Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 4, pp. 755-774, Warsaw 2003

ON THE MODELLING OF THIN UNIPERIODIC
CYLINDRICAL SHELLS

Barbara Tomczyk
Department of Structural Mechanics, University of Technology, Łódź

e-mail: btomczyk@ck-sg.p.lodz.pl

Theaimof this contribution is topropose a newaveragedmodel of dyna-
mic problems for thin linear-elastic cylindrical shells having a periodic
structure along one direction tangent to the shell midsurface. In con-
trast with the known homogenized models, the proposed one makes it
possible to describe the effect of the periodicity cell size on the global
dynamic shell behavior (a length-scale effect). In order to derive gover-
ning equations with constant or slowly varying coefficients, the known
tolerance averaging procedure is applied. The comparison between the
proposed model and the model without the length-scale effect as well
as the known length-scale model for cylindrical shells with the periodic
structure in both directions tangent to the shellmidsurface is presented.

Key words: shell, modelling, dynamics, cell

1. Introduction

In thispaperanewmodel of cylindrical shells havingaperiodic structure (a
periodically varying thickness and/or periodically varying elastic and inertial
properties) along one direction tangent to the undeformed shell midsurface
M is presented.
Cylindrical shells under consideration are composed of a large number of

identical elements which are periodically distributed along one direction tan-
gent to M.Moreover, every such element is treated as a shallow shell. Itmeans
that the period of inhomogeneity is very large compared with the maximum
shell thickness and very small as compared to themidsurface curvature radius
as well as the smallest characteristic length dimension of the shell midsurface.
Structures like that are termed uniperiodic.



756 B.Tomczyk

It should be noted that in the general case, on the shellmidsurfacewe deal
withnot a periodic structurebutwithwhat is called a locally periodic structu-
re in directions tangent to M. FollowingWoźniak (1999), it means that every
small piece of the shell constituting a shallow shell, with sufficient accuracy,
can be described as having a periodic structure related to the Cartesian coor-
dinates on a certain plane tangent to M. Hence, to every point x belonging
to M we assign the plane Tx tangent to M at this point and periods lα(x),
α=1,2 in the direction of unit vectors eα(x) on Tx. On every plane Tx a local
periodicity cell spanned on the vectors (lαeα)(x) is defined. For locally uni-
periodic shells, the index α is equal to either 1 or 2. For cylindrical shells, the
Gaussian curvature is equal to zero, and hence on the developable cylindrical
surface we can separate a cell which can be referred to as the representative
cell for thewhole shell midsurface. It means that on the cylindrical surface we
deal with not a locally periodic but with periodic structure.

Problems of periodic (or locally periodic) structures are investigated by
means of different methods. The exact analysis of shells and plates of this
kind within solid mechanics can be carried out only for a few special pro-
blems. In the most cases, the exact equations of the shell (plate) theory are
too complicated to constitute the basis for investigations of most engineering
problems because they involve highly oscillating and often discontinuous co-
efficients. Thus, many different approximated modelling methods for periodic
(locally periodic) shells and plates have been formulated.

Structures of this kind are usually described using homogenized models
derived by means of asymptotic methods. These models from a formal point
of view represent certain equivalent structureswith constant or slowly varying
stiffnesses and averaged mass densities. In the case of periodic plates, these
asymptotic homogenization methods were presented by Caillerie (1984) (in
this contribution two small parameters – thickness of a plate and the charac-
teristic size of a periodicity cell – are used to investigate periodic plates),Kohn
andVogelius (1984) (this paper dealswith thin plates having a rapidly varying
thickness), Lewiński (1992) (in this contribution homogenized stiffnesses are
analyzed) and others. The asymptotic approach to periodic shells was propo-
sedbyLutoborski (1985),Kalamkarov (1987), Lewiński andTelega (1988); the
discussion of the above approach can be found inWoźniak (1999). The formu-
lation of mathematical models of shells by using the asymptotic expansions
is rather complicated from the computational point of view. That is why the
asymptotic procedures are restricted to the first approximation. Within this
approximation, we obtain models which neglect the effect of periodicity cell
length dimensions on the global structure behavior (the length-scale effect).



On the modelling of thin uniperiodic... 757

This effect plays an important role mainly in the vibration and wave propa-
gation analysis. To formulate the length-scale models in the framework of the
asymptotic homogenization we could find higher-order terms of the asympto-
tic expansions, cf. Lewiński and Kucharski (1992). Models of this kind have
a complicated analytical form, and applied to the investigation of boundary-
value problems often lead to a large number of boundary conditions, which
may be not well motivated from the physical viewpoint.
The alternative nonasymptotic modelling procedure based on the notion

of tolerance and leading to the so-called length-scale (or tolerance) models
of dynamic and stationary problems for micro-periodic structures was pro-
posed by Woźniak in a series of papers, e.g. Woźniak (1993, 1997), Woźniak
andWierzbicki (2000). These tolerance models have constant coefficients and
take into account the effect of the periodicity cell size on the global body be-
havior (the length-scale effect). This effect is described by means of certain
extra unknowns called internal or fluctuation variables and by known func-
tions which represent oscillations inside the periodicity cell, and are obtained
as approximate solutions to special eigenvalue problems for free vibrations on
the separated cell with periodic boundary conditions. The averaged models
of this kind have been applied to analyze certain dynamic problems of perio-
dic structures, e.g. for Hencky-Reissner periodic plates (Baron and Woźniak,
1995), forKirchhoffperiodic plates (Jędrysiak, 1998, 2000), for periodic beams
(Mazur-Śniady, 1993), for periodicwavy-plates (Michalak, 1998, 2000), for cy-
lindrical shells with a two-directional periodic structure (Tomczyk, 1999) and
others.
A general modelling method based on the concept of internal variables

and leading from 2D equations of thin shells with a two-directional locally
periodic structure to the averaged equations with slowly varying coefficients
depending on the local cell length dimensions has been proposed byWoźniak
(1999). However, these internal variable models are not sufficient to analyze
problems of shells with a locally periodic (or periodic) structure in only one
direction tangent to the udeformed shell midsurface. Shells of this kind, called
the locally uniperiodic shells, in general are not special cases of those with a
locally periodic structure in both directions tangent to M.
The aim of this contribution is three-fold:
• First, to derive anaveragedmodel of auniperiodic cylindrical shellwhich
has constant coefficients in the direction of periodicity and describes the
effect of a cell size on the overall shell behavior. The length scales will
be introduced to the global description of both inertial and constitutive
properties of the shell under consideration. This model will be derived
by using the tolerance averaging procedure proposed by Woźniak and



758 B.Tomczyk

Wierzbicki (2000), and hence will be called the tolerance fluctuation va-
riable model for uniperiodic cylindrical shells.

• Second, to derive a simplified (homogenized)model inwhich the length-
scale effect is neglected.

• Third, to compare the proposed here tolerance fluctuation variable mo-
del with the homogenized one and with the known tolerance model of
cylindrical shells having a periodic structure in both directions tangent
to M.

Basic denotations and starting equations of the shell theory will be pre-
sented in Section 2. To make considerations more clear, the general line of
the tolerance averaging approach, following the monograph by Woźniak and
Wierzbicki (2000), will be presented in Section 3. In the subsequent section,
the tolerance model with the fluctuation variables for dynamic problems in
linear-elastic thin cylindrical shells with a periodic structure along one direc-
tion tangent to M and a slowly varying structure along the perpendicular one
tangent to M will be shown. For comparison, the governing equations of a
certain homogenized model will be presented in Section 5. Final remarks will
be formulated in the last section.

2. Preliminaries

In this paper, we will investigate thin linear-elastic cylindrical shells with
a periodic structure along one direction tangent to M and a slowly varying
structure along the perpendicular direction tangent to M. Cylindrical shells
of this kind will be termed uniperiodic. Examples of such shells are presented
in Fig.1.

Fig. 1. Examples of uniperiodic shells



On the modelling of thin uniperiodic... 759

Denoteby Ω⊂R2 a regular regionofpoints Θ≡ (Θ1,Θ2) on the OΘ1Θ2-
plane, Θ1, Θ2 being the Cartesian orthogonal coordinates on this plane, and
let E3 be the physical space parametrized by the Cartesian orthogonal coor-
dinate system Ox1x2x3. Let us introduce the orthogonal parametric represen-
tation of the undeformed smooth cylindrical shell midsurface M bymeans of
M := {x≡ (x1,x2,x3)∈E3 : x= x(Θ1,Θ2), Θ ∈Ω}, where x(Θ1,Θ2) is
a position vector of a point on M having coordinates Θ1,Θ2.

Throughout the paper, the indices α,β,... run over 1,2 and are related to
the midsurface parameters Θ1,Θ2; the indices A,B,... run over 1,2, ...,N,
the summation convention holds for all aforesaid indices.

To every point x=x(Θ), Θ ∈Ω we assign covariant base vectors aα =
x,α and covariant midsurface first and secondmetric tensors denoted by aαβ,
bαβ, respectively, which are given as follows: aαβ = aα ·aβ, bαβ = n ·aα,β,
where n is a unit vector normal to M.

Let δ(Θ) stand for the shell thickness.We also define t as the time coor-
dinate.

Taking into account that coordinate lines Θ2 = const are parallel on the
OΘ1Θ2-plane and that Θ2 is an arc coordinate on M, we define l as the pe-
riod of the shell structure in the Θ2-direction. The period l is assumed to be
sufficiently large compared with themaximum shell thickness and sufficiently
small as compared to the midsurface curvature radius R as well as the cha-
racteristic length dimension L of the shell midsurface along the direction of
shell periodicity, i.e. supδ(·)≪ l≪min{R,L}. On the given above assump-
tions for the period l, the shell under consideration will be referred to as a
mezostructured shell, cf. Woźniak (1999), and the period l will be called the
mezostructured length parameter.

We shall denote by Λ ≡ {0}× (−l/2, l/2) the straight line segment on
the OΘ1Θ2-plane along the OΘ2-axis direction, which can be taken as a
representative cell of the shell periodic structure (the periodicity cell). To
every Θ∈Ω an arbitrary cell on the OΘ1Θ2-plane will be defined bymeans
of: Λ(Θ) +Λ, Θ ∈ ΩΛ, ΩΛ := {Θ ∈ Ω : Λ(Θ) ⊂ Ω}, where the point
Θ∈ΩΛ is a center of a cell Λ(Θ) and ΩΛ is a set of all the cell centers which
are inside Ω.

A function f(Θ) defined on ΩΛ will be called Λ-periodic if for arbitrary
but fixed Θ1 and arbitrary Θ2,Θ2± l it satisfies the condition: f(Θ1,Θ2)=
f(Θ1,Θ2± l) in the whole domain of its definition, and it is not constant.

It is assumed that the cylindrical shell thickness as well as its material
and inertial properties are Λ-periodic functions of Θ2 and slowly varying



760 B.Tomczyk

functions of Θ1. Shells like that are called uniperiodic, moreover, on the given
above assumptions for the period l they are referred tomezostructured shells.
For an arbitrary integrable function ϕ(·) defined on Ω, followingWoźniak

andWierzbicki (2000), we define the averaging operation, given by

〈ϕ〉(Θ)≡
1
l

∫

Λ(Θ)

ϕ(Θ1,Ψ2) dΨ2 Θ=(Θ1,Θ2)∈ΩΛ (2.1)

For a function ϕ, which is Λ-periodic in Θ2, formula (2.1) leads to
〈ϕ〉(Θ1). If the function ϕ is Λ-periodic in Θ2 and is independent of Θ1,
its averaged value obtained from (2.1) is constant.
Our considerations will be based on the simplified linear Kirchhoff-Love

theory of thin elastic shells in which terms depending on the second metric
tensor of M are neglected in the formulae for curvature changes.
Let uα(Θ, t), w(Θ, t) stand for the midsurface shell displacements in di-

rections tangent and normal to M, respectively. We denote by εαβ(Θ, t),
καβ(Θ, t) the membrane and curvature strain tensors and by n

αβ(Θ, t),
mαβ(Θ, t) the stress resultants and stress couples, respectively. Theproperties
of the shell are described by 2D-shell stiffness tensors Dαβγδ(Θ), Bαβγδ(Θ),
and let µ(Θ) stand for the shell mass density per midsurface unit area. Let
fα(Θ, t), f(Θ, t) be external force components per midsurface unit area, re-
spectively tangent and normal to M.
Functions µ(Θ),Dαβγδ(Θ),Bαβγδ(Θ) and δ(Θ) are Λ-periodic functions

of Θ2 and are assumed to be slowly varying functions of Θ1.
The equations of the shell theory under consideration consist of:

— the strain-displacement equations

εγδ =u(γ,δ)− bγδw κγδ =−w,γδ (2.2)

— the stress-strain relations

nαβ =Dαβγδεγδ m
αβ =Bαβγδκgd (2.3)

— the equations of motion

nαβ,α −µa
αβüα+f

β =0
(2.4)

m
αβ
,αβ
+bαβn

αβ −µẅ+f =0

In the above equations, thedisplacements uα =uα(Θ, t) and w=w(Θ, t),
ϕ, are the basic unknowns.



On the modelling of thin uniperiodic... 761

For mezostructured shells, µ(Θ), Dαβγδ(Θ) and Bαβγδ(Θ), Θ ∈ Ω, are
highly oscillating Λ-periodic functions; that is why equations (2.2)-(2.4) can-
not be directly applied to the numerical analysis of special problems. From
(2.2)-(2.4), an averaged model of uniperiodic cylindrical shells having coeffi-
cients, which are independent of the Θ2-midsurface parameter, and are slowly
varying functions of Θ1 as well as describing the length-scale effect will be
derived. In order to derive it, the tolerance averaging procedure given byWoź-
niak andWierzbicki (2000), will be applied. Tomake the analysis more clear,
in the next section we shall outline the basic concepts and the main kinema-
tic assumption of this approach, following the monograph by Woźniak and
Wierzbicki (2000).

3. Basic concepts

Following the monograph by Woźniak and Wierzbicki (2000), we outline
below the basic concepts, which will be used in the course of the modelling
procedure.
The fundamental concepts of the tolerance averaging approach are that

of a certain tolerance system, slowly varying functions, periodic-like functions
and periodic-like oscillating functions. These functions will be defined with
respect to the Λ-periodic shell structure defined in the foregoing section.
By a tolerance system we shall mean a pair T = (F,ε(·)), where F is a

set of real valued bounded functions F(Θ) defined on Ω and their derivatives
(includingalso timederivatives),which represent theunknowns in theproblem
under consideration (such as unknown shell displacements tangent andnormal
to M), and forwhich the tolerance parameters εF beingpositive real numbers
and determining the admissible accuracy related to computations of values of
F(·) are given; by ε the mapping F ∋F → εF is denoted.
A continuous bounded differentiable function F(Θ, t) defined on Ω is cal-

led Λ-slowly varying with respect to the cell Λ and the tolerance system T ,
F ∈SVΛ(T), if roughly speaking, can be treated (togetherwith its derivatives)
as constant on an arbitrary periodicity cell Λ.
The continuous function ϕ(·) defined on Ω will be termed a Λ-periodic-

like function, ϕ(·) ∈ PLΛ(T), with respect to the cell Λ and the tolerance
system T , if for every Θ=(Θ1,Θ2)∈ΩΛ there exists a continuous Λ-periodic
function ϕΘ(·) such that (∀Ψ =(Θ1,Θ2) [‖Θ−Ψ‖¬ l⇒ϕΘ(Ψ)],Ψ ∈Λ(Θ),
and similar conditions are also fulfilled by all its derivatives. Itmeans that the
values of the periodic-like function ϕ(·) in an arbitrary cell Λ(Θ), Θ ∈ ΩΛ,



762 B.Tomczyk

can be approximated, with sufficient accuracy, by corresponding values of a
certain Λ-periodic function ϕΘ(·). The function ϕΘ(·) will be referred to as
a Λ-periodic approximation of ϕ(·) on Λ(Θ).
Let µ(·) be a positive value Λ-periodic function.Theperiodic-like function

ϕ is called Λ-oscillating (with the weight µ), ϕ(·) ∈ PLµ
Λ
(T), provided that

the condition 〈µϕ〉(Θ)∼=0 holds for every Θ∈ΩΛ.
If F ∈ SVΛ(T), ϕ(·) ∈ PLΛ(T) and ϕΘ(·) is a Λ-periodic appro-

ximation of ϕ(·) on Λ(Θ), then for every Λ-periodic bounded function
f(·) and every continuous Λ-periodic differentiable function h(·) such that
sup{|h(Ψ1,Ψ2)|, (Ψ1,Ψ2)∈Λ}¬ l, the following tolerance averaging relations
hold for every Θ∈ΩΛ:

(T1) 〈fF〉(Θ)∼= 〈f〉(Θ)F(Θ) for ε= 〈|f|〉εF

(T2) 〈f(hF),2〉(Θ)∼= 〈fFh,2〉(Θ) for ε= 〈|f|〉(εF + lεF,2)

(T3) 〈fϕ〉(Θ)∼= 〈fϕΘ〉(Θ) for ε= 〈|f|〉εϕ

(T4) 〈h(fϕ),2〉(Θ)∼=−〈fϕh,2〉(Θ) for

{
ε= εF + lεF,2

F = 〈hfϕ〉

where ε is a tolerance parameter which defines the pertinent tolerance ∼=.
In the tolerance averaging procedure, the left-hand sides of formulae (T1)-

(T4)will be approximated by their right-hand sides, respectively – this ope-
ration will be called theTolerance Averaging Assumption.
In the subsequent considerations, the following lemma will be used:

(L1) If ϕ(·) ∈ PLΛ(T) and f is a bounded Λ-periodic function then
〈fϕ〉(·)∈SVΛ(T)

(L2) If ϕ(·) ∈ PLΛ(T) then there exists the decomposition ϕ(·) = ϕ0(·)+
ϕ̃(·), where ϕ0(·) ∈ SVΛ(T) and ϕ̃(·) ∈ PL

µ
Λ(T), moreover, it can be

shown that ϕ0(·)∼= 〈µϕ〉(·)〈µ〉−1

(L3) If F ∈SVΛ(T) and f is a bounded continuous Λ-periodic function then
〈fF〉 ∈PLΛ(T)

(L4) If F ∈SVΛ(T), G∈SVΛ(T), kF +mG∈F for some reals k,m, then
kF +mG∈SVΛ(T).

Themainkinematic assumptionof the tolerance averagingmethod is called
the Conformability Assumption and states that in every periodic solid the



On the modelling of thin uniperiodic... 763

displacement fields have to conform to the periodic structure of this solid. It
means that thedisplacement fields are periodic-like functions andhence canbe
represented by a sumof the averaged displacements, which are slowly varying,
and by highly oscillating periodic-like disturbances, caused by the periodic
structure of the solid.
The aforementioned Conformability Assumption together with the Tole-

rance AveragingAssumption constitute the foundations of the tolerance avera-
ging technique.Using this technique, the tolerancemodel of dynamicproblems
for uniperiodic cylindrical shells will be derived in the subsequent section.

4. The tolerance model of dynamic problems for uniperiodic
cylindrical shells

Let us assume that there is a certain tolerance system T =(F,ε(·)), where
the set F consists of the unknown shell displacements tangent and normal
to M and their derivatives (also time derivatives). From the Conformability
Assumption, it follows that the unknown shell displacements uα(·, t), w(·, t)
in Eqs (2.2)-(2.4) have to satisfy the conditions: uα(·, t)∈ PLΛ(T), w(·, t) ∈
PLΛ(T). Itmeans that in every cell Λ(·),Θ∈ΩΛ, the displacement fields can
be represented, within a tolerance, by their periodic approximations.
Taking into account Lemma (L2), we obtain what is called themodelling

decomposition

uα(·, t)=Uα(·, t)+dα(·, t) w(·, t) =W(·, t)+p(·, t)

Uα(·, t), W(·, t)∈SVΛ(T) dα(·, t),p(·, t)∈PL
µ
Λ
(T)

(4.1)

which becomes under the normalizing condition 〈µdα(·, t)〉= 〈µp(·, t)〉=0 (in
dynamic problems) or 〈dα(·, t)〉= 〈p(·, t)〉=0 (in quasi-stationary problems).
It can be shown, cf. Woźniak and Wierzbicki (2000), that the unknown

Λ-slowly varying averaged displacements Uα(·, t), W(·, t) in (4.1) are given
by: Uα(·, t)≡〈µ〉−1(Θ1)〈µuα〉(·, t), W(·, t)≡〈µ〉−1(Θ1)〈µw〉(·, t).
Theunknowndisplacementdisturbances dα(·, t), p(·, t) in (4.1) beingoscil-

lating periodic-like functions are caused by the highly oscillating character of
the shell mezostructure.
Substituting the right-hand side of (4.1) into (2.4), and after the tolerance

averaging of the resulting equations, we arrive at the equations



764 B.Tomczyk

[〈Dαβγδ〉(Θ1)(Uγ,δ − bγδW)+ 〈D
αβγδdγ,δ〉(Θ, t)+

−bγδ〈D
αβγδp〉(Θ, t)],α−〈µ〉(Θ

1)aαβÜα =−〈f
β〉(Θ, t)

(4.2)

[〈Bαβγδ〉(Θ1)W,γδ + 〈B
αβγδp,γδ〉(Θ, t)],αβ − bαβ[〈D

αβγδ〉(Θ1)(Uγ,δ − bγδW)+

+〈Dαβγδdγ,δ〉(Θ, t)− bγδ〈D
αβγδp〉]+ 〈µ〉(Θ1)Ẅ = 〈f〉(Θ, t)

which must hold for every Θ∈ΩΛ and every time t.
Bymeans ofLemma (L4), the left-hand sides ofEqs (4.2) canbe treated as

slowly varying functions; hence from Lemma (L1) it follows that 〈fβ〉(Θ, t),
〈f〉(Θ, t) ∈ SVΛ(T). This situation takes place if the shell external loadings
satisfy the condition: fβ(Θ, t),f(Θ, t)∈PLΛ(T). This condition is called the
Loading Restriction.
From the Loading Restriction and Lemma (L2) it follows that the shell

external loadings can be presented as the sum of Λ-slowly varying loadings
and Λ-oscillating periodic-like loadings, i.e.

fβ(·, t)= fβ0 + f̃
β(·, t) f(·, t)= f0(·, t)+ f̃(·, t)

f
β
0 (·, t),f0(·, t)∈SVΛ(T) f̃

β(·, t), f̃(·, t)∈PL1Λ(T)
(4.3)

where 〈f̃β〉(Θ, t)= 〈f̃〉(Θ, t)∼=0.
MultiplyingEqs (2.4) by arbitrary Λ-periodic test functions d∗,p∗, respec-

tively, such that 〈µd∗〉 = 〈µp∗〉 = 0, integrating these equations over Λ(Θ),
Θ∈ΩΛ, and using the Tolerance Averaging Assumption, as well as denoting
by d̃α, p̃ the Λ-periodic approximations of dα, p, respectively, on Λ(Θ), we
obtain the periodic problemon Λ(Θ) for functions d̃α(Θ1,Θ2, t), p̃(Θ1,Θ2, t),
(Θ1,Θ2)∈Λ(Θ)=Λ(Θ1,Θ2), given by the following variational conditions

−〈d∗,2D
2βγδd̃γ,δ〉+ 〈d

∗(D1βγδd̃γ,δ),1〉− bγδ[−〈d
∗

,2D
2βγδp̃〉+

+〈d∗(D1βγδp̃),1〉]−〈d
∗µ
¨̃
d〉aαβ =

=−〈d∗fβ〉+ 〈d∗,αD
αβγδ〉(Uγ,δ − bγδW)− [〈d

∗D1βγδ〉(Uγ,δ − bγδW)],1
(4.4)

〈p∗,22B
22γδp̃,γδ〉−2〈p

∗

,2(B
21γδp̃,γδ),1〉+ 〈p

∗(B11γδp̃,γδ),11〉+

−bαβ[〈p
∗Dαβγδd̃γ,δ〉− bγδ〈p

∗Dαβγδp̃〉]+ 〈p∗µ¨̃p〉=

= 〈p∗f〉+ bαβ〈p
∗Dαβγδ〉(Uγ,δ − bγδW)−〈p

∗

,22B
22λδ〉W,γδ +

+2[(〈p∗,2B
21γδ〉,1−〈p

∗

,21B
21γδ〉)W,γδ + 〈p

∗

,2B
21γδ〉W,γδ1]+

−{[(〈p∗B11γδ〉,1−2〈p
∗

,1B
11λδ〉),1+ 〈p

∗

,11B
11γδ〉]W,γδ +2(〈p

∗B11γδ〉,1+

−〈p∗,1B
11γδ〉)W,γδ1+ 〈p

∗B11γδ〉W,γδ11}



On the modelling of thin uniperiodic... 765

Conditions (4.4) must hold for every Λ-periodic test function d∗ and for
every Λ-periodic test function p∗, respectively.
Equations (4.2) and (4.4) represent the basis for obtaining the tolerance

models for analyzing quasi-stationary and dynamic problems of linear elastic
uniperiodic cylindrical shells. In this work, the model of dynamic problems
will be derived.
In order to obtain solutions to the periodic problems on Λ(Θ), given by

variational equations (4.4), we can apply the orthogonalizationmethod known
in the dynamics of elastic shells and plates.
The right-hand sides of Eqs (4.4) can be interpreted as certain time de-

pendent loadings on the cell Λ(Θ). In the absence of these loadings we obtain
from (4.4) a periodic problem on Λ(Θ) given by

(D2βγ2d̃γ,2−D
2β22b22p̃),2−µa

αβ¨̃dα =0
(4.5)

(B2222p̃,22),22− b22(D
22γ2d̃γ,2− b22D

2222p̃)+µ¨̃p=0

which on the assumption that d̃α(Θ1,Ψ2, t) = hα(Θ1,Ψ2)cos(ωt),
p̃(Θ1,Ψ2, t) = g(Θ1,Ψ2)cos(ωt), (Θ1,Ψ2)∈Λ(Θ), Θ=(Θ1,Θ2)∈ΩΛ, leads
to the periodic eigenvalue problem of finding Λ-periodic functions hα, g given
by the equations

[D2βγ2(Θ1,Ψ2)hγ,2(Θ
1,Ψ2)],2+µ(Θ

1,Ψ2)[ω(Θ1)]2aαβhα(Θ
1,Ψ2)= 0

(4.6)

[B2222(Θ1,Ψ2)g22(Θ
1,Ψ2)],22−µ(Θ

1,Ψ2)[ω(Θ1)]2g(Θ1,Ψ2)= 0

and by the periodic boundary conditions on the cell Λ(Θ) together with the
continuity conditions inside Λ(Θ). By averaging the above equations over
Λ(Θ), we obtain 〈µhα〉(Θ1)= 〈µg〉(Θ1)= 0.
Let h1α(Θ

1,Ψ2),g1(Θ1,Ψ2),h2α(Θ
1,Ψ2),g2(Θ1,Ψ2), ..., be a sequence of

eigenfunctions related to the sequence of eigenvalues [ω2α,ω
2]1, [ω2α,ω

2]2, ... .
For arbitrary Θ1 and (Θ1,Ψ2) ∈Λ(Θ), Θ = (Θ1,Θ2) ∈ΩΛ we can look for
solutions to the periodic problem (4.4) in the form of the finite series

d̃α(Θ
1,Ψ2, t)=hA(Θ1,Ψ2)QAα(Θ

1,Θ2, t)
(4.7)

p̃(Θ1,Ψ2, t)= gA(Θ1,Ψ2)VA(Θ1,Θ2, t) A=1,2, ...,N

in which the choice of the number N of terms determines different degrees of
approximations.



766 B.Tomczyk

The functions hA(Θ1, ·), gA(Θ1, ·), A = 1, ...,N, are called the mode-
shape functions and are assumed to be known in every problem un-
der consideration. They are linear independent, l-periodic and such that
hA, lhA,2, l

−1gA, gA,2, lg
A
,22 ∈ O(l) and max |h

A(Θ1,Ψ2)| ¬ l,
max |gA(Θ1,Ψ2)| ¬ l2 as well as 〈µhA〉(Θ1) = 〈µgA〉(Θ1) = 0 for every A
and 〈µhAhB〉(Θ1)= 〈µgAgB〉(Θ1)= 0 for every A 6=B.
Inmost problems, the analysiswill be restricted to the simplest case N =1

inwhich we take into account only the lowest natural vibrationmodes (in the
direction tangent and normal to M) related to Eqs (4.6).
The functions QAα(Θ

1,Θ2, t), VA(Θ1,Θ2, t) in (4.7) represent new unk-
nowns, called the fluctuation variables. Because the functions d̃α(Θ1,Ψ2, t),
p̃(Θ1,Ψ2, t) are the Λ-periodic approximations of dα(Θ1,Ψ2, t), p(Θ1,Ψ2, t),
respectively, on the cell Λ and dα(Θ1,Ψ2, t),p(Θ1,Ψ2, t) ∈ PL

µ
Λ
(T), then

from (4.7) and fromLemma (L3) it follows that the functions QAα(Θ
1,Θ2, t),

VA(Θ1,Θ2, t), A = 1,2, ...,N, are Λ-slowly varying functions in Θ2, i.e.
QAα,V

A ∈SVΛ(T).
Substituting the right-hand sides of (4.7) into (4.2) and (4.4), setting

d∗ = hA(Θ1,Ψ2), p∗ = gA(Θ1,Ψ2), A = 1,2, ...,N, in (4.4) and taking into
account (4.3), on the basis of the Tolerance Averaging Assumption we arrive
at the tolerance fluctuation variable model of dynamic problems for unperiodic
cylindrical shells. Under extra denotations

D̃αβγδ ≡〈Dαβγδ〉 DAαβγ ≡〈DαβγδhA,δ〉

D
Aαβγ

≡ l−1〈Dαβγ1hA〉 LAαβ ≡ l−2bγδ〈D
αβγδgA〉

B̃αβγδ ≡〈Bαβγδ〉 KAαβ ≡〈BαβγδgA,γδ〉

K
Aαβ
≡ l−1〈Bαβ1δgA,δ〉 K̆

Aαβ ≡ l−2〈Bαβ11gA〉

CABβγ ≡〈DαβγδhA,αh
B
,δ〉 C

ABβγ
≡ l−1〈Dαβγ1hA,αh

B〉

FABβ ≡ l−2bγδ〈D
αβγδhA,αg

B〉 C̃ABβγ ≡ l−2〈D1βγ1hAhB〉

F
ABβ
≡ l−3bγδ〈D

1βγδhAgB〉 RAB ≡〈BαβγδgA,αβg
B
,γδ〉

L
AB
≡ l−4bαβbγδ〈D

αβγδgAgB〉 R̆AB ≡ l−1〈B1βγδgA,βg
B
,γδ〉

R̃AB ≡ l−2〈B11γδgA,γδg
B〉 R

AB
≡ l−3〈B1β11gA,βg

B〉

R̂AB ≡ l−4〈B1111gAgB〉 S̃AB ≡ l−2〈B1γ1δgA,γg
B
,δ〉

µ̃≡〈µ〉 µ̃AB ≡ l−2〈µhAhB〉

µAB ≡ l−4〈µgAgB〉 P̃Aβ ≡ l−1〈f̃βhA〉

P̃A ≡ l−2〈f̃gA〉

(4.8)

this model is represented by:



On the modelling of thin uniperiodic... 767

— the constitutive equations

Nαβ = D̃αβγδ(Uγ,δ − bγδW)+D
BαβγQBγ + lD

Bαβγ
QBγ,1− l

2LBαβVB

Mαβ = B̃αβγδW,γδ +K
BαβVB +2lK

Bαβ
VB,1 + l

2K̆BαβVB,11

HAβ =DAβγδ(Uγ,δ − bγδW)+C
ABβγQBγ + lC

ABβγ
QBγ,1− l

2FABβVB

H
Aβ
≡ lD

Aβγδ
(Uγ,δ − bγδW)+ lC

ABβγ
QBγ + l

2C̃ABβγQBγ,1− l
3F
ABβ
VB
(4.9)

GA ≡−l2LAγδ(Uγ,δ − bγδW)+K
AαβW,αβ − l

2FABγQBγ − l
3F
ABγ
QBγ,1+

+(RAB + l4L
AB
)VB +2lR̆ABVB,1 + l

2R̃ABVB,11

G̃A = l2KAαβW,αβ + l
2R̃ABVB +2l3R

AB
VB,1 + l

4R̂ABVB,11

G
A
= lK

Aαβ
W,αβ + lR̆

ABVB +2l2S̃ABVB,1 + l
3R
AB
VB,11

— the system of three averaged partial differential equations of motion for the
averaged displacements Uα(Θ, t),W(Θ, t)

Nαβ,α − µ̃a
αβÜα+f

β
0 =0

(4.10)

M
αβ
,αβ
− bαβN

αβ + µ̃Ẅ −f0 =0

— the system of 3N partial differential equations for the fluctuation variables
QBα(Θ, t), V

B(Θ, t),B=1,2, ...,N, called the dynamic evolution equations

l2µ̃ABaγβQ̈Bγ +H
Aβ −H

Aβ
,1 − lP̃

Aβ =0
(4.11)

l4µABV̈B +GA+ G̃A,11−2G
A
,1− l

2P̃A =0 A,B=1,2, ...,N

The above model has a physical sense provided that the basic unknowns
Uα(Θ, t),W(Θ, t),QAγ (Θ, t), V

A(Θ, t)∈SVΛ(T),A=1,2, ...,N, i.e. they are
Λ-slowly varying functions of the Θ2-midsurface parameter.
Taking into account (4.1) and (4.7), the shell displacement fields can be

approximated bymeans of formulae

uα(·, t)≈Uα(·, t)+h
A(·)QAα(·, t)

(4.12)

w(·, t)≈W(·, t)+gA(·)VA(·, t) A=1,2, ...,N

where the approximation ≈ depends on the number of terms hA(·)QAα(·, t),
gA(·)VA(·, t).



768 B.Tomczyk

The characteristic features of the derived model are:

• The model takes into account the effect of the cell size on the overall
dynamic shell behavior; this effect is described by the underlined coeffi-
cients dependent on the mezostructure length parameter l.

• Themodel equations involve averaged coefficientswhich are independent
of the Θ2-midsurface parameter (i.e. they are constant in direction of
periodicity), and are slowly varying functions of Θ1.

• The number and form of the boundary conditions for the averaged di-
splacements Uα(Θ, t), W(Θ, t) are the same as in the classical shell
theory governed by equations (2.2)-(2.4). The boundary conditions for
the fluctuation variables QAγ (Θ, t), V

A(Θ, t) should be defined only on
the boundaries Θ1 = const.

• It is easy to see that in order to derive governing equations (4.9)-(4.11),
we have to obtain the mode-shape functions hA(Θ1,Ψ2), gA(Θ1,Ψ2),
A = 1,2, ...,N, as solutions to the periodic eigenvalue problem given
by (4.6). In practice, derivation of these exact solutions is possible only
for cells with a structure which is not too complicated. In most cases,
these eigenfunctions have to be obtained byusing approximatemethods.
Moreover, for uniperiodic shells, the mode-shape functions are periodic
in only one direction; in this work they are l-periodic functions only of
the Θ2-midsurface parameter.

Assuming that the cylindrical shell under consideration has material and
geometrical properties independent of Θ1, we obtain governing equations
(4.9)-(4.11) with constant averaged coefficients. Moreover, in this case the
mode-shape functions hA, gA, A = 1,2, ...,N, are also independent of the
Θ1-midsurface parameter.
For a homogeneous shell, µ(Θ), Dαβγδ(Θ) and Bαβγδ(Θ), Θ ∈ Ω are

constant, and because 〈µhA〉 = 〈µgA〉 = 0 we obtain 〈hA〉 = 〈gA〉 = 0, and
hence 〈hA,α〉 = 〈g

A
,α〉 = 〈g

A
,αβ〉 = 0. In this case, equations (4.10) reduce to

the well known linear-elastic shell equations of motion for the averaged di-
splacements Uα(Θ, t),W(Θ, t), and independently for QAα(Θ, t),V

A(Θ, t) we
arrive at a system of N differential equations. In this case, under the condi-
tion f̃β = f̃ =0 and for the initial conditions QAα(Θ, t0) = V

A(Θ, t0) = 0,
A = 1,2, ...,N, we obtain QAα = V

A = 0; hence constitutive equations (4.9)
and equations of motion (4.10) reduce to starting equations (2.3) and (2.4),
respectively.
At the end of this section let us compare the obtained above tolerance

fluctuation variable model of uniperiodic cylindrical shells with the tolerance



On the modelling of thin uniperiodic... 769

internal variable model of shells having a locally periodic structure in both
directions tangent to M, whichwereproposedbyWoźniak (1999), andused to
analyze dynamic problems of cylindrical shellswith two-directional periodicity
byTomczyk (1999). In the sequel, cylindrical shells having aperiodic structure
in both directions tangent to M will be termed biperiodic, cf. Woźniak and
Wierzbicki (2000). An example of such a shell is presented in Fig.2.

Fig. 2. Example of a biperiodic shell

Following Tomczyk (1999), the governing equations of the tolerance inter-
nal variable model of cylindrical biperiodic shells is represented by:
— the constitutive equations (A,B =1,2, ...,N)

Nαβ = 〈Dαβγδ〉(Uγ,δ − bγδW)+ 〈D
αβγδhB,δ〉Q

B
γ −〈D

αβγδgB〉bγδV
B

Mαβ =−〈Bαβγδ〉W,γδ −〈B
αβγδgB,γδ〉V

B

(4.13)

HAβ = 〈DαβγδhA,α〉(Uγ,δ − bγδW)+ 〈D
αβγδhA,αh

B
,δ〉Q

B
γ − bγδ〈D

αβγδhA,αg
B〉VB

GA =−bαβ〈D
αβγδgA〉(Uγ,δ − bγδW)+ 〈B

αβγδgA,γδ〉W,αβ +

−bαβ〈D
αβγδgAhB,δ〉Q

B
γ +(〈B

αβγδgA,αβg
B
,γδ〉+ bαβ〈D

αβγδgAgB〉bγδ)V
B

— the system of three averaged partial differential equations of motion for the
averaged displacements Uα(Θ, t),W(Θ, t)

Nαβ,α −〈µ〉a
αβÜα+f

β
0 =0

(4.14)

M
αβ
,αβ
+ bαβN

αβ −〈µ〉Ẅ +f0 =0

— the systemof 3N ordinary differential equations for the fluctuation variables
QAα(Θ, t), V

A(Θ, t) called the dynamic evolution equations



770 B.Tomczyk

〈µhAhB〉aγβQ̈Bγ +H
Aβ + 〈f̃βhA〉=0

(4.15)

〈µgAgB〉V̈B +GA+ 〈f̃gA〉=0 A,B=1,2, ...,N

where the basic unknowns Uα(Θ, t), W(Θ, t), QAα(Θ, t), V
A(Θ, t),

A = 1,2, ...,N, are slowly varying functions (with respect to the two-
dimensional periodicity cell and tolerance system) in Θ1 and Θ2 alike, and
also the mode-shape functions hA, gA are l-periodic functions in both Θ1

and Θ2. Equations (4.13)-(4.15) have constant coefficients; the underlined
terms depend on the mezostructure length parameter l, and hence describe
the effect of the cell size on the overall shell behavior.
Comparing Eqs (4.13)-(4.15) and (4.9)-(4.11) it can be seen that Eqs

(4.13)-(4.15) for biperiodic shells can be obtained from Eqs (4.9)-(4.11) by
neglecting in (4.9) the singly underlined terms; it means that the tolerance
model of biperiodic cylindrical shells is a special case of that for uniperiodic
shells proposed in this paper. Themain differences between bothmodels are:

• in the model of a uniperiodic shell we deal with functions which are
slowly varying or periodic-like (with respect to the cell and tolerance
system) in only one direction, while in the other one these functions are
slowly-varying or periodic-like in two directions

• within the framework of the model of uniperiodic shells, the unknowns
QAα(Θ, t), V

A(Θ, t), A = 1,2, ...,N, are governed by the system of 3N
partial differential equations (4.11), while within the framework of the
model of biperiodic shells these unknowns are governed by the system of
3N ordinary differential equations involving only time derivatives; hence
there are no extra boundary conditions for these functions, and that
is why they play the role of kinematic internal variables, cf. Woźniak
(1999).

In the next section the homogenizedmodel of uniperiodic cylindrical shells
will be derived as a special case of Eqs (4.9)-(4.11).

5. Homogenized model

The simplified model of uniperiodic cylindrical shells can be derived di-
rectly from the tolerance model, (4.9)-(4.11), by the limit passage l→ 0, i.e.



On the modelling of thin uniperiodic... 771

by neglecting the underlined termswhich depend on themezostructure length
parameter l. Hence, Eqs (4.11) yield

CABβγQBγ =−D
Aβγδ(Uγ,δ − bγδW)

(5.1)

RABVA =−KBγδW,γδ

Fromthepositive definiteness of the strain energy it follows that the N×N
matrix of the elements RAB is non-singular, and the linear transformation de-
terminedby thecomponents CABβγ is invertible.Hencea solution toequations
(5.1) can be written in the form

QBγ =−G
BC
γη D

Cηµϑ(Uµ,ϑ− bµϑW)
(5.2)

VA =−EABKBγδW,γδ

where GABαβ and E
AB are defined by

GABαβ C
BCβγ = δγαδ

AC EABRBC = δAC

Setting

D
αβγδ
eff
≡ D̃αβγδ −DAαβηGABηξ D

Bξγδ

B
αβγδ
eff
≡ B̃αβγδ −KAαβEABKBγδ

and substituting expression (5.2) into constitutive equations (4.9)1,2, in which
the underlined terms are neglected, we arrive at the homogenized shell model
governed by:
— equations of motion

D
αβγδ
eff
(Uγ,δα− bγδW,α)−〈µ〉a

αβÜα+f
β
0 =0

(5.3)

B
αβγδ
eff
W,αβγδ − bαβD

αβγδ
eff
(Uγ,δ − bγδW)+ 〈µ〉Ẅ −f0 =0

— constitutive equations

Nαβ =Dαβγδ
eff
(Uγ,δ − bγδW) M

αβ =−Bαβγδ
eff
W,γδ (5.4)

where Dαβγδ
eff
,Bαβγδ
eff
are called the effective stiffnesses.

The obtained above homogenizedmodel governed byEqs (5.3), (5.4) is not
able to describe the length-scale effect on the overall dynamic shell behavior
being independent of the mezostructure length parameter l.
Inorder to showdifferencesbetween the resultsobtained fromthe tolerance

uniperiodic shell model, (4.9)-(4.11), and from the homogenized model, (5.3)
and (5.4), free vibrations of the uniperiodic cylindrical shell will be analyzed
in the second part of this paper.



772 B.Tomczyk

6. Final remarks

The subject-matter of this contribution is a thin linear-elastic cylindrical
shell having a periodic structure in one direction tangent to the undeformed
shell midsurface M. Shells of this kind are termed uniperiodic. For these
shells, equations governed by the Kirchhoff-Love shell theory involve highly
oscillating periodic coefficients.
In order to simplify theKirchhoff-Love shell theory to the formwhich can

be applied to engineering problems and also can take into account the effect
of the periodicity cell on the overall dynamic shell behavior a new model of
thin uniperiodic cylindrical shells has been proposed. In order to derive it,
the tolerance averaging procedure given by Woźniak and Wierzbicki (2000),
has been applied. This model, called the tolerance model, is represented by
a system of partial differential equations (4.10) and (4.11) with coefficients
which are constant in the direction of periodicity. The basic unknowns are:
the averaged displacements Uα, W and the fluctuation variables QAα, V

A,
A=1,2, ...,N, which have to be slowly varying functions with respect to the
cell and certain tolerance system. This requirement imposes certain restric-
tions on the class of problems described by the model under consideration.
In order to obtain the governing equations, themode-shape functions hA, gA,
A = 1,2, ...,N, should be derived as approximated solutions to eigenvalue
problems on the periodicity cell with periodic boundary conditions.
In contrastwith the homogenizedmodels, the proposed onemakes it possi-

ble todescribe the effect of theperiodicity cell on theoverall shell behavior (the
length-scale effect). The length scale is introduced to the global description of
both inertial and constitutive properties of the shell under consideration.
Comparing the proposed here tolerance fluctuation variable model for uni-

periodic cylindrical shells given by Eqs (4.9)-(4.11), and the known tolerance
internal variable model for biperiodic cylindrical shells (i.e. shells with a perio-
dic structure in both directions tangent to M) governed byEqs (4.13)-(4.15),
it is seen that the equations for uniperiodic shells contain the singly underli-
ned terms which have no counterparts in the equations for biperiodic shells.
Moreover, for uniperiodic shells, the unknows QAα, V

A, A = 1,2, ...,N, are
governed by a systemof 3N partial differential equations (4.11), and hence do
not play the role of kinematic internal variables, unlike the unknows QAα, V

A,
A=1,2, ...,N in Eqs (4.15). It means that the tolerance model of biperiodic
shells is a special case of that describing the uniperiodic shells proposed in this
paper, and hence the biperiodic shell model is not sufficient to analyze dyna-
mic problems of shellswith aperiodic structure in one direction tangent toM.



On the modelling of thin uniperiodic... 773

The problems related to various applications of proposed Eqs (4.9)-(4.11)
to the dynamics of uniperiodic cylindrical shells are reserved for a separate
paper.

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Modelowanie cienkich powłok walcowych o jednokierunkowej periodyce

Streszczenie

Celempracy jestwyprowadzenie uśrednionegomodelu służącegodo analizy dyna-
miki cienkich liniowo-sprężystychpowłokwalcowychmającychperiodyczną strukturę
w jednym kierunku stycznym do powierzchni środkowej powłoki. Proponowany mo-
del, w przeciwieństwie do znanych modeli zhomogenizowanych, umożliwia badanie
wpływuwielkości komórki periodyczności na dynamikę powłoki walcowej (wpływ ten
zwany jest efektem skali).W celu wyprowadzenia równań o stałych lub wolnozmien-
nych współczynnikach zastosowano znaną metodę tolerancyjnego uśredniania. Wy-
prowadzonymodel porównano zmodelem dla powłoki walcowej z periodykąwdwóch
kierunkach wzajemnie prostopadłych i stycznych do powierzchni środkowej powłoki
oraz z modelem bez efektu skali.

Manuscript received July 1, 2003; accepted for print July 21, 2003