Flow-driven orientation dynamics of semiflexible fiber systems

We study the flow-induced orientation dynamics of semiflexible fibers in dilute fiber suspensions. Starting from the equations of motion for a two-rod model of flexible fibers in Stokes flow, the Smoluchowski equation for a connected monomer orientation distribution function is derived. We then obtain a set of equations for the time dependence of the first and second moments of the orientation distribution function, thus extending the Folgar Tucker equations for short rigid fiber suspensions to flexible fiber suspensions. The resulting generalized equations for the orientation dynamics of a suspension of flexible fibers are solved for simple channel flow. It is shown that all qualitative effects of bending and straightening of fibers and their influence on the orientation of flexible fibers are captured within our model. A scalar measure for the distribution of bending in a flow is introduced, which allows to detect the degree of bending of fibers. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Investigation of the abrupt contraction flow of fiber suspensions in polymeric fluids

Numerical simulations of the flow of rigid fibres through a 4:1 planar contraction, and the predicted flow pattern and fiber orientation are presented. Entirely new is the examination of the nature of the suspending matrix which may consist of either a Newtonian fluid or a polymer melt. In the case of a polymer matrix three rheological models, the Phan-Thien–Tanner, FENE-CR, and Carreau models have been used to investigate the effects of shear-thinning and elasticity on the flow and the orientation of the fibers. The effects of inertia are neglected, and the governing equations for the flow field, polymer stress, and fiber orientation are coupled and simultaneously solved. A parametric study is used to explore the effects of different dimensionless parameters on the velocity field, the fiber orientation, the pressure drop, as well as the vortex size measured by the dimensionless reattachment length. We particularly focus on the role of the fibers aspect ratio, volume fraction, and interaction coefficient which measures the intensity of fiber interaction in the suspension. Furthermore, we evaluate and compare the results of four different closure approximations: the quadratic, linear, hybrid A and T, and natural closures.     

Numerical simulation of flow kinematics for suspensions containing continuously dispersed aspect ratio fibers

Fiber suspension flow and fiber orientation through a parallel-plate channel were numerically simulated for fiber suspensions including continuously dispersed aspect ratios from 10 to 50. In the simulations, both the fiber–fiber and fiber–wall interactions were not taken into account. A statistical scheme that proceeds by evaluating the orientation evolution of a large number of fibers from the solution of the Jeffery equation along the streamlines was confirmed to be a very useful and feasible method to accurately analyze the orientation distribution of fibers with continuously dispersed aspect ratios. For monodisperse suspensions with small-aspect-ratio fibers, flip-over or oscillation phenomenon of the orientation ellipsoid caused the wavy patterns of the velocity profile and the streamlines as well as the abrupt and complex variation of the shear stress and the normal stress difference near the channel wall as proven in one of our former works. On the other hand, continuous dispersions containing from small- to large-aspect-ratio fibers were able to induce smoother evolutions of the fiber orientation and the flow kinematics. In the processing of fiber composites, the length of suspended fibers is always continuously distributed because of fiber breakage during processing; thus, the smooth evolutions of the flow kinematics and the stress distribution can be attained.This paper was presented at the Annual Meeting of the European Society of Rheology, Grenoble, April 2005.     

Non Linear Rheology for Long Chain Branching characterization,comparison of two methodologies : Fourier Transform Rheology and Relaxation.

In this study we compare three rheological ways for Long Chain Branching (LCB) characterization of a broad variety of linear and branched polyethylene compounds. One method is based on dynamical spectrometry in the linear domain and uses the van Gurp Palmen plot. The two other methods are both based on non linear rheology (Fourier Transform Rheology (FTR) and chain orientation/relaxation experiments). FTR consists in the Fourier analysis of the shear stress signal due to large oscillatory shear strains. In the present work we focus on the third and the fifth harmonics of the shear stress response. Chain orientation/relaxation experiment consists in the analysis of the polymer relaxation after a large step strain obtained by squeeze flow. In this method, relaxation is measured by dynamical spectrometry and is characterized by two relaxation times related to LCB. All methods distinguish clearly the group of linear polyethylene from the group of branched polyethylene. However, FTR and Chain orientation/relaxation experiments show a better sensitivity than the van Gurp Palmen plot. Non linear experiments seem suitable to distinguish long branched polyethylene between themselves.     

Structure and rheology of fiber-laden membranes via integration of nematodynamics and membranodynamics

This paper presents a study of the structure and dynamics of rigid fiber-laden deformable curved fluid membranes based on an viscoelastic model that integrates the statics of anisotropic membranes, the planar nematodynamics of fibers and the dynamics of isotropic membranes. Fiber-laden membranes arise frequently in biological systems, such as the plant cell wall and in protein–lipid bilayers. Based on the membrane's force and torque balance equations and the fiber's balance of molecular fields, a viscoelastic anisotropic model that provides the governing equations for the membrane's velocity and curvature and the fiber structure (fiber orientation and order) is found. A Helmholtz free energy that incorporates the tension/bending/and torsion membrane elasticity, the Landau–de Gennes fiber ordering, and fiber order-membrane curvature interactions is used to derive elastic moments, torques, and stresses. The corresponding viscous stresses and moments include the Boussinesq–Scriven contributions as well as bending, torsion, and rotational dissipation. A spectral decomposition leads to the main viscoelastic material functions for anisotropic fluid membranes. Applications of the rheological model to cylindrical growth and cylindrical axial stretching show that competing curvo-phobic, curvo-philic interactions under extensional flow predict transitions between axial and azimuthal fiber arrangements, of interest to cellulose fiber orientation in plant morphogenesis.     

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.     

Elasticity solution for free vibration analysis of four-parameter functionally graded fiber orientation cylindrical panels using differential quadrature method

In this paper, three-dimensional free vibrations analysis of a four-parameter functionally graded fiber orientation cylindrical panel is presented. The panel is simply supported at the edges and assumed to have an arbitrary variation of fiber orientation in the radial direction. A generalization of the power-law distribution presented in literature is proposed. Symmetric and asymmetric fiber orientation profiles are studied in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method to obtain the natural frequency. The main contribution of this work is to illustrate the influence of the power-law exponent, of the power-law distribution choice and of the choice of the four parameters on the natural frequencies of continuous grading fiber orientation cylindrical panels. Numerical results are presented for a cylindrical panel with arbitrary variation of fiber orientation in the shell’s thickness and compared with discrete laminates composite panels. It is shown maximum natural frequencies will be obtained by using symmetric fiber orientation profiles.     

旋转SMA纤维混杂复合材料薄壁梁的自由振动

研究具有SMA主动纤维的旋转复合材料单闭室薄壁截面梁的耦合自由振动问题.基于Hamilton原理并结合SMA纤维复合材料薄壁梁的二维截面内力(矩)与位移(转角)关系方程,导出旋转单闭室截面薄壁复合材料梁的1D耦合自由振动分析模型.该模型还考虑薄壁梁调矩角和预锥角的作用.采用Galerkin法求解振动模型,获得梁耦合振动...     

Orientation tensors in simple flows of dilute suspensions of non-Brownian rigid ellipsoids,comparison of analytical and approximate solutions

General analytical solutions are obtained for the planar orientation structure of rigid ellipsoid of revolutions subjected to an arbitrary homogeneous flow in a Newtonian fluid. Both finite and infinite aspect ratio particles are considered. The orientation structure is described in terms of two-dimensional, time-dependent tensors that are commonly employed in constitutive equations for anisotropic fluids such as fiber suspensions. The effect of particle aspect ratio on the evolution of orientation structure is studied in simple shear and planar elongational flows. With the availability of analytical solutions, accuracies of quadratic closure approximations used for nonhomogeneous flows are analyzed, avoiding numerical integration of orientation distribution function. In general, fourth-order orientation evolution equations with sixth-order quadratic closure approximations yield more accurate representations compared to the commonly used second-order evolution equations with fourth-order quadratic closure approximations. However, quadratic closure approximations of any order are found to give correct maximum orientation angle (i.e., preferred direction) results for all particle aspect ratios and flow cases.     

Transient shear flow behavior of concentrated long glass fiber suspensions in a sliding plate rheometer

In order to eventually predict the behavior of long fiber suspensions in complex flows commonly found in processing operations, it is necessary to understand their rheology and its connection to the evolution of fiber orientation and configuration in well defined flows. In this paper we report the transient behavior at the startup of shear flow of a polymer melt containing long glass fibers with a length (L) >1 mm, using a sliding plate rheometer (SPR). The operation of the SPR was confirmed by comparing the transient shear viscosity (η⁺) for a polymer melt and a melt containing short glass fibers (L < 1 mm) with measurements obtained from a cone-and-plate device, using a modified sample geometry that was designed to avoid wall effects. For the long fiber systems, measurements could only be obtained in the SPR because these systems would not stay within the gap of the rotational rheometer. Transient stress growth behavior of the long fiber systems was obtained as a function of shear rate and fiber concentration for samples prepared with three different initial orientations. Results showed that, unlike short fiber systems (with a random planar initial orientation) that usually exhibit a single overshoot peak followed by a steady state, η⁺ of the long fiber suspensions often passed through multiple transient regions, depending on the fiber concentration and applied shear rate. Additionally, η⁺ of the long fiber suspensions was found to be highly dependent on the initial orientation of the sheared samples. Finally, the initial and final fiber orientations of the long glass fiber samples were measured and used to initiate an explanation of the viscosity behavior. The results obtained in this research will be useful for future assessment of a quantitative correlation between transient rheology and the evolution of fiber orientation.     

Fiber orientation kinetics of a concentrated short glass fiber suspension in startup of simple shear flow

The common approach for simulating the evolution of fiber orientation during flow in concentrated suspensions is to use an empirically modified form of Jeffery's equation referred to as the Folgar–Tucker (F-T) model. Direct measurements of fiber orientation were performed in the startup of shear flow for a 30 wt% short glass fiber-filled polybutylene terephthalate (PBT-30); a matrix that behaves similar to a Newtonian fluid. Comparison between predictions based on the F-T model and the experimental fiber orientation show that the model over predicts the rate of fiber reorientation. Rheological measurements of the stress growth functions show that the stress overshoot phenomenon approaches a steady state at a similar strain as the fiber microstructure, at roughly 50 units. However, fiber orientation measurements suggest that a steady state is not reached as the fiber orientation continues to slowly evolve, even up to 200 strain units. The addition of a “slip” parameter to the F-T model improved the model predictions of the fiber orientation and rheological stress growth functions.     

Isothermal extrusion of non-dilute fiber suspensions

The extrusion of a rod-like fiber suspension is a Newtonian solvent, as a first step to the fast and inexpensive production of composite materials, is investigated. The analysis is carried out by means of an integral constitutive equation for a non-dilute suspension, streamlined finite element for liquid with memory, and Newton iteration of nonlinear integro-differential equations. The predictions show substantial differences between dilute and nondilute fiber suspension regarding the processing conditions (pressure drop, velocity distribution, die-swell) and the resulting fiber orientation. Nondilute fiber suspensions exhibit substantial shear-thinning and negligible elasticity as evidenced by the small die-swell, and fiber concentration viscosity-thickening as evidenced by the large pressure drop. The fiber orientation is computed by solving the orientation distribution function along selected streamlines of the complex velocity field. It is shown that the fiber orientation far downstream can be made independent of the random fiber orientation at the inlet.     

Mechanical response of a short fiber-reinforced thermoplastic: Experimental investigation and continuum mechanical modeling

The present work can be regarded as a first step toward an integrated modeling of mold filling during injection molding process of polymer composites and the resulting material behavior under service loading conditions. More precisely, the emphasis of the present paper is laid on how to account for local fiber orientation in the ground matrix on the prediction of the mechanical response of the composite at its final solid state. To this end, a set of experiments which captures the mechanical behavior of an injection molded short fiber-reinforced thermoplastic under different strain histories is described. It is shown that the material exhibits complex response mainly due to non-linearity, anisotropy, time/rate-dependence, hysteresis and permanent strain. Furthermore, the relaxed state of the material is characterized by the existence of an equilibrium hysteresis independently of the applied strain rate. A three-dimensional phenomenological model to represent experimentally observed response is developed. The microstructure configuration of the material is simplified and assumed to be entirely represented by a distributed fiber orientation in the ground matrix. In order to account for distributed short fiber orientations in a continuum sense, a concept of (symmetric) generalized structural tensor (tensor of orientation) of second order is adopted. The proposed model is based on assumption that the strain energy function of the composite is given by a linear mixture of the strain energy of each constituent: an isotropic part representing Phase 1 which is essentially related to the ground matrix and an anisotropic part describing Phase 2 which is mainly related to the fibers and the interphase as a whole. Hence, taking into account the fiber content and orientation, the efficiency of the model is assessed and perspectives are drawn.     

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

Processing,characterisation and rheology of transparent concentrated fibre-bundle suspensions

Highly concentrated planar fibre-bundle suspensions with a transparent PMMA matrix were processed with various initial bundle contents and orientations. They were submitted to simple compression and plane strain compression deformation modes. First rheological measurements are presented. They highlight the role of the bundle content and orientation on recorded stress levels. The transparent matrix allows the observation of fibrous microstructures before and after compressions: The in-plane deformation of bundles (flattening and bending) as well as the evolution of their orientation are analysed and discussed. This paper was presented at the 3rd Annual European Rheology Conference (AERC) 2006, held in Hersonisos, Crete, Greece, April 27–29, 2006.     

Experimental and theoretical study of short fiber orientation in diverging flows

Two types of experiments have been carried out to study the fiber orientation in flow through a divergent channel. First, a reinforced polyamid mold sprue containing two types of orientation was investigated: near the center, the fibers are mostly oriented perpendicular to the flow lines, whereas on the periphery, they are oriented parallel to them. Second, direct observation of copper fibers moving in a corn syrup was performed in a transparent diverging device: the fibers rapidly become oriented transverse to the flow lines. The solution of Stokes equations for the undisturbed fluid motion gives the shear rate and elongation rate, which are then substituted in Jeffery's orientation equations. The resolution shows two types of behavior: in a large area in the center, the fiber tends to a stable equilibrium position which depends strongly on the flow line on which it moves. On the periphery, the fiber follows a shear-like behavior. The strong influence of the elongational component relative to the shear component is demonstrated and the time necessary for orientation is calculated. The theoretical results are found to be in agreement with the observations.     

The stress-microstructure relationship in an evolving mixing layer of fiber suspensions

The equations for fiber suspensions in an evolving mixing layer were solved by the spectral method, and the trajectory and orientation of fibers were calculated based on the slender body theory. The calculated spatial and orientation distributions of fibers are consistent with the experimental ones that were performed in this paper. The relationship between the microstructure of fibers and additional stress was examined. The results show that the spatial and orientation distributions of fibers are heterogeneous because of the influence of coherent vortices in the flow, which leads to the heterogeneity of the additional stress. The degree of heterogeneity increases with the increasing of St number and fiber aspect ratio. The fibers in the flow make the momentum loss thickness of the mixing layer thicker and accelerate the vorticity dispersion.The project supported by the Doctoral Program of Higher Education in China (20030335001)     

Simulation of fiber reinforced composite materials mold filling process and mechanical properties analysis

A dynamic simulation of fiber reinforced composite materials mold filling process with double inlets is presented based on the gas–solid–liquid model proposed by Yang et al. [B.X. Yang, J. Ouyang, J. Tao, C.T. Liu, Modeling and simulation of fiber reinforced polymer mold filling process by level set method, CMES – Computer Modeling in Engineering and Sciences 63 (3) (2010) 191–222]. Numerical results show that the fibers far away from the melt interface are in skin-core-skin structure, while those near the interface are almost parallel to the arc of the interface. When the two streams of melts meet, the weld line will be formed, where the orientation of fibers is perpendicular to the flow direction. The orientation of fibers of the numerical result shows well agreement with the experimental results. Finally, the mechanical properties of fiber reinforced composite materials are analyzed. The composite materials with skin-core-skin structure are regarded as laminated orthogonal plywood and the elastic modulus, the shear modulus and Poisson’s ratio are predicted under different slenderness ratios and fiber volume fractions.     

Nonlinear elastic effects in graphite/epoxy: An analytical and numerical prediction of energy flux deviation

Manipulating acoustic wave propagation through a material have several interdisciplinary applications. Here we predict shift in energy flux deviation for acoustic waves propagating in unidirectional graphite/epoxy due to applied normal and shear stresses using both an analytical model, using acoustoelastic continuum theory, and a finite element discrete numerical model. The acoustoelastic theory predicts that the quasi-transverse (QT) wave exhibits larger shifts in energy flux deviation compared to quasi-longitudinal (QL) or the pure transverse (PT) due to an applied shear stress for fiber orientation angle ranging from 0° to 60°. Due to an applied shear stress the QT wave exhibits a shift in energy flux deviation at 0° fiber orientation angle as compared to normal stress case where the flux deviation and its load induced shift are both zero. A finite element model (FEM) is developed where equations of motion include the effect of nonlinear elastic coefficients. Element equations were integrated in time using Newmark’s method to determine the shift in energy flux deviations in graphite/epoxy for different loading cases. The energy flux shift of QT waves predicted by FEM for fiber orientation angles from 0° to 60° for applied shear stress case is in excellent agreement with acoustoelastic theory. Because energy shift magnitudes are not small, it is possible to experimentally measure these shifts and calibrate shifts with respect to load type (normal/shear) and magnitude.     

Motion of a slender body in quiescent polymer solutions

The motion of a slender body falling in quiescent polymer solutions is investigated experimentally. It represents the simplest model of motion of single fibers in the flow of fiber suspensions. The fall behavior in quiescent polymer solutions is compared with that in water. It is demonstrated that a slender body falling in Newtonian liquids rotates to adopt a horizontal orientation, whereas in non-Newtonian liquids it rotates towards a vertical orientation but for less concentrated solutions is not able to reach the vertical orientation and moves sideways with a constant orientation angle. The effects of shear thinning and elasticity on the motion of the body are discussed.