CONTACT MECHANICS

Prepared by:-

Kumar Ankur

Contact Between Solid Surfaces

Introduction

When two nominally flat surfaces are placed in contact, surface roughnesscauses contact to occur at discrete contact spots (junctions). The sum of theareas of all the contact spots constitutes the real (true) area of contact orsimply contact area, and for most materials with applied load, this will be onlya small fraction of the apparent (nominal) area of contact (that which wouldoccur if the surfaces were perfectly smooth).

Figure 1. Schematic representation of an interface, showing the apparent andreal areas of contact.

• The real area of contact is a function of the surface texture, materialproperties and interfacial loading conditions. The proximity of theasperities results in adhesive contacts caused by interatomic interactions.

• When two surfaces move relative to each other, the friction force iscontributed by adhesion of these asperities and other sources of surfaceinteractions. Repeated surface interactions and surface and subsurfacestresses, developed at the interface, result in formation of wear particlesand eventual failure.

• A smaller real area of contact results in a lower degree of interaction,leading generally to lower wear.

• The problem of relating friction and wear to the surface texture andmaterial properties generally involves the determination of the real areaof contact. Therefore, understanding of friction and wear requiresunderstanding of the mechanics of contact of solid bodies.

• During the contact of two surfaces, contact will initially occur at only a fewpoints to support the normal load (force). As the normal load is increased,the surfaces move closer together, a larger number of higher asperities onthe two surfaces come into contact, and existing contacts grow to supportthe increasing load.

• Deformation occurs in the region of the contact spots, establishingstresses that oppose the applied load. The mode of surface deformationmay be elastic, plastic, viscoelastic or viscoplastic, and depends onnominal normal and shear stresses.

• The local stresses at the contact spots are much higher than the nominal stresses. Although nominal stresses may be in the elastic range, the local stresses may exceed the elastic limit (yield strength) and the contact will yield plastically.

• In most contact situations, some asperities are deformed elastically, while others are deformed plastically; the load induces a generally elastic deformation of the solid bodies but at the tips of the asperities, where the actual contact occurs, local plastic deformation may take place.

Analysis of the Contacts Single Asperity Contact of Homogeneous and Frictionless Solids

A single asperity contact reduces to a problem of deformation of two curvedbodies in contact. For the analysis of a single asperity contact, it is convenientto model an asperity as a small spherically shaped protuberance.

Elastic Contact

The first analysis of the deformation and pressure at the contact of twoelastic solids with geometries defined by quadratic surfaces is due to Hertz(1882) and such contacts are referred to as Hertizian contact.

His analysis is based on the following assumptions:

(1) the surfaces are continuous, smooth and nonconforming,

(2) the strains are small,

(3) each solid can be considered as an elastic half-space in the proximity ofthe contact region,

4) the surfaces are frictionless.

Figure 2. Schematic of two frictionless solids of general shape (but chosen convex

for convenience) in static contact.

Two solids of general shape (but chosen convex for convenience) loaded together are shown in cross section after deformation in Figure 2.

• The x-y plane is the contact plane. The point of first contact is taken as the origin of a Cartesian coordinate system in which the x-y plane is the common tangent plane to the two surfaces and the z axis lies along the common normal directed positively into the lower solid.

• The separation between the two surfaces at radius r before loading is z1 + z2. • During the compression by a normal force W, distant points in the two bodies

T1 and T2 move towards O, parallel to the z axis, by vertical displacements δ1 and δ2, respectively.

• If the solids did not deform their profiles would overlap as shown by the dotted lines in Figure 2.

• The elastic deformation results in displacement of the surface outside the footprint such that the contact size (2a) is less than the overlap length resulting from intersection of the dotted lines.

• Due to the contact pressure the surface of each body is displaced parallel to Oz by an amount u ̄z1 and u ̄z2 (measured positive into each body), relative to the distant points T1 and T2, and after displacement points S1 and S2 become coincident.

• The total displacement δ = δ1 + δ2 is called total interference or normalapproach which is defined as the distance by which points on the twobodies remote from the deformation zone move together on applicationof a normal load; it arises from the flattening and displacement of thesurface within the deformation zone.

Now consider the problem of elastic deformation of two spheres of radii R1and R2 in solid contact with an applied normal load W. The contact area iscircular, having a radius a and the contact pressure is elliptical with p(r) at aradius r in the contact zone. From Hertz analysis, we have the contact radius

(1a)

The area of contact for the elastic case is

Are = πa2 = π Rδ (1b)

The displacements within the contact case can be expressed as

(2a)

And (2b)

The pressure distribution is elliptical with the maximum pressure at the contact center, (3a)

with the maximum contact pressure p0 being 3/2 times the mean contact pressure, pm , given as

(3b)

where the composite or effective modulus

(4)

and the composite or effective curvature,

(5)

The parameters E and ν are Young’s modulus of elasticity and the Poisson’sratio, respectively; subscripts 1 and 2 refer to the two bodies. Note that thereal area of contact in Eq 1b is exactly half the area covered by intersection ofdotted lines (=2π Rδ). From Eq 1b also note that the area of contact increasesas (normal load)2/3.

Next we examine the stress distributions at the surface and within the twosolids, for the Hertz pressure exerted between two frictionless elastic spheresin contact.

The Cartesian components of the stress field are given by Hamilton and Goodman (1966).

For pressure applied to a circular region, the expressions for the stress field in polar coordinates. The polar components of the stress field in the surface z = 0, inside the loaded circle (r < a) are (Johnson, 1985)

(6a)

(6b)

(6c)

and outside the circle

(7)

• They are all compressive except at the very edge of contact where theradial stress is tensile having a maximum value of (1−2ν)p0/3 at the edgeof the circle at r = a.

• This is the maximum tensile stress occurring anywhere in the contact andit is held responsible for the ring cracks which are observed to form whenbrittle materials such as glass are pressed into contact (Lawn, 1993).

• At the center the radial stress is compressive and of value (1 + 2ν)p0/2.

• Thus, for an incompressible material (ν = 0.5) the stress at the origin ishydrostatic.

• Outside the contact area, the radial and hoop (circumferential) stressesare of equal magnitude and are tensile and compressive, respectively.

.

Limit of Elastic Deformation

• As the normal load between two contacting bodies is applied, they initiallydeform elastically according to their Young’s moduli of elasticity.

• As the load is increased, one of the two bodies with lower hardness maystart to deform plastically.

• As the normal load is further increased, the plastic zone grows until theentire material surrounding the contact has gone through plasticdeformation.

• Metals, alloys and some nonmetals and brittle materials deformpredominantly by “plastic shear” or “slip” in which one plane of atomsslides over the next adjacent plane.

• The load at which the plastic flow or plastic yield begins in the complexstress field of two contacting solids is related to the yield point of thesofter material in a simple tension or pure shear test through anappropriate yield criterion.

Two of the yield criteria most commonly employed for most ductile materials as well as sometimes for brittle materials are described here :-

• In Tresca’s maximum shear stress criterion, the yielding will occur when themaximum shear stress (half the difference between the maximum andminimum principal stresses) reaches the yield stress in the pure shear or halfof yield stress in simple tension,

(8)

Here σ 1 , σ 2 and σ 3 are the principal stresses in the state of complex stress. Theyield point in pure shear k is half the yield stress in simple tension (orcompression) Y.

• In the von Mises shear strain energy criterion, yielding will occur when thedistortion energy equals the distortion energy at yield in simple tension orpure shear.

Therefore yielding occurs when the square root of the second invariant of the stress deviator tensor (Sij) reaches the yield stress in simple shear or (1/√3) of yield stress in simple tension,

( (9)

√J2 and √3J2 are referred to as von Mises stress in shear and in tension,respectively.

• Note that the yield stress in pure shear is (1/√3) times the yield stress insimple tension. Thus the von Mises criterion predicts a pure shear yieldstress which is about 15% higher than predicted by the Tresca criterion

• Based on Lode’s experiments (Lode, 1926), the von Mises criterion usuallyfits the experimental data of metallic specimens better than othertheories.

• However, the difference in the predictions of the two criteria is not large.

• Tresca’s criterion is employed for its algebraic simplicity to determine the limit of elastic deformation. However, this criterion often does not permit continuous mathematical formulation of the resulting yield surface,while von Mises criterion does. Therefore, von Mises criterion is employed more often than Tresca’s in plasticity analyses.

• In the case of axisymmetric contact of two spheres, maximum shear stress occurs beneath the surface on the axis of symmetry, z axis (Figure 1). Along this axis, σ r , σ θ , and σ z are principal stresses and σr = σθ. For ν = 0.3, the value of 1/2|σz −σr| is 0.31p0 at a depth of 0.48 a. Thus, by the Tresca criterion, the value of p0 for yield is given by

– (10)

while by the von Mises criterion

(11)

The load to initiate yield Wy is given by Equations 3b and 10,

(12)

The maximum normal approach before the onset of plastic deformation is given by Equations 2b, 3b and 10 or 11,

(13)

• Note that yielding would occur in one of the two solids with a lower yieldstress or hardness. Further note that to carry a high load (highinterference) without yielding it is desirable to choose a material with ahigh yield strength or hardness and with a low elastic modulus.

Table. Stress and Deformation Formulas for Normal Contact of Elastic Solids (Hertz Contact).

Example Problem 3.2.1

A ceramic ball with a radius of 5 mm is pressed into a hemispherical recess of10 mm in radius in a steel plate. (a) What normal load is necessary to initiateyield in the steel plate; (b) what is the radius of the contact; and (c) at whatdepth does yield first occur? The given parameters are: Eceramic = 450 GPa, Esteel

= 200 GPa, νceramic = 0.3, νsteel = 0.3, Hceramic = 20 GPa and Hsteel = 5 GPa. Assumethat H ∼ 2.8 Y.

Solution

The composite modulus is given by

E∗ = 152.2 GPa

The composite radius is given by

(a) Yield will occur when

and

(b)

(c) Yield occurs at a depth of 0.48 a

= 0.48 × 0.29 mm = 0.14 mm.