When the soil slips away
Continuum-mechanical models help understand natural disasters
by Timo Reisner
January 31, 2014
Due to the complex character of the events, it is almost impossible to gather data on geophysical mass flows that occur in nature. At the Chair of Continuum Mechanics, we therefore investigate those flow processes theoretically. We study mixtures of grains and fluids under clearly defined and controlled conditions, for example by considering the flow of a gravel-water mixture from a three-metre tall container (fig. 1). Our work is conducted in close collaboration with a team of geotechnical engineers from the TU Darmstadt who are studying this very flow process in experiments.
The test rig in Darmstadt: water level and water pressure as well as container weight are permanently measured by pressure sensors in the side walls and force sensors underneath the test rig. As the project progresses, optical measurement methods are to be deployed to determine the velocity and the trajectory of the individual grains. © RUBIN
The first step in the experiment carried out by our Darmstadt colleagues is to open a hatch at the bottom of the filled container. The gravel-water mixture sets in motion with full force and streams out within a period of ten seconds. The processes inside the container are monitored with state-of-the-art measurement equipment.
In the field of continuum mechanics, we use mathematical models to describe processes like those happening in the test rig. In the first place, these models help us gain a better understanding of the physical laws that govern natural processes; secondly, they can be used to predict how a system will behave once the conditions change, for example if we alter the size of the hatch or the grain size of the gravel. We are developing such a model together with mathematicians from the TU Bucharest, Romania. The objective of this project is to mimic the flow process that was observed in the Darmstadt laboratory as precisely as possible and, eventually, to understand the motion of mass flows. We simulate the velocity of the fluid and granulate flow as they leave the container, the percentage they make up within the mixture and the pressure at different points inside the container. The results should correspond with the values determined in the experiments. The model development is based on mixture theory.
Thought experiment dealing with the material behaviour of water. The two glass plates are infinite in order to prevent the water from flowing out at the edges. © RUBIN
From the mathematical point of view, it does initially not matter what type of fluid and what type of grains are used, i.e. if we are dealing with gravel, sand or glass beads. We therefore use the general term granular-fluid mixture. We derive general equations for the respective mixture that are based on elementary physical principles. They are known as conservation equations, because they express that, for example, the mass inside a closed system such as an air-tight room can neither increase nor decrease. In the next step, in order to be able to solve the equations at all, we require information pertaining to the actual behaviour exhibited by the deployed materials in experiments. For example, we can visualise the material behaviour of water in the following thought experiment (fig. 2): We fill the space between two infinite parallel glass plates with water. We hold the bottom plate in place, whilst sliding the top plate in one direction. The distance between the plates remains constant. Because of the friction inside the water layer, the top plate moves at a constant velocity, and the water layer is accordingly deformed. If we apply more force to push the top plate, the velocity increases. The force used for pushing the plate and the velocities inside the water layer stand in linear correlation to each other. Such behaviour of fluids is referred to as Newtonian rheology. The viscosity of a Newtonian fluid determines how strongly it resists being deformed: the more viscous it is, the higher the resistance against which the glass plate has to be moved. In mathematical terms, such material behaviour is described by a so-called constitutive relation, which is incorporated into our mathematical model.
Now, we have to consider the material behaviour of the grains. For this purpose, the water in our thought experiment is replaced by granulate, i.e. sand, gravel or glass beads. If the top plate is only lightly pushed, the grain cluster behaves like a solid body – the glass plate does not move at all. The “viscosity” is virtually infinitely large. Once a specific force is applied, however, the granulate starts flowing and the top plate is set in motion. Rather than behaving like a solid body, the material now behaves like a fluid, comparable with the sand in an hourglass that is just being flipped. Once we increase the force pushing the glass plate, the velocities inside the granulate layer change – however, unlike in a Newtonian fluid, the correlation is non-linear. In our model, this complex material behaviour, too, is described by a constitutive relation.
The water content determines the material properties of a granulate composition made up of grains of sand: moist sand behaves like a solid body. Dry sand can attain the properties of a solid body or of a fluid. If the pores between the grains are fully water-saturated, the mixture becomes flowable. © RUBIN
Furthermore, we have to include answers to the following questions into our constitutive equations: In its capacity as a “lubricant” in the pore space, how does the fluid influence the flow behaviour of the granulate (fig. 3)? In what way does grain size matter? The larger the grains, the weaker the internal cohesion of the grain cluster – it is impossible to build a sandcastle with moist gravel. How are the shapes of the individual grains relevant? Angular gravel grains do not flow as well as perfectly round glass beads.
It is hardly possible to determine figures such as the minimal force required for moving the glass plate in our thought experiment and the viscosity of a water-saturated granulate in the laboratory. Therefore, the process is as follows: we make plausible assumptions, determine solutions on the basis of our system of equations and compare them with the results of the experiment. If our solutions deviate from the figures gathered in the experiment, we modify the equations that describe material behaviour accordingly. Consequently, our research aims at establishing those constitutive laws: we wish to modify them in such a manner that they will reflect the real flow behaviour of granular-fluid mixtures, in order to thus gain an understanding of the mixture’s material behaviour.
Simulation of granulate sedimentation. On the left, the granulate is distributed evenly; in the centre, it has partially sedimented. On top, a clear-water layer is created. On the right, the granulate is fully sedimented. © RUBIN
Before this can be attempted, a further obstacle has to be overcome: the equations are so complicated that they cannot be solved with the help of pen and paper; rather, we require a numerical method, i.e. a powerful arithmetic technique for calculating solutions approximately at the computer. Because a software in which our equation system may be implemented does not exist, we have to develop one first – a time-consuming endeavour. Deploying our model and the relevant software, we have already simulated simple processes such as sedimentation, i.e. the descent of granulate under gravity (fig. 4). Here, we examined only one space dimension, as the grains move in only one direction: downwards. In qualitative terms, the simulation results resemble the results of experimental sedimentation tests. This indicates that the principles of our model and the numerical method are essentially sound. For example, they predict sharp boundaries, such as the boundary between fully sedimented granulate and clear water. Most numerical methods have a problem with such abrupt junctions. Following the first positive tests, the programme is now ready to be translated into another programming language so that it may be deployed on high-performance computers with many processors to facilitate simulations in two or even three dimensions. Thus, we will get another step closer to establishing the correct constitutive relations and, consequently, to identifying the laws that govern debris flows.
Continuum mechanics and mixture theory
The matter that surrounds us is made up of micro components: a magazine, for example, is composed of individual paper and ink molecules which, in turn, are comprised of single atoms. In order to describe the magazine’s mechanical behaviour, we may try describing each individual molecule and the way it interacts with all other molecules – doubtlessly an extremely time-consuming endeavour. In the field of continuum mechanics, we therefore simplify the process by assuming that matter is spread evenly (continuously) throughout a body and fills out the body completely. Examining one small section of that body, we no longer “see” individual molecules. We do, however, know the average properties of the molecules in that section. As long as the examined body is large enough and contains a sufficient amount of individual components, this approach leads to excellent results.
Mixture theory constitutes, in a way, an extension of that continuum approach: it states that we can deploy the same concept for macroscopically distinguishable components of a mixture, for example gravel grains and water, as long as we do not examine individual grains but rather the average of a large number of grains. Applying this method, we lose information pertaining to the individual grain. However, we know the percentage composition of the mixture at any given point within that space and we know how the components behave on average.