On a Stress Resultant Geometrically Exact Shell Model Part i

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 72 (1989) 267-304NORTH-HOLLAND

ON A STRESS RESULTANT GEOMETRICALLY EXACT SHELL MODEL.PART I: FORMULATION AND OPTIMAL PARAMETRIZATION

J.C. SIMO and D.D. FOXDi vi sion of Appl ied M echanics, Department of M echanical Engineeri ng, St anford Uni versit y, Stanf ord,

CA, U.S.A.Received 2 October 1987

Revised manuscript received 7 June 1988

1. Introduction and overview1.1. Overview

Over the past two decades computational shell analysis has been, to a large extent,dominated by the so-called degenerat ed soli d appr oach, which finds its point of departure inthe paper of Ahmad, Irons and Ziekiewicz [3]. The works of Ramm [40], Parish [39], Hughesand Liu [28,29], Hughes and Carnoy 1301, Bathe and Dvorkin [13], Hallquist, Benson andGoudreau [27], Parks and Stanley [38], and Liu, Law, Lam and Belytschko [33], among manyothers, constitute representative examples of this methodology carried over in its fullgenerality to the nonlinear regime. The thesis of Stanley [48], and the books of Bathe [12],Hughes 1211, and Crisfield {20], offer comprehensive overviews of the degenerated solidapproach and related methodologies which involve some type of reduction to a resultantformulat ion. An alternative approach to the development of shell elements is found in thepioneering work of Argyris et al. [&lo], which makes use of the classical matrix displacementmethod with high order interpolations (5th- and Ith-order polynominals) within the context ofthe authors natural approach.By contrast, the present work, the first part of a series of papers, constitutes a departurefrom the aforementioned methodology. In a sense, the proposed approach represents a returnto the origins of classical nonlinear shell theory, which has its modern point of departure in thepioneering work of the Cosserats 1191, subsequently rediscovered by Ericksen and Truesdell[21], and further elaborated upon by a number of authors; notably Green and Laws 1231,Green and Zerna [25], or Cohen and DaSilva [18]. We refer to [35] for a compehensive reviewincluding many references to the classical literature and historical overviews, to [4,5] for acareful analysis the mathematical foundations of classical shell theory, and to [44] for adiscussion of the underlying Hamiltonian structure.Although the hypothesis underlying the degenerated solid approach and classical shelltheory are essentially the same, the reduction to resultant form is typically carried outnume~~ally in the former, and analytically in the latter. conceptually, this appears to be theonly essential difference between the two approaches. A point frequently made concerning thedegenerated approach is that it avoids the mathematical complexities associated with classical~45- 7825/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-HoIland)


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J.C. Simo, D. D. Fox, St ress redtant geometr i call y exact shel l model. Part I 271

2.1. The rot at i on group and it s Li e al gebraFollowing standard notation, we denote by SO(3) the group of orthogonal transformations,i.e.

SO(3) := {A: R3+ R3 1 A = A- and det A = t-1) . (2-l)Any A E SO(3) possesses an eigenvector + E R3 such that A$ = a,k. Geometrically, Arepresents a rotation about +. The tangent space to SO(3) at the identity, denoted byT,S0(3), is SO(~), the set of skew-symmetric tensors, and is defined as

so(3) = { 6: R3 -& ~~+8=0}. (2.2)Any & E SO(~) processes an eigenvector 0 E R3 such that 60 = 0. This defines an isomorph-ism A: SO(~)+ R3 by the relation

& h = 0 x h for any h E R3 , cmIn what follows, we denote by {E,},=,,,,, an inertial (fixed) basis, chosen to be the standardbasis, i.e. E1=[Bi,;LIB}, ;={Hi. (2.4)Thus, E, = E. For any & E SO(~) we have the matrix representation (relative to {E,})

[&,:]= o3 -p[ -02 01

->l], (O> f]. (2.5)

Any 6 E SO(~) represents an i~~~ites~rna~ otation about 0 E R. The tangent space at anyA E SO(3) is defined (by either left or right translations of SO(~)) as

T,S0(3) := {A& l = h4 1 6 E SO(~) or 6 = A&A E SO(~)} . (2.6)Note that A& ET,S0(3) can be Athought of as a finife rotation superposed onto aninfinitesimal rotation. Alternatively, f?A E T,S0(3), where 8 = A@A E SO(~) may be thoughtof as an infinitesimal rotation 0 superposed onto a finite rotation,

Finally, the exponential mapping exp: T,S0(3)+ SO(3) maps infinitesimal rotations intofinite rotations according to4 E so(3) - exp[&] : = kgo $ 0 E SOf3) . (2.7)

One speaks of iA E T,S0(3) and A& E T,S0(3)SO(3) at A. as the right and left representation of the tangent space of


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274 J.C. Simo, D.D. Fox, Stress result ant geomet ri call y exact shell model. Part I

COROLLARY 2.2. T,,Si and T,S* are in one-t o-one correspondence t hrough t he i somorphi sm&A ET,& t-+ GETS* wi t h W = w x t . (2.18)

PROOF. (i ) Let &A ET,Si. Hence w - AE = 0. Define W E R3 through the relationO:=w X t, where t=AE. Then t -6=0 and GETS*.(ii) Conversely, let 6 E T,S*, so that W l t = 0. Define w E R3 through the relation o := t x0. Since

m.AE=w.t=(tX&)nt=o + cjllCJ&, (2.19)the result follows. ElREPARK 2.3. By particularizing (2.18) to the identity, we obtain the one-to-one corre-spondence between T,Si and T,S*

jl! ET& r--) &T,S* with fi = a X E , (2.20)where n*E=a*E=O. q

A geometric illustration of the preceding concepts is contained in Fig. 5.2.3. The exponent i al map i n Sz

Making use of the correspondence established in Proposition 2.1 between the unit sphere S*and the subset S: (or Si)CSO(3), we derive below the closed-form expression for theexponential map in S.Let t E S2 be any fixed (but otherwise arbitrary) vector in S*. We have the following.PROPOSITION 2.4. The fol l ow ing closed-f orm expression for t he exponent i al mapexp,: T,S2 -+ S* holds:

Fig. 5. The S2-sphere and its tangent spaces. A E S, is the orthogonal transformation mapping E ++ = AE E S.


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276 J.C. Si rno, D. D. Fox, St ress resul t ant geometr i call y exact shel l model. Part I

denoted by 5 E SL We setg = tlEl + t*E, , ([l, () E R* . (3.2)

The basic kinematic assumption is that of an i next ensibl e one-di rect or Cosserut surface.Accordingly, any configuration of the shell is described by a pair (9, t) E %, where:(i) The map 9: ~4--, [w3defines the position of the mid-surface of the shell.(ii) The map t : & + S* defines a unit vector field at each point of the surface, referred toas the director (or fiber) field.One is then led to the following kinematic hypothesis.BASIC ASSUMPTION. Any configuration of the shell Y C R3 is assumed to be defined as

Y:={xE[W3]X=~+& where (q,t)E% and ~E[!z-,h]}. (3.3)In particular, the reference configuration is exactly described as*

9 := {x0 E Iw31x,, = q,, + &, with (q,,, to) E % and 5 E [h-, h]} . P-4)Here, [h-, h+] C R, with h > h- and h = h - h- is the thickness of the shell. ClWe shall often use the notation

x = @(,l, 5*, > := 10(5,) +ml, 5).It follows that @: L&X [h-, h]+ R. A deformation of the shell, then, is a mapping

W)

An illustration of the basic kinematics of the shell model is contained in Fig. 6.3.2. The t angent map at a confi gurat ion. Deformat ion gradient

Given a configuration @: & X [h-, h+]* R3 the tangent map is the Frechet derivative,denoted by V@. Relative to {E,},=,,,,, one has (with t3 = 6)V@:=$@E=g,@EE. (3.7)

One refers to g, = ~@/a( as the convected basis. By the chain rule, the deformation gradientassociated with a deformation x: 93 + Y isF = T,y := V@o(V@ . (3.8)

An explicit expression for V@ is contained in the following.*This parameterization is often termed normal coordinate chart; i.e. see [3O].


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PROPOSITION 3.1. Th.e tangent mq V@ associated with Qs: A? x [h-, II+]-+ Iw3 s given by

(3.9)(3.10)

PROP, From (3.5) and (3.7) onr: has immediatelyV~=fvJ,,+rt,,)~E~t~PE. (3.11)

Since by Proposition 2-5. we have 1= MC, where 6, = E3 =E, substitution of (3.10) into (3.31)yields the result. q

R~~~AR~~ 3.ii?.1) In the assumpt~u~ of smaff strains anst large rotations one has VQ, = A.,i.e. an ~rth~g~na~ tra~sf~rmat~u~_ Such an assumption wit1 not be made here.(2) In addition to the convected basis {gI}l=1_2,3basis {gL,,~J

(where g3 = t) one defines the recipracorlby the standard relation gr - gJ = tii- Thus


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278 J.C. Simo, D.D. FQX, St ress result ant geometr ical ly exact shell model. Part I

(3) We shall use the following notation:j : = de@@] , j. : = de@@,,] , J = j/j 0 , (3.13)

where J := det[F] is the Jacobian of the deformation gradient which is given by (3.8).(4) Observe that while g, = t, g3 is given by

g3 = ; g, x g, . (3.14)Thus, llttl = Ikll = 1, andj=[glxg2]*g3, but llgl1+1- q3.3. Reference frames on t he mi d-surface

In addition to the fixed inertial frame {E,},,,,,,,, we define two reference frames on themid-surface that play an important role in the subsequent developments.(i) Surface convected frame. Denote by {a,}I=1,2,3 the surface convected frame defined asa, =g,(5_0. Note that {a,},= 1,2 span the tangent space to the mid-surface. It follows from(3.5) and (3.7) that

a, = pa and a3 = g, =: t . (3.15)The metric tensor (first fundamental form) on the reference surface is then

whereWL1,2,3enotes the dual surface convected basis defined by the standard relationa, l a = Si.(ii) Director orthogonal frame. Denote by (t,},=l,,,, the director orthogonal frame definedthrough the orthogonal transformation A: & C Iw 2-- ,Si as

t, = AE, , t,= AE,=t. (3.17)Note that (t,},=, 2 3 is the o~tho~~rrna~ basis which, by virtue of (3.9), is the closest to~%L1,2,3 In addition, {tr 2,) span T,S2, the tangent space normal to t, since t * , = 0.REMARK 3.3. Our definition of the orthogonal basis {tl}I_,,2,3 is intrinsic, and is motivatedby expression (3.9) for V@. This definition bypasses ad-hoc constructions often made in thecomputational literature; see [31, Ch. 61. Cl

These frames are illustrated in Fig. 7. Finally, we recall that the element of mid-surface areais given by the differential (two-form)d& := a, x a2 dtl dt2. (3.18a)


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280 J.C. Simo, D . D . Fox, St ress resul t ant geometr i call y exact shel l model. Part3.4.2. Densi t y and i nert i a

Let pO(el, t*, 5) and p( [l, S, 5, t) be the mass density in the reference and currentconfigurations, % and Sp, respectively. Recall that conse~ation of mass requires p0 = Jp,where J = det F, = j/j0 is the Jacobian of the deformation gradient F, = [V@t][V@O]- as-sociated with the motion xt = @t 0 @i: ?$ --, 9. We select the mid-surface v,,: sllz R3 (whichdoes not necessarily correspond to the geometric center of the cross-section) such that

(3.22)Define the surface densities &: &+ Iw and p: SQ Iw by the expressions

1Poj,,dt and @:= TIIn addition, define the surface inertias joP: &+ R and fP := &--, Iw as

(3.23)

(3.24)The significance of these definitions will become apparent in what follows. By (3.23), (3.24)and the conservation of mass relation p0 = Jp, we have

Is, = .&S and &, = J, ,which constitutes the statement of conservation of mass for shells.

3.4.3. ~esui t ant i near, a~gu~ar~ nd dir ect or ~o~ent ~~For a motion et: & x [h-, k+] X II %+ R3, time di~erentiation of

Define the result ant inear ~o~en~~ pt, and the resul t ant angularexpressions

(3.25)

expression (3.5) yields(3.26)

momentum m, by the

(3.27)Making use of (3.22), (3.26), the kinematic hy~thesis (3.5) and introducing definitions (3.23)and (3.24) we have

(3.28)where the relation w, l tt = 0 was employed.


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J.C. Simo, D.D . Fox, Stress resul t ant geometr i call y exact shel l model. Part I 281

Since mZ , = 0, by Corollary 2.2, there is a ~~~~~e +, E T,S* such that tit = rr*x t,; in fact- .Gt := Tr, x t, = zpw* x t, = zptt . (3.29)

#f is referred to as the resultant director momentum.

4. Stress resultants and stress couples. Local balance lawsIn this section we define stress resultants and stress couple resultants from the three-dimensional theory and develop the balance laws in terms of these resultants.Given a rnot ~o~ xl: $32 IR, -+ Y, where x, = @!0 @P,, we denote by ft the symmetricCauchy stress tensor in Y, and by P the nonsymmetric (first) Piola-Kirchhoff stress tensorrelative to 5%and Y.

4.1, Defini t i ons for t he t hree-di mensi onal t heoryConsider sections in the current configuration

Y := {Xf R3 1x = @(5=const,} , Q =The (one-form) field normal to Y is given by

Y C iR3 defined as1,2. (4.1)

dY* := j[V@]-E 65 d,$ = jg dt2 dr , (4.2)and the analogous expressioncoordinate length 5 is

R :=

holds for dY2. Consequently, the force acting on .Yi per unit of

Similarly, the torque acting on Y1 per unit of coordinate length 5 isy-l:= (x - 40)x u $ = h:x - 4p)x ugj dt . (4.4)

We define the stress result ant n, and the str ess coupl e ma, by normalizing R* and T with thesurface Jacobian i= [/al x a2 11.Accordingly, we set1

Ih +

n * := =j h - 4 j de ,

1ma := -I f ::(X-yl)xMjdL

(4Sa)

(4Sb)


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282 J.C. Simo, D. D. Fox, Stress resul t ant geometr i call y exact shel l model . Part I

where x = @( 5 , r*, 6). Finally, one defines the across-the-thickness stress resultunt, denotedby 1, by the expression

(4.6)where y = (g, x g2) / 11 , x g, 11= ( jg3) / IIg, x g, II is the (one-form) normal to the laminaesurface.REMARKS 4.1. (1) For the stress resultant n we have the equivalent expressions

1.I-

h+na= y

I h-ug" j dt = f ,,y+Pgtj, dt ,I (43

the second of which follows by recalling the relation P = JaF-, where F =V@[WO]- is thedeformation gradient and J : = det F = /jO, and noting thatjuVW_ = jOPV@ . (4.8)

(2) Similarly, noting that (X - 9) = [t, for the stress couple tnn we have the alternativeexpressions in terms of the Piola-Kirchhoff stress1m=tx 7I

f h: &rgci d& = t x r, l+j h- @gGjo dS . @*9)(3) In terms of the Piola-Kirchhoff stress, the across-the-thickness resultant can be written

1 1 h+ = y I h- ag3j d[ = ) fh+

h- &j, dS . (4.10)

(4) Observe that ma * = 0. Hence, there is no component of he stress couple along thedirector t. This is at variance with some formulations of shell theory employing the rotationvector, as in [32]. AIternatively, one may define the director stress couple, &, according tothe expression

It should be noted that ~6 * # 0. However, because of the constraint t E S, he component of&* along t does not enter explicitly in the subsequent developments. In fact, this componentcould be eliminated completely by defining ti := hi - (6 l t)t, so that iii * = 0, and (4.11)would be of the form ma = t X ii. C


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J.C. Simo, D .D. Fox, St ress resul t ant geometr i call y exact shel l model. Part 283

Starting with the momentum balance equations of the three-dimensional theory, it can beshown (see Appendix A) that the resultant local form of the momentum balance equationstake the following form:

(4.12a)

(4.12b)

where $ and niare the applied resultant force and applied resultant coupler per unit length asdefined in Appendix A. We note that

m.t=O and m*f=O. (4.13)Equations (4.12) are in the form considered by several authors; see Green and Zerna [25,pp+ 379-3801 or Libai and Simmonds 1321. These authors, however, immediately proceed toderive component equations relative to the Gauss frame {a,, a2, t} leading, inevitably, to theexplicit appearance of the Christoffel symbols associated with the Riemannian connection of

the mid-surface. In this regard, see also ]35], We show below that the direct use of the vectorequations (4.12) is ali that is needed in the weak formulation of the equations. TheRiemannian connection of the mid-surface does not enter ~~~Zicitly in the formulation.

The balance of angular momentum of the three-dimensional theory is expressed by thesymmetry of the Cauchy stress tensor, CT= a: In what follows, this balance law is interpretedas a restriction balance of angular momentum places on the admissible form of the constitutiveequations for the resultants 1~&,ma (or equivalently m), and 1.

Expressing q in components relative to the convected basis g, wehave o = crzJg, $r g,. Thesymmetry condition CI = 0 implies g, X gJ arJ = g, X rrg =0. Integration of the latter relationover [h-, h] C R and use of the expressions g, = 8, + St,, and g, = t yieldsk*crgjd[ + E,, x @gjd& + tx h_ crgjdt==O.f (4.14)

Introducing definitions (4.5a), (4.6) and (4.11), (4.14) reduces toq&3~Xnn+t,*X~iia+tX1=0. (4.15)

Equation (4.15) is the restriction that balance of angular momentum places on the admissibleform of the constitutive equations.


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288 .I. C. Simo, D. D. Fox, St ress result ant geomet ri call y exact shell model. Part I

REMARK 5.3. As defined, the convected time derivative is a particular form of the Liederivative? IJ5.2. General elastic constitutive equations

Within the context of the purely mechanical theory, hyperelastic constitutive equations maybe formulated by postulating the existence of a stored energy function of the form@( 5, a, Y,7 ,, ; *o, yo, P 0,a) or equivalently q( 5, E, S, p; a,, yO,, to,,). We now reduce theform of !P by exploiting standard invariance requirements under superposed rigid-bodymotions. Given any superposed rigid-body rotation t I.+ Q(t) E SO(3), E, 8, p, $, n, 6, &, and6 must transform according to

&+= Q(+Q"(t) , ii = Q(t)ZQ(t) , (5.13a)6 = Q(t)6 , 4+= Q(t)@, (513b)P+= Q(t)pQ'(t) , 2 = Q(+Q'(t) , (513c)

(513d)Since the base vectors a, and the dual vectors an must transform according to

4 =Q(t)aa , aa+ = Q(t)u , (5.14)it follows from (5.13a) and (5.14) that

EC= &ipaCl+ @a P+= Q(t)(c&an @ aP)Q(t)= Q(t)(&~~u~ C3~~)Qt(t) . (5.15)

Result (5.15) implies E$ = E,~. Following the same argument for S and p we conclude that(5.16)

Thus w is expressed in invariant form as(5.17)

REMARKS 5.4. (1) $ depends on the relative measures E,@,8, and paa.(2) The dependence of 3/ on the reference quantities aOup A& and Ai, is discussed in detailin [16].(3) The dependence of $ on Ai, could be eliminated as Ai, = -r,,A&. ClFor a further discussion of Lie derivatives see [29, Section 1.61.


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J. C. Simo, D. D. Fox, Stress resul t ant geometr i call y exact shel l model. Part I 289

Using the Clausius-Duhem inequality and following standard arguments (see [34, Section2.53 and references cited there), we write the properly invariant hyperelastic constitutiverelations(5.18)

5.3. ExampEe: I sotropi c const i t ut i ve rel at i onsThe simplest properly invariant isotropic constitutive relations for the effective membraneand shear stress resultants 6 and F, and for the stress couple resultant Grip s given by

nP* - pEhl- Y2 fpWe YS&e-w = pEh3

12(1 - V) @aySPcy8) 7 (5.19)

where E is Youngs modulus, Y is Poissons ratio, K is the shear reduction coefficient, p = p/his the three dimensional mass density taken to be independent of the through-the-thicknessTable 1Basic kinematicsl Definition of the kinematics

4p: dc682+R3) mid-surfacet: .!2cc2-+s2 ( director field{E,, E2,Q t inertial frame

l Orthogonal transformation such that t = At(t # -t)A = (t- t )1+ [Gal + & (t x t)@J(tx t)

l Exact relation between director and rotation vector 8 (exponential map)t = COSl~Gllf ~ -, where0*t=O, 8=@xt

l Surface convected frame and surface Jacobiana,, = PO,, t a03 = t 0 T i, = Ita,, x a0211

a* = 44a 3 a3 = t , i= /I@, x %/I

l Relative strain measures


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290 J.C. Simo, D. D. Fox, St ress resul t ant geometr i call y exact shel l model . Part 1

variable, andHPaYa= 1IU$u; + i(l - zJ>> . (5.20)

REMARK 5.5. (1) The constitutive equation for the resultant couple given above is in termsof the symmetric part of mMpa. It is assumed here and in the following developments that theskew-symmetric part of fiP *s zero, i.e.

&aPl = 0. (5.21)(2) Constitutive equations (5.19) can be justified on the basis of an asymptotic expansion ofthe three-dimensional Saint Venant-Kirchhoff model that employs definitions (AB4c) and(AB5) from Appendix B and retains terms up to order O(hlR), where h = h - h- is the

thickness of the shell and R is the radius of curvature. Since terms involving products of aa,A& and Ai are of higher order, in classical shell theory the resulting constitutive equations aretypically assumed to depend on the refence surface only though the first fundamental form(i.e., a;P); see [25; 35, p. 606; 37]:The general dependence of the constitutive equations onthe reference surface is discussed in detail in [16]. 0

For convenience, the basic theory presented in Sections l-5 is summarized in Tables 1and 2.Table 2Local momentum equations. Constitutive relationsl Balance of linear and director momentum

l Stress resultants and stress couplesna = npaap + qts = gBa, + fi 3tI =&+11= At + Ta, + A,Gipa,

0 Effective stress resultants$ = nPu _ @y+Q = qa - +a

= q + A ; ya6?0 Constitutive equations: !P = +( 5, E,@,8, , pap aOap , Yo,, A$, Aim


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In what follows, with a slight abuse in notationse write = (9, t) E %:,where $6: d-+ 11;8defines the p~si~i*~ of the mid-surface, and 1: A$-+-_*defines the dire&x fiefd, We assumedisplacement boundary conditions of the fen-m

where CM is the b~u~da~ of & C R, The tangent space at Q = (gcr,b), denoted by T, 552,s thelinear space af admissible variations defined as

Since &( 5 > E T,S2 for each & E &, we have the following alternative useful characterization ofthe variations 22: ti--+ T$ associated with the director field:fi) S~~~~~~ e~&~~p~~~~.ecalling the ane-to-one correspondence between T,S2 and TJt$-;we can write

REMARK 61. The rotated rep~ese~tati~~ (6Sb) is pa~tie~la~l~ useful in a ~Qrnputati~~alcontext. The reasun for this lies in the relation 62 - E = 0. ~~~luw~~~ Remark 25 we can write


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292 f. C. Simo, D. D. Fox, Str ess result ant geomet ri cal& exact shell model. Part I& T-E =0 3 6T=6Tl& -+6T2E,. 66)

Consequently, only two co~~o~e~~ (relative to the inertial frame {Ef}f=1,2,3) need to beconsidered. Thus, this representation automatically excludes the drill degree of freedomalong E, = E. In the spatial description (6Sa), however, this constraint is only implicit in thecondition 6t t = 0. q6.2. Weak form of momentum ~ffl u~ce

By considering arbitrary variations 6c;D= (&4p,8t) E T, %, and integrating over thesurface q: ,s?+ lR3, he weak form of the momentum equations in Table 2 becomes

G~~(~, 8Qi) := G(@, S@) + j-&[+ * 69 + I,& lit] dp ,

current

(6.7)

where dp = jdtl ds2 is the (two-form) current surface measure and G(@, ;SQi) is the staticweak form or virtual work expression defined in the following subsection.

6.2.1. Stat ic w eak formUsing the divergence theorem for surfaces (integration by parts), a straightforward manipu-lation gives the following expression:

where G,,,(G@) is the virtual work of the external loading given by(6-9)

Here rf-and 6 are the prescribed force and torque on the boundary of the shell; i.e.n= nv, on d,& and 6 = &%, on a,& , (6.10)

where v = ~,a is the (one-form) field normal to the boundary +~(a&). Of course, a,& na, &=@ and a,,,&r&aa,s42=0.6.2.2. Component expressionWe now introduce the explicit expression for I and the component relations (4.20) and(4.22) into expression (6.8). Performing manipulations identical to those in the proof ofCorollary 5.2, it can be shown that the weak form (6.8) is expressed in components as


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294 J.C. Simo, D. D. Fox, St ress resul t ant geometr i call y exact shel l model. Part I

With these preliminary results at hand, we proceed to the linearization of the strainmeasures followed by the linearization of the weak form.

6.3.2. Li neari zed strai n measuresLet (a& rt , K$) be the kinematic quantities associated with the one-parameter family ofconfIgurations E H @, = (pE, t,) defined by (6.13). For convenience, we introduce the notation

(6.18)By making use of the results in Section 6.3.1, we readily obtain the expressions

AY, := tt*b,a + Aw,J ,Aq3 : = A+Q~ t,, -t v,4n l At,, -

We now proceed to the linearization of

6.4. Li neari zed w eak form. Symmet ry

(6.19)

the weak form.

The conceptual procedure is standard; see e.g. [34, Ch. 61. One substitutes the one-parameter family of configurations (6.13) into (6.11) and then one makes use of the relationbetween the Frechet directional derivative given by the standard formula

DG-A@ 2 E=OG(@$, 8~). (6.20)The linearization of (6.20) can be broken into two parts, D,G - A@ and D,G .A@, each ofwhich possesses classical interpretations.

6.4.1. Opera t or expr essi onsFor subsequent developments, it proves convenient to introduce a more compact notation.To this end, we define the following resultant stress and stress couple vectors. As is stated inRemark 5.5, we assume %ifpul = 0, i.e. 6 = G! We thus have

Furthermore, utilizing expression (6Sb) for the director variation field, set

(6.21)

(6.22)


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.l. C. Simo, D.D . Fox, Stress resul t ant geometr i call y exact shell model. Part f

We define the matrix differential operators2%

t aV,l ag 1

BbIn = t at.2 2att a, Bbb = ii 3x2 )

where j is the (3 x 2) matrix obtain by deleting the third column of -4, i.e.

4x2 =

(623a)

(623b)

(6.23~)

(6.24)

With this notation at hand, we can rewrite the weak form (6.11) as

- G,t@Q,) (6.25)where dpO = jO de dt2 is the reference surface measure.Aiternatively , defining the total resultant stress vector as

R=Z )L-1 8X1and the differential operator

the weak form (6.25) can be written asG(@, SD) := /& B{ fiT) R d,u,, - G&j@) .

(6.26)

(6.27)

(6.28)


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296 J.C. Simo, D.D. Fox, Stress result ant geomet ri call y exact shell model, Part IWe emphasized once more that by employing the material representation of the director fieldvariation, the weak form (628) is in a five degree uf freedom format with the constraint~~~ditiu~ f l 66 = 0 aut~rnati~~~y satisfied through the material refation ST * E3 = 0,~~~A~~ 62, For the case of hyperelastic response with properly invariant stored energyft.mtion tlr(S, q+, a,, Pap; aomPTTo,, &L Al)> G(@, S@), as given by (6.28), is the firstvariation of the functionat

(629)(6.30)

is he ~tentia~ energy of the external goading (we assume dead ~uading~~)_ iIWe ~CSW roceed to the ~inea~~ati~~ of the weak form (6.28).

6.42. M aterial partThe material part, D,G l A@, of the tangent operator arises as a resuit of Iinearizing thesubstitutive equation at fixed ~~~rnet~. For illustrative purposes, we consider elastic re-

sponse . Accordingly,

where

aP*243%

(6.31)

f3P128P22 aPdPl2

is the symme~~ tangent elasticity tensor.


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298 J.C. Simo, D.D . Fox, St ress resul t ant geometr i call y exact shel l model. Part I7. Closure and overview of subsequent research

We have presented a careful development of the continuum basis of a geomet~cally exactshell theory obtained by reduction of the three-dimensional theory by means of the oYteinextensible director kinematic assumption. Our subsequent work will be concerned withnumerical analysis and computational aspects involved in the implementation of the presentformulation within the framework of the finite element method. In particular, we plan to focuson the following aspects:(a) The linearized theory. The relevant field equations are obtained simply by particulariz-ing the results developed above to the reference configuration. The linear theory provides anideal framework to address, in detail, issues concerned with interpolation and convergence ofthe finite element procedure. We provide a detailed account of the proposed interpolation,develop explicit matrix expressions, and assess the performance of the model in standardbenchmark problems, as well as new test problems. Our results indicate that the presenttheory exactly matches results obtained with the degenerated solid approach.

(b) Computat ional aspects of t he ull y nonl inear theory. The central topic to be addressed isthe development of discrete update procedure for the director field which is exact andsingularity free regardless of the magnitude of the rotational increments. In contrast withcurrent procedures, often only second-order accurate, the proposed approach is exact, andpreserves objectivity for arbitrary large displacement and rotation increments. Computational-ly, the need for using quaternion parameters to avoid singularities is bypassed. Detailedaccount of the interpolation procedure, and expressions for the residual and consistent tangentmatrix are given. The formulation is assessed through numerical simulations involvinginstability and bi~rcation phenomena.(c) Enhunced model s includi ng t hrough t he t hi ckness str et ch. The model considered in thispaper can be easily extended to accommodate thickness changes in the shell by means of a oneextensible director kinematic assumption. It is shown that this additional effect adds very littlecomplexity to the model while allowing for an entirely new class of problems to be addressed.The advantages of such a model include: exact recovery of the plane stress constitutiveequations in the thin shell limit, explicit account of the through-the-thickness-stress forproblems dominated by behavior such as delamination and avoidance of numerical problemsassociated with the undetermined pressure-like inextensible constraint.(d) Inelasti cit y i n st ress result ants at ini te strui ns. Elastoplastic constitutive models at finitestrains that incorporate classical kinematic and isotropic hardening rules are formulatedentirely in stress resultants. The formulation completely bypasses the need for the costlyintegration-through-the-thickness procedure in the traditional degenerated solid formulation.The integration of the elastoplastic equations is performed by means of recently proposedgeneral return mapping algorithms that can accommodate non-smooth yield surfaces.(e) Nonlinear dynamics. Computational aspects. The present geometric setting, and theassociated Hamiltonian structure considered in [39] can be exploited to implement timestepping algorithms that exactly preserve the constants of motions. Recent work indicates thatone can develop time stepping algorithms which are, in fact, discrete canonical transforma-tions, thus preserving the discrete Hamiltonian structure of the model. We plan to addressthese and related issues in our subsequent work.


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300 J.C. Simo, D.l ). Fox, St ress resuhrt t geome3ri & ~l l y exact shel l model. Part I

and using definitun (3.231, we have

(A-6)Since the choice of Y (or equivalently &) is arbitrary, assuming enough smoothness so thatthe standard locahzation result holds; see [26, p. 383, we obtain the resultant form of the localbalance of linear momentum (4.12a)

The three-dimensional integral equation of balance of angular momentum is written

SI @xoiidY+ ~~~ @xpBdCP=$ iii Q,xp&dclr, (A-8)aqr gr Iwhere the definitions in (A.11 apply. Again employing the kinematic assumption (J.3), thenormal map relation ii dY = jV@ -&dI, and the change of variables relation, (A.8) becomes

IRecal! that the normals to the lateral, top, and bottom surfaces are given by & = v,E, fi = E3,and IV = -E3, respectively. Thus, with the definition of the reciprocal basis (3.12) and thedefinition of the mid-surface (3.22) we can write

(A.10)Using the stress and stress couple resultant definitions (4.5), (AJ), (3.23), and (3.24) anddefining the external couple as


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J. C. Simo, D. D. Fox, Stress resul t ant geometr i call y exact shell model . Part I 301

we can rephrase. (A.10) as(A.ll)

(A.12).a J a

where the divergence theorem, balance of linear momentum (A.7), and the relation ti, = t X twere used. Since V (or equivalently L$) can be chosen arbitrarily, assuming enough smooth-ness we obtain the local resultant form of the balance of angular momentum (4.12b) as1 TLIT(,rn ),p+~,llXnn+m=zPi,. (A.13)

Appendix B. Stress resultant componentsIn this appendix we derive explicit component expressions for the stress and stress coupleresultants from the three-dimensional theory. First, we present some preliminary results that

are helpful in the following development:g, = @,,I I$ g, = a, + O,, , g, = t 7t,, = AEa, + Ait, ag = aSgp + u39. (B.1)

From definitions (4.7) and (4.11) we recall that1I I

h+no1= - 1h- agjdt, fi=: I I :_ Sagj d5 .

Substitution of (B.l) into (B.2) yields1na = -I I +h_ aPa(ap + [hga,)jd[ + L I h+j h- (cT~~+ & i iaP)t j dl ,

(B-2)

From component expressions (4.22) relative to the surface convected basis {a,, t } w e obtainp= 1 I

h+j h- (f ? + eAEaPa)j dt , (B.4a)

(03a + tAiaP)jd[, (B.4b)


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302 .I. C. Simo, D . D. Fox, St ress resul t ant geometr i call y exact shell model . Part I

(B .4c)

(B.4d)From (4.25) and (4.27) we now obtain component relations for the effective membrane andshear stress resultants as

AcknowledgmentThis work was performed under the auspices of the Air Force Office of Scientific Research.Support was provided by grant nos. 2-DJA-544 and 2-DJA-771 with Stanford University. Thissupport is gratefully acknowledged. We wish to thank Professors S. Antman, T. Hughes, P.

Krishnaprassad, 3. Marsden, C. Steele and R. Taylor for many helpful discussions.

ReferencesPIPIt31141PIbl[71PIf91

R. Abraham and J.E. Marsden, Foundations of Mechanics (Addison-Wesley, Reading, MA, 2nd ed., 1978).R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications (Addison-Wesley,Springer, New York, 2nd ed., 1987).S. Ahmad, B.M. Irons and O.C. Zienkiewicz, Analysis of thick and thin shell structures by curved finiteelements, Internat. J. Numer. Meths. Engrg. 2 (1970) 419-451.S.S. Antman, Ordinary differential equations of nonlinear elasticity I: Foundations of the theories ofnon-linearly elastic rods and shell, Arch. Rat. Mech. Anal. 61 (4) (1976) 307-351.S.S. Antman, Ordinary differential equations of nonlinear elasticity II: Existency and regularity theory forconservative boundary value problems, Arch. Rat. Mech. Anal. 61 (2) (1976) 353-393.J.H. Argyris and D.W. Scharpf, The SHEBA family of shell elements for the matrix displacement method.Part I. Natural definition of geometry and strains, Aeron. J. Roy. Aeron. Sot. 72 (1968) 873-878.J.H. Argyris and D.W. Scharpf, The SHEBA family of shell elements for the matrix displacement method,Part II. Interpolation scheme and stiffness matrix, Aeron. J. Roy. Aeron. Sot. 71 (1968) 878-883.J.H. Argyris and D.W. Scharpf, The SHEBA family of shell elements for the matrix displacement method,Part IIl. Large displacements, Aeron. J. Roy. Aeron. Sot. 73 (1969) 423-426.J.H. Argyris, M. Haase and G.A. Malejannakis, Natural geometry of surfaces with specific reference to thematrix dispIacement analysis of shells, Proc. Kon. Nederl. Akad. Wet. Ser. B 76 (5) (1973).


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304 J.C. Simo, D.D. Fox, Str ess result ant geomet ri call y exact shell model. Part I[40] E. Ramm, A plate/shell element for large deflection and rotations, in: K.J. Bathe, J.T. Oden and W.

Wunderlich, eds. (Formulations and Computational Algorithms in Finite Element Analysis) (MIT Press,Cambridge, MA, 1977).

[41] J.G. Simmonds and D.A. Danielson, Nonlinear shell theory with a finite rotation vector, Kon. Ned. Ak. Wet.B73 (1970) 460-478.

[42] J.G. Simmonds and D.A. Danielson, Nonlinear shell theory with finite rotation and stress-function vectors, J.Appl. Mech. 39 (1972) 1085-1090.[43] J.C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Comput. Meths.Appl. Mech. Engrg. 49 (1985) 55-70.[44] J.C. Simo, J.E. Marsden and P.S. Krishnaprassad, The Hamiltonian structure of elasticity. The convectiverepresentation of solids, rods and plates, Arch. Rat. Mech. Anal. (1987) (to appear).

[45] J.C. Simo and L.V. Quoc, A three-dimensional finite-strain rod model. Part II: Geometric and computationalaspects, Comput. Meths. Appl. Mech. Engrg. 58 (1986) 79-116.

[46] J.C. Simo and L. Vu-Quoc, A beam model including shear and torsional warping distortions based on an exactgeometric description of nonlinear deformations, Internat. J. Solids and Structures (1987) (to appear).

1471 J.C. Simo and L. Vu-Quoc, On the dyn~i~ in Space of rods undergoing large motions--A geometri~llyexact approach, Comput. Meths. Appl. Mech. Engrg. (to appear).[48] G. Stanley, Continuum-based shell elements, Ph.D. Dissertation, Applied Mechanics Division, StanfordUniversity, Stanford, CA, 1985.

On a Stress Resultant Geometrically Exact Shell Model Part i

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