Download Supplemental Methods: Mechanical testing and analysis of

Supplemental Methods: Mechanical testing and analysis of Arabidopsis stems.
Mechanical testing
Mechanical testing of stems was performed in a custom-built rig (Figure S3a), which
consisted of a static, lower part and a moveable upper part (Figure S3b). Both parts were
made by folding sheet metal to produce two faces separated by an air gap, which allowed the
stem to move freely in the gap during testing.
Stems were clamped inside tabs comprising two, square (10 x 10 mm) steel plate pieces
(thickness 1 mm). A semi-circular channel (diameter 1mm), was routed into each plate.
Stems were laid into this channel, so that the pairs of plates could be bonded together using
standard fast-setting superglue to create a rigid end-piece, without crushing the stem ends.
Accuracy of tab assembly was provided using an assembly jig (not shown), which gave rise
to a precise 10 mm gauge-length of stem.
The schematic diagram in Figure S3b indicates the tab-specimen-tab resting on the static
base, which was fixed to the bottom plate of a standard INSTRON tensometer. The upper
part of the rig was connected by a long threaded rod to a moving 100 N load cell, which
measured the downwards force, f, applied through the upper part to the corner of each tab.
The load cell also recorded the synchronous vertical displacement of the tabs, u.
This testing arrangement results in what is commonly known as “four-point” bending, as
explained in Figure S3c. The tab-stem-tab specimen is shown as a free-body diagram, and
the four contact faces applied to it are as shown as external forces, each being f/2. If the
length of the tab is b, then the moment, M, initially applied to each end of the stem is the
couple, (f/2)b. These opposing couples create a constant bending moment along the length of
the stem that, in turn engenders a uniform radius of curvature, R, during its deformation
(Figure S3d).
During testing, the moveable part displaced vertically downwards at a rate of no more than
2mm/minute. A continuous read-out of the force-displacement curve on screen enabled us to
observe when the specimen had failed and, after some further displacement for completeness,
the test was then terminated; the unloaded performance was not captured. The data of f and u
for each stem were converted into values of material stiffness and strength as described
below.
Four point bending analysis
The moment, M, applied initially to each end of the stem, in equal and opposite directions, is
given by (f/2)b. As each tab rotates further, the effective lever arm of the applied force, f/2,
decreases, where, denoting as b’, Pythagorean trigonometry gives
and the current moment is M = (f/2)b’. The rotation of each tab from the horizontal is
denoted by θ, in which tanθ = u/b, and is well approximated by θ = u/b over the range of
vertical displacements performed here. The curvature, C, of the bent stem is uniform
everywhere along its length under four-point bending, and is given by
where the gauge-length of the specimen is l, as shown in Figure S3c.
In order to determine the material stiffness, first, the initial bending stiffness must be
calculated. As for engineering materials, it is assumed that a simple linear constitutive
behaviour is appropriate, in which the applied moment is directly proportional to the
curvature it induces, according to:
The bending stiffness, B, is equal to the product of the material stiffness, denoted usually by
E, and the second moment of the cross-section, I, which is a geometric property that describes
the distribution of material within the stem cross-section, about the axis of bending. In the
present case, I, is equal to
which assumes that the stem has a solid circular cross-section, of outside diameter, a. In
practice, the cellularity of cross-section demands that this formula should be multiplied by a
factor, less than the unity, of the relative density of actual material contained within a givencross section. However, in the absence of any direct measurements, this factor is difficult to
estimate; and, as seen in other related studies (Patterson, 1992; Turner and Somerville, 1997;
Ryder et al. 2003; Bosca et al. 2006), this factor is often ignored. If the same formula for I is
used to calculate the strength of section, the material stiffness and strength values for each
specimen are consistently calculated, and then, as noted before, their ratio is not affected (for
the I terms in appropriate formulae effectively cancel in order to produce this ratio). This
point reinforces the importance of strength-to stiffness ratio as a measurement of mechanical
consistency across a range of stems.
The initial bending stiffness can be found by re-arranging and substituting for terms as
follows:
where (f/u)initial is the initial slope of the raw data when f is plotted against u, and this is
measured by hand in each case for accuracy.
The material strength of the stem is taken to be the maximum value of bending stress, σ,
reached during a given test, where the standard engineer’s formula applies:
with y being the radius of the stem, a/2, and ([f/2]b’)max being the first occurrence of
maximum moment within the raw data.
In summary: the bending stiffness, B, is calculated from Eqn 5, and the material stiffness is
B/I in which I is taken to be πa4/64 for all specimens. The material strength is given by Eqn
6. For both data, an average value of stem diameter, a, was obtained by using digital
callipers at several points along the stem, immediately after testing.
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