International Journal of Solids and Structures

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International Journal of Solids and Structures 6 (9) 8 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homeage: www.elsevier.

Barber c a School of Mechanical Engineering, Yonsei University, Shinchon-dong, Seodaemun-gu, Seoul -9, Reublic of Korea b Orrick, Herrington & Sutcliffe LLP, Marsh Road, Menlo Park, CA 9, USA c

1 International Journal of Solids and Structures 6 (9) 8 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homeage: The thermoelastic Hertzian contact roblem Yong Hoon Jang a, *, Hanbum Cho b, J.R. Barber c a School of Mechanical Engineering, Yonsei University, Shinchon-dong, Seodaemun-gu, Seoul -9, Reublic of Korea b Orrick, Herrington & Sutcliffe LLP, Marsh Road, Menlo Park, CA 9, USA c Deartment of Mechanical Engineering and Alied Mechanics, University of Michigan, Ann Arbor, MI 89-, USA article info abstract Article history: Received Aril 9 Received in revised form June 9 Available online August 9 Keywords: Hertzian contact Thermoelastic contact Thermal contact resistance Thermoelasticity A numerical solution is obtained for the steady-state thermoelastic contact roblem in which heat is conducted between two elastic bodies of dissimilar materials at different temeratures with arbitrary quadratic rofiles. Thermoelastic deformation causes the initially ellitical contact area to be reduced in size and to become more nearly circular as the temerature difference is increased. There is also a small but identifiable deviation from exact elliticity at intermediate temerature differences. An aroximate analytical solution is obtained, based on aroximating the contact area by an ellise. Ó 9 Elsevier Ltd. All rights reserved.. Introduction When two conforming bodies are laced in contact, the contact ressure distribution is sensitive to comaratively small changes in surface rofile. Thermoelastic deformations, though generally small, can therefore have a significant effect on systems involving contact. For examle, Clausing (966) showed exerimentally that the thermal contact resistance between two contacting bodies varied with the transmitted heat flux as a result of thermoelastically driven changes in the extent of the contact area. If the contacting bodies are small, their surfaces can be aroximated by quadratic functions in the vicinity of the contact area and in the absence of thermoelastic deformation, the solution of the elastic contact roblem is given by the classical Hertz theory (Johnson, 98). In articular, the contact area is an ellise whose elliticity and orientation are unique functions of the coefficients defining the quadratic surfaces and whose linear dimensions vary with P =, P is the contact force. If the extremities of the two bodies are now raised to different temeratures T T, heat will flow through the contact area and the resulting thermoelastic deformation will influence the contact area and the contact ressure distribution. This roblem was solved by Barber (9) for the secial case the bodies are axisymmetric. In this case, the contact area is always circular and its radius a for a given contact force decreases with increasing temerature difference, being given by * Corresonding author. Tel.: +8 8 fax: address: jyh@yonsei.ac.kr (Y.H. Jang). ^a þ H^a ¼ ^a ¼ a H ¼ ðd d ÞðT T ÞKR a H a H R ¼ þ R R K ¼ K þ K a H is the radius of the isothermal (Hertzian) elastic contact area for the same contact force P R R are the radii of the two contacting bodies and the distortivity d is defined as d ¼ að þ mþ ðþ K a m K are the coefficient of thermal exansion, Poisson s ratio and thermal conductivity. This solution is strictly only alicable when the heat flows into the body with the higher distortivity and hence the dimensionless arameter H >, since for the oosite direction of heat flow, a small annulus of imerfect thermal contact is develoed at the edge of the contact area (Barber, 98 Kulchytsky-Zhyhailo et al., ). In this aer, we use a numerical method to determine the effect of thermoelastic deformation for the more general Hertzian case the bodies have general quadratic shaes and the isothermal contact area is ellitical. We shall show that the contact area becomes smaller and also more nearly circular as the temerature difference is increased. We shall also develo an aroximate analytical solution to the roblem, using an aroach roosed by Yevtushenko and Kulchytsky-Zhyhailo (996). ðþ ðþ -68/$ - see front matter Ó 9 Elsevier Ltd. All rights reserved. doi:.6/j.ijsolstr.9.8.

2 Y.H. Jang et al. / International Journal of Solids and Structures 6 (9) 8. Statement of the roblem We consider the roblem in which two thermally conducting elastic bodies are ressed together by a force P, whilst their extremities are maintained at temeratures T and T, resectively. Frictionless contact conditions are assumed and heat flow between the elastic bodies is only ermitted to take lace by conduction through the contact area A. As in the axisymmetric case, we restrict attention to the case the heat flows into the more distortive material and hence ðt T Þðd d Þ <... The heat conduction roblem The temerature at the oint defined by coordinates ðx yþ on the surface of body i can be written as T i ðx yþ ¼ qðn gþdndg þ T i ðþ K i r q is the heat flux directed into the body and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðx nþ þðy gþ : ðþ In the absence of surface tractions, this heat flux would also cause thermoelastic dislacement w i in the inward normal direction given by (Barber, 9) w i ðx yþ ¼ d i qðn gþ lnðrþdndg we have omitted a rigid-body dislacement. Continuity of heat flux and temerature at the contact area then leads to the integral equation DT T T ¼ K A ð6þ qðn gþdndg ðþ r K is defined in Eq. (). This equation serves to determine the heat flux q, which is here taken as ositive in the direction from body to body. Once q is determined, the differential thermoelastic exansion can then be determined from Eq. (6) as w ðx yþþw ðx yþ ¼ ðd d Þ.. The contact roblem qðn gþ lnðrþdndg: We suose that the two contacting bodies have rofiles defined by the functions g ðx yþ g ðx yþ, as shown in Fig., so that the initial ga between the undeformed bodies is g ðx yþ ¼g ðx yþþg ðx yþ: As in the Hertzian theory, we assume that the contact area is sufficiently small to ermit this exression to be reresented by the quadratic function ð8þ ð9þ g ðx yþ ¼ x R I þ y R II ðþ R I R II are the rincial radii of curvature of the combined rofile, as defined by Eq. (.) of Johnson (98). If the bodies are now ressed together, a contact ressure ðx yþ will be develoed in the contact area, which will generate elastic normal surface dislacements u i ðx y Þ ¼ m i ðn gþdndg ðþ E i r directed into the resective bodies i ¼, E i is Young s modulus of the contacting body i. The final ga between the bodies is given by gðx yþ ¼g ðx yþþu ðx yþþu ðx yþþw ðx yþþw ðx yþþd ðþ d is an unknown rigid body dislacement. The contact roblem can then be stated by noting that the ga is zero by definition in the contact area and ositive outside, leading to the unilateral contact roblem gðx yþ¼ ðx yþ > ðx yþa ðþ ðx yþ ¼ gðx yþ > ðx yþ R A: ðþ Combining Eqs. (8), () (), we then have ðn gþdndg E þ ðd d Þ qðn gþ lnðrþdndg A r A ¼ d g ðx yþ ðx yþ A > d g ðx yþ ðx yþ R A ðþ E ¼ m E þ m E : ð6þ The corresonding total force is then given by P ¼ ðn gþdndg: A ðþ The integral equations () and () serve to determine the heat flux q and the contact ressure, whilst Eq. () and the inequality in Eq. () determine the extent of the contact area A... Dimensionless formulation The number of indeendent arameters can be reduced by using an aroriate dimensionless reresentation. We first define two length scales R a H through the relations R ¼ R I þ R II rffiffiffiffiffiffiffiffi PR a H ¼ E ð8þ Fig.. Initial ga between two bodies.

3 Y.H. Jang et al. / International Journal of Solids and Structures 6 (9) 8 and then define dimensionless coordinates ^x ¼ x=a H ^y ¼ y=a H. Notice that a H is the isothermal Hertzian contact radius for the axisymmetric case R I ¼ R II ¼ R. The governing equations can then be written as ^ð^n ^gþd^nd^g þ ^r ¼ ^d ^x þ R ^y ð þ R Þ ð^x ^yþ ^qð^n ^gþ lnð^rþd^nd^g > ^d ^x þ R ^y ð þ R Þ ð^x ^yþ R ð9þ R ¼ R I R II ^qð^n ^gþd^nd^g ¼ H ^r ^ð^n ^gþd^nd^g ¼ ^ ¼ R E a H H ¼ ðd d ÞKRðT T Þ a H > : Rd ^d ¼ ^q ¼ðd a d ÞRq H ðþ ðþ ðþ Notice that with this formulation, the roblem is comletely defined by the dimensionless arameters H R, since ^d must be chosen so as to satisfy the equilibrium condition (). We also note that in the isothermal case H ¼, Eq. (9) reduces to the classical Hertzian equation with solution (Johnson, 98) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ð^n ^gþ ¼ ^x ^y ðþ ^a^b ^a ^b and the contact area is an ellise of semi-axes ^a ^b determined by the two simultaneous equations ð þ R Þ ¼ e ^a ½KðeÞ EðeÞŠ ðþ R ð þ R Þ ¼ EðeÞ e ^a KðeÞ ðþ ^b ffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ^a e ¼ k k ð6þ and KðeÞ and EðeÞ are the comlete ellitic integrals of the first and second kind, resectively.. Numerical imlementation For the numerical solution, we use a strategy based on Hartnett s solution to the isothermal contact roblem (Hartnett, 99). Suose that a lane rectangular region, referred to as the blanket region, is chosen larger than the exected contact area and this region is divided into N rectangular segments j ¼ N over which the dimensionless ressure ^ j and heat flux ^q j are assumed to be constant. The otimal size of the blanket for a given run of the rogram and given comutational resources is only just large enough to contain the contact area. The contact area is defined in the numerical solution by the finite set, A, of rectangular segments in contact. If this set were known, the corresonding heat fluxes could be found from the discrete form of Eq. () which can be written as X C ij^q j ¼ H i A ðþ ja C ij is a set of influence coefficients defined in Aendix A. However, the contact set A is determined by the inequalities in the discrete form of Eq. (9) which we write X C ij^ j þ X D ij^q j ¼ ^d ^x i R ^y i ð þ R Þ i A > ^d ^x i R ^y i ð þ R Þ i R A ja ja ð8þ D ij is a set of influence coefficients aroriate to the second integral in Eq. (9) and is defined in Aendix A. We therefore adot an iterative solution to the roblem in which Eqs. (9) and () are solved alternately, the contact set at the latest iteration of Eq. (9) being used in Eq. () and the heat fluxes q j from the solution of Eq. () being taken as known in the next iteration of Eq. (9). Notice also that in the iterative solution of Eq. (9), the arameter ^d must be chosen to satisfy the equilibrium condition () which has the discrete form X ^ j A j ¼ ð9þ ja A j is the area of the segment j... Numerical validation and convergence To validate the numerical rogram and also to exlore the mesh refinement required to give a good descrition of the thermoelastic contact behaviour, we first aly the numerical method to the axisymmetric Hertizan contact roblem solved analytically by Barber (9), for which the contact area is given by Eq. () and the contact ressure distribution is ^ð^rþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^a ^r þ H ( 8 v ^a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!) ^a ^r ^a þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ ^a ^r v ðxþ ¼ Z x ln þ y dy y y ¼ X m¼ x m ðm Þ : ðþ Fig. shows the ercentage difference between the analytical maximum contact ressure ^ðþ and the numerical value as a function of the number N of elements in the contact area, for a dimensionless temerature difference H ¼. In the same figure, we also show the ercentage error in the contact radius ^a which is estimated in the numerical solution by equating the total contact area P A j j A to ^a. In the following numerical results, the number of elements in the contact area is above,.. Non-axisymmetric results Numerical results were obtained for various values of the dimensionless arameters R H. Fig. shows ressure contours and the extent of the contact area for R ¼ = and various values of the temerature difference H. In the isothermal case H ¼, the classical Hertzian analysis alies and the contact area is ellitical. As H is increased, the contact area gets smaller and its elliticity is reduced. The solution can be seen as a trade-off between the elastic and thermoelastic terms in Eq. (9). The limit H!can be aroached either by allowing the temerature difference to increase without limit or by allowing R!, which corresonds to the contact between two bodies with lane surfaces. In the latter case, it is clear that the ratio R becomes irrelevant and the contact area is circular, being given by Eq. () as rffiffiffiffiffiffiffi ^a ¼ ðþ H or in dimensional terms

4 6 Y.H. Jang et al. / International Journal of Solids and Structures 6 (9) 8 Error (%) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P a ¼ E : ðþ ðd d ÞðT T ÞK. Ellitical aroximation to the contact area Fig. shows that the contact area remains aroximately ellitical for all values of H and this suggests an alternative aroximate analytical aroach to the roblem. Suose we assume the contact area to be an ellise of rescribed dimensionless semi-axes ^a ^b. Eq. () then defines a classical roblem in otential theory with solution H ^qð^n ^gþ ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^bkðeþ ^n : ðþ =^a ^g =^b We next calculate the second integral term in Eq. (9), which we write as ^w ð^x ^y Þ ¼ N (x ) Fig.. Percentage difference between the analytical and numerical values of maximum contact ressure () and contact radius () as a function of the number of elements defining the contact area. H ^bkðeþ lnð^rþd^nd^g q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^n : ðþ =^a ^g =^b This integral is evaluated in Aendix B, after which we aroximate ^wð^x ^yþ by the quadratic function ^w ð^x ^y Þ ¼ C þ C ^x þ C ^y : ð6þ The curvatures C C are chosen so as to agree with the exact result at the center and at the ends of the major and minor axes of the ellise ð^a Þ ð ^bþ, giving C ¼ ^wð^a Þ ^wð Þ ^a C ¼ ^wð ^bþ ^wð Þ ^b : ðþ The constant C can be wraed into ^d in Eq. (9). This technique was used by Yevtushenko and Kulchytsky-Zhyhailo (996) for the related roblem of thermoelastic contact heat is generated at the interface due to frictional sliding. With this aroximation, Eq. (9) once again defines a classical Hertzian contact roblem for ^ and the semi-axes of the contact ellise are defined by the modified equations ð þ R Þ þ H k^akðeþ U ¼ k e ^a ½ KðeÞ EðeÞ Š ð8þ R ð þ R Þ þ H k^akðeþ UðkÞ ¼ EðeÞ e ^a KðeÞ ð9þ k UðkÞ ¼ Z = cos uarctanhðcos uþþ lnð cos uþ du : ðþ k sin u þ cos u In the axisymmetric limit R ¼ k ¼ e ¼ and Eqs. (8) and (9)both reduce to ^a þ HUðÞ^a ¼ ðþ with UðÞ ¼=ð lnðþþ. Comaring with the exact exression (), we find that the error in the multilier on the second term is.9%. 6. Results Fig. shows the dimensionless major axis ^a as a function of H for various values of R as redicted by the numerical solution and by the aroximate solution of Eqs. (8) and (9). To comare redictions of the shae of the contact area, we resent values of the ratio k ¼ ^b=^a as a function of H for various values of R in Fig.. It is clear that thermoelastic effects tend to reduce the elliticity of the contact area. In the limit H!, the contact area becomes circular and the solution is adequately described by the axisymmetric theory. Notice that the aroximate theory consistently overestimates k (and hence underestimates the elliticity e) at larger values of H. Numerical redictions of k in the isothermal (Hertzian) case H ¼ are extremely good, so we conclude that this discreancy is a real effect and not attributable to discretization. To obtain a more robust characterization of the shae of the contact area redicted by the numerical solution, we first defined the oints on the discrete boundary in the form of a iecewise continuous function ^rðhþ in olar coordinates. If the contact area were a true ellise, we would have ^r cos h þ ^r sin h ^a k ^a ¼ ðþ and hence Θ= Θ=. Θ=.8 Θ=. Θ=.6 Fig.. Contact ressure distribution and extent of the contact area for R ¼ = and various values of the dimensionless temerature difference H.

5 λ Y.H. Jang et al. / International Journal of Solids and Structures 6 (9) 8 a.... R * =/ R * =/ R * =/ R * =/ Table Comarison of Fourier coefficients for the series () with the aroximate analytical solution. H.9.89 Hertzian Numerical c c c.9.6. c c Conclusion 6 8 Fig.. Dimensionless major axis of the contact area (^a) as a function of dimensionless temerature difference H. The circles and the solid lines reresent the numerical and aroximate analytical solutions, resectively R * =/ R * =/ R * =/ R * =/ 6 8 Θ Fig.. The ratio of major/minor axes of the contact area (k ¼ b=a) as a function of dimensionless temerature difference H. The circles and the solid lines reresent the numerical and aroximate analytical solutions, resectively. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^rðhþ ¼ ^a k : ð þ k Þ ð k Þ cosðhþ ðþ It then follows that the Fourier series reresentation of the function X f ðhþ ¼ c ^r n cosðnhþ: n¼ ðþ would have only two non-zero coefficients, c c. Table shows the first four coefficients of this series for R ¼ and several values of H. In the first two columns, we comare the numerical values of the coefficients with the exact (Hertzian) values from Eqs. () and (). The magnitude of the higher coefficients c c 6... in the numerical solution for H ¼ rovides an indication of the error due to discretization. For H, the third coefficient c is significantly larger than this error and hence describes a real effect. Its sign is such as to indicate an elongation of the contact area on the major and minor axes and a reduction in ^rðhþ at h ¼. However, the higher coefficients are still much smaller than c c indicating that the contact area remains redominantly ellitical. We have resented a numerical solution and an aroximate analytical solution to the roblem of the general thermoelastic Hertzian contact roblem with heat flow through the contact area driven by a temerature difference between the extremities of the two contacting bodies. The solution is characterized by only two dimensionless arameters, the ratio R of rincial curvatures of the bodies which governs the elliticity of the contact area in the isothermal (Hertzian) case, and a dimensionless temerature difference H. The contact area remains substantially ellitical for all values of H, but the elliticity decreases with increasing H, aroaching the limiting axisymmetric solution as H!. The analytical aroximation, based on a technique due to Yevtushenko and Kulchytsky- Zhyhailo (996), underestimates the elliticity at intermediate values of H. Aendix A. C ij and D ij of Eq. (8) If x j y j are the coordinates of the center of the rectangular contact element j of dimensions h l, the influence coefficients C ij D ij of Eqs. () and (8) are C ij ¼ð^x j ^x i þ ^hþln 6 ð^y j ^y i þ^lþþ ð^y j ^y i þ^lþ þð^x j ^x i þ ^hþ ð^y j ^y i ^lþþ ð^y j ^y i ^lþ þð^x j ^x i þ ^hþ þð^x j ^x i ^hþln 6 ð^y j ^y i ^lþþ ð^y j ^y i ^lþ þð^x j ^x i ^hþ ð^y j ^y i þ^lþþ ð^y j ^y i þ^lþ þð^x j ^x i ^hþ þð^y j ^y i þ^lþln ð^x j ^x i þ ^hþþ 6 ð^y j ^y i þ^lþ þð^x j ^x i þ ^hþ ð^x j ^x i ^hþþ ð^y j ^y i þ^lþ þð^x j ^x i ^hþ þð^y j ^y i ^lþln ð^x j ^x i ^hþþ 6 ð^y j ^y i ^lþ þð^x j ^x i ^hþ ð^x j ^x i þ ^hþþ ð^y j ^y i ^lþ þð^x j ^x i þ ^hþ D ij ¼ ^h ð^x j ^x i Þ ^lþð^yj ^y i Þ lnfð^h ð^x j ^x i ÞÞ þð^lþð^y j ^y i ÞÞ g þðþð^lþð^y j ^y i ÞÞlnfðÞ þ ^lþð^y g j ^y i Þ þ ^hþð^x j ^x i Þ ^l ð^yj ^y i Þ lnfðþ þ ^l ð^y g j ^y i Þ þ ^h ð^x j ^x i Þ ^l ð^yj ^y i Þ lnfð^h ð^x j ^x i ÞÞ þð^l ð^y j ^y i ÞÞ g þðþ ^lþð^yj ^y i Þ ^l ð^yj ^y i Þ arctan þarctan ðþ

6 8 Y.H. Jang et al. / International Journal of Solids and Structures 6 (9) 8 ^h ð^x þð^lþð^y j ^y i ÞÞ j ^x i Þ arctan þarctan ^lþð^yj ^y i Þ ^lþð^yj ^y i Þ þ ^l ð^y ^h ð^xj ^x i Þ j ^y i Þ arctan þarctan ^l ð^yj ^y i Þ ^l ð^yj ^y i Þ þð^h ð^x j ^x i ÞÞ ^lþð^yj ^y i Þ ^l ð^yj ^y i Þ arctan þarctan ^h^l ^h ð^x j ^x i Þ ^h ð^x j ^x i Þ ð6þ is bounded. Using the change of variable cos u ¼ ^as sin u ¼ ^bt, we then obtain Z Z cos udu J = ðþ sinðwþd I ¼ ^b ^a sin u þ ^b cos u = ^b ffiffiffi Z wfð = = w Þ cos udu ¼ ffiffiffi ^a sin u þ ^b ðþ cos u from Gradshteyn and Ryzhik (98), , Aendix B. Evaluation of Eq. () w ¼ x^a cos u þ y^b sin u ðþ Following Yevtushenko and Kulchytsky-Zhyhailo (996), we note that I ð^x^yþ Z Z f ðs tþ ¼ Z Z f ð^n ^gþlnð^rþd^nd^g ¼ Z Z f ð^n ^gþe ıð^nsþ^gtþ d^nd^g is the double Fourier transform of f ð^n ^gþ. For the integral in Eq. (), we have e ıð^nsþ^gtþ d^nd^g f ðstþ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^n ¼ =^a ^g =^b f ðstþe ıð^xsþ^ytþ dsdt ðþ ðs þ t Þ cosð^ns þ ^gtþd^nd^g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^n =^a ^g =^b ð8þ ð9þ and the ellise can be maed to the unit circle using the change of variable ^n ¼ ^a cos u ^g ¼ ^b sin u, giving Z Z d f ðs tþ ¼^a^b ffiffiffiffiffiffiffiffiffiffiffiffiffi cos ð^as cos u þ ^bt sin uþ du ffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^a^b ¼ ^a s þ ^b = J = ^a s þ ^b t ðþ t using Gradshteyn and Ryzhik (98),.9. and Using this result in Eq. (), we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z I ¼ ffiffiffiffiffiffi Z J = ^a s þ ^b t cosð^xs þ ^ytþds dt ^a^b ðs þ t Þ ^a s þ ^b = ðþ t but this integral is unbounded because of the behaviour of the integrand at the origin. This is to be exected, since if the net heat flow into a body is not zero, the rigid-body thermoelastic dislacement of the heated region is unbounded relative to the oint at infinity (Barber, 9). However, only the shae of the thermoelastically distorted surface lays a rôle in the contact roblem, and the integral I ¼ Z Z J = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^a s þ ^b t sinð^xs þ ^ytþsdsdt = ðþ ðs þ t Þ ^a s þ ^b t and Fða b c xþ is the Gauss hyergeometric function. Using Gradshteyn and Ryzhik (98), 9.., 9.., and 9.. and results from Yevtushenko and Kulchytsky-Zhyhailo (996), we obtain wfð = = w Þ¼arctanhðwÞ ¼ ln þ w ðþ w and hence we ¼ ffiffiffiffiffiffi Z ^a^bi ¼ ^a^b arctanhðwþ cos udu ^a sin u þ ^b cos u : ð6þ A similar rocedure can be used to =@^y after which we deduce that the integral I has the form I ¼ ^a ^b Z ½warctanhðwÞþ lnð w ÞŠdu ^a sin u þ ^b ðþ cos u aart from an infinite constant which can be wraed into ^d. References Clausing, A.M., 966. Heat transfer at the interface between dissimilar metals the influence of thermal strain. International Journal of Heat and Mass Transfer 9, 9 8. Barber, J.R., 9. The solution of heated unch roblems by oint source methods. International Journal Engineering Science 9, 6. Barber, J.R., 9. Indentation of the semi-infinite elastic solids by a hot shere. International Journal of Mechanical Science, Barber, J.R., 98. Contact roblems involving a cooled unch. Journal of Elasticity 8, 9. Kulchytsky-Zhyhailo, R.D., Olesiak, Z.S., Yevtushenko, O.O.,. On thermal contact of two axially symmetric elastic solids. Journal of Elasticity 6,. Gradshteyn, I.S., Ryzhik, I.M., 98. Tables of Integrals Series and Products. Academic Press, New York. Hartnett, M.J., 99. Analysis of contact stresses in rolling element bearings. Journal of Lubrication Technology Transaction of ASME, 9. Johnson, K.L., 98. Contact Mechanics. Cambridge University Press, Cambridge, MA. Yevtushenko, A.A., Kulchytsky-Zhyhailo, R.D., 996. Aroximation solution of the thermoelastic contact roblem with frictional heating in the general case of the rofile shae. Journal of the Mechanics and Physics of Solids,.

Implementation and Validation of Finite Volume C++ Codes for Plane Stress Analysis

Implementation and Validation of Finite Volume C++ Codes for Plane Stress Analysis CST0 191 October, 011, Krabi Imlementation and Validation of Finite Volume C++ Codes for Plane Stress Analysis Chakrit Suvanjumrat and Ekachai Chaichanasiri* Deartment of Mechanical Engineering, Faculty