Engineering Rock Mechanics

Time-dependency 2 1 9

However, for constant strain, d&ldt=O, the basic differential equation
reduces to 0 = EE, with the result that

o1 = o[,[l- e;'' ] stepped relaxation.

These two types of behaviour are illustrated in Fig. 13.8.
The description which we have given of the Maxwell and Kelvin basic
rheological models have only incorporated two components-in series
for the Maxwell model and in parallel for the Kelvin model. One can
consider three, four or indeed any number of such rheological elements
connected in series and parallel networks: for example, combining the
Maxwell and Kelvin models in series produces the Burger's substance
illustrated in Fig. 13.6. These rheological models are models of one-
dimensional behaviour and to use them to analyse the behaviour of three-
dimensional continua, it is necessary to assume that the viscoelastic
response is due only to the deviatoric and distortional components of stress
and strain, respectively, with the spherical and dilatational components
causing time-independent volume change. The fundamental differential
equation for an isotropic Maxwell material in terms of the distortional and
deviatoric components is

0': 0:

E': =- +-

2,u 2G

where the superscripted asterisk denotes deviatoric and distortional
components, the overdot represents the first derivative with respect to time,
and ,u = F13, G = E/2(1 + v).
In order to write this in terms of total strain, stress and stress rate, we make
use of the relations between total, spherical and deviatoric components, i.e.

Stress in viscous element

Stress in elastic element 01 = 00 (1 e

Figure 13.8 Non-linear creep and stepped relaxation for the Kelvin substance.