Viscoelastic Wave (Under Development)¶
Mathematical Model¶
For isothermal viscoelastic material, the model equations consist conservation of mass and momentum as follows,
(1)
where are the Cartesian component of the velocity, the density, the stress tensor, the internal variables, and the Kronecker delta. Subscripts are for the Cartesian tensors. , and are the constants of the standard linear solid (SLS) model with . is the number of the employed SLS model components.
Equation (1) can be further organized to a vector form:
(2)
where is the solution variable, , , and flux functions, and the source term.
Jacobian Matrices¶
By applying the chain rule to Eq. (2), we can derive the Jacobian matrices:
(3)
where , , and are are the Jacobian matrices:
(4)
where
and
(5)
(6)
(7)
, , and are matrices. , , and are matrices.
Hyperbolicity¶
The left hand side of the model equation Eq. (3) can be proved as a hyperbolic system. The method of proof is similar to the Hydro-Acoustics (Under Development). The list of the eigenvalues is provided:
(8)
where , and . The , and are the components of a direction vector, as used in Hydro-Acoustics (Under Development).