4
Resolution Methods
4.1. Assessment
The study of deformable solids consists of determining the following at each point of the structure:
There are 15 unknown functions.
There are three relations that are:
- – the relation between the displacement and the strain;
- – the equilibrium equation;
- – the material’s behavior law:
These three relations have completely different natures:
- – The first stems directly from the definition of the strain tensor. In this form, it only varies with SPH, but it is completely possible to write it without too much difficulty in other instances as well. This is an exact relation!
- – The second one is a principle and it is therefore true all of the time. We can evidently adapt it to the dynamic.
- – The third is a model and it depends on the material. It is given here in its most simple form possible, which is of a linear elastic homogeneous isotropic material; of course, it is generally much more complex than this. The problem of how to write these behavior laws is broadly unresolved, which is the subject of much research at present.
There are therefore 15 scalar equations. This problem of multiple equations can therefore be resolved by adding the boundary conditions in order to determine the integration constants.
The displacement boundary conditions where the displacements are imposed can be written as:
And the stress boundary conditions where the external forces are imposed can be written as:
Evidently, this problem is far from simple to solve and the problems which can be solved manually are exceptional cases! All the same, we are lucky to know that the solution to our problem does exist and that it is unique, so if we manage to find one, then we can be sure that it is correct!
Finally, we must not lose sight of the objective of studying deformable solids, which is to size and design structures. Once the stress, strain and displacement fields have been determined, we can then apply the sizing criterion to the structure (rupture criterion, non-permanent deformation criterion, etc.) and to determine whether or not the current structure meets the specifications. Incidentally, the sizing criterion used needs to be pertinent, which is not necessarily a given.
From there:
- – either the structure does not meet the specifications and therefore the shape or material needs to be modified, then the previous calculations need to be redone with the new structure;
- – or the structure meets the specifications and perhaps it can be built, or perhaps we deem that it resists external forces too well and that it is possible to make it lighter (aeronautics). We can modify its shape or its material, and then redo the previous calculations with the new structure.
Obviously, this is an iterative approach.
4.2. Displacement method
This method consists of postulating a form for the displacement field. This field obviously needs to verify the displacement boundary conditions.
We then verify that this field verifies the equilibrium equation which, in the instance of linear elastic isotropic behavior, can be written via the intermediary of Navier’s equation:
We then determine the strain, then the stress, and finally, we verify the stress boundary conditions.
Once again, as the solution to our problem exists and is unique (i.e. if the boundary limits are properly laid out), then if we find a solution, it must be the correct one! All that remains now is to have some good ideas for postulating a pertinent displacement field.
4.3. Stress method
This method consists of postulating a form for the stress field. This field obviously needs to verify the stress boundary conditions.
We then verify that the stress field verifies the equilibrium equation:
We then determine the strain via the intermediary of the behavior law. It now therefore remains to integrate this field in order to determine the displacement, but in practice, in order for this field to be integrable, it must verify the compatibility equation, either written in strain as:
or written in stress, which gives the case of linear elastic isotropic behavior:
It can be shown that these two scripts are equivalent.
The strain can also be integrated to determine the displacement field, and it therefore remains to verify the displacement boundary conditions.
Once again, as the solution to our problem exists and is unique (i.e. if the boundary limits are properly laid out), then if we find a solution, it must be the correct one! All that remains now is to have some good ideas for postulating a pertinent stress field.
4.4. Finite element method
As we are unable to resolve the problem in an exact manner for the majority of cases, we have to resort to solving them in an approximate manner. This is, for example, the objective of the finite element method, which shall be introduced throughout the following chapter. Currently, this method is the most commonly used one by far and it enables us to size almost all types of structure.