ENERGY AND MOMENTUM CONSERVING VARIATIONAL BASED TIME INTEGRATION OF ANISOTROPIC HYPERELASTIC CONTINUA

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1 ECCOMAS Cogress 06 VII Europea Cogress o Computatioal Methods i Applied Scieces ad Egieerig M. Papadrakakis, V. Papadopoulos, G. Stefaou, V. Plevris (eds.) Crete Islad, Greece, 5 0 Jue 06 ENERGY AND MOMENTUM CONSERVING VARIATIONAL BASED TIME INTEGRATION OF ANISOTROPIC HYPERELASTIC CONTINUA Michael Groß, Rajesh Ramesh ad Julia Dietzsch 3 Techische Uiversität Chemitz, Professorship of applied mechaics ad dyamics Reichehaier Straße 70, D-096 Chemitz michael.gross@mb.tu-chemitz.de, rajesh.ramesh@mb.tu-chemitz.de, 3 tmd@mb.tu-chemitz.de Keywords: Time steppig schemes, fiite elasticity, aisotropy, fiber-reiforced cotiua. Abstract. For may years, the importace of fiber-reiforced polymers is steadily icreasig i mechaical egieerig. Accordig to the high stregth i fiber directio, these composites replace more ad more traditioal homogeeous materials, especially i lightweight structures. Fiber-reiforced material parts are ofte maufactured from carbo fibers as pure attachmet parts, or from steel for trasmittig forces. Whereas attachmet parts are mostly subjected to small deformatios, force trasmissio parts usually suffer large deformatios i at least oe directio. For the latter, a geometrically o-liear formulatio of these aisotropic cotiua is idispesable. A familar example is a rotor blade, i which the fibers possess the fuctio of stabilizig the structure i order to couteract large cetrifugal forces. For log-ru umerical aalyses of rotor blade motios, we have to apply umerically stable ad robust time itegratio schemes for aisotropic cotiua. This paper is a extesio of Referece, which is i tur a extesio of Referece 3 to a special aisotropic material class, amely a trasversely isotropic hyperelastic material based o the wellkow cocept of structural tesors. I Referece 3, higher-order accurate time-steppig schemes are developed systematically with the focus o umerical stability ad robustess i the presece of stiffess combied with large rotatios for computig large motios. I the former work, these advatages over covetioal time steppig schemes are combied with highly o-liear aisotropic material formulated with polycovex free eergy desity fuctios 4. The correspodig time itegrators preserve all coservatio laws of a free motio of a hyperelastic cotiuum, which meas the total liear ad the total agular mometum coservatio law as well as the total eergy coservatio law. Both are umerically advatageous, because it guaratees that the discrete cofiguratio vector is embedded i the physically cosistet solutio space. I order to guaratee the preservatio of the total eergy, the trasiet approximatio of the aisotropic stress tesor is superimposed with a algorithmic stress field based o a assumed strai field. The preseted umerical examples show the behaviour of the o-liear aisotropic material i Referece 4 uder static ad trasiet loads, their coservatio laws ad the higher-order accuracy of the variatioal based time approximatio.

2 INTRODUCTION We begi by summarizig the kiematical aspects of the cosidered trasversely isotropic cotiuum. I Fig. o the right-had side, we show the referece cofiguratio of the cosidered fiber-reiforced cotiuum body B. The cofiguratio = B M 0 B F 0 is defied u F x 3 X a 0 x B t x Figure : Referece ad curret cofiguratio of a trasversely isotropic cotiuum. as the homogeized uio of the set B M 0 for the matrix ad the set B F 0 for the fibers. The imagiary fiber at ay poit X is directed alog the ormalized vector a 0. Sice we assume that both subsets are perfectly coected, the correspodig stretched vector a i the deformed cofiguratio B t is give by where a = F a 0 () F := u + I () deotes the deformatio gradiet of ad u the displacemet vector field. The tesor I desigates the secod-order uit tesor. The symbol deotes the partial derivative with respect to the material poit X. The deformatio gradiet F F of the fiber cotiuum B0 F the takes the form F F := a a 0 = F a 0 a 0 = F A 0 (3) where A 0 := a 0 a 0 (4) desigates the structural tesor of the fiber referece cofiguratio B F 0. The correspodig right CAUCHY-GREEN tesor C F the reads C F := F T F F F = F a 0 a 0 T F a 0 a 0 = a 0 a 0 C a 0 a 0 = A 0 C A 0 = C : A 0 A 0 (5) where C := F T F deotes the right CAUCHY-GREEN tesor of. Based o these deformatio measures, we cosider the strai eergy fuctio W of the cosidered trasversely isotropic elastic cotiuum o the oe had (i) as the upartitioed fuctio W (C; A 0, κ 0 ), where the semicolo i the argumet separates the parameter A 0 ad κ 0, actig at ay X, from the variable C, ad o the other had (ii) as the partitioed fuctio W (C; A 0, κ 0 ) = W M (C; κ 0M ) + W F (C F ; κ 0F ) (6)

3 The parameter κ 0, κ 0M ad κ 0F are vectors icludig material costats with respect to. The secod PIOLA-KIRCHHOFF stress tesor S correspodig to Eq. (6) is give by S W (C; A 0, κ 0 ) C = S M + S F (7) = DW M (C; κ 0M ) + DW F (C F ; κ 0M ) : C F C = DW M (C; κ 0M ) + A 0 DW F (C F ; κ 0M ) A 0 (8) S = DW M (C; κ 0M ) + DW F (C F ; κ 0M ) : A 0 A 0 (9) The otatio D( ) deotes the FRÉCHET derivative of a volume desity with respect to its argumet. The strai eergy fuctios W of, W M of B M 0 or W F of B F 0, respectively, directly depeds o the ivariats of the correspodig right CAUCHY-GREEN tesors. We assume (i) the upartitioed case ad (ii) the partitioed case where W (C; A 0, κ 0 ) = Ŵ (I, I, I 3, I 4 ; κ 0 ) (0) W M = ŴM(I, I, I 3 ; κ 0M ) W F = ŴF (I 4 ; κ 0F ) () I := C : I I := (I ) C : I I 3 := det C () deotes the tesor ivariats of the right CAUCHY-GREEN tesors C, ad I 4 C F : A 0 = A 0 C A 0 : A 0 = a 0 C a 0 a 0 a 0 = a 0 C a 0 = C : A 0 = a a =: C F (3) the squared fiber stretch C F λ F. Usig the fourth ivariat I 4 =: C F, the right CAUCHY- GREEN tesor C F ad the secod PIOLA-KIRCHHOFF stress tesor S F of the partitioed strai eergy fuctio, respectively, ca be simply writte as C F = C F A 0 S F = DŴF (C F ; κ 0F ) C F : A 0 A 0 = DŴF (C F ; κ 0F ) A 0 (4) C F Hece, the directios of the fiber deformatio tesor C F ad fiber stress tesor S F are uiquely prescribed by the structure tesor A 0, as expected. EULER-LAGRANGE EQUATIONS With regard to the umerical time itegratio, we ow itroduce variatioally cosistet temporally cotiuous assumed strais C ad C F, as well as temporally discotiuous superimposed stresses S ad S F, respectively. The former are ecessary for a exact aalytical time itegratio of approximated strai eergy fuctios 6, ad the later for their exact umerical time itegratio 8. Hece, the superimposed stresses S ad S F are resposible for the eergy cosistecy of the discrete EULER- LAGRANGE equatios, but they have to vaish idetically for guarateeig eergy cosistecy of the cotiuous EULER-LAGRANGE equatios. We cosider the strai eergy fuctio 3

4 . W (C; A 0, κ 0 ) o (upartitioed strai eergy), or. W M (C; κ 0M ) ad W F (C F ; κ 0F ) separately (partitioed strai eergy). We derive the cotiuous equatios of motio by usig a mixed priciple of virtual power or priciple of Jourdai, respectively, a differetial variatioal priciple 9. The modivatio for applyig this priciple from the outset is to satisfy the total eergy balace i both the cotiuous as well as the discrete settig. Deotig by a superimposed dot the partial time derivative, i the upartitioed case, this balace takes the form T ( u, v, ṗ; ρ 0 ) + Π it ( u, C, S; A0, κ 0, S) = with the kietic power where T ( u, v, ṗ; ρ 0 ) := ρ 0 b u dv + t u da + h ( u ū ) da t u (5) ρ 0 v p v dv ṗ v u dv + p ü dv (6) p = ρ 0 v ad v = u (7) deotes the liear mometum vector ad the material velocity vector, respectively. The scalar ρ 0 deotes the mass desity field i. The stress power Π it is writte i depedece o the secod PIOLA-KIRCHHOFF stress tesor S, the superimposed stress tesor S ad the assumed strai tesor C as Π it ( u, C, S; A0, κ 0, S) := DW ( C; A 0, κ 0 ) + S S : C dv Ṡ : C C(u) dv + S : Ċ( u) dv B 0 = Ẇ dv (8) where S = DW ( C; A 0, κ 0 ) + S C = C(u) := ( u + I) T ( u + I) (9) The superimposed stress tesor S = O, with the zero tesor O, has to vaish for eergy cosistecy. O the right-had side of Eq. (5), there is the exteral power depedig o the body force vector b per uit mass o, the tractio force vector t per uit area o the NEUMANN boudary t ad the LAGRANGE multiplier vector h eforcig the costrait u ū = 0 o u (0) of a prescribed displacemet ū o the DIRICHLET boudary u. Both boudary sets satisfy the coditios = t u ad t u =, where desigates the boudary of the referece cofiguratio. 4

5 . The upartitioed strai eergy fuctio I the upartitioed case, we itroduce the assumed strai field C ad the superimposed stress field S for the etire referece cofiguratio variatioally cosistet by cosiderig the virtual power priciple δ Ḣ( u, v, ṗ, C, S; ρ0, A 0, κ 0, b, t, ū, S) := δ T ( u, v, ṗ; ρ 0 ) + δ Πit ( u, C, S; A0, κ 0, S) + δ Πext ( u; ρ 0, b, t, ū) = 0 () with the virtual kietic power δ T ( u, v, ṗ; ρ 0 ) := ρ 0 v p δ v dv δ ṗ v u dv + ṗ δ u dv () the virtual exteral power δ Πext ( u; ρ 0, b, t, ū) := ρ 0 b δ u dv t δ u da t h δ u da u (3) ad the virtual iteral power δ Πit ( u, C, S; A0, κ 0, S) := DW ( C; A 0, κ 0 ) + S S : δ C dv δ S : C Ċ( u) dv + S : δ Ċ( u) dv (4) The symbol δ deotes the variatio with respect to the variables (ot the parameter behid the semicolo) i the fuctio argumet. Itegratio by parts i the last term of Eq. (4) furishes S : δ Ċ( u) dv = FS : (δ u) dv = FSN δ u da DIVFS δ u dv (5) t The vector N deotes the ormal field o the NEUMANN boudary t, ad DIV the divergece operator with respect to X. Rearragig termes i Eq. () accordig to the variatios δ ṗ, δ v, δ S, δ C ad δ u, we obtai the variatioal form 0 = ρ 0 v p δ v dv δ ṗ v u dv DIVFS + ρ 0 b ṗ δ u dv S DW ( C; A 0, κ 0 ) S : δ C dv δ S : C Ċ( u) dv t FSN δ u da h δ u da (6) t u Owig to the fudametal theorem of variatioal calculus, the correspodig cotiuous EU- LER-LAGRANGE equatios read v = u with u(t 0 ) = u 0 (7) ρ 0 v = p t t 0 (8) DIVFS + ρ 0 b = ṗ with p(t 0 ) = p 0 ρ 0 v 0 (9) Ċ( u) = C with C(t0 ) = C(u 0 ) ( u 0 + I) T ( u 0 + I) (30) DW ( C; A 0, κ 0 ) + S = S t t 0 (3) 5

6 with the correspodig iital coditios i as well as the boudary coditios FSN = t t t 0 o t (3) δ u = 0 u = ū with u(t 0 ) = ū(t 0 ) o u (33) Cosequetly, the PIOLA-KIRCHHOFF stress tesor field S is temporally discotiuous, but the displacemet vector field u, the material velocity field v, the liear mometum field p as well as the assumed strai field C are temporally cotiuous. The superimposed stress tesor S has to vaish i these EULER-LAGRANGE equatios for satisfyig the total eergy balace i Eq. (5).. The partitioed strai eergy fuctio I the partitioed formulatio, we itroduce the assumed strai tesor C ad the superimposed stress tesor S M o B0 M ad the assumed strai C F ad the superimposed fiber stress S F o B0 F by cosiderig the virtual power priciple δ Ḣ( u, v, ṗ, C, SM, CF, S F ; ρ 0, κ 0M, κ 0F, A 0, b, t, ū, S M, S F ) := (34) δ T ( u, v, ṗ; ρ 0 ) + δ Πext ( u; ρ 0, b, t, ū) + δ Πit ( u, C, SM, CF, S F ; κ 0M, κ 0F, A 0, S M, S F ) = 0 The virtual kietic power is idetical to Eq. () ad the virtual exteral power is idetical to Eq. (3). But the virtual iteral power is ow give by with δ Πit ( u, C, SM, CF, S F ; κ 0M, κ 0F, A 0, S M, S F ) := DW M ( C; κ 0M ) + S M S M : δ C dv B 0 DŴF ( C F ; κ 0F ) + S F S F : A 0 δ CF dv δ S M : C Ċ( u) dv + S M : δ Ċ( u) dv δ S F : CF A 0 ĊF ( u) dv + S F : δ Ċ F ( u) dv (35) C F (u) := ( u + I) A 0 T u + I A 0 = A 0 ( u + I) T ( u + I) A 0 (36) Bearig i mid the idetity S F : δ Ċ F ( u) = S F : A 0 δ Ċ( u) A 0 = F A 0 S F A 0 : u = F F S F : A 0 : u (37) itegratio by parts leads to S F : δ Ċ F ( u) dv = F F S F : A 0 N δ u da DIV F F (S F : A 0 ) δ u dv (38) t 6

7 Agai, by rearragig the terms i Eq. (34) due to the idepedet variatios δ ṗ, δ v, δ S M, δ C, δ S F, δ CF ad δ u, we obtai the variatioal form 0 = ρ 0 v p δ v dv δ ṗ v u dv DIVFS M + F F (S F : A 0 ) + ρ 0 b ṗ δ u dv t (FS M + F F (S F : A 0 )) N δ u da t h δ u da u S M DW M ( C; κ 0M ) S M S F : A 0 DŴF ( C F ; κ 0F ) S δ CF F : δ C dv dv C Ċ( u) : δ S M dv CF A 0 ĊF ( u) : δ S F dv (39) Takig the fudametal theorem of variatioal calculus ito accout, we arrive at the EULER- LAGRANGE equatios v = u with u(t 0 ) = u 0 (40) ρ 0 v = p t t 0 (4) DIV FS M + F F (S F : A 0 ) + ρ 0 b = ṗ with p(t 0 ) = p 0 ρ 0 v 0 (4) Ċ( u) = C with C(t0 ) = C(u 0 ) (43) Ċ F ( u) : A 0 = CF with CF (t 0 ) = C F (u 0 ) : A 0 (44) DW M ( C; κ 0M ) + S M = S M t t 0 (45) DŴF ( C F ; κ 0F ) + S F A 0 = S F t t 0 (46) with S M := O ad S F := 0 for satisfyig the total eergy balace, ad the boudary coditios FS M + F F (S F : A 0 ) N = t t t 0 o t (47) δ u = 0 u = ū with u(t 0 ) = ū(t 0 ) o u (48) Cosequetly, the PIOLA-KIRCHHOFF stress fields S M ad S F are temporally discotiuous, but the assumed strai fields C ad C F are agai temporally cotiuous. 3 FULLY-DISCRETE WEAK FORMULATION Next, we derive the temporally ad spatially discrete weak variatioal formulatio. I this sectio, we restrict ourselves to a liear piecewise cotiuous time approximatio i u, v, p (compare Referece 5) as well as C ad C F, i order to demostrate the cosistecy of the variatioal derivatio of the assumed strai approximatios ad the superimposed stress tesors S ad S F with Referece 8. But ote that i the upartitioed case both approximatios are ew for higher-order time approximatios. 7

8 3. The upartitioed strai eergy fuctio The time itegrator preseted here follows from cosiderig the virtual power priciple at collocatio poits ξ i of the time iterval t 0, t N of iterest, itroduced by a time itegratio tn δ Ḣ( u(t), v(t), ṗ(t), C(t), S(t); ρ0, A 0, κ 0, b(t), t(t), ū(t), S(t)) dt = t 0 N t+ δ Ḣ( u (t), v (t), ṗ (t), C (t), S (t); ρ 0, A 0, κ 0, b (t), t (t), ū (t), S (t)) dt =0 t N =0 =0 δ Ḣ( u h(α), v h(α), ṗ h(α), C h(α), S h(α); ρ 0, A 0, κ 0, b h(α), t h(α), ū h(α), S h(α))h dα 0 N δ Ḣ( u h(ξ ), v h(ξ ), ṗ h(ξ ), C h(ξ ), S h(ξ ); ρ 0, A 0, κ 0, b h(ξ ), t h(ξ ), ū h(ξ ), S h(ξ )) h N =0 δ Ḣ d (u +, v +, p +, C +, S + ; ρ 0, A 0, κ 0, b +, t + ad the ormalized time α 0, via the liear trasformatio, ū +, S. + ) h = 0 (49) τ : t, t + t t + α (t + t ) = t + α h (50) with respect to the time step size h, ad after applyig the midpoit rule with the oe Gauss poit ξ = to the metioed piecewise liear time approximatios u h(α) := u + α (u + u ) v h(α) := v + α (v + v ) (5) p h(α) := p + α ( p + p ) C h(α) := C + α (C + C ) (5) I the followig, we use the commo fiite differece otatio ( ) + for symbols ( ) h ( ). Without itegratig by parts but rearragig termes i Eq. (49) accordig to the idepedet variatios, we obtai the semi-discrete variatioal form 0 = ρ 0 v + p + δ v + dv δ p + v + u + u dv h S + DW ( C + ; A 0, κ 0 ) S + : δ C + dv C + C (F + + F ) T (F + F ) : δ S + dv p+ p + + B T S + h + ρ 0 b + δ u + dv δ u + da δ u + da (53) t t + with the liearized strai operator B + u h + defied by 8 B + δ u + := F T (δ + u + ) + (δ u + ) T F + (54) 8

9 ad bearig i mid the differetiatio rule ( ) = d( ) dα dα dt = d( ) dα h (55) as well as the vaishig variatios δ u, δ v, δ p ad δ C due to the iitial coditios u(t 0 ) = u 0 p(t 0 ) = ρ 0 v 0 (56) v(t 0 ) = v 0 C(t0 ) = ( u 0 + I) T ( u 0 + I) (57) i the first time step t 0, t. At this poit, we are able to derive spatially local relatios of tesor fields at each poit X, which ca be used to elimiate variables i the discrete system of equatios of motio without takig spatial itegrals. O the other had, we may keep all the variables ad solve a multifield formulatio if we are iterested i these variables for postprocessig purposes, or we may elimiate the variables after the spatial itegrals have bee take. The first lie of Eq. (53) leads to p = ρ 0 v p + = ρ 0 v + h v + = u + u (58) where Eq. (58.) is obvious from the iitial coditio i Eq. (56.), ad Eq. (58.3) ca be see as first equatio of motio. The secod lie of Eq. (58) furishes the discrete costitutive relatio S + = DW ( C + ; A 0, κ 0 ) + S + But ote that the superimposed secod PIOLA-KIRCHHOFF stress tesor S + must ot vaish for eergy cosistecy as i the cotiuous settig (compare Referece 7). We derive it below i a separate variatioal problem. I the third lie of Eq. (53), we take ito accout the symmetry of δ S +, leadig to the idetity ad cosequetly to the local relatio F T + F F T F + : δ S + (59) = 0 (60) C + C = F T +F + F T F C + = F T +F + (6) accordig to the iitial coditio i Eq. (57.). Hece, we arrive at the followig variatioally cosistet assumed strai approximatio, which is proposed for eergy cosistet time steppig schemes at least sice the publicatio of Referece 6: C + := C + C + (6) The spatial approximatio i the variatioal formulatio is based o triliear shape fuctios for a eight-ode brick elemet for the volume ad biliear shape fuctios for a four-ode quadrilateral elemet for the boudaries, which approximate the geometry i, the displacemet vector u ad the material velocity vector v at the cosidered discrete time poits t. Hece, followig the otatio i Referece 0, we apply the approximatios u = N u δ u = N δ u (63) v = N v δ v = N δ v (64) 9

10 where N deotes the matrix of the triliear shape fuctios ad the vector u combies the odal displacemets. A aalogous otatio is used for the odal velocities ad odal variatios, respectively. O the boudary, we apply the approximatios u = N u δ u = N δ u (65) where N deotes the matrix of the biliear shape fuctios. The last two lies of Eq. (53) the leads to the discrete variatioal formulatio δ u T + M v + v + S h + dv = δ u T + H t t + + H u h + + M b + (66) B T + with the system matrices M := ρ 0 N T N dv H t := N dv t NT Hu := N dv u NT (67) as well as the matrix represetatios B + ad S + of the liearized strai operator ad the secod PIOLA-KIRCHHOFF stress tesor, respectively. Fially, we apply the fudametal theorem of variatioal calculus ad arrive at the discrete system of equatios of motio M v + v h + B T + S + = H t t + + H u h + + M b + If we ow multiply Eq. (68) o both sides from the left by the velocity vector (68) v + the first term o the left had side takes the form v T M v + v = + h = v + + v (69) v T h + M v + v T M v = T + T h (70) which deotes the discrete time derivative of the total kietic eergy. O the righthad side of Eq. (68), we obtai the discrete exteral power v T + H t t + + H u h + + M b + = ut + u T H t t h + + H u h + + M b + = Πext + Π ext (7) h where Eq. (58.3) have bee take ito accout. Accordigly, we arrive at the discrete total eergy balace if the relatio v + B T + u + u B T + h DW ( C + DW ( C + ; A 0, κ 0 ) + S + ; A 0, κ 0 ) + S + 0 dv = dv = Πit + Π it (7) h

11 or W + W = DW ( C + ; A 0, κ 0 ) + S + : B + u + u (73) is fulfilled. O the other had, employig the assumed strai tesor i Eq. (6) i the defiitio of the liearized strai operator i Eq. (54), we obtai the idetity B + u + u = F T F + + F + F + F + F T = C+ C + C T + C T = C + C (74) which i the ed leads to the scalar-valued costrait G( S + ) o the superimposed stress field at each poit X, give by S + G( S + ) := W + W DW ( C + ; A 0, κ 0 ) + S + : C + C = 0 (75) As the superimposed stress tesor is symmetric, dim ( dim + )/ compoets of the tesor S + has to be uiquely determied such that the scalar-valued costrait i Eq. (75) is satisfied. Therefore, we solve the separate costrait variatioal problem with L(µ, S + ) := C + S + δ L(µ, S + ) = 0 (76) : S + C + usig the correspodig discrete EULER-LAGRANGE equatios + µ G( S + ) (77) L S + C + S + C + µ C + C = O L µ G( S + ) = 0 (78) Note that i Eq. (77) the right CAUCHY-GREEN tesor C + operates as metric tesor as i the physically cosistet deviator stress i Referece. Therefore, this costrait variatioal problem could be also pushed forward to the curret cofiguratio B t, ad formulated with the KIRCHHOFF stress tesor τ ad the metric g i B t. After isertig Eq. (78.) i Eq. (78.), we arrive at the two spatially local discrete EULER-LAGRANGE equatios for the superimposed stress field S + ad the scalig factor µ at each poit X, give by S + = µ G(O) = µ C + C + C C + C + C + C : C + C C + (79) (80) We are able to search umerically for the LAGRANGE multiplier µ, but usually it is elimiated aalytically. This leads to the stress tesor 8 S + = G(O) C + C + C : C + C C + C + C + C C + (8)

12 Cosequetly, usig the superimposed discrete stress tesor i Eq. (8), the discrete equatio of motio i Eq. (68) leads, by desig, to the discrete eergy balace T + T + Π it + Π it + Π ext + Π ext = 0 H + = H (8) which idicates exact algorithmic total eergy coservatio. But ote that i a umerical implemetatio, the exact algorithmic total eergy coservatio is idicated by H + H < tol (83) where tol deotes the tolerace of the applied NEWTON-RAPHSON method for solvig the oliear discrete EULER-LAGRANGE equatios. Further, you should be careful with solutio steps where C + C whe applyig the stress formula i Eq. (8). You should geerally implemet a prestep formula for the displacemets u + takig ito accout all applied loads ad iitial coditios 3. If the tesor S + the strai eergies W ad W + at the assumed strai tesor C + is eglected the a TAYLOR series expasio of, give by DW ( C + ; A 0, κ 0 ) : C + C = W + W + O ( C + C 3) (84) shows that Eq. (83) ca be guarateed oly for C + C < tol. Therefore, we coclude that eergy cosistecy of the discrete EULER-LAGRANGE equatios is oly give if the discrete superimposed stress tesor S + is o-vaishig. 3. The partitioed strai eergy fuctio The time steppig scheme for the partitioed strai eergy also follows from a time itegratio of the correspodig virtual power priciple o the time iterval t 0, t N of iterest. Thus, we obtai a discrete variatioal coditio at the collocatio poit ξ, give by δ Ḣ d (u +, v +, p +, C +, S M+, C F+, S F+ ; ρ 0, κ 0F, κ 0M, A 0, b +, t +, ū +, S M+, S F+ ) h = 0 (85) The assumed squared fiber stretch C F is also liear piecewise cotiuous approximated by C F h (α) := C F + α ( C F+ C F ) (86) with the iitial coditio o the time step t, t +, give by C F = C F : A 0 = F T F F F : A 0 (87)

13 Rearragig termes i Eq. (85) accordig to the idepedet variatios δ p +, δ v +, δ u +, δ S M+, δ C +, δ S F+ ad δ C F+, we obtai the semi-discrete variatioal form 0 = ρ 0 v + p + δ v + dv δ p + v + u + u dv h S M+ DW M ( C + ; κ 0M ) S M+ : δ C + dv B 0 S F+ : A 0 DŴF ( C F+ ; κ 0F ) S F+ : δ C F+ dv B 0 C + C (F + + F ) T (F + F ) : δ S + dv C F+ A 0 C F A 0 ( ) T ( ) F F+ + F F FF+ F F : δ S F+ dv B 0 p+ p ) + + B T S + h M+ + (S F+ : A 0 A 0 ρ 0 b + δ u + dv δ u + da δ u + da (88) t t + u h + The first lie of Eq. (88) also furishes the Eqs. (58). The secod ad third lie leads to the discrete costitutive stress relatios S M+ = DW M ( C + ; κ 0M ) + S M+ (89) S F+ = DŴF ( C F+ ; κ 0F ) + S F+ A 0 (90) The fourth ad fivth lie of Eq. (88) determies the right CAUCHY-GREEN strais at the time poit t + by the equatios C + = F T +F + C F+ = C F+ : A 0 = F T F + F F+ : A 0 (9) leadig to the approximatios C + CF + C F+ := C + C + CF+ := (9) Hece, the full symmetric tesor C F has ot to be stored at the midpoit t +, but merely the scalar strai C F+. Aalogous to the upartitioed case, the last lies of Eq. (88) gives the discrete system of equatios of motio M v + v + DW M ( C h + ; κ 0M ) + DŴF ( C F+ ; κ 0F ) A 0 (93) B T + + S M+ + S F+ A 0 dv = H t t + + H u h + + M b + by takig ito accout the Eq. (89) ad (90). Accordigly, we arrive at exact algorithmic total eergy coservatio i the sese of Eq. (8), if for the matrix cotiuum B0 M the costrait G M ( S M+ ) := W M+ W M DW M ( C + ; κ 0M ) + S M+ : C + C = 0 (94) 3

14 is fulfilled, ad if the relatio v + B T + u + u B T + h DŴF ( C + ; A 0, κ 0F ) + DŴF ( C + DŴF ( C + DŴF ( C + ; A 0, κ 0F ) + ; A 0, κ 0F ) + S + ; A 0, κ 0F ) + S + A 0 dv = A 0 dv = S + A 0 : B + u + u dv = ŴF+ ŴF dv S + A 0 : C + C = ŴF + ŴF (95) or G F ( S F+ ) := ŴF + ŴF DŴF ( C F+ ; κ 0F ) + S F+ C F+ C F = 0 (96) is fulfilled. Accordig to Eq. (8), the superimposed stress tesor S M for the matrix cotiuum is give by S M+ = G M (O) C + C + C : C + C C + C + C + C C + (97) The superimposed scalar stress field S F i the equatio of motio is defied such that we have to take ito accout the idetity Ŵ F+ ŴF C F+ C F A 0 = DŴF ( C F+ ; κ 0F ) + S F+ A 0 (98) which elimiates completely a FRÉCHET derivative DŴF of the strai eergy ŴF i the equatio of motio. 4 NUMERICAL EXAMPLE As umerical example, we cosider a trasversely isotropic blade discretized i space by eight-ode brick elemets. I the iitial cofiguratio, the ceter of the blade s hub is positioed i the origi of the three-dimesioal EUCLIDEAN space (see Fig. ). The material is described by the upartitioed strai eergy fuctio with the fuctios Ŵ (I, I, I 3, I 4 ; κ 0 ) = Ŵ isotr (I, I, I 3 ; c, c, c 3 ) + Ŵ aiso (I 3, I 4 ; c 3, c 4 ) (99) Ŵ isotr (I, I, I 3 ; c, c, c 3 ) = c (I 3 3 I 3) + c (I 3 ) (00) Ŵ aiso (I 3, I 4 ; c 3, c 4 ) = c ( ) 3 exp c 4 I /3 3 I 4 (0) c 4 4

15 Simulatio parameter spatial mesh Eight-ode bricks elemet umber el 00 ode umber o 38 mass desity ρ 0 strai eergy W = W isotr + W aiso soft material c 300 c 00 c 3 40 c 4 80 stiff material c 3000 c 000 c c fiber vector a 0,, T / 3 iitial velocity v A 0 = v T + ω 0 q A 0 velocity vector v T, 0, 0. T agular velocity ω 0 0, 0.7, 0.7 T NEWTON tolerace tol 0 8 Figure : Left: Iitital cofiguratio of the fiber-reiforced blade. The colours idicate the VON MISES stress at the temporal Gauss poit t 0+/ determied by the cg() method i the o-stiff case. The arrows show the iitial velocity field of the free flight. Right: Simulatio parameter of the motio ad of the algorithm. the parameters c, c, c 3 ad the dimesioless parameter c 4 (compare Referece 4). The applied material parameter values are summarized i Fig. o the right. We distiguish betwee soft ad stiff material. The blade are i free flight due to its iitial traslatioal velocity field ad its iitial agular velocity field (see Fig. ). We compare two umerical methods: (i) the variatioal cosistet discrete method preseted above, referred to as eg() method i the followig, ad (ii) the cotiuous Galerki cg() method or midpoit rule, respectively, give by Eq. (68) ad the correspodig secod PIOLA-KIRCHHOFF stress tesor S mid + based o a temporally discotiuous strai approximatio. = DW (F T F + + ; A 0, κ 0 ) (0) Cosiderig soft material, both methods show similar curret cofiguratios for a moderate costat time step size. Therefore, we show oly the motio of the cg() method i Fig. 3. But, by chagig the time step size durig the simulatio, the NEWTON-RAPHSON method i the time loop of the cg() method aborts after some time steps. This ca be show by plottig the total eergy of the blade versus time (see Fig. 4 o the left). I cotrast to the eg() method, the cg() method shows a oscillatig total eergy with a eergy blow-up after the time step size chage. Cosiderig stiff material, o time step size chage is ecessary for illustratig the ustable behaviour of the cg() method i cotrast to the eg() method (see Fig. 4 o the right). 5

16 Figure 3: Curret cofiguratios B t of the blade with the o-stiff material determied by the cg() method, startig at t 0 = 0 o the left. The colour idicates the VON MISES stress at the temporal Gauss poit t +/. The arrows shows the curret Lagragia velocity field cg()-method eg()-method cg()-method eg()-method total eergy H(t) total eergy H(t) time t time t Figure 4: Compariso of the total eergy H of the cg()-method ad the eg()-method, respectively, usig the soft material (left) ad stiff material (right). The time step size is 0. for t 0 ad 0. for t > 0. 5 SUMMARY I this paper, we cosider trasversely isotropic materials from two perspectives. We examie. the geeral case of formally oe free eergy fuctio with o separatio of tesor ivariats (upartitioed free eergy fuctio), ad. a partitio of the free eergy fuctio ito two separate terms correspodig to isotropic ad aisotropic ivariats, respectively (partitioed free eergy fuctio). The reaso is that the fudametal theorem of calculus correspodig to partitioed free eergy fuctios ca be split ito separate equatios as it is well-kow from the kietic ad potetial eergy of atural systems with respect to iertial referece frames. As the fudametal theorem of calculus serves as a desig criterio for eergy cosistet time steppig algorithms, a separatio ito two criteria therefore allows to modify the algorithm i a more targeted maer. Further, already implemeted eergy cosistet time steppig algorithms for isotropic materials could be exteted rather tha modified to trasversely isotropic materials or composite materials with more tha oe family of fibers. I this work, we start the desig of eergy cosistet time steppig algorithms by discretizig a mixed variatioal priciple, because we aim at a uified desig procedure for these importat algorithms. Such a uified procedure already exists for mometum cosistet time steppig 6

17 algorithms leadig to the so-called variatioal itegrators 5. I this variatioal framework, the cosistecy with the mometum balaces does ot deped o the umerical quadrature as i usual fiite differece or GALERKIN-based schemes. Eergy cosistet time steppig algorithms, however, are hitherto desiged for specific mechaical problems, 3, 8,, 3, although the correspodig desig procedures exhibit may commo features. I each of these refereces, the discrete total eergy balace is satisfied by itroducig temporally cotiuous approximatios of the idepedet argumet tesors of the total eergy, together with temporally discotiuous superimposed work cojugate tesor fields emaatig from spatially local formulatios of the fudametal theorem of calculus i time. But, the applicatio of this cocept for desigig higher-order accurate time steppig schemes as geeralisatio of existig secod-order accurate schemes raises questios i the details, 3, startig with the temporally cotiuous approximatio of the idepedet argumet tesors of the strai eergy. I these refereces, a mixture of strai tesor approximatios has to be used for satisfyig eergy cosistecy, which is ot obvious from a physical perspective. These problems ad the eed of a uified framework have led to the herewith preseted idea of discretizig a mixed variatioal priciple, providig. a proof of existig adhoc time approximatios, ad. ew higher-order accurate eergy cosistet time approximatios (see Appedix). The first adhoc time approximatio is the midpoit evaluatio of the GREEN-LAGRANGE strai tesor i Referece 6, or later called assumed strai approximatio i Referece 5, respectively. This approximatio is ofte used as a physically modivated assumptio (frameivariace of discrete strais), or as a iheret part of eergy cosistet discrete gradiets of strai eergy fuctios i fiite differece schemes 8. I Referece 6, this temporally cotiuous approximatio of the GREEN-LAGRANGE strai tesor is modivated by the exact quadrature of the approximated strai eergy fuctio pertaiig to the quadratic SAINT-VENANT KIRCHHOFF model. The discrete total eergy balace correspodig to this strai eergy fuctio is therefore fulfilled without a superimposed stress field, or i other words, a superimposed stress field vaish for this strai eergy fuctio as i the temporally cotiuous equatios of motio. Hece, ispired by three-field variatioal fuctioals of EAS methods 6, we here itroduce a temporally cotiuous strai tesor with the correspodig atural ordiary differetial equatio i time by meas of a mixed variatioal priciple. I this way, we actually prove the variatioal cosistecy of the assumed strai approximatio for well-kow secod-order accurate methods, ad derive a ew assumed strai approximatio for higher-order accurate schemes which avoids uphysical approximatio mixtures as i Referece 3 (see Appedix). The secod adhoc time approximatio is the superimposed stress tesor i Refereces, 3, based o the well-kow discrete gradiet i Referece 7. This superimposed stress tesor is derived from a costrait variatioal problem at each poit X i Refereces 3, which therefore provides a proof of the uiqueess of this superimposed stress tesor. However, this variatioal problem is ot physically modivated ad therefore ot ivariat with respect to a push-forward i the spatial cofiguratio B t. This problem has bee caused by ot usig a coordiate free ad metric idepedet geometric formulatio of cotiuum mechaics. I fact, the Euclidea metric δ AB is assumed i the referece cofiguratio from the outset. Therefore, 7

18 i this paper, we arrive at the right CAUCHY-GREEN tesor as metric tesor by usig a covariat tesor formulatio. The obtaied costrait variatioal problem is therefore ivariat uder a push-forward i the spatial cofiguratio B t, ad leads to a equivalet variatioal problem with respect to the KIRCHHOFF stress tesor. As special case for a liear approximatio i time, we obtai the superimposed stress field i Referece 8. Note, however, that a further improvemet of the computatioal performace i compariso to the superimposed stress tesor i Refereces 3 is ot recogized by cosiderig free flights of stiff materials. But if the algorithm has to be pushed forward i a computatioal more efficiet spatial settig, the ew superimposed stress field is ecessary. Further ote that the costrait variatioal problems are still separate variatioal problems of parameters of the mixed variatioal priciple. A APPENDIX I this appedix, we show a iterestig cosequece of the above theory for the ew assumed strai approximatio of higher-order accurate time itegratio schemes, i.e. schemes which take ito accout ier time poits t +αi, α i 0,, beside the time step boudary poits t ad t +. This ier time poits are usually equidistat distributed over the time step (see Table ). Here, we have to start i the upartitioed case with the discrete priciple N =0 k i= δ Ḣ( u h(ξ i ), v h(ξ i ), ṗ h(ξ i ), C h(ξ i ), S h(ξ i ); ρ 0, A 0, κ 0, b h(ξ i ), t h(ξ i ), ū h(ξ i ), S h(ξ i )) w i h = 0 (03) where ξ i, w i, i =,..., k, deote the quadrature poits ad weights, respectively, ad k the degree of the shape fuctios M j (α), j =,..., k + i time. Usually, the Lagragia shape fuctios ad the Gaussia quadrature rules are used (see Table ad Table, respectively). Accordig to this priciple, we obtai the weak equatio k δ S d C h(ξ i ) : h(ξ i ) dα C( u h (ξ i )) w i dv = 0 (04) i= with the assumed strai approximatio d C h(α) dα = k+ j= M j+ (α) C j k i= M i (α) C i (05) ad the shorthad otatio ( ) for the differetiatio with respect to α. The tesors C j ad C i desigate the idepeded odal values of the assumed strai approximatio ad its derivative, respectively, at the correspodig time poits α i ad α i (see Table ). Havig agai a elimiatio of the assumed strai field i mid, we arrive at the spatially local relatio k+ j= M j+ (ξ i ) C j C( u h (ξ i )) = O (i =,..., k) (06) After further algebraic trasformatios, the ukow odal values C l, l =,..., k, becomes C l := k i= m li C( u h (ξ i )) + C (07) 8

19 by takig ito accout the iitial coditio C := (F ) T F C. The coefficiets A li are the compoets of the k k matrix M (ξ )... M k+ (ξ ) m =. M (ξ k ).... M k+ (ξ k ) I the case of liear time approximatios (k = ), these relatios lead to the odal values (08) C C = (F ) T F C + C = (F ) T F (09) (compare Eq. (6). For quadratic time approximatios (k = ), we arrive at the odal values C C := (F ) T F (0) C := F + F T 3 F F + F 3 F + (F 3 ) T F () C + C 3 := (F 3) T F 3 () For higher degrees of shape fuctios, we obtai aalogous results. Accordigly, the odal values at the time step boudaries t ad t + are solely determied by u ad u +, respectively, but the odal values at the ier time poits depeds o the displacemets of all time poits. This is i cotrast to the (frame-idifferet) assumed strai approximatio C h(α) defied i Referece 3 by the extrapolatio k+ C h(α) = M j (α) (F j ) T F j (3) j= of the formula C h(α) for liear time approximatios (k = ) kow from Referece 5. As the first term i Eq. () is ukowigly eglected i Eq. (3), the authors of Referece 3 have to itroduce a mixture of time approximatios i the superimposed stress tesor for eergy cosistecy. O the other had, lookig at Eq. (), we may recogize a possibility to simplify the relatios for the odal values C,... C k by itroducig a assumed deformatio gradiet field, which for k = takes the form with the odal value F h(α) = M (α) F + M (α) ˆF + M 3 (α) F 3 (4) ˆF := F + F 3 u + I u + + I (5) Thus, the approximated displacemet field u h (α) is here coected to the deformatio gradiet field oly at the boudaries of the time step t, t + with the liear approximatio The correspodig assumed strai field reads F h(α) = M (α) F + M (α) F 3 (6) C h(α) = M (α) (F ) T F + M (α) (ˆF ) T ˆF + M 3 (α) (F 3) T F 3 (7) = (α ) (F ) T F + α(α ) (F ) T F 3 + (F 3) T F + α (F 3) T F 3 (8) 9

20 However, we have to examie this possibility with respect to the accuracy order of the resultig time itegratio scheme. Further, a liear time approximatio of the deformatio gradiet field ad a quadratic time approximatio of the displacemet field seems to be icosistet, i cotrast to a aalogous space approximatio 4. Nevertheless, the accuracy order of the time itegratio scheme correspodig to the approximatio C h(α) with the odal values i Eq. (07) is guarateed for shape fuctios of degree up to four (k = 4). We have eve implemeted this approximatio i the fiite elemet code associated with the thermo-mechaical problem i Referece 3, ad obtaied the umerical results i Fig. 5. This approximatio is therefore eve recommeded for mechaically coupled problems. The higher-order approximatio of the superimposed stress tesor of the upartitioed strai eergy fuctio at the temporal Gauss poit is give by S h(ξ i ) := k C h(ξ l ) l= G(O) C h(ξ i ) C h (ξ l ) : C h (ξ l ) C h(ξ l ) w l C h (ξ i ) C h(ξ i ) (9) with G(O) := W + W k DW ( C h(ξ l ); A 0, κ 0 ) : C h (ξ l ) w l = 0 (0) l= The superimposed stress S F correspodig to the partitioed strai eergy is due to the scalarvalued argumet C F aalogous to the dyamical problem of a particle system i Referece 3 (compare Eq. (98) with the case k = ). Hece, we obtai the relatio S F h (ξ i ) := k l= G(0) C F h (ξ l ) C F h (ξ l ) w l C F h (ξ i ) () with G F (0) := ŴF + ŴF ad the fiber assumed strai approximatio k DŴF ( C F h (ξ l ); κ 0F ) l= C F h (ξ l ) w l = 0 () k+ C F h (α) = M j+ (α) CF j (3) j= where the odal values C F l, l =,..., k, take the form C F l := k i= A li C( u h (ξ i )) : A 0 + C : A 0 (4) 0

21 0 Square block, Nel= 4, T= s 0 Square block, Nel= 4, T= s Relative L error i positio Number of time steps Relative L error i positio Total CPU time Relative L error i velocity Square block, Nel= 4, T= s Number of time steps Relative L error i ielastic strai Square block, Nel= 4, T= s Number of time steps Square block, Nel= 4, T= s 0 4 Square block, Nel= 4, T= s Relative L error i temperature Relative error i total eergy Number of time steps Number of time steps Figure 5: Accuracy orders of the eergy cosistet time steppig scheme preseted above usig the ew assumed strai approximatio, determied with a flyig stiff square discretized by four four-ode quadrilateral elemets. We ivestigated the thermo-mechaically problem i Referece 3. The plots show the relative L errors at the fial simulatio time T =. For shape fuctios of degree k i the mechaical ad thermal fields (labels kkk ), we obtai the accuracy order k. The order of the total eergy has the order k +, because the temperature is calculated with a eergy cosistet discotiuous GALERKIN method. Through the strog thermo-mechaical couplig, the temperature shows the same order as the displacemets or curret positios, respectively.

22 Table : Lagragia shape fuctios i time of degree k ad k with respect to the paret domai 0,. k M j (α) α j Mi (α) α i α 0 α (α )(α ) 0 α 0 4α (α ) α (α ) α 3 9 (α )(α )(α ) 0 (α )(α ) (α )(α ) α 4α (α ) (α )(α ) α (α ) α (α )(α ) α (α )(α )(α 3)(α ) 0 9 (α )(α )(α ) (α )(α 3)(α )α 7 (α )(α ) α (α )(α 3)(α )α 7 (α )(α ) α (α )(α )(α )α (α )(α )(α 3)α (α )(α ) α 3 3 Table : Gaussia quadrature with N qp Gauss poits with respect to the temporal paret domai 0,. N qp ξ l w l / ( / 3)/ / ( + / 3)/ / 3 ( 3/5)/ 5/8 / 4/9 ( + 3/5)/ 5/8 4 ( 3/7 + 6/5/7)/ (3 5/6)/ ( 3/7 6/5/7)/ / (3 5/6)/ ( + 3/7 6/5/7)/ / (3 5/6)/ ( + 3/7 + 6/5/7)/ (3 5/6)/

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