Fiber Suspensions in Complex Geometries: Flow/Orientation Coupling

A Galerkin finite element solution is developed for the flow of fiber suspensions. Primary variables are velocity, pressure, and a second‐order tensor describing the fiber orientation. The model treats the orientation as three‐dimensional, includes fiber—fiber interaction effects, and uses an orthotropic closure approximation. The flow and orientation are strongly coupled through an orientation‐dependent constitutive equation. We explore the effect of this coupling on the fluid mechanics of fiber suspensions by studying three flows: an axisymmetric contraction, an axisymmetric expansion, and a center‐gated disk. Coupling enhances the corner vortex in the contraction, in quantitative agreement with published experiments and calculations. The expansion results demonstrate that the aligned‐fiber approximation is not valid for this flow. In the center‐gated disk the effects of coupling are modest and are only noticeable near the center of the disk. This supports the use of decoupled models for injection molding in thin cavities.     

Numerical simulation of three‐dimensional fiber orientation in injection molding including fountain flow effect

It is essential to predict the nature of flow field inside mold and flow‐induced variation of fiber orientation for effective design of short fiber reinforced plastic parts. In this investigation, numerical simulations of flow field and three‐dimensional fiber orientation were carried out in special consideration of fountain flow effect. Fiber orientation distribution was described using the second‐order orientation tensor. Fiber interaction was modeled using the interaction coefficient C_I. Three closure approximations, hybrid, modified hybrid, and closure equation for C_I=0, were selected for determination of the fiber orientation. The fiber orientation routine was incorporated into a previously developed program of injection mold filling (CAMPmold), which was based on the fixed‐grid finite element/finite difference method assuming the Hele‐Shaw flow. For consideration of the fountain flow effect, simplified deformation behavior of fountain flow was employed to obtain the initial condition for fiber orientation in the flow front region. Comparisons with experimental results available in the literature were made for film‐gated strip and centergated disk cavities. It was found that the orientation components near the wall were were accurately predicted by considering the fountain flow effect. Test simulations were also carried out for the filling analysis of a practical part, and it was shown that the currently developed numerical algorithm can be effectively used for the prediction of fiber orientation distribution in complex parts.     

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     

纤维增强复合材料的力学性能预测的数值模拟

纤维增强复合材料的力学性能和热物理性能依赖于纤维的取向状态.在注射成型过程中纤维最终的取向状态依赖于充填过程的速度场,因此最终的产品性质依赖于成型的详细过程.研究发现,注塑成型制品的结构呈层状分布,层数依赖于模具几何和成型条件,不过大多数的结构在成型表面为沿流动方向取向,而在中心层为横向排列,有时在制件表面还有一层薄的介于二者之间排列的取向层.本文主要给出两个简单模型中纤维取向预测的理论和数值方法,这两个模型分别为:中心浇口圆盘和边浇口长条.     

Application and evaluation of the method of ellipses for measuring the orientation of long,semi‐flexible fibers

The method of ellipses (MoE) is a common experimental technique utilized to quantitatively determine the orientation state of a population of rigid fibers within a fiber–polymer composite. In this research, the validity of applying the MoE to long, semi‐flexible fiber systems in which the majority of fibers are flexible is discussed. The components of the orientation tensor were first determined for a composite formed by a homogenous, simple shear field. The minimum acceptable image analysis width, or bin width, for the selected geometry was found to be ∼5.5 mm, or 1.4 times the average fiber length. This modified bin width was then used to determine the orientation at multiple percentages of flow within an injection‐molded, center‐gated disc, and compared to orientation values obtained utilizing the traditional, 0.7‐mm bin width. The results show that the traditional, 0.7‐mm bin width is sufficient for analysis of the center‐gated geometry. This fortuitous result is attributed to the axisymmetric nature of the center‐gated geometry, and the highly transverse fiber alignment seen within the samples, especially at moderate to high percentages of flow. In more complex flows, it is expected that the conventional bin width will not apply. POLYM. COMPOS., 2013. © 2013 Society of Plastics Engineers     

Numerical prediction of residual stresses and birefringence in injection/compression molded center‐gated disk. Part I: Basic modeling and results for injection molding

The present study attempted to numerically predict both the flow‐induced and thermally‐induced residual stresses and birefringence in injection or injection/compression molded center‐gated disks. A numerical analysis system has been developed to simulate the entire process based on a physical modeling including a nonlinear viscoelastic fluid model, stress‐optical law, a linear viscoelastic solid model, free volume theory for density relaxation phenomena and a photoviscoelasticity and so on. Part I presents physical modeling and typical numerical analysis results of residual stresses and birefringence in the injection molded center‐gated disk. Typical distribution of thermal residual stresses indicates a tensile stress in the core and a compressive stress near the surface. However, depending on the processing condition and material properties, the residual stress sometimes becomes tensile on the surface, especially when fast cooling takes place near the mold surface, preventing the shrinkage from occurring. The birefringence distribution shows a double‐hump profile across the thickness with nonzero value at the center: the nonzero birefringence is found to be thermally induced, the outer peak due to the shear flow and subsequent stress relaxation during the filling stage and the inner peak due to the additional shear flow and stress relaxation during the packing stage. The combination of the flow‐induced and thermally‐induced birefringence makes the shape of predicted birefringence distribution quite similar to the experimental one.     

Experimental study and calculations of short glass fiber orientation in a center gated molded disc

Short glass fiber orientation in a center gated molded disc of polyamide is studied using optical microscopy techniques. The very different orientation between the core and the surface of the molding is quantified with an orientation function. The influence of the molding conditions is investigated. A numerical scheme is used for modeling the mold filling with a viscous melted polymer. A computation method is introduced to describe the fiber movement during the flow. The theoretical results are in good agreement with the experimental ones. In particular, the very different orientation between the skin and the core of the disc is well predicted.     

Fiber orientation in injection molding with rotating flow

The development of fiber orientation in injection molding was manipulated by a special molding tool, the RCEM mold, which imposes a rotation action by one of the cavity surfaces during the filling stage. Center‐gated disc moldings were produced from glass fiber reinforced polypropylene with different cavity rotation velocities, inducing distinct distributions and levels of fiber orientation. The morphologies of the moldings were characterized by optical and electronic microscopy. The through‐thickness profiles of fiber orientation were assessed by means of the orientation tensor, and the relationship between the processing thermo‐mechanical environment and the fiber orientation was established. At high rotation velocities, the resulting fiber orientation pattern is mainly controlled by the rotational motion, inducing a much more homogeneous through‐the‐thickness fiber orientation distribution, with a preferential alignment on the circumferential direction. POLYM. ENG. SCI., 2008. © 2007 Society of Plastics Engineers     

Improved through‐plane electrical conductivity in a carbon‐filled thermoplastic via foaming

Flow‐induced orientation of the conductive fillers in injection molding creates parts with anisotropic electrical conductivity where through‐plane conductivity is several orders of magnitude lower than in‐plane conductivity. This article provides insight into a novel processing method using a chemical blowing agent to manipulate carbon fiber (CF) orientation within a polymer matrix during injection molding. The study used a fractional factorial experimental design to identify the important processing factors for improving the through‐plane electrical conductivity of plates molded from a carbon‐filled cyclic olefin copolymer (COC) containing 10 vol% CF and 2 vol% carbon black. The molded COC plates were analyzed for fiber orientation, morphology, and electrical conductivity. With increasing porosity in the molded foam part, it was found that greater out‐of‐plane fiber orientation and higher electrical conductivity could be achieved. Maximum conductivity and fiber reorientation in the through‐plane direction occurred at lower injection flow rate and higher melt temperature. These process conditions correspond with foam flow during filling of the mold cavity, indicating the importance of shear stress on the effectiveness of a fiber being rotated out‐of‐plane during injection molding. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers     

Mechanical properties of glass fiber reinforced polypropylene injection molded with a rotation mold

A special mold (Rotation, Compression, and Expansion Mold) was used to impose a controlled shear action during injection molding of short glass fiber reinforced polypropylene discs. This was achieved by superimposing an external rotation to the pressure‐driven advancing flow front during the mold filling stage. Central gated discs were molded with different cavity rotation velocities, inducing distinct levels of fiber orientation through the thickness. The mechanical behavior of the moldings was assessed, in tensile and flexural modes on specimens cut at different locations along the flow path. Complete discs were also tested in four‐point flexural and in impact tests. The respective results are analyzed and discussed in terms of relationships between the developed fiber orientation level and the mechanical properties. The experimental results confirm that mechanical properties of the moldings depend strongly on fiber orientation and can thus be tailored by the imposed rotation during molding. POLYM. ENG. SCI. 46:1598–1607, 2006. © 2006 Society of Plastics Engineers.     

Coupled analysis of injection molding filling and fiber orientation,including in-plane velocity gradient effect

On injection molding of short fiber reinforced plastics, fiber orientation during mold filling is determined by the flow field and the interactions between the fibers. The flow field is, in turn, affected by the orientation of fibers. The Dinh and Armstrong rheological equation of state for semiconcentrated fiber suspensions was incorporated into the coupled analysis of mold filling flow and fiber orientation. The viscous shear stress and extra shear stress due to fibers dominate the momentum balance in the coupled Hele-Shaw flow approximation, but the extra in-plane stretching stress terms could be of the same order as those shear stress terms, for large in-plane stretching of suspensions of large particle number. Therefore, a new pressure equation, governing the mold filling process, was derived, including the stresses due to the in-plane velocity gradients. The mold filling simulation was then performed by solving the new pressure equation and the 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 effects of stresses due to the in-plane velocity gradient on pressure, velocity, and fiber orientation fields were investigated in the center-gated radial diverging flow in the cases of both an isothermal Newtonian fluid matrix and a nonisothermal polymeric matrix. In particular, the in-plane velocity gradient effect on the fiber orientation was found to be significant near the gate, and more notably for the case of a nonisothermal polymer matrix.     

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.     

Fiber orientation in divergent/convergent flows in expansion and compression injection molding

This study compares the fiber orientation patterns of short glass fiber‐reinforced polypropylene developed in conventional and nonconventional injection molding, the latter using a special mold, RCEM (rotation, compression, and expansion mold). This mold allows for a wide variety of operating modes during mold filling, which leads to a great versatility in obtaining different fiber orientation patterns. By incorporating through‐thickness convergent and divergent flows during the filling stage (compression and expansion modes, respectively), the fiber orientation can be tailored. These linear dynamics can be superimposed with a simultaneous rotational movement of one of the mold surfaces. These combined actions induce a high fiber orientation transversely to the radial flow direction, this effect being more pronounced in the expansion mode. POLYM. COMPOS. 27:539–551, 2006. © 2006 Society of Plastics Engineers     

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.     

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.     

Modeling material property heterogeneity in fiber‐reinforced injection molded plastic parts

Fiber reinforced plastic parts manufactured by injection molding have heterogeneous stiffness and strength behavior due to the molding process influence on the fiber orientations. This paper presents a methodology for determining the process‐dependent anisotropic and inhomogeneous mechanical properties of injection‐molded parts using a thickness‐wise layered homogenization technique. This technique produces an equivalent laminated meso‐scale representation at any location in the part and enables point‐wise application of the existing laminated plate and shell theories. The methodology is demonstrated by illustrating property variations in edge‐gated and center‐gated plaques. Spatial variations of elastic moduli, shear modulus, and Poisson's ratio are modeled. The model can be conveniently embedded within finite element structural analyses accounting for the process‐dependent material heterogeneities within the structure. POLYM. COMPOS., 26:98–113, 2005. © 2004 Society of Plastics Engineers     

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.     

Numerical investigation of three-dimensional fiber suspension flow by using finite volume method

Fiber suspension flow is common in many industrial processes like papermaking and fiber-reinforcing polymer-based material forming. The investigation of the mechanism of fiber suspension flow is of significant importance, since the orientation distribution of fibers directly influences the mechanical and physical properties of the final products. A numerical methodology based on the finite volume method is presented in the study to simulate three-dimensional fiber suspension flow within complex flow field. The evolution of fiber orientation is described using different formulations including FT model and RSC model. The pressure implicit with splitting of operators algorithm is adopted to avoid oscillations in the calculation. A laminate structure of fiber orientation including the shell layer, the transition layer and the core layer along radial direction within a center-gated disk flow channel is predicted through a three-dimensional simulation, which agrees well with Mazahir’s experimental results. The evolution of fiber orientation during the filling process within the complex flow field is further discussed. The mathematical model and numerical method proposed in the study can be successfully adopted to predict fiber suspension flow patterns and hence to reveal the fiber orientation mechanism.     

Numerical prediction of residual stresses and birefringence in injection/compression molded center‐gated disk. Part II: Effects of processing conditions

The accompanying paper, Part I, has presented the physical modeling and basic numerical analysis results of the entire injection molding process, in particular with regard to both flow‐induced and thermally‐induced residual stress and birefringence in an injection molded center‐gated disk. The present paper, Part II, investigates the effects of various processing conditions of injection/compression molding process on the residual stress and birefringence. The birefringence is significantly affected by injection melt temperature, packing pressure and packing time. However, the thermally‐induced birefringence in the core region is insignificantly affected by most of the processing conditions. On the other hand, packing pressure, packing time and mold wall temperature affect the thermally‐induced residual stress rather significantly in the shell layer, but insignificantly in the core region. The residual stress in the shell layer is usually compressive, but could be tensile if the packing time is long, packing pressure is large, and the mold temperature is low. The lateral constraint type turns out to play an important role in determining the residual stress in the shell layer. Injection/compression molding has been found to reduce flow‐induced birefringence in comparison with the conventional injection molding process. In particular, mold closing velocity and initial opening thickness for the compression stage of injection/compression molding have significant effects on the flow‐induced birefringence, but not on the thermal residual stress and the thermally‐induced birefringence.     

Birefringence in injection‐compression molding of amorphous polymers: Simulation and experiment

The influence of the processing variables on the residual birefringence was analyzed for polystyrene and polycarbonate disks obtained by injection‐compression molding under various processing conditions. The processing variables studied were melt and mold temperatures, compression stroke, and switchover time. The modeling of flow‐induced residual stresses and birefringence of amorphous polymers in injection‐compression molded center‐gated disks was carried out using a numerical scheme based on a hybrid finite element/finite difference/control volume method. A nonlinear viscoelastic constitutive equation and stress‐optical rule were used to model frozen‐in flow stresses in moldings. The filling, compression, packing, and cooling stages were considered. Thermally‐induced residual birefringence was calculated using the linear viscoelastic and photoviscoelastic constitutive equations combined with the first‐order rate equation for volume relaxation and the master curves for the Young's relaxation modulus and strain‐optical coefficient functions. The residual birefringence in injection‐compression moldings was measured. The effects of various processing conditions on the measured and simulated birefringence distribution Δn and average transverse birefringence <n_rr?n_θθ> were elucidated. Comparison of the birefringence in disks manufactured by the injection molding and injection‐compression molding was made. The predicted and measured birefringence is found to be in fair agreement. POLYM. ENG. SCI., 2013. © 2013 Society of Plastics Engineers