Example

A cylindrical steel sample is tested using the conventional tensile test. The initial effective length of the sample is $25$ mm $,$ and its diameter is $10$ mm $.$ At the end of the stretch, the length of the sample is $30$ mm $.$ This steel grade has work hardening behavior, and its true stress$-$strain relation follows the equation below:

$=200+20**{}^{0\mathrm{.}5},$

where $$ is the true stress, and $$ is the true strain. The stress unit is MPa.

Assumptions: $1)$ The material is isotropic; $2)$ The sample has perfect cylindrical shape at the end of the test $($no necking$)$; $3)$ The elastic strain is negligible, so the volume of the sample has no change during testing; $4)$ the strain rate hardening is negligible.

Find the engineering stress, engineering strain, true stress, true strain, and the tension load at the end of the test.