Biomedical Engineering Reference
In-Depth Information
Fig. 6.11 Four typical creep
functions J ( t ). The vertical
scale is the creep function J ( t )
that has the dimensions of one
upon stress. The horizontal
scale is time
Fig. 6.12 A three-parameter
viscoelastic model
η
F
k D
k
Voigt model, the transverse or horizontal bars or connecting elements are
required not to rotate, so the two parallel elements are constrained to always
have the same extension in both elements at any instant of time. In these spring
and dashpot models, in general, the deflection represents the strain and the
force represents the stress. In this particular illustration the deflection
represents the shearing strain and the force the shearing stress. The differential
equation for a Voigt model is formed by adding together the stress in the two
parallel branches, T 12 ¼ (1/ J o ) E 12 for the spring and, T 12 ¼ ( t r / J o )( d /d t )( E 12 )
for the dashpot, thus
d E 12
d t ¼
d T 12
d t
1
t r G o T 12 þ
1
G o
:
The problem is to determine the response of the Voigt model to a step loading
in shearing stress T 12 . The magnitude of the step loading is T o . This problem is
formally similar to the Example 6.5.2, the only change being in the model. The
solution of the problem requires the solution of a first order ordinary differen-
tial equation subject to the case when the stress history is given by T 12 ¼
T o h
( t ) and no other loading is applied to the model. Determine the stress response
to this strain history loading and show that, for this model, the creep function is
given by J
. Typical creep functions are
e t= t
ð
t
Þ¼ð
1
=
T o Þ
E 12 ¼
J o h
ð
t
Þ
1
c
shown in Fig. 6.11 .
6.5.5 A three-parameter viscoelastic model is shown in the Fig. 6.12 . The model
consists of two branches, the lower branch with a spring and the upper branch
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