**1 . Analysis of the Orthotropic Lamina**

**1.1. Introduction**

**1.2. Hooke’s Law**

**1.3. Relationships between elastic constants and Matrix of ****Elasticity**

**1.4. Matrix of Elasticity**

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**2. Classical theory of Laminates**

**2.1. Introduction**

**2.2. Basic Formulas**

**2.3. Laminate stiffener matrix**

**2.4. Calculation of Stress and deformation**

**2.5. Thermal Stress**

**2.6. Calculation of Elastic Constants**

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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 *σ _{ij} *(

*i, j*= 1,2,3). The same applies to the state of deformation, described by the nine components

*ε*(

_{kl}*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 *E _{ijkl }* (

*i, j, k, l*= 1,2,3). Since the tensors σ

_{ij }and

*ε*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:

_{kl}Therefore deriving this with respect to the generic component of deformation *ε _{kl} *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):

*E _{klij}=E_{ijkl} * (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 *σ _{ii} *(

*i*=1,2,3) doesn’t produce distortions

*ε*(

_{lj}*i ≠ j*), it must also be:

*E _{ijkl}* = 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 *σ _{ij}* (

*i, j*= 1,2,3 and i ≠ j ) does not produce distortion

*ε*

_{kl}_{ }(

*k ≠ l*) in other plans (

*ij ≠ kl*) , ie it must also be:

* E*_{ijkl}=0 if *i* ≠*j*, *k* ≠ *l* e *ij* ≠ *kl* (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 (*σ _{33}=σ_{13}=σ_{23}*) 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*).