|Coordinate Transformation is Necessary
The generalized Hooke's law of a fiber-reinforced lamina for the principal directions is not always the most convenient form for all applications. Usually, the coordinate system used to analyze a structure is based on the shape of the structure rather than the direction of the fibers of a particular lamina.
For example, to analyze a bar or a shaft, we almost always align one axis of the coordinate system with the bar's longitudinal direction. However, the directions of the primary stresses may not line up with the chosen coordinate system. For instance, the failure plane of a brittle shaft under torsion is often at a 45° angle with the shaft. To fight this failure mode, layers with fibers running at ± 45° are usually added, resulting in a structure formed by laminae with different fiber directions. In order to "bring each layer to the same table," stress and strain transformation formulae are required.
|Coordinate Transformation of Stress-Strain Relations for Lamina
If we define the coordinate transformation matrix as
The coordinate transform of plane stress can be written in the following matrix form:
Similarly, the strain transform becomes
Please notice that the tensor shear strain is used in the above formula. Suppose we define the engineering-tensor interchange matrix [R]
The stress-strain relations for a lamina of an arbitry orientation can therefore be derived as detailed below.
where the stiffness matrix is defined as
The complicance matrix is therefore
The individual components of the stiffness and compliance matrices can be found here.