Radially Symmetric Steady State Thermal and Mechanical Stresses of a Poro FGM Hollow Sphere

Abstract

A general solution for the one-dimensional steady-state thermal and mechanical stresses in a hollow thick sphere made of porous functionally graded material (FGPM) is presented. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. The material properties, except Poisson's ratio, are assumed to vary along the radius r according to a power law function. The analytical solution of the heat conduction equation and the Navier equation lead to the temperature profile, radial displacement, radial stress, and hoop stress as a function of radial direction.

1. Introduction

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, and biomaterials [1]. Thick hollow sphere analysis made of FGM under mechanical and thermal loads and in asymmetric and two-dimensional (𝑟,𝜃) state was conducted investigating navier equations and using legendre polynomials [2]. The analytical solution for the stresses in spheres and cylinders made of functionally graded materials are given by Lutz and Zimmerman [3, 4]. They considered thick spheres and cylinders under radial thermal loads, where radially graded materials with linear composition of the constituent materials were considered. Obata and Noda [5] studied one-dimensional steady thermal stresses in a functionally graded circular hollow cylinder and a hollow sphere using the perturbation method. The transient thermal stresses in a plate made of FGM are presented by the same authors [6]. By introducing the theory of laminated composites, Ootao and Tanigawa derived the three-dimensional transient thermal stresses of a nonhomogeneous hollow sphere with a rotating heat source [7]. Jabbari et al. presented the analytical solution of one and two-dimensional steady state thermoelastic problems of the FGM cylinder [8]. Two-dimensional nonaxisymmetric transient mechanical and thermal stresses in a thick hollow cylinder is presented by Jabbari et al. [9]. Shao and Wang presented three-dimensional solution to obtain stress fields in an FGM cylindrical plate with a finite length which is mechanical and thermal load tests and showed the results an a graph [10]. The transient thermal stress problem in a hollow sphere with homogeneous and isotropic properties is analytically solved by Cheung et al. [11]. Their assumed thermal boundary conditions are varied along the 𝛾 direction and the problem is solved with the potential function method. Takeuti and Tanigawa have employed the potential function method to obtain the analytical solution of a homogeneous spherical vessel with rotating heat source [12]. They considered the general form of thermal boundary conditions and analyzed the problem in transient condition with the potential functions method.

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 [13, 14]. The study of the thermomechanical response of fluid saturated porous materials is important for several branches of engineering [15–19]. Some of the important cases are the disposal of high-level radioactive waste, the extraction of oil or geothermal energy, the storage of hot fluids, and road subgrade or furnace foundations, which are usually subjected to cyclic changes of temperature. Wang and Papamichos [20, 21] presented analytical solutions for the temperature, pore pressure and stresses around a cylindrical wellbore and a spherical cavity subjected to a constant temperature change and a constant fluid flow rate by coupling the conductive heat transfer with the pore-fluid flow. Using the so-called heat source function method. Kurashige [22] analyzed the heat and fluid flow problem of a spherical cavity subjected to a constant heat flux in an impermeable medium.

Inspite of conducted studies on spherical and cylindrical vessel made of FGM to obtain mechanical displacements and mechanical and thermal stresses, there is not any study on composition of poro and FGM materials. This study investigates the effect of mechanical and thermal stresses on one-dimensional steady state in the poro FGM hollow sphere that fluid trapped in the poro medium which is located in undrained conditions.

2. Analysis

Consider a thick hollow sphere of inside radius 𝑎 and outside radius 𝑏 made of poro FGM. The sphere material is graded through the radial 𝑟-direction. Thus, the material properties are functions of 𝑟. Let 𝑢 be the displacement component along the radial direction. The strain-displacement relations are 𝜀_𝜃𝜃𝜀_𝑟𝑟=𝑢_,𝑟,=𝜀_𝜙𝜙=𝑢𝑟.(1) The stress-strain relations one-dimensional in the poro FGM hollow sphere with fluid trapped in the poro pure are 𝜎𝜎_𝑟𝑟=𝐶₁₁𝜀_𝑟𝑟+2𝐶₁₂𝜀_𝜃𝜃−𝛾𝑝𝛿_𝑟𝑟−𝑧_𝑟𝑇(𝑟),_𝜃𝜃=𝜎_𝜙𝜙=𝐶₁₂𝜀_𝑟𝑟+𝐶₂₂+𝐶₂₃𝜀_𝜃𝜃−𝛾𝑝𝛿_𝜃𝜃−𝑧_𝜃𝑇(𝑟),(2) where 𝑧_𝜃𝑧_𝑟=𝐶₁₁𝛼_𝑟+2𝐶₁₂𝛼_𝜃,=𝐶₁₂𝛼_𝑟𝛼+_𝑟𝐶₂₂+𝐶₂₃𝛼_𝜃,=𝛼_𝜃,(3) where 𝜎_𝑖𝑗 and 𝜀_𝑖𝑗(𝑖,𝑗=𝑟,𝜃) are the stress and strain tensors, 𝑇(𝑟) is the temperature distribution determined from the heat conduction equation, 𝛼_𝑖 is the coefficient of thermal expansion of effective stress, and 𝐶_𝑖𝑗, 𝛾, 𝑝, and 𝛿_𝑖𝑖(𝑖=𝑟,𝜃) are elastic constants, Biots coefficient of effective stress, poro pressure, and kronecker delta.

Equations in a porous material are 𝑝=𝑀𝑀=(𝜁𝑘_𝑢−𝛾∈),−𝑘𝛾²𝐵=𝑘_𝑢,−𝑘𝛾𝑘_𝑢𝑘_𝑢,=𝑘1+𝛾²𝑘_𝑓(1−𝛾)𝛾−𝜙_𝑝𝑘_𝑓+𝜙_𝑝𝑘_𝑓.(4) For fluid in undrained condition, 𝑝=−𝛾𝑀∈=−𝜁𝛾𝑀=0,𝜀_𝑟𝑟𝜀_𝜃𝜃+𝜀_𝜃𝜃+𝜀_𝜙𝜙,=𝜀_𝜙𝜙,(5) where 𝜁, ∈, 𝑀, 𝑘_𝑢, 𝑘, 𝑘_𝑓, 𝐵, and 𝜙_𝑝are variation of fluid content, volumetric strain, Biots modulus, undrained bulk modulus, drained bulk modulus, bulk modulus of fluid, compressibility coefficient, and porosity.

The equilibrium equation in the radial direction, disregarding the body force and the inertia term, is 𝜎_{𝑟𝑟,𝑟}+2𝑟𝜎^𝑟𝑟−𝜎_𝜃𝜃=0.(6) Substituting (5) into (2) leads to 𝜎𝜎_𝑟𝑟=𝑐^∗1𝜀_𝑟𝑟+𝑐^∗2𝜀_𝜃𝜃−𝑧_𝑟𝑇(𝑟),_𝜃𝜃=𝜎_𝜙𝜙=𝑐^∗3𝜀_𝑟𝑟+𝑐^∗4𝜀_𝜃𝜃−𝑧_𝜃𝑇(𝑟),(7) where 𝑐𝑐^∗1=𝐶₁₁+𝛾²𝑚,_∗2𝑐=_∗32𝐶₁₂+𝛾²𝑚,𝑐=𝐶₁₂+𝛾²𝑚,_∗4=𝐶₂₂+𝐶₂₃+2𝛾²𝑚.(8) The sphere’s material is assumed to be described with a power law function of the radial direction as 𝑧𝑐^∗𝑖_𝑖=𝑐^∗𝑖𝑟^𝑚,=𝑧_𝑖𝑟^2𝑚,(9) where 𝑚 is the power law index of the material.

Using relations (1)–(9), the Navier equation in term of the displacement is 𝑢_,𝑟𝑟+1𝑟(𝑚+2)𝑐^∗1+𝑐^∗2−2𝑐^∗3𝐶^∗1𝑢^,𝑟+1𝑟²(𝑚+1)𝑐^∗2−2𝑐^∗4=𝑐^∗1𝑧_𝑟𝑐^∗1(𝑚+1+2𝐿)𝑟^𝑚−2+2𝑐^∗2(𝑚+1−𝐿)^{𝑟2𝑚−1},(10) where 𝐿=𝑧_𝜃𝑧_𝑟.(11)

3. Heat Conduction Problem

The heat conduction equation in the steady-state condition for the one-dimensional problem in spherical coordinates and the thermal boundary conditions for a poro FGM hollow sphere are given, respectively, as 1𝑟²𝑐𝑟²𝑘(𝑟)𝑇^(𝑟)^=0,₁₁𝑐₂₁𝑇^(𝑎)+𝑐₁₂𝑇(𝑎)=𝑓₁,𝑇^(𝑏)+𝑐₂₂𝑇(𝑏)=𝑓₂,(12) where 𝑘=𝑘(𝑟) is the thermal conduction coefficient and 𝑐_𝑖𝑗 are either thermal conduction coefficient 𝑘, or convection coefficient ℎ, depending on the type of thermal boundary conditions. The terms 𝑓₁ and 𝑓₂ are known constants on the inside and outside radii. It is assumed that the thermal conduction coefficient 𝑘(𝑟) is a power function of 𝑟 as 𝑘=𝑘(𝑟)𝑟^𝑚.(13) Using (13), the heat conduction equation becomes 1𝑟²𝑟^𝑚+2𝑇^(𝑟)^=0.(14) Integrating (12) twice yields 𝑇(𝑟)=𝑐₁𝑟^−(𝑚+1)+𝑐₂.(15) Using the boundary conditions (16) and (17) to determine the constants 𝑐₁ and 𝑐₂ yields𝑐₁=𝑐²²𝑓₁−𝑐₁₂𝑓₂𝑐₁₂(𝑚+1)𝑐₂₁𝑏^−(𝑚+2)−𝑐₂₂𝑏^−(𝑚+1)−𝑐₂₂(𝑚+1)𝑐₁₁𝑎^−(𝑚+1)−𝑐₁₁𝑎^−(𝑚+1),(16)𝑐₂=𝑓¹(𝑚+1)𝑐₂₁𝑏^−(𝑚+2)𝑐₂₂𝑏^−(𝑚+1)−𝑓₂(𝑚+1)𝑐₁₁𝑎^−(𝑚+2)−𝑐₁₂𝑎^−(𝑚+1)𝑐₁₂(𝑚+1)𝑐²¹𝑏^−(𝑚+2)𝑐₂₂𝑏^−(𝑚+1)−𝑐²²(𝑚+1)𝑐₁₁𝑎^−(𝑚+2)−𝑐₁₂𝑎^−(𝑚+1).(17)

4. Solution of the Navier Equation

The Navier equation for the radial displacement 𝑢 was given in (10). Equation (10) is the nonhomogeneous Euler differential equation with the general and particular solutions. The general solution, 𝑢_𝑔, is obtained by assuming𝑢_𝑔(𝑟)=𝐵𝑟^𝜂.(18) Substituting (18) into the homogeneous form of (10) yields 𝜂²+(𝑚+1)𝑐^∗1+𝑐^∗2−2𝑐^∗3𝑐^∗1𝜂+(𝑚+1)𝑐^∗2−2𝑐^∗4𝑐^∗1=0,(19) where (19) has two real roots 𝜂₁ and 𝜂₂ as𝜂_1,2=−(𝑚+1)𝑐^∗1+𝑐^∗2+2𝑐^∗3±(𝑚+1)𝑐^∗1+𝑐^∗2+2𝑐^∗3²−4𝑐^∗1(𝑚+1)𝑐^∗2−2𝑐^∗42𝑐^∗1.(20)Thus, the general solution is 𝑢_𝑔(𝑟)=𝐵₁𝑟^𝜂1+𝐵₂𝑟^𝜂2.(21) The particular solution 𝑢_𝑝(𝑟)is assumed to have the form 𝑢_𝑝(𝑟)=𝐷₁𝑟^𝑚+𝐷₂𝑟^2𝑚+1.(22)

Substituting (22) in (10) yields 𝑚(𝑚−1)+(𝑚+2)𝑐^∗1+𝑐^∗2−2𝑐^∗3+(𝑚+1)𝑐^∗1𝑚𝑐^∗2+2𝑐^∗4𝑐^∗1𝐷¹𝑟^𝑚−2+(2𝑚+1)(2𝑚)+(𝑚+2)𝑐^∗1+𝑐^∗2+2𝑐^∗3×(2𝑚+1)+(𝑚+1)𝑐^∗1𝑐^∗2+2𝑐^∗4𝑐^∗1𝐷²𝑟^2𝑚−1=𝑧_𝑟𝑐^∗1(𝑚+1−2𝐿)𝑟^𝑚−2+2𝑐^∗2(𝑚+1−𝐿)𝑟^2𝑚−1.(23) Equating the coefficients of the identical powers yields𝐷₁=𝑧_𝑟𝑐^∗1(𝑚+1)−2𝐿𝑚(𝑚−1)𝑐^∗1+(𝑚+2)𝑐^∗1+𝑐^∗2+2𝑐^∗3𝑚+(𝑚+1)𝑐^∗2+2𝑐^∗4,(24)𝐷₂=𝑧_𝑟2𝑐^∗2(𝑚+1−𝐿)(2𝑚+1)(2𝑚)𝑐^∗1+(𝑚+2)𝑐^∗1+𝑐^∗2+2𝑐^∗3(2𝑚+1)+(𝑚+1)𝑐^∗2+2𝑐^∗4.(25) The complete solution for 𝑢(𝑟) is the sum of the general and particular solutions as 𝑢(𝑟)=𝑢_𝑔(𝑟)+𝑢_𝑝(𝑟).(26) Thus, 𝑢(𝑟)=𝐵₁𝑟^𝜂1+𝐵₂𝑟^𝜂2+𝐷₁𝑟^𝑚+𝐷₂𝑟^2𝑚+1.(27) Substituting (27) into (1) and (2), the strains and stresses are obtained as 𝜀𝜀_{𝑟𝑟𝜃𝜃}=𝐵₁𝜂₁𝑟^𝜂1−1+𝐵₂𝜂₂𝑟^𝜂2−1+𝐷₁𝑚𝑟^𝑚−1+𝐷₂(2𝑚+1)𝑟^2𝑚,𝜎_𝑟𝑟=𝐵₁𝑟^𝜂1−1+𝐵₂𝑟^𝜂2−1+𝐷₁𝑟^𝑚−1+𝐷₂𝑟^2𝑚,=𝑐^∗1𝐵₁𝜂₁+ℎ𝑟^𝜂1−1+𝐵₂𝜂₂+ℎ𝑟^𝜂2−1+𝐷₁(𝑚+ℎ)𝑟^𝑚−1+𝐷₂(2𝑚+1+ℎ)𝑟^2𝑚𝜎−𝑆𝑐^∗1𝑟^−(𝑚+1)+𝑐^∗2_𝜃𝜃,=𝑐^∗1𝐵₁𝜂₁+𝑓𝑟^𝜂1−1+𝐵₂𝜂₂+𝑓𝑟^𝜂2−1+𝐷₁(𝑚+𝑓)𝑟^𝑚−1+𝐷₂(2𝑚+1+𝑓)𝑟^2𝑚−𝑔𝑐^∗1𝑟^−(𝑚+1)+𝑐^∗2,(28) where ℎ=𝑐^∗2𝑐^∗1𝑆=,𝑧_𝑟𝑐^∗1𝑓=,𝑐^∗4𝑐^∗3𝑔=,𝑧_𝜃𝑐^∗3.(29) To determine the constants 𝐵₁ and 𝐵₂, the boundary conditions for stresses may be used. Consider the mechanical boundary conditions on the inside and outside radii as 𝜎_𝑟𝑟(𝑎)=−𝑃_𝑎,𝜎_𝑟𝑟(𝑏)=−𝑃_𝑏.(30)

5. Numerical Results and Discussion

The analytical solution obtained in the previous section may be checked for a one of example. Then, consider a thick hollow sphere of inner radius 𝑎 = 1 m, and the outer radius 𝑏 = 1.2 m of poro FGM material with properties are given in Table 1. The boundary conditions for temperature are taken as 𝑇(𝑎)=10^∘C and 𝑇(𝑏)=0^∘C. The hollow sphere may be assumed to be under internal pressure of 50 MPa and zero external pressure (𝜎_𝑟𝑟(𝑎) = −50 MPa and𝜎_𝑟𝑟(𝑏) = 0 MPa). For different values of 𝑚, temperature profile, radial displacement, radial stresses, and hoop stresses along the radial direction are plotted in Figures 1–4. Figure 1 shows that as the power law increases, the temperature decreases. Figure 2 shows that for higher values of 𝑚, radial displacement decreases. Figure 3 represents the radial stress along the radial direction and decreases as the power law index increases. The circumferential stress versus the radial direction is shown in Figure 4. It is seen that for 𝑚<1, the circumferential stress decreases along the radial direction. When 𝑚>1, the situation is reversed, and the circumferential stress increases along the radial direction. The curve associated with 𝑚=1 shows that the variation of circumferential stress along the radial direction is minor and is almost uniform across the radius. Effective porosity and compressibility on mechanical radial displacement in Figures 5 and 6, radial stress in Figures 7 and 8 and hoop stress in Figures 9 and 10 for 𝑚=1 are shown. As noted from Figures 5–10, distribution that with increasing porosity increases radial displacement, and all mechanical stresses decrease. Also, with increasing compressibility, radial displacement decreases, and all mechanical stresses increases.

Table 1 

Matrial constants.

Figure 1 

Radial temperature distribution.

Figure 2 

Radial displacement distribution.

Figure 3 

Radial stress distribution.

Figure 4 

Circumferential stress distribution.

Figure 5 

Effective porosity on mechanical radial displacement for 𝑚=1.

Figure 6 

Effective compressibility on mechanical radial displacement for 𝑚=1.

Figure 7 

Effective porosity on radial stress for 𝑚=1.

Figure 8 

Effective compressibility on radial stress for 𝑚=1.

Figure 9 

Effective porosity on circumferential stress for 𝑚=1.

Figure 10 

Effective compressibility on circumferential stress for 𝑚=1.