Contribution to a formulation of integral type of the mechanics of Cosserat continua ( * ).

Transcription

1 ontributo per una formuazione di tipo integrae dea meccanica dei continui di osserat Ann Mat pura ed App (976) ontribution to a formuation of integra type of the mechanics of osserat continua ( * ) GIUSEPPE GRIOLI (Padua) Transated by D H Dephenich Abstract With the goa of making a contribution to a formuation of integra type of the mechanics of osserat continua with finite deformations we estabish: ) A variationa property of stress ) A condition of integra type on strain I recenty estabished a variationa property of stress for finite deformations of a cassica continuum of hypereastic type [] [] I treated a property of stationarity ie of minimum action of a potentia energy that not ony characterized the rea stress but aso permitted one to give an existence theorem and to conceive of a pure integration procedure in the presence of uniatera surface constraints In the case of more compex continua viz ones with micro-structure it is ony in the inearized case that it is possibe to estabish a variationa property that is anaogous to this and of the same magnitude [3] On the other hand this is not possibe in the case of finite deformation and the fundamenta reason consists in the fact that it does not seem possibe to express the fied equation without making the deformation intervene in it unike what happens in the cassica case in which one appeas to Kirchhoff s asymmetric representation of stress as one can do A fina difficuty is then connected with the extreme compexity of the constitutive equations With the goa of expanding upon the question whie considering the case of a hypereastic osserat continuum that is subject to finite deformations we show how in reaity the stress is not characterized by the stationarity of a potentia energy B and aso estabish what the fit variation of B wi equa that corresponds to rea stress More precisey we estabished that the fit variation wi prove to be equa to a quantity that is annued when one inearizes the probem as with sma deformations [3] in which it has higher order than the fit variation of B The integra property that is estabished for rea stress cannot have the operationa significance of the anaogue that was estabished for the cassica case but certainy can be considered to be a fit contribution to the formuation of the mechanics of osserat continua in integra form With the goa of extending that contribution and aso because the question is directy inked to the integra property on which the variationa property that we estabished is founded (which is however not invertibe) I woud ike to point ( * ) Submitted for editing on 4 August 975

2 Grioi A formuation of integra type in the mechanics of osserat continua out the possibiity of giving a goba form to the integrabiity conditions for the strain variabes Introduction Let and be the reference configuration and the current one resp for a osserat continuum et P and P be corresponding points and et y i x i be their coordinates with respect to the same oriented tri-rectanguar reference From the purey geometric and kinematic viewpoint the transformation from to is defined by the knowedge of the reations: () x i = x i (y y y 3 ; t) R = R (y y y 3 ; t) where R is the matrix that expresses the rotation that is associated with P One supposes that () satisfies a of the required anaytica conditions for the one-to-one character of the correspondence between and ; eg continuity etc The state of tension (ie stress) in is defined by the knowedge of two asymmetric matrices X ψ the former of which corresponds to the usua stress in auchy form whie the atter one expresses the density of contact coupes If a comma denotes the derivative with respect to y i and one supposes as is egitimate that D = Det x rs > 0 then one sets: () X = K x r s D ψ = λ x r D s The stress variabes K λ whose significance is obvious satisfy the fied equations: (3) K = F r λ + ε rms K s x m = M r (in ) (4) K r n = f r λ r n = m r (on σ) where F r M r denote the voume force and coupe densities resp when referred to which consist of the inertia force in the dynamica case whie f r m r consist of the corresponding surface force and coupe resp that are defined on the boundary σ of In (3) e rms denotes the Ricci indicator of three-dimensiona Eucidian space whie in (4) n r is the interior norma to σ If we et a denote the matrix of components x rs then the strain is characterized by the four matrices: (5) ε = (a(t) a ) v = a (T) R v (s) = a (T) R s One observes that if one sets: (6) Z () = R(T) R then it not ony resuts that:

3 Grioi A formuation of integra type in the mechanics of osserat continua 3 (7) Z = R pr R ps = R pr R ps R ps = Z ( ) R pr but aso that: (8) ε = (vv(t) ) v (s) = v Z (s) One concudes that utimatey the strain is characterized by the knowedge of the matrices v Z (s) A particuar form for the constitutive equations The new quantity R can be expressed by means of three paramete Q i One sets: (9) B i (Q s ) = e isp R pt Q R st Under the hypothesis of hypereasticity there exists a potentia energy density W from which the stress is derived It wi depend upon the fundamenta variabes x rs R R t which characterize the geometrica behavior of a osserat continuum ony through the agency ( ) of the matrices v Z and one has ( ): (0) K = Rrp = vsp x λbr = Q s In the seque it wi be convenient to give a more appropriate form to the constitutive equations (0) To that end one observes that on the basis of (7) (9) it resuts that: () Z = Z ( ) = e rqs R B p Q t sr Since W can depend upon Q r s ony by means of the ( ) Z from (0 ) it foows that: () λ B r = s Z rt Z Q ( p) rt s = ( s) Z rt e rqt R B p One knows that Det B r > 0 One therefore deduces from () that: ( ) One can treat this as a consequence of the principe of materia indifference Naturay W wi depend on other variabes in genera eg temperature etc Nevertheess such circumstances wi not be considered here ( ) See (40) in [5] if one assumes that the couping K = x rm Y m is vaid in it

4 Grioi A formuation of integra type in the mechanics of osserat continua 4 (3) λ = epmqrrm ( s) Z Set: (4) τ = e pmq R rm λ λ = e ( s) pmq R rm τ ( s ) so from (3) one utimatey has that: (5) ( s ) τ = ( s) Z A convenient transformation of (0 ) can be obtained from: (6) N sp = K R rp K = R rp N sp It foows immediatey that: (7) N sp = v sp Equations (5) (7) constitute a particuar form that is adapted to the context that appies to the constitutive equations of a osserat continuum with free rotations One easiy convinces onesef that reations (5) (7) are invertibe One can set ( 3 ): (8) v = α (N; τ () ) Z = β (N; τ () ) where the α β are functions of the variabes N τ that are deduced from the inveion of (5) (7) Set: (9) W = W[α(N; τ () ); β(n; τ () ( ) ( ) )] α N β Z so W defines a second form for the potentia energy and one easiy deduces that: (0) v = N Z = ( ) τ 3 A variationa property of rea stress Suppose that a surface force and coupe are given on some part σ of σ whie some transations and rotations are given on the remaining part σ On σ one has x r x r R R where x r R denote functions ( 3 ) This does not begin to address the compex and interesting question of the possibe a priori breakdown of uniqueness in the α β that are deduced from the inveion (8) Furthermore the same question presents itsef in the case of cassica continua

5 Grioi A formuation of integra type in the mechanics of osserat continua 5 that are defined on σ Let V be the cass of possibe reactions and reaction coupes whie φ r and µ r are aowabe constraints Let K λ be the matrices that express the rea stress that corresponds to the current configuration which are characterized by the vaues of x r R at the instant t vaues that wi be indicated by x r R Any other stress that satisfies (3) (4) on σ and corresponds to the current configuration and to the voume inertia and surface (on σ ) forces at the instant t is obtained by adding increments K λ to K λ that satisfy the equations: () K = 0 λ + e rms K s x m = 0 () = 0 (on σ) Krn = φr (on σ ) = 0 (on σ) λrn = µ r (on σ ) where φ r µ r denote increments in the φ r µ r that are aowed by the constraints From () it foows immediatey that: K x e K x e R d = 0 (3) { [ ] r r λ r + rms s m rqp qp} Taking () into account one deduces from (3) that: (4) { r r [ rqp qp λr + qp rms s m ]} K x e R R e K x d + [ φ r xr erqp µ rrqp ] dσ σ = 0 Letting R [qp] denote the anti-symmetric part of R qp and keeping (7) (4) in mind it foows from (4) after some cacuations that: ( ) ( ) (5) { R Z ps τ ts R x [ sm] m K s + K x r r } d + [ φ x e R r r rqp µ r qp ] d σ σ = 0 where v and Z indicate the expressions for v and when one identifies x r and R in them with x r and After some fina cacuations (5) becomes: Z that are provided by (5) (7) R respectivey ( ) ( ) (6) { R Z ps τ ts + [ δ p R [ p ]] v t N p } d + [ φ x e R r r rqp µ r qp ] d σ σ = 0 Taking into account the fact that the rea stress satisfies the constitutive equations (0) it utimatey foows from (6) that:

6 Grioi A formuation of integra type in the mechanics of osserat continua 6 (7) τ [ ts Rps + Np δ p R[ ps] d τ N t φ r xr erqp µ rrqp dσ σ = 0 Set R = R δ and additionay: (8) B = W d [ ] φ x e µ R dσ σ r r p r sp One immediatey recognizes that (7) can be presented in the form: (9) B = R ps τ ( ) ts R [ sm] Ns d τ pt N m about which one asserts that as opposed to what happens for cassica continua the rea stress does not render the potentia energy B stationary for the cass of stresses that are in equiibrium with the given voume inertia and surface forces which can give rise to constraint reactions that are aowed by the constraints Nevertheess one can observe that in the inearized case B wi resut in stationarity propery speaking that corresponds to the rea stress as was observed in [3] Indeed one easiy recognizes that the righthand side of (9) is set equa to zero in the case of sma deformations that are consistent with the inearization of the probem in which regarding as is necessary the quantity R δ to be of fit order it foows that it is of higher order with respect to the efthand side 4 A possibe integra formuation of the compatibiity conditions for the strain ( ) matrices Suppose that the strain matrices ϕ γ are given consider a rotation matrix ρ and set: (30) ϕ = ξ mr ρ ms mp p γ = ρ η where the matrices ξ mr η are uniquey determined and the γ are assumed to be skew-symmetric (emi-simmetrica) in the ower indices In addition one sets: mt mp (3) N sp = K ρrp λ ( s ) τ = e pmq ρ rm where K λ represent an arbitrary soution of the differentia system: (3) r (33) Krn K = 0 λ r + ermsk sξm = 0 (in ) = 0 λ rn = 0 (on σ) One conside a system of integra equations:

7 Grioi A formuation of integra type in the mechanics of osserat continua 7 (34) N spϕspd = 0 ρ γ τ ρ ϕ ( ) [ ps ts [ pt ] tnp] d = 0 One has the theorem: A necessary and sufficient condition for the matrices ϕ γ ( ) p the second of which is skew-symmetric to represent a strain in a osserat continuum is that it satisfy (34) for any choice of N τ that are constructed on the basis for (3) with any possibe sp st soution of the system of equations (3) (33) In that case the transformation from which the strain is derived descends precisey from the rotation ρ [and a suitabe dispacement vector] The condition is necessary: Suppose that: (35) ξ mr = ξ m r η = ρ mp mp It foows from (3) (33) that: K r ξrd = 0 (36) [ λ r + ermsk sξm ] d = 0 One deduces (34 ) from (36 ) immediatey When one takes (30) (3) into account (36 ) becomes: (37) [ e λ ρ + rmp r mp ρ N ϕ [ pt] p t ] d = 0 which again on the basis of (30) (3) transforms into (34 ) The condition is sufficient ( 4 ): Assuming (30) (3) one suppose that (34) is satisfied by the ϕ γ for any choice of N p τ st that are constructed from soutions of (3) Introduce an arbitrary doube system of functions χ that are differentiabe and zero on the boundary of and set: (38) K = e sti χ rti N sp = e sti ρ rp χ rti One easiy recognizes that the K that are defined in (38) satisfy (3) (34 ) then becomes: (39) ξ χ d sti rt i = 0 e ( 4 ) Observe that the hypothesis that (33) is satisfied never ente into the proof of sufficiency It is enough to suppose that the χ are zero on the boundary of but arbitrary everywhere ese

8 Grioi A formuation of integra type in the mechanics of osserat continua 8 and from the arbitrariness of χ this impies that: (40) e sti ξ i = 0 (40) shows the existence of three functions ξ r that satisfy (35 ) As a consequence it then resuts that: (4) ϕ = ξ m r ρ ms Taking (40) into account one recognizes that the functions that were defined by (38 ) as we as: (4) λ = e rpm e σqs ξ mσ χ satisfy (3 ) for arbitraryχ It then foows that: (43) τ = e pst ρ e rvm e σq ξ mσ χ vq pt and on the basis of (30) (38) (4) and (43) (34 ) becomes: e e e e d = 0 (44) { ρ ρ ρ η ξ χ sϕt rpm σq rϕ εs τt τε mσ + ρ ξ χ [ pm] qi m i} (44) simpifies to: e e e d = 0 (45) { ξ [ η σ q m σ ε rτ rpm τε ρ [ pm] i] χ } which upon taking into account the equivaence: (46) ρ [pm] = e rpm e rτε ρ τε and the arbitrariness of the χ gives: (47) e rpm e σqi e rτε ξ mσ ( η i τε ρ τε i ) = 0 Set: (48) c ri = e rpm e σqi ξ mσ g ri = e rτε ( η i ρ τε i ) so (47) assumes the form: (49) c ri g ri = 0 This constitutes a new homogeneous inear system of equations in the new unknowns g ri One can prove that the determinant of the coefficients is non-zero in genera: ie Det c ri 0 It then foows that: (50) η = ρ τε i + ( i ) τε ( i) L τε τε

9 Grioi A formuation of integra type in the mechanics of osserat continua 9 ( i) where the L τε constitute an arbitrary system for any i that is symmetric with respect to the ower indices From (30 ) one obtains: (5) γ = ( i) mt ( mp i Lmp ) ρ ρ + and the condition of skew-symmetry for the γ impies that: (5) ρ L + ρ L = 0 ( i) ( i) mt mp mp mt ( i) The system of six equations (5) for any vaue of i in the six unknowns L admits the zero soution as its unique soution if one is given that the determinant of the coefficients is non-zero ( 5 ) Upon taking (4) and (5) into account and setting L i 0 one thus concudes that the matrices ϕ γ define an effective strain It is provided by the deformation that is characterized by the dispacement ξ r and the rotation ρ QED BIBLIOGRAPHY [] G GRIOLI Proprietà variazionai nea Meccanica dei ontinui Nota I Rend Acc Naz dei Lincei s VIII vo 54 fasc 5 (May 973) [] G GRIOLI Proprietà variazionai nea Meccanica dei ontinui Nota II ibid fasc 6 (June 973) [3] G GRIOLI Una proprietà variazionai nea teoria dee Microstrutture Rend di Mathematica s VI vo 8 (975) [4] G GRIOLI Microstrutture Nota I Rend Acc Naz dei Lincei s VIII vo 49 fasc 5 (November 970) [5] G GRIOLI Microstrutture Nota II ibid fasc 6 (December 970) ( 5 ) Under the correspondence of the eement P of one assumes that the reference triad has its third axis parae to the rotationa axis that is defined by the matrix ρ : Let θ be the ange of rotation so one has: ρ = ρ = cos θ ρ 33 = ρ = ρ = sin θ ρ i3 = ρ 3i = 0 (i = ) One easiy recognizes the vaidity of what we asserted in such a situation