8. Contact Mechanics - WordPress.com · 08/11/2017 · DE2-EA 2.1: M4DE Dr Connor Myant 2017/2018 8. Contact Mechanics Comments and corrections to [email protected] Lecture

DE2-EA 2.1: M4DE Dr Connor Myant

2017/2018

8. Contact Mechanics

Comments and corrections to [email protected]

Lecture resources may be found on Blackboard and at http://connormyant.com

http://connormyant.com/

Dr Connor Myant DE2-EA2.1 M4DE 2

Contents Introduction ............................................................................................................................................ 3

Contact Mechanics .................................................................................................................................. 3

Types of contacts .................................................................................................................................... 5

Bearing contacts ..................................................................................................................................... 6

Case of a cylinder-cylinder contact ..................................................................................................... 6

1. Negligible clearance and rigid bodies ..................................................................................... 7

2. Negligible clearance and elastic bodies .................................................................................. 7

3. Clearance and elastic bodies ................................................................................................... 8

Case of a conformal sphere-sphere contact ....................................................................................... 9

Hertz theory of elastic contact .............................................................................................................. 10

Simplifying Assumptions to Hertz's Theory ...................................................................................... 11

The tangent plane ................................................................................................................................. 12

Contact Radii and Elastic Modulus .................................................................................................... 12

Equations for various contacts.......................................................................................................... 13

The effect of surface roughness........................................................................................................ 15

Adhesive contacts ................................................................................................................................. 15

References ............................................................................................................................................ 16


Introduction In this part of the module we are going to look at the contact between solid bodies, such as bearings,

gears, wheels on the road or human joints. For now we are going to limit ourselves to contacts where

no lubricant is present or when the conditions do not give rise to a lubricating film. This may be

typically either because there is no lubricant present (‘dry’ contacts) or, because the surface speeds

are too low (‘static’ contacts).

In particular, we shall examine the motion and deformation near the area of contact between solids

(‘contact mechanics’). In practice, many of the results can also be used for situations in which a thin

lubricant film is present.

Contact Mechanics Contact mechanics is the study of the deformation of solids that touch each other at one or more

points [1, 2]. The physical and mathematical formulation of the subject is built upon the mechanics of

materials and continuum mechanics and focuses on computations involving elastic, viscoelastic, and

plastic bodies in static or dynamic contact. Central aspects in contact mechanics are the pressures and

adhesion acting perpendicular to the contacting bodies' surfaces (known as the normal direction) and

the frictional stresses acting tangentially between the surfaces.

The original work in contact mechanics dates back to 1882 with the publication of the paper "On the

contact of elastic solids"[3] by Heinrich Hertz. Hertz was attempting to understand how the optical

properties of multiple, stacked lenses might change with the force holding them together. Hertzian

contact stress refers to the localized stresses that develop as two curved surfaces come in contact and

deform slightly under the imposed loads. This amount of deformation is dependent on the modulus

of elasticity of the material in contact. It gives the contact stress as a function of the normal contact

force, the radii of curvature of both bodies and the modulus of elasticity of both bodies. Hertzian

contact stress forms the foundation for the equations for load bearing capabilities and fatigue life in

bearings, gears, and any other bodies where two surfaces are in contact. It is recorded that Hertz

completed his analysis whilst still a student, during the Christmas vacation of 1881!

Design Engineering Example: Some examples of everyday engineering or biological contacts where no lubricant is present:

Road, rail, wheel contacts (no effective lubricant)

Wood working

Gripping a tennis racket

Slow gears in a wrist watch


Design Engineering Example: Contact mechanics provides necessary information for the safe and energy efficient design of technical systems and is a key component in the study of Tribology. Principles of contacts mechanics can be applied in areas such as train wheel-rail contact, braking systems, road tires, bearings, gears, combustion engines, mechanical linkages, human joints, contact lenses, seals, metalworking, electrical contacts, sports ball and racket interactions and many others. Current challenges faced in the field may include stress analysis of contact and coupling members and the influence of lubrication and material design on friction and wear. Applications of contact mechanics further extend into the micro- and nano-technological realm.

Design Engineering Example: Tribology is defined by the IMechE as;

"science and technology of interacting surfaces in relative motion", Originating from the Greek, "tribos" which means rubbing or attrition. First coined in the 19060s and has since emerged as a multidisciplinary subject with important applications in Engineering and Design. In brief Tribology is the study of friction, wear and lubrication. You can break down nearly any assembly, part, or mechanical system and identify elements where friction, wear and lubrication are key factors in it’s performance.


Types of contacts We can divide contacts into two distinct types;

Conforming (or conformal) contacts: between a convex surface (male cylinder or sphere) and a

concave surface (female cylinder or sphere: bore or hemispherical cup).

Examples Characteristics

• Shoe and floor • Pin-in-hole • Journal bearing • Ball-in-socket • Eye and eye lid

• Low pressure between the surfaces (~ Pa) • Large area of contact • Small elastic deformations

Figure 8.1 Classic example of a highly conformal contact from a universal ball joint.

Non-conforming (or counterformal) contacts: between two convex surfaces. Examples Characteristics

• Ball or roller bearings • Gear teeth • Cam and tappet • Baseball bat and ball

• High local pressures (~GPa) • Area of contact small compared with other dimensions • Significant elastic (or plastic) deformations


Figure 8.2. The contact point between two touching snooker balls is a perfect example of highly non-

conformal contact.

Bearing contacts Bearing contacts are a particular case of contact mechanics often occurring in conformal contacts. A

contact between a male part (convex) and a female part (concave) is considered when the radii of

curvature are close to one another. There is no tightening and the joint slides with no friction

therefore, the contact forces are normal to the tangent of the contact surface.

Moreover, bearing pressure is restricted to the case where the charge can be described by a radial

force pointing towards the centre of the joint.

Case of a cylinder-cylinder contact In the case of a revolute joint or of a hinge joint, there is a contact between a male cylinder and a

female cylinder. The complexity depends on the situation, Figure 8.3 shows the three cases which can

be distinguished by:

1. the parts have negligible clearance and are rigid bodies,

2. the parts have negligible clearance and are elastic bodies;

3. the clearance cannot be ignored and the parts are elastic bodies.

In all three cases the axes of the cylinders are along the z-axis, and two external forces apply to the

male cylinder:

- a force 𝐹 along the y-axis, the load;

- the action of the bore (contact pressure).

The main concern is the contact pressure with the bore, which is uniformly distributed along the 𝑧-

axis.


Figure 8.3. Bearing pressure for a cylinder-cylinder contact

1. Negligible clearance and rigid bodies In this first modelling, the pressure is uniform. It is equal to;

𝑃 =𝑟𝑎𝑑𝑖𝑎𝑙 𝑙𝑜𝑎𝑑

𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑟𝑒𝑎=

𝐹

𝐷 × 𝐿

Where 𝐷 is the nominal diameter of both male and female cylinders and 𝐿 the guiding length.

Figure 8.4. Uniform bearing pressure: case of rigid bodies when the clearing can be neglected.

2. Negligible clearance and elastic bodies If it is considered that the parts deform elastically, then the contact pressure is no longer uniform and

transforms to a sinusoidal repartition;


𝑃(𝜃) = 𝑃𝑚𝑎𝑥 × 𝑐𝑜𝑠𝜃

Where

𝑃𝑚𝑎𝑥 =4

𝜋×

𝐹

𝐿𝐷

Figure 8.5. Bearing pressure with a sinusoid repartition: case of elastic bodies when the clearing can

be neglected.

3. Clearance and elastic bodies In cases where the clearance cannot be neglected, the contact between the male part is no longer the

whole half-cylinder surface but is limited to a 2𝜃0 angle. The pressure follows Hooke's law:

𝑃(𝜃) = 𝐾Φ𝛼(𝜃)

Where;

- 𝐾 is the rigidity coefficient; a positive real number that represents the rigidity of the

materials

- Φ(θ) is the radial displacement of the contact point at the angle 𝜃

- 𝛼 is a coefficient that represents the behaviour of the material:

• 𝛼 = 1 for metals (purely elastic behaviour),

• 𝛼 > 1 for polymers (viscoelastic or viscoplastic behaviour).

The maximum pressure is defined as;

𝑃𝑚𝑎𝑥 =4𝐹

𝐿𝐷×

1 − 𝑐𝑜𝑠𝜃0

2𝜃0 − 𝑠𝑖𝑛2𝜃0


the angle 𝜃0 is in radians.

Figure 8.6. Bearing pressure in case of elastic bodies when the clearance must be taken into account.

Case of a conformal sphere-sphere contact A sphere-sphere contact corresponds to a spherical joint (socket/ball), such as the one shown in Figure

8.1. It can also describe the situation of bearing balls.

For cases of uniform pressure we can model the contact like the cases above: i.e. when the parts are

considered as rigid bodies and the clearance can be neglected (Figure 8.7.a), then the mean pressure

is supposed to be uniform. It can also be calculated considering the projected area;

𝑃 =𝑟𝑎𝑑𝑖𝑎𝑙 𝑙𝑜𝑎𝑑

𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑟𝑒𝑎=

𝐹

𝜋𝑅2

As in the case of cylinder-cylinder contact, when the parts are modelled as elastic bodies with a

negligible clearance (Figure 8.7.b), then the mean pressure can be modelled with a sinusoidal

repartition, and simplified to;

𝑃 =3𝐹

2𝜋𝑅2


Figure 8.7. Bearing pressure in the case of a sphere-sphere contact.

When the clearance cannot be neglected (Figure 8.7.c), it is then necessary to know the value of the

half contact angle 𝜃0, which cannot be determined in a simple way and must be measured. When this

value is not available, the Hertz contact theory can be used.

Hertz theory of elastic contact Hertz was concerned with the contact between optical lenses, but his analysis, the results of which

are summarised here, is widely used for most non-conformal contacts and is especially important for

those in rolling bearings, gears, railways (rail-and-wheel) and engine valve-gear (cam-and-tappet).

The Hertz theory is normally only valid when the surfaces are non-conformal; one surface must be

convex, the other one must be also convex or plane. However, we can, and do, apply Hertz theory for

conformal contacts but the results must be considered with great care. It is an approximation only,

and only valid when the inner radius of the container 𝑅1 is far greater than the outer radius of the

content 𝑅2, in which case the surface container is then seen as flat by the content. However, in all

cases, the pressure that is calculated with the Hertz theory is greater than the actual pressure (because

the contact surface of the model is smaller than the real contact surface), which affords designers with

a safety margin for their design.


Figure 8.8. Hertz contact stress in the case of a male cylinder-female cylinder contact.

Figure 8.9. Hertz contact stress in the case of a male sphere-female sphere contact.

Simplifying Assumptions to Hertz's Theory Hertz's model of contact stress is based on the following simplifying assumptions:

The materials in contact are homogeneous and the yield stress is not exceeded,

Contact stress is caused by the load which is normal to the contact tangent plane,

The contact area is very small compared with the dimensions of the contacting solids,

The contacting solids are at rest and in equilibrium,

The effect of surface roughness is negligible,

There is now adhesive force acting between the two contacting surfaces.


The tangent plane At the point of contact the surfaces are tangent to a plane and have a common normal. This means

the stress experienced by the contacting bodies is caused by the load which is normal to the contact

tangent plane which effectively means that there are no tangential forces acting between the solids.

Figure 8.10. Contact tangent plane for a non-conformal contact.

Contact Radii and Elastic Modulus In general, if the elements in contact have three-dimensional shapes, such as balls in a rolling-element

bearing, the contact can be simplified into that of an equivalent sphere and a rigid half space (flat

plane). The contact in this idealised model is said to have a radius or relative curvature, 𝑅’, and a

contact elastic modulus of 𝐸’.

Suffices 1, 2 refer to the two bodies. 𝑥 and 𝑦, refer to the directions of principal (i.e. maximum or

minimum) curvature, which are assumed to coincide for the two bodies. Thus 𝑅𝑥1 is the radius of body

1 in the 𝑥 direction etc.

We define the following:

Radii of relative curvature (convex positive, concave negative):

1

𝑅𝑥=

1

𝑅𝑥1+

1

𝑅𝑥2

And,


1

𝑅𝑦=

1

𝑅𝑦1+

1

𝑅𝑦2

1

𝑅′=

1

𝑅𝑥+

1

𝑅𝑦

If either 𝑅𝑥 or 𝑅𝑦 = ∞ , then the contact patch is rectangular (a ‘line’ contact). If 𝑅𝑥 = 𝑅𝑦, then the

contact patch is circular. We will not be considering elliptical contacts here.

Contact modulus:

𝐸′ = (1 − 𝑣1

2

𝐸1+

1 − 𝑣22

𝐸2)

−1

where 𝑣 is the Poisson's ratio of the material and 𝐸 its Young's modulus.

Equations for various contacts Semi

contact width or radius, 𝑎

Maximum contact pressure, 𝑃𝑚𝑎𝑥

Mean pressure, �̅� Depth of

indentation, 𝛿

Contact between a ball (sphere) and flat surface (a half-space)

𝑎

= (3𝐹𝑅′

4𝐸′)

13


2𝜋𝑎2

= (1

𝜋) (

6𝐹𝐸′2

𝑅′2)

13

�̅� =𝐹

𝜋𝑎2

𝛿 =𝑎2

𝑅′

= (9𝐹2

16𝐸′2𝑅′)

13

Contact between two spheres

For contact between two spheres of radii 𝑅1 and 𝑅2, the area of contact is a circle of radius 𝑎. The equations are the same as for a ball on flat (above) except that the effective radius 𝑅’ has changed.


Contact between two crossed cylinders

This is equivalent to contact between a ball on flat. Consider rotating the image until you are looking straight down one cylinder!

Contact between a rigid cylinder with flat-ended and an elastic half-space

𝑎 is the radius of the cylinder

𝑃𝑚𝑎𝑥 =1

𝜋𝐸′

𝛿

𝑎 �̅� =

𝐹

𝜋𝑎2 𝛿 =

𝐹

2𝑎𝐸′

Contact between a rigid conical indenter and an elastic half-space

𝑎 = 𝑧𝑡𝑎𝑛𝜃 Where 𝑧 is the depth of the contact region

The stress has a logarithmic singularity at the tip of the cone

�̅� =4𝐹

𝜋𝑎2 𝛿 =

𝜋

2𝑧


Contact between two parallel cylinders

𝑎

= (4𝐹𝑅′

𝜋𝐿𝐸′)

12


𝜋𝑎𝐿

= (𝐹𝐸′

𝜋𝐿𝑅′)

12

�̅� =𝐹

2𝑎𝐿

𝛿

=2𝑃

𝜋[1 − 𝑣1

2

𝐸1

𝑙𝑛 (4𝑅1

𝑎

− 0.5)

+1 − 𝑣2

2

𝐸2

𝑙𝑛 (4𝑅2

𝑎

− 0.5)]

The effect of surface roughness The above solutions and methods for calculating contact size/pressure/stress is only applicable to

smooth surfaces. In practice all surfaces are rough, to some degree. This causes the actual area of

contact to be less than that predicted by Hertz and the local pressures and subsurface stresses to be

higher - often much higher. Normally surface roughness is ignored and for many bearing contacts

(which have very smooth surface) this is fine.

Adhesive contacts There are several engineering systems that involve the rubbing contact of highly elastic bodies,

including windscreen wipers, tyres and elastomeric seals. In biological systems this situation is almost

universal since one or both of the contacting bodies is generally formed of human, animal or plant

tissue. With a highly elastic body in contact, the contact area becomes large, mainly because the two

surfaces are able to fully conform despite their roughness. When this occurs, adhesive forces become

significant compared to the applied load and so must be included in the contact model.


The Hertzian model of contact does not consider adhesion possible. However, in the late 1960s,

several contradictions were observed when the Hertz theory was compared with experiments

involving contact between rubber and glass spheres.

It was observed [5] that, though Hertz theory applied at large loads, at low loads;

• the area of contact was larger than that predicted by Hertz theory,

• the area of contact had a non-zero value even when the load was removed, and

• there was strong adhesion if the contacting surfaces were clean and dry.

References 1. Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.

2. Popov, Valentin L., 2010, Contact Mechanics and Friction. Physical Principles and

Applications, Springer-Verlag, 362 p., ISBN 978-3-642-10802-0.

3. H. Hertz, Über die berührung fester elastischer Körper (On the contact of rigid elastic

solids). In: Miscellaneous Papers. Jones and Schott, Editors, J. reine und angewandte

Mathematik 92, Macmillan, London (1896), p. 156 English translation: Hertz, H

4. Bradley, RS., 1932, The cohesive force between solid surfaces and the surface energy of

solids, Philosophical Magazine Series 7, 13(86), pp. 853--862.

5. K. L. Johnson and K. Kendall and A. D. Roberts, Surface energy and the contact of elastic

solids, Proc. R. Soc. London A 324 (1971) 301-313

6. Derjaguin, BV and Muller, VM and Toporov, Y.P., 1975, Effect of contact deformations on

the adhesion of particles, Journal of Colloid and Interface Science, 53(2), pp. 314-326

Design Engineering Example: There are numerous biological contacts, or bio-tribological systems, that can be studied using the contact mechanics discussed in this chapter. Many of these are of great interest to design engineers. Human-related:

• Natural and replacement human joints • Tooth wear • Hair conditioners • Skin creams • Contact lens lubrication • Friction and “feel” of fabrics • Oral processing (eating) • Catheters and surgical instruments

Non human-related:

• Mobility of simple organisms • Gecko adhesion and motion • Snail lubrication • Leaf cleansing processes


7. Muller, VM and Derjaguin, BV and Toporov, Y.P., 1983, On two methods of calculation of the

force of sticking of an elastic sphere to a rigid plane, Colloids and Surfaces, 7(3), pp. 251-

259.

8. Tabor, D., 1977, Surface forces and surface interactions, Journal of Colloid and Interface

Science, 58(1), pp. 2-13

Bearing pressure images attributed to:

Cdang (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia

Commons

Documents

8. Contact Mechanics - WordPress.com · 08/11/2017 · DE2-EA 2.1: M4DE Dr Connor Myant 2017/2018 8. Contact Mechanics Comments and corrections to [email protected] Lecture