Analysing variable density structures in plant stems

Nature is always an excellent inspiration for engineering designs. My objective is to develop strong and light weight structures automatically, using topology optimisation.

The traditional Solid Isotropic Material with Penalisation (SIMP) way of topology optimisation is to assign a density value [0,1] to each element of the 3D mesh and use an optimiser to vary the density values. In addition, there is a penalisation function that encourages the optimiser to converge to either a density of either 0 or 1. A density of 0 means the removal of the element while 1 keeps the element in the final design. The objective function to minimise is Compliance, the opposite of stiffness. The more compliant a structure, the less stiff it is.

Compliance is a measure of TOTAL elastic potential energy in the structure. When considering a single spring, the elastic potential energy is calculated by \frac{1}{2}kx^{2} . We can consider every element in as a spring under strain. An element under greater strain, \Delta x, will have a greater potential energy. When you add up all the potential energy for each element, you have a total elastic potential energy for the whole structure. Minimising compliance is the target.

Image result for spring hookes law potential energy
Diagram taken from Hyperphysics.

I have seen the capabilities of Toplogy optimisation algorithms in the Fusion360 package. It seems to be robust and manages to converge most of the time. The results are usually solid models. However, we see in nature that many lightweight and strong structures have a internal lattice structure that brings down weight even further. For example, avian bone structure is very very light and yet strong. You may see such structures present in these bones.

Avian bone. Source

Initially during my investigation, I decided to look into the structure of bamboo stems. In the lifetime of a bamboo, it experiences lateral loading from winds blowing against it or possibly a Panda climbing on it! It needs to handle high transverse loading forces without yielding and killing the plant. From beam bending theory, in cantilever loading, we have the highest stress on the region furthest away from the neutral axis, as seen in the image below.

Diagram showing stress distribution on a cantilever beam, loaded at the free end. Diagram taken from strenghtandstiffness.com
Transverse loading of a beam, like the bamboo stem.
Image taken from Module 7 Simple Beam Theory, MIT Structural Mechanics

Bamboo is well known for its functionally graded properties. The natural design concentrates high tensile strength vascular bundles where it is most required(highest load concentrations at regions furthest away from neutral axis) while spreading out these bundles in areas where there are less strain.

I decided to look at cross sections of views of more generic plant stems from other species because bamboo has been extensively already analysed by others. I was adviced to look at other species. So I got a collection of images of stem cross section stock photos from the internet. These sections have been created by using a microtome to slice the stem to ~30 microns, dyed, and placed under the microscope.

My approach to find the density distribution of high strength fibrous materials was to plot the distribution of dark pixels in each micrograph as a function of stem radius. I assumed the dark areas were high strength fibrous material. This method is a very simple, rule of thumb approach. The dark regions are not necessarily a homogeneous material. It only transmits less light than other emptier regions.

Image processing of a stock image of the cross section of a cotton stem.

This is the steps I took to calculating the area fraction of dense materials:

  1. I thresholded the image using the adaptive gaussian threshold function in OpenCV. I used that values of blockSize = 201 and C = 1. It automatically thresholds the image to set the dark pixels to a value of 255 and light to 0.
  2. I manually select the points around the edges of the stem cross section and the algorithm fits a circle around the points using the Least Squares optimisation algorithm found in scipy.optimise. This yields the centre point and diameter of circle. This is approximate because most stem cross sections are not exactly circular. Hand selected edge points are shown by red points in circle overlay plot.
  3. I look at every pixel within the circle and measure its distance from the centre of circle. I log the color of each pixel and distance from the centre.
  4. I split the radius into 50 bins.
  5. I then plot the fraction of dark pixels to total pixels in each radius bin.

The final result is a plot showing the density distribution in the stem from the center to the outer edge of the stem image.

Plot showing the density distribution from center to outer ring of Cotton stem, same as the one shown above.

I did this for 8 different stem images. The common trend is indeed that the density increases from the centre to the outer edge of the stem. Some trends appear to be linear but some seem to be exponential. The 10 stems are here:

There are weakness of this approach.

  1. The stems are not perfectly circular. The approach I use assumes a completely circular stem with features arranged in concentric circles.

All the stems analyses, bamboo and the various stems all have similar structures where the high strength material is concentrated around the edges. Can we actually emulate this structure to develop very light weight structures for critical applications such as Drones, aircraft, race cars or even furniture? I try to generate those structures here.

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