1

Hertzian vs. non-HertzianElastic Contacts

Hertzian contacts (conformal and non-conformal) = curved (spheres, cylinders and ellipsoids) surfaces

Non-Hertzian contacts = pointed (wedge , cone, indenter) or flat (planar) surfaces

Conforming contacts: a is large & P is low Non-conforming contacts: a is small & P is high

Exs: rolling bearings, gears, cam and tappets, wheel-on-rail, etc.

2

Loading Beyond the Elastic Limit (Plastic Deformation)

•As normal load between two contacting bodies is applied, they initially deform elastically according to their Elastic moduli.

•As the load is further increased, one of the two bodies with lower hardness may start to deform plastically:

•If the contact load is sufficiently high then the maximum shear stress will exceed the yield stress of the material, i.e., (max>k), and plastic deformation takes place.

•For wedge or cone contacts, the maximum shear stress lies adjacent to the apex, whereas for curved bodies, plastic enclave lies beneath the contact surface.

•With increasing load, the plastic zone grows until the entire material surrounding the contact has gone through plastic deformation.

•The load at which the plastic flow or plastic yield begins in the complex stress field of two contacting solids is related to the yield point of softer material in a simple tension/compression or pure shear test through an appropriate yield criterion.

http://www.doitpoms.ac.uk/tlplib/metal-forming-1/printall.phpGood (interactive) website for review of stress analysis, Mohr's circle, yield criteria, etc.:

3

• Returning to example of two cylinders in Hertzian line load contact only carrying a normal load, we determined the maximum shear stress occurs at depth of 0.78a and of magnitude0.30po , which is approximately equal to 0.20W/La , since from last class

• There must be some value of W that weaker cylinder starts to plastically deform.• In order to find WY , we need to apply yield criterion to the surface.• The yield of ductile metals and some brittle materials are governed by either Tresca

maximum shear stress criterion or von Mises strain energy criterion. • Suppose a metal yields in simple tension by a stress Y, then principle shear stresses are 1=Y, 2=3=0. The maximum shear stress is =1/2 (which is equal to ½ the greatest difference between the principle stresses) and thus =Y/2.

• The Tresca criterion suggests that in pure shear the material will yield at a shear yield stresswhose magnitude k is =Y/2. (the yield point in pure shear, k, is half the yield stress in simple tension, or compression).

2||

21|,|

21|,|

21max 133221

Yk

3 2 1

Tresca

2τ 31

max

maxTresca 2

Plastic Deformation (continued)

k is yield stress in pure shearY is yield stress in tension (or compression)

aLWpo

2

4

•In the von Mises shear strain energy criterion, yield will occur when the distortion energy equals the distortion energy at yield in simple tension/compression or pure shear.

•Note, the von Mises criterion is based on the strain energy density associated with a change in shape (with a zero volume change) at a material point.•The von Mises criterion depends on the value of the expression:

•In uniaxial tension 1 = Y, 2 = 3 = 0, then the constant is given by 2Y2

•We can also define the yield stress in terms of a pure shear, k. A pure shear stress can be represented in a Mohr’s Circle: In pure shear, the principle stresses are

1 = k, 2 = –k, 3 = 0.So that the von Mises condition predicts

k and Y are related by 6k2=2Y2 or k=Y/√3.

3

)()()(61 2

2213

232

221

Yk

•The difference between the two criterion is not large ~15%, von Mises predicts the larger pure yield stress:•Tresca criterion is employed whenever algebraic simplicity is required. However it does not permit continuous mathematical formulation of resulting yield surface, while von Mises does:

The criterion simply states that when:

the material point is elastic

the material point is yielding

Y

Y

v.m.

v.m.

Plastic Deformation (continued)

Comparison of von Misesand Tresca yield criterion for biaxial stress states.For plane stress, let the principal stresses be 1and 2, with 3 = 0.

5

•Applying the Tresca criterion to two deformable cylinders which plastically deform involves equating the maximum principle shear stress,1 (= 0.30po ) to k (=Y/2).•Thus the critical value of the peak pressure is:

and the mean pressure is:

•The load per unit length of the contact WY/L required to bring material to the point of yieldcan be found by substituting the value of po

Y into the expression for W/L in equation thus:

•We can do the same calculations for two axisymmetric contact of two deformable spheres......

YkpYo 67.13.3

om pp4

YkpYm 3.16.2 since for line contact

RLWEpo

*

*

2

* 76.82

ERYp

ER

LW Y

o

Y

(last class):

(last class):

Example: A TiC ball with a diameter of 5 mm is pressed into a hemispherical recess of 10 mm in diameter on a steel plate. Calculate (a) the normal load necessary to initiate yield in the steel plate (b) the radius of the contact, (c) depth when yielding first occurs. The given parameters are: ETiC = 450 GPa, Esteel = 200 GPa, TiC = 0.3, steel = 0.3 and HTiC = 20 GPa, Hsteel=5 GPa. Assume H~2.8Y.

Plastic Deformation (continued)

6

Elastic-Plastic Contacts

•In (a), the plastic zone is still totally surrounded by region in which the stresses and strains are still elastic. This limits the extent of plastic deformation, since plastic strains must be of the same order as the adjacent elastic strains (contained plasticity due to elastic expansion of surrounding).•In (b), if normal load on contact is increased further, beyond WY/L, then the plastic zone size grows until it breaks out at the free surface (no more contained plasticity/very large plastic strains). Provided there is no strain hardening, the material may be idealized as a rigid perfectly plastic solid which flows under a constant shear flow stress, k.

Elastic-plastic contact: Onset of plasticity below the surface at an indentation pressure pm~1.1Y~H/3 where H is hardness of softer material.

Plastic contact:At a higher load full plasticityis reached and plastic flowextends to the surface at an indentation pressure pm~2.8Y~H

(a) (b)

Figure. Indentation of an elastic-perfectly plastic solid by a spherical indenter

•The mean contact pressure, pm, can be thought of as a measure of the indentation hardness, H, of the material, that is the smallest value of normal pressure required to bring about significant plastic deformation under a rigid indenter. pm~2.8Y~H~5.6k

7

•Within the core the materialis in state of some hydrostatic compression at a pressure, or pm

•Outside core, plastic flow spreads intothe surrounding material, the plastic strain gradually diminishing until they match the elastic strains in the hinterland at some radius c.•For Figs. the elastic-plastic boundary lies at radius c (>a) while at the interface r=a.•The stress in plastic zone (a≤r≤c) are given by and in elastic zone (r≥c):

•At the boundary of core (r=a), the core pressure is•For this model, two conditions must be satisfied, hydrostatic core pressure/stress must equal radial stress component in the plastic region and core is incompressible (further penetration causes volume it displaces to be accommodated by the radial displacements at r=a).

Elastic-Plastic Contact (continued)FYI only

Figs. Cavity model of elastic-plastic contact between a rigid indenter and a deforming material.

p

•The elastic-plastic stress field simplified model produces subsurface displacements by any blunt indenter which are approximately radial from the point of first contact with roughly hemispherical contours of equal strain.

32ln2

rc

Yr

3

32

rc

Yr

32ln2

ac

YYp rm

8

•For cylindrical line contacts within the elastic-plastic regime

where is the value of angle between the local tangent and the surface at the edge of the contact.

•For point contacts within the elastic-plastic regime: (full plasticity ~3Y).

•Thus in both cases the pressure beneath the indenter is a function of the single non-dimensional group (Etan/Y), which is a ratio of the strain imposed by the indenter (which is related to its sharpness/or the value of tan) to the strain capacity of the material as measured by elastic strain at yield (i.e. Y/E).

•Thus it’s a measure of the severity of the loading.

Y3

tanE4ln13

1Yp

Y3

tanEln132

Yp p

Elastic-Plastic Contact (continued)FYI only

9

pm~1.1Y

pm~2.8Y

•Three regions of behavior (elastic, elastic-plastic, and plastic) can be displayed on below graphs

The elasticity of indenter can be taken into account byreplacing E with E*

Fig. Indentation of half-space by sphere and cone. Pm/Y is ratio of the mean contact pressure to the material yield stress and E*tan/Y is a measure of severity of loading.

The same gradual transition from purely elastic contact to fully plastic behavior will occur when the contact is subjected to a tangential load applied simultaneously with normal pressure.

Ra

YE

Ypm

31ln1

32Or another

relationship

size of indentation

pm/Y~2.8

Elastic-Plastic Contact (continued)FYI only

10

Contours of maximum principal stressin half space during normal loading with =0.25. Three minimum principal stresstrajectories initiating at x=-0.8a,x=-a, x=1.2a along which penny-shapedfractures grow.

Contours of von Mises shear stress.Maximum shear stress occurs at value of 0.37po at depth of 0.5a.=0.25.

Maximum tensile stress occurs at r=a

Elastic-Plastic Contact (continued)FYI only