This document discusses Mohr's circle and its representation of different states of stress, including uniaxial tension and compression, biaxial tension and compression, triaxial tension and compression, and combined tension and compression. It also covers engineering stress-strain curves and how they are obtained from tensile testing. Key parameters like yield strength, tensile strength, ductility measures, and how the curve is influenced by material properties and prior processing are summarized. Videos are embedded to demonstrate some of the stress states and a wire drawing process.

1. Mohr’s circle & Formability
Dr.R.Narayanasamy,B.E.,M.Tech.,M.Engg.,Ph.D.,(D.Sc.)
Professor,
Department of Production Engineering,
National Institute of Technology, Tiruchirappalli – 620 015.
Tamil Nadu, India

2. σ2
σ3
σ1
σ
τ1
τ3
τ2
τ
Mohr’s circle representation of
a three-dimensional state of
stress.
σ2

3. Mohr’s circle (three – dimensional) for the state of
stress
Uniaxial tension
- σ σ
σ1
τ max=σ1/ 2
τ
σ2 = σ3 = 0
(a)
3
2
https://www.youtube.com/watch?v=IIWSgApn
IpM

4. Uniaxial tension (video)

5. Mohr’s circle (three – dimensional) for the state of
stress
Uniaxial compression
σ1 = σ2 = 0
σ3
τ max
2
1
https://www.youtube.com/watch?v=53QjIamB
W3w

6. Uniaxial compression (video)

7. Uniaxial tension/Compression
• Tensile/Compressive stress is applied in one
direction
• A maximum shear stress is achieved in the
positive region for tensile stress and vice versa
for the compressive stress
• The maximum shear stress is uniform for
tension & compression and which is larger
than biaxial tension/compression.

8. Mohr’s circle (three – dimensional) for the state of
stress
biaxial tension
σ3 = 0
σ2
σ1
τ max = τ2
τ1 τ3
3
σ2
σ3= 0
http://youtube.com/watch?v=FlOjVZis4SA

9. Mohr’s circle (three – dimensional) for the state of stress
biaxial tension
• σ2 - Tensile stress right angle to σ1 (Tensile
stress)
• Decrease in principal shear stress on two sets
of planes out of three
• Maximum shear stress is decreased from what
would be for uniaxial tension

10. biaxial tension simulation (video)

11. Mohr’s circle (three – dimensional) for the state of
stress
biaxial compression
σ1
σ3 = 0
τ1τ3
σ2
σ2
3
τ max = τ2
σ3 = 0
https://www.youtube.com/watch?v=h_bL40T5
dl4

12. biaxial compression simulation (video)

13. Mohr’s circle (three – dimensional) for the state of
stress
triaxial tension (unequal)
σ1
σ2 = σ3
τ max = τ2= τ3
σ3
σ2
σ1 = 2 σ2 = 2σ3

14. Mohr’s circle (three – dimensional) for the state of
stress
triaxial tension (equal)
σ1
σ2 = σ3
τ max = τ2 =τ3
σ3
σ2
σ1 = σ2 = σ3
σ1 = σ2 = σ3

15. Mohr’s circle (three – dimensional) for the state of stress
triaxial tension (unequal)&(Equal)
• Tensile stress applied in the three directions
• Maximum shear stress is reduced
• For equal triaxial tension Mohr’s circle will be
a point and no shear stress acts on the body.
This will reduce the ductility, because plastic
deformation is produced by shear stress
• Equal triaxial state of stress will lead to brittle
failure

16. Mohr’s circle (three – dimensional) for the state of
stress
triaxial compression (unequal)
σ2 = σ3
σ1
τ max = τ2 = τ3
σ2
σ3

17. triaxial compression simulation (video)

18. Mohr’s circle (three – dimensional) for the state of
stress
combined tension and compression
σ1
τ max = τ2 = τ3
σ3
σ2
σ1 = -2 σ2 = -2σ3
σ2 = σ3 σ1
τ max = τ2 = τ3
Example: Wire drawing

19. Mohr’s circle (three – dimensional) for the state of stress
combined tension and compression (Wire drawing)
• Compressive stresses are applied to the lateral
surface of tensile stress, which gives large
maximum shear stress than uniaxial stress.
• Materials subjected to tension + compression will
have good formability before fracture.
• It is applied in plastic working of metals. (Ex:
More ductility is achieved in wire drawing
through a die. Here applied load is tensile in
nature whereas the reaction between the die and
metal is compressive in nature).

20. Drawing operation (video)

21. Drawing operation (video)

22. Engineering Stress Strain Curve
• Engineering tension test provide basic design
information on strength of material.
• In tension test uniaxial tensile force is applied
on the specimen and its elongation was
observed.
• Engineering stress strain curve was formed
using load vs elongation
• The stress used in this curve is average
longitudinal stress = load/original c/s area

23. Engineering stress strain curve
•Tensile strength & yield
strength (strength parameters)
•Percentage elongation and
reduction of area (ductility
parameters)

24. Engineering stress strain curve
• The strain used for engineering stress strain is
average linear strain. (elongation of gauge
length/original length)
• Both stress and strain are obtained by dividing
the load and elongation by constant factors
the (load-elongation) curve will have same
shape as engineering stress strain curve.

25. Engineering stress strain curve
• The shape and magnitude of stress strain
curve depends on:
1. chemical composition of material
2. heat treatment
3. prior history of plastic deformation
4. strain rate
5. temperature
6. state of stress imposed during testing.

26. Engineering stress strain curve
• Stress is proportional to strain (Elastic region)
• As load increases the value corresponding to
yield strength the specimen undergoes plastic
deformation
• The stress to produce continued plastic
deformation increases with increasing plastic
strain (metal strain hardens)
• The volume of specimen remains constant during
plastic deformation

27. Engineering stress strain curve
• As the specimen elongates it decreases uniformly
along the gauge length in cross sectional area.
• Initially, strain hardening compensates for
decrease in area and the engineering stress in
proportional to engineering strain (engineering
stress is proportional to load)
• At certain point decrease in cross sectional area is
> than increase in deformation load arising from
strain hardening (this condition occurs in the
specimen which is weaker than the rest portion).
The specimen begins to neck/thin down locally.

28. Reference
• Mechanical Metallurgy by George E.Dieter,
McGraw – Hill Publication, London,1988.

29. Thank You