Folgar

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

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... 

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. 

Folgar F, Tucker CL (19S4). J Reinforced Plast Composites 3 98. 

Cersosimo MG, Lasala B, Folgar S, MicheU F. Epidural lipomatosis secondary to indinavir in an HIV-positive patient. Chn Neuropharmacol 2002 25(1) 51. ... 

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... 

Folgar, F. P., and Tucker, C. L., Orientation behavior of fibers in concentrated suspensions, J. Reinf. Plast. Compos., 3, 98-119 (1984). 

Folgar, R, Scott, B. R., Walsh, S. M., Wolbert, J. Thermoplastic Matrix Combat Helmet With Graphite-Epoxy Skin , 23rd International Symposium On Ballistics, Tarragona, Spain, 16-20 April 2007. 

Analysis of the final properties of injection-molded short-fiber composite parts requires accurate prediction of flow-induced fiber orientation. Several different fiber suspension theories and numerical methods are available for the calculation of the motion of fibers during flow. Only the works of Folgar and Tucker and Fan are briefly reviewed here since they are the most relevant. The reader is referred to Phan-Thien and Zheng for additional information for other constitutive theories of fiber suspensions. In what follows we assume the fibers are rigid rods of circular cross section. 

Jeffery s equation was extended to concentrated solutions by Folgar and Tucker who added a diffusion term to account for the fiber-fiber interaction. In terms of the orientation tensor, the Tucker-Folgar equation has the form... 

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. 

Folgar and Tucker (1984) have considered the evolution of the orientation state of non-dilute fiber suspensions as a diffusive process. The fiber-fiber interactions are treated as a random force effect superposed with the hydrodynamic effect of the... 

Fig. 5.3 A comparison of the simulated C/ for different aspect ratios (filled symbols) with experimental data of Folgar and Tucker (open symbols). The solid line represents Eq. 5.26 (From Phan-Thien et al. (2002), with permission from Elsevier)...

The results of the direct simulation and the empirical equation 5.26 are comparable with the data of Folgar and Tucker (1984), as shown in Fig. 5.3. 

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,. 

Folgar FP, Tucker CL IB (1984) Orientation behavior of fiber in concentrated suspensions. J Reinforced Plastic Compos 3 98-119... 

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... 

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