Theory

It is very normal to test a material based on pressure along one axis, which we also did. The question is now how to compare this characterisation to a more complex case, where pressure exists on one axis, and shearing (dragging the surface paralel with itself) along another. For this the so called von Mises yield criterion (1) was utilized. The basic assumption is that it is not volumetric change that breaks an object, but uneven stress. An example is a balloon; if you put it under water, there is pressure on all sides and it is pushed into a smaller volume, but does not fracture. But if you squeeze a balloon between your hands, it will eventually pop. In more technical terms, the cause of this phenomena, is the distortion energy which only increases based on differences in stress.

Newton’s second law tells us that if we know the stress on one of the sides, we also know the forces on the opposite side to get the observed acceleration, e.g. if the block stands still and is pushed on one side the block will not move as it must be pushed on the other, where the faces perpendicular to an axis form a pair. Stress only needs to be described on one of the faces in the pair. The stress on a face can be split into two types. If you have a paper tissue, and you are either pulling the paper apart or pushing (crumbling), it is one type ($\sigma$), but if you start to shear the paper that will be a different type of stress($\tau$). The same view can be applied in 3D, where the shearing can happen in two directions on each facepair per axis, but the pull and push forces can occur on all three axes. The stress on each of the sides (faces) will be a combination of the forces described above. A more complex solid than the paper example can be modelled as an infinite combination of these boxes.
The von Mises stress is an attempt of calculating the size of the combined stress differences, analogous to vector length, which in the 2D push/pull scenario reduces to the pull/push stress.This is exactly what we measured in our characterisation tests, that is how high stress can be before cracking. Since it is an infinitesimally small box, if you sheer on one side, the other side in that plane will also be sheared, e.g. shear on the x face in y direction also gives shear on the y face. Generally these are the shears with the same subscript but different order. This means some of the stresses are identical giving a simpler formula. The formula for the von Mises stress is (1) (for the notation fig. 1): $$\sigma_{von Mises}=\sqrt{\frac{(\sigma_{xx}-\sigma_{yy})^2+(\sigma_{yy}-\sigma_{zz})^2+(\sigma_{zz}-\sigma_{xx})^2+6(\tau_{xy}^2-\tau_{yz}^2+\tau_{zx}^2)}{2}}$$

STOOOOOOp Tip extension rate is calculated by using the equation below, which considers growth kinetics for the fungi and the amount of substrate available. It essentially outputs the accelerated growth dependent on the amount of substrate available, where the accelerated growth equation depends on fungal kinetics and branch lengths [1, 3]. \begin{equation} r_{tip, i} = \bigg(k_{tip,1} + k_{tip,2}\cdot \frac{l_{br,i}}{l_{br, i} + K_t}\bigg)\cdot\bigg(\frac{S}{S+K_s}\bigg) \end{equation} $k_{tip,1}$ is the initial tip extension rate of the branch and $k_{tip,2}$ is the difference between the maximum extension rate and $k_{tip,1}$ [1, 3]. The length of branch $i$ described by $l_{br,i}$. $S$ is the substrate concentration and $K_S$ corresponds to the substrate concentration to reach half of the maximum growth level [4]. As the starting coordinates of this simulation are $(x_0, y_0)$ and end coordinates $(x, y)$, the length of branch $i$ can be calculated as the distance between two points: \begin{equation} l_{br, i} = \sqrt{(x-x_0)^2 + (y-y_0)^2} \end{equation} By dividing the growth area into a grid of $w\cdot h$ areas, it is possible to investigate the uptake of substrate, hyphal movement and the development of biomass through the simulation.

At the start of the simulation, it can be assumed that the initial substrate concentration $S_0$ will be distributed evenly across the grid. When the spores are placed randomly, the branches start to develop by using the substrate available around the hyphal tip. So for each branch, it is checked which tip is in which grid and whether there is any substrate available. That means that one can track the substrate depletion in each grid, and when there is no more substrate in one of the areas of the grid, the hyphal tips located there can no longer grow and can be considered as inactive hyphae. If the total area around the mycelium no longer contains substrate, the growth will simply stop.

In the same way as tracking substrate depletion, biomass production can also be viewed in a grid. But instead of calculating a loss of substrate, each new branch extension is considered a gain in biomass or the density $d$. It should show the same mechanics as the substrate depletion, as these two are directly related to each other by the equation $l_{br,i}$.