A dual-reciprocity boundary element method for axisymmetric thermoelastodynamic deformations in functionally graded solids

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1 APCOM & ISCM th Decembe, 013, Singapoe A dual-ecipocity bounday element method fo axisymmetic themoelastodynamic defomations in functionally gaded solids *W. T. Ang and B. I. Yun Division of Engineeing Mechanics, School of Mechanical and Aeospace Engineeing, Nanyang Technological Univesity, Singapoe. *Coesponding autho: wtang@pmail.ntu.edu.sg Abstact A dual-ecipocity bounday element method is outlined fo solving a class of initial-bounday value poblems concening axisymmetic themoelastodynamic defomations in functionally gaded mateials. The time deivatives of the tempeatue and the displacement, which appea in the govening patial diffeential equations, ae suppessed by using the Laplace tansfomation technique. In the Laplace tansfom domain, the poblem unde consideation is fomulated in tems of integal equations which contain both bounday integals and domain integals. The dualecipocity method is used togethe with suitably constucted intepolating functions to educe the domain integals appoximately into bounday integals. The poblem unde consideation is eventually educed to linea algebaic equations which may be solved fo the numeical values of the Laplace tansfoms of the tempeatue and the displacements at selected points in space. The tempeatue and the displacement in the physical time domain ae appoximately ecoveed by using a numeical method fo inveting Laplace tansfoms. To check that the numeical pocedue pesented is valid, it is applied to solve a specific test poblem which has a closed-fom analytic solution. Keywods: Bounday element method, Dual-ecipocity method, Intepolating functions, Laplace tansfomation, Axisymmetic themoelasticity, Functionally gaded mateials. Intoduction In ecent yeas, thee has been consideable inteest in the analysis of axisymmetic mateials possessing mateial popeties that ae gaded continuously along the axial and adial diections. Fo example, Clements and Kusuma (011) studied the axisymmetic defomation of an elastic half space having elastic moduli that vay as a quadatic function of the axial coodinate; Matysiak, Kulchytsky-Zhyhailo and Pekowski (011) consideed the Reissne-Sagoci poblem fo a homogeneous laye bonded to an elastic half space with a shea modulus that vaies axially in accodance with a simple powe law; and Keles and Tutuncu (011) calculated the dynamic displacement and stess fields in hollow cylindes and sphees with mateial popeties that ae functionally gaded along the axial diection by a simple powe law. In the pesent pape, the dual-ecipocity bounday element appoach and the intepolating functions poposed in Yun and Ang (01) fo solving an axisymmetic themoelastostatic poblem involving functionally gaded mateials is extended to themoelastodynamic defomations. The mateial popeties vay with the axial and adial coodinates following sufficiently smooth functions in geneal foms. It may be of inteest to note that a bounday element solution of the coesponding two-dimensional themoelastodynamic poblem fo functionally gaded solids may be found in a vey ecent pape by Ekhlakov. Khay, Zhang, Sladek and Sladek (01). 1

2 Basic equations of axisymmetic themoelastodynamics With efeence to the cylindical pola coodinates, and, the tempeatue T and the displacement u in an isotopic solid that is symmetical about the axis is independent of and the only non-eo components of the displacement u ae given by u and u. If the mateial popeties of the solid ae adially and axially gaded using sufficiently smooth functions of and, the govening patial diffeential equations of axisymmetic themoelastodynamics ae given by u u u T ( T) Q T0 [ ] c, t t (1) u 1 u u u axisu ( ) 1 1 T u { ( 0 ) ( u T T F ) u u u u u [ ( )] }, 1 t () 1 u u u axisu ( ) 1 1 T u { ( 0 ) ( u T T F ) u u u u u [ ( )] }, 1 t (3) 1 whee axis, t is the time coodinate, T 0 is a constant efeence tempeatue at which the body does not expeience any themally induced stess, the coefficients,,, c, and ae espectively the themal conductivity, stess-tempeatue coefficient, density, specific heat capacity, Poisson's atio and shea modulus of the isotopic body, F and F ae espectively the and the components of the body foce, and Q is the intenal heat geneation tem. Note that,,, c and ae, in geneal, functions of and and the Poisson s atio is assumed to be constant. The body foce components F and F and the intenal heat geneato Q ae, in geneal, functions of the axisymmetic coodinates and and the time coodinate t. Details on the basic equations of themoelasticity may be found in Nowacki (1986).

3 Bounday-domain integal equations The govening patial diffeential equations in (1), () and (3) in tems of the bounday-domain integal equations (, ) (, ) T(,, t) and { Tt (,, )[ ( G, ) 1(, ; 0, 0; n, n) G0(, ; 0, 0)] n (, G ) 0(,; 0, 0)(, qtn,;, n)} ds (, ) Qt (,,) G0(,; 0, 0)[ T(,,) t axis( (, )) (, c )(, ) Tt (,, ) t (, T ) 0 u(, t,) u(, t,) u(, t,) [ ] dd t fo (, ), 0 0 (4) (, ) u (,, t) 0 0 K 0 0 ( (, ;, ) p( tn,, ;, n) K 0 0 (,;, ; n, n) u(, t,)) ds (, ) K K (,; 0, 0){ [ Tt (,,)] x [ (, )]( Tt (,, ) T0 ) F ( t,, ) x x [ (, )] [ u ( t,, )] (1 ) XN [ un (,,)] t YN [ un (,,)] t uk (,,) t (, ) } dd t fo (, ) ( K, ), 0 0 (5) whee is the solution domain on the O plane, is the bounday of (excluding the pat that lies on the axis), n and n ae espectively the and components of the unit nomal outwad vecto to cuve at the point (, ), G (,;, ) is the fundamental solution of axisymmetic Laplace s equation, G1(,; 0, 0; n, n) is the nomal deivative of G0(,; 0, 0) along the diection of the vecto [ n, n ], the uppecase Latin subscipts (such as K ) ae assigned values and and 3

4 summation ove those values ae implied fo epeated subscipts, (,; K 0, 0) is the fundamental solution of the patial diffeential equations fo axisymmetic elastostatics, K (,; 0, 0; n, n ) is the taction function coesponding to (,; K 0, 0), and p( tn,, ;, n ), XN (, ) and Y (, ) N ae defined by u u u u p (,, t; n, n ) ( [ ]) n (, ) 1 u u ( ) n (, ), u u p (,, t; n, n ) ( ) n (, ) u u u u ( [ ]) n (, ), 1 (6) X (, ), X (, ), 1 (1 ) X (, ), X (, ), 1 (1 ) Y (, ), Y (, ), 1 (, ) (, ) Y (, ), Y (, ). 1 (7) The functions p,, t; n, n ) ae elated to the axisymmetic tactions t,, t; n, n ) though ( ( (8) t(, tn,;, n) p(, tn,;, n) [ Tt (,,) T] n, 0 L L whee N is the Konecke-delta. The bounday-domain integal equations in (4) and (5) fo the coesponding case of axisymmetic themoelastostatic defomations ae given in Yun and Ang (01) whee the details of the functions G0(,; 0, 0), G1(,; 0, 0; n, n), (,; K 0, 0) and K (,; 0, 0; n, n ) ae explicitly witten out. Dual-ecipocity bounday element method The dual-ecipocity method in Patidge, Bebbia and Wobel (199) may be employed to appoximate the domain integals ove in the integal equations (4) and (5) in tems of bounday integals ove the cuve by using intepolating functions centeed about selected collocation 4

5 points in. As in Yun and Ang (01), the collocating functions centeed about the n-th collocation point, denoted by,, and ), ae assumed to be sufficiently smooth and ae equied to satisfy the patial diffeential equations ) axis, ) ) axis ) 1 ) ) ( [ (, )] [ (, )]) (, ), 1 ) axis ) 1 ) ) ( [ (, )] [ (, )]) (, ). 1 (9) In Agnantiais, Polyos and Beskos (001) and Wang, Mattheij and te Mosche (003), the intepolating functions,, and ) ae constucted by integating axially selected adial basis functions in thee-dimensional space. The intepolating functions thus constucted ae well defined at 0, but they ae in highly complicated foms and ae expessed in tems of special functions given by the elliptic integals. To constuct intepolating functions expessed in tems of elatively simple elementay functions, one may choose and ) to be sufficiently smooth functions of ( ) ) 0 0 ) 0 0 ( ) ( ), whee (, ) is the n-th collocation point, and detemine ) and ) using (9). Nevetheless, the intepolating functions and ) constucted in this manne ae not well defined at 0. This poses a poblem if the axis is pat of the solution domain. In Yun and Ang (01), the singula behavios of and ) at 0 ae emoved by modifying and in such a way that and ) behave as O ( ) fo small. Specifically, and ) ae taken to be ) 1 ) 3 ) 3 {[ (,; 0, 0 )] [ (,; 0, 0 )]}, 9 ) ) 3 [ (0,; 0, 0 )], 9 ) (, ) (, ) 0, (, ) (, ), (10) whee (,,, ) ( ) ( ) Fo a numeical pocedue fo solving initial-bounday value poblems govened by (1), () and (3), we apply the Laplace tansfomation on the bounday-domain integal equations (4) and (5) to suppess the time deivatives of, u and u, use the dual-ecipocity method togethe with the T 5

6 intepolating functions constucted using (10) to appoximate the domain integals in the esulting bounday-domain integal equations in tems of bounday integals, and discetie the bounday into elements to develop a bounday element pocedue fo finding the tempeatue and the displacement in the Laplace tansfom domain. The tempeatue and the displacement in the physical domain may be ecoveed by using a numeical method fo inveting Laplace tansfoms. Test poblem The coefficients of the patial diffeential equations in (1), () and (3) ae chosen to be given by, c,,, ( ), 3/10, and Qt t t 3 3 (,,) {sin()[ ] cos( ) 16 4 }, 1 F (,,) t {cos()[4 t ] t (1 sin( ))[ 8 4 ] 4 }, 1 F (,,) t {cos()[ t ] 4 3 t 5 3 (1 sin( ))[ 6 ] 4 }. 3 It is easy to check that a solution of the patial diffeential equations is given by Tt t (,,) (sin() 1), u t t (,,) ( )cos(), u (,,) t ( )cos(). t (11) Fo a specific initial-bounday value poblem as a test poblem, take the solution domain to be 1, 0 1, which is a ectangula egion on the O plane, and use the solution in (11) to u geneate the following initial and bounday data (a) initial values of T, u, u, t u t at time t 0 at points ( in, ), (b) bounday values of the displacement ( u, u) on the entie bounday of fo time t 0, (c) bounday values of T on the sides of the ectangula egion whee 0 and 1 fo time 0, T on the sides of the ectangula egion whee 1 and fo time t 0. 6

7 Fo the bounday element pocedue, the sides of the ectangula egion ae discetied into 80 staight line elements. The Laplace tansfoms of the tempeatue, heat flux, displacement and taction on the bounday elements ae appoximated using discontinuous linea functions. As many as 11 well distibuted collocation points in (including those on the bounday elements) ae used in the dual-ecipocity method fo conveting appoximately the domain integals in the integal fomulation of the initial-bounday value poblem into bounday integals. We use the numeical method in Stehfest (1970) to invet the Laplace tansfoms in ode to ecove the tempeatue and the displacement in the physical domain. Numeical values of T, u and u obtained using the dual-ecipocity bounday element method (DRBEM) ae plotted against t (0t 6) at (, ) (1.5, 0.5) in Figues 1, and 3 espectively. The numeical values agee well with the analytical solution in (11), showing that the intepolating functions given in (9) and (10) ae employed successfully to teat the domain integals in the bounday-domain integal equations in (4) and (5). Figue 1. A compaison of the numeical and exact T at (, ) (1.5,0.5) fo 0 t 6. 7

8 Figue. A compaison of the numeical and exact u at (, ) (1.5,0.5) fo 0 t 6. Figue 3. A compaison of the numeical and exact u at (, ) (1.5,0.5) fo 0 t 6. Refeences Agnantiais,. P., Polyos, D. and Beskos, D. E. (001), Fee vibation analysis of non-axisymmetic and axisymmetic stuctues by the dual-ecipocity BEM, Engineeing Analysis with Bounday Elements, 5, pp Clements, D. L. and Kusuma,. (011), Axisymmetic loading of a class of inhomogeneous tansvesely isotopic halfspaces with quadatic elastic moduli, Quately ounal of Mechanics and Applied Mathematics, 64, pp Ekhlakov A. V., Khay, ). M., Zhang Ch., Sladek,. and Sladek V. (01), A DBEM fo tansient themoelastic cack poblems in functionally gaded mateials unde themal shock, Computational Mateial Science, 57, pp

9 Keles, I. and Tutuncu, N. (011), Exact analysis of axisymmetic dynamic esponse of functionally gaded cylindes (o disks) and sphees, ounal of Applied Mechanics, 78, Matysiak, S.., Kulchytsky-Zhyhailo, R. and Pekowski, D. M. (011), Reissne-Sagoci poblem fo a homogeneous coating on a functionally gaded half-space, Mechanics Reseach Communications, 38, pp Nowacki, W. (1986), Themoelasticity, Wasaw and Pegamon Pess, Oxfod. Patidge, P. W., Bebbia, C. A. and Wobel, L. C. (199), The Dual Recipocity Bounday Element Method, Computational Mechanics Publications, London. Stehfest, H. (1970), Numeical invesion of the Laplace tansfom, Communications of ACM, 13, pp (see also p64). Wang, K., Mattheij, R. M. M. and te Mosche, H. G. (003), Altenative DRM fomulations, Engineeing Analysis with Bounday Elements, 7, pp Yun, B. I. and Ang, W. T. (01), A dual-ecipocity bounday element method fo axisymmetic themoelastostatic analysis of nonhomogeneous mateials, Engineeing Analysis with Bounday Elements, 36, pp