Before discussing the mechanics of laminated composites, we need to understand the mechanical behavior of a single layer -- lamina. Since each lamina is a thin layer, one can treat a lamina as a plane stress problem. This simplification immediately reduces the 6×6 stiffness matrix to a 3×3 one.
Since each lamina is constructed by unidirectional fibers bonded by a metal or polymer matrix, it can be considered as an orthotropic material. Thus, the stress-strain relations on the principal axes can be expressed by the compliance matrix [S] such that
] = [S
or by the stiffness matrix [C] such that
] = [C
Please note that the engineering shear strain is used in the stress-strain relations, and, the notation S for the compliance matrix and C for the stiffness matrix are not misprints. Please consult this page for more information.
For both stiffness and compliant matrices are symmetric, i.e.,
only four of , , , , and are independent material properties. Again, the shear modulus G12 corresponds to the engineering shear strain which is twice the tensor shear strain .
Please note that there can be many fibers across the thickness of a lamina and these fibers may not be arranged uniformly in most industrial practice. However, the combination of the matrix and the fibers forms an orthotropic and homogeneous material from a marcomechanics standpoint. Some literature therefore schematically illustrates a lamina with only one layer of uniformly distributed fibers as shown below.