## 1.2. Hooke’s Law

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As is known from the theory of elasticity, the state of tension present in a generic material at around of the point is uniquely described by 9 stress components σij (i, j = 1,2,3). The same applies to the state of deformation, described by the nine components εkl (k, l = 1,2,3). Consequently, the assumption of linear elastic behavior, the relationship between stresses and deformations (generalized Hooke’s law) can be written as:

In the case of fully anisotropic material, the bond stress-deformation, involves 9×9 = 81 elastic constants Eijkl  (i, j, k, l = 1,2,3). Since the tensors σij   and εkl are symmetric, only 6 components are independent, the independent elastic constants that describe the behavior of an anisotropic material are 6×6 = 36. Thermodynamic considerations also help to reduce furtherly these constants to 21. Indicated with “U” the elastic potential, infact we have that is:

Therefore deriving this with respect to the generic component of deformation εkl we obtain:

Then inverting the order of derivation and taking into account the continuity of U with respect to the functions of deformation, then we get (Schwartz theorem):

Eklij=Eijkl      (4)

The ( 4 ) constitute a system of 15 independent equations that allows precisely to reduce  constants from 36 to 21 . If the material is orthotropic , ie admits three planes of symmetry mutually orthogonal, then the constitutive laws involving only 9 independent elastic constants. In fact, indicating with 1,2,3 the three principal axes of the material, since the application of σii (i=1,2,3) doesn’t produce distortions εlj (i ≠ j), it must also be:

Eijkl = 0 if  k ≠ l    (5)

The (5) represents a system of nine equations that allows to reduce the constants from 21 to 12 .

In addition due to the symmetry with respect to the plans 1-2,1-3,2-3 , the application of a shear stress σij (i, j = 1,2,3 and i ≠ j ) does not produce distortion εkl (k ≠ l) in other plans (ij ≠ kl) , ie it must also be:

Eijkl=0 if ij, kl e ijkl     (6)

For the principle of reciprocity of shear stress (σij  = σji) the (6) represents a system of 3 equations that reduces furtherly the elastic constants from 12 to just 9. The elastic constants of an orthotropic material may be advantageously rappresented in a 6×6 symmetric matrix (matrix of elasticity) that allows you to write Hooke’s law in the matrix form :

When we have plane tensions (σ331323) this relationship can be simplified:

From (8) it is possible, by simple inversion of the matrix of elasticity, obtaining the relationship between stress and strain:

The matrix [S] is called the inverse matrix of elasticity. The significant terms of [S] are related by the terms of the matrix of elasticity by the relations of inversion:

In conclusion the constitutive equations of an anisotropic material involving 21 elastic constants (symmetric matrix 6×6), those of the elastic constants of an orthotropic material 9 (6×6 sparse matrix, see Eq. (7)) that in the case plan reduced to only 4 (3×3 sparse matrices, see eq. (7-8). In any case it has a greater complexity than in the case of isotropic materials that involve, for being two-dimensional and three-dimensional, only two elastic constants (Ε, ν).    