Civil Engineering Reference
In-Depth Information
The dot above the
d
defines this as the rate of deformation with respect
to time. If
d =
0
at time
t
=
0
when a constant force
F
is applied, the defor-
mation at time
t
is
Ft
b
d =
(1.6)
When the force is removed, the specimen retains the deformed shape.
There is no recovery of any of the deformation.
The
St. Venant
element, as seen in Figure 1.10(c), has the characteristics
of a sliding block that resists movement by friction. When the force
F
ap-
plied to the element is less than the critical force there is no movement.
If the force is increased to overcome the static friction, the element will slide
and continue to slide as long as the force is applied. This element is unreal-
istic, since any sustained force sufficient to cause movement would cause
the block to accelerate. Hence, the St. Venant element is always used in com-
bination with the other basic elements.
The basic elements are usually combined in parallel or series to model
material response. Figure 1.11 shows the three primary two-component
models: the Maxwell, Kelvin, and Prandtl models. The Maxwell and Kelvin
models have a spring and dashpot in series and parallel, respectively. The
Prandtl model uses a spring and St. Venant elements in series.
In the
Maxwell
model [Figure 1.11(a)], the total deformation is the sum
of the deformations of the individual elements. The force in each of the ele-
ments must be equal to the total force Thus, the equation for
the total deformation at any time after a constant load is applied is simply
F
O
,
1
F
=
F
1
=
F
2
2
.
F
M
+
Ft
b
d = d
1
+ d
2
=
(1.7)
F
F
d
1
F
1
t
t
0
0
d
2
F
2
d
1
F
1
d
2
F
2
d
d
F
F
t
t
F
0
0
(a)
(b)
F
F
O
F
O
F
d
0
(c)
FIGURE 1.11
Two-element rheological models: (a) Maxwell, (b) Kelvin, and
(c) Prandtl.
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