2. Reference solutions#

2.1. Calculation method used for reference solutions#

The whole of this demonstration can be read in more detail in the [bib1] document.

In the case of a linear isotropic viscoelastic material, the behavior over time can be described using two functions

and

so that the deformations and the stresses can be written as:

where \({I}_{3}\) refers to the identity matrix of rank 3

and \(\text{*}\) the convolution product:

The equivalent thermoelastic problem, using the Laplace transform, is:

By eliminating the « + » sign:

either,

According to the equilibrium equation, we have

, we get:

which when integrating with respect to r gives:

Boundary conditions

give:

Using the initial notations, we therefore have:

Or, taking the inverse transform,

We deduce

and \(w\):

Move \(\mathrm{DX}\) on node \(B\)

\(\text{0\%}\): analytical solution

Ph. Of BONNIERES, two analytical solutions to axisymmetric problems in linear viscoelasticity and with unilateral contact, Note HI-71/8301