Why study theory of elasticity before FEM? Well, because all of the major universities have it as the first chapter or at least as a part of the first unit of their FEM course plan. But, why is that so? Well, keep reading to know.
Consider mild steel. We have all performed the infamous destructive test to determine the yield strength of mild steel. That was a fun experiment. Now consider this, if in case you had not loaded up to the ultimate stress, or until the yield stress and just loaded the test piece until the stress Vs strain graph stayed in the linear region. What would happen then? The experiment definitely would not have been fun but if you observed the material carefully, the extended test piece after removing the load, would come back to its initial length. This is known as ELASTIC behaviour. So, guess what theory of elasticity deals with... yeah right it’s about the behaviour of bodies within the elastic range.
The theory of elasticity assumes that the bodies are perfectly elastic along with this there are a few other assumptions. They are:
· Molecular Structure: The molecular structure of bodies will not be considered here. It will be assumed that matter is homogeneous and continuously distributed over its volume. Hence if you cut out a chunk out of the body, it will have the same physical properties as that of the remaining body.
· Isotropic: it is assumed that the molecular structure is the same in all the directions.
These assumptions are not really accurate for real world materials (E.g.: consider steel, the properties of steel far away from being homogenous when studied at the microscopic level.) but there is proof that the values obtained from this theory and that from the experiment are in agreement. So the results depend on the size of the chunk that you take. If it has micro dimensions, the theory will not hold good.
What are the forces acting on the element? There are two kinds, body forces and surface forces body forces are the one which act on the volume say the forces of gravity acting on your body right now. Surface forces could be the air pressure acting on the surface of your skin.
This is how surface forces and stresses are denoted. The letter σa denotes normal stresses parallel to the coordinate axes where a is the direction of the coordinate axis it is acting along. Similarly, τab denotes shear stresses acting on a surface where a is the surface the shear force is perpendicular to and b is the direction the force is parallel to. I feel it is the easiest way to remember. The directions shown in the image are taken as positive. If the forces act in the opposite direction, the sign changes.
Let us consider strain,
I have this cube and let us say it is not free to move around (stuck to the table with superglue) but, it is free to deform when force is applied. Now the displacement of the particles are resolved into u, v, and w which are parallel to the axes x, y and z, respectively. Since I have a cube of dimensions dx, dy and dz as shown in the figure, the displacement would be something like this (in 2D) you just have to have a look at the diagram to get the idea. Totally self-explanatory.
Let us use the letter ϵ for strains that are parallel to the coordinate axes and ϒ for shear strains,(i.e. the ones that are parallel to the surface of the body). Determining the directions of strains parallel to the coordinate axes is easy as there is only one subscript (like ϵx or ϵy or ϵz) and the shear strains have the same naming pattern as that of the shear stresses. So now we know what theory of elasticity is, the notations used to denote stresses, strains and the directions. In my next post, I will complete the basic equilibrium equations involved in the theory of elasticity.
References:
Theory of elasticity-Timoshenko & J.N. Goodier
