Advances in Engineering Software 41 (2010) 604–610

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Advances in Engineering Software

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Thermo-elastic stress of a metal-matrix composite disc underlinearly-increasing temperature loading by analytical and FEM analysis

Gürkan Altan *, Muzaffer TopçuDepartment of Mechanical Engineering, Faculty of Engineering, Pamukkale University, Kınıklı, 20070 Denizli, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 July 2009Received in revised form 11 November 2009Accepted 24 November 2009Available online 23 December 2009

Keywords:Metal-matrix compositeDiscThermo-elastic stress analysis

0965-9978/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.advengsoft.2009.11.007

* Corresponding author. Tel.: +90 258 296 31 63; fE-mail address: gurkanaltan@pau.edu.tr (G. Altan)

In this study, the elastic stress analysis of a composite disc with aluminium matrix subjected to a linearly-increasing temperature distribution has been carried out. The values of tangential and radial stress com-ponents that have occurred under the effect of the temperature from the inner surface of the disc towardsthe outer surface have been obtained by two different methods, numerical and analytical. In the analyt-ical analysis, a computer program has been developed to get the values of thermal stresses. But in thenumerical study carried out with the finite-element method, Abaqus 6.8 package program has been used.As a result of these analyses, it has been observed that the stress values obtained from both methods sup-port each other. It has been determined under this temperature distribution that tangential stress com-ponents are always on the condition of compression in the inner surface of the disc and of tensile in theouter surface. It has also been found out that radial stress components are always in the state of tensilealong the whole profile of the disc. Finally, the stress analysis of the same composite disc subjected to thistemperature distribution, but with a reduced mass, has also been examined numerically.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In recent times, composite materials which are lighter and havehigher specific strength than other materials have a wide field ofuse, especially in the aviation industry and in the automotiveindustry. Because of that, load-carrying capacities of the reinforceddiscs are higher than those of the isotropic steel discs with thesame geometry. The discs with hole have been the subject of manyscientific studies under different conditions.

Sayman [1] used an analytical method to investigate thermo-elastic stress analysis on an orthotropic composite disc with alu-minium metal-matrix reinforced towards x and y direction. Singhand Ray [2] determined the creep on an orthotropic aluminium sil-icon carbide composite rotating disc according to the Tsai–Hillyield criterion. The results obtained from this study were com-pared with the ones from the Von Mises yield criterion used forisotropic composites. Stanley and Garroch [3] developed a new testmethod for the composite discs reinforced by moulded fiber andobtained thermo-elastic data from fiber-reinforced orthotropicdisc in this study to find out the values of the orientation tensor.Karakuzu and Sayman [4] used the finite-element method andmade an elasto-plastic stress analysis in the orthotropic rotatingdiscs with holes. Bektas� et al. [5] realized the elastic–plastic stressanalysis on a composite disc with an aluminium metal-matrix un-

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ax: +90 258 296 32 62..

der internal pressure loads. Altan et al. [6] carried out an elastic–plastic thermal stress analysis of an orthotropic composite discwith a hole by using an analytical solution. The thermal load distri-bution was chosen as parabolically. Çallıoglu et al. [7] studiedstress analysis analytically on a circular orthotropic rotating discunder mechanical and thermal load. They obtained the solutionsunder parabolic temperatures along a radial section from the innersurface of the disc to the outer surface. Güven and Altay [8] studiedthermal stresses in a linear hardening solid disc under a uniformheat source. Jain et al. [9] designed a constant thickness compositedisc of uniform strength by tailoring the anisotropic elastic con-stants. Sen and Aldas [10] obtained thermal stresses in a thermo-plastic composite disc unidirectionally reinforced by steel fibers.They used finite-element method to calculate the thermal stressdistributions within the composite disc. Sen [11] investigated theeffect on thermal and residual stresses of parabolic temperatureloading in a thermoplastic composite disc.

In this study, the elastic stress analysis of the hollow composite discunder the linearly-increasing distribution of temperature has beencarried out analytically with the help of a computer program [12].At the same time, finite-element method with the Abaqus 6.8 programhas been used to calculate the magnitude of thermal stresses.

2. Thermal elastic stress solution

As steel fiber-reinforced orthotropic aluminium composite discis thin, the case of plane stress is valid. In this case, strain–stress

T0

a

b

θ

r

r

T

Fig. 1. Linearly increasing temperature distribution.

G. Altan, M. Topçu / Advances in Engineering Software 41 (2010) 604–610 605

correlation that takes place on an orthotropic composite disc underthe influence of temperature can be written as follows:

er ¼dudr¼ arrrr þ arhrh þ arT ð1Þ

eh ¼ur¼ arhrr þ ahhrh þ ahT ð2Þ

Here ar and ah are thermal expansion coefficients in radial andtangential directions. T represents the magnitude of the tempera-ture distribution. Constants of the arr, arh and ahh elasticity matrixare as follows:

ahh ¼1Eh; arr ¼

1Er

and arh ¼ ahr ¼ �mrh

Er¼ � mhr

Eh

where Er and Eh are the modulus of elasticity in the radial and tan-gential directions, respectively. If the stress distribution is, as seenin Fig. 4, symmetrical to an axis that passes from 0 and that is per-pendicular to x–y plane, stress components cannot be dependent on‘h’ and become the functions of ‘r’ only. Because of the symmetry,the value of srh shear stress is equal to zero. The equilibrium equa-tion for the plane stress case can be written as follows:

rdrr

drþ rr � rh ¼ 0 ð3Þ

Stress components as in Eq. (4), obtained from an F stress func-tion, secure the equilibrium equation in Eq. (3).

rr ¼Fr

and rh ¼dFdr

ð4Þ

er ¼dudr

and eh ¼ur

ð5Þ

and the relation between strains can be written as follows:

er ¼ddrðr � ehÞ ð6Þ

Accordingly, Eqs. (1) and (2) can be written as follows:

dudr¼ arr

Frþ arh

dFdrþ arT ð7Þ

ur¼ arh

Frþ ahh

dFdrþ ahT ð8Þ

If these equations are put in their place in the compatibilityequation (Eq. (6)), differential equation of the F stress function isfound as follows.

Considering k2 ¼ arrahh¼ Eh

Er;

r2F 00 þ rF 0 � k2F ¼ ðar � ahÞTahh

r � ahT 0

ahhr2 ð9Þ

As seen in Fig. 1, to find the distribution of the stresses whenT = 0 �C in the internal surface and under temperature distributionlinearly increasing towards the outer surface T = T0, differentialequality in Eq. (9) is demoted for the F stress function. The stressfunction F can be obtained by using the transform of r = et.

Relationship between linearly-increasing temperature andequation, and F stress function are as follows:

TT0¼ r � a

b� a; T ¼ T0

r � ab� a

ð10Þ

F ¼ C1rk þ C2r�k þ Er þ Dr2 ð11Þ

Here C1 and C2 are arbitrary integral constants and E and D con-stants are as follows:

D ¼ T0ðar � 2ahÞahhð4� k2Þðb� aÞ

and E ¼ T0aðah � arÞahhð1� k2Þðb� aÞ

ð12Þ

rr and rh stress components can be found from F stress functionas follows:

rr ¼ C1rk�1 þ C2r�k�1 þ Eþ Dr ð13Þ

rh ¼ C1krk�1 � C2kr�k�1 þ Eþ 2Dr ð14Þ

C1 and C2 arbitrary integral constants can be found with the useof limit conditions mentioned below:

rr ¼ 0 at the r = a, rr = 0 at the r = b

C1 ¼ Eðakþ1 � bkþ1Þ

b2k � a2kþ Dðakþ2 � bkþ2Þ

b2k � a2kð15Þ

C2 ¼ Ea2kbkþ1 � b2kakþ1

b2k � a2kþ Dða2kbkþ2 � akþ2b2kÞ

b2k � a2kð16Þ

Tangential and radial stress components under the load of line-arly-increasing temperature can be found in this way.

3. Composite disc and mechanical properties

The composite disc has been manufactured at the MechanicalLaboratory of the Faculty of Engineering, Pamukkale University[12]. Aluminium matrix has been reinforced by the steel fibers ofa circular form.

Composite disc manufacture has been carried out under pres-sure by means of a hydraulic press. The moulds have been heatedup to 550 �C with the use of electrical resistances in the course ofmanufacture and 30 MPa pressure has been applied with the presson them for 10 min. Under these conditions, aluminium plateshave amalgamated and made up the matrix of the manufacturedcomposite disc. The steel wires within the aluminium matrix makeup the fibers of the manufactured composite disc. Volume propor-tion of the steel fibers in the composite disc is Vf = 20% and that ofthe matrix is Vm = 80%. Mechanical properties of the composite discmanufactured are given in Table 1, [12]. These mechanical proper-ties of the composite disc have been experimentally obtained byusing strain-gauges.

4. Results and discussion

In the stress analysis that has been done, the elastic thermalstress analysis of a composite disc has been carried out under thelinearly-increasing temperature distribution. To determine theaccuracy of the stress analysis carried out, analytic and numericalsolution methods have been used and compared with each other.Disc dimensions have been taken as a = 15 mm and b = 75 mm inthe analysis. During the analysis, the thermal stresses at 0 �C havebeen taken as zero. At the same time, it has been assumed thatmechanical properties of composite materials do not change athigh temperatures. According to these assumptions, the values of

606 G. Altan, M. Topçu / Advances in Engineering Software 41 (2010) 604–610

radial and tangential stress components that have occurred on thedisc at the linearly-increasing temperature distribution have beenfound out. Analytical analysis has been carried out with Fortran

-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

40

45

15 20 25 30 35 40 45 50 55 60

r (mm)

Tan

gent

ial s

tres

s (M

Pa)

Analytical Numerical

σθ (MPaa

b

Fig. 2. (a) The thermal tangential stresses obtained with the numerical m

Table 1Mechanical properties of aluminium composite disc.

Eh (MPa) Er (MPa) Ghr (MPa) mhr X (MPa)

89,500 71,500 32,000 0.28 97

computer program by using the thermal elastic stress equations.Abaqus 6.8 CAE package program has been used in the numericalanalysis carried out with the finite-element method. The disc

65 70 75

r (mm)

)

ethod and (b) the comparison with analytical method for DT = 50 �C.

Y (MPa) S (MPa) ah (1/�C) ar (1/�C)

36 48 18.6 � 10�6 21.6 � 10�6

G. Altan, M. Topçu / Advances in Engineering Software 41 (2010) 604–610 607

model performed as finite-element model consists of 9243 ele-ments and 9306 nodes. The disc model generated for temperatureloadings has been divided into small circular pieces from internaldiameter to the external diameter. Thus, the numerical analysishas been performed.

The comparison between the thermal stresses obtained withthe numerical method under the temperature DT = 50 �C and thethermal stresses obtained with the analytical method is shown inFigs. 2 and 3 in detail. As seen in the figures, it has been determinedthat the results of both methods have remarkably close values and

0

5

10

15

15 20 25 30 35 40 45 50 55 60r (mm)

Rad

ial s

tres

s (M

Pa)

Analytical Numerical

a

b

Fig. 3. (a) The radial stresses obtained with the numerical method

have the same distributions. Thus, it could be judged that bothmethods support each other.

The stress distributions of the disc model from the inner surfaceto the outer surface, that is, all along the profile, are shown in Figs.2 and 3a chromatically. Again in the same figures, the changes ofthe stress components obtained with the analytical analysis andthe stress components obtained with the numerical analysis allalong the profile are shown graphically in the same figures. Theanalysis techniques displaying the same graphical change haveconfirmed each other in this way. To understand the changes of

65 70 75

and (b) the comparison with analytical method for DT = 50 �C.

20

25

608 G. Altan, M. Topçu / Advances in Engineering Software 41 (2010) 604–610

radial stress components on the numerical disc model in a betterway, graphical change configurations of the stress componentshave been placed all along disc profile.

The results of thermal elastic stress analysis carried out withthe analytical analysis and numerical method under the linearly-increasing temperatures of 50, 60, 70 and 80 �C are given compar-atively in Table 2. The stress values obtained in the inner and outersurface limits have been compared analytically and numericallyand it has been determined that the results of these two analysessupport each other. As it is seen in Table 2, it has been designatedthat tangential stresses under various linearly-increasing tempera-tures are always in the form of tensile in the inner surface of thecomposite disc and of compression in the outer surface of the com-posite disc. It has also been observed that the values of stress com-ponents both in the inner surface and outer surface increase withthe temperature. Radial stress values have been designated to bezero in the inner and outer surfaces of the disc.

Distributions of tangential stresses obtained from the analyticalanalysis towards the (r) radial direction are displayed graphicallyin Fig. 4. As is seen in the figure, it has been determined that tan-gential stress components are in the form of tensile in the innersides of the disc and of compression in the outer sides of the disc.Moreover, it has been determined that tangential stress values areapproximately zero around the middle part of the disc. It has alsobeen found that the intensity of the tangential stress in the inner

-75-70-65-60-55-50-45-40-35-30-25-20-15-10-505

1015202530354045505560657075

15 20 25 30 35 40 45 50 55 60 65 70 75

r (mm)

Tan

gen

tial

str

ess

(MP

a)

50°C 60°C 70°C 80°C

Fig. 4. Distributions of tangential stresses towards the (r) radial direction.

Table 2Tangential stress components at the inner and outer disc surfaces under linearly-increasing temperature distributions.

Temperature DT (�C) Surfaces rh (MPa) rh (MPa)Analytical Numerical

50 Inner 43.88 42.21Outer �27.88 �26.60

60 Inner 52.65 50.65Outer �33.45 �31.92

70 Inner 61.43 59.09Outer �39.03 �37.24

80 Inner 70.21 67.53Outer �44.60 �42.55

surface is higher than that of the tangential stress in the outer sur-face. That the values of tangential stress components increase onall the profile of the disc with the increase in temperature has beendetermined, as well.

The changes of radial stresses obtained from thermal elasticstress analysis towards the (r) radial direction are shown inFig. 5. As it is seen in the figure, radial stress components have beendetermined to be in the form of tensile on every part of the disc.The values of radial stress components have been obtained asremarkably lower, when compared with the values of tangentialstress components. Depending on the limit conditions, the valuesof radial stress components at the radius r = 15 mm andr = 75 mm have always been obtained as zero. It has been deter-mined that the value of radial stress component reaches its maxi-mum level through the parts of (r) radial axis close to the internaldiameter and that it stands at its maximum point at hightemperatures.

The effects of symmetrical holes generated for lesseningthe weight on the disc, a finite-element model to which the

0

5

10

15

15 20 25 30 35 40 45 50 55 60 65 70 75r (mm)

Rad

ial s

tres

s (

MP

a)

50°C 60°C 70°C 80°C

Fig. 5. The changes of radial stresses towards the (r) radial direction.

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 0.1 0.2 0.3 0.4 0.5 0.6

d/r

Tan

gent

ial s

tres

s (M

Pa)

Inner Outer

Fig. 6. Tangential stress components of lessened weight-hollow disc for DT = 50 �C.

Table 3Tangential stress component at the inner and outer surfaces of lessened weight-hollow disc under linear increasing temperature distributions.

DT (50 �C) d/r 0 0.1 0.2 0.3 0.4 0.5

Surfaces Inner rh (MPa) 42.21 42.21 42.16 42.08 41.88 41.46Numerical

Outer rh (MPa) �26.60 �26.77 �26.96 �27.44 �28.05 �29.19Numerical

Fig. 7. Numerical analysis of the tangential stress components of hollow disc for DT = 50 �C.

G. Altan, M. Topçu / Advances in Engineering Software 41 (2010) 604–610 609

analytically-supported numerical stress analysis has beenapplied, on the tangential stress components at 50 �C areshown in Table 3. We have also examined the effects of thegeometric change of the d diameter of the symmetrical holesgenerated for lessening the disc’s weight according to themaximum hole diameter that can be formed on the disc pro-file (r = 45 mm).

Here the maximum diameter of the hole that can be perforatedon the disc profile has been taken as r = 45 mm. The change of thetangential stresses that occur in the inner and outer surfaces of thedisc with the change in this diameter by the ratio of 0.1–0.5 isshown in Fig. 6.

As can be seen from Table 3 and Fig. 6, it has been found that tan-gential stress components decrease in the inner surfaces and in-crease in the outer surfaces with the increase in (d/r) ratio. When(d/r) ratio is 0.1 and 0.5, the distribution of the tangential stresscomponents of a holed disc obtained from the numerical analysisat 50 �C is shown in Fig. 7. It has been determined that in such smallvalues of the diameter as 0.1 the stress components do not changemuch, while in the big values of the diameter such as 0.5 the stresscomponents change. As a result of the analysis carried out, it hasbeen concluded that the stress values are nearly the same untilthe (d/r) ratio of 0.2, while the stresses after this value change.

5. Conclusions

The following points are obtained from the analytical andnumerical thermal elastic stress analysis of a composite disc with

aluminium metal-matrix under the load of linearly-increasingtemperature.

� It has been observed that the results obtained from these twosolution methods are very close. Therefore, both of thesesolution methods have been found to support each other.

� As orthotropic composite discs have different thermal expan-sions, tangential and radial stress components occur.

� The magnitudes of tangential stress components are biggerthan those of the radial stress components.

� Tangential stress components remain as tensile in the innerparts of the composite disc and as compressive in its outerparts.

� Radial stress components are zero in the inner and outer sur-faces of the composite disc and remain as tensile in the mid-dle parts of it.

� Radial and tangential thermal stresses increase by increasinglinearly-increasing thermal loads.

� It has been concluded that the stress values are nearly thesame until the (d/r) ratio of 0.2.

� It has been found that tangential stress components decreasein the inner surfaces and increase in the outer surfaces withthe increase in (d/r) ratio.

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