UNCOMPLICATED TORSION AND BENDING THEORIES FOR MICROPOLAR ELASTIC BEAMS

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1 11th Word Congress on Computationa Mechanics WCCM XI 5th European Conference on Computationa Mechanics ECCM V 6th European Conference on Computationa Fuid Dynamics ECFD VI E. Oñate J. Oiver and. Huerta Eds UNCOMPICTED TORSION ND BENDING THEORIES FOR MICROPOR ESTIC BEMS SOROOSH HSSNPOUR ND G. R. HEPPER Mechanica and Mechatronics Engineering University of Wateroo Wateroo ON Canada N G1 e-mai: soroosh@uwateroo.ca Systems Design Engineering University of Wateroo Wateroo ON Canada N G1 e-mai: hepper@uwateroo.ca Key words: Eastic Beam Micropoar Easticity Torsion Bending bstract. simpified mode for torsion and bending of D micropoar eastic beams is formuated first. The obtained static governing equations are soved numericay by using a finite eement approach and numerica exampes are provided. Speciay the conditions under which the resuts of the cassica beam modes are recovered wi be presented. 1 INTRODUCTION The origins of micropoar easticity begin with Voigt s work on adding an independent coupe stress vector to the cassica force stress vector which was deveoped further by E. and F. Cosserat by suggesting independent dispacement and microrotation fieds i.e. six DOFs for every eement of the body. Eringen extended the Cosserat theory to incude microinertia effects and renamed it the micropoar theory of inear easticty. detaied review of the micropoar theory of easticity can be found in [1. s in cassica easticity theory it is usefu to simpify the genera micropoar theory of inear easticity to the specia case of a micropoar beam and to deveop the micropoar torsion and bending modes for a micropoar beam. However most of the existing micropoar beam modes whie being of interest do not seem as convenient as cassica beam torsion and bending modes to a practitioner. For exampe the torsion probem of micropoar eastic beams has been soved for both circuar and noncircuar cross sections [ 5 and there are aso papers on bending of the micropoar eastic beams [6 8; yet the compexity of these modes is far beyond what is needed in a typica engineering probem. simpe mode for the micropoar beam bending probem was deveoped by Haung [9 who as an extension of the cassica Euer-Bernoui bending mode assumed 1

2 Soroosh Hassanpour and G. R. Hepper that the shear deformations are negigibe and the microrotations are equa to the bending rotations of the beam pane sections. nother simpe but more genera mode for the micropoar beam bending probem was addressed by Ramezani [1 who as an extension of the cassica Timoshenko bending mode incuded the shear deformations and considered the microrotations to be independent from the bending rotations. However due to using an incorrect definition for the strain tensor there are some errors in the equations presented by Ramezani. In this ight the focus of this manuscript is to deveop an uncompicated but comprehensive mode to represent a micropoar beam deforming in D space. Such a mode wi empoy a simpe ongitudina deformation theory combined with a generaized form of Dueau torsion theory and an extended Timoshenko bending theory to characterize the torsiona and bending deformations of micropoar beams. BCKGROUND s noted previousy in micropoar easticity mode the dispacement fied vector u and the force stress tensor σ are compemented by an independent microrotation fied vector ϑ and a coupe stress tensor χ. The potentiay asymmetric strain tensor ε and twist wryness or torsion tensor τ are defined as: ε ij = u ji ɛ ijk ϑ k τ ij = ϑ ji 1 and are reated to the force and coupe stresses via the micropoar constitutive reations: σ ij = µ + κ ε ij + µ κ ε ji + λ ε kk 1 ij χ ij = γ + β τ ij + γ β τ ji + α τ kk 1 ij where 1 is the second-order Kronecker deta tensor. The six eastic constants in Eq. are the amé coefficients µ and λ and four micropoar eastic constants κ γ β and α. Finay in the micropoar easticity theory the eastic energy U takes the form: U = U V dv U V = 1 σ ij ε ij + 1 χ ij τ ij. KINEMTICS V Consider the uniform beam in Figure 1 with an attached beam frame F c at its eft end. The frame F c is parae to the inertia frame F o and its first axis is coincide with the beam neutra axis with boundary points P 1 and P. The beam has ength cross section area poar moment of area I 1 and principa second moments of area I and I. By assuming very sma deformations through which the pane sections of the beam remain pane the torsiona warping effects on the cross section are negigibe and: σ = σ = χ = χ = χ = χ = ϑ 1 = ϑ 1 = 4

3 Soroosh Hassanpour and G. R. Hepper o x c x P1 P o x o c x c o x1 c x1 Figure 1: micropoar eastic beam. and by defining the resutant torsiona and bending rotations as: θ 1 = 1 u d = 1 u d θ = 1 u 1 d = 1 u 1 d 5 the dispacement and microrotation fieds in a micropoar beam can be written as [11: u 1 = ū 1 t c x 1 c x t c x 1 + c x θ t c x 1 u = ū t c x 1 c x ν ū 11 t c x 1 c x t c x 1 u = ū t c x 1 + c x t c x 1 c x ν ū 11 t c x 1 6 and: ϑ 1 = ϑ 1 t c x 1 ϑ = ϑ t c x 1 c x ξ ϑ 11 t c x 1 ϑ = ϑ t c x 1 c x ξ ϑ 11 t c x 1 which are indeed the first-order expansions of the genera dispacement and microrotation fieds on the beam s cross section and around the beam s neutra axis. In Eqs. 6 and 7 the variabes ū i and ϑ i i = 1 are deformations i.e. dispacements and microrotations of the beam s neutra axis. In addition ν is the strain Poisson s ratio reating norma strains to each other in a pane force stress probem and ξ is the twist Poisson s ratio reating norma twists torsions to each other in a pane coupe stress probem. These Poisson s ratios are reated to the other materia eastic constants as: ν = λ ξ = µ + λ 7 α. 8 γ + α Based on Eqs. 6 and 7 one can concude that a micropoar beam has nine independent continuous generaized coordinates or generaized dispacements i.e. three beam neutra axis dispacements ū i three resutant rotations of the beam s pane section θ i and three beam neutra axis microrotations ϑ i i = 1.

4 Soroosh Hassanpour and G. R. Hepper 4 ESTIC ENERGY EXPRESSION Based on the resuts obtained in the previous section and by utiizing Eqs. 1 and the micropoar beam eastic energy expression takes the form [11: U = 1 E ū 11 ū 11 d + 1 E I θ 1 d + 1 E I θ 1 1 d + 1 µ + 1 µ I 1 ū 1 θ d + 1 µ θ 11 1 d ū 1 + θ d + 1 κ + 1 κ I E ū 1 + θ ϑ 1 d + κ 1 ξ ϑ 11 d + κ ϑ 11 1 d + 1 γ + β ū 1 θ + ϑ d ϑ 1 d + ϑ 1 ϑ1 d 9 where µ is the shear moduus E is the tensie Young s moduus reating the norma force stresses to the norma strains in a pane force stress probem and E is the tortie torsiona moduus reating the norma coupe stresses to the norma twists torsions in a pane coupe stress probem. The tensie and tortie modui are defined as: E = µ µ + λ µ + λ = µ 1 + ν E = 5 VIRTU WORK EXPRESSION γ γ + α γ + α = γ 1 + ξ. 1 ssume that the micropoar beam shown in Figure 1 is under the action of externa voume and boundary surface forces and moments f V m V f S and m S where boundary surface forces and moments are appied ony on the most eft and right beam cross sections. The virtua work expression for such a beam can be derived as [11: δw = f Vi δū i + m i δ θ i + m Vi δ ϑ i d + f S i P 1 δū i P 1 + m P i P 1 δ θ i P 1 + m S i P 1 δ ϑ i P 1 + f S i P δū i P + m P i P δ θ i P + m S i P δ ϑ i P 4 11

5 Soroosh Hassanpour and G. R. Hepper where: f V i = m 1 = m 1 P = fi V d f S i = c x f V c x f V fi S d m i V = d m = c x f S c x f S d m P = mi V d m i S = mi S d c x f V 1 d m = c x f S 1 d m P = c x f V 1 d c x f S 1 d. 1 6 STTIC GOVERNING EQUTIONS Upon appying the simpified Hamiton s principe or the principe of virtua work on the eastic energy and virtua work expressions and after adding the required correction factors i.e. incuding k s and k s as the shear correction factors and k t as the torsion correction factor the static governing equations for a micropoar beam wi be obtained as the foowing nine equations [11: f V 1 + E ū 111 = f V + k s µ ū 11 θ 1 + κ ū 11 + θ 1 ϑ 1 = f V + k s µ ū 11 + θ 1 + κ ū 11 θ 1 + ϑ 1 = m 1 + k t µ I κ I 1 11 ξ ϑ κ ϑ 1 = m + E I 1 k s µ ū 1 + θ + κ ū 1 θ + ϑ = m + E I 11 + k s µ ū 1 θ κ ū 1 + θ ϑ = m 1 V + E ϑ 111 ξ κ I 1 11 ξ ϑ κ ϑ 1 = m V + γ + β ϑ 11 κ ū 1 θ + ϑ = m V + γ + β ϑ 11 + κ ū 1 + θ ϑ =. 1 It is noteworthy that four modes of deformation are characterized by the reations in Eq. 1; ongitudina dispacement aong the c x 1 axis by the first reation torsiona rotation around the c x 1 axis by the fourth and seventh reations atera deformation in the c x c 1 x pane by the second sixth and ninth reations and atera deformation in the c x c 1 x pane by the third fifth and eighth reations. The static equations given by Eq. 1 shoud be soved aong with the foowing BCs at boundary points P 1 and P where the + cases correspond to P 1 and the cases 5

6 Soroosh Hassanpour and G. R. Hepper correspond to P : δū 1 = or f S 1 ± E ū 11 = δū = or f S ± k s µ ū 1 θ ± κ ū 1 + θ ϑ = δū = or f S ± k s µ ū 1 + θ ± κ ū 1 θ + ϑ = δ θ 1 = or m P 1 ± k t µ I 1 1 ± κ I 1 1 ξ ϑ 11 = δ θ = or m P ± E I = δ θ = or m P ± E I 1 = δ ϑ 1 = or m 1 S ± E ϑ 11 ± ξ κ I 1 ξ ϑ 11 θ 11 = δ ϑ = or m S ± δ ϑ = or m S ± γ + β γ + β ϑ 1 = ϑ 1 = FINITE EEMENT FORMUTION common numerica approach for soving the static equations in Eq. 1 is to use the variationa or weak form of the dispacement-based FEM which is founded on the eastic energy and virtua work expressions. Here an isoparametric four-node eement with C continuity and cubic agrange poynomia basis functions [1 which are integrated exacty wi be used to generate the consistent dispacement-based finite eement matrices. With these seections the shear ocking is avoided [1 14 and the competeness and compatibiity requirements of the FEM monotonic convergence are fufied [1. The seected four-node eement has four basis shape functions as: H 1 e x 1 = 9 e x 1 e x 1 e x 1 H e x 1 = + 7 e x e 1 x 1 e x 1 H e x 1 = 7 e e x 1 x 1 e x 1 H 4 e x 1 = + 9 e e x 1 x 1 e x 1 where and e x 1 e x 1 are the eement s ength and oca frame coordinate respectivey. Denoting the noda coordinates by c x j i and the noda generaized dispacements by q j i one can interpoate the within-eement coordinates c x i generaized dispacements q i and the first space derivative of generaized dispacements q i1 as: 15 c x i = H 1 c x 1 i + H c x i + H c x i + H 4 c x 4 i q i = H 1 q 1 i + H q i + H q i + H 4 q 4 i q i1 = H 1 q 1 i + H q i + H q i + H 4 q 4 i 16 6

7 Soroosh Hassanpour and G. R. Hepper where prime i.e. denotes the first spatia derivative with respect to e x 1. Now one can define the matrix of noda generaized dispacements q j and the matrix of eementa generaized dispacements q e as: q j = [ [ q j i = ū j 1 ū j ū j [ q e = [ q j = q 1 T q T θ j 1 q T θ j q 4 T ϑ j 1 ϑ j ϑ j T T 17 θ j which can be used to rewrite the expansion of q i in Eq. 16 as: q i = H qi q e 18 where H qi are the shape function matrices of the forms: Hū1 = [ H H 1 8 H 1 8 H Hū = [ 1 1 H H 1 8 H 1 8 H Hū = [ 1 H H 1 8 H 1 8 H H = [ 1 H H 1 8 H 1 8 H H θ = [ 1 4 H H 1 8 H 1 8 H H = [ 1 5 H H 1 8 H 1 8 H 4 1 H = [ 1 6 H H 1 8 H 1 8 H 4 1 H ϑ = [ H 1 H H H H ϑ = [ H 1 H H H By recaing Eqs. 9 and 11 and using the expansions of the form given by Eq. 18 and the shape functions given in Eq. 19 the virtua work and eastic energy expressions corresponding to a micropoar beam eement can be written as: 19 and: where Q e δw e = δ q e T Q e U e = 1 q e T K e q e 1 is the finite eement generaized force matrix: Q e = H T ū i + P V f i + H T θi m i + H T ϑi m i V d H T ū i S f i + H T θi m i P + H T ϑi m i S 7

8 Soroosh Hassanpour and G. R. Hepper and K e is the finite eement stiffness matrix: K e = E H T ū 1 H ū 1 d + E I H θ T H θ d + E I + k s µ H T ū H H ū H d H T H d + k s µ H T ū + H H ū θ + H θ d + k t µ I 1 + κ H T ū + H H H ū ϑ + H H ϑ d H T H d + κ + κ I 1 H T ū H θ + H H ū ϑ H θ + H ϑ d H T ξ H H ξ H d + 4 κ H T H H H d + E H T H d + γ + β H ϑ T H ϑ + H ϑ T H ϑ d. 8 RESUTS ND CONCUSIONS To provide numerica exampes and make a comparison with the we-known cassica beam modes the herein deveoped micropoar beam mode is impemented in MTB [15 as a FEM mode with 16 eements 4 nodes per eement and 9 DOFs per node by using the nondimensiona parameters in Tabe 1. The nondimensiona parameters given in Tabe 1 and the nondimensiona dispacements used for reporting the resuts are reated to the micropoar beam s dimensiona parameters and dispacements as: ˆR i = I i ˆµ = µ E ˆκ = κ E ˆγ = γ E ˆβ = β E ˆū i = ūi. 4 The variations of the tip torsiona rotations of a cantievered micropoar beam subjected to the action of an externa body moment m 1 V with the variations of nondimensiona parameters ˆκ and ˆγ are shown in Figure. naogous pots for the tip bending deformations of a cantievered micropoar beam under the action of externa body force and moment 8

9 Soroosh Hassanpour and G. R. Hepper Tabe 1: Dimensioness parameters used in the numerica beam mode. Parameter ˆR ˆµ k t k s k s ˆκ ˆγ = ˆβ ξ Vaue [1 1 1 [ f V and m V are depicted in Figures and 4. In Figures 4 a the deformations are normaized with respect to the deformations of a cassica beam mode based on Dueau torsion and Timoshenko bending theories under the action of the same oad obtained from a FEM-based numerica mode with 16 eements 4 nodes per eement and 6 DOFs per node. The figures show the size-effect phenomenon [4 and the effect of choice of micropoar materia constants ˆκ and ˆγ on the torsiona and bending deformations of a micropoar beam. s expected arge vaues of ˆκ and ˆγ resut in more significant differences between the micropoar and cassica beam modes. arge ˆκ vaues and sma ˆγ vaues on the other hand resut in a micropoar beam mode that coincides with the cassica beam mode. The transition region is we defined and reativey abrupt notice that the parameter scaes are ogarithmic. It is aso interesting that the micropoar beam mode with very sma ˆκ vaues shows singuar behaviors when subjected to the action of an externa voume moment. Note that the noisy behavior of the surface pots in the region where ˆκ 1 and ˆγ 1 is due to an i-conditioned FEM stiffness matrix and the incusion of numerica precision errors. Out of the many or indeed infinite possibe micropoar beam modes which can be obtained by varying ˆκ and ˆγ five specific micropoar beam modes are seected for a more thorough examination of the static deformations. In Figures 5 and 6 the compete torsiona and bending deformations of five specific micropoar beam modes under the action of externa moments m V 1 and m V are potted aongside with the torsiona and bending deformations of a cassica beam mode based on Dueau torsion and Timoshenko bending theories under the action of the same oad. The magnitude of the appied oad in each of Figures 5 and 6 set of six pots is seected to resut in a maximum torsiona rotation or bending nondimensiona dispacement of. in the cassica beam mode. The singuar behavior of a micropoar beam mode with sma ˆκ vaues and the coincidentwith-cassica behavior of a micropoar beam mode with arge ˆκ vaues and sma ˆγ vaues are more apparent in Figures 5 and 6. REFERENCES [1 W. Nowacki. Theory of asymmetric easticity transated by H. Zorski. Poish Scientific Pubishers PWN & Pergamon Press Warsaw Poand & Oxford United Kingdom [ D. Iesan. Torsion of micropoar eastic beams. Int. J. of Eng. Sci. 911:

10 Soroosh Hassanpour and G. R. Hepper ˆR = 5 ˆR = 5 mp/c ogˆκ.51 - ogˆγ -1 mp/c ogˆκ.51 - ogˆγ -1 Figure : Reative tip torsiona deformations of a cantievered micropoar beam under voume moment m 1 V vs. micropoar eastic constants. ˆR = 5 ˆR = 5 ˆR = 5 ˆūmp/ˆūc ogˆκ.51 - ogˆγ -1 mp/c ogˆκ.51 - ogˆγ -1 ϑmp/c ogˆκ.51 - ogˆγ -1 Figure : Reative tip bending deformations of a cantievered micropoar beam under voume force f V vs. micropoar eastic constants. ˆR = 5 ˆR = 5 ˆR = 5 ˆūmp/ˆūc ogˆκ.51 - ogˆγ -1 mp/c ogˆκ.51 - ogˆγ -1 ϑmp/c ogˆκ.51 - ogˆγ -1 Figure 4: Reative tip bending deformations of a cantievered micropoar beam under voume moment m V vs. micropoar eastic constants. 1

11 Soroosh Hassanpour and G. R. Hepper.4 Cassica Mode ˆR = Micropoar Mode; ˆκ = 1 ˆγ = 1 1 ˆR = Micropoar Mode; ˆκ = 1 1 ˆγ = 1 1 ˆR = Micropoar Mode; ˆκ = 1 ˆγ = 1 ˆR = Micropoar Mode; ˆκ = 1 1 ˆγ = 1 ˆR = Micropoar Mode; ˆκ = 1 4 ˆγ = 1 4 ˆR = Figure 5: Torsiona deformations of a cantievered beam under voume moment ˆ m V 1 different cassica and micropoar modes. obtained from.4 Cassica Mode ˆR = Micropoar Mode; ˆκ = 1 ˆγ = 1 1 ˆR = Micropoar Mode; ˆκ = 1 1 ˆγ = 1 1 ˆR = ˆū.51 ˆū.51 & ϑ ˆū.51 & ϑ ˆū Micropoar Mode; ˆκ = 1 ˆγ = 1 ˆR = 5.8 ˆū ϑ Micropoar Mode; ˆκ = 1 1 ˆγ = 1 ˆR = 5.8 ˆū ϑ Micropoar Mode; ˆκ = 1 4 ˆγ = 1 4 ˆR = ˆū.51 & ϑ ˆū.51 & ϑ ˆū.51 & ϑ ˆū ϑ ˆū ϑ ˆū ϑ Figure 6: Bending deformations of a cantievered beam under voume moment ˆ m V different cassica and micropoar modes. obtained from 11

12 Soroosh Hassanpour and G. R. Hepper [ R.D. Gauthier and W.E. Jahsman. quest for micropoar eastic constants. Tran. of the SME Series E-J. of pp. Mech. 4: [4 H.C. Park and R.S. akes. Torsion of a micropoar eastic prism of square crosssection. Int. J. of So. and Str. 4: [5 S. Potapenko and E. Shmoyova. Weak soutions of the probem of torsion of micropoar eastic beams. Zeitschrift für ngewandte Mathematik und Physik 61: [6 R.D. Gauthier and W.E. Jahsman. Bending of a curved bar of micropoar eastic materia. Trans. of the SME Series E-J. of pp. Mech. 4: [7 G.V.K. Reddy and N.K. Venkatasubramanian. On the fexura rigidity of a micropoar eastic cyinder. Trans. of the SME J. of pp. Mech. 45: [8 G.V.K. Reddy and N.K. Venkatasubramanian. On the fexura rigidity of a micropoar eastic circuar cyindrica tube. Int. J. of Eng. Sci. 179: [9 F.Y. Huang B.H. Yan J.. Yan and D.U. Yang. Bending anaysis of micropoar eastic beam using a -D finite eement method. Int. J. of Eng. Sci. 8: [1 S. Ramezani R. Naghdabadi and S. Sohrabpour. naysis of micropoar eastic beams. Eur. J. of Mech.-/Soids 8: 8 9. [11 S. Hassanpour. Dynamics of Gyroeastic Continua. Ph.D. thesis Mechanica and Mechatronics Engineering Department University of Wateroo Wateroo ON Canada 14. [1 K.J. Bathe. Finite eement procedures in engineering anaysis. Prentice-Ha Upper Sadde River NJ United States 198. [1.H. Vermeuen and G.R. Hepper. Predicting and avoiding shear ocking in beam vibration probems using the b-spine fied approximation method. Comp. Meth. in pp. Mech. and Eng. 158: [14 C.C. Richter and G.R. Hepper. -spine finite eements. In SME Int. Mech. Eng. Cong. and Exp. pages Orando F US November [15 MTB. 64-bit win64 version R1a. The MathWorks Inc. Natick M United States 1. [Computer software. 1

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