# Transient Thermoelastic Analysis of Pressurized Rotating Disks Subjected to Arbitrary Boundary and Initial Conditions

Authors: Mohammad Zamani Nejad; Azam Afshin
Journal: Chinese Journal of Engineering
 (2014)

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## Abstract

This paper focuses on exact analytical solution of transient thermoelastic behaviors of rotating pressurized disks subjected to arbitrary boundary and initial conditions. The pressure, inner radius, and outer radius are considered constant. The basic thermoelasticity theory under generalized assumptions is used to solve the thermoelastic problem. Using the method of the separation of variables, the relations of temperature and transient thermal stresses in the radial direction are obtained. In the case study, the disk is considered under heat flux. Some useful discussions and numerical examples are presented. The analytical results were compared with those of the finite element method and good agreement was found. The relations obtained in this paper can be applied to any arbitrary boundary and initial conditions.


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## Body

## 1. Introduction

The rotating disks subjected to mechanical and transient thermal loads are widely used in many engineering fields such as aerospace, mechanical, naval, nuclear energy, chemical plant, electronics, and biomaterials. Therefore, much research has been conducted in this field.Cheung et al. [1] investigated the problem of transient thermal stresses in a solid elastic homogeneous and isotropic sphere for uniform and nonuniform local surface heating. Sugano [2] analyzed the problem of transient thermal stresses in a homogeneous transversely isotropic finite cylinder due to an arbitrary internal heat generation. A transient response of one-dimensional axisymmetric quasistatic-coupled thermoelastic problems was studied by Yang et al. [3]. Transient thermal stresses in cylindrically orthotropic tubes were obtained by Kardomateas [4]. The transient thermal stresses in a homogeneous cylindrically orthotropic hollow cylinder due to a constant temperature were imposed on one surface and heat convection into a medium at the other surface was studied by Kardomateas [5]. Ashida et al. [6] presented a general solution technique for transient thermoelasticity of transversely isotropic solids in cylindrical coordinates. For quasistatic problems, and without considering the effect of inertial term, Yee and Moon [7] obtained the closed-form solutions of the orthotropic hollow cylinder subjected to an arbitrary axisymmetric temperature distribution. Assuming that the material properties vary nonlinearly in the radial direction and that Poisson’s ratio is constant, Zamani Nejad and Rahimi [8] obtained closed-form solutions for one-dimensional steady-state thermal stresses in a rotating functionally graded pressurized thick-walled hollow circular cylinder. In another study, they obtained a complete and consistent three-dimensional set of field equations by tensor analysis to characterize the behavior of FGM thick shells of revolution with arbitrary curvature and variable thickness along the meridional direction [9]. In this work, transient thermoelastic analysis of homogeneous and isotropic rotating disks under general boundary conditions was obtained.


## 2. Temperature Analysis

Consider a homogeneous isotropic hollow disk of inner and outer radiiri and ro, respectively. One-dimensional transient heat conduction equation of disk, in the absence of heat source, for axisymmetric case is
(1)1α∂T∂t=∂2T∂r2+1r∂T∂r,α=kρc,
where T(r,t) is the temperature distribution, α is thermal diffusivity, and k, ρ, c are the thermal conductivity, mass density, and specific heat capacity, respectively.The initial and boundary conditions are as follows:(2)B.C:{C11T(ri,t)+C12∂T∂r|ri=g1,C21T(ro,t)+C22∂T∂r|ro=g2,I.C:T(r,0)=Ti(r),
where Cij  (i,j=1,2) and gi  (i=1,2) are the constants dependent on the thermal boundary condition, and Ti(r) is the given initial condition.The solution of (1) may be obtained by the method of separation of variables, generalized Bessel function, and eigenfunction method as
(3)T(r,t)=Th(r,t)+Ts(r),(4)1r∂∂r(r∂Ts∂r)=0,(5)1α∂Th∂t=∂2Th∂r2+1r∂Th∂r.
The boundary conditions are defined as follows:
(6)C11Ts(ri,t)+C12∂Ts∂r|ri=g1,C21Ts(ro,t)+C22∂Ts∂r|ro=g2.
Integrating (4) twice yields
(7)Ts=C1ln⁡(r)+C2.
Using the boundary conditions (see (6)) to determine the constants C1 and C2 yields
(8)C1=C11g2-C21g1C21C11ln⁡(ro/ri)-(C21C12/ri)+(C22/ro),C2=1C11(g1-(C11g2-C21g1)(C11ln⁡ri+(C12/ri))C21C11ln⁡(ro/ri)-(C21C12/ri)+(C22/ro))
or
(9)C2=1C21(g2-(C11g2-C21g1)(C21ln⁡ro+(C22/ro))C21C11ln⁡(ro/ri)-(C21C12/ri)+(C22/ro)).
For Th(r,t), boundary and initial conditions are as follows:
(10)B.C:{C11Th(ri,t)+C12∂Th∂r|ri=0,C21Th(ro,t)+C22∂Th∂r|ro=0,I.C:Th(r,0)=Ti(r)-Ts(r)=Ti*(r).
Thus, the solution for Th(r,t) is expressed as
(11)Th(r,t)=∑n=1∞1∥f(r,βn)∥2×(+C1λn2((f(ro,βn)-f(ri,βn)))(roβn2(C1ln(ro)+C2)∂f(r,βn)∂r|r=ro)]∫riroTi(r)f(r,βn)rdr+[(roβn2(C1ln⁡(ro)+C2)∂f(r,βn)∂r|r=ro)-(riβn2(C1ln⁡(ri)+C2)∂f(r,βn)∂r|r=ri)+C1λn2((f(ro,βn)-f(ri,βn)))(roβn2(C1ln(ro)+C2)∂f(r,βn)∂r|r=ro)])×f(r,βn)e-αtβn2,
where
(12)∂f(r,βn)∂r=-βn(AJ1(βnr)+BY1(βnr)).
Here βn are eigenvalue and are positive roots of the following equation:
(13)(C11Y0(βnri)-βnC12Y1(βnri))×(C21J0(βnro)-βnC22J1(βnro))-(C21Y0(βnro)-βnC22Y1(βnro))×(C11J0(βnri)-βnC12J1(βnri))=0.
Here J0 and Y0 are Bessel functions of the first and the second kinds.Eigenfunctionf(r,βn) is as follows:
(14)f(r,βn)=AJ0(βnr)+BY0(βnr).A and B parameters are constant and are defined as
(15)A=C11Y0(βnri)-C12βnY1(βnri),B=-C11J0(βnri)+C12βnJ1(βnri)
or
(16)A=C21Y0(βnro)-C22βnY1(βnro),B=-C21J0(βnro)+C22βnJ1(βnro).∥f(r,βn)∥2 is the norm of eigenfunction and are obtained as
(17)∥f(r,βn)∥2=∫rirorf(r,βn)2dr=ro22((AJ0(βnro)+BY0(βnro))2+(AJ1(βnro)+BY1(βnro))2)-ri22((AJ0(βnri)+BY0(βnri))2+(AJ1(βnri)+BY1(βnri))2).


## 3. Transient Thermoelastic Analysis

Consider thick hollow disk in the previous section at a constant angular velocityω about its central axis. In the cylindrical coordinate system (r,θ,z), for the axisymmetric problem, radial and circumferential strains (εrr,εθθ) are as follows:
(18)εθθ=urr,εrr=∂ur∂r.
Here, radial displacement ur is the only nonzero component of the displacement and is only the function of radial coordinate r.For homogenous and isotropic disk, the linear thermoelastic constitutive relations can be expressed as follows:(19)σrr=E(1-ν2)(εrr+νεθθ-(1+ν)αT),(20)σθθ=E(1-ν2)(εθθ+νεrr-(1+ν)αT),(21)T=T(r,t)-T0,
where E is modulus of elasticity, α is the coefficient of linear thermal expansion, ν is Poisson’s ratio and σrr and σθθ are components of stresses in radial and hoop directions, respectively, and T0 is the reference temperature. Although, in this paper the reference temperature is taken as zero, the analysis would be valid for any nonzero value.The equilibrium equation of the rotating hollow disk, in the absence of body forces, is expressed as(22)∂σrr∂r+σrr-σθθr=-ρrω2,
where ω is angular velocity.Substituting (18) into (19)-(20) and utilizing (22) yield
(23)∂∂r(1r∂(urr)∂r)=(1+ν)α∂T∂r-(1-ν2)Eρrω2.
Integrating (23) twice yields
(24)ur=Ar+Br+αr(1+ν)∫rirTrdr-(1-ν2)8Eρr3ω2.
Substituting (24) into (19)-(20), the radial and circumferential stresses of the rotating thick hollow disk are obtained as
(25)σrr=C1*-C2*r2-αEr2∫rirTrdr-3+ν8ρr2ω2,(26)σθθ=C1*+C2*r2+αE(1r2∫rirTrdr-T)-1+3ν8ρr2ω2,
where
(27)C1*=AE1-ν,C2*=BE1+ν.
The mechanical boundary conditions are expressed as
(28)σrr(r=ri)=-Pi,σrr(r=ro)=-Po.
Substituting boundary conditions into (25) gives
(29)C1*=1ro2-ri2(Piri2-Poro2+αE∫riroTrdr∫riroTrdr+3+ν8ρω2(ro4-ri4))∫riroTrdr+3+ν8ρω2(ro4-ri4)),C2*=ri2ro2ro2-ri2(Pi-Po+αEro2∫riroTrdr∫riroTrdr+3+ν8ρω2(ro4-ri4))∫riroTrdr+3+ν8ρω2(ro2-ri2)).
Therefore,
(30)σrr=-Pi-1-(ri/r)21-(ri/ro)2(Po-Pi)-3+ν8ρω2(r2-ri2)(1-(ror)2)+αEr2(r2-ri2ro2-ri2∫riroTrdr-∫rirTrdr),(31)σθθ=-Pi-1+(ri/r)21-(ri/ro)2(Po-Pi)-1+3ν8ρω2(r2-1+3ν3+ν(ri2+ro2+ro2ri2r2))+αEr2(∫rirTrdr+r2+ri2ro2-ri2∫riroTrdr-r2T),
where
(32)∫rirT(r,t)rdr=C12(ri2ln⁡(ri)-r2-ri22r2ln⁡(r)-ri2ln⁡(ri)-r2-ri22)+C22(r2-ri2)+∑n=1∞e-αtβn2∥f(r,βn)∥2×(∫riroTi(r)f(r,βn)rdr(roβn2(C1ln⁡(ro)+C2)∂f(r,βn)∂r|r=ro)+[(roβn2(C1ln⁡(ro)+C2)∂f(r,βn)∂r|r=ro)-(riβn2(C1ln⁡(ri)+C2)∂f(r,βn)∂r|r=ri)(roβn2(C1ln⁡(ro)+C2)∂f(r,βn)∂r|r=ro)+C1βn2(f(ro,βn)-f(ri,βn))])×(riβn2∂f(r,βn)∂r|r=ri-rβn2∂f(r,βn)∂r).


## 4. Results and Discussion

Consider a disk withri=0.4 m and ro=0.6 m. The modulus of elasticity and Poisson’s ratio, respectively, have values of E=200 GPa and ν=0.3. The applied internal pressure is 70 MPa. The mass density is ρ=7854 Kg/m3. The thermal diffusivity, thermal conductivity, and specific heat capacity are α=1.17(10-5)1/°C,  k=60.5 W/m·K, and c=434 J/Kg·K, respectively. In the following, the analytical solution described in the preceding section may be checked at two examples.Example 1.
Assume that the inner surface of disk is under heat flux and the outer surface of disc is under convection. The initial condition is a linear function in terms of the radius. Thus, thermal boundary condition is expressed as follows:(33)B.C:{k∂T∂r|r=ri=q=500W/m2-k∂T∂r|r=ro=h(T∞-T(ro,t))=25(26.85-T(ro,t))I.C:T(r,0)=Ti(r)=20r.
Temperature distribution for this thermal boundary condition is
(34)T(r,t)=T∞+qrik(ln⁡rro-kroh)+π∑n=1∞q+khT∞βnGnλnk(1-Gn2(h2+k2βn2))f(r,βn)e-αtβn2-π22∑n=1∞βn2f(r,βn)e-αtβn21-Gn2(h2+k2βn2)∫riroTi(η)f(η,βn)ηdη,
where
(35)Gn=Y1(βnri)(hY0(βnro)-kβnY1(βnro))=J1(βnri)(hJ0(βnro)-kβnJ1(βnro)).Figure1 indicates temperature distribution for the course of 20 seconds for r=0.45 m. Figure 2 illustrates temperature distribution in the radial direction at t=3 sec.Figure 1
Temperature distribution (r=0.45 m).Figure 2
Temperature distribution in the radial direction (t=3 sec).Figure3 shows the comparison of the temperature distribution for different radii. As can be seen, temperature increases as radius increases. Figure 4 compares the temperature distribution for different times at point r=0.45 m. When t→∞, the variation of temperature versus dimensionless radius becomes linear.Figure 3
Comparison of the temperature distribution for different radius.Figure 4
Comparison of the temperature distribution for different times.Figures5, 6, and 7 show the distribution of radial displacement, radial stress, and hoop stress versus radial direction. Meanwhile, this result is compared with FEM.Figure 5
Radial displacement distribution (t=3 sec).Figure 6
Radial stress distribution (t=3 sec).Figure 7
Hoop stress distribution (t=3 sec).Figures8, 9, and 10 show the distribution of radial displacement, radial stress, and hoop stress versus time.Figure 8
Radial displacement distribution (r=0.45 m).Figure 9
Radial stress distribution (r=0.45 m).Figure 10
Hoop stress distribution (r=0.45 m).Figures11, 12, 13, 14, 15, and 16 show comparison of the displacement and stresses distribution for different time and different radius. Figures 14 and 16 show that variation of radial and hoop stresses into variation of time is very small. Figures 17, 18, 19, 20, 21, and 22 show comparison of the displacement and stresses distribution for different angular velocity. (In these figures, ω is angular velocity and its dimension is radian per second (rad/sec).) In Figure 20 it is seen that angular velocity increases as radial stress decreases and if angular velocity increases more, radial stress changes from pressure to tension.Figure 11
Comparison of the radial displacement distribution for different radius.Figure 12
Comparison of the radial displacement distribution for different times.Figure 13
Comparison of the radial stress distribution for different radius.Figure 14
Comparison of the radial stress distribution for different times.Figure 15
Comparison of the hoop stress distribution for different radius.Figure 16
Comparison of the hoop stress distribution for different times.Figure 17
Comparison of the radial displacement distribution for different angular velocity (r=0.45 m).Figure 18
Comparison of the radial displacement distribution for different angular velocity (t=3 sec).Figure 19
Comparison of the radial stress distribution for different angular velocity (r=0.45 m).Figure 20
Comparison of the radial stress distribution for different angular velocity (t=3 sec).Figure 21
Comparison of the hoop stress distribution for different angular velocity (r=0.45 m).Figure 22
Comparison of the hoop stress distribution for different angular velocity (t=3 sec).Example 2.
Thermal boundary and initial conditions for the second example are(36)B.C:{T(ri,t)=T1=6°C,-k∂T∂r|r=ro=q=-800W/m2,I.C:T(r,0)=Ti(r)=20r.
Temperature distribution for this thermal boundary condition is
(37)T(r,t)=T1-qrokln⁡rri-π∑n=1∞(kβnT1J1(βnro)-qJ0(βnri))kβn(J12(βnro)-J02(βnri))×J0(βnri)f(r,βn)e-αtβn2-π22∑n=1∞βn2J02(βnri)J12(βnro)-J02(βnri)×f(r,βn)∫riroTi(η)f(η,βn)ηdηe-αtβn2.Figure23 indicates temperature distribution for the course of 20 seconds and Figure 24 illustrates temperature distribution in thickness; meanwhile the result of this state is compared with FEM.Figure 23
Temperature distribution (r=0.45 m).Figure 24
Temperature distribution in the radial direction (t=3 sec).Figure25 shows comparison of the temperature distribution for different radius. As can be seen, temperature increases as radius increases. Figure 26 shows comparison of the temperature distribution for different times at point r=0.45  m. In the initial seconds and toward the inner radius the variation of temperature is intensive.Figure 25
Comparison of the temperature distribution for different radius.Figure 26
Comparison of the temperature distribution for different times.Whent→∞ the variation of temperature versus dimensionless radius becomes linear.Figures27, 28, 29, 30, 31, and 32 show radial displacement and radial stress and hoop stress distribution versus time and thickness; meanwhile this result is compared with FEM. Figure 32 shows that in the course of 20 seconds hoop stress graphs have a minimum point. Figures 33, 34, 35, 36, 37, and 38 show comparison of the displacement and stresses distribution for different time and different radius. Figures 36 and 38 show that variation of radial and hoop stresses into variation of time is very small. Figures 39, 40, 41, 42, 43, and 44 show comparison of the displacement and stresses distribution for different angular velocity. (In these figures, ω is angular velocity and its dimension is radian per second (rad/sec).) In Figure 42 it is seen that angular velocity increases as radial stress decreases and if angular velocity increases more, radial stress changes from pressure to tension.Figure 27
Radial displacement distribution (t=3 sec).Figure 28
Radial displacement distribution (r=0.45 m).Figure 29
Radial stress distribution (t=3 sec).Figure 30
Radial stress distribution (r=0.45 m).Figure 31
Hoop stress distribution (t=3 sec).Figure 32
Hoop stress distribution (r=0.45 m).Figure 33
Comparison of the radial displacement distribution for different radius.Figure 34
Comparison of the radial displacement distribution for different times.Figure 35
Comparison of the radial stress distribution for different radius.Figure 36
Comparison of the radial stress distribution for different times.Figure 37
Comparison of the hoop stress distribution for different radius.Figure 38
Comparison of the hoop stress distribution for different times.Figure 39
Comparison of the radial displacement distribution for different angular velocity (r=0.45 m).Figure 40
Comparison of the radial displacement distribution for different angular velocity (t=3 sec).Figure 41
Comparison of the radial stress distribution for different angular velocity (r=0.45 m).Figure 42
Comparison of the radial stress distribution for different angular velocity (t=3 sec).Figure 43
Comparison of the hoop stress distribution for different angular velocity (r=0.45 m).Figure 44
Comparison of the hoop stress distribution for different angular velocity (t=3 sec).


## 5. Conclusions

In the present study, an exact solution procedure has been presented for transient thermoelastic analysis of isotropic and homogeneous hollow disks under general initial and boundary conditions. Some numerical results are shown in the figures. The distribution of the transient temperature, radial displacement, and thermal stresses through the radial direction of the disk is plotted and the results are compared with the solution using finite element method (FEM), which showed good agreement.



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