Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

3, 39, 2001

INTERFACE INCLUSION PROBLEMS IN LAMINATED

MEDIUM

Bogdan Rogowski

Marcin Pawlik

Department of Mechanics of Materials, Technical University of Łódź

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

Amoduli perturbation method is used to construct the solution for the
contact stiffness in the inclusion problem of a composite laminate un-
der a state of torsional deformation. The inclusion is considered to be
embedded at the interface of the laminate. The following solutions ha-
ve been obtained: (i) the exact solution for the inclusion in a medium
consisting of two layers and two half- spaces, (ii) a first-order accurate
solution for a layeredmediumconsisting of (n+m) constituents, (iii) an
approximative solution for a physically inhomogeneous medium as the
limiting case of the layeredmedium.

Key words: anisotropy, layered media, inclusion, perturbation method,
torsion interface

1. Introduction

The stress analysis of laminated fiber-reinforced composite materials has
been a subject of increasing importance due to the expanded use of suchma-
terials in diverse modern engineering applications. This analysis is not easy
because composite laminates very often contain interlaminar inclusions, cracks
or delaminations, which has been observed as a common and unavoidable oc-
currence inmany practical situations. It is well known that studies of the inc-
lusions problems for a bondedmultiphasemediumrequire special physical and
analytical considerations that are not encountered in those corresponding to
their homogeneous counterpars. In the multiphase system, the solution arises
not only from a geometric discontinuity but also from the material disconti-
nuity. The plane inclusion problem for two isotropic planes solved by means
of singular integral equations methods was considered by Grilickǐı and Sulim



622 B.Rogowski, M.Pawlik

(1975). Some inclusion problems and a comprehensive list of references up
to the year 1982 are presented in the book by Aleksandrov and Michitarian
(1983). Jevtushenko et al. (1995) presented some plane contact problems for
a layered medium with an interface inclusion in the framework of the homo-
genisation theory. The problem of an interface inclusion between dissimilar
orthotropic half-spaces was considered by Rogowski (1993).

In this paper, the problem of an interface inclusion in a layered composite
or in a physically nonhomogeneous medium is considered in the framework of
themodulus perturbation approach. Thismethodwas applied byGao (1991),
Fan ey al. (1992) to the solution to some fracture and inhomogeneity pro-
blems, respectively. The analysis differs from the previous studies in a number
of aspects. First, by using the modulus perturbation approach a closed-form
solution for the torsional contact stiffness of a composite which consists of
two dissimilar layers and two dissimilar half-spaces is obtained. This solution
has also an advantage over the previous analyses by simplicity and analytical
form of the results. Further, we show that the perturbation analysis actually
provides the first-order accuracy solution to the contact stiffness of an elasti-
cally nonhomogeneousmediumwith arbitrary, piecewise constant (layered) or
continuously varying elastic moduli in the depthwise direction.

2. Reference state

We use cylindrical coordinates and denote them by (r,ϑ,z). Consider
an N-layered composite laminate containing an interfacial rigid circular thin
inclusion of the radius a located between the two layers and twisted by a
small angle ϕ by means of the torque T applied to the disc. The inclusion
may represent the resinous or cementing material, which is used to transfer
the anchoring loads to the geological medium, for instance.

As the reference state we choose the solution to the two-phase counterpart
problem obtained by Rogowski (1993). If the circumferential axisymmetric
displacement of the rigid disc is

ν(r,0)=ϕ0r r¬ a (2.1)

then the contact stresses are

σzϑ0(r,0)=
4

π
ϕ0(−1)iµi

r√
a2−r2

r <a (−1)iz¬ 0 (2.2)



Interface inclusion problems in laminated medium 623

and the rigid rotation ϕ0 of the disc is given by

ϕ0 =
3T

16(µ1+µ2)a
3

(2.3)

In equations (2.2) and (2.3) µi is the average shearmodulus of the orthotropic
material, i.e. µi =

√
GriGzi, Gri and Gzi are thematerial shearmoduli, and

i = 1,2 refers to bodies 1 and 2, respectively. The stress σzϑ0(r,z) and the
displacement ν0(r,z) inside the bimaterial medium are given by formulae
(Rogowski, 1993)

ν
(i)
0 (ξi,ηi)=

2

π
ϕ0r
(π

2
− tan−1ξi−

ξi
1+ξ2i

)

(2.4)

σ
(i)
zϑ0
(ξi,ηi)=−

4

π
ϕ0µi

ηi
ξ2i +η

2
i

√

1−η2i
1+ ξ2i

The two sets of the oblate spheroidal coordinates ξi,ηi) are related to the
cylindrical coordinates (r,z) by the equations

r2 = a2(1+ ξ2i )(1−η2i ) siz= aξiηi
ξi ­ 0
−1¬ ηi ¬ 1

(2.5)

where si =
√

Gri/Gzi is themeasure of the orthotropy and si =1 represents
an isotropicmaterial. Solutions (2.1)-(2.4) correspondto thebimaterial infinite
medium, which is chosen as the reference state or the 0th order solution.

3. Moduli perturbation analysis

3.1. Composite consisting of twodissimilar layers and twodissimilarhalf-

spaces

Now the question is how the torque T and the rotation ϕwill be related
with each other in the case of a layered medium. The exact answer to this
question would require the exact solution. A review of the existing literature
reveals that the exact analytical solution to this problem is not available. In
this paperweobtain a closed-form solution to theproblemwithout restrictions
on the range of applicability, using themoduli perturbation analysis. Consider
an interface inclusion in a system of two layers bonded to the inclusion and
to two half-spaces. The range of convergence of the perturbation solution will



624 B.Rogowski, M.Pawlik

be without an restriction if we choose a bimaterial infinite medium with two
averagemoduli corresponding to the upper and the lower film as the reference

state, i.e. µi = (µ
(i)
f
+µ

(i)
s )/2, where µ

(i)
f
and µ

(i)
s are the average shear

moduli of the ith filmandof the ith substrate, respectively. Then introducing
material parameter

κi =
µ
(i)
f
−µ(i)s

µ
(1)
f
+µ
(1)
s +µ

(2)
f
+µ
(2)
s

(3.1)

we have κi ∈ (−1,1). The parameter κi denotes the ratio of the change of the
average shear modulus of the ith layer (film) to the modulus of the reference
bimaterial medium. Alternatively −κi denotes the ratio of the change of the
average shear modulus of the ith substrate to the modulus of the reference
medium. During the transformation, the applied torque T is kept constant
but the twist angle ϕ and the strain energy are allowed to change. The energy
conservation law requires that the extra work done by the torque T be equal
to the energy change in the whole body. To the first-order accuracy in the
moduli variations κi for 0< |z|<hi and −κi for |z|>hi the equation of
the energy conservation reads

1

2
Tδϕ0 =−

1

2

2
∑

i=1

κi
(

∫

A
(i)
f

σ
(i)
zϑ0
ν
(i)
0 dAi+

∫

A
(i)
s

σ
(i)
zϑ0
ν
(i)
0 dAi

)

(3.2)

where δϕ0 is the rotation change, σ
(i)
zϑ0
and ν

(i)
0 are the known stresses and

displacements for the reference bimaterial medium, and A
(i)
f
,A
(i)
s denote the

boundary surfaces of the transforming regions. For the present of geometry

A
(i)
f
consists of the three horizontal planes: z = 0+ and z = h−1 , z = −h

−

2

and A
(i)
s consists of the two planes: z = h+1

+ and z = −h+2 . The area
integral on the plane z = 0 is equal Tϕ0. The right-hand side of Eq. (3.2)
denotes the first order energy variation due to themoduli transformation. Eq.
(3.2) reduces to the following expression for δϕ0/ϕ0

δϕ0
ϕ0
=−

2
∑

i=1

κi
[

1+
2π

Tϕ0

∞
∫

0

(σ
(i)
zϑ0
ν
(i)
0 )z=±h±

i
r dr
]

(3.3)

Substituting Eq. (2.4)1 and (2.5) with µi = (µ
(i)
f
+µ
(i)
s )/2 and noticing that

siz is equal s
(1)
f
for 0 < z ¬ h1 and s

(1)
s z+ h1(s

(1)
f
− s(1)s ) for z ­ h1

i.e. ξ
(1)
0 = s

(1)
f
h1/a, and alternatively ξ

(2)
0 = s

(2)
f
h2/a for the planes z = h1



Interface inclusion problems in laminated medium 625

and z = −h2, respectively, and then integrating we arrive at the following
expression for δϕ0/ϕ0

δϕ0
ϕ0
=−κ1

[

2I
(s
(1)
f
h1

a

)

−1
]

−κ2
[

2I
(s
(2)
f
h2

a

)

−1
]

(3.4)

where the integral I(ξ
(i)
0 , ξ

(i)
0 = s

(i)
f
hi/a, is found to be

1+
2π

Tϕ0

∞
∫

0

(σ
(i)
zϑ0
ν
(i)
0 )z=±h±

i
r dr=

=1−
6

π
ξ
(i)
0

∞
∫

ξ
(i)
0

(π

2
− tan−1ξi−

ξi
1+ ξ2i

)( 1

ξ2i
−
(ξ
(i)
0 )
2

ξ4i

)

dξi =2I(ξ
(i)
0 )−1

(3.5)

I(ξ
(i)
0 )=

2

π
tan−1ξ

(i)
0 +

ξ
(i)
0

π

[(

3+2(ξ
(i)
0 )
2
)

ln
(1+(ξ

(i)
0 )
2

(ξ
(i)
0 )
2

)

−2
]

ξ
(i)
0 =

s
(i)
f
hi

a

The first order approximation of ϕ,ϕ1 =ϕ0+δϕ0 is 0(κi), 1st order solution

ϕ1 =ϕ0
[

1−κ1
(

2I(ξ
(1)
0 )−1

)

−κ2
(

2I(ξ
(2)
0 )−1

)]

(3.6)

The corresponding higher order solutions are obtained recursively from the
previous solutions, so that ϕ2 =ϕ0+ δϕ1, is 0(κ

2
i), 2nd order solution

ϕ2 = ϕ0
{

1−κ1
(

2I(ξ
(1)
0 )−1

)

−κ2
(

2I(ξ
(2)
0 )−1

)

+
(3.7)

+
[

κ1
(

2I(ξ
(1)
0 )−1

)

+κ2
(

2I(ξ
(2)
0 )−1

)]2
}

In general, one has 0(κni ), nth order solution, n> 1

ϕn =ϕ0

[

1+
n
∑

k=1

(−1)k
[

κ1
(

2I(ξ
(1)
0 )−1

)

+κ2
(

2I(ξ
(2)
0 )−1

)]k
]

(3.8)

It is seen that the sum converges to the exact solution as n→∞ in the range
of convergence, i.e.

ϕ= lim
n→∞
ϕn =ϕ0

[

1+κ1
(

2I(ξ
(1)
0 )−1

)

+κ2
(

2I(ξ
(2)
0 )−1

)]−1
=

(3.9)

=
3T

16a3

[

µ(1)s

(

1− I(ξ(1)0 )
)

+µ
(1)
f
I(ξ
(1)
0 )+µ

(2)
s

(

1− I(ξ(2)0 )
)

+µ
(2)
f
I(ξ
(2)
0 )
]−1



626 B.Rogowski, M.Pawlik

if
∣

∣

∣κ1
(

2I(ξ
(1)
0 )−1

)

+κ2
(

2I(ξ
(2)
0 )−1

)
∣

∣

∣< 1 (3.10)

The weighting functions I(ξ
(i)
0 ), ξ

(i)
0 = s

(i)
f
hi/a, vanish as hi → 0 and appro-

ach unity as hi → ∞, and monotonically increase with ξ
(i)
0 . The inequality

in Eq. (3.10) is satisfied since |κ1+κ2|< 1 and 2I(ξ
(i)
0 )−1∈ (−1,1) for any

values of the material and geometric parameters.

Theperturbation solution for ϕ inEq. (3.9) perfectlymatches theexact so-

lution inboth limitingbimaterial cases, i.e. as s
(i)
f
hi/a→ 0and s

(i)
f
hi/a→∞,

corresponding to the cases of the bimaterial infinitemedium,with the interfa-
ce inclusion, made entirely of the half-space materials or, alternatively, of two
layers by transforming to the two half space regions |z| > hi, respectively.
It can be seen from Eqs (3.5) and (3.9) that the weighting functions I(ξ

(i)
0 )

and 1−I(ξ(i)0 ) give the ratios of the strain energy stored in the transforming
regions of the films and of the substrata, respectively, to the total strain ener-
gy stored in the reference bimaterial medium. The effective contact stiffness
should have the form

µeff =µ
(1)
s

(

1−I(ξ(1)0 )
)

+µ
(1)
f
I(ξ
(1)
0 )+µ

(2)
s

(

1−I(ξ(2)0 )
)

+µ
(2)
f
I(ξ
(2)
0 ) (3.11)

Hence, it can be said that µeff is the average shearmodulusweighted by the
strain energy density distribution.

3.2. Nonhomogeneous medium with piecewise constant moduli

Themodulus perturbation approach provides a natural channel for exten-
sion to more complicated problems. Using the previously presented analysis,
it can be shown that the twist angle of the interface inclusion in the (n+m)th
layer of the composite laminate can be obtained as

ϕ=
T

16a3µeff
(3.12)

with the effective stiffness

µeff =µ
(1)
s

[

1− I
(zns

(1)
n

a

)]

+µ(2)s

[

1− I
(zms

(2)
m

a

)]

+

(3.13)

+
n
∑

i=1

µ
(1)
i

[

I
(zis

(1)
i

a

)

− I
(zi−1s

(1)
i−1

a

)]

+
n
∑

i=1

µ
(2)
i

[

I
(|zi|s

(2)
i

a

)

− I
(|zi−1|s

(2)
i−1

a

)]



Interface inclusion problems in laminated medium 627

Here I(·) is defined by Eq. (3.5), zi is the z coordinate of the interface
between the ith and (i+1)th layer (film), so that the layer thickness is given
by hi = zi− zi−1, i=1,2, ...n, with z0 =0. A rigorous analysis of the error
bounds for the perturbation formula in Eq. (3.13) is not yet available. Hence,
it is necessary to examine the following inequality

∣

∣

∣

∣

1

µ
(1)
s +µ

(2)
s

{

n
∑

i=1

µ
(1)
i

[

I
(zis

(1)
i

a

)

− I
(zi−1s

(1)
i−1

a

)]

+

+
n
∑

i=1

µ
(2)
i

[

I
(|zi|s(2)i
a

)

− I
(|zi−1|s

(2)
i−1

a

)]}

− (3.14)

−
µ
(1)
s

µ
(1)
s +µ

(2)
s

I
(zns

(1)
n

a

)

−
µ
(2)
s

µ
(1)
s +µ

(2)
s

I
(|zm|s(1)m
a

)

∣

∣

∣

∣

< 1

3.3. Nonhomogeneous medium with continuously varying moduli

When the elastic modulus is a function of spatial coordinates then the
problem becomes more complicated compared to the homogeneous case. It
is noted that the perturbation solution can be obtained in that case under
some restrictions. The solution to the problem of a rigid interface disc in
the bimaterial nonhomogeneous medium with continuously varying moduli

G
(i)
z (z), i = 1,2, and constant parameters s

(i), G
(i)
r (z) = s

2
iG
(i)
z (z), can be

constructed by taking the limits as zi → zi−1 and |zi|→ |zi−1| and n→∞
in Eq. (3.13). The effective contact stiffness has then the integral form

µeff = s
(1)

∞
∫

0

dI(s(1)z/a)

dz
G(1)z (z) dz+s

(2)

∞
∫

0

dI(s(2)|z|/a)
d|z|

G(2)z (z) dz (3.15)

where

dI(s(1)z/a)

dz
=
3s(i)

πa

[(

1+2
(s(i))2z2

a2

)

ln
(a2+(s(i))2z2

(s(i))2z2

)

−2
]

(3.16)

When the elastic medium consists of two nonhomogeneous layers of the

thickness h1 and h2 and the shear moduli G
(1)
zf
(z) and G

(2)
zf
(z′) with

G
(i)
rf
(z) = (s

(i)
f
)2G
(i)
zf
(z), s

(i)
f
= const bonded to two homogeneous half-spaces

with the moduli G
(1)
zs , G

(i)
rs = (s

(i)
s )
2G
(i)
zs then the solution for the contact

stiffness can be obtained as



628 B.Rogowski, M.Pawlik

µeff =µ
(1)
s

[

1− I
(s
(1)
f
h1

a

)]

+µ(2)s

[

1− I
(s
(2)
f
h1

a

)]

+

(3.17)

+s
(1)
f

h1
∫

0

dI(s
(1)
f
z/a)

dz
G
(1)
zf
(z) dz+s

(2)
f

h2
∫

0

dI(s
(2)
f
|z′|/a)
d|z′|

G
(2)
zf
(z′) dz′

When a rigid inclusion is embedded in a nonhomogeneous infinite medium
with the shear moduli Gz(z) and Gr(z)= s

2Gz(z) then the contact stiffness
is given by the equation

µeff =
3s2

πa

∞
∫

−∞

[(

1+2
s2z2

a2

)

ln
(

1+
a2

s2z2

)

−2
]

Gz(z) dz (3.18)

Thedifficultywith the perturbationmethod is that in special cases the expres-
sion for the contact stiffness given byEqs (3.15) or (3.18)may reduce to diver-
gent integrals. We expect that these formulae have similar ranges of validity
as Eq. (3.13).

3.4. Two-constituent composites

Rogowski (1995) performed an integral equation analysis to study an axi-
symmetric torsion interface inclusion,whichappearsbetween aboundary layer
anddissimilar half-space (Fig.2). In this analysis the solution of the first-order
accuracy has the form of Eq. (3.12), where (in our notations)

µeff =(µ1+µ2)
[

1− η(3)
3πξ30

+
η(5)

5πξ50
+ ...
]

ξ0 =
hs1
a
(3.19)

η(2m+1)=
µ1
µ1+µ2

∞
∑

n=1

(µ1−µ2
µ1+µ2

)n−1 1

n2m+1

In particular, if µ1 =µ2 then Eq. (3.19) yields

µeff =2µ1
[

1−
a3

6πh3s31
+

a5

10πh5s51
+ ...
]

(3.20)

Solutions (3.18) and (3.19) are valid if hs1/a > 1. The moduli perturbation
approach, see Eq. (3.11), gives

µeff =µ2+µ1I
(s1h

a

)

(3.21)



Interface inclusion problems in laminated medium 629

Fig. 1. Inclusions geometry, coordinates and notation

Fig. 2. Two-phase counterpart of the interface inclusion problem



630 B.Rogowski, M.Pawlik

Fig. 3. Torsional stiffness of the rigid disc on a two-phase orthotropic half-space;

Eq. (3.11) for µ
(2)
s =0=µ

(2)
f
, µ
(1)
f
=µf, µ

(1)
s =µs, ξ

(1)
0 = ξ0 = sfh/a

or for µ1 =µ2

µeff =µ1
[

1+ I
(s1h

a

)]

(3.22)

Equations (3.21) and (3.22) can be rewritten as follows

µeff =(µ1+µ2)
{

1− µ1
µ1+µ2

[

1− I
(s1h

a

)]}

(3.23)

and for µ1 =µ2

µeff =2µ1
{

1− 1
2

[

1− I
(s1h

a

)]}

(3.24)

For a thick boundary layer (hs1/a) > 1 Eqs (3.23) and (3.19) or (3.24) and
(3.20) give the resultswhichdiffer less thanonepercent.Perturbation solutions
(3.21), (3.22) are valid also for small values of hs1/a. For example, Eq. (3.21)
yields

µeff =















µ2+0.0880µ1 for s1h/a=0.01

µ2+0.4434µ1 for s1h/a=0.10

µ2+0.9666µ1 for s1h/a=1.00

(3.25)

If the inclusion is embedded at the finite distance h from the bimaterial
interface of the infinite medium with the shear moduli µ1 and µ2 (in the
material ”1”) then Eq. (3.11) yields

µeff =µ1
[

1+ I
(s1h

a

)]

+µ2
[

1− I
(s1h

a

)]

(3.26)



Interface inclusion problems in laminated medium 631

This equation yields

µeff =















1.0880µ1 +0.9120µ2 for s1h/a=0.01

1.4434µ1 +0.5566µ2 for s1h/a=0.10

1.9666µ1 +0.0334µ2 for s1h/a=1.00

(3.27)

4. Concluding remarks

We have presented a modulus perturbation scheme for determining ela-
stic contact stiffness distributed by inhomogeneities. Although finite element
methods, see Laursen and Simo (1992) for instance, or boundary elementme-
thods, see e.g. Telles andBrebbia (1981), can handle these types of inhomoge-
neity problems, the present perturbation procedure still shows its advantages
by simplicity and analytical form of the results. The presented closed-form
perturbation solution (3.11) may be applicable without any restrictions, i.e.
for any combinations of the elastic constants of material constituents, while
the first-order accurate solutions (3.13), (3.15), and (3.17) may be applied in
amoderate range ofmaterial combinations of practical significance. The effec-
tive elastic constants of the laminatedmediumare given byAchenbach (1975)
and Christensen (1979). If we consider a laminated composite consisting of
alternating plane parallel layers of two homogeneous isotropic materials, then
Achenbach’s and Christensen’s results give

Gz =
G1G2

G2δ1+G1(1− δ1)
Gr =G1δ1+G2(1− δ1)

where δ1 = h1/(h1 + h2) and where h1, h2 are the thicknesses and G1,
G2 the shear moduli of the two elastic layers. Parallel to the above studies of
elastic contact problems a homogenizedmodelwithmicrolocal parameters has
also been developed to evaluate the effective stiffness of a layered body. The
related works are given by Kaczyński and Matysiak (1988, 1992), Matysiak
andWoźniak (1987).

Fromthe solutions presented in this paperweconclude, that elastic proper-
ties of the boundary layer influence strongly the effective stiffness of a layered
body and the solution depends also on the ratio of the layer thickness to the
radius of contact region. The formulae mentioned in the literature do not de-
scribe these effects. By replacing the shear modulus with the effective shear
modulus in equations (2.1), (2.2) and (2.3) we obtain the solution.



632 B.Rogowski, M.Pawlik

The mechanical torsion fields of a laminated orthotropic medium under
and above the interface inclusion are given by:

— rotation

ϕ=
3T

16(µ1 eff +µ2 eff)a
3

(4.1)

— displacement

ν(i)(ξi,ηi) = ϕr
[

1−
2

π

(

tan−1ξi+
ξi
1+ ξ2i

)]

=

(4.2)

=















ϕ z=0± r¬ a

2

π
ϕr
(

sin−1
a

r
−
a

r

√

1−
a2

r2

)

z=0± r­ a

—stresses

σ
(i)
zϑ
(ξi,ηi) = −

4

π
ϕµi eff

ηi
ξ2i +η

2
i

√

1−η2i
1+ ξ2i

=

(4.3)

=











4

π
(−1)iϕµi eff

r√
a2−r2

z=0± r<a

0 z=0± r>a

σ
(i)
rϑ
(ξi,ηi) = −

4

π
ϕG(i)r

r2

a2
ξi

(1+ ξ2i )
2+(ξ2i +η

2
i )
2
=

(4.4)

=















0 z=0± r<a

−4
π
ϕG(i)r

a3

r2
√
r2−a2

z=0± r>a

—stress concentration factors (i=1,2)

L
(I)
zϑ
=
3T

4π
√
a5
(−1)i

µi eff
µ1 eff +µ2 eff

z=0± r→ a−

L
(I)
rϑ
=−

3T

4π
√
a5

G
(i)
r

µ1 eff +µ2 eff
z=0± r→ a+

(4.5)

In the above solutions µi eff, i = 1,2, is the effective contact stiffness of
the lower and upper nonhomogeneous half-spaces, respectively. The oblate



Interface inclusion problems in laminated medium 633

spheroidal coordinates associated with the material parameters si by Eq.
(2.5) are continuous at the interfaces since the plane z= zi, r­ 0 is given by

ξi ­ ξ′i =
sizi
a

ηi =
ξ′i
ξi

(4.6)

The displacement ν and the stress σzϑ are continuous in the laminated
medium, but the stress σrϑ has jumps in the interfaces, which are given by

[[σ
(i)
rϑ
]] =−4

π
ϕ[G(i+1)r −G(i)r ]

r2

a2
ξ3i

(1+ ξ2i )
2+(ξ4i + ξ

′2
i )

ξi ­ ξ′i =
sizi
a
(4.7)

We observe that both stress components have square root singularities as
r → a− or r → a+, respectively. These singularities are presented by the
stress concentration factors, see Eq. (4.5).

When the torsional forces are distributed along the circle (r= r′, z= z′)
in the interior of the ith layer (zi−1 < z ¬ zi) then the rotation ϕ of the
rigid inclusion will be

ϕ=
3T

16(µ1 eff +µ2 eff)a
3

[

1−
2

π

(

tan−1ξi+
ξi
1+ ξ2i

)]

(4.8)

where ξ1, ηi, i = 1,2, ...,n are related to r = r
′ and z = z′ and si, as

shows Eqs (2.5), and T is the resultant torque of the applied torsional forces.
The similarity between formulae (4.2) and (4.8) result fromBetti’s reciprocal
theorem (compare the solution in the paper byRogowski (1998)). If the upper
layered half-space is also loaded in the samemanner (symmetrically) then the
rotation ϕ of the rigid inclusion will be

ϕ=
3T

16(µ1 eff +µ2 eff)a
3

[

2−2
π

(

tan−1ξi+
ξi
1+ ξ2i

+tan−1ξj+
ξj
1+ξ2j

)]

(4.9)

where ξj is associated with si, i= 1,2, ...,n, and ξj is associated with sj,
j=1,2, ...,m.

References

1. Aleksandrov V.M., Michitarian S.M., 1983,Kontaktnye zadachi dla tiel
s tonkimi pokrytyami i proslonkami, Nauka,Moskwa



634 B.Rogowski, M.Pawlik

2. Achenbach J.D., 1975,A Theory of Elasticity with Microstructure for Direc-
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3. Christensen R.M., 1979, Mechanics of Composite Materials, Whiley, New
York

4. Fan H., Keer L.M., Mura T., 1992, Inhomogeneity Problem Revisited via
the Modulus Perturbation Approach, Int. J. Solids Structures, 29, 20, 2593-
2594

5. Gao H., 1991, Fracture Analysis in Nonhomogeneous Material via a Moduli-
Perturbation Approach, Int. J. Solids Structures, 27, 20, 1663-1682

6. Grilickǐı D.V., SulimG.T., 1975,Pieriodicheskaya zadachadla uprugǒı plo-
skosti s tonkostennymi vkluchenyami,Prikl. Matem. i Mekh., 39, 3, 520-529

7. Jevtushenko A.A., Kaczyński A., Matysiak S.J., 1995, Napryazhen-
noe sostoyanie sloistovo uprugovo kompozita s tonkim liněınym vklyucheniem,
Prikl. Matem. i Mekh., 59, 4, 698-703

8. Kaczyński A., Matysiak S.J., 1988, Plane Contact Problems for a Periodic
Two-LayeredElastic Composites, Ingenieur Archiv, 58, 137-147

9. Kaczyński A., Matysiak S.J., 1992,Modelling of Mechanical Behaviour of
Some Layered Soils,Bull. of the Int. Assoc. of Eng. Geology

10. Laursen T.A., Simo J.C., 1992, A Study of Mechanics of Microindentation
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12. Rogowski B., 1993, Internal PointTorque in aTwo-PhaseMaterial. Interface
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105-119

13. Rogowski B., 1995, Interface Crack or Inclusion in a Two-Phase Half-Space
UnderConcentratedTorque,Zeszyty Naukowe PŁ, Ser. Budownictwo, 46, 5-31

14. Rogowski B., 1998, A Statical Problem of Reissner-Sagoci Type of an Inter-
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15. Telles J.C.F., Brebbia C.A., 1981, Boundary Element Solution for Half-
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Interface inclusion problems in laminated medium 635

Zagadnienia międzypowierzchniowej inkluzji w ośrodkach warstwowych

Streszczenie

Za pomocąmetody perturbacjimodułów znaleziono rozwiązania określające kon-
taktową sztywność w zagadnieniach inkluzji umieszczonej na powierzchni ”sklejenia”
warstwowego kompozytu bedącego w stanie skrętnej deformacji. Otrzymano nastę-
pujące rozwiązania: (i) dokładne rozwiązanie dla inkluzji w ośrodku składającym się
z dwóch warstw i dwóch półprzestrzeni, (ii) rozwiązanie o dokładności pierwszego
rzędu dla ośrodka (n+m)-warstwowego, (iii) przybliżone rozwiązanie dla ośrodka
fizycznie niejednorodnego z modułami ścinania zmieniającymi się w sposób ciągły,
otrzymane jako przejście graniczne w rozwiązaniu dla ośrodka warstwowego.

Manuscript received December 5, 2000; accepted for print January 24, 2001