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

Some damping models where the actual stress does not depend on the actual s train but also on the entire strain history are studied. Basic requirements in the frequency and time domain significant for the choice of damping mod el are outlined. A one-dimensional linear constitutive viscoelastic equatio n is considered. Three different equivalent constitutive equations describi ng the viscoelastic model are presented. The constitutive relation on the c onvolution 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 kernel's relation to the fr actional derivative operator is discussed in particular and the fractional derivative of the convolution term is derived. The fractional derivative mo del is also given by two coupled equations using an "internal variable". Th e inclusion of the fractional derivative constitutive equation in the equat ions of motion for a viscoelastic structure is discussed. We suggest a form ulation of the structural equations that involves the convolution integral description of the fractional derivative model of viscoelasticity. This for m is shown to possess several mathematical advantages compared to an often used formulation that involves a fractional derivative operator form of con stitutive relation. An efficient time discretization algorithm, based on Ne wmark's 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, whic h avoids the actual evaluation of the memory kernel. (C) 1998 Elsevier Scie nce Ltd. All rights reserved.