# Mechanical and Thermal Stresses in a FGPM Hollow Cylinder due to Radially Symmetric Loads

Authors: M. Jabbari; M. Meshkini; M. R. Eslami
Journal: ISRN Mechanical Engineering
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

Poroelasticity is a theory that models the interaction of deformation and fluid flow in a fluid-saturated porous medium. The deformation of the medium influences the flow of the fluid and vice versa. The theory was proposed by Biot [1, 2] as a theoretical extension of soil consolidation models developed to calculate the settlement of structures placed on fluid-saturated porous soils. The historical development of the theory is sketched by De Boer [3]. The theory has been widely applied to geotechnical problems beyond soil consolidation, most notably problems in rock mechanics. Detournay and Cheng [4] survey both these methods with special attention to rock mechanics. These include familiar analytical methods (displacement potentials, method of singularities) and computational methods (finite element and boundary element). Sandhu and Wilson [5] are acknowledged for pioneering the application of finite element techniques to poroelasticity. Detournay and Cheng [6] presented fundamentals of poroelasticity.Abousleiman and Ekbote [7] presented the analytical solutions for inclined hollow cylinder in a transversely isotropic material subjected to thermal and stress perturbations, and they systematically evaluated the effect of the anisotropy of the poromechanical material parameters as well as thermal material properties on stress and porous pressure distributions. Chen [8] presented and analyzed the problems of linear thermo elasticity in a transversely isotropic hollow cylinder of finite length by a direct power series approximation through the application of the Lanczos-Chebyshev  method. Bai [9] presented then derived an analytical method solving the responses of a saturated porous media subjected to cyclic thermal loading by the Laplace transform and the Gauss-Lengender method of Laplace transform inversion. Wang and Papamichos [10, 11] presented analytical solution for the temperature, pore pressure, and stresses around a cylindrical well bore and a spherical cavity subjected to a constant fluid flow rate by coupling the conductive heat transfer with the pore-fluid flow.  Ghassemi and Tao [12] presented influence of coupled chemo-poro-thermoelastic processes on pore pressure and stress distributions around a wellbore in swelling shale. Wirth and sobey [13] presented an axisymmetric and fully 3-D poroelastic model forth evolution of hydrocephalus. Yang and Zhang [14] presented poroelastic wave equation including the Biot/squirt mechanism and the solid/fluid coupling anisotropy. Arora and Tomar [15] presented the elastic waves along a cylindrical borehole in a poroelastic medium saturated by two immiscible fluids. Hamiel et al. [16] presented the coupled evolution of damage and porosity in poroelastic media theory and applications to the deformation of porous rocks. Ghassemi [17] presented stress and pore prepressure distribution around a pressurized, cooled crack in hollow permeability rock. Youssef [18] theory of generalized porothermoelasticity was presented. Jourine et al. [19] presented modeling poroelastic hollow cylinder experiments with realistic boundary conditions.Functionally graded materials (FGMs) are heterogeneous materials in which the elastic and thermal properties change from 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 high strength, ceramic-Metal FGM may work at super high-temperature or under high-temperature difference field. In effect, the governing equation for the temperature and stress distributions are coordinate dependent as the material properties are functions of position. Classical method of analysis is to combine the equilibrium equations with the stress-strain and strain-displacement relations to arrive at the governing equation in terms of the displacement components called the Navier equation. There are some analytical thermal and stress calculations for functionally graded material in the one-dimensional case for thick cylinders and spheres [20, 21]. The authors have considered the nonhomogeneous material properties as liner function of r. Jabbari et al. [22] studied a general solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to nonaxisymmetric steady-state load. They applied separation of variables and complex Fourier series to solve the heat conduction 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 wave propagation analyses of thick temperature-dependent FGM cylinders, using a second-order point-collocation method. Lü et al. [25] presented elastic mechanical behavior of nanoscaled FGM films incorporating surface energies. Afsar and Sekine [26] presented inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media. Zhang and Zhou [27] presented a theoretical analysis of FGM thin plates based on physical neutral surface. Fazelzadeh and Hosseini [28] presented aerothermoelastic behavior of supersonic rotating thin-walled beams made of functionally graded materials. Ootao and Tanigawa [29] presented the transient thermoelastic problem of functionally graded thick strip due to nonuniform heat supply. They obtained the exact solution for the two-dimensional temperature change in a transient state, and thermal stress of a simple supported strip under the state of plane strain. Jabbari et al. [30] presented and studied the mechanical and thermal stresses in functionally graded hollow cylinder due to radially symmetric loads. They assumed the temperature distribution to be a function of radial direction. They applied a method to solve the heat conduction and Navier equations. Farid et al. [31] presented three-dimensional temperature dependent-free vibration analysis of functionally graded material curved panels resting on two-parameter elastic foundation using a hybrid semianalytic, differential quadrature method. Bagri and Eslami [32] presented Generalized coupled thermoelasticity of functionally graded annular disk considering the Lord-Shulman theory. Shariat and Eslami [33] presented buckling of thick functionally graded plates under mechanical and thermal loads. Jabbari et al. [34] studied an axisymmetric mechanical and thermal stresses in thick short length functionally graded material cylinder. They applied the separation of variables and complex Fourier series to solve the heat conduction and Navier equation. Thieme et al. [35] presented titanium powder sintering for preparation of a porous FGM destined as a skeletal replacement implant.In this work, a direct method of solution of the Navier equations presented which does not have limitation of the potential function method as to handle the general type of mechanical and thermal under one-dimensional  steady-state temperature distribution  with general type of thermal and mechanical boundary conditions is considered. The functionally graded porous material properties of the cylinder are assumed to be expressed by power functions inr. The Naviear equation terms of displacements are derived and solved analytically by the direct method, so any boundary conditions for stresses and displacements can be satisfied.


## 2. Heat Conduction Problem

Consider a hollow circular cylinder of inner radiusa, outer radius b made of functionally graded porous material (FGPM) respectively. Axisymmetric cylindrical coordinates (r) are considered along the radial direction. The cylinder’s material graded through the r direction, thus the material properties are porous and functions of r. The first law of thermodynamics for energy equation in the steady-state condition for the FGPM on dimensional cylinder is:(1)1r∂∂r[rk(r)(∂T∂r)]=0⟶∂2T∂r2+(k/(r)k(r)+1r)∂T∂r=0,a≤r≤b,
where T(r) is temperature distribution, k(r) is the thermal conduction coefficient, and symbol (/) denotes derivative with respect to r.The thermal boundary is assumed as(2)S11T(a)+S12T,r(a)=f1,S21T(b)+S22T,r(b)=f2,
where (,) denotes partial derivative, and Sij are the constant thermal parameters related to conduction and convection coefficients. We assume that nonhomogeneous thermal conduction coefficient k(r) is power function of r as k(r)=k0rm3, where k0 and m3 material parameter. Using the definition for the material properties, the temperature equation becomes(3)∂2T∂r2+(m3+1)1r∂T∂r=0.
Integrating (4) twice yields(4)β2+(m3+1)β=0⟶{β1=-(m3+1)β2=0⟶β=-(m3+1),(5)T(r)=aβ+1rβ+1+b⟶T(r)=-am3r-m3+b.
Using the boundary conditions (2) to determine the constants a and b yields                                                       (6)A1=e4f1-e2f2e1e4-e2e3,A2=e1f2-e3f1e1e4-e2e3,
where constants e1 to e4 are given in Appendix A.


## 3. Stress Analysis

Letu displacement components in the radial direction. Then strain-displacement relations are(7)εrr=∂u∂r,εθθ=ur,
and stress-strain relations of a functionally graded porous cylinder for nonaxisymmetric condition are(8)σrr=C11εrr+C12εθθ-γpδrr-Z1T(r),σθθ=C22εrr+C12εθθ-γpδθθ-Z2T(r),C11+Mγ2=C*11,C22+Mγ2=C*22,C12+Mγ2=C*12,
where σij, ∈ij (i,j=r,θ), M, γ, α, λ, μ, and p are  stress tensors, strain tensors, Biot’s modulus, Biot’s coefficient of effective stress, thermal expansion coefficient, lame’s coefficient, and the pore pressure, respectively, p related to the Biot’s modulus, volumetric  strain and the variation of fluid content.We assume that pore-cylinder  if undrained condition then (ζ=0) as:(9)p=M(ζ-γ(εrr+εθθ))=-Mγ(εrr+εθθ),
where:(10)M=2μ(νu-ν)γ2(1-2ν)(1-2νu).
Thus,(11)σrr=C*11εrr+C*12εθθ-Z1T(r),σθθ=C*22εθθ+C*21εrr-Z2T(r).
The equilibrium equation in the radial direction, disregarding body force and the inertia terms, is (12)∂σrr∂r+1r(σrr-σθθ)=0,Z1=C*11αr+2C*12αθ,Z2=2C*21αr+C*22αθ.
To obtain the equilibrium equations in terms of the displacement components for the FGPM cylinder, the functional relationship and pore of the material properties must be known. Because the cylinder material is assumed to be graded along the r-direction, the modulus of elasticity and coefficient of thermal expansion are material constantassumed to be described with the power laws as(13)αr=α01rm1,αθ=α02rm2,K=k0rm3,Cij=C̅ijrm4,
where the coefficients are described as(14)α01=α01′am1,α02=α02′am2,K=k0′am3,C̅ij=C̅ij′am4,
and a is the inner radius.Using the relations (7) to (14), the Navier equations in terms of the displacement components are (15)(∂2u∂r2+(m4+1)1r(∂u∂r)+((m4+1)C̅12-C̅22C̅11)1r2u)×rm4=({[(m1+m4-1)C̅11-2C̅12C̅11]×rm1-1α01+[2(m2+m4+1)C̅12+C̅22C̅11]×rm2-1α02}T(r,θ)+[C̅11rm1α01+2C̅12rm2α02C̅11]dTdr)×rm4.
The Navier equation (15) is nonhomogeneous system of partial 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 general and particular solutions.The general solution is assumed to have the form(16)ug(r)=Brη.
Substituting (16) into (15) yields
(17)[η(η-1)+(m4+1)η+1C̅11[(m4+1)C̅12-C̅22]]=0.
Equation (17) has two roots η1 to η2. Thus, the general solutions are
(18)η1,2=-m42±(m424-(m4+1)C̅12-C̅22C̅11)1/2.
Thus, the general solution is(19)ug(r)=D1rη1+D2rη2.
The particular solutions up(r) are assumed as (20)up(r)=(I1+I2)rβ+m2.
Substituting (20) into (18) yields(21)d1I1rβ+m2-1+d2I2rm2-1=d3rβ+m2-1+d4rm2-1.
The complete details for solution of (21) is presented in Appendix B.The complete solutions foru(r) is sum of the general and particular solutions and are
(22)u(r)=ug(r)+up(r).
Thus(23)u(r)=D1rη1+D2rη2+(I1+I2)rβ+m2.
Substituting (23) into (1) and (2), the strains and stresses are obtained as(24)εrr=η1D1rη1-1+η2D2rη2-1+(β+m2)(I1+I2)rβ+m2-1,εθθ=D1rη1-1+D2η2rη2-1+(I1+I2)rβ+m2,σrr=(C̅11[(rβ+m4+m2-1)η1D1rη1+m4-1+η2D2rη2+m4-1+(β+m2)(I1+I2)rβ+m2+m4-1+α01[A1+A2](rβ+m4+m2-1)]+C̅12[D1rη1+m4-1+D2η2rη2+m4-1+(I1+I2)rβ+m4+m2+2α02[A1+A2](rβ+m4+m2-1)])einθ,σθθ=(C̅22[(rβ+m4+m2)D1rη1+m4-1+D2η2rη2+m4-1+(I1+I2)rβ+m4+m2+α01[A1+A2](rβ+m4+m2)]+C̅21[(rβ+m4+m2)η1D1rη1+m4-1+η2D2rη2+m4-1+(β+m2)(I1+I2)rβ+m2+m4-1+2α02[A1+A2](rβ+m4+m2)])einθ.To determine the constantsD1 and D2, consider the boundary conditions for stresses given by
(25)σrr(a1)=-pi,σrr(a2)=-p0.


## 5. Numerical Results and Discussion

Consider a thick hollow cylinder of inner radiusa = 1 (m) and outer radius b = 1.2 (cm), shown properties are given in Table 1. For simplicity of analysis, we consider that the power law of material properties is the same as m1=m2=m3=m. To examine the proposed solution method, two example problems are considered. The example problem may have some physical interpretation.Table 1ParametersC11C12C22γE0ννuα01,α02Numerical139 GPa78 GPa139 GPa0.47200 GPa0.2 GPa0.3 GPa0.000012(1/C°)As the example, consider a thick hollow cylinder where the inside boundary is traction free with given temperature distribution of Table2. The outside boundary is assumed to be radially fixed with zero temperature. Therefore, the assumed boundary conditions yield of Table 2.Table 2ParametersT(a)T(b)σrr(a)σrr(b)mNumerical50°C0−500−1, 0, 1Figure1 shows the variations of the temperature along the radial direction for different values of the power law index. The figure shows that as the power law index m increases, the temperature decreased.Figure 1
Temperature distribution in the cross-section of cylindrical.Figure2 shows the plot of the radial displacement along the radius. The magnitude of the radial displacement is decreased as the power index m is increased.Figure 2
Radial displacement in the cross-section of cylindrical.The radial and circumferential stresses are plotted along the radial direction and shown in Figures3 and 4, and the magnitude of the radial stress is increased as m is increased. The hoop stress along the radius decreases for m,1 (similar to thick cylinders made of isotropic materials), due to the acting internal pressure and zero external pressure. For m<1, the hoop stress increases as the radius increases, since the modulus of elasticity is an increasing function of the radius. Physically, this means that the outer layers of the cylinder are biased to maintain the stress due to their higher stiffness. There is a limiting value for m, where the hoop stress remains almost a constant along the radius. For low values of the ratio b/a (Figures 7 and 8). Figures 5 and 6 show the radial and hoop thermal stresses in the cross-section of the cylinder, respectively, where the pore compressibility coefficient (B) is changed, the other parameters are fixed. Figures 5 and 6 show these stresses based on the pore volume fraction; (ϕ) is pore volume per total volume.Figure 3
Radial distribution of radial stress.Figure 4
Radial distribution of hoop stress.Figure 5
Radial thermal stress in the cross-section of cylindrical based on the pore volume fraction (B) changing.Figure 6
Radial thermal stress in the cross-section of cylindrical based on the pore volume fraction (ϕ) changing.Figure 7
Hoop thermal stress in the cross section of cylindrical based on the compressibility coefficient (B) changing.Figure 8
Hoop thermal stress in the cross section of cylindrical based on the pore volume fraction (ϕ) changing.Figure9 shows the radial displacements in the cross-section of the cylinder based on the pore compressibility coefficient (B) changing. Figure 10 also shows these displacements based on the pore volume fraction (ϕ) changing.Figure 9
Radial displacement in the cross-section of cylindrical based on the  compressibility coefficient (B) changing.Figure 10
Radial displacement in the cross-section of cylindrical based on the compressibility coefficient (ϕ) changing.


## 6. Conclusions

In the present work, an attempt has been made to study the problem of general solution for the thermal and mechanical stresses in a thick FGPM hollow cylinder due to the one-dimensional axisymmetric steady-state loads. The method of solution is based on the direct method and uses power series, rather than the potential function method. The advantage of this method is its mathematical power to handle both simple and complicated mathematical function for the thermal and mechanical stresses boundary conditions. The potential function method is capable of handling complicated mathematical functions as boundary condition. The proposed method does not have the mathematical limitations to handle the general types of boundary conditions which are usually countered in the potential function method.



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