The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites

The properties of a set of even-order tensors, used to describe the probability distribution function of fiber orientation in suspensions and composites containing short rigid fibers, are reviewed. These tensors are related to the coefficients of a Fourier series expansion of the probability distribution function. If an n-th-order tensor property of a composite can be found from a linear average of a transversely isotropic tensor over the distribution function, then predicting that property only requires knowledge of the n-th-order orientation tensor. Equations of change for the second- and fourth-order tensors are derived; these can be used to predict the orientation of fibers by flow during processing. A closure approximation is required in the equations of change. A hybrid closure approximation, combining previous linear and quadratic forms, performs best in the equations of change for planar orientation. The accuracy of closure approximations is also explored by calculating the mechanical properties of solid composites with three-dimensional fiber orientation. Again the hybrid closure works best over the full range of orientation states, Tensors offer considerable advantage for numerical computation because they are a compact description of the fiber orientation state.