It is known that at point of instability a triaxial state of stress exists.However I do not understand how we arrive at the relation where the strain hardening exponent is equal to the ultimate value of strain.Please clarify whether this is true strain or engineering strain.
Let us define some terms before we start answering this question.
A0= Initial cross-section area
A= instantaneous cross section area
P = load
L0= Initial length
L= instantaneous length
sigma_n is the true stress = P/A
S_n is the engineering stress = P/A0
epsilon_n is the true strain = Integral (dL/L); limits Lo -> L = ln (L/L0)
e_n is the engineering strain = (L-L0)/L0
n = strain hardening exponent
In light of these definitions, let us examine your question(s)
1. why is n = epsilon_n at the point of instability? I presume you are implicitly referring to the necking instability.
2. You also have a doubt if the RHS of the equation should contain epsilon_n (true strain) or e_n (engineering strain).
The exact statement in Considere's criterion for necking is:
"Necking begins when the increase in stress due to decrease in the cross-sectional area is greater than the increase in load bearing capacity of the specimen due to work hardening."
Since the statement mentions that the cross-section area decreases, the correct stress term will be the true stress here. I will later talk about how engineering stress can be directly used to evaluate the point of necking, but let us first focus on true stress here.
The statement clearly mentions that the load bearing capacity of the specimen actually reduces beyond this point. Remember that the load bearing capacity was increasing due to strain hardening before necking.
Hence mathematically it implies that the change in load at this point is zero. This is written as:
dP = 0;
since P = sigma_n * A
dP = A*dsigma_n + sigma_n * dA = 0
This can now be rewritten as
dsigma_n/sigma_n = -dA/A - (eq 1)
Since plastic flow is volume conservative (note that the yield stress has been exceeded much before necking starts), we can use the relation:
AL = Constant; On differentiating this
dA * L + dL* A = 0
-dA/A = dL/L - (eq 2)
combing (eq 1) and (eq 2)
dsigma_n/sigma_n = dL/L = depsilon_n
Note that dL/L is the derivative of the true strain, this can be rewritten as:
dsigma_n/depsilon_n = sigma_n -(eq 3)
Which means that the strain hardening rate is equal to the true stress, when necking occurs.
The relationship between n and epsilon_n will come once we incorporate the power law hardening during plasticity:
sigma_n = K* (epsilon_n) ^ n -(eq 4)
This is called power law hardening and it is applicable for metals which strain harden when they undergo plastic deformation. Note that power law hardening is obtained from a true stress-true strain curve. This equation (eq4) is NOT valid for engineering stress- engineering strain curve. I will explain this later.
Once you differentiate eq 4:
dsigma_n = n*K* (epsilon_n)^n-1 depsilon_n
rearranging this equation we get:
dsigma_n/depsilon_n = n*K* (epsilon_n)^n-1 -(eq 5)
comparing this with (eq 3)
sigma_n = n*K* (epsilon_n)^n-1
apply (eq 4) again and you get
K* (epsilon_n) ^ n = n*K* (epsilon_n)^n-1
most of the terms on either side cancel and we finally get
n = epsilon_n
This result cannot be applied to engineering stress and engineering strains curves because beyond necking, the engineering stress- strain curve has a negative slope. Since hardening in (eq 4) follows a power law equation, with no negative components, the the fitting curve has to keep increasing. This law is only valid for the true stress-strain curve as the latter keeps increasing even beyond necking.
However, the necking stress can still be found from the engineering stress strain curve very easily. The point of necking in the engineering stress strain curve is the maxima of the curve. This is because, beyond this point the specimen cant bear higher loads and the engineering stress would reduce. The other way to define this is:
dS_n/de_n = 0
That is the slope of the engineering stress strain curve is zero.
Hope this explanation helps.
Another clarification: At the onset of necking, the stress state of a sample subjected to tension is still uniaxial. It only becomes triaxial beyond necking.
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