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                   True Stress-Strain Curve (or) Flow Curve :

The conventional engineering stress-strain curve for ductile material is based on original cross-sectional area. Thus, it doesn’t explain the deformation characteristics of a metal properly. Whereas the true stress-strain curve is based on instantaneous cross-sectional area of the material. In engineering stress-strain curve we can see that the curve drops after maximum load(necking). This is due to the fact that the stress is based on the original area. This stress is known as engineering stress and is given by :




After necking the load required to deform the material decreases and area being constant the stress also decreases, thus the curve drops until fracture.

But here in true stress-strain curve the stress is based on instantaneous area during the application of load. Thus, during necking we know that the area decreases and in turn the stress increases since the stress and area are inversely proportional. Thus, the curve slopes upwards in a true stress-strain curve. This stress is known as true stress. It is given by :




The trues strain is also known as logarithmic strain or natural strain. It is given by :

True stress at maximum load :

The true stress at maximum load corresponds to the true tensile strength(i.e. UTS in the case of engineering stress-strain curve). The necking begins from the maximum load for most of the material. The true stress at maximum load is given by :

Here, the strain is given in terms of area. This is due to the fact that the strain after the onset of necking(i.e. beyond maximum load) must be expressed in terms of area or diameter only.

Equating both of them,


True fracture stress and strain:

The true fracture stress is the load at fracture divided by the cross-sectional area at fracture. And the true fracture strain is the strain based on the original cross-sectional area and the area after fracture.


Power law :

The flow curve of many metals in the plastic deformation region can be expressed by a relation known as power law. It is given by :

Where, K is the strength coefficient, n is the strain-hardening coefficient.

Significance of power law :

·       The n value determines the shape of the true stress-strain curve. As n value varies the shape of the curve also varies.

·       In n is low the initial work hardening rate is high and decreases rapidly with strain and vice versa.

·       n=0 for perfectly plastic solid, n=1 for elastic solid and n=1.5 for 0.1<n<0.5 for most of the metals.


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