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 characterization to a more complex case, where pressure exists on one axis, and shearing (dragging the surface parallel 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.

A complex solid can be modeled as an infinite combination of infinitesimally small boxes. Stress only needs to be described on one of the faces in the pair because of the infinitesimally small size and Newtons third law (3). 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.
The von Mises stress is an attempt to calculate 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 shear 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, see fig. 1): $$\sigma_{vM}=\sqrt{\frac{(\sigma_{xx}-\sigma_{yy})^2+(\sigma_{yy}-\sigma_{zz})^2+(\sigma_{zz}-\sigma_{xx})^2+6(\tau_{xy}^2-\tau_{yz}^2+\tau_{zx}^2)}{2}}$$

Using COMSOL®, a type of modeling software combining different physics in one simulation, we can run a simulation and get the von Mises stress out over the entire structure and ensure that it is below the max value measured at DTU Civil Engineering. The simulation is based on the FEM (Finite Element Method), where you first construct a 3D representation of the structure. Then you split the structure into many smaller elements (see fig. 2), which can theoretically be of any shape. You can then either do a temporal solution or a stationary one. If you want to do a temporal solution you take the governing physics (in our case mechanical physics) and calculate the time derivative from the corresponding equations. Technically discretizing would force us to redefine gradients as the difference between the value at the evaluation element and the values of the neighboring elements, and the same principle applies to other vector operators. Now knowing the temporal change of the physical properties of any element, take a step forward in time, thus changing the values by the just calculated temporal change times the amount of time. This discretization of time and space is not a complete representation and leads to an error, but the smaller the elements and time step, the better the approximation. If you want a stationary solution, you can do the same, but continue until the temporal change is zero, and then the steps do not matter anymore. A stationary solution only depends on the boundary conditions, which are the values at the boundary. For instance, the velocity being zero on a face means that the face is stationary all the time.

Results

Changing Young’s modulus does not change the maximum stress significantly when looking at the range of Young's modulus of our best bricks; it changes only how much the walls are displaced. To see how these are related, see fig. 3.

Increasing the thickness of the shell gives more material to spread the stress over. The largest stress is on the inside. Since fungi are very light and filled with air, they insulate quite well, and we might need rather thick walls for protection from the cold Martian environment. But neglecting this structurally, we only need a thickness of 0.34 m.

Since the maximum stress is on the inside, it decreases slower as thickness increases since the added thickness gets further and further away from the stress zone. This is shown in fig. 5.

As mentioned, 1 atm pressure inside makes it easy for plants to grow and does not alter the operation of vital machinery. But it does require a thicker wall – according to our simulations, approximately double the thickness. Fungi have the advantage of growing in a couple of weeks, so the extra thickness can quickly be grown. But in scenarios where only humans need to be there and resources (e.g. heat) are scarce, a lower internal pressure could be preferable to lessen stress and strain on the structure. See fig. 6 for the stress distribution for these cases.

Another way to minimise the stress is by adding weight on top to counteract the outwards force. This could either be just mass eg. regolith or useful mass in the form of solar cells, or a protective layer against micrometeorites. This effect is less pronounced than the others, but also limits the displacement. The effect of this on stress and displacement is shown in fig. 7.

Results

In conclusion using FEM we have showed that choosing Aspergillus oryzae is strong enough for our use case and a wall thickness of a 3.4 m is enough to ensure structural integrity. This gives us grounds for choosing this over Ganoderma resinaceum which is used by Ecovative, though it can still be used for more demanding parts of a colony. The cold on mars can also be a factor requiring a thicker wall for isolation. Which will also give more secure walls for holding up to winds.