This document discusses plastic instability in uniaxial tension testing of materials. It defines true stress and strain, which account for changes in cross-sectional area during tension, and explains how they relate to engineering stress and strain. The condition for plastic instability is derived as the maximum in the true stress-strain curve where the rate of true stress increase with respect to true strain is equal to the true stress. This occurs at a uniform true strain value called the instability strain. Examples of true stress-strain curves and determinations of the strain hardening exponent are also provided.

1. Plastic Instability in Uniaxial
Tension
By
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. True Stress vs Engineering Stress
• Engineering Stress(s)=Load(P)/A0
A0 is the initial area of cross section of
tensile sample.
• True Stress(σ) =Load(P)/Ai
Ai is the instantaneous area of cross section
• In the Load-Extension graph, volume constancy
principle can be applied upto the maximum load
point.

3. Engineering Strain vs True Strain
• Engineering Strain(e)= Change in length(Δl)/𝑙0
l0 is the initial gauge length of tensile sample.
lf is the final or intermediate length of
sample during tensile test.
• Natural (or) True Strain(ϵ) = 𝑙0
𝑙 𝑓 𝑑𝑙
𝑙
• True Strain(ϵ) =ln(lf/l0) = ln((l0 + Δl)/l0) = ln(1+e)

4. Plastic Instability in Uniaxial Tension
• Load(P) = σ A
where σ is True Stress and A is instantaneous
area of cross section.
• Differentiating on the both sides, we get
dP=dσ A + σdA
• At maximum load, the value of dP = 0 .
• The equation becomes: (dσ/σ) = -(dA/A).

5. Plastic Instability in Uniaxial Tension
cont..
• For Constant Volume : A l = Constant .
• Differentiating on both sides, we get
dA l +A dl =0
This can be written as:
-(dA/A) = (dl/l) = dϵ
• Therefore, the above equation becomes:
(dσ/σ) = dϵ
This can be written as: (dσ/dϵ) = σ
This is the condition for plastic instability.

6. True Stress vs True Strain Plot

7. Plastic Instability in Uniaxial Tension

8. True Stress vs True Strain
• True Stress(σ) and True Strain(ϵ) can be related
using Power Law equation according to Ludwik as
follows:
where
σ is True Stress, ϵ is True Strain, n is strain
hardening exponent and K is the strength
coefficient.
• Unit for True Stress and Strength coefficient is
MPa (metric unit).
• K and n are to be determined by curve fitting.
σ = 𝐾ϵ 𝑛

9. Determination of K and n- values

10. Determination of K and n- values
Strain hardening exponent (n) is the ratio of the
physical distance (mm) of “a” by the physical
distance (mm) of “b”.
n – value has no unit.
n – value represents the strain hardening ability
of the material.
n – value varies from 0.1 to 0.5 for conventional
metals.

11. The effect of Strain hardening
exponent on True stress -True strain

12. Typical K and n values for various
metals

13. Plastic Instability in uniaxial tension
We know that:
Differentiating on both sides, we get:
(Or)
This can be written as: (Or)

14. Plastic Instability in uniaxial tension
cont…
At maximum load, (Or)
Hence, the above expression becomes:
Therefore, n = ε
Where ε is the true uniform strain, which is
denoted by the symbol έ.

15. Plastic Instability in uniaxial tension
cont…
It is important to note that rate of strain
hardening is not identical with strain
hardening exponent (n)value.

16. Plastic Instability in uniaxial tension
The rate of strain hardening can be written as
follows:

17. Plastic Instability in uniaxial tension
Considere’s construction for the determination of the point of maximum load

18. Necking in uniaxial tension test
illustration of diffuse necking and localized
necking in case of sheet metal tensile
specimen

19. Diffused necking & Localized necking
• Necking in cylindrical specimen is symmetrical around
tensile axis in case of isotropic
• For sheet tensile specimen width is greater than
thickness and two type of tensile instability occurs
• Diffused necking: Its extension is greater than
thickness. It will end in fracture and also it is followed
by second instability (localized necking)
• Localized necking: Neck is narrow width is
approximately equal to thickness inclined at an angle
to the specimen axis.
• Localized necking corresponds to plane-strain
deformation

20. Diffused necking & Localized necking
cont…

21. Diffused necking & Localized necking
cont…

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

23. Thank you