The ordinary stress and strain used in introductory tensile testing are engineering measures: they refer deformation to the specimen's original dimensions.

Engineering stress is

$$\sigma_e=\frac{F}{A_0},$$

and engineering strain is

$$\varepsilon_e=\frac{L-L_0}{L_0}.$$

True stress

As a tensile specimen stretches, its cross-sectional area usually decreases. True stress uses the instantaneous area:

$$\sigma_t=\frac{F}{A}.$$

After necking begins, this distinction becomes especially important because the local area changes substantially.

True strain

True strain accumulates many small relative changes in length. For uniform uniaxial deformation,

$$\varepsilon_t=\ln\left(\frac{L}{L_0}\right)=\ln(1+\varepsilon_e).$$

Why the distinction matters

Engineering measures are convenient because they use dimensions known before the test. True measures more closely represent the instantaneous mechanical state during large deformation.

For small strains the two descriptions are very close; they diverge as deformation becomes large.