Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 2, pp. 325-335, Warsaw 2007

SINGULARITY OF STRESSES IN A PERIODIC LAMINATED

SEMI-SPACE WITH A BOUNDARY NORMAL TO THE

LAYERING

Stanisław J. Matysiak

Faculty of Geology, University of Warsaw

e-mail: s.j.matysiak@uw.edu.pl

Dariusz M. Perkowski

Faculty of Mechanical Engineering, Białystok University of Technology

The paper deals with the plane problem of stress distributions in a non-
homogeneous elastic semi-space caused by concentrated loadings acting
on its boundary. The body is composed of periodically repeated two-
constituent laminae, and its boundary is assumed to be normal to the
layering. The solution to the problem is presentedwithin the framework
of the homogenized model with microlocal parameters (Woźniak, 1987;
MatysiakandWoźniak, 1987).Analytical results for stressesareobtained
and shown in figures.

Key words: composite, displacement, stresses, elasticity, homogenized
model, concentrated loading

1. Introduction

Layered elasticmaterials with periodic structures represent an important type
of composites, which can be found in nature (varved clays, sandstone-slates,
sandstone-shales, thin-layered limestones) or are made by man and used in
various engineering technologies. The behavior of systems, which are made
of a large number of repeated laminae, is described within the framework of
elasticity theory by partial differential equations with discontinuous, oscilla-
ting coefficients. Thus formulated problems are too complicated for analytical
and numerical approaches. For this reason, applications of some approximate
models seem to be useful. One of the models is the homogenized model with
microlocal parameters devised byWoźniak (1987) and developed formicrope-
riodic layered elastic composites byMatysiak andWoźniak (1987).



326 S.J. Matysiak, D.M. Perkowski

Thismodel satisfies continuity conditions for displacements and stress vec-
tors on interfaces. The homogenized model with microlocal parameters was
applied to solve many problems, which were partially reassumed in papers of
Matysiak (1995),Woźniak andWoźniak (1995). The two-dimensional problem
of stress distribution in a layered semi-space caused by a concentrated loading
acting on the boundary was solved by Kaczyński and Matysiak (1987) in the
case of a boundaryplane parallel to the layering. The problemof concentrated
moving loads on the boundarywith steady supersonic velocity for the lamina-
ted semi-space was discussed within the framework of the ”effective stiffness
theory” by Sve and Hermann (1974).
In this paper, a periodically two-layered elastic semi-space in the plane

state of strain is considered. The boundary plane is normal to the layering.
The concentrated loading with the intensity σ0 acting at an arbitrary point
of the boundary and inclined with the angle θ to the boundary is considered.
The problem is solvedwithin the framework of the homogenizedmodel, which
satisfies continuity conditions on the interfaces. The component of normal
stresses to the boundary is discontinuous on the interfaces, what leads to some
discontinuous, oscillating boundary condition. In this paper, that condition is
approximatedbyaveraged ones.Thus, the stressdistributions in the laminated
elastic semi-space are determined and presented in form of graphs.
The results obtainedwithin the framework of the homogenizedmodelwith

microlocal parameters for certain heat conditionproblemswere comparedwith
the adequate solutions determined by the classical description byKulchytsky-
Zhyailo and Matysiak (2006). Moreover, the analogous analysis of bounda-
ry value problems of periodically layered elastic composites was derived by
Kulchytsky-Zhyhailo et al. (2006). In both papers, a good agreement of the
solutions obtained within the framework of the homogenized model and the
classical approach is observed.

2. Formulation of the problem

Consider a two-dimensional static problem of a half-space with the layering
normal to the boundary plane, (see Fig.1). Let λi, µi, i = 1,2 be Lamés
constants and li, i = 1,2 be the thicknesses of subsequent layers being the
composite constituents, and l, l = l1 + l2 be the thickness of fundamental
lamina. The perfect bonding between the layers is assumed. Referring to the
Cartesian coordinate system (x,y) with the y – axis directed parallel to the
layering and the x – axis being one of the interfaces, denote at the point
(x,y) the displacement vector [u(x,y),v(x,y),0] and the stress component

σ
(j)
xx , σ

(j)
xy , σ

(j)
yy , σ

(j)
zz , j = 1,2, in the j-th kind of composite constituent. Let



Singularity of stresses in a periodic laminated semi-space... 327

thenonhomogeneoushalf-space be loaded at thepoint (a,0) bya concentrated
force with the intensity σ0 inclined with the angle θ to the x-axis, see Fig.1.

Fig. 1. A cross-section of the periodically laminated half-space

To solve this problem, the homogenizedmodel withmicrolocal parameters
will be applied. This model, presented in the general case of periodic ther-
moelastic composites byWoźniak (1987) and adopted to layered structures by
Matysiak andWoźniak (1987), was employed inmanyproblems of periodically
stratified elastic and thermoelastic bodies.We recall only a brief outline of the
governing equations in the case of plane state of strains.

The displacement vector is postulated in the form

u(x,y)=U(x,y)+h(x)qx(x,y)
(2.1)

v(x,y)=V (x,y)+h(x)qy(x,y)

where U, V are unknown functions interpreted as macrodisplacements, and
qx, qy stand formicrolocal parameters and are related to the periodic structure
of the body. The function h is known a priori l-periodic function given in the
form (see for example, Kaczyński andMatysiak, 1987)

h(x) =





x−
1

2
l1 for 0¬x¬ l1

−ηx
1−η

−
1

2
l1+

l1
1−η

for l1 ¬x¬ l

(2.2)

h(x+ l)=h(x) η=
l1
l



328 S.J. Matysiak, D.M. Perkowski

Since for every x |h(x)|< l, then for small l the underlined terms in (2.1)
are small too andwill be neglected. The derivative h′(x) is not small however,
so it determines the following approximations

u≈U v≈V ∂u
∂x
≈ ∂U
∂x
+hjqx

∂v

∂x
≈
∂V

∂x
+hjqy

∂u

∂y
≈
∂U

∂y

∂v

∂y
≈
∂V

∂y

(2.3)

where hj, j = 1,2 are derivatives of the function h(x) in the jth kind of
composite component

h1 =1 h2 =−
η

1−η
(2.4)

The homogenizedmodelwithmicrolocal parameters in the plane state of stra-
ins is describedby the following equations (seeKaczyński andMatysiak, 1987)

A1
∂2U

∂x2
+C
∂2U

∂y2
+(B+C)

∂2V

∂x∂y
=0

(2.5)

C
∂2V

∂x2
+A2

∂2V

∂y2
+(B+C)

∂2U

∂x∂y
=0

and the stresses in the jth kind of composite constituent (j=1,2)

σ(j)xy =C
(∂U
∂y
+
∂V

∂x

)
σ(j)xx =A1

∂U

∂x
+B
∂V

∂y

σ(j)yy =Dj
∂U

∂x
+Ej
∂V

∂y
σ(j)zz =

λj
λj +2µj

(σ(j)xx +σ
(j)
yy )

(2.6)

where

A1 = λ̃+2µ̃−
([λ]+2[µ])2

λ̂+2µ̂
=

(λ1+2µ1)(λ2+2µ2)

(1−η)(λ1+2µ1)+η(λ2+2µ2)
> 0

A2 = λ̃+2µ̃−
[λ]2

λ̂+2µ̂
=A1+

4η(1−η)(µ1−µ2)(λ1−λ2+µ1−µ2)
(1−η)(λ1+2µ1)+η(λ2+2µ2)

> 0

B= λ̃− [λ]([λ]+2[µ])
λ̂+2µ̂

=
(1−η)λ2(λ1+2µ1)+ηλ1(λ2+2µ2)
(1−η)(λ1+2µ1)+η(λ2+2µ2)

> 0 (2.7)

C = µ̃−
[µ]2

µ̂
=

µ1µ2
(1−η)µ1+ηµ2

> 0 Dj =
λj

λj +2µj
A1

Ej =
4µj(λj +µj)

λj +2µj
+

λj
λj +2µj

B



Singularity of stresses in a periodic laminated semi-space... 329

and

λ̃= ηλ1+(1−η)λ2 [λ] = η(λ1−λ2)

λ̂= ηλ1+
η2

1−η
λ2 µ̃= ηµ1+(1−η)µ2

[µ] = η(µ1−µ2) µ̂= ηµ1+
η2

1−η
µ2

(2.8)

It should be emphasized that the continuity conditions on the interfaces are

satisfied because the relations for stress components σ
(j)
xx , σ

(j)
xy , j = 1,2 (see

Eqs. (2.6)) are independent of the kind of composite constituent. The stresses

σ
(j)
yy have jumps on the interfaces, also on the boundary of the half-plane. It
leads to a rather complicated boundary value problem described by equations
(2.5) with the following boundary conditions

σ
(j)
yy (x,0)=−Pδ(x−a) P =σ0 sinθ

σ
(j)
xy (x,0)=−Qδ(x−a) Q=σ0cosθ

(2.9)

and regularity conditions at infinity

σ(j)xx ,σ
(j)
yy ,σ

(j)
xy → 0 for x2+y2 →∞ (2.10)

where δ(x) is the Dirac function and P,Q are known positive constants, see
Fig.1.

Boundary condition (2.9)1 can be approximated by replacing the stress

components σ
(j)
yy , j=1,2 by averaged ones as follows

B
∂U

∂x
+A2

∂V

∂y
=−Pδ(x−a) (2.11)

It can be shown that

D̃=D1η+D2(1−η)=
λ1η

λ1+2µ1
A1+

λ2(1−η)
λ2+2µ2

A2 =B

Ẽ =E1η+E2(1−η)= η
[4µ1(λ1+µ1)
λ1+2µ1

+
λ1

λ1+2µ1

(
λ̃−
[λ]([λ]+2[µ])

λ̂+2µ̂

)]
+

+(1−η)
[4µ2(λ2+µ2)
λ2+2µ2

+
λ2

λ2+2µ2

(
λ̃−
[λ]([λ]+2[µ])

λ̂+2µ̂

)]
=A2

Thus, the considered problem is defined by equations (2.5), boundary con-
ditions (2.9)2, (2.11) and regularity conditions (2.10).



330 S.J. Matysiak, D.M. Perkowski

3. Solution to the problem

The system of partial differential equations (2.5) can be separated by intro-
ducing the following potentials Ψ1 and Ψ2 (see Kulchytsky-Zhyhailo et al.,
2006)

U =κ1
∂Ψ1
∂x
+κ2
∂Ψ2
∂x

V =
∂Ψ1
∂y
+
∂Ψ2
∂y

(3.1)

where

κi =
A2γ

2
i −C
B+C

i=1,2 (3.2)

and γi are the roots of the characteristic equation

A2Cγ
4
i +(B

2+2BC−A1A2)γ2i +A1C =0 (3.3)
Thus, we obtain the following equations for the unknown potentials

γ2i
∂2Ψi
∂x2
+
∂2Ψi
∂y2
=0 i=1,2 (3.4)

Algebraic equation (3.3) in the case of µ1 6=µ2 has four real roots ±γ1,±γ2,
where

γ1 =

√
A1A2−2BC−B2−

√
∆

2A2C
γ2 =

√
A1A2−2BC−B2+

√
∆

2A2C
(3.5)

∆=(B2+2BC−A1A2)2−4A1A2C2 > 0
Let f̃ denote the Fourier transform of an integrable function f with respect
to the variable x

f̃(s,y)=
1√
2π

∞∫

−∞

f(x,y)exp(−ixs) dx (3.6)

By using equations (3.4), (3.1) and regularity conditions (2.10), we obtain

Ũ = is[κ1a1(s)exp(−|s|γ1y)+κ2a2(s)exp(−|s|γ2y)]
(3.7)

Ṽ =−|s|[γ1a1(s)exp(−|s|γ1y)+γ2a2(s)exp(−|s|γ2y)]
where a1(s), a2(s) should be determined from boundary conditions (2.9)2,
(2.11) together with (2.6). It leads to the following system of algebraic equ-
ations

(A2γ
2
1 −κ1B)a1(s)+(A2γ22 −κ2B)a2(s)=

−P exp(−ias)
s2
√
2π

(3.8)

−γ1(1+κ1)a1(s)−γ2(1+κ2)a2(s)=
iQexp(−ias)
s|s|
√
2π



Singularity of stresses in a periodic laminated semi-space... 331

The solution to equations (3.8) takes the form

a1(s)=
(γ2|s|P − isQ)exp(−ias)
|s|s2(γ1−γ2)

√
2π

B+C

(A2γ
2
1 +B)C

(3.9)

a2(s)=
(isQ−γ1|s|P)exp(−ias)
|s|s2(γ1−γ2)

√
2π

B+C

(A2γ
2
2 +B)C

Knowing the functions a1(s), a2(s) and using equations (3.1), (2.6), after
some calculations of the inverse Fourier transforms for the obtained stress
components, we find a closed form of the solution

σ(j)xx(x,y)= (γ
2
1B−κ1A1)G1

Q(x−a)+Pγ1γ2y
π[(a−x)2+y2γ21]

+

−(γ22B−κ2A1)G2
Q(x−a)+Pγ1γ2y
π[(a−x)2+y2γ22]

σ(j)yy (x,y)= (γ
2
1Ej −κ1Dj)G1

Q(x−a)+Pγ1γ2y
π[(a−x)2+y2γ21)

+

(3.10)

−(γ22Ej −κ2Dj)G2
Q(x−a)+Pγ1γ2y
π[(a−x)2+y2γ22]

σ(j)xy (x,y)=−γ1(1+κ1)CG1
Qγ1y+P(a−x)γ2
π[(a−x)2+y2γ21]

+

+γ2(1+κ2)CG2
Qγ2y+P(a−x)γ1
π[(a−x)2+y2γ22]

and

G1 =
1

γ1−γ2
B+C

(A2γ
2
1 +B)C

G2 =
1

γ1−γ2
B+C

(A2γ
2
2 +B)C

(3.11)

4. Discussion of the results

Let us introduce the following dimensionless variables

x′ =
x

l
y′ =
y

l
(4.1)

The distributions of dimensionless stresses σ
(j)
xx(x

′,y′)/σ0 due to the concen-
trated force acting at the point (0,0) for two special cases

(1◦) P =σ0 Q=0

(2◦) P =0 Q=σ0
(4.2)



332 S.J. Matysiak, D.M. Perkowski

are presented in Figures 2a (case 1◦) and 2b (case 2◦) for the ratio of Young’s
moduliE1/E2 =4, η=0.5 the Poissons ratios ν1 = ν2 =0.3 and a=0.

Fig. 2. Lines of constant dimensionless stresses σ
(j)
xx(x

′,y′)/σ0 for two cases:
(a) P =σ0,Q=0, (b) P =0,Q=σ0 (E1/E2 =4, η=0.5, ν1 = ν2 =0.3, a=0)

Figures 3a and 3b show lines of constant dimensionless stresses of

σ
(j)
xy (x

′,y′)/σ0 for case (1
◦) and case (2◦), respectively.

Fig. 3. Lines of constant dimensionless stresses σ
(j)
xy (x

′,y′)/σ0 for two cases:
(a) P =σ0,Q=0, (b) P =0,Q=σ0 (E1/E2 =4, η=0.5, ν1 = ν2 =0.3, a=0)

The component of stresses σ
(j)
yy (x

′,y′)/σ0 are presented in Fig.4 andFig.5

The distributions of σ
(j)
yy (x

′,y′)/σ0 for the case (1
◦), (see Eq. (4.2)) are shown



Singularity of stresses in a periodic laminated semi-space... 333

in Fig.4a in the plane y′ = 0.25, and in Fig.4b in the plane y′ = 0.5. It is
seen that the values of jumps on the interfaces decrease together with gro-
wing distance from the boundary. Figures 5a,b present distributions of dimen-

sionless stresses σ
(j)
yy (x

′,y′)/σ0 for case (2
◦) for two depths y′ = 0.25; 0.5,

respectively.

Fig. 4. Lines of constant dimensionless stresses σ
(j)
yy (x

′,y′)/σ0 for P =σ0,Q=0;
(a) y′ =0.25, (b) y′ =0.5 (E1/E2 =4, η=0.25, ν1 = ν2 =0.3, a=0)

Fig. 5. Lines of constant dimensionless stresses σ
(j)
yy (x

′,y′)/σ0 for ) P =0,Q=σ0;
(a) y′ =0.25, (b) y′ =0.5 (E1/E2 =4, η=0.25, ν1 = ν2 =0.3, a=0)

5. Final remarks

The stress distribution given by equations (3.10) stands for the fundamental
solution for a periodically laminated half-spacewith the layering normal to the
boundary, and canbeused to solve someboundaryvalue problems of a nonho-
mogeneous body (for example: a crack normal to the layering in the composite



334 S.J. Matysiak, D.M. Perkowski

space, contact of the half-spacewithpunches).The analysis presented above is
strictly connected with the case of shear modulus µ1 6=µ2. If µ1 =µ2, calcu-
lations of stresses should be carried out starting from characteristic equation
(3.3). The same investigations were presented for an analogical problemof the
fundamental solution for a laminated half-spacewith the boundary parallel to
the layering, see Kaczyńsk andMatysiak (1987).

Acknowledgements

The investigations described in this paper are a part of the research project BW

realized at the University ofWarsaw and the research projectW/WM/2/05 realized

at Bialystok University of Technology.

References

1. Kaczyński A., Matysiak S.J., 1987, The influence of microlocal effects on
singular stress concentrations in periodic two-layered elastic composites, Bull.
Ac. Pol.: Techn. Sci., 35, 371-382

2. Kulchytsky-ZhyhailoR., Matysiak S.J., 2006,On temperature distribu-
tions in a semi-infinite periodically stratified layer,Bull. Polon. Ac. Techn. Sci.,
54, 45-49

3. Kulchytsky-Zhyhailo R., Matysiak S.J., Perkowski D.M., 2006, On
displacement and stresses in a semi-infinite laminated layer: comparative re-
sults,Meccanica (in press)

4. Matysiak S.J., Woźniak C., 1987, Micromorphic effects in a modeling of
periodic multilayered elastic composites, Int. J., Eng. Sci., 25, 549-559

5. Matysiak S.J., 1995, On the microlocal parameter method in modeling of
periodically layered thermoelastic composites, J. Theor. Appl. Mech., 33, 481-
487

6. Sve C., Hermann G., 1974,Moving load on a laminated composite, J. Appl.
Mech., 41, 663-667

7. Woźniak C., 1987, A nonstandard method of modeling of thermoelastic pe-
riodic composites, Int. J. Eng. Sci., 25, 483-499

8. Woźniak C., Woźniak M., 1995, Modeling of Composites. Theory and Ap-
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Singularity of stresses in a periodic laminated semi-space... 335

Osobliwość naprężeń w periodycznie warstwowej półprzestrzeni

z brzegiem prostopadłym do lamin

Streszczenie

W pracy rozpatrzono płaskie zagadnienie dotyczące rozkładu naprężeń w niejed-
norodnej sprężystej półprzestrzeni wywołanych obciążeniami skupionymi działający-
mi na jej brzegu.Ośrodek jest złożony z periodycznie powtarzającymi się dwuskładni-
kowymi laminami, a jego brzeg jest prostopadły do uwarstwienia.Otrzymano rozwią-
zaniew ramachmodelu homogenizowanegoz parametramimikrolokalnymi (Woźniak,
1987;Matysiak iWoźniak, 1987). Zostały podane analitycznewyrażenia dla naprężeń
i następnie przedstawione w postaci wykresów.

Manuscript received January 24, 2007; accepted for print February 19, 2007