Closed-form solutions of the Kramers-Kronig dispersion relations in linear dissipative continua

Student: Kristel Meza
Supervisor: Dr C. Lai

Abstract

Closed-form solutions of the Kramers-Kronig equations relating damping ratio and phase velocity have been derived within the framework of the theory of linear viscoelasticity. The only assumption made states that the relaxation function of the material exponentially decays with time towards a constant non-zero value, corresponding to a solid viscoelastic behavior. The explicit expression for phase velocity as a function of damping ratio was found by means of the theory of linear singular integral equations, in particular the solution of the associated homogeneous Riemann Boundary Value problem. A non-explicit form was also obtained reducing the linear singular integral equation to a linear non-singular Fredholm equation of the second kind, ready to be solved with a Nystrom method. The explicit solution for damping as a function of phase velocity was found through the components of the complex wavenumber. The results were validated and illustrated with numerical examples for difference types of soils and compared to other simplified solutions that have been preiously proposed in the literature.