Past exam of the elasticity and plasticity course of the University of Oxford / 2023 / 3 / a

Assume homogeneous Lamé parameters, with $\mu >0$ and positive bulk modulus $K=\lambda +2\mu /3$. For spherical symmetry, write $q={\tau }_{rr}$ and $t={\tau }_{\theta \theta }={\tau }_{\varphi \varphi }$. Radial force balance is $q^{\prime }+2(q-t)/r=0$. The given linear elasticity relations implySubstitution into radial force balance givesThusThe volumetric strain is $e=u^{\prime }+2u/r$, and the trace of the stress tensor isSince $\lambda +2\mu =K+4\mu /3>0$ is constant, $e^{\prime }=0$. The radial elastic compatibility condition follows:This condition is local to a region governed by homogeneous linear elasticity; it is not a constitutive condition for a plastically deforming region.