research article contact problem for an elastic layer on
TRANSCRIPT
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 137427, 14 pageshttp://dx.doi.org/10.1155/2013/137427
Research ArticleContact Problem for an Elastic Layer on an Elastic Half PlaneLoaded by Means of Three Rigid Flat Punches
T. S. Ozsahin and O. TaskJner
Civil Engineering Department, Karadeniz Technical University, 61080 Trabzon, Turkey
Correspondence should be addressed to T. S. Ozsahin; [email protected]
Received 21 December 2012; Revised 27 February 2013; Accepted 13 March 2013
Academic Editor: Safa Bozkurt Coskun
Copyright ยฉ 2013 T. S. Ozsahin and O. Taskฤฑner.This is an open access article distributed under theCreative CommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.
The frictionless contact problem for an elastic layer resting on an elastic half plane is considered. The problem is solved by usingthe theory of elasticity and integral transformation technique. The compressive loads P and Q (per unit thickness in ๐ง direction)are applied to the layer through three rigid flat punches. The elastic layer is also subjected to uniform vertical body force due toeffect of gravity. The contact along the interface between elastic layer and half plane is continuous, if the value of the load factor, ๐,is less than a critical value, ๐cr. In this case, initial separation loads, ๐cr and initial separation points, ๐ฅcr are determined. Also therequired distance between the punches to avoid any separation between the punches and the elastic layer is studied and the limitdistance between punches that ends interaction of punches is investigated for various dimensionless quantities. However, if tensiletractions are not allowed on the interface, for ๐ > ๐cr the layer separates from the interface along a certain finite region. Numericalresults for distance determining the separation area, vertical displacement in the separation zone, contact stress distribution alongthe interface between elastic layer and half plane are given for this discontinuous contact case.
1. Introduction
Contact between deformable bodies abounds in industryand everyday life. Because of the industrial importance ofthe physical processes that take place during contact, aconsiderable effort has beenmade in theirmodeling, analysis,and numerical simulations.
The range of application in contact mechanics startswith problems like foundations in civil engineering, wherethe lift off the foundation from soil due to eccentric forcesacting on a building, railway ballasts, foundation grillages,continuous foundation beams, runaways, liquid tanks restingon the ground, and grain silos is considered. Furthermore,foundation including piles as supporting members or thedriving of piles into the soil is of interest. Also classicalbearing problem of steel constructions and the connectionof structural members by bolts or screws are areas in whichcontact analysis enters the design process in civil engineering[1].
A complete analysis of the interaction problem for elasticbodies generally requires the determination of stresses andstrain within the individual bodies in contact, together with
information regarding the distribution of displacements andstresses at the contact regions.
The contact problem for an elastic layer has attractedconsiderable attention in the past because of its possibleapplication to a variety of structures of practical interest.The layer usually rests on a foundation which may be eitherelastic or rigid and the body force due to gravity may beneglected [2โ27] or effect of gravitymay be taken into account[28โ35]. Studies regarding the frictionless contact along theinterface may be found in [2โ15]. Also contact region mayexhibit frictional characteristics, giving rise to normal andshear traction at the contact surface [16โ27].
While initial contact is determined by the geometricfeatures of the bodies, the extent of the contact generallychanges not only by the particular loads applied to the bodiesbut also with the elastic constants of the materials. Due tothe bending of the layer under local compressive loads, inthe absence of gravity effects, the contact area would decreaseto a finite size which is independent of the magnitude ofthe applied load [2โ7]. If the effect of gravity was taken intoaccount, the normal stress along the layer subspace interfacewill be compressive and the contact is maintained through
2 Mathematical Problems in Engineering
2
1 I IIII๐๐๐๐๐๐
๐๐
๐๐ ๐
๐ฆ
๐ฅ
โ
๐๐
๐ 1, ๐1, ๏ฟฝ1
๐ 2, ๐2, ๏ฟฝ2
Figure 1: Elastic layer resting on an elastic half plane loaded bymeans of three rigid flat punches.
the frictionless interface unless the compressive applied loadexceeds a certain critical value. When the magnitude of thecompressive external load exceeds a certain value, a separa-tion will take place between the layer and the foundation.Thelength of separation region along the interface is unknown,and the problem becomes a discontinuous contact problem[28โ35].
Interaction between an elastic medium and a rigid punchforms another group of contact problems. Rigid punchesmay be structural elements such as foundations, beams, andplates of finite or infinite extent resting on idealized linearlydeformable elastic media. Here, the shape of the contactregion may be known a priori and remains constant, or con-tact region may be changed due to the shape of punch profile[8โ27]. The problem of flat-ended rigid punch has importantapplications in soil mechanics, particularly in estimating thesafety of foundations. The application of the three punchesfor an elastic layer resting on an elastic half plane in soilmechanics is obvious; for example, the punches can be takenas foundations placed on layered soil. When the foundationsare placed on soil, there is a possibility of pressure isobarsof adjacent foundations overlapping each other. The soil ishighly stressed in the zones of overlapping, or the differenceof settlement between two adjacent foundations, commonlyreferred to as differential settlement may cause damage tothe structure. It is possible to avoid overlapping of pressuresor differential settlement by installing the foundations atconsiderable distance apart from each other.
In this study, contact problem of the three punches foran elastic layer resting on an elastic half plane is consideredaccording to the theory of elasticity with integral transfor-mation technique. The compressive loads ๐ and ๐ (per unitthickness in ๐ง direction) are applied to the layer throughthree rigid flat punches. The width of midmost punch canbe different from the other two punches and thickness of thelayer is constant, โ. The layer is subjected to homogeneousvertical body force due to gravity, ๐
1๐. All surfaces are
frictionless. The layer remains in contact with the elastichalf plane where the magnitude of the load factor ๐ is lessthan a critical value, ๐cr (๐ = ๐/๐
1๐โ2). If ๐ > ๐cr,
the contact is discontinuous and a separation takes placebetween the layer and the half plane. A numerical integration
procedure is performed for the solution of the problems, anddifferent parameters are researched for various dimensionlessquantities for both continuous and discontinuous contactcases. Finally, numerical results are analyzed and conclusionsare drawn.
2. General Expressions for Stresses andDisplacements
Consider a frictionless elastic layer of thickness h lying on anelastic half plane. The geometry and coordinate system areshown in Figure 1. The governing equations are
๐๐โ2๐ข๐+
2๐๐
(๐ ๐โ 1)
๐
๐๐ฅ
(
๐๐ข๐
๐๐ฅ
+
๐V๐
๐๐ฆ
) = 0, (1a)
๐๐โ2V๐+
2๐๐
(๐ ๐โ 1)
๐
๐๐ฆ
(
๐๐ข๐
๐๐ฅ
+
๐V๐
๐๐ฆ
) = ๐๐๐,
(๐ = 1, 2)
(1b)
where ๐๐๐ is the intensity of the body force acting vertically in
which ๐๐and ๐ are mass density and gravity acceleration. ๐ข
๐
and V๐are the๐ฅ and๐ฆ components of the displacement vector,
๐๐and ๐ ๐represent shear modulus and elastic constant of the
layer and the half plane, respectively. ๐ ๐= (3 โ ๐พ
๐)/(1 + ๐พ
๐)
for plane stress and ๐ ๐= (3 โ 4๐พ
๐) for plane strain. ๐พ
๐is the
Poisson ratio (๐ = 1, 2). Subscript 1 indicates the elastic layerand subscript 2 indicates the elastic half plane.
๐ข๐and V
๐represent the displacements for the case in
which gravity forces are considered. ๐ขโand V
โare the
displacements when the gravity forces are ignored, and totalfield of displacements may be expressed as
๐ข = ๐ข๐+ ๐ขโ, (2a)
V = V๐+ Vโ. (2b)
Observing that๐ฅ = 0 is a plane of symmetry, it is sufficientto consider the problem in the region 0 โค ๐ฅ โค โ only. Usingthe symmetry consideration, the following expressions maybe written:
๐ข1(๐ฅ, ๐ฆ) = โ๐ข
1(โ๐ฅ, ๐ฆ) , (3a)
V1(๐ฅ, ๐ฆ) = V
1(โ๐ฅ, ๐ฆ) , (3b)
๐ข1(๐ฅ, ๐ฆ) =
2
๐
โซ
โ
0
๐1(๐ผ, ๐ฆ) sin (๐ผ๐ฅ) ๐๐ผ, (3c)
V1(๐ฅ, ๐ฆ) =
2
๐
โซ
โ
0
ฮจ1(๐ผ, ๐ฆ) cos (๐ผ๐ฅ) ๐๐ผ, (3d)
where ๐1and ฮจ
1functions are inverse Fourier transforms
of ๐ข1and V1respectively. Taking necessary derivatives of (3c)
and (3d), substituting them into (1a) and (1b), and solving the
Mathematical Problems in Engineering 3
second-order differential equations, the following equationsmay be obtained for displacements:
๐ข1โ(๐ฅ, ๐ฆ) =
2
๐
โซ
โ
0
[(๐ด + ๐ต๐ฆ) ๐โ๐ผ๐ฆ
+ (๐ถ + ๐ท๐ฆ) ๐๐ผ๐ฆ] sin (๐ผ๐ฅ) ๐๐ผ,
(4a)
V1โ(๐ฅ, ๐ฆ) =
2
๐
โซ
โ
0
[[๐ด + (
๐ 1
๐ผ
๐ + ๐ฆ)๐ต] ๐โ๐ผ๐ฆ
+ [โ๐ถ + (
๐ 1
๐ผ
โ ๐ฆ)๐ท] ๐๐ผ๐ฆ] cos (๐ผ๐ฅ) ๐๐ผ.
(4b)
Using Hookeโs law and (4a) and (4b), stress componentswhich do not include the gravity force may be expressed asfollows:
๐๐ฅ1โ
(๐ฅ, ๐ฆ) =
4๐1
๐
โซ
โ
0
[[๐ผ (๐ด + ๐ต๐ฆ) โ (
3 โ ๐ 1
2
)๐ต] ๐โ๐ผ๐ฆ
+ [๐ผ (๐ถ + ๐ท๐ฆ) + (
3 โ ๐ ๐
2
)๐ท] ๐๐ผ๐ฆ]
ร cos (๐ผ๐ฅ) ๐๐ผ,(5a)
๐๐ฆ1โ
(๐ฅ, ๐ฆ) =
4๐1
๐
โซ
โ
0
[โ [๐ผ (๐ด + ๐ต๐ฆ) + (
๐ 1+ 1
2
)๐ต] ๐โ๐ผ๐ฆ
+[โ๐ผ (๐ถ + ๐ท๐ฆ) +(
๐ 1+ 1
2
)๐ท] ๐๐ผ๐ฆ]
ร cos (๐ผ๐ฅ) ๐๐ผ,(5b)
๐๐ฅ๐ฆ1โ
(๐ฅ, ๐ฆ) =
4๐1
๐
โซ
โ
0
[โ [๐ผ (๐ด + ๐ต๐ฆ) + (
๐ ๐โ 1
2
)๐ต] ๐โ๐ผ๐ฆ
+ [๐ผ (๐ถ + ๐ท๐ฆ) โ (
๐ ๐โ 1
2
)๐ท] ๐๐ผ๐ฆ]
ร sin (๐ผ๐ฅ) ๐๐ผ.(5c)
For the case in which gravity force exist, particular partof the displacement components corresponding to ๐
1๐, the
following expressions are obtained, that is, special solution ofthe Navier equations for a layer with a height โ:
๐ข1๐=
3 โ ๐ 1
8๐1
๐1๐โ
2
๐ฅ, (6a)
V1๐=
๐1๐๐ฆ
2๐1
[
(๐ 1โ 1)
(๐ 1+ 1)
(๐ฆ โ โ) โ
(๐ 1+ 1)
8
โ] , (6b)
๐๐ฆ1๐
= ๐1๐ (๐ฆ โ โ) , (6c)
๐๐ฅ1๐
= ๐1๐(๐ฆ โ
โ
2
)
(1 + ๐ 2)
(1 + ๐ 2) + 2๐
2(1 + ๐
1)
, (6d)
๐๐ฅ๐ฆ1๐
= 0. (6e)
Considering the orthogonal axes shown in Figure 1,displacements will be zero for ๐ฆ = โโ, and if ๐
2, ]2are the
elastic constants of the half plane, then the homogenous fieldof displacements and stresses of the elastic half plane may beobtained as
๐ข2โ(๐ฅ, ๐ฆ) =
2
๐
โซ
โ
0
[(๐ธ + ๐น๐ฆ) ๐๐ผ๐ฆ] sin (๐ผ๐ฅ) ๐๐ผ, (7a)
V2โ(๐ฅ, ๐ฆ) =
2
๐
โซ
โ
0
[[โ๐ธ + (
๐ 2
๐ผ
โ ๐ฆ)๐น] ๐๐ผ๐ฆ] cos (๐ผ๐ฅ) ๐๐ผ,
(7b)
๐๐ฅ2โ
(๐ฅ, ๐ฆ) =
4๐2
๐
โซ
โ
0
[[๐ผ (๐ธ + ๐น๐ฆ) + (
3 โ ๐ 2
2
)๐น] ๐๐ผ๐ฆ]
ร cos (๐ผ๐ฅ) ๐๐ผ,(7c)
๐๐ฆ2โ
(๐ฅ, ๐ฆ) =
4๐2
๐
โซ
โ
0
[[ โ ๐ผ (๐ธ + ๐น๐ฆ)
+ (
๐ 2+ 1
2
)๐น] ๐๐ผ๐ฆ] cos (๐ผ๐ฅ) ๐๐ผ,
(7d)
๐๐ฅ๐ฆ2โ
(๐ฅ, ๐ฆ) =
4๐2
๐
โซ
โ
0
[[๐ผ (๐ธ + ๐น๐ฆ) โ (
๐ 2โ 1
2
)๐น] ๐๐ผ๐ฆ]
ร sin (๐ผ๐ฅ) ๐๐ผ,(7e)
subscript 2 indicates the elastic half plane. Note that the bodyforce acting in the foundation is neglected since it does notdisturb the contact pressure distribution. A, B, C, D, E, and Fare the unknown constants, which will be determined fromthe boundary and continuity conditions at ๐ฆ = 0 and ๐ฆ = โ.
3. Case of Continuous Contact
An elastic layer with a height of โ resting on an elastic halfplane, shown in Figure 1, is analyzed for unit thickness in๐ง direction. Widths of punches at both sides are similar,(๐โ๐)/โ, and each of these punches transmits a concentratedload of ๐ to the elastic layer. The width of the midmostpunch is different, 2๐/โ and it is subjected to a concen-trated load, ๐. All surfaces are frictionless. Particularly, theinitial separation load (๐cr) and point (๐ฅcr) where the layerseparated from the elastic half plane and the variation ofthe stress distribution between elastic layer and elastic halfplane is examined depending on material properties, widthof punches, and magnitude of the external loads, ๐ and ๐.Due to the different settlement of punches, a separation takesplace between punch I and elastic layer, if punches are closeenough. Therefore, the critical distance between the punchesindicating the initiation of separation between the punch Iand the elastic layer is researched and also limit distancebetween the punches where the interaction of punches endsis investigated.
If load factor (๐) is sufficiently small, then the contactalong the layer-subspace, ๐ฆ = 0, 0 < ๐ฅ < โ, will be
4 Mathematical Problems in Engineering
continuous, andA, B, C, D, E, and Fmust be determined fromthe following boundary and continuity conditions:
๐๐ฆ1(๐ฅ, โ) =
{{
{{
{
โ๐ (๐ฅ) , 0 < ๐ฅ < ๐
โ๐ (๐ฅ) , ๐ < ๐ฅ < ๐
0, ๐ < ๐ฅ < ๐, ๐ < ๐ฅ < โ,
(8a)
๐๐ฅ๐ฆ1(๐ฅ, โ) = 0, 0 < ๐ฅ < โ, (8b)
๐๐ฅ๐ฆ1(๐ฅ, 0) = 0, 0 < ๐ฅ < โ, (8c)
๐๐ฅ๐ฆ2(๐ฅ, 0) = 0, 0 < ๐ฅ < โ, (8d)
๐๐ฆ1(๐ฅ, 0) = ๐
๐ฆ2(๐ฅ, 0) , 0 < ๐ฅ < โ, (8e)
๐
๐๐ฅ
[V2(๐ฅ, 0) โ V
1(๐ฅ, 0)] = 0, 0 < ๐ฅ < โ, (8f)
๐
๐๐ฅ
[V1(๐ฅ, โ)] = 0, 0 < ๐ฅ < ๐, (8g)
๐
๐๐ฅ
[V1(๐ฅ, โ)] = 0, ๐ < ๐ฅ < ๐, (8h)
in which subscripts 1 and 2 indicate relation to the elasticlayer and the elastic half plane, respectively. ๐(๐ฅ) is theunknown contact pressure under punch I and ๐(๐ฅ) is theunknown contact pressure under punch II, which have notbeen determined yet. If a separation occurs between theelastic layer and elastic half plane, this will give rise to adiscontinuous contact position and the following results forformer solution will no longer be valid and new solution willbe attained for the latter case.
Equilibrium conditions of the problem may be expressedas
โซ
๐
0
๐ (๐ฅ) ๐๐ฅ =
๐
2
, (9a)
โซ
๐
๐
๐ (๐ฅ) ๐๐ฅ = ๐. (9b)
Displacement and stress expressions (4a), (4b), (5a)โ(5c), (6a)โ(6e), and (7a)โ(7e) are substituted into boundaryconditions (8a)โ(8f), and unknown constants A, B, C, D, E,and F are determined in terms of unknown functions ๐(๐ฅ)and ๐(๐ฅ). By making use of (8g) and (8h), after some simplemanipulations, onemay obtain the following singular integralequations for ๐(๐ฅ) and ๐(๐ฅ) [36, 37]:
โ
1
๐๐1
โซ
๐
0
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
+
1
๐ก โ ๐ฅ
)]๐ (๐ก) ๐๐ก
โ
1
๐๐1
โซ
๐
๐
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
+
1
๐ก โ ๐ฅ
)]
ร ๐ (๐ก) ๐๐ก = 0, 0 < ๐ฅ < ๐,
(10a)
โ
1
๐๐1
โซ
๐
0
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
+
1
๐ก โ ๐ฅ
)]๐ (๐ก) ๐๐ก
โ
1
๐๐1
โซ
๐
๐
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
+
1
๐ก โ ๐ฅ
)]
ร ๐ (๐ก) ๐๐ก = 0, ๐ < ๐ฅ < ๐,
(10b)
where
๐1(๐ฅ, ๐ก) = โซ
โ
0
{{
๐ผ3
8 (1 + ๐ 1)
ร [[(1 + ๐ 2) +
๐2
๐1(1 + ๐
1)
]
+ ๐โ2๐ผโ
[4๐ผโ (1 + ๐ 2) โ
2๐2
๐1(1 + ๐
1)
]
+ ๐โ4๐ผโ
[โ1 โ ๐ 2+
๐2
๐1(1 + ๐
1)
]]}(ฮ)โ1
โ
(1 + ๐ 1)
4
}
ร {sin๐ผ (๐ก + ๐ฅ) โ sin๐ผ (๐ก โ ๐ฅ)}๐๐ผ,(11)
in which
ฮ = ๐ผ3(2{(โ1 โ ๐
2โ
๐2
๐1(1 + ๐
1)
)
+ ๐โ4๐ผโ
[โ1 โ ๐ 2+
๐2
๐1(1 + ๐
1)
]
+ ๐โ2๐ผโ
[ (1 + ๐ 2) (2 + 4๐ผ
2โ2)
+
4๐ผโ๐2
๐1(โ1 โ ๐
1)
]})
โ1
.
(12)
If evaluated values of A, B, C, and D in terms of ๐(๐ฅ) and๐(๐ฅ) are substituted into (5b), the expression of the contactstress between elastic layer and half plane may be obtained as
๐๐ฆ1(๐ฅ, 0) =โ ๐
1๐โ โ
1
๐
โซ
๐
0
๐2(๐ฅ, ๐ก) ๐ (๐ก) ๐๐ก
โ
1
๐
โซ
๐
๐
๐2(๐ฅ, ๐ก) ๐ (๐ก) ๐๐ก, 0 < ๐ฅ < โ,
(13)
Mathematical Problems in Engineering 5
in which ๐1and ๐ are mass density and gravity acceleration,
respectively, where
๐2(๐ฅ, ๐ก) = โซ
โ
0
{
๐ผ3๐2
๐1(1 + ๐
1)
ร [๐โ3๐ผโ
(โ1 + ๐ผโ) + ๐โ๐ผโ
(1 + ๐ผโ)] } (ฮ)โ1
ร {cos๐ผ (๐ก + ๐ฅ) + cos๐ผ (๐ก โ ๐ฅ)}๐๐ผ.(14)
To simplify the numerical analysis, the following dimen-sionless quantities are introduced:
๐ฅ1= ๐๐1, (15a)
๐ก1= ๐๐ 1, (15b)
๐ฅ2=
๐ โ ๐
2
๐2+
๐ + ๐
2
, (15c)
๐ก2=
๐ โ ๐
2
๐ 2+
๐ + ๐
2
, (15d)
๐1(๐ 1) =
๐ (๐๐ 1)
๐/โ
, (15e)
๐2(๐ 2) =
๐ (((๐ โ ๐) /2) ๐ 2+ (๐ + ๐) /2)
๐/โ
, (15f)
๐ผ = ๐คโ, (15g)
๐ =
๐
๐1๐โ2. (15h)
Substituting from (15a)โ(15h), (9a), (9b) and (10a), (10b) maybe expressed as
โซ
1
0
๐1(๐ 1)
๐
โ
๐๐ 1=
1
2
, (16a)
โซ
1
โ1
๐2(๐ 2)
๐ โ ๐
2โ
๐๐ 2=
๐
๐
, (16b)
โ
1
๐
โซ
1
0
๐1(๐ 1)
๐
โ
๐๐ 1[๐1(๐1, ๐ 1) +
(1 + ๐ 1)
4
ร (
1
๐ (๐ 1+ ๐1)
โ
1
๐ (๐ 1โ ๐1)
)]
โ
1
๐
โซ
1
โ1
๐2(๐ 2)
๐ โ ๐
2โ
๐๐ 2
ร [๐2(๐1, ๐ 2) +
(1 + ๐ 1)
4
ร (
1
(((๐ โ ๐) /2) ๐ 2+ (๐ + ๐) /2) + ๐๐
1
โ
1
(((๐ โ ๐) /2) ๐ 2+(๐ + ๐) /2) โ ๐๐
1
)]= 0,
0 < ๐1< 1,
(16c)
โ
1
๐
โซ
1
0
๐1(๐ 1)
๐
โ
๐๐ 1
ร [๐3(๐2, ๐ 1) +
(1 + ๐ 1)
4
ร (
1
๐๐ 1+ (((๐ โ ๐) /2) ๐
2+ (๐ + ๐) /2)
โ
1
๐๐ 1โ (((๐ โ ๐) /2) ๐
2+ (๐ + ๐) /2)
)]
โ
1
๐
โซ
1
โ1
๐2(๐ 2)
๐ โ ๐
2
๐๐ 2
ร [๐4(๐2, ๐ 2) +
(1 + ๐ 1)
4
ร (
1
((๐ โ ๐) /2) (๐ 2+ ๐2) + (๐ + ๐)
โ
1
((๐ โ ๐) /2) (๐ 2โ ๐2)
)] = 0,
โ 1 < ๐2< 1,
(16d)
๐๐ฆ1(๐ฅ, 0)
๐/โ
= โ
1
๐
โ
1
๐
โซ
1
0
๐5(๐1, ๐ 1) ๐1(๐ 1)
๐
โ
๐๐ 1
โ
1
๐
โซ
1
โ1
๐6(๐2, ๐ 2) ๐2(๐ 2)
๐ โ ๐
2โ
๐๐ 2,
(16e)
where
๐1(๐1, ๐ 1) = ๐1(๐ฅ1, ๐ก1) , (17a)
๐2(๐1, ๐ 2) = ๐1(๐ฅ1, ๐ก2) , (17b)
๐3(๐2, ๐ 1) = ๐1(๐ฅ2, ๐ก1) , (17c)
๐4(๐2, ๐ 2) = ๐1(๐ฅ2, ๐ก2) , (17d)
๐5(๐1, ๐ 1) = ๐2(๐ฅ1, ๐ก1) , (17e)
๐6(๐2, ๐ 2) = ๐2(๐ฅ2, ๐ก2) . (17f)
The index of the integral equations in (16a)โ(16e) is +1;so the functions ๐
1(๐ 1) and ๐
2(๐ 2) may be expressed in the
following forms:
๐1(๐ 1) = ๐บ1(๐ 1) (1 โ ๐
2
1)
โ1/2
, (0 < ๐ 1< 1) , (18a)
๐2(๐ 2) = ๐บ2(๐ 2) (1 โ ๐
2
2)
โ1/2
, (โ1 < ๐ 2< 1) , (18b)
where ๐บ1(๐ 1) is bounded in [0, 1] and ๐บ
2(๐ 2) is bounded in
[โ1, 1].Then using appropriate Gauss-Chebyshev integrationformula given in [36], (16a)โ(16d) are replaced by the follow-ing algebraic equations:
๐
โ
๐=1
๐๐๐
๐
โ
๐บ1(๐ 1๐) =
1
2
, (19a)
6 Mathematical Problems in Engineering
๐
โ
๐=1
๐๐๐
๐ โ ๐
2โ
๐บ2(๐ 2๐) =
๐
๐
, (19b)
โ
๐
โ
๐=1
๐๐๐บ1(๐ 1๐)
๐
โ
[
[
๐1(๐1๐, ๐ 1๐) +
(1 + ๐ 1)
4
ร (
1
๐ (๐ 1๐+ ๐1๐)
โ
1
๐ (๐ 1๐โ ๐1๐)
)]
]
โ
๐
โ
๐=1
๐๐๐บ2(๐ 2๐)
๐ โ ๐
2โ
ร [
[
๐2(๐1๐, ๐ 2๐) +
(1 + ๐ 1)
4
ร (
1
(((๐ โ ๐) /2) ๐ 2๐+ (๐ + ๐) /2) + ๐๐
1๐
โ
1
(((๐ โ ๐) /2) ๐ 2๐+(๐ + ๐) /2) โ ๐๐
1๐
)]
]
= 0,
(๐ = 1, . . . , ๐ โ 1) ,
(19c)
โ
๐
โ
๐=1
๐๐๐บ1(๐ 1๐)
๐
โ
ร [
[
๐3(๐2๐, ๐ 1๐) +
(1 + ๐ 1)
4
ร (
1
๐๐ 1๐+ (((๐ โ ๐) /2) ๐
2๐+ (๐ + ๐) /2)
โ
1
๐๐ 1๐โ (((๐ โ ๐) /2) ๐
2๐+ (๐ + ๐) /2)
)]
]
โ
๐
โ
๐=1
๐๐๐บ2(๐ 2๐)
๐ โ ๐
2โ
ร [
[
๐4(๐2๐, ๐ 2๐) +
(1 + ๐ 1)
4
ร (
1
((๐ โ ๐) /2) (๐ 2๐+ ๐2๐) + (๐ + ๐)
โ
1
((๐ โ ๐) /2) (๐ 2๐โ ๐2๐)
)]
]
= 0,
(๐ = 1, . . . , ๐ โ 1) ,
(19d)
๐1= ๐๐=
1
2๐ โ 2
, ๐๐=
1
๐ โ 1
,
(๐ = 2, . . . , ๐ โ 1) ,
(19e)
๐ 1๐= cos( ๐ โ 1
2๐ โ 1
๐) , ๐ 2๐= cos( ๐ โ 1
๐ โ 1
๐) ,
(๐ = 1, . . . , ๐) ,
(19f)
๐1๐= cos(
2๐ โ 1
4๐ โ 2
๐) , ๐2๐= cos(
2๐ โ 1
2๐ โ 2
๐) ,
(๐ = 1, . . . , ๐ โ 1) .
(19g)
The unknowns๐บ1(๐ 1๐) and๐บ
2(๐ 2๐) (๐ = 1, . . . , ๐) are deter-
mined from the system (19a)โ(19d). By using (18a), (18b), sub-stituting the results into (16e), and using Gauss-Chebyshevintegration formula, the contact stress ๐
๐ฆ1(๐ฅ, 0)โ/๐ is evalu-
ated. It should be observed that the integral equations (16c)and (16d) are valid provided the contact stress ๐
๐ฆ1(๐ฅ, 0)โ/๐
is compressive everywhere; that is, 0 < ๐ฅ < โ. The criticalload factor, ๐cr and the corresponding location of interfaceseparation, ๐ฅcr can be determined through the use of thefollowing condition for various dimensionless quantities:
๐๐ฆ1(๐ฅ, 0)
๐/โ
= 0. (20)
4. Case of Discontinuous Contact
Since the interface cannot carry tensile tractions for ๐ > ๐cr,there will be separation between the elastic layer and theelastic half plane in the neighborhood of ๐ฅ = ๐ฅcr on thecontact plane ๐ฆ = 0, as shown in Figure 1. Assuming thatthe separation region is described by ๐ < ๐ฅ < ๐, ๐ฆ = 0,where ๐ and ๐ are unknowns and functions of ๐, boundaryand continuity conditions for the discontinuous contact caseare defined as follows:
๐๐ฆ1(๐ฅ, โ) =
{{
{{
{
โ๐ (๐ฅ) , 0 < ๐ฅ < ๐
โ๐ (๐ฅ) , ๐ < ๐ฅ < ๐
0, ๐ < ๐ฅ < ๐, ๐ < ๐ฅ < โ,
(21a)
๐๐ฅ๐ฆ1(๐ฅ, โ) = 0, 0 < ๐ฅ < โ, (21b)
๐๐ฅ๐ฆ1(๐ฅ, 0) = 0, 0 < ๐ฅ < โ, (21c)
๐๐ฅ๐ฆ2(๐ฅ, 0) = 0, 0 < ๐ฅ < โ, (21d)
๐๐ฆ1(๐ฅ, 0) = ๐
๐ฆ2(๐ฅ, 0) , 0 < ๐ฅ < โ, (21e)
๐
๐๐ฅ
[V2(๐ฅ, 0) โ V
1(๐ฅ, 0)] = {
๐ (๐ฅ) , ๐ < ๐ฅ < ๐
0, 0 < ๐ฅ < ๐, ๐ < ๐ฅ < โ,
(21f)
๐๐ฆ1(๐ฅ, 0) = ๐
๐ฆ2(๐ฅ, 0) = 0, ๐ < ๐ฅ < ๐, (21g)
Mathematical Problems in Engineering 7
๐
๐๐ฅ
[V1(๐ฅ, โ)] = 0, 0 < ๐ฅ < ๐, (21h)
๐
๐๐ฅ
[V1(๐ฅ, โ)] = 0, ๐ < ๐ฅ < ๐. (21i)
After utilizing the boundary and continuity conditionsdefined in (21a)โ(21f), new values for the constantsA, B, C, D,E, and F which appear in (4a), (4b), (5a)โ(5c), and (7a)โ(7e)may be obtained in terms of new unknown functions ๐(๐ฅ),๐(๐ฅ), and ๐(๐ฅ). Unknown functions are then determinedfrom the conditions (21h)-(21i) which have not yet beensatisfied. These conditions give the following system ofsingular integral equations:
โ
1
๐๐1
โซ
๐
0
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
โ
1
๐ก โ ๐ฅ
)]๐ (๐ก) ๐๐ก
โ
1
๐๐1
โซ
๐
๐
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
โ
1
๐ก โ ๐ฅ
)]๐ (๐ก) ๐๐ก
+
1
๐
โซ
๐
๐
๐2(๐ฅ, ๐ก) ๐ (๐ก) ๐๐ก = 0, 0 < ๐ฅ < ๐,
(22a)
โ
1
๐๐1
โซ
๐
0
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
โ
1
๐ก โ ๐ฅ
)]๐ (๐ก) ๐๐ก
โ
1
๐๐1
โซ
๐
๐
[๐1(๐ฅ, ๐ก) +
(1 + ๐ 1)
4
(
1
๐ก + ๐ฅ
โ
1
๐ก โ ๐ฅ
)]๐ (๐ก) ๐๐ก
+
1
๐
โซ
๐
๐
๐2(๐ฅ, ๐ก) ๐ (๐ก) ๐๐ก = 0, ๐ < ๐ฅ < ๐,
(22b)
1
๐
โซ
๐
0
๐2(๐ฅ, ๐ก) ๐ (๐ก) ๐๐ก +
1
๐
โซ
๐
๐
๐2(๐ฅ, ๐ก) ๐ (๐ก) ๐๐ก
โ
๐1
๐
โซ
๐
๐
[๐3(๐ฅ, ๐ก) โ
4๐2/๐1
(1 + ๐ 2) + ๐2/๐1(1 + ๐
1)
ร (
1
๐ก + ๐ฅ
+
1
๐ก โ ๐ฅ
)]๐ (๐ก) ๐๐ก โ ๐1๐โ = 0,
๐ < ๐ฅ < ๐,
(22c)
where kernels ๐1(๐ฅ, ๐ก) and ๐
2(๐ฅ, ๐ก) are given by (11) and (14)
and
๐3(๐ฅ, ๐ก) = โซ
โ
0
{{โ
2๐ผ3๐2
๐1
{๐โ2๐ผโ
(2 + 4๐ผ2โ2) โ ๐โ4๐ผโ
โ 1}}
ร (ฮ)โ1+
4๐2/๐1
(1 + ๐ 2) + ๐2/๐1(1 + ๐
1)
}
ร {sin๐ผ (๐ก + ๐ฅ) + sin๐ผ (๐ก โ ๐ฅ)} ๐๐ผ,(23)
in which ฮ is given by (12).The index of integral equations (22a) and (22b) is +1. On
the other hand, the index of the singular integral equation(22c) is โ1 due to the physical requirement of smooth contact
at the end points ๐ and ๐ [37]. Thus, in solving the problemthe two conditions which would account for the unknowns ๐and ๐ are the consistency condition of integral equation (22c)and the single-valuedness condition
โซ
๐
๐
๐ (๐ฅ) ๐๐ฅ = 0. (24)
Defining the following dimensionless quantities
๐ฅ3=
๐ โ ๐
2
๐3+
๐ + ๐
2
, ๐ก3=
๐ โ ๐
2
๐ 3+
๐ + ๐
2
, (25a)
๐3(๐ 3) =
๐1๐ (((๐ โ ๐) /2) ๐
3+ (๐ + ๐) /2)
๐/โ
; (25b)
by making use of (15a)โ(15h), the integral equations (22a)โ(22c) may be expressed as follows:
โ
1
๐
โซ
1
0
๐1(๐ 1)
๐
โ
๐๐ 1[๐โ
1(๐1, ๐ 1) +
(1 + ๐ 1)
4
ร (
1
๐ (๐ 1+ ๐1)
โ
1
๐ (๐ 1โ ๐1)
)]
โ
1
๐
โซ
1
โ1
๐2(๐ 2)
๐ โ ๐
2โ
๐๐ 2
ร [๐โ
2(๐1, ๐ 2) +
(1 + ๐ 1)
4
ร (
1
(((๐ โ ๐) /2) ๐ 2+ (๐ + ๐) /2) + ๐๐
1
โ
1
(((๐ โ ๐) /2) ๐ 2+ (๐ + ๐) /2) โ ๐๐
1
)]
+
1
๐
โซ
1
โ1
๐3(๐ 3)๐โ
3(๐1, ๐ 3)
๐ โ ๐
2โ
๐๐ 3= 0,
0 < ๐1< 1,
(26a)
โ
1
๐
โซ
1
โ1
๐1(๐ 1)
๐
โ
๐๐ 1
ร [๐โ
4(๐2, ๐ 1) +
(1 + ๐ 1)
4
ร (
1
๐๐ 1+ (((๐ โ ๐) /2) ๐
2+ (๐ + ๐) /2)
โ
1
๐๐ 1โ (((๐ โ ๐) /2) ๐
2+ (๐ + ๐) /2)
)]
โ
1
๐
โซ
1
โ1
๐2(๐ 2)
๐ โ ๐
2โ
๐๐ 2[๐โ
5(๐2, ๐ 2) +
(1 + ๐ 1)
4
ร (
1
((๐โ๐) /2) (๐ 2+๐2)+(๐+๐)
โ
1
((๐ โ ๐) /2) (๐ 2โ ๐2)
)]
+
1
๐
โซ
1
โ1
๐3(๐ 3)๐โ
6(๐2, ๐ 3)
๐ โ ๐
2โ
๐๐ 3= 0,
โ 1 < ๐2< 1,
(26b)
8 Mathematical Problems in Engineering
โ
1
๐
โซ
1
โ1
๐1(๐ 1)๐โ
7(๐3, ๐ 1)
๐
โ
๐๐ 1
โ
1
๐
โซ
1
โ1
๐2(๐ 2)๐โ
8(๐3, ๐ 2)
๐ โ ๐
2โ
๐๐ 2
โ
1
๐
โซ
1
โ1
๐3(๐ 3)
๐ โ ๐
2โ
๐๐ 3
ร [๐โ
9(๐3, ๐ 3) โ
4๐2/๐1
(1 + ๐ 2) + ๐2/๐1(1 + ๐
1)
ร (
1
((๐ โ ๐) /2) (๐ 3+ ๐3) + (๐ + ๐)
+
1
((๐ โ ๐) /2) (๐ 3โ ๐3)
)] โ
1
๐
= 0,
โ 1 < ๐3< 1,
(26c)
where
๐โ
1(๐1, ๐ 1) = ๐1(๐ฅ1, ๐ก1) , (27a)
๐โ
2(๐1, ๐ 2) = ๐1(๐ฅ1, ๐ก2) , (27b)
๐โ
3(๐1, ๐ 3) = ๐2(๐ฅ1, ๐ก3) , (27c)
๐โ
4(๐2, ๐ 1) = ๐1(๐ฅ2, ๐ก1) , (27d)
๐โ
5(๐2, ๐ 2) = ๐1(๐ฅ2, ๐ก2) , (27e)
๐โ
6(๐2, ๐ 3) = ๐2(๐ฅ2, ๐ก3) , (27f)
๐โ
7(๐3, ๐ 1) = ๐2(๐ฅ3, ๐ก1) , (27g)
๐โ
8(๐3, ๐ 2) = ๐2(๐ฅ3, ๐ก2) , (27h)
๐โ
9(๐3, ๐ 3) = ๐3(๐ฅ3, ๐ก3) . (27i)
Similar to (16a), (16b), additional condition (24) may beexpressed as
โซ
1
โ1
๐3(๐ 3) ๐๐ 3= 0. (28)
To solve the system of integral equations, it is found tobe more convenient to assume that (26c) as well as (26a) and(26b) has an index +1 [29]; consequently, the function ๐
๐(๐ ๐)
(๐ = 1, . . . , 3)may be expressed in the form
๐1(๐ 1) = ๐บ1(๐ 1) (1 โ ๐
2
1)
โ1/2
, 0 < ๐ 1< 1, (29a)
๐๐(๐ ๐) = ๐บ๐(๐ ๐) (1 โ ๐
2
๐)
โ1/2
,
โ 1 < ๐ ๐< 1, (๐ = 2, 3) ,
(29b)
where๐บ๐(๐ ๐) is a bounded function. In order to insure smooth
contact at the end points of the separation region, we thenimpose the following conditions on ๐บ
3(๐ 3):
๐บ3(โ1) = 0, ๐บ
3(1) = 0. (30)
Equations (26a)โ(26c), (16a), (16b), and (28) can easily bereduced to the following system of linear algebraic equationsby employing the appropriate Gauss-Chebyshev integrationformula [36]:
โ
๐
โ
๐=1
๐๐๐บ1(๐ 1๐)
๐
โ
[
[
๐โ
1(๐1๐, ๐ 1๐) +
(1 + ๐ 1)
4
ร
1
๐ (๐ 1๐+ ๐1๐)
โ
1
๐ (๐ 1๐โ ๐1๐)
]
]
โ
๐
โ
๐=1
๐๐๐บ2(๐ 2๐)
๐ โ ๐
2โ
ร [
[
๐โ
2(๐1๐, ๐ 2๐) +
(1 + ๐ 1)
4
ร (
1
(((๐ โ ๐) /2) ๐ 2๐+ (๐ + ๐) /2) + ๐๐
1๐
โ
1
(((๐ โ ๐) /2) ๐ 2๐+ (๐ + ๐) /2) โ ๐๐
1๐
)]
]
+
๐โ1
โ
๐=2
๐๐๐บ3(๐ 3๐)
๐ โ ๐
2โ
๐โ
3(๐1๐, ๐ 3๐) = 0,
(๐ = 1, . . . , ๐ โ 1) ,
(31a)
โ
๐
โ
๐=1
๐๐๐บ1(๐ 1๐)
๐
โ
ร [
[
๐โ
4(๐2๐, ๐ 1๐) +
(1 + ๐ 1)
4
ร (
1
๐๐ 1๐+ (((๐ โ ๐) /2) ๐
2๐+ (๐ + ๐) /2)
โ
1
๐๐ 1๐โ (((๐ โ ๐) /2) ๐
2๐+ (๐ + ๐) /2)
)]
]
โ
๐
โ
๐=1
๐๐๐บ2(๐ 2๐)
๐ โ ๐
2โ
[
[
๐โ
5(๐2๐, ๐ 2๐) +
(1 + ๐ 1)
4
ร (
1
((๐โ๐) /2) (๐ 2๐โ๐2๐)+(๐+๐)
ยฑ
1
((๐ โ ๐) /2) (๐ 2๐โ ๐2๐)
)]
]
+
๐โ1
โ
๐=2
๐๐๐บ3(๐ 3๐)
๐ โ ๐
2โ
๐โ
6(๐2๐, ๐ 3๐) = 0
(๐ = 1, . . . , ๐ โ 1) ,
(31b)
Mathematical Problems in Engineering 9
โ
๐
โ
๐=1
๐๐๐บ1(๐ 1๐)
๐
โ
๐โ
7(๐3๐, ๐ 1๐)
โ
๐
โ
๐=1
๐๐๐บ2(๐ 2๐)
๐ โ ๐
2โ
๐โ
8(๐3๐, ๐ 2๐)
โ
๐โ1
โ
๐=2
๐๐๐บ3(๐ 3๐)
๐ โ ๐
2โ
ร [
[
๐โ
9(๐3๐, ๐ 3๐) โ
4๐2/๐1
(1 + ๐ 2) + ๐2/๐1(1 + ๐
1)
ร (
1
((๐ โ ๐) /2) (๐ 3๐โ ๐3๐) + (๐ + ๐)
+
1
((๐ โ ๐) /2) (๐ 3๐โ ๐3๐)
)]
]
โ
1
๐
= 0,
(๐ = 1, . . . , ๐ โ 1) ,
(31c)
๐
โ
๐=1
๐๐๐
๐
โ
๐บ1(๐ 1๐) =
1
2
, (31d)
๐
โ
๐=1
๐๐๐
๐ โ ๐
2โ
๐บ2(๐ 2๐) =
๐
๐
, (31e)
๐โ1
โ
๐=2
๐๐๐๐บ3(๐ 3๐) = 0, (31f)
where ๐๐, ๐ ๐, and ๐
๐are given by (19e)โ(19g) (๐
2= ๐3, ๐ 2=
๐ 3). It was shown in [36] that the consistency condition is
automatically satisfied if the Gauss-Chebyshev integrationformula is used for solving integral equations. Thus, (31a)โ(31d) and (31e)-(31f) give 3๐ equation for 3๐ unknowns๐บ
1(๐ ๐),
๐บ2(๐ ๐), ๐บ3(๐ ๐) (๐ = 1, . . . , ๐), (๐ = 2, . . . , ๐ โ 1), ๐ and ๐. The
equation system is nonlinear in ๐ and ๐; so an interpolationscheme is required for the solution. Selected values of ๐ and๐ are substituted into (31a)โ(31d), and ๐บ
1(๐ ), ๐บ2(๐ ), and ๐บ
3(๐ )
are obtained which must satisfy (31e), (31f) at the same timefor known ๐ > ๐cr. If (31e), (31f) are not satisfied, thensolution must be repeated with new values of ๐ and ๐ untilthe (31e), (31f) are satisfied at the same time.
It should be noted that (22c) gives the ๐๐ฆ1(๐ฅ, 0)โ/๐
outside as well as inside the separation region (๐, ๐). Thus,once the functions ๐บ
1(๐ ), ๐บ2(๐ ), and ๐บ
3(๐ ) and the constants
๐ and ๐ are determined, contact stress ๐๐ฆ1(๐ฅ, 0)โ/๐ may be
easily evaluated.The displacement component Vโ(๐ฅ, 0) in theseparation region (๐, ๐), referring to (21f) and (25b), may beobtained from
Vโ(๐ฅ, 0) = V
2(๐ฅ, 0) โ V
1(๐ฅ, 0) = โซ
๐ฅ
๐
๐3(๐ก) ๐๐ก,
๐ < ๐ฅ < ๐
(32a)
Table 1: Variation of minimum value of distance between twopunches (๐ โ ๐)/โ to avoid separation under first punch with ๐/โand (๐ โ ๐)/โ for various values of ๐/๐ (๐
2/๐1= 6.48).
(๐ โ ๐)/โ ๐/โ(๐ โ ๐)/โ
2๐ 4๐ 6๐ 8๐ 10๐ 12๐
0.250.5 0.2773 0.5465 0.7263 0.8564 0.9569 1.03830.75 0.4645 0.7676 0.9485 1.075 1.1716 1.2501.0 0.603 0.9107 1.089 1.2138 1.3105 1.3893
0.50.5 0.185 0.4403 0.617 0.7466 0.8474 0.92910.75 0.3617 0.6597 0.8405 0.9675 1.065 1.1441.0 0.4983 0.8035 0.983 1.1086 1.206 1.2856
0.750.5 0.108 0.3432 0.516 0.645 0.746 0.82850.75 0.2715 0.5613 0.742 0.8699 0.9685 1.04851.0 0.4045 0.7073 0.8875 1.015 1.1134 1.4445
1.00.5 0.0357 0.2504 0.4195 0.548 0.6497 0.73320.75 0.191 0.4694 0.6502 0.7797 0.8798 0.96131.0 0.320 0.6188 0.8005 0.930 1.030 1.1123
or
๐1
๐/โ
Vโ(๐ฅ, 0) =
๐ โ ๐
2โ
โซ
๐3
โ1
๐บ3(๐ 3) ๐๐ 3,
โ 1 < ๐3< 1,
(32b)
where
๐ฅ =
๐ โ ๐
2
๐3+
๐ + ๐
2
. (32c)
Also using appropriate Gauss-Chebyshev integration for-mula and taking +1 as the index of (32b), the followingexpressionmay be written for the vertical displacement in theseparation region:
๐1
๐/โ
Vโ(๐ฅ, 0) =
๐ โ ๐
2โ
๐โ1
โ
๐=2
๐๐๐บ3(๐ 3๐) ,
(๐ = 2, . . . , ๐ โ 1) ,
(33)
where๐๐and ๐ ๐are given by (19e)โ(19g).
5. Results and Discussion
Some of the calculated results obtained from the solutionof the continuous and discontinuous contact problems forvarious dimensionless quantities such as ๐
2/๐1, ๐/โ, (๐โ๐)/โ,
๐/๐, ๐, and (๐ โ ๐)/โ are presented in Figures 2, 3, 4, 5, and6 and Tables 1, 2, 3, 4, and 5. First separation point betweenelastic layer and elastic half plane is determined and first sep-aration region is investigated. Contact pressure๐
๐ฆ1(๐ฅ, 0)โ/๐ is
presented. Depending on load factor ๐, possibilities of otherseparation regions between elastic layer and elastic half planeare determined. Also possibility of separation between firstpunch and elastic layer is researched. Besides, the distance(๐ โ๐)/โ, that ends the interaction of punches is examined. Itis assumed that ๐/โ โฅ ๐/โ.
10 Mathematical Problems in Engineering
Table 2: Variation of distance between two punches (๐ โ ๐)/โ that ends interaction of punches with ๐/๐ (๐2/๐1= 2.75, ๐/โ = 0.25, and
(๐ โ ๐)/โ = 0.5).
๐ (๐ โ ๐)/โPunch I Punch II
๐crright (๐ฅcrright โ ๐)/โ ๐crleft = ๐crright (๐ โ ๐ฅcrleft )/โ = (๐ฅcrright โ ๐)/โ
๐ 10.5585 96.6748 2.091 96.6175 2.0922๐ 9.5874 96.0544 2.091 48.3612 2.0924๐ 8.6032 94.2026 2.094 24.1974 2.0926๐ 8.1313 91.6959 2.100 16.1360 2.0928๐ 7.8582 88.6769 2.111 12.1037 2.09110๐ 7.6742 85.2495 2.128 9.6838 2.09112๐ 7.5378 81.4924 2.153 8.0704 2.09114๐ 7.4308 81.4924 2.192 6.9178 2.090
Table 3: Variation of distance between two punches (๐ โ ๐)/โ that ends interaction of punches with ๐2/๐1(๐ = 6๐, ๐/โ = 0.25, and
(๐ โ ๐)/โ = 0.5).
๐2/๐1
(๐ โ ๐)/โPunch I Punch II
๐crright (๐ฅcrright โ ๐)/โ ๐crleft = ๐crright (๐ โ ๐ฅcrleft )/โ = (๐ฅcrright โ ๐)/โ
0.15 14.8404 164.5971 5.309 36.6977 4.7950.36 12.1858 156.6651 3.678 30.6651 3.5960.61 10.8897 143.5372 3.066 26.8566 3.0271.65 9.2688 114.5372 2.418 20.5185 2.4022.75 8.1313 91.6959 2.100 16.1360 2.0926.48 6.4981 65.5235 1.812 11.4136 1.793
Figure 2 shows limit distance between punches that initi-ates separation under first punch for various values of mate-rial constant,๐
2/๐1with๐/๐.Thedistance (๐โ๐)/โ increases,
maintaining continuous contact between first punch andelastic layer with ๐/๐. Besides, for bigger values of ๐
2/๐1,
limit distance that initiates separation under first punchdecreases. In such a case, elastic half plane gets stiffer and itbecomes easy to separate first punch from the elastic layer.
Variation of critical distance between punches with ๐/โand (๐ โ ๐)/โ for various values of ๐/๐ is presented inTable 1. For fixed values of second punch width, (๐ โ ๐)/โ,an increase in first punch width requires longer distancebetween punches to avoid separation under first punch. Onthe contrary, for fixed values of first punch width, ๐/โ,an increase in second punch width decreases (๐ โ ๐)/โ
and punches can be placed closer to each other withoutseparation.
Interaction between punches ends for a definite value of(๐ โ ๐)/โ. Tables 2โ4 show the critical value of the distancethat ends interaction between punches with dimensionlessquantities ๐
2/๐1, ๐/โ, (๐ โ ๐)/โ, and ๐/๐. In such a case,
there is no need to consider punches together. Also thesetables show the values of the load factor that cause separationbetween elastic layer and elastic half plane, ๐cr. For ๐ = ๐cr,๐๐ฆ1(๐ฅ, 0)โ/๐ is zero. Contact between punches and elastic
layer is continuous.Table 3 shows the critical distance between punches that
ends interaction of punches with elastic constant, ๐2/๐1. For
small values of ๐2/๐1; that is, it is easy to bend elastic layer,
interaction between punches ends in a longer distance. Initialseparation point ๐ฅcr between elastic layer and elastic halfplane from the origin ๐ฅ = 0 decreases with an increase in๐2/๐1. Critical load factor also decreases in this situation.
Distance (๐โ๐)/โ that ends interaction between punchesincreases with a decrease in second punch width while firstpunch width is fixed. If both first and second widths areincreased, interaction of punches ends in a shorter distance.This situation is presented in Table 4. In this case, critical loadfactor, ๐cr increases but initial separation point, ๐ฅcr decreaseswith increment in punchwidths. Separation also occurs at theright-hand side of the second punch.
For fixed values of ๐/โ = 0.5, (๐ โ ๐)/โ = 1, ๐ = 2๐,and ๐
2/๐1= 1.65, variations of critical load factor, ๐cr and
initial separation point, ๐ฅcr are given in Table 5. For smallvalues of (๐ โ ๐)/โ, initial separation point between elasticlayer and elastic half plane appears at the right-hand side ofsecond punch. If (๐ โ ๐)/โ increases, in this case separationtakes place between two punches. Keeping on increasing,the distance (๐ โ ๐)/โ ends the interaction of punches. For(๐ โ ๐)/โ = 7.9446, there is no need to consider punchestogether.
In Figures 3โ5, the normalized contact stress distribution๐๐ฆ1(๐ฅ, 0)โ/๐ at the interface of elastic layer and elastic half
plane is given for the problems described in Section 3, andSection 4. Different scales have been used for continuous anddiscontinuous contact cases in order to include the entirepressure distribution and to give sufficient details in compactforms.
Mathematical Problems in Engineering 11
Table 4: Variation of distance between two punches (๐โ๐)/โ that ends interaction of punches with ๐/โ and (๐โ๐)/โ (๐ = 4๐, ๐2/๐1= 0.36).
๐/โ (๐ โ ๐)/โ (๐ โ ๐)/โPunch I Punch II
๐crright (๐ฅcrright โ ๐)/โ ๐crleft = ๐crright (๐ โ ๐ฅcrleft )/โ = (๐ฅcrright โ ๐)/โ
0.25 1.5 10.8345 150.2590 3.775 54.1240 3.38160.25 1.0 11.8775 164.4214 3.655 49.0259 3.44950.25 0.5 12.7008 169.3000 3.628 45.9626 3.59680.50 1.0 11.6754 173.4645 3.523 49.0235 3.44940.75 1.5 10.4960 176.6011 3.421 54.1289 3.3810
0
0.6
1.2
1.8
2.4
0.3
0.9
1.5
2.1(1)
(2)
(3)
(4)
(๐โ๐)/โ
๐/๐
5 10 15 207.5 12.5 17.5
(3) ๐2/๐1 = 2.75(1) ๐2/๐1 = 0.61(2) ๐2/๐1 = 1.65 (4) ๐2/๐1 = 6.48
Figure 2: Variation of minimum value of distance between twopunches (๐ โ ๐)/โ to avoid separation under first punch with ๐/๐for various values of ๐
2/๐1(๐/โ = 0.5, (๐ โ ๐)/โ = 1).
Table 5: Variation of load factor values with distance between twopunches (๐ โ ๐)/โ (๐ = 2๐, ๐
2/๐1= 1.65, ๐/โ = 0.5, and (๐ โ ๐)/โ =
1).
(๐ โ ๐)/โPunch I Punch II
๐crright ๐ฅcrright ๐crleft ๐ฅcrleft ๐crright ๐ฅcrright
0.5 30.6814 4.2941 32.4279 4.8063 34.0783 6.8085 28.4331 3.167 28.4331 3.167 34.2756 8.8036 32.0727 4.184 32.0727 4.184 34.3055 9.8017 33.8531 5.211 33.8531 5.211 34.3175 10.8007.9446 123.1714 2.835 34.3182 6.148 34.3182 11.743
Variation of the contact stress distribution ๐๐ฆ1(๐ฅ, 0)โ/๐
with load factor ๐ = ๐/๐1๐โ2 for fixed values of ๐ = 2๐,
(1)
(3) (3) (2)
0 1.5 3 4.5 6 7.5 9 10.5 12๐ฅ/โ
0โ0.011โ0.022โ0.05
โ0.4
โ0.6
โ0.8
โ1
โ1.2
โ1.4
โ1.6(1), (2), (3)
(2) ๐ = ๐cr = 45.894 (๐ฅcr = 4.327)(๐/โ = 3.66, ๐/โ = 5.926)(1) ๐ = 90 > ๐cr
(3) ๐ = 20 < ๐cr
(๐๐ฆ1(๐ฅ,0))/(๐/โ)
Figure 3: Contact stress distribution between elastic layer andelastic half plane for the cases of continuous (๐ < ๐cr) anddiscontinuous contact (๐ > ๐cr) (๐ = 2๐, ๐
2/๐1= 2.75, ๐/โ = 0.25,
(๐ โ ๐)/โ = 1.5, and (๐ โ ๐)/โ = 0.5).
๐2/๐1= 2.75, ๐/โ = 0.25, (๐โ๐)/โ = 1.5, and (๐โ๐)/โ = 0.5 is
presented in Figure 3, ๐ < ๐cr and ๐ > ๐cr show contact stressdistribution for the continuous and discontinuous contactcases, respectively. It can be seen from the figure that anotherseparation region is possible if distance between punches(๐ โ ๐)/โ or load factor ๐ is increased. Contact pressure haspeaks around the edges of the rigid punches.
In Figure 4, the variation of the normalized contact stress๐๐ฆ1(๐ฅ, 0)โ/๐ with ๐/๐ is given for the discontinuous contact
case (๐ = 75 > ๐cr). Figure 4 shows that both contact stressand separation zone (๐ โ ๐)/โ increase with an increase in๐/๐. Variation of the normalized contact stress ๐
๐ฆ1(๐ฅ, 0)โ/๐
for the discontinuous contact case shows three differentregions between elastic layer and elastic half plane. Theseare continuous contact region, separation zone, and alsocontinuous contact region, where the effects of external load๐/โ and ๐/โ decrease and disappear infinitely.
Variation of contact stress ๐๐ฆ1(๐ฅ, 0)โ/๐ with ๐
2/๐1= 1.65
is shown in Figure 5. Separation zone increases as elastic halfplane gets stiffer than the elastic layer. In this case, peak valueof the contact stress also increases.
12 Mathematical Problems in Engineering
(3)
(2)
(1)
(3) (1)
(2) (1) (3)
(2)
0 2 4 6 8 10 12 14 16๐ฅ/โ
0
โ0.013
โ1
โ1.5
โ2
โ2.5
โ3
(2) ๐ = 3๐(3) ๐ = 4๐
(1) ๐ = 2๐ (๐โ = 3.634, ๐/โ = 6.085)(๐โ = 3.396, ๐/โ = 7.245)(๐โ = 3.27, ๐/โ = 8.153)
(๐๐ฆ1(๐ฅ,0))/(๐/โ)
Figure 4: Contact stress distribution between elastic layer andelastic half plane for the case of discontinuous contact (๐
2/๐1= 0.36,
๐/โ = 0.125, (๐ โ ๐)/โ = 0.5, (๐ โ ๐)/โ = 0.5, and ๐ = 75 > ๐cr).
Results calculated from (33) giving the displacementVโ(๐ฅ, 0) in the separation region ๐ < ๐ฅ < ๐ as a functionof ๐ฅ are shown in Figure 6. Also results are compared withthose of [35]. It is seen that separation region ๐ < ๐ฅ < ๐ anddisplacement values increase with an increase in๐/๐ ratio asthis is the case in [35]. Separation zone is nearly the same, butdisplacement values are smaller than those of [35] for fixedvalues of ๐
2/๐1= 6.48, ๐/โ = 0.75, (๐ โ ๐)/โ = 1, and
(๐ โ ๐)/โ = 1.5.
6. Conclusion
In this paper, continuous and discontinuous contact prob-lems for an elastic layer resting on an elastic half planeloaded by means of three rigid flat punches are considered.Numerical procedures developed in this study can be usedto find approximate solutions to problems of engineeringinterest.
Numerical results show that the punch widths, elasticconstants, external loads, distance between punches play avery important role in the formation of the continuous anddiscontinuous contact areas, initial separation point and load,separation displacement, limit distance that ends the interac-tion of punches, critical distance that causes separation underfirst punch, and the contact pressure distribution.
From the study, the following conclusions may be drawn:
(i) it is easy to separate first punch from elastic layer if๐/๐ or ๐/โ increases. This results also decrease in๐2/๐1or (๐ โ ๐)/โ.
(3)
(2) (3) (2)
(3)
(2)
(1)
(1)
(1) (3) 0 2 4 6 8 10 12 14 16
0โ0.01
โ0.31
โ0.46
โ0.61
โ0.76
โ0.92
โ1.07
โ1.22
(1) ๐2/๐1 = 0.36(2) ๐2/๐1 = 0.61(3) ๐2/๐1 = 1.65
(๐โ = 6.498, ๐/โ = 7.612)(๐โ = 5.869, ๐/โ = 7.299)(๐โ = 5.175, ๐/โ = 7.003)
๐ฅ/โ
(๐๐ฆ1(๐ฅ,0))/(๐/โ)
Figure 5: Contact stress distribution between elastic layer andelastic half plane for the case of discontinuous contact (๐ = 2๐,๐/โ = 0.5, (๐ โ ๐)/โ = 2, (๐ โ ๐)/โ = 1, and ๐ = 100 > ๐cr).
(1b) ref [35]
(1a)
(2b) ref [35]
(2a)
(3b) ref [35]
(3a) 0
0.03
0.06
0.09
0.105
0.12
0.075
0.045
0.015
4 4.5 5 5.5 6 6.5 7 7.5 8๐ฅ/โ
(1) ๐ = 4๐(a) ((๐ โ ๐)/โ = 2.001)(b) ((๐ โ ๐)/โ = 2.033)โ
(2) ๐ = 6๐(a) ((๐ โ ๐)/โ = 2.854)(b) ((๐ โ ๐)/โ = 2.857)โ
(3) ๐ = 8๐(a) ((๐ โ ๐)/โ = 3.494)(b) ((๐ โ ๐)/โ = 3.483)โ
(๐๏ฟฝโ(๐ฅ,0))/(๐/โ)
Figure 6: Separation displacement Vโ(๐ฅ, 0) between elastic layer andelastic half space as a function of ๐ฅ for various values of secondpunch load๐ (๐
2/๐1= 6.48, ๐/โ = 0.75, (๐โ๐)/โ = 1, (๐โ๐)/โ = 1.5,
and ๐ = 50 > ๐cr), (โ: [35]).
Mathematical Problems in Engineering 13
(ii) Increment in ๐/๐, ๐/โ or (๐ โ ๐)/โ ends interactionof punches in a shorter distance. Also an increase in๐2/๐1causes the same result.
(iii) First separation between elastic layer and elastic halfplane occurs between punches or in the region ๐ <
๐ฅ < โ depending on ๐. If (๐ โ ๐)/โ is big enough,interaction of punches disappears.
(iv) Size of separation region is not affected much with๐/๐, but separation displacement is increasedwith anincrease in ๐/๐.
Nomenclature
๐: Compressive load per unit thickness in ๐งdirection on punch I, (N/m)
๐: Compressive load per unit thickness in ๐งdirection on punch II and punch III,(N/m)
โ: Thickness of the layer, (m)๐: Load factor๐ข: ๐ฅ-component of the displacement, (m)V: ๐ฆ-component of the displacement, (m)๐๐: Mass density, (kg/m3)
๐: Gravity acceleration, (m/sn2)๐: Shear modulus, (Pa)๐ : Elastic constant๐พ: Poissonโs ratio(๐ โ ๐): Width of punch II and punch III, (m)๐: Half width of punch I, (m)๐(๐ฅ): Contact pressure under punch I, (Pa)๐(๐ฅ): Contact pressure under punch II and
punch III, (Pa)(๐ โ ๐): Separation length, (m).
References
[1] P. Wriggers, Computational Contact Mechanics, John Wiley &Sons, Chichester, UK, 2002.
[2] M. R. Gecit, โThe axisymmetric double contact problem fora frictionless elastic layer indented by an elastic cylinder,โInternational Journal of Engineering Science, vol. 24, no. 9, pp.1571โ1584, 1986.
[3] V. Kahya, T. S. Ozsahin, A. Birinci, and R. Erdol, โA recedingcontact problem for an anisotropic elastic medium consistingof a layer and a half plane,โ International Journal of Solids andStructures, vol. 44, no. 17, pp. 5695โ5710, 2007.
[4] I. Comez, A. Birinci, and R. Erdol, โDouble receding contactproblem for a rigid stamp and two elastic layers,โ EuropeanJournal of Mechanics, A/Solids, vol. 23, no. 2, pp. 301โ309, 2004.
[5] L. M. Keer, J. Dondurs, and K. C. Tsai, โProblems involving areceding contact between a layer and a half space,โ Journal ofApplied Mechanics, Transactions ASME, vol. 39, no. 4, pp. 1115โ1120, 1972.
[6] F. Erdogan andM. Ratwani, โThe contact problem for an elasticlayer supported by two elastic quarter planes,โ Journal of AppliedMechanics, Transactions ASME, vol. 41, no. 3, pp. 673โ678, 1974.
[7] M. B. Civelek and F. Erdogan, โThe axisymmetric doublecontact problem for a frictionless elastic layer,โ InternationalJournal of Solids and Structures, vol. 10, no. 6, pp. 639โ659, 1974.
[8] Q. Lan, G. A. C. Graham, and A. P. S. Selvadurai, โCertain two-punch problems for an elastic layer,โ International Journal ofSolids and Structures, vol. 33, no. 19, pp. 2759โ2774, 1996.
[9] M. J. Jaffar, โFrictionless contact between an elastic layer on arigid base and a circular flat-ended punch with rounded edgeor a conical punch with rounded tip,โ International Journal ofMechanical Sciences, vol. 44, no. 3, pp. 545โ560, 2002.
[10] G. Fu andA.Chandra, โNormal indentation of elastic half-spacewith a rigid frictionless axisymmetric punch,โ Journal of AppliedMechanics, Transactions ASME, vol. 69, no. 2, pp. 142โ147, 2002.
[11] Y. Pinyochotiwong, J. Rungamornrat, and T. Senjuntichai,โAnalysis of rigid frictionless indentation on half-space withsurface elasticity,โ in Proceedings of the 12th East Asia-PasificConference on Structural Engineering and Construction, vol. 14,pp. 2403โ2410, 2011.
[12] G. M. L. Gladwell, โOn some unbonded contact problems inplane elasticity theory,โ Journal of Applied Mechanics, Transac-tions ASME, vol. 43, no. 2, pp. 263โ267, 1976.
[13] O. I. Zhupanska, โContact problem for elastic spheres: applica-bility of the Hertz theory to non-small contact areas,โ Interna-tional Journal of Engineering Science, vol. 49, no. 7, pp. 576โ588,2011.
[14] M. Porter and D. A. Hills, โA flat punch pressed against anelastic interlayer under conditions of slip and separation,โInternational Journal of Mechanical Sciences, vol. 44, no. 3, pp.465โ474, 2002.
[15] M. R. . Gecit and S. Gokpฤฑnar, โFrictionless contact between anelastic layer and a rigid rounded support,โ Arabian Journal ofScience and Engineering, vol. 10, no. 3, pp. 243โ251, 1985.
[16] F. Yang, โAdhesive contact between an elliptical rigid flat-endedpunch and an elastic half space,โ Journal of Physics D, vol. 38, no.8, pp. 1211โ1214, 2005.
[17] M. I. Porter and D. A. Hills, โNote on the complete contactbetween a flat rigid punch and an elastic layer attached toa dissimilar substrate,โ International Journal of MechanicalSciences, vol. 44, no. 3, pp. 509โ520, 2002.
[18] M. E. R. Shanahan, โAdhesion of a punch to a thin membrane,โComptes Rendus de lโAcademie des Sciences IV, vol. 1, no. 4, pp.517โ522, 2000.
[19] A. Klarbring, A. Mikelic, and M. Shillor, โThe rigid punchproblem with friction,โ International Journal of EngineeringScience, vol. 29, no. 6, pp. 751โ768, 1991.
[20] V. I. Fabrikant andT. S. Sankar, โConcentrated force underneatha punch bonded to a transversely isotropic half-space,โ Interna-tional Journal of Engineering Science, vol. 24, no. 1, pp. 111โ117,1986.
[21] S. Karuppanan, C. M. Churchman, D. A. Hills, and E. Giner,โSliding frictional contact between a square block and anelastically similar half-plane,โ European Journal of Mechanics,A/Solids, vol. 27, no. 3, pp. 443โ459, 2008.
[22] I. Comez and R. Erdol, โFrictional contact problem of a rigidstamp and an elastic layer bonded to a homogeneous substrate,โArchive of Applied Mechanics, vol. 83, no. 1, pp. 15โ24, 2013.
[23] A. P. S. Selvadurai, โMechanics of a rigid circular disc bonded toa cracked elastic half-space,โ International Journal of Solids andStructures, vol. 39, no. 24, pp. 6035โ6053, 2002.
[24] A. P. S. Selvadurai, โBoussinesq indentation of an isotropicelastic halfspace reinforced with an inextensible membrane,โInternational Journal of Engineering Science, vol. 47, no. 11-12, pp.1339โ1345, 2009.
14 Mathematical Problems in Engineering
[25] R. Artan and M. Omurtag, โTwo plane punches on a nonlocalelastic half plane,โ International Journal of Engineering Science,vol. 38, no. 4, pp. 395โ403, 2000.
[26] R. S. Dhaliwal, โPunch problem for an elastic layer overlying anelastic foundation,โ International Journal of Engineering Science,vol. 8, no. 4, pp. 273โ288, 1970.
[27] I. Comez, โFrictional contact problem for a rigid cylindricalstamp and an elastic layer resting on a half plane,โ InternationalJournal of Solids and Structures, vol. 47, no. 7-8, pp. 1090โ1097,2010.
[28] M. R. Gecit, โA tensionless contact without friction between anelastic layer and an elastic foundation,โ International Journal ofSolids and Structures, vol. 16, no. 5, pp. 387โ396, 1980.
[29] A. O. Cakiroglu and F. L. Cakiroglu, โContinuous and discon-tinuous contact problems for strips on an elastic semi-infiniteplane,โ International Journal of Engineering Science, vol. 29, no.1, pp. 99โ111, 1991.
[30] M. R. Gecit and F. Erdogan, โFrictionless contact problem for anelastic layer under axisymmetric loading,โ International Journalof Solids and Structures, vol. 14, no. 9, pp. 771โ785, 1978.
[31] A. Birinci and R. Erdol, โContinuous and discontinuous contactproblem for a layered composite resting on simple supports,โStructural Engineering and Mechanics, vol. 12, no. 1, pp. 17โ34,2001.
[32] M. B. Civelek and F. Erdogan, โThe frictionless contact problemfor an elastic layer under gravity,โ Journal of Applied Mechanics,Transactions ASME, vol. 42, no. 97, pp. 136โ140, 1975.
[33] D. Schmueser, M. Comninou, and J. Dundurs, โSeparation andslip between a layer and a substrate caused by a tensile load,โInternational Journal of Engineering Science, vol. 18, no. 9, pp.1149โ1155, 1980.
[34] M. B. Civelek, F. Erdogan, and A. O. Cakฤฑroglu, โInterface sepa-ration for an elastic layer loaded by a rigid stamp,โ InternationalJournal of Engineering Science, vol. 16, no. 9, pp. 669โ679, 1978.
[35] T. S. Ozsahin, โFrictionless contact problem for a layer onan elastic half plane loaded by means of two dissimilar rigidpunches,โ Structural Engineering and Mechanics, vol. 25, no. 4,pp. 383โ403, 2007.
[36] F. Erdogan and G. D. Gupta, โOn the numerical solution ofsingular integral equations,โ Quarterly of Applied Mathematics,vol. 29, pp. 525โ534, 1972.
[37] N. I. Muskhelishvili, Singular Integral Equations, Noordhoff,Leyden, The Netherlands, 1958.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of