Flowinduced anisotropic viscosity in short FRPs
 Róbert Bertóti^{1} and
 Thomas Böhlke^{1}Email author
DOI: 10.1186/s4075901600167
© The Author(s) 2017
Received: 12 October 2016
Accepted: 19 December 2016
Published: 17 January 2017
Abstract
Background
The commonly used flow models for Fiber Reinforced Polymers (FRPs) often neglect the flowinduced anisotropy of the suspension, but with increasing fiber volume fraction, this plays an important role. There exist already some models which count on this effect. They are, however, phenomenological and need a fitted model parameter. In this paper, a micromechanicallybased constitutive law is proposed which considers the flowinduced anisotropic viscosity of the fiber suspension.
Methods
The introduced viscosity tensor can handle arbitrary anisotropy of the fluidfiber suspension which depends on the actual fiber orientation distribution. Assuming incompressible material behaviour, a homogenization method for unidirectional structures in contribution with orientation averaging is used to determine the effective viscosity tensor. The motion of rigid ellipsoidal fibers induced by the flow of the matrix material is described based on Jeffery’s equation. The reorientation of the fibers is modeled in two ways: by describing them with fiber orientation vectors, and by fiber orientation tensors. A numerical implementation of the introduced model is applied to representative flow modes.
Results
The predicted effective stress values depending on the actual fiber orientation distribution through the anisotropic viscosity are analyzed in transient and stationary flow cases. In the case of the assumed incompressibility, they show similar effective viscous material behaviour as the results obtained by the use of the DinhArmstrong constitutive law.
Conclusions
The introduced model is a possible way to describe the flowinduced anisotropic viscosity of a fluidfiber suspension based on the mean field theory.
Keywords
Anisotropic viscosity Homogenization Jeffery’s equationBackground
The orientation distribution of discontinuous fibers in a composite has a considerable impact on the mechanical properties of the reinforced material. To forecast the final fiber orientation distribution of a produced part, the flow process has to be simulated. Polymercomposites containing short fiber reinforcements are often produced through injection molding. Nowadays, with the compression molding technology also discontinuous long fiber reinforced polymer parts can be produced.
The motivation of this work is to set up a micromechanicallybased rheological model which can describe the flowinduced effective anisotropic viscous behaviour of a fiber suspension, depending on the fiber orientation distribution in the fluid.
To describe the orientation state and the dynamics of short rigid fibers in a numerically efficient way, Advani and Tucker (1987) introduced the fiber orientation tensors of the first kind (Kanatani 1984). Based on Jeffery’s Equation (Jeffery 1922; Junk and Illner 2007), the motion of a single rigid fiber in a Newtonian fluid and the change of the orientation tensors can be described. In the literature, for many fiber systems, the fiberfiber interactions are modeled in different ways. The most often used models are the FolgarTucker (1984), the Reduced Strain Closure (Wang et al. 2008) and the Anisotropic Rotary Diffusion (Phelps and Tucker III 2009). All of them are already implemented in commercial software packages.
The rheology of fiber suspensions is also described by various models in the literature. If the molten composite is considered to be linear and isotropic, then the viscosity model used for Newtonian fluids is applicable. It would be then sufficient to determine the effective scalar viscosity of the mixture which can be measured, e.g., with a shear rheometer.
But polymer melts show a strongly nonlinear behaviour, see, e.g., (Ferry 1980). The stress is not directly proportional to the strain rate, even not for pure molten polymers. Considering filled polymer melts, the nonlinear stress–strain rate relation is described, e.g., by (Poslinski et al. 1988). In commercial software, for modeling the flow of fiber reinforced polymer melts, the shear thinning behaviour of the fiberfluid mixture is implemented, but the anisotropy of the suspension is mostly neglected.
There are also constitutive laws which consider the anisotropic flow properties of fiber suspensions. One of them is the Transversely Isotropic Fluid (TIF) model from Ericksen (1960). The PhanThien–Graham model (PhanThien and Graham 1991) is a modification of the TIF model and it has the same structure as the DinhArmstrong model (Dinh and Armstrong 1984). Both of them depend linearly on the fourth order fiber orientation tensor.
The objective of this paper is to propose a flowinduced anisotropic viscosity model which depends on the complete orientation information of the fibers. The introduced method is based on a homogenization scheme (Willis 1977), combined with orientation averaging (Advani and Tucker III 1987). The fully coupled fluidfiber interaction is described. The evolution of the fiber orientation distribution, caused by the flow, is investigated. The number of model parameters is kept minimal in order to reduce (future) measurement costs, fitting efforts and uncertainties. A threedimensional numerical implementation of the presented model can be done similar to (Ospald 2014) or (Schneider et al. 2015). The foregoing aspects contribute to the development of the mold filling simulations of fiber reinforced materials.
Methods
Model equations
Effective stress in the case of incompressibility
In this paper, only the pressureindependent deviatoric part of the Cauchy stress \(\bar {\boldsymbol {\sigma }}'\) is investigated, since \(\bar {p}\) is a reaction force and follows from the boundary conditions.
Effective viscosity tensor
Homogenization for the effective viscosity tensor
The short fibers are considered to be rigid, which is ensured in the model by infinite fiber viscosities.
where c _{f} is the fiber and c _{m} is the matrix volume fraction. \(\mathbb {P}_{\alpha }\) is the polarization tensor described, e.g., by Walpole (1969) and Ponte Castaneda (1998) depending on the matrix material properties and on the geometry and the orientation of the fiber α. The properties of the matrix viscosity \(\mathbb {V}^{\mathrm {m}}\) are discussed later.
The advantage of the HashinShtrikman homogenization method is that it is derived based on the variational principle (Willis 1977), and the lower bound gives a finite effective viscosity for the unidirectional pseudodomain, also in the case of infinitely viscous, i.e. rigid, fibers. The disadvantage of this method is that we get only a lower bound. The orientation averaging (4) gives a Voigttype average which is an upper bound for the dissipation.
Matrix behaviour
where η _{v} is the volume and η _{s} is the shear viscosity of the matrix material. The tensor \(\mathbb {P}_{1}=\boldsymbol {I} \otimes \boldsymbol {I} /3\) and \(\mathbb {P}_{2}=\mathbb {I}^{\mathrm {s}}  \mathbb {P}_{1}\) are the first and the second projector tensors of fourth order with major and minor symmetry, I is the second order unit tensor, and \(\mathbb {I}^{\mathrm {s}}\) is the symmetric part of the fourth order unit tensor. Because of the assumed incompressibility, only the \(\mathbb {P}_{2}\) term of the matrix viscosity tensor is relevant, the \(\mathbb {P}_{1}\) term cancels out through \(\bar {\boldsymbol D}'\). This induces that the volume viscosity η _{v} in \(\mathbb {V}^{\mathrm {m}}\) has no influence in the case of incompressibility. From now on, the notation \({\mathbb {V}^{\mathrm {m}}}'=\eta _{\mathrm {s}}\mathbb {P}_{2}\) is used. The polarization tensor \(\mathbb {P}_{\alpha }\) depends in general also on η _{v}. For the case of incompressibility, the effect of η _{v} can not be filtrated from \(\mathbb {P}_{\alpha }\) as simply as from \(\mathbb {V}^{\mathrm {m}}\). To ensure the incompressibility, the limit of \(\mathbb {P}_{\alpha }\) is calculated for the case that η _{v} goes to infinity. This limit results a singular polarization tensor \(\mathbb {P}_{\alpha }'\) which is inverted on the nonsingular subspace on traceless tensors for (5). The consequent notation of \(\bar {\mathbb {V}}\) for the incompressible case would be \(\bar {\mathbb {V}}'\), but the prime is dropped for simpler notation.
Fiber orientation and reorientation
where α=1,2,…,M is the index of the fibers. \(\bar {\boldsymbol {W}}\) is the effective vorticity tensor, and \(\bar {\boldsymbol {D}}\) is the effective strain rate tensor, being the antisymmetric and the symmetric parts of the effective velocity gradient \(\bar {\boldsymbol {L}}=\partial \bar {\boldsymbol {v}}/\partial \boldsymbol {x}\), respectively. \(\bar {\xi }\) is a geometry parameter of a representative fiber, determined through the aspectratio \(\bar {a}\), the length \(\bar {l}\), and the diameter \(\bar {d}\). With the use of the dilute distribution approximation, which assumes that the fibers do not have any interaction with each other, Jeffery’s equation separately describes the motion of all M pieces of fibers.
In the evolution equation of the second order fiber orientation tensor \(\bar {\boldsymbol {N}}\), the fourth order fiber orientation tensor \(\bar {\mathbb {N}}\) appears. Analogously, in the evolution equation of \(\bar {\mathbb {N}}\), the next higher even order fiber orientation tensor appears. Instead of the evolution equation of \(\bar {\mathbb {N}}\), the invariantbased optimal fitting (IBOF) closure (Chung et al. 2002) is used to approximate the values of \(\bar {\mathbb {N}}\) in each time step, based on the invariants of \(\bar {\boldsymbol {N}}\).
With the use of the fiber orientation tensors, the orientation evolution (7) can be calculated more efficiently (9). Similarly, the orientation averaging (4) can also be done numerically in a more efficient way with the help of the fiber orientation tensors, as it is described in (Advani and Tucker III 1987).
The evolution of \(\bar {\boldsymbol {N}}\) (9) can be calculated also with the Fast Exact Closure (FEC) (MontgomerySmith et al. 2011). With the FEC, the exact solution of Jeffery’s equation is obtained, if there are no fiberfiber interactions and the initial fiber orientation distribution is isotropic. Although all test cases considered in this paper fulfil the mentioned two criteria, we used the numerical IBOF closure to solve Jeffery’s equation.
Due to the assumed incompressibility, only isochoric deformations are allowed for the introduced model. That means that \(\bar {\boldsymbol {D}}\) is identical with its deviatoric part \(\bar {\boldsymbol {D}}'\). Jeffery’s Eq. (7) and its extension to second order tensors (8) are valid for compressional cases, as well. Therefore, (7) and (8) are denoted here in the general form with \(\bar {\boldsymbol {D}}\).
Deformation, anisotropy, dimensionless and reduced measures
are the isotropic parts of the corresponding tensors. It is assumed that \(\mathbb {T}\) has the major and the minor symmetries. The scalar products T·I and \(\mathbb {T} \cdot \mathbb {P}_{2}\) are calculated in Caresian coordinate systems by T _{ ij } I _{ ij } and T _{ ijkl } P _{2ijkl }, respectively, with the use of Einstein’s summation convention. Since the threedimensional space is considered here, the summation indices are running from 1 to 3. . denotes the Frobenius norm.
Phenomenological model for comparison
where N _{p} is the coupling parameter and \(\bar {\eta }_{\mathrm {s}}\) is the effective scalar shear viscosity of the mixture, determined for the comparison through (14). Please note, the difference lies in the fact that (16) is linear in \(\bar {\mathbb {N}}\), but (15) is a function of \(\bar {\boldsymbol {N}}\) and \(\bar {\mathbb {N}}\) through (4) as it is described in (Advani and Tucker III 1987).
for every time step. \(\bar {\boldsymbol {\sigma }}^{*}_{\text {ph}}\) calculated with the optimal coupling parameter \(N^{\text {opt}}_{\mathrm {p}}\) is compared to \(\bar {\boldsymbol {\sigma }}^{*}_{\text {mf}}\) in the Results section.
Numerical implementation
Only one point of the macrocontinuum is investigated with stationary velocity gradient. M pieces of randomly generated, equally distributed discrete orientation vectors (Mardia and Jupp 2000), immersed in the liquid matrix, are investigated.
The model calculated based on (18), (7) and (8) is called a discrete model. To calculate (19) with the use of (9) (and with isotropic initial fiber orientation tensor \(\bar {\boldsymbol {N}}(0)\)), a closure approximation is needed to approximate \(\bar {\mathbb {N}}\) based on \(\bar {\boldsymbol {N}}\) in each time step, since otherwise the expression (9) is undetermined. The models based on (19) calculated with quadratic closure and with IBOF closure are called here quadratic model and IBOF model, respectively.
is exact, because the velocity gradient is considered to be constant. The time step size Δ t was decreased until the computations with the time increments 2Δ t, Δ t and Δ t/2 gave the same results.
Material parameters for the simulations
Symbol  Value  Unit 

M  1000   
\(\bar {a}\)  ∞   
η _{v}  ∞  Pa s 
η _{s}  1  Pa s 
c _{f}  0.3   
Investigated, representative flow modes
where b is an arbitrary scalar factor which determines also the magnitude of the strain rate. b has the dimension s^{−1}. The simulations are carried out with b=1 s^{−1}.
Results
Discrete pole figures
Component plots
In the following, the component plots of the second (\(\bar {\boldsymbol {N}}\)) and fourth (\(\bar {\mathbb {N}}\)) order fiber orientation tensor, of the dimensionless effective viscosity tensor (\(\bar {\mathbb {V}}^{*}\)) and of the reduced effective stress tensor (\(\bar {\boldsymbol {\sigma }}^{*}\)) are shown and described, for the four investigated flow cases.
Shear flow
The component plot curves of \(\bar {\mathbb {N}}\) graphically do not represent more information for the reader than the component plot curves of \(\bar {\boldsymbol {N}}\), although \(\bar {\mathbb {N}}\) contains more information with respect to the anisotropy of the orientation distribution than \(\bar {\boldsymbol {N}}\). For this reason, for the other three investigated flow cases, only the component plots of \(\bar {\boldsymbol {N}}\), \(\bar {\mathbb {V}}^{*}\) and \(\bar {\boldsymbol {\sigma }}^{*}\) are given in the following.
Comparison of the flow cases
The reorientation of the fibers in the investigated elongational flow is faster than in the shear flow. The stationary state is reached in this case already at about \(\bar {\varepsilon }_{\mathrm {M}}=6\), as Fig. 3 c and d show. The transient change is transversely isotropic, and the stationary case is unidirectional considering the fiber orientation.
In the case of the compressional flow, the reorientation of the fibers is even faster than in the flow cases mentioned before. The stationary state is already reached at about \(\bar {\varepsilon }_{\mathrm {M}}=4\), as it is shown in Fig. 3 e and f. The transient change is, here, also transversely isotropic, as in the elongational flow, but the end state is, here, a planar isotropic one.
The investigated planar flow is a combination of an elongational and a compressional flow. As Fig. 3 g and h show, the reorientation of the fibers is slower than in the two investigated cases before, the stationary state is reached only at about \(\bar {\varepsilon }_{\mathrm {M}}=8\). The transient change is, here, not transversely isotropic similarly as in the shear flow. In the stationary case, the fiber orientation is unidirectional.
Comparison of the two models
Discussion
Relative calculation times and closure approximation
Sensitivity of the viscosity
Note that \(\bar {\mathbb {V}}^{*}\) is singular, because of the assumed incompressibility and it is inverted on its nonsingular subspace, analogously as \(\mathbb {P}_{\alpha }\) is inverted for (5). The dimensionless effective scalar viscosity \(\bar {\eta }_{\mathrm {s}}^{*}(\boldsymbol {p},\boldsymbol {d})\) can be imagined as a fourdimensional surface. The three independent coordinates of \(\bar {\eta }_{\textrm {s}}^{*}(\boldsymbol {p},\boldsymbol {d})\) are the two angles determining p and one angle determining d in the plane. The fourth, dependent variable is \(\bar {\eta }_{\mathrm {s}}^{*}\). Instead of the fourdimensional surface, a twodimensional representative slice of it is depicted in Fig. 6 d which represents \(\bar {\eta }_{\mathrm {s}}^{*}(\boldsymbol {p},\boldsymbol {d})\) for the case that both p and d lie in the xy plane. In Fig. 6 d, p is the radial direction, and d is tangential to it (p·d=0). The three curves represent the matrix (black circle), the effective isotropic (c _{f}=30%) and the effective unidirectional (in xdirection) scalar viscosity belonging to the proper ps and ds. The introduced model predicts that due to adding 30V/V% fibers to the matrix material, the effective isotropic viscosity increases to 1.7fold of the matrix viscosity (blue circle). Considering the unidirectional fiber orientation state in xdirection, two maximum and two minimum “directions” are observed (orange curve). The maximum is about 1.9fold matrix viscosity and the minimum is about 1.1fold matrix viscosity. The model predicts the two maximum “directions” in fiber direction and perpendicular to it. The minimum “directions” have the angle ±45° to the fiber direction.
Beside the actual fiber orientation distribution and the velocity field of the flow, the local fiber volume content, the spatial and temporal change of the temperature, together with the curing grade, have also remarkable effects on the effective viscosity. These effects can be taken into account with the extension of the introduced model by making V ^{m} and \(\bar {\boldsymbol {V}}^{\text {ud}}\) dependent on the listed effects.
Conclusions
A mean field theory based effective linear viscosity model is introduced for short fiber reinforced polymers based on a twostep homogenization method. The presented model can handle arbitrary anisotropy of the effective viscosity which is caused by the evolving orientation distribution of the fibers. The numerical implementation of the introduced model with fiber orientation tensors calculates about eight times faster than the same model calculating with 1000 fiber orientation vectors. The use of the fiber orientation tensors is highly recommended for an efficient calculation even with the arising closure approximation. The effective direction dependent scalar shear viscosity is 1.1 to 1.8 times higher than the matrix shear viscosity at 30% fiber volume fraction. For each matrixfiber suspension, the effect of the direction dependency has to be compared to the effect of the temperature change and to the effect of the deformation rate dependency of the matrix shear viscosity to decide whether the direction dependency has a remarkable effect on the flow of the suspension, or not. With the assumptions that the matrix material is incompressible and the fibers are rigid, the presented model gives qualitatively the same results as the phenomenological model used for comparison. A possible way to consider the anisotropic viscosity of fiber reinforced polymers is given in this paper.
Abbreviations
 FODF:

Fiber orientation distribution function
 FRP:

Fiber reinforced polymer
 IBOF:

Invariant based optimal fitting
 TIF:

Transversely isotropic fluid
Declarations
Acknowledgements
This work was partially supported by the German Research Foundation (DFG) within the International Research Training Group “Integrated engineering of continuousdiscontinuous long fiber reinforced polymer structures” (GRK 2078).
Funding
German Research Foundation, Grant Number: DFG GRK 2078/1.
Authors’ contributions
RB carried out this study and drafted the manuscript. TB supervised the research, planned the study, contributed to the interpretation of results and writing of the manuscript. Both the authors read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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