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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; talat@ktu.edu.tr
Received 21 December 2012; Revised 27 February 2013; Accepted 13 March 2013
Academic Editor: Safa Bozkurt Coskun
Copyright Β© 2013 T. S. Ozsahin and O. TaskΔ±ner.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 π§ 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, π,is less than a critical value, πcr. In this case, initial separation loads, πcr and initial separation points, π₯cr 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 π > πcr 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 [2β27] or effect of gravitymay be taken into account[28β35]. Studies regarding the frictionless contact along theinterface may be found in [2β15]. Also contact region mayexhibit frictional characteristics, giving rise to normal andshear traction at the contact surface [16β27].
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 [2β7]. 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 IIIIππππππ
ππ
ππ π
π¦
π₯
β
ππ
π 1, π1, οΏ½1
π 2, π2, οΏ½2
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[28β35].
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[8β27]. 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 π and π (per unitthickness in π§ 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, β. The layer is subjected to homogeneousvertical body force due to gravity, π
1π. All surfaces are
frictionless. The layer remains in contact with the elastichalf plane where the magnitude of the load factor π is lessthan a critical value, πcr (π = π/π
1πβ2). If π > πcr,
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
ππβ2π’π+
2ππ
(π πβ 1)
π
ππ₯
(
ππ’π
ππ₯
+
πVπ
ππ¦
) = 0, (1a)
ππβ2Vπ+
2ππ
(π πβ 1)
π
ππ¦
(
ππ’π
ππ₯
+
πVπ
ππ¦
) = πππ,
(π = 1, 2)
(1b)
where πππ is the intensity of the body force acting vertically in
which ππand π are mass density and gravity acceleration. π’
π
and Vπare theπ₯ andπ¦ components of the displacement vector,
ππand π πrepresent shear modulus and elastic constant of the
layer and the half plane, respectively. π π= (3 β πΎ
π)/(1 + πΎ
π)
for plane stress and π π= (3 β 4πΎ
π) for plane strain. πΎ
πis the
Poisson ratio (π = 1, 2). Subscript 1 indicates the elastic layerand subscript 2 indicates the elastic half plane.
π’πand V
πrepresent the displacements for the case in
which gravity forces are considered. π’βand V
βare the
displacements when the gravity forces are ignored, and totalfield of displacements may be expressed as
π’ = π’π+ π’β, (2a)
V = Vπ+ Vβ. (2b)
Observing thatπ₯ = 0 is a plane of symmetry, it is sufficientto consider the problem in the region 0 β€ π₯ β€ β only. Usingthe symmetry consideration, the following expressions maybe written:
π’1(π₯, π¦) = βπ’
1(βπ₯, π¦) , (3a)
V1(π₯, π¦) = V
1(βπ₯, π¦) , (3b)
π’1(π₯, π¦) =
2
π
β«
β
0
π1(πΌ, π¦) sin (πΌπ₯) ππΌ, (3c)
V1(π₯, π¦) =
2
π
β«
β
0
Ξ¨1(πΌ, π¦) cos (πΌπ₯) ππΌ, (3d)
where π1and Ξ¨
1functions are inverse Fourier transforms
of π’1and 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:
π’1β(π₯, π¦) =
2
π
β«
β
0
[(π΄ + π΅π¦) πβπΌπ¦
+ (πΆ + π·π¦) ππΌπ¦] sin (πΌπ₯) ππΌ,
(4a)
V1β(π₯, π¦) =
2
π
β«
β
0
[[π΄ + (
π 1
πΌ
π + π¦)π΅] πβπΌπ¦
+ [βπΆ + (
π 1
πΌ
β π¦)π·] ππΌπ¦] cos (πΌπ₯) ππΌ.
(4b)
Using Hookeβs law and (4a) and (4b), stress componentswhich do not include the gravity force may be expressed asfollows:
ππ₯1β
(π₯, π¦) =
4π1
π
β«
β
0
[[πΌ (π΄ + π΅π¦) β (
3 β π 1
2
)π΅] πβπΌπ¦
+ [πΌ (πΆ + π·π¦) + (
3 β π π
2
)π·] ππΌπ¦]
Γ cos (πΌπ₯) ππΌ,(5a)
ππ¦1β
(π₯, π¦) =
4π1
π
β«
β
0
[β [πΌ (π΄ + π΅π¦) + (
π 1+ 1
2
)π΅] πβπΌπ¦
+[βπΌ (πΆ + π·π¦) +(
π 1+ 1
2
)π·] ππΌπ¦]
Γ cos (πΌπ₯) ππΌ,(5b)
ππ₯π¦1β
(π₯, π¦) =
4π1
π
β«
β
0
[β [πΌ (π΄ + π΅π¦) + (
π πβ 1
2
)π΅] πβπΌπ¦
+ [πΌ (πΆ + π·π¦) β (
π πβ 1
2
)π·] ππΌπ¦]
Γ sin (πΌπ₯) ππΌ.(5c)
For the case in which gravity force exist, particular partof the displacement components corresponding to π
1π, the
following expressions are obtained, that is, special solution ofthe Navier equations for a layer with a height β:
π’1π=
3 β π 1
8π1
π1πβ
2
π₯, (6a)
V1π=
π1ππ¦
2π1
[
(π 1β 1)
(π 1+ 1)
(π¦ β β) β
(π 1+ 1)
8
β] , (6b)
ππ¦1π
= π1π (π¦ β β) , (6c)
ππ₯1π
= π1π(π¦ β
β
2
)
(1 + π 2)
(1 + π 2) + 2π
2(1 + π
1)
, (6d)
ππ₯π¦1π
= 0. (6e)
Considering the orthogonal axes shown in Figure 1,displacements will be zero for π¦ = ββ, and if π
2, ]2are the
elastic constants of the half plane, then the homogenous fieldof displacements and stresses of the elastic half plane may beobtained as
π’2β(π₯, π¦) =
2
π
β«
β
0
[(πΈ + πΉπ¦) ππΌπ¦] sin (πΌπ₯) ππΌ, (7a)
V2β(π₯, π¦) =
2
π
β«
β
0
[[βπΈ + (
π 2
πΌ
β π¦)πΉ] ππΌπ¦] cos (πΌπ₯) ππΌ,
(7b)
ππ₯2β
(π₯, π¦) =
4π2
π
β«
β
0
[[πΌ (πΈ + πΉπ¦) + (
3 β π 2
2
)πΉ] ππΌπ¦]
Γ cos (πΌπ₯) ππΌ,(7c)
ππ¦2β
(π₯, π¦) =
4π2
π
β«
β
0
[[ β πΌ (πΈ + πΉπ¦)
+ (
π 2+ 1
2
)πΉ] ππΌπ¦] cos (πΌπ₯) ππΌ,
(7d)
ππ₯π¦2β
(π₯, π¦) =
4π2
π
β«
β
0
[[πΌ (πΈ + πΉπ¦) β (
π 2β 1
2
)πΉ] ππΌπ¦]
Γ sin (πΌπ₯) ππΌ,(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 π¦ = 0 and π¦ = β.
3. Case of Continuous Contact
An elastic layer with a height of β resting on an elastic halfplane, shown in Figure 1, is analyzed for unit thickness inπ§ direction. Widths of punches at both sides are similar,(πβπ)/β, and each of these punches transmits a concentratedload of π to the elastic layer. The width of the midmostpunch is different, 2π/β and it is subjected to a concen-trated load, π. All surfaces are frictionless. Particularly, theinitial separation load (πcr) and point (π₯cr) 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, π and π.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 (π) is sufficiently small, then the contactalong the layer-subspace, π¦ = 0, 0 < π₯ < β, will be
4 Mathematical Problems in Engineering
continuous, andA, B, C, D, E, and Fmust be determined fromthe following boundary and continuity conditions:
ππ¦1(π₯, β) =
{{
{{
{
βπ (π₯) , 0 < π₯ < π
βπ (π₯) , π < π₯ < π
0, π < π₯ < π, π < π₯ < β,
(8a)
ππ₯π¦1(π₯, β) = 0, 0 < π₯ < β, (8b)
ππ₯π¦1(π₯, 0) = 0, 0 < π₯ < β, (8c)
ππ₯π¦2(π₯, 0) = 0, 0 < π₯ < β, (8d)
ππ¦1(π₯, 0) = π
π¦2(π₯, 0) , 0 < π₯ < β, (8e)
π
ππ₯
[V2(π₯, 0) β V
1(π₯, 0)] = 0, 0 < π₯ < β, (8f)
π
ππ₯
[V1(π₯, β)] = 0, 0 < π₯ < π, (8g)
π
ππ₯
[V1(π₯, β)] = 0, π < π₯ < π, (8h)
in which subscripts 1 and 2 indicate relation to the elasticlayer and the elastic half plane, respectively. π(π₯) is theunknown contact pressure under punch I and π(π₯) 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
β«
π
0
π (π₯) ππ₯ =
π
2
, (9a)
β«
π
π
π (π₯) ππ₯ = π. (9b)
Displacement and stress expressions (4a), (4b), (5a)β(5c), (6a)β(6e), and (7a)β(7e) are substituted into boundaryconditions (8a)β(8f), and unknown constants A, B, C, D, E,and F are determined in terms of unknown functions π(π₯)and π(π₯). By making use of (8g) and (8h), after some simplemanipulations, onemay obtain the following singular integralequations for π(π₯) and π(π₯) [36, 37]:
β
1
ππ1
β«
π
0
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
+
1
π‘ β π₯
)]π (π‘) ππ‘
β
1
ππ1
β«
π
π
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
+
1
π‘ β π₯
)]
Γ π (π‘) ππ‘ = 0, 0 < π₯ < π,
(10a)
β
1
ππ1
β«
π
0
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
+
1
π‘ β π₯
)]π (π‘) ππ‘
β
1
ππ1
β«
π
π
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
+
1
π‘ β π₯
)]
Γ π (π‘) ππ‘ = 0, π < π₯ < π,
(10b)
where
π1(π₯, π‘) = β«
β
0
{{
πΌ3
8 (1 + π 1)
Γ [[(1 + π 2) +
π2
π1(1 + π
1)
]
+ πβ2πΌβ
[4πΌβ (1 + π 2) β
2π2
π1(1 + π
1)
]
+ πβ4πΌβ
[β1 β π 2+
π2
π1(1 + π
1)
]]}(Ξ)β1
β
(1 + π 1)
4
}
Γ {sinπΌ (π‘ + π₯) β sinπΌ (π‘ β π₯)}ππΌ,(11)
in which
Ξ = πΌ3(2{(β1 β π
2β
π2
π1(1 + π
1)
)
+ πβ4πΌβ
[β1 β π 2+
π2
π1(1 + π
1)
]
+ πβ2πΌβ
[ (1 + π 2) (2 + 4πΌ
2β2)
+
4πΌβπ2
π1(β1 β π
1)
]})
β1
.
(12)
If evaluated values of A, B, C, and D in terms of π(π₯) andπ(π₯) are substituted into (5b), the expression of the contactstress between elastic layer and half plane may be obtained as
ππ¦1(π₯, 0) =β π
1πβ β
1
π
β«
π
0
π2(π₯, π‘) π (π‘) ππ‘
β
1
π
β«
π
π
π2(π₯, π‘) π (π‘) ππ‘, 0 < π₯ < β,
(13)
Mathematical Problems in Engineering 5
in which π1and π are mass density and gravity acceleration,
respectively, where
π2(π₯, π‘) = β«
β
0
{
πΌ3π2
π1(1 + π
1)
Γ [πβ3πΌβ
(β1 + πΌβ) + πβπΌβ
(1 + πΌβ)] } (Ξ)β1
Γ {cosπΌ (π‘ + π₯) + cosπΌ (π‘ β π₯)}ππΌ.(14)
To simplify the numerical analysis, the following dimen-sionless quantities are introduced:
π₯1= ππ1, (15a)
π‘1= ππ 1, (15b)
π₯2=
π β π
2
π2+
π + π
2
, (15c)
π‘2=
π β π
2
π 2+
π + π
2
, (15d)
π1(π 1) =
π (ππ 1)
π/β
, (15e)
π2(π 2) =
π (((π β π) /2) π 2+ (π + π) /2)
π/β
, (15f)
πΌ = π€β, (15g)
π =
π
π1πβ2. (15h)
Substituting from (15a)β(15h), (9a), (9b) and (10a), (10b) maybe expressed as
β«
1
0
π1(π 1)
π
β
ππ 1=
1
2
, (16a)
β«
1
β1
π2(π 2)
π β π
2β
ππ 2=
π
π
, (16b)
β
1
π
β«
1
0
π1(π 1)
π
β
ππ 1[π1(π1, π 1) +
(1 + π 1)
4
Γ (
1
π (π 1+ π1)
β
1
π (π 1β π1)
)]
β
1
π
β«
1
β1
π2(π 2)
π β π
2β
ππ 2
Γ [π2(π1, π 2) +
(1 + π 1)
4
Γ (
1
(((π β π) /2) π 2+ (π + π) /2) + ππ
1
β
1
(((π β π) /2) π 2+(π + π) /2) β ππ
1
)]= 0,
0 < π1< 1,
(16c)
β
1
π
β«
1
0
π1(π 1)
π
β
ππ 1
Γ [π3(π2, π 1) +
(1 + π 1)
4
Γ (
1
ππ 1+ (((π β π) /2) π
2+ (π + π) /2)
β
1
ππ 1β (((π β π) /2) π
2+ (π + π) /2)
)]
β
1
π
β«
1
β1
π2(π 2)
π β π
2
ππ 2
Γ [π4(π2, π 2) +
(1 + π 1)
4
Γ (
1
((π β π) /2) (π 2+ π2) + (π + π)
β
1
((π β π) /2) (π 2β π2)
)] = 0,
β 1 < π2< 1,
(16d)
ππ¦1(π₯, 0)
π/β
= β
1
π
β
1
π
β«
1
0
π5(π1, π 1) π1(π 1)
π
β
ππ 1
β
1
π
β«
1
β1
π6(π2, π 2) π2(π 2)
π β π
2β
ππ 2,
(16e)
where
π1(π1, π 1) = π1(π₯1, π‘1) , (17a)
π2(π1, π 2) = π1(π₯1, π‘2) , (17b)
π3(π2, π 1) = π1(π₯2, π‘1) , (17c)
π4(π2, π 2) = π1(π₯2, π‘2) , (17d)
π5(π1, π 1) = π2(π₯1, π‘1) , (17e)
π6(π2, π 2) = π2(π₯2, π‘2) . (17f)
The index of the integral equations in (16a)β(16e) is +1;so the functions π
1(π 1) and π
2(π 2) may be expressed in the
following forms:
π1(π 1) = πΊ1(π 1) (1 β π
2
1)
β1/2
, (0 < π 1< 1) , (18a)
π2(π 2) = πΊ2(π 2) (1 β π
2
2)
β1/2
, (β1 < π 2< 1) , (18b)
where πΊ1(π 1) is bounded in [0, 1] and πΊ
2(π 2) is bounded in
[β1, 1].Then using appropriate Gauss-Chebyshev integrationformula given in [36], (16a)β(16d) are replaced by the follow-ing algebraic equations:
π
β
π=1
πππ
π
β
πΊ1(π 1π) =
1
2
, (19a)
6 Mathematical Problems in Engineering
π
β
π=1
πππ
π β π
2β
πΊ2(π 2π) =
π
π
, (19b)
β
π
β
π=1
πππΊ1(π 1π)
π
β
[
[
π1(π1π, π 1π) +
(1 + π 1)
4
Γ (
1
π (π 1π+ π1π)
β
1
π (π 1πβ π1π)
)]
]
β
π
β
π=1
πππΊ2(π 2π)
π β π
2β
Γ [
[
π2(π1π, π 2π) +
(1 + π 1)
4
Γ (
1
(((π β π) /2) π 2π+ (π + π) /2) + ππ
1π
β
1
(((π β π) /2) π 2π+(π + π) /2) β ππ
1π
)]
]
= 0,
(π = 1, . . . , π β 1) ,
(19c)
β
π
β
π=1
πππΊ1(π 1π)
π
β
Γ [
[
π3(π2π, π 1π) +
(1 + π 1)
4
Γ (
1
ππ 1π+ (((π β π) /2) π
2π+ (π + π) /2)
β
1
ππ 1πβ (((π β π) /2) π
2π+ (π + π) /2)
)]
]
β
π
β
π=1
πππΊ2(π 2π)
π β π
2β
Γ [
[
π4(π2π, π 2π) +
(1 + π 1)
4
Γ (
1
((π β π) /2) (π 2π+ π2π) + (π + π)
β
1
((π β π) /2) (π 2πβ π2π)
)]
]
= 0,
(π = 1, . . . , π β 1) ,
(19d)
π1= ππ=
1
2π β 2
, ππ=
1
π β 1
,
(π = 2, . . . , π β 1) ,
(19e)
π 1π= cos( π β 1
2π β 1
π) , π 2π= cos( π β 1
π β 1
π) ,
(π = 1, . . . , π) ,
(19f)
π1π= cos(
2π β 1
4π β 2
π) , π2π= cos(
2π β 1
2π β 2
π) ,
(π = 1, . . . , π β 1) .
(19g)
The unknownsπΊ1(π 1π) andπΊ
2(π 2π) (π = 1, . . . , π) are deter-
mined from the system (19a)β(19d). By using (18a), (18b), sub-stituting the results into (16e), and using Gauss-Chebyshevintegration formula, the contact stress π
π¦1(π₯, 0)β/π is evalu-
ated. It should be observed that the integral equations (16c)and (16d) are valid provided the contact stress π
π¦1(π₯, 0)β/π
is compressive everywhere; that is, 0 < π₯ < β. The criticalload factor, πcr and the corresponding location of interfaceseparation, π₯cr can be determined through the use of thefollowing condition for various dimensionless quantities:
ππ¦1(π₯, 0)
π/β
= 0. (20)
4. Case of Discontinuous Contact
Since the interface cannot carry tensile tractions for π > πcr,there will be separation between the elastic layer and theelastic half plane in the neighborhood of π₯ = π₯cr on thecontact plane π¦ = 0, as shown in Figure 1. Assuming thatthe separation region is described by π < π₯ < π, π¦ = 0,where π and π are unknowns and functions of π, boundaryand continuity conditions for the discontinuous contact caseare defined as follows:
ππ¦1(π₯, β) =
{{
{{
{
βπ (π₯) , 0 < π₯ < π
βπ (π₯) , π < π₯ < π
0, π < π₯ < π, π < π₯ < β,
(21a)
ππ₯π¦1(π₯, β) = 0, 0 < π₯ < β, (21b)
ππ₯π¦1(π₯, 0) = 0, 0 < π₯ < β, (21c)
ππ₯π¦2(π₯, 0) = 0, 0 < π₯ < β, (21d)
ππ¦1(π₯, 0) = π
π¦2(π₯, 0) , 0 < π₯ < β, (21e)
π
ππ₯
[V2(π₯, 0) β V
1(π₯, 0)] = {
π (π₯) , π < π₯ < π
0, 0 < π₯ < π, π < π₯ < β,
(21f)
ππ¦1(π₯, 0) = π
π¦2(π₯, 0) = 0, π < π₯ < π, (21g)
Mathematical Problems in Engineering 7
π
ππ₯
[V1(π₯, β)] = 0, 0 < π₯ < π, (21h)
π
ππ₯
[V1(π₯, β)] = 0, π < π₯ < π. (21i)
After utilizing the boundary and continuity conditionsdefined in (21a)β(21f), new values for the constantsA, B, C, D,E, and F which appear in (4a), (4b), (5a)β(5c), and (7a)β(7e)may be obtained in terms of new unknown functions π(π₯),π(π₯), and π(π₯). Unknown functions are then determinedfrom the conditions (21h)-(21i) which have not yet beensatisfied. These conditions give the following system ofsingular integral equations:
β
1
ππ1
β«
π
0
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
β
1
π‘ β π₯
)]π (π‘) ππ‘
β
1
ππ1
β«
π
π
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
β
1
π‘ β π₯
)]π (π‘) ππ‘
+
1
π
β«
π
π
π2(π₯, π‘) π (π‘) ππ‘ = 0, 0 < π₯ < π,
(22a)
β
1
ππ1
β«
π
0
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
β
1
π‘ β π₯
)]π (π‘) ππ‘
β
1
ππ1
β«
π
π
[π1(π₯, π‘) +
(1 + π 1)
4
(
1
π‘ + π₯
β
1
π‘ β π₯
)]π (π‘) ππ‘
+
1
π
β«
π
π
π2(π₯, π‘) π (π‘) ππ‘ = 0, π < π₯ < π,
(22b)
1
π
β«
π
0
π2(π₯, π‘) π (π‘) ππ‘ +
1
π
β«
π
π
π2(π₯, π‘) π (π‘) ππ‘
β
π1
π
β«
π
π
[π3(π₯, π‘) β
4π2/π1
(1 + π 2) + π2/π1(1 + π
1)
Γ (
1
π‘ + π₯
+
1
π‘ β π₯
)]π (π‘) ππ‘ β π1πβ = 0,
π < π₯ < π,
(22c)
where kernels π1(π₯, π‘) and π
2(π₯, π‘) are given by (11) and (14)
and
π3(π₯, π‘) = β«
β
0
{{β
2πΌ3π2
π1
{πβ2πΌβ
(2 + 4πΌ2β2) β πβ4πΌβ
β 1}}
Γ (Ξ)β1+
4π2/π1
(1 + π 2) + π2/π1(1 + π
1)
}
Γ {sinπΌ (π‘ + π₯) + sinπΌ (π‘ β π₯)} ππΌ,(23)
in which Ξ 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 β1 due to the physical requirement of smooth contact
at the end points π and π [37]. Thus, in solving the problemthe two conditions which would account for the unknowns πand π are the consistency condition of integral equation (22c)and the single-valuedness condition
β«
π
π
π (π₯) ππ₯ = 0. (24)
Defining the following dimensionless quantities
π₯3=
π β π
2
π3+
π + π
2
, π‘3=
π β π
2
π 3+
π + π
2
, (25a)
π3(π 3) =
π1π (((π β π) /2) π
3+ (π + π) /2)
π/β
; (25b)
by making use of (15a)β(15h), the integral equations (22a)β(22c) may be expressed as follows:
β
1
π
β«
1
0
π1(π 1)
π
β
ππ 1[πβ
1(π1, π 1) +
(1 + π 1)
4
Γ (
1
π (π 1+ π1)
β
1
π (π 1β π1)
)]
β
1
π
β«
1
β1
π2(π 2)
π β π
2β
ππ 2
Γ [πβ
2(π1, π 2) +
(1 + π 1)
4
Γ (
1
(((π β π) /2) π 2+ (π + π) /2) + ππ
1
β
1
(((π β π) /2) π 2+ (π + π) /2) β ππ
1
)]
+
1
π
β«
1
β1
π3(π 3)πβ
3(π1, π 3)
π β π
2β
ππ 3= 0,
0 < π1< 1,
(26a)
β
1
π
β«
1
β1
π1(π 1)
π
β
ππ 1
Γ [πβ
4(π2, π 1) +
(1 + π 1)
4
Γ (
1
ππ 1+ (((π β π) /2) π
2+ (π + π) /2)
β
1
ππ 1β (((π β π) /2) π
2+ (π + π) /2)
)]
β
1
π
β«
1
β1
π2(π 2)
π β π
2β
ππ 2[πβ
5(π2, π 2) +
(1 + π 1)
4
Γ (
1
((πβπ) /2) (π 2+π2)+(π+π)
β
1
((π β π) /2) (π 2β π2)
)]
+
1
π
β«
1
β1
π3(π 3)πβ
6(π2, π 3)
π β π
2β
ππ 3= 0,
β 1 < π2< 1,
(26b)
8 Mathematical Problems in Engineering
β
1
π
β«
1
β1
π1(π 1)πβ
7(π3, π 1)
π
β
ππ 1
β
1
π
β«
1
β1
π2(π 2)πβ
8(π3, π 2)
π β π
2β
ππ 2
β
1
π
β«
1
β1
π3(π 3)
π β π
2β
ππ 3
Γ [πβ
9(π3, π 3) β
4π2/π1
(1 + π 2) + π2/π1(1 + π
1)
Γ (
1
((π β π) /2) (π 3+ π3) + (π + π)
+
1
((π β π) /2) (π 3β π3)
)] β
1
π
= 0,
β 1 < π3< 1,
(26c)
where
πβ
1(π1, π 1) = π1(π₯1, π‘1) , (27a)
πβ
2(π1, π 2) = π1(π₯1, π‘2) , (27b)
πβ
3(π1, π 3) = π2(π₯1, π‘3) , (27c)
πβ
4(π2, π 1) = π1(π₯2, π‘1) , (27d)
πβ
5(π2, π 2) = π1(π₯2, π‘2) , (27e)
πβ
6(π2, π 3) = π2(π₯2, π‘3) , (27f)
πβ
7(π3, π 1) = π2(π₯3, π‘1) , (27g)
πβ
8(π3, π 2) = π2(π₯3, π‘2) , (27h)
πβ
9(π3, π 3) = π3(π₯3, π‘3) . (27i)
Similar to (16a), (16b), additional condition (24) may beexpressed as
β«
1
β1
π3(π 3) ππ 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 π
π(π π)
(π = 1, . . . , 3)may be expressed in the form
π1(π 1) = πΊ1(π 1) (1 β π
2
1)
β1/2
, 0 < π 1< 1, (29a)
ππ(π π) = πΊπ(π π) (1 β π
2
π)
β1/2
,
β 1 < π π< 1, (π = 2, 3) ,
(29b)
whereπΊπ(π π) is a bounded function. In order to insure smooth
contact at the end points of the separation region, we thenimpose the following conditions on πΊ
3(π 3):
πΊ3(β1) = 0, πΊ
3(1) = 0. (30)
Equations (26a)β(26c), (16a), (16b), and (28) can easily bereduced to the following system of linear algebraic equationsby employing the appropriate Gauss-Chebyshev integrationformula [36]:
β
π
β
π=1
πππΊ1(π 1π)
π
β
[
[
πβ
1(π1π, π 1π) +
(1 + π 1)
4
Γ
1
π (π 1π+ π1π)
β
1
π (π 1πβ π1π)
]
]
β
π
β
π=1
πππΊ2(π 2π)
π β π
2β
Γ [
[
πβ
2(π1π, π 2π) +
(1 + π 1)
4
Γ (
1
(((π β π) /2) π 2π+ (π + π) /2) + ππ
1π
β
1
(((π β π) /2) π 2π+ (π + π) /2) β ππ
1π
)]
]
+
πβ1
β
π=2
πππΊ3(π 3π)
π β π
2β
πβ
3(π1π, π 3π) = 0,
(π = 1, . . . , π β 1) ,
(31a)
β
π
β
π=1
πππΊ1(π 1π)
π
β
Γ [
[
πβ
4(π2π, π 1π) +
(1 + π 1)
4
Γ (
1
ππ 1π+ (((π β π) /2) π
2π+ (π + π) /2)
β
1
ππ 1πβ (((π β π) /2) π
2π+ (π + π) /2)
)]
]
β
π
β
π=1
πππΊ2(π 2π)
π β π
2β
[
[
πβ
5(π2π, π 2π) +
(1 + π 1)
4
Γ (
1
((πβπ) /2) (π 2πβπ2π)+(π+π)
Β±
1
((π β π) /2) (π 2πβ π2π)
)]
]
+
πβ1
β
π=2
πππΊ3(π 3π)
π β π
2β
πβ
6(π2π, π 3π) = 0
(π = 1, . . . , π β 1) ,
(31b)
Mathematical Problems in Engineering 9
β
π
β
π=1
πππΊ1(π 1π)
π
β
πβ
7(π3π, π 1π)
β
π
β
π=1
πππΊ2(π 2π)
π β π
2β
πβ
8(π3π, π 2π)
β
πβ1
β
π=2
πππΊ3(π 3π)
π β π
2β
Γ [
[
πβ
9(π3π, π 3π) β
4π2/π1
(1 + π 2) + π2/π1(1 + π
1)
Γ (
1
((π β π) /2) (π 3πβ π3π) + (π + π)
+
1
((π β π) /2) (π 3πβ π3π)
)]
]
β
1
π
= 0,
(π = 1, . . . , π β 1) ,
(31c)
π
β
π=1
πππ
π
β
πΊ1(π 1π) =
1
2
, (31d)
π
β
π=1
πππ
π β π
2β
πΊ2(π 2π) =
π
π
, (31e)
πβ1
β
π=2
ππππΊ3(π 3π) = 0, (31f)
where ππ, π π, and π
πare given by (19e)β(19g) (π
2= π3, π 2=
π 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)β(31d) and (31e)-(31f) give 3π equation for 3π unknownsπΊ
1(π π),
πΊ2(π π), πΊ3(π π) (π = 1, . . . , π), (π = 2, . . . , π β 1), π and π. The
equation system is nonlinear in π and π; so an interpolationscheme is required for the solution. Selected values of π andπ are substituted into (31a)β(31d), and πΊ
1(π ), πΊ2(π ), and πΊ
3(π )
are obtained which must satisfy (31e), (31f) at the same timefor known π > πcr. If (31e), (31f) are not satisfied, thensolution must be repeated with new values of π and π untilthe (31e), (31f) are satisfied at the same time.
It should be noted that (22c) gives the ππ¦1(π₯, 0)β/π
outside as well as inside the separation region (π, π). Thus,once the functions πΊ
1(π ), πΊ2(π ), and πΊ
3(π ) and the constants
π and π are determined, contact stress ππ¦1(π₯, 0)β/π may be
easily evaluated.The displacement component Vβ(π₯, 0) in theseparation region (π, π), referring to (21f) and (25b), may beobtained from
Vβ(π₯, 0) = V
2(π₯, 0) β V
1(π₯, 0) = β«
π₯
π
π3(π‘) ππ‘,
π < π₯ < π
(32a)
Table 1: Variation of minimum value of distance between twopunches (π β π)/β to avoid separation under first punch with π/βand (π β π)/β for various values of π/π (π
2/π1= 6.48).
(π β π)/β π/β(π β π)/β
2π 4π 6π 8π 10π 12π
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
π1
π/β
Vβ(π₯, 0) =
π β π
2β
β«
π3
β1
πΊ3(π 3) ππ 3,
β 1 < π3< 1,
(32b)
where
π₯ =
π β π
2
π3+
π + π
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:
π1
π/β
Vβ(π₯, 0) =
π β π
2β
πβ1
β
π=2
πππΊ3(π 3π) ,
(π = 2, . . . , π β 1) ,
(33)
whereππand π πare given by (19e)β(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 π
2/π1, π/β, (πβπ)/β,
π/π, π, and (π β π)/β 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 pressureπ
π¦1(π₯, 0)β/π is
presented. Depending on load factor π, 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(π βπ)/β, that ends the interaction of punches is examined. Itis assumed that π/β β₯ π/β.
10 Mathematical Problems in Engineering
Table 2: Variation of distance between two punches (π β π)/β that ends interaction of punches with π/π (π2/π1= 2.75, π/β = 0.25, and
(π β π)/β = 0.5).
π (π β π)/βPunch I Punch II
πcrright (π₯crright β π)/β πcrleft = πcrright (π β π₯crleft )/β = (π₯crright β π)/β
π 10.5585 96.6748 2.091 96.6175 2.0922π 9.5874 96.0544 2.091 48.3612 2.0924π 8.6032 94.2026 2.094 24.1974 2.0926π 8.1313 91.6959 2.100 16.1360 2.0928π 7.8582 88.6769 2.111 12.1037 2.09110π 7.6742 85.2495 2.128 9.6838 2.09112π 7.5378 81.4924 2.153 8.0704 2.09114π 7.4308 81.4924 2.192 6.9178 2.090
Table 3: Variation of distance between two punches (π β π)/β that ends interaction of punches with π2/π1(π = 6π, π/β = 0.25, and
(π β π)/β = 0.5).
π2/π1
(π β π)/βPunch I Punch II
πcrright (π₯crright β π)/β πcrleft = πcrright (π β π₯crleft )/β = (π₯crright β π)/β
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,π
2/π1withπ/π.Thedistance (πβπ)/β increases,
maintaining continuous contact between first punch andelastic layer with π/π. Besides, for bigger values of π
2/π1,
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 π/βand (π β π)/β for various values of π/π is presented inTable 1. For fixed values of second punch width, (π β π)/β,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, π/β,an increase in second punch width decreases (π β π)/β
and punches can be placed closer to each other withoutseparation.
Interaction between punches ends for a definite value of(π β π)/β. Tables 2β4 show the critical value of the distancethat ends interaction between punches with dimensionlessquantities π
2/π1, π/β, (π β π)/β, and π/π. 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, πcr. For π = πcr,ππ¦1(π₯, 0)β/π 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, π2/π1. For
small values of π2/π1; that is, it is easy to bend elastic layer,
interaction between punches ends in a longer distance. Initialseparation point π₯cr between elastic layer and elastic halfplane from the origin π₯ = 0 decreases with an increase inπ2/π1. Critical load factor also decreases in this situation.
Distance (πβπ)/β 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, πcr increases but initial separation point, π₯cr decreaseswith increment in punchwidths. Separation also occurs at theright-hand side of the second punch.
For fixed values of π/β = 0.5, (π β π)/β = 1, π = 2π,and π
2/π1= 1.65, variations of critical load factor, πcr and
initial separation point, π₯cr are given in Table 5. For smallvalues of (π β π)/β, initial separation point between elasticlayer and elastic half plane appears at the right-hand side ofsecond punch. If (π β π)/β increases, in this case separationtakes place between two punches. Keeping on increasing,the distance (π β π)/β ends the interaction of punches. For(π β π)/β = 7.9446, there is no need to consider punchestogether.
In Figures 3β5, the normalized contact stress distributionππ¦1(π₯, 0)β/π 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 (πβπ)/β that ends interaction of punches with π/β and (πβπ)/β (π = 4π, π2/π1= 0.36).
π/β (π β π)/β (π β π)/βPunch I Punch II
πcrright (π₯crright β π)/β πcrleft = πcrright (π β π₯crleft )/β = (π₯crright β π)/β
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)
(πβπ)/β
π/π
5 10 15 207.5 12.5 17.5
(3) π2/π1 = 2.75(1) π2/π1 = 0.61(2) π2/π1 = 1.65 (4) π2/π1 = 6.48
Figure 2: Variation of minimum value of distance between twopunches (π β π)/β to avoid separation under first punch with π/πfor various values of π
2/π1(π/β = 0.5, (π β π)/β = 1).
Table 5: Variation of load factor values with distance between twopunches (π β π)/β (π = 2π, π
2/π1= 1.65, π/β = 0.5, and (π β π)/β =
1).
(π β π)/βPunch I Punch II
πcrright π₯crright πcrleft π₯crleft πcrright π₯crright
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 ππ¦1(π₯, 0)β/π
with load factor π = π/π1πβ2 for fixed values of π = 2π,
(1)
(3) (3) (2)
0 1.5 3 4.5 6 7.5 9 10.5 12π₯/β
0β0.011β0.022β0.05
β0.4
β0.6
β0.8
β1
β1.2
β1.4
β1.6(1), (2), (3)
(2) π = πcr = 45.894 (π₯cr = 4.327)(π/β = 3.66, π/β = 5.926)(1) π = 90 > πcr
(3) π = 20 < πcr
(ππ¦1(π₯,0))/(π/β)
Figure 3: Contact stress distribution between elastic layer andelastic half plane for the cases of continuous (π < πcr) anddiscontinuous contact (π > πcr) (π = 2π, π
2/π1= 2.75, π/β = 0.25,
(π β π)/β = 1.5, and (π β π)/β = 0.5).
π2/π1= 2.75, π/β = 0.25, (πβπ)/β = 1.5, and (πβπ)/β = 0.5 is
presented in Figure 3, π < πcr and π > πcr 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(π β π)/β or load factor π is increased. Contact pressure haspeaks around the edges of the rigid punches.
In Figure 4, the variation of the normalized contact stressππ¦1(π₯, 0)β/π with π/π is given for the discontinuous contact
case (π = 75 > πcr). Figure 4 shows that both contact stressand separation zone (π β π)/β increase with an increase inπ/π. Variation of the normalized contact stress π
π¦1(π₯, 0)β/π
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 loadπ/β and π/β decrease and disappear infinitely.
Variation of contact stress ππ¦1(π₯, 0)β/π with π
2/π1= 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 16π₯/β
0
β0.013
β1
β1.5
β2
β2.5
β3
(2) π = 3π(3) π = 4π
(1) π = 2π (πβ = 3.634, π/β = 6.085)(πβ = 3.396, π/β = 7.245)(πβ = 3.27, π/β = 8.153)
(ππ¦1(π₯,0))/(π/β)
Figure 4: Contact stress distribution between elastic layer andelastic half plane for the case of discontinuous contact (π
2/π1= 0.36,
π/β = 0.125, (π β π)/β = 0.5, (π β π)/β = 0.5, and π = 75 > πcr).
Results calculated from (33) giving the displacementVβ(π₯, 0) in the separation region π < π₯ < π as a functionof π₯ are shown in Figure 6. Also results are compared withthose of [35]. It is seen that separation region π < π₯ < π anddisplacement values increase with an increase inπ/π ratio asthis is the case in [35]. Separation zone is nearly the same, butdisplacement values are smaller than those of [35] for fixedvalues of π
2/π1= 6.48, π/β = 0.75, (π β π)/β = 1, and
(π β π)/β = 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 ifπ/π or π/β increases. This results also decrease inπ2/π1or (π β π)/β.
(3)
(2) (3) (2)
(3)
(2)
(1)
(1)
(1) (3) 0 2 4 6 8 10 12 14 16
0β0.01
β0.31
β0.46
β0.61
β0.76
β0.92
β1.07
β1.22
(1) π2/π1 = 0.36(2) π2/π1 = 0.61(3) π2/π1 = 1.65
(πβ = 6.498, π/β = 7.612)(πβ = 5.869, π/β = 7.299)(πβ = 5.175, π/β = 7.003)
π₯/β
(ππ¦1(π₯,0))/(π/β)
Figure 5: Contact stress distribution between elastic layer andelastic half plane for the case of discontinuous contact (π = 2π,π/β = 0.5, (π β π)/β = 2, (π β π)/β = 1, and π = 100 > πcr).
(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 8π₯/β
(1) π = 4π(a) ((π β π)/β = 2.001)(b) ((π β π)/β = 2.033)β
(2) π = 6π(a) ((π β π)/β = 2.854)(b) ((π β π)/β = 2.857)β
(3) π = 8π(a) ((π β π)/β = 3.494)(b) ((π β π)/β = 3.483)β
(ποΏ½β(π₯,0))/(π/β)
Figure 6: Separation displacement Vβ(π₯, 0) between elastic layer andelastic half space as a function of π₯ for various values of secondpunch loadπ (π
2/π1= 6.48, π/β = 0.75, (πβπ)/β = 1, (πβπ)/β = 1.5,
and π = 50 > πcr), (β: [35]).
Mathematical Problems in Engineering 13
(ii) Increment in π/π, π/β or (π β π)/β ends interactionof punches in a shorter distance. Also an increase inπ2/π1causes the same result.
(iii) First separation between elastic layer and elastic halfplane occurs between punches or in the region π <
π₯ < β depending on π. If (π β π)/β is big enough,interaction of punches disappears.
(iv) Size of separation region is not affected much withπ/π, but separation displacement is increasedwith anincrease in π/π.
Nomenclature
π: Compressive load per unit thickness in π§direction on punch I, (N/m)
π: Compressive load per unit thickness in π§direction on punch II and punch III,(N/m)
β: Thickness of the layer, (m)π: Load factorπ’: π₯-component of the displacement, (m)V: π¦-component of the displacement, (m)ππ: Mass density, (kg/m3)
π: Gravity acceleration, (m/sn2)π: Shear modulus, (Pa)π : Elastic constantπΎ: Poissonβs ratio(π β π): Width of punch II and punch III, (m)π: Half width of punch I, (m)π(π₯): Contact pressure under punch I, (Pa)π(π₯): Contact pressure under punch II and
punch III, (Pa)(π β π): Separation length, (m).
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