MechanicalandThermalStressesinaFGPMHollowCylinder ... · axisymmetric mechanical and thermal stresses in thick short length functionally graded material cylinder. They applied the

International Scholarly Research NetworkISRN Mechanical EngineeringVolume 2011, Article ID 291409, 9 pagesdoi:10.5402/2011/291409

Research Article

Mechanical and Thermal Stresses in a FGPM Hollow Cylinderdue to Radially Symmetric Loads

M. Jabbari,1 M. Meshkini,1 and M. R. Eslami2

1 Islamic Azad University, South Tehran Branch, Tehran, Iran2 Department of Mechanical Engineering, Academy of Sciences, Amirkabir University of Technology, Tehran, Iran

Correspondence should be addressed to M. Jabbari, [email protected]

Received 25 May 2011; Accepted 21 June 2011

Academic Editor: J. Seok

Copyright © 2011 M. Jabbari et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The general solution of steady-state on one-dimensional Axisymmetric mechanical and thermal stresses for a hollow thick made ofcylinder Functionally Graded porous material is developed. Temperature, as functions of the radial direction with general thermaland mechanical boundary-conditions on the inside and outside surfaces. A standard method is used to solve a nonhomogenoussystem of partial differential Navier equations with nonconstant coefficients, using complex Fourier series, rather power functionsmethod and solve the heat conduction. The material properties, except poisson’s ratio, are assumed to depend on the variable r,and they are expressed as power functions of r.

1. Introduction

Poroelasticity is a theory that models the interaction ofdeformation and fluid flow in a fluid-saturated porousmedium. The deformation of the medium influences theflow of the fluid and vice versa. The theory was proposedby Biot [1, 2] as a theoretical extension of soil consolidationmodels developed to calculate the settlement of structuresplaced on fluid-saturated porous soils. The historical devel-opment of the theory is sketched by De Boer [3]. Thetheory has been widely applied to geotechnical problemsbeyond soil consolidation, most notably problems in rockmechanics. Detournay and Cheng [4] survey both thesemethods with special attention to rock mechanics. Theseinclude familiar analytical methods (displacement potentials,method of singularities) and computational methods (finiteelement and boundary element). Sandhu and Wilson [5]are acknowledged for pioneering the application of finiteelement techniques to poroelasticity. Detournay and Cheng[6] presented fundamentals of poroelasticity.

Abousleiman and Ekbote [7] presented the analyticalsolutions for inclined hollow cylinder in a transverselyisotropic material subjected to thermal and stress pertur-bations, and they systematically evaluated the effect of the

anisotropy of the poromechanical material parameters aswell as thermal material properties on stress and porouspressure distributions. Chen [8] presented and analyzedthe problems of linear thermo elasticity in a transverselyisotropic hollow cylinder of finite length by a direct powerseries approximation through the application of the Lanczos-Chebyshev method. Bai [9] presented then derived ananalytical method solving the responses of a saturatedporous media subjected to cyclic thermal loading by theLaplace transform and the Gauss-Lengender method ofLaplace transform inversion. Wang and Papamichos [10,11] presented analytical solution for the temperature, porepressure, and stresses around a cylindrical well bore and aspherical cavity subjected to a constant fluid flow rate bycoupling the conductive heat transfer with the pore-fluidflow. Ghassemi and Tao [12] presented influence of coupledchemo-poro-thermoelastic processes on pore pressure andstress distributions around a wellbore in swelling shale.Wirth and sobey [13] presented an axisymmetric and fully3-D poroelastic model forth evolution of hydrocephalus.Yang and Zhang [14] presented poroelastic wave equationincluding the Biot/squirt mechanism and the solid/fluidcoupling anisotropy. Arora and Tomar [15] presented theelastic waves along a cylindrical borehole in a poroelastic

2 ISRN Mechanical Engineering

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r/a

m = 1m = 0m = −1

0

1

2

3

4

5

6

7

8

9

10

T/T

a

Figure 1: Temperature distribution in the cross-section of cylindri-cal.

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

m = 1m = 0m = −1

r (m)

u(m

)

0.5

1

1.5

2

2.5

3

3.5

×10−4

Figure 2: Radial displacement in the cross-section of cylindrical.

medium saturated by two immiscible fluids. Hamiel etal. [16] presented the coupled evolution of damage andporosity in poroelastic media theory and applications to thedeformation of porous rocks. Ghassemi [17] presented stressand pore prepressure distribution around a pressurized,cooled crack in hollow permeability rock. Youssef [18] theory

of generalized porothermoelasticity was presented. Jourineet al. [19] presented modeling poroelastic hollow cylinderexperiments with realistic boundary conditions.

Functionally graded materials (FGMs) are heterogeneousmaterials in which the elastic and thermal properties changefrom one surface to the other, gradually and continuously.The material is constructed by smoothly changing materials.Since ceramic has good heat resistance and metal has highstrength, ceramic-Metal FGM may work at super high-temperature or under high-temperature difference field. Ineffect, the governing equation for the temperature and stressdistributions are coordinate dependent as the material prop-erties are functions of position. Classical method of analysisis to combine the equilibrium equations with the stress-strain and strain-displacement relations to arrive at the gov-erning equation in terms of the displacement componentscalled the Navier equation. There are some analytical thermaland stress calculations for functionally graded material inthe one-dimensional case for thick cylinders and spheres[20, 21]. The authors have considered the nonhomogeneousmaterial properties as liner function of r. Jabbari et al.[22] studied a general solution for mechanical and thermalstresses in a functionally graded hollow cylinder due tononaxisymmetric steady-state load. They applied separationof variables and complex Fourier series to solve the heatconduction and Navier equation. Poultangari et al. [23]presented Functionally graded hollow spheres under non-axisymmetric thermomechanical loads. Shariyat et al. [24]presented nonlinear transient thermal stress and elastic wavepropagation analyses of thick temperature-dependent FGMcylinders, using a second-order point-collocation method.Lu et al. [25] presented elastic mechanical behavior ofnanoscaled FGM films incorporating surface energies. Afsarand Sekine [26] presented inverse problems of materialdistributions for prescribed apparent fracture toughness inFGM coatings around a circular hole in infinite elastic media.Zhang and Zhou [27] presented a theoretical analysis of FGMthin plates based on physical neutral surface. Fazelzadehand Hosseini [28] presented aerothermoelastic behavior ofsupersonic rotating thin-walled beams made of functionallygraded materials. Ootao and Tanigawa [29] presented thetransient thermoelastic problem of functionally graded thickstrip due to nonuniform heat supply. They obtained theexact solution for the two-dimensional temperature changein a transient state, and thermal stress of a simple supportedstrip under the state of plane strain. Jabbari et al. [30]presented and studied the mechanical and thermal stressesin functionally graded hollow cylinder due to radiallysymmetric loads. They assumed the temperature distributionto be a function of radial direction. They applied a method tosolve the heat conduction and Navier equations. Farid et al.[31] presented three-dimensional temperature dependent-free vibration analysis of functionally graded material curvedpanels resting on two-parameter elastic foundation using ahybrid semianalytic, differential quadrature method. Bagriand Eslami [32] presented Generalized coupled thermoe-lasticity of functionally graded annular disk consideringthe Lord-Shulman theory. Shariat and Eslami [33] pre-sented buckling of thick functionally graded plates under

ISRN Mechanical Engineering 3

mechanical and thermal loads. Jabbari et al. [34] studied anaxisymmetric mechanical and thermal stresses in thick shortlength functionally graded material cylinder. They appliedthe separation of variables and complex Fourier series tosolve the heat conduction and Navier equation. Thieme etal. [35] presented titanium powder sintering for preparationof a porous FGM destined as a skeletal replacement implant.

In this work, a direct method of solution of the Navierequations presented which does not have limitation ofthe potential function method as to handle the generaltype of mechanical and thermal under one-dimensionalsteady-state temperature distribution with general type ofthermal and mechanical boundary conditions is considered.The functionally graded porous material properties of thecylinder are assumed to be expressed by power functions in r.The Naviear equation terms of displacements are derived andsolved analytically by the direct method, so any boundaryconditions for stresses and displacements can be satisfied.

Consider a hollow circular cylinder of inner radius a,outer radius b made of functionally graded porous material(FGPM) respectively. Axisymmetric cylindrical coordinates(r) are considered along the radial direction. The cylinder’smaterial graded through the r direction, thus the materialproperties are porous and functions of r. The first lawof thermodynamics for energy equation in the steady-statecondition for the FGPM on dimensional cylinder is:

1r

∂

∂r

[rk(r)

(∂T

∂r

)]= 0 −→ ∂2T

∂r2+

(k/(r)k(r)

+1r

)∂T

∂r= 0,

a ≤ r ≤ b,(1)

where T(r) is temperature distribution, k(r) is the thermalconduction coefficient, and symbol (/) denotes derivativewith respect to r.

2. Heat Conduction Problem

The thermal boundary is assumed as

S11T(a) + S12T,r(a) = f1,

S21T(b) + S22T,r(b) = f2,(2)

where (,) denotes partial derivative, and Si j are the constantthermal parameters related to conduction and convectioncoefficients. We assume that nonhomogeneous thermalconduction coefficient k(r) is power function of r ask(r) = k0rm3 , where k0 and m3 material parameter. Usingthe definition for the material properties, the temperatureequation becomes

∂2T

∂r2+ (m3 + 1)

1r

∂T

∂r= 0. (3)

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r/a

m = 1m = 0m = −1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

σ rr/P

i

Figure 3: Radial distribution of radial stress.

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r/a

m = 1m = 0m = −1

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

σ θθ/P

i

Figure 4: Radial distribution of hoop stress.

Integrating (4) twice yields

β2 +(m3 +1)β=0 −→⎧⎨⎩β1 = −(m3 + 1)

β2 = 0−→ β=−(m3 +1),

(4)

T(r) = a

β + 1rβ+1 + b −→ T(r) = −a

m3r−m3 + b. (5)


1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r/a

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

B = 0.85B = 0.65B = 0.5

σ rr/P

i

Figure 5: Radial thermal stress in the cross-section of cylindricalbased on the pore volume fraction (B) changing.

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r/a

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

φ = 0.25φ = 0.15φ = 0.05

σ rr/P

i

Figure 6: Radial thermal stress in the cross-section of cylindricalbased on the pore volume fraction (φ) changing.

Using the boundary conditions (2) to determine the con-stants a and b yields

A1 = e4 f1 − e2 f2e1e4 − e2e3

, A2 = e1 f2 − e3 f1e1e4 − e2e3

, (6)

where constants e1 to e4 are given in Appendix A.

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r/a

B = 0.85B = 0.65B = 0.5

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

σ θθ/P

i

Figure 7: Hoop thermal stress in the cross section of cylindricalbased on the compressibility coefficient (B) changing.

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r/a

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

φ = 0.25φ = 0.15φ = 0.05

σ θθ/P

i

Figure 8: Hoop thermal stress in the cross section of cylindricalbased on the pore volume fraction (φ) changing.

3. Stress Analysis

Let u displacement components in the radial direction. Thenstrain-displacement relations are

εrr = ∂u

∂r, εθθ = u

r, (7)


1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r (m)

×10−3

u(m

)

0.5

1

1.5

2

2.5

3

B = 0.85B = 0.65B = 0.5

Figure 9: Radial displacement in the cross-section of cylindricalbased on the compressibility coefficient (B) changing.

and stress-strain relations of a functionally graded porouscylinder for nonaxisymmetric condition are

σrr = C11εrr + C12εθθ − γpδrr − Z1T(r),

σθθ = C22εrr + C12εθθ − γpδθθ − Z2T(r),

C11 + Mγ2 = ∗C11,

C22 + Mγ2 = ∗C22,

C12 + Mγ2 = ∗C12,

(8)

where σi j , ∈i j (i, j = r, θ), M, γ, α, λ, μ, and p are stresstensors, strain tensors, Biot’s modulus, Biot’s coefficientof effective stress, thermal expansion coefficient, lame’scoefficient, and the pore pressure, respectively, p related tothe Biot’s modulus, volumetric strain and the variation offluid content.

We assume that pore-cylinder if undrained conditionthen (ζ = 0) as:

p =M(ζ − γ(εrr + εθθ)

) = −Mγ(εrr + εθθ), (9)

where:

M = 2μ(νu − ν)γ2(1− 2ν)(1− 2νu)

. (10)

Thus,

σrr =∗C11εrr +

∗C12εθθ − Z1T(r),

σθθ =∗C22εθθ +

∗C21εrr − Z2T(r).

(11)

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

r (m)

×10−3

u(m

)

0.5

1

1.5

2

2.5

3

φ = 0.25φ = 0.15φ = 0.05

Figure 10: Radial displacement in the cross-section of cylindricalbased on the compressibility coefficient (φ) changing.

The equilibrium equation in the radial direction, disregard-ing body force and the inertia terms, is

∂σrr∂r

+1r

(σrr − σθθ) = 0,

Z1 =∗C11αr + 2

∗C12αθ ,

Z2 = 2∗C21αr +

∗C22αθ.

(12)

To obtain the equilibrium equations in terms of the displace-ment components for the FGPM cylinder, the functionalrelationship and pore of the material properties must beknown. Because the cylinder material is assumed to begraded along the r-direction, the modulus of elasticity andcoefficient of thermal expansion are material constant

assumed to be described with the power laws as

αr = α01rm1 , αθ = α02r

m2 , K = k0rm3 , Cij = Cijr

m4,(13)

where the coefficients are described as

α01 = α′01

am1, α02 = α′02

am2, K = k′0

am3, Cij =

C′i j

am4, (14)

and a is the inner radius.


Table 1

Parameters C11 C12 C22 γ E0 ν νu α01, α02

Numerical 139 GPa 78 GPa 139 GPa 0.47 200 GPa 0.2 GPa 0.3 GPa 0.000012 (1/ ◦C)

Using the relations (7) to (14), the Navier equations interms of the displacement components are(∂2u

∂r2+ (m4 + 1)

1r

(∂u

∂r

)+

((m4 + 1)C12 − C22

C11

)1r2u

)×rm4

=({[

(m1 + m4 − 1)C11 − 2C12

C11

]× rm1−1α01

+

[2(m2 + m4 + 1)C12 + C22

C11

]× rm2−1α02

}T(r, θ)

+

[C11rm1α01 + 2C12rm2α02

C11

]dT

dr

)× rm4 .

(15)

The Navier equation (15) is nonhomogeneous system ofpartial differential equations with non-constant coefficients.We assume that m1 = m2.

4. Solution of the Navier Equation

Equation (15) is the Euler differential equation with generaland particular solutions.

The general solution is assumed to have the form

ug(r) = Brη. (16)

Substituting (16) into (15) yields[η(η − 1

)+ (m4 + 1)η +

1C11

[(m4 + 1)C12 − C22

]]= 0.

(17)

Equation (17) has two roots η1 to η2. Thus, the generalsolutions are

η1,2 = −m4

2±(m2

4

4− (m4 + 1)C12 − C22

C11

)1/2

. (18)

Thus, the general solution is

ug(r) = D1rη1 + D2r

η2 . (19)

The particular solutions up(r) are assumed as

up(r) = (I1 + I2)rβ+m2 . (20)

Substituting (20) into (18) yields

d1I1rβ+m2−1 + d2I2r

m2−1 = d3rβ+m2−1 + d4r

m2−1. (21)

The complete details for solution of (21) is presented inAppendix B.

The complete solutions for u(r) is sum of the general andparticular solutions and are

u(r) = ug(r) + up(r). (22)

Thus

u(r) = D1rη1 + D2r

η2 + (I1 + I2)rβ+m2 . (23)

Table 2

Parameters T(a) T(b) σrr(a) σrr(b) m

Numerical 50◦C 0 −50 0 −1, 0, 1

Substituting (23) into (1) and (2), the strains and stresses areobtained as

εrr = η1D1rη1−1 + η2D2r

η2−1 +(β + m2

)(I1 + I2)rβ+m2−1,

εθθ = D1rη1−1 + D2η2r

η2−1 + (I1 + I2)rβ+m2 ,

σrr =(C11

[η1D1r

η1+m4−1 + η2D2rη2+m4−1

+(β + m2

)(I1 + I2)rβ+m2+m4−1

+α01[A1 + A2](rβ+m4+m2−1

)]

+ C12

[D1r

η1+m4−1 +D2η2rη2+m4−1 +(I1 +I2)rβ+m4+m2

+2α02[A1 + A2](rβ+m4+m2−1

)])einθ ,

σθθ =(C22

[D1r

η1+m4−1 + D2η2rη2+m4−1

+(I1 + I2)rβ+m4+m2 + α01[A1 + A2](rβ+m4+m2

)]

+ C21

[η1D1r

η1+m4−1 + η2D2rη2+m4−1

+(β + m2

)(I1 + I2)rβ+m2+m4−1

+2α02[A1 + A2](rβ+m4+m2

)])einθ.

(24)

To determine the constants D1 and D2, consider theboundary conditions for stresses given by

σrr(a1) = −pi, σrr(a2) = −p0. (25)

5. Numerical Results and Discussion

Consider a thick hollow cylinder of inner radius a = 1 (m)and outer radius b = 1.2 (cm), shown properties are given inTable 1. For simplicity of analysis, we consider that the powerlaw of material properties is the same as m1 = m2 = m3 = m.To examine the proposed solution method, two exampleproblems are considered. The example problem may havesome physical interpretation.

As the example, consider a thick hollow cylinder wherethe inside boundary is traction free with given temperaturedistribution of Table 2. The outside boundary is assumedto be radially fixed with zero temperature. Therefore, theassumed boundary conditions yield of Table 2.

Figure 1 shows the variations of the temperature alongthe radial direction for different values of the power law


index. The figure shows that as the power law index mincreases, the temperature decreased.

Figure 2 shows the plot of the radial displacement alongthe radius. The magnitude of the radial displacement isdecreased as the power index m is increased.

The radial and circumferential stresses are plotted alongthe radial direction and shown in Figures 3 and 4, and themagnitude of the radial stress is increased as m is increased.The hoop stress along the radius decreases for m, 1 (similar tothick cylinders made of isotropic materials), due to the actinginternal pressure and zero external pressure. For m < 1,the hoop stress increases as the radius increases, since themodulus of elasticity is an increasing function of the radius.Physically, this means that the outer layers of the cylinderare biased to maintain the stress due to their higher stiffness.There is a limiting value for m, where the hoop stress remainsalmost a constant along the radius. For low values of the ratiob/a (Figures 7 and 8). Figures 5 and 6 show the radial andhoop thermal stresses in the cross-section of the cylinder,respectively, where the pore compressibility coefficient (B) ischanged, the other parameters are fixed. Figures 5 and 6 showthese stresses based on the pore volume fraction; (φ) is porevolume per total volume.

Figure 9 shows the radial displacements in the cross-section of the cylinder based on the pore compressibilitycoefficient (B) changing. Figure 10 also shows these displace-ments based on the pore volume fraction (φ) changing.

6. Conclusions

In the present work, an attempt has been made to study theproblem of general solution for the thermal and mechanicalstresses in a thick FGPM hollow cylinder due to the one-dimensional axisymmetric steady-state loads. The method ofsolution is based on the direct method and uses power series,rather than the potential function method. The advantage ofthis method is its mathematical power to handle both simpleand complicated mathematical function for the thermaland mechanical stresses boundary conditions. The potentialfunction method is capable of handling complicated math-ematical functions as boundary condition. The proposedmethod does not have the mathematical limitations tohandle the general types of boundary conditions which areusually countered in the potential function method.

Appendices

A. Compressibility Coefficients and PoreVolume Fraction

e1 =(S12a

β − S11

m3Aβ+1

), e2 = (S11),

e3 =(S22A

β − S21

m3Aβ+1

), e4 = (S21),

μ = E0

2(1 + ν).

(A.1)

B: compressibility coefficient, sometimes called the Skemp-ton pore pressure coefficient.

B = 3(νu − ν)(1− 2ν)(1 + νu)

, 0 ≤ B ≤ 1 (A.2)

φ: pore volume fraction is pore per unite total volume.

φ =γ(B − k f

)

B[(1− α) + k], (A.3)

k f and k are bulk modulus of the fluid phase and bulkmodulus of the poroelastic medium under the drainedcondition, respectively.

B. Constants Material

I1 = d4d5 − d2d6

d1d4 − d2d3, I2 = d1d6 − d3d5

d1d4 − d2d3, (B.1)

where constants d1 to d6 are given

d1 = (m2 + 1)(m2) + (m4 + 1)(m2 + 1) +(m4 + 1)C12 − C22

C11

d2 = (m2 −m3)(m2 −m3 − 1) + (m4 + 1)(m2 −m3)

+(m4 + 1)C12 − C22

C11,

d3 ={[

(m2 + m4 −m3 − 2)− 2C12

C11

]α01

+

[2(m+m +1)C12 +C22−2(m3 − 1)C12

C11

]α02

}An1,

d4 = 1k0

{[(m2 + m4 −m3 − 2)− 2C12

C11

]α01

+

[2(m1 + m4 +1)C12 +C22−2(m3 +1)C12

C11

]α02

}An2,

I1 = d3

d1, I2 = d4

d2

d1 = C11(η1 + 1

)aη1+m4−1,

d2 = C22(η2 + 1

)aη1+m4−1,

d1 = C11(η1 + 1

)bη1+m4−1,

d2 = C22(η2 + 1

)bη2+m4−1.

(B.2)


Nomenclature

a: Inner radiusan: Thermal constantb: Outer radiusbn: Thermal constantSi j : Constant temperature parametersdi: Mechanical and thermal constantsei: Mechanical and thermal constantsDij : Constant mechanical parametersf1, f2: Inner and outer temperature boundary

conditionsg1, g2, . . . , g8: Inner and outer mechanical boundary

conditionsk: Thermal conduction coefficientk0: Material parameterE: Yong’s modulusE0: Material constantm1, m2, m3: Material parameter(r, θ): Cylinder coordinateT : Cylinder temperatureTn: Coefficient of sine Fourier seriesu, v: Displacement componentsα: Thermal expansion coefficientα0: Material constantμ: Lame coefficientν: Poisson’s ratioνu: Undrained Poisson’s ratiop: The pore pressureM: Biot’s modulusγ: Biot’s coefficient of effective stressδi j : Delta caranckerζ : The variation of fluid content

(undrained ζ = 0)εi j : Strain tensor (i, j) = (r, θ)∈: Volumetric strain (∈ = εrr + εθθ)σi j : Stress tensor (i, j) = (r, θ)B: Compressibility coefficientφ: Pore volume fraction is pore per unite

total volume.

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MechanicalandThermalStressesinaFGPMHollowCylinder ... · axisymmetric mechanical and thermal stresses in thick short length functionally graded material cylinder. They applied the