On nth order linear strain gradient viscoelasticity

Abstract Considering small deformations, we introduce strain-gradient viscoelasticity of the nth order. According to this theory, stresses and hyperstresses depend on the history of deformations, including strain and strain gradients up to the n th order. We also introduce consistent kinetic energy. We discuss the correspondence between n th order strain gradient viscoelasticity and lattice dynamics. In particular, we show that considering the dispersion properties of a one-dimensional (1D) viscoelastic chain leads to the use of Padé approximations of order $$[2n,2n-2]$$ [ 2 n , 2 n - 2 ] , which are only admissible for odd n . These correspond exactly to the n th order continuum model.