Elasticity constants

In deformable solid mechanics, it is demonstrated that the deformation tensor and stress tensor are symmetrical. These tensors are of rank 2 and order 3, and can be written as:

Accordingly, we can number the indexes in the following manner:

However, the theory of linear elasticity is written strictly as:

Eljkl is a rank 4 tensor.

Using the notations above, we can simply write out:

We will show that matrix Eqr is symmetrical.

For two stress-deformation couples, the Helmoltz energy of the solid varies in the following manner:

Let us introduce potential Ф (that is a state function):

By combining [1.3] and [1.4], we obtain:

However, Ф is a state function. Consequently:

Hence:

In other words, the elasticity constant matrix E;j is a symmetrical matrix of rank 2 and order 6. Therefore, its maximum number of elements is 21.

The Eij elements of this matrix have properties that depend on the symmetrical properties of the crystal in question [NOV 95].

Therefore, for the triclinic crystal system with the lowest number of symmetries, the 21 elements are all different. On the other hand, for the cubic system, only three elements are required to characterize a crystal’s elastic properties. These elements are En, E44 and EJ2.

A complete table of the various crystal systems may be found in the work of Novick [NOV 95].

These Young models are measured in GPa (1 gigaPa = 109 Pa) and, for a perfect crystal, their value is

Sanquer et al. [SAN 87] established that there is identity between the measured values of sound wave velocity and those calculated based on crystal potential.

This calculation is based on the fact that a displacement has two consequences:

  • - the apparition of deformation tensor;
  • - the apparition of a modification in crystal potential with the apparition of stress tensor as a corollary.

For organic compounds, Eij does not exceed 30 GPa but can reach 200 GPa for metals such as iron or for minerals.

 
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