True Stress And True Strain

In Section the idea of true stress and true strain was introduced. It is emphasised here again that the stress and strain referred to so far were “engineering” stress and strain and it is customary to drop the prefix “engineering”.

 “True stress” at any point in the tension test specimen is defined as the ratio of load to corresponding area of cross-section, or true stress,
                                                      σ= P /A
where, A is the current area of cross-section corresponding to load P during the tension test.

 “True strain” is defined as summation of incremental strain over the entire load range from zero to P. If the initial length L0 changes to length L during load change of 0 to P, true strain

Because of the final form of the true strain, it is often called logarithmic strain.

Since,                                           L = L0 + Δ L

where, Δ L is the extension in length L  so that engineering strain ε= Δ L / L0

Hence true strain = ε′ = ln ((L  + Δ L )/ L0)  = ln (1 + ε) .

One confusion that sometimes arises regarding the nature of the stress-strain curve is the decreasing of stress while the strain keeps on increasing. This is actually because of the definition of stress which is based upon original area of cross-section. Once the necking, begins (at ultimate point) the area of cross-section starts reducing fast. In this even the engineering stress calculated as ratio of load to original area of cross-section falls much short of actual stress, and hence apparently diagram behaves as shown earlier. If true

stress and true strain are plotted instead of σ and ε then the curve will not go down but stress will keep in increasing with reduction in area. However, where σ′ is very small as in elastic region the σ − ε and σ′ − ε′ diagrams will almost coincide. This is shown in Figure .

Figure 13 : Comparison of Engineering Stress-Strain Diagram with True Stress-Strain Diagram

It is well known fact that volume of material remains constant during plastic deformation, that is following

A₀ L₀ = A_L or  A₀  =  L  /L_o                ------------ (14)

With this fact it can be shown that

ε′ = ln  L / L₀ = = ln A₀ /A                 --------- (15)