2D Finite Element Formulation for 3D Temperature Analysis of Layered Hybrid Structures

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1 D Finite Element Formultion for 3D emperture Anlysis of Lyered Hyrid Structures Rimund Rolfes, Jn eßmer Institut für Struturmechni, DLR e.v., Brunschweig, Germny Summry: hree-dimensionl (3D) finite element formultions re usully pplied for the nlysis of temperture fields for hyrid nd conventionl composite structures. his leds to high effort with respect to discretistion nd computtion time, especilly for trnsient clcultions. Using new formultion sing on two-dimensionl (D) finite element discretistion it is now possile to pproximte the three-dimensionl temperture field very effectively. hrough comintion of lyerwise shpe functions in thicness direction (liner or qudrtic) nd suitle het trnsfer conditions t the interfces the nodl numer of degrees of freedom remins independent from the numer of lyers. In conjunction with shell elements tht re generlly pplied for the stress nlysis of thin composite structures it is lso esier to trnsfer temperture dt fields due to the sme two-dimensionl discretistion. his leds to fst nd ccurte formultion for the temperture nlysis of hyrid composite structures. Keywords: Finite elements; therml nlysis; hyrid composite structures; lyerwise theory Numericl Simultion of Het rnsfer My 9-0, 00

2 Introduction One min ojective of the therml nlysis is to control if structure fulfils the therml requirements nd to supply the full three-dimensionl temperture distriution s input for the thermo-mechnicl nlysis. In the cse of composites nd sndwich structures the lyers hve different therml conductivities in different directions. Figure shows different composites nd typicl sndwich structure. Fce Sheet Fier/resin Composite structure Hyrid composite structure Aluminum sheet Honeycom Core Sndwich structure Figure : Exmples of composite nd sndwich structures For the evlution of the full three-dimensionl temperture distriution 3D finite element or finite difference methods re commonly used. his leds to high modelling nd numericl effort, which is not cceptle within design process. Besides, for thermo-mechnicl clcultions twodimensionl model is sufficient, since most commercil finite element codes provide two-dimensionl finite elements for composite nd sndwich structures. herefore it is desirle to hve finite elements, sed on two-dimensionl model, which cn clculte the full three-dimensionl temperture distriution reducing the modelling nd numericl effort drsticlly. For lminted composites (CFRP) Rolfes [7] hs proposed liner therml lmintion theory which is nlogous to the first order sher deformtion theory (FSD). For locl effects or trnsient prolems the sme uthor hs suggested qudrtic therml lmintion theory. Susequently, new D finite element formultions re outlined which re sed on lyerwise liner or qudrtic temperture distriutions in thicness direction. his llows for composite structures with different therml conductivity in thicness direction for ech lyer. hey cn e used especilly for lightweight structures in very cold or very hot environments where lyers with different therml nd stiffness properties re comined in one ly-up. It should e mentioned tht these finite element formultions fcilitte the possiility to e used within concurrent integrted engineering process. Due to two-dimensionl dt structure of the therml model it cn e coupled much esier to mechnicl models consisting of shell elements thn conventionl three-dimensionl therml models. his is importnt for fst nd ccurte nlysis within the preliminry design phse of structurl prts. Finite Element Formultion CFRP, hyrid composites nd sndwich structures cn e idelised s lyered structures, see figure. For lyers in which ll modes of het trnsfer (het conduction, rdition nd convection) occur (for exmple honeycom cores) therml homogenistion is necessry. his homogenistion is not specific requirement for D finite elements, ut is eqully needed if 3D finite element or finite difference model is pplied. A lyerwise discretistion with 3D finite elements is very costly. herefore different pproches hve een mde to reduce the modelling effort. For n overview of different methods see [4] nd [8]. An useful method for CFRP structures ws developed y Rolfes [7]. It descries the therml lmintion theory (L), ssuming either liner or qudrtic temperture distriutions over the whole lminte. his theory holds for the following conditions: - Identicl therml conductivity of ll lyers in the thicness direction - No het-trnsfer resistnce t the interfces he liner L cn then e formulted s xyz (,, ) = ( xy, ) + z (, xy). () ( ) ( ) 0 0, z Non-liner temperture distriutions in thicness direction cn occur in the presence of Numericl Simultion of Het rnsfer My 9-0, 00

3 - lrge temperture grdients in the thicness direction in conjunction with temperture-dependent thermo-physicl properties - trnsient prolems with rpid heting - sptilly concentrted therml lods In such cses, the qudrtic L is etter suited. It ssumes ( ) ( ) z ( ) xyz (,, ) = 0 (, xy) + z 0, z ( xy, ) + 0, zz( xy, ). () For modelling hyrid structures (e.g. metllic multiwll PS, hyrid composites (GLARE), sndwiches or hot structures) it is necessry to give up the first condition stted ove, nd therefore ssuming - different therml conductivity of ech lyer in thicness direction. his leds to the need of lyerwise theories. A liner lyered theory (LL) (conf. Figure ) ws first used y Sipetov [9] for stedy stte therml prolems. It ssumes for ech lyer ( xyz,, ) = ( xy, ) + z ( xy, ) (3) ( ) ( ) ( ) 0 0, z Using two het trnsfer equilirium conditions t ech lyer interfce for the - temperture, - het flux in trnsverse direction, the numer of functionl degrees of freedom cn e mde independent from the numer of lyers. his theory ws extended to trnsient prolems y Noc nd Rolfes [3]. For the sme resons s stted for the L, qudrtic lyered theory ws formulted for trnsient therml prolems nd locl het lods. It reds ( ) ( ) ( ) z ( ) ( xyz,, ) = 0 ( xy, ) + z 0, z ( xy, ) + 0, zz(, xy). (4) By the use of third het trnsfer equilirium condition t ech lyer interfce for - the chnge of the het flux in trnsverse direction, gin the numer of functionl degrees of freedom cn e mde independent from the numer of lyers. Bsed on these theories the finite elements QUADLL nd QUADQL were developed, showing liner or qudrtic temperture distriutions in thicness direction for ech lyer. N (z) z z = 0 t t z +d -z z () 0 () 0,z LL Figure : Lyered design of composite nd sndwich structures Numericl Simultion of Het rnsfer 3 My 9-0, 00

4 Strting with Fourier s lw for lyer of nisotropic mteril ( ) ( ) ( ) q = K (grd ) (5) equtions (3) nd (4) my lterntively e used for the pproximtion of the temperture. Introducing the ove mentioned het trnsfer equilirium conditions t the lyer interfces leds to for the liner lyerwise pproximtion nd ( xyz,, ) = ( xy, ) + z () z ( xy, ) (6) ( ) ( ) ( ) 0 0, z ( xyz,, ) = ( xy, ) + z () z (, xy) + z ( zz, ) ( xy, ) (7) ( ) ( ) ( ) ( ) 0 0, z 0, zz for the qudrtic lyerwise pproximtion. he index denotes the reference lyer. For the vlues of z nd z, which re functions of the thicness-coordinte z, see [3] nd [4]. It is importnt tht for oth formultions the functionl degrees of freedom remin independent of the numer of lyers which is crucil for the needed computtionl effort. Integrting these equtions into the three-dimensionl we formultion for liner stedy-stte het trnsfer Ω grd ) ( v KgrddΩ+ qn vdγ= 0 (8) it is possile to split the integrtion in z-direction from the integrtion in x- nd y-direction. Performing the integrtion in z-direction nlyticlly, the finite element method hs to e pplied to the remining two-dimensionl we formultion A Γ NKN ϑ da+ qnvdγ= 0. (9) In eqution (8) v is the test function. he oundry conditions t the edge Γ, considering free convection q nd het flux q, red c Γ qn = qc + q. (0) he integrtion in z-direction leds to the modified het conduction mtrix K. he shpe functions nd their derivtives re summrised in the mtrix N nd the nodl degrees of freedom in the vector ϑ. Eqution (9) cn now e implemented into n ordinry D finite element formultion. For detils of the referred vlues see [3] nd [4]. 3 Exmples A squre plte with locl het flux of q = 00 W/m ws considered. At the ottom of the plte convection oundry condition with n mient temperture of = 0 C nd α c = 30 W/m K ws pplied. he geometry is shown in figure 3. het flux point p Figure 3: Exmple prolem for the therml nlysis Numericl Simultion of Het rnsfer 4 My 9-0, 00

5 wo different composites were nlysed. he temperture distriutions in trnsverse direction t point P re shown in figure 4. he results show good greement etween D nd 3D nlysis. Especilly, qudrtic lyered theory nd 3D results mtch excellently. Numericl nd modelling effort re drsticlly reduced y the new finite elements. Sndwich Hyrid Composite (CARE) 6 4 D LL D QL 3D MSC/NASRAN 0, ,6 D LL D QL 3D MSC/NASRAN 0,3 z [mm] 0 - z [mm] 0,0-0,3-4 -0,6-6 -0, emperture [ C] emperture [ C] Figure 4: Comprison of D nd 3D therml nlysis (trnsverse temperture distriution t point P) 4 References [] Argyris, J.; ene, L.; Öerg, F.: A Multilyer Composite ringulr Element for Stedy Stte Conduction/Convection/Rdition Het rnsfer in Complex Shells, Computer Methods in Applied Mechnics nd Engineering 0, 995, 7-30 [] Diers, A.; Rolfes, R.: Berücsichtigung von Strhlungsrndedingungen ei der hermlnlyse von Fserverundstruturen mit Hilfe der Methode der finiten Elemente, DLR Report IB3-96/3, Brunschweig, 996 [3] Noc, J.; Rolfes, R.: Efficient therml nlysis of hot structures using new finite elements, Proceedings Europen Conference on Spcecrft Structure, Mterils nd Mechnicl esting, Brunschweig, Germny, 998, 7-0 [4] Noc, J.: Eine schichtweise heorie und Numeri für Wärmeleitung in Hyridstruturen, Disserttion, FB Mschinenu, U Brunschweig, 000 [5] Noor, A.K.; Burton, W.S.: Stedy-Stte Het Conduction in Multilyered Composite Pltes nd Shells, Computers & Structures. 39, 99, [6] Reddy, J.N.: A Simple Higher-Order heory for Lminted Composite Pltes, Journl of Applied Mechnics 5, 984, [7] Rolfes, R.: Higher Order heory nd Finite Element for Het Conduction in Composites, in: Lewis, R.W.; Chin, J.H.; Homsy, G.M. (ed): Numericl Methods in herml Prolems, Proceedings of the Seventh Interntionl Conference. vol. VII, 99, [8] Rolfes, R., Noc, J., eschner: "High performnce 3D-nlysis of thermo-mechniclly loded composite structures", Composite Structures 46, 999, [9] Sipetov, V.S; Krpilovsii, V.S; Demchu, O.N.: Appliction of the Finite Element Method to Solve the Sttionry Het Conduction Prolem of Piecewise Inhomogeneous Systems, in: J. Eng. Phys. 55, Nr. 6, 989, Numericl Simultion of Het rnsfer 5 My 9-0, 00