Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 2, pp. 289-304, Warsaw 2003

ON ELASTODYNAMICS OF BIPERIODIC COMPOSITE

MEDIA1

Małgorzata Woźniak

Department of Civil and Environmental Engineering, Kielce University of Technology

Czesław Woźniak

Institute of Mathematics and Informatics, Częstochowa University of Technology

e-mail: wozniak@matinf.pcz.czest.pl

The aim of this paper is twofold. First, we formulate a mathematical
model for the analysis ofwavespropagating in a linear-elastic composite,
which in every plane normal to a certain straight line has an identical
periodic structure. Second, we apply the derived model equations to
the investigations of waves propagating across a laminatedmediumwith
periodically folded laminae. Lowerandhigherpropagationspeeds for the
longitudinal and transversalwaves are calculated and for the longwaves
represented in the form of simple asymptotic formulas.

Key words: modelling, composites, waves, dispersion

1. Introduction

By a biperiodic composite we understand a heterogeneous solid which has
a periodic structure in a certain plane, and the properties of which are con-
stant in the direction normal to this plane. A formulation of an approximate
theory describing the dynamic response of a biperiodic composite solid was
detailed in a book byWoźniak andWierzbicki (2000), and applied to the ana-
lysis of some initial-boundary-value problems in papers by Wierzbicki et al.
(2001), andWoźniak et al. (2002). This formulation was based onwhat is cal-
led the tolerance averaging of differential equations with periodic coefficients.

1
This contribution is an extended version of two lectures delivered on the Second Sym-

posium on Composites and Layered Structures, PTMTS, Wrocław-Karpacz, 7-9 November,

2002.



290 M.Woźniak, C.Woźniak

Various applications of the tolerance averaging technique to the investigation
of selected dynamic problems for composite solids and structures can be found
in a series of papers by Baron and Woźniak (1995), Dell’ Isola et al. (1998),
Ignaczak (1998), Jędrysiak (1999, 2000),Michalak (1998, 2000),Mazur-Śniady
(2001),Woźniak (1999),Woźniak andWierzbicki (2002),Woźniak (1996) and
others.
The approximate theory of biperiodic composites, based on the tolerance

averaging of the equations of elastodynamics, which has been formulated in
Woźniak andWierzbicki (2000), cannotbeapplied to theanalysis ofwaves pro-
pagating in an arbitrary direction. The aim of this contribution is to provide
the reader with a certainmodified version of this theorywhich is free from the
above drawback. The proposed version makes it possible to investigate some
dispersion phenomena related to the propagation of waves in an arbitrary di-
rection. The obtained equations are applied to the analysis of harmonic waves
in a laminated medium with periodically folded laminae. The considerations
are restricted to biperiodic composites made of perfectly bonded linear-elastic
constituents.
Tomake thispaper self-consistent, in the subsequent section the fundamen-

tal ideas and assumptions of the tolerance averaging technique are outlined;
for details the reader is referred toWoźniak andWierzbicki (2000).

Denotations. Considerations are carried out in the orthogonal Carte-
sian coordinate system 0x1x2x3. Partial derivatives with respect to x1, x2,
x3 are denoted by ∂1, ∂2, ∂3, respectively, and the time derivative is de-
noted by the overdot. The gradient operators are introduced in the form
∇ = (∂1,∂2,∂3), ∇ = (∂1,∂2,0) and ∂ = (0,0,∂3). We also denote
x = (x1,x2,x3), x = (x1,x2); hence x = (x,x3). Superscripts A,B run
over 1, ...,N, summation convention holds. We also use the index notation;
subscripts k,l, ... run over 1,2,3, subscripts α,β run over 1,2 and partial
derivatives are indicated by a comma.

2. Modelling technique

In this section we shall assume that the biperiodic composite solid under
consideration occupies in the reference configuration a region Ω=Π×(0,L),
where Π is a region on the 0x1x2-plane. Define ∆ = (−l1/2, l1/2) ×
(−l2/2, l2/2)× {0} as a cell on the 0x1x2-plane, where l1 and l2 are the
periods of inhomogeneity in directions of the x1- and x2-axes, respectively.



On elastodynamics of biperiodic composite media 291

Wealso assume that the smallest characteristic length dimension of the region
Π is sufficiently large when compared to the periods l1, l2. For every position
vector x=(x1,x2,x3) we define ∆(x) := x+∆ and

Ω0 := {x∈Ω : ∆(x)⊂Ω}

For every x∈Ω0 and for an arbitrary integrable function f defined in Ω we
introduce the averaging operator given by

〈f〉(x)= 1
l1l2

∫

∆(x)

f(y,x3) dy x∈Ω0 (2.1)

Subsequently, the function f can also depend on the time coordinate t.
The philosophy of the proposed modelling approach is based on the sup-

position that to every physical field Φ can be assigned a positive number εΦ
called the tolerance parameter such that every two values Φ1,Φ2 of this field
satisfying the condition |Φ1 −Φ2| < εΦ can be treated as indiscernible. Fol-
lowing Fichera (1992) we say that the values of Φ which do not exceed εΦ
cannot be detected by instruments. Hence, εΦ represents a certain degree of
accuracy in performing the measurement or calculations. The above philoso-
phy has been applied in Fichera (1992), where εΦ was referred to as an upper
bound for negligibles.

Setting l =
√
l21+ l

2
2, denoting by T a set of all tolerance parameters

regarded in themodelling procedure and by ‖x−y‖ the distance between the
points x, y we shall introduce two important definitions.

Definition 1. The function F defined on Π will be called slowly-varying,
F ∈ SVl(T), if for every x,y ∈ Π the condition ‖x−y‖ ¬ l implies
|F(x)−F(y)| ¬ εF .

Definition 2. The function ϕ defined on Π will be called periodic-like,
ϕ ∈ PLl(T), if for every x ∈ Π there exists a ∆-periodic function
ϕx such that for every y ∈ Π the condition ‖x − y‖ ¬ l implies
|ϕ(y)−ϕx(y)| ¬ εϕ.

Function ϕx will be referred to as the ∆-periodic approximation of ϕ in
the vicinity of the point x. It can be shown that if ϕ∈PLl(T) then 〈ϕ〉 is a
slowly-varying function.

Remark. Subsequently, the functions F,ϕwill also depend on x3 ∈ (0,L)
(and on time t); that is why instead of ϕx we shall write ϕx.



292 M.Woźniak, C.Woźniak

The tolerance averaging technique of differential equations with periodic
coefficients is based on twomodelling assumptions. The first is strictly related
to the concepts of slowly-varying and periodic-like functions.

Tolerance Averaging Approximation (TAA). For every ∆-periodic
integrable function f and every F ∈ SVl(T), ϕ ∈ PLl(T), the following
approximations are assumed to hold

〈fF〉(x)≈〈f〉F(x) 〈fϕ〉(x)≈〈fϕx〉(x) x∈Ω0 (2.2)

From (2.2)1 it follows that in the course of averaging the increments
F(y)−F(x), y∈∆(x), of the slowly varying function F(·) can be neglected.
Before formulating the secondmodelling assumption, let us recall the well

known equation of the linear elastodynamics

∇· (C :∇u)−ρü+ρf =0 (2.3)

where u is a displacement field, f is a body force, and where the elasticity
tensor field C as well as the mass density scalar field ρ are always assumed
to be the known ∆-periodic functions independent of the x3-coordinate. The
above equationhas tobesatisfied for every time t in the region Ω=Π×(0,L),
and holds together with the known continuity conditions on the interfaces
between the constituents of the composite, and with the prescribed boundary
and initial conditions.
The second modelling assumption is based on heuristic premises and re-

stricts the class of elastodynamic problems under consideration to those in
which a typical wavelength of what is called a macroscopic deformation pat-
tern is sufficiently large when compared to the diameter l of cell ∆.

Conformability Assumption (CA). The displacement field
u = u(x,x3, t), x ∈ Π, x3 ∈ (0,L), away from the boundary ∂Π
of Π, conforms to the ∆-periodic heterogeneous structure of the composite,
i.e., u=u(·,x3, t) is for every x3 ∈ (0,L) and for every time t a periodic-like
function.
The tolerance averaging procedure related to equation (2.3)will be realized

in five steps.

1. Setting w = 〈ρ〉−1〈ρu〉 and defining r = u−w, we introduce the
decomposition of the displacement field

u(x,x3, t)=w(x,x3, t)+r(x,x3, t) (x,x3)∈Ω0 (2.4)

where bymeans of (CA) we conclude that w(·,x3, t) is a slowly-varying
function and 〈ρr〉(x, t) = 0. Hence w and r represent the averaged



On elastodynamics of biperiodic composite media 293

and oscillating (residual) parts of u, respectively. At the same time the
values of r have to be quantities of an order l, r(x, t)∈O(l).

2. Substituting (2.4) into (2.3), averaging the resulting equation over ∆(x)
and using (2.2), we obtain a variational ∆-periodic cell problem for the
∆-periodic approximation rx of r in ∆(x). This problem is governed
by

〈ρr · r̈x〉(x, t)+ 〈∇r :C :∇rx〉(x, t)−∂ · 〈r ·C :∇rx〉(x, t)=
=−〈∇r :C〉 :∇w(x, t)+∂ · [〈r ·C〉 :∇w(x, t)]+ 〈ρr ·f〉(x, t)

(2.5)

〈ρrx〉(x, t)= 0 x∈Ω0
where equation (2.5)1 has to hold for every integrable ∆-periodic test
function r of y = (y1,y2) satisfying the conditions 〈ρr〉 = 0 and
r(x1,x2)∈O(l).

3. We look for an approximate solution to (2.5) in the form

rx(y,x3, t)=h
A(y)vA(x,x3, t) (y,x3)∈∆(x) (2.6)

where hA(·) are certain postulated a priori linear-independent conti-
nuous ∆-periodic functions satisfying the conditions 〈ρhA〉 = 0 and
hA(y) ∈ O(l). Moreover, vA(·,x3, t) are assumed to be slowly-varying
functions. The functions vA represent new unknowns which will be ter-
med fluctuation variables. The functions hA(·) can be assumed as the
interpolation functions related to the periodic FEMdiscretization of the
cell ∆, Żmijewski (1987), Augustowska andWierzbicki (2002).

4. Substituting (2.6) into (2.5)1 and assuming r = hA(y)cA, y ∈ ∆(x)
where cA are arbitrary constant vectors, we obtain a systemof N vector
equations

〈hAhBρ〉v̈B + 〈∇hA ·C ·∇hB〉 ·vB + 〈hBC ·∇hA−hAC ·∇hB〉 : ∂vB −
(2.7)

−∂ · (〈hAhBC〉 :∇vB)=−〈∇hA ·C〉 :∇w+∂ · (〈hA ·C〉 :∇w)+ 〈fhAρ〉

which together with (2.6) represent a certain approximation to periodic
cell problem (2.5).

5. Substituting (2.4) into (2.3) and averaging the resulting equation over
∆(x), after using (2.2) and (2.6), we obtain the vector equation

∇· (〈C〉 :∇w+ 〈C ·∇hB〉 ·vB+ 〈hBC〉 : ∂vB)−〈ρ〉ẅ+ 〈fρ〉=0 (2.8)



294 M.Woźniak, C.Woźniak

Equations (2.7) and (2.8) for the unknowns w, vA, A = 1, ...,N, have
constant coefficients and hence they represent the averagedmodel of a biperio-
dic composite for the analysis of dynamic problems restricted by the heuristic
hypothesis (CA).
It has to be emphasized that the solutions w, vA to equations (2.7), (2.8)

have a physical sense only if the functions w(·,x3, t),vA(·,x3, t) together with
their derivatives are slowly-varying (possibly except for the vicinity of the
boundary ∂Π). The above requirement can be used as a certain a posteriori
condition for the evaluation of tolerance parameters,Woźniak andWierzbicki
(2000).
We have stated above that the functions hA can be derived by the perio-

dic discretization of ∆, and hence they are periodic interpolation functions
satisfying extra conditions 〈ρhA〉=0. In most cases, the number N of these
functions has to be large and hence in model equations (2.7), (2.8) we deal
with a large number N of the unknownfluctuation variables vA. To eliminate
this drawback we shall introduce into the modelling technique the problem
of free periodic vibrations of cell ∆. This is an eigenvalue problem of finding
a continuous function h(y), y ∈ ∆, which is ∆-periodic and satisfies the
condition 〈ρh〉=0 as well as the variational condition

〈∇h :C :∇h〉−λ〈ρh ·h〉=0 (2.9)

which holds for every ∆-periodic test function h=h(y) such that 〈hρ〉=0.
The eigenvalues λ of (2.9) represent here the squares of the free periodic
vibration frequencies of the cell ∆.We shall look for an approximate solution
to this eigenvalue problem in the form

h(y)=hA(y)aA y∈∆ (2.10)

where hA have the same meaning as before (being derived by the periodic
FEM discretization of ∆) and aA are arbitrary constant vectors. Combining
(2.10) and (2.9), we obtain a new eigenvalue problem

(〈∇hA ·C ·∇hB〉− Iλ〈ρhAhB〉) ·aB =0 (2.11)

where I stands for a unit tensor in R2. Let (a1a, ...,a
N
a ), a=1, ...,n, n<N,

be the first n eigenvectors related to problem (2.11). In this case, from (2.10)
we obtain

ha(y)=h
A(y)aAa a=1, ...,n (2.12)

and instead of (2.6) we shall look for an approximate solution to (2.5) in the
form

rx(y,x3, t)=hb(y)vb(x,x3, t) (y,x3)∈∆(x) (2.13)



On elastodynamics of biperiodic composite media 295

where here and hereafter the summation convention over b = 1, ...,n holds.
The functions va(·), a=1, ...,n, are new unknowns which are assumed to be
slowly-varying functions of x=(x1,x2). Using approximation (2.13), instead
of (2.7), (2.8), we obtain the following system of equations for w and va

〈ha ·hbρ〉v̈b+ 〈∇ha :C :∇hb〉vb+ 〈hb ·C :∇ha−ha ·C :∇hb〉 ·∂vb−
−∂ · [〈(ha⊗hb) :C〉 ·∇vb] =−〈∇ha :C〉 :∇w+∂ · (〈ha ·C〉 :∇w)+
+〈f ·ha〉 (2.14)
∇· (〈C〉 :∇w+ 〈C :∇hb〉vb+ 〈hb ·C〉 : ∂vb)−〈ρ〉ẅ+ 〈f〉=0

The number n of the unknowns va can be small and that is why equations
(2.14) represent what will be called the reduced order averaged model of a
biperiodic composite.
In many special cases the form of the functions ha as well as hA can be

also based on a heuristic assumption that ha and hAeα (where eα, α=1,2,
constitute a vector base in E2) approximate the expected form of the free
periodic vibrations of the cell ∆ and satisfy the conditions 〈ρha〉 = 0 and
〈ρhA〉=0, respectively.
In the subsequentpart of this contribution, the considerationswill bebased

on equations (2.7), (2.8), but the application of the reduced ordermodel leads
to similar conclusions.

3. Averaged wave-type model

It can be observed that the averaged model of biperiodic composites, re-
presented by equations (2.7), (2.8), in the general case cannot be applied
to the analysis of waves propagating along the 0x3-axis. In order to ob-
tain the wave-type averaged equations we introduce an extra assumption
that the fields w, vA together with their derivatives are slowly-varying func-
tions not only with respect to x = (x1,x2) but also with respect to the
x3-coordinate. It means that for every x3, y3 the condition |x3−y3| ¬ l im-
plies ‖w(x,x3, t)−w(x,y3, t)‖ ¬ εu and ‖vA(x,x3, t)−vA(x,y3, t)‖ ¬ εu,
where εu is a tolerance parameter assigned to the evaluation of displacements;
similar conditions hold also for the derivatives of w and vA. For the sake of
simplicity the set of these slowly-varying functions will be also denoted by
SVl(T). Hence, averaging the aforementioned slowly-varying functions over
the three dimensional cell V (x) ≡ ∆(x,x3)× (x3 − l/2,x3 + l/2), we shall
neglect in V (x) increments of w, vA also in the 0x3-axis direction. In this



296 M.Woźniak, C.Woźniak

case, all underlined terms in (2.7), (2.8) can be neglected, and in the absense
of body forces we obtain

∇· (〈C〉 :∇w+ 〈C ·∇hB〉 ·vB)−〈ρ〉ẅ=0
(3.1)

〈hAhBρ〉v̈B + 〈∇hA ·C ·∇hB〉 ·vB + 〈∇hA ·C〉 :∇w=0

Equations (3.1) represent the wave-type averaged model of a biperiodic com-
posite. It has to be remembered that solutions to (3.1) have a physical sense
only if the following conditions

w(·, t)∈SVl(T) vA(·, t)∈SVl(T) (3.2)

as well as similar conditions for the derivatives of w, vA occurring in (3.1)
hold.
The final conclusion is that in a biperiodic medium waves can propagate

also in the 0x3-axis direction provided that the pertinent averaged parts of
the displacements w and the fluctuation variables vA are slowly-varying with
respect to all spatial coordinates.
Independentlyof this general statement,we can also dealwith some special

situations in which wave equations (3.1) can be obtained directly from equ-
ations (2.7), (2.8). They are situationswhere the functions hA can be assumed
in the form satisfying identically the extra conditions

〈hAC〉=0 〈hAC ·∇hB〉=0

Hence, after neglecting the body forces, equations (2.7), (2.8) yield

∇· (〈C〉 :w+ 〈C ·∇hB〉 ·vB)−〈ρ〉ẅ=0
(3.3)

〈hAhBρ〉v̈B + 〈∇hA ·C ·∇hB〉 ·vB −∂ · (〈hAhBC〉 :∇vB)+
+〈∇hA ·C〉 :∇w=0

In this case, instead of conditions (3.2) we shall deal with weaker conditions
of the form

w(·,x3, t)∈SVl(T) vA(·,x3, t)∈SVl(T) (3.4)

Itmeans that the solutions w,vA to (3.3) have to be slowly-varying functions
only with respect to the x1- and the x2-coordinates.
Subsequently, we shall take into account the averaged wave equations in

the form (3.3) bearing in mind that in the case described by equations (3.1)



On elastodynamics of biperiodic composite media 297

the terms depending on the coefficients 〈hAhBC〉 have to be neglected, and
the fields w,vA are slowly-varying functionswith respect to all spatial coordi-
nates.We shall also assume that the constituents of a composite are isotropic
with the moduli λ, µ and mass density ρ as ∆-periodic functions of x1, x2.
In this case, equations (3.3) in the index notation take the form

〈λ+µ〉wk,kα+ 〈µ〉wα,kk+ 〈λhA,γ〉vAγ ,α+ 〈µh
A
,γ〉vAα,γ +

+〈µhA,α〉vAk ,k−〈ρ〉ẅα =0

〈λ+µ〉wk,k3+ 〈µ〉w3,kk+ 〈λhA,γ〉vAγ ,3+ 〈µh
A
,γ〉vA3 ,γ −〈ρ〉ẅ3 =0

〈ρhAhB〉v̈Bα + 〈λhA,αhB,β〉vBβ + 〈µhA,γhB,γ〉vBα + 〈µhA,γhB,α〉vBγ − (3.5)
−〈µhAhB〉vBα ,3α =−〈λh

A
,α〉wk,k−〈µhA,γ〉(wα,γ +wγ,α)

〈ρhAhB〉v̈B3 + 〈µhA,γhB,γ〉vB3 −〈(λ+2µ)hAhB〉vB3 ,33−〈λh
AhB〉vBα,α3 =

=−〈µhA,α〉(w3,α+wα,3)

Let us investigate the plane wave propagating along the x3-axis. Setting
wk =wk(x3, t), v

A
k = v

A
k (x3, t) we obtain the longitudinal wave equations

〈λ+2µ〉w3,33+ 〈λhA,γ〉vAγ ,3−〈ρ〉ẅ3 =0
(3.6)

〈ρhAhB〉v̈Bα + 〈λhA,αhB,β〉vBβ + 〈µhA,γhB,γ〉vBα −〈µhAhB〉vBα ,33 =
=−〈λhA,α〉w3,3

for w3, vAα and the transversal wave equations

〈µ〉wα,33+ 〈µhA,α〉vA3 ,3−〈ρ〉ẅα =0
(3.7)

〈ρhAhB〉v̈B3 + 〈µhA,γhB,γ〉vB3 −〈(λ+2µ)hAhB〉vB3 ,33 =−〈µh
A
,α〉wα,3

for wα,vA3 . Letusobserve that to theaveraged displacements w3 and wα there
are assigned the displacement fluctuations hAvAα and h

AvA3 , respectively, in
the directions normal to the pertinent averaged displacements.
For a cylindrical wave propagating in the direction normal to the x3-axis,

by setting wk = wk(x1,x2, t), v
A
k = v

A
k (x1,x2, t), we obtain from (3.5) the

independent equations for w3, vA3 representing transversal waves in the form

〈µ〉w3,αα+ 〈µhA,γ〉vA3 ,γ −〈ρ〉ẅ3 =0
(3.8)

〈ρhAhB〉v̈B3 + 〈µhA,γhB,γ〉vB3 =−〈µhA,α〉w3,α



298 M.Woźniak, C.Woźniak

For an arbitrary biperiodic compositewith isotropic constituents, in the direc-
tions normal to the x3-axis, the longitudinal wave as well as the transversal
wave in the 0x1x2 plane cannot propagate.

4. Example of application

Theobjective of the analysiswill benowabiperiodic two componentunbo-
unded laminated medium in which the laminae are isotropic and periodically
slightly folded in the direction of the x2-axis. A fragment of a cross section
x3 = const of this laminate is shown in Fig.1.

Fig. 1. Cross-section of a biperiodic laminated medium

In order to describe the biperiodic material structure under consideration
we denote by x1 =α(x2) the periodic function with the period l2, the mean
value of which in (0, l) is equal to zero and its amplitude A satisfies the
condition A/l2 ≪ 1. We also assume that α(0) = 0. The interfaces between
components are assumed to be cylindrical and given by

x1 =α(x2)+nl1±
g(x2)
2

n=0,±1,±2, ...

where g(x2), l2 − g(x2) are the thicknesses of laminae measured along the
x1-axis. By means of the condition A/l2 ≪ 1 we shall assume that g ≈ g0,
where now g0, and l1−g0 are themean thicknesses of the laminae.We define



On elastodynamics of biperiodic composite media 299

ν′ = g0/l1, ν′′ =(l1−g0)/l1 and denote by ρ′, λ′, µ′ and ρ′′, λ′′, µ′′ the mass
densities and Lame’s moduli in the laminae with the mean thickness g0 and
l1−g0, respectively. Moreover, let ϕ=ϕ(x1) be a periodic saw-like function
the diagram of which is shown in Fig.2.

Fig. 2. Diagram of function ϕ=ϕ(x1)

We shall introduce an approximatedmathematical model of the biperiodic
laminate under consideration by assuming N = 1 and defining a function
h=h1(x1,x2), (x1,x2)∈R2, in the form

h(x1,x2)=ϕ(x1−α(x2))

We deal here with a certain generalization of the known approach to the mo-
delling of a layered mediumwhich was applied in papers by Ignaczak (1998),
Matysiak andNagórko (1995),Wierzbicki et al. (2001),Woźniak (1996),Woź-
niaket al. (2002) andmanyothers.Denotingby η anarbitrary fromthemoduli
λ, µ, we obtain the following values of coefficients in equations (3.5)-(3.8)

〈η〉= η′ν′+η′′nu′′ 〈ηh,1〉=2
√
3(η′′−η′)

〈ηh,2〉=0 〈η(h,1)2〉=12
(η′

ν′
+
η′′

ν′′

)

〈ηh,1h,2〉=0 〈η(h,2)2〉=12ξ
(η′

ν′
+
η′′

ν′′

)

〈ρ〉= ν′ρ′+ν′′ρ′′ 〈ρ(h)2〉=(l1)2〈ρ〉
〈η(h)2〉=(l1)2〈η〉

(4.1)

From equations (3.5) under the extra denotation

ξ=

l2∫

0

α′(x2) dx2



300 M.Woźniak, C.Woźniak

we obtain

〈λ+2µ〉w3,33+ 〈λh,1〉v1,3−〈ρ〉ẅ3 =0
(4.2)

(l1)
2〈ρ〉v̈1+ 〈(λ+µ+ ξµ)(h,1)2〉v1− (l1)2〈µ〉v1,33 =−〈λh,1〉w3,3

and without the loss of generality we can assume v2 =0. From (3.6) we get

〈µ〉w1,33+ 〈µh,1〉v3,3−〈ρ〉ẅ1 =0
(4.3)

(l1)
2〈ρ〉v̈3+ 〈µ(1+ ξ)(h,1)2〉v3− (l1)2〈(λ+2µ)〉v3,33 =−〈µh,1〉w1,3

and
〈µ〉w2,33−〈ρ〉ẅ2 =0 (4.4)

It means that in the laminated biperiodic medium under consideration three
kinds of waves can propagate in the x3-axis direction: the longitudinal wave
described by (4.2) and two transversal waves governed by (4.3) and (4.4), for
which free vibrations take place in the directions of the 0x1- and 0x2-axis,
respectively. The wave described by (4.4) is a nondispersive wave. In order to
investigate thesewaveswe shallwrite equations (4.2) and (4.3) in the following
form valid for both of them

aw,33+ bv,3−〈ρ〉ẅ=0
(4.5)

l2〈ρ〉v̈+ev− l2dv,33 =−bw,3

where w=w(x3, t), v= v(x3, t), l= l1 and a,b,e,d are thepertinent constant
coefficients occurring either in (4.2) or in (4.3), satisfying the conditions a> 0,
e> 0, d> 0 and ae− b2 > 0.
Let us investigate the propagation of harmonic waves by substituting to

(4.5) the right- hand sides of the formulae

w=Aw exp[ik(x3− ct)] v=Av exp[ik(x3− ct)] (4.6)

where k=2π/L is the wave number (here L is the wavelength) and Aw,Av
are the amplitudes. Substituting the right-hand sides of (4.6) into (4.5), we
obtain the following dispersion relation

l2〈ρ〉2k2c4−〈ρ〉(e+k2l2d+k2l2a)c2+ae−b2+k2l2da=0 (4.7)

which yields two propagation speeds c1 and c2.



On elastodynamics of biperiodic composite media 301

Let us observe that in the problemunder considerationwe dealwith the si-
tuation describedby equations (3.3) and conditions (3.4). Hence, the functions
in (4.2)-(4.6)maynotbe slowly varyingwith respect to the x3-coordinate, and
dispersion relation (4.7) has a physical sense for an arbitrary wave number k.
However, if thewavelength L is largewhen compared to theperiod l= l1 (i.e.,
if functions (3.4) are slowly varying) then the nondimensional wave number
q= kl=2πl/L is small when compared to 1. Transforming equation (4.7) to
the form

〈ρ〉q2c4−〈ρ〉[e+(a+d)q2]c2+ae− b2+adq2 =0

and restricting considerations to the long waves (when compared to the pe-
riod l), after denotation

ã= a− b
2

e
+
ad

e

we obtain the solutions c1, c2 to dispersion relation (4.7) in the asymptotic
form

(c1)
2 =
ã

〈ρ〉
+O(q2)

(c2)
2 =

e

〈ρ〉
1
q2
− ad− b

2−ed
〈ρ〉e

+O(q2)

Thus, we conclude that in the biperiodic laminated medium under consi-
deration the following kinds of waves can propagate along the x3-axis:

• the longitudinal and transversal wave (with vibrations in the 0x1x3-
plane) propagating with two different speeds c1, c2 determined by di-
spersion relation (4.7),

• the transversal nondispersive wave described by (4.4).

Aswehave stated above, if thebiperiodicmediumismodelledby equations
(3.3), then the waves propagating in the x3-axis direction can have arbitrary
lengths; this situation takes place in the above problem.

5. Conclusions

The results obtained in this contribution can be summarized by the follo-
wing conclusions.



302 M.Woźniak, C.Woźniak

• Dynamic problems of biperiodic linear-elastic composites, in which di-
splacement fields are represented by periodic-like functions with respect
to the x1- and x2-coordinates, can be investigated in the framework of
the averagedmodel governed by equations (2.7), (2.8). Using thismodel
we can satisfy, with a required accuracy, the initial conditions for every
constituent of the biperiodic composite as well as the boundary condi-
tions for this constituent on the boundaries x3 =0,L. On the remaining
part ∂Π×(0,L) of the solid boundary, the displacement conditions can
be imposed only on the averaged displacement field. Some remarks on
this subject can also be found inWoźniak andWierzbicki, (2000).

• In the general case the waves of an arbitrary length, represented by the
periodic-like functions of x1, x2, cannot propagate across the biperiodic
medium in the direction of the 0x3-axis. The wave propagation in this
direction is possible only in special situations which are described by
equations (3.3).

• In the biperiodic linear-elastic medium only the waves can propagate
which are represented by functions being periodic-like not only with
respect to the x1- and x2-coordinates but also with respect to the x3-
coordinate. In this case the propagation of waves is described by equ-
ations (3.1).

• The main difficulty in the formulation of the proposed models lies in
finding proper approximation (2.6) to periodic cell problem (2.5). An
approximate solution to this problem can be found on the basis of a
certain heuristic hypothesis as it was done in Section 4, where only one
∆-periodic function h(x1,x2) described the form of displacement fluc-
tuations. In general, the functions hA(x1,x2) can be derived from a
periodic discretizations of the cell ∆, cf. Augustowska and Wierzbicki
(2002), but the modelling approach can lead to a large number of the
unknowns vA in the model equations. In these situations we have to
apply the reduced order models represented by equations (2.14).

• By a formal limit passage l→ 0, differential equations (2.7) are reduced
to a system of linear algebraic equations for vA; this passage is due to
the fact that hA → 0 together with l → 0 but ∇hA remain finite. In
this limit case, the unknowns vA can be eliminated from (2.8) and we
arrive at a single equation for the averaged displacement w. At the same
time, periodic cell problem (2.5) reduces to the well known periodic cell
problemof thehomogenization theory, cf.Bensoussan et al. (1978), Jikov



On elastodynamics of biperiodic composite media 303

et al. (1994). It follows that from the physical point of view the model
obtainedby the tolerance averaging technique canbe treated as a certain
generalization of thehomogenizedmodel of a linear-elastic periodic solid.

• The example discussed in Section 4 shows that the obtainedmodel equ-
ations can be successfully applied to the analysis of wave propagation
problems including the dispersion phenomena caused by the heterogene-
ous biperiodic structure of a solid.

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O elastodynamice dwuperiodycznych kompozytów

Streszczenie

W artykule przedstawiono dwa problemy. Po pierwsze, sformułowano matema-
tyczny model umożliwiający badanie propagacji fal w liniowo-sprężystych kompozy-
tach, dla których każda płaszczyzna prostopadła do pewnej prostej ma taką samą
dwuwymiarową strukturę periodyczną. Po drugie, otrzymanymodel zastosowano do
analizypropagacji falw laminacie operiodyczniepofałdowanychwarstwach.Dlaprzy-
padku tegowyznaczononiższą iwyższą prędkość fazowąpropagacji, którewprzypad-
ku fal długich dają się wyrazić za pomocą prostych formuł asymptotycznych.

Manuscript received December 16, 2002; accepted for print January 21, 2003