David A. Jack

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.

Electrical conductivity modeling and experimental study of densely packed SWCNT networks

Single-walled carbon nanotube (SWCNT) networks have become a subject of interest due to their ability to support structural, thermal and electrical loadings, but to date their application has been hindered due, in large part, to the inability to model macroscopic responses in an industrial product with any reasonable confidence. This paper seeks to address the relationship between macroscale electrical conductivity and the nanostructure of a dense network composed of SWCNTs and presents a uniquely formulated physics-based computational model for electrical conductivity predictions. The proposed model incorporates physics-based stochastic parameters for the individual nanotubes to construct the nanostructure such as: an experimentally obtained orientation distribution function, experimentally derived length and diameter distributions, and assumed distributions of chirality and registry of individual CNTs. Case studies are presented to investigate the relationship between macroscale conductivity and nanostructured variations in the bulk stochastic length, diameter and orientation distributions. Simulation results correspond nicely with those available in the literature for case studies of conductivity versus length and conductivity versus diameter. In addition, predictions for the increasing anisotropy of the bulk conductivity as a function of the tube orientation distribution are in reasonable agreement with our experimental results. Examples are presented to demonstrate the importance of incorporating various stochastic characteristics in bulk conductivity predictions. Finally, a design consideration for industrial applications is discussed based on localized network power emission considerations and may lend insight to the design engineer to better predict network failure under high current loading applications.

Assessing the use of Tensor Closure Methods with Orientation Distribution Reconstruction Functions

Orientation tensors are widely used to describe fiber orientations in mold filling simulations of short-fiber-reinforced composite systems. In these flow calculations, a closure is employed that approximates the fourth-order orientation tensor as a function of the second-order orientation tensor. Sixth-order closures have also been proposed. This paper assesses the effect of using closure approximations in fiber orientation predictions by reconstructing the fiber orientation distribution function from successively higher order orientation tensors in a Fourier series representation. This approach recognizes that the orientation tensors are related to the series expansion coefficients of the distribution function. An error metric is introduced and applied that makes it possible to compare closures of varying order. Errors associated with several fourth-order closures and a sixth-order closure are investigated and compared with the truncation error that results from a reconstruction of the exact second-, fourth-, and sixth-order orientation tensors. Examples are provided over a range of interaction coefficients and flow fields typical of injection molded fiber-reinforced composites to illustrate the proposed closure assessment method.

Numerical Evaluation of Single Fiber Motion for Short-Fiber-Reinforced Composite Materials Processing

This paper presents a computational approach for simulating the motion of a single fiber suspended within a viscous fluid. We develop a Finite Element Method (FEM) for modeling the dynamics of a single rigid fiber suspended in a moving fluid. Our approach seeks solutions using the Newton-Raphson method for the fiber’s linear and angular velocities such that the net hydrodynamic forces and torques acting on the fiber are zero. Fiber motion is then computed with a Runge-Kutta method to update the fiber position and orientation as a function of time. Low-Reynolds-number viscous flows are considered since these best represent the flow conditions for a polymer melt within a mold cavity. This approach is first used to verify Jeffery’s orbit (1922) and addresses such issues as the role of a fiber’s geometry on the dynamics of a single fiber, which were not addressed in Jeffery’s original work. The method is quite general and allows for fiber shapes that include, but are not limited to, ellipsoidal fibers (such as that studied in Jeffery’s original work), cylindrical fibers and bead-chain fibers. The relationships between equivalent aspect ratio and geometric aspect ratio of cylindrical and other axisymmetric fibers are derived in this paper.

Anisotropic Diffusion Model for Suspensions of Short‐Fiber Composite Processes

Fiber orientation kinematic models of non‐dilute suspensions have relied on the Folgar and Tucker (1984) model for diffusion for over two decades. Recent research, however, has exposed the propensity of this fiber collision model to over‐predict the rate of alignment. To promote the advancement of light‐weight, high strength composites, a new fundamental approach is needed to accurately capture fiber interactions within the melt flow. We present our initial work in the development of an objective anisotropic diffusion model for fiber collisions. This paper modifies the Jeffery model (1922) to incorporate two new effects, (1) local directionally dependent effects assumed proportional to the probability of collision between two fibers, and (2) large scale volume averaged diffusion behavior analogous to shear rate dependant Brownian motion. Extensional flow results demonstrate a scalable rate of alignment for the transient solution while retaining the desired steady state solution. Conversely, results for sh...

Assessing the Use of Tensor Closure Methods With Orientation Distribution Reconstruction Functions

Orientation tensors are widely used to describe fiber distri-butions in short fiber reinforced composite systems. Although these tensors capture the stochastic nature of concentrated fiber suspensions in a compact form, the evolution equation for each lower order tensor is a function of the next higher order tensor. Flow calculations typically employ a closure that approximates the fourth-order orientation tensor as a function of the second order orientation tensor. Recent work has been done with eigen-value based and invariant based closure approximations of the fourth-order tensor. The effect of using lower order tensors tensors in process simulations by reconstructing the distribution function from successively higher order orientation tensors in a Fourier series representation is considered. This analysis uses the property that orientation tensors are related to the series expansion coefficients of the distribution function. Errors for several closures are investigated and compared with errors developed when using a reconstruction from the exact 2nd , 4th , and 6th order orientation tensors over a range of interaction coefficients from 10−4 to 10−1 for several flow fields.Copyright

Spherical Harmonic Solutions of Fiber Orientation Probability Distributions for Composite Processes

This paper presents a method to numerically solve partial differential equations such as the Jefferys equation, which calculates the orientation of fiber in a slow moving fluid. Our method relies on spherical harmonics. This method is equivalent to using the higher order moment tensors with the linear closure, but using tensors of very high order, even as high as 400. However, using spherical harmonics and its related theory, it is possible to create methods that compute surprisingly quickly.In fact we created a general process, whereby many kinds of partial differential equation that describes a distribution of directions of fibers, can be automatically converted into a computer program capable of solving the differential equation. This automated process is carried out using a script, which converts the partial differential equation into a threaded C program.

A Statistical Method to Obtain Material Properties From the Orientation Distribution Function for Short-Fiber Polymer Composites

Material behavior of short-fiber composites can be found from the fiber orientation distribution function, with the only widely accepted procedure derived from the application of orientation/moment tensors. The use of orientation tensors requires a closure, whereby the higher order tensor is approximated as a function of the lower order tensor thereby introducing additional computational errors. We present material property expectation values computed directly from the fiber orientation distribution function, thereby alleviating the closure problem inherent to orientation tensors. Material properties are computed from statistically independent unidirectional fiber samples taken from the fiber orientation distribution function. The statistical nature of the distribution function is evaluated with Monte-Carlo simulations to obtain approximate stiffness tensors from the underlying unidirectional composite properties. Examples are presented for simple analytical distributions to demonstrate the effectiveness of expectation values and results are compared to properties obtained through orientation tensors. Results yield a value less than 1.5% for the coefficient of variation and suggest that the orientation tensor method for computing material properties is applicable only for the case of non-interacting fibers.Copyright

Numerical Evaluation of Single Fiber Motion for Short Fiber Composites Materials Processing

This paper presents a numerical approach for calculating the single fiber motion in a viscous flow. This approach addresses such issues as the role of axis ratio and fiber shape on the dynamics of a single fiber, which was not addressed in Jeffery’s original work. We develop a Finite Element Method (FEM) for modeling the dynamics of a single rigid fiber suspended in a moving fluid. Low Reynolds number viscous flows are considered since these best represent the flow conditions for a polymer melt within a mold cavity. Our approach seeks the fiber angular velocities that zero the hydrodynamic torques acting on the fiber using the NewtonRaphson method. Fiber motion is then computed with a Runge-Kutta method to update the position, i.e. the angle of the fiber as a function of time. This method is quite general and allows for fiber shapes that include, but are not limited to, ellipsoidal fibers (such as that studied in Jeffery’s original work), cylindrical fibers and beads-chain fibers. The relationships between equivalent axis ratios and geometrical axis ratios for cylindrical and beads-chain fibers are derived in this paper.

Investigating the Use of Sixth‐Order Orientation Tensors in Polymer Composite Melt Flow Simulations

Short fiber reinforced polymer composite systems are often represented by the application of orientation tensors. These orientation tensors, along with the required closures, capture the stochastic nature of concentrated fiber suspensions in a compact form suitable for numerical computation and make it possible to evaluate fiber orientation states during complex polymer composite melt flow simulations. Recent research on fitted closures has provided improved approximations of the fourth‐order orientation tensor in simple flow fields, but has not addressed the need for using higher‐order tensors during fiber orientation simulations. The accuracy of closures is investigated using a Fourier series expansion of the fiber distribution function, and reconstructing the distribution function from the orientation tensors. It is demonstrated that fourth‐order orientation tensors are not sufficient to represent the higher‐order information contained in distribution functions typical of polymer processing application...