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Solid mechanics - Basics

Solid mechanics requires very basics knowledge to be explained. This concerns as well the technical terms, as the material physics. This first page introduces the very basics of what you need to have a useful knowledge. It describes the principles of stress and strain, the Young modulus, and the Poisson ratio.

Stress and Strain

For the explanations about stress and strain, let's consider a mesh structure (see figure below). Each small ball is an atom, and all the atoms make a net, that is spatially periodic. Consider also that the structure is elastic.
The energy linking the atoms in the net can be redistributed following different configurations, depending on the external applied forces, temperature, pressure, etc.

Atoms mesh
Schematic of a network of atoms, forming a mesh

Now, if you pull part of the structure along its length, you will strain the structure. The length will be extended, but the global volume of the structure will not be changed so strongly, because the other dimensions will reduce (along the width and the thickness direction).

Strain of the mesh
Strain of the structure, the width and thickness are reduced following the extension of the length

The schematics and the mesh structure are example to help you understand. Not all of the materials have their atoms forming a cristal. The principle is the same for amorph materials.

The force needed to strain the structure depends on its mechanical properties. If you «pull» a surface of a cantilever, you apply a stress on it, that we will note σ, that has the dimension of a pressure. To express the strain in a mathematical way, we will talk about ε, with:


As you can guess, ε have no unit.
And so the relation between stress and strain is given by Hooke's law:


where E is called Young's modulus of the material. The Young's modulus express the difficulty to strain the material. Soft materials have a low Young's modulus, while tough materials have a high one.

As I said earlier, if you extend a structure in a direction, its sizes along the other directions will change also. The changing depends also of material properties, and it is not constant from a material to another one!
This is expressed by the Poisson's ratio, and we note it ν. Thus, if you consider now that the strain of the material along the stress direction is noted εL for «longitudinal», and in the other directions, is noted εT, for «transversal», we have, for the Poisson's ratio:


Of course, the deformation along transversal directions will be of opposite sign to the one in the stress direction. The Poisson's ratio is smaller than 0.5, but depends on the material.

Note that I've considered an isotropic material. This makes things far simpler, but is not always true. If you consider a non-isotropic material, the Young modulus E becomes a vector, and the Poisson ratio ν becomes a matrix. Rigorous calculations in solid mechanics require tensor mathematics. But for simple cases, considering isotropic material, you can use these ratios directly.

Now that we have these basics, let's see the primary stresses, from which you can compose any kind of stress!