2.2  Mechanical Theory

Atoms of a solid object move as response to an applied force. This movement changes the object’s shape and size, and it is called deformation [39]. A deformation can be temporary or permanent. A temporary deformation is reversible and linearly dependent on the applied force. When the force ceases to exist, the object returns to its original size and shape. This behavior of solids is called elasticity. On the other hand, permanent deformations are usually non-linearly dependent on the applied force and, as the name suggests, are irreversible. This behavior is called plasticity. The vast majority of materials presents both elastic and plastic behavior depending on how big the applied force is. A typical evolution of deformation regarding force is shown in Fig. 2.1.


Figure 2.1.: Relationship between force and deformation. The linear region is known as the elastic regime and any deformation in this regime is temporary and ceases to exist after unloading. Beyond the elastic limit the deformations become permanent and the material enters the plastic regime.

In order to quantitatively study the deformation of solids, a traditional approach is to look at the material as a continuum [39]. This means that the solid is not considered as a set of arranged atoms, but rather it is treated as a body of matter without empty spaces and every point has the same properties, even if it is sub-divided infinitesimally. Therefore, the solid is treated as a single material defined by its geometry and confined to a single set of parameters. With these considerations, it is possible to describe mathematically the elastic portion of the graphics in Fig. 2.1 for small deformations.

2.2.1 Infinitesimal Strain Theory
 Strain Tensor Derivation
2.2.2 Hooke’s Law
 Coefficient Determination for the Normal Components
 Coefficient Determination for the Shear Components
2.2.3 Thermal Expansion