Finite element modelling
This project is almost entirely based on the use of numerical finite element modelling using the Abaqus software. In this section some important aspects of this type of computational modelling will be presented. A Finite Element Method (FEM) is a way of producing approximate numerical solutions that can be adopted in engineering disciplines such as Solid and Fluid Mechanics. The basic principle involves separating an otherwise complex system into smaller elements, with a simple geometry, defined by nodes and analyzing these separate elements. This allows approximate solutions to be found by calculating stresses and strains at the integration points (points at locations within the element, specific for different element types) and evaluating the displacements at the nodes.
Software
The software that is going to be used mainly in this project is Abaqus. This software is a powerful tool that can produce approximate numerical solutions to complicated engineering problems. Abaqus provides two alternatives in terms of user navigation. The first option is to use the built in user software to define the geometry, the boundary conditions, the material properties and the loading (or this case dynamic conditions) of the problem. All these steps can alternatively be defined by creating an input file using the Abaqus keywords and commands. Both approaches have advantages and disadvantages and can be used together for the same model.
Element Selection
The element that is going to be adopted, at least for the first models, is called C3D8R. This is a 3D brick element with eight nodes that uses reduced integration. It was chosen for the initial model as it is suitable for the geometry of the problem and will result in low running times. One of the disadvantages of this element is that due to its only integration point being in the centre of the element, a relatively fine mesh (small elements) is required to produce accurate estimations of the stresses and strains at the nodes. The mesh convergence test however will help us take this into account when selecting the size of the grid.
Mesh Convergence Test
The size of the mesh that is used for the model is crucial for the accuracy of the results. A finer mesh will have a larger amount of smaller elements, which means that the nodes will be closer to the integration points. This will result in a higher accuracy of the results that are read at the nodes. The finer the mesh however the more computational power will be required, which can dramatically increase the running time for the model. Therefore it is crucial to find the exact size of the mesh at which the model starts to give sufficiently accurate results. This is done by starting with a coarse grid of nodes and selecting a node at the edge of the plate so that it is always there as the grid is refined. The model is then run and a value (stress for instance) is noted down for the selected node. The distance between the nodes is then reduced in half and the same result is recorded. This process is then repeated until the values at this node start to converge with what seems like a reasonable error percentage. When a new mesh size gives a fairly similar result the previous size can be used for the model with this geometry. As the project progresses a non-homogeneous mesh can be adopted. This would involve having a finer mesh in the centre of the target where all plastic deformations will occur and a coarser mesh in all other regions. This would be a big step towards building a more efficient model as it reduces the computational power that is directed towards regions of the model which are of little significance to the research and concentrating it on the regions that are crucial.
Symmetry
Exploiting symmetry can significantly reduce the running time of a model. There are several conditions that apply if a model is to be constructed by exploiting symmetry. The problem that is being analyzed has to have symmetrical geometry, symmetrical boundary conditions and the projectile has to be accelerated towards the target on the line of symmetry for the previous two conditions. If any of these three conditions do not apply the entire model has to be constructed. Then exploiting symmetry it is essential to apply the appropriate boundary conditions to the faces where the geometry has been cut. These boundary conditions have to take into account that during impact the system will remain symmetrical and therefore the target and the projectile have to be restricted in moving in directions they wouldn't move int if the whole system was analyzed. This means that all faces lying on the planes of symmetry (where the model has been cut) will remain on these planes.
Material Models
The most important factor, that allows the Abaqus model to give results that can match experimental values, is the implementation of models and equations that account for the material behavior and the failure mechanisms.
The Mie-Gruneisen equation of state – this equation allows modelling materials at high pressure.
The Johnson-Cook plasticity model – allows the Abaqus model to take into account strain rates, thermal effects and compressibility.
The Johnson-Cook dynamic failure model – defines the Abaqus ductile damage initiation criterion.
The Progressive damage framework
The Mie-Gruneisen equation of state – this equation allows modelling materials at high pressure.
The Johnson-Cook plasticity model – allows the Abaqus model to take into account strain rates, thermal effects and compressibility.
The Johnson-Cook dynamic failure model – defines the Abaqus ductile damage initiation criterion.
The Progressive damage framework