m + fj„2) + ((m + 2)pr2e"21)

C33 C33 C23 C33 C33

(m + 0, 2+1)- „(„ + 1)(Cf), d 7 - -=r{(2m + 1- f) + fi^, dt= ^((2m +1-/) + fi„ 2)A„ 2, d9 = (m +A+^ + ,0+ (m + 2)

C C C C

(m + /?„1+1)-((m + 2) + „(„ + 1) + C^), dw = m + A 2+Km + fi„ 2)+ (m + ^m + fi„ 2+1)-((m + 2) + „(„ + 1) Ci +

C44 C44 C44 C44

C' C* + C C C'+ C

dn= -(1 + crX.m + P*i+1)-(.(m + 2) + 11C 12X dn = -(1 +C1)^ +A2 +1)" ((m +2) + 1C 12)

C44 C44 C44 C44

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