Steady state thermoelastic contact problem in a functionally graded

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International Journal of Engineering Science 46 (2008) 775–789 www.elsevier.com/locate/ijengsci

Steady state thermoelastic contact problem in a functionally graded material Sakti Pada Barik a, M. Kanoria b, P.K. Chaudhuri b,* a

Department of Mathematics, Gobardanga Hindu College, Khantura, 24-Parganas (N), West Bengal, India b Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700 009, India Received 19 November 2007; received in revised form 3 February 2008; accepted 11 February 2008 Available online 1 April 2008

Abstract This paper is concerned with the stationary plane contact of a functionally graded heat conducting punch and a rigid insulated half-space. The frictional heat generation inside the contact region due to sliding of the punch over the half-space surface and the heat radiation outside the contact region are taken into account. Elastic coefficient l, thermal expansion coefficient at and coefficient of thermal conductivity k are assumed to vary along the normal to the plane of contact. With the help of Fourier integral transform the problem is reduced to a system of two singular integral equations. The equations are solved numerically. The effects of nonhomogeneity parameters in FGMs and thermal effect are discussed and shown graphically. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Functionally graded material; Fourier transform; Frictional heating; Heat radiation; Singular integral equation; Fredholm integral equation

1. Introduction The importance of contact problems in solid mechanics stems from the very basic fact that load application to deformable bodies are mainly done through contact between two bodies. There are of course exceptions; such as loading of the boundary by fluid pressure or various kinds of body forces such as gravitational or magnetic forces. When two solids are in contact, the determination of the state of stress in the media has been the subject of study in literature for many years and the problems are usually termed as contact problems. Usually the contact problems are of two types: (a) the bodies in contact are bonded together and consequently, the contact regions are known a priori, and the main task is to determine the stress distribution in the media;

*

Corresponding author. Tel.: +91 3323373633. E-mail address: [email protected] (P.K. Chaudhuri).

0020-7225/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2008.02.003

776

S.P. Barik et al. / International Journal of Engineering Science 46 (2008) 775–789

(b) the bodies are in contact without bond so that the region of contact is not known. In such class of problems determination of the contact region (which depends upon geometric features of the bodies as well as upon the load distribution on the system) becomes an additional task. Due to the application of load the area of contact may increase, decrease or may even remain stationary. Accordingly, contact problems are classified as advancing, receding or stationary. Contact problems in which frictional forces at the contact surface are not taken into account have been studied by many investigators. Among several works done we may mention a few: Comez et al. [8], Chaudhuri and Ray [5], El-borgi et al. [9], Jing and Liao [17], Fabrikant [12], Barik et al. [4], Avci and Yapici [2]. Another factor plays a vital role in the study of contact problem. If the surfaces of the solid in contact are not smooth, and the contact surface changes due to relative motion of the solids, there will be generation of heat due to frictional effects at the contact region. The generated heat produces significant thermoelastic distortion of the contacting surfaces, which in turn, can effect the contact pressure distribution. The notable points in this kind of problem are (i) moving source of heat (ii) normal and tangential loadings between the solids and (iii) the unknown contact area. Problems of this type have been discussed by many investigators [1,3,6,7,13,14,34,36,15] considering solids as homogeneous elastic material. But over the last few decades various problems in solid mechanics are being studied where the elastic coefficients are no longer constants but they are function of position. The investigations result from the fact that idea of nonhomogeneity in elastic coefficients is not at all hypothetical, but more realistic. Elastic properties in soil may vary considerably with positions. The earth crust itself is nonhomogeneous. Besides these, some structural materials such as functionally graded materials (FGMs) have distinct nonhomogeneous character. For example, in graded composite materials, graded regions are treated as series of perfectly bonded composite layers, each layer being assigned slightly different properties. In FGMs the material properties vary gradually with location within the body. In many applications FGMs are found to be better substitutes for conventional homogeneous materials. Among several uses of FGMs, one such is the use of FGMs in automotive brakes and clutches [10,20,35,37,38] where the effect of frictional heat generation is the subject of concern to the scientists. When brakes are applied to a moving system, the kinetic energy produced at the wheel is transformed into heat energy, which does not dissipate fast enough into the air stream from the brake surface into the brake disk and as a result, the high temperatures and thermal stresses that accompany them produce a number of disadvantageous effects such as surface cracks or permanent distortions. The thermal effect also affects the contact pressure between the surfaces. In order to avoid such type of damage FGMs have been considered as protecting coatings between the contact surfaces. From various studies on contact problems [16,18,23] it has been observed materials with controlled gradients in mechanical properties of compositions and structures exhibit resistance to contact deformation and damage that cannot be realized in conventional homogeneous materials. The conventional ferrous material used in a friction brake and clutch causes thermoelastic instability. A coated layer of properly graded ceramic metal FGMs has the advantage of the heat and corrosion resistance of ceramic and the mechanical strength of metal, associated with thermal efficiency of the system and bonding strength along the coating substrate interfaces. The use of FGM layer would reduce the magnitude of residual and thermal stresses. The feasibility of FGMs on the brake and clutch system has been investigated by Jang and Ahn [16]. FGMs made from ceramics and metals are also suitable for use in high temperature generated systems, for example, the use of ceramic composite microstructure in gas-turbines can protect metals and improve the life and reliability of the thermal barrier coatings. FGMs formed by appropriately combining two or more materials in a perfectly designed manner can be shown to be more resistant to crack initiation and propagation [11]. FGMs are used as interfacial zones to reduce stresses arising from matching different material properties, to increase bonding strength and to provide protection against thermal and chemical environment [21]. In some special environments where additional load bearing capacity is needed, properly designed FGMs could be useful. Study of thermoelastic problems in a functionally graded media is quite involved because of the fact that in addition to the complicated form of the basic equations due to arbitrary variations of elastic moduli with position, the heat conduction equation too becomes complicated when thermal conductivity coefficient is position dependent. Basically due to these constraints analytical study of thermoelastic problems in functionally graded media has been restricted to problems with special types of variation of elastic and thermal moduli.


777

If the contact surface is not smooth then any point in the contact surface will have one of the two states, namely the ‘stick’ state in which there is no relative motion and the resultant tangential traction is fp where f is the coefficient of friction and p is the normal traction, and the’slip’ state during which there is a relative motion and the tangential traction is of magnitude fp and opposes instantaneous direction of slip. Pauk [25,26,29–31], Pauk and Wozniak [27], Pauk and Zastrau [28], Marzeda et al. [22] discussed second kind of problem by taking the material as elastically isotropic and homogeneous. The main objective of the present problem is to study the stationary plane contact of a functionally graded elastic heat conducting punch sliding over the rigid insulated half-space. Following the technique discussed in Pauk [25] we have found two integral equations involving surface temperature and contact pressure, which have been solved numerically. Numerical computations have been done to assess the effects of graded parameters considered in the problem on the surface temperature, temperature inside the punch and contact pressure. 2. Formulation of the problem We consider a cylindrical punch of elastic material lying on a rigid insulated half-space and moving on the surface with a uniform sliding velocity V in a direction which we take as the negative x-direction. The material of the cylinder is assumed to be elastically isotropic but functionally graded and thermally heat conducting with position dependent conductivity. The punch is pressed to the half-space by a resultant normal force P. The problem is formulated in a moving co-ordinate system ðx; yÞ in which y-axis is fixed in the punch along the outward drawn normal to the half-space and moving with the punch. It follows that the contact area C ¼ ða; bÞ will remain stationary with respect to the punch. Clearly a; b depend on the applied load and the nature of the elastic material. The constants a; b are at present unknown. Due to the sliding motion of the punch frictional heating will take place and the generated heat flux qðxÞ is assumed to be directed only into the punch. It is supposed that the thermoelastic process in the punch is two dimensional and stationary. The geometry of the problem is shown in Fig. 1. The sliding motion of the punch will generate Coulomb friction at the interface, giving tangential tractions which are everywhere proportional to the normal traction pðxÞ through a coefficient of friction f and outside the contact region heat radiation is described by Newton’s law. In deriving analytical solution in the present study the elastic parameters k, l, thermal expansion coefficient at and coefficient of thermal conductivity k have been assumed to vary exponentially in the direction perpendicular to the plane of contact, that is k ¼ k0 eay ;

l ¼ l0 eay ;

at ¼ a0t egy

and

k ¼ k 0 edy ;

where a, g and d are real numbers. Mathematically, the problem under consideration is reduced to the solution of thermoelasticity equations [24]: (i) Equilibrium equations: o2 u o2 u o2 v ou ov oT 2ð1 mÞ 2 þ ð1 2mÞ 2 þ þ að1 2mÞ þ ; ¼ b1 ox oy oxoy oy ox ox o2 v o2 v o2 u ov ou oT ð1 2mÞ 2 þ 2ð1 mÞ 2 þ þ a 2ð1 mÞ þ 2m þ ða þ gÞT : ¼ b1 ox oy oxoy oy ox oy

ð2:1Þ

(ii) Steady state heat conduction equation: o2 T o2 T oT ¼0 þ þd ox2 oy 2 oy with the boundary conditions at y ¼ 0 oT ¼ qðxÞ; x 2 C; k0 oy ¼ hT ðxÞ; x 62 C;

ð2:2Þ

ð2:3Þ

778


V P

y

x a

b

Fig. 1. Geometry of the contact.

ryy ðxÞ ¼ pðxÞ;

x 2 C;

¼ 0; x 62 C; rxy ðxÞ ¼ fpðxÞ; x 2 C;

ð2:4Þ

¼ 0; x 62 C; qðxÞ ¼ fVpðxÞ; x 2 C; dv x ¼ ; x 2 C; dx R

ð2:5Þ ð2:6Þ ð2:7Þ

where b1 ¼ 2ð1 þ mÞat is thermal modulus; u; v are displacements; rxx ; rxy ; ryy are stresses; T is a temperature; p is a contact pressure; m is Poisson’s ratio; qðxÞ is a heat flux; h; f ; are heat radiation coefficient, coefficient of Coulomb friction respectively and R is radius of the cylinder. 3. Method of solution Taking Fourier transforms of the heat conduction equation (2.2) and using boundary conditions in Eq. (2.3), we get T ðn; yÞ ¼

IðnÞep1 ðnÞy ; p1 ðnÞ þ c1

ð3:1Þ

where T ðn; yÞ denotes the Fourier transform of T ðx; yÞ, c1 ¼ kh0 , sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d d2 p1 ðnÞ ¼ þ þ n2 2 4 and I¼

1 pffiffiffiffiffiffi k 0 2p

Z

b

½qðxÞ þ hT ðx; 0Þeinx dx:

a

Applying Fourier inversion on Eq. (3.1) we find that the temperature in the punch has to satisfy the integral equation Z b 1 T ðx; yÞ ¼ ½qðx0 Þ þ hT ðx0 ; 0ÞLðx x0 ; yÞdx0 ; ð3:2Þ pk 0 a where Lðz; yÞ ¼

1 2

Z

1 1

eðinzÞ ep1 ðnÞy dn: p1 ðnÞ þ c1

ð3:3Þ


779

To solve the partial differential equations of (2.1), Fourier transform is applied to Eq. (2.1) with respect to the variable x. The transformed equations in terms of Fourier transform parameter n can be written as o2 u ou inv ¼ inb1 T ; 2ð1 mÞn2 u þ ð1 2mÞ 2 inv þ að1 2mÞ oy oy ð3:4Þ 2 ov ou ov oT 2! þ ða þ gÞT ; ð1 2mÞn v þ 2ð1 mÞ 2 in þ a 2ð1 mÞ i2mnu ¼ b1 oy oy oy oy where u and v are the Fourier transforms of u and v, respectively. Now elimination of u from Eq. (3.4) yields a differential equation in v; oT þ f2 ðnÞT ; F ðDÞv ¼ b1 f1 ð3:5Þ oy where F ðDÞ ¼ 2ð1 mÞD4 þ 4ð1 mÞaD3 þ 2ð1 mÞða2 2n2 ÞD2 4ð1 mÞan2 D þ 2ð1 mÞn4 þ 2ma2 n2 ; 2

2

f1 ¼ 2ð1 2mÞad þ a d þ gð2a þ g dÞ

ð3:6Þ ð3:7Þ

and f2 ðnÞ ¼ ð1 2mÞdn2 þ 2magða þ gÞ

gn2 : 1 2m

ð3:8Þ

The solution of differential equation (3.5) is found to be vðn; yÞ ¼

4 X

Ak emk y þ b1 HðnÞT 0 ðnÞep1 y ;

ð3:9Þ

k¼1

where HðnÞ ¼

f2 ðnÞ f1 p1 ðnÞ F ½p1 ðnÞ

and the functions Ak ðk ¼ 1; . . . ; 4Þ are arbitrary unknowns, mk ðk ¼ 1; . . . ; 4Þ are the roots of following characteristic equation: F ½m ¼ 0: The characteristic equation (3.10) may easily be expressed as m 2 a2 n2 : ðm2 þ am n2 Þ ¼ 1m

ð3:10Þ

ð3:11Þ

It follows that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi ffi a2 m 2 n þ þi an; 1m 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi ffi 2 a a m m2 ¼ m4 ¼ n2 þ þ i an: 2 1m 4 a m1 ¼ m3 ¼ þ 2

ð3:12Þ ð3:13Þ

It is obvious that Reðm1 ; m3 Þ > 0 and Reðm2 ; m4 Þ < 0, to satisfy the condition of convergency of v at y ¼ 1 in the solution given by (3.9), we must put A1 ¼ A3 ¼ 0 for y > 0: Thus for y > 0, vðn; yÞ ¼ A2 em2 y þ A4 em4 y þ HðnÞep1 y b1 T 0 ðnÞ:

ð3:14Þ

Now,with the help of Fourier transformed equations of (2.4), (2.5) and (3.14), the normal surface displacement v can be expressed as the sum of the elastic displacement ve and thermal displacement vth . So we can write

780


dv dve dvth ¼ þ : dx dx dx

ð3:15Þ

Following [33] the elastic displacement ve ðxÞ is determined from the formula Z b Z b Z b dve 1 m pðx0 Þ 0 ð1 2mÞf 0 0 0 0 0 0 ¼ dx L ðx x Þpðx Þdx pðxÞ þ L ðx x Þpðx Þdx 1 2 0 pl0 2l0 dx a xx a a and the thermal displacement vth ðxÞ is determined from Z b dvth b1 ¼ ½qðx0 Þ þ hT ðx0 ÞL3 ðx x0 Þdx0 ; dx 2k 0 p a where

ð3:17Þ

1 2m nG1 ðnÞ 1 sinðznÞdn; 2ð1 mÞ DðnÞ 0 Z 1 1 2n2 G2 ðnÞ L2 ðzÞ ¼ 1 cosðznÞdn; p 0 ð1 2mÞ DðnÞ Z 1 N 3 ðnÞ sinðznÞ L3 ðzÞ ¼ dn; c1 þ p1 ðnÞ 0

L1 ðzÞ ¼

Z

1

ð3:16Þ

ð3:18Þ

DðnÞ; G1 ðnÞ, G2 ðnÞ and N 3 ðnÞ are shown in Appendix 1. Substituting (3.16) and (3.17) into (3.15) and then utilizing the boundary condition (2.7) we get the following singular integral equation: Z b Z b Z b 1m pðx0 Þ 0 ð1 2mÞf 0 0 0 0 0 0 dx L1 ðx x Þpðx Þdx pðxÞ þ L2 ðx x Þpðx Þdx 0 pl0 2l0 a xx a a Z b b x ð3:19Þ þ 1 ½qðx0 Þ þ hT ðx0 ÞL3 ðx x0 Þdx0 ¼ : R 2k 0 p a Also, use of integral equation (3.2) for y ¼ 0 yields a Fredholm type integral equation of second kind Z b 1 T ðxÞ ½qðx0 Þ þ hT ðx0 ÞLðx x0 Þdx0 ¼ 0: ð3:20Þ pk 0 a The contact pressure must satisfy the equilibrium condition Z b pðxÞdx ¼ P :

ð3:21Þ

a

The system of Eqs. (3.19)–(3.21) in terms of dimensionless variables s¼

x b0 ; a0

r¼

x0 b0 ; a0

t¼

y ; a0

a0 ¼

ba ; 2

b0 ¼

bþa ; 2

p ¼

a0 p; P

T ¼

k0 T fPV

becomes fBp ðsÞ þ

1 p

Z

1 1

1 2fP e HL3 ðs rÞ L1 ðs rÞ pfBL2 ðs rÞ p ðrÞdr sr

Z 2fP e HBi 1 A2 s þ C P H ; jsj 6 1; T ðrÞL3 ðs rÞdr ¼ p 2pa0 b0 P 1 Z Z Bi 1 1 1 T ðsÞ T ðrÞL ðs rÞdr þ p ðrÞL ðs rÞdr ¼ 0; p 1 p 1 Z 1 p ðrÞdr ¼ 1; þ

ð3:22Þ ð3:23Þ ð3:24Þ

1 12m 0 d1 j where A ¼ aaH0 ; H ¼ l1m ; d1 ¼ 2kb10 ; C ¼ bbH0 ð1 2a0 ÞA; Bi ¼ hak00 (Biot’s number), B ¼ 2ð1mÞ (Dundurs’ parameter), Va0 P e ¼ 2j (Peclet’s number), j is thermal diffusivity, aH = half-width in corresponding isothermal problem,


781

bH = contact zone location in corresponding isothermal problem, PH = load in corresponding isothermal problem and 1 1 1 1 a0 ¼ arctan ð3:25Þ þ N 0 ; b0 ¼ arctan þ M 0; p fB p fB where N 0 and M 0 are arbitrary integers and are determined from the physics of the problem. The quantities aH ; bH ; P H are connected by the relations [32] a2H ¼

bH ¼ ð1 2a0 ÞaH ;

P H Rð1 mÞ : 2pa0 b0 l0

The kernels L1 ðzÞ, L2 ðzÞ, L3 ðzÞ and L ðzÞ of the integral equations (3.22) and (3.23) are given by Z 1 1 2m fG1 ðfÞ 1 sinðzfÞdf; L1 ðzÞ ¼ 2ð1 mÞ DðfÞ 0 Z 1 1 2 f2 G2 ðfÞ L2 ðzÞ ¼ 1 cosðzfÞdf; p 0 ð1 2mÞ DðfÞ Z 1 N ðfÞ sinðzfÞ 3 L3 ðzÞ ¼ df; p ðfÞ þ Bi 0 Z 1 1 cosðfzÞ L ðzÞ ¼ df: p1 ðfÞ þ Bi 0

ð3:26Þ ð3:27Þ ð3:28Þ ð3:29Þ

4. Solution of integral equations The singular integral equation (3.22) is a Cauchy-type singular integral equation for an unknown p ðrÞ expressed in terms of T ðrÞ and a certain known function. The integral equation (3.23) is a Fredholm-type integral equation of second kind for an unknown T ðrÞ expressed in terms of p ðrÞ and a certain known function. For the contact pressure p ðrÞ we assume p ðrÞ ¼ /ðrÞð1 rÞa0 ð1 þ rÞb0 ; jrj < 1; ð4:1Þ where /ðrÞ is a regular function neither vanishing at r ¼ 1 nor tending to infinity at r ! 1 and a0 ; b0 are defined in Eq. (3.25). Because of the smooth contact at r ¼ 1, the physics of the problem demands that both a0 and b0 be positive. This, by considering Eq. (3.25), may be fulfilled by setting N 0 ¼ 0 and M 0 ¼ 1 i.e. j (index of the singular integral equation (3.22)) ¼ ða0 þ b0 Þ ¼ ðN 0 þ M 0 Þ ¼ 1. Using the method developed in Krenk [19] we have the discretized form of the (3.22)–(3.24) as N 1X 1 /ðrk ÞW Nk þ L1 ðsm rk Þ þ pfBL2 ðsm rk Þ þ 2fP e HL3 ðsm rk Þ p k¼1 r k sm

N 2fP e HBi X 2 A2 s m þ C P H T ðqk ÞL3 ðsm qk Þ ¼ ; p N 2pa0 b0 P k¼1

T ðqm Þ N X

m ¼ 1; . . . ; N þ 1;

N N Bi X 2 1X T ðqk ÞL ðqm rk Þ þ /ðrk ÞW Nk L ðqm rk Þ ¼ 0; p k¼1 N p k¼1

ð4:2Þ

m ¼ 1; . . . ; N ;

W Nk /ðrk Þ ¼ 1;

ð4:4Þ

k¼1

where ða ;b Þ

ða ;b0 Þ

PN0

ðrk Þ ¼ 0;

m ¼ 1; . . . ; N þ 1

p P N j0 0 ðrk Þ ; k ¼ 1; . . . ; N ; sinðpa0 Þ P Nða0 ;b0 Þ0 ðrk Þ 2k 1 k ¼ 1; . . . ; N ; qk ¼ 1 þ N

W Nk ¼ 2j

ð4:3Þ

ða ;b0 Þ

P N j0

ðsm Þ ¼ 0;

782


Table 1 I 2 ðAÞ for z ¼ 0:7, d ¼ :01, Bi ¼ :01 and cut-off point s ¼ 25 A

I 2 ðAÞ

A

I 2 ðAÞ

30 35 40 45 50 55 60 65 70 71 72 73 74 75 76 77 78 79

0.169562 0.103283 0.139541 0.130334 0.117192 0.146828 0.107100 0.150572 0.109099 0.117558 0.130941 0.142905 0.147936 0.143888 0.132879 0.120201 0.111784 0.111427

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

0.119093 0.131022 0.141568 0.145864 0.142064 0.132125 0.120811 0.113413 0.113282 0.120314 0.131079 0.140487 0.144192 0.140593 0.131521 0.112313 0.114745 0.114797

T* 4.5

α=0

4.3

α=0.1 4.1

α=0.5 3.9

α=1.0 3.7 -3

-2

-1

0

1

s

3

2

Fig. 2. Effect of a on surface temperature for fPeH = 0.5, Bi = .01, d = .01 and g = 0.2.

T* 2

1.6

δ =0 1.2

δ =.1

0.8

δ =.4 δ =1

0.4

0 -3

-2

-1

0

1

2

3

s

Fig. 3. Effect of d on surface temperature for a = 0.1, g = 0.001, Bi = 0.01 and fPeH = 0.5.


783

ða ;b Þ

P N 0 0 ðÞ denotes the Jacobi polynomial of degree N with indices a0 and b0 . The relations (4.2)–(4.4) constitute the system of 2N þ 2 linear algebraic equations to determine ð2N þ 1Þ unknowns /ðrk Þ; T ðqk Þ; k ¼ 1; . . . ; N and PPH . For (4.2) it is sufficient to choose only N of the ðN þ 1Þ possible collocation points. This is consistent with fact that in actual applications the extra equation is used to normalize the interval of integration. 5. Numerical results and discussion The main objective of the present discussion is to study the effects of nonhomogeneity in FGMs and/or in thermal conductivity on the thermoelastic contact problems. The presence of the nonhomogeneity parameters a; d; g in FGMs does not yield a complete analytical solution. Solution of the problem can be obtained using numerical methods. The numerical process involves evaluation of a number of improper integrals whose convergence needs to be assured. We have noted that zero is not a singularity for the infinite integrals present in Eqs. (3.26)–(3.29), and as the integrals are regular for finite f, in numerical evaluation of these integrals, behaviours are to be studied at infinity only. To examine the convergence of the improper integrals in Eqs. (3.26)–(3.29) let us take any one, say, the improper integral T* 4.5

η= 0.0 4.3

η= 0.2 4.1

η= 0.5 η= 1.0

3.9

s

3.7 -3

-2

-1

0

1

2

3

Fig. 4. Effect of g on surface temperature for fPeH = 0.5, Bi = .01, d = .01 and a = 0.1.

T* 4.5 4 3.5

-Bi=.01

3

-Bi=.08

2.5

-Bi=0.6

2

-Bi=1.0 1.5 1 0.5 -3

-2

-1

0

1

2

3

s

Fig. 5. Effect of Biot’s number on surface temperature for fPeH = 0.5, a = 0.1, d = .01 and g = 0.001.

784


Z 0

1

cosðfzÞ df p1 ðfÞ þ Bi

in Eq. (3.29). We write Z 1 Z A cosðfzÞ cosðfzÞ df ¼ lim df: A!1 p1 ðfÞ þ Bi 0 0 p 1 ðfÞ þ Bi For large f, the integral Z A cosðfzÞ df 0 p 1 ðfÞ þ Bi is written as the sum of two integrals I 1 and I 2 such that Z s cosðfzÞ df; I1 ¼ 0 p 1 ðfÞ þ Bi Z A cosðfzÞ I 2 ¼ I 2 ðAÞ ¼ df; p s 1 ðfÞ þ Bi T* 4.5 4.4 4.3

ν= 0.

4.2

ν=.3

4.1

ν=.45

4

ν=.49

3.9 3.8 3.7 -3

-2

-1

s 0

1

2

3

Fig. 6. Effect m on surface temperature for a = .1, a = .01, g = .2, Bi = .01 and fPeH = .5.

Fig. 7. Temperature contour for fPeH = .5, Bi = .01, a = .1, d = .01 and g = .001.


785

where s is a suitably chosen cut-off point. The first integral I 1 is numerically evaluated using GaussianQuadrature technique. For evaluation of second integral I 2 we express the integrand of I 2 in asymptotic form, that is Z A 1 v1 v2 v3 þ þ cosðzfÞdf: I2 f f2 f3 f4 s

σyy(s,0) 0.7

0.6

0.5

0.4

(1) 0.3

(2) 0.2

0.1

s

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 8. Temperature effect on normal stress distribution over the contact region: (1) smooth contact; (2) frictional contact causing heat generation.

σyy (s,0) 0.7

0.6

0.5

α=.2 0.4

α=.3

0.3

α=.5

0.2

α=.8

0.1

0 -1

-0.8

-0.6

-0.4

-0.2

s 0

0.2

0.4

0.6

0.8

1

Fig. 9. Effect of graded parameter a on normal stress distribution over the contact region for d = 0.01, g = 0.001, Bi = 0.01 and fPeH = 0.5.

786


The coefficients v1 , v2 , v3 are given in Appendix 1. It is expected that for sufficiently large A, the integral I 2 ðAÞ should converge. The behavior of I 2 as A increases is shown in the Table 1. We have noted that for a given tolerance , the upper limit A can be determined numerically, and the value of the integral I 2 ð1Þ can be approximated as I 2 ðAÞ at tolerance . The numerical solution of the system of algebraic equations (4.2)– PH/P

α=.1

1

α=.15

0.8

α=.2 0.6

α=.5

0.4

0.2

0

fPeH 1

0

2

3

Fig. 10. Effect of a on the ratio PH/P for d = .01, g = .001 and Bi = .01.

PH /P 1

η=0

0.8

η=.0005

0.6

η=.0008 0.4

η=.001 0.2

0 0

0.5

1

1.5

2

2.5

3

fPeH

Fig. 11. Effect of g on the ratio PH/P for a = 0.1, d = .01 and Bi = .01.

PH /P 1 0.8

-Bi=0.03

0.6

-Bi=0.07

0.4

-Bi=0.08 -Bi=0.09

0.2 0 0

0.5

1

1.5

2

2.5

3

fPeH

Fig. 12. Effect of Biot’s number on the ratio PH/P for a = .1, d = .01 and g = .001.


787

PH /P 1 0.8

δ=0

0.6

δ=.005

0.4

δ=.008

0.2

δ=.01

0 0

0.5

1

1.5

2

2.5

3

fPeH

Fig. 13. Effect of d on the ratio PH/P for a = .1, g = .001 and Bi = .01.

(4.4) was done by using a standard computational method. We assume as in Pauk [25] that the contact zone in the present case is the same as that in the isothermal case i.e. aaH0 ¼ bbH0 ¼ 1. The input parameters for calculations are f, P e , H, Bi. In our numerical evaluation we take m ¼ 0:3, f ¼ 0:4 and fP e H ¼ 0:5 as in Pauk [25]. Surface temperature distribution with varied m has also been investigated (Fig. 6). The variation of surface temperature for various values of the nonhomogenity parameters a, d and g are shown in Figs. 2–4. Radiation effect on surface temperature distribution obtained by varying Biot’s number is shown in Fig. 5. The effect is similar to that in homogeneous material. Temperature contour in Fig. 7 shows the temperature distribution in the punch. Fig. 8 displays the temperature effect on normalized stress distribution over the contact region. The effect of nonhomogeneity parameter in FGMs on normalized stress distribution over the contact region is observed in Fig. 9. Figs. 10–13 depict the ratio PPH against fPeH . It is observed that the value of PPH decreases linearly with the parameter fP e H . 6. Conclusion In the present study of the problem of the stationary plane contact of a functionally graded heat conducting punch and a rigid insulated half-space the following observations are made: (a) The surface temperature is observed to increase with elastic parameter a in the contact region, whereas the effect is opposite outside the contact region (Fig. 2). (b) Increase in conductivity results decrease in surface temperature (Fig. 3). (c) Fig. 4 shows that greater value of linear thermal expansion coefficient at causes relatively lower surface temperature in the contact zone. (d) The temperature level decreases with the increase of radiation on the surface (Fig. 5). (e) Fig. 6 displays the variation of surface temperature for different values of m. It is observed that surface temperature decreases in the contact region with increase of m, whereas the behavior outside the contact region is almost reverse. (f) Fig. 7 shows that with some specific values of nonhomogeneity parameters in the graded material, as expected, the temperature decreases in the graded material. The temperature distribution is observed to be nonuniform on the contact surface. (g) The effect of temperature on the normal stress has been shown in Fig. 8. It indicates that temperature effect due to friction causes more stress in the neighbourhood of the origin and the effect is almost reverse in the remaining part of the contact region. Fig. 9 shows that with increase of a the contact pressure decreases in the neighbourhood of origin and increases in the remaining part. (h) The dependence of PPH with fP e H is studied for various values of nonhomogeneity and radiation parameters (Figs. 10–13). The dependence is linear in all cases and PPH decreases with fP e H . The critical value of fP e H (where PPH is zero) is shown to decrease with increasing a or increasing Biot’s number; on other hand with increasing g or d the critical value of fP e H increases.

788


Appendix 1 DðnÞ ¼ a01 an2 ½b1 an2 2cðb21 n2 þ b22 a2 Þ b3 að4c2 r2 Þ þ a1 n2 ½ðb01 a2 n2 þ b02 n4 Þ r2 ðb21 n2 þ b22 a2 Þ 2cb3 ar2 þ a2 a½2cðb01 a2 n2 þ b02 n4 Þ þ b1 an2 r2 b3 ar4 þ a3 ½ðb01 a2 n2 þ b02 n4 Þð4c2 r2 Þ þ 2cb1 an2 r2 þ ðb21 n2 þ b22 a2 Þr4 ; 2

2 2

2

2 2

2

2

2

2

ðA1Þ 2

G1 ðnÞ ¼ ½ð1 mÞn þ mð1 2mÞ a ½b1 an þ 2cðb21 n þ b22 a Þ þ b3 að4c r Þ; 2

2

ðA2Þ

2

G2 ðnÞ ¼ ½ð1 mÞn þ mð1 2mÞ a ½a1 n þ a3 ð4c r Þ þ 2ca2 a; pffiffiffiffiffiffiffiffiffi 2 qðnÞ½qðnÞ þ n2 þ a4 a ffi; c ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 m ðanÞ2 þ ½qðnÞ þ n2 þ a 2 1m

r2 ¼ qðnÞ þ

4

h a2 ai a cþ ; 2 4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a4 1 þ m ðanÞ ; qðnÞ ¼ n4 þ þ 16 1 m 2

a01 ¼ 2m2 ð1 mÞð1 2mÞ; a1 ¼ ð1 m2 Þð1 2mÞ; a2 ¼ 2ma1 ; a01 ; b01 ¼ 2m2 ð1 2mÞ; b02 ¼ 2mð1 mÞ; b1 ¼ 2a1 ; a3 ¼ 2m 2 b21 ¼ 2ð1 mÞ ; b22 ¼ 2a3 ; b3 ¼ b22 ; G1 ðnÞ N 3 ðnÞ ¼ 2HðnÞ þ ð1 þ gHðnÞ þ 2ð1 mÞHðnÞp1 ðnÞ mN 1 ðnÞÞ DðnÞ G ðnÞ 2 þðN 2 ðnÞ þ 2HðnÞn2 Þ n; DðnÞ

ðA3Þ

ðA4Þ

½ð1 mÞn2 þ mð1 2mÞ2 a2 N 1 ðnÞ ¼ 2ð1 mÞð1 2mÞHðnÞp21 ½p1 2ð1 mÞa 4mð1 mÞð1 2mÞHðnÞan2 2

þ 2ð1 mÞHðnÞðp1 gÞ½ð1 2mÞ a2 þ 2mn2 2ð1 mÞð1 2mÞg½2ð1 mÞaðg 2p1 Þ þ ðg2 3gp1 þ 3p21 ÞHðnÞ þ ð1 2mÞp1 ½d 2ð1 mÞa þ ð1 2mÞ2 a2 þ 2ð1 2mÞn2 ; 2

ðA5Þ

2 2

½ð1 mÞn þ mð1 2mÞ a N 2 ðnÞ ¼ 2p1 ½ð1 mÞn2 ð1 2mÞmaða dÞ 2ð1 mÞð1 2mÞan2 2ð1 2mÞn2 HðnÞ½ð1 mÞn2 þ ma2 þ 2ð1 m2 Þap1 þ 4ð1 m2 Þð1 2mÞagn2 þ 2ð1 mÞ½2ð1 mÞn2 2mð1 2mÞa2 gðg 2p1 ÞHðnÞ 4ð1 mÞð1 2mÞmagðg2 3gp1 þ 3p21 ÞHðnÞ þ 4ð1 mÞHðnÞp21 ½ð1 mÞn2 mð1 2mÞa2 þ 2ð1 2mÞmap1 ; d v1 ¼ Bi þ ; 2

ðA6Þ

2

v2 ¼ BiðBi þ dÞ þ

d ; 4

v3 ¼ v1 BiðBi þ dÞ:

ðA7Þ

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