When the displacements and displacement gradients are small, the second Piola-Kirchhoff stress is approximately equal to the Cauchy stress. Thus, for small vibrations of an elastic system, the motion equation reduces to:
ΔXi
Assuming Hooke's law T = Xe + 2Gei, where e = ε and X and G are the Lame elastic moduli, show that the equation of motion then leads to the following displacement equation of motion:
a^2u