Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 4, pp. 735-753, Warsaw 2003

ON CONTINUUM MODELLING OF DYNAMIC PROBLEMS

IN COMPOSITE SOLIDS WITH PERIODIC

MICROSTRUCTURE

Jowita Rychlewska

Czesław Woźniak

Institute of Mathematics and Computer Sciences, Częstochowa University of Technology

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

In this contribution we propose an approach to the macroscopic mo-
delling of periodic composites which is based on the periodic simplicial
subdivision of the unit cell. This approach makes it possible to derive
a hierarchy of continuum models which describe dynamic behaviour of
periodic composites on different levels of accuracy. The obtained results
are compared and applied to the analysis of a certain wave propagation
problem.

Key words: composites, dynamics, modelling

1. Introduction

The investigations of wave dispersion problems in the elastodynamics of
solids with a periodic microstructure can be carried out either on the basis
of the Floquet-Bloch wave theory, cf. Lee (1972), Stoker (1950), Tolf (1983),
or in the framework of different simplified continuum theories and models of
these solids. We can mention here the effective stiffness theories, Achenbach
and Sun (1972), Achenbach et al.(1968), Herrmann et al. (1976), the mixture
theories, Bedford and Stern (1971), the interacting continuum theories, Hege-
meier (1972), Lee (1972), the asymptoticmodels, Boutin andAuriault (1993),
Fish and Wen Chen (2001), or the tolerance averaging models, Woźniak and
Wierzbicki (2000). All aforementioned continuum theories andmodels descri-
bedispersionphenomena inwhat are called low-frequency vibration problems.
They are problems in which the macroscopic wavelength of the deformation
pattern is sufficiently large when compared to the diameter of the unit cell



736 J.Rychlewska, C.Woźniak

of a periodic composite. In Rychlewska et al. (2000) it was proposed a di-
screte model of a periodic solid which can be also applied to the analysis of
high-frequency vibration problems. The idea of the above approachwas based
on special simplicial subdivision of the unit cell and resulted in the finite-
difference form of the governing equations.

The aim of this paper is to show that after introducing smoothing ope-
rations to the finite difference equations of a discrete model proposed in Ry-
chlewska et al. (2000), it is possible to obtain a hierarchy of continuummodels
describing the macroscopic behaviour of a micro- periodic solid on different
levels of accuracy. The simplest from the aforementioned models leads to the
equations of a homogeneous equivalent medium which is not dispersive and
can be obtained in the framework of the homogenization theory, Bensoussan
et al. (1978), Jikov et al. (1994), Sanchez-Palencia (1980). It is also shown
that continuum models derived in this paper on a higher level of accuracy
constitute a proper tool for the analysis of dispersion phenomena in a compo-
sitemedium.The general results are illustrated, compared and verified on the
example of the wave propagation in a certain periodic composite medium.

Tomake this paper self-consistent in the subsequent section we summarise
the main concepts introduced in Rychlewska et al. (2000).

2. Preliminaries

Let the composite solid under consideration occupies a region Ω in E3,
has perfectly bounded linear-elastic constituents and a periodic structure de-
termined by a vector basis d1,d2,d3 in E

3. We denote by ∆ a polyhedron
in E3 such that for every x∈ ∂∆ and some dα, α = 1,2,3, we have either
x+dα ∈ ∂∆ or x−dα ∈ ∂∆ (but not both). Let us also assume that the
diameter l of ∆ is sufficiently small when compared to the smallest charac-
teristic length dimension of the region Ω. In this case the polyhedron ∆ will
be referred to as the unit cell.

Let Λ be the Bravais lattice in E3

Λ :=
{
z∈ E3 : z= η1d1+η2d2+η3d3, ηα =0,±1,±2, ..., α =1,2,3

}

and let us denote ∆(z) := z + ∆, Λ0 := {z ∈ Λ : ∆(z) ⊂ Ω} and
Ω0 := {x ∈ ∆(z) : z ∈ Λ0}, where Ω0 is a regular subregion of Ω. A
simplicial division of E3 will be called ∆-periodic if it implies the simplicial
subdivision of every ∆(z), z ∈ Λ, into simplexes Tk(z), k = 1, ...,m, such



On continuum modelling of dynamic problems... 737

that Tk(z) = Tk + z, z ∈ Λ where Tk, k = 1, ...,m, are simplexes in ∆.
Let {pa0 ∈ ∆ : a = 1, ...,n +1}, n ­ 1, be the smallest set of vertexes (no-
dal points) of Tk, k = 1, ...,m, such that {pa0 +z : a = 1, ...,n +1,z ∈ Λ}
is the set of all nodal points in E3 related to a certain ∆-periodic simpli-
cial division of E3. We shall also introduce a system of vectors dA ∈ Λ,
A =0,1, ...,N, such that d0 =0 and every vertex related to T

k, k =1, ...,m,
can be uniquely represented by the sum pa0 + dA. It can be seen that
N = 7 for the spatial problem and N = 3 for the plane problem. Setting
I := {(a,A) ∈ {1, ...,n +1}× {0,1, ...,N} : pa0 + dA ∈ ∆} and denoting
paA :=p

a
0+dA for every (a,A)∈ I, we conclude that {p

a
A : (a,A)∈ I} is the

set of all nodal points in ∆ which is related to the ∆-periodic simplicial divi-
sion of E3. Hence, every simplex Tk can be represented by Tk =paAp

b
Bp
c
Cp
d
D

where (a,A), ...,(d,D) ∈ I. Setting I0 := {(a,A) ∈ I : A 6= 0} we see that
paA ∈ ∂∆ if and only if (a,A) ∈ I0. Here and hereafter it is assumed that a
certain ∆-periodic simplicial division of E3 is known.

Fig. 1. Simplicial division of the 0x1x2-plane with the cell ∆ and a system of
vectors dA and nodal points p

a

A
in ∆



738 J.Rychlewska, C.Woźniak

In Fig.1 a simple example of a ∆-periodic simplicial division of the plane
(hence A runs over 0,1,2,3) with points paA, a = 1,2, and vectors dA, is
shown.
For an arbitrary function f(·) defined on Λ0 we shall define the finite

differences

∆Af(z)= f(z+dA)−f(z) ∆Af(z)= f(z)−f(z−dA) (2.1)

provided that z,z+dA,z−dA ∈ Λ0.
Throughout the paper itwill be assumed that the superscripts a,b,c,d run

over 1, ...,n +1, n ­ 1, and the subscripts A,B run over 0,1, ...,N, unless
otherwise stated. We shall also introduce superscripts p, q which run over
1, ...,n. The summation convention with respect to all aforementioned indices
holds.
Let w(x, t), x∈ Ω, stand for a displacement field at time t for the solid

under consideration. Let us denote

uaA(z, t) :=w
(
paA(z), t

)
(a,A)∈ I0 z∈ Λ0

Subsequently, we shall interpret the simplexes Tk, k = 1, ...,m, as finite ele-
ments of the unit cell ∆ which are subjected to uniform strains. Hence paA(z)
are nodal points of these elements. Let us also ”approximate” the region Ω
by Ω0. In this case the displacement field w(·, t) in every cell ∆(z),z ∈ Λ0
will be uniquely determined by the displacements uaA(z, t) of the nodal points
paA(z), (a,A)∈ I. Bearing inmind (2.1), we see that these displacements can
be uniquely represented in the form

u
a
A(z, t)=u

a(z, t)+∆Au
a(z, t) (a,A)∈ I

where for A = 0 we obtain ua0(z, t) = u
a(z, t). Let u be a certain averaged

value of u1, ...,un+1, given by

u= νau
a

where νa > 0 and ν1+ ...+νn+1 =1. The values νa will be specified in the
subsequent section.Under the above denotations, the strain andkinetic energy
functions for the solid under consideration are respectively represented by

U = U(∆Au
a,up−νau

a) K = K(∆Au̇
a,u̇b) (2.2)

where (a,A)∈ I0, b =1, ...,n+1, p =1, ...,n. The coefficients of forms (2.2)
canbeuniquelydetermined for everyperiodic solid. Introducing thedifferences



On continuum modelling of dynamic problems... 739

up−u as arguments of the strain energy function we have taken into account
the translational invariance of U(·). It can be shown,Woźniak (1971), that for
the unknowns ua(z, t), a =1, ..,n+1, z∈ Λ0, in the absence of body forces,
we obtain a system of ordinary differential equations which can be expressed
in the following finite-difference form

∆As
a
A−

∂U

∂ua
=

d

dt

(∂K
∂u̇a
−∆Aj

a
A) a =1, ...,n+1 (2.3)

where

saA =
∂U

∂∆Au
a

jaA =
∂K

∂∆Au̇
a (a,A)∈ I0 (2.4)

Equations (2.3), (2.4) are assumed to hold for every z ∈ Λ0 such that
z ± dA ∈ Λ0 for A = 1, ...,N, and represent a finite difference model of
the periodic composite under consideration. It has to be emphasised that
this model has a physical sense only if the diameters lk of simplexes T

k,
k = 1, ...,m are small as compared to the typical wavelength of the defor-
mation pattern in the problem under consideration. Thus, for high-frequency
vibration problems, the number m of simplexes Tk and hence also the num-
ber n of the unknowns ua(z, t) for every z∈ Λ0 can be very large.

Equations of the form (2.3), (2.4) have been derived and applied in
Rychlewska et al. (2000); in this paper they constitute the foundations of
the subsequent analysis leading to different continuum models of the micro-
periodic solids under consideration.

3. Simplified finite difference models

For the given ν1, ...,νn+1 let us denote

ũ
a :=ua−u=ua−νbu

b

It follows that

νaũ
a =0

and hence the fields ũa(z, t), z ∈ Λ0, are linear dependent. In order to sa-
tisfy the above condition we shall introduce new linear independent fields
vq =vq(z, t), z∈ Λ0, such that

ũ
a = lhaqvq



740 J.Rychlewska, C.Woźniak

where l is the diameter of ∆ and haq are elements of the (n+1)×n matrix
of an order n, satisfying conditions

νah
aq =0

Hence
u
a =u+ lhaqvq (3.1)

and we shall take u and vq as the basic unknowns. It can be seen that the
above formula represents a decomposition of the displacement field ua into
the averaged u= νau

a and fluctuating ũa parts.
Subsequently, we shall restrict ourselves to problems in which the incre-

ments ∆Aũ
a of fluctuations can be neglected as small with respect to the

increments ∆Au of the averaged displacements. Thus, we shall apply to (2.2)
an approximation

∆Au
a ∼= ∆Au (3.2)

whichholds for every (a,A)∈ I0.Theabove formulastates that inanarbitrary
but fixed periodicity cell ∆(z), z∈ Λ0, the displacement fluctuations can be
treated as periodic

ũ
a(z+dA, t)∼= ũ

a(z, t) (a,A)∈ I0

Subsequently, for the sake of simplicity, we shall also approximate the
mass distribution in the periodicmediumbyaperiodic systemof concentrated
masses ma, a =1, ...,n+1, assigned to the nodal points. Setting m = m1+
...+mn+1 we shall assume that νa = m

a/m.Hence, the kinetic energy function
will take the form

K =
1

2|∆|

n+1∑

a=1

mau̇a · u̇a

where |∆| is the measure of the cell ∆. Taking into account formula (3.1) we
obtain the kinetic energy function in the form

K̃ =
1

2
ρu̇ · u̇+

1

2
l2Mpqv̇p · v̇q (3.3)

where

ρ =
m

|∆|
Mpq =

1

|∆|

n+1∑

a=1

mahaqhap

Taking into account (3.1) and (3.2), we obtain from (2.2) the strain energy
function

Ũ =
1

2
aAB∆Au ·∆Bu+

1

2
bpqvp ·vq + c

q
Av
q ·∆Au (3.4)



On continuum modelling of dynamic problems... 741

which is the density per unit measure of ∆. Because of ∆Au∈ O(l) we have
aAB ∈ O(l

−2), c
q
A
∈ O(l−1) and bpq ∈ O(1), i.e., all terms in (3.4) are of the

same order.
Using (3.3) and (3.4), we shall transform equations (2.3), (2.4) to the form

∆AsA+c
q
A
∆Av

q = ρü sA = aAB∆Bu
(3.5)

l2Mpqv̈q+ bpqvq+ c
p
A∆Au=0

The above equations hold for every z ∈ Λ0 and time t, and represent a
simplified finite differencemodel of the periodic compositemediumunder con-
sideration.
Let us observe that l2Mpqv̈q ∈ O(l2) and the values of all other terms in

(3.5) are independent of l. Hence, for a sufficiently small l we can apply the
limit passage l → 0. In this case, the first term in the second from equations
(3.5) will be neglected and we arrive at the equations

bpqvq =−c
p
A
∆Au

Since bpq represent a non-singular n ×n matrix, then denoting by Bpq the
elements of the inverse matrix and setting

a0AB := aAB − c
q
A
Bqpc

p
B

the first from equations (3.5) yields

a0AB∆A∆Bu= ρü (3.6)

Thus, we have arrived at the single equation for the averaged displacement
field u(z, t), z∈ Λ0. The above equation together with the formulae

vq =−Bqpc
p
A
∆Au (3.7)

represent what will be called the asymptotic discrete finite element model of
the periodic composite under consideration. Let us observe that for stationary
problems the last from equations (3.5) coincide with equations (3.7).
Discrete models governed by equations (3.5) and (3.6) will be treated in

subsequent analysis as a basis for the formulation of continuum models. The
main advantage of the aforementioned equations is that they involve finite
differences with respect to only one unknown field u, in contrast to equations
(2.3), (2.4).This factwill implya relatively simple formofpertinent continuum
model equations which will be derived in the subsequent section. It has to
be remembered that equations (3.5), (3.6) can be applied exclusively to the
analysis of the long wave propagation problems.



742 J.Rychlewska, C.Woźniak

4. Formulation of continuum models

Let u(·, t), sA(·, t) be arbitrary sufficiently smooth fields defined on Ω,
which after restricting their domain Ω to Λ0 reduce to fields u(z, t), sA(z, t),
z∈ Λ0, occurring in (3.5). Inorder toobtainacontinuummodel of theperiodic
solid under consideration, we shall assume that for every z∈ Λ0 and every y
such that |y| ¬ l and z+y ∈ Ω, the aforementioned smooth fields can be
approximated bymeans of the formulae

w(z+y, t)∼=w(z, t)+y ·∇w(z, t)+
1

2
(y⊗y) : (∇⊗∇)w(z, t)

where w stands for u and sA. From the above approximation we also obtain

w(z, t)∼=w(z−y, t)+y ·∇w(z, t)−
1

2
(y⊗y) : (∇⊗∇)w(z, t)

Hence, under denotations (no summation over A!)

eA :=dAl
−1

EA :=
1

2
eA⊗eA

we conclude that the following approximations

∆Au(z, t)∼= leA ·∇u(z, t)+ l
2
EA : (∇⊗∇)u(z, t)

(4.1)

∆AsA(z, t)∼= leA ·∇sA(z, t)− l
2EA : (∇⊗∇)sA(z, t)

hold for every z∈ Λ0. Substituting the right-hand sides of the above formulae
into (3.5) and denoting

G := aABEA⊗EBl
2 C := aABeA⊗eBl

2

Hq := c
q
AEAl h

q := c
q
AeAl

(4.2)

after simple manipulations we obtain

−l2(∇⊗∇) : [G : (∇⊗∇)u]+∇·(C·∇u)− lHq : (∇⊗∇)vq+hq ·∇vq = ρü
(4.3)

l2Mpqv̈q + bpqvq+hq ·∇u+ lHq : (∇⊗∇)u=0 q =1, ...,n

Because u(·, t), vq(·, t) are functions defined for every time t on the region Ω
wehave arrivedat the systemof n+1differential equations (4.3) for n+1unk-
nown vector fields u, vq. The aforementioned equations represent what will



On continuum modelling of dynamic problems... 743

be called the second order continuummodel of the periodic compositemedium
under consideration. It has to be emphasized that for the averaged displace-
ment field u we have obtained the partial differential equation and for the
displacement fluctuations vq the system of n ordinary differential equations.
It follows that the boundary conditions can be imposed only on the averaged
displacement field; we deal here with a situation similar to that occurring in
the tolerance averaging model equations,Woźniak andWierzbicki (2000).
Applying approximations (4.1) to equation (3.6) and denoting

G0 := a
0
ABEA⊗EBl

2 C0 := a
0
ABeA⊗eBl

2 (4.4)

we obtain

−l2(∇⊗∇) : [G0 : (∇⊗∇)u]+∇· (C0 ·∇u)= ρü (4.5)

The above equation represent the asymptotic second order continuum model
of the medium under consideration.
Now assume that instead of (4.1) we introduce the linear approximations

∆Au(z, t)∼= leA ·∇u(z, t) ∆AsA(z, t)∼= leA ·∇sA(z, t) (4.6)

In this case equations (3.5) reduce to the form

∇· (C ·∇u)+hq ·∇vq = ρü
(4.7)

l2Mpqv̈q+ bpqvq+hq ·∇u=0 q =1, ...,n

where C and hq are defined by formulae (4.2). The above equations represent
the first order continuum model of the periodic composite medium. Similarly,
from (3.6) we derive the equation

∇· (C0 ·∇u)= ρü (4.8)

representing the asymptotic first order continuummodel of themediumunder
consideration.
By introducing higher order derivatives into approximations of the form

(4.1), it is possible to formulate higher-order continuummodels of the periodic
medium under consideration; these models have a rather complicated form
and will not be discussed here. Subsequently, we shall apply the obtained
model equations only to the analysis of thewave propagation in an unbounded
medium; that is why in this paper we shall not discuss the physical meaning
of boundary conditions related to equations (4.3), (4.5), (4.7) and (4.8). It can



744 J.Rychlewska, C.Woźniak

be shown that the aforementioned equations together with pertinent natural
boundaryconditions can alsobederived fromtheprinciple of stationary action
for the action functional

t1∫

t0

∫

Ω

(K −U) dxdt

where K and U are obtained from (3.3) and (3.4), respectively, by using the
approximations introduced at the beginning of this section.
Summarising the obtained results, we shall state that themacroscopic dy-

namic behaviour of the elastic composites with a periodic microstructure can
be analysed in the framework of different continuummodels describedby inde-
pendent systems of equations (4.3), (4.5), (4.7) and (4.8). The above equations
have constant coefficients which depend on the geometric andmaterial struc-
ture of the unit cell, i.e. on the vectors eA, A = 1, ...,N, and coefficients of
the quadratic forms (3.3), (3.4) related to the discrete model. Solutions to
these equations have a physical sense only if approximation formulae (4.1) or
(4.6) are satisfiedwith a sufficient accuracy. Obviously, the derived continuum
models describe the dynamic behaviour of the composite on different levels
of accuracy. Thus, the problem arises what is the interrelation between the-
se models and their accuracy when compared to the discrete model given by
equations (2.3), (2.4). The above problemwill be discussed in the subsequent
section.

5. Comparison and reliability of continuum models

The aim of the subsequent analysis is to compare the results obtained
from the second and first order continuum models (represented respectively
by equations (4.3), (4.5) and (4.7), (4.8)) with the results derived from the
discrete models described by equations (2.3), (2.4). This comparison will be
carried out on the example of the analysis of a harmonic wave propagating in
an unbounded linear elastic homogeneous mediumwhich is reinforced by two
parallel families of fibres.We shall assume that theaxes of all fibres are parallel
to the x3-axis of theCartesian orthogonal coordinate system 0x1x2x3, and the
cross-sections of fibres belonging to the first and second family are periodically
distributed on the 0x1x2-plane as shown in Fig.2.
The Lamé moduli andmass density of the mediumwill be denoted by λ,

µ and ρ0, respectively. Themass densities and areas of cross sections of fibres
belonging to thefirst and second family of fibres are denotedby ρ1,A1 and ρ2,



On continuum modelling of dynamic problems... 745

Fig. 2. Cross section of a composite reinforced by two families of parallel fibres

A2, respectively. The values A1, A2 are assumed to be small when compared
with the area A of the unit cell, and hence the masses ρ1A1 and ρ2A2 will
be treated as concentrated masses on the 0x1x2-plane. To simplify conside-
rations, we shall deal with the longitudinal wave propagating in the x1-axis
direction. Introducing the periodic simplicial division of the plane as shown in
Fig.1, with a =1,2 and A =0,1,2,3, we obtain the scheme of displacements
of nodal points of ∆ as in Fig.3, where the finite-difference operator ∆1 is
denoted by ∆. We also denote l := l1. For the sake of simplicity we have
introduced here the simplest ∆-periodic simplicial division of the plane, and
hence all subsequent results can be treated only as a rough approximation of
the problem under consideration.

Fig. 3. Scheme of displacements for the nodal points of cell ∆

Webeginwith thediscretemodel governed by equations (2.3), (2.4).Under
denotations

η := λ+2µ ε1 :=
A1
A

ε2 :=
A2
A



746 J.Rychlewska, C.Woźniak

the strain energy function is given by

U =
1

2
α(∆u1)2+

1

2
β(u2−u1)2+γ(u2−u1)∆u1

where

α =
2η

l2
β =
4η

l2
γ =−

2η

l2

and the kinetic energy function has the form

K =
1

2
ψ(∆u̇1)2+

1

2
ν(u̇1)2+

1

2
ξ(u̇2)2+ ζu̇1∆u̇1+ δu̇2∆u̇1+ϑu̇1u̇2

where

ψ =
1

6
ρ0 ν =

1

3
ρ0+ρ1ε1 ξ =

1

3
ρ0+ρ2ε2

ζ =
1

6
ρ0 δ =

1

12
ρ0 ϑ =

1

6
ρ0

Governing equations (2.3), (2.4) are given by

∆s1+β(u2−u1)+γ∆u1 = νü1+ ζ∆ü1+ϑü2−
d

dt
∆j1

(5.1)

−β(u2−u1)−γ∆u1 = ξü2+ δ∆ü1+ϑü1

where

s1 = α∆u1+γ(u2−u1) j1 = ψ∆u̇1+ δu̇2+ ζu̇1

In order to obtain equations (3.5), we assign the concentrated masses m1, m2
to points p10, p

2
0, respectively, where

m1 =
1

2
ρ0A+ρ1A1 m2 =

1

2
ρ0A+ρ2A2

Setting

h1 =
m2

m1+m2
h2 =−

m1
m1+m2

decomposition (3.1) will be taken in the form

u1 = u+ lh1v u2 = u+ lh2v (5.2)

Hence, the kinetic energy function is given by

K =
1

2
ν(u̇1)2+

1

2
ξ(u̇2)2+ϑu̇1u̇2



On continuum modelling of dynamic problems... 747

Using (5.2) we obtain (3.3) in the form

K̃ =
1

2
ρ(u̇)2+

1

2
l2M(v̇)2

where

ρ = ρ0+ρ1ε1+ρ2ε2

M =
(1
2
ρ0+ρ1ε1)(

1
2
ρ0+ρ2ε2)

ρ0+ρ1ε1+ρ2ε2
−
1

6
ρ0 =

=
ρ20+4ρ0(ρ1ε1+ρ2ε2)+12ρ1ε1ρ2ε2

12(ρ0+ρ1ε1+ρ2ε2)

The strain energy function (3.4), reduces to the form

Ũ =
1

2
a(∆u)2+

1

2
b(v)2+cv∆u

where

a =
2η

l2
b =4η c =

2η

l

In order to formulate continuum models for the problem under consideration
we shall use the denotation

∂k(·)≡
∂k(·)

∂xk
k =1,2,3,4

Second order continuummodel equations (4.3) yield

−l2G∂4u+C∂2u− lH∂2v+h∂v = ρü
(5.3)

l2Mv̈+ bv+h∂u+ lH∂2u =0

where bymeans of (4.2) we obtain

G =
1

2
η C =2η H = η h =2η

The asymptotic second order continuummodel equation reduces to the form

−l2G0∂
4u+C0∂

2u = ρü (5.4)

where

G0 =
1

4
η C0 = η



748 J.Rychlewska, C.Woźniak

For the pertinent first order continuum model equations we have to neglect
the underlined terms in equations (5.3), (5.4).

In order to compare the results obtained from the proposedmodels, let us
investigate propagation of a harmonic wave, which for discrete model (5.1) is
given by

u1 = A1exp[ik(nl−ct)] u
2 = A2exp[ik(nl− ct)] n =0,±1,±2, ...

where A1, A2 are thewave amplitudes, k =2π/L is thewave number and c is
thepropagation speed. Substituting the right-hand sides of the above formulae
into (5.1), we obtain the dispersion relation between the wave number k and
the vibration frequency ω = kc in the form

(
−
ρ20
72
coskl+r+

ρ20
72

)
ω4+α

(
−
ρ0
3
coskl−2ρ+

ρ0
3

)
ω2−2α2 coskl+2α2 =0

(5.5)
where

r1 :=
1

2
ρ0+ρ1κ1 r2 :=

1

2
ρ0+ρ2κ2 r := r1r2−

1

6
ρ0ρ

Introducing the nondimensional wave number q = kl and bearing in mind
that the analysis is restricted to the wavelengths L sufficiently large when
compared to the microstructure length l, we have to assume that q ≪ 1.
Hence q can be treated as a small parameter, and wemay set

cosq ∼=1−
1

2
q2

In this case, we obtain from (5.5) the following asymptotic formulae for lower
and higher free vibration frequencies

(ω
−
)2 =

α

2ρ
q2−αρ0

12r+ρ0ρ

288rρ2
q4+O(q6)

(5.6)

(ω+)
2 =
2αρ

r
−
αr21r

2
2

2r2ρ
q2+αρ0

36r2+ρ20ρ
2+3ρ0ρr

864ρ2r2
q4+O(q6)

where the terms O(q6) are small and can be neglected.

On passing to the analysis of harmonic waves in the framework of the
proposed continuum models we assume

u = Aexp[ik(x− ct)] v = Bexp[ik(x− ct)]



On continuum modelling of dynamic problems... 749

Substituting the above formulae into second order continuummodel equations
(5.3), we obtain the dispersion relation of the form

rω4−α
[r
ρ
q2
(
1+
1

4
q2
)
+2ρ
]
ω2+α2q2

(
1+
1

4
q2
)
=0 (5.7)

where q = kl is the nondimensional wave number. By means of q ≪ 1, we
obtain from (5.7) the asymptotic formulae

(ω
−
)2 =

α

2ρ
q2+

α

8ρ

(
1−

r

ρ2

)
q4+O(q6)

(5.8)

(ω+)
2 =
2αρ

r
+
α

2ρ
q2+

α

8ρ

(
1+

r

ρ2

)
q4+O(q6)

Thepertinent formulae obtained in the framework of the first order continuum
model equations are

(ω
−
)2 =

α

2ρ
q2−

αr

8ρ3
q4+O(q6)

(5.9)

(ω+)
2 =
2αρ

r
+
α

2ρ
q2+

αr

8ρ3
q4+O(q6)

For the asymptotic model equations, we assume

u = Aexp[ik(x− ct)]

and after substituting the right-hand side of this equation into (5.4), we obtain

ω2 =
α

2ρ

(
q2+
1

4
q4
)

(5.10)

The pertinent formula obtained in the framework of the first order asymptotic
model equations is

ω2 =
α

2ρ
q2 (5.11)

At the end of this contribution we present some numerical results. We
introduce the following dimensionless coefficients

κ1 :=
ρ1
ρ0

κ2 :=
ρ2
ρ0

where ρ0, ρ1, ρ2 aremass densities of themedium and fibres belonging to the
first and the second family of fibres, respectively. The free vibrations frequen-
cies for discrete model (5.6), second order continuum model (5.8), first order



750 J.Rychlewska, C.Woźniak

continuummodel (5.9), asymptotic second order continuummodel (5.10) and
asymptotic first order continuummodel (5.11), wewill write in the dimension-
less form

ω̃2 :=
ρ0
α
(ω0)

2

where α = 2ηl−2 and ω0 is the frequency described by (5.6), (5.8)-(5.11).
The calculations were carried out for κ1 = 5, λ1 = 0.1, λ2 = 0.05 and
κ2 = 5; 10; 15. The diagrams of spectral lines obtained in the framework
of the proposed models coincide if q ¬ 0.1; for lower frequencies are shown
in Fig.4, and for higher frequencies in Fig.5. Diagrams of spectral lines for
frequencies in the asymptotic first and second order continuum models also
coincide for q ¬ 0.1 and are presented in Fig.6.

Fig. 4. Diagrams of spectral lines for lower frequencies in the finite difference models
and the first and second order continuummodels

Fig. 5. Diagrams of spectral lines for higher frequencies in the finite difference
models and the first and second order continuummodels



On continuum modelling of dynamic problems... 751

Fig. 6. Diagrams of spectral lines for frequencies in the asymptotic first and second
order continuummodels

6. Concluding remarks

The aim of this contribution was to derive a certain hierarchy of continu-
ummodels for the dynamics of a linear elastic composite solid with a periodic
microstructure and to compare different results obtained in the framework
of these models with the results derived from the discrete finite-difference
models given by equations (2.3), (2.4). Restricting the considerations to the
low-frequency vibration problems and introducing successively a series of ap-
proximations into the aforementioned discrete model, we formulated models
given by independent equations (4.3), (4.5), (4.7), (4.8) which describe the dy-
namic problemsondifferent levels of accuracy.The simplest fromthesemodels
is represented by equation (4.8) which can be also obtained by the asymptotic
homogenization of a periodic medium.
Among new results obtained in this contribution, the following ones seem

to bemost important.

• The proposed discrete model makes it possible to obtain independent
systems of equations for displacement fluctuations vq(z, t), q =1, ...,n,
in every cell ∆(z), z∈ Λ0.

• Theproposed continuummodels are governed bypartial differential equ-
ations only for themean displacement field u(·). The displacement fluc-
tuation fields vq(·) are governed by ordinary differential equations invo-
lvingonly timederivatives of vq(·). It follows that in stationaryproblems
the fields vq(·) are governed by linear algebraic equations and can be
eliminated from the governing equations.



752 J.Rychlewska, C.Woźniak

• The displacement fluctuations vq are governed by a system of linear
algebraic equations also in dynamic problems provided that we apply
the asymptotic approximation both to discrete and continuum model
equations.

• Apart from the asymptotic first order continuum model, all proposed
models take into account the effect of the microstructure size on the
dynamic behaviour of a composite solid, which plays an important role
in the dispersive analysis of dynamic problems.

• From a formal point of view, the second order continuum model, (4.3),
corresponds to that obtained in the framework of the tolerance averaging
method,Woźniak andWierzbicki (2000).

• Comparing formulae (5.6) related to the discrete model and formulae
(5.8), (5.9) obtained in the framework of the second and first order con-
tinuummodels, it can be seen that the first terms coincide for lower and
higher frequencies, respectively.

• Differences between the values of free vibration frequencies calculated
within the first and second order continuum models are negligible.

It has to mentioned that in most engineering problems the number n of
displacement fluctuations can be large, and solution to these problems requires
applications of computational methods.

References

1. AchenbachJ.D., SunC.T., 1972,TheDirectionally ReinforcedComposite as
a Homogeneous Continuum with Microstructure, in: E.H. Lee (Ed.),Dynamics
of Composite Materials, Am. Soc. Mech. Eng., NewYork

2. Achenbach J.D., Sun C.T., Herrmann G., 1968, On the vibrations of a
laminated body, J. Appl. Mech., 35, 467-475

3. Bedford A., Stern M., 1971, Toward a diffusing continuum theory of com-
posite materials, J. Appl. Mech., 38, 8-14

4. Bensoussan A., Lions J.L., Papanicolau G., 1978, Asymptotic Analysis
for Periodic Structures, North-Holland, Amsterdam

5. Boutin C., Auriault J.L., 1993, Rayleigh scattering in elastic composite
materials, Int. J. Eng. Sci., 12, 1669-1689

6. Fish J., Wen Chen, 2001, Higher-order homogenization of Initial/boundary-
value Problem, J. of Eng. Mech., 127, 1223-1230



On continuum modelling of dynamic problems... 753

7. Hegemeier G.A., 1972,On a Theory of Interacting Continua for Wave Pro-
pagation in Composites, in: E.H. Lee (Ed.),Dynamics of Composite Materials,
Am. Soc. Mech. Eng., NewYork

8. Herrmann G., Kaul R.K., Delph T.J., 1976, On continuum modelling of
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9. Jikov V.V., Kozlov S.M., Oleinik O.A., 1994, Homogenization of Diffe-
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10. Maewal A., 1986, Construction of models of dispersive elastodynamic beha-
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11. LeeE.H., 1972,ASurvey ofVariationalMethods forElasticWavePropagation
Analysis in Composites with Period Structures, in: E.H. Lee (Ed.), Dynamics
of Composite Materials, Am. Soc. Mech. Eng., NewYork

12. Rychlewska J., Szymczyk J., Woźniak C., 2000, A simplicial model for
dynamic problems in periodic media, J. Theor. Appl. Mech., 38, 3-13

13. Sanchez-PalenciaE., 1980,Non-HomogeneousMedia andVibration Theory,
Lecture Notes in Physics, 127, Springer-Verlag, Berlin

14. StokerJ.J., 1950,NonlinearVibrations inMechanical andElectrical Systems,
Interscience Publ. Inc.

15. Tolf G., 1983, On dynamical description of fibre reinforced composites [in:]
NewProblems inMechanics of Continua, Proc. Third Swedish-Polish Symp. on

Mechanics, Eds. Brulin O. and Hsish R.K.T., Univ.Waterloo Press

16. Woźniak C., Wierzbicki E., 2000,Averaging Techniques in Thermomecha-
nics of Composite Solids, Wydawnictwo Politechniki Częstochowskiej, Często-
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17. Woźniak C., 1971, Discrete Elasticity,Arch. Mech., 23, 801-816

Modele ciągłe zagadnień dynamiki kompozytów z periodyczną

mikrostrukturą

Streszczenie

W pracy zaproponowano nowe podejście do modelowania zagadnień dynamiki
w liniowo-sprężystych mikroperiodycznych ośrodkach ciągłych, które za punkt wyj-
ścia przyjmuje periodyczny podział symplicjalny elementu reprezentatywnego struk-
tury. Podejście to umożliwia wyprowadzenie pewnej hierarchiimodeli ciągłych, które
opisują zagadnienia dynamiki z różną dokładnością. Otrzymane wyniki zastosowano
do analizy propagacji fali w nieograniczonymośrodku o budowiemikroperiodycznej.

Manuscript received February 4, 2003; accpeted for print April 15, 2003