Folgar-Tucker model

PREDICTING FIBER ORIENTATION - THE FOLGAR-TUCKER MODEL 443... 

To illustrate the effect of fiber orientation on material properties of the final part, Fig. 8.60 [5] shows how the fiber orientation distributions that correspond to 67 50 and 33% initial mold coverage affect the stiffness of the finished plates. The Folgar-Tucker model has been implemented into various, commercially available compression mold filling simulation programs and successfully tested with several realistic compression molding applications. 

An empirical constant called the interaction coefficient Cj is introduced in the diffusion term. The constant C/ for a given suspension is assumed to be isotropic and independent of the orientation state, as a first approximation. The Folgar-Tucker model has extended the fiber orientation simulations into nondilute regimes. It is widely used to determine the orientation of fibers in injection molding. 

Motion of a fiber in flow is described by Jeffery s model [3]. It is assumed that the fiber is a single rigid ellipsoidal partide suspended in a viscous fluid, the flow is a creeping flow of a Newtonian and incompressible fluid, and Brownian motion and inertia terms of the fiber are neglected. Jeffery s model was used for prediction of fiber orientation in the early period of injection molding CAE. Since it is, however, for dilute suspension, the model is replaced with the Folgar-Tucker model for concentrated suspension. 

Modifications to Folgar-Tucker Model 5.5.1 Anisotropic Rotary Diffusion Model... 

As experimental evidence has shown that the standard Folgar-Tucker model predicts a faster transient orientation evolution than that observed experimentally. Tucker et al. (2007), Wang et al. (2008) and Phelps and Tucker (2009) have proposed a new evolution equation, i.e., the so-called reduced-strain closure (RSC) model, to slow down the fiber orientation kinetics. Their approach is based on the spectral decomposition theorem. The theorem states that if T is a symmetric second-order tensor, then there is a basis e, i — 1, 2, 3 consisting entirely of eigenvectors of T and the corresponding eigenvalues Aj, i — 1, 2, 3 forming the entire spectrum of T, thus T can be represented by T = A,e,e,. 

Here, 7 is the magnitude of the strain rate tensor and C/ is a phenomenological coefficient which models the interactions between the fibers, usually referred to as the Folgar-Tucker interaction coefficient. The coefficient varies between 0, for a fiber without interaction with its neighbors, and 1, for a closely packed bed of fibers. For a fiber reinforced polyester resin mat with 20-50% volume fiber content, CV is usually between 0.03 and 0.06. When eqn. (8.153) is substituted into eqn. (8.152), the transient governing equation for fiber orientation distribution with fiber interaction built-in, becomes... 

Phan-Thien N, Fan XJ, Tanner RI, Zheng R (2002) Folgar-Tucker constant for a fiber suspension in a Newtonian fluid. J Non-Newtonian Fluid Mech 103 251-260 Phan-Thien N, Fan XJ, Zheng R (2000) A numerical simulation of suspension flow using a constitutive model based on anisotropic interparticle interactions. Rheol Acta 39 122-130 Phan-Thien N, Graham AL (1991) A new constitutive model for fiber suspensions flow past a sphere. Rheol Acta 30 44-57... 

The second model used by Letwimolnun et al. [2007] is an extension of that used by Sepehr et al. [2004] for short fiber suspensions. A hydrodynamic diffusive term related to the Brownian motion, Dr, was added to the diffusive term of the Folgar and Tucker [1984] equation Dr = Cj + Dr in... 

Because many SMC parts have a thickness much less than the length in the direction of flow, an adequate model may be constructed based on classical Hele-Shaw flow the flow field is essentially a flat sheet of a constant thickness. Tucker and Folgar... 

Tucker CL, Folgar F. A model of compression mold filling. Polym Eng Sci 1983 23 69-73. 

---