Vibrations of inhomogeneous anisotropic viscoelastic bodies described with fractional derivative models

The dynamic response of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated. The mechanical behavior of the viscoelastic material is described by differential constitutive equations with fractional order derivatives. The governing equations, which are derived by considering the dynamic equilibrium of the plane body element, are two coupled linear fractional evolution partial differential equations in terms of the displacements, whose order is in general greater than two with respect to time derivatives. A method is presented to establish the additional required initial conditions beside the described initial displacements and velocities. Using the Analog Equation Method (AEM) in conjunction with the Domain Boundary Element Method (D/BEM) the governing equations are transformed into a system of multi-term ordinary fractional differential equations (FDEs), which are solved using the numerical method for multi-term FDEs developed recently by Katsikadelis. Numerical examples are presented, which not only demonstrate the efficiency of the solution procedure and validate its accuracy, but also permit a better understanding of the dynamic response of plane bodies described by different viscoelastic models.