A new CNT-oriented shell theory

European Journal of Mechanics A/Solids 35 (2012) 75e96

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A new CNT-oriented shell theory Antonino Favata*, Paolo Podio-Guidugli Dipartimento di Ingegneria Civile, Università di Roma TorVergata, Via Politecnico 1, 00133 Rome, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 April 2011 Accepted 21 January 2012 Available online 1 February 2012

A theory of linearly elastic orthotropic shells is presented, with potential application to the continuous modeling of Carbon NanoTubes. Two relevant features are: the selected type of orthotropic response, which should be suitable to capture differences in chirality; the possibility of accounting for thickness changes due to changes in inter-wall separation to be expected in multi-wall CNTs. A simpler version of the theory is also proposed, in which orthotropy is preserved but thickness changes are excluded, intended for possible application to single-wall CNTs. Another feature of both versions of the present theory is that boundary-value problems of torsion, axial traction, uniform inner pressure, and rim flexure, can be solved explicitly in closed form. Various directions of ongoing further research are indicated.
2012 Elsevier Masson SAS. All rights reserved.

Keywords: Shell theory Single- and multi-wall carbon nanotubes Torsion, traction and pressure problems Rim-flexure problems

1. Introduction The application that motivated this work is the modeling of carbon nanotubes (CNTs). When CNTs are employed as nanodevice components, they are regarded as elastic beam-like or shell-like objects and their mechanical response is characterized in terms of an as-small-as-possible number of stiffness and inertia parameters. To define and evaluate these parameters is the common goal of all modelers; a way to achieve it is to try and bridge between the microscopic scale of molecular mechanics and the macroscopic scale of continuous structure mechanics, by way of a mesoscopic scale, at which concepts from discrete structure mechanics apply. At the onset of putting together a bottom-up model of this sort (Bajaj et al., forthcoming), we realized that ordinary shell theories, which presume an isotropic three-dimensional response of the material comprising the shell, could not possibly guarantee an accurate macroscopic account of the mesoscopic texture of singlewall carbon nanotubes (SWCNTs): a glance to armchair and zig-zag CNTs (Fig. 1) suggests instead an orthotropic response in planes orthogonal to radial directions (see Fig. 2, where the three little cylinders suggest what probes one should cut out of a cylindrical shell-like body in order to determine its material moduli). Moreover, when modeling multi-wall carbon nanotubes (MWCNTs), it seems to us important to allow for thickness distension: we conjecture that thickness changes are essentially due to changes in the inter-wall distances, as a consequence of the interplay of the

* Corresponding author. E-mail address: [email protected] (A. Favata). 0997-7538/$ e see front matter
2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2012.01.006

applied loads with the van der Waals interactions between adjacent walls. A search of the literature convinced us that we better produced such a shell theory ourselves. This paper describes the results of our efforts, results that have been partly anticipated in (Favata and Podio-Guidugli, 2011) and that e so we believe e may find application also in contexts different from the mechanics of nanotubes. In the next section, we expound the general lines of a theory of linearly elastic orthotropic shells of constant referential thickness 23 , whose geometry (Section 2.1) is dictated by a piecewise smooth referential model surface S. Imitating the classic approach of Kirchhoff, we specify the admissible kinematics (Section 2.2) by choosing for the displacement field in the tubular region GðS; 3 Þ a representation parameterized by a few fields defined over S. In the deformations we envisage, (i) thickness may change, if the applied loads require and the boundary conditions permit; (ii) material fibers orthogonal to S must remain orthogonal to the deformed shape of S itself, an internal constraint we refer to as unshearability. Both the balance and the constitutive equations of our shell theory (Sections 2.3 and 2.4, resp.) are inherently consistent with the corresponding equations of three-dimensional linear elasticity: - the balance equations follow from a two-dimensional Principle of Virtual Powers that is a direct consequence of stating the corresponding three-dimensional Principle for all virtual velocity fields in the linear space to which admissible displacements belong; they are expressed in terms of a pair of two-dimensional stress measures that are defined as weighted thickness averages of the three-dimensional stress field in GðS; 3 Þ (Section 2.5);

76

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

Fig. 1. Roll-up and chiral axes of armchair and zigzag carbon nanotubes. Note the orthogonality of zigzag (red) and armchair (blue) atom sequences. (For interpretation of reference to color in this figure caption, please refer to web version of this article.)

be defined for a cylindrical shell, regarded as a traction or torsion probe (Sections 5.3 and 5.4). Next, in Section 6, we take up orthotropic shells whose thickness is constitutively immutable, a class that we designate by the names of Kirchhoff and Love by analogy with the corresponding classic plate theory. We adapt to the simpler case of Kirchhoff-Love cylindrical shells all the formulas derived in Sections 4 and 5, both for whatever thinness and in the small thickness limit; in the latter case, we show how the four constitutive moduli characterizing the mechanical response could be determined on the basis of simple real or computer experiments. In our final Section 7, we briefly recapitulate our main findings, and we indicate the directions of our future research, with special attention to the application to CNTs of the concepts and methods developed in the present paper.

2. General theory - the constitutive equations are arrived at when the threedimensional constitutive equations for unshearable orthotropic materials are inserted in the definitions of the twodimensional stress measures. The remaining part of the paper is dedicated to cylindrical shells. We begin with shells whose thickness can change. In Sections from 3.1 to 3.4, we parallel and specify the developments of Section 2 as to, respectively, geometry, kinematics, balance laws, and constitutive equations. Then, we confine attention to axisymmetric equilibrium problems, and solve explicitly and exactly those of torsion and axial traction (Sections 4.1 and 4.2) e the cases for which experimental tests and numerical simulations seem to be especially easy to set up for CNTs e as well as the problems of pressure and rim flexure (Sections 4.3 and 4.4). Finally, we lay down natural geometrical notions of thinness and slenderness, and we show how remarkably the analytical solutions derived in the previous section simplify for slender shells (Sections 5.1 and 5.2), and how effective contraction moduli and effective stiffnesses can

2.1. Geometry In this opening section we recapitulate some well known notions, with the main purpose of introducing our notation and terminology. Following the approach to construct a shell theory proposed in (Podio-Guidugli, 1991), we let S denote a compact, regular, orientable and oriented surface embedded in the three-dimensional Euclidean space E, and we let x denote its typical point and n (x) the value of its normal vector field at x, with jnðxÞjh1. We choose an origin o˛E, and denote by x: ¼ xo the position vector of x with respect to o. We assume that S admits a tubular 3  neighborhood GðS; 3 Þ (see Section 2.2 of (Do Carmo, 1976)) and a global parametrization



R2 ISH z1 ; z2 1x z1 ; z2 ˛S3E (here S is an open set). A point p˛GðS; 3 Þ has position vector

p : ¼ p  o ¼ x  o þ znðxÞ;

x˛S; z˛I : ¼ ð3 ; þ3 Þ;

with respect to o; jzj is the distance of p from x, the point where the straight line through p perpendicular to S intersects S itself. The mapping




 

z1 ; z2 ; z 1p z1 ; z2 ; z : ¼ x z1 ; z2 ; z þ zn x z1 ; z2 is a global parametrization of GðS; 3 Þ, with ðz1 ; z2 ; zÞ the triplet of normal curvilinear coordinates of p (Fig. 3). We term the region GðS; 3 Þ of E a shell-shaped region, of model surface S and constant thickness 23 . The thickness of GðS; 3 Þ can be visualized as the length, whatever x˛S one picks, of the material fiber F ðxÞ through x perpendicular to S; clearly, F ðxÞ : ¼ fp˛GðS; 3 Þjp ¼ x þ znðxÞg.

Fig. 2. Shell axis is chosen parallel to roll-up axis.

Fig. 3. Geometrical equipment of a typical shell-shaped region.

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

The chosen parametrization induces a system of coordinate curves on S, described by the mappings za 1xðz1 ; z2 Þða ¼ 1; 2Þ. The tangent space T x : ¼ spanfea ðxÞg to S at x h (z1,z2) is spanned by the tangent vectors to the coordinate curves at that point:




1

ea z ; z

2






1

2

: ¼ x;a z ; z



:

The metric tensor G is defined to be:

G : ¼ g i 5g i ¼ g i 5g i ; with s

On taking the normal field to be n(x) ¼ vers (e1(x)  e2(x)), and on setting

2ðxÞ : ¼ e1 ðxÞ  e2 ðxÞ$nðxÞ>0; the covariant and contravariant bases at x are, respectively, {e1(x),e2(x),n(x)} and {e1(x),e2(x),n(x)}, where a

2ðxÞea ðxÞ : ¼ ð1Þ nðxÞ  eaþ1 ðxÞ ðmodulo 2; a not summedÞ: For the rest of this subsection we leave the indication of the typical point x˛S tacit. Accordingly, we write

77

G : ¼ g a 5g a

its surface part. The covariant bases {ei} at x˛S and {gi} at x þ znðxÞ ¼ p˛GðS; 3 Þ are related by the shift tensor  briefly, the shifter  A:

Aðx; zÞ ¼ g i 5ei 5Aek ¼ g k ; it can be shown (Podio-Guidugli, 1991) that the ratio of the volume measures at p and x is equal to the determinant of A:

dvolðpÞ ¼ aðx; zÞdvolðxÞ; a : ¼ detA:

(2)

The shifter

Bðx; zÞ : ¼ g i 5ei ¼ s B þ n5n;

for the metric tensor and

maps the controvariant basis at x into the controvariant basis at p. We have that

s

P : ¼ ea 5ea ¼ ea 5ea

for the surface metric tensor. A vector field y defined over S can be represented both in the covariant basis and in the controvariant basis:

v ¼ yi ei ¼ yj ej ;

yj : ¼ v$ej ;

with

in terms of its covariant, contravariant, or mixed components Tij ¼ T$ei5ej, Tij ¼ T$ei5ej, or Tji ¼ T$ei 5ej and Tij ¼ T$ei 5ej ði; j ¼ 1; 2; 3Þ. Remark. The physical dimensions of covariant and contravariant basis vectors, and hence of the corresponding components of vectors and tensors, may differ. To circumvent this difficulty is easy, whenever it so happens that

e1 $e2 ¼ 0: Simply, one introduces the so-called physical basis at x, that is to say, the orthonormal basis

g 3 : ¼ p;3 ¼ n;

(1)

where

is the curvature tensor of the oriented surface S. We have here denoted by sV the operation of taking the surface gradient of a smooth vector field y over S: sVv ¼ v,a 5 ea. Likewise, we denote by V the gradient of a vector field v defined over GðS; 3 Þ: Vv ¼ v,i 5 gi. Two divergence operators are associated with the gradient operators, Div v: ¼ Vv$G and sDiv v: ¼ sVv$sP, where the field v is defined, respectively, over GðS; 3 Þ and over S. The surface divergence of a tensor field T over S is defined as follows: sDiv(TTv) ¼ :sDiv T$v, for all constant vectors y. ~ the fiber-wise constant extension to GðS; 3 Þ of a vector field Let a ~ , we have that a defined over S. When taking the gradient of a

In the following, we will not make any notational distinction between a field defined over S and its fiber-wise constant extension to GðS; 3 Þ: e.g., we shall write:

Va ¼ ðs VaÞs BT :

(4)

2.2. Kinematics The three-dimensional strain measure we use is the standard symmetrized gradient of the displacement field:

and

2 g a : ¼ ð1Þa n  g aþ1 ðmodulo 2; a not summedÞ;

(3)

~ ¼ a ~ ;a 5g a ¼ a ~ ;a 5ðs B ea Þ ¼ ðs VaÞs BT : Va

ea ea ¼ a : jea j je j

We shall be making use of physical bases and components (e.g., v :¼ v$e) in Section 3, where we deal with shells whose model surface is a right circular cylinder. At a point of GðS; 3 Þ, the covariant and contravariant basis vectors are, respectively,

g a : ¼ p;a ¼ ea þ zn;a ;

Aðx; zÞ : ¼ g a 5ea

W : ¼ s Vn ¼ n;a 5ea

j

e : ¼

s

¼ s PðxÞ  zWðxÞ;

T ¼ T ij ei 5ej ¼ Tij ei 5ej ¼ Tji ei 5ej ¼ Ti ei 5ej ;

2 : ¼ g1  g2 $n:

Note that

ði; j ¼ 1; 2; 3Þ:

fe<1> ðxÞ; e<2> ðxÞ; nðxÞg;

ABT ¼ G:

Aðx; zÞ ¼ s Aðx; zÞ þ nðxÞ5nðxÞ;

yi : ¼ v$ei ;

Analogously, a second-order tensor field T can be represented as

where

BT A ¼ P;

s

B : ¼ g a 5ea ;

P : ¼ ei 5ei ¼ ei 5ei

g 3 ¼ n;

EðuÞ ¼ sym Vu : ¼

 1
Vu þ VuT : 2

It follows from this definition that the covariant components of E are:

78

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96



2Eij ¼ 2E$g i 5g j ¼ u;k 5g k þ g k 5u;k $ g i 5g j  

¼ ðu$g i Þ;j þ u$g j ;i  u$ g i;j þ g j;i ;

kinematic Ansatz typical of the linear theory of Kirchhoff-Love shells (Podio-Guidugli, 1991); interestingly, the displacement field:

uKL ðx; zÞ ¼ s Aðx; zÞaðxÞ þ wðxÞnðxÞ  zs VwðxÞ;

aðxÞ$nðxÞ ¼ 0; (13)

whence

can be shown to be the general solution of the system of three PDEs that expresses the Kirchhoff-Love constraint, namely,


1
Eij ¼ ui;j þ uj;i  u$ g i;j þ g j;i : 2

(5)

We restrict attention to shell-shaped bodies GðS; 3 Þ whose admissible deformations may induce thickness changes but must keep the material fibers orthogonal to the model surface, in the sense that, whatever x˛S, the deformed material fiber uðF ðxÞÞ must be found orthogonal to the deformed model surface uðSÞ. This pointwise internal constraint  a restriction on admissible displacement fields that, as anticipated, we refer to as unshearability  can be expressed in terms of the linear strain measure E in the following form:

EðuÞn$y ¼ 0

for all y such that y$n ¼ 0;

(6)

or rather, equivalently, as

EðuÞn$g a ¼ 0

in GðS; 3 Þ;

(7)

EðuÞn ¼ 0

in GðS; 3 Þ

(14)

(cf. (7)). Remark. That the fiber stretch g defined by (12)3 be constant at any point of GðS; 3 Þ to satisfy the second of (10) is a counterintuitive consequence of expressing the unshearability constraint in terms of the linear measure of deformation EðuÞ. In fact, on adopting one of the standard exact deformation measures, namely,

D :¼

 1
ðVf ÞT ðVf Þ  I ; 2

where

f ðpÞ : ¼ p þ uðpÞ;

one finds that

1 D ¼ E þ H T H; 2

H : ¼ Vu:

with the use of (1) and (5), (7) becomes: Thus, the exact conterpart of (7) is:

ðu$g a Þ;3 þ ðu$nÞ;a  2u$n;a ¼ 0;

(8)

with the same quantification. We look for solutions of this system of two PDEs having the following form: ð0Þ

ð1Þ

uðx; zÞ ¼ u ðxÞ þ z u ðxÞ;

(9)

note, in particular, that

1 0 ¼ Dn$g a ¼ En$g a þ Hn$Hg a ; 2 where the term quadratic in jHj does not vanish in general. In particular, for u of the form 9, we find that ð1Þ ð0Þ

ð1Þ ð1Þ

Hn$Hg a ¼ u $ u ;a þ z u $ u ;a ;

ð1Þ

u ðxÞ$nðxÞ ¼ EðuðxÞÞ$nðxÞ5nðxÞ

hence, the second of (10) is replaced by

is the stretch of the material fiber F ðxÞ, uniform all along it. Substituting (9) into (8), and exploiting the quantification with respect to the variable z, we find the following restrictions on the

ð1Þ2 
ð1Þ   u $n þ  u 

ð0Þ ð1Þ

;a

¼ 0;

and intuition rings no bell.

choice of the fields u ; u over S: ð1Þ

ð0Þ

u $ea ¼  u ;a $n;

2.3. Balance Assumptions


ð1Þ  u $n ;a ¼ 0:

(10)

The second of (10) tells us that, for (7) to have a solution of type (9), the fiber stretch must in fact be uniform all over GðS; 3 Þ; ð1Þ

moreover, the tangential part of u must be expressed as follows in

Given a material body occupying a three-dimensional open and bounded region U and a velocity field y over U, the internal power expenditure over a part P of U associated with y is:

Z

ð1Þ

terms of u and the curvature tensor of S:

P


ð1Þ 
ð0Þ  ð0Þ u  u $n n ¼ s V u $n  W u :

ð1Þ

where S denotes the stress field in U; and, the external virtualpower expenditure over P is:

With the use of 3, we conclude that we should pick:

uS ðx; zÞ : ¼ s Aðx; zÞaðxÞ þ wðxÞnðxÞ þ zð  s VwðxÞ þ gnðxÞÞ;

(11)

where we have set: ð0Þ

a :¼ u 


ð0Þ  u $n n;

Z P

ð0Þ

w : ¼ u $n;

S$Vy

ð1Þ

g : ¼ u $n:

(12)

Note that the displacement field (11) is parameterized by two fields over S, the tangential vector field a and the scalar field w, and by the constant g.1 Note also that, by taking g ¼ 0, we recover the

1 A vector field a (a tensor field T) defined over S is termed tangential if it so happens that a(x),n(x)h0 (Tn h 0).

do $y þ

Z

c o $y;

vP

where (do, co) denote, respectively, the distance force for unit volume and the contact force per unit area exerted on P by its own complement with respect to U and by the environment of the latter. These representations of power expenditures are those typical in the theory of the so-called simple material bodies. In that theory, a part is customarily a subset of non-null volume of U (which makes it for a part collection deemed sufficiently rich), and a virtual velocity field is a smooth vector field whose support is a part; the Principle of Virtual Powers is the stipulation that

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

Z

Z

S$Vy ¼

U

do $y þ

Z

co $y

and

(15)

vU

U

f ð3Þ ðxÞ : ¼

for all virtual velocities in a collection modeled after a chosen collection of admissible motions, an alternative stipulation being

Z

Z

S$Vy ¼

P

do $y þ

Z

co $y

for all parts P of U and for all velocity fields y consistent with the admissible motions. The virtual-power principle is interpreted as a balance statement for the internal and external fields S and do, co. We use here below a restricted version of the three-dimensional statement (16) to deduce a two-dimensional Principle of Virtual Powers and the associated balance laws for shells in terms of twodimensional stress measures and applied loads. The first restriction we introduce has to do with the special shape of the threedimensional bodies we consider: they must be shell-shaped in the sense of Section 2.1. The second restriction stems from the special class of admissible body parts we choose: they all must have the same thickness of the shell-like body. Thirdly and lastly, we pick a special class of virtual velocities, consistent with the representations of admissible displacements discussed in the previous section.

ð1Þ

and the standard divergence theorem that

Z

Z

S$Vy ¼

þ

Z

(17)

do $y ¼

P

Z

ð1Þ

co $y ¼

P

ð1Þ

F$s V y þ s M$s V y þ f

Z

1

0 ð0Þ

ado A$ y þ @

I

Z

1

1 ð1Þ

azdo A$ y A

I

aðx; zÞzSðx; zÞs Bðx; zÞdz

Z





ð0Þ




ð1Þ

  þ þ   aþ c þ o þ a co $ y þ 3 a co þ a co $ y

Z Z

   Z ð0Þ ð1Þ zco $ y co $ y þ

I

I

(20) for the contact force, where we have set:

a ðxÞ : ¼ aðx; 3 Þ;

$y :

c o ðxÞ : ¼ c o ðx; 3 Þ;

x˛P:

We are now in a position to define the load fields over S induced by the three-dimensional fields (do, co). These are:

 

I Z

IZ

I

@@

aðx; zÞSðx; zÞg a ðx; zÞdz 5ea ;

aðx; zÞSðx; zÞs Bðx; zÞdz ¼

:¼

00

3

 ð3Þ ð1Þ

Z

Z

s MðxÞ

 ð1Þ  s Divs M þ f ð3Þ $ y

ð0Þ ð1Þ Fm$ y þ s Mm$ y :

vP

Here we have made use of the following definitions:

:¼

s

þ

P

s FðxÞ

Z

P

ð1Þ

so that, having recourse to Fubini theorem and recalling (2), the internal power expenditure reads: s

Z





for the distance force, and

S$Vy ¼ ðS s BÞ$s V y þ zðS s BÞ$s V y þ Sn$ y ;

S$Vy ¼

ð0Þ

 s Divs F$ y þ

P

vP

ð0Þ

Div T$y;

P

2.3.2. External power expenditure. applied loads As to power expenditure of the applied forces, we find:

ð0Þ ð1Þ ð1Þ s V y þ zs V y s B þ y 5n:

Z


s

vP

Hence,

Z

Z
P

P

y . On recalling 4, we have that:

ð0Þ

Z

where, if t is the tangential vector to the curve vP in the tangent plane to the surface S, m ¼ t  n is the normal to vP. Thus,

2

Vy ¼

Tm$y 

vP

ð0Þ




Z

T$s Vy ¼

P

with P the intersection of the supports of the vector fields y and ð1Þ

(19)


T$s Vy ¼ s Div T T y  s Div T$y

2.3.1. Internal power expenditure. Stress measures The first two restrictions we listed imply that a typical part of UhGðS; 3 Þ can be identified with the Cartesian product P ¼ PI of P, an open subset of S, and the interval I. As to virtual velocities, we choose them of the form (9): ð0Þ

aðx; zÞSðx; zÞnðxÞdz;

for the two-dimensional stress measures that we call, respectively, force tensor, moment tensor, and shear vector. It is important to observe that the force and moment tensors are tangential. Now, for a tangential tensor field T over S, it follows from the general identity

(16)

yðx; zÞ ¼ y ðxÞ þ z y ðxÞ;

Z I

vP

P

79

(18)

aðx; zÞzSðx; zÞg a ðx; zÞdz 5ea ;

¼ I

2 Choosing instead virtual velocities of the less general, constrained form (11) would preclude the appearance of reactive terms in the balance equations following from (16). We shall return on this important issue after completion of the generating procedure we are now about to start, in the remark at the end of Section 2.5.

3

Here we have made use of the fact that vP ¼ fPf3 ggWfvPIg.

80

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

 the distance force and distance couple per unit area

Z

must also vanish (yielding boundary equations of Neumann type), because localization of (24) leads to:

  aðx; zÞdo ðx; zÞdz þ aþ ðxÞcþ o ðxÞ þ a ðxÞc o ðxÞ;

qo ðxÞ : ¼ ZI ro ðxÞ : ¼

aðx; zÞzdo ðx; zÞdz þ 3

ð0Þ

ðs Fm  lo Þ$ y ¼ 0

   aþ ðxÞcþ o ðxÞ  a ðxÞc o ðxÞ ;



ð0Þ

(21)  the contact force and contact couple per unit length

lo ðxÞ : ¼

f ðiÞ ðxÞ : ¼

mo ðxÞ : ¼

zco ðx; zÞdz:

P

Z

c o $y ¼

Z


ð0Þ

ð1Þ

qo $ y þ r o $ y

þ

P

vP

Z


Z
P

þ

ð0Þ ð1Þ ðs Fm  lo Þ$ y þ ðs Mm  mo Þ$ y ;

s

F ¼ f ð Þ 5ea ; a

aÞ

¼ s F ea ;

ð1Þ

(29)

s

M ¼ mðaÞ 5ea ;

(30)

mðaÞ ¼ s M ea :

(31)

aÞ

þ qo ¼ 0; (32)

ðaÞ

m;a þ gbab mðaÞ  f ð3Þ þ ro ¼ 0:

We call membrane forces the components Fab :¼ F$ea 5 b e ¼ f(b)$ea ¼ sF$ea 5 eb e respectively, normal m. f. for a ¼ b and shear m.f. for a s b; and we call Maa : ¼ M$ea 5 ea ¼ m(a)$ea ¼ sM$ea 5 ea (a ¼ 1,2) the bending moments and Mab: ¼ M$ea 5 eb ¼ m(a)$eb ¼ sM$ea 5 eb, (a, b ¼ 1,2, a s b) the twisting moments. Finally, we call F3a ¼ f(3)$ea the transverse shears, F33 ¼ f (3)3 ¼ f (3)$n the thickness shear, and M3a ¼ m(3)$ea the thickness moments. Component-wise, the general field Eq. (23) can be written in the following form:

Granted (23), what remains of (22) is: ð0Þ

M : ¼ mðiÞ 5ei ;

In terms of force and moment vectors, Eq. (23) become:

f ;ða Þ þ gbab f ð

ð1Þ

(23)

 f ð3Þ þ ro ¼ 0:

(28)

with

a

þ qo ¼ 0;

aðx; zÞzSðx; zÞg i ðx; zÞdz:

With these definitions, we may further set:

fð

a statement to hold for every pair of vector fields y and y on S and for every part P of S. Under the standard blanket assumptions of smoothness, and with the use of a standard localization lemma, (22) yields the (two-dimensional) field equations to hold at any interior point of S:

Z


Z I

ð22Þ ð0Þ

s Divs M

(27)

and

vP

s Divs F

aðx; zÞSðx; zÞg i ðx; zÞdz;

F : ¼ f ðiÞ 5ei ¼ s F þ f ð3Þ 5n;

vP

 ð1Þ ð0Þ
ð  s Divs F  qo Þ$ y þ  s Divs M þ f ð3Þ  r o $ y

Z


mðiÞ ðxÞ : ¼

ð0Þ ð1Þ lo $ y þ mo $ y :

2.3.3. Principle of Virtual Powers. Field equations. Boundary equations The two-dimensional Principle of Virtual Powers we arrive at is:

0¼

Z

the moment vectors are:

All in all, the external virtual-power expenditure takes the following form:

do $y þ

ð1Þ

I

I

Z

(26)

Remark. The shear vector f ð3Þ, defined by (19) and entering the last of Eq. (33), is one of the force vectors:

Z co ðx; zÞdz;

ð1Þ

ðs Mm  mo Þ$ y ¼ 0;

for all admissible choices of y and y .

I

Z

and

for all admissible variations y and y . Localization of (24) yields different results according to where it is performed. At an interior point of S, where it can be combined with arbitrariness in the choice of P, we have that:

  F da   Wad F 3a þ qdo ¼ 0 ðd ¼ 1; 2Þ; a  F 3a a þ Wba F ba þ q3o ¼ 0;   M da   F 3d þ rod ¼ 0ðd ¼ 1; 2Þ; a  M 3a a þ Wba Mba  F 33 þ ro3 ¼ 0;

s

where a vertical bar denotes covariant differentiation (given a tensor

ðs Fm  lo Þ$ y þ ðs Mm  mo Þ$ y

¼ 0

(24)

vP ð0Þ

Fm ¼ lo ;

s

Mm ¼ mo ;

ð1Þ

(25)

two relations that parallel the classic relation between stress tensor and contact-force vector for three-dimensional Cauchy bodies.4 At a point of vS where a Dirichlet boundary condition prevails e that is to say, where one or more components of the boundary trace of the displacement field are prescribed e the corresponding components of the admissible variations must vanish; accordingly, the complementing components of both vectors (sFm  lo) and (sMm  mo)

b da da d a d ba field T over S, T da ja : ¼ T;da a þ gba T þ gab T , with T :¼ T$e 5 e )

and where gdba : ¼ ed $eb;a are the surface Christoffel symbols. Remark. A shell-shaped body GðS; 3 Þ is in a membrane regime if it so happens that the following conditions:

f ð Þ $n ¼ 0; a

f ð3Þ ¼ 0;

(34)

hold identically in GðS; 3 Þ. The first two of (34) imply that F the fourth that Mia ¼ 0. Consequently, Eq. (33) reduce to:

a

See (Podio-Guidugli and Vianello, 2010) for a discussion of this issue that covers second-gradient materials as well.

mðaÞ ¼ 0 3a

  F da  þ qdo ¼ 0; 4

(33)

Wba F ba þ q3o ¼ 0;

¼ 0,

(35)

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

together with the following compatibility condition on the data:

ro ¼ 0: 2.4. Constitutive Assumptions The penultimate step in our construction of a shell theory consists in specifying how the two-dimensional stress measures (18) depend on the admissible deformations (11). We wish to come up with the simplest theory accommodating both an orthotropic response and the unshearability constraint discussed in Section 2.2. To this end, we confine attention to cases when the material response is uniform all over GðS; 3 Þ. Moreover, all along each fixed material fiber F ðxÞ we take the orthotropy plane orthogonal to that fiber, so that the orthotropy and the tangent planes coincide at x˛S; in particular, whenever use is made of orthogonal curvilinear coordinates, as is the case for the cylindrical shells treated in Section 3, we take the orthotropy axes tangent to both the coordinate lines za ¼ const. 2.4.1. Orthotropic elasticity tensors Let C denote the fourth-order tensor of elasticity, a linear transformation of the space of symmetric tensors, that specifies the stress response to deformations in the parent three-dimensional theory we are going to select. An orthogonal tensor Q is a symmetry transformation for the linearly elastic material described by C if it so happens that

Q C½EQ

T

i h ¼ C QEQ T ;

for all symmetric tensors E;

(cf. (Gurtin, 1972)); the orthotropic material class is then parameterized by the 9 elastic moduli C1111 ; .; C2233 . 2.4.2. The elasticity tensor of unshearable orthotropic materials As a direct consequence of the fact that the shell model we are after incorporates a kinematic constraint, the appropriate threedimensional response is captured by an elasticity tensor somewhat simpler than (37). To see why, and to derive such an elasticity tensor, we apply to our present case a general representation result in constrained linear elasticity (Podio-Guidugli and Vianello, 1992; Podio-Guidugli, 2000). We make the shell geometry agree with the geometry intrinsic to the material response, in the sense that, at any fixed point x˛S, we identify c3 with n(x). With this identification, the internal constraint (6) can be read as the requirement that all admissible strains be orthogonal to the following subspace of Sym:

Dt : ¼ spanðV a ; a ¼ 1; 2Þ:

(38)

Accordingly, the space Sym is split into the direct sum of two orthogonal subspaces:

Sym ¼ D4Dt ;

(39)

and the stress is split into reactive and active parts:

S ¼ S ðRÞ þ S ðAÞ ;

(40)

with ðRÞ S ðRÞ ¼ jðRÞ a V a ; ja ˛R; ðAÞ ~ ~ S ¼ C½E; C : D/D;

i.e., that

QC ¼ CQ; where the fourth-order tensor Q is such that Q½A : ¼ QAQ T for every A in the space the second-order tensors Lin; the symmetry group of C is the set

(41)

here the coefficients jðRÞ a are constitutively unspecified, and the constraint space D implicitly defined by (38) and (39) can be identified as

D ¼ spanðV 3 ; W i ; i ¼ 1; 2; 3Þ:

GC : ¼ fQ ˛OrthjQC ¼ CQg: Given a subgroup G of the orthogonal group (or, for what it matters here, of the group of all rotations), one seeks a representation formula for all elasticity tensors C such that GC IG, i.e., for all elasticity tensors sharing a given symmetry group. A linearly elastic material is called orthotropic when its stress response is insensitive to a rotation of p about a given axis c, i.e., when the symmetry group of its elasticity tensor includes that rotation. We give here below a general representation formula for the elasticity tensors in question. Let {ci (i ¼ 1,2,3)} be an orthonormal basis of vectors. Consider the following orthonormal basis for the linear space Sym of all symmetric tensors:

1 V a ¼ pffiffiffi ðc a 5c 3 þ c 3 5c a Þða ¼ 1; 2Þ; 2 W a ¼ c a 5c a ða not summedÞ;

81

W3

(42)

On applying a general result proved in (Podio-Guidugli and Vianello, 1992), a representation for the desired elasticity tensor ~ can be deduced from the one given for C in (37): C

~ ¼ PD Cj ; C D

PD : ¼ I  V a 5V a :

where PD denotes the orthogonal projector of Sym on D. One finds the 7-parameter representation

~ ¼ C1111 W 1 5W 1 þ C2222 W 2 5W 2 þ C3333 V 3 5V 3 C þ C1212 W 3 5W 3 þ C1122 ðW 1 5W 2 þ W 2 5W 1 Þ þ C1133 ðW 1 5V 3 þ V 3 5W 1 Þ þ C2233 ðW 2 5V 3 þ V 3 5W 2 Þ:

V 3 ¼ c 3 5c3 ;

(43)

~ is positive-definite, i.e., that As is customary, we assume that C

1 ¼ pffiffiffi ðc 1 5c 2 þ c 2 5c1 Þ: 2 (36)

~ E$C½E>0

for all E˛Dnf0g:

(44)

With the use of this basis, any orthotropic elasticity tensor can be written in the following form:

Remark. In case the internal constraint (6) is reinforced à la Kirchhoff-Love as specified by (14), so as to exclude thickness changes, the above procedure yields the elasticity tensor

C ¼ C1111 W 1 5W 1 þ C2222 W 2 5W 2 þ C3333 V 3 5V 3

^ ¼ C1111 W 1 5W 1 þ C2222 W 2 5W 2 þ C1212 W 3 5W 3 C

þ C1212 W 3 5W 3 þ C3131 V 1 5V 1 þ C2323 V 2 5V 2

þ C1122 ðW 1 5W 2 þ W 2 5W 1 Þ;

þ C1122 ðW 1 5W 2 þ W 2 5W 1 Þ þ C1133 ðW 1 5V 3 þ V 3 5W 1 Þ þ C2233 ðW 2 5V 3 þ V 3 5W 2 Þ

(37)

a linear transformation of ^ ¼ spanðW ; i ¼ 1; 2; 3Þ into itself. D i

(45) the

constraint

space

82

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

2.4.3. The technical moduli In technical applications of classic isotropic elasticity, the two Lamé constants are replaced by the technical moduli E and n of Young and Poisson, plus the shear modulus G, under the condition that E ¼ 2(1 þ n) G. Likewise, we here replace the seven Lamé-like constants C1111 ; .; C2233 in (43) by an equivalent list of ten technical moduli e three of them being Young-like, six Poisson-like, and one shear-like e that must satisfy three independent algebraic conditions. The technical moduli in question are precisely those that enter the following representation of the compliance tensor ~ 1 : C

n E1 ¼ 13 ; n31 E3

establishes the nature of G as a shear modulus. It is straightforward to see that G ¼ C1212 . The other technical ~ moduli can be written as follows in terms of the components of C:

C1111 C2222 C3333  C3333 C21122  C2222 C21133  C1111 C22233 þ 2C1122 C1133 C2233 C1111 C3333  C21133 C1111 C2222 C3333  C3333 C21122  C2222 C21133  C1111 C22233 þ 2C1122 C1133 C2233

E3 ¼

C1111 C2222  C21122

v12 ¼ v13 ¼ v23 ¼

C3333 C1122  C1133 C2233 C2222 C3333  C22233 C2222 C1133  C1122 C2233 C2222 C3333 

C22233

C1111 C2233  C1122 C1133 C1111 C3333 

C21133

;

v21 ¼

;

v31 ¼

;

v32 ¼



n12 E1

n23 E2

ðW 1 5W 2 þ W 2 5W 1 Þ 

n13 E1

C1111 C3333  C21133 C2222 C1133  C1122 C2233 C1111 C2222  C21122 C1111 C2233  C1122 C1133 C1111 C2222  C21122

ðW 1 5V 3 þ V 3 5W 1 Þ

ðW 2 5V 3 þ V 3 5W 2 Þ

(46)

1

½W 1  ¼

s E1

ðW 1  n12 W 2  n13 V 3 Þ:

S$W 1 ; E$W 1

(47)

the ratio between the axial stress and the corresponding axial deformation. Moreover, we find that

E$W 2 ; E$W 1

;

; ; :

To sum up, the 7-parameter representation (43) can be rewritten in the form:

~ ¼ D1 ðE1 ð1  n23 n32 ÞW 1 5W 1 þ E2 ð1  n13 n31 ÞW 2 5W 2 C þ E3 ð1  n12 n21 ÞV 3 5V 3 þ 2DGW 3 5W 3 þ E1 ðn21 þ n23 n31 ÞðW 1 5W 2 þ W 2 5W 1 Þ þ E1 ðn31 þ n21 n32 Þ

n13 ¼ 

E$V 3 ; E$W 1

þ V 3 5W 2 ÞÞ; (51) where

D : ¼ 1  n12 n21  n13 n31  n23 n32  2n12 n23 n31 :

(52)

2.5. Force and moment vectors and tensors

Then, we find that

n12 ¼ 

;

 ðW 1 5V 3 þ V 3 5W 1 Þ þ E2 ðn32 þ n31 n12 ÞðW 2 5V 3

(as is well known, the positivity assumption (44) guarantees ~ invertibility of C). To confirm that the three moduli E1, E2 and E3 are Young-like, imagine to induce a state of uniaxial traction in the direction, say, c1 in a specimen made of the material under examination, so that the stress is S ¼ s W1 and the corresponding strain is

~ E ¼ sC

;

(50)

C3333 C1122  C1133 C2233

~ 1 ¼ 1 W 1 5W 1 þ 1 W 2 5W 2 þ 1 V 3 5V 3 þ 1 W 3 5W 3 C E1 E2 E3 2G12 

(49)

S$W 3 ; E$W 3

2G ¼

C2222 C3333  C22233

E2 ¼

n E2 ¼ 23 : n32 E3

Finally, the formula

C1111 C2222 C3333  C3333 C21122  C2222 C21133  C1111 C22233 þ 2C1122 C1133 C2233

E1 ¼

E1 ¼

n E1 ¼ 12 ; n21 E2

(48)

the Poisson-like negative ratios of the transverse deformations in the directions c2 and c3 and the axial deformation in the direction c1. The defining formulae for E2,n21,n23 and E3,n31,n32 are completely ~ it analogous to (47) and (48). Due to the built-in symmetries of C, turns out that

The final step in the assemblage of our shell theory e the posing of initial- and boundary-value problems e demands that the balance Eqs. (33) are written in terms of the parameters involved in the general representation (11) for an admissible displacement field. All we have to do to construct the corresponding parametric representations for those components of the force and moment tensors that enter the balance equations is to insert in the definitions (27) and (28) the general parametric representation for the stress field in GðS; 3 Þ, and then make use of definitions (30). Now, the required stress representation is obtained by combining (40)e(41) e with c3 h n(x) e and (41) e with E ¼ E (uS), and uS given by (11); one finds:

~ Sðx; zÞ ¼ sðRÞ ðx; zÞ5nðxÞ þ nðxÞ5sðRÞ ðx; zÞ þ C½Eðu S ðx; zÞÞ; (53)

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

where the restriction to the fiber F ðxÞ of the reactive field s(R) is a vector field perpendicular to n (x).5 Consequently, the force and moment vectors have both reactive and active parts, namely, R ðaÞ

f

Z ðxÞ ¼

R ð3Þ

f

AðiÞ

f

ZI ðxÞ ¼







aðx; zÞ sðRÞ ðx; zÞ$g a ðx; zÞ dz nðxÞ;

aðx; zÞsðRÞ ðx; zÞdz;

I

Z






i ~ aðx; zÞ C½Eðu S ðx; zÞÞg ðx; zÞ dz;

ðxÞ ¼

3. Cylindrical shells: generalities

and

3.1. Geometry

Z

R

mðaÞ ðxÞ ¼ m

ZI ðxÞ ¼

A

mðiÞ ðxÞ ¼

Z

   aðx; zÞz sðRÞ ðx; zÞ$g a ðx; zÞ dz nðxÞ;

We now restrict our attention to shells whose model surface S is a portion of a right circular cylinder, that we parameterize as usual by means of cylindrical coordinates:

aðx; zÞzsðRÞ ðx; zÞdz;

z1 ¼ x1 ;

I i ~ aðx; zÞzC½Eðu S ðx; zÞÞg ðx; zÞdz:

(55)

I

It is not difficult to check that R ðiÞ

f

A ðiÞ

ðxÞ$f

ðxÞ ¼ 0;

R

A

mðiÞ ðxÞ$mðiÞ ðxÞ ¼ 0:

(56)

It is also easy to see, in the light of (29), that the force and moment tensors sF and sM have active and reactive parts as well. Thus, the balance equations one arrives at are not pure, in the sense that, in addition to the parameter fields, they also include reactive terms. In fact, the transverse shears are reactive, and Eq. (33)3 relate them to the active bending moments6:

  d F 3d ¼ M da  þ ro:

(57)

a

The thickness moments are also reactive, and (33)5 relates their divergence to the active bending moments and the active thickness shear:

  M 3a  ¼ Wba M ba þ F 33  ro3 : a

(58)

These observations suggest a sequential strategy to solve a shell problem within our present theory, where the unknowns are (the fields a, w and the constant g that parameterize) the displacement field uS and the reaction force and moment fields: firstly, by the use of the projection operator PD defined in the Subsection 2.4.2, one derives a set of reaction-free consequences of the balance equations; secondly, one solves such ‘purified’ system of equations for uS; thirdly, one returns to the full balance equations, where the active terms can now be computed explicitly, and solves them for the reactive fields.7 Remark. As anticipated in footnote 2, reactive forces and moments are found in the balance Eq. (33), because the class of variations (17) we used to derive those equations from the Principle of Virtual Powers (16) is visibly larger than the class (11) of

5 It follows from the first of (41) and the first pffiffiffi S(R) ¼ s(R) 5 c3 þ c3 5 s(R), with sðRÞ ¼ ð1= 2ÞjðRÞ a ca .

R

ð3Þ

two of

(36)

z2 ¼ w

(Fig. 4). For ro the radius of the directrix of S and n the normal to S, and for 3 < ro the half thickness of the shell-shaped region GðS; 3 Þ, the position vectors with respect to the origin o of two typical points p˛GðS; 3 Þ and x˛S are, respectively,

p  o ¼ pðx1 ; w; zÞ ¼ xðx1 ; wÞ þ znðwÞ   z nðwÞ: ¼ x1 c1 þ ro 1 þ

ro

and

x  o ¼ xðx1 ; wÞ ¼ x1 c 1 þ ro nðwÞ;

nðwÞ ¼ sin wc2 þ cos wc 3 :

The relative covariant and contravariant bases can both be represented in terms of the orthonormal basis fc 1 ; n0 ðwÞ; nðwÞg, that serves as physical basis (recall the remark in Section 2.1):

g 1 ðpÞ ¼ e e1 ðxÞ ¼ c1 ; 1 ðxÞ;  z e2 ðxÞ; e2 ðxÞ ¼ ro n0 ðwÞ; g 2 ðpÞ ¼ 1 þ

ro

(59)

g 3 ðpÞ ¼ e3 ðxÞ ¼ nðwÞ; and

g 1 ðpÞ ¼ e1 ðxÞ; e1 ðxÞ ¼ c1 ;   z 1 2 0 e ðxÞ; e2 ðxÞ ¼ r1 g 2 ðpÞ ¼ 1 þ o n ðwÞ;

ro

g 3 ðpÞ ¼ e3 ðxÞ ¼ nðwÞ: The surface shifter defined by (3)2 turns out to be:

that

A ðdÞ

In fact, F 3d ¼ f ð3Þ $ed ¼ f ðxÞ$ed ; and, M da ¼ mðdÞ $ea ¼ m ðxÞ$ea : The annihilation procedure of the reactive terms occurring in the threedimensional balance equations of constrained linear elasticity is discussed in (Podio-Guidugli, 2000), Section 17.2. 6

admissible displacements (this will not be the case in the next section). Although we here do not pursue this issue any further, we recall that having at one’s disposal the reactive dynamical descriptors associated with the kinematical Ansatz adopted to construct a lower-dimensional structure theory can be proved beneficial to improve the pointwise approximation of the relative three-dimensional stress field (Podio-Guidugli, 1989; Lembo and Podio-Guidugli, 2001; Lembo and Podio-Guidugli, 2007; PodioGuidugli et al., 2008).

(54)

I

R ð3Þ

83

7

Fig. 4. Geometrical equipment of the model surface of a cylindrical shell.

(60)

84

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

  z 0 s n ðwÞ5n0 ðwÞ; Aðw; zÞ ¼ c1 5c 1 þ 1 þ

ð0Þ

y ¼ y1 c1 þ y2 n0 þ y3 n;

ro

with the scalar fields yi ¼ yi ðx1 ; wÞ (i ¼ 1,2,3) compactly supported in S and with y4 a constant.

z : ro

Finally, the only non-null Christoffel symbols on S are:

3.3.1. Field equations With a view toward deriving the field equations to which the general balances (23) reduce in the present situation, we firstly take y4 ¼ 0 in (65). In this instance, on setting:

g223 ¼ g232 ¼ r1 o :

g322 ¼ ro ;

3.2. Kinematics

ð0Þ

f : ¼ ðs Divs F þ qo Þ;

The displacement field (11) now reads:

   z 0 n ðwÞ5n0 ðwÞ aðx1 ; wÞ c 1 5c 1 þ 1 þ ro
þ ðwðx1 ; wÞ þ zgÞnðwÞ  z w;1 ðx1 ; wÞc 1  0 þ r1 o w;2 ðx1 ; wÞn ðwÞ ;



uC ðx1 ; w; zÞ ¼

(65)



y ¼ y3;1 c1 þ r1 y2  y3;2 n0 þ y4 n; o

whence

aðzÞ ¼ 1 þ



ð1Þ

the formulation (22) of the Principle of Virtual Powers reads:

0 ¼

uC ðx1 ; w; zÞ ¼ u<1> ðx1 ; w; zÞc 1 þ u<2> ðx1 ; w; zÞn0 ðwÞ (62)

with components:

u<1> ¼ a<1>  zw;1 ;   z z a<2>  w;2 ; u<2> ¼ 1 þ

ro

(63)

ro

u<3> ¼ w þ zg: Since

VuC ¼ u;i 5g i ¼ u;1 5c1 þ



1

r0

1þ

z ro

1

z ro

z ro

1þ  w;21 þ r1 o 0

0

E<22> ¼ VuC $n 5n ¼

r1 o



z 1þ ro

E<33> ¼ VuC $n5n ¼ u<3>;3

ro ¼ g:

ð0Þ

f $c1 ¼ 0; ð0Þ ð0Þ 0 f þ r1 o m $n ¼ 0;
ð0Þ ð0Þ  ð0Þ 0   f þ m $c 1 ;1 þ r1 m $n ;2 $n ¼ 0: o

To find the balance law that follows from testing the equality of internal and external powers by way of virtual velocity fields of the form: ð1Þ

y ¼ y4 n;

u<2>;2 þ u<3>

(67)

ro

and that y4 can be chosen arbitrarily, (15) and (67) yield:

  a<1>;2  zw;12 ; 

(66)

  z Vy ¼ y4 n5n þ n0 5n0 ;

Zþl Z2pZ



     z 1 z z zg 1 þ 1 þ a ;  w þ w þ ¼ r1 <2>;2 ;22 o

ro

Hence, three field equations must hold at each interior point of S, namely,

ro

1

;2

with y4 an arbitrary real number, we turn to the formulation (15) of the Principle: indeed, y4 must be taken constant over the whole of U, because it is meant to be a variation of the constant thickness strain E<33> ¼ g in the representation (63) of the admissible displacement.8 Given that, for any v as in (67),

2E<12> ¼ 2E<21> ¼ VuC $ðc1 5n0 þ n0 5c1 Þ ¼ u<2>;1     z 1 z 1þ a<2>;1 þr1 u<1>;2 ¼ 1 þ o 1

;1

ð1Þ

E<11> ¼ VuC $c 1 5c 1 ¼ u<1>;1 ¼ a<1>;1  zw;11 ;



P


ð0Þ
ð0Þ 
ð0Þ    m $n0 $n : þ y3 f þ m $c 1 þ r1 o

y ¼ zy;

u;2 5n0 þ u;3 5n;

the non-null physical components of the strain tensor are:

ro

Z

ð0Þ  Z
ð0Þ
ð0Þ ð0Þ ð1Þ ð0Þ ð0Þ 0 y1 f $c1 þ y2 f þ r1 f $ y þ m$ y ¼ o m $n P

(61)

its representation in the physical basis fg<1> ¼ c1 ; g<2> ¼ n0 ðwÞ; g<3> ¼ nðwÞgis:

þ u<3> ðx1 ; w; zÞnðwÞ;


m : ¼  s Divs M  f ð3Þ þ ro ;

ð0Þ

ro

(64)

3.3. Balance Assumptions We exploit the virtual-power procedure detailed in Section 2.3, with the difference anticipated in the remark at the end of Section 2.5: the variations we now employ have the same structure as the admissible displacements (62)e(63); hence, no reactive contributions are going to enter the field and boundary equations that the procedure delivers. Precisely, the variations in question have the following form:

l 0

I







z a S$n5n þ S$n0 5n0  zdo $n dz dx1 dw ¼ 0: ro

(68)

In terms of physical components of the force and moment tensors, the system of Eqs. (66) and (68) can be written as:

F þ r1 o F<12>;2 þ qo<1> ¼ 0;   <11>;1 1 F 1 F<21> þ r1 <22> þ ro M<22> ;2 o M<21> ;1 þ ro þqo<2> þ r1 o ro<2> ¼ 0;

8 More generally, we point out that, whenever the admissible motions are such as to suggest the use of test fields of the form v ¼ yow, with y0 an arbitrary constant and w a given vector field, then (15) is the appropriate formulation of the Principle of Virtual Powers, and the associated balance information is an integral relation of R R R the form: U S$Vw ¼ U do $w þ vU c o $w:

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96 2 M<11>;11 þ r1 o ðM<12> þ M<21> Þ;12 þ ro M<22>;22

0 ¼

þ r1 o F<22>

l


  y2  y3;2 n0 þ y4 n    y3;1 c 1 þ r1 o



r1 o M<22> þ F<33>  ro<3> dx1 dw ¼ 0;

(69)

¼

0

þ

22 F;1

þ

þ

r1 o

22 M;22

Zþl Z2p
l

dw

y1 ðF<11>  lo<1> Þ


 1 þy2 F<21> þ r1 o M<21>  lo<2>  ro mo<2>
 1 þy3 F<31>  r1 o M<21>;2  lo<3>  ro mo<2>;2

11 þ F 12 þ q1 ¼ 0; F;1 o ;2

11 M;11

Zw1


x1 ¼l1

wo

the following version in terms of contravariant components may easy a comparison with (33):

21 F;1

ðs Fc1  lo Þ$ðy1 c1 þ y2 n0 þ y3 nÞ þ ðs Mc 1  mo Þ

wo

þqo<3> þ ro<1>;1 þ r1 o ro<2>;2 ¼ 0; Zþl Z2p


Zw1


85

þ






21 M;1

M21

þ

þ

22 M;2

M 12

þ



ro2



þ

 ro ;12

q2o

F 22

 y3;1 ðM<11>  mo<1> Þ þ y4 ðM<31>  mo<3> Þjx1 ¼l1 dw ðl1 ;w1 Þ

 ; þ y3 M<21>  r1  o mo<2>

¼ 0;

þ

q3o

þ

ðl1 ;w0 Þ

1 ro;1

þ

2 ro;2

ð70Þ

¼ 0; over g 1, where m ¼ n0 ðwÞ, (24) becomes:



ro M22 þ F 33  ro3 dx1 dw ¼ 0: 0 ¼

0

Zl1


ðs Fn0  lo Þ$ðy1 c 1 þ y2 n0 þ y3 nÞ þ ðs Mn0  mo Þ

lo


   y2  y3;2 n0 þ y4 n    y3;1 c1 þ r1 o

3.3.2. Boundary equations We start from the general weak statement (24), repeated here for the reader’s convenience:

Z


ð0Þ

ð1Þ

ðS Fm  lo Þ$ v þ ðS Mm  mo Þ$ v

¼

d

a

y1 ðF<12>  lo<1> Þ


 1 þy2 F<22> þ r1 o M<22>  lo<2>  ro mo<2>
 1 þy3 F<32>  r1 o M<12>;1  lo<3>  ro mo<2>;1   þ  y3;2 ðM<22>  mo<1> Þ þ y4 ðM<32>  mo<3> Þ 

¼ 0;

where we now insert variations of type (65). For simplicity, we restrict attention to parts of the model surface whose boundary consists of the union of two identical parts d 0, d 1 of directrices at different axial abscissae and the relative two segments of generatrices g 0 , g 1 , that is to say, with reference to Fig. 5, parts P such that

[

dx1

lo

vP

vP ¼

Zl1


w¼w1

W g a ða ¼ 0; 1Þ;

a



ðl1 ;w1 Þ  þ y3 M<12>  r1 ;  o mo<2> ðl0 ;w1 Þ

Over d 1, where m ¼ c1, the integral condition (24) takes the form:

dx1 (71)

needless to say, relations completely similar to (70) and (71) hold, respectively, at d 0 and g 0. ð1Þ

with

d a : ¼ fx˛Sjxhðla ; wÞ; w˛ðwo ; w1 Þg; g a : ¼ fx˛Sjxhðx1 ; wa Þ; x1 ˛ðl0 ; l1 Þg:

w¼w1

As mentioned in Section 5, if u 3 or anyone of the components of ð0Þ

ð0Þ

u and V u 3 is assigned at a boundary point, then y4 or the correð0Þ

ð0Þ

sponding component of v and V y 3 must vanish at that point; on the other hand, the arbitrary variation of each of the remaining parameters in (65) induces at the same point a boundary equation of Neumann type: for example, inspection of (70) shows that, if ð0Þ

u $c 1 ¼ a<1> is assigned over d a, then y1 must be taken there identically null, while the Neumann boundary equations 1 F<21> þ r1 o M<21> ¼ lo<2> þ ro mo<2> etc:

must hold; in other words, at a point of d a admissible boundary conditions must consist of a list of mutually exclusive assignments of the one or the other element of the following five powerconjugate pairs:

  ðF ;a Þ; F<21> þ r1 ; o M<21> ; a<2>    <11> <1> M ; w ; M ; w ; ðM F<31>  r1 <21>;2 <11> ;1 <31> ; gÞ: o

Fig. 5. A typical part of the model surface of a cylindrical shell, bounded by generatrices and directrices.

(72)

Likewise, from (71) we deduce that the boundary conditions at a point of g a should consist of mutually exclusive assignments of the pairs:

86

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

  ðF ;a Þ; F<22> þ r1 ; o M<22> ; a<2>    <12> <1> F<32>  r1 o M<12>;1 ; w ; M<22> ; w;2 ; ðM<32> ; gÞ:

(73)

3.3.3. Inertial interactions. Evolution equations The time evolution of an unshearable orthotropic shell is ruled by the partial differential equations that follow from (23) when the inertial force is separated from the rest of the distance force per unit volume of GðS; 3 Þ. To this effect, we set:

€ din o : ¼ do u

and

in dni o : ¼ do  do ;

for, respectively, the inertial and noninertial distance forces (here do denotes the mass density per unit reference volume and a superposed dot signifies time differentiation). We then set:

qin o :¼ 

Z

€ ðx; z; tÞdz and aðx; zÞdo ðx; zÞu

ZI

€ ðx; z; tÞdz; aðx; zÞzdo ðx; zÞu

r in o :¼ 

(74)

: ¼ qo 

Needless to say, these equations are to be equipped with a set of initial conditions for the unknown fields a<1>, a<2>, w, and g, and for their time rates.

3.4. Constitutive Assumptions The components of force and moment tensors that appear in (69) depend on the active part of the three-dimensional stress field. The latter has the following expression in terms of the constitutive law (51) and the strain tensor (64):

~ Sðx; zÞ ¼ C½Eðu C ðx; zÞÞ;

qin o

rni o

and

: ¼ ro 

rin o ;

qin o<2> ¼ qin o<3> ¼ in ro<1>

¼



1 €<1>  3 2 r1 € ;1 ; w do a 3 o    1 32 1 32 €<2>  € ;2 ; w 1þ a do 3 r2o 3 r2o   1 32 €þ g€ ; do w 3 ro
 € € do r1 o a<1>  w;1 ;

(76)



3

2

3ro

1 € € € ;11 þ r1 g€ þ a€<1>;1  ro w o a<2>;2  ro w;2

¼ M<11>;11 þ

þ M<21> Þ;12 þ

D

D

ðn21 þ n23 n31 Þ E<22>

E2

D

E1

D

ðn21 þ n23 n31 Þ E<11>

ðn32 þ n31 n12 ÞE<33> ;

ð1  n12 n21 Þ E<33> þ

E1

D

ðn31 þ n21 n32 Þ E<11>

ðn32 þ n31 n12 ÞE<22> ;



r2 o M<22>;22

(79)

where the constitutive modulus D is defined as in (52), the components E as in (64). On inserting these relations in the appropriate consequences of definitions (29) and (30) we find:



1þ

z S ro <11>

  1 ð1  n23 n32 Þ ro a<1>;1  3 2 w;11 ro D 3    þ ðn21 þ n23 n32 Þ a<2>;2 þ w þ ðn31 þ n21 n32 Þro g ; (80) ¼ 2

;2

r1 o ðM<12>

E3

E2

I






S<33> ¼

ð1  n13 n31 Þ E<22> þ

Z 

1 2 1 ni € ;1 ¼ F<11>;1 þ r1 w 3 r o F<12>;2 þ qo < 1>; 3 o    
2 32 2 32 € ;2 ¼ F<21> þ r1 do 1 þ 2 a€<2>  2 w o M<21> ;1 3 ro 3 ro
 1 1 ni 1 ni þro F<22> þ ro M<22> þ qo < 2> þ ro ro < 2>; €þ do w

D

F<11> ¼

where do : ¼ 23 do is the uniform mass density per unit area of the model surface S. Hence, the evolution equations corresponding to the balance Eq. (69) are:

do a€<1> 

D

S<12> ¼ S<21> ¼ 2G E<12> ;

 1 32
€ € ;2 ; a w 3 ro <2>  2
3 € þ ro g € ; w ¼ do 3ro



E2

þ

in ro<2> ¼ do in ro<3>

E1

D

þ



ð1  n23 n32 Þ E<11> þ

D

(75)

In the case of cylindrical shells whose mass distribution is uniform, definitions (74) yield:

¼

(78)

E þ 1 ðn31 þ n21 n32 ÞE<33> ; S<22> ¼

s s ni qin o ¼ Div F þ qo ; in s s ro ¼ Div M  f ð3Þ þ rni o :

E1

S<11> ¼

for the relative noninertial loadings. In conclusion, the balance Eq. (23) take the evolutionary form:

qin o<1>

(77)

in components, this equation reads:

I

for the inertial force and the inertial couple per unit area of S. Finally, we return to definitions (21)1,2 and set:

qni o

ni ni 1 ni r1 o F<22> þ qo < 3> þ ro < 1>;1 þ ro ro < 2>;2 ;  2
3 ni € þ ro g € ¼ r1 do w o M<22> þ F<33>  ro < 3>: 3ro

3

E1

Z F<22> ¼

S<22> I

0 0 2 3 1 1þ E2 6 1 ro C B B ¼ 2 4ð1  n13 n31 Þ@a<2>;2  @1  3 log 3 A ro D 1 2 3

3

1

ro

ro

1þ   1 ro C  w;22  ro g þ 3 log wA 3 2 1

ro

ro

3

r 7 þ o ðn21 þ n23 n31 Þa<1>;1 þ ðn32 þ n31 n12 Þro g5; h

(81)

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

Z 



1þ

F<33> ¼ I



displacement fields whose physical components u are all independent of the circumferential coordinate w, that is to say, in view of (63), if  

z 3 E3 S ð1  n12 n21 Þro g ¼ 2 ro D ro <33> 

n31 þ n21 n32 1 ro a<1>;1  3 2 w;11 l 3   n32  n31 n12  þ a<2>;2 þ w ; m



u<1> ¼ a<1>  zw0 ;

þ

Z  I

F<12>

(82)



1þ

F<21> ¼

 

 1 32 1 32 1 a<2>;1  ¼ 23 G 1 þ w;21 þ a<1>;2 ; 2 2 ro 3 ro 3 ro 2 ε Z 1þ 1 ro 6 ¼ S<12> ¼ 2εG4a<2>;1 þ log ε a<1>;2 2ε 1 I ro 0 3 ε1 1þ 1 ro C B 7  @1  ε log ε Aw;12 5; 2 1

and

Z 

(83)

I

 z 4 33  zS<21> ¼ G a<2>;1  w;12 ; ro 3 ro

I

0 1B a<1>;2 3 2 @1log 3

r2o Z M<22> ¼

zS<22> ¼ 23 I

where

1þ 1

2

3

1

3

ro

7 7 Aw;12 7; 5

ro C 3

ro

(86)

0

1þ

ro

n E2 1 ¼ 23 ¼ : : n32 m E3

ro

(87)

4. Cylindrical shells: axisymmetric boundary-value problems A boundary-value problem for a cylindrical shell is axisymmetric if the load and confinement data induce solution

E1



00 2 3 1 1þ E2 6 1 ro C BB r g ¼ 2 4ð1  n13 n31 Þ@@1  3 log 3 A o ro D 2 1 ro ro 1 3 1þ n þ n23 n31 0 1 ro C þ 3 log wA þ ro 21 a<1> 3 h 2 1 ro ro 3 3

  1 32 0 a ; F<21> ¼ 23 G 1 þ 3 r2o <2>

n þ n21 n32 3 E3 ð1  n12 n21 Þro g þ 31 F<33> ¼ 2

ro D

l

   2 n þ n31 n12 3  ro a0<1>  w00 þ 32 w ; m 3

(88)

The equations governing the equilibria of unshearable cylindrical shells are arrived at when the above constitutive equations are inserted into the balances (69). In their general form, those equations are complicated to solve analytically; we choose not to list them here. However, certain highly symmetric problems admit simple and explicit solutions. We deal with such problems in the next section, while the noticeable simplifications obtained when the approximations judged appropriate for thin and slender shells will be introduced and exemplified in Section 5.

3

7 þðn32 þ n31 n12 Þro g5

1

3

E2 6 1 ro C B 4ð1  n13 n31 Þ@1  3 log 3 A D 2 1

n E1 1 ¼ 13 ¼ : ; n31 l E3

F<22>

ro

3   1 2 n12 þ n23 n31 7 w;11 5;  w  w;22 þ ro g þ  3 h 3

n E1 1 ¼ 12 ¼ : ; n21 h E2

(90)

  1 ð1  n23 n32 Þ ro a0<1>  3 2 w00 ro D 3  þðn21 þ n23 n31 Þw þ ðn31 þ n21 n32 Þro g ;

F<11> ¼ 2

(85)

0 3 1 1þ ro B 2 33 6 1 ro C 6 zS<12> ¼ G6a w;21 þ3 2 @1 3 log 3 A 3 ro 4 <2>;1 3 1

M<12> ¼

(i) the strain components (64) take the simpler form:

2

I

Z

(89)

(ii) the constitutive Eqs. (83)e(87) for the force and moment components become:



1þ

u<3> ¼ w þ zg;

E<33> ¼ g;

z 2 3 3 E1 zS<11> ¼ ð1  n23 n32 Þ ro 3 ro D

Z 

z a ; ro <2>

ro

ro

     a<1>;1  ro w;11 þ ðn21 þ n23 n31 Þ a<2>;2  w;22

(84) þ ðn31 þ n21 n32 Þg ;

M<21> ¼

1þ

E<11> ¼ a0<1>  zw00 ;   z 0 1 1þ a ; E<12> ¼ E<21> ¼ ro <2> 2 1   z ðw þ zgÞ; E<22> ¼ ro 1 þ



1þ

M<11> ¼

u<2> ¼

where a prime denotes differentiation with respect to x1, the onlyspace variable from which all of the parameter fields a<1>, a<2>, and w, may depend. When the displacement field has the form (89),

z S ro <21>

ro

87

(91)

and

 

2 3 3 E1 M<11> ¼  ð1 n23 n32 Þ ro w00 a0<1> ðn31 þ n21 n32 Þg ; 3 ro D 4 33 0 2 33 0 Ga<2> ; M<12> ¼ Ga ; 3 ro 3 ro <2> 0 2 3 1 1þ E 6 1 ro C B M<22> ¼ 23 2 4ð1 n13 n31 Þ@1 3 log ðwþ ro gÞ 3 A D 2 1 ro ro 3

M<21> ¼

1 n þn n 7  3 2 12 23 31 w00 5; h 3

(92)

88

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

(iii) the reaction-free Eqs. (69) reduce to the field equations: 0 F<11> 

þ qo<1> ¼ 0; 0 1 F<21> þ r1 o M<21> þqo <2> þ ro ro<2> ¼ 0; 00 1 0 M<11>  ro F<22> þ qo<3> þ ro<1> ¼ 0;

When the only applied load is a distribution of end tractions statically equivalent to two mutually balancing torques of magnitude

(93)



r1 o M<22> þ F<33>  ro<3> dx1 ¼ 0:




 2pr2o t;

with

t ¼ Oð3 Þ;

(100)

the thickness stretch g and the axial displacement a<1> vanish, and (99), the only relevant equation, reduces to a0<2> ¼ a constant, whose value is determined by the boundary condition:

holding in the interval (l, þ l), plus the integral relation:

Zþl


T ¼

(94)

l

 2 3 0 1 þ G; F<21> þ r1 M ¼ t ¼ r a ; r : ¼ 2 3 T <2> T <21> o 2

(101)

ro

(iv) the reaction-free boundary conditions consist in specifications at  l of one of the elements in each of the pairs (72)1, (72)2, and (72)4:

ðF<11> ; a<1> Þ;




F<21> þ r1 o M<21> ; a<2>



0

; ðM<11> ; w Þ;

(95)

Insertion of (91) and (92) into (93) and (94) yields the system of equations ruling axisymmetric boundary-value problems in our theory. We are going to solve this system for assignments of Neumann data corresponding, respectively, to problems of torsion, axial traction, pressure, and rim flexure. Interestingly, as we shall quickly demonstrate, this system splits into one equation for the circumferential displacement a<2> plus a system of three equations for the axial and radial displacements a<1> and w and the thickness stretch g. Remark. As mentioned in closing Section 2.5, once the equilibrium displacement field has been found, the axial distributions of reactive stress measures F<31> and M<31> consistent with boundary conditions compatible with (72)3 and (72)5 can be determined by a use of, respectively, the balance Eq. (57)1 and the balance Eq. (58), keeping into account (94). The situations of our present interest occur when homogeneous Neumann data are prescribed at the boundary, so that the problems to solve are: 0 F<31> ¼ M<11> þ ro<1> inðl; þlÞ;

F<31> ðlÞ ¼ 0;

(96)

and 0 M<31> ¼ 0 inðl; þlÞ; M<31> ðlÞ ¼ 05M<31> h0 in ½l; þl:

(97) Note that Eq. (96)1 has the familiar structure of the moment balance in Bernoulli-Navier rod theory, with F<31> playing the role of the shear resultant, M<11> of the bending moment, and ro<1> of the diffused applied couples; both F<31> here and the shear resultant in that classic rod theory have reactive nature, as a consequence of one and the same unshearability constraint. 4.1. Torsion From (91)3 and (92)2, we have that

 2 3 1 þ Ga0<2> ; F<21> þ r1 M ¼ 2 3 <21> o 2

ro

ro

0 Q : ¼ r1 o a<2>

(102) 1

proportional to (rorT) . Moreover, given that the function a<2> must be odd,

a<2> ðx1 Þ ¼

1 1þ

1 t x : 2G 1

3

2 3

(103)

r2o

Remark. In line with (76)2, set

  1 32 € a qo<2> ¼ qin o<2> ¼ do 1 þ 3 r2o <2> in Eq. (99), so that it takes the form of the classical wave equation: 2

2

v a<2> ðx1 ; tÞ v a<2> ðx1 ; tÞ  c2 ¼ 0; vt 2 vx21

with

c2 ¼

G

do

1þ 1þ

3

2

r2o

2 32 3 r2o

:

(104) Then,

a<2> ðx1 ; tÞ ¼ 4ðx1 þ ctÞ þ jðx1  ctÞ; and two twist waves propagate along the axis with speed jcj, the one in the positive direction the other in the negative direction. Remark. In all boundary-value problems we shall solve next e we recall, axial traction, uniform pressure, and rim flexure e the twisting loads are null. As a consequence, in all three cases, the second equation of system (93) and the boundary equations:


 F<21> þ r1 o M<21> ðlÞ ¼ 0 together imply that the construct ðF<21> þ r1 o M<21> Þ is identically null in [l, þl]. Hence, by the constitutive relations (91)3 and (92)2, the circumferential displacement a<2> has to have constant value, as is the case for a rotation about the x1-axis; we note that this, because of (92)2, implies that

M<21> ðlÞ ¼ 0; (98)

with this, Eq. (93)2 takes the form of a second-order equation for a<2> :

 2 3 23 1 þ 2 Ga00<2> þ qo<2> þ r1 o ro<2> ¼ 0:

it follows from (101) that a twisted shell of the type we study undergoes a rotation per unit length

(99)

This equation accounts for whatever twisting about its axis an unshearable cylindrical shell may have; it can be associated with boundary conditions specifying the values at x1 ¼ l of either a<2> or a0<2>.

and we take a<2> h 0. In fact, as is customary in elasticity with Neumann data, in all three cases we expect to arrive at a displacement solution being unique to within an ignorable rigid motion. Moreover, given the common built-in symmetries, there will be no loss of generality in searching for solutions with a<1> an odd function of x1 ˛ [l, þl], and w even. 4.2. Traction Let us take all distance forces and couples null and all boundary conditions of Neumann type and homogeneous, except for

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

a distribution of tractions at the ends equivalent to two mutually balancing axial forces of magnitude

P ¼ ð2pro Þp;

with p ¼ Oð3 Þ:

(105)

Two of the balance Eq. (93) are in force:

(106)

the accompanying boundary equations are:

M<11> ðlÞ ¼ 0;

0 M<11> ðlÞ

¼ 0

(107)

the last one following from (96). Now, (106)1 and (107)1 imply that the membrane force F<11> is constant and equal to p in the closed interval [l, þl]; then, making use of (91)1, we obtain that the differential relation:

  1 ð1  n23 n32 Þ ro a0<1>  3 2 w00 þ ðn21 þ n23 n31 Þ w 3 1 p 2E1

3

þ ðn31 þ n21 n32 Þ ro g ¼ ro D

The general solution of the homogeneous equation associated with (111) has the form:

wh ðx1 Þ ¼ c1 ðexpða1 x1 Þ þ expða1 x1 ÞÞ þ c2 ðexpða2 x1 Þ þ expða2 x1 ÞÞ;

0 F<11> ¼ 0; 00 M<11>  r1 o F<22> ¼ 0;

F<11> ðlÞ ¼ p;

89

(108)

must hold in [l, þl]. Moreover, with the use of (92)1 and under the parity assumptions we made for a<1> and w, we see that the boundary conditions (107)2 take the common form:

  ð1  n23 n32 Þ a0<1> ðlÞ  ro w00 ðlÞ þ ðn31 þ n21 n32 Þg ¼ 0:

(109)

(113)

where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi r2o 1 32 2 a þ a  4b 2 1  3 r2o 3 a21 : ¼ ;   2 13 2r2o 1  3 r2o sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi r2o 1 32 2 a  a  4b 2 1  3 r2o 3 a22 : ¼ :   2 13 2r2o 1  3 r2o

(114)

With this, we write:

wðx1 Þ ¼ wh ðx1 Þ þ wp ;

wp : ¼ ro

! c 3 1 p d D þ g ; b 2E1 b

(115)

Finally, with the use of (92)1 again and of (91)2, and under the provisional assumption that the traction problem admits a solution with g a constant, (106)2 becomes:

with wp the constant solution of (111). With a view to determining the coefficients c1 and c2, we firstly return to the boundary condition (107)3, that, when combined with (92)1, reads:

00 1 2 1  n23 n32
2 00 ro w  ro a0<1> 3 h 3 00

ro w000 ðlÞ  a00<1> ðlÞ ¼ 0;

3

1

3

1

1þ 1þ 1 1 ro C ro C BB r g þ 3 log þ ð1  n13 n31 Þ @@1  3 log wA 3 A o 3 2 2 1 1

ro

þ

ro

ro

n21 þ n23 n31 0 ro a<1> þ ðn32 þ n31 n12 Þro g ¼ 0: h

ro

(110)

On eliminating a0<1> by means of (108), (110) yields: 3

2 2 ro

  1 p 1 32 3 1 w0000 þ 3 2 aw00 þ bw þ cro D þ dro g ¼ 0; 2 3 ro 2E1

n21 þ n23 n31 0 1 2 000 3 w ðlÞ  w ðlÞ; 3 1  n23 n32

the last two relations together imply that



2ðn21 þ n23 n31 Þ a :¼ ; 1  n23 n32 0

1

1 3 1þ 1 ro ðn21 þ n23 n31 Þ2 C B b :¼  @hð1  n13 n31 Þ 3 log A; 3 1  n23 n32 1  n23 n32 2 1 3

ro

d :¼

ro a00<1> ðlÞ ¼

(111)

where

c :¼

in addition, by differentiating (108) and invoking continuity up to the boundary of the resultant expression, we obtain that

3ðn21 þ n23 n31 Þ ; ð1  n23 n32 Þ2 0

 n þ n23 n31 1 0 1 32 r w000 ðlÞ þ 21 w ðlÞ ¼ 0: 1  n23 n32 ro 3 r2o o

(116)

Secondly, on eliminating a0<1> ðlÞ in (109) by means of (108), we obtain:

ro

0

3



1

1

1þ 1 ro C B B @hð1  n13 n31 Þ@1  3 log 3 A 1  n23 n32 2 1 3

ro

 1 p n þn n 1 1 32 1 3 r w00 ðlÞ þ 21 23 31 wðlÞ  D ¼ 0: 1  n23 n32 ro 3 r2o o ð1  n23 n32 Þ 2E1 (117)

ro

ðn þ n21 n32 Þðn21 þ n23 n31 Þ  31 þ hðn32 þ n31 n12 Þ: 1  n23 n32

(112)

On taking (115) into account, the system of Eqs. (116) and (117) determines the coefficients ca in (113):

c1 ¼ k1

expð2a1 lÞ  1 ; a1 ðexpð2a1 lÞ  1Þðexpð2a2 lÞ þ 1Þ  a2 ðexpð2a1 lÞ þ 1Þðexpð2a2 lÞ  1Þ

c2 ¼ k2

expð2a2 lÞ  1 ; a2 ðexpð2a2 lÞ  1Þðexpð2a1 lÞ þ 1Þ  a1 ðexpð2a2 lÞ þ 1Þðexpð2a1 lÞ  1Þ

(118)

90

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

(iii) in view of (115), we end up with: with

a2 k1 : ¼ 

a1 k2 : ¼ 

!   1 n21 þ n23 n31 c 3 p n21 þ n23 n21 d D g 1 þ b 2E1 1  n23 n32 1  n23 n32 b ;   1 32 1 n21 þ n23 n31 2 a1 ro 1 þ 2 þ ro 3 ro 1  n23 n32 !   1 n21 þ n23 n31 c 3 p n21 þ n23 n21 d D g 1 þ 1  n23 n32 1  n23 n32 b b 2E1 ;   n21 þ n23 n31 1 32 a22 ro 1 þ 2 þ r1 o 1  n23 n32 3 ro

 
d a  3 1 p a1 þ a2 g ¼  1 þ c 1 D b b 2E1  2
1
a1 c1 a1 þ 3 2 ao a21 sinhða1 lÞ þ lro
  2 2 þ a1 2 c2 a1 þ 3 ao a2 sinhða2 lÞ ;

ð121Þ

an implicit equation for g. Figs. 6e8 help visualizing some relevant features of the analytic solution we just constructed. 4.3. Pressure

(119) note that both the coefficients ca depend on g. Having found the form of the radial displacement w in [l, þl], we revert to Eq. (108) to find the axial displacement a<1>. A simply calculation yields:

"

We now let the cylindrical shell we study be subject to a uniform pressure qo<3> ¼ 6 ¼ Oð3 Þ, all the other applied loads being null. Accordingly, the field and boundary Eqs. (106) and (107) are replaced by, respectively,

 #





n þ n23 n31 c 3 1 p d D a<1> ðx1 Þ ¼ 1 þ 21 þ ð1  n23 n32 Þ1 ðn21 þ n23 n31 Þ  ðn31 þ n21 n32 Þ g x1 : 1  n23 n32 b 2E1 b

  n þn n 1 a c 3 2 a21  3 21 23 31 ðexpða1 x1 Þ  expða1 x1 ÞÞ þ 3ro a1 a2 2 1 1  n23 n32    n21 þ n23 n31 2 2 ðexpða2 x1 Þ  expða2 x1 ÞÞ þ a1 c2 3 a2  3 1  n23 n32 (needless to say, the parity condition a<1>(0) ¼ 0 is satisfied). The one task remaining is to find the constant g. This we do by a sequence of manipulations of the integral balance Eq. (94):

0

F<11> ðlÞ ¼ 0;

1

3

Zþl 1þ 1 ro C B B ðwþ ro gÞ @ð1 n13 n31 Þ@1 3 log 3 A 2 1

l

ro

M<11> ðlÞ ¼ 0;

0 M<11> ðlÞ ¼ 0:

(123)

Eq. (122)1 and (123)1 imply that F<11> is identically null, whence, in view of (91)1, that

ro

ro a0<1> 

1 n þn n  3 2 12 23 31 w00 þ mð1 n12 n21 Þ ro g h 3 

(122)

and

(i) using (91)4 and (92)3, we give Eq. (94) the form:

0

0 F<11> ¼ 0; 00 M<11>  r1 o F<22> þ 6 ¼ 0;

1



n þ n21 n32 1 ro a0<1>  3 2 w00 þðn32  n31 n12 ÞwC þ 31 A ¼ 0; ð120Þ h 3 (ii) on expunging the construct ðro a0<1>  ð1=3Þ3 2 w00 Þ by means of (108), we have:

n21 þ n23 n31 1 2 00 3 w ¼  w 3 1  n23 n32 n þ n21 n32 r g  31 1  n23 n32 o

over½l; þl:

(124)

With this, (92)1 and (91)2, Eq. (122)2 takes the form:

 3

2 2 ro

1

1 32 3 r2o



3

w0000 þ 3 2 aw00 þ bw  ~cr2o D

1 6

2E1

þ dro g ¼ 0; (125)

23 2 a0 w0h ðlÞ þ a1 where

a0 : ¼

Zþl

!

1 p þ a2 g 2E1

3

wh ¼ 2l ro D

l

!

where

 a1 wp ; ~c : ¼

3 ; 1  n23 n32

1 1  n23 n31 ; 3 ðn31 þ n21 n32 Þðn12 þ n23 n31 Þ 0

0 3 11 1þ 1  n23 n32 B 1 ro CC B a1 : ¼ h ; @n þ n32 n31  n23 þ n31 n12  ð1  n13 n31 Þ@1  3 log 3 AA n31 þ n21 n32 21 2 1 0

0

3

1

ro

1

ro

1þ 1  n23 n32 B 1 ro C B C a2 : ¼ h þ mð1  n12 n21 ÞA þ ðn31 þ n21 n32 Þ; @ð1  n13 n31 Þ@1  3 log 3 A n31 þ n21 n32 2 1

ro

ro

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

Fig. 7. Axial Traction Problem: radial displacement w.

Fig. 6. Axial Traction Problem: qualitative diagrams of bending moment M<11> and shear force F<31>.

the other constants being defined by (112)1,2,4. The general solution of the homogeneous equation associated to (125) has the form (113):

wh ðx1 Þ ¼ ~c1 ðexpða1 x1 Þ þ expða1 x1 ÞÞ þ ~c2 ðexpða2 x1 Þ þ expða2 x1 ÞÞ;

(126)

with the coefficients ad given by (114); then,

wðx1 Þ ¼ wh ðx1 Þ þ wp ;

wp : ¼ ro

1

 n þ n23 n31 1 1 32 r w00 ðlÞ þ 21 wðlÞ ¼ 0; 1  n23 n32 ro 3 r2o o

  n þ n23 n31 1 0 000 1 32 r w ðlÞ þ 21 w ðlÞ ¼ 0; 1 1  n23 n32 ro 3 r2o o

Lastly, we consider the case when a uniform distribution of bending couples m ¼ O (3 ) per unit length is applied at both rims of the cylinder. This time, the field equations we have to satisfy are:

(133)

(127) with the boundary conditions:

with wp the constant solution of (125). The boundary conditions (123)2,3 become:



4.4. Rim flexure

0 F<11> ¼ 0; 00 M<11>  r1 o F<22> ¼ 0;

!

1 6 ~c 3 d ro D  g ; b 2E1 b

91

(128)

F<11> ðlÞ ¼ 0;

M<11> ðlÞ ¼ m;

0 M<11> ðlÞ ¼ 0:

(134)

Just as in the pressure case, Eq. (133)1 and (134)1 imply that (124) holds; moreover, with the use of (92)1, (91)2, and (124), Eq. (133)2 takes the form:



(129) 3

2 2 ro

1

and are expedient to determine the constants ~c1 and ~c2 :

 1 32 w0000 þ 3 2 aw00 þ bw þ dro g ¼ 0 3 r2o

expð2a1 lÞ  1 ; a1 ðexpð2a1 lÞ  1Þðexpð2a2 lÞ þ 1Þ  a2 ðexpð2a1 lÞ þ 1Þðexpð2a2 lÞ  1Þ expð2a2 lÞ  1 ~c2 ¼ ~ k2 ; a2 ðexpð2a2 lÞ  1Þðexpð2a1 lÞ þ 1Þ  a1 ðexpð2a2 lÞ þ 1Þðexpð2a1 lÞ  1Þ

(135)

~c1 ¼ ~ k1

(130)

with



a2 ~ k1 : ¼

a1 ~ k2 : ¼



1 6 3

!

~c n þ n23 n31 n þ n23 n21 d ro D g 1  21  21 b 2E1 1  n23 n32 1  n23 n32 b ;   n21 þ n23 n31 1 32 a21 ro 1 þ 2 þ r1 o 3 ro 1  n23 n32 !   1 6 ~c n21 þ n23 n31 n21 þ n23 n21 d 3 ro D g 1  b 2E1 1  n23 n32 1  n23 n32 b :   n21 þ n23 n31 1 32 a22 ro 1 þ 2 þ r1 o 3 ro 1  n23 n32

(cf. (125)), where the constants are defined in (112). We set:

d wðx1 Þ ¼  ro g þ wh ðx1 Þ; b with

wh ðx1 Þ ¼ ^c1 ðexpða1 x1 Þ þ expða1 x1 ÞÞ þ ^c2 ðexpða2 x1 Þ þ expða2 x1 ÞÞ;

(136)

(131) Finally, by a sequence of steps completely analogous to the one leading to (121), the integral balance (94) yields an implicit equation for the constant g:



 1 6 ~c d 3 a1 þ a2 g ¼ a1 ro D b b 2E1  2
1
a1 c1 a1 þ 3 2 ao a21 sinhða1 lÞ þ lro
  2 2 þ a1 2 c2 a1 þ 3 ao a2 sinhða2 lÞ :

ð132Þ

Fig. 8. Axial Traction Problem: cartoon visualization of deformed and undeformed shapes.

92

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

the constants a1, a2 being given by (114). On writing the boundary conditions (134)2,3 in terms of displacements:

 1

 n þ n23 n31 1 1 32 3 mr r w00 ðlÞ þ 21 wðlÞ ¼  D 3 o ; 3 r2o o 2 3 E1 1  n23 n32 ro

  n þ n23 n31 1 1 32 r w000 ðlÞ þ 21 1 w0ðlÞ ¼ 0; 3 r2o o 1  n23 n32 ro

(137)

loads as a tubular membrane. On the other hand, for both 3 and ro fixed, longer and longer shells become slender and slender, without loosing their shell-like response. To exemplify the simplifications ensuing from taking largethinness limits, we take the integral balance (94), that has the same displacement form (120) in all three boundary-value problems of traction, pressure, and rim flexure. Since

(138) lim

3 = o /0

r

we determine the constants ^c1 and ^c2 :

1 2

3

ro

1þ log

1

3

ro 3

¼ 1;

(140)

ro

  a2 ðexpð2a2 lÞ  1Þexpða1 lÞ ^1 3D mro  dg ^c1 ¼ k ; 1 a1 ðexpð2a2 lÞ þ 1Þðexpð2a1 lÞ  1Þ  a2 ðexpð2a1 lÞ þ 1Þðexpð2a2 lÞ  1Þ 2 3 3 E1 b   a1 ðexpð2a1 lÞ  1Þexpða2 lÞ ^1 3D mro  dg ^c2 ¼ k ; 2 a2 ðexpð2a1 lÞ þ 1Þðexpð2a2 lÞ  1Þ  a1 ðexpð2a2 lÞ þ 1Þðexpð2a1 lÞ  1Þ 2 3 3 E1 b where

^ k1 ¼ ^ k2 ¼

 1  1

1 32 3 r2o



2

13 3 r2o

ro a21 þ r1 o ro a22 þ r1 o

we see that (120) reduces to

n21 þ n23 n31 ; 1  n23 n32

mð1  n12 n21 Þ ro g þ h1 ðn31 þ n21 n32 Þ

n21 þ n23 n31 : 1  n23 n32

þ ðn32  n31 n12 Þ

Once again, the integral condition balance (94) yields an implicit equation for g:





2







d a1 c a þ 3 2 ao a21 sinhða1 lÞ a1 þ a2 g ¼ b lro 1 1 1
  2 2 þ a1 2 c2 a1 þ 3 ao a2 sinhða2 lÞ :

ð139Þ

5. Cylindrical shells: thinness, slenderness, contraction moduli, and stiffnesses As all figures from 6 to 13 make evident, a phenomenon of boundary localization takes place, whatever the shell’s thickness, in the boundary-value problems of traction, pressure and rim-flexure we have solved analytically. This phenomenon is more and more pronounced as the shell’s length grows, ceteris paribus. In our opinion, this fact amply justifies the use of the much simpler formulas for slender shells that we derive in this section. 5.1. Thin shells and slender shells We term thin a cylindrical shell of diameter 2(ro þ 3 ) and length 2l if 3 /ro  1, slender if ro/l  1; most of times, ro is indeed smaller than l, so that a thin shell is slender as well (and 3 /l1). For ro/l fixed, a thinner and thinner shell looses its bending and twisting stiffness more and more; in the limit for 3 /ro / 0, it responds to

Fig. 9. Pressure Problem: qualitative diagrams of bending moment M<11> and shear force F<31>.

1 2l

ro l

a<1> ðlÞ

Zþl w ¼ 0; l

R þl where of course the values of both a<1>(l) and l w are problemdependent. Remark. CNTs, no matter if single- or multi-wall, are as a rule slender. However, they may be thin or not, in the sense of the above definition (Goldstein et al., 2010). Of course, all shell theories concern thin objects, but the thinness notions they are constructed upon may differ (see the discussions in (Podio-Guidugli et al., 2007) and (Podio-Guidugli et al., 2008)); in particular, those notions need not be expressed in terms of one purely geometrical aspect ratio. Our shell theory works whatever that ratio, because its subtler and more complex notion of thinness is the one typical of the method of internal constraints, a method to derive the mathematical models of linear structure mechanics firstly sketched in (Podio-Guidugli, 1989). 5.2. Axisymmetric equilibria of slender shells Hereafter, we display the slenderness approximations of the solutions to the fundamental problems analyzed in the previous section, except for the torsion problem, where no such approximation is in order, because the solution does not depend on the cylinder’s length.

Fig. 10. Pressure Problem: radial displacement w.

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

93

Fig. 11. Rim Flexure Problem: qualitative diagrams of bending moment M<11> and shear force F<31>. Fig. 13. Rim Flexure Problem: cartoon visualization of deformed and undeformed shapes.

(i) Traction. Looking at (118), it easy to conclude that

and

lim c1 ¼ lim c2 ¼ 0;

l/N

l/N

then, (113) takes the form:

~q wðx1 Þhw

ro 3 1 6 2E1

~ q : ¼ ro w

;

wh ðx1 Þh0:

g ¼ g

c b g :¼ D: d a2 þ a1 b 1 þ a1

1 p

2E1

;

(141)

wðx1 Þh  wp

2E1

;

c 1 þ a1 C Bc b B CD: w p : ¼ ro @ þ dA b a2 þ a1 b

2E1

;

g~ : ¼

1 6 ~ ~ r1 o wq ðn21 þ n23 n31 Þ þ gðn31 þ n21 n32 Þ ro 3 : 1  n23 n23 2E1

~c b

a2 þ a1

d b

D;

l/N

(142) 5.3. Effective contraction moduli

(ii) Pressure. This time, we look at (130) to conclude that, in the limit for l / N, both constants ~ca tend to zero. Thus, once again, w(x1) h wp, and we find that

g ¼ g~

(145)

everywhere null.

(143)

a1

a0<1> ðx1 Þh 

l/N

1 p D þ r1 o wp ðn21 þ n23 n31 Þ þ gðn31 þ n21 n32 Þ 3 a0<1> ðx1 Þh : 1  n23 n32 2E1

ro 3 1 6

D:

that in the large-slenderness limit the displacement field is

1

To arrive to a large-slenderness approximation for the function x1 1a < 1 > ðx1 Þ, the remaining unknown of our problem, we turn to (108). Under the present circumstances, that equation yields the value of the axial strain in a slender shell:

L :¼

d b

(iii) Rim Flexure. Given that lim ^c1 ¼ lim ^c2 ¼ 0, it easy to show

0

1 p

a2 þ a1

(146)

With this, one finds that

3

~c b

Finally, a use of (124) yields:

Moreover, (121) yields that: 3

a2

(144)

Let us now regard a slender cylindrical shell as a threedimensional rod-like body, in short, a probe. On defining the cross-section strain measure:

Eðx1 Þ : ¼

1 23

Zþ3

aðzÞEðx1 ; zÞ;

(147)

3

we have from (90) that

E<11> ¼ a0<1> ;

E<12> ¼ E<21> ¼

E<22> ¼ r1 o w;

E<33> ¼ g:

1 0 ; a 2 <2> (148)

Moreover, (89)3 implies that the deformed external radius of a shell of undeformed external radius r0ext ¼ r0 þ 3 is rext ¼ r0 þ w þ 3 g, so that

G :¼

r ext  r0ext ¼ E<22> þ Oð3 Þ: r0ext

Motivated by this observation, by an effective contraction modulus we mean:  in case of axial traction,

nCA : ¼ 

E<22> E<11>

¼

Dþ

r1 o wp ð1  n23 n32 Þ 1 ro wp ðn21 þ n23 n31 Þ þ gðn31

þ n21 n32 Þ

;

(149) Fig. 12. Rim Flexure Problem: radial displacement w.

~ p and g ~ are given by, respectively, (142) and (141) (note where w that this definition is directly reminiscent of the laboratory

94

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

procedure followed to measure Poisson’s modulus for isotropic materials);

this measure has to be accompanied by a few other adjustments that we now categorize and detail.

 in case of uniform inner pressure,

nCP : ¼ 

E<11> E<22>

¼

Formally, to recover the representation (14), one only has to take

g ¼ 0 in (11). To adjourn the developments of Sections 3, 4, and 5,

(i) (Balance Assumptions) The variations entering the Principle of Virtual Powers must have the following form, to be compared with (65):

~ ~ ðn31 þ n21 n32 Þ r1 o wq ðn21 þ n23 n31 Þ þ g ; ~ r1 o wq ð1  n23 n32 Þ (150)

~ p and g ~ are given by, respectively, (145) and (144). where w Both nCA and nCP depend on thickness, through, respectively, ~ p; g ~. wq ; g and w 5.4. Effective stiffnesses It remains for us to introduce suitable notions of effective traction and torsion stiffnesses for a slender cylindrical shell regarded as a probe. This we do by mimicking the relative familiar formulas from one-dimensional rod theory. As to effective traction stiffness, given that the traction stiffness of a rod is defined to be (axial load)/(axial strain), we set:

sA : ¼

P ; L

(151)

whence

sA ¼

ð1  n23 n32 Þ E1 Að3 Þ; D þ r1 o wp ðn21 þ n23 n31 Þ þ gðn31 þ n21 n32 Þ

T

y ¼ y1 c1 þ y2 n0 þ y3 n; 

ð1Þ

(155)



y ¼ y3;1 c1 þ r1 y2  y3;2 n0 ; o

Consequently, the integral balance Eq. (69)4 disappears, and the same happens with the last power-conjugate pair in each of the boundary conditions (72) and (95). (ii) (Constitutive Assumptions) The constraint space (42) reduces to

D ¼ spanðW i ; i ¼ 1; 2; 3Þ; S<33> is then reactive and only four constitutive moduli survive. It is not difficult to show that (51) and (52) must be replaced, respectively, by

Að3 Þ : ¼ 4pro 3 ;

where we have made use of (105), (143), and (148)1, and where A(3 ) is the area of the shell’s cross-section (needless to say, sA depends on 3 also through wp and g). As to effective torsion stiffnesses, we recall that, for a twisted rod, one takes it to be (torsion moment)/(twist per unit length); accordingly, we set:

sT : ¼

ð0Þ

(152)

~ ¼ D1 ðE1 W 1 5W 1 þ E2 W 2 5W 2 þ 2DGW 3 5W 3 C þ E1 n21 ðW 1 5W 2 þ W 2 5W 1 ÞÞ;

(156)

and

D : ¼ 1  n12 n21 : ;

(153)

Q

so that, on recalling (100)e(102), we have that

sT ¼ GJð3 Þ;

 2 3 Jð3 Þ : ¼ 4pr3o 1 þ 2 ;

(154)

ro

where Jð3 Þ is the polar inertia moment of the cross section.9

(157)

This result is achieved by the first of various applications to follow of a procedure that consists in taking the limits for E3, l, and m, tending to infinity, and in setting na3 ¼ n3a ¼ 0 (a ¼ 1,2). It is important to realize that, due to the first of (88) that we here repeat for the reader’s convenience:

n E1 ¼ 12 ; n21 E2

(158)

(156) and (157) integrate a 4-parameter constitutive representation.

6. Cylindrical Kirchhoff-Love shells Recall from Section 2.2 the Kirchhoff-Love representation (14) for the displacement field: s

uKL ðx; zÞ ¼ s Aðx; zÞaðxÞ þ wðxÞnðxÞ  z VwðxÞ;

aðxÞ$nðxÞ ¼ 0;

and compare it with the more general representation (11): in both cases the unshearability constraint is imposed implicitly, together with, but only in the first case, inextensibility of the fibers orthogonal to the middle surface. Thus, the thickness of these shells does not change whatever the applied loads, making the simpler KirchhoffLove theory suitable, in our opinion, for application to single-wall CNTs.

(iii) (Torsion Problem) The solution given in Section 4.1 does not change, because it does not involve any of the constitutive moduli that take singular values in the case of Kirchhoff-Love shells. (iv) (Traction Problem) The two governing equations are, mutatis mutandis, (108) and (111); specifically, the equation corresponding to (108) is:

ro a0<1> 

(159)

while the one corresponding to (111) reads:

 3

9 Note that (154) holds whatever the slenderness of the shell under consideration.

1 p 1 2 00 3 3 w þ n21 w ¼ ro D ; 3 2E1

2 2 ro

1

1 32 3 r2o



3

w0000 þ 3 2 aw00 þ bw þ cro D

1 p

2E1

¼ 0;

(160)

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

where the constants have the following expressions, that can be recovered from (112):

0

a : ¼ 2n21 ;

1 3 1þ ro B 1 C b : ¼ 3h@ 3 log  n12 n21 A; 3 2 1

ro

c : ¼ 3n21 :

ro

(161) As to Eq. (115), it is replaced by

wðx1 Þ ¼ wh ðx1 Þ þ wp ;

wp ¼ ro n12 dð3 Þ

1 p ; 2E1

3

(162)

where

ð1  n12 n21 Þ

dð3 Þ : ¼ 1 2

3

1þ log

ro

1

3

ro 3

95

inner pressure and the measured or computed values of Q, L, and G. In the large-thinness limit, one would have:

GJð3 Þ ¼ sT ;

E1 Að3 Þ ¼ sA ;

n12 ¼ nCA ;

n21 ¼ nCP ;

so that the values of the four constitutive moduli G, E1, n12, and n21 would follow, provided one could measure or evaluate the values of the geometrical parameters ro and 3 . Now, given the well-known difficulties, commonly referred to as the Yakobson’s Paradox (Shenderova et al., 2002), in choosing a representative value for the wall thickness of a SWCNT, we think it best to characterize the mechanical response of a Kirchhoff-Love shell by the contraction ~ : ¼ G3 , moduli n12, n21 and the effective constitutive moduli G ~a : ¼ Ea 3 ða ¼ 1; 2Þ. E


 note for later use that lim dð3 Þ ¼ 1 :

(163)

3 = o /0

r

 n12 n21

ro

For slender shells, w(x1) ¼ wp and (159) allows to conclude that

a<1> ðx1 Þ ¼ ð1  n12 n21 ð1  dð3 ÞÞÞ

3

1 p

2E1

x1 :

(164)

(v) (Pressure Problem) The two relevant Eqs. (124) and (125), become, respectively:

Remark. We find it appropriate to expand a little on this last point. For slender and thin cylindrical shells, we find that: -when subject to end traction,

wT ¼ ro n12

p ~ 2E 1

;

a0<1>T ¼

p ~ 2E 1

;

(171)

by way of Eqs. (162) and (164);

ro a0<1>

1  3 2 w00 ¼ n21 w; 3

(165)

- when subject to inner pressure,

and

 3

2 2 ro

1 32 1 3 r2o



3

w0000 þ 3 2 aw00 þ bw  3r2o D

1 6

2E1

¼ 0;

wP ¼ r2o

wðx1 Þ ¼ wh ðx1 Þ þ wq ; wq ¼ r2o dð3 Þ

3

1 6

2E2

:

(167)

ro 3 1 6 2E2

x1 :

dð3 Þ n 1  n12 n21 ð1  dð3 ÞÞ 12

(168)

and

nCP ¼ n21 :

ro 6 ~ 2E 2

;

(172)

by way of Eqs. (167) and (168). Eqs. (171) and (172) can be regarded as a system of four equa~ , n12, n21, whose solution is: ~ ,E tions in the unknowns E 1 2

~ ¼ E 1

0 ~ ¼ r2 6 ; n ¼  wT ; n ¼ r a<1>P : ; E 2 12 21 o o ro a0<1>T 2a0<1>T 2wP wP

p

Interestingly, the consistency condition (88)1, which can now be written as

(vi) (Effective Contraction Moduli. Effective Stiffnesses) As to contraction moduli, formulas (149) and (150) become:

nCA ¼

a0<1>P ¼ n21

(173)

By using (165), we conclude that, for slender shells,

a<1> ¼ n21 dð3 Þ

2

(166)

where the constants a and b are the same as in Eq. (161). Eq (127) is replaced by

6 ; ~ 2E

(169)

~ n E 1 ¼ 12 ; ~ n21 E 2 implies that

p 6 ¼ 0 ; wT a<1>P a relation that can be used to check whether the present simplified form of our shell theory is applicable.

As to traction stiffness, (152) reduces to

1 sA ¼ E Að3 Þ; 1  n12 n21 ð1  dð3 ÞÞ 1

7. Conclusions and further developments

(170)

torsion stiffness continues to be given by formula (154). (vii) (Determination of Effective Constitutive Moduli) In principle, a set of torsion, traction, and pressure, experiments or simulations allows to deduce the values of sT, sA, nCA, and nCP, from the prescribed values of applied torque, axial load, and

We have given a detailed presentation of a theory of linearly elastic orthotropic shells with potential application to the continuous modeling of carbon nanotubes. Linearity is of course an intrinsic limitation of our theory (for an assessment of how nonlinearities affect elastic properties and strength, see Pugno, 2006; Pugno et al., 2006; Wu et al., 2008); its novelty resides in two features: (1) the type of orthotropic response we have selected seems suitable, with minimal tuning, to capture chirality, not only in

96

A. Favata, P. Podio-Guidugli / European Journal of Mechanics A/Solids 35 (2012) 75e96

the extreme cases of zig-zag and armchair SWCNTs, but also when it varies in an essentially undetectable manner from wall to wall of a MWCNT; (2) the possibility of accounting for overall thickness changes, that should be almost exclusively due to changes in interwall separation. As a matter of fact, the referential thickness of an ideal MWCNT is dictated by inter-wall forces of van der Waals type, whose action may be modeled essentially in two ways: either they can be regarded small with respect to applied loads, and ignored altogether; or they can be thought of as inducing a referential equilibrated stress state that should be taken into account when studying the effects of boundary conditions, no matter if hard or soft: our sounding of this latter approach is encouraging. In addition, we have proposed a simpler version of the theory, in which orthotropy is preserved but thickness changes are excluded in all admissible deformational vicissitudes; we believe this simpler theory to fit SWCNTs, whose effective thickness when regarded as cylindrical shells should not change appreciably when loaded, no matter how evaluated. Presuming, as we do, a rather complex material response requires specification of a number of constitutive parameters e seven when thickness changes are allowed, four when they are not. Luckily, another feature of our present theory is that, in both its versions, it leads to a number of significant boundary-value problems that can be solved explicitly in closed form. These problems are: torsion, axial traction, uniform inner pressure, and rim flexure; were their solutions coupled with the corresponding measurements and/or simulation results, applicability of our theory could be unequivocally assessed and all constitutive parameters in it uniquely determined. It is not difficult to paste explicit solutions of the type we here derived for two or more coaxial shells, both in statical and dynamical situations; we are currently developing this line of research with a view to a better understanding of van der Waals forces and the role and evolution of defects (Di Carlo et al., forthcoming; Pugno, 2007). Acknowledgments We gratefully acknowledge a useful discussion with Professor M. Bertsch.

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