Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 96

Increassing stress applied Stress released

Final state

Newtonian

(a) Elastic ( h constant)

h = t
d

e /d t

(b) Viscous

non-Newtonian

( h variable)

Strain rate d e /d t

(c) Plastic

Fluid h (Pa)

Water (30º)
Oil
Basalt lava
Rhyolite lava
Salt
Asthenosphere

0.8 10 -3
0.08
10 2
10 8
10 16
10 22

s < s y

s > s y

T1 T4 T5

T2 T3

TIME

Fig. 3.98 Viscosity is the resistance of a fluid to deform or flow: it is
the slope of the curve stress/strain rate. Fluids showing linear
relations (constant viscosity) are Newtonian. Fluids with nonlinear
relation (

Fig. 3.97 Strain of different materials with time (stages T 1 to T 5)
applying increasing levels of stress: (a) Elastic solids show discrete
strain increments with increasing stress levels (linear relation);
strain is reversible once the stress is removed ( T 5); (b) Viscous
fluids flow faster (higher strain rates) with increasing stress; the
deformation is permanent once the stress is released; (c) Plastic
solids will not deform until a critical threshold or yield stress is
overpassed (at T 4 in this case). Deformation is nonreversible
(at T 5).

variable) are non-Newtonian. The table shows the values
of viscosity ( ) for some viscous materials.

level of stress is required to start deformation, as the mate-
rial has an initial resistance to deformation. This stress
value is called yield stress

y (Fig. 3.91c). After the yield
stress is reached the body of material will be deformed a
big deal instantaneously, and the deformation will be per-
manent and without a loss of internal coherence. So, two
important differences with respect to elastic behavior are
that the strain is not directly proportional to the stress, as
there is an initial resistance, and that the strain is not
reversible as in elastic behavior (Fig. 3.97). An analogical
model for plastic deformation is that of a heavy load rest-
ing on the floor (Fig. 3.92c). If the force used to slide the
load along a surface is not big enough, the load will not
budge. This would depend on the frictional resistance
exerted by the surface. Once the frictional resistance, and
so the yield stress, is exceeded, the load will slide easily and
the movement can be maintained indefinitely as long as
the force is sustained at the same level over the critical
threshold or yield stress. The load will not go back on its
own! So the deformation is not reversible (Fig. 3.92c).

measured in Pascals. Fluids that show a linear relation between
the stress and the strain rate, and so have a constant viscosity,
are called Newtonian. Fluids, whose viscosity changes with the
level of stress are called non-Newtonian (Fig. 3.98). Viscous
behavior is generally compared to a piston or a dashpot con-
taining some hydraulic fluid (Fig. 3.92b). The fluid is pressed
by the piston (creating a stress or loading) and the fluid moves
up and down a cylinder, producing permanent deformation;
the quicker the piston moves the more rapid the fluid deforms
or flows up and down. The viscosity can be described as the
resistance of the fluid to movement. High viscosity fluids are
more difficult to displace by the piston up and down the cylin-
der. For non-Newtonian fluids (Fig. 3.98) as the piston is
pushed more and more strongly in equal increments of added
stress the rate of movement or strain rate rapidly increases in a
non-linear fashion.

3.15.4 Plastic model 3.15.5 Combined rheological models

Plastic deformation is characteristic of materials which do
not deform immediately when a stress is applied. A certain

Elastic, viscous, and plastic models correspond to simple
mathematical relationships which apply to materials under