Finite element modeling of transient ultrasonic waves in linear viscoelastic media

Linear viscoelasticity offers a minimal framework within which to construct a causal model for wave propagation in absorptive media. Viscoelastic media are often described as media with `fading memory,' that is, the present state of stress is dependent on the present strain and the complete time history of strain convolved with appropriate time-dependent shear and bulk stress relaxation moduli. An axisymmetric, displacement-based finite element method for modeling pulsed ultrasonic waves in linear, homogeneous, and isotropic (LHI) viscoelastic media is developed that does not require storage of the complete time history of displacement at every node. This is accomplished by modeling stress relaxation moduli as discrete or continuous spectra of decaying exponentials and relaxation times. Details of the construction and computation of the time-dependent stiffness matrix are presented. As an application of the finite element method, a finite number of exponentials (amplitudes and relaxation times) are employed to represent a typical model for a continuous relaxation spectrum. It is demonstrated that a small number of discrete exponentials are required to model ultrasonic wave propagation of a typical band-limited pulse in a model material accurately. Previous work has shown this model to be consistent with other analytic models for wave propagation in viscoelastic media