A Geometrically Nonlinear ShearDeformation Theory for Composite Shells

Wenbin Yu and Dewey H. Hodges

Georgia Institute of Technology, Atlanta, Georgia 30332-0150

A geometrically nonlinear shear deformation theory has been developed for elastic shells to ac-commodate a constitutive model suitable for composite shells when modeled as a two-dimensionalcontinuum. A complete set of kinematical and intrinsic equilibrium equations are derived for shellsundergoing large displacements and rotations but with small, two-dimensional, generalized strains.The large rotation is represented by the general finite rotation of a frame embedded in the unde-formed configuration, of which one axis is along the normal line. The unit vector along the normalline of the undeformed reference surface is not in general normal to the deformed reference sur-face because of transverse shear. It is shown that the rotation of the frame about the normal lineis not zero and that it can be expressed in terms of other global deformation variables. Basedon a generalized constitutive model obtained from an asymptotic dimensional reduction from thethree-dimensional energy, and in the form of a Reissner-Mindlin type theory, a set of intrinsic equi-librium equations and boundary conditions follow. It is shown that only five equilibrium equationscan be derived in this manner because the component of virtual rotation about the normal is notindependent. It is shown, however, that these equilibrium equations contain terms that cannot beobtained without the use of all three components of the finite rotation vector.

IntroductionFor an elastic three-dimensional (3-D) continuum, there are two types of nonlinearity: geomet-

rical and physical. A theory is geometrically nonlinear if the kinematical (strain-displacement)relations are nonlinear but the constitutive (stress-strain) relations are linear. This kind of theoryallows large displacements and rotations with the restriction that strain must be small. A physi-cally (or materially) nonlinear theory is necessary for biological, rubber-like or inflatable structureswhere the strain cannot be considered small, and a nonlinear constitutive law is needed to relatethe stress and strain. Although this classification seems obvious and clear for a structure modeledas a 3-D continuum, it becomes somewhat ambiguous to model dimensionally reducible structures structures that have one or two dimensions much smaller than the other(s) such as beams, plates

Post Doctoral Fellow, School of Aerospace Engineering. Presently, Assistant Professor, Department of Mechani-cal and Aerospace Engineering, Utah State University, Logan Utah 84322-4130.

Professor, School of Aerospace Engineering. Member, ASME.

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and shells [1] using reduced one-dimensional (1-D) or two-dimensional (2-D) models. A nonlin-ear constitutive law for the reduced structural model can in some circumstances be obtained fromthe reduction of a geometrically nonlinear 3-D theory. For example, in the so-called Wagner ortrapeze effect [25], the effective torsional rigidity is increased due to axial force. This physicallynonlinear 1-D model stems from a purely geometrically nonlinear theory at the 3-D level. On theother hand, the present paper focuses on a geometrically nonlinear analysis at the 3-D level whichbecomes a geometrically nonlinear analysis at the two-dimensional 2-D as well. That is, the 2-D generalized strain-displacement relations are nonlinear while the 2-D generalized stress-strainrelations turn out to be linear.

A shell is a 3-D body with a relatively small thickness and a smooth reference surface. Thefeature of the small thickness attracts researchers to simplify their analyses by reducing the original3-D problem to a 2-D problem by taking advantage of the thinness. By comparison with theoriginal 3-D problem, an exact shell theory does not exist. Dimensional reduction is an inherentlyapproximate process. Shell theory is a very old subject, since the vibration of a bell was attemptedby Euler even before elasticity theory was well established [6]. Even so, shell theory still receivesa lot of attention from modern researchers because it is used so extensively in so many engineeringapplications. Moreover, many shells are now made with advanced materials that have only recentlybecome available.

Generally speaking, shell theories can be classified according to direct, derived and mixed ap-proaches. The direct approach, which originated with the Cosserat brothers [7], models a shelldirectly as a 2-D orientated continuum. Naghdi [8] provided an extensive review of this kind ofapproach. Although the direct approach is elegant and able to account for transverse and normalstrains and rotations associated with couple stresses, it nowhere connects with the fact that a shellis a 3-D body and thus completely isolates itself from 3-D continuum mechanics. This could bethe main reason that this approach has not been much appreciated in the engineering community.One of the complaints of these approaches that they are difficult for numerical implementationhas been answered by Simo and his co-workers by providing an efficient formulation free frommathematical complexities and suitable for large scale computation [9, 10]. And more recentlya similar theory was developed by Ibrahimbegovic [11] to include drilling rotations so that not-so-smooth shell structures can be analyzed conveniently. However, the main complaint remainsthat these approaches lack a meaningful way to find the constitutive models which can only beexperienced and formulated properly in our 3-D real world [12]. Reissner [13] developed a verygeneral nonlinear shell theory introducing twelve generalized strains by considering the dynamicsof stress resultants and couples on the reference surface as the basis. He gracefully avoided theawkwardness of finding a proper constitutive model by pointing out two possible means to estab-lish them. It is recommended in [13] that one could either design experiments to determine the

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constitutive constants without explicit reference to the 3-D nature of the structure or derive an ap-propriate 2-D model from the given knowledge of the constitutive relations for the real 3-D modelof the structure.

Derived approaches reduce the original 3-D elasticity problem into a 2-D problem to be solvedover the reference surface. Such reductions are usually carried out in one of two ways. The mostcommon approach is to assume a priori the distribution of 3-D quantities through the thickness andthen to construct a 2-D strain energy per unit area by integrating the 3-D energy per unit volumethrough the thickness. Remarkably, classical (also known as Kirchhoff-Love type theory), first-order shear deformation (also known as Reissner-Mindlin type theory), higher-order, and layer-wise shell theories all fall into this category, including the theories proposed by Reddy [14], forexample. Another approach is to apply an asymptotic method to expand all quantities into anasymptotic series of the thickness coordinate, so that a sequence of 2-D problems can be solvedaccording to the different orders.

The mixed approach is used in [15] based on the argument that all the 3-D elasticity equationsexcept the constitutive relations are independent of the material properties, such as the kinematicalrelations, equilibrium of momentum and forces. The constitutive law must be determined experi-mentally, and hence it is avoidable that it is approximate. Libai and Simmonds [15] obtain exactshell equations for the balance of momentum, heat flow and an entropy inequality from the 3-Dcontinuum mechanics via integration through the thickness. An analogous 2-D constitutive law ispostulated due to the fact that even 3-D constitutive laws are inexact.

There is a sense in which the present approach can also be considered as mixed. The 2-Dconstitutive model is obtained by the Variational Asymptotic Method (VAM) [16] such that the2-D energy is as close to an asymptotic approximation of the original 3-D energy as possible[17]. The process of constructing the constitutive model defines the reference surface and thekinematics of this surface are geometrically exact formulated in an intrinsic format. The 2-Dequilibrium equations are obtained from the 2-D energy with the knowledge of the variations ofthe generalized strains. The only approximate part of our 2-D shell theory is the constitutive lawwhich is not postulated but is mathematically obtained by VAM.

Shell KinematicsThe equations of 2-D shell theory are written over the domain of the reference surface, on

which every point can be represented by a position vector r in the undeformed configuration andR in the deformed configuration (see Fig. 1) with respect to a fixed point O in the space. A setof two curvilinear coordinates, x, are required to located a point on the reference surface. Thecoordinates are so-called convected coordinates such that every point of the configuration has thesame coordinates during the deformation. (Here and throughout the paper Latin indices assume 1,

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2, 3; and Greek indices assume values 1 and 2. Dummy indices are summed over their range exceptwhere explicitly indicated.) Without loss of generality, x are chosen to be the lines of curvaturesof the surface to simplify the formulation. For the purpose of representing finite rotations, anorthonormal triad bi is introduced for the initial configuration, such that

b = a/A b3 = b1 b2 (1)

where a is the set of base vectors associated with x and A are the Lame parameters, defined as

a = r, A =a a (2)

From the differential geometry of the surface and following [13] and [18] one can express thederivatives of bi as

bi, = Ak bi (3)

where k is the curvature vector measured in bi with the components

k = bk2 k1 k3cT (4)

in which k refers to out-of-plane curvatures. We note that k12 = k21 = 0 because the coordinatesare the lines of curvatures. The geodesic curvatures k3 can be expressed in terms of the Lameparameters as

k13 = A1,2A1A2

k23 =A2,1A1A2

(5)

When the shell deforms, the particle that had position vector r in the undeformed state nowhas position vector R in the deformed shell. The triad bi rotates to be Bi. The rotation relatingthese two triads can be arbitrarily large and represented in the form of a matrix of direction cosinesC(x) so that

Bi = Cijbj Cij = Bi bj (6)

A definition of the 2-D generalized strain measures is needed for the purpose of formulatingthis problem in an intrinsic form. Following [13] and [18], they can be defined as

R, = A(B + B + 23B3) (7)

andBi, = A(K2B1 +K1B2 +K3B3)Bi (8)

where are the 2-D in-plane strains, and Kij are the curvatures of the deformed surface, which

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are the summation of curvatures of undeformed geometry kij and curvatures introduced by thedeformation ij , and 3 are the transverse strains because B3 is not constrained to be normal tothe reference surface after deformation. Please note that the 2-D generalized strain measures aredefined by Eqs. (7) and (8) in an intrinsic fashion, the symmetry of the inplane strain measuressuch that 12 = 21 does not hold automatically. Nevertheless, one is free to set 12 = 21, i.e.

B1 R,2A2

=B2 R,1A1

(9)

which is a constraint used in [17] to make the 3-D formulation unique.At this point sufficient preliminary information has been obtained to develop a geometrically

nonlinear shell theory.

Compatibility EquationsIt is well known that a rigid body in 3-D space has only six degrees of freedom. Thus, the

kinematics of an element of the deformed shell reference surface can be expressed in terms of atmost six independent quantities: three measures of displacement, say u bi, and three measures ofthe rotation of Bi (since the global rotation tensor C, which brings bi into Bi, can be expressedin terms of three independent quantities). However, we have the eleven 2-D strain measures 11,212, 22, 23, , and 3 as defined in Eqs. (7) and (8). Thus, they are not independent; thereare some compatibility equations among these eleven quantities. In [19] and [13] appropriatecompatibility equations are derived by first enforcing the equalities

R,12 = R,21 (10)

andBi,12 = Bi,21 (11)

These two vector equations lead to six independent compatibility equations equivalent to a formof those found in [13]. These equations are rewritten here for convenience in the present notation.First, from the B3 components of Eq. (10), we obtain

(1 + 22)12 (1 + 11)21 = (A2223),1A1A2

(A1213),2A1A2

+ 12(K22 K11) (12)

Next, from the B components of Eq. (10) we obtain two equations for =1 and 2, respectively, as

(1 + 22)K13 12K23 = (A212),1A1A2

[A1(1 + 11)],2A1A2

21321 + 223K11

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(1 + 11)K23 12K13 =[A2(1 + 22)],1

A1A2 (A112),2

A1A2 213K22 + 22312 (13)

Finally, from the three components of Eq. (11) we have nine identities. However, there are onlythree independent equations, given by

(A1K11),2A1A2

(A221),1A1A2

+K13K22 12K23 =0(A112),2A1A2

(A2K22),1A1A2

+K23K11 21K13 =0(A2K23),1A1A2

(A1K13),2A1A2

+K11K22 1221 =0 (14)

There are now eleven quantities which are related by six compatibility equations. This means thatthese strain measures can be determined in terms of only five independent quantities not six.

In the process of dimensional reduction of [17] to find an accurate constitutive model for com-posite shells, the authors encountered the question whether one should include 21 and 12 as twodifferent generalized strain measures. This was determined by the following argument. Let usdenote a new twist measure 2 = 12+21. From Eq. (12) the difference between 21 and 12 canbe obtained as

12 212

=

(A2223),1(A1213),2A1A2

+ 12(K22 K11) + (11 22)(2 + 11 + 22)

(15)

This difference is clearly O( h`2) or O(

R) disregarding the nonlinear terms ( is the order of gen-

eralized strains, h is the thickness of the shell, l is the wavelength of in-plane deformation and Ris the characteristic radius of shell). One can show that it contributes terms that are O( h

2

`2hR) or

O( h2

R2) to the three-dimensional strains. Clearly, such terms will not be counted in a physically

linear theory with only correction up to the order of h/R and (h/l)2.Eqs. (13) can be solved for the in-plane curvatures 13 and 23, and Eq. (15) can be used

to express 12 and 21 in terms of . Now, using these expressions, one can rewrite the threeEqs. (14) entirely in terms of the eight strain measures 11, 212, 22, 213, 223, 11, 2, and 22.This confirms that only five independent measures of displacement and rotation are necessary todefine these strain measures as we will demonstrate conclusively below by deriving such measures.

Global Displacement and Rotation VariablesThere is no unique choice for the global deformation variables. For this reason, the importance

(not to mention the beauty) of an intrinsic formulation is widely appreciated. On the other hand, forthe purpose of understanding the displacement field more fully, for practical computational algo-

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rithms, and for easy derivation of virtual strain-displacement relations, it is expedient to introducea suitable set of displacement measures.

The displacement measures we choose are derived by expressing R in terms of r plus a dis-placement vector so that

R(x1, x2) = r(x1, x2) + uibi (16)

Differentiating both sides of Eq. (16) with respect to x, and making use of Eq. (7), one can obtainthe identity

B + B + 23B3 = b + ui;bi + uik bi (17)

where ( ); = 1A( )

. The above formula allows the determination of the strain measures and23 in terms of C, ui and the derivatives of ui. Introducing column matrices u = bu1 u2 u3cT ,e1 = b1 0 0cT , e2 = b0 1 0cT , 1 = b11 12 213cT , and 2 = b21 22 223cT , we can obtain thefollowing identity in matrix form

e + = C(e + u; + ku) (18)

where C is the matrix of direction cosines from Eq. (6) and k is defined in Eq. (4), and ( )ij =eijk( )k.

Rodrigues parameters [20] can be used as rotation measures to allow a compact expression ofC. These are derived based on Eulers theorem, which shows that any rotation can be representedas a rotation of magnitude about a line parallel to a unit vector e. Defining the Rodriguesparameters i = 2e bi tan

(2

)and arranging these in a column matrix = b1 2 3cT , the

matrix C can simply be written as

C =

(1 T

4

)I + T

2

1 + T 4

(19)

Let us also denote the direction cosines of B3 by

C3i = 3i + i (20)

Hodges [21] has shown that, given the third row of C, the Rodrigues parameters can be uniquely

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expressed in terms of i as

1 =31 222 + 3

2 =32 + 212 + 3

3 =2 tan

(32

)(21)

where 3 can be understood as a change of variables to simplify later parts of the derivation. Lateron we will discuss the meaning of 3 for a special case. Finally, it is noted that the three rotationalparameters i are not independent but instead satisfy the constraint

21 + 22 + (1 + 3)

2 = 1 (22)

When Eq. (21) is substituted into Eq. (19), the resulting elements of C can be expressed asfunctions of i and 3

C11 =(2 + 3 21) cos3 12 sin3

2 + 3

C12 =(2 + 3 22) sin3 12 cos3

2 + 3

C13 = 1 cos3 2 sin3C21 =

(2 + 3 21) sin3 12 cos32 + 3

C22 =(2 + 3 22) cos3 + 12 sin3

2 + 3

C23 =1 sin3 2 cos3C31 =1

C32 =2

C33 =1 + 3 (23)

This representation reduces to those of [22] when considering small, finite rotations. There is anapparent singularity in the present scheme when 3 = 2 (i.e., when the shell deforms in such away that B3 is pointed in the opposite direction of b3). This should pose no practical problem,however, since 1 = 2 = 0 for that condition, and none of the kinematical relations becomeinfinite in the limit as 3 2.

When these expressions for the direction cosines are substituted into Eq. (18), explicit expres-

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sions for the strain measures can be found as

11 =[(2 + 3 21)(1 + u1;1 k13u2 + k11u3) 12(u2;1 + k13u1)

2 + 3+ 1(k11u1 u3;1)

]cos3

+[(2 + 3 22)(u2;1 + k13u1) 12(1 + u1;1 k13u2 + k11u3)

2 + 3+ 2(k11u1 u3;1)

]sin3 1

22 =[(2 + 3 22)(1 + u2;2 + k23u1 + k22u3) 12(u1;2 k23u2)

2 + 3 2(u3;2 k22u2)

]cos3

+[(2 + 3 21)(k23u2 u1;2) + 12(1 + u2;2 + k23u1 + k22u3)

2 + 3+ 1(u3;2 k22u2)

]sin3 1

12 =[(2 + 3 22)(u2;1 + k13u1) 12(1 + u1;1 k13u2 + k11u3)

2 + 3+ 2(k11u1 u3;1)

]cos3

[(2 + 3 21)(1 + u1;1 k13u2 + k11u3) 12(u2;1 + k13u1)

2 + 3+ 1(k11u1 u3;1)

]sin3

21 =[(2 + 3 21)(u1;2 k23u2) + 12(1 + u2;2 + k23u1 + k22u3)

2 + 3+ 1(u3;2 k22u2)

]cos3

+[(2 + 3 22)(1 + u2;2 + k23u1 + k22u3) 12(u1;2 k23u2)

2 + 3 2(u3;2 k22u2)

]sin3

213 = 1(1 + u1;1 k13u2 + k11u3) + 2(u2;1 + k13u1) + (1 + 3) (u3;1 k11u1)223 = 1(u1;2 k23u2) + 2(1 + u2;2 + k23u1 + k22u3) + (1 + 3) (u3;2 k22u2)

(24)

These expressions explicitly depend on sin3 and cos3. It is evident that one can choose 3 sothat 12 = 21, yielding

tan3 =n1 + 22(u2;1 + k13u1) 21(u1;2 k23u2) + 12 [u1;1 u2;2 + (k11 k22)u3 k23u1 k13u2]n2 + 21(u1;1 + k11u3 k13u2) + 22(u2;2 + k22u3 + k23u1) + 12(u1;2 + u2;1 k23u2 + k13u1)

(25)

where

n1 = (2 + 3) [u1;2 u2;1 1(u3;2 k22u2) + 2(u3;1 k11u1) k13u1 k23u2]n2 = (2 + 3)[1(u3;1 k11u1) + 2(u3;2 k22u2) u1;1 u2;2 2 3 k23u1 (k22 + k11)u3 + k13u2] (26)

It is now clear that once the functions u1, u2, u3, 1 and 2 are known, the entire deformation isdetermined. Because of this, one should expect that a variational formulation would yield only fiveequilibrium equations not six.

For small displacement and small strain, one can obtain 3 as

3 =(A2u2),1 (A1u1),2

2A1A2(27)

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which is half the angle of rotation about B3, the same as obtained in [23].Although one can now find exact expressions for 11, 212, 22, 213 and 223 which are inde-

pendent of 3, such expressions are rather lengthy and are not given here. Alternatively, one couldleave 3 in the equations and regard Eq. (25) as a constraint. This would allow the construction ofa shell finite element which would be compatible with beam elements which have three rotationaldegrees of freedom at the nodes.

Expressions for the curvatures can be found in terms of C as

K = C;CT + CkCT (28)

whereK = bk2 k1 k3cT + b2 1 3cT (29)

Following [24], the curvature vector can also be found using Rodrigues parameters

K =I

2

1 + T 4

; + Ck (30)

Using the form of C from Eqs. (23), the curvatures become

1 = 1; cos3 + 2; sin3 3; (1 cos3 + 2 sin3)2 + 3

+ k1 k1

2 = 1; sin3 + 2; cos3 + 3;(1 sin3 2 cos3)2 + 3

+ k2 k2

3 = 3; +1;2 12;

2 + 3+ k3 (31)

where

k1 = (k3 k212 + 3

+k122 + 3

)(1 sin3 2 cos3) + k2 sin3 + k1 cos3

k2 = (k3 k212 + 3

+k122 + 3

)(1 cos3 + 2 sin3) + k2 cos3 k1 sin3k3 = k21 + k12 + k33 (32)

As before, 3 can be eliminated from these expressions, so that all six curvatures can be expressedin terms of five independent quantities. Note that 3 are not independent 2-D generalized strains.They will, however, appear in the equilibrium equations because of their appearance in the virtualstrain-displacement relations derived later.

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2-D Constitutive LawTo complete the analysis for an elastic shell, a 2-D constitutive law is required to relate 2-

D generalized stresses and strains. As mentioned before the constitutive law can not be exact,however, one should try to avoid introducing any unnecessary approximation in addition to thealready-approximate 3-D constitutive relations.

Among many approaches that have been proposed to deal with dimensional reduction, the ap-proach in [17] stands out for its accuracy and simplicity. In that work, a simple Reissner-Mindlintype energy model is constructed that is as close as possible to being asymptotically correct. More-over, the original 3-D results can be recovered accurately. The resulting model can be expressedas

2 = TA+ TG + 2TF (33)

where = b11 212 22 11 12 + 21 22cT and = b213 223cT . It is noticed that there isonly one in-plane shear strain 12 in Eq. (33). This is possible only after one uses the constraintsin Eq. (9). Moreover, the strain energy is independent of 3 so that the rotation about the normalonly appears algebraically, making it possible for it to be eliminated.

This simple constitutive model is rigorously reduced from the original 3-D model for multilayershells, each layer of which is made with an anisotropic material with monoclinic symmetry. Thevariational asymptotic method [16] is used to guarantee the resulting 2-D shell model to yield thebest approximation to the energy stored in the original 3-D structure by discarding all the insignif-icant contribution to the energy higher than the order of (h/l)2 and h/R. The stiffness matricesA and G obtained through this process carry all the material and geometry information throughthe thickness (see Eqs. (63) and (73) in Ref. [17] for detailed expressions). The term containingthe column matrix F is produced by body forces in the shell structure and tractions on the top andbottom surfaces, and it is very important for the recovery of the original 3-D results. Interestedreaders can refer to Ref. [17] for details of constructing the model in Eq. (33) for multilayeredcomposite shells.

Having obtained the 2-D constitutive law from 3-D elasticity, one can derive all the otherrelations over the reference surface of the shell, a 2-D continuum.

Virtual Strain-Displacement RelationsIn order to derive intrinsic equilibrium equations from the 2-D energy, it is necessary to express

the variations of generalized strain measures in terms of virtual displacements and virtual rotations.

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The variation of the energy expressed in Eq. (33) can be written as

=

1111 +

1212 +

2222 +

1313

+

2323 +

1111 +

+

2222 (34)

It is now obvious that one must express 11, . . ., 22, in terms of virtual displacements androtations in order to obtain the final Euler-Lagrange equations of the energy functional in theirintrinsic form. Following Ref. [24], we introduce measures of virtual displacement and rotationthat are compatible with the intrinsic strain measures. For the virtual displacement, we note theform of Eq. (18) and choose

q = Cu (35)

Similarly, for the virtual rotation, we note the form of Eq. (28) and write

= CCT (36)

where is a column matrix arranged similarly as the curvature column matrix in Eq. (4) =b2 1 3cT . The bars indicate that these quantities are not necessarily the variations offunctions. Using these relations it is clear that

u = CT q (37)

andC = C (38)

Let us begin with the generalized strain-displacement relationship, Eq. (18). A particular in-planestrain element can be written as

= eT

[C(e + u; + ku) e

](39)

Taking a straightforward variation, one obtains

= eT

[C(e + u; + ku) + C(u; + ku)

](40)

The right hand side contains u; and u;, which must be eliminated in order to obtain variations ofthe strain that are independent of displacements. These are needed to derive intrinsic equilibriumequations.

Premultiplying both sides of Eq. (18) by CT , making use of Eq. (36), and finally using a

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property of the tilde operator that, for arbitrary column matrices Y and Z, Y Z = ZY , one canmake the first term in brackets on the righthand side independent of u;. After all this, one obtains

C(e + u; + ku) = CCT (e + ) = (e + ) = (e + ) (41)

An expression for the second term in brackets on the right hand side of Eq. (40) can now beobtained by differentiating Eq. (37) with respect to x and premultiplying by C. This yields

C(u; + ku) = C(CT q); + Cku = q; + Kq (42)

Substituting Eqs. (41) and (42) into Eq. (40), one obtains an intrinsic expression for the variationof the in-plane strain components as

= eT

[q; + Kq + (e + )

](43)

where eT e vanishes when = . This matrix equation can be written explicitly as four scalarequations

11 = q1;1 K13q2 +K11q3 2131 + 12312 = q2;1 +K13q1 + 12q3 2132 (1 + 11)321 = q1;2 K23q2 + 21q3 2231 + (1 + 22)322 = q2;2 +K23q1 +K22q3 2232 123 (44)

The variations 12 and 21 should be equal due to Eq. (9); hence, one can solve for the virtualrotation component about B3 as

3 =q2;1 q1;2 +K13q1 +K23q2 + (12 21)q3 2132 + 2231

2 + 11 + 22(45)

It is now possible to write the variations of all strain measures in terms of three virtual displacementand two virtual rotation components as

11 = q1;1 K13q2 +K11q3 2131 + 12322 = q2;2 +K23q1 +K22q3 2232 123212 = q2;1 + q1;2 +K13q1 K23q2 + 2q3

2132 2231 + (22 11)3 (46)

with 3 taken from Eq. (45).

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Let us now consider the transverse shear strains

23 = eT3

[C(e + u; + ku) e

](47)

Following a procedure similar to the above, one can obtain the virtual strain-displacementequation for transverse shear strains as

23 = eT3

[q; + Kq + (e + )

](48)

Explicit expressions for the variations of the shear strain components are now easily written as

23 = q3; + + Kq (49)

Finally, variations of the curvatures are found. First, taking the straightforward variation of Eq. (28),one obtains

= C,CT

A C,C

T

A+ CkC

T + CkCT (50)

In order to eliminate C,, we differentiate Eq. (36) with respect to x

, = C,CT CCT, (51)

In order to eliminate C, we can use Eq. (38). Then, Eq. (50) becomes

= ; + K K (52)

Using another tilde identity (Y Z = Y ZZY ) one can find the virtual strain-displacement relationas

= ; + K (53)

In explicit form

11 =1,1A1

K132 + 123

22 =2,2A2

+K231 213

2 =1,2A2

+2,1A1

+K131 K232 + (K22 K11)3 (54)

where 3 can again be eliminated by using Eq. (45).

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Intrinsic Equilibrium EquationsIn this section, we will make use of the virtual strain-displacement relations in the variation of

the internal strain energy in order to derive the intrinsic equilibrium equations. Here we define thegeneralized forces as

11= N11

22= N22

1

2

12= N12

11=M11

22=M22

1

2

=M12

1

2

13= Q1

1

2

23= Q2 (55)

To use the principle of virtual work to derive the equilibrium equations, one needs to knowthe applied loads. In addition to the applied loads used in the modeling process, iBi at the topsurface, iBi at the bottom surface and body force iBi [17], one can also specify appropriatecombinations of displacements, rotations (geometrical boundary conditions), running forces andmoments (natural boundary conditions) along the boundary around the reference surface. It istrivial to apply the geometrical boundary conditions. Although it is possible in most cases thatnatural boundary conditions can be derived from Newtons law, the procedure is tedious and noteasily applied here because the physical meanings for some of the generalized forces are not clear.Thus, natural boundary conditions are best derived from the principle of virtual work.

Suppose on boundary (see Fig. 2), we specify a force resultant N and moment resultantM along the outward normal of the boundary curve tangent to the reference surface , N andM along the tangent of the boundary curve , N3 along the normal of the reference surface.Then the principle of virtual work (strictly speaking, the principle of virtual displacements) can bestated as:

s

( qifi m)A1A2dx1dx2

(Nq + Nq + N3q3 + M + M )d = 0 (56)

where fi and m are taken directly from [17].It is now possible to obtain intrinsic equilibrium equations and consistent edge conditions by

use of the principle of virtual work and the virtual strain-displacement relations derived in the

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previous section. The equilibrium equations are

(A2N11),1A1A2

+[A1(N12 +N )],2

A1A2K13(N12 N )K23N22 +Q1K11 +Q221 + f1 = 0

(A1N22),2A1A2

+[A2(N12 N )],1

A1A2+K23(N12 +N ) +K13N11 +Q112 +Q2K22 + f2 = 0

(A2Q1),1A1A2

+(A1Q2),2A1A2

K11N11 K22N22 2N12 + (12 21)N + f3 = 0(A2M11),1A1A2

+(A1M12),2A1A2

Q1(1 + 11)

Q212 + 213N11 + 223(N12 +N )M12K13 M22K23 +m1 = 0(A2M12),1A1A2

+(A1M22),2A1A2

Q2(1 + 22)

Q112 + 213(N12 N ) + 223N22 +M11K13 +M12K23 +m2 = 0 (57)

where

N = (N22 N11)12 +N12(11 22) +M2221 M1112 +M12(K11 K22)2 + 11 + 22

(58)

The associated natural boundary conditions on are

N = 21N11 + 212N12 +

22N22

N = 12(N22 N11) + (21 22)N12 NN =

21N11 + 212N12 +

22N22

N3 = 1Q1 + 2Q2

M = 21M11 + 21n2M12 +

22M22

M = 12(M22 M11) + (21 22)M12 (59)

where 1 = cos, 2 = sin, and is the angle between the outward normal of the boundary andthe x1 direction as shown in Fig. 2. The terms containing N stem from consistent inclusion of thefinite rotation from undeformed triad to deformed triad although the nonzero rotation about B3 isexpressed in terms of other kinematical quantities. Similar terms are found in the shell equationsderived by Berdichevsky [16] where only five equilibrium equations are derived.

In a mixed formulation, N can be shown to be the Lagrange multiplier that enforces Eq. (45).To further understand the nature of N one can undertake the following exercise: Setting Pi = 0and 12 = 21 for the equilibrium equations given in [13], N21N122 can be solved from Reissnerssixth equilibrium equation. This shows that Reissners N21N12

2is the same as our N , and Reiss-

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ners N21+N122

is the same as our N12. Finally, substitution of this sixth equation into the other fiveyields the five equilibrium equations given here in Eqs. (57). It is noted that Reissners equilib-rium equations are derived based on the basis of Newtons law of motion without consideration ofeither constitutive law or strain-displacement relations. However, the present derivation is purelydisplacement-based. The reproduction of those equilibrium equations by the present derivationillustrates that, as long as the formulation is geometrically exact, one can derive exact equilibriumequations.

A few investigators have noted an apparent conflict between the symmetry of the stress resul-tants and the satisfaction of moment equilibrium about the normal. In reality there is no conflict,but one must be careful. We have shown herein that the triad Bi can always be chosen so that12 = 21. If this relation is enforced strongly, there is only one in-plane shear stress resultant,N12, that can be derived from the energy. In that case the physical quantity associated with theantisymmetric part of Reissners in-plane stress resultants, while it is not available from the con-stitutive law, is nevertheless available as a reactive quantity from the moment equilibrium equationabout the normal. However, it must be stressed that the moment equilibrium equation about thenormal is not available from a conventional energy approach, in which the virtual displacementsand rotations must be independent.

In a somewhat similar vein, not being able to obtain the antisymmetric part of the moment stressresultants from derivatives of the 2-D strain energy is a result of the approximate dimensional re-duction process in which it was determined, based on asymptotic considerations and geometricallynonlinear 3-D elasticity, that the antisymmetric term 12 21 does not appear as an independentgeneralized strain measure in the 2-D constitutive law with correction only to the order of h/R.However, if a more refined theory with respect to h/R is required, then 12 21 would appear asa generalized strain in the 2-D constitutive law and a new generalized moment would be definedbased on the constitutive law.

For practical computational schemes, equilibrium equations and boundary conditions need touse the constitutive law to relate with the generalized 2-D strains. Finally a set of kinematicalequations is needed. Depending on how this part is done, the analysis can be completed in eitherof two fundamentally different ways: a purely intrinsic form, relying on compatibility equations,and a mixed form relying on explicit strain-displacement relations.

In the intrinsic form we have five equilibrium equations, Eqs. (57); six compatibility equations,Eqs. (12) (14); and the eight constitutive equations a total of 19 equations. The 19 unknowns arethe eight stress resultants, N11, N12, N22, Q1, Q2, M11, M12, and M22; and the 11 strain measures11, 212, 22, 213, 223, 11, 212, and 22, along with 13, 23, and 1221. The last three strainmeasures appear in the equilibrium equations but not in the constitutive law.

In a mixed formulation one would use the same five equilibrium equations and eight con-

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stitutive equations. One would also need a set of strain-displacement relations among the 11generalized strain measures 11, 212, 22, 213, 223, 11, 2, and 22, along with 13, 23, and12 21, and the five global displacement and rotational variables u1, u2, u3, 1, and 2. Onepossible set of such equations is as follows: use five of Eqs. (24), using either 12 or 21; use thesix Eqs. (31). There are also the two other rotational variables 3 and 3, which are governedby Eqs. (22) and (25), respectively. This way there are 26 equations and 26 unknowns. Thismixed formulation is capable of handling boundary conditions on 2-D stress resultants and dis-placement/rotation variables. At least in principle, one could recover a displacement formulationby eliminating all the unknowns except the displacement and rotation variables.

Eqs. (57) and (58) contain terms that could be disregarded because of the original assumptionof small strain. We will not undertake this simplification here, because it is out of the scope of thepresent study to actually implement the 2-D nonlinear theory. Therefore, our equilibrium equationsand kinematical equations are geometrically exact; all approximations stem from the dimensionalreduction process used to obtain the 2-D constitutive law.

The present work is a direct extension of [18] to treat shells. If one sets kij = 0 and A = 1, allthe formulas developed here will reduce to those in [18], which indirectly verifies that derivation.

ConclusionsA nonlinear shear-deformable shell theory has been developed to be completely compatible

with the modeling process in [17]. The compatibility equations, kinematical relations and equi-librium equations are derived for arbitrarily large displacements and rotations under the restrictionthat the strain must be small. The resulting formulae are compared with others in the literature.The following conclusions can be drawn from the present work:

1. The variational asymptotic method can be used to decouple the original 3-D elasticity prob-lem of a shell into a 1-D, through-the-thickness analysis [17] and a 2-D, shell analysis. Thethrough-the-thickness analysis provides both an accurate 2-D constitutive law for the nonlin-ear shell theory and accurate through-the-thickness recovery relations for 3-D displacement,strain, and stress. This way, an intimate relation between the shell theory and 3-D elasticityis established.

2. A full finite rotation must be applied to fully specify the displacement field. However, sincethe strain energy on which the formulation is based is independent of 3, the rotation aboutthe normal is not independent and can be expressed in terms of other quantities. Thus, itcan be chosen so that the two-dimensional, in-plane shear strain measures are equal. Thisway all the strain measures can be expressed in terms of five independent quantities: threedisplacement and two rotation measures, and only one stress resultant for in-plane shear canbe derived from the 2-D energy.

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3. Only five equilibrium equations are obtainable in a displacement-based variational formula-tion. Moment equilibrium about the normal is satisfied implicitly. If one does not includethe full finite rotation, but rather sets the rotation about the normal equal to zero, the cor-rect equilibrium equations cannot be obtained. This should shed some light on the nature ofdrilling degrees of freedom.

References1W. Yu. Variational Asymptotic Modeling of Composite Dimensionally Reducible Structures.

PhD thesis, Aerospace Engineering, Georgia Institute of Technology, May 2002.2A. Campbell. On vibration galvanometer with unifilar torsional control. Proceedings of the

Physical Society, 25:203 205, April 1913.3H. Pealing. On an anomalous variation of the rigidity of phosphor bronze. Philosophical

Magazine, 25(147):418 427, March 1913.4J. C. Buckley. The bifilar property of twisted strips. Philosophical Magazine, 28:778 785,

1914.5H. Wagner. Torsion and buckling of open sections. NACA TM 807, 1936.6A. E. H. Love. Mathematical Theory of Elasticity. Dover Publications, New York, New York,

4th edition, 1944.7B. Cosserat and F. Cosserat. Theorie des Corps Deformables. Hermann, Paris, 1909.8P. M. Naghdi. The theory of shells and plates. Handbuch der Pyhsik, 6 a/2:425640, 1972.

Springer-Verlag, Berlin.9J.C. Simo and D. D. Fox. On a stress resultant geometrically exact shell model. part i: For-

mulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering,72:267304, 1989.

10J.C. Simo and D. D. Fox. On a stress resultant geometrically exact shell model. part ii: Thelinear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering,73:5392, 1989.

11A. Ibrahimbegovic. Stress resultant geometrically nonlinear shell theory with drillingrotations-part i. a consistent formulation. Computer Methods in Applied Mechanics and Engi-neering, 118:265284, 1994.

12W. B. Kratzig. best transverse shearing and stretching shell theory for nonlinear finiteelement simulations. Computer Methods in Applied Mechanics and Engineering, 103(1-2):135160, 1993.

13E. Reissner. Linear and nonlinear theory of shells. In Y. C. Fung and E. E. Sechler, editors,Thin Shell Structures, pages 29 44. Prentice Hall, 1974.

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14J. N. Reddy. Mechanics of Laminated Composite Plates: Theory and Analysis. CRC Press,Boca Raton, Florida, 1997.

15A. Libai and J. G. Simmonds. The Nonlinear Theory of Elastic Shells. Cambridge UniversityPress, 2nd edition, 1998.

16V. L. Berdichevsky. Variational-asymptotic method of constructing a theory of shells. PMM,43(4):664 687, 1979.

17Wenbin Yu, Dewey H. Hodges, and Vitali V. Volovoi. Asymptotic generalization of reissner-mindlin theory: Accurate three-dimensional recovery for composite shells. Computer Methods inApplied Mechanics and Engineering, 191(44):49715112, 2002.

18D. H. Hodges, A. R. Atilgan, and D. A. Danielson. A geometrically nonlinear theory ofelastic plates. Journal of Applied Mechanics, 60(1):109 116, March 1993.

19J. G. Simmonds and Donald A. Danielson. Nonlinear shell theory with finite rotation andstress-function vectors. Journal of Applied Mechanics, pages 10851090, December 1972.

20Thomas R. Kane, Peter W. Likins, and David A. Levinson. Spacecraft Dynamics. McGraw-Hill Book Company, New York, New York, 1983.

21D. H. Hodges. Finite rotation and nonlinear beam kinematics. Vertica, 11(1/2):297 307,1987.

22E. Reissner. On finite deflections of anisotropic laminated elastic plates. International Jour-nal of Solids and Structures, 22(10):1107 1115, 1986.

23E. L. Axelrad. Theory of Flexible Shells. Elsevier Science Publishers, North-Holland, 1986.24D. H. Hodges. A mixed variational formulation based on exact intrinsic equations for dy-

namics of moving beams. International Journal of Solids and Structures, 26(11):1253 1273,1990.

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List of Figure Captions

Figure 1: Schematic of shell deformation

Figure 2: Schematic of an arbitrary boundary

21 OF 23

Undeformed State

Deformed State

O

B1 1 2

( , )x x

B2 1 2

( , )x x

r( , )x x1 2

R( , )x x1 2

u( , )x x1 2

b2 1 2

( , )x xb3 1 2

( , )x x

b1 1 2

( , )x x B3 1 2

( , )x x

FIGURE 1: 22 OF 23

G2x

1x

n

t

f

FIGURE 2: 23 OF 23