Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

Fiber orientation induced by injection mold filling of short-fiber-reinforced thermoplastics (FRTP) causes anisotropy in material properties and warps molded parts. Predicting fiber orientation is important for part and mold design to produce sound molded parts. A numerical scheme is presented to predict fiber orientation in three-dimensional thin-walled molded parts of FRTP. Folgar and Tucker's orientation equation is used to represent planar orientation behavior of rigid cylindrical fibers in concentrated suspensions. The equation is solved about a distribution function of fiber orientation by using a finite difference method with input of velocity data from a mold filling analysis. The mold filling is assumed to be nonisothermal Hele-Shaw flow of a non-Newtonian fluid and analyzed by using a finite element method. To define a degree of fiber orientation, an orientation parameter is calculated from the distribution function against a typical orientation angle. Computed orientation parameters were compared with measured thermal expansion coefficients for molded square plates of glass-fiber-reinforced polypropylene. A good correlation was found.     

Numerical Analysis of the Coupling Between the Flow Kinematics and the Fiber Orientation in Eulerian Simulations of Dilute Short Fiber Suspensions Flows

This paper focuses on an accurate evaluation of short fibers suspensions models coupling the flow kinematics with the fiber orientation evolution. In coupled models the flow kinematics is usually solved using the finite element method, where the fiber orientation is introduced in the constitutive equation through its value in some points (nodes or integration points). In this paper we will compare in a simple steady shear flow, the exact solutions of the extra‐stresses associated with the fibers' presence with the numerical simulations obtained using both the method of characteristics and the discontinuous Galerkin's method to solve the equation governing the generalized gradient evolution, in order to avoid the introduction of any closure relation. The error introduced if a quadratic closure relation is considered in the constitutive equation will be also quantified.     

Numerical simulation of fiber orientation in injection molding of short-fiber-reinforced thermoplastics

The present study develops a numerical simulation program to predict the transient behavior of fiber orientations together with a mold filling simulation for short-fiber-reinforced thermoplastics in arbitrary three-dimensional injection mold cavities. The Dinh-Armstrong model including an additional stress due to the existence of fibers is incorporated into the Hele-Shaw equation to result in a new pressure equation governing the filling process. The mold filling simulation is performed by solving the new pressure equation and energy equation via a finite element/finite difference method as well as evolution equations for the second-order orientation tensor via the fourth-order Runge-Kutta method. The fiber orientation tensor is determined at every layer of each element across the thickness of molded parts with appropriate tensor transformations for arbitrary three-dimensional cavity space.     

纸桨泵叶轮中纤维悬浮湍流的流动特性(英文)

A numerical method for predicting fiber orientation is presented to explore the flow properties of turbu-lent fiber suspension flowing through a stock pump impeller. The Fokker-Planck equation is used to describe the distribution of fiber orientation. The effect of flow-fiber coupling is considered by modifying the constitutive mode. The three-dimensional orientation distribution function is formulated and the corresponding equations are solved in terms of second-order and fourth-order orientation tensors. The evolution of fiber orientation, flow velocity and pressure, additional shear stress and normal stress difference are presented. The results show that the evolutions of fiber orientation are different along different streamlines. The velocity and its gradient are large in the concave wall region, while they are very small in the convex wall region. The additional shear stress and normal stress difference are large in the inlet and concave wall regions, and moderate in the mid-region, while they are almost zero in most downstream regions. The non-equilibrium fiber orientation distribution is dominant at the inlet and the concave wall regions. The flow will consume more energy to overcome the additional shearing losses due to fibers at the inlet and the concave wall regions. The change of flow rates has effect on the distribution of additional shear stress and normal stress difference. The flow structure in the inlet and concave wall regions is essential in the resultant rheological properties of the fiber suspension through the stock pump impeller, which will directly affect the flow efficiency of the fiber suspension through the impeller.     

Rheological properties of fiber suspensions flowing through a curved expansion duct

A numerical method based on finite volume method for solving Fokker‐Planck equation on a unit sphere was first developed to simulate numerically the fiber orientation distribution of fiber suspensions flowing through a curved expansion duct. The momentum equation with fiber extra stress was used to explore the rheological properties of the fiber suspensions. The second‐order fiber orientation tensors in present simulation are compared with the previous numerical results for validating the model and computational code. The results showed that the pressure increases from inlet to outlet, and changes along the lateral direction nonmonotonically except at outlet. The extra shear stress and normal stress difference decrease from inlet to outlet, and changes along the lateral direction nonmonotonically. The effect of fiber aspect ratio on the extra shear stress and normal stress difference is negligible near the wall region. The extra shear stress and normal stress difference increase with the fiber aspect ratio near the centerline. The lateral distribution curves of extra shear stress and normal stress difference become flat with increasing the distance from the inlet. POLYM. ENG. SCI., 50:1994–2003, 2010. © 2010 Society of Plastics Engineers     

Modeling and simulation of fiber‐reinforced polymer mold‐filling with phase change

A gas‐solid‐liquid three‐phase model for the simulation of fiber‐reinforced composites mold‐filling with phase change is established. The influence of fluid flow on the fibers is described by Newton's law of motion, and the influence of fibers on fluid flow is described by the momentum exchange source term in the model. A revised enthalpy method that can be used for both the melt and air in the mold cavity is proposed to describe the phase change during the mold‐filling. The finite‐volume method on a non‐staggered grid coupled with a level set method for viscoelastic‐Newtonian fluid flow is used to solve the model. The “frozen skin” layers are simulated successfully. Information regarding the fiber transformation and orientation is obtained in the mold‐filling process. The results show that fibers in the cavity are divided into five layers during the mold‐filling process, which is in accordance with experimental studies. Fibers have disturbance on these physical quantities, and the disturbance increases as the slenderness ratio increases. During mold‐filling process with two injection inlets, fiber orientation around the weld line area is in accordance with the experimental results. At the same time, single fiber's trajectory in the cavity, and physical quantities such as velocity, pressure, temperature, and stresses distributions in the cavity at end of mold‐filling process are also obtained. © 2015 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016 , 133, 42881.     

Nonisothermal transient filling simulation of fiber suspended viscoelastic liquid in a center‐gated disk

The present study numerically investigates a fiber orientation in injection‐molded short fiber reinforced thermoplastic composite by using a rheological model, which includes the nonlinear viscoelasticity of polymer and the anisotropic effect of fiber in the total stress. A nonisothermal transient‐filling process for a center‐gated disk geometry is analyzed by a finite element method using a discrete‐elastic‐viscous split stress formulation with a matrix logarithm for the viscoelastic fluid flow and a streamline upwind Petrov–Galerkin method for convection‐dominated problems. The numerical analysis result is compared to the experimental data available in the literature in terms of the fiber orientation in center‐gated disk. The effects of the fiber coupling and the slow‐orientation kinetics of the fiber are discussed. Also, the effect of the injection‐molding processing condition is discussed by varying the filling time and the mold temperature. POLYM. COMPOS., 2011. © 2010 Society of Plastics Engineers     

Model resin permeation of fiber reinforcements after shear deformation

In liquid composite molding processes such as resin transfer molding and structural reaction injection molding, fiber reinforcements are formed with automated processes to conform to the complex shape of the mold cavity. Deformation of the fiber reinforcement during the forming operation can be characterized by factors such as the local surface curvature of the mold and the type of reinforcement. For bidirectional fiber fabrics, simple shear is the major deformation mode in the forming process. Deformation of the fiber reinforcement after being formed to the mold cavity shape results in variations of local fiber content. In addition, the network structure of the fiber reinforcement is also rearranged. This may cause some significant effects on the fiber permeability and result in a mold filling pattern quite different from that expected. Therefore, a good understanding and measurement of the permeabilities for the deformed fiber reinforcements is of great importance. In the flow simulation of the filling process, the success of the prediction depends greatly on the correct values of in-plane permeabilities. A change of the in-plane permeability of the fabric after shear deformation must be well understood before an accurate flow simulation can be obtained. This article investigates the permeability of fiber reinforcements in relation to different shear angles. Several flow experiments were conducted on bidirectional woven roving fabrics at different shear angles. Two relevant factors—the ratio of principal permeabilities and the direction of principal axes with respect to the orientation of the fabric—are studied to investigate their variations with respect to shear deformation of the fiber reinforcements. It is found that the angle shift of the principal axes increases with the shear angle. At the same time, the in-plane permeability ration may decrease with the shear angle.     

Orientation Effect of Clay Platelets in Transfer Molded Polymer Nanocomposites

Transfer molding has been increasingly used to process polymer composites with various shaped nanoparticles, including platelet type nanoparticles. Platelet nanoparticles exhibit high aspect ratios (length to thickness); therefore their distributions in polymer matrix can be greatly affected by the flow trajectories in the mold. In present study, the clay platelet-polyethylene nanocomposite was prepared by transfer molding. The orientation of clay platelets developed during molding process was analyzed and measured with wide angle X-ray diffraction. Due to large velocity and shear stress gradients in the mold, the platelets caught in surface regions were rotated towards the flow direction and formed the oriented morphologies. With weaker shear stresses at the central region, the platelets were mostly randomly distributed. The shear stresses may be amplified locally at regions near the mold walls, which can further lead to or accelerate the orientations of clay particles. The orientation distribution was found to depend upon the clay fraction and sample size. The oriented platelets can be re-randomized through the annealing process. The orientation of clay platelets enhanced the orientation of polyethylene lamellae and caused shear band formation in composites when deformed.     

Numerical simulation of fiber orientation distribution in round turbulent jet of fiber suspension

The Reynolds averaged Navier–Stokes equation was solved numerically with the Reynolds stress model to get the mean fluid velocity and the turbulent kinetic energy in a round turbulent jet of fiber suspension. The fluctuating fluid velocity was described as a Fourier series with random coefficients. Then the slender-body theory was used to calculate the fiber orientation distribution and orientation tensor. Numerical results of mean axial velocity and turbulent shear stress along the lateral direction were validated by comparing with the experimental ones. The results show that most fibers are aligned with the flow direction as they go downstream, and fibers are more aligned with the flow direction within the region near the jet core. The fibers with high aspect ratio tend much easier to align with the flow direction, and the fiber orientation distribution is not sensitive to fiber aspect ratio when fiber aspect ratio is larger than 5. Fiber density has no obvious influence on the fiber orientation distribution and fiber orientation tensor. The randomizing effect of turbulence is insignificant in the regions near outside and jet core, and becomes significant in the region between outside and jet core. The randomizing effect increases with the increasing of the distance from the jet exit. Different components of fiber orientation tensor show a similar distribution pattern.     

Dynamic simulation of microstructure and rheology of fiber suspensions

A numerical simulation method has been developed to predict the motion of fiber suspensions in various flows by using the particle simulation method (PSM), in which a fiber is modeled by an array of spheres. The hydrodynamic interaction among fibers is considered by decomposing into intra-fiber and inter-fiber interactions. For the intra-fiber case, the many-body problem is solved by calculating the mobility matrix for each fiber to obtain the hydrodynamic force and torque exerted on each sphere. For the inter-fiber case, only the near-field lubrication force is considered between spheres belonging to different fibers. The methodology was applied to predicting the microstructure and rheological properties of rigid fiber suspensions in simple shear flow and fiber orientation in two-dimensional diverging flow by combining this method with the finite element method (FEM). In the former, the overshoot of suspension viscosity, due to the transient change of the microstructure, was observed in semi-dilute to concentrated systems. In the latter, the calculated fiber orientations agreed with expectations of theory.     

Fiber orientation in simple injection moldings. Part I: Theory and numerical methods

This paper sets out the theory and numerical methods used to simulate filling and fiber orientation is simple injection moldings (a film-gated strip and a center-gated disk). Our simulation applies to these simple geometry problems for the flow of a generalized Newtonian fluid where the velocities can be solved independently of fiber orientation. This simplification is valid when the orientation is so flat that the fibers do not contribute to the gapwise shear stresses. A finite difference solution calculates the temperature and velocity fields along the flow direction and through the thickness of the part, and fiber orientation is then integrated numerically along pathlines. Fiber orientation is three-dimensional, using a second-rank tensor representation of the orientation distribution function. The assumptions used to develop the simulation are not valid near the flow front, where the recirculating fountain flow complicates the problem. We present a numerrical scheme that includes the effect of the fountain flow on temperature and fiber orientation near the flow front. The simulation predicts that the orientation will vary through the thickness of the part, causing the molding to appear layered. The outer “skin” layer is predicted only if the effects of the fountain flow and heat transfer are included in the simulation.     

The prediction of flow and orientation behavior of short fiber reinforced melts in simple flow systems

A model for the prediction of changes in fiber orientation in simple flows of fiber suspensions is proposed. Fiber interactions are modeled as randomizing forces over the rotation of fibers in closed orbits in simple shear. The resulting Fokker-Planck type convection-diffusion equation in orientation space is solved using a finite difference technique. The solution technique permits the use of periodic boundary condition for the convection-diffusion equation and different initial conditions for the orientation distribution. The model predictions for simple shear flow demonstrate the interaction between the structural changes and the bulk rheological properties. The effect of non-Newtonian fluid properties on the orientation distribution was also incorporated at the slow flow limit. Structural changes are assumed to be irreversible. The irreversibility is incorporated through an orientation distribution dependent diffusion coefficient.     

纤维粘弹性流体悬浮液流变性质研究

描述了纤维Oldroyd-B流体半浓悬浮液的本构方程,计算了定常剪切流动突然开始后的纤维取向和应力的发展过程,并将计算结果与其它本构方程的计算结果和实验结果进行比较,结果表明,本文的本构方程能合理地描述应力的发展过程,尤其能刻划出应力峰值后的弹性增长,这是其它本构方程所不能描述的。     

Rheology of LDPE‐based semiflexible fiber suspensions

Molten LDPE suspensions containing fibers of different flexibilities have been investigated in simple shear and small and large amplitude oscillatory shear (LAOS) flow. The suspensions exhibited viscosity and normal stress overshoots in stress growth experiments, and the magnitude and width of the overshoots became larger as the fiber flexibility increased. LAOS was used to help understanding the relationship between stress growth and fiber orientation. For all composites, the stress signal decreased with time in LAOS, and this behavior was more pronounced in the case of the more rigid fibers. The energy dissipated per LAOS cycle was evaluated for each composite, and it showed that less energy was dissipated as fiber flexibility decreased. In addition, the dissipated energy decreased with time and this has been interpreted in terms of a reduction of fiber contacts. The first normal stress difference showed a nonsinusoidal periodic response, and fast Fourier transform analysis indicated the presence of a first harmonic corresponding to the applied frequency for the fiber‐filled systems, in addition to the second harmonic observed for the neat LDPE. It resulted in asymmetrical strain‐normal force Lissajou curves for the suspensions, with this asymmetry being more pronounced in the case of the more rigid fibers. This has been attributed to a more extensive fiber orientation for the latter. POLYM. COMPOS., 31:1474–1486, 2010. © 2009 Society of Plastics Engineers     

Analytical solutions for fiber orientation in 2-D flows of dilute suspensions

Analytical solutions for fiber orientation in dilute solutions are presented for both 2-D planar and axisymmetric flow. The viscosity is assumed to be independent of position along the flow direction, but varies through the thickness of the mold cavity. For the particular case of a laminate of two fluids, an analytical solution is derived. Based upon the assumption that the fluid adjacent to the mold wall exhibits reduced viscosity due to non-isothermal considerations, the fluid kinematics simplify, and an analytical solution to Jeffery's orientation equation is derived. Since the fiber orientation analysis is based on Jeffery's equation, it is valid only for dilute suspensions, i.e. for fiber volume fraction < 1/(fiber aspect ratio)². Qualitatively, the analytical results obtained provide superior correlation with experimental results, indicating that non-isothermal fluid flow may play an important role in the development of the fiber orientation distribution in the molded component.     

Nonisothermal crystallization of poly(ethylene terephthalate) and its blends in the injection-molding process

We investigated the nonisothermal crystallization during the cooling process of injection molding of poly(ethylene terephthalate) (PET), PET/talc, and PET/Surlyn blends. We applied the isothermal crystallization parameters obtained by the Hoffman–Lauritzen theory to the kinetics of nonisothermal crystallization and then calculated the relative crystallinity χ/χ_c as a function of the mold temperature. χ/χ_c were nicely interpreted by calculation without effect of the pressure history on crystallization in PET and PET/talc (1 wt %) blends. In contrast, in the PET/Surlyn (3 wt %) blend, crystallization occurred at a lower mold temperature than predicted by our calculation. The transmission electron micrograph near the surface of the injection-molded PET/Surlyn blend showed deformation and stretching of dispersed Surlyn particles, suggesting that orientation of the PET matrix proceeds with the flow in processing. The orientation of the PET matrix resulted in acceleration of the crystallization in the injection molding. © 1995 John Wiley & Sons, Inc.     

Prediction of fiber orientation in the injection molding of long fiber suspensions

The properties of long glass fiber reinforced parts are highly dependent on the fiber orientation generated during processing. In this research, the orientation of concentrated long glass fibers generated during the filling stage of a center‐gated disk (CGD) mold was simulated. The orientation of the fibers was calculated using both the Folgar‐Tucker model and a recently developed semiflexible Bead‐Rod model. Rheologically consistent model parameters were used in these simulations, as determined from a previously proposed method, using a sliding plate rheometer and newly modified stress theory. The predicted CGD orientations were compared with experimentally measured values obtained from the parts. Both models performed very well when using model parameters consistent with the independent rheological study, and the results provide encouragement for the proposed method. Comparatively, the Folgar‐Tucker model provided slightly better orientation predictions up to 20% of the fill radius, but above 20% the Bead‐Rod model predicted better values of the orientation in both the radial and circumferential directions. The Folgar‐Tucker model, however, provided better orientation values perpendicular to the flow direction. Lastly, both models only qualitatively represented the orientation above 70% of the fill radius where frontal flow effects were suspected to be non‐negligible. The uniqueness of this research rests on a method for obtaining model parameters needed to predict fiber orientation which are independent of the experiments being simulated and a method for handling long semiflexible fiber suspensions. POLYM. COMPOS., 2012. © 2012 Society of Plastics Engineers     

Rheological modeling of carbon nanotube suspensions with rod–rod interactions

To explain the shear‐thinning behavior of untreated carbon nanotube (CNT) suspensions in a Newtonian matrix, a new set of rheological equations is developed. The CNTs are modeled as rigid rods dispersed in a Newtonian matrix and the evolution of the system is controlled by hydrodynamic and rod–rod interactions. The particle–particle interactions is modeled by a nonlinear lubrication force, function of the relative velocity at the contact point, and weighted by the contact probability. The stress tensor is calculated from the known fourth‐order orientation tensor and a new fourth‐order interaction tensor. The Fokker‐Planck equation is numerically solved for steady simple shear flows using a finite volume method. The model predictions show a good agreement with the steady shear data of CNTs dispersed in a Newtonian epoxy matrix as well as for suspensions of glass fibers in polybutene,¹ demonstrating its ability to describe the behavior of micro‐ and nanoscale particle suspensions. © 2013 American Institute of Chemical Engineers AIChE J, 60: 1476–1487, 2014