On the Folding of a Viscoelastic Medium with Adhering Layer under Compressive Initial Stress*

The exact solution is given for the folding by compression of a viscoelastic layer embedded in a viscoelastic medium, under the assumption that there is perfect adherence between layer and medium. This solution agrees closely with the earlier result obtained by Biot which was based on the assumption that layer and medium could slip over each other. In a previous publication, Biot [l] has discussed the case of folding a layered viscoelastic medium under initial stress. An exact solution was presented for the folding, due to instability, of a viscoelastic layer of thickness h embedded in a viscoelastic medium extending infinitely far in both directions perpendicular to the layer; the assumption was made that layer and medium did not adhere and could slip over one another as if perfectly lubricated. An approximate solution for the case of perfect adherence was developed in a second publication of the same year by Biot [2]. The influence of the adherence was found to be small. The purpose of this note is to check this conclusion by an exact solution for the case when perfect adherence exists between layer and medium. The method is the same as that in Ref. [l] and it combines two distinct developments contained in earlier work by the same author. One of these developments is the theory of elasticity of a medium under initial stress (1934-1941). The other was introduced as a correspondence principle (1954-56) by which elastic moduli are replaced by corresponding operators [3, 4, 51. Results are therefore applicable to either elastic or viscoelastic media of a very general nature. It was shown at the same time that the correspondence extends to problems which involve anisotropy, dynamics, wave propagation, and variational procedures. The mathematical restrictions on the operators were also derived from thermodynamics [3]. The horizontal and vertical displacements u and v, in the layer representing the departure of the displacement from an initial steady state, satisfy the equations