Damping described by fading memory—analysis and application to fractional derivative models

Abstract

Some damping models where the actual stress does not depend on the actual strain but also on the entire strain history are studied. Basic requirements in the frequency and time domain significant for the choice of damping model are outlined. A one-dimensional linear constitutive viscoelastic equation is considered. Three different equivalent constitutive equations describing the viscoelastic model are presented. The constitutive relation on the convolution integral form is studied in particular. A closed form expression for the memory kernel corresponding to the fractional derivative model of viscoelasticity is given. The memory kernel is examined with respect to its regularity and asymptotic behavior. The memory kernels relation to the fractional derivative operator is discussed in particular and the fractional derivative of the convolution term is derived. The fractional derivative model is also given by two coupled equations using an internal variable. The inclusion of the fractional derivative constitutive equation in the equations of motion for a viscoelastic structure is discussed. We suggest a formulation of the structural equations that involves the convolution integral description of the fractional derivative model of viscoelasticity. This form is shown to possess several mathematical advantages compared to an often used formulation that involves a fractional derivative operator form of constitutive relation. An efficient time discretization algorithm, based on Newmarks method, for solving the structural equations is presented and some numerical examples are given. A simplification of the fractional derivative of the memory kernel, derived in the present study, is then employed, which avoids the actual evaluation of the memory kernel.