Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 1, pp. 5-14, Warsaw 2007

ASYMPTOTIC STUDY OF ELASTIC HALF-PLANE WITH

EMBEDDED PUNCH

Igor V. Andrianov

Institut für Allgemeine Mechanik, RWTH Aachen, Germany

e-maik: igor andrianov@hotmail.com

Jan Awrejcewicz

Department of Automatics and Biomechanics, Technical University of Lodz, Poland

e-mail: awrejcew@p.lodz.pl

Yurii A. Kirichek

Department of Soil Mechanics and Foundation Engineering, PGASA, Dnepropetrovsk, Ukraine

e-mail: yu@kir.dp.ua

Steve G. Koblik

USA; e-mail: s.koblik@worudnet.att.net

Acontactproblem for an elastic half-plane andanembedded rigidpunch
is studied. The employed mathematical model describes the behavior
of a soil with embedded foundation. The analytical solution governing
the stress field behaviour is derived. Singular perturbation and complex
analysis techniques are used.

Key words: mixed boundary value problem, singular asymptotics, com-
plex variable theory

1. Introduction

The problem under consideration is important for engineers, especially from
the point of view of earthquake engineering (Zeng andCakmak, 1984; Tyapin,
1990). Usually, engineers and designers face with the dilemmawhether to use
numerical or analytical methods. The use of standard procedures of the finite
or boundary element method (Zeng andCakmak, 1984; Tyapin, 1990) results
in theneed for further processingof numerical files used to extract the required
information. Analysis of available analytical approaches, for example, theme-
thod of reduction to a system of integral equations (Glushkov andGlushkova,
1990;Vorovich et al., 1974) has indicated, however, that none of themmakes it
possible to solve the problem in an exact or, at least, in a simple approximate



6 I.V. Andrianov et al.

way. As a rule, one must use complicated numerical procedures and, in addi-
tion, deal with ill-posed problems. These reasons stimulated our choice of the
perturbationprocedure earlier proposed inManevitch et al. (1970),Manevitch
et al. (1979), Manevitch and Pavlenko (1991), Shamrovskii (1997).
The remaining part of the paper is organized as follows. In Section 2 we

present the governing relations. A significant simplification of the input boun-
dary value problem through a singular perturbation technique is proposed in
Section 3. In Section 4, by using a complex variable technique, we compute
an analytical solution to the problem. Finally, we discuss and comment on the
results obtained (Section 5).

2. Statement of the problem

Westudyan elastic half-planewith an embeddedpunch (seeFig.1).According
to the linear theory of elasticity, the equilibrium equations gain the following
form (Muskhelishvili, 1953)

eUxx+GUyy +(eµ+G)Vxy =0
(2.1)

GVxx+eVyy +(eµ+G)Uxy =0

where: e=E/(1−µ2), E is Young’s modulus, G is the shear modulus, µ is
Poisson’s coefficient.

Fig. 1. Half-plane with embedded punch

In Cartesian coordinates, the components of the stress tensor have the
following form (Muskhelishvili, 1953)

σy = e(Vy +µUx) σx = e(Ux+µVy) τxy =G(Vx+Uy) (2.2)

where: σx, σy is the stress in the x, y direction, respectively and τxy is the
shear stress.



Asymptotic study of elastic half-plane with embedded punch 7

Now let us consider an interaction between the punch and the elastic half-
plane, assuming that the punch is rigidly coupled with the half-plane. Owing
to the axial symmetry with respect to the line x = a, we can restrict our
considerations to the zone x¬ a. The conditions of symmetry give

U =0
τxy =0

}

at x= a ∧ y> 0 (2.3)

The boundaries of the half-plane (y =−b, x < 0 and x > 2a) are free from
loading, hence

σy = τxy =0 at

{

x< 0 ∧ y=−b
x> 2a ∧ y=−b (2.4)

The conditions of the punch and the halfplane contact follow

U =0
V = d

}

at

{

x=0 ∧ − b< y¬ 0
0<x<a ∧ y=0 (2.5)

where d is the displacement of the punch.
Equation (2.1) and conditions (2.2)-(2.4) give us a biharmonic mixed bo-

undary value problem. Since the exact solution to this problem is unknown,
we are going to present how the asymptotic technique can be used.

3. Reduction to the Laplace equations

Muskhelishvili (1953) successfully solved the plane elasticity problem for an
isotropic body through the application of complex analysis. Meanwhile, the
transition to the anisotropic case generally involves much higher complexity
(Lekhnitskii, 1968; Ting, 1996). For a slight deviation from the isotropic case,
it is possible to introduce a small parameter γ=(Ba−Bi)/Bi ≪ 1, where Ba
and Bi are corresponding properties of the anisotropic and isotropic media,
respectively. Then an asymptotic solution (with respect to γ) can be develo-
ped (Lekhnitskii, 1968). On the contrary, for a strong anisotropy, one can take
into account the smallness of the parameter 1/γ. It is obvious that loading in
the 0X (0Y ) direction causes mainly displacement U (V ). These reasonable
approximations have been used for a long time in the aircraft (Kuhn, 1956)
and rocket (Balabukh et al., 1969) engineering. Furthermore, they were suc-
cessfully applied to the theory of composite and nonhomogeneous materials
(Everstine and Pipkin, 1971, 1973; Spencer, 1974; Christensen, 1979; Kosmo-
damianskii, 1975, 1976).Most of the authors referred above used only the zero



8 I.V. Andrianov et al.

order approximation of the procedure. However, further development of this
partially empirical engineering approach was restricted by the evident draw-
backs: boundary conditions were not satisfied, the choice of an appropriate
approximationwas not unique and clear, therewas a lack of error estimations,
etc.
Beginning from the paper by Manevitch et al. (1970), a special asympto-

tic technique using expansions with respect to γ was developed. A singular
character of the asymptotic solution was detected, and the input biharmo-
nic equation was reduced to two Laplace equations. It gave possibility to use
the theory of potential. In Manevitch et al. (1979), Manevitch and Pavlenko
(1991), Shamrovskii (1997), higher order approximations were derived, and it
was also shown that even in the isotropic case (when the small parameter has
themaximumvalue) the error of the first approximation is rather low.Mathe-
matical foundations of the described asymptotic technique were also studied
by Bogan (1983, 1987).
Now let us briefly describe the asymptotic procedure used (for technical

detailswe refer the readers to references:Awrejcewicz et al. (1998),Bauer et al.
(1994), Shamrovskii (1979)).We introducedimensionless parameters ε=G/e,
µ= ε−1µ. Then input equations (2.1) can be rewritten as follows

Uxx+εUyy +ε(µ+1)Vxy =0
(3.1)

εVxx+Vyy +ε(µ+1)Uxy =0

As soon as ε< 1, µ≈ 1, we can use the parameter ε for asymptotic splitting
of system (3.1). We pose expansions for the functions V ,U of the forms

U =u0+εu1+ε
2u2+ . . .

(3.2)

V = v0+εv1+ε
2v2+ . . .

The derivatives of the displacements are estimated as follows

∂(U;V )

∂x
≈ εα1(U;V ) ∂(U;V )

∂y
≈ εα2(U;V ) (3.3)

The order of the function U with respect to the function V can be estimated
in the following manner

U ≈ εα3V (3.4)
Substituting formulas (3.2) and (3.3),(3.4) into equations (3.1), and compa-
ring the coefficients, one may conclude that asymptotics strongly depend on
the parameters αk. Therefore, all possible values of αk are sought, when the
asymptotics (ε → 0) have both mathematical (well-posedness) and physical



Asymptotic study of elastic half-plane with embedded punch 9

meanings. It is remarkable that by this simple (but laborious) procedure we
obtain only two systems which are analyzed below:
— α1 =0.5, α2 =0, α3 =1.5

u
(1)
0xx+εu

(1)
0yy =0 v

(1)
0y =−ε(µ+1)u

(1)
0x (3.5)

— α1 =−0.5, α2 =0, α3 =1.5

εv
(2)
0xx+v

(2)
0yy =0 u

(2)
0x =−ε(µ+1)v

(2)
0y (3.6)

Analysis of stress relations (2.2) leads to the following results. For the first
and second type of stress-strain state (3.5) and (3.6), the main stresses read

σ
(1)
0x = eu

(1)
0x τ

(1)
0xy =Gu

(1)
0y (3.7)

σ
(2)
0y = ev

(2)
0y τ

(2)
0xy =Gv

(2)
0x

Simplified equations (3.5), (3.7)1,2 and (3.6), (3.7)3,4 describe all possible
stress-strain states. Links between these states can be established after split-
ting the input boundary conditions, see Awrejcewicz et al. (1998), Bauer et
al. (1994), Shamrovskii (1979). Equations (3.5)1 and (3.6)1 can be reduced to
Laplace equations due to a simple affine transformation of the independent
variables (see below). So, we can use the highly developed theory of complex
variables.

4. Analytical solution

Werestrict our investigation only to the function v
(2)
0 , since thepunchpressure

can be expressed by it. For equations (3.6)1 and (3.7)2, the affine transforma-

tion x1 = x
√
ε is used. Let us denote v

(2)
0 ≡ v, σ

(2)
0y ≡ σy, τ

(2)
oxy ≡ τxy. Then

one obtains the following Laplace equation

vx1x1 +vyy =0 (4.1)

and the expressions for the stresses

σy = evy τxy =
√
Gevx (4.2)

Boundary conditions for equation (4.1) can be written as follows

vy=0 at x1 < 0 ∧ y=−b
vx =0 at x= a1 ∧ 0<y<∞

v= d at

{

x1 =0 ∧ − b< y< 0
0<x1 <a1 ∧ y=0

(4.3)

where a1 = a
√
ε.



10 I.V. Andrianov et al.

We replace boundary conditions (4.3)3 by the expressions

vy =0 at x1 =0 ∧ − b< y< 0
vx =0 at 0<x1 <a1 ∧ y=0

(4.4)

Let us introduce a complex variable z = x1 +iy and an analytical function
Φ(z) = vx − ivy. One can write the following expressions for the boundary
values of the complex variable function Φ

ImΦ(z)= 0 at (x1 < 0,y=−b)∪ (x1 =0,−b< y< 0)

ReΦ(z)= 0 at (0<x1 <a1,y=0)∪ (x1 = a1,0<y<∞)

Φ(z)→ 0 when |z|→∞

(4.5)

Now wemap our governing area (see Fig.1) onto the upper half-plane by the
Schwarz-Christoffel transformation (Walker, 1964)

z= iC

ξ
∫

0

√

ϕ

(ϕ−1)(ϕ+B)
dϕ (4.6)

where B,C are constants.
The correspondence of the points can be written as follows:

z ξ

−∞− ib −∞− i0
−0− ib −B+i0
0+i0 0+i0

a1+i0 1+i0

a1+i∞ ∞+i0

The boundary values for Φ(z(ξ)) read

ImΦ(z(ξ))= 0 at −∞<ξ< 0

ReΦ(z(ξ))= 0 at 0<ξ<∞)

Φ(z(ξ))→ 0 when |ξ|→ 0

(4.7)

The solution to boundary value problem (4.7) has the form

Φ(z(ξ))=
Ai√
ξ

(4.8)

where A is a constant.



Asymptotic study of elastic half-plane with embedded punch 11

For the shear stress σxy, one obtains

τxy =
√
Gevx1 =−

√
GeReΦ(z)=

√
GeA√
−ξ

(4.9)

at x1 =0, −b< y< 0, where

y=−
2C

k′
[E(δ,k)−k′2F(δ,k)]+2C

√

ξ(1− ξ)′
ξ+B

−β< ξ< 0 δ=arcsin
√

B− ξ
1+ ξ

(4.10)

k=

√

B

1+B
k′ =

√

1−k2 =
1√
1+B

The stress σy has the form

σy = evy =−eImΦ(z)=−
eA√
ξ

at 0<x1 <a1

x1 =
2C

k
[E(δ′,k′)−k2F(δ′,k′)]−2C

√

ξ(1− ξ)
ξ+B

0<ξ< 1

(4.11)
where

δ′ =arcsin

√

ξ(1+B)

ξ+B

and F(·, ·),E(·, ·) are elliptic integrals of thefirstandsecondkind, respectively.
Schwarz-Christoffel integral (4.6) is calculated through formulas (3.141.9)-

(3.141.11) taken from the handbook by Gradshteyn and Ryzhik (1965).
For the determination of the unknown constants A,B,C, one can use the

following conditions:
— the equilibrium condition

1

2
P =

0
∫

−b

τxy dy−
a
∫

0

σy dx=G
[

0
∫

−b

ReΦ(z) dy+

a1
∫

0

ImΦ(z) dx1
]

=

(4.12)

=GAC

1
∫

B

dξ
√

(1− ξ)(ξ+B)
=πAGC

—conditions of the x1 and y correspondence

a1 =x1
∣

∣

ξ=1
=
2C

k′

[

E
(π

2
,k′
)

−k2F
(π

2
,k′
)]

(4.13)

−b= y
∣

∣

ξ=−B
=
−2C
k

[

E
(π

2
,k
)

−k′2F
(π

2
,k
)]



12 I.V. Andrianov et al.

and, therefore, one gets

A=
1
2
P

πGC
C =

bk′

2
[

E
(

π
2
,k
)

−k′2F
(

π
2
,k
)] (4.14)

Equations (4.13) yield a transcendental equation for the constant B

b

a1
=
E
(

π
2
,k
)

−k′2F
(

π
2
,k
)

E
(

π
2
,k′
)

−k2F
(

π
2
,k
) (4.15)

Since (4.15) can be solved numerically by application of a routine procedure,
formulas (4.9)-(4.11) and (4.14) yield an analytical solution.

5. Conclusion

In this paper, a novel approximate solution to the contact problem for the
elastic half-planewithanembeddedpunch isproposed.Althoughourapproach
is based on the approximate method, it seems that the obtained accuracy is
acceptable in the engineeringpractice.Thisquestion is going tobe investigated
numerically in our future research.

References

1. Awrejcewicz J., Andrianov I.V.,Manevitch L.I., 1998,Asymptotic Ap-
proaches in Nonlinear Dynamics: New Trends and Applications, Heidelberg,
Springer-Verlag

2. Balabukh L.I., Kolesnikov K.S., Zarubin V.S., Alfutov N.A., Usy-
ukinB.I.,ChizhovV.F., 1969,Principles of StructuralMechanics of Rockets,
Moscow, Viszhaya Shkola [in Russian]

3. BauerS.M.,Filippov S.B., SmirnovA.L.,TovstikP.E., 1994,Asympto-
ticmethods inmechanicswithapplications to thinplates andshells.Asymptotic
methods in mechanics,CRMP Proceedings and Lectures Notes, 3, 3-140

4. Bogan Yu.A., 1983, Singular perturbed boundary value problem in plane
theory of elasticity,Dinamika Sploshnoi Sredi, 61, 13-24 [in Russian]

5. Bogan Yu.A., 1987, A certain class of singular perturbed boundary value
problem in two-dimensional theory of elasticity, J. Appl. Mech. Tech. Phys.,
28, 2, 138-143



Asymptotic study of elastic half-plane with embedded punch 13

6. Christensen R.M., 1979,Mechanics of Composite Materials, NewYork,Wi-
ley Interscience

7. EverstineG.C., PipkinA.C., 1971, Stress channeling in transversely isotro-
pic elastic composites, ZAMP, 22, 5, 825-834

8. Everstine G.C., Pipkin A.C., 1973, Boundary layers in fiberreinforcedma-
terials, J. Appl. Mech., 40, 518-522

9. GlushkovE.V.,GlushkovaN.V., 1990,Determination of the dynamic con-
tact stiffness of an elastic layer,PMM, J. Appl. Math. Mech., 54, 3, 393-397

10. Gradshteyn I.S.,Ryzhik I.M., 1965,Table of Integrals, Series andProducts,
New York-San Francisco-London, Academic Press

11. Kosmodamianskii A.S., 1975,Plane Problems of the Theory of Elasticity for
Plates with Holes, Cut-outs, and Bulges, Kiev, Visha Schkola [in Russian]

12. Kosmodamianskii A.S., 1976, Stress State of Anisotropic Bodies with Holes
or Cavities, Kiev-Donezk, Visha Schkola [in Russian]

13. Kuhn P., 1956, Stresses in Aircraft Shell Structures, New York,McGraw-Hill

14. Lekhnitskii S.G., 1968, Anisotropic Plates, 2 ed., New York, Gordon and
Breach

15. Manevitch L.I., Pavlenko A.V., 1991, Asymptotic Methods in Microme-
chanics of Composite Materials, Kiev, Naukova Dumka [in Russian]

16. Manevitch L.I., Pavlenko A.V., Koblik S.G., 1979,Asymptotic Methods
in the Theory of Elasticity of Orthotropic Body, Kiev-Donezk, Visha Schkola
[in Russian]

17. Manevitch L.I., PavlenkoA.V., Shamrovskii A.D., 1970,Application of
the group theory methods to the dynamical problems for orthotropic plates,
Proceedings of the 7th All-Union Conference on Plates and Shells, Dneprope-

trovsk, 1969, Moscow, Nauka, 408-412 [in Russian]

18. MuskhelishviliN.I., 1953,SomeBasic Problems of theMathematical Theory
of Elasticity, Groningen, Noordhoff

19. Shamrovskii A.D., 1997, Asymptotic-Group Analysis of Differential Equ-
ations in the Theory of Elasticity, Zaporozhie, State Engineering Academy [in
Russian]

20. ShamrovskiiA.D., 1979,Asymptotic integrationof static equation of the the-
ory of elasticity in Cartesian coordinates with automated search of integration
parameters,PMM, J. Appl. Math. Mech., 43, 5, 925-934

21. Spencer A.J.M., 1974, Boundary layers in highly anisotropic plane elasticity,
Int. J. Solids and Struct., 10, 1103-1123

22. Ting T.C.T., 1996,Anisotropic Elasticity: Theory and Applications, Oxford,
Oxford University Press



14 I.V. Andrianov et al.

23. Tyapin A.G., 1990, Influence of foundation depth location on reaction of con-
struction under seismic excitation, Structural Mechanics and Calculation of
Structures, 4, 32-37 [in Russian]

24. Vorovich I.I., Alexandrov V.M., Babeshko V.A., 1974, Non-classical
Mixed Boundary Value Problem of the Theory of Elasticity, Moscow,Nauka [in
Russian]

25. Walker M., 1964, The Schwarz-Christoffel Transformation and its Applica-
tions – a Simple Exposition, NewYork, Dower

26. Zeng X., Cakmak A.S., 1984, Dynamic structure-soil interaction under the
action of vertical ground motion. Part I: The axisymmetric problem, Soil Dy-
namics and Earthquake Engineering, 3, 4, 200-210

Analiza asymptotyczna zagadnienia kontaktowego stempla

i półprzestrzeni sprężystej

Streszczenie

W pracy rozważono zagadnienie kontaktowe związane z oddziaływaniem sztyw-
nego stempla i półprzestrzeni sprężystej. Analizowany problem może np. modelować
fundamentybudowli usytuowanewpodatnymgruncie. Sformułowanowpostaci anali-
tycznejpolenaprężeńprzyużyciumetodyperturbacji osobliwych i techniki zmiennych
zespolonych.

Manuscript received August 3, 2006; accepted for print October 18, 2006