The most general linear relation (more details are found in the Structural Mechanics Theory section of the
Structural Mechanics Module User’s Guide) between the stress and strain tensors in solid materials can be written as
here, σ is the Cauchy’s stress tensor,
ε is the strain tensor,
and
cijkl is a fourth-order elasticity tensor. For small deformations, the strain tensor is defined as
where u represents the displacement vector.
here, ρ is the medium density, and
s0 and
F represent source terms.
with f (SI unit: Hz) denoting the frequency and
ω = 2π f (SI unit: rad/s) the angular frequency. Assuming the same time-harmonic dependency for the source terms
s0 and
F, the wave equation for linear elastic waves reduces to an inhomogeneous Helmholtz equation:
