The stress functions of the Cosserat continuum

Transcription

1 De Spannungsfuncktonen des Cosserat-Kontnuum ZAMM 47 (967) 9-6 The stress functons of the Cosserat contnuum By S KESSE Transated by D H Dephench The equbrum condtons of the Cosserat contnuum are satsfed dentcay by a compete stress functon representaton of the stresses For a knematcay compatbe stress state n an sotropc eastc contnuum the stress functons can be expressed n terms of potenta functons and soutons of the Hemhotz equatons Introducton One can mathematcay formuate an eastostatc probem n cassca near eastcty theory as ether a boundary-vaue probem for the dspacement vector fed or a boundary-vaue probem for the stress functon tensor fed In both cases the soutons of the fed equatons can be expressed wth the hep of certan Ansätze regardng potenta functons; the Ansatz for the dspacement vector fed was found by NEUBER [] and PAPKOWICH [] whe SCHAEFER [] found the stress functon Ansatz Anaogousy the eastostatc probem for the near sotropcay-eastc COSSERAT contnuum can be formuated as ether a boundary-vaue probem for the dspacement and rotaton vector feds or as a boundary-vaue probem for the two tensor feds of the stress functons NEUBER [4] has shown that one can convert the soutons of the sx couped dfferenta equatons for the knematc feds by certan Ansätze on the soutons of potenta equatons and Hemhotz equatons In the present paper the correspondng Ansätze for the stress functons w be derved and compared to the NEUBER Ansätze Preface We compute n Cartesan coordnates and empoy the summaton conventon that Greek ndces are to be summed over from to It w be assumed that a scaar vector and tensor feds are defned n a smpy-connected but possby mutpybounded regon G wth an outer surface G and that they are contnuousy-dfferentabe as many tmes as s requred The equatons of knematcs and statcs of COSSERAT contnuum can be wrtten qute smpy wth the use certan we-defned dfferenta operators Certan reatons between these dfferenta operators characterze the anaoges that exst wth the dfferenta operators of vector anayss; ther nterpretaton n the cacuus of aternatng dfferenta forms was recognzed by SCHAEFER [5]

2 S KESSE The stress functons of the COSSERAT contnuum We combne the tensor feds V k and ( W k e V ) whch are defned by the two vector feds V and W and defne them as the resut of appyng a dfferenta operator Grad to the vector feds V and W: () Grad V W Vk Wk e V Smary we defne a Dv operator that acts on two tensor feds of rank two Q and R: () Dv Q R Q k R k + ek Q β β and a Rot operator that ewse acts on two tensor feds: () Rot Q R eβ Qβ k eβ ( Rβ k + ekγ Qβγ ) In ths e s the permutaton symbo that s skew-symmetrc n a ndces In addton to the operators defned n () to () we aso defne the operators Grad * Dv * and Rot * : () * Grad * V W Vk Wk + e V () * Dv * Q R Q k R k ekβ Qβ () * Rot * Q R eβ Qβ k eβ ( Rβ k ekγ Qβγ ) One may easy check that the foowng denttes exst: (4) (5) Dv Grad * Dv * Grad (6) (7) Dv Rot Dv * Rot * (8) (9) Rot Grad Rot Rot * Grad * Dv Knematcs statcs and the matera aw of the near sotropcay-eastc Cosserat contnuum We assume: Any pont of the COSSERAT contnuum s orentabe and has the possbe motons of a rgd body We descrbe the sx functona degrees of freedom of the contnuum by a dspacement vector fed u(x ) and for sma rotatons by a

3 S KESSE The stress functons of the COSSERAT contnuum rotaton vector fed ϕ(x ) The deformaton state of the COSSERAT contnuum s descrbed by the two deformaton tensors χ and ε: () χ ε Grad ϕ u The asymmetrc tensor χ s the tensor of curvature deformaton the symmetrc part of ε s the deformaton tensor of cassca near eastcty theory and the skew-symmetrc part of ε measures the dfference between the oca rgd rotaton that s determned by the dspacement vector fed and the absoute rotaton of the ponts of contnuum If the deformaton tensor feds χ and ε are gven n a smpy-connected regon G then we can cacuate a rotaton vector fed and a transaton vector fed unquey from them up to a rgd rotaton when the compatbty condtons: () Rot χ ε are fufed n G The deformaton tensor ε can be assocated wth a force stress tensor σ and the curvature tensor χ wth a moment-stress tensor µ under the prncpe of vrtua dspacements and the AGRANGE beraton prncpe [6] For a surface eement df wth the externa unt norma vector n that s oaded wth a force p df and a moment m df one has: () n σ p n µ m The dfferenta equbrum condtons for a voume eement of the contnuum that s oaded wth the voume force X and the voume moment Y read: () Dv σ µ X Y The matera aw for the near sotropcay-eastc body s [7]: (4) (5) c c ν σ 4 ε ε δ ε 4 ν c c µ G + χ + χ + c δ χ 4 4 G + + k + k ; c χ + χ + µ k δ µ G c c + c ν ε + σ + σ k δσ G c c ν

4 S KESSE The stress functons of the COSSERAT contnuum 4 ncudes sx matera constants: the shear moduus G the transverse contracton number ν a matera constant wth the dmenson of a ength and the three-dmensona matera constants c c c For brevty we wrte: (6) σ µ M() χ () M ε χ ε M () M () σ µ n whch M ( ) and M are sotropc tensors of rank four ( ) The basc eastc equatons and the souton Ansatz of Neuber For the determnaton of the dspacement and rotaton vector fed n a body G for whch the knematc degrees of freedom are restrcted on the outer surface G or that outer surface G s oaded wth force and moment stresses we have the basc eastc equatons: () (7) Dv M Grad ϕ () M u wth consderaton to the correspondng boundary condtons to be soved specfcay these equatons read: More (8) c c + u + grad dv u + c rot ϕ ν c c c c grad dv + rot ; ϕ ϕ u ϕ t w be soved by an Ansatz [4] that corresponds to the NEUBER-PAPKOWICH Ansatz of cassca eastcty theory: (9) u grad Φ + r Φ + dv ψ + + 4( ν ) Φ ψ + c rot + ϕ Φ ψ + grad χ c () () Φ Φ () ψ ( + c)( + c) 8c

5 S KESSE The stress functons of the COSSERAT contnuum 5 () χ + c c The stress functon souton The ntegraton of the eastostatc probem wth the hep of stress functons comes from an Ansatz for the stresses that the equbrum condtons satsfy dentcay THEOREM: Any equbrum system of force and moment stresses may be represented n the form: σ F S (4) Rot + Grad µ G T n whch [8]: (5) (6) Dv F G S T X Y Proof: et (7) (8) H K σ dv 4π G r Dv σ µ µ X Y Then from (7) one has: H (9) K σ µ and wth (9): H H () Rot Rot + Grad Dv K K σ µ We now set: () Dv F G Dv* Rot * H K wth: S (4) T Dv H Dv H K K Dv σ µ X Y Wth that we have shown that we can fnd a representaton of the form (4) for any equbrum stress state For the foowng we assume that the voume force X and the voume eement Y are zero The vectors S and T n the stress functon Ansatz (4) w then be harmonc vectors: (5) S T

6 S KESSE The stress functons of the COSSERAT contnuum 6 We consder the auxary condton (5) n whch the stress functon tensors of frst order F and G are expressed n terms of stress functon tensors of second order F and G: (6) F G F Rot* G The stress functon Ansatz (4) then reads: (7) or (8) (8 ) σ µ Rot F Rot* G + S Grad* T σ µ F S F Grad* Dv + G T G σ F k + Sk F µ G k + ekβ Fβ + Tk + ekβ S G Snce the body G s smpy connected we can excude the exstence of proper stresses; the stress functons must then be determned n such a way that the equbrum stress state (8) s compatbe so from () t must satsfy the condton: (9) Rot Ths means that: M () F S F Grad Dv + () M G T G M () F S F Grad Dv + () M G T G Grad ϕ u On the eft-hand sde of ths equaton after a seres of conversons one may spt off a gradent : () () N ( F G) (4) Grad () + () N ( F G) Thus one has: () (4) N G ( G + F + T ) Grad ϕ u () N G ( F + S ) () (F G) and () (F G) are tensora dfferenta expressons n the stress functons F and G We set:

7 S KESSE The stress functons of the COSSERAT contnuum 7 (4) ϕ N () u N () and (4) () (F G) () (F G) Equatons (4) are 8 couped parta dfferenta equatons of second order for the 8 stress functons F and G When we spt them nto ther symmetrc and ant-symmetrc parts: ( ) (44) m ( m) ( m) ( ) + e m ( m) ( ) ( m ( ) F G ) ( F G) they read: () c (45) ( ) G( ) δ ( G + βg( β ) + F + T ) G + c (46) (47) (48) δ ( e G G + e T ) G + ( F( ) + S ) e β Fβ c c () β γ ( β ) β G c () ν ( ) F( ) δ ( G β F( β ) S ) G ν δ ( e F F + e S ) F + ( G( ) + F + T + e β Gβ ) c () β γ ( β ) β G c n whch have ntroduced: (49) F F () + e F G G () + e G The dfferenta equatons for the stress functons may now be decouped by means of certan Ansätze and turn nto potenta and POISSON equatons as we as homogeneous and nhomogeneous HEMHOTZ dfferenta equatons We next sove the equaton (47) by the Ansatz: (5) F () f () + δ f wth (5) f ()

8 S KESSE The stress functons of the COSSERAT contnuum 8 By substtutng ths n (47) we obtan the dfferenta equaton for f: ν (5) f ( β fβ S ) ν + whose souton from (5) and (5) reads: (5) f ν f ( x β f( β ) + x S ) ( ν ) wth: (54) Equaton (47) s thus fufed We next take the trace of (45): f (55) G( ) βg( β ) F T c substtute n (45) and obtan the dfferenta equaton for G () : (56) G( ) δg( ) whch we sove wth the Ansatz: (57) (58) G () g () + δ g g () ; for the moment the functon g s st arbtrary Wth (57) t foows from (55) that: + c (59) F g g( ) T + ε β Aβ c wth a st-undetermned vector fed A One deduces the foowng dfferenta equaton for g by takng the dvergence of the vector equaton (48): c (6) g g + T + c + c c Up to a potenta functon whch can be assumed aong wth δ g n g () g must satsfy the dfferenta equaton: c (6) g g T + c

9 S KESSE The stress functons of the COSSERAT contnuum 9 Due to (5) the souton reads: (6) (6) g c g+ T c g We substtute a of the resuts that we have obtaned up to now n (48) and obtan the equaton: (64) e β G β + c c e β A A e T h e T β + c β βλµ λ µ + c β βλµ λ µ n whch: (65) h f () + e β γ g (γβ) + S + e β T β s a potenta vector From (64) t foows that: (66) G A c c + A e T + + h e T + ρ β β β β c c wth a st-undetermned scaar functon ρ Fnay t foows from equaton (46) for the determnaton of A and ρ: (67) (A A ) + h + c ( c ) ( c ) ρ f A A + A + c ( + c ) ( + c) and we set: (68) A B + C + D (69) B ( + c)( + c) 8c (7) (7) C D h (7) ρ f + A + c ( c )( c ) A h + c ( c )( + c ) n whch not ony ρ but aso ρ w be requred for the computaton of u ϕ σ and µ The compatbe stress functons read thus:

10 S KESSE The stress functons of the COSSERAT contnuum F () f () + δ f G () g () + δ g wth: + c F g g( ) T + e β Aβ c + c c G A A + h eβ Tβ + ρ c c c S T f () g () f C D h B g and the abbrevatons: f ν f ( x β f( β ) + x S ) g c g + T ( ν ) c A B + C + D h f( ) + e β γ g( γβ ) + S + e β Tβ Equaton (7) s true for ρ Comparson of the stress functon souton wth the Neuber souton We substtute the compatbe stress functons nto the equatons (4) and obtan: h η (7) u + B + x h + B G G G 4( ν ) G G hβ + c (74) ϕ g + eβ + B β G c G c G wth: (75) η ν f x h + C + D x eβγ β ( λ g( λγ ) + Tγ ) 4( ν ) and one has η A comparson wth the NEUBER Ansatz:

11 S KESSE The stress functons of the COSSERAT contnuum (9 ) u Φ + ψ Φ + x Φ + ψ 4( ν ) + c ϕ χ e β β ψ + Φ + β c yeds: (76) η ϕ G h χ g Φ G c G ψ G B Wth that we have shown that the partcuar souton to the equaton (4) that was constructed n the prevous secton s compete f the competeness of the NEUBER Ansatz (9) s known Exampe We determne the stress state that beongs to the speca Ansatz: (77) g g ( z ) f S T C B ; f () g () Snce g ( z ) must be a souton to the dfferenta equaton: (78) one must have: (79) g g g( z ) z / z / Ae + Be wth the two ntegraton constants A and B From (8) t then foows that: (8) c z / z / σ ( Ae Be ) c c z / z / µ ( Ae + Be ) ( + c) + c c We thus sove the boundary-vaue probem for an nfntey extended amna of thckness d (Fgure ): z d: n e z n σ n µ m e z ;

12 S KESSE The stress functons of the COSSERAT contnuum and obtan: σ z d: n e z n σ n µ m e z ; m snhζ cosh δ mc coshζ µ ζ + c cosh δ c + c z δ d d ζ d We see from Fgure that the stresses n a amna whose thckness s arge compared to the matera constant : (δ ) drop off very qucky away from outer surface I woud e to warmy thank Herrn Professor Dr H SCHAEFER for hs support durng the producton of ths paper and Herrn Professor Dr W GÜNTHER for many stmuatng dscussons terature H NEUBER En neuer Ansatz zur ösung räumcher Probeme der Eastztätstheore ZAMM 4 (94) P F PAPKOWICH Souton générae des equatons dfférentees fondamentaes d eastcté exprmée par tros fonctons harmonques Compt Rend Acad Sc Pars 95 (9) H SCHAEFER De Spannungsfunktonen des dredmensonaen Kontnuum; statsche Deutung und Randwerte Ing-Arch 8 (959) 4 H NEUBER On the genera souton of near-eastc probems n sotropc and ansotropc Cosserat contnua Proc of the XI Int Cong App Mech Sprnger- Verag (966) 5 H SCHAEFER Anayss der Motorfeder m Cosserat-Kontnuums ZAMM 47 pp 9-8 (967) 6 W GÜNTHER Zur Stat und Knemat des Cosserat-Kontnuums Abhandung d Brschw Wss Ges X (958) 7 S KESSE neare Eastztätstheore des anstropen Cosserat-Kontnuum Abhandung d Brschw Wss Ges XVI (964) 8 H SCHAEFER De Spannungsfunktonen enes Kontnuums mt Momentenspannungen I and II Buetn de Académe Poonase de Scences Ser X XI 5/ (967) Manuscrpt submsson: 4966

13 S KESSE The stress functons of the COSSERAT contnuum Address: Dr SIEGFRIED KESSE ehrstuh für Theoretsche Mechan Technsche Hochschue 75 Karsruhe Kaserstr