Long fiber-reinforced thermoplastics (LFTs) are an attractive design option for many engineering applications due to their excellent mechanical properties and processability. When processing these materials, the length of the fibers inevitably decreases, which ultimately affects the mechanical performance of the finished part. Since none of the existing modeling techniques can accurately predict fiber damage of LFTs during injection molding, a new phenomenological approach for modeling fiber attrition is presented. First, multiple controlled studies employing a Couette rheometer are performed to determine correlations between processing conditions, material properties, and fiber length reduction. The results show shear stress and fiber concentration impact fiber damage. Based on these findings, a phenomenological model to predict breaking rate and unbreakable length of a fiber under giving conditions is developed. The model is based on the beam theory with distributed hydrodynamic stresses acting on a fiber. Fiber–fiber interactions are accounted for and correlated with the fiber volume fraction via a fitting parameter. The model tracks both the number-average and weight-average fiber length during processing, which can in turn be used to extract the fiber length distribution.

The use of injection-molded long-fiber thermoplastic composites (or LFTs) has been rapidly increasing in the past decade, mainly due to the market push for more fuel-efficient vehicles as well as electric vehicles. While the transportation industry constitutes 80% of LFTs' world usage, their use is also increasing in applications, such as durable consumer appliances, electronics, and sporting goods.1,2 Since their advantage is the high aspect ratio of the fibers, a primary concern for manufacturers is preserving the fibers' length throughout the molding process.3 Initial fiber length in LFT pellets typically ranges from 10 to 15mm. Fiber length measurements suggested a remaining average fiber length in the molded part in the 1–3mm range.4–8 

The impact of processing variables on fiber attrition at different stages of the molding process has been the focus of numerous studies. Bailey and Kraft found that most damage occurs during plastication, as did Lafranche et al. and others.4–7 Most notably, they found higher fiber lengths in the core region than the skin region in molded parts, also observed in other studies, which suggests that the characteristic flow regime during mold filling causes uneven fiber breakage.5,9 These findings go hand in hand with the recent findings on the inhomogeneous fiber density distribution in injection molding of LFT.10 

Rohde et al. performed a full factorial DOE varying processing parameters, such as injection speed, screw back pressure, holding pressure, and screw speed, where their results clearly show a significant impact of back pressure on fiber length.7 Most processing variables had no significant statistical impact on fiber length, while an increase in backpressure from 50 to 80bar reduced the fiber length in the mold cavity by approximately 30%. Nevertheless, the authors state that it is difficult to isolate the mechanisms that cause fiber attrition due to the complex phase change and shear history present in the injection molding process.

Inoue et al. found that the screw design in the compression zone has a substantial impact on fiber breakage.11 They observed that an optimized screw design (Dulmadge12 with a variable pitch) could reduce the fiber breakage compared to a standard screw. Von Turkovich et al. conducted compounding experiments with short-fiber thermoplastic composites (or SFTs) and found that most of the fiber damage occurred in the compression zone.13 Based on their results, they concluded that fiber concentration and initial fiber dispersion had no visible impact on the fiber length.

More fundamental studies on fiber motion were conducted by Forgacs and Mason, employing a Couette device and subjecting single fibers to simple shear flow.14,15 They used Burger's formulation along with Euler's beam theory to derive an equation to estimate the critical product γ̇η at which a fiber would buckle under compressive stresses.16 By increasing either the aspect ratio of the fibers or the product γ̇η, Forgacs and Mason identified various orbits of rotation; Fig. 1 summarizes their findings.

FIG. 1.

Types of fiber motion in simple shear flow as flexibility increases from (a) to (e): (a) springy rotation, (b) and (c) snake orbit, and (d) and (e) coiled rotation (adapted from Ref. 17).

FIG. 1.

Types of fiber motion in simple shear flow as flexibility increases from (a) to (e): (a) springy rotation, (b) and (c) snake orbit, and (d) and (e) coiled rotation (adapted from Ref. 17).

Close modal

Using a similar device, Salinas and Pittman performed experiments with reinforcing fibrous materials, including glass fibers with aspect ratios ranging from 280 to 680.17 They found that fibers fractured in the snake orbit regime when the minimum radii of curvature were reached, and that only very stiff or brittle materials (carbon fiber, asbestos) were predicted to break in the springy regime. They derived an empirical expression for the critical product γ̇η that would cause sufficient deformation to cause rupture. These early studies on fiber motion and failure are the basis for most modeling approaches to the fiber attrition phenomena.

Shon et al. evaluated various combinations of mixing elements and their impact on fiber length reduction as a function of axial distance in a counter-rotating twin screw extruder.18 They introduced a kinetic model in which an average fiber length, L, decreases exponentially toward a residual, or unbreakable length, L, as follows:

dLdt=kf(LL),
(1)

where kf is a breaking rate coefficient. They did not attempt to derive expressions for the parameters (L,kf) in their model but approached the problem by simply fitting the curve to the experimental data. Bumm et al. later extended the work of Shon et al. and derived expressions for both parameters based on Euler's buckling theory.19 

Durin et al. and Phelps et al. proposed similar models for fiber attrition during compounding and injection molding, respectively.20,21 Rather than tracking the average fiber length changes over time, they wrote an equation of conservation for the complete fiber length distribution (FLD) based on a breakage probability. Following Forgacs and Masons' analysis, fiber buckling is determined to be the mechanism by which fibers break. A dimensionless buckling number, Bi, defines the critical conditions under which a fiber would have the highest possibility of buckling and eventually failing. This number is a function of the product γ̇η, the fiber's stiffness and aspect ratio, given by

Bi=4ζηmli4π3Efdf4,
(2)

where ηm is the matrix viscosity, Ef is the fiber's Young's modulus, and li and df are the fiber's length and diameter. The variable ζ is the fiber's dimensionless drag coefficient and is used as a fitting parameter. These models introduce a normal or Weibull probability distribution that determines the location along the fiber's axis where failure is likely to occur.

Kang et al., proposed a model similar to Durin's.22 In their approach, fiber buckling leads to damage, which is based on the pure bending theory of simply supported beams. They compared the prediction capability to fiber breakage experiments performed with glass fiber-reinforced polypropylene (PPGF) in a parallel-plate rheometer. The experiments were purposely conducted with low fiber concentration (10 wt. %) to minimize the effect of fiber–fiber interactions on fiber damage. Malatyali et al. also employed a similar approach to predict breakage of carbon fibers along a twin screw extruder.23 For model verification, 35 wt. % carbon fiber-reinforced polypropylene was used. The common denominator between these models is that fiber buckling is presumed to be the cause or initiator of failure. Also, none of these models consider fiber–fiber interactions to have influence on the fiber breakage phenomena. The model of Phelps et al. is currently the standard method for predicting fiber damage during injection molding in commercial software. The model of Durin et al has also been implemented in commercial software for twin-screw extrusion.

Chen et al. studied the breakage on a capillary rheometer and proposed a breakage model based on the fibers' semi-flexible orientation.24 Following Salinas and Pittman's conclusions, they argue fibers break due to large deformations when the minimum radius of curvature is reached. Their model indirectly accounts for fiber–fiber interactions by using the isotropic rotary diffusion parameter Ci in the orientation model.25 Moritzer et al. conducted fiber breakage experiments under simple shear employing a Couette device and pre-compounded polypropylene with short fibers.26 Their results corroborate Shon et al.'s proposed kinetics and showed that fiber concentration influenced both the breakage rate and the residual length L. After performing a dimensional analysis, they proposed a phenomenological model that accounts for hydrodynamic stresses, fiber concentration, and fiber properties.

In recent years, direct particle simulation approaches have been used to study in detail the fiber breakage mechanisms. Sasayama et al. simulated fiber breakage under simple shear flow considering fiber–fiber interaction.27 Their results show that the fiber length decreases with the product γ̇η. They present an interesting depiction of how fiber–fiber interactions can lead to breakage. Chang et al. built a failure criterion based on a critical radius distribution obtained via loop test measurements, similar to Salinas and Pittman.28 The introduction of this failure distribution in their direct particle simulation improved their fiber length prediction compared to Couette experiments. These approaches are somewhat limited due to their computational cost, so it is unlikely they will be used to predict fiber length in real components. However, they have great potential as numerical rheometers since the coupled effects of fiber orientation, fiber length, and fiber concentration can be studied.

The main objective of this study is to develop a new model to predict fiber length during injection molding of long fiber-reinforced thermoplastics. First, a set of fiber damage experiments were conducted under highly controlled conditions employing a Couette rheometer. Important process and material variables are studied, and their impact on fiber damage is measured. Based on the experimental observations, this work derives a constitutive equation for fiber breakage kinetics, and expressions for the model parameters based on the hydrodynamic stress and mechanical failure are deduced.

The material used in this work is a commercially available long glass fiber-reinforced polypropylene. The material is supplied as coated fiber pellets with a length of 15mm, nominal glass fiber content of 20%, 30%, and 40% by weight (PPGF20, PPGF30, and PPGF40). Due to their fiber length, these materials are widely used for automotive panels and multi-wall structured panels in the construction industry. Table I lists the pertinent material properties.

TABLE I.

Typical long fiber PPGF material properties.

Material propertyValue
Nominal fiber length Pellet length 
Fiber diameter (μm) 14–2429  
Density of fibers (g/cm32.5630  
Density of polypropylene (PP) (g/cm30.930  
Modulus of fibers (GPa) 74–8031  
Ultimate strength of fibers (MPa) 2000–250031  
Material propertyValue
Nominal fiber length Pellet length 
Fiber diameter (μm) 14–2429  
Density of fibers (g/cm32.5630  
Density of polypropylene (PP) (g/cm30.930  
Modulus of fibers (GPa) 74–8031  
Ultimate strength of fibers (MPa) 2000–250031  

Investigating the underlying physics of fiber breakage experimentally in an injection molding process or during compounding in a twin-screw extruder is challenging because of the complex and changing flow conditions that the fibers are exposed to in the various stages of the process. In this work, a Couette rheometer was used to study fiber breakage under simple shear and controlled processing conditions.32 The Couette rheometer dimensions were selected to be characteristic of the conditions during plastication in injection molding, with the inner cylinder diameter being 35mm, the annular gap being 5mm, and the length of the annular volume being 80mm (Fig. 2). The temperature is controlled through insulated heater bands surrounding the outer cylinder and a thermocouple measuring the melt temperature. The inner cylinder is driven by a Plasti CorderTM torque rheometer (C.W. Brabender Instruments Inc., Hackensack, NJ), controlling the rotational speed.

FIG. 2.

Illustration of the Couette rheometer setup for the study of fiber breakage.

FIG. 2.

Illustration of the Couette rheometer setup for the study of fiber breakage.

Close modal

Based on the fiber breakup model suggested by Shon et al.,18 a set of studies were conducted with the Couette rheometer to understand the kinetics of fiber length degradation and to isolate the effects that processing conditions have on the steady-state fiber length (L) and the breakage rate coefficient (kf). After each experiment, the material was removed from the Couette, and the fiber length was measured employing a technique developed at the Polymer Engineering Center at the University of Wisconsin-Madison, which is based on the method developed by Kunc et al. to determine the fiber length distribution in LFT materials.33,34

1. Length decay over time

The first study explored the fiber length degradation over time for different processing speeds while the fiber content and melt temperature were kept constant (Table II). The material was sheared for increasing intervals of time, starting with 20s until 300s. This set of experiments allow the characterization of fiber length degradation as a function of residence time. Hence, the overall kinetic can be observed and general conclusions can be drawn.

TABLE II.

Experimental plan 1: impact of residence time and shear rate on length decay.

VariableLevels
Fiber content (wt. %) 30 
Melt temperature (°C) 250 
Rotational speed (rpm) 50, 100 
Residence time (s) 20–300 
VariableLevels
Fiber content (wt. %) 30 
Melt temperature (°C) 250 
Rotational speed (rpm) 50, 100 
Residence time (s) 20–300 

2. Steady-state length

The second set of experiments aimed to study the different impact variables have on the residual fiber length, that is, the length at which no additional amount of shearing time will cause additional fiber breakage, or as described by Shon et al., the fiber length at which there is no more buckling. For each variant in this study, the residence time and processing speed were selected to ensure L was reached based on the results of the first experimental plan (300s). This study was designed as a full factorial DOE with three factors and three levels, as shown in Table III. The melt temperature was varied between 220 °C and 280 °C, representing the limits of the processing temperature range suggested by the material supplier. Hence, the obtained measurements represent the most severe impact that can be expected from this factor.

TABLE III.

Experimental plan 2: impact of process variables on equilibrium length.

VariableLevels
Fiber content (wt. %) 20, 30, 40 
Melt temperature (°C) 220, 250, 280 
Rotational speed (rpm) 50, 100, 150 
Residence time (s) 300 
VariableLevels
Fiber content (wt. %) 20, 30, 40 
Melt temperature (°C) 220, 250, 280 
Rotational speed (rpm) 50, 100, 150 
Residence time (s) 300 

3. Attrition rate

This third set of experiments aim to identify the influence processing conditions have on the initial breakage rate. Based on the exponential decay kinetics observed in previous studies, the most significant fiber length change occurs early in the shearing cycle. Hence, fiber length measurements at the start of shearing would provide the best insights into the dynamic behavior.

The goal of modeling fiber attrition in this work is aimed for a homogeneous suspension. Consequently, the fact that the fibers and matrix are not mixed in the initial pellet introduces additional variability into the study. To illustrate this fact, Fig. 3 shows a sequence of x-ray micro-computed tomography (μCT) images of PPGF40 pellets before shearing and after 5s, 12s, and 20s of shearing. It is evident that perhaps a heterogeneous mixture of fiber bundles and polymer matrix is present before 12s rather than a suspension as observed at 20s.

FIG. 3.

μCT slices of fiber dispersion for PPGF40 exposed to a simple shear flow at 50s1 for different residence times.

FIG. 3.

μCT slices of fiber dispersion for PPGF40 exposed to a simple shear flow at 50s1 for different residence times.

Close modal

To address the issue of initial heterogeneity, a pre-dispersion step was introduced to disperse the bundle of fibers without causing excessive damage. This step involved shearing the polymer melt for 2s with a shear rate of 50s−1. After pre-dispersion, the sample was sheared for two additional seconds at higher shear rates as part of the actual experiment. The experiments were performed using a high acceleration rate Couette rheometer at SABIC Technology Center in Geleen, The Netherlands. With this equipment, high deformation rates could be imposed on the polymer melt for short periods. The temperature remained constant, while fiber content and processing speed were varied, as shown in Table IV.

TABLE IV.

Experimental plan 3: impact of shear rate and fiber concentration on attrition rate.

VariableLevels
Fiber content (wt. %) 20, 30, 40 
Melt temperature (°C) 250 
Shear rate (s–1300, 500, 700 
Residence time (s) 
VariableLevels
Fiber content (wt. %) 20, 30, 40 
Melt temperature (°C) 250 
Shear rate (s–1300, 500, 700 
Residence time (s) 

The results are presented in three sections corresponding to the three experimental studies. When evaluating the results, fiber length is presented in terms of the weight-average LW. The number-average fiber length (LN) is also needed to reconstruct the FLD; however, as both variables show the same trends with respect to the processing parameters, the results are presented in terms of LW as it is more representative of the length variable in LFTs.

Figure 4 shows the fiber length reduction as a function of residence time for PPGF30 with a melt temperature of 250 °C at different shearing speeds. For these processing conditions, LW decreased from 15 mm down to 1.6 and 0.75mm with Couette rotational speeds of 50 and 100rpm, respectively. Overall, the results confirm the expected exponential decay with a severe length reduction of the fibers occurring within the first 50s of processing. As expected, an increment in processing speed increases the fibers break rate and decreases the residual/equilibrium length.18,26

FIG. 4.

Fiber length reduction over time for PPGF30 at 250 °C.

FIG. 4.

Fiber length reduction over time for PPGF30 at 250 °C.

Close modal

The investigation of the processing conditions and fiber concentration on the unbreakable fiber length, L, allows direct analysis of the mechanisms that drive fiber damage because it excludes the transient attrition at lower residence times. Figure 5 presents the results of the DOE showing the unbreakable length as the weight-average fiber length.

FIG. 5.

Outcome of the DOE for the steady-state length showing the obtained weight-average fiber length, LW.

FIG. 5.

Outcome of the DOE for the steady-state length showing the obtained weight-average fiber length, LW.

Close modal

An analysis of variance was applied to the measurements to determine differences in means between the factors and the level of the DOE factors. Figure 6 shows the main effect plots with respect to LW. All three factors influence LW and have a statistically significant impact on the process-induced fiber breakage.

FIG. 6.

Result of the statistical analysis showing the main effect plots on LW.

FIG. 6.

Result of the statistical analysis showing the main effect plots on LW.

Close modal

The melt temperature affects the suspension viscosity and, consequently, the stresses that the fibers are exposed to during processing. This dependency is evident in the obtained results, as the increase in temperature results in longer equilibrium length across all other factors. Additionally, processing speed also had the expected effect on the equilibrium length as the rate of deformation increases the hydrodynamic stress experienced by the fibers, forcing the equilibrium length to go down. In all trials, an increased fiber concentration resulted in reduced LW. An increment in fiber content implies an increase in one of the breakage mechanisms, namely, fiber–fiber interactions. This result is essential since it indicates that fiber concentration is an essential variable in the kinetics of fiber attrition, and the controlled conditions of the experimental setup allow for a clean incorporation into the modeling.

Figure 7 shows the decrease in length for a short period for PPGF20, PPGF30, and PPGF40. Here, the pre-dispersion step's impact is evident; the length was reduced from its initial value of 15 to 7mm approximately. However, it is still constant for each fiber concentration which is necessary for the subsequent analysis.

FIG. 7.

Outcome of experimental plan 3 showing the effect of shear rate on fiber attrition rate.

FIG. 7.

Outcome of experimental plan 3 showing the effect of shear rate on fiber attrition rate.

Close modal

As observed in the first study, the attrition rate increases with the rate of deformation, and the fiber length after shearing the sample at 700s−1 is nearly half of the length after shearing at 300s−1. Although the slope depends on the initial length and the initial length changes between fiber concentrations, it can be observed that the lines are nearly parallel, which suggests that the fiber content might not have a strong impact on the fibers' initial break rate.

Plotting the absolute value of the average initial slope as a function of the shear rate shows a linear proportionality between the variables, especially when including the data point at the origin; since if the fibers are subjected to no shear, there will be no damage, and the slope will be zero (Fig. 8).

FIG. 8.

Average initial slope as a function of shear rate. Dashed line shows linear fit with R2 = 0.998.

FIG. 8.

Average initial slope as a function of shear rate. Dashed line shows linear fit with R2 = 0.998.

Close modal

Based on the analysis of the experimental results and reviewed previews work, the main arguments for the development of a new modeling approach for stress-induced fiber damage are presented.

  • Von Turkovich et al. conducted studies on the compounding of SFTs employing a single screw extruder.13 One main conclusion was that fiber content does not have a significant effect on fiber attrition. Partly based on their conclusion, Phelps et al. neglected fiber interactions as a source of damage and defined buckling as the sole mechanism for triggering failure.20,35 However, high stresses and complex flow conditions present during the extrusion process make it hard to establish direct correlations between fiber length and individual factors confidently.

  • While the hydrodynamic stresses will cause buckling, which might lead to failure in short fibers; on average, much larger deformations are needed to cause breakage of the E-glass fibers commonly used in LFTs (D ≈ 20 μm). Salinas and Pittman show that fibers can reach large deformations, far past the point of buckling before finally breaking.17 Additionally, critical radii measurements via loop test for E-glass fibers showed that very small curvatures could be reached before a fracture occurs.28 

  • The experiments under controlled processing conditions employing the Couette rheometer showed a significant correlation between fiber volume fraction and the steady-state length.26 While Moritzer concluded that fiber content impacts both the attrition rate and the steady-state length, the experimental result of this work suggests the fiber volume fraction has a weak impact on the attrition rate.

We postulate the average fiber length reduction over time follows an exponential decay toward an equilibrium value as suggested by Shon et al.18 A state equation can then be written for each length average LN and LW. This will allow for the reconstruction of the fiber length distribution at any point in time,

dLNdt=kN,fLNLN,
(3)
dLWdt=kW,fLWLW.
(4)

We now move on to deriving expressions for both parameters of the state equation L and kf.

Describing the fundamental attrition mechanism as a single fiber undergoes bending deformation caused by the hydrodynamic drag of the moving melt, the fiber could be assumed as a cantilever beam under a distributed load. This fiber held by perhaps a group of neighboring fibers or the walls of the barrel or screw can be at the brink of rupture at the length, L, due to the distributed load, as depicted in Fig. 9.

FIG. 9.

Illustration of a fiber experiencing distributed load due to drag force.

FIG. 9.

Illustration of a fiber experiencing distributed load due to drag force.

Close modal

The hydrodynamic load exerted by the melt flow on the cylindrical body is based on Stokes law by approximating the cantilever length as a chain of spheres with equal diameter.36 The load w per unit length is calculated as

w=3πηmUo,
(5)

where ηm is the matrix viscosity and Uo is the uniform speed of the melt. This leads to the maximum hydrodynamic stress at the base of

σmax=48ηmUoL2df3,
(6)

with L and df as the cantilever length and fiber diameter, respectively. When the stress reaches the fiber's ultimate strength, failure occurs. This leads to the following expression for the critical or unbreakable length L,

L=σutdf348ηmUo1/2.
(7)

The characteristic shear rate can be defined as γ̇=U0/h, where h is introduced to describe a characteristic distance, such as channel depth, runner size, or cavity thickness,

L=σutdf348ηmγ̇h1/2.
(8)

Assuming that h is proportional to df, hdf, so that

Lσutdf248ηmγ̇1/2.
(9)

The proportionality can be resolved by introducing a dimensionless constant, λ, which gives

L=λσudf2ηmγ̇1/2.
(10)

The coefficient λ is a material-dependent property and a measure of fiber interactions that cause fiber attrition during processing. The parameter is assumed to capture the effects of fiber concentration (fiber–fiber interactions) and fiber–wall interactions. Figure 10 shows the interaction coefficient λ as a function of fiber content as obtained from the measured LW values of the Couette rheometer experiments.

FIG. 10.

Average interaction coefficient λ as function of fiber concentration.

FIG. 10.

Average interaction coefficient λ as function of fiber concentration.

Close modal

The parameter kf can also be calculated from the Couette rheometer experiments. As seen in Fig. 11, an analysis of variance was applied to kf to determine its variation with respect to fiber concentration and shear rate. While the fiber concentration does not have a statistically significant impact on the breakage rate coefficient, the shear rate does.

FIG. 11.

Result of the statistical analysis showing main effect plots on kf.

FIG. 11.

Result of the statistical analysis showing main effect plots on kf.

Close modal

In the current modeling approach, kf represents how often fibers will reach the critical conditions described in the formulation of L. Similar to the fiber–fiber interaction in the Folgar–Tucker model,37 it can be argued that this frequency is proportional to the amount of fiber motion, kfγ̇, caused by the flow as described by Forgacs and Salinas.17 The linear correlation between γ̇ and kf shown in Fig. 11 leads to a straightforward expression for the breakage rate coefficient

kf=ξγ̇,
(11)

where ξ is a scale factor for the rate of deformation.

In the work by Wolf and Gupta et al., fiber length was measured along the melting zone of a single screw extruder.38,39 Both researchers observed that a large population of very short fibers originated from the damage occurring at the interface between the bed of solids and the melt pool. In contrast, moderate fiber damage was observed inside the melt pool where fibers were already dispersed and fully surrounded by the matrix. This points to the presence of different damage mechanisms during plastication. When full bundles are present, fiber damage happens mainly at the ends of these bundles, when the drag flow shears off the tips of the fibers [Fig. 12(a)]; but when fibers are fully surrounded by matrix, drag forces cause deformation, which leads to damage [Fig. 12(b)].

FIG. 12.

Illustration of fiber breakage mechanisms under different dispersion stages: (a) undispersed fiber bundles and (b) fully dispersed fibers.

FIG. 12.

Illustration of fiber breakage mechanisms under different dispersion stages: (a) undispersed fiber bundles and (b) fully dispersed fibers.

Close modal

The attrition rate coefficient, kf, is a measurement of how regularly breakage occurs. Therefore, in the presence of two different damage mechanisms, the attrition rate coefficient kf will undergo a transition as the flow condition evolves from a heterogeneous mix of bundles and matrix, to a fully dispersed suspension. To evaluate the effect fiber dispersion can have on the breakage rate, the results obtained from the Couette experiments are used. First, the value for ξW can be recovered from the attrition rate experiment, where the pre-dispersion step ensures damage occurred predominantly due to fiber motion and deformation as illustrated in Fig. 12(b). This results in a value of ξW = 4.66×10−4. Similarly, by fitting Eq. (1) to the experimental data presented in Fig. 4, the value for kf,W and ξW can be calculated. In this case, the shearing process started when the fiber bundles were undispersed, mainly leading to the type of damage illustrated in Fig. 12(a). This results in a value of ξW = 1.21×10−3. This simple comparison suggests that fiber breakage occurs much more often when the fibers are not well dispersed.

The proposed model uses two fitting parameters: ξ, which scales γ̇ in the breakage rate formulation and the interaction coefficient λ, which encapsulates the effects of fiber interactions and is a function of the fiber concentration. An independent set of fitting parameters must be used for the constitutive equations in terms of LN and LW. The definition of both parameters in terms of processing conditions and material properties allows for the constitutive equations to be written in terms of the total derivative and implemented in a cavity filling simulation where the flow field information is used to calculate the model constants to predict the fiber length,

dLNdt+uLN=kf,N(LNLN,),
(12)
dLWdt+uLW=kf,W(LWLW,).
(13)

To test the performance of the proposed model, its prediction can be compared to fiber length measurements from the Couette studies. Since the fiber length measurements from various Couette studies were used to find the fitting parameters λ(ϕ) and ξ, an independent set Couette experiments, similar to those presented in Fig. 4 will be used as reference for comparison. However, since the initial sample is composed of undispersed bundles, the parameters ξN and ξW used in this validation correspond to the values determined from the experiments where the fibers were initially undispersed at the start of the shearing cycle. The processing conditions for the reference Couette experiments are listed in Table V.

TABLE V.

Process conditions of Couette experiment for model validation.

ParameterValue
Melt temperature (°C) 222 
Residence time (s) 30–300 
Fiber concentration (wt. %) 30 
Speed (rpm) 160 
ParameterValue
Melt temperature (°C) 222 
Residence time (s) 30–300 
Fiber concentration (wt. %) 30 
Speed (rpm) 160 

These processing conditions are rather extreme since the temperature is near the lowest value recommended by the material supplier and the rotational speed was set to the higher limit of the Brabender Plasti CorderTM torque rheometer. This places the processing conditions right at the bound of the design space from the fiber breakage DOE. When modeling the flow in the Couette rheometer, the velocity profile is constant, and there are no changes in the z and θ directions; therefore, the convective terms in Eqs. (12) and (13) are not included and the length decay follows Eqs. (3) and (4).

The comparison between experimental and predicted fiber length is presented in Fig. 13. As expected, the equilibrium length is well captured by the model for both length averages LN and LW. It is difficult to establish comparisons in the dynamic portion of the length decay since there are a few experimental points. However, it can be observed the rate of attrition shows a good agreement for LN, but it seems to be overpredicted for LW. Normally, LN measurements have lower standard deviation than LW, and this is reflected in the estimation of the fitting parameter ξW. Nevertheless, the model captures the overall behavior well.

FIG. 13.

Comparison between fiber length decay obtained with the Couette rheometer and fiber predicted with the proposed model.

FIG. 13.

Comparison between fiber length decay obtained with the Couette rheometer and fiber predicted with the proposed model.

Close modal

Using LN and LW as two moments of the fiber length distributioin function, a predicted fiber length distribution is computed using a lognormal probability density function (PDF) given by

f(x)=1xσ2πexplnxμ22σ2.
(14)

The resulting PDF is presented in Fig. 14. The work by Nguyen et al. shows that this PDF can closely reproduce the FLD and yield elastic constants very close to those obtained from experimental measurements.40 However, the authors noted that when using this probability distribution, the quantity of mid-range fiber lengths is often overpredicted. This can also be observed in our results, as the continuous distribution diverts form the experimental data near the 2-mm mark. This discrepancy has little to no effect in the predicted elastic constants; however, it can have more impact in the prediction of the ultimate strength and impact strength.

FIG. 14.

Recovered fiber length distribution employing lognormal PDF vs experimental fiber length distribution.

FIG. 14.

Recovered fiber length distribution employing lognormal PDF vs experimental fiber length distribution.

Close modal

Fiber breakage under simple shear experiments was conducted by employing a Couette rheometer and investigating the influence of fiber concentration, rate of deformation, and viscosity in breakage kinetics. Based on the experimental observations, a continuum approach is developed to predict the fiber length averages LN and LW. A model for the equilibrium length (L) is deduced, using the concept that fibers break by bending under hydrodynamically induced forces and fiber–fiber interactions. The breakage rate coefficient (kf) is associated with the fiber motion and, therefore, modeled proportionally to the rate of deformation. Finally, both length averages are used to recover the fiber length distribution employing a Weibull or lognormal PDF.

Some of the obtained experimental results contradict previous assertions on fiber–fiber interactions' role in fiber breakage phenomena. Bailey and Kraft observed an increase in fiber length with a 20% wt increment in fiber concentration in moldings with PA66 and PP, while von Turkovich et al. concluded the fiber concentration had no impact on fiber damage. Using a Couette device to impose simple shear on the suspension, this work looks to remove the complexities present during molding and extrusion processes, allowing us to isolate individual factors' role on fiber breakage. However, it is important to identify the different breakage mechanisms present as the fibers disperse into a homogeneous suspension.

There are similarities between the model presented in this work and the model by Phelps et al. In their work, they derive an unbreakable length, L (or Lub),

L=π3Efdf44ζηmγ̇1/4.
(15)

This expression represents an equilibrium between internal resistance of the fiber (Efdf) and external stresses (ηmγ̇), as does the expression of Eq. (10). However, the model presented in this work assigns more weight to the product ηmγ̇ on account of the exponent ½ vs ¼ in their model. In their model, the overall rate of reduction of fiber length scales with CBγ̇, which is equivalent to the expression kf=ξγ̇ used in the model presented in this work. Furthermore, the values obtained for ξ after fitting kf to the Couette results (1×10−3–4×10−4) fall within the range suggested for CB in previous work (2×10–2–2×10−4).20,35

Reducing the number of fitting parameters introduced when developing a model is beneficial since this makes the approach more robust and potentially reduces the number of experiments needed to determine such parameters.41 Modeling LN and LW independently for this case, each constitutive equation has its set of two fitting parameters that are determined from the experimental data. Additionally, a single length measurement provides data for both sets of model parameters.

The proposed model in this work can be implemented into a flow solver for either injection molding or extrusion compounding. Implementation in a mold filling simulation using COMSOL Multiphysics, experimental validation and comparison with other modeling approaches will be the subject of a following publication.

The authors wish to thank the National Science Foundation for financially supporting this work (Award No. 1633967). The authors also thank SABIC Global Technologies B.V. for their financial support, technical input, and for providing the materials used in this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request. The data are not publicly available due to company restrictions.

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