Relation between True stress and Engineering stress

When a ductile material is subjected to tensile stress, beyond a certain stress, the cross sectional area of the material decreases at a particular position in the material; i.e. a construction develops at a particular position. This is called necking. The area of the specimen at the neck changes continuously as the load is increased. The true stress at any time of loading, is the

force divided by the instantaneous cross sectional area A, at the instant of time (at the neck); i.e.

True stress (σ_T) = F/A_i

The engineering stress, on the other hand, is the force divided by the original area of cross-section AO; i.e.

Engineering stress (σ) = F/A_o

The true strain is defined by

True strain (ε_T) = ln (L_i/L_o)

Where l_i is the instantaneous length of the specimen and l_o is the original length.

The Engineering strain is given by

ε = (L_i-L_o)/L_o

Relation between engineering strain ε and true strain ε_T :

ε = L_i/L_o -1 => L_i/L_o = ε + 1

ln (L_i/L_o) = ln (ε + 1) => ε_T = ln (ε + 1)

Relation between engineering stress σ and true stress σ_T :

Assuming that there is no volume change during deformation A_oL_o = A_iL_i

A_i = A_o (L_o/L_i) so that

σ_T = F/A_i = (F×L_i)/(A_o×L_o) and therefore

σ_T = σ(ε + 1)