Determination of the Steady State Fiber Orientation Tensor States in Homogeneous Flows with Newton–Raphson Iteration Using Exact Jacobians

Fiber orientation is an important descriptor of the microstructure for short fiber polymer composite materials where accurate and efficient prediction of the orientation state is crucial when evaluating the bulk thermo-mechanical response of the material. Macroscopic fiber orientation models employ the moment-tensor form in representing the fiber orientation state, and they all require a closure approximation for the higher-order orientation tensors. In addition, various models have more recently been developed to account for rotary diffusion due to fiber-fiber and fiber-matrix interactions which can now more accurately simulate the experimentally observed slow fiber kinematics in polymer composite processing. It is common to use explicit numerical initial value problem-ordinary differential equation (IVP-ODE) solvers such as the 4^th- and 5^th-order Dormand Prince Runge–Kutta (RK45) method to predict the transient and steady-state fiber orientation response. Here, we propose a computationally efficient method based on the Newton-Raphson (NR) iterative technique for determining steady state orientation tensor values by evaluating exact derivatives of the moment-tensor evolution equation with respect to the independent components of the orientation tensor. We consider various existing macroscopic-fiber orientation models and several closure approximations to ensure the robustness and reliability of the method. The performance and stability of the approach for obtaining physical solutions in various homogeneous flow fields is demonstrated through several examples. Validation of our orientation tensor exact derivatives is performed by benchmarking with results of finite difference techniques. Overall, our results show that the proposed NR method accurately predicts the steady state orientation for all tensor models, closure approximations and flow types considered in this paper and was relatively faster compared to the RK45 method. The NR convergence and stability behavior was seen to be sensitive to the initial orientation tensor guess value, the fiber orientation tensor model type and complexity, the flow type and extension to shear rate ratio.