Biomedical Engineering Reference
In-Depth Information
For vibration analysis and modeling, the most important value of the Young's modulus is at
zero strain, which is simply a 1 . Therefore, the constitutive relation for vibration models is
 
(22)
where E = a 1 from the test above.
In the bending test where the rind is treated as a simply supported beam, the equation for
the young's modulus can be derived accordingly. If the force is applied in the middle of the
span, then the deflection at that point is
3
FL
x
(23)
48
EI
If the material is linear, the constant Young's modulus is
3
LF
E
(24)
48
Ix
Where L is the span, and I is the moment of inertia of the rind sample, which is given by
3
base height
.
I
(25)
3
For nonlinear materials, E can be obtained by taking the derivative of the force with respect
to x. In particular, the Young's modulus at zero strain is
3
LdF
Ex
()
(26)
48
Idx
x
0
The modulus of elasticity of a melon was measured statically using an UTM machine, which
can record the force as a function of deflection. For the flesh, cylindrical core samples were cut
out of the melon; whereas for the rind, beam-like samples were cut out (Ehle, 2002). The force-
displacement curve for the cylindrical core compression test was transformed into a stress-
strain curve for small deflections by dividing the force by the cross-section area and dividing
the deflection by the length. The small changes in cross-section area and the length were
neglected because the displacement was small. Cubic regression of the stress-strain curve
obtained the coefficients in Equation 20. Then Equation 21 was used to calculate the slope,
which was the Young's modulus. The value at zero strain is the value used for modeling the
vibration properties of the melon. Figure 12 shows that a cubic function fits the data very well,
and that the slope at zero strain can be obtained accurately. In the above compression test of
the flesh samples, the Young's modulus could also be obtained from the load-displacement
curve without transforming into the stress-strain curve numerically. A little algebra gives:
df
L
Ex
()
(27)
Adx
x
0
For the bending test, the force-displacement data were processed according to Eq. 24 to
obtain the Young's modulus of the rind. The result shows characteristics similar to those
illustrated in Figure 12.
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