J[ Mech[ Phys[ Solids\ Vol[ 35\ No[ 09\ pp[ 0858Ð0879\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved  Pergamon Printed in Great Britain 9911Ð4985:87 ,*see front matter PII ] S9911Ð4985"87#99934Ð2 APPLICATION OF THE COSSERAT SPECTRUM THEORY TO VISCOELASTICITY XANTHIPPI MARKENSCOFF\$% WENSEN LIU\%& and MICHAEL PAUKSHTO' % University of California\ San Diego\ Department of Applied Mathematics and Engineering Sciences\ La Jolla CA 81982\ U[S[A[ & San Diego State University\ College of Engineering\ San Diego CA 81071\ U[S[A[ ' Saint Petersburg State University\ Institute of Mathematics and Mechanics\ Saint Petersburg\ Russia "Received 19 December 0886 ^ in revised form 10 February 0887# ABSTRACT The Cosserat Spectrum theory is applied in the theory of linear viscoelasticity[ The solution for the Laplace transform of the displacement of the viscoelastic problem is expressed in a series of the Cosserat eigenfunctions\ which are dependent only on position\ and the coe.cients are expressed as convolutions of the time dependent body force or surface loading provided that the inverse Laplace transforms of the viscoelastic moduli are known[ This renders the Cosserat Spectrum theory advantageous for the solution of viscoelastic problems[ Several examples are shown[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords ] A[ creep\ B[ viscoelastic material\ C[ Cosserat Spectrum[ 0[ INTRODUCTION The homogeneous Navier equations Du l ¦v99 = u l  9\ ðv "l¦m#:m  "0:"0−1n#\ l and m being the Lame |s constants\ n the Poisson|s ratioŁ\ with homogeneous boundary conditions of displacement or traction admit nontrivial solutions when v takes values in a set of points lying outside the physical range of Poisson|s ratio called the Cosserat Spectrum[ The Cosserat Spectrum theory was introduced by Cosserat and Cosserat "0787# and then fully developed by Mikhlin "0862#\ who proved the completeness and orthogonality of the Cosserat eigenfunctions and represented the displacement _eld u l as summation of the Cosserat eigenfunctions for the boundary value problems of displacement or traction[ Pobedria "0869# applied the Cosserat Spectrum theory to 1!D viscoelastic problems[ In a recent paper\ Markensco} and Paukshto "0887# applied it to problems in elasticity and thermoelasticity[ They also developed a variational principle in thermoelasticity within the frame of the Cosserat Spectrum theory[ In the present article we develop a theory of linear viscoelasticity based on the Cosserat Spectrum theory[ We show that in the Laplace transformed space the Navier equations hold and the Cosserat Spectrum theory can be applied[ The solution for $ To whom correspondence should be addressed[ E!mail ] xmarkensÝames[ucsd[edu 0858