A one-dimensional nonlinear elastic bar has stored energy per unit reference volume…

A one-dimensional nonlinear elastic bar has stored energy per unit reference volume \(W(\lambda)=(\mu/2)(\lambda^2-1)-\mu\ln\lambda\), with stretch \(\lambda=\partial x/\partial X>0\), shear modulus \(\mu=355\,\mathrm{kPa}\), and uniform reference density \(\rho_0=965\,\mathrm{kg/m^3}\). The first Piola–Kirchhoff stress is \(P=W'(\lambda)\). Body force is absent, and the motion is purely longitudinal. Derive \(P(\lambda)\) and the tangent modulus \(P'(\lambda)\). Determine every \(\lambda>0\) for which \(P'(\lambda)>0\), and write the Lagrangian characteristic speeds \(\pm c(\lambda)\) with \(c(\lambda)=\sqrt{P'(\lambda)/\rho_0}\). At the uniform rest state \(\lambda=1.48\), compute \(c\). A right-going simple wave then connects that rest state \((\lambda,v)=(1.48,0)\) to a uniformly stretched region with \(\lambda=1.86\), where \(v=\partial x/\partial t\) is the particle velocity. Using the characteristic relations \(dv=\pm c(\lambda)\,d\lambda\) on the two families, identify which Riemann invariant is constant through the simple wave, compute the particle velocity in the stretched region, and find the time at which the head of the wave reaches the reference station \(X=0.65\,\mathrm{m}\).

\(P=W'\) is elementary; both terms in \(1+\lambda^{-2}\) are positive, so \(P'>0\) for all \(\lambda>0\). A right-going simple wave holds the opposite-family Riemann invariant constant.