Heterogeneous structure of modes and Kramers–Kronig relationship in anisotropic viscoelastic materials

Abstract

Composite materials made of fibers and a viscoelastic matrix, exhibit an orthotropic viscoelastic behavior that is described by a tensor with nine independent complex viscoelastic moduli. This tensor makes it possible to compute the velocity and attenuation of heterogeneous or homogeneous modes in any direction. In experiments, the immersion method of a plate shaped sample insonified by plane ultrasonic waves is used to measure this complex tensor. The liquid/solid interface generates heterogeneous quasilongitudinal and quasishear bulk modes that propagate through the plate with velocities and attenuations that depend on the frequency. In a viscoelastic material, velocity and attenuation are linked by the Kramers–Kronig relations. For heterogeneous modes, the attenuation that needs to be used is the projection of the damping vector on the wave vector. This paper shows that these relations limited to a “local” frequency band can be experimentally verified and permit one to link anisotropic velocity and attenuation dispersion of quasilongitudinal and quasishear modes only if their heterogeneous structure is taken into account. The extrapolation of the material properties determined by ultrasonic measurements, towards low frequencies, relies on this feature in combination with a model of the attenuation evolution versus frequency.