Augmented Hooke's law in frequency domain. A three dimensional, material damping formulation

Abstract

An isothermal, fully three-dimensional material damping modelling technique, to some extent alternative to classic viscoelasticity, is proposed. The method is formulated in the frequency domain as an augmented Hooke's law (AHL) with a constitutive matrix in which material damping is introduced by adding frequency dependent, complex valued terms to the classical material modulus matrix of Hooke's generalized law. The derivations are based on linear, irreversible thermodynamics and the concept of hidden coordinates as introduced by Biot [(1955) Phys. Rev.97, 1463–1469]. The one-dimensional concept of multiple augmenting thermodynamic fields by Lesieutre [(1992) Int. J. Solids Structures29, 1567–1579] is generalized to a suitable three-dimensional continuum form through the introduction of a special type of hidden coordinate vectors with linear, first order, time domain relaxation equations. Consistent with the hidden coordinates, a free energy density function, assuming isothermal conditions, is introduced as the time domain basis of the augmented Hooke's law. Through a time-domain model of the coupled evolution of the mechanical displacements and the thermodynamical variables, issues of causality are avoided completely in the final frequency domain formulation. The general time-domain model used is shown to be equivalent to a three-dimensional, multiple anelastic displacement field model. An isotropic augmented Hooke's law with both dilatational and shearing damping has been implemented and tested using a 20-node volume element in the finite element code ASKA Acoustics. A close agreement between finite element calculations and the corresponding analytically exact results for the studied rod and beam cases is obtained.