Biomedical Engineering Reference
In-Depth Information
functions can be obtained by solving the stress-strain-time (σ-ε-
t
)
relationship for a condition of step stress
= σ
0
H(
t
)
(5-22)
σ
(
t
)
or step strain
ε (
t
) = ε
0
H(
t
)
(5-23)
where the H(
t
) is the Heaviside step function. From this time-dependent
response to a step input, the relaxation function
G
(
t
) and creep function
J
(
t
) are defined as
σ
(
t
)
ε
0
G
(
t
)
(5-24)
=
ε
(
t
)
σ
0
J
(
t
)
(5-25)
=
These functions typically have a quite simple summed exponential
functional form for linearly viscoelastic solids (
i.e
. those defined by a
combination of linear springs and dashpots) even when many elements
are present, such that they can be generalized as
(
)
G
(
t
)
=
C
0
+
C
i
exp
−
t
/
τ
i
(5-26)
(
)
(5-27)
J
(
t
)
=
C
0
−
C
i
exp
−
t
/
τ
i
where
are material time constants.
The creep and relaxation functions are often taken as a simple
summed exponential functional form:
τ
1
,
G
(
t
) ≠
J
(
t
) (5-28)
In many cases, it is of limited value to try and relate the coefficients
C
k
back to individual spring and dashpot terms in a 1-D mechanical model
either an instantaneous or infinite time modulus value [
i.e
.
G
(0) or
G
(
G
(0) =
J
(0)
−
1
,
G
(∞) =
J
(∞)
−
∞
)].
6.1.2.
Linear viscoelastic indentation correspondence
The development of viscoelastic functions here has been for relationships
between single scalar stress and strain components. In adapting this
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