Biomedical Engineering Reference
In-Depth Information
From extensive finite element analyses, the representative strain was
found to be a function of α
(from 60° to 80°):
.
(6-7)
ε
=
0.0319 cot
α
R
.
Comparing with other definitions, 4,49,53,67 the formulation of the
representative stress-strain given in Eq. 6-6 , which was derived from an
optimization process, not only has a better physical meaning, but also
leads to the best apparent independence of n in Eq. 6-5 (and thus the best
accuracy). 68 For
Specified for a Berkovich indenter (
),
α =
70.30
o
ε
=
0.0115
R
α =
70.30
o
, the dimensionless loading curvature can be
fitted as 5,68 :
3
2
E
E
E
E
(6-8)
Π= −
0.66 ln
+
8.40 ln
12.31 ln
+
9.21
σ
σ
σ
σ
R
R
R
R
Similar polynomial fittings were widely adopted in the literature, e.g .
Refs. 4, 49, 53, 55 and 69 and its variations were used for spherical
indentations. 67,70 Such high-order fitting function often makes it very
difficult to explicitly solve for material properties during the reverse
analysis. Moreover, the fitted coefficients of the polynomial function
(such as those in Eq. 6-8 ) do not have clear physical meaning, and the
fitting function may behave incorrectly for extremely elastic or plastic
materials used outside the ma te rial space of forward analysis. A simple
relationship between C and
need to be established on a more
E
/
σ
R
physical basis.
Note that the specimen is essentially elastic when the variable
, whereas the material ap proaches to rigid plastic when
E
/
σ
0
R
(
)
should incorporate the limits
of both mechanisms, such that it remains valid for all materials
regardless of the range of data used for fitting. In the representative case
of
- the functional form
E
/
σ
→∞
Π
E
/
σ
R
R
α ( Fig. 6-3 ) , both of the elastic and rigid plastic limits can be
well define d:
When
70.3
(
)
(elastic),
varies linearly with
E
/
σ
0
C
/
σ
E
/
σ
R
R
R
:
E
/
σ
R
C
E
as
.
(6-9)
=⋅
m
E
/
σ
0
e
R
σ
σ
R
R
 
 
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