Radial elastic compatibility ($\frac{d}{dr}\mathrm{tr}\tau =0$)

In a homogeneous isotropic solid without body force, a radial displacement field satisfies $(\lambda +2\mu )(u^{\prime }+2u/r{)}^{\prime }=0$. For nonzero $\lambda +2\mu$, its volumetric strain is constant. Since Hooke's law gives $\mathrm{tr}\tau =(3\lambda +2\mu )(u^{\prime }+2u/r)$, its stress tensor has constant trace. After plasticity in continuum mechanics has occurred, this conclusion applies to purely elastic increments during elastic unloading, and need not apply to the total residual stress.