Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 1, pp. 41-57, Warsaw 2004

ON THE PROBLEM OF SOME INTERFACE DEFECT FILLED
WITH A COMPRESSIBLE FLUID IN A PERIODIC

STRATIFIED MEDIUM1

Andrzej Kaczyński

Faculty of Mathematics and Information Science, Warsaw University of Technology

e-mail: akacz@alpha.mini.pw.edu.pl

Bohdan Monastyrskyy

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU, Lviv, Ukraine

e-mail: labmtd@iapmm.lviv.ua

A periodic two-layered elastic space containing an interface defect filled
withabarotropic compressiblefluid is considered.At infinity, the compo-
site is subjected to a uniformly distributed load applied perpendicularly
to the layering. Faces of the defect are under action of constant internal
fluid pressure. An approximate solution to this problem is given within
a certain homogenizedmodel. The resulting singular integro-differential
equation is obtained and solved for two types of defects by using an ana-
logue of Dyson’s theorem. The influence of the filler on the mechanical
behaviour of the considered body is analysed and illustrated graphically.

Key words: periodic two-layered space, interface defect, barotropic com-
pressible fluid, singular integro-differential equation

1. Introduction

Natural geological structures are not usually homogeneous.Many types of
soils exhibit features of periodic layering (see, for example, Amadei, 1983).
In real conditions a soil contains a lot of cracks or cavities, some of which
may be filled with a gas or fluid. That is why the investigation of mechanical
behaviour of layered structures containing such defects is a very important

1
Thispaperwas presentedonSymposiumDamageMechanics ofMaterials and Structures,

June 2003, Augustów, Poland



42 A.Kaczyński, B.Monastyrskyy

problem involvingmany geotechnical applications in mining engineering, gas-
and oil-producing industry.

Many researchers have studied problems of solids possessing cavities filled
with certain substances. It seems that the first article on a general model of a
crack filled with a heat-conducting medium was written by Pidstryhach and
Kit (1967). Based on thismodel, the results of the investigation of temperatu-
re and stress fields were summarised in twomonographs byKit andKryvtsun
(1983), andKit andKhay (1989). However, in all these studies, themechanical
influence of the defect filler was neglected. One of the first attempts to take
into account this effect was made by Yevtushenko and Sulym (1980). In this
work it is suggested that themechanical action of the filler can be simulaterd
by a constant pressure dependent on the crack opening and determined from
the equation of state of the fluid. Other researchers used this idea. Kuznetsov
(1988) considered static contact of two isotropic bodieswith surface gaps filled
with a compressible fluid. Recently, the contact interaction of bodies having
defects filled with a gas was studied by Martynyak (1998), Machyshyn and
Martynyak (2000) and Machyshyn and Nagórko (2003). A combined thermal
andmechanical effect of the ideal gas filling a crackwas analysed byMatczyń-
ski et al. (1999) in the case of plane strain.

This contribution is devoted to a three-dimensional problem for a bima-
terial periodically layered space containing an interface cavity filled with a
barotropic compressible fluid. In Section 2, this problem is formulated and the
use of the homogenized model of the composite is demonstrated. Section 3
presents the resulting boundary-value problem and its reduction to a singular
integro-differential equation. A detailed analysis is performed and illustrated
by graphs for two special types of defects in Section 4. Conclusions are given
in the last section.

2. Description of the problem

2.1. Formulation

Let us consider a stratified space, the middle cross of which is given in
Fig.1. A repeated fundamental lamina of a small thickness δ consists of two
homogeneous isotropic layers (denoted by 1 and 2) of thicknesses δ1 and δ2
(δ = δ1 + δ2), and characterised by the Lamé constants λ1, µ1 and λ2, µ2,
respectively. Let refer thebody to theCartesian coordinate system (x1,x2,x3)
with its center on the interface plane and the x3-axis normal to the layering.



On the problem of some interface defect... 43

We suppose that there is an interface defect (a cavity) occupying the re-
gular region Vd defined as

Vd =
{
(x1,x2,x3) : (x1,x2)∈Sa ∧ −

1

2
f(x1,x2)¬x3 ¬

1

2
f(x1,x2)

}

(2.1)

(x1,x2)∈Sa ⇔ r2 ≡x21+x22 ¬ a2

In the above, Sa is the circular median (equatorial) section of the defect with
the radius a, and f is a smooth function describing the small initial height of
the defect (before applying the load) such that

f
∣∣
r=a
=0

0¬ f(x1,x2)<max(δ1,δ2) ∀(x1,x2)∈Sa
f(x1,x2)≪ a ∀(x1,x2)∈Sa

(2.2)

This defect is assumed to be completely filled with a barotropic compres-
sible fluid, whose state is governed by the equation (Galanov et al., 1985)

V =
m∗
ρ0
exp
(
−
Pfl
β

)
(2.3)

where ρ0 stands for the density of the compressed fluid, V is the volume of
the fluid of a fixedmass m∗ at the internal pressure Pfl, and β stands for the
coefficient of volume compressibility.

The composite is subjected to a uniform load p applied at infinity and di-
rected perpendicularly to the interface (see Fig.1). Moreover, perfect bonding
between the subsequent layers (excluding the defect region) is assumed.

We shall consider two cases separately:

Case (i): The defect degenerates into a crack. Itmeans that its initial height
is equal to zero, i.e. f(x1,x2)≡ 0, and the region of occupation Sa does
not alter under the external load. The surfaces of the crack are under
the action of the fluid pressurePfl.

Case (ii): We deal with a cavity (described by Eqs (2.1) and (2.2)), filled
with a barotropic compressible fluid, and a situation when owing to the
external load p its faces are in frictionless contact in a certain annulus
b < r ¬ a. Hence, Sb = {(x1,x2) : x21 +x22 ¬ b2} is the circular non-
contacting domainwith the radius b, subjected to the fluidpressure Pfl.



44 A.Kaczyński, B.Monastyrskyy

Fig. 1. Periodic two-layer space weakened by an interface defect filled with
a compressible barotropic fluid

Notice that Pfl is an unknown parameter and is constant (in view of the
Pascal law). Besides, we assume that the fluid does not transmit tensile forces,
i. e. the pressure of the fluid Pfl may be compressive only.

The problem lies in the determination of the fields of displacements and
stresses in the stratified space. Of prime interest is to analyse the effect of the
filler on the stresses. The unknown fluid pressure Pfl is found during solving
the problem by using Eq. (2.3). Moreover, in Case (ii) the parameter b is
determined by the condition of finiteness of the normal stress on r= b.

2.2. Governing equations

To solve the problem under study (with an infinite number of thin lay-
ers), a direct analytical approach becomes more intricated and the numerical
formulation more unstable. A classic idea is the use of the homogenization
process to replace the heterogeneousmediumby a continuous, equivalent one,
which gives the average behaviour of themediumat themacroscopic scale.We
base on a specific non-asymptotic procedure called the microlocal modelling
(cf Woźniak, 1987; Matysiak and Woźniak, 1988). The final governing equ-
ations (in a static case with neglecting body forces) and constitutive relations



On the problem of some interface defect... 45

for the stresses σ
(l)
ij of the homogenizedmodel, given in terms of the unknown

macrodisplacements wi, take the form
1 (for details, see Kaczyński, 1993)

1

2
(c11+ c12)wγ,γα+

1

2
(c11− c12)wα,γγ +c44wα,33+(c13+ c44)w3,3α =0

(2.4)

(c13+ c44)wγ,γ3+ c44w3,γγ + c33w3,33 =0

σ
(l)
α3 = c44(wα,3+w3,α) σ

(l)
33 = c13wγ,γ + c33w3,3

σ
(l)
12 =µl(w1,2+w2,1) σ

(l)
11 = d

(l)
11w1,1+d

(l)
12w2,2+d

(l)
13w3,3

σ
(l)
22 = d

(l)
12w1,1+d

(l)
11w2,2+d

(l)
13w3,3

(2.5)

All coefficients appearing above are given inAppendixA. They depend on
thematerial andgeometrical characteristics of subsequent layers. It is notewor-
thy here that the condition of perfect bonding between the layers is satisfied
(the components σ3j do not depend on l implying the continuity of the stress
vector at the interfaces). Finally, by assuming λ1 = λ2 ≡ λ, µ1 = µ2 ≡ µ we
get c11 = c33 = λ+2µ, c12 = c13 = λ, c44 = µ, passing directly to the well-
known equations of elasticity for a homogeneous isotropic body with Lamé’s
constants λ, µ.

3. The boundary-value problem and method of its solution

Bearing Eqs (2.1) and (2.2) in mind, the problem under consideration is
now posed within the homogenized model as follows: find the fields wi, σij
symmetric about the interface plane x3 =0 and suitably smooth on R

3−Sa
such that Eqs (2.4) holds, subject to the following boundary conditions

σ33(x1,x2,+∞)= p
σ31(x1,x2,+∞)=σ32(x1,x2,+∞)= 0

∀(x1,x2)∈R2 (3.1)

1Throughout thispaper, theLatin indices i,j runover1,2,3while theGreek indices
α,β run over 1,2, and the summation over repeated subscripts is taken for granted.
Subscripts preceded by a comma indicate partial differentiation with respect to the
corresponding coordinates. The index l or (l), assuming values 1 or 2, is associated
with layer 1 and 2, respectively.



46 A.Kaczyński, B.Monastyrskyy

—Case (i)

σ33(x1,x2,0
+)=−Pfl ∀(x1,x2)∈Sa

σ31(x1,x2,0
+)=σ32(x1,x2,0

+)= 0 ∀(x1,x2)∈R2

w3(x1,x2,0
+)= 0 ∀(x1,x2) /∈Sa

(3.2)

—Case (ii)

σ33(x1,x2,0
+)=−Pfl ∀(x1,x2)∈Sb

σ31(x1,x2,0
+)=σ32(x1,x2,0

+)= 0 ∀(x1,x2)∈R2

w3(x1,x2,0
+)=

1

2
f(x1,x2) ∀(x1,x2)∈Sa−Sb

w3(x1,x2,0
+)= 0 ∀(x1,x2) /∈Sa

(3.3)

The procedure for obtaining the solution follows along the same line of
reasoning as that used in the classical crack theory. Applying the principle of
superposition, we construct the solution as the sum of a trivial solution for
the homogenized space without any defects, loaded by the given external load
σ33 = p at infinity, and a corrective solution to the problem involving pertur-
bations caused by the defect Vd, which tends to zero at infinity. Attention will
be drawn then on finding the corrective solution. In view of the symmetry, we
can reduce the perturbedproblem (having tilde) to a boundary-value problem
for the half-space x3 ­ 0 defined by the boundary conditions

σ̃33(x1,x2,+∞)= 0
σ̃31(x1,x2,+∞)= σ̃32(x1,x2,+∞)= 0

∀(x1,x2)∈R2 (3.4)

—Case (i)

σ̃33(x1,x2,0
+)=−p−Pfl ∀(x1,x2)∈Sa

σ̃31(x1,x2,0
+)= σ̃32(x1,x2,0

+)= 0 ∀(x1,x2)∈R2

w̃3(x1,x2,0
+)= 0 ∀(x1,x2) /∈Sa

(3.5)

—Case (ii)

σ̃33(x1,x2,0
+)=−p−Pfl ∀(x1,x2)∈Sb

σ̃31(x1,x2,0
+)= σ̃32(x1,x2,0

+)= 0 ∀(x1,x2)∈R2

w̃3(x1,x2,0
+)=

1

2
f(x1,x2) ∀(x1,x2)∈Sa−Sb

w̃3(x1,x2,0
+)= 0 ∀(x1,x2) /∈Sa

(3.6)



On the problem of some interface defect... 47

Themethod of solving the above problem is based on the results given by
Kaczyński (1993). A convenient representation of the solution is expressed by
a single harmonic function denoted by

ϕ=

{
ϕ̂ for µ1 6=µ2
ϕ for µ1 =µ2, λ1 6=λ2

as follows2:

—Case µ1 6=µ2

w̃α(x1,x2,x3)=
2∑

γ=1

(−1)γ[tγ(1+mγ)]−1ϕ̂,α(x1,x2,zγ)

w̃3(x1,x2,x3)=
2∑

γ=1

(−1)γmγ(1+mγ)−1
∂

∂zγ
ϕ̂(x1,x2,zγ)

(3.7)

σ̃3α(x1,x2,x3)=
2∑

γ=1

(−1)γc44
[ ∂
∂zγ
ϕ̂(x1,x2,zγ)

]

,α

σ̃33(x1,x2,x3)=
2∑

γ=1

(−1)γc44t−1γ
∂2

∂z2γ
ϕ̂(x1,x2,zγ)

—Case µ1 =µ2 ≡µ, λ1 6=λ2

w̃α =
µ

B+µ
ϕ,α w̃3 =−

B+2µ

B+µ
ϕ,3+x3ϕ,33

σ̃3α =2µx3ϕ,α33 σ̃33 =2µ(−ϕ,33+x3ϕ,333)
(3.8)

where

B=
λ1λ2+2µ[(1−η)λ2+ηλ1]
(1−η)λ1+ηλ2+2µ

Expressions for the remaining stresses have been omitted.

Note that the above suitable representation automatically satisfies the con-
ditionsappearing inEqs (3.5)2 and (3.6)2.Thequantities of immediate interest
are

w̃3(x1,x2,0
+)=Lϕ,3

∣∣∣
x3=0+

σ̃33(x1,x2,0
+)=Mϕ,33

∣∣∣
x3=0+

(3.9)

2All constants appearing in the following equations are given in Appendix B;
z
α
= t
α
x3, α=1,2



48 A.Kaczyński, B.Monastyrskyy

where

L=






m2
1+m2

−
m1
1+m1

for µ1 6=µ2

−B+2µ
B+µ

for µ1 =µ2 ≡µ
(3.10)

M =





c44
( 1
t2
−
1

t1

)
for µ1 6=µ2

−2µ for µ1 =µ2 ≡µ

The problem described by Eqs (3.4)-(3.6) is then reduced to amixed pro-
blem of finding a harmonic function ϕ in the half-space x3 ­ 0, decaying at
infinity (in view of (3.4)) and satisfying the boundary the conditions:
—Case (i)

Mϕ,33
∣∣∣
x3=0+

=−p−Pfl ∀(x1,x2)∈Sa

Lϕ,3
∣∣∣
x3=0+

=0 ∀(x1,x2) /∈Sa
(3.11)

—Case (ii)

Mϕ,33
∣∣∣
x3=0+

=−p−Pfl ∀(x1,x2)∈Sb

Lϕ,3
∣∣∣
x3=0+

=
1

2
f(x1,x2) ∀(x1,x2)∈Sa−Sb

Lϕ,3
∣∣∣
x3=0+

=0 ∀(x1,x2) /∈Sa

(3.12)

The harmonic function ϕ can be represented as a potential of a single
layer:
—Case (i)

ϕ(x1,x2,x3)=−
1

2πL

∫∫

Sa

h(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2+x23

(3.13)

—Case (ii)

ϕ(x1,x2,x3) = −
1

2πL

∫∫

Sa

h(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2+x23

+

(3.14)

+
1

2πL

∫∫

Sb

f(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2+x23



On the problem of some interface defect... 49

where h is the unknown layer density that can be identified as the defect face
displacement: w̃3(x1,x2,0

+) in Case (i) and w̃3(x1,x2,0
+)+0.5f(x1,x2) in

Case (ii). Expressions (3.13) and (3.14) satisfy boundary conditions (3.11)2
and (3.12)2,3 identically due to the property of the normal derivative of the
single-layer potential. Substitution (3.13) and (3.14) into remaining conditions
(3.11)1 and (3.12)1 leads to the following integro-differential singular equation
for the function h
—Case (i)

∇2
∫∫

Sa

h(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2

=
2πL(−p−Pfl)

M
(3.15)

—Case (ii)

∇2
∫∫

Sb

h(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2

=

(3.16)

=
2πL(−p−Pfl)

M
+∇2

∫∫

Sa

f(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2

where ∇2 ≡ ∂2/∂x21 +∂2/∂x22 stands for the two-dimensional Laplace ope-
rator.
To find the two unknown parameters of our problem, namely the pressure

Pfl and the radius of the non-contacting zone b, we have to use state equation
(2.3) and the condition of boundedness of the normal stress σ33|r=b, respecti-
vely. Once the function h is known from the solution to Eqs (3.15) and (3.16),
the stresses and displacements in any point of the composite can be found
from the harmonic potential ϕ, determined from Eqs (3.13) or (3.14), and
formulas (3.7) or (3.8).

4. Examples

In this Section, we present and discuss solutions to the boundary-value
problem defined by (3.4)-(3.6) for Case (i) and Case (ii).

4.1. Case (i)

In this extreme case, when the defect is a circular crack Sa, the problem
can be solved in elementary functions.



50 A.Kaczyński, B.Monastyrskyy

The solution to Eq. (3.15) is known (see, for example, Khay, 1993), and
given as

h(x1,x2)=
2L(p+Pfl)

πM

√
a2−r2 (4.1)

SubstitutingEq. (4.1) into Eq. (3.13) and evaluating the integral, as outli-
ned by Fabrikant (1989), one obtains the governing potential ϕ in the form

ϕ(x1,x2,x3)=−
p+Pfl
2πM

[
(2a2+2x23−r2)sin−1

a

l+
−
2a2−3l2−
a

√
l2+−a2

]

(4.2)
where

l∓ ≡
1

2

[√
(r+a)2+x23∓

√
(r−a)2+x23

]
(4.3)

To find Pfl, we take into account Eq. (2.3), in which

V =

∫∫

Sa

[w3(x1,x2,0
+)−w3(x1,x2,0−)] dx1dx2 =2

∫∫

Sa

h(x1,x2) dx1dx2

(4.4)

Substituting Eq. (4.1) into Eq. (4.4), and performing integration, we get

V =
8L(p+Pfl)a

3

3M
(4.5)

and, according to Eq. (2.3), one arrives at the following transcendental equ-
ation for the unknown pressure Pfl

8L(p+Pfl)a
3

3M
=
m∗
ρ0
exp
(
−
Pfl
β

)
(4.6)

Now we can discuss the singular behaviour at the crack edge in the above
symmetrical problem. By means of Eqs (3.9) and (4.2), one finds

σ̃33 =
2(p+Pfl)

π

( a√
r2−a2

− sin−1
a

r

)
r >a, x3 =0 (4.7)

If the stress intensity factor (SIF) is defined as

KI = lim
r→a+

√
2π(r−a)σ̃33(r,0) (4.8)

then we have

KI =2

√
a

π
(p+Pfl) (4.9)



On the problem of some interface defect... 51

Fig. 2. Dependence of the stress intensity factor of mode I on the external load
(1−γ=0.5, η=0.1, 2−γ=0.5, η=0.5, 3−γ=1, η=1 corresponds to the
homogeneous isotropic material, 4−γ=2, η=0.5, 5−γ=2, η=0.1)

It is interesting to note that this factor turns out to be independent of the
elastic properties of the laminated body.

Figure 2 demonstrates the change of the SIF KI under the loading. The
calculations were performed on the simplifying assumptions λ1 = µ1 and
λ2 =µ2 for the following dimensionless quantities

η=
δ1
δ

γ=
µ2
µ1

β=
β

µ1
=5 ·10−4

p=
p

µ1
Pfl =

Pfl
µ1

m∗
a3ρ0

=10−3
(4.10)

It can be seen that even for negative values of the parameter p, i.e. for
a compressive pressure at infinity, the stress intensity factor has a positive
magnitude. As the compressive load increases, SIF KI monotonically decre-
ases and asymptotically tends to zero. For this range of the external load,
the dependence of KI as a function of p is strictly nonlinear. This fact is a
consequence of the presence of the filler in the defect. For the tensile external
pressure p the internal pressure of the fluid Pfl decreases, andhence the effect
of the defect filler weakens. There is some critical value of the tensile external
load underwhich the pressure of the filler becomes zero. If the external pressu-
re is greater than this critical value, the filler of the cavity does not affect the
mechanical behaviour of the body. The dashed lines in Fig. 2 correspond to
this range of the external load. Then the dependence of KI on p is linear and
all curves coincide. It means that this stress intensity factor does not depend
on mechanical and geometrical parameters of the body. This result is similar



52 A.Kaczyński, B.Monastyrskyy

to that of the classical theory of homogeneous isotropic solid (see, for example,
Khay, 1993).

4.2. Case (ii)

Let us consider the cavity Vd (see (2.1) and (2.2)), the initial shape of
which is of a specific form, given by the function

f(x1,x2)=h0

√
(
1− x

2
1+x

2
2

a2

)3
h0 ≪ a (4.11)

In this case, the faces of this defect may contact through an unknown

ring-shaped zone b< r=
√
x21+x

2
2 ¬ a adjacent to its rim.

Having calculated the integral in the right-hand sidewith the use of (4.11)
(see Khay, 1993), integro-differential equation (3.16) takes the form

∇2
∫∫

Sb

h(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2

=
2πL(−p−Pfl)

M
−
3π2h0
2a3

[
a2−

3

2
(x21+x

2
2)
]

(4.12)
To solve it, an analogue of Dyson’s theorem (Khay, 1993) will be applied.

Thus, we seek a solution to Eq. (4.12) in the following form

h(x1,x2)=
√
b2−x21−x22 (c00+c10x1+c01x2+c20x

2
1+c11x1x2+c02x

2
2) (4.13)

where cij are unknown coefficients.
Substituting (4.13) into (4.12) and calculating the resulting integrals (see

Khay, 1993), we arrive at the equality of two polynomials of the second order.
Hence, the coefficients cij can be easily found by comparing the coefficients,
and the sought function turns out to be

h(x1,x2)=
√
b2−r2

[2L(p+Pfl)
πM

+
3h20(a

2+ b2)

2a3
+
h0(b

2−r2)
a3

]
(4.14)

This expression contains two unknown parameters: b and Pfl. Using the
requirement that thenormal stresses σ̃33 areboundedat r= bor, equivalently,
the corresponding SIF KI has to be zero, we obtain the equation

2

√
b

π
(p+Pfl)+

3h0M
√
πb

2a3L
(a2+ b2)= 0 (4.15)

The parameter Pfl is found from state equation (2.3), similarly to Case (i)
with one exception. Here we assume that the fluid fills thewhole initial defect



On the problem of some interface defect... 53

without any internal pressure. Itmeans that the volume of the unpressed fluid
of the mass m∗ is equal to the volume of the initial defect. In other words,
the following relationship takes place

m∗
ρ0
≡V0 =

∫∫

Sa

f(x1,x2) dx1dx2 (4.16)

The calculation of the volume of the cavity after application of the load, and
insertion into Eq. (2.3), yields the following transcendental algebraic equation

4π
[ 2L
πM
(p+Pfl)+

3h0
2a3
(a2+ b2)

]
+4π
h0b
5

5a3
=4π

h0a
2

5
exp
(
−
Pfl
β

)
(4.17)

Using the system of two equations (4.15) and (4.17), we find b and Pfl
and then, substituting them into formula (4.14), we get the solution via the
harmonic function ϕ, given by (3.14).

Some results of numerical calculations are shown in Fig.3 and Fig.4 by
using the following parameters as in Case (i), see Eq. (4.10)

β=
β

µ1
p=
p

µ1
Pfl =

Pfl
µ1

σ33 =
σ̃33
µ1

b=
b

a
η=
δ1
δ
=1

γ=
µ2
µ1
=1 h0 =

h0
a
=10−3

(4.18)

The simplifying assumptions λ1 =µ1 and λ2 =µ2 have been used as well.

Fig. 3. Dependence of the inner radius of the contacting zone on the external load
(1−β=0, 2−β=2.5 ·10−5, 3−β=10−4, 4−β=10−3, 5−β=∞)



54 A.Kaczyński, B.Monastyrskyy

Figure 3 illustrates the dependence of the inner radius of contacting zone b
on the external load p. The graphs obtained show that this zone decreases
monotonically as the tensile external load increases. The less is the coefficient
of compressibility of the fluid β, the faster decreases the curve. For the range
0 < β < ∞, while the parameter p tends to infinity, the curves show an
asymptotic decrease to zero, non-intersecting the abscissa axis. This effect is
a consequence of the assumption on the fixed mass of the fluid in the defect.
If β = ∞ (the fluid is imcompressible), the shape and size of the cavity do
not change under the loading.

Fig. 4. Normal stress distribution on the interface x3 =0 versus x1 = r/a
(1−β=2.5 ·10−5, 2−β=10−3, 3−β=∞)

Thenormal stressdistributionon the interfaceplane is shown inFig.4.The
curves correspond to the value of the tensile external load p=−3.142 ·10−3
anddifferentvalues of thecoefficient of compressibilityof thefluid.Referring to
Fig.4, it can be noted that the difference between the greatest and the least
magnitudes of the normal stress, so-called dispersion, is less for the bigger
value of the parameter β. In the limit case β =∞, which corresponds to an
imcompressible fluid, the dispersion of the normal stresses is equal to zero. In
this case, the normal stresses are distributed uniformly on the interface.

5. Conclusions

As a result of the analysis made it was revealed that the compressible
barotropic fluid filling the interface defect induces severe non-linear behaviour
of the composite structure. It is true for such a range of the external load,
under which the internal pressure of the filler is greater than zero.



On the problem of some interface defect... 55

A. Appendix

Denoting by bl = λl +2µl (l = 1,2), b = (1− η)b1 + ηb2, the positive
coefficients in governing equations (2.4) and (2.5) are given by the following
formulae

c11 =
b1b2+4η(1−η)(µ1−µ2)(λ1−λ2+µ1−µ2)

b

c12 =
λ1λ2+2[ηµ2+(1−η)µ1][ηλ1+(1−η)λ2]

b

c13 =
(1−η)λ2b1+ηλ1b2

b
c33 =

b1b2
b

c44 =
µ1µ2

(1−η)µ1+ηµ2
d
(l)
11 =

4µl(λl+µl)+λlc13
bl

d
(l)
12 =

2µlλl+λlc13
bl

d
(l)
13 =

λlc33
bl

B. Appendix

The constants appearing first in Eqs. (3.7) are given as follows

t1 =
1

2
(t+− t−) t2 =

1

2
(t++ t−)

mα =
c11t
−2
α −c44
c13+ c44

∀α∈{1,2}

where

t± =

√
(A±±2c44)A∓
c33c44

A± =
√
c11c33± c13

Note that

t1t2 =

√
c11
c33

m1m2 =1

References

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56 A.Kaczyński, B.Monastyrskyy

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On the problem of some interface defect... 57

O zagadnieniu pewnego defektu międzywarstwowego wypełnionego
ściśliwym płynem w periodycznym ośrodku warstwowym

Streszczenie

W pracy rozważono zagadnienie periodycznej dwuwarstwowej przestrzeni zawie-
rającej defekt na granicy warstw, który wypełniony jest ściśliwym płynem. Zakłada
się, że kompozyt jest pod działaniem stałego obciążenia w nieskończoności prosto-
padłego do uwarstwienia, a powierzchnie defektu są pod wpływem stałego i niezna-
nego wewnętrznego ciśnienia płynu. Analizowany problem sprowadza się w ramach
pewnego zhomogenizowanego modelu do poszukiwania rozwiązania osobliwego rów-
nania całkowo-różniczkowego.Wyniki otrzymano dla dwóch typówdefektu z użyciem
analogu twierdzenia Dysona. Zbadano i zilustrowano wpływ wypełniacza defektu na
mechaniczne zachowanie rozważanego ciała.

Manuscript received October 13, 2003; accepted for print November 4, 2003