This paper examines the influence of adhesive bonding on the flexuralinteraction between a thin plate and an isotropic elastic halfspace.The modelling is developed for the possible application of the resultsto the study of functionally stiffness graded elastic media where thegrading is restricted to the near surface region.
1 Introduction
The problem of the elastically supported infinite plate whose flexuralbehaviour is described by the Germain-Poisson-Kirchhoff thin platetheory, is a classical problem in contact mechanics and structuralmechanics. The elastic support can be idealized as either a set ofindependent Winkler springs or a Vlazov-type elastic layer with aninextensibility constraint or as an elastic medium of semi-infinite orfinite extent. The classical study by Hertz [1] deals with the case of
the Winkler support. Vlazov and Leontiev [2] present the case wherethe plate is supported on a constrained elastic medium. The studies byWoinowsky-Krieger [3], Marguerre [4], Hogg [5] and Roll [6] considerthe axisymmetric problems related to the smooth contact between theplate and either an elastic halfspace or an elastic layer. The class ofproblems examined are axisymmetrical and the interaction is inducedby axisymmetric external loads. Also, according to Korenev [7], studiesby Shekhter [8] and Leonev [9] also examine the problem of a thin plateelastically supported by an elastic halfspace.
In these classical studies dealing with the interaction betweena thin plate and an elastic halfspace it is invariably assumed that theinterface is smooth and capable of sustaining tensile tractions to avoidseparation at the plate-elastic halfspace interface. In a noteworthypaper by Sneddon et al. [10], a number of contact problems related to aplate and an elastic halfspace or an elastic layer have been examined.Although the title of this paper refers to a 'bonded contact', theanalysis essentially deals with a smooth contact where the normaltractions could be tensile. In the context of this paper, this type ofcontact is defined as a bilateral contact. Hie problem of the unbondedcontact between an infinite plate and an elastic halfspace considers thesituation where the contact surface is void of both shear tractions andtensile normal tractions. In this case the extent of the zone of contact isan unknown in the problem, which needs to be determined by imposingzero traction boundary conditions at the point of separation between theplate and the elastic halfspace. This class of problems has beenexamined by Jung [11], Gladwell [12] and Laermann [13].
Literature on elastostatic contact mechanics problems involvingthe interaction between plates and an elastic halfspace region is quiteextensive and no attempt will be made to review the subject matter indepth. Noteworthy reviews of the general topic of contact problemsinvolving the interaction of plates resting on elastic media are given byKorenev [7], Timoshenko and Woinowsky-Krieger [14], Hetenyi [15], dePater and Kalker [16], Selvadurai [17], Gladwell [18] and Johnson [19].
This paper deals with the axisymmetric problem involving theadhesive or bonded contact between a thin plate and an isotropicelastic halfspace(Figure 1). The problem is of technological interestparticularly with adhesive bonding between a thin wafer and asubstrate. Such problems are of current interest in connection withapplications in functionally graded materials and surface treatedsolids. The thin plate is an idealization of the near surface materialwhich can exhibit properties significantly different from the substrate.Although the modelling of the problem can be made by representing thenear surface material as a non-homogeneous elastic layer, thesimplified model of representing the graded region as a plate is a usefulfirst approximation which gives reasonable results which could be used
shall consider the axisymmetric problem related to the surface loadingof an isotropic elastic halfspace by a normal traction q(r), in thepresence of constraints related to bonding and smooth bilateral contact.The axisymmetric elastostatic problem can be examined in a variety ofways; in this presentation we employ the strain function approachproposed by Love [20] (see also Sneddon [21]). In the absence of bodyforces, the solution to axisymmetric problems in elastostatics can berepresented in terms of a single function <&(r,z) which satisfies thebiharmonic equation
= 0 (1)
where
i di I (2)
the axisymmetric form of Laplace's operator referred to the cylindricalpolar coordinate system (r,0,z). The displacement and stresscomponents can be expressed in terms of the derivatives of <P(r,z) . Thecomponents of interest to this analysis are
(3)
d_
dz
_d_
~fr
(4)
(5)
where G and v are the linear elastic shear modulus and Poisson's ratio,respectively.
We consider the problem where the surface of an elastichalfspace region(r e (0, oo); z £ (0, °o)) is subjected to either the set of traction boundaryconditions
where c\(r) is a compressive normal stress. Gaul and Plenge [22] havealso considered these classes of boundary conditions in the analysis ofthe dynamically loaded halfspace. In relation to the interactionbetween the infinite plate and the elastic halfspace we can identifythe set of boundary conditions (6) as being applicable to the situationinvolving smooth contact between the plate and the elastic halfspaceand the set of boundary conditions (7) are applicable to the caseinvolving fully bonded contact between the plate and the elastichalfspace. The zero radial displacement boundary conditions arisefrom the fact that in the modelling of the plate as a thin plate whichexperiences only flexure, the extensional displacements are assumed tobe negligible. It is possible to modify the plate theory to includemembrane effects; in which case the radial displacement at the bondedregion has to satisfy a consistent compatibility condition applicable toextensional displacements associated with membrane effects. For theanalysis of the boundary value problems described by (6) and (7) it isconvenient to use a Hankel transform analysis of the governing partialdifferential equation (1). It can be shown that the solution of (1)applicable to the halfspace region z e (0,°°) is given by
0( z) = } %z);/ ^ (8)o
where <P°(%,z) is the zeroth order Hankel transform of <P(r,z) given by
]e (9)
where A(%) and B(%) are arbitrary functions (chosen to satisfy the setsof boundary conditions (6) and/or (7)) and the form of (8) gives regulardisplacement and stress fields as r,z -> <*>. Also, in (8) ] (fy) is thezeroth-order Bessel function of the first kind. Avoiding details, it can
be shown that the relationship between u (%,0) and a°(%,0) derived by
satisfying the boundary conditions (6) and (7) can be written as
Consider the problem of the flexure of a thin plate which interactswith an isotropic elastic half space region and exhibits either fullybonded conditions or frictionless bilateral contact conditions at theinterface. The deflection of the plate is denoted by w(r) and the plateis subjected to an external normal stress p(r) and the reactive stress atthe plate-elastic half space interface is denoted by q(r). We note thatif there is no separation at the plate-elastic halfspace interface
%/r,0) = w(r) ; ojr,0) =- %W (12)
The differential equation governing axisymmetric flexure of the plateis (see, e.g., Selvadurai [17])
q(r) = pM (13)
where
and D (=Ept3/12(l-Vp2)) is the flexural stiffness of the plate, E*, and Vpare the elastic constants of the plate region and t its thickness.
Operating on (13) with the zeroth-order Hankel transform weobtain
From (10) and (15) we can determine expressions for \v °(£) andi.e.
The flexural stresses and the shear force in the plate can be obtainedfrom the relationships
M = -D
M«=-D
dw
I dw
~ dr
dr
d w
(21)
(22)
(23)
The integral expressions for the flexural moments and the shear forcecan be obtained by substituting the expression (19) for w(r). Thisformally completed the generalized analysis of the contact betweenthe infinite plate and the elastic halfspace. The constant C defines thenature of contact between the plate and the elastic halfspace over theentire region r e (0,°°). The interaction is induced only by p(r) whosezeroth-order Hankel transform is given by
Although the generalized results presented in the previous section areapplicable to any arbitrary form of axisymmetric loading p(r), it isuseful to consider the specific case where the bonded plate is subjectedto a concentrated force of magnitude F which acts at the origin. Thelocalized loading problem is a useful model for examining thebehaviour of the bonded plate problem which is subjected to aspherical rigid indentor (Figure 2). The contact stresses between theplate and the indentor can be approximated by an equivalentconcentrated force. From (24) we obtain
(25)
sphericalindentor
thin plate-
isotropic elastichalfspace (G,v)
Figure 2. Indentation of the bonded plate.
The expressions (19) and (20) evaluated at the origin (r = 0) give
The integrals occurring in (26) and (27) can be evaluated in exact formsince
m ix ax
n sin[(m+l)n/n]; 0 < m + l < n (28)
We obtain
w(0) =J3PC
9G(R*/(29)
)*\2/3 (30)
The results corresponding to the contact conditions involving adhesivebonding and smooth bilateral contact conditions can be obtained byassigning the separate values of C defined by (11). We have
[W(U) \ijondedcontact~D3/3
(3-4v)
4G(l-v)
2/3(31)
[w(0)\\smooth bilateralcontact
(3-WG
213(32)
A useful estimate is the ratio of these displacements which indicatesthe effectiveness of the bonded contact, i.e.,
w' =(3-4v)
4(1- vY
213(33)
In the limit of material incompressibility of the halfspace region v =1/2 and w* = 1. Hence the flexural deflections of the plate areuninfluenced by the nature of the bonding between the plate and thehalfspace provided there is no separation at the interface. This is acharacteristic feature in all elastostatic bonded contact problemsincluding mixed boundary value problems. The underlying rationale forthis behaviour stems from the fact that in the limit of material
incompressibility the surface radial displacement of the halfspace isindependently zero, irrespective of the nature of the normal tractions.The same rationale prevails in the correspondence between the resultfor Kelvin's solution for the interior loading of an infinite space by aconcentrated load and Boussinesq's solution for the surface loading of ahalfspace by a concentrated normal force.
For the lower limit of v = 0, we have
(34)
which indicates that the plate bonded to the halfspace experiences asmaller deflection than the corresponding case involving smoothbilateral contact.Similarly,
and
bondedcontact~
\smooth bilateralcontact
and
D(3-4v)
G(l-v)
D
2/3
(35)
(36)
(3-4v)
2/3(37)
For the material incompressibility q*= *Jl6 = 2.52 and when Poisson's
ratio is zero q* = 16/9 - 2.08, indicating that the contact stresses areincreaseddue to the imposed bonded conditions for all values of v £ (0,1/2),
5 Concluding remarks
It is shown that the contact problem for a thin plate which is resting incomplete contact with an isotropic elastic halfspace can be presented ina general fashion from which results for the cases involving completeadhesive bonding and smooth bilateral contact can be recovered asspecial cases. The possible application of the contact model for theexamination of functionally stiffness graded material surfaces is alsoindicated. The flexural response of the plate can be modified to take
into account possible gradual variations in the elastic modulus of thesurface region which will result in the representation of D by anequivalent estimate for a thin plate laminate in which Ep, Vp and t canbe varied. The results developed for the problem of the homogeneousplate which is subjected to a concentrated external load indicates thatthe deflection of the surface layer is reduced by adhesive bonding andthat the contact stresses beneath the load are increased. It is alsoimportant to note that the surface displacement of the adhesivelybonded plate at the point of application of the load is finite. Incontrast, the modelling of the functionally graded region by an elasticlayer will give rise to unbounded displacements at the point ofapplication of the load. This result will be particularly useful in theinterpretation of indentation tests which can be used to establish thestate of adhesion at the interface.
AcknowledgementThe work associated with this paper was completed during the tenureof a Preistrager Fellowship awarded by the Alexander von HumboldtFoundation. One of the authors (APSS) is grateful to the Foundationand for the hospitality at the Institut A fur Mechanik at theUniversitat Stuttgart.
6 References
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