To obtain the stiffness and strength of a laminate, exists different types of methods. The most recognized, although it has its limitations, is the Classical Laminate Theory or CLT. With the Classical Laminate Theory can be obtained an approximation of the composite material laminate mechanical properties, as well as the tendency in deformation and stresses. This method is used because of the anisotropic behavior of composite materials. In the case of a laminate the orientations of the plies can vary among them.
Remember that a typical ply thickness is between 0.1mm and 0.3mm. So, a laminate with 50 plies will be considered as a thin ply because of its thickness.
Classical laminate theory simplifications
The following assumptions are made for the classical laminate theory:
- A perfect interlaminar bond exists between various laminas.
- All laminas are macroscopically homogeneous and behave in a linearly elastic manner.
- Individual layer properties can be isotropic, transverse isotropic or orthotropic.
- Each layer is in a state of plane stress.
- The displacements are supposed small to ensure the linearity of the problem to apply Hook’s law.
- Normal to half plane displacements are constant which allows us to discard the normal deformations.
- The shapes of the deformed plate surface are small compared to unity.
- Normal to the undeformed plate surface remain normal to the deformed plate surface.
- Vertical deflection does not vary through the thickness.
- Stress normal to the plate surface is negligible.
A disadvantage of the classical laminate theory is that it does not cover the possibility of delamination which can occur, particularly at free edges. Thus, the analysis is limited to in-plane failures.
Applied force and moment resultant on a laminate are related to the midplane strains and curvatures by the following matrix equation:
Where N is the normal force in the x and y direction and shear force. M is the bending moment in the yz and xz plane and twisting moment resultant. The A, B and D matrix, are the stiffness matrix of the laminate. Depending on the values of each matrix, the laminate will have a different performance. This can be achieved varying the laminate stacking sequence.
Extensional stiffness matrix A (N/m)
A16 and A26 values are zero indicates that there is no extension/shear coupling. To achieve this the laminate stacking sequence should be balanced, unidirectional or cross-ply, A16 and A 26 become zero, eliminating extension/shear coupling.
Extension-bending coupling matrix B (N)
Nonzero values indicates that there is coupling between bending/twisting curvatures and extension/shear loads. All values are zero for symmetric laminates. Symmetric laminates are the most common used because include structural dimensional stability.
Bending stiffness matrix D (Nm)
D16 and D26 have nonzero values indicates that there is bending/twisting coupling. These terms will vanish only if a laminate is balanced and if, for each ply oriented at +θ above the laminate mid-plane, there is an identical ply (in material and thickness) oriented at -θ at an equal distance below the midplane. Such a laminate cannot be symmetric, unless it contains only 0° and 90° plies (cross-ply).
