Complex variable boundary integral method for linear viscoelasticity: Part I—basic formulations

The basic formulations (direct and indirect) of the complex variable boundary integral method for linear viscoelasticity are presented. Complex variable temporal integral equations for the formulations are obtained for viscoelastic solids whose behavior in shear is governed by a Boltzmann model while the bulk behavior is purely elastic. The functions involved in the integral equations are the time-dependent complex boundary tractions and displacements for the direct approach and the unknown time-dependent complex density functions for the indirect approaches. The temporal integral equations give the displacements and stresses at a point inside a viscoelastic region in terms of time convolution and space integrals over the boundary of this region. The equations are valid for the boundaries of arbitrary shapes provided that these boundaries are sufficiently smooth. Complex variable temporal boundary equations are obtained by taking the inner point to the boundary. Numerical treatment of spatial and time convolution integrals involved in the boundary equations is discussed.