Optics11 CHIARO NANOINDENTER, User Manual, Page: 42 / 65

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In both models the analysis returns the effective or reduced Young's modulus (
𝐸
∗
), in which the
Poisson ratio has not been taken into account. The Piuma software also enables to directly calculate
the ‘bulk’ Young’s Modulus ( 𝐸 ), for which a Poisson’s ratio can be set in the software (default is 0.5
for perfect incompressible materials). If the Poisson’s ratio of the sample is unknown, it is advised to
report the effective Young's modulus (
𝐸
∗
), since the Poisson’s ratio can have a slight impact on the
calculated Young's modulus (
𝐸
).
DMA Analysis
Dynamic Mechanical Analysis (DMA) uses a cyclic motion with frequency,
𝑓 , while controlling
displacement or load with an amplitude of
ℎ
0
or
𝑃
0
, respectively. The frequency-dependent storage
modulus,
𝐸′
, and loss modulus,
𝐸′′
, are calculated with the following equations:
𝐸
′
(𝑓)
(1 − 𝜈
2
)
=
𝑃
0
ℎ
0
cos(𝜙)
1
√ℎ𝑅
𝐸′′(𝑓)
(1 − 𝜈
2
)
=
𝑃
0
ℎ
0
sin(𝜙)
1
√ℎ𝑅
If the amplitude of the input oscillation is small enough, the measurements can be considered
within the material linear viscoelastic region (LVR). Within the LVR, the response is assumed to be
independent of the input amplitude and sinusoidal with the same input frequency and a phase lag,
Figure 35: The loading curve is used to derive the effective Young's modulus from the Hertzian contact (left).
A
sphere of radius 𝑅
𝑖
indents an elastic half-space to depth ℎ
𝑚𝑎𝑥
, and thus creates a contact area of radius
𝑎 = √𝑅
𝑖
ℎ
𝑚𝑎𝑥
(right).