Determination of Preferred Fiber Orientation State based on Newton-Raphson Method using Exact Jacobian
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. Recent macroscopic fiber orientation models have employed the moment-tensor form in representing the fiber orientation state which all require a closure approximation for the higher order orientation tensors. In addition, various models have 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. Traditionally explicit numerical IVP-ODE transient solvers like the 4th order Runge-Kutta method have been used to predict the steady-state fiber orientation state. Here we propose a computationally efficient method based on the Newton-Raphson iterative technique for determining steady state orientation tensor values by evaluating the 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 ap-proximations 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 the obtained exact derivatives of the orientation tensor is performed by benchmarking with results of finite difference techniques