Contact mechanics

1. Hertzian Contact Analysis
---Analysis of deformation and pressure at the contact of two
elastic solids
Hertzian contact analysis
118CR0138
CR4102 Assignment
by
Suraj Bhan Singh
Course mentor- Prof. Debashish Sarkar

2. The first analysis of the deformation and pressure at the contact of two elastic
solids was given by Hertz and hence such contacts are referred as Hertzian
contact.
When two nominally surfaces are placed in contact , due to surface roughness contact
occurs at discrete junctions . The sum of all contact spots area gives real area of
contact .
As surfaces are not perfectly smooth .
Real area of contact <<<<<< Apparent area of contact

3. In the contact of two rough surfaces , a large number of asperities of different
shapes and sizes are pressed against each other .
But here in this analysis ,tips of surface asperities are considered spherically
shaped.
Hertz considered simpler idealized case of a single asperity and generalized the
results for all asperities.
Hertzian Analysis 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 of the
contact region
(4) The surfaces are frictionless
(5) Static contact ,no sliding

4. Real area of contact is function of :
 Surface texture
 Material properties ( mainly mechanical and thermal )
Mechanical properties – E , H , K ic
Thermal properties – K
 Interfacial Loading conditions

5. What happens when two bodies are in contact :
 Contact will initially occur at only a few points to support the normal load.
 As the normal load is increased, the surfaces move closer together.
 A larger number of higher asperities come into contact, and existing contacts
grow.
 Deformation occurs in the region of the contact spots.
Mode of deformation depends on applied load .
 With increasing load deformation changes from elastic , plastic to viscoelastic.

6. Hertzian contact analysis talks about elastic limit or limit after which plastic
deformation starts .
Schematic of two frictionless solids under static contact

7. The x-y plane in this figure is contact plane
 Solid line – represents initial contact of solids which is at origin
 Dashed line – solids overlapping in presence of load
The separation between two surfaces at radius r before loading is Z1+Z2.
Now considering elastic deformation of two spheres of radius R1 and R2, the
effective modulus is given by :
Where V1 and V2 are Poisson ratio and E1 ,E2 are elastic modulus of each body .

8. Hertzian analysis results for point contact.
The contact area is circular, having a radius a and the contact pressure is elliptical
with p(r) at a radius r in the contact zone. From Hertz analysis, we have the contact
radius as
Area of contact in elastic case is given by:
The pressure distribution is elliptical with maximum pressure at contact center

9. Maximum contact pressure -- P0
Mean contact pressure - Pm
Mean contact pressure – Pm
With , Po =3/2 Pm
Line contact
Contact radius –
Contact pressure –
Maximum shear stress occurs at 0.30 P0 at Z = 0.78a.
Maximum shear stress occurs at 0.31 P0 at Z= 0.48a for a point contact.

10. Onset of plastic deformation
 When normal load is applied bodies initially deform elastically .
 when load increases one of the two body starts deforms plastically which has lower
hardness.
 Plastic zone grow until the entire material surrounding the contact area has gone
through plastic deformation .
 metals , alloys and some non- metal deform predominantly by plastic shear or slip .
 The load at which plastic flow begins in stress field of two contact bodies is called
yield point of softer material .
Two of the yield criteria are most commonly employed .
1. Tresca’s maximum shear stress criteria
2. Von Mises shear strain energy criteria

11. Contours of principal shear stress in sub- surface.
Stress along Z- axis caused by Hertz pressure
acting on circular area of radius a .

12. 1. Tresca’s criteria
In Tresca’s maximum shear stress criterion, the yielding will occur when the
maximum shear stress (half the difference between the maximum and
minimum principal stresses) reaches the yield stress in the pure shear or half
of yield stress in simple tension.
The value of Po for yield is given by ,
2. Von Mises criteria
In the von Mises shear strain energy criterion, yielding will occur when the
distortion energy equals the distortion energy at yield in simple tension or
pure shear.
The value of Po for yield is given by ,

13. Conclusion
1. Hertzian contact analysis is applicable for static contact and tells about load at
which plastic deformation starts .
2. It gives the value of contact radius when two bodies are in contact .
3. It tells about yield value of materials in contact .
4. Using Hertzian contact analysis we can get at what distance on z- axis stress
generating is maximum.