|Mechanical Behaviors of a Lamina
A continuous, unidirectional fiber reinforced composite lamina is an orthotropic material. As discussed in Stress-Strain Relations of Materials, there are 9 independent material constants for an orthotropic material. For a thin plate such as a lamina, the plane stress assumption holds and the number of independent constants can be further reduced from 9 to 4 (see this section for details).
The stress-strain relations can be written as
only four of , , , , and are independent material constants.
Due to the large number of possible fiber-matrix combinations and their volume fraction ratios, these constants are usually not available without conducting a series of experiments. Nontheless, estimated values can be obtained, assuming that the properties of both the matrix and the fibers are known.
|Determination of E1
Suppose the bonding between the fibers and the matrix is perfect, the strain of the fibers and the strain of the matrix have to be the same in the fiber direction (i.e.,) when the lamina is subjected to a uniaxial force along the fiber direction.
The total force applied on the lamina is
where Af and Am are the the cross section areas of the fibers and the matrix, respectively. The Young modulus E1 can then be written
where V is the volume fraction and L is the length of the lamina. Notice that based on the no-void assumption.
One can visualize the fibers and the matrix as two springs in parallel as illustrated below.
|Determination of E2
Again, assuming perfect fiber-matrix bonding, the stress of the fiber and the stress of the matrix are the same in the transverse direction of the fiber () when the lamina is subjected to a uniaxial force:
The transverse strain is the sum of the contributions from the fibers and the matrix which are in proportion to their respective volume fractions:
The Young's modulus E2 can be calculated using the serial-spring model:
In this case, the fibers and the matrix act like two springs in series:
|Determination of 12
The major Poisson's ratio 12 is defined as
As shown in the Determination of E1 section, we have
The major Poisson's ratio can then be written as
|Determination of G12
Based on the same argument used in the Determination of E2 section, we assume that the shear stress of the fibers and that of the matrix are the same, that is, .
The shear strain is the sum of the contributions from the fibers and the matrix, which are proportional to their respective volume fractions:
The shear modulus G12 can therefore be calculated using the serial-spring model:
Material constants calculated from the above formulae are merely estimates and should not be trusted without further verification. The true material properties can only be obtained through experiments.