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Elastic Properties of Short-fiber Polymer Composites, Derivation and Demonstration of Analytical Forms for Expectation and Variance from Orientation Tensors

Existing methods for predicting elastic properties of short-fiber polymer composites from fiber orientation tensors are based on the orientation average of a transversely isotropic stiffness tensor. These evaluations focus solely on average properties and have yet to include a quantitative measure of property variation. Recognizing the statistical nature of fiber orientations within the composite commonly defined through the fiber orientation distribution function, analytical expressions are developed here to predict both expectation and variance of the material stiffness tensor from a fiber orientation distribution function. The fiber orientation distribution function is expanded through the Laplace series of complex spherical harmonics and results demonstrate that material stiffness tensor expectation is a function of orientation tensors up through fourth-order and the corresponding variance requires orientation tensors up through eighth-order. Numerical simulations obtained with the method of Monte-Carlo for sample sets generated from statistically independent unidirectional samples belonging to the fiber orientation distribution function from the accept—reject generation algorithm are shown to agree with the analytic expressions for material expectation and variance.