1. Imagine a thin cylinder of initial internal diameter Do, initial wall thickness To and initial fixed length (fixed by two side walls) Lo is bulging slowly by increasing internal fluid (oil) pressure till it becomes unstable (plastic instability) and bursts.
Consider this as a plane stress problem (through-thickness stress is zero). The radius of the circular arc of the cylinder wall (due to plastic bulging) is R and the final dimensions of the cylinder at the crown/pole (mid-length) at instability are Df, Tf and Lf (= Lo).
Find a mathematical expression for each of the following:
a. Effective True Stress considering plane stress conditions
b. Effective True strain considering true stress conditions; and simplify this expression using Volume Constancy assumption.