CPA Help Pages
					Help for Fiber Orientation Parameters
		The thermoelastic properties of a composite material are determined by the distribution of fibers at a material point as well as by the composition, effective aspect ratio and constituent properties of the components. In general, individual fibers will be oriented in 3-dimentions. Three Eulerian angles are required to define their orientation as illustrated in the Figure.
		Consequently, vector and tensorial averages result in only two independent orientation parameters being required to specify the general state of fiber orientation. The orientation parameters corresponding to program input are defined below,
		The subscript "p" has been introduced to emphasize that fp describes the planar orientation in the "1-2" or "L-T" plane. The subscript "a" emphasizes that fa quantifies the axial orientation about the "3" or "p" (perpendicular) axis in the figure. Evaluation of the orientation parameter from experimental data requires histograms to be constructed from microstructural observation. A typical histogram is presented for the determination of the planar orientation parameter, fp in this Figure. The integrals defined above can be numerically integrated in a straight forward manner.
		The values of fp range from zero to one. Values of zero (fp = 0) correspond to a random distribution in the "1-2" plane. Values of one (fp = 1) indicate a totally aligned distribution parallel to the 1 axis. Intermediate values of fp correspond to intermediate fiber orientation distributions as illustrated in this Figure. The values of fa range from –1/2 to one and quantify the tendency for fibers to tilt out of the 1-2 plane and align parallel to the normal (3) direction. For fa = -1/2, all the fibers lie perpendicular to the 3 axis, fa = 0 corresponds to a random distribution in the azimuthal angle, and fa = 1 implies that all fibers are aligned along the 3 axis. In general, any 3 dimensional fiber orientation distribution can be represented by a combination of these orientation parameters. The important special case where fp = 0 and fa = 0 corresponds to a "true" 3-dimensionally random distribution.