Draft:Tensor of elastic constants

Components of the tensor of tension, corresponding to the components of rigid tensors.

The constant elastic tensor, or the rigid tensor, is a mathematical tool used with the concept of elasticity. It is a symmetric tensor of the 4th order that intervenes inside the expression of Hooke's Law of Elasticity, generalizing anisotropic materials. For the most part, it contains 21 independent coefficients that link the 6 components of the deformation tensor with the 6 components of the tension tensor. These components all have dimensional pressures, that is, they are expressed in pascals in the international system, as Young's module generalizes. The inverse of the constant elasticities tensor is called the flexibility tensor.

The expressions used vary according to the contexts and authors. The tensor components can be demonstrated as such:

$c_{ijkl}$, $C_{ijkl}$ o $E_{ijkl}$.

According to Hooke's rule of Elasticity, the constant elasticities tensor is given by the deformation tensor 𝜀𝑘𝑙 and by the tension tensor 𝜎𝑖𝑗l, of course referencing the Einstein convention summation:

$\sigma _{ij}=c_{ijkl}\,\varepsilon _{kl}.$

It is a tensor of order 4, as $3^{4}=81$ coefficients. Being that the tensors $\varepsilon _{ij}$ and $\sigma _{kl}$ are symmetrical, this tensor verifies that $c_{ijkl}=c_{jikl}=c_{ijlk}$. Besides assuming that the tension tensor can deduct one potential energy, it can demonstrate that the constant elasticities tensor is half-invariant regarding the permutation of the pair of indices: $c_{ijkl}=c_{klij}$. The existence of these relationships reduces the number of independent coefficients to 21. This is the maximum number, validated with a triclinic crystalline structure.

See Also

• Elastic modulus
• Transverse isotropic