SAMPE Journal - May/June 2024 - 33
Figure 3. Plots comparing the mode shapes of the metabeam FE models with beams of equivalent rigidity. The insets depict the shape as seen on
the metabeam.
not properly account for some of the complexities found in the
metabeam. Figure 3 shows the mode shapes for all six modes in
both fully expanded and fully collapsed cases.
4. CLOSED-FORM EXPRESSION OF HOMOGENEOUS
SECTION FLEXURAL PROPERTIES
Having verified the beam-like behavior of the metabeam, the
question that remains is how to predict its rigidity using a closedform
solution depending on the properties of its constituents
and its geometry. The material properties of the honeycomb were
previously discussed in Refs.[8,7]
. The method is revisited here
for the unit cell parameters present in this metabeam iteration.
An equivalent beam cross-section is then proposed in order to
calculate the second moment of area, I, about each axis.
4.1 Honeycomb Properties via Finite Element Analysis
Calculating the elastic modulus of the honeycomb is achieved
in this work through the use of a FE model comprising a 10-by10
array of unit cells (Figure 4a). A simulated uniaxial loading
test is conducted to measure the force-displacement response
about each state (Figure 4b). The elastic modulus can then be
calculated using the following equation:
[13]
where k is the slope of the force-displacement curve, A is the
average, projected cross-sectional area of the honeycomb (here
A = 1710 mm2
), and Lo
is redefined for deflections about each state (i.e., Lo
the fully expanded state and Lo
= 2.714 MPa.
is the undeformed length. In this work, Lo
= 286.6 mm in
state). This results in an expanded elastic modulus of Eexp
MPa and a collapsed modulus of Ecol
= 184.7 mm in the fully collapsed
= 5.758
4.2 Flexural Properties
To calculate the flexural rigidity of the equivalent homogeneous
beam, the cross-section of the metabeam is reduced to a model
(shown in Figure 1d) consisting only of the two honeycomb
plates with an effective width, bHC
, which can be calculated as:
[14]
Figure 4. Material properties for the honeycomb are derived by a) analyzing a 10-by-10 matrix of cells in both expanded (left) and collapsed (right)
states, b) extracting a localized force-displacement curve about the respective state.
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SAMPE Journal - May/June 2024
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