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This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three-dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of Cintra & Tucker (J.

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a b s t r a c tJeffery's equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold's numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a 'closure' that approximates the distribution function's fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery's equation. Numerical examples for dense fiber suspensions are provided with both a Folgar-Tucker (1984) [3] diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009) [9]. Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.

Recent advances in Fused Filament Fabrication (FFF) include large material deposition rates and the addition of chopped carbon fibers to the filament feedstock. During processing, the flow field within the polymer melt orients the fiber suspension, which is important to quantify as the underlying fiber orientation influences the mechanical and thermal properties. This paper investigates the correlation between processing conditions and the resulting locally varying thermal-structural properties that dictate both the final part performance and part dimensionality. The flow domain includes both the confined and unconfined flow indicative of the extruder nozzle within the FFF deposition process. The resulting orientation is obtained through two different isotropic rotary diffusion models, the model by Folgar and Tucker and that of Wang et al., and a comparison is made to demonstrate the sensitivity of the deposited bead's spatially varying orientation as well as the final processed part's thermal-structural performance. The results indicate the sensitivity of the final part behavior is quite sensitive to the choice of the slowness parameter in the Wang et al. model. Results also show the need, albeit less than that of the choice of fiber interaction model, to include the extrudate swell and deposition within the flow domain.

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.

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