The motion of a variable-density rod (Q1102786)

A semi-infinite bar of variable density \(\rho\) (x) is loaded on the edge \(x=0\) by given normal stresses \(\sigma_ 0(t)\) (t\(\geq 0)\). In the assumed case of nonlinear constitutive equation \(\epsilon =f(\sigma)\) the resulting partial differential equation is nonlinear. To the simplified version of this equation \((\rho =\rho (\gamma x)\), \(0<\gamma \leq 1\), \(f(\sigma)=a\sigma +b\sigma^ 2)\) the Green-Liouville transformation is applied. A simple closed form solution is obtained after asymptotic analysis.