2D Problem for a Long Cylinder in the Fractional Theory of Thermoelasticity

Transcription

1 596 D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity Abstact In this manuscipt, we solve an asymmetic D poblem fo a long cylinde. The suface is assumed to be taction fee and subjected to an asymmetic tempeatue distibution. A diect appoach is used to solve the poblem in the Laplace tansfomed domain. A numeical method is used to invet the Laplace tansfoms. Gaphically esults ae given and discussed. Keywods Factional Calculus; Infinitely Long Cylinde; Themoelasticity Hany H. Sheief a W. E. Raslan b a Depatment of Mathematics, Alexandia Univesity, Alexandia, Egypt. hanysheief@gmail.com b Depatment of Mathematics and engineeing physics, Mansoua Univesity, Mansoua, Egypt. w_aslan@yahoo.com Received In evised fom Accepted Available online NOMENCLATURE t time T absolute tempeatue ij Stess tenso components density, Lamé's constants = (3+) t αt coefficient of linea themal expansion k themal conductivity T 0 efeence tempeatue assumed to be such that ( T T0)/ T0, 0 constants such that 0 > 0, 0 c E specific heat pe unit mass in the absence of defomation

2 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 597 INTRODUCTION In 967 Lod and Shulman (Lod & Shulman, 967) wee the fist to genealize Biot s theoy of coupled themoelasticity. This theoy ensues finite speeds of popagation fo waves. Shama and Pathania studied wave popagation (Shama & Pathania, 006), Sheief and Anwe solved a two dimensional poblem fo an infinitly long cylinde (Sheief & Anwa, 994), Sheief and Ezzat obtained the fundemental solution in the fom of seies of functions (Sheief & Ezzat, 994) and Sheief and Saleh solved a geneilized themoelastic poblem fo an infinite body with a spheical cavity using complex counto integation (Sheief & Saleh, 998). An ingoing pocess is the use of factional calculus to ceate a eplacement fo many physical models (Hilfe et al., 000; Machado, Galhano, & Tujillo, 03). Sheief et al used factional deivative to genealized Hodgkin and Huxley model (Sheief, El-Sayed, Behiy, & Raslan, 0). Povstenko used factional deivatives to deive new models fo the conduction of heat (Povstenko, 009). The factional theoy of themoelasticity was intoduced in 00 (H. H. Sheief, El-Sayed, & Abd El-Latief, 00). The main eason behind the intoduction of this theoy is that it pedicts etaded esponse to physical effects, as is found in natue, as opposed to instantaneous esponse pedicted by the genealized theoy of themoelasticity. This etaded esponse stems fom the fact that factional deivatives ae in fact integals ove time. Physically this esults fom the weak van de Walles foces. In the following, some applications of the factional ode theoy of themoelasticity ae intoduced. Raslan has solved a poblem fo a cylindical cavity (Raslan, 04). El-Kaamany and Ezzat applied factional ode theoy to pefect conducting themoelastic medium (El-Kaamany & Ezzat, 0; Ezzat & El-Kaamany, 0), Sheief and Abd El-Latief studied the effect of vaiable themal conductivity on a half-space unde the factional ode theoy of themoelasticity (Sheief & Abd El- Latief, 03), also they applied the theoy to a D poblem fo a half space (Sheief & Abd El Latief, 04), Tiwai and Mukhopadhyay intoduced Bounday Integal Equations Fomulation fo Factional Ode Themoelasticity ( Tiwai & Mukhopadhyay, 04). FORMULATION OF THE PROBLEM In this manuscipt, we conside a homogeneous isotopic cylinde of adius a and infinite length. We shall use the cylindical coodinates (,,z). The initial conditions ae taken to be homogeneous. The suface of the cylinde is assumed to be taction fee and subjected to an asymmetic tempeatue distibution. The physics of the medium unde discussion ensues that all quantities ae independent of z. all functions depend on and. The displacement vecto has the non-zeo components u and v in and diections, espectively. The govening equations ae gad div gadt u u u () t k T 0 c ET T0e t t () Latin Ameican Jounal of Solids and Stuctues 3 (06)

3 598 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity whee Applying the divegence opeato to both sides of equation (), we obtain e e T (3) t whee e is the cubical dilatation given by div v e u u (4) The constitutive equations can be witten as u e T T0 (5a) v u e T T0 (5b) zz e T T 0 (5c) v v u (5d) We shall use the following non-dimensional quantities z z 0 (5e) c, u cu, t c t, v cv T T0 ij θ,, ij τ 0 c τ0, λ μ λ μ ρc whee c, η E. ρ k These non-dimensional vaiables wee fist intoduced by Sheief (980) in his PhD thesis. They wee obtained by tial and eo. They ae useful because the solution obtained using these vaiables does not depend on the units used. Using the above non-dimensional quantities, (dopping the asteisk fo convenience), the govening equations take the fom gade gad u u (6) t Latin Ameican Jounal of Solids and Stuctues 3 (06)

4 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 599 whee. Equation (6) gives the following two equations e v u u t v e u v t e e t (7) (8) (9) 0 e t t u e v u e (0) (a) (b) zz e (c) v v u (d) z z 0 (e) whee T 0 /( ) k. We note that in the above tansfomed equations, all the vaiables and constants (,, 0) ae non dimensional. The factional deivative used in equation (0) is the Caputo deivative. The bounday conditions can be expessed as: a,, t 0 (a) a,, t 0 (b) a,, t f, t (c) The bounday conditions on the stess components means that the component of the stess in the nomal diection ( diection) ae zeo. This follow fom the fact that thee ae no suface foces Latin Ameican Jounal of Solids and Stuctues 3 (06)

5 600 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity affecting the bounday so the nomal component of stess that act to neutalize these foces ae also zeo. The stess component is the esultant of intenal foces and not necessaily zeo. 3 SOLUTION IN THE TRANSFORM DOMAIN Applying the Laplace tansfom with paamete s (denoted by an ove ba) to both sides of equations (7), (9-), we get the following equations e v u u t (3) s e (4) 0 0 s s s s e (5) u e v u e (6a) (6b) zz e (6c) v v u (6d) Equations () tansfom to: a,, s 0 (7a) a,, s 0 (7b) a,, s f, s (7c) Applying the opeato ( s ) to both sides of equation (5) and multiplying both sides of equation (4) by s 0s and subtacting, we obtain 4 s ( )( s τ 3 0s ) s ( τ 0s ) 0 Equation (8) can be witten in the fom: (8) ( k ) ( k ) 0 (9) Latin Ameican Jounal of Solids and Stuctues 3 (06)

6 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 60 whee k and k ae the complex oots which have positive eal pats of the following chaacteistic equation k 4 k s ( )( s τ 3 0s ) s ( τ 0s ) 0 (0) The solution of equation (9) can be witten in the fom () i i whee i is the solution of k i i 0, i =,. (a) o i i k 0 i i (b) The solution of equation (9), bounded at the oigin, may be witten as n0i Ain sk s In ( ki ) cosn (3) i whee Ain ae some paametes depends on s only and I n( ki ) is the modified Bessel function of fist kind of ode n. In a simila manne, the solution fo e compatible with equation (4) can be witten as e A s k i I k n n0i in n ( i ) cos (4) The Laplace tansfoms of equations (4) and (7) can be combined to give s u e e (5) Substituting fom (3) and (4) into (5), we obtain s u A s n n0 i i n i k i s kiin ki in cos (6) n k n s I k We have used the following elations of the modified Bessel functions (Bell, 986) Latin Ameican Jounal of Solids and Stuctues 3 (06)

7 60 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity di m( x ) m di ( ) ( ) ( ) and m x m I x I x m m Im( x) Im ( x) (7) dx x dx x Afte some manipulations, the solution of equation (6) takes the fom ni n k i u Ain s ki In ki n n0i Bn sin s cosn n cos whee Bn s ae some paametes depending on s only. We note that we have set B0 = 0 because I0( s) lim is not bounded. 0 Substituting fom equations (4) and (8) into (3), and integating with espect to, we obtain whee n s v A s I k B s I s I s n n n0i in n i n n n sin (9) We expand the function f, s Fn in a Fouie cosine seies in as f, s Fn ( s)cos( n) n 0 s ae the Fouie coefficient given by F0 s f, sd 0 Fn s f, scosnd 0 We have chosen to expand the function in a cosine seies to facilitate the computations. This means that we take the tempeatue as an even function of. A full expansion in tems of sine and cosine will add nothing to the physical meaning of the poblem consideed. Substituting fom equations (3), (4), and (8) into Equation (6a), and applying the bounday condition (7a), we get fo n = 0 while fo n =,,3, i k a (8) i s I 0 kia I kiaai 0 s 0 (30) Latin Ameican Jounal of Solids and Stuctues 3 (06)

8 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 603 s a n n In kia i n i in k ai k a A s i n In sasain sa Bn s0 Similaly, bounday equation (7b) yields fo n =,,3, i n I k ak ai k ana s n i i n i in s a sa n In sa In sabn s0 n n (3) (3) Finally the bounday condition (7c) leads to fo n = 0 and fo n =,,3, Ai0sk s I0( kia) F0s (33) i i Ain sk s In ( ki a) Fn s (34) i Equations (30) and (33) can be solved to obtain A 0 and A 0, i A0 F 0( a I0kas - Ikak) F 0(Ikak- a I0kas ) A0 whee ( a I 0kaI0kas ( k - k )-( I0kaIkak( k - s ) I 0kaIkak( s - k ))) Equations (3), (3) and (34), can be witten as: aan aan a3bn 0 aan aan a3bn 0 a3an a3an Fn whee Latin Ameican Jounal of Solids and Stuctues 3 (06)

9 604 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity Solving the above equations, we obtain a s a n n In ka kai n ka a s a n n In ka kain ka a3 n I n sa sai n sa a nn I n ka nkai n ka a nn I n ka nkai n ka s a sa a3 n In sa In sa n n a 3 k s In ( ka ) a 3 k s In ( ka ) aa3 a3a Fn An a3a aa3 Fn An a3a aa3 Bn whee a a a a a a a a a a NUMERICAL RESULTS AND DISCUSSION We shall apply ou esults to a medium composed of the coppe mateial. The paametes of the poblem ae k = 386 W/(m K), t =.78 (0) -5 K -, ce = 38 J/(kg K), = , = 3.86 (0) 0 kg/(m s ), = 7.76 (0) 0 kg/(m s ), = 8954 kg/m 3, T0 = 93 K, 0 /, a = m, 0 = 0.05 s and = The above values wee obtained fom ((Thomas, 980) except fo 0 which was assumed. We shall conside two cases of the applies heating Case f t o 0 othewise,, t 0 0 Fn ae thus given by 0 F0 and Fn sin n0 n n Latin Ameican Jounal of Solids and Stuctues 3 (06)

10 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 605 Case f t,, 0 othewise t 0 The Fouie coefficients Fn ae thus given by 0 0sin n0 cosn0 F0, Fn n n n Two methods wee tied to solve the poblem. Fistly, the Laplace tansfom of the tems of the seies wee inveted tem by tem and then summed up as a seies of eal numbes. Secondly, the seies was summed up as a complex-tem seies and then the invese Laplace tansfom was applied. It was found that the fist method is bette. It achieves highe ode of convegence. Figue () shows the solution fo diffeent values of N (maximum numbe of tems taken in the seies). It was found that the solution stabilized afte N = 9.The pogamming was done using the Fotan language on an I7 coe compute. The numeical invesion of the Laplace tansfom was done using a method outlined in (Honig & Hides, 984) N = 5 N = 7 N = N = N = = -4/5 = / Figue : Convegente gaph fo tempeatua at = 0.5 (case ) fo t = 0. Figs to 5 epesent case while figues 6 to 9 epesent case. We did the evaluations using 3 values of, which ae: = 0.5, 0.95 and fo t = The esults ae shown in Fig., 6 fo the tempeatue distibution, Fig. 3, 7 fo the adial displacement distibution, Fig. 4, 8 fo the tangential displacement distibution and Fig. 5, 9 Fo stess distibution. Latin Ameican Jounal of Solids and Stuctues 3 (06)

11 606 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity = 0.5 = 0.95 = 0.0 = -4/5 = / Figue : Tempeatue distibution fo diffeent (case ) fo t = u * (0) -3 = 0.5 = 0.95 = 0.5 = -4/5 = / Figue 3: Radial displacement distibution fo diffeent (case ) fo t = 0.06 v * (0) -4.5 = 0.5 = 0.95 = 0.5 = -4/5 = / Figue 4: Tangential displacement distibution fo diffeent (case ) fo t = 0.06 Latin Ameican Jounal of Solids and Stuctues 3 (06)

12 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 607 = -4/ = / = 0.5 = 0.95 = -0.5 Figue 5: Radial stess distibution fo diffeent (case ) fo t = =.5 =.95 = = -4/5 = / Figue 6: Tempeatue distibution fo diffeent (case ) fo t = u * (0) -4 =.5 =.95 = = -4/5 = / Figue 7: Radial displacement distibution fo diffeent (case ) fo t = 0.06 Latin Ameican Jounal of Solids and Stuctues 3 (06)

13 608 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity v * (0) -5.5 =.5 =.95 = 0.5 = -4/5 = / Figue 8: Tangential displacement fo diffeent (case ) fo t = = -4/ = / =.5 =.95 = Figue 9: Radial stess distibution fo diffeent (case ) fo t = 0.06 Figs 0 to 7 epesent the time evolution of the diffeent functions when t = 0., 0. and 0.3 fo = 0.5. Case is shown in figues 0 to 3 while figues 4 to 7 epesent case t = 0. t = 0. t = 0.3 = -4/5 0. = / Figue 0: Tempeatue distibution fo diffeent t (case ) fo = 0.5 Latin Ameican Jounal of Solids and Stuctues 3 (06)

14 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 609 u * (0) - t = 0. t = 0. t = 0.3 = -4/5 = / Figue : Radial displacement distibution fo diffeent t (case ) fo = 0.5 v * (0) -4.5 t = 0. t = 0. t = = -4/5 = / Figue : Tangential distibution fo diffeent t (case ) fo = 0.5 = -4/ = / t = 0. t = 0. t = 0.3 Figue 3: Radial stess distibution fo diffeent t (case ) fo = 0.5 Latin Ameican Jounal of Solids and Stuctues 3 (06)

15 60 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity t = 0. t = 0. t = = -4/5 = /5 Figue 4: Tempeatue distibution fo diffeent t (case ) fo = u * (0) -3 3 t = 0. t = 0. t = = -4/5 - = /5 Figue 5: Radial displacement distibution fo diffeent t (case ) fo = 0.5 v * (0) -4 6 t = 0. t = 0. 4 t = 0.3 = -4/5 = / Figue 6: Tangential distibution fo diffeent t (case ) fo = 0.5 Latin Ameican Jounal of Solids and Stuctues 3 (06)

16 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity = -4/5 = /5 0.5 t = 0. t = 0. t = Figue 7: Radial stess distibution fo diffeent t (case ) fo = 0.5 Figues 8 and 9 epesent tempeatue vesus fo case and espectively at t = 0. and = 0.4. All these figues epesent the functions as functions of on the diagonal /5 and 4 / * (0) -3 = 0.5 = 0.95 = - 0 Figue 8: Tempeatue vs fo diffeent (case ) fo t = 0., = 0.4 Latin Ameican Jounal of Solids and Stuctues 3 (06)

17 6 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 6 * (0) =.5 =.95 = - 0 Figue 9: Tempeatue vs fo diffeent (case ) at t = 0., = 0.4 The computations show that:. Fo α = 0.5, we can see fom the gaphs that the waves in the medium popagate with infinite speeds like the coupled theoy of themoelasticity. The pogam was un with α = 0 (coesponding to the coupled theoy of themoelaciticity), the esults wee almost identical to those when α = 0.5. Fo α =, the solution exhibits finite speeds since it is that of the genealized theoy. The heat Equation associated with the couple theoy of themoelasticity ( = 0) is of paabolic type and pedicts infinite speed of popagation fo heat waves. The solution is nonzeo (though it may be vey small) at points fa emoved fom the souce of heating. The heat equation of the genealized theoy of themoelasticity ( = ) is of hypebolic type and pedicts finite speed fo heat waves. This means that heat popagates fom the souce of heating with a finite velocity. The solution is identically zeo at points fathe than the wave font. The location of the wave fonts and the value of the velocities fo heat and elastic waves wee discussed in (Sheief, H., & Hamza,994).. Fo α, the situation is somewhat difficult to detemine. The solution seems to tavel with finite speeds. Of couse, this is based on numeical evaluations only. This aspect would be vey impotant when poved theoetically. The same conjectue was expessed in (Povstenko, 0). Refeences Bell, W. Special Functions fo Scientists and Enginees. 968: D. Van Nostand Company Ltd, London. El-Kaamany, A. S., & Ezzat, M. A. (0). On factional themoelasticity. Mathematics and Mechanics of Solids, 6(3), Ezzat, M. A., & El-Kaamany, A. S. (0). Factional ode theoy of a pefect conducting themoelastic medium. Canadian Jounal of Physics, 89(3), Hilfe, R., et al. (000). Applications of factional calculus in physics (Vol. 5): Wold Scientific. Latin Ameican Jounal of Solids and Stuctues 3 (06)

18 H.H. Sheief and W.E. Raslan / D Poblem fo a Long Cylinde in the Factional Theoy of Themoelasticity 63 Honig, G., & Hides, U. (984). A method fo the numeical invesion of Laplace tansfoms. Jounal of Computational and Applied Mathematics, 0(), 3-3. Lod, H. W., & Shulman, Y. (967). A genegeealized dynamical theoy of themoelasticity. Jounal of the Mechanics and Physics of Solids, 5(5), Machado, J. T., Galhano, A. M., & Tujillo, J. J. (03). Science metics on factional calculus development since 966. Factional Calculus and Applied Analysis, 6(), Povstenko, Y. (009). Themoelasticity that uses factional heat conduction equation. Jounal of Mathematical Sciences, 6(), Povstenko, Y. (0). Factional Cattaneo-type equations and genealized themoelasticity. Jounal of Themal Stesses, 34(), Raslan, W. (04). Application of factional ode theoy of themoelasticity to a D poblem fo a cylindical cavity. Achives of Mechanics, 66(4), Shama, J., & Pathania, V. (006). Themoelastic waves in coated homogeneous anisotopic mateials. Intenational jounal of mechanical sciences, 48(5), Sheief, H., & Abd El-Latief, A. (03). Effect of vaiable themal conductivity on a half-space unde the factional ode theoy of themoelasticity. Intenational Jounal of Mechanical Sciences, 74, Sheief, H. H., & Abd El Latief, A. (04). Application of factional ode theoy of themoelasticity to a D poblem fo a half space. ZAMM Jounal of Applied Mathematics and Mechanics/Zeitschift fü Angewandte Mathematik und Mechanik, 94(6), Sheief, H. H., & Anwa, M. N. (994). Two-dimensional genealized themoelasticity poblem fo an infinitely long cylinde. Jounal of themal stesses, 7(), 3-7. Sheief, H. H., El-Sayed, A., & Abd El-Latief, A. (00). Factional ode theoy of themoelasticity. Intenational Jounal of Solids and stuctues, 47(), Sheief, H. H., El-Sayed, A., Behiy, S., & Raslan, W. (0). Using Factional Deivatives to Genealize the Hodgkin Huxley Model Factional Dynamics and Contol (pp. 75-8): Spinge. Sheief, H. H., & Ezzat, M. A. (994). Solution of the genealized poblem of themoelasticity in the fom of seies of functions. Jounal of themal stesses, 7(), Sheief, H. H., & Hamza, F. A. (994). Genealized themoelastic poblem of a thick plate unde axisymmetic tempeatue distibution. Jounal of themal stesses, 7(3), Sheief, H. H., & Saleh, H. A. (998). A poblem fo an infinite themoelastic body with a spheical cavity. Intenational jounal of engineeing science, 36(4), Tiwai, R., & Mukhopadhyay, S. (04). Bounday Integal Equations Fomulation fo Factional Ode Themoelasticity. Computational Methods in Science and Technology, 0. Thomas, L., (980). Fundamentals of Heat Tansfe. Pentice-Hall Inc., Englewood Cliffs, New Jesey. Latin Ameican Jounal of Solids and Stuctues 3 (06)