Engineering Rock Mechanics

Time-dependency 2 1 5

Rheological models
These are analogues of different material behaviour, formed from
assemblages of mechanical components, usually springs, dashpots and
sliders. They assist in understanding the material behaviour and allow the
formulation of the various constitutive relations.
Using just three rheological elements-spring (or Hookean substance),
dashpot (or Newtonian substance) and slider (or St. Venant substance)-
it is possible to produce a bewildering array of rheological models,
depending on whether the elements are connected in series, parallel or a
mixture. In Fig. 13.5, all of the two-element models are illustrated, together
with their names. In Fig. 13.6, some multi-element models are shown.
In Fig. 13.6, the top left-hand rheological model is a viscoelastic material
(element 2,Z in Fig. 13.5) with an additional spring in series, producing what
is known as the generalized Kelvin substance. In the top right-hand model
of Fig. 13.6, another dashpot has been added in series to the generalized
Kelvin model, producing Burger’s substance. Note that this is an
elastoviscous material in series with a viscoelastic material, i.e. a Maxwell
model in series with a Kelvin model (cf. Fig. 13.5). The two lower substances
of Fig. 13.6 are the behavioural models associated with the leading diagonal
of Fig. 13.5, first in series (i.e. formed by working down the leading
diagonal) and second in parallel. Note that a clockwise convention is used
when connecting the elements on the leading diagonal in this matrix form

Elastic

Hookean substance

Spring

Viscoelastic

Kelvin model

Plastoelastic

Elastoviscous

Maxwell model

Viscous

Newtonian substance

Dashpot

Plastoviscous

Elastoplastic

Viscoplastic

Plastic

St. Venant substance

Slider

Figure 13.5 Two-component rheological models.