Non Linear Shells

N O N L I N E A R G E O M E T R I C AND M A T E R I A L B E H A V I O R

OF S H E L L S T R U C T U R E S W I T H L A R G E S T R A I N S

By S. A. Schimmels I and A. N. Palazotto, z Fellow, ASCE

ABSTRACT: This paper presents an approach for a general laminated shell geometry describable by orthogonal curvilinear coordinates. The theory includes through-the-thickness parabolic distribution of transverse shear stress, transformation of Cauchy stress-strain relations into Lagrangian coordinates, and a layered elastic-plastic analyses. Additionally, a simplified approach that allows large displacements and rotations is incorporated. The theory is cast into a displacement- based finite-element formulation and then specialized to cylindrical and spherical geometry. The theory is then applied to the isotropic and transversely isotropic laminated shells. Results incorporating the Cauchy-Lagrangian transformation with through-the-thickness strain show a slightly more flexible response than in published results that are based on inextensible assumptions. These problems also indicate that the usual locking associated with shell elements is eliminated.

INTRODUCTION

The nonlinear behavior of transversely loaded isotropic and composite shell panels has been studied by Alwar and Narasimhan (1991), Dennis and Palazotto (1990), Muc (1992), Palazotto and Dennis (1992), Parish (1991), Simo and Kennedy (1992), Tsai and Palazotto (1990, 1991), Yang and Wu (1989), and Yuan and Liang (1989). Analytical models using shell finite elements with Donnell approximations, combined with a layered elastic- plastic material model, have been able to predict the shell response to transverse loading accurately up to local rotations of approximately 25 degrees (Paris 1981; Yuan and Liang 1984). The purpose of this paper is to apply a transformation between Cauchy and Lagrangian coordinate systems considering isotropic and composite shells under transverse loading while considering elastic and elastic-plastic material behavior. The approach includes several features of composite shell behavior. Through the thickness, parabolic transverse shear distribution is assumed, similar to that proposed by Reddy (1984). Additionally, a simplified large rotation capability is included. A modified von Mises yield criteria is incorporated for the elastic- plastic analysis, and a large-strain formulation is included in the material model. The resulting finite element is spherically or cylindrically shaped, thus capturing shell bending-membrane coupling.

THEORETICAL APPROACH

General Although nonlinear shell theory can be based entirely on surface defi-

nitions, transverse shear strains and stresses are not easily included. One way to define strain-displacement relations that can easily incorporate

1Ph.D. Candidate, Dept. of Aeronautics and Astronautics, Air Force Inst. of Technol. (AU), WPAFB, OH 45433.

2Prof., Dept. of Aeronautics and Astronautics, Air Force Inst. of Technol. (AU), WPAFB, OH.

Note. Discussion open until July 1, 1994. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on August 3, 1992. This paper is part of the Journal of Engineering Mechanics, Vol. 120, No. 2, February, 1994. �9 ISSN 0733-9399/94/0002-0320/$1.00 + $. 15 per page. Paper No. 4533.

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three-dimensionality (3D) is to specialize the general 3D strain-displacement relations expressed in arbitrary orthogonal curvilinear coordinates. Follow- ing the development in Soada (1989) and Washizu (1982), the physical strains, eij, are then found from

Y6 ei~ = h~j (no sum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where hi = scale factors defined from the metric tensor of the undeformed coordinate system; and -& are shown in (32), where ui = coordinates of the displacement vector, u. Consider a shell geometry that is described by orthogonal curvilinear coordinates at the shell mid-surface: ~ and (2, surface normal (, and the radii of curvature, R~ and R2 (Saada 1989; Washizu 1982). For this geometry, the scale-factor terms of (1) are given by

h, = %(1 - (/R~) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2a)

h2 = a2(1 - (/R2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2b)

h3 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2c)

where % (y = 1, 2) are related to the elements of the surface metric. For a body of volume V in equilibrium with prescribed forces on a part

of its surface, S~, and prescribed geometric boundary conditions on the remaining surface, $2, that has undergone an infinitesimal virtual displacement, 8u, we have

fv (cruSeu - ekSu,) dV - fs, FkSUk dS = 0 . . . . . . . . . . . . . . . . . . . . . (3)

where or u = elements of the second Piola-Kirchhoff stress tensor; su are defined in (1); Pk = components of prescribed body forces; and F~ are components of prescribed surface traction (Washizu 1982). Assuming the existence of a strain energy density function, W*, smallincremental strain deformation, and a incrementally linear constitutive law, the principle of virtual work is stated as

8IIp = fv W*(ui) dV - fs Fi dui dS = 0 . . . . . . . . . . . . . . . . . . . . . . . . (4) 1

where W*(ui) = e . e = 1/2C~jk~e~jejk, and Qjkl Cauchy elastic constants. In (4), the strain energy density is written in terms of the displacement

components by using (1). The integrals are taken over the original, undeformed volume and surfaces consistent with a Lagrangian viewpoint. Ad- ditionally, the array of elasticity constants, E C~jkl, are defined with respect to the material axis. All of the necessary tools have been introduced for a Lagrangian displacement-based energy approach to solve elastic geometrically nonlinear shell problems. To develop shell strain-displacement relations, one should first identify, respectively, the general 3D curvilinear coordinates y~, Y2, and Y3 of (1) with the shell coordinates ~ , ~2, and (. Then, by defining kinematics that relate the middle surface displacements to the continuum displacements of the shell, strain-displacement relations are written, and finally, the total potential energy expressed in terms of displacement, using (4), is extremized with respect to displacement, resulting in the equilibrium equations. The definition of the kinematics for these types of formulations is obviously very important. The assumptions that will

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eventually lead to a general shell kinematics and a potential energy expression are next discussed.

Basic Assumptions A geometrically nonlinear theory governing the shell is based on the

following assumptions:

1. The shell is comparatively thin and we therefore may assume the transverse normal stress is approximately equal to zero. This assumption effectively reduces the generally 3D behavior of the shell so that it can be described by the behavior of only a datum surface; this results in a two- dimensional (2D) approach.

2. The transverse shear stress distribution is parabolic through the shell thickness and vanishes on the top and bottom surfaces of the shell. Because the shell is relatively thin, in-plane stress tends to dominate the shell response under loading, i.e. the transverse stresses are of lesser importance. However, it is well known that laminated thin flat plate response can be significantly influenced by transverse shear deformation. Similar results have been found for laminated shells (Dennis and Palazotto 1989; Libresca 1987). Therefore, transverse shear stress is included in the present model, via a 2D approach, whereas transverse normal stress is not. This approach is consistent with remarks made by John (1965) and Koiter (1967), who per- formed order-of-magnitude studies on the stresses in general shell geome- tries. Additionally, their results are seen to hold generally true in the elasticity solutions of both laminated flat plates by Pagano (1969, 1970) and laminated shells in cylindrical bending by Ren (1987).

3. Exact Green's strain-displacement relations are assumed for the in- plane Ell , E22 , and e12 of (1) and (2) and linear strain-displacement relations are assumed for the transverse strains E23 and ex3.

4. As the shell is assumed to be thin, the normals to the middle surface are approximately inextensible, hence the transverse displacement is constant through the thickness. Consistent with the preceding assumption, re- tention of only linear strain-displacement terms for e.g. ez3 yields e33 = 0.

5. The shell is assumed to consist of perfectly bonded layers. Each layer may enter the plastic regime independently from other layers (Owen and Hinton 1980; Parish 1981; Yang and Wu 1989). However, since each layer is relatively thin, plastic strains only occur in-plane. The transverse shear strains are assumed to remain elastic. Librescu (1987), as mentioned, develops general shell equations based upon varying approximations in the magnitudes of both strain and the rotation differential elements undergo during deformation. His consistent small strain, moderate rotation theory has nonlinear (not exact) strain-displacement relations for the in-plane strains yet linear relations for the transverse shear strains. The present approach is therefore viewed as a large strain (plasticity), simplified large rotation theory due to the exact in-plane strain assumption. The accuracy in rotation is limited by the linear (in displacement) assumption on the transverse shear strains.

CONSTITUTIVE RELATIONS

Elastic Constitutive Relations Materials consisting of unidirectional fibers embedded in a matrix are of

interest in this study. For fibers aligned with the 1 material axis, we can

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assume that the material is transversely isotropic (engineering constants E2 = E3 and v12 = v13) with respect to planes parallel to the 2-3 plane. In this case, applying assumption 1, the constitutive relation used in (4) becomes

{~i} = [QiJ{ej} i, j = 1, 2, 4, 5, 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where Qq = Cij - (Ci3CjJC33) , and for transverse isotropy it is easily shown that Qla = Ex/A; Q12 = v21E2/A; Q22 = E2/A; Q66 = G12; Q44 : G23; Qss = G13; and A = 1 - ]2121)21.

We now consider a shell that is constructed of layers of the transversely isotropic material described by (5). Generally, the fibers of the kth individual layer or ply are oriented at an angle with respect to the middle surface shell coordinates ~1 and {2. Therefore, the constitutive relations of (5) for that ply must be transformed into shell coordinates, resulting in

; 2 / = Q_ 6/ = . . . . . . . . . . . . . . . . . (6o Q66J ~ ~6

-1 {t {} 0"4 = 044 Q_45 I~ 4 = [C, ]~ e4 . . . . . . . . . . . . . . . . . . . . . . ( 6 b )

where Qq (i, j = 1, 2, 6) and Qm, (m, n = 4, 5) are elements of symmetric arrays of transformed stiffnesses for the kth ply and ~i, ej, 0"m, and ~, are measured with respect to shell coordinates {~ and ~ (Dennis and Palazotto 1990).

Elastoplastic Constitutive Relations Mindlin plate bending relations are formulated from Owen and Hinton

(1980) to include transverse shear and plasticity using multiple layers. This layered approach is used to capture the spread of plasticity over the depth of the plate, and can be extended to shells. Thus, the shell considered herein is divided into a number of layers, each of which may become plastic sep- arately. As the number of layers is increased, the model provides a more realistic representation of the gradual spread of plasticity over the shell cross section. Owen and Hinton (1980), Parish (1981), and Yuan and Liang (1989) suggest that a minimum of eight layers are needed.

The assumption is that as soon as the stress in the middle of the outer layers reaches the yield value, then the outer layers become plastic, while the rest of the layers remain elastic, as shown in Figs. l(a and b). Then as more stress is applied, more layers become plastic, until the whole cross section becomes plastic [see Figs. l (b) - l (d) ] .

After initial yielding, the material behavior will be partly elastic and partly plastic. During any increment of strain, the complete elastoplastic incremental stress-strain relation is written as

{d~} = [Ce.]~{d~} = de} . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where de = de e + de p = sum of the elastic and plastic components; and [Cep] k = elastoplastic constitutive relations. It should be noted that because the shell is thin, it is assumed that the transverse shear strains remain small and thus never enter the plastic regime. Using the associated flow rule and

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<0o

<0o

~a

Oo

Oo

( c ) ~ (d)

FIG. 1.

O' ~ 0 ' ~

O'Q

/

Yielding of Layered Shell (Owen and Hinton 1980)

Prandtl-Reuss theory (Hill 1950; Kachanov 1971; Owen and Hinton 1980), [(Cep)i] k is defined by

[ ( c . ) 3 ~ = [ g l ~ dDd~

n + d~{a} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8a)

where

do = [Cf]{a} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8b)

OF OF )r [ OF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8c) {a} = Leo-i, 00.22, 00"12

To calculate the [(Cep)y] k matrix, the explicit form of the scalar term A and the flow vector {a} must be determined. From the work-hardening hypothesis and yield function (Owen and Hinton 1980), scalar A is determined to be H' , which is the local slope of the effective stress-plastic strain curve and can be determined by

d6- H' = - - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9) dgp

The flow vector {a} can be determined from the selected yield criterion and then used to define the yield surface. For the purposes of this paper, the writers are only considering layered isotropic materials. The plastic potential function, F(0-), for isotropic material is assumed to follow the von Mises yield criteria given by

v~ v~ F(0-) = 8 2 = ~ 0-i'/0-i'j = - ~ [(0-a - 0-2) 2 + (0-2 - 0-3) 3 + (0-3 - o-a) 2]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

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LARGE-STRAIN FORMULATION

According to Washizu (1982), if one defines a local Eulerian coordinate system (x~, x2, xa) at a point P~~ before deformation, one can formulate the transformation law between the Green's strain tensor and Cauchy strain tensor, and the second Piola-Kirchhoff stress tensor and the Cauchy stress tensor (see Figs. 2 and 3). Let the strain tensor defined with respect to the

X 3 (N) O

o13

o~2 +A o12

L

Xt

X2

FIG. 2. Definition of Kirchhoff Stress Tensor (Washizu 1982)

X 3

fl B

% #

pe,§

~+I) B E

o13 + A o is

X �9 2

C ~ X 1

FIG. 3. Definition of Cauchy Stress Tensor (Washizu 1982)

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local Eulerian coordinates be denoted by ex~. Then the Eulerian (Cauchy) strain components defined with respect to the general Lagrangian coordinates are given by

OxK Oxp e ~ = OXx dX~ eK~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

In (11), the Ox~/OX~ terms are elements of the Jacohian matrix for transformation between Eulerian and Lagrangian coordinates.

A similar transformation between the Cauchy stress tensor, ff~, and the second Piola-Kirchhoff stress tensor, a~0 , is given by

%E _ 10X{ , OX~ (12) D Ox~ Ox~ {rx~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where

.... 0u~ oxo = x ,~ . , = 8, . , + u,~., = 8~,., + - - . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 3 ) a x v ' Ox~,

a n d

D - o(X, , X~, X~) 0(X1 ' X2, X3 ) = de t (Xi4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

When the strain is sufficiently small, the material and structural axis systems are assumed to be colocated. Thus, the material is assumed to be linearly elastic and the strain tensor is related to the stress tensor by

~ , = C ~ , e k , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 5 )

where C~k, = elastic stiffness constants in the Eulerian coordinate system [see (5)]. However, when the strains are no longer insignificant, the material and structural axis systems no longer coincide and the relationship between stress and strain becomes complex. The relationship is given by

1 1 [Tz]{e} (16) [ T , ] { , ~ } = [ C q ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and rewriting yields

1 {{r} = ~ [T1]-'[Ce][T2I{e} = [CL]{~} . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

where [C L] = Lagrangian constitutive matrix, and

I (1 + uia) 2 (ua2) z 0 0 2(1 + ula)ul,2 /

U~ 2 0 0 2U2,1(1; U2,2) ,1 (1 +0U2,2) (1 + U2,2) J [Td = u2a

0 ua, 2 (1 Jr- Ul,1) 0 L_(1 + ula)u2,1 u1,2(1 + u2,2) 0 0 (1 + u,a)(1 + u2,2) + /'/1,2/'/2,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

L (I + u2,2) 2 u{2 0 0 -2u12(1 + u2,2) q

ui,l (1 0 0 - 2 ( 1 0 u~a)uza / +0 u~1)2 (1 "]- Ul,1) -u2aD

IT2] 00 0 -Ul,2D (1 q- u2,2)D /'/2.t(1 + //2.2) --(1 Jc UI.1)Ul.2 0 0 (1 + //1,1)(i + U2,2) + //I,2U2.1J

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

3 2 6


D = (1 + u,.0(1 + u2.2) - UlaU2., . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

Thus, it is necessary to determine ITs], [T2], and D in order to generate the Lagrangian constitutive matrix, [CL]. The assumption is made that the transformation of the material axis of the shell mid-surface applies to all layers of the shell.

KINEMATICS

Consider the following kinematics for the arbitrary shell described by orthogonal curvilinear coordinates, referring to Saada (1989) for conven- tions:

/'/1(61, 62, ~ ) = / . / ( 1 - ~11) -~ ~Jl ~- ~20}11 Jr- ~3yl n t- ~401 . . . . . . . . (21a)

/ U2(~1, ~2, ~)"=~ "U ~ 1 - ~22)"3t-~1.~.12-{ - ~2~b2 "1-~3y2-~-~402 . . . . . . . . . (21b)

u3(6,, {2) = w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21c)

where u, v, w, q,~, +~, %, and 0a = functions of 61 and {2; 0~ = rotations of the normals and +~, %, and 0~ = functions to be determined such that ~4 and ~5 vanish on the top and bot tom surfaces of the shell.

The assumed displacements of (21) are slightly more involved than those introduced for flat plates by Reddy (1984). For a plate, only third-order terms in the thickness parameter , {, were necessary to give the desired parabolic transverse shear stress distribution. Shell structures, because of their curved surfaces, have coupling between displacements that plates do not and the fourth-order terms become necessary. The kinematics will ul- timately closely resemble the plate kinematics that have been used in the literature. Additionally, Reddy and Liu (1985) arrived at identical general shell kinematics as derived here but used Sanders ' shell relations.

Following the developments in Dennis and Palazotto (1989) and Palazotto and Dennis (1992), the kinematics of (22) represent the continuum displacements of the shell, where the transverse shears are zero on the lateral surfaces and are parabolic through the thickness. Thus

u,(~, , 62, ~) = u 1 - ~ + { t~ + {3k 01 + . . . . . . . . . . . (22a)

u3(6,, 62) = w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22c)

where k = - 4/3h 2. The rotation of the normal, 0~, is a result of bending deformation and

therefore, the slope of the elastic curve, w~, is always a combination of 0~ and shear, [3~ (Palazotto and Dennis 1992). For a flat plate, ~/R~, = 0 and % = 1, (22) reduces to that used in the literature.

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SHELL STRAIN-DISPLACEMENT RELATIONS

The kinematics of (22) next define the general shell strain-displacement relations used in (32), as given in (23)

g~ = go + 4pX~p; i = 1, 2, 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23a)

gi = go + 4pX~p; p = sum 1 - 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23b)

g,,, = go + ~2• m = 4, 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23c)

where eo, X i l - - Xi7, go, Xm2 are shown in (33)-(40) , specialized for spherical coordinates. The first subscript of • refers to the strain component 1, 2, or 6, and the second subscript refers to the power of ~ it multiplies. The go and the • - • terms consist of displacement expressions that are dependent only on the surface parameters ~1 and ~2.

SHELL POTENTIAL ENERGY

From (4), let

Hp -- U + V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

where U = internal strain energy; and V = work done by external forces. From the development in Palozotto and Dennis (1992), the internal strain energy of the arbitrary shell can be developed in terms of in-plane energy, out-of-plane energy and through-the-thickness energy, which takes on the form convenient for finite-element development. The functions can be considered as

gl = 2 (/AI -~ 1"/2 -[- I'/3) df~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

and

if. U 2 = ~ ( e~176 4- 2g~ q- Xn2Xm2Fmn) d• . . . . . . . . . . . . (26)

[Am"' Omn' Fmn] = fh Qm,[1, 42, 44] a4 . . . . . . . . . . . . . . . . . . . . . . . . . (27)

where expressions of U1 are of the form

[Aq, Bij, Dq, Eq, F~, Gq, Hij, I~j, Jq, Kq, Lq, Pij, Rq, S,, Tq]

= fh Oij[l"~ ~' 42, ~3, ~4, 45, 46, 47, 48 49 ~10 411, 412 413, ~14] d4 . . . . (28)

[see (41)-(42) for more details]. For the case of plasticity, or the large- strain formulation, (25)-(28) are mo_dified by replacing t_he elastic Qq matrix with the appropriate elastic-plastic Q~e or large-strain Q} matrix.

Primarily because of the assumed nonlinearity, many new elasticity arrays are developed. The energy terms that contain the higher-order elasticity constants are also multiplied by powers of k = - 4 / 3 h 2 [see (22)]. Because of the squared shell thickness term in the denominator of k, the energy terms are not as many orders of magnitude apart as (26) may at first indicate. For example, Dq, Fq, and Hq energy terms are all the same order of mag-

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nitude in h or ~ despite four orders of magnitude difference in the definitions of the elasticity arrays given in (26).

In a general geometrically nonlinear analysis, the energy due to the ex- ternally applied loads contains nonlinear displacement terms. These arise when, because of large rotations, the applied loading develops higher-order coupling components in coordinate directions where no loading originally existed. As discussed by Brush and Almroth (1975), these higher-order terms give potential energy contributions that are negligibly small for the cases of intermediate nonlinearity, i.e. moderate rotations. However, the large rotational case generally must account for them. If the loading is actually a prescription of displacement, the higher-order nonlinear loading terms need not be included, as applied loading is zero. This alternative simpler approach is taken here.

Thus, (25) and (26) with (23), (27), (28), and (33)-(42) give the total potential energy for of the arbitrary shell, where a specific shell geometry is defined by specifying the scale factor terms in (2). This expression is then extremized with respect to displacements, giving the equilibrium equations. The result of the extremization is five coupled nonlinear partial differential equations. The present approach gives nonlinear extensional terms, i.e. Aq, as well as nonlinear terms in Dij through T~j due to the nonlinearity in the curvature terms (Xip) of (23). The moderate rotational theories retain, in the strain-displacement relations, nonlinear displacement terms only in the transverse displacement, w, as a result of assuming that the rotations are small compared with unity (Novozhilov 1953). This results in nonlinearity in the equilibrium equations in the extensional Aij terms only.

FINITE-ELEMENT SOLUTION

The variation of the potential energy, Hp, of the discretized domain gives coupled nonlinear algebraic equations where the unknowns are the values of the displacement at the nodes. In a generally nonlinear analysis, these equations are not usually solved directly, but instead, are linearized and solved by incremental/iterative methods. The linearized equations are found by effectively carrying out an additional differentiation of lip with respect to the displacements. The manipulations that follow give a convenient form

FIG. 4. 36-DOF Rectangular Shell Element

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for lip such that variation and linearization are straightforward to accom- plish. Afterward, a 36-degree,of-freedom (DOF) curved element (Fig. 4) is discussed.

Element Independent Formulation The shell domain is discretized such that the continuum displacements

are approximated by nodal values and interpolation functions. The displacement gradient vector d is defined based on the unique displacement terms of the strain-displacement relations. As seen in Dennis and Palazotto (1989) and Palazotto and Dennis (1992), the potential energy is then given for the discretized domain in (29). The first variation of the energy, ~IIp, gives the equilibrium equations F(q), shown in (30).

13/, = -~- [DI r g + -~- + [D l df~q - q r R . . . . . . . . . . . . . (29a)

lip = -~- K + ~ + d•q - qrR . . . . . . . . . . . . . . . . . . . . . (29b)

~llp = aqr [Dlr K + _~_ + [D l dOq - aqrR . . . . . . . . . . . (30a)

glIp = gqr K + -~- + daq - gqrR = gqrF(q) --- 0 . . . . . (30b)

where [D] = an array of interpolation functions and their derivatives; R = a column array of the nodal loads; K = an array of constant stiffness coefficients; N1 = an array of stiffness coefficients that are linear in displacement; N2 = an array of stiffness coefficients that are quadratic in displacement. The symmetric 18 x 18 arrays, K, NI, and N2 are defined from (29) and shown in (31) using a generalization of the approach described by Rajasekaran and Murray (Dennis and Palazotto 1989; Palazotto and Dennis 1992)

g = f~l [Dlr/~[DI dlI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31a)

N1 = [DIT [DI d a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31b)

N2 = fn [D]~N2[D] dO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31c)

For the case of plasticity or large-strain, the Cijp+ term is replaced with the appropriate CTp~§ or C~ +. Thus, (29) and (30) are simply redefined for the appropriate m~terial m~deled.

VALIDATION OF APPROACH

The approach is verified by comparing present results to those due to Sabir and Lock (1972), Argyris et al. (1980), and Gould (1988). Sabir and Lock analytically traced the equilibrium path of an isotropic cylindrical panel subjected to a transverse point load. A 4 x 6 mesh (longitudinal by cir-

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16e.ee p 0b)

2 e . ~

E 0 . ~

' l e . ~

,,,~, Numed~ - "~mt'ormad~ I I , l l l l l I l l l l l l l l t l l l I I I I I 1 , 111111111111 I 1 , 111111111 ,11111 I

~.i 0 ~,20 ~.4~ ~.(~ ~.8~ 1.00 1.2e

w(~-)

FIG. 5. Point-Loaded Isotropic Cyl indrical Shell Response with Material Analysis

4e.ee.

3e.ee.

lS.ee-

B . ~ O.

- 1 8 . ~

P ~ h

x

P

. . . . . . . . . . . . . . . . . . . . .

w (in)

FIG. 6. Point-Loaded Isotropic Spherical Shell Response

cumferential) modeled one quadrant of the panel due to the point load [E = 4.5e6 psi (3.103e10 Pa); v = 0.3; R = 100 in. (2.54e3 ram); h = 0.5 in. (12.7 mm); L = 20 in. (508 mm); 0 = 0.1 rad; ~/B = 0.05; longitudinal edges hinged; circumferential edges free]. Fig. 5 shows the transverse response of the center of the panel due to the point load. As can be seen, the present approach duplicates the referenced results beyond collapse. Due

331


to the shallowness of the shell (~/B is small), small rotations occur (20 0.2 rad). Thus, with the isotropic material and small rotations, the large strain formulation model has a negligible effect to the global response of the shell. Argyris et al. (1980) analytically traced the equilibrium path of an isotropic spherical panel subjected to a transverse point load. A 6 • 6 mesh modeled one quadrant of the panel [E = le7 psi (6.985e10 Pa); v = 0.3; R = 4.76 in. (120.9 mm); h = 0.01576 in. (0.4003 mm); L = 1.811 in. (46.0 mm); 0 = 0.1902 rad; ~/B = 0.0474; edges hinged]. Fig. 6 shows the transverse response of the center of the panel due to the point load. The numerical results are more flexible due to Argyris et al. (1980) incorporating only classical Donnell shell relations for a four noded element. The finite-element model used herein has eight nodes (for quadratic approximations of u and v) and has the complete Green's strain terms for the in- plane strains. To verify the large-strain formulation for laminated anisotropic materials, present results were compared to Gr/PEEK[ _ 45]4. tensile coupon experimental results (Gould 1988).

Gould (1988) experimentally traced the equilibrium path of a 16-ply Gr/ PEEK tensile coupon that underwent large fiber rotations (~18~ A 2 • 20 mesh modeled the entire coupon [El = 1.95e7 psi (1.362ell Pa); E2 = 1.55e6 psi (1.07el0 Pa); v12 = 0.31; G12 = G13 = 8.125e5 psi (5.602e9 Pa); G23 = 6.5e5 psi (4.482e9 Pa); a = 1.0 in. (25.4 ram); b = 10.0 in. (254 mm); h = 0.084 in, (2.134 mm)]. Fig. 7 shows the load-displacement response for the numerical model without and with the constitutive transformation. The numerical model without the constitutive transformation begins to deviate from the other model when the global displacement is approximately 15 times the coupon thickness. This model overestimates the experimental failure load by =9%. The model with the constitutive transformation is 25% above experimental results. As can be seen, the present approach models the experimental results beyond the linear regime.

EO00.~,

3 ~ . 0 0 ,

:2000.~,

1 ~ . ~

0 . ~ 0.

FIG. 7.

X " * b/2:

$ " *8/2:

/ ;

j / ..-;:. 2"-,'Z"

;~ . . . . . ~ , . ~ . . . . . i : ~ . . . . . i : ~ . . . . . ~,,.~ . . . . . ~,,.~ D i s p l a c ~ ( ~ )

u - v - w - w , , - w ~ - p l - ~ 2 - 0

free

E 1 == 1.95 x 107

E 2 - 1.55 x 10 s

Glz ,, 0.8125 x 10 e

Gla = 0.8125x 10 =

G . - o.e5 x lo '

Axially Loaded [-+ 4514s GR/PEEK Tensile Coupon Response

332


RESULTS

Isotropic Cylindrical Shell with Transverse Load An isotropic cylindrical shell, similar to the one shown in Fig. 8 [h =

0.25 in. (6.35 mm)] is analyzed. The results are shown in Fig. 8. The equilibrium path is significantly changed due to the reduced shell thickness. As before, small rotations occur (i.e. 20 ~ 0.2 tad). Thus, due to the isotropic material and the small rotations, the large strain formulation model has a negligible effect on the global response of the shell.

lsotropic Deep Cylindrical Arch with Point Load Larger displacements and rotations can be examined by studying deep

circular arches. The simply supported arch subjected to a center point load is similar to the previous hinge-free cylindrical shell but is simpler since the arch is effectively only a one-dimensional problem. Consider a deep arch with the following geometrical quantities: R = 100 in. (2,540 mm); h = 1.0 in. (25.4 ram); 0 = 1 rad; and ~ / B = 0.50. For this case, ~ / B and 0 are over 10 times larger than the cylindrical shell values. The boundary conditions and loading will result in a symmetric response where the arch crown displaces only radially. Results based on Donnell approximation, full nonlinear in-plane assumptions, and the large-strain formulation are shown in Fig. 9. The full nonlinear model is significantly more flexible than the Don- nell approximation. This is attributed to the higher-order representation of the deformation of the midsurface of the arch. Aimroth and Brogan (1980) saw a similar effect compared to the inextensible solution of a deep arch. The transverse shear degrees of freedom will have a small effect but will also increase the displacement under load compared to the Donnell (1992). The large-strain formulation model shows even greater flexibility than the

/~o.- "

p /~'"

4@.@@~

::~. @@

2@.@@

1@.@@

@.@@ @,

-1@.@@

-L::~.@@- O

..~Ur ~ ~ck - ~ . ~ " o o o o o N U n l ~ n c l l

O �9 �9 ~ N u r n e d c ~ l - T rans fo rmat ion

- ~ . ~

FIG. 8. Point-Loaded I$otropic Cylindrical Shell Response with Material Trans- formation Analysis

333


I Y

t2ze.m P (Ibs)

X

8~e.6e

Numer,caJ Transformation e. ~'"' ~: ~""1'~:~'"i~:~'"~.'~" ~ :~ ' "~ ' :~

WOn)

FIG. 9. Deep Simply Supported Circular Arch Crown Displacement versus Load with Material Transformation Analysis

full nonlinear model. This is due to the large local rotations occurring in the deformation of the arch. At the collapse load point, A, over 26% of the surface of the arch is undergoing rotations of 22 ~ (0.384 rad) or better. As the load increases, so do the local rotations. At the end of the analysis, a maximum local rotation of 39.5 ~ (0.689 rad) occurs and 34% of the arch surface has rotations >28 ~ (0.489 rad). Due to these large rotations, the material axis is beginning to move away from the structural axis and the assumption of Cauchy (Eulerian) material modeling no longer is valid.

Laminated Composite Cylindrical Shells with Transverse Load The theory is next applied to the shell geometry shown in Figs. 10 and

11 with the following values: E1 = 20.466e6 psi (1.411ell Pa); E2 = 1.34e6 psi (9.239e9 Pa); l , '12 = 0.313; Glz = G13 = 8.63e5 psi (5.95e9 Pa); Gz3 = 5.18e5 psi (3.57e9 Pa); R = 100 in. (2,540 ram); h = 0.04 in. (1.106 ram); L = 10 in. (254 mm); 0 = 0.1 rad; ~/B = 0.05; longitudinal edges hinged; circumferential edges free). Fig. 10 shows the transverse response of the center of the panel due to a point load for [0]8 and [0,0,90,90]s ply layups with and wihout the large strain formulation model. For the [0]8 ply layup, the large strain model had no effect on the global response of the shell. Since the fibers are longitudinal in orientation, they add little stiffness to the shell (under a transverse load) because of no coupling with the shell's curvature (in the 90 ~ direction). Combining this with the shallowness of the shell (~/B < 0.1), the material axis undergoes little rotation for the analysis. Only 9% of the shell underwent rotations larger than 10.8 ~ (0.195 rad). The same applies for the [0,0,90,90]s ply layup. Although the shell's stiffness is increased (due to the 90~ fibers), the 90 ~ layers are near the mid- surface of the shell and the coupling effect of fiber orientation and shell curvature is minimized. Thus, the large strain model only decreases the collapse load by 4% where rotations of 20 ~ (0.349 rad) occur over 18% of

334


8 . 0 0 .

6 . 0 0 �9

4.00,

2.00

O.00 0

-2.00

-4~

FIG. 10.

P (Ibs)

W (in)

[0,0,90,90], with Trans format ion

Point Loaded Composite Cylindrical Shell Response with Material Trans- formation Analysis [0], [0/90]

,0.0oe~176176176 P (lbs)

2 0 . 0 0

0.00 0 0.20 0.40 0.60 0

W (in) -~o.oo [90]8 _ ii ~:~ LOOLg0,0,0J. -40 .00 "" " [90J8 w i t h T r a n s f o r m a t i o n

* ** [90,90,000J= w i t h T r a n s f o r m a t i o n

-80.00

FIG. 11. Point-Loaded Composite Cylindrical Shell Response with Material Trans- formation Analysis [90], [90/0]

the shell. Fig. 11 shows the transverse response of the center of the panel due to a point load for [90]8 and [90,90,0,0]~ ply layups are modeled with and without the large strain formulation model. Here the coupling effects of fiber orientation and shell curvature, combined with the large-strain

335


formulation model are enhanced. For the [90]8 model, the peak collapse load is decreased by 6% as 15% of the shell underwent rotations larger than 20 ~ Similar results occurred for the [90,90,0,0], model. The peak load is decreased by 8% and over 26% of the shell underwent rotations larger than 20 ~ . It should be noted the first peak in the load-displacement curve represents localized snapping occurring in the center of the shell (Tsai and

T A B L E 1. Results for Laminated Cyl indrical Shells

% of Shell >20 ~ Laminate a Peak load Maximum rotation (0.349 rad)

(1) (2) (3) (4)

0 [0]~

[O]s-T

[90]s

[90].-T

[02/902]2

[02/902]z-T

[902/0212

[90_,/0212-T

2.33546 lb 1.6103e4 Pa 2.33546 lb 1.6103e4 Pa 65.3096 lb 4.5030e6 Pa 61.3725 lb 4.2315e5 Pa 7.88188 lb 5.4344e5 Pa 7.56474 lb 5.2157e4 Pa 59.4528 lb 4.0992e5 Pa 54.7075 lb 3.7720e5 Pa

12.5 ~ 0.218 rad

12.5 ~ 0.218 rad

25.2 ~ 0.440 rad

29.4 ~ 0.513 rad

22.1 ~ 0.386 rad

26.7 ~ 0.466 rad

27.1 ~ 0.473 rad

33.5 ~ 0.585 rad

0

8.6

15.1

10.8

18.2

19.3

25.9

"T implies large-strain formulation model.

* U t.t Prl~.nt(,E.JI.Pl~...tic) in. zran. .~wrw t,- m,s~cl

- - Iwesent(Ela~c) in. blm. P (lhs)

1

r iV, , .................................. ',r ..............

1

- l e . ~ in. tnm. - Indudee mmd~ tmnskmmUbn

FIG. 12. Point-Loaded Isotropic Spherical Shell Response Elastic and Elastic- Plastic Analysis

336


ez, o.oo P ( l b s

900.00

/

160.00

100.00

w f o r m a U o n w (in)

o.oo . . . . . . . . 6.~' . . . . . . . 6.t6 . . . . . . . 6.t~ . . . . . . . 6./;o

FIG. 13. Point-Loaded Composite Spherical Shell Response with Material Trans- formation Analysis

e~o.oo P ( l b s )

100.00

[0 ,0 ,90,90] . [90,90,0 oo.oo / . . . . . [0,0,90,90J., [90,90,0,0J, w i t h TreLus fomat ion

/ w 0n) 0.00 i i i i i i i i i i i i i i i i i i I I i i i i i i i i i i i i i i i i i i i i i i i i i i i I

0, 0.64 0.08 0.12 0.18 0.20

FIG. 14. Point-Loaded Composite Spherical Shell Response, with Material Trans- formation Analysis for [0,90] and [90,0] Layups

337


P.JIO.~

800,00

180.00

I00,00

00.00

0.00 0.I

1> h

. . . . . . . . ~ . ~ . . . . . . . ~.t0' . . . . . . . ~ . t~ . . . . . . . ~3.0

FIG. 15. Point-Loaded Composite Spherical Shell Response with Material Trans- formation Analysis, for QuasMsotropic Layup

TABLE 2. Results for Laminated Spherical Shells

% of Shell Maximum >20 ~

Laminate a Peak load rotation (0.349 tad) (1) (2) (3) (4)

[01~, [9018, [45]8, [-4518

[0]~-T, [9018-T, [4518-T, [-4518-T

239.3840 lb 1.6505e6 Pa 225.0209 lb 1.5515e6 Pa

24.1 ~ 0.421 rad

27.9 ~ 0.487 rad

[0f1902]z, [902/0212

[0_,/90q,-T, [902/0q,-T

[0/-45/45/90],

[0/- 45/45/90],-T

230.9580 lb 1.5924e6 Pa 217.1129 lb 1.4970e6 Pa 247.9620 ib 1.7096e6 Pa 225.3241 lb 1.5536e6 Pa

24.1 ~ 0.421 rad

27.9 ~ 0.487 rad

28.2 ~ 0.492 rad

35.1 ~ 0.613 rad

8.7

15.1

8.7

15.1

17.7

28.3

aT implies large-strain formulation model.

Palazotto 1990, 1991), and the second peak represents the critical load for the ent i re shell. This effect occurs for the large-s t ra in model , bu t it is minimized by the ro ta t ion of mater ia l axis. Tab le 1 is a compi la t ion of the results for the different l amina ted cylindrical shells.

Isotropic Spherical Shell with Transverse Load (Elastic and Elastic-Plastic Analysis)

A spherical cap is loaded at the apex and suppor ted on a fixed h inge at the c i rcumference, where the geomet ry and mater ia l proper t ies are discussed

338


7 . 0 0 /

/ 6 . 0 0 /

/ 5.00 /

4.00

3.00 /

/ 2.00

Sixno & Kenne.dy 1.00 _ __ Preaent - Sch~mwele / _ ~ ~

Transvea-so Displacement 0.00 . . . . . . . . . j . . . . . . . . . ~ . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . ,

0.00 50.00 100.00 150.00 200.00 250.00 300.00

FIG. 16. Pinched Elastic-Perfectly Plastic Isotropic Cylinder

in the validation section and shown in Fig. 6. For the elastic-plastic analysis, a yield stress of 2e4 psi (1.379e5 Pa) is used and the material is assumed to behave as elastic-perfectly plastic (Argyris et al. 1980). Because the height of the spherical cap is shallow, the displacement is gradually increased until collapse occurs and into the post-collapse regime. The spherical cap is modeled by a 6 • 6 mesh for one quadrant and the load-displacement curves for both elastic and elastic-plastic are shown in Fig. 12. The results are compared to the TRUMP solution of (Argyris et al. 1980), where this example has been studied in detail. Both the elastic and elastic-plastic solutions agree quite well. However, in both cases, the numerical solution of the full nonlinear model is more flexible in terms of both load and displacement. This is due primarily to the that the TRUMP model ignores transverse shear (a minor difference for isotropic materials) and ignores many of the higher-order terms to describe the mid-surface deflection of the shell that the full in-plane nonlinear model incorporates. Also, the 36-DOF element used herein is more flexible than the TRUMP element.

Laminated Composite Spherical Shells with Transverse Load The theory is applied to the shell geometry shown in Figs. 13-15 with

the following values: E1 = 20.466e6 psi (1.411ell Pa); E 2 = 1.34e6 psi (9.239e9 Pa); Via = 0.313; G12 = G13 = 8.63e5 psi (5.95e9 Pa); G23 = 5.18e5 psi (3.57e9 Pa); R = 4.76 in. (120.9 mm); h = 0.04 in. (1.106 mm); L = 1.811 in. (46 mm); 0 = 0.1902 rad; ~/B = 0.0474; edges hinged. Fig. 13 shows the transverse response of the center of the shell due to a point load for [0]8, [90]8, [45]8, [-45]8, layups with and without the large strain formulation. Because of identical curvature in both 0 ~ and 90 ~ orientations, all four models behave similarly. Approximately 15% of the shell underwent rotations of at least 20 ~ (0.349 rad) and the large strain model decreases the peak load by 6%. The same results occurred for the [0,0,90,90L and

339


(a)

~ ~ z

R P

s2 ~ s~ free

30.00

25.00

20.00

15.00

I0.00

5.00

0.00 0.00

(x 1/1000) ~ , .-

f./ / / /

/ / / / / ~

/ . . . . e f l y / / ~ -- ~ Pr-es~ --Schimmels

i i I l l I l l , [ I I I I I 1 1 l l l l l l , I I I I I [ l l l ' l I I I I I ] I I I I I I I I I I

z.oo 4.00 8.00 8.00 zo.oo Transverse Displacement

FIG. 17. Pinched Elastic-Perfectly Plastic Isotropic Sphere

[90,90,0,0]s models shown in Fig. 14. When the quasi-isotropic model is analyzed (see Fig. 15), the large-strain formulation reduces the peak load by 9%, and 28% of the shell underwent rotations of 20 ~ or more. Table 2 is a compilation of the results for the different laminated spherical shell models.

As a final set of solutions, the writers compared their results for large- strain analyses with Simo and Kennedy (1992). This reference used a com- pletely different approach to shell analysis incorporating a stress resultant technique which did not require through-the-thickness integration. Fig. 16 is a comparison of deflection for a pinched cylindrical shell with nondi- mensional E = 103; h = 3; v = 0.3; and yield stress O-y = 2.7 • 102. Fig. 17 compares results for a pinched spherical shell with E = 10; h = 0.5; v = 0.2; and % = 0.65. In both cases, the results compare fairly well considering the difference in the theory used for the analyses from the force- displacement as well as the plasticity point of view.

340


CONCLUSIONS

A shell theory that includes parabolic transverse shear, a layered plasticity model, and a large strain formulation was applied to isotropic and laminated orthotropic shells with cylindrical or spherical mid-surfaces. The numerical results showed good agreement with previous literature and experimental results for both global and local shell response. The numerical results dem- onstrated the effectiveness in reducing stiffness for laminated composite shells by including the large-strain formulation. For a laminated composite shell, when the shews displacement reaches three times the shell thickness, the fiber movement (rotation) becomes large enough to affect the material transformation matrices. This stiffness reduction becomes significant for global shell response when at least 15% of the shell is experiencing mod- erately large rotations (>20~ The layered plasticity model accurately pre- dicted the effect of the variation of plasticity through the thickness for shells undergoing large movements (snapping).

APPENDIX I.

0 0 2%j = a i - (ule 1 q- u2e 2 -F u3e3) -t- a j ~ (ule 1 -I- u2e 2 q- u3e3)

Oyj oyi

0 0 -]- - - ( / . ~ l e l -1- u2e 2 -1- u 3 e 3 ) - S T ( u l e 1 -}- u2e 2 -~- b/3e3) . . . . . . . . . . . . . . (32)

ioy oy/

where ei = ai/h~. It should be noted that spherical coordinates are used, i.e. c% = 1, c = 1/R1 = 1/R2 = 1/R.

el = E~ = ~P• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (33)

1 2 c [uwl (1 + u,)w] ~ = . , + ~ (u~, + v , + w~,) +

C 2 .~_ ~ ( . 2 _~_ W2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 4 a )

XH = c 3w2 + c2[u(w,t + ' 1 ) - (1 + u ,)w] + c[uw , (w , + *1)

- w*, . ,] + v , * 2 . , + (1 + u l)t~,., . . . . . . . . . . . . . . . . . . . . . . . . . . (34b)

C 2 C 2 X,2 = c3uO, + ~ 0 , ( 0 , + 4 w , ) - ~ (3-o21 -}- 4W,l,1 ) q- c[2u.1,2,1

1 0~,1) + (1 + u) , , . , ] + } (,12, + . . . . . . . . . . . . . . . . . . . . . . . . . . . . (34c)

X13 = C31~ 2 "~ k [ ( 1 + U l ) ( W 11 -}- I~/1,1) q- 73,l(W 12 -~ I]/2.1) 1

+ c(t~,, + ,~ . , ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (34d)

X14 -}- c 3 k ( w . l -~- I]ll) - c2k[2w(w .u + I~/1.1) -l- (2w, + I.[ll)(W l

+ ~1)] + ck[2v,,(w.12 + ~2,1) -{" (1 + /,/1)(W, l l ~t_ i~1,1) ]

-~- kl~Ii, l(W , 1 -~- I~1.1 ) -[- k •2 ,2 (w 12 -i I- I~J2,1) . . . . . . . . . . . . . . . . . . . . . ( 3 4 e )

341


X15 = 2c3k*l(W,1 Jr- iIJ1) + 2ck[+~,,(w,n + ' 1 ,1 ) "~ '2,1(w,12 -~- 1~2,1)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (34f)

(ck) 2 k = • = 2 ( w l + .1) 2 + - ~ - [ ( W l l + * 1 , 1 ) 2 + (w l= + .2,1) 2 ] . . . (34g)

• = c3k2(w,1 + ' 1 ) 2 + ck2[(W,n + ' 1 , 0 2 + (W.le + '2,1) 2] . . . . (34h)

22 = e~ = ~VXe p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (35)

1 e _ w.2) + c [vw2 (1 + V e)W] e ~ = ~ , ~ + ~ ( u ~ + v ~ , -

c 2 + ~ - ( v 2 + w 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36a)

• = c 3w2 + c2[v(w,2 + *2) - (1 + v2)w] + c [ v w 2 ( w 2 + *2)

- w'2,2] + u,2'1,2 + (1 + v.2)+2,2 . . . . . . . . . . . . . . . . . . . . . . . . . . (36b)

c 2 c 2 X22 = c3v '2 + ~ '2(t~2 + 4w z) - -~ (3u22 + 4w'2 .2) + c[2u.2*,.2

1 + (1 + v).2,21 + ~ (*2,2 + *2,2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36c)

• = c3022 + k[(1 + v,2)(W2z + *2,2) + u ,2 (w l z + '1 ,2)]

+ c(*~,2 + *2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36d)

X24 = c3k(w.2 Jr- *2) -- c2k[2w(w.= + '2 .2) + (2w.2 + ,2)(w.2 + *2)]

+ ck[2u,2(w12 + *1.2) + (1 + v 2 ) ( w = + '2,2)1 + k*l,2(w,12 -t- '1 ,2)

+ k*2,2(w = + *2,2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36e)

X2s = 2c3k*z(W.2 + *2) + 2ck[+L2(w.12 + '1.2) + t~z.2(w.22 + *2.2)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 q )

(ck) 2 k 2 • = --E--- (w2 + $2) 2 + T [(w12 + 01,2) ~ + (w22 + ,~,2)q . . . (36g)

X27 = c3k2(w,2 dr *2) 2 "~ ck2[(w,12 + . 1 .2 ) 2 + (w,22 -F t1/2,2) 2] . . . . ( 3 6 h )

26 = eg = ~P• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (37)

E~ = U 2 "l- "U1 "}- C2UV + C(UW 2 + "UW,1 -- blz,W -- t / , lW)

+ u lu2 + V aV2 + w a w 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (38a)

X61 = c2[u(w,2 "}- *2) nt" ?)(w,1 -t- I]/1) - w ( u , 2 -1- 72,1)]

+ C[W,2(Nw.1 "~ ' 1 ) -- W(*l ,1 ''1" '2 ,1)1 -1- (1 + U 1 ) ' 1 , 2 -1- U.201,1

+ (1 + v.2)*2.a + v,1'2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (38b)

X62 = C2[2(W,2dq + W, ldd2) + ' 1 ' 2 - 2 w ( ' 1 , 2 + 1~2,1)]

+ c[(1 + u.1)'1,2 + (1 + v,2)'2,1 + u,2'1,2 + u ,1 '2 ,2]

342


+ 1111,11~1,2 "]- ~J2,111/2, 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o ' . , (38c)

•63 = ck[W,l(W,2 "~- Ill2) "~ w,2(w,1 ~- I~1) - w(w,12 -t- 11/2,1) ]

+ 2cOkl.l~t.z + ~2.1tkz.2) + k[ul(w12 + ~l.2) + u 2 ( w , + +1.~)

+ V l(W22 + ,2.2) + v 2 ( w , 2 + ,2.1)] . . . . . . . . . . . . . . . . . . . . . . . . (38d)

• = 2c2k[w.l(w.2 + ~2) + w.2(w.1 + I'~1)] + ck[l~Jl(W.2 "~ ~2)

A c 1~2(W, 1 ..~ iI/1 ) ..~ U l(Wl2 .~ i~J1,2).. ~ U 2(W, ll ..~ i~1,1) q!- V, l (W22 _~- i~J2,2 )

+ V.z(W 12 + ~2.1) + 2 w l e + +~.2 + +2.1] + k[~ l .~ (w.2 + +1.2)

+ 1~1,2(W 11 + I]/1,1) + 1112,1(W,22 "1- 1~2,2 ) + 1~/2,2(W 12 n L I~J2,1) ] . . . . . (38e)

X65 : Ck[l~1.l(W,12 @ 11/1,2) -~- 1~1,2(W, l l @ IIJl,,) -~- 1112,1(w,22 -1- 1~/2,2)

-~- d~2,2(w 12 --1- 1~2,1)1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 8 f )

• = k2[ (w. , + tkl.1)(w 12 + ~ . 2 ) + (w~2 + ~2.1)(w.22 + tk2.2)] . . . (38g)

X67 ~-- ck2[(W, ll "~- I~11,1)(w,12 -~ 1~/1,2) "~- (w,12 "~ IJ~2,1)(w,22 q- I]/2,2)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (38h)

E 4 E~ -~" 2 o ~_ ~--- E X42,~;4 w,2 + 42; X42 3 k e ] . . . . . . . . . . . . . . . . (39)

E 5 = E~ + E2X52, E~ ~--- W1 n L ItJ1; X52 = 3ke~ . . . . . . . . . . . . . (40)

U 1 = ~ (U 1 "Jr /~2 ~- U3)d~'~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (41)

u, = fh E~176 dE = fh e~176162 dE = efe~ (42a) j i : r . . . . . . . . . . . . . . . . .

u 2 = = 2 ~ O i j X i p E p dE ~--" 2 ~ ( X i l B i j -}- xi2Dij

+ • o + • + x,-sG;j + • + X;~L;) . . . . . . . . . . . . . . . . . . . (42b)

U3 ~ ZT(KrOK)Z dE fh - ,+r ~- = XipXirQijE dE = XjlXilOij + 2XjlXi2Eij

~- 2(X]lXi 3 "Jr" Xj2Xi2)fi] -~ 2(Xjl•i 4 -~ Xj2Xi3)Gij + 2(Xjl + Xi5)nij

+ 2(XjlXi 6 -t- Xj2Xi5 "~- Xj3Xi4)Ii] + (2XjlXi7 q- 2Xj2Xi6 Jr- 2Xj3Xi5

+ Xj4Xi4)Jij + 2(Xj2Xi7 + Xy3Xi6 + Xj4Xi5)gij + (2Xj3Xi7 + 2Xj4Xi6

+ Xj5Xi5)Lij + 2(Xj4Xi 7 -t- Xj5Xi6)Pu q- (2XjsXi7 + Xj6Xi6)Rij + 2Xj6Xi7S U

+ Xj7xi7Zij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (42c) where i , j = 1, 2, 6; a n d p , r = 1 - 7

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