Numerical Solution for a FG Cylinder Problem Using Finite-Difference Method

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1 Aled Matheats 3-3 do:.436/a..4 Publshed Onlne January (htt:// ueral Soluton or a FG Cylnder Proble Usng Fnte-Derene Method Abstrat Daoud S. Mashat Deartent o Matheats Faulty o Sene Kng AbdulAzz Unversty Jeddah Saud Araba E-al: dashat@kau.edu.sa Reeved August 8 ; revsed oveber 4 ; aeted oveber 8 A rened nte-derene aroah s resented to solve the theroelast roble o untonally graded ylnders. Materal roertes o the resent ylnder are assued to be graded n the radal dreton aordng to a ower-law dstrbuton n ters o the volue ratons o the etal and era onsttuents. The governng seond-order derental equatons are derved ro the oton and the heat-onduton equatons. ueral results or densonless teerature radal dslaeent ehanal stresses and eletroagnet stress are dstrbuted along the radal dretons. The eets o te araeter and the untonally graded oeent are nvestgated. Keywords: Funtonally Graded Annular Cylnder Eletro-Magneto-Theroelast Feld. Introduton Magneto-theroelast untonally graded ateral s a knd o agneto-theroelast ateral wth ateral ooston and roertes varyng ontnuously along ertan dretons. It s the ooste ateral ntentonally desgned so that they ossess desrable roertes or soe se alatons. The advantage o ths new knd o aterals an rove the relablty o le san o agnet deves. Reently there has been growng nterest n aterals delberately abrated so that ther eletr agnet theral and ehanal roertes vary ontnuously n sae on the aroso sale. Ths researh subjet s so new that only a ew results an be ound n the lteratures. In reent years the theory o agneto -thero -elastty that deals wth the nteratons aong stran teerature and eletroagnet elds has drawn the attenton o any researhers. Ths theory s extensve uses n dverse eld suh as geohyss or understandng the eet o the Earth s agnet eld on ses waves dang o aoust waves n a agnet eld and essons o eletroagnet radatons ro nulear deves. It s also develoed a hghly senstve suerondutng agnetoeter eletral ower engneerng and ots. In the eld o agneto-elastty or agneto-theroelastty any studes have been onduted on an analytal treatent o an nteraton between elast eletroagnet and teerature elds (e.g. Para [] Wang et al. [] Wang and Da [3] Wang and Dong [4] Banerjee and Royhoudhur [5]). The agneto-theroelast roble o untonally graded ateral (FGM) hollow strutures subjeted to ehanal loads s onsdered by Da and Fu [6]. Hou and Leung [7] analyzed the lane stran dyna roble o a agneto -eletro -elast hollow ylnder by vrtue o the searaton o varables orthogonal exanson tehnque and the nterolaton ethod. Buhanan [8] onsdered the ree vbraton roble o an nnte agneto-eletro-elast ylnder. The requeny behavour o laed-laed agneto -eletro -elast ylndral shells s analysed by Annger et al. [9] usng the se-analytal nte eleent aroah. Hou et al. [] studed the transent resonses o a seal non-hoogeneous agneto-eletro-elast hollow ylnder or axsyetr lane stran roble. The agnetother-oelast roble n non-hoogeneous sotro ylnder n a rary agnet eld s dsussed by Abd-El-Sala et al. []. The dyna and quas-stat behavors o agneto -thero -elast stresses n a ondutng hollow rular ylnder subjeted to an arbtrary varaton o agnet eld are nvestgated by Hguh et al. []. Bhangale and Ganesan [3] studed the ree vbraton on FGM agneto-eletro-elast ylndral shells usng a seres soluton. Kong et al. [4] resented an analytal ethod to nvestgate thero -agneto-elast stresses Coyrght SRes.

2 4 D. S. MASHAT and erturbaton o the agnet eld vetor n a ondutng non-hoogeneous hollow ylnder under theral shok. Wang and Da [5] resented agneto-therodyna stresses and erturbaton o agnet eld vetor n an orthotro theroelast ylnder. Da and Wang [6] resented an analytal soluton or agneto-thero-eletro-elast robles o a ezoeletr hollow ylnder laed n an axal agnet eld subjeted to arbtrary theral shok ehanal load and transent eletr extaton. The objetve o ths nvestgaton s to generate dslaeent stresses teerature and agnet eld n a FGM annular ylnder. Materal roertes o the resent ylnder are assued to be graded n the radal dreton aordng to a ower-law dstrbuton n ters o the volue ratons o the etal and era onsttuents. The governng oton and the heat-onduton equatons are obtaned n onservaton ors and solved nuerally usng nte derene ethod. A rened ntederene aroah s resented to solve the resent roble. ueral results or the varaton o teerature dslaeent and stresses are resented or a etalera FG ylnder. To ake the study reasonably teerature dslaeent and stresses are gven or derent hoogenzaton shees and exonents n the ower-law that desrbes along-the-thkness varaton o the onsttuents. The eets o any araeters are nvestgated.. Matheatal Model Let us onsder an annular ylnder o outer radus r = b nner radus r = a and ade o a untonally graded ateral. The ylndral oordnates syste r z or the axally syetr roble s used wth z-axs ondng wth the axs o the ylnder. We have only the radal dslaeent u r whh s ndeendent o and z. The ylnder s laed n a onstant rary agnet eld H as shown n Fgure. In a generalzed lane stran we suose that the lanes erendular to the z-axs and u r s a unton o the radal dreton r and te t only. The Cauhy s relatons are ur ur err e ezz ezr er ez r r () where e j are the stran oonents. The ehanal stress oonents rr and and Maxwell s eletroagnet stress oonent rr are gven resetvely by rr rh e e. rr r err e r err r T r e e r e r T rr rr () Fgure. Teerature dstrbuton through the radal dreton o the FGM annular ylnder at derent tes and or derent values the gradaton exonent. Coyrght SRes.

3 D. S. MASHAT 5 where T s the absolute teerature and Laé s oeents the agnet ereablty 3 the stress-teerature odulus n whh s the lnear theral exanson and H s the rary agnet eld. The agneto-elasto-dyna equaton n the radal dreton o the FG annular ylnder s gven by rr rr ur rr r (3) r r r t where s the ateral densty o the ylnder. The heat onduton equaton n the absene o heat soure an be wrtten n the or [7] r C r T rt u r r r r t tr r (4) where T s the reerene teerature C the se heat at onstant volue and s the theral ondutvty. The ateral roertes lke and o the FGM ylnder are assued to be unton o the volue raton o the onsttuent aterals. The untonally graded between the hysal roertes and the radal dreton r or era and etal FG ylnder s gven by j r a PrP P P (5) b a where P and P are the orresondng roertes o era and etal resetvely and j s the volue raton exonent whh takes values greater than or equal to zero. The value o j equal to zero reresents a ully etal annular ylnder. The above ower-law assuton relets a sle rule o xtures used to obtan the eetve roertes o the era-etal FGM annular ylnder. The rule o xtures ales only to the radal dreton. ote that the volue raton o the etal s hgh near the nner surae o the ylnder and that o era hgh near the outer surae. In addton Equaton (3) ndates that the nner surae o the annular ylnder s etal-rh whereas ts outer surae s era-rh. The densty and other hysal oonents o the annular ylnder aordng to the ower law and the ower-law exonent ay be vared to obtan derent dstrbutons o the oonent aterals through the radal dreton o the ylnder. 3. Soluton o the Proble Introdung the ollowng densonless varables ay be slyng the solvng roess: rr rr T HT r a t T r t ur r t R A T U b b b T bt where s densonless te and s the lnear theral exanson onstant. The eet o ateral roertes varaton o the FG ylnder an be taken nto aount n Equatons (3) and (4) usng Equaton (5). The substtuton o Equatons () and () nto Equatons (3) and (4) wth the ad o the densonless varables gven n Equaton (6) rodues the governng equatons or the FG ylnder as ollows: U U U R R U R VR R R R T RV RU R V RR T R U R T U R R R R R where H H j R H V R T C C where the letter n the exresson V R. A (6) (7) (8) (9) () s gven nstead o the araeters C or. ote that the re () n Equaton (7) denotes derentaton wth reset to R. Also C and ( = or ) are Laé s onstants theral odulus theral ondutvty se heat agnet ereablty and densty o the hoogeneous etal or era ateral resetvely and the araeter n Equaton () ay be used to reresent one o these onstants. The densonless stresses ndued by the teerature T and the eletroagnet stress are related to the densonless radal dslaeent U by U U U R R R T U U U R R R T () () Coyrght SRes.

4 6 D. S. MASHAT U U. (3) R R One agan the letter n Equaton () s gven nstead o the araeters or. The elast soluton or the FG hollow ylnder s oleted by the alaton o the ntal and boundary ondtons. The ntal ondtons an be exressed as T U T U at. (4) The boundary ondtons at the nner and outer rad o the FG ylnder ay be exressed as T e at R A (5) T at R where s an exonent o the deayed heat lux. 4. ueral Shee A nte eleent shee s used here to get the teerature and radal dslaeent. The nte derene grds wth satal ntervals h (esh wdth) n the radal dreton and k as the te ste or the axu te ax and use the subsrts and n to denote the th dsrete ont n the R dreton and the nth dsrete te. A esh s dened by A R Ah h. (6) n ax kn n 3 K k. K The dslaeent and teerature ay be gven or ostve ntegers and K at any nodal loaton by n n n n U R U T R T. (7) The equaton o oton and the heat onduton equaton gven n Equatons (7) and (8) ay be exressed n the nte derene as where U U U U U n n n n n 3 n n n 4 T T 5T T gt g T g T T n n n n n 3 h A h n n n U U U n n n n U U U U h k h A h V (8) (9) k V A h h ( A h) k h A h 3 h k 4 5 h V () k V (d) k h h g g g3. (e) h A h A h ote that j Ah V V A j j Ah V V V. A A () The ehanal stresses and eletroagnet stress are gven aordngly by n n n n U U U T k A h k A h n n n n U U U T n n n U U U. k A h () (3) (4) The ntal ondtons n Equaton (4) ay be wrtten as T T T U U U. (5) Puttng n = n Equatons (8) and (9) one obtans U U U T T 3 4 T gt g T U A h 4h 3 U UUU (6) (7) where 3. The boundary ondtons gven n Equaton (5) at R = A ay be wrtten as n kn n n h n n T e U U U T A (8) and at R = we get n n n h n n T U U U T. (9) Coyrght SRes.

5 D. S. MASHAT 7 Puttng = n Equaton (8) and usng the orward derene aroxaton or the teerature s dervatve wth the ad o Equaton (8) one obtans n n h n n U 3 U 3 U U A n n h kn 4 T 4T e where (3) k h h A (3a) k h A (3b) k 3 h h A (3) k 4. h (3d) Puttng = n Equaton (8) and usng the bakward derene aroxaton or the teerature s dervatve wth the ad o Equaton (9) one obtans n h n n U U 3 U (3) n U n T n 4 T where 4 5. ueral Results k h (33a) h h k h (33b) 3 4 h h k h k. (33) The teerature dslaeent and stresses or the resent ylnder are obtaned usng the above nte eleent shee. The results are resented n the non -densonal or: 6 8 u U. All results o ths artle are or alunu as nner etal surae and aluna as outer era surae. Generally the agnet ereablty K ( = or ) s gven n ters o the ereablty o sae 7 4 A and the relatve ereablty or both alunu K and aluna K. The ateral roertes are assued to be as: Metal (alunu): E 3 7 GPa.35 7 Kg 6 3. K K.3 37 W K. Cera (aluna): 3 E 6 GPa.33 3 Kg K K..78 W K. ote that the roertes o and or etal or era are graded through the radal dreton aordng to the ollowng relatons: E E 3. Results are resented n Fgures -5 or teerature radal dslaeent radal stress ruerental stress and eletroagnet stress aordng to the xed onstants b A T H K Oersted.5. The senstvty o the te araeter and the FGM exonent j are dsussed through the gures. Fgure reresents the varaton o the densonless teerature T through the radal dreton o the FGM annular ylnder. Four values o the te araeter wth j = 3 are used n Fgure whle our values o the FGM exonent j wth.3 are used n Fgure. Slar results or the densonless dslaeent u the radal stress the ruerental stress and the eletroagnet stress are lotted n Fgures -5 resetvely. Fgure shows that the absolute value o the teerature dereases as the te araeter nreases and the FGM exonent j dereases. It s to be noted that the teerature s axu or the hoogeneous etal ylnders. The sae behavor ours or dslaeent and stress quanttes. The soluton satsed the boundary ondtons (see Fgures and 3) and the derene between hoogeneous and FGM ases s shown. The varaton o teerature dslaeent and stresses are due to the eet o nerta and agnet eld. It s seen that the nluene o the FGM on teerature dslaeent and stresses s very ronouned. Fnally t s nterested to see that all quanttes y by vanshed near and at the external era surae o the annular ylnder. Coyrght SRes.

6 8 D. S. MASHAT Fgure. Dslaeent dstrbuton through the radal dreton o the FGM annular ylnder at derent tes and or derent values the gradaton exonent. Fgure 3. Radal stress dstrbuton through the radal dreton o the FGM annular ylnder at derent tes and or derent values the gradaton exonent. Coyrght SRes.

7 D. S. MASHAT 9 Fgure 4. Cruerental stress dstrbuton through the radal dreton o the FGM annular ylnder at derent tes and or derent values the gradaton exonent. Fgure 5. Eletroagnet stress dstrbuton through the radal dreton o the FGM annular ylnder at derent tes and or derent values the gradaton exonent. Coyrght SRes.

8 3 D. S. MASHAT 6. Conlusons The an ontrbuton n ths aer s to desrbe seondorder exlt nte-derene shee. Ths shee hels us to solve the ouled hyerbol equatons on a unor grd and s qute eent or outaton theral stresses. The results obtaned show that behavor o the teerature dslaeent and stresses ay hange sgnantly by reason o nluene o exonent heat lux and rary agnet eld n hoogeneous and untonally graded ases. These results are se or the exale onsdered but the exale ay have derent trends beause o the deendene o the results on the agnet and theral onstants o the etal-era untonally graded ateral. 7. Reerenes [] G. Para Magneto-Elastty and Magneto-Thero-Elastty Advanes n Aled Mehans Vol do:.6/s65-56(8) [] X. Wang G. Lu and S. R. Gullow Magnetotherodyna Stress and Perturbaton o Magnet Feld Vetor n a Sold Cylnder Journal o Theral Stresses Vol. 5 o do:.8/ [3] X. Wang and H. L. Da Magnetotherodyna Stress and Perturbaton o Magnet Feld Vetor n an Orthotro Theroelast Cylnder Internatonal Journal o Engneerng Sene Vol. 4 o do:.6/j.jengs.3.8. [4] X. Wang and K. Dong Magnetotherodyna Stress and Perturbaton o Magnet Feld Vetor n a on- Hoogeneous Theroelast Cylnder Euroean Journal o Mehans-A/Solds Vol. 5 o do:.6/j.euroehsol [5] S. Banerjee and S. K. Royhoudhur Magneto-Thero- Elast Interatons n an Innte Isotro Elast Cylnder Subjeted to a Perod Loadng Internatonal Journal o Engneerng Sene Vol. 35 o do:.6/s-75(96)7-5 [6] H. L. Da and Y. M. Fu Magnetotheroelast Interatons n Hollow Strutures o Funtonally Graded Materal Subjeted to Mehanal Loads Internatonal Journal o Pressure Vessels and Png Vol. 84 o do:.6/j.jv.6.. [7] P. F. Hou and A. Y. T. Leung The Transent Resonses o Magneto -Eletro-Elast Hollow Cylnders Sart Materals and Strutures Vol. 3 o do:.88/964-76/3/4/4 [8] G. R. Buhanan Free Vbraton o an Innte Magneto-Eletroelast Cylnder Journal o Sound and Vbraton Vol. 68 o do:. 6/S-46X(3)357- [9] A. R. Annger. Ganesan and S. Swarnaan Free Vbratons o Claed-Claed Magneto-Eletro-Elast Cylndral Shells Journal o Sound and Vbraton Vol. 9 o do:.6/j.jsv [] P. F. Hou H. J. Dng and A. Y. T. Leung The Transent Resonses o a Seal on-hoogeneous Magneto- Eletro-Elast Hollow Cylnder or Axsyetr Plane Stran Proble Journal o Sound and Vbraton Vol. 9 o do:.6/j.jsv [] M. R. Abd-El-Sala A. M. Abd-Alla and H. A. Hosha A ueral Soluton o Magneto -Theroelast Proble n on-hoogeneous Isotro Cylnder by the Fnte- Derene Method Aled Matheatal Modellng Vol. 3 o do:.6/j.a [] M. Hguh R. Kawaura and Y. Tangawa Dyna and Quas-Stat Behavors o Magneto-Thero-Elast Stresses n a Condutng Hollow Crular Cylnder Subjeted to an Arbtrary Varaton o Magnet Feld Internatonal Journal o Mehanal Senes Vol. 5 o [3] R. K. Bhangale and. Ganesan Free Vbraton Studes o Sly Suorted on-hoogeneous Funtonally Graded Magneto -Eletro-Elast Fnte Cylndral Shells Journal o Sound and Vbraton Vol. 88 o do:.6/j.jsv [4] T. Kong D. X. L and X. Wang Thero-Magneto- Dyna Stresses and Perturbaton o Magnet Feld Vetor n a on-hoogeneous Hollow Cylnder Aled Matheatal Modellng Vol. 33 o do:.6/j.a.8..3 [5] X. Wang and H. L. Da Magneto-Thero-Dyna Stress and Perturbaton o Magnet Feld Vetor n a Hollow Cylnder Journal o Theral Stresses Vol do:.8/ [6] H. L. Da and X. Wang Magneto-Thero-Eletro- Elast Transent Resonse n a Pezoeletr Hollow Cylnder Subjeted to Colex Loadngs Internatonal Journal o Solds and Strutures Vol. 43 o do:.6/j.jsolstr [7] S. B. Mukhoadhyay Theroelast Interatons wthout Energy Dssaton n an Unbounded Body wth a Sheral Cavty Subjeted to Haronally Varyng Teerature Mehans Researh Counatons Vol. 3 o do:.6/s93-643(3) 8-X Coyrght SRes.