Fundamentals of Mechanics – Linear Isotropic Elasticity Part 1

This know-how article deals with the fundamentals of linear-isotropic elasticity. This is the simplest material law used in FEM simulations. It is suitable for the simple description of the material behavior of steel and plastics in the linear range. In addition, it is the basis for many other material models and therefore of outstanding importance.

The stiffness of a component is not only determined by the geometry but also by the material. The stiffness of a material is the resistance to elongation under external forces A measure of this, which is used in FEM simulations, is the Youngs modulus. It describes the relationship between the technical stress σ and the elongation ε in the material. Since it is a material constant, the value of the Youngs modulus does not change under the assumption of constant temperature and pressure.

For understanding, first some basics about the definition of stress and strain are necessary. For understanding, first some basics about the definition of stress and strain are necessary. The stress is defined as the external acting force F related to the initial surface A0:

#### $\sigma = \frac{\mathrm{F}}{A_0}$

For a round cross-section with initial diameter D0 , the base area is calculated as follows:

#### $A = \frac{\mathrm{\pi \cdot D_0^2}}{4}$

The elongation ε describes the relative elongation of the material and is calculated by the absolute length change Δl in relation to the initial length l0.

#### $\epsilon = \frac{\mathrm{\Delta l}}{l_0} =\frac{\mathrm{l_1 - l_0}}{l_0}$

The figure shows the relationship between the stress and strain of a material. Such stress-strain diagrams can be determined in tensile tests and are the basis for the determination of material characteristics. For the linear-isotropic elasticity only the linear section of the curve is relevant. Since no plasticity, i.e. no permanent deformation, occurs in this section, this is also the elastic part of the stress-strain diagram. To describe the graph in this area, only the straight line gradient has to be determined:

#### $E = \frac{\mathrm{\Delta\sigma}}{\Delta\epsilon}$

The correlation is called the Youngs modulus and is the most important key figure to characterize the material behavior of materials.