Fourth-order Orientation Tensor Research Articles

On the influence of the fourth order orientation tensor on the dynamics of the second order one

Abstract The consequences of introducing the fourth order orientation tensor as an independent variable in addition to the second order one are investigated. In the first part consequences of the Second Law of Thermodynamics are exploited. The cases with the second order alignment tensor in the state space on one hand and with the second and fourth order alignment tensors on the other hand are analogous. In the latter case differential equations for the second and fourth order tensors result from linear force-flux relations with a coupling arising due to coupling terms in the free energy. In the second part the differential equations for the second order orientation tensor or the second and fourth order orientation tensors, respectively are given explicitly in the special case of a rotation symmetric orientation distribution. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. The fourth order parameter is eliminated later. The resulting equation for the second order parameter is a Duffing equation with a behavior of solutions completely different from the solutions of the Riccati equation.

Maximum Entropy Closure Relation for Higher Order Alignment and Orientation Tensors Compared to Quadratic and Hybrid Closure

A closure relation expresses the fourth order orientation tensor as a function of the second order one. Two well-known closure relations, the hybrid closure and the maximum entropy closure, are compared in the case of a rotation symmetric orientation distribution function. The maximum entropy closure predicts a positive fourth order parameter in the whole range of the second order parameter, whereas the hybrid closure results in negative fourth order parameters for small values of the second order one. For the maximum entropy closure quadratic fit polynomials are presented. For a general distribution without rotation symmetry, the expression for the entropy is exploited to derive an explicit form for the maximum entropy distribution. Lowest order approximation of this distribution function leads to simple closure forms for the fourth order alignment tensor and also for higher order alignment tensors.

Planar fiber orientation: Jeffery, non-orthotropic closures, and reconstructing distribution functions

Many models for fiber suspensions use the second-order orientation tensor to represent the orientation state. This requires that the fourth-order orientation tensor be approximated using a closure formula. The prediction of non-linear properties also requires reconstructing an approximate orientation distribution function. All existing fourth-order closures and most reconstructions have orthotropic symmetry, which limits their accuracy. In this paper, closures and reconstruction are explored for planar fiber orientation, using a recent parameterization of fourth-order orientation tensors by Bauer and Böhlke (2021, 2022a, 2022b). We show that the orientation distribution for fibers that follow Jeffery’s equation, together with the associated natural closure, form the central axis of the space of fourth-order orientation tensors. A new family of non-orthotropic planar closure approximations is created by including the rate-of-deformation tensor as an additional independent variable. These closures are shown to be substantially more accurate than any planar orthotropic closure. We also explore the Jeffery, Bingham, ellipse radius, and fourth-order maximum entropy (ME4) functions for reconstructing the orientation distribution. When compared to distributions from the Folgar-Tucker model, ME4 using the exact fourth-order orientation tensor gives the most accurate reconstruction, while ME4 with the fourth-order tensor from the new non-orthotropic closure is nearly as good. • The Jeffery distribution is central to planar fourth-order orientation tensors. • The first-ever non-orthotropic closure approximations are developed. • Non-orthotropic closures are more accurate than any previous planar closure. • Fourth-order maximum entropy distributions give the most accurate reconstruction.

Novel closure approximation for prediction of the effective elastic properties of composites with discontinuous reinforcement

The paper introduces a novel closure approximation method for the determination of the fourth-order orientation tensor in terms of known second-order tensor. The fourth-order orientation tensor must be known to predict the effective elastic properties of composites with misaligned fibres but very often only the second-order orientation is available. The proposed method operates on a set of outcomes of optimization based on a genetic algorithm. In consequence, it leads to obtaining distributions of effective elastic constants instead of deterministic quantities. Results obtained by using the novel approach have been compared with results based on exact fourth-order orientation tensor, tensor known from experimental data and determined by using popular closure approximations from literature. Furthermore, the obtained results have been compared with results of finite element analysis based on geometrical models representing the microstructure which contains several hundred fibres. The new method provides high accuracy of elastic constant predictions for all studied fibre orientation distributions, which distinguishes it from the closure approximation methods known from the literature. Moreover, results obtained in form of distributions give an insight into the uncertainty of the predictions.

Exploring the applicability of a simplified fully coupled flow/orientation algorithm developed for polymer composites extrusion deposition additive manufacturing

Abstract Prior literature employed a finite element based fully coupled algorithm that solved the flow/orientation coupling effects in extrusion deposition additive manufacturing, i.e., a key component for identifying the material properties of solidified deposited polymer composites. However, the complex iterative solution protocol reduces the computational capability of the algorithm. This paper simplifies the finite element based fully coupled algorithm by employing a scalar representation of the fourth order orientation tensor affected viscosity tensor. The revised code predicts similar fiber orientation results as compared to a related prior study. With the simplified fully coupled algorithm, effects of material deposition rate and fiber aspect ratio on a 2D planar large area extrusion deposition flow are explored, where the power law rheology model is employed for the flow simulation. Computed results show that an increased material deposition rate reduces the extrudate-swell ratio of the melt front and varies the local fiber orientation of the deposited bead especially near the flow region in contact with the substrate. Higher fiber aspect ratio enhances the principal fiber alignment of the extrusion and deposition flows, where the maximum local difference of the principal A 11 component is 23% with increasing fiber aspect ratio from 10 to 25.

Variety of fiber orientation tensors

Fiber orientation tensors are established descriptors of fiber orientation states in (thermo-)mechanical material models for fiber-reinforced composites. In this paper, the variety of fourth-order orientation tensors is analyzed and specified by parameterizations and admissible parameter ranges. The combination of parameterizations and admissible parameter ranges allows for studies on the mechanical response of different fiber architectures. Linear invariant decomposition with focus on index symmetry leads to a novel compact hierarchical parameterization, which highlights the central role of the isotropic state. Deviation from the isotropic state is given by a triclinic harmonic tensor with simplified structure in the orientation coordinate system, which is spanned by the second-order orientation tensor. Material symmetries reduce the number of independent parameters. The requirement of positive-semi-definiteness defines admissible ranges of independent parameters. Admissible parameter ranges for transversely isotropic and planar cases are given in a compact closed form and the orthotropic variety is visualized and discussed in detail. Sets of discrete unit vectors, leading to selected orientation states, are given.

The rotation of rigid spheroids in a viscous fluid under mean-field effects

An analytical model capturing the effects of mean-field considerations on the orientation evolution of a plurality of rigid spheroids suspended in a Newtonian matrix is developed. Three approaches are taken with increasing sophistication: a mesoscale view involving the analysis of several distinct layers of material subjected to simple shear flow, a traditional micromechanics inspired approach involving the Reuss assumption that each heterogeneity sees an equivalent stress, and the application of the Mori-Tanaka assumption in the viscous inclusion framework of Wetzel and Tucker (JFM 426, 2001). The resulting model behaviors are explored in simple shear flow and extensional flow. The behavior of the Mori-Tanaka model is qualitatively equivalent to that seen in a recent investigation by Gilormini and Chinesta (JNNFM 270, 2019) which considered the effects of the suspending medium being anisotropic in nature. With increase in concentration, the rate of orientation evolution towards collimation is increased as compared to the dilute case, but the relevant fourth-order orientation tensor component in simple shearing flow is shown to see an increased peak value.

Development of the layered structure in a double-gated glass fiber-reinforced polypropylene injection molding: Experimental and simulated results

The present study aims to experimentally validate numerical simulation of fiber orientation distribution performed by molding simulation software Moldex3D in a double-gated injection-molded glass fiber-filled (40 wt%) polypropylene box, by making a detailed comparison of predicted and experimentally measured fiber orientation distribution data. The modeling approach evaluated in this work consists in the implementation of the Folgar–Tucker rotary diffusion model with the invariant-based optimal fitting closure approximation for the fourth-order orientation tensor. The specimen used has a weld line in the center and sharp corners. This investigation characterizes in detail the development of the through-thickness layered structure at distinctive locations of the specimen. The sensitivity of fiber orientation distribution and the layered structure to changes upon injection time and melt temperature is also evaluated. The boxes display the typical layered laminate structure, with fibers aligned in the main flow direction near the walls (shell layer) and less oriented in the middle plane (core layer). The boxes injected at the lowest melt temperature display an additional skin layer. Unfortunately, simulation fails in predicting the five layers structure developed under these latter conditions. The grade of fiber orientation is deemed to be independent of process parameters but not the layered structure.

Planar and Orthotropic Closures for Orientation Tensors in Fiber Suspension Flow Models

This paper presents a bottom-up approach to derivation of orientation tensor closures for fiber suspension flow models. To begin with, we consider polynomial approximations based on the two-dimensional (2D) versions of the linear, quadratic, natural, and orthotropic smooth closures for reconstruction of the fourth-order orientation tensor. A numerical study is performed for simple flows. The investigation of planar closures provides new insights and boundary conditions for the design of orthotropic closures in three dimensions. The proposed extensions use finite element shape functions to interpolate the data at principal orientation states and additional points. The results for 3D simple flows indicate that natural closures based on (extended) quadratic and piecewise-linear interpolation provide a far better description of the 3D orientation dynamics than any other orthotropic closure considered in this study.

Analysis of the Folgar & Tucker model for concentrated fibre suspensions in unconfined and confined shear flows via direct numerical simulation

The classical Jeffery model allows for the prediction of the flow-induced orientation in dilute fibre suspensions. In most industrial applications, however, fibre suspensions are concentrated and fibre-fibre interactions cannot be ignored any longer. These interactions have been traditionally modelled at the mesoscopic and macroscopic scales by introducing a phenomenological diffusion term inducing a randomizing effect. In the so-called Folgar & Tucker (F&T) model, widely used in applications, the diffusion coefficient is assumed to scale linearly with the flow intensity, the latter being described by the second invariant of the rate of strain tensor. Modifications and alternatives to the F&T model have been proposed in view of the difficulty for the F&T model to explain an apparent orientation delay observed experimentally in injection-moulded parts. The noticed deviations were attributed to the intense fibre-fibre interactions, thus pointing to the limitations of a phenomenological diffusion term for describing them. In the present work, we revisit the F&T model and compare its predictions with those obtained bysimplified yet state-of-the-art direct numerical simulation (DNS) in unconfined and confined simple shear flows for a range of shear rates and concentrations, the latter ensuring intense fibre-fibre interactions. In unconfined flows, we find that the F&T model agrees quantitatively with the DNS results once an adequate closure relation is considered for approximating the fourth-order orientation tensor involved in the F&T model. Thus, the results seem to confirm, at least in simple shear flows, the F&T assumption for the form of the isotropic rotary diffusion function scaling linearly with the magnitude of the scalar rate of deformation. Also, a linear scalingof the diffusivity with the fibre concentration is observed. This conclusion remains unexpectedly valid under moderately-confined flow conditions as soon as an advanced fitted closure, like the IBOF, is considered within the F&T model. Other simpler closures (e.g. quadratic or hybrid), however, definitively fail to address confinement issues as also reported in our former work for the dilute regime. Obviously, these conclusions rest on the validity of the considered state-of-the-art DNS, which remains at present an open question.

Prediction of effective elastic properties of fiber reinforced composites using fiber orientation tensors

The main objective of this work is to give an answer to the question: is it sufficient to consider only the second-order fiber orientation tensor as microstructure variable describing the orientation distribution of short-fiber reinforced composites (SFRCs) in the prediction of effective elastic properties? This question is addressed in the context of SFRCs on the one hand with an overall transversal symmetric orientation distribution of fibers and, hence, effective transversally isotropic properties, and on the other hand with experimentally determined microstructure data using micro-computed tomography. Applying the maximum entropy principle, it is shown, how the fiber orientation distribution function (FODF) can be estimated by relying on the second and/or the fourth-order orientation tensor, only. Both estimates are used within the self-consistent and the interaction direct derivate approach to calculate the effective linear elastic properties. It is shown, that the predicted stiffness tensors significantly depend on the estimation of the FODF. Relative deviations of up to 20% in terms of stiffnesses and up to 46% in terms of Young's modulus are observed. For the experimentally determined microstructure, small deviations of up to 4.3% are found.

Flow property of fiber suspension in a turbulent channel flow under considering both Brownian and turbulent diffusions

The issue of fiber suspension flow has received great substantial attention in the last decades. In contrast with the abundant researches of normal size fiber suspensions flow, this paper is devoted to the small size fiber suspension composed of water and polyarmide fiber where Brownian motion plays an important role and thus cannot be neglected. The spatial distribution and orientation of fiber, streamwise mean velocity profile, turbulent kinetic energy, Reynolds stress and rheological property of fiber suspension in a turbulent channel flow are obtained and analyzed both numerically and experimentally. To simulate the small fiber suspension flow well, the well-known Reynolds averaged Navier-Stokes (N-S) equation governing the suspension flow is modified in consideration of the effect of fibers on base flow. The equation describing the probability density functions for fiber orientation is derived in view of the rotary Brownian diffusion. The general dynamic equation for fibers is reshaped in the effect of spatial Brownian diffusion. For the sake of the closure of the N-S equation, the turbulence kinetic energy and turbulence dissipation rate equations with fiber term are employed. The conditional finite difference method is adopted to discrete these partial differential equations. And the diffusion term and convective term are discretized by the central finite differences and the second-order upwind finite difference schemes, respectively. The second and fourth-order orientation tensors are integrated by the Simpson formula. Experiment is also performed to validate some numerical results. The results show that most fibers tend to align parallelly to the flow direction in the flow, especially in regions near the wall. Such a phenomenon is more obvious with the decreases of Reynolds number and fiber concentration, and with the increase of fiber aspect ratio. Fiber volume fraction distribution is non-uniform across the channel, and becomes more uniform with increasing Reynolds number, and with reducing fiber aspect ratio. The changes of fiber orientation distribution and spatial distribution are not sensitive to fiber aspect ratio for 5. Streamwise mean velocity profile in fiber suspension has a steeper slope than that in single phase flow, and the steepness increases as the fiber concentration and fiber aspect ratio increase, and as the Reynolds number decreases. The presence of fiber will reduce the turbulence kinetic energy and Reynolds stress. The effect of fiber on the turbulence suppression becomes more obvious with the increases of fiber concentration and aspect ratio, and with the decrease of Reynolds number. The first normal stress difference is less than 0.05 and much less than the shear stress. From the wall to the center of the channel, the shear stress increases while the first normal stress difference decreases. Both the shear stress and the first normal stress difference increase with increasing fiber concentration and aspect ratio. Shear stress increases while the first normal stress difference decreases with increasing Reynolds number. The effects of fiber concentration on the shear stress and the first normal stress difference are larger than the fiber aspect ratio.

Numerical simulation of the rheological properties of fiber suspensions in turbulent pipe flows

Purpose – The purpose of this paper is to study numerically the rheological properties of fiber suspensions flowing through turbulent pipe flows. Design/methodology/approach – The work presented in this paper is derived the fluctuating equation for fiber orientation distribution function (FODF) in turbulent flows and solved using the method of characteristics. The FODF is predicted numerically. The numerical results of root-mean-square velocities generated by kinetic simulation sweeping model and are compared with the experimental data. Findings – The fiber orientation distribution becomes wider with increasing Re. The components of the fourth-order orientation tensor increase with the increase of Re, and also increase along the radial direction and reach the maximum at the center line. The first normal stress difference is much less than the shear stress. For different Re the shear stress increases rapidly in the region far from the pipe center, and reaches its maximums at center, while the first normal stress difference decreases rapidly in the region far from the pipe center, and reaches its minimum at center finally. Originality/value – By solving numerically the equation in a turbulent pipe flow with Reynolds number ranging from 2,500 to 1,000, the authors obtain the mean FODF which is in agreement with the experimental one qualitatively. Then the shear stress and first normal stress difference of suspensions are calculated based on the mean FODF.

On the use of interaction tensors to describe and predict rod interactions in rod suspensions

Recently, Ferec et al. (2009a) proposed a model for nondilute rod-like suspensions, where particle interactions are taking into account via a micromechanical approach. The derived governing equation used the well-known second- and fourth-order orientation tensors (a2 and a4) and novel second- and fourth-order interaction tensors (b2 and b4). To completely close the model, it is necessary to express a4, b2, and b4 in terms of a2. This paper gives the general framework to elaborate these new relations. Firstly, approximations for b2 are developed based on linear combinations of a2 and a4. Moreover, a new closure approximation is also derived for b4, based on the orthotropic fitted closure approach. Unknown parameters are determined by a least-square fitting technique with assumed exact solutions constructed from the probability distribution function (PDF). As numerical solutions for the PDF are difficult to obtain given the nonlinearity of the problem, a combination of steady state solutions is used to generate PDF designed to cover uniformly the entire domain of possible orientations. All these proposed approximations are tested against the particle-based simulations in a variety of flow fields. Improvements of the different approximations are observed, and the couple iORW-CO4P3 gives efficient results.

Validation of the modified rule of mixtures using a combination of fibre orientation and fibre length measurements

The goal of this study was to investigate the fibre orientation distribution (FOD), and subsequent mechanical properties, of an injection moulded plate with two different number averaged fibre lengths, termed in this paper medium (1.35mm) and long (2.40mm). Fibre orientation measurements (FOD) were made using the 2D elliptical section method and an in-house developed image analyser. The samples were injected from a pin gate located at the centre and top of the plate. Expansion flow on the divergent flow front from this pin gate resulted in a core region with circumferential alignment, while through thickness shear resulted in the usual realignment of fibres in the flow direction either side of the core, termed the shell layers. Two interesting aspects were discovered from these measurements. First, and most importantly, the FOD was found to be independent of the two fibre lengths in this study, and so predominantly controlled by the mould shape and the interaction with the flow front. Second, the fibres in the core region were found to be much closer packed than those in the shell regions.The interaction between the flow front and the mould shape resulted in a range of FOD across the moulded plate, from equal in-plane orientation at the centre of the plate, to highly aligned at the plate edge. This gave a very useful set of samples from which to test out the well known modified rule of mixtures (MROM). Often the fibre orientation distribution cannot be measured directly, but indirectly using the modified rule of mixtures model in reverse. The samples from this moulding (at two different average fibre lengths) gave an excellent opportunity to validate this often used approach.Both the tensile modulus and strength (measured parallel to the injection direction) were found to show a strong correlation with the measured fibre orientation, with a significant increase in both measures between the centre and the edge of both plates. The increased length of the ‘long’ fibre plate was found to give only a small increase in tensile modulus but a much larger increase in tensile strength.The tensile modulus showed a linear dependence with the measured fourth order orientation tensor average, 〈cos4θ〉, with respect to the injection direction of the plate, as predicted by the modified rule of mixtures. Excellent agreement was found between the measured modulus and the predictions from the modified rule of mixtures, based only on measured quantities (matrix modulus, fibre fraction and average fibre length) for both plates.

Numerical modelling of non-isothermal flow of fibre suspensions: prediction of fibre orientation in three-dimensional cavities

The most important phenomenon observed in fibre suspensions is flow induced fibre orientation. Fibre size is the key parameter in prediction of fibre orientation which in turn describes the microstructures of injection moulded parts. This paper focuses on developing a three-dimensional numerical model for the analysis of fibre orientation prediction incorporating the effect of fibre size distribution using the finite volume method FVM. The flow was considered to be incompressible, and behaves as non-Newtonian fluid flowing under non-isothermal condition. The hybrid closure model of Advani and Tucker was used to approximate the evolved fourth order orientation tensor. To validate the developed simulation model, several cases were modelled and compared with available experimental data for rectangular and cylindrical geometries. The simulation results showed that they are in good agreement with the experimental data. Hence, the numerical model could assist in decisions regarding the design of composite products.

Flow Properties of Turbulent Fiber Suspensions in a Stock Pump Impeller

A numerical method for predicting fiber orientation is presented to explore the flow properties of turbulent 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.

A study of fiber orientation in short fiber-reinforced composites with simultaneous mold filling and phase change effects

Most of the properties of injection molded short fiber-reinforced composites are highly dependent on the patterns of their fiber orientation, which are induced by the flow. On the other hand, in most practical injection molding processes, both filling and solidification of the molten suspension takes place simultaneously. This behavior indicates that both filling and phase change for solidification can occur at the same time and therefore affect the flow behavior of the suspension, hence the fiber orientation. The aim of the current work is to present a numerical analysis of fiber orientation prediction in a three-dimensional rectangular cavity considering simultaneous mold filling and phase change of the suspending polymer. To trace the flow front during the filling process, the volume of fluid method (VOF) has been used, while an enthalpy-based approach was used to model the solidification. The standard Hybrid closure model of Advani and Tucker was applied to approximate the evolved fourth order orientation tensor during the fiber orientation calculation. To validate the developed numerical model, the results of the simulation model were compared with available experimental data for the rectangular cavity. The simulation results showed that they are in good agreement with the experimental data. Hence, the numerical model could assist in decisions regarding the design of polymer composite products.

A new expression for the velocity profile in the turbulent pipe flows of fiber suspension

The equation of Reynolds averaged Navier-Stokes with the term of additional stress resulted from fibers and the equation of probability distribution function for mean fiber orientation with the second- and fourth-order orientation tensor of the fiber were derived and solved to explore the characteristics of turbulent fiber suspension in the turbulent pipe flows of fiber suspension. The effect of Reynolds number, fiber concentration and fiber aspect-ratio on the mean velocity were presented and analyzed. It was found that the velocity profile of fiber suspension becomes sharper with decreasing the flow rate and/or increasing the fiber concentration, which agrees with the corresponding experimental ones. The velocity profile becomes steeper as fiber aspect-ratio increases. Fibers at this range of concentration and scales play a role in the increase of viscosity. Finally, the prediction formula of velocity profile involving the effect of Reynolds number, fiber concentration, and aspect-ratio for the turbulent pipe flows of fiber suspension was derived.

Fiber suspension flow in a tapered channel: The effect of flow/fiber coupling

A numerical model for predicting the flow and orientation state of semi-dilute, rigid fiber suspensions in a tapered channel is presented. The effect of the two-way flow/fiber coupling is investigated for low Reynolds number flow using the constitutive model of Shaqfeh and Fredrickson. An orientation distribution function is used to describe the local orientation state of the suspension and evolves according to a Fokker–Plank type equation. The planar orientation distribution function is determined along streamlines of the flow and is coupled with the fluid momentum equations through a fourth-order orientation tensor. The coupling term accounts for the two-way interaction and momentum exchange between the fluid and fiber phases. The fibers are free to interact through long range hydrodynamic fiber–fiber interactions which are modeled using a rotary diffusion coefficient, an approach outlined by Folgar and Tucker. Numerical predictions are made for two different orientation states at the inlet to the contraction, namely a fully random and a partially aligned fiber orientation state. Results from these numerical predictions show that the streamlines of the flow are altered and that velocity profiles change from Jeffery–Hamel, to something resembling a plug flow when the fiber phase is considered in the fluid momentum equations. This phenomenon was found when the suspension enters the channel in either a pre-aligned, or in a fully random orientation state. When the suspension enters the channel in an aligned orientation state, fiber orientation is shown to be only marginally changed when the two-way coupling is included. However, significant differences between coupled and uncoupled predictions of fiber orientation were found when the suspension enters the channel in a random orientation state. In this case, the suspension was shown to align much more quickly when the mutual coupling was accounted for and profiles of the orientation anisotropy were considerably different both qualitatively and quantitatively.