Finite Difference Solution to Thermoelastic Field in a Thin Circular FGM Disk with a Concentric HoleAcademic research paper on "Materials engineering"

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Abstract of research paper on Materials engineering, author of scientific article — B. Arnab, S.M.R. Islam, A.A. Khalak, A.M. Afsar

Abstract This study pays attention to the analysis of thermoelastic filed in a thin circular functionally graded material (FGM) disk with a concentric hole subjected to thermal loads. The material distribution in the disk is assumed to vary in the radial direction only. The mechanical and thermal properties of the FGM disk are simulated by the power function and exponential variation with the radius of the disk. As the range of variation of Poisson's ratio is small and its effect on the thermoelastic characteristics is negligible, it is reasonably assumed to be constant throughout the disk. Based on the two dimensional thermoelastic theories, the problem is reduced to the solution of a second order differential equation which is converted to a system of algebraic linear equations with the help of finite difference discretization. Solution of these equations is obtained by Gauss elimination method for an Al2O3/Al FGM disk. Numerical results of different components of stress and displacement are presented and analyzed. The analysis of the results reveals that the thermoelastic characteristics of an FGM disk are substantially dependant on material distribution as well as difference of temperatures at the inner and outer surfaces of the disk.

Academic research paper on topic "Finite Difference Solution to Thermoelastic Field in a Thin Circular FGM Disk with a Concentric Hole"

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Procedía Engineering 90 (2014) 193 - 198

Procedía Engineering

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10th International Conference on Mechanical Engineering, ICME 2013

Finite difference solution to thermoelastic field in a thin circular FGM disk with a concentric hole

B. Arnab, S. M. R. Islam, A. A. Khalak, A. M. Afsar*

Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh

Abstract

This study pays attention to the analysis of thermoelasticfiled in a thin circular functionally graded material (FGM) disk with a concentric hole subjected to thermal loads. The material distribution in the disk is assumed to vary in the radial direction only. The mechanical and thermal properties of the FGM disk are simulated by the power function and exponential variation with the radius of the disk. As the range of variation of Poisson's ratio is small and its effect on the thermoelastic characteristics is negligible, it is reasonably assumed to be constant throughout the disk. Based on the two dimensional thermoelastic theories, the problem is reduced to the solution of a second order differential equation which is converted to a system of algebraic linear equations with the help of finite difference discretization.Solution of these equations is obtained by Gauss elimination method for an Al2O3/Al FGM disk. Numerical results of different components of stress and displacement are presented and analyzed.The analysis of the results revealsthat the thermoelasticcharacteristics of an FGM disk are substantially dependant on material distribution as well as difference of temperatures at the inner and outer surfaces of the disk. © 2014TheAuthors.PublishedbyElsevierLtd. Thisisan open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Selection andpeer-reviewunder responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering andTechnology (BUET)

Keywords: Functionally graded material; material distribution;plane stress;thermoelastic field;finite difference method.

1. Introduction

Functionally graded materials (FGMs) are a special category of composites, which are microscopically inhomogeneous. Consequently,their mechanical and thermal properties vary continuously from one surface to another. The key feature is that gradation of properties reduces thermal stress, residual stress, and stress

* Corresponding author. Tel.: +880-2-9665636; fax: 880-2-8613046. E-mail address: mdafsarali@me .buet .ac.bd

Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering

and Technology (BUET)

doi: 10.1016/j.proeng.2014.11.836

concentration usually found in traditional composites. Typically, these materials are made from a mixture of metal and ceramic or a combination of different metals. FGMs have the outstanding advantages of being able to survive in adverse environment of higher temperature gradient, higher rate of wear and corrosion while maintaining their structural integrity ensuring toughness, machinability, and thermal conductivity. This stimulated the interests of researchers towards the development of new potential structural applications of FGMs. Structural components such as FGM beams [1-3], plates [4-5], and cylinders [6-7] have been studied under various thermal and mechanical loading conditions. FGM circular cylinders were considered by Obata and Noda [6] and Liew et al.[7] to analyze thermal stresses. Yongdong et al.[8] and Zhong and Yu[9] carried out the stress analysis of FGM beams of rectangular cross section under mechanical loads. Nigh and Olson [10] employed finite element method for rotating disks in body-fixed or a space-fixed coordinate system.

As the FGMs have continuously varying co-efficient of thermal expansion, an eigenstrain [11] is developed in the materials when they undergo a change in temperature. The incompatibility of thiseigenstrain has substantial effect on the overall thermoelastic field as pointed out by Afsar and Go [12] in the analysis of a rotating FGM disk. However, the above analysis was carried out for assumed distributions of temperature which might predict characteristics different from those corresponding to actual temperature distribution. Thus, this study considers actual temperature distributions for analyzing the thermoelastic field in an FGM disk with a concentric hole under thermal loads only.

2. Modeling of the FGM Disk

Figure 1 shows an FGM circular disk with a concentric hole of radius R. The outer radius of the disk is designated by Ro whereas r and 0 denote any radial and angular distance, respectively. The inner surface of the disk is assumed to be consisting of 100% material Awhile the outer surface has 100% material B. However, their distribution varies continuously only along a radial line from one surface to the other.Therefore, it is legitimate to assume that any property of the FGM disk varies in the radial direction only.However, the Poison's ratio (v) is assumed to remain constant throughout the disk as its effect is negligible on the characteristics. The material properties,namely Young's modulus (E), thermal conductivity (K), and coefficient of thermal expansion (a), of the disk are modeled by two different cases of power function and exponential variation as shown below.

Case I (power function variation): E = E1 + E0rn ; K = Kx + K0rn ; a = a1 +a0rn

Case II (exponential variation): E = E2epr; K = K2e^ ; a = a.^

The constants E1, E0, K1, K0, ai, a0, E2, p, K2,^, a2, and y can be determined from the following boundary conditions.

(i) r = Rh E = Ea, K = Ka, a = aA

(ii) r = Ro, E = EB, K = KB, a = aB

Here, the subscripts A and B represent the corresponding properties of materials A and B, respectively. As the thickness of the disk is small, the analysis is carried out under plane stress condition.

Fig.1. A thin circular FGM disk with a concentric hole.

3. Thermoelastic Formulation

Eigenstrain[11] is the generic name of non-elastic strains or free expansion strains that may develop in a body due to various reasons, such as change in temperature, phase transformation, precipitation, etc. The incompatibility of an eigenstraininduceseigenstressin the body. Therefore, the incompatible eigenstrain should be taken into account in the analysis of thermoelastic characteristics of a body. In the present case, the incompatible eigenstrain is given by

e' = a(r)T (r) (1)

whereT(r) is the change in temperature at any pointr. The eigenstrain in Eq. (1) is the same in all the directions as the FGM disk is isotropic. The total strain is the sum of elastic strain and eigenstrain. Thus, the components of strain are given by

sr = er + s ; se = ee+s (2)

where er and ee are the radial and tangential components of the total strain, and er and ee are the radial and tangential components of elastic strain. The elastic strains are related to stresses by Hooke's law. Thus, using of Hooke's law and combination of Eqs. (1) and (2) yield

er = -1 (or -vae) + T(r)a(r) E

ee = -1 ( ae - var ) + T (r Mr )

For an axisymmetric problem, strain-displacement relationships are given by

(3a) (3b)

whereM is the displacement in radial direction of the disk. The two dimensional equilibrium equationin polar coordinate is given by

+ - (ar - a, ) = 0

Combination of Eqs. (3) - (5) yields

E^ + ^^ | -(1+v)T(r)a(r) dr r

E ,, du u || u du dr r ) 1 r dr

For cases I and II mentioned earlier, Eq. (6) can be rearranged into the following two forms. Case I:

(Ei + Eo rn )

d u dr 2

nE0r- + * + {„E0rur-2 -(Ei + Eo^

r dr r

= — {(Ei + Eo r" ) (1 + v )T (r)(«i +«o r" )} dr

Case II:

du 1 du p u u d r r

) + ep {vp----) {^ (1 + v)T(r)a2e' }

dr dr r dr r r dr

For the steady state axisymmetric condition without heat generation, the heat conduction equation for a nonhomogeneous material reduces to

1 d f rxdL V 0

r dr ^ dr )

The solution of Eq. (9) can readily be obtained for the above two cases as shown below. Case I:

T(r) = TA +(Ta -TB) Case II:

T (r ) = TA + (Ta - TB ) -

r 1 ( J + Ko Ri )|

RJ (JK, + K o r"" m-ßr ) - Ei (-ßR, )

Ei{-PR) - Ei (-PR) whereEi is the exponential integral.

i "(K, + Ko R0" ) ~

4. Numerical Formulation

It is found that the analytical solutions of Eqs. (7) and(8) are quite involved that leads to adoption of a suitable numerical approach for their solution. Here, finite difference method is used for this purpose. Applying the central difference discretization technique to Eqs. (10) and (11),respectively, one obtains

(+E0rn ^(u+i";;"'2+"'-i)+(u^i){nE0rn-1+^n+uior2

At-2 2 r ri (12)

= j- [{1 + v)T(r)(El + Eorf )(«1 +«0r")]

m . ,— 2m. + m . , u, — m . , 1 v £> 1 d ^

e^ i '+1 2 ; ''') + ( '+' . ''')epr' 2p+ -) + u,e c--= d[(1+v)T{f''epr i (13)

At 2 2Ar r. r. r dr

Equations (12) and (13) represent two systems of linear algebraic equations for the case I and II, respectively. These equations can be solved by using any suitable numerical technique of solution.

5. Results and Discussion

To demonstrate the finite difference scheme developed in the study, an Al/Al2O3 FGM disk consisting of aluminum (Al) and alumina (Al2O3) is considered. The ingredient materials A and B correspond to Al andAl2O3,respectively. The mechanical and thermal properties of the ingredient materials are displayed in Table 1.To generate numerical results, the inner and outer radii of the disk are assumed to 1.5 cm and 15.0 cm, respectively.

Figure 2 exhibits the calculated temperature distributions in the disk for the assumed inner and outer surface temperatures of 35°C and 500°C, respectively, for both the cases I and II. For this temperature distribution (Fig. 2),

Table l.Properties of aluminum and alumina.

Properties Al AI2O3

E (GPa) 70 375

a (/0C) 22.2x10-6 8.4x10-6

K (W/mk) 214.52 35

400 300

8 200 &

00 0.2 0.4 0.6 0.8 1 Non-dimensional distance, (r-R)/(R„-R)

Fig. 2.Temperature distributions for case I and case II.

0.2 0.4 0.6 0.8

Non-dimensional distance, (r-R)/(Ro~R) Fig. 3.Displacement vs. normalized radial distance.

0.2 0.4 0.6 0.8

Non-dimensional distance, (r-Ri )/(Ro-Ri) Fig. 4.Displacementvs.normalized radial distance.

. x 10°

Ei fc> 2

tr ial

0 0.2 0.4 0.6 0.8 Non-dimensional distance, (r-R.)/(Ro-R.) Fig. 5.Radial stress vs. normalized radial distance.

0.2 0.4 0.6 0.8 1

displacement and stresses are evaluated as shown in Figs. 3 to 8.Figures 3 and 4 display the displacement as a function of normalized radial distance for the cases I and II, respectively. For case I, displacement varies smoothly from zero at the inner surface to the maximum value at the outer surface. Further, displacement is found to increase with the increase of the exponent n of the power function. In other words, the linear variation of the material properties yields the minimum displacement. For case II corresponding to the exponential variation of material properties, displacement increases sharply from zero at the inner surface to the maximum value at around 20% thickness of the disk. In the next region of the disk, the displacement remains almost uniform.The sharp change of the displacement causes higher strains as can be confirmed from Eq. 4.These higher strains are responsible for the higher magnitude of stress as seen in Figs. 5 to 8.

In the inner region of the disk, the radial stress shown in Fig. 5 is much lower than that shown in Fig. 6. It is noted that the radial stress of Fig. 5 corresponds the displacement of Fig. 3 while the radial stress of Fig. 6 corresponds the displacement of Fig. 4.The similar scenario is observed for the circumferential stress component as shown in Figs. 7 and 8, i.e.,this stress is also lower for case I corresponding to the smooth variation of displacement (Fig. 3). For both the radial and circumferential stresses, smooth variation of displacement (case I, Fig. 3) induces stresses which are lower by two order than the stresses corresponding to case II. Thus, case I, i.e., the power function variation of material properties is advisable compared to case II (exponential variation of material properties) when stress is the main concern in designing an FGM disk.

- x 10

0.2 0.4 0.6 0.8 1

Non-dimensional distance, (r-R)!(Ro~R)

Fig. 7.Circumferential stress as a function of normalized radial distance.

0.2 0.4 0.6 0.8 1

Non-dimensional distance, (r-Ri)/(Ro-Ri)

Fig. 8.Circumferential stress as a function of normalized radial distance.

The radial stress corresponding to case I decreases with the increase of the exponent n of the power function as can be observed from Fig. 5. Thus, greater exponent is advisable in terms of radial stress which is in conflict with the displacement. In terms of displacement, smaller exponent is found to be suitable.The circumferential stress at the outer surface increases as the exponent increases as seen from Fig. 7.

6. Conclusions

A finite difference scheme is developed for the analysis of thermoelastic field in an FGM disk subjected to thermal loads. The scheme can be helpful for the design of a circular cutter or a grinding disk with an FGM coating at the outer cutting or grinding surface. The scheme is demonstrated for an Al/Al2O3 FGM disk to analyze some numerical results. From the analysis of the numerical results, it is found that the power function variation of the material properties is advisable in comparison with the exponential variation of the same. Although, the greater exponent of the power function is better for the radial stress, it is not suitable in terms of displacement and circumferential stress.

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