**1 . Analysis of the Orthotropic Lamina**

**1.1. Introduction**

**1.2. Hooke’s Law**

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

**1.4. Matrix of Elasticity**

** **

**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**

Consider an element of the laminate as depicted in Figure below, suppose that composite is subjected to deformation εL in the direction of the fibers. The fiber will be stressed by a tension:

*σ*_{f}= *E _{f}*

*·ε*

_{L}

while the matrix is subjected to a stress tension equal to:

σ_{m}= *E _{m}*

*·ε*

_{m}

where *E _{f}* and

*E*are respectively the Young module of the fibers and matrix.

_{m}The area section A of unit volume of laminate (ortogonal to L direction, see figure above) is subjected to stress tension σ_{L }in order to this equivalence is valid:

*P=**σ*_{f}* *·*A** _{f}* =

*σ*

_{f}*·*

*A*

*+*

_{f}*σ*

*m*·

*A*

_{m}Since *σ*_{f}= *E _{f}*

*·ε*

_{L}, then:

This formula of Young module rapresents a linear variation of *E _{L}*, from

*E*when

_{m }to E_{f},*V*varies from 0 to 1 (see figure below).

_{f }Young’s module in the T direction can be considered considering the same fundamental volume element of the of the plate, loaded with a stress *σ**T *(Figure below).

The deformation in the T direction for the matrix is:

The deformation for the fiber is:

The trasversal area where *ε _{L}* is applied is approximatively

*V*, while for

_{f}W*ε*is

_{m}*VmW.*The total deformation is:

*ε*

_{T}*·*

*W*=

*V*

_{f }*·*

*W*

*·*

*ε*

_{L}+ Vm*·*

*W*

*·*

*ε*è

_{L }

*ε*=

_{T}*V*

_{f }*·*

*ε*

_{L}+ Vm*·*

*ε*

_{L}Then using the previous formulas of *ε _{L }and ε_{T}*:

Thus:

By means can be calculated the Young module in the T direction:

With similar considerations, we can derive the values of the Poisson coefficients and the tangential module: