On a consistent finite-strain shell theory based on 3-D nonlinear elasticity

International Journal of Solids and Structures 97–98 (2016) 137–149

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On a consistent finite-strain shell theory based on 3-D nonlinear elasticity Zilong Song, Hui-Hui Dai∗ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

a r t i c l e

i n f o

Article history: Received 24 May 2016 Revised 18 July 2016 Available online 28 July 2016 Keywords: Shell theory Nonlinear elasticity Soft materials Asymptotic analysis

a b s t r a c t This paper presents a general finite-strain shell theory, which is consistent with the principle of stationary three-dimensional (3-D) potential energy. Based on 3-D nonlinear elasticity and by a series expansion about the bottom surface, we deduce a vector shell equation with three unknowns, which preserves the local force-balance structure. The key in developing this consistent theory lies in deriving exact recursion relations for the high-order expansion coefficients from the 3-D system. Appropriate 2-D boundary conditions and associated 2-D weak formulations are also proposed, including various practical cases on the edge. Then, to demonstrate its validity, axisymmetric deformations of spherical and circular cylindrical shells are investigated, and comparisons with the exact solutions are made. It is found that the present shell theory produces second-order correct results for the general dead-load case and internally pressurized case. The advantages of the present shell theory include consistency, high accuracy, incorporating both stretching and bending effects, no involvement of higher-order stress resultants and its applicability to general loadings. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Shells are ubiquitous structures in nature and engineering, which have inspired extensive research interests on shell theory and its applications (see, e.g., Timoshenko et al., 1959; Ventsel and Krauthammer, 2001; Pietraszkiewicz and Gorski, 2013). Shell theories with small strains can be dated back at least 120 years ago, and the literature is vast, including both direct and derived shell theories. For direct theories, we refer to Altenbach et al. (2010) for a recent review. Here, we only give a brief review on some selected derived shell theories, which are more relevant to the present work. Early attempts on shell theories relied on a priori hypotheses of a geometrical or mechanical nature, mostly motivated by engineering intuition. Based on Kirchhoff’s assumptions (Kirchhoff, 1850) and thin shell approximation, the classical shell theory was developed by Love (1888) to provide the leading approximation. For the “best” equivalent variants of first approximation theory, we refer to Sanders (1959) and Koiter (1960). Later works relaxed some assumptions, with an attempt to include certain nonlinear and/or rotational effects, e.g., incorporating geometrically nonlinear effect and postulating a linear distribution for displacements. These fur-

∗

corresponding author. Fax: +852 27888561. E-mail address: [email protected] (H.-H. Dai).

http://dx.doi.org/10.1016/j.ijsolstr.2016.07.034 0020-7683/© 2016 Elsevier Ltd. All rights reserved.

nished some well-known shell theories with famous names attached to them, such as Koiter (1966), Reissner (1974) and Donnell (1976). For detailed reviews of early developments, we further refer to Koiter and Simmonds (1973), Pietraszkiewicz (1979) and Wan and Weinitschke (1988). Also based on assumptions, it is worth mentioning the hierarchic method (Naghdi, 1972) and constraint method (Podio-Guidugli, 1990), where admissible displacement fields are restricted. Although these theories have found a huge success in engineering applications, due to the hypotheses involved their consistency with the three-dimensional (3-D) theory cannot be expected under general loadings. Present research efforts concerning such theories are to justify them mathematically and to find out the range of applicability. Some mathematical approaches have been utilized to derive consistent shell theories or to justify existing shell theories under certain circumstances. Gol’denveizer (1963) first developed the idea of applying formal asymptotic expansions to the 3-D differential formulation to derive shell theory, and Hamdouni and Millet (20 03a; 20 03b) extended it to more general cases. An alternative way of applying asymptotic method is based on the 3-D weak formulation. This approach was initiated by Ciarlet and Destuynder (1979a; 1979b) for plates and was first attempted by Destuynder (1980) for shells, and we refer to Ciarlet (20 0 0) and Ciarlet and Mardare (2008) for a comprehensive account. A welcome feature is its capability of providing convergence results for some linear shell theories. Convergence results, even for some nonlinear theory,

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can also be achieved by Gamma convergence method, which is concerned with the limiting 2-D variational problem of vanishing thickness (Friesecke et al., 2003; Le Dret and Raoult, 1996). Although quite impressive in terms of rigorous derivations and technical sophistication, these approaches do rely on a priori scalings, assuming applied loads (or deformation) be of certain orders of shell thickness. Nevertheless, applied loads are external, which should not be directly related to the thickness in various situations. As a consequence of scalings, these methods do not lead to a single shell model incorporating both bending and stretching effects, which is probably most needed in applications. Based on more general expansions of displacements, some higher-order (refined) shell theories have also been derived by 2-D variational principle method, see Reddy and Liu (1985) and Reddy and Arciniega (2004) for linear case and Meroueh (1986) for nonlinear case. Recently, starting from 3-D nonlinear elasticity, Steigmann (2008; 2010; 2013) derived a notable shell theory which extends Koiter’s shell theory and dictates an optimal third-order approximation for the 3-D potential energy. However, his derivation is restricted to the nearly traction-free case. In this work, we aim to generalize the previous plate theory (Dai and Song, 2014; Song and Dai, 2016; Wang et al., 2016) to a finite-strain shell theory, which is consistent with the 3-D energy principle and with no special restrictions on applied loads. Specifically, we start with a general series expansion for deformed positions about the bottom surface, and make asymptotically consistent truncations on the 3-D differential system. The key in the derivation lies in obtaining the recursion relations for all highorder expansion coefficients, leading to a final vector shell equation for the leading coefficient, which preserves the local force balance in all three dimensions. Suitable 2-D boundary conditions, as well as the associated 2-D weak form, are also consistently derived, accommodating various practical situations. The specific forms of shell equations for cylindrical and spherical shells are also provided, and concrete problems concerning these structures with axisymmetric deformations are investigated to verify the theory. The distinctive features of the present shell theory, not all enjoyed by existing ones, include: (i) finite strains; (ii) consistency with 3-D energy principle; (iii) general loadings; (iv) incorporating bending and stretching effects simultaneously; and (v) without any ad hoc hypotheses or any a priori scaling assumptions. In short, it is an “all-purpose” shell theory. The manuscript is arranged as follows. In Section 2, we recall the kinematics of a shell and the 3-D formulation. Section 3 derives the finite-strain shell theory, which is consistent with the 3D weak formulation. Subsequently in Section 4, the associated 2-D weak formulation is discussed in detail, which also yields various natural boundary conditions. Section 5 presents the shell equations and some concrete problems for two common shell structures, to ease application and to substantiate the consistency. Finally, some concluding remarks are made.

to parametrize the base surface  of the shell in the reference configuration, where a generic point is denoted by r = r(θ α ). The covariant base (tangent) vectors along the coordinate lines are given by gα = r,α = ∂ r/∂ θ α , and their contravariant counterparts are denoted by gα . Then, the unit normal n to  is well-defined according to n = g1 ∧ g2 /|g1 ∧ g2 | to form a right-handed basis, where ∧ means the cross product. In component forms, we will identify g3 = g3 = n, and components with subscripts and superscripts are respectively associated with bases gi and gi . Subsequently, the 3-D reference position is decomposed as

2. Kinematics and the 3-D formulation

where ∇ = gα ∂ /∂ θ α is the in-plane 2-D gradient on the base surface (∇ x = x,α  gα ). Keep in mind that the tensor μ and its determinant μ depend on Z and the curvatures of the shell, which degenerate to 1 and 1 for plates. For a hyperelastic material with strain energy function (F), the nominal stress tensor is defined by

We consider a homogeneous thin shell of constant thickness composed of a hyperelastic material. A material point in the reference configuration  × [0, 2h] is denoted by X, where the thickness 2h of the shell is assumed to be small compared with the other two dimensions of the bottom (base) surface  as well as the local radius of curvature. The deformed position in the current configuration is denoted by x. In the following notations, Greek letters (α , β , γ ...) run from 1 to 2, whereas Latin letters (i, j, k...) run from 1 to 3, repeated summation convention is employed and a comma preceding indices means differentiation. Following Ciarlet (2005) and Steigmann (2012), we first recall the kinematics of the shell. Curvilinear coordinates θ α are utilized

X = r(θ α ) + Zn(θ α ),

0 ≤ Z ≤ 2h,

(1)

where Z is the shell thickness coordinate perpendicular to . A distinct feature from plates is that the normal n depends on the coordinates of the base surface. By the Weingarten equations, the variation of n is described by the curvature map κ

dn = n,α dθ α ,

dn = −κdr,

dr = gα dθ α .

(2)

Equivalently, we may write curvature tensor as

κ = −n,α  gα = καβ gβ  gα = κβα gβ  gα .

(3)

Then, the mean and Gaussian curvatures are defined by

H=

1 1 tr(κ ) = καα , 2 2

β

K = det(κ ) = [(2H )2 − κα κβα ]/2.

(4)

By choosing orthogonal principal directions as the base vectors, the nonzero components κ11  κ1 and κ22  κ2 are called the two principal curvatures, and consequently H = (κ1 + κ2 )/2 and K = κ1 κ2 . Accordingly, the differential of the reference position is given by

dX = dr + Zdn + ndZ = μdr + ndZ  gˆ α dθ α + ndZ,

μ = ( 1 − Z κ ),

(5)

where 1 is the rank-2 identity tensor within the tangent plane, and we have defined the covariant and contravariant base vectors at an arbitrary point inside the shell

gˆ α = μgα ,

gˆ α = μ−T gα ,

(6)

which are also orthogonal to n. Note that previous assumption on geometry implies that |2hκ α | < 1, and hence the inverse μ−1 is well-defined. The volume element is calculated as

dV = μ(Z )dAdZ,

μ(Z ) = det(μ ) = 1 − 2HZ + KZ 2 ,

(7)

with the area element dA on the base surface defined by

dA = |g1 ∧ g2 |dθ 1 dθ 2 =

√ gdθ 1 dθ 2 ,

(8)

where g is the determinant of the metric tensor (gαβ ) induced by the base vectors gα . The current position is written as x = x(θ α , Z ), then the deformation gradient is given by

F = x,α  gˆ α +

S=

∂ , ∂F

∂x ∂Z



Si j =

n=

(∇ x )μ−1 +

 ∂ , ∂ Fji

∂x ∂Z

 n,

(9)

(10)

and the associated first and second-order elastic moduli are defined by

A1 ( F ) =

∂ 2 ∂ 3 , A2 ( F ) = . ∂ F∂ F ∂ F∂ F∂ F

(11)

Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

It is assumed that the strain energy function for the deformations concerned satisfies the strong-ellipticity condition

a  b : A 1 ( F )[ a  b ] > 0 ,

for all a  b = 0,

(12)

The principle of stationary potential energy requires the first variation of E to be zero, which is expressed as

δE = −

A1 [A]

i j

= (A1 )i jkl Alk ,



A2 [A, B]

i j

= (A2 )i jklmn Alk Bnm .

(13)



 +

For the case of dead-loading and in the absence of body forces, the 3-D potential energy E is given by

 

 −





∂ q 0

2h

where are the applied tractions on the top and bottom surfaces of the shell, q is the applied traction on the lateral surface (the edge), and da (in distinction from dA in (8)) is the area element on the edge. The boundary ∂  of the base surface is divided into two parts, the position boundary ∂ 0 and traction boundary ∂ q . On ∂ , let s be the arclength variable, and τ and ν be the unit tangent and outward normal vectors, in a way that ν = τ ∧ n. By these definitions, we easily get dr = τ ds along the boundary. Then, it follows from (5) that the local differential consists of dXτ = μτ ds along tangential direction and dXn = ndZ along normal direction. The outward normal of the lateral surface is denoted as N, and we obtain

Nda = dXτ ∧ dXn = (μτ ) ∧ ndZds,

μτ = (1 − Z κ )τ = (τ β − Z τ α καβ )gβ , |(μτ ) ∧ n|2 = |μτ|2 = τ · τ − 2Z τ · (κτ ) + Z 2 (κτ ) · (κτ )

(15)

= 1 − 2Z τ α καβ τ β + Z 2 τ α κασ κσ β τ β  1 − 2κτ Z + cτ Z 2 , where the coefficients of Z in last expression are connected with the three fundamental forms of the base surface (note cαβ  κασ κσ β are the components of the third fundamental form). Immediately, we deduce

√ gτ dZds,

ν1  −κτ ∧ n,

(16)

Remark. If principal directions are utilized and the edge is along θ 1 , then

√

√

√ gτ = 1 − Z κ1 ,

(17)

and the vector ν1 has the same direction as ν, which implies N = ν. If we follow Steigmann (2012) and locally adopt a different representation for curvature tensor like

κ = κτ τ  τ + κν ν  ν + κτ ν (τ  ν + ν  τ ),

(18)

then the area element is computed as

Nda = (μτ ) ∧ ndZds = [(1 − Z κτ )ν + Z κτ ν τ ]dZds, da =



(1 − Z κτ )2 + (Z κτ ν )2 dZds.



2h

∂ q 0

ST N · δ x(s, Z )da





ST N − q · δ x(s, Z )da = 0.

It follows that the 3-D field equations together with boundary conditions are

in  × [0, 2h], − S n = −q , in , Z=0 T S n = q+ , in , Z=2h x = b(s, Z ),

on

ST N = q(s, Z ),

on

(21)

∂ 0 × [0, 2h], ∂ q × [0, 2h],

where b is the prescribed position on part of the lateral surface. 3. The 2-D shell theory In this section, we intend to reduce the previous 3-D system to a consistent 2-D shell theory, consisting of a fourth-order vector shell equation and appropriate 2-D boundary conditions. The same consistency criterion as in Dai and Song (2014) is employed, i.e., each of the five terms in the 3-D weak formulation (20) should be of a required asymptotic order, say O(h4 ), separately for the shell approximation. The starting point is a generic series expansion of the current position vector, followed by proper truncations of the 3-D differential system (21) carried out with the consistency criterion in mind. 3.1. Series expansions Suppose the current position x(X) and the strain energy (F) are both C5 in their arguments. Then, the current position is expanded about the bottom surface Z = 0 in the form

1 2 (2 ) α 1 Z x ( θ ) + Z 3 x (3 ) ( θ α ) 2 6 (22) 1 4 (4 ) α 5 + Z x (θ ) + O(Z ), 0 ≤ Z ≤ 2h, 24

x(X ) = x(0) (θ α ) + Zx(1) (θ α ) +

where gτ represents the preceding quadratic form.

τ = g1 /|g1 | = g1 / g11 , ds = g11 dθ 1 ,

2h

T

q±



(20)

Z=2h



Div S = 0,

q(s, Z ) · x(s, Z )da,

da = 1 − 2κτ Z + cτ Z 2 dZds  √ gτ N = (μτ ) ∧ n = ν + Z ν1 ,

2h

∂ 0 0

 +

2h

¯ + V¯ , E = ¯ = (F )μ(Z )dZdA,  0    q− (θ α ) · x(θ α , 0 ) + μ(2h )q+ (θ α ) · x(θ α , 2h ) dA (14) V¯ = −

 

(Div S ) · δ x μ(Z )dZdA   0

− ST n + q− · δ x(θ α , 0 )dA Z=0  

+ ST n − q+ · δ x(θ α , 2h ) μ(2h )dA

where the colon between second-order tensors means the trace of their tensor product A : B = tr(AB ), and the square brackets after a modulus tensor represent the operations



139

where (· )(n ) = ∂ n (· )/∂ Z n |Z=0 (n = 1, · · · , 4 ). Accordingly, the deformation gradient has a similar expansion

F = F(0) (θ α ) + ZF(1) (θ α ) +

1 2 (2 ) α 1 Z F ( θ ) + Z 3 F ( 3 ) ( θ α ) + O ( Z 4 ). 2 6 (23)

It follows from the definition (9) that

F = (∇ x )μ−1 +

∂x ∂Z

n=

( ∇ x )[ 1 + Z κ + Z 2 κ 2 + · · · ] +

∂x ∂Z

 n. (24)

(19)

Comparing this representation with (16), we can identify ν1 = κτ ν τ − κτ ν and cτ = κτ2 + κτ2ν .

Substituting (22) into (24) and comparing with (23), we obtain the relations

F (m ) =

m  m! (∇ x(i) )κm−i + x(m+1)  n, i! i=0

m = 0, 1, 2, 3,

(25)

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Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

which indicates that F(m) depends linearly on x(m+1 ) . Based on the smoothness of , the nominal stress S can be expanded as

S (F ) = S

(0 )

1 (θ α ) + ZS(1) (θ α ) + Z 2 S(2) (θ α ) (26)

By the chain rule and using the elastic moduli in (11), we obtain

S ( 0 ) = S ( F ( 0 ) ), S

(2 )

= A (F 1

(0 )

S ( 1 ) = A 1 ( F ( 0 ) )[ F ( 1 ) ] , (2 )

)[ F ] + A ( F 2

(0 )

(1 )

(27)

(1 )

)[ F , F ] .

3.2. Derivation of the vector shell equation By substituting the preceding series expansions into the differential system (21), it seems that there are too many unknowns x(i) (i = 0, . . . , 4) involved. But they are necessary in the sense that they form a closed system by consistently truncating the 3-D system to O(h4 ). Some equations in (21) will serve to deduce recursion relations for the high-order coefficients x(i) (i = 1, . . . , 4), leading to a final shell equation for x(0) . Note that some works (e.g. Naghdi, 1972) in the literature often treat these expansion coefficients as independent unknowns. The approach adopted here is different. We observe that these coefficients are correlated if one substitutes them into the 3-D differential system (21). In the following, we manage to explore these intrinsic relations by using the 3-D pointwise information. First of all, we analyze the recursion relation for x(1) . Substituting (26) into the bottom traction condition (21)2 leads to

S (0 ) ( ∇ x (0 ) + x (1 )  n )

T

n = −q− .

(28)

The strong-ellipticity condition, together with the implicit function theorem, guarantees that x(1) can be uniquely solved in terms of ∇ x(0) (cf. Steigmann, 2013). Next, the 3-D field Eq. (21)1 dictates

Div S = gˆ α · S,α +

∂ T ( S n ) = 0, ∂Z

(29)

∀ a,





Bi j = A13i3 j ,

(33)

where both the acoustic tensor B and the vector f(2) only involve x(0) and x(1) . The strong-ellipticity condition ensures that B is invertible, and the above equation yields that

x(2) = −B−1 f(2) .

(34)

gˆ α = [1 + Z κ + Z 2 κ2 + · · · ]T gα .

Substituting the expansions (26,30) into (29), the vanishing of the coefficients of Zn leads to

with the vector

f (3 ) =



κT gα · S,(α0) + ∇ · S(1)

+

A1

2  2 (∇ x(i) )κ2−i i!



T + A2 [F ( 1 ) , F ( 1 ) ]

n.

(36)

i=0

The coefficient x(4) is an intermediate quantity, whose explicit expression is not needed, however the third Eq. (32)3 as a whole will be utilized to eliminate it later. In brief, by repeatedly substituting the recursion relations, all x(i) (i = 1, 2, 3) can essentially be expressed by x(0) and its derivatives. Finally, the top traction condition (21)3 states

S ( 0 )T n + 2 hS ( 1 )T n + 2 h2 S ( 2 )T n +

4 3 ( 3 )T h S n + O ( h4 ) = q + . 3

(37)

Alternatively, here we take into consideration the factor μ(2h ) = 1 − 4hH + 4h2 K as in the third term of (20), then the above equation is equivalent to (with a slightly different remainder)1

μ(2h )S(0)T n + 2hμ(2h )S(1)T n + 2h2 (1 − 4hH )S(2)T n +

(38)

4 3 ( 3 )T h S n + O ( h4 ) = μ ( 2 h )q + . 3

Clearly this equation and (28,32) form a closed system for the five unknown vectors x(i) (i = 0, . . . , 4). Substituting (28,32) as a whole into (38) and neglecting the remainder, we deduce the vector shell equation

(31)

+ h(1 − 4hH )[κT gα ] · S,(α0) + +

In particular, the first three are explicitly expressed by

(32)

where ∇ · S(0 ) = gα · S,(α0 ) denotes the 2-D divergence. For plates κ = 0, these reduce to two-term relations as in Dai and Song (2014). In principle, the three relations in (32) can be utilized respectively to solve x(i) (i = 2, 3, 4), since S(m+1 ) (m = 0, 1, 2) only depends linearly on x(m+2 ) . For example, the first Eq. (32)1 can be

4 2 T α h [κ g ] · S,(α1) 3

4 2 h [(κ2 )T gα ] · S,(α0) 3 = −(1 − 4hH + 4h2 K )(q+ + q− )/(2h ). (39)

i=0

∇ · S ( 0 ) + S ( 1 )T n = 0,  T α (0 ) κ g · S,α + ∇ · S(1) + S(2)T n = 0, 



2 (κ2 )T gα · S,(α0) + 2 κT gα · S,(α1) + ∇ · S(2) + S(3)T n = 0,

(35)

2 3

(30)

m = 0, 1, 2, . . .

x(3) = −B−1 f(3) ,

(1 − 4hH + 4h2 K )∇ · S(0) + h(1 − 4hH )∇ · S(1) + h2 ∇ · S(2)

where gˆ α is given in (6) and has the expansion

m  m! [(κm−i )T gα ] · S,(αi ) + S(m+1)T n = 0, i!

Ba  (A1 [a  n] )T n,

Similarly, from the second Eq. (32)2 it follows that

The formula for S(3) is omitted for brevity since it is not needed in the derivations to follow. An observation is that S(m) (m = 1, 2) also depends linearly on x(m+1 ) , which is one key point of the reductions in the next subsection.



Bx(2) + f(2) = 0,

f ( 2 ) = ∇ · S ( 0 ) + ( A 1 [ ( ∇ x ( 0 ) )κ + ∇ x ( 1 ) ] )T n,

2

1 + Z 3 S ( 3 ) ( θ α ) + O ( Z 4 ). 6

recast as

This 2-D vector shell equation preserves the 3-D force balance/equilibrium in three directions in the sense of throughthickness average, since formally we can directly deduce this equation by an integration of (29) multiplied by μ(Z) over Z (with the aid of top and bottom traction conditions). By some delicate manipulations (see Appendix A), the above shell equation can be recast in a compact form



∇ · 1(S˜ + hκSˆ ) = −q˜ ,

(40)

1 We have tacitly assumed that the radius of curvature has similar order as or larger than planar length scale, i.e., Rα = 1/|κα | ≥ O(L ), or O(h/Rα ) ≤ O(h/L).

Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

where

8 2 S˜ = (1 − 2hH )S(0) + h(1 − hH )S(1) + h2 S(2) , 3 3 4 Sˆ = S(0) + hS(1) , 3  1  q˜ = (1 − 4hH + 4h2 K )q+ + q− . 2h

(41)

For plates (i.e., κ = 0 ), S˜ + hκSˆ reduces to S¯ (averaged stress tensor over the thickness) defined previously in Dai and Song (2014). Therefore, the quantity S˜ + hκSˆ is considered as the averaged stress, and q˜ is the weighted average of surface tractions (the weight depends on the surface area). The component form is given by

[S˜α j + hκδα Sˆδ j ];α = −q˜ j ,

Remark 1. Now, we show that the shell equations are actually local force equilibrium equations for a shell element. Take a shell element with a base surface R ⊂  and thickness 2h. The force balance requires that



∂ R×[0,2h]

ST Nda = −



R



q− + μ(2h )q+ dA.

(43)

The left-hand side can be written as





∂ R×[0,2h]

ST Nda =  =  

∂ R×[0,2h] ∂ R×[0,2h]

∂R

ST [(1 − Z κτ )ν + Z κτ ν τ ]dZds ST [(1 − 2HZ )1 + Z κ]νdZds

(44)

N νds,

where use has been made of (19)1 and (18), and N is the so-called stress resultant tensor defined in Libai and Simmonds (1998, p460) and given by



N =

2h 0

satisfied (in a pointwise manner), the local form of the 3-D moment equilibrium equations is T = TT in the referential description, where T = SF−T is the second Piola-Kirchhoff stress tensor. The principle of frame indifference requires the strain energy function to be a function of the Green-Lagrangian strain tensor E = 1 T 2 (F F − I ), which is symmetric. Thus, by definition, T = ∂ /∂ E is also symmetric, which implies that the 3-D moment equilibrium equations are automatically satisfied. For the present shell theory based on general expansions, it is easy to check that the symmetries of E and T are retained up to O(Z3 ). So, the 3-D moment equilibrium equations in local form are satisfied up to O(Z3 ). Note that the shell equations only have three unknowns, but they are supplemented with recursion relations (from which one can recover the 3-D displacement field).

(42)

where the covariant derivatives, indicated by a semicolon, are defined in Appendix B. The shell equation only entails up to x(3) , and by recursion relations, results in a fourth-order differential equation for x(0) . Although only the leading coefficient x(0) appears in above shell equation, the other expansion coefficients x(k) (k = 1, . . . , 4) can be calculated from recursion relations once x(0) is solved. Therefore, the distribution of 3-D current position vector can be recovered. Now we briefly analyze the asymptotic order concerning above truncations to derive (40). Since the three equations in (32) are utilized, the error for the 3-D field equation is Div S = O(Z 3 ). Also, the two traction conditions (28,38) are kept correct to O(h3 ). Therefore, the present shell equation, together with the recursion relations, guarantees an O(h4 ) error for each of the first three terms in the variation δ E in (20). In the next subsection, we shall introduce proper shell boundary conditions to make the error of the two edge terms in (20) also O(h4 ).



141

ST [(1 − 2HZ )1 + Z κ]dZ.

(45)

One can easily check by substituting the expansions of S that

1 T N = 1(S˜ + hκSˆ ) + O(h3 ). 2h

(46)

The local form of the shell element force equilibrium equations can be obtained by using the divergence theorem, also see (F.2) of Libai and Simmonds (1998, p460). Then, dividing the factor 2h, one obtains exactly (40) except the remainder O(h3 ). Remark 2. In the 3-D formulation, one only needs 3 force equilibrium equations with three unknowns (displacement components), and the 3 moment equilibrium equations are automatically satisfied. Here is the reason. When the force equilibrium equations are

Remark 3. It may be worth recalling two previous famous formulations on shell theory. Naghdi (1972) also started with a general series expansion of 3-D current position vector, but treated all the expansion coefficients as independent unknowns. The system of 2-D shell equations can be obtained by direct integration of 3D field equations and moment equilibrium equations after multiplication by Zn (Naghdi, 1972, p. 519). The final shell system has a nice structure, but contains many independent unknowns since the higher-order coefficients (unknowns) cannot be expressed in terms of lower-order coefficients by using the 2-D shell equations there. It should be noted that Naghdi’s approach does not rule out that the high-order expansion coefficients can be represented in terms of the lower-order ones directly from the 3-D equations in a pointwise manner, since in deriving his shell theory the 3-D equations were only used in the manner of through-thickness averages. In the present formulation, we manage to express all higher-order expansion coefficients in terms of leading coefficient x(0) by utilizing 3-D pointwise information, including the bottom traction condition and the expansions of 3-D field equations. This leads to a simpler vector shell equation for only x(0) . Under the assumption of smoothness, the derivation is rigorous. The idea of simplifying higher-order coefficients by 3-D information has also been utilized in recent papers of Steigmann (e.g. Steigmann, 2013). The difference is that his derivation was restricted to the nearly traction-free case with the purpose of obtaining an optimal O(h3 ) 2-D energy, while the present work deals with the general case and intends to derive a theory consistent with the 3-D energy principle. In the mixed approach (between a direct shell theory and a derived one) adopted in Libai and Simmonds (1998), averaged 2-D quantities including displacement and stress are suitably defined and treated as variables instead of being expanded in series. One elegance of such an approach is that the balances of translational and rotational momentums are exactly formulated for the shell structure. In this theory, like a direct shell theory, the constitutive relations for the 2-D variables need to be specified, which may not be an easy task. In the present work, the constitutive relations for S(i) are naturally inherited from the original 3-D one and need not be postulated again. Also, once x(0) is solved, the distribution of 3D current position can be recovered by recursion relations and the rotation can then be calculated. Remark 4. The above shell equation can be easily adapted to some non-dead loading cases, when the base surface is subjected to a pressure or sitting on a foundation. For instance, when the pressure P is given in the current configuration, the traction q− now depends on the deformation, in the form q− = P det(F(0 ) )(F(0 ) )−T n. In this case, as long as the derivative of S(0 )T n + q− with respect to x(1) is invertible, one can similarly solve x(1) from the condition (28). The final shell Eq. (40) is still valid. This case will be mentioned in the examples of spherical and cylindrical shells in Section 5.

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Remark 5. The explicit formulas for x(2) and x(3) involve moduli A1 and A2 , which make the formulas inconvenient to use directly. Thus, these formulas have more theoretic value than practical. In many concrete examples, one can readily get the explicit expression of S(i) (from the strain energy function), and directly derive the recursion relations from the Eq. (32) without calculating such moduli.

0, 1, 2, 3). In this case, we define the averaged traction q0 (s) as

q0 =

1 2h

=

1 2h

3.3. Boundary conditions



2h

0

x(s, Z )dZ = x(0) + hx(1) +

x ( 0 ) = b ( 0 ) ( s ),

x¯ = b¯ ,

on

∂ 0 ,

2 1 x(1) + hx(2) + O(h2 ) = (b¯ − b(0) ), 3 h

(48)

(49) Recall that ν1 = κτ ν τ − κτ ν in the remark following (16), thus we get N = ν for the general case. Nevertheless, when the edge is along the principal direction, we indeed have ν1 = −κτ ν, N = ν √ and gτ = 1 − Z κτ . Based on the definition of p, we get

0

√ ST N · δ x gτ dZ = 

=

2h

0



+

p(0) · δ xdZ + 2h

0



1 2 (2 ) Z q + ··· 2



1 − 2κτ Z + cτ Z 2 dZ

 2 2  (2 ) h q − 2κτ q(1) + κτ2ν q(0) + O(h3 ). 3



2h 0



2h 0

p · δ xdZ Zp(1) · [δ x(0) + Z δ x(1) ]dZ

2 2 (2 ) h p + O ( h3 ) = q 0 , 3 on ∂ q ,

S ( 0 )T ν = q ( 0 ) ,

(52)

p(i)

where are defined in (49). Alternatively, we may define a stress resultant (or moment about the middle line) as

 2h √ 1 q(s, Z ) gτ (Z − h )dZ 2h 0  1  = h2 q(1) − κτ q(0) 3  1  + h3 q(2) − 2κτ q(1) + κτ2ν q(0) + O(h4 ), 3

m (s ) =

(53)

1 2h



2h 0

1 3

1 3

(Z − h )pdZ = h2 p(1) + h3 p(2) + O(h4 ) = m(s ). (54)

√ 1 p = g τ S T N = [ S ( 0 ) + Z S ( 1 ) + Z 2 S ( 2 ) + · · · ] T [ ν + Z ν1 ] 2 1 ( 0 )T ( 1 )T ( 0 )T =S ν + Z[S ν + S ν1 ] + Z 2 [S(2)T ν + 2S(1)T ν1 ] + O(Z 3 ) 2 1 2 (2 ) (0 ) (1 ) 3 p + Z p + Z p + O ( Z ) 2

2h



q (0 ) + Z q (1 ) +

and the second condition in (52) can be suitably replaced by

where b(0 ) = b|Z=0 and b¯ is defined in the same way as x¯ . The second condition contains up to the second-order derivatives of x(0) upon using the recursion relations. To check the consistency, we should examine the asymptotic order of the fourth term in (20). For simplicity (see (16) for da), we define



0



p¯  p(0) + hp(1) +

2 2 (2 ) h x + O ( h3 ), 3

and adopt the following two conditions

⇔

2h

√ q(s, Z ) gτ dZ

And, we can adopt the following two conditions2

(47)

x ( 0 ) = b ( 0 ) ( s ),

0

(51)

In this subsection, we aim to reduce the 3-D boundary conditions to appropriate ones for the preceding 2-D vector shell Eq. (40). Since the shell equation is of fourth-order, two conditions regarding x(0) and its derivatives are needed, either on the position boundary ∂ 0 or on the traction boundary ∂ q . Case 1. Prescribed position in the 3-D formulation On ∂ 0 × [0, 2h], the position b is prescribed as in (21)4 . In this case, we define

1 2h



2h

= q(0) + h q(1) − κτ q(0) +

x¯ (s ) =



(50)

1 2 (2 ) Z p · δ x(0) dZ + O(h4 ). 2

Obviously the first term in (50) is zero due to the condition (48)2 , and the third term is zero as δ x(0 ) = 0. Also, the second term is of O(h4 ) since it is easily seen from (48) that δ x(1 ) = O(h ). Therefore, the overall error of this integral, or the fourth term in δ E, is O(h4 ). Case 2. Prescribed traction in the 3-D formulation On ∂ q × [0, 2h], the traction q is specified as in (21)5 . Denote the first four coefficients of its Taylor expansion by q(i) (i =

The conditions in (52,54) involve up to third-order derivatives of x(0) by virtue of recursion relations. To check the consistency, we examine the asymptotic order of the fifth term in (20). For convenience, we denote

qˆ =

√ √ gτ ( S T N − q ) = p − gτ q ,

(55)

and use qˆ (i ) (i = 0, 1, 2 . . .) to represent the coefficients of its Taylor expansion over Z. Then, the inner integral of that term in (20) is given by



2h 0

qˆ · δ xdZ =

 0

+

2h

 0

qˆ · δ x(0) dZ + 2h

 0

2h





Z qˆ (0) + Z qˆ (1) · δ x(1) dZ

1 2 (0 ) Z qˆ · δ x(2) dZ + O(h4 ). 2 (56)

On the right-hand side, the first term is zero due to the condition (52)1 . By simple manipulations of the two conditions in (52) (or (52)1 and (54)), one can show that qˆ (0 ) = O(h2 ) and qˆ (1 ) = O(h ), and consequently the remaining terms are of O(h4 ). To sum up, the 2-D vector shell Eq. (40) together with recursion relations and boundary conditions (48) and (52) (or (48), (52)1 and (54)), is consistent with the 3-D weak formulation, with an O(h4 ) error. Note that no higher-order stress resultants are involved in above boundary conditions, in contrast to some higher-order shell theories. The derivation of the above 2-D boundary conditions directly from 3-D ones is very natural, since these boundary conditions satisfy the proposed consistency criterion. Other types of common 2-D boundary conditions in existing shell theories are often based on 2-D variational principle. In the present formulation, one can similarly propose such 2-D boundary conditions based on a 2-D weak form, as to be done in the next section. 2 Note that since N depends on Z, from the 3-D condition ST N = q, in general we can not derive S(1)T N = q(1) or S(1)T ν = q(1) , which is true only for principal directions.

Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149



4. Associated weak formulations In this section, we present the associated 2-D weak formulation for the previous shell system, for the ease of numerical calculations. Another direct consequence of such a weak form is that one can propose suitable boundary conditions for various practical cases when the 3-D edge conditions are unclear. First of all, multiplying the 2-D shell Eq. (40) by ξ = δ x(0 ) and integrating over  lead to



 



 ∇ · 1(S˜ + hκSˆ ) · ξ dA = −





q˜ · ξ dA.



∇ · wdA =



∂

w · νds.

(58)

Applying this to w = 1Sξ , we readily get





[∇ · (1S )] · ξ dA =





∇ · wdA −





 [(1S )T ν] · ξ ds − (1S ) : ∇ξ dA   ∂  T = (S ν ) · ξ ds − S : ∇ξ dA.

(59)



And, with the above S replaced by S˜ + hκSˆ , it follows from (57,59) that





(W1 + W2 − q˜ · ξ )dA = =





∂  ∂



S˜ T ν + h(κSˆ )T ν · ξ ds (60)

p¯ · ξ ds,

where the terms in square brackets correspond to exactly p¯ by comparing (18,41) with (49,52) and neglecting the O(h3 ) remainder. The two scalar functions are defined by



W1 = (1 − 2hH )S(0) + h(1 −

 ∂

 (S(1) − 2HS(0) + κS(0) )T ν · η

+ (S0T ν ) · x(2) ds  2 = h2 p(1) · η + (S0T ν ) · x(2) ds, 3 ∂

(65)

where

 2  W3 = − h2 (S(1) − 2HS(0) + κS(0) ) : ∇η + [∇ · (1S0 )] · x(2) . 3 (66)



2 2  2 (1 ) (1 ) h A [F , F ] : ∇ξ + A1 [F1(2) ] : ∇ξ , 3 2 2 W2 = h2 A1 [F2(2) ] : ∇ξ = h2 A1 [∇ξ ] : F2(2) , 3 3





(W1 + W3 − q˜ · ξ )dA  =

∂

p¯ · ξ −

(67)

2 2  (1 ) h p · η + (S0T ν ) · x(2) ds. 3

(1) 2-D weak form for the previous cases of boundary conditions On ∂ 0 , it is easy to deduce from (48) that ξ = δ x(0 ) = 0 and η = δ x(1 ) = O(h ), which, together with (63), further imply that ∇ξ = O(h ) and S0 = O(h ). As a result, the boundary integral on ∂ 0 in (67) is of O(h3 ). In this context, we can readily replace ∂  by ∂ q in (67). While on ∂ q , it follows from conditions (52,54) that S0T ν = δ [S(0)T ν] = O(h2 ). Thus, the third term inside the boundary integral can be neglected. Also, replacing p(1) by the condition in (54) only causes a higher-order correction. Therefore, the 2-D weak form (67) reduces to



 

8 hH )S(1) + hκSˆ : ∇ξ 3

+

In summary, coupling (60) and (65) furnishes the following 2-D weak form

In the following, we will adapt this to two distinct types of boundary conditions, i.e., the previous cases in Section 3.3 based on 3-D information, and some practical cases when the 3-D edge conditions are unclear.

(1S ) : ∇ξ dA

=

∂



(57)

To tackle the left-hand side, we need to utilize the 2-D divergence theorem. It is well-known that for a vector w = wα gα on the base surface , the divergence theorem dictates





2 3

(W2 − W3 )dA = h2

143



W1 + W3 − q˜ · ξ dA =

 =



 ∂ q ∂ q



p¯ · ξ −



2 2 (1 ) h p · η ds 3



q0 · ξ − 2m · η ds,

(61)

(68)

where we have used the decomposition F(2 ) = F1(2 ) + F2(2 ) , see Appendix C for the explicit expressions. In principle, the weak formulation for the fourth-order Eq. (40) should only involve the second-order derivatives of x(0) (especially regarding the functional space in finite-element calculations). However, the term F2(2 ) contains the third-order derivative of x(0) , which we intend to eliminate by conducting divergence theorem once more. To this end, we further recast W2 as

where we have used conditions (52)1 and (54) in the last equality. If the two conditions in (52) are adopted, we should replace m by a combination of q0 and q(0) . (2) 2-D weak form for some practical edge conditions In a number of practical situations, one does not know the edge traction distribution (e.g., a pinned edge) or displacement distribution (e.g., a clamped edge). In these cases, one should resort to the weak formulation to propose the socalled 2-D natural boundary conditions. To this end, we would like to recast the boundary terms in (67) as a combination of ξ and its normal derivative ξ , ν . For convenience, we introduce a third-order (moment) tensor M through

  2  W2 = h2 S0 : ∇ x(2) + η · ∇ · (1S(1) − 2H 1S(0) + κS(0) ) , 3 (62) where

η = −B−1 (A1 [∇ξ ] )T n, S0 = A1 [∇ξ + η  n].

(63)

In fact, it can be proved by (28) and (27)1 that

η = δ x ( 1 ) , S0 = δ S ( 0 ) . Subsequently, integration by parts leads to

(64)

−

2 2  (1 ) h p · η + S0T ν · x(2) 3 

2  = h2 A1 [B−1 p(1) + (A1 [x(2)  ν] )T n  n 3 

(69)

−x(2)  ν] : ∇ξ  (M[ν] ) : ∇ξ , T

where p(1) also contains ν as in (65). Furthermore, we introduce the decomposition

∇ξ = ξ,s  τ + ξ,ν  ν,

(70)

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Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

where subscripts s, ν preceded by a comma are tangential and normal derivatives on ∂ . Substituting (69,70) into (67) followed by an integration by parts provides the 2-D weak form







W1 + W3 − q˜ · ξ dA   = {p¯ − (M[ν]τ ),s } · ξ + {M[ν]ν} · ξ,ν ds,

(71)

∂

where the terms before ξ and ξ , ν are respectively the generalized traction and moment on the edge. Here, the smoothness of ∂  is tacitly assumed. Based on (71), one can propose boundary conditions regarding the following conjugate variables

x (0 ) ,

primary variables :

secondary variables : p¯ − (M[ν]τ ),s ,

x,(ν0) ,

(72)

M[ν ]ν .

One might specify the primary or secondary variables or a mixture of them. As in the variational approach, various boundary conditions such as clamped and simply-supported conditions can be suitably proposed (see Dai and Song, 2014, for details).

From the assumptions of the present theory on geometry and curvature, we need the condition that 2h/A  (B − A )/A is small. Now we illustrate the procedure of writing down the shell equations for a practical spherical shell problem. First, the deformation gradient F follows from (9), and the components Fij are obtained by expressing x in terms of the base gi . Once the strain energy function is given, the nominal stress S is calculated according to (10). Often for isotropic materials, S can be easily expressed in terms of base (eR , e , e ), and one can rewrite it in terms of the base gi to get the components Sij . The vector x(1) or its components xi(1 ) are solved in terms of x(0) by (28). Note that with known explicit expression of S, the recursive relations for x(2) and x(3) will be obtained directly from (32)1, 2 instead of using the moduli A1 and A2 . With the above data in (77,78), the final shell equations in (42) take the form

1 13 (S¯ + S¯23 ) − cos  sin S¯22 = −q˜1 , A 1 S¯,αα2 + (2S¯12 + S¯21 ) cot  + (S¯13 + S¯23 ) = −q˜2 , A 2 S¯,αα3 + S¯13 cot  − A(S¯11 + S¯22 sin ) = −q˜3 , S¯,αα1 + S¯11 cot  +

(79)

where

h αj Sˆ . A

5. Examples

S¯α j  S˜α j −

In this section, we examine in detail two commonly used structures, spherical shell and circular cylindrical shell, in order to facilitate applications and to validate the present theory. For axisymmetric deformations, some exact solutions are available in the literature, which will be utilized to compare with the approximate solutions obtained from the present shell theory.

In the following, we consider a concrete example with axisymmetric deformations, defined by

Conventionally for a spherical shell, the spherical coordinates (R, , ) are used with the domains

0 ≤  ≤ π,

0 ≤  ≤ 2π ,

(73)

X2 = R sin  sin ,

X3 = R cos .

(74)

In the notations of current shell theory, we have the corresponding relations

θ 1 = , θ 2 = , Z = R − A.

(75)

The base vectors on the base surface  (i.e., the inner surface of spherical shell) are calculated as

g1 = A(cos  cos e1 + cos  sin e2 − sin e3 ) = Ae = A g , 2 1

2

(76) where ei (i = 1, 2, 3) are cartesian base vectors and e , e , eR are the commonly adopted physical base vectors along the coγ β ordinates. The associated nonzero constants for αβ , καβ , κα in Appendix B are given by

 =  = cot ,

q− = q− eR ,

q+ = q+ eR .

(84)

As an illustrative example for comparison, we set μ = as a result the constants are explicitly given by

C2 =



A3 B3 q+

2 B3 − A3



, ν

1 C1 = 1+ 2



1 =− , A



B3 − A3

1 A

1 . A2

ν

= 0, and

 .

(85)

We point out that the exact solution given by (83,85) is valid provided that q+ > 12 ν (A3 /B3 − 1 ), which corresponds to the condition q > −3hν /A asymptotically as h/A tends to 0. By the present shell theory, we write r = r (Z ) (recall that R = A + Z in (75)) and expand it as

r ( Z ) = r0 + r1 Z +

1 1 r2 Z 2 + r3 Z 3 + · · · , 2 6

0 ≤ Z ≤ 2h = B − A, (86)

(77)

where the second line implies

κ = − 1, H = − , K =

q−

B3 ( 2q+ + ν ) − A3 ν

1 22

κ11 = −A, κ22 = −A sin2 , κ11 = κ22 1 A

(83)

 = − cos  sin ,

2 21

(82)

where the two constants C1 and C2 are to be determined by boundary conditions on the inner and outer surfaces of the spherical shell. On the two surfaces, we consider the dead-load case

2

g3 = sin  cos e1 + sin  sin e2 + cos e3 = eR = g3 = n,

2 12

1 (2ν − μ )I13 − ν I2 + μI3 + ν, 27

where√I1 , I2 , I3 are the three principal invariants of the stretch tensor FT F, and ν ≥ 0, μ ≤ 0 are two material constants. By the 3-D field equation, the exact general solution found in Ogden (1997) takes the form

g2 = A sin (− sin e1 + cos e2 ) = A sin e = A sin g , 2

(81)

r (R ) = C1 R + C2 /R2 ,

and they are related to rectangular cartesian coordinates by

X1 = R sin  cos ,

A ≤ R ≤ B.

As in Ogden (1997), we adopt the strain-energy function

(I1 , I2 , I3 ) =

5.1. The spherical shell

A ≤ R ≤ B,

x ( X ) = r ( R )eR ,

(80)

(78)

where for clarity subscripts rather than superscripts are used in the unknown coefficients ri . The nonzero components of Fij are given by

F11 =

A2 r , A+Z

A2 r sin  , A+Z 2

F22 =

F33 = r (Z ),

(87)

Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

and the three principal stretches of the stretch tensor are

λ1 = λ2 =

r , A+Z

see Appendix D. Therefore for this problem, the shell theory produces pointwise O(h2 )-correct results for the deformation r(Z).

λ3 = r (Z ).

(88)

The nonzero stress components are calculated as

∂ 1 1 = [ (2ν − μ )I12 ∂λ1 A2 9 − ν (λ2 + λ3 ) + μλ2 λ3 ], ∂ 1 S33 = = (2ν − μ )I12 − ν (λ1 + λ2 ) + μλ1 λ2 , ∂λ3 9 S11 = S22 sin  = 2

1 A2

(89)

where I1 = λ1 + λ2 + λ3 . Then, by expanding them with respect to Z, we easily get the explicit expressions for components S(0)ij , S(1)ij , S(2)ij , which are given in Appendix D. Now, we present the recursion relations and final equation for the general case (82,84). By the bottom traction condition (28), r1 can be solved as



r1 =



−2Ar0 (μ − 2ν ) − 3 A2 (μ − 2ν ) q− A2 − 2r0 ν A + r02 μ A2 ( μ − 2ν )

145

,

Remark. Although we only choose an illustrative example for q− = 0, the final conclusion that r(Z) is correct to O(h2 ) holds for the general dead-load case (82,84). Another widely investigated case (e.g., Chung et al., 1986) is that the inner surface of the spherical shell is subjected to a pressure P, i.e.,

q− = P λ1 λ2 = P (r0 /A )2 .

(97)

Although this is not a dead-load case, the above procedure can be readily applied to this case with only slight modifications and also produces O(h2 )-correct results. 5.2. The circular cylindrical shell For this structure, the cylindrical coordinates (R, , X3 ) are related to the cartesian coordinates by

X1 = R cos ,

X2 = R sin ,

X3 = X3 ,

(98)

0 ≤ X3 ≤ L.

(99)

with the domains (90)

A ≤ R ≤ B,

0 ≤  ≤ 2π ,

and the other unreasonable solution is discarded. For the recursion relation of r2 , the Eq. (32)1 reduces to

According to the present shell theory, we identify them as

S(0)11 κ11 + S(0)22 κ22 + 2Hq− + S(1)33 = 0,

θ 1 = , θ 2 = X3 , Z = R − A.

(91)

The base vectors on the inner surface of cylinder are utilized

which yields that

 r2 = −

(μ − 2ν )r12 + 9q− A2 − 2r0 (r1 (μ − 2ν ) + 9ν )A + r02 (μ + 16ν ) . A2 (2r0 + Ar1 )(μ − 2ν ) (92)

Similarly, one can derive the recursive formula for r3 (see Appendix D). The nontrivial shell Eq. (79)3 has the form

−2AS˜11 + 2hSˆ11 = −q˜3 ,





A2 q+ ( A + 2 h ) 2 − 4 h 3 A2 + 4 h2 ν r0 √ 

√ + 4 Ah 3 A 2 + 4 h 2 ν r 0 = 0 , which yields that

 1+

A ( A + 2h )2 q+



g1 = A(− sin e1 + cos e2 ) = Ae = A2 g1 , g2 = e3 = g2 ,

4 h3 + 3 A2 h

ν

(94)

2 .

(95)

Finally we make a comparison between the exact result by (83,85) and the approximate one from the shell theory. By setting R = A in the solution (83,85), we get the exact r0 . Then, expanding the exact and approximate r0 in terms of h provides the same first three terms

 √   1 √  1 3 r0 = A 6 + q + 2 3 q + 3 + q 1 +  h 12 3 q+3 

√ q 2 ( q + 3 ) 3/2 − 3 ( q − 6 ) h2 + + O ( h3 ), 9 A ( q + 3 ) 3/2

(96)

where we have utilized the scale q+ = qhν /A (note that this scale already gives finite deformations since the thickness is small). Clearly, the present r0 in (95) is correct to O(h2 ). From the recursion relations, the other coefficients ri (i = 1, 2, 3) can be recovered. By comparing them with those from the exact solution (83), we conclude that all ri (i = 1, 2, 3) have the correct terms up to O(h2 ),

(101)

g3 = cos e1 + sin e2 = eR = g = n. 3

The

associated

nonzero

constants

for

Appendix B are given by

(93)

which, by substituting all the recursion relations, provides an algebraic equation for r0 , see (D.3) in Appendix D. For the illustrative example with μ = q− = 0, it reduces to

1 r0 = A 1 + 4

(100)

1 A

κ11 = −A, κ11 = − ,

σ , κ , κβ αβ αβ α

in

(102)

which implies H = −1/2A and K = 0. For this structure, the final shell equations in (42) have the form

h 11 1 13 h 13 Sˆ + (S˜ − Sˆ ) = −q˜1 , A ,1 A A h 12 α 2 2 ˜ ˆ S,α − S,1 = −q˜ , A h S˜,αα3 − Sˆ,13 − AS˜11 + hSˆ11 = −q˜3 . A 1 S˜,αα1 −

(103)

In parallel with the previous subsection, we consider the axisymmetric deformations defined by

r = r ( R ),

θ = , x3 = λX3 ,

(104)

where (r, θ , x3 ) denote the current coordinates. And we still adopt the strain-energy function in (82). The exact general solution for r(R) found in Ogden (1997) takes the form

r (R ) =

1 (C1 − λ )R + C2 /R. 2

(105)

where the integrating constants C1 and C2 are to be determined by the boundary conditions on the inner and outer surfaces of the cylindrical shell. And, the parameter λ should be determined by the resultant axial load (the quantity q0 in (51) on the edge in our notations), but for simplicity we set λ = 1 in following illustrative examples. As the first example similar to the previous subsection, we set

μ = 0, λ = 1 , q − = q− eR = 0, q+ = q+ eR .

(106)

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Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

The constants in the exact solution (105) are given by

 9 3 + 8 8

C1 =

B2 (32q+ + 25ν ) − 25A2 ν



B2 − A2 ν

,

C2 =



From the present shell theory, the recursion relations are similar to the previous example (see Appendix D), and the nontrivial shell  equation leads to the following algebraic equation for r˜0 = (ν − P )r0 + Aν ,

A2 B2 q+ B2 − A2

. ν



(107) By the present shell theory, we also write r = r (Z ) and expand it as in (86). Accordingly, the nonzero components Fij and principle stretches are given by

F11 = A2 λ1 =

A2 r , A+Z

F22 = λ2 = λ,

1 A2

(108)

∂ ∂ ∂ , S22 = , S33 = . ∂λ1 ∂λ2 ∂λ3

(109)

For the case (106), the explicit expressions of their expansion coefficients S(k)ij (k = 0, 1, 2) are given in Appendix D. With a similar procedure as the previous subsection, the recursive formulas for ri (i = 1, 2, 3) can be obtained, see Appendix D. Finally, the nontrivial shell Eq. (103)  3 provides an algebraic equation for r0 or equivalently r˜0 = r0 + A,

√  √ 



− 4 2h A2 − hA + 2h2 ν r˜02 + 6 Ah A2 − hA + 2h2 ν r˜0 √ 

+ 2A q+ A3 + 2h(q+ + ν )A2 − 2h2 ν A + 4h3 ν = 0,

(110)

which yields the solution

⎛

r˜0 =

√ A √ ⎝ 4 2

 

h A2 − hA + 2h2 ν

+ 3⎠. (111)

By Taylor expansion, we get the three-term approximation for r0







 1 3 r0 = A 4q + 3 8q + 25 + 1 + q 1 + 16 4 

+

 



q (8q + 25 )3/2 − 30q + 75 h2 4A(8q + 25 )3/2

3 8q + 25

h (112)

+ O ( h3 ),

where we have also made use of the scale q+ = qhν /A. The expansion (112) is exactly the same as that from the exact solution in (105,107). Also, one can easily verify that the other coefficients ri (i = 1, 2, 3) are all correct to O(h2 ), therefore we conclude that the shell theory produces an O(h2 )-correct result for the deformation. We remark in passing that this conclusion is also true for the general dead-load case. In Fourney and Stern (1968), the stability of an inflated and extended cylindrical shell has been studied within finite deformation theory for incompressible materials. The equilibrium states are constructed by a finite membrane state and a superposed linearized deformation. Here, as the second example, we similarly consider the case subjected to an internal pressure but for above compressible material (82). We set

μ = 0, λ = 1, q− = q− eR = (Pr0 /A )eR , q+ = 0,

(113)

where P represents the pressure inside the cylindrical shell. For this case, the constants C1 and C2 in (105) are calculated as C1 = −

3





3 ( P − ν )A2 − 3B 2 P + 3B 2 ν

8 ν A2 + B 2 ( P − ν )  



+ 9P 2 − 2ν P + 25ν 2 A4 − 2B2 P 2 − 26ν P + 25ν 2 A2 + 25B4 (P − ν )2 ,

C2 = −

A2 B2 ((C1 − 1 )P + 2q+ ) 2



ν A2 + B 2 ( P − ν )



(115)



+ Aν −PA + 2hν A − 2h (P + ν )A + 4h (P + ν ) = 0, 3

2

2

. (114)

1



r˜0 = −

3



2 A2 (A + 2h )P − 4h A2 − hA + 2h2  √ √ 

√ 3 2 Ah A2 − hA + 2h2 ν (ν − P )





+ 18Ah2 A2 − hA + 2h2



2 



ν

ν ( ν − P )2

− 4Aν A2 (A + 2h )P − 4h A2 − hA + 2h2



(116)



ν

−PA3 + 2hν A2 − 2h2 (P + ν )A + 4h3 (P + ν )

1/2 

.

By Taylor expansion, one obtains the three-term approximation





2 A(2 p + 3 p˜ + 1 ) 3 p 8 p − 37 p + (5 p − 2 ) p˜ + 74 h r0 = − 2 ( p − 4) ( p − 4 )3 p˜

+

⎞

8q+ A3 + h(16q+ + 25ν )A2 − 25h2 ν A + 50h3 ν



which gives

F33 = λ3 = r (Z ).

The nonzero stress components can be computed according to

S11 =



A2 (A + 2h )P − 4h A2 − hA + 2h2 ν r˜02 √ √ 

√ + 3 2 Ah A2 − hA + 2h2 ν (ν − P )r˜0

 ph2 ( p(70 p − 47 ) + 40 ) p˜3 A( p − 4 )4 p˜3

(117)

+ 3( p( p( p( p(63 p − 598 ) + 2389 ) − 4308 )



+ 2689 ) + 2600 ) + O(h3 ), where we have used the scale P = phν /A and denoted p˜ =  2( p − 6 ) p + 25. Again this expansion (117) is the same as that

from the exact solution (105,114), and all the coefficients ri (i = 1, 2, 3) are all correct to O(h2 ) (omitted for brevity). Therefore for this internally pressurized case, the shell theory also produces pointwise O(h2 )-correct results for the deformation. In summary, the illustrative examples have convinced us that the present theory is asymptotically correct, and can produce highorder results in some cases. Besides the dead-load case, the theory also has the flexibility to deal with the pressurized case, and is expected to have wide applications. 6. Conclusions In this paper, a consistent finite-strain shell theory for a general shell of arbitrary geometry is developed, with no special restrictions on external loadings (like scalings with thickness). It includes the previous plate theory (Dai and Song, 2014) as a special case, and is also consistent with the 3-D weak formulation with an O(h4 ) error for each term. The derivation procedure is similar to that of a previously-derived plate theory in that recursion relations are derived from the 3-D differential system, but is not trivial considering the complicated effects due to the presence of curvatures. The present shell equation naturally preserves the local force balance. Proper 2-D boundary conditions and associated weak formulations are proposed, accommodating either the reduced cases from the 3-D edge conditions or various practical cases. The validity of the shell theory is substantiated by the concrete examples regarding axisymmetric deformations of spherical and circular cylindrical shells. In recent years, soft materials (e.g., gels, elastomers and polymers) have found broad applications. A basic characteristic of those materials is their ability to undergo finite-strain deformations. The present shell theory is established in the framework of finite-strains with a general strain energy function, which incorporates bending and stretching simultaneously, thus it may

Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

provide a general mathematical framework for studying deformations of thin structures composed of soft materials. Acknowledgments The work described in this paper was supported by a GRF grant (Project No.: CityU 11303015) from the Research Grants Council of Hong Kong SAR, China and a grant from the National Nature Science Foundation of China (Project No.: 11572272). Appendix A. Simplification of the shell equation Here we intend to simplify the vector shell Eq. (39) to the form (40), by utilizing (28,32), Cayley–Hamilton (C-H) formula κ2 = 2H κ − K1 and the following identities

∇ · S(i) = ∇ · (1S(i) ) − 2HS(i)T n, [κT gα ] · S,(αi ) = [κT gα ] · (1S(i ) ),α − (gα · κ2 gα )S(i )T n = [κ

T

gα ] · (1S(i ) )

,α

− ( 4H − 2K )S 2

( i )T

n,



8 LHS = 1 − 4hH + h2 K ∇ · S(0) + h(1 − 4hH )∇ · S(1) 3 4 2 2 (2 ) + h ∇ · S + h(1 − hH )[κT gα ] · S,(α0) 3 3 4 + h2 [κT gα ] · S,(α1) 3     8 8 (A.1 )1 ,(32 )2 = 1 − 4hH + h2 K ∇ · S(0) + h 1 − hH ∇ · S(1) 3 3 2 2 4 2 T α (0 ) (2 ) T α + h ∇ · (1S ) + h[κ g ] · S,α + h [κ g ] · S,(α1) 3 3   8 2 (A.1 )1 ,(32 )1 2 (0 ) = C0 + h (K − 2H ) ∇ · S + hC1 ∇ · (1S(1) ) 3 2 2 4 + h ∇ · (1S(2) ) + h[κT gα ] · S,(α0) + h2 [κT gα ] · S,(α1) 3 3 2 2 (A.1 )2 (1 ) (0 ) = C0 ∇ · S + hC1 ∇ · (1S ) + h ∇ · (1S(2) ) 3 4 2 T α (0 ) T α + h[κ g ] · S,α + h [κ g ] · (1S(1) ),α 3 2 (A.1 )1,2 ,(28 ) (0 ) = C0 ∇ · (1S ) + hC1 ∇ · (1S(1) ) + h2 ∇ · (1S(2) ) 3 4 2 T α T α (0 ) + h[κ g ] · (1S ),α + h [κ g ] · (1S(1) ),α 3 + [2HC0 + h(4H 2 − 2K )]q− (A.1 )3

=

∇ · (1S˜ ) + 2hSˆ T ∇ H + h[κT gα ] · (1Sˆ ),α + (2H − 2hK )q−

=∇

· (1S˜ + hκSˆ ) + h(1Sˆ )T [2∇ H − ∇ · κ] + (2H − 2hK )q−

(B.9 )

=

∂ gα σ g + κ n, with  σ = ∂ gα · gσ , = αβ σ αβ αβ ∂θ β ∂θ β ∂n β β = −καβ g = −κα gβ , ∂θ α

∇ · (1S˜ + hκSˆ ) + (2H − 2hK )q− , (A.2)

∂ gi β β 3 = ipβ g p , with 3α = −κα , αβ = καβ , 33β = 0. ∂θ β (B.3) Then, the covariant derivative for a tensor under base gi is defined as follows δ g  g + Sβ j  p g  g 1(1S ),α = S,α gβ  g j + Sβ j βα p j δ jα β



βj



β

(B.4)

βj

 S ;α g β  g j , which immediately implies

∇ · (1S ) =S;ααj g j , [κT gα ] · (1S ),α = κδα gδ · (1S ),α = κδα S;δαj g j . (B.5) With S replaced by S˜ + hκSˆ in (B.5)1 , one obtains the component form of the first term in the shell Eq. (40). Similarly, we define the covariant derivative for a tensor with β mixed base, like κ = κγ gβ  gγ , through β

β

β

γ

δ g  gγ − κ  g  gδ 1(κ ),α = κγ ,α gβ  gγ + κγ βα γ αδ β δ β γ

+ κγ κα gβ  n =



 β β δ β γ κγβ,α + κγδ δα − κδ αγ gβ  gγ + κγ κα gβ  n β

β γ

α gσ + κ α n. Then, it follows where we have used ∂ gα /∂θ β = −βσ β that

∇ · κ =κγα;α gγ + κγα καγ n.

The Mainardi–Codazzi equations can be written as (Steigmann, 2012)

κ11;2 = κ21;1 , κ12;2 = κ22;1 ,



(B.8)

then one can easily check

1[2∇ H − ∇ · κ] = [(2H ),β − κβα;α ]gβ = [(2H ),β − καα;β ]gβ = 0. (B.9)

It follows from (25,35) that

The stress tensor S has the decomposition like



(B.7)

Appendix C. Decomposition of F(2)

Appendix B. Covariant derivatives



(B.6)

 κγ ;α gβ  gγ + κγ κα gβ  n,

where (B.9) can be found in Appendix B, C0 = 1 − 2hH, C1 = 1 − 8 ˜ ˆ 3 hH, and the two stresses S and S are defined in (41). Then, the Eq. (40) follows by rearranging the terms.

S =Si j gi  g j = 1S + n  ST n = gα  ST gα + n  ST n

(B.2)

which can be universally written as

= S,α + Sσ j σ α + Sβ k kjα gβ  g j

for i = 0, 1, 2 and any scalar function C. More precisely, the lefthand side of (39) is calculated as



On the base surface, the equations of Guass and Weingarten read

βj

(A.1)

C ∇ · (1S(i ) ) = ∇ · (C1S(i ) ) − S(i )T ∇ C,

C−H

147



= Sαβ gα  gβ + Sα 3 gα  n + S3α n  gα + S33 n  n .

(B.1)

F(2) =2(∇ x(1) )κ + 2(∇ x(0) )κ2 + ∇ x(2) + x(3)  n, x(3) = − B−1 f(3) .

(C.1)

148

Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

To get a decomposition of F(2) , we first decompose f(3) in (36) as follows



f (3 ) =

α

κT g · S,(α0) + ∇ · S(1) 2 

A1

+

i=0



2 (∇ x(i) )κ2−i i!

A1

+

+ A2 [F ( 1 ) , F ( 1 ) ]



+ A2 [F ( 1 ) , F ( 1 ) ]

n

(C.2)

where (A.1)1 has been used in the second equality and the two braces define respectively f2(3 ) and f1(3 ) . In the above derivation, the term ∇ · (2H1S(0) ) is added and subtracted to get the desired form. Then we define (2 )



F1 =2(∇ x(1) )κ + 2(∇ x(0) )κ2 − B (2 )

F2 =∇ x

(2 )



− B

−1

 (3 )

f2

−1 (3 )

f1



(D.3) where



 n,

(C.3)

 n.

Appendix D. Expressions in the examples

(D.4)

The asymptotic expansions for ri (i = 1, 2, 3) from both the exact solution and present shell theory are given by





√  1 qh r1 = 3−q+ 3 q+3 + 6 3A

 

q 4 q + 3q +

√ 3 

q+3



−2

 √ √  3q + 12 q + 3 − 6 3 h2

9 A2 ( q + 3 ) 3/2

+ O ( h3 ),

q 2qh 4qh2 + 2 + + O ( h3 ), 2A A 3A3 2q 8qh 16qh2 r3 = − 2 − 3 − + O ( h3 ). A A 3A4 r2 =

D1. The spherical shell The expanded coefficients of nonzero stresses in (89) are given by

(D.5) −r1 (r1 (μ − 2ν ) + 9ν )A2 + r0 (5r1 μ + 8r1 ν − 9ν )A − 4r02 (μ − 2ν ) , 9 A4

1  = 5 (5μ + 8ν )r12 + (−2Ar2 μ + 4Ar2 ν − 9ν )r1 − 9Ar2 ν A2 9A  + r0 (−13r1 μ + 5Ar2 μ + 8r1 ν + 8Ar2 ν + 9ν )A + 8r02 (μ − 2ν ) ,

S(0)11 =

 1  = 6 (−18μr12 + −2r3 (μ − 2ν )A2 + 3r2 (5μ + 8ν )A + 18ν r1 9A



  + A A −2(μ − 2ν )r22 − 9r3 ν − 9r2 ν )A2 − 24r02 (μ − 2ν ) 



+ r0 r3 (5μ + 8ν )A2 − 18r2 μA + 6r1 (7μ − 8ν ) − 18ν A , −9μr02 + 18Aν r0 + (2r0 + Ar1 )2 (μ − 2ν ) , 9 A2 2  =− r1 (2r1 (μ − 2ν ) + Ar2 (μ − 2ν ) + 9ν )A2 9 A3  + r0 (2Ar2 (μ − 2ν ) − 9ν − r1 (7μ + 4ν ))A + r02 (5μ + 8ν ) ,

S(0)33 = − S(1)33





A2 (μ − 2ν ) q− A2 − 2r0 ν A + r02 μ .

−

S(2)11

(D.2)

  (q− + q+ )t (μ − 2ν )A5 + 4h(μ − 2ν ) − 3q− ν A4 + 3r0 2ν 2 + q− μ A3

 

 + −9μν r02 + q− t + q+ t A2 + 3 r03 μ2 − r0 t ν A + 3r02 t μ A2   + h2 − 72q− r0 μ(μ − 2ν )A4 − 4 2q− t (4μ + ν ) 

−(μ − 2ν ) 36μν r02 + q+ t A3 2

 − 72r0 μ r0 μ(μ − 2ν ) − t ν A2 − 72r02 t μ(μ − ν )A

  + 4h3 − 7q− (μ − 2ν )ν A4 + r0 (μ − 2ν ) 8ν 2 + 79q− μ A3

 + 2 q− t (19μ − 2ν ) − 78r02 μ(μ − 2ν )ν A2 

+ 4r0 19r02 (μ − 2ν )μ2 + t ν (2ν − 19μ ) A  + 4r02 t μ(19μ − 20ν ) = 0,

t=

The purpose of such a decomposition is to make sure that all the third-order derivatives of x(0) (after using recursion relations) are kept in the part F2(2 ) .

S(1)11



The algebraic equation for r0 for the general dead-load case takes the form

f2(3) + f1(3) ,

(2 )







F(2) =F1 + F2 ,

 1 − 2r22 (μ − 2ν )A4 (2r0 + Ar1 )(μ − 2ν )

+ 4r0 (4r1 (μ − 2ν ) − 9ν )A + 2r02 (32ν − 7μ ) .

n

∇ · [κS(0) + 1S(1) − 2H 1S(0) ] + (A1 [∇ x(2) ] )T n 

T + A1 [2(∇ x(1) )κ + 2(∇ x(0) )κ2 ] + A2 [F(1) , F(1) ] n  − (∇ · κ ) · S(0) − 2HS(1)T n + ∇ · (2H 1S(0) )

(2 )

2A3

− 12r1 r2 (μ − 2ν )A3 + 45q− − 2r12 (μ − 2ν ) A2

T

i=0



=

2  2 (∇ x(i) )κ2−i i!

r3 =

T

=∇ · (κS(0) ) − (∇ · κ ) · S(0) + ∇ · (1S(1) ) − 2HS(1)T n



The recursion relation for r3 is given by

 2  (−9μr12 + r3 (μ − 2ν )A2 + 6r2 (μ − 2ν )A − 18ν r1 9 A4 + Ar2 (Ar2 (μ − 2ν ) + 9ν ))A2 − 3r02 (5μ + 8ν )

The expanded coefficients of nonzero stresses in (109) are given by

S

(0 )11

 =

S(2)11 =

S(0)22 =

+ r0 2r3 (μ − 2ν )A2 − 9r2 μA + 18ν + 24r1 (μ + ν ) A ,

, 9A4

A2 (4r0 + A(4r1 − 5 ))r2 − 4(r0 − Ar1 )(r1 A + A + r0 ) ν

ν 



9A5

4r22 + (4r1 − 5 )r3 A4 + 4(3r1 r2 + r2 + r0 r3 )A3 9A6  −4(r1 (r1 + 2 ) + r0 r2 )A2 − 8r0 (r1 − 1 )A + 12r02 ,

(r0 + A(r1 − 2 ))(2r0 + A(2r1 − 1 ))ν

ν 

9A4



4r22 + (4r1 − 5 )r3 A4 + ((12r1 − 5 )r2 + 4r0 r3 )A3



−2(r1 (2r1 − 5 ) + 2r0 r2 )A2 − 2r0 (4r1 + 5 )A + 12r02 ,



(D.1) and S(k )22 = S(k )11 / sin2 , (k = 0, 1, 2).

(r1 + 1 )(2r1 − 7 )A2 + 4r0 (r1 + 1 )A + 2r02 ν

, 9A2 (4r0 + A(4r1 − 5 ))(A(r1 + Ar2 ) − r0 )ν = , 9A3

S(2)22 =





S(1)11 =

S(1)22

S(2)33 = −



D2. The circular cylindrical shell

S

(0 )33

=

(2r1 (r1 + 2 ) − 7 )A2 + r0 (4r1 − 5 )A + 2r02 ν 9A2

,

,

Z. Song, H.-H. Dai / International Journal of Solids and Structures 97–98 (2016) 137–149

 S

(1 )33

S

(2 )33

= =

4(r1 + 1 )r2 A3 + (r1 (4r1 − 5 ) + 4r0 r2 )A2 + 5r0 A − 4r02

ν  



9A3

ν

,

4 + r1 r3 + r3 A + ((12r1 − 5 )r2 + 4r0 r3 )A3 9A4  −2(r1 (2r1 − 5 ) + 2r0 r2 )A2 − 2r0 (4r1 + 5 )A + 12r02 . (D.6) r22

4

The recursion relations for ri (i = 1, 2, 3) are given by

√  −2ν A2 − 2r0 ν A + 3 2 A3 ν (r0 ν + A(ν − q− )) , 2  −  2

22A ν 9q − 2r1 + 7 ν A + r0 (4r1 − 1 )ν A + 6r02 ν r2 = , 4A2 ( r1 A + A + r0 )ν  1 r3 = 3 − 4r22 ν A4 − 4(3r1 + 1 )r2 ν A3 4A ( r1 A + A + r0 )ν r1 =

+



(D.7)



ν − 9q− A2 

2r12 + 4r1 + 4r0 r2 + 7

+ r0 (4r1 − 3 )ν A − 14r02 ν .

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