Hello,
According to the power law, Stress=K*strain^n. Could someone please explain why an increase in strain hardening exponent n results in less material necking from this equation?
Please see the attached plot, here I used K=1 to ignore it's effect and I have varied n for n different values.
It seems that for the same stress put into the material, the curve at n=0.1 produces the smallest strain, where as the curve at n=0.5 produces the largest strain, up to strain=1. If the stiffest material is desired, is n=0.1 not more favourable here?
This behaviour changes at strains greater than 1, this has also left me confused.
Any clarification on this will be very much appreciated.
Thank you!
Hi Hamid Reza Attar ,
according to the change of the behaviour at strain=1 this is purely mathematical since the exponent is n
Hi Dimitris Dimitriou ,
Thanks for your reply.
When you mention more elastic behaviour, do you mean that, for example, to get the same displacement (strain), a material with a lower strain hardening exponent would result in a higher induced stress? And this higher stress means more necking?
Thanks!
Yes Hamid Reza Attar somethink like that. The way i understand this is that with a lower strain hardening exponent the behaviour is more "plastic" and that reflects to the plateau of the blue line in your plot. This plateau itself shows the larger necking.
Hi Hamid Reza Attar,
First, the stress material can carry can be expressed \sigma=\sigma_yield+K*\epsilon_plastic^n. So materials with different n values have also different yield stresses also different K constants as well. Secondly, if n=1 the equation would be linear, similar to Hooke's Law of elasticity. On the other side, if n=0, it means that the material flow under constant stress (like creep) and you would get a perfect plasticity (stress-strain curve becomes flat after yielding). Thus, as n increases you would get more "elastic" or "rigid" plastic behaviour of material meaning it would show less "plastic" necking".
Further explanation would be this: Necking occurs when material strain hardening is less than the cross-sectional area decrease. (cross-sectional area decrease due to elongation in 1 direction in tensile test due to volume preserved in metal plasticity) As the slope of stress-strain curve decreases, at one point the increase of stress due to strain hardening won't be enough for cross-sectional area decrease due to Poisson's ratio. Therefore, strain will be localized causing a further localized decrease in cross-sectional area causing neck. Since the decrease of the slope of the stress strain hardening curve for bigger "n" values is more gentle, necking behaviour would likely to happen at higher strains (if material would not fail by then) If n=1 then the slope of the stress-strain hardening behaviour will never change so no necking would occur. (it is what mean as "elastic" or rigid" plastic behaviour. In case of n=0, further increase of stress is not possible and local heterogeneities (local fluctuations in stress strain) would exhibit necking immediately (at the onset of yielding).
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