Stress and Strain - Quantitative Plate Tectonics - page 233

Fig. 7.10 Response of a
linear viscous system to a
stress step. An analog
model for this system is the
dashpot. There is
continuous deformation at
constant rate under
constant stress, and the
strain © is permanent after
stress removal

Another theoretical response of a material to
stress steps, which can eventually combine with
the elastic response, is associated with the vis-
cous behavior . In this instance, the constitutive
equation relating stress and stress expresses pro-
portionality between stress and strain rate rather
than strain:

£.t/ D 2ǜ P ©.t/ p (7.54)

Fig. 7.11 Analog model for the Maxwell rheology

where p is the hydrostatic pressure and the con-
stant parameter ǜ, which depends from the ma-
terial and is a decreasing function of temper-
ature, is called viscosity and has units [Pa s].
The time derivative of strain, P ©, ha units [s 1 ]
and gives a measure of the rapidity of deforma-
tion. In the case of non-solid materials, where
© represents pure shear strain, the constitutive
law ( 7.54 ) describes the behavior of Newtonian
fluids .
The linear viscous behavior is illustrated
in Fig. 7.10 . These materials do not display
strain recovery and deform continuously under
constant stress. Just as a spring furnishes an
analog model for elastic systems, a dashpot
can be used as a paradigm for linear viscous
materials. However, many materials of geologic
interest have a mechanical behavior that can be
considered as a combination of linear viscous and
elastic rheologies. They are termed viscoelastic
materials . A simple description of their behavior
can be obtained using analog models. These
models are built combining springs and dashpots
into

to describe some simple rheological models
commonly used in geodynamic modelling. The
first of them is the Maxwell rheology model,
which is a two-element system consisting of a
linear spring element and a dashpot connected in
series, as shown in Fig. 7.11 .
In this model, the total strain is partitioned into
a spring strain, © S , and a dashpot strain, © D ,so
that:

P ©.t/ D © S .t/ C © D .t/

(7.55)

where © S and © D are related to the applied stress £
by the following equations:

£.t/ D Y© S .t/
£.t/ D 2ǜ P © D .t/ p (7.56)

Inserting these expressions into ( 7.55 )gives:

1
Y P £.t/ C

1
2ǜ Œ£.t/ C p

P ©.t/ D (7.57)

complex systems. Here we are going