Unbounded complex modulus of viscoelastic materials and the Kramers–Kronig relations

The Kramers–Kronig (K–K) dispersion relations developed for the complex modulus of elasticity of solid viscoelastic materials connect the frequency dependences of the dynamic modulus and loss modulus. Whether the boundedness of the complex modulus at high (“infinite”) frequency is required, or not, for the applicability of the K–K relations is investigated in this paper. The derivation of the K–K relations developed for the complex modulus is presented by examining the physical background of the relations. It is shown that the K–K relations can be applied even if the complex modulus is unbounded at high frequencies. The fractional derivative Kelvin model is used to demonstrate the application of the K–K relations for a class of unbounded complex modulus.