Fiber Orientation Tensor Research Articles

Characterization of multilayer structures in fiber reinforced polymer employing synchrotron and laboratory X-ray CT

Abstract Specimens of carbon or glass fiber reinforced polymer can be imaged using both conventional laboratory X-ray micro-computed tomography equipment and synchrotron light sources. The image quality when using intense (partially) coherent synchrotron light is still superior, especially when applying phase-retrieval algorithms. In the resulting volume images, the fiber direction distribution and other mechanically relevant parameters such as volume fractions or layer thickness can be determined. In this contribution, we will demonstrate how fiber direction results can be used to detect regions with locally different fiber orientations in carbon or glass fiber reinforced polymer which arise in the molding process of such samples. To this end, we evaluate the three-dimensional fiber orientation tensor locally across the thickness of different specimens. For each resulting individual layer, we can automatically detect the layer thickness and the preferred fiber direction. These methods have been successfully applied to various commercial specimens. We will demonstrate results on volume images of samples from both synchrotron and laboratory micro-computed tomography and discuss the specific advantages and disadvantages in this application.

Material Modelling of Short Fiber Reinforced Thermoplastic for the FEA of a Clinching Test

In modern car body construction, multi-material and hybrid design is used, whereby short fibre reinforced plastics combined with light metals represent an interesting class of work-piece materials. In order to realize modern hybrid construction, suitable joining techniques are therefore required. Clinching represents a cost-effective and easy to implement joining method. In this paper the material modelling of the short fibre reinforced thermoplastic sheets considering the fibre orientation tensor for the FEA of the clinching process is presented.

Fiber/matrix interfacial thermal conductance effect on the thermal conductivity of SiC/SiC composites

Abstract SiC/SiC composites used in fusion reactor applications are subjected to high heat fluxes and require knowledge and tailoring of their in-service thermal conductivity. Accurately predicting the thermal conductivity of SiC/SiC composites as a function of temperature will guide the design of these materials for their intended use, which will eventually include the effects of 14-MeV neutron irradiations. This paper applies an Eshelby–Mori–Tanaka approach (EMTA) to compute the thermal conductivity of unirradiated SiC/SiC composites. The homogenization procedure includes three steps. In the first step EMTA computes the homogenized thermal conductivity of the unidirectional (UD) SiC fiber embraced by its coating layer. The second step computes the thermal conductivity of the UD composite formed by the equivalent SiC fibers embedded in a SiC matrix, and finally the thermal conductivity of the as-formed SiC/SiC composite is obtained by averaging the solution for the UD composite over all possible fiber orientations using the second-order fiber orientation tensor. The EMTA predictions for the transverse thermal conductivity of several types of SiC/SiC composites with different fiber types and interfaces are compared to the predicted and experimental results by Youngblood et al. [J. Nucl. Mater. 307–311 (2002) 1120–1125, Fusion Sci. Technol. 45 (2004) 583–591, Compos. Sci. Technol. 62 (2002) 1127–1139.]

유리섬유가 강화된 필름 삽입 사출품의 섬유배향 및 휨

필름 삽입 사출 시편의 휨은 비대칭적인 잔류응력 분포에 기인한다. 비대칭적 잔류응력과 온도 분포는 삽입된 필름 표면의 수직방향으로 지연되는 열 전달에 의해 발생한다. 사출 공정조건 최적화를 통해 필름 삽입 사출 시편의 휨을 억제할 수 없었기 때문에, 필름 삽입 사출 시편의 휨을 최소화하기 위해서 유리 섬유가 강화된 복합재료를 기판으로 사용하였다. 유리 단섬유의 분포를 마이크로 CT 장비를 이용하여 평가하였다. 복합재료로 구성된 기판을 이용한 필름 삽입 사출 시편의 배향 텐서와 휨을 계산하기 위해서는 적절한 마이크로 역학, 이방적 열팽창계수 및 닫힌 어림법 모델이 선택되어야만 한다. 여섯 종류의 마이크로 역학모델, 세 종류의 열 팽창 계수 모델 및 다섯 종류의 닫힌 어림법 모델을 고려한 후, Mori-Tanaka 모델, Rosen and Hashin 모델 및 third orthotropic 닫힌 어림법 모델을 선택하였다. 수치적으로 계산된 섬유 배향 텐서와 휨에 관한 결과들은 실험결과와 잘 일치하였고, 유리 섬유의 보강효과가 필름 삽입 사출 복합 재료 시편의 휨에 미치는 영향을 파악하였다. Warpage of the film insert molded (FIM) part is caused by an asymmetric residual stress distribution. Asymmetric residual stress and temperature distribution is generated by the retarded heat transfer in the perpendicular direction to the attached film surface. Since warpage was not prevented by controlling injection molding conditions, glass fiber (GF) filled composites were employed as substrates for film insert molding to minimize the warpage. Distribution of short GFs was evaluated by using micro-CT equipment. Proper models for micro mechanics, anisotropic thermal expansion coefficients, and closure approximation should be selected in order to calculate fiber orientation tensor and warpage of the FIM part with the composite substrate. After six kinds of micro mechanics models, three models of the thermal expansion coefficient and five models of the closure approximation had been considered, the Mori-Tanaka model, the Rosen and Hashin model, and the third orthotropic closure approximation were selected in this study. The numerically predicted results on fiber orientation tensor and warpage were in good agreement with experimental results and effects of GF reinforcement on warpage of the FIM composite specimen were identified by the numerical results.

Effectiveness of recent fiber-interaction diffusion models for orientation and the part stiffness predictions in injection molded short-fiber reinforced composites

Two fiber interaction models for predicting the fiber orientation and resulting stiffness of a short-fiber reinforced thermoplastic composite are investigated, the isotropic rotary diffusion of Folgar and Tucker (1984) [1] and the anisotropic rotary diffusion of Phelps and Tucker (2009) [6]. This study employs several fiber orientation tensor closure approximations for both diffusion models and results are compared to those from the numerically exact spherical harmonic approach. Results are presented for variations in the fiber orientation and the processed part stiffness. A significant difference was observed between the stiffness predicted by both rotary diffusion models. It is worth noting that not all closures behave the same between the diffusion models, thus encouraging further studies to refine and validate the new fiber interaction models and solution approaches. A study of the predicted flexural modulus is presented, and results suggest that flexural modulus experiments may aid in further refining the fiber interaction models.

Numerical simulation of fiber orientation distribution in round turbulent jet of fiber suspension

The Reynolds averaged Navier–Stokes equation was solved numerically with the Reynolds stress model to get the mean fluid velocity and the turbulent kinetic energy in a round turbulent jet of fiber suspension. The fluctuating fluid velocity was described as a Fourier series with random coefficients. Then the slender-body theory was used to calculate the fiber orientation distribution and orientation tensor. Numerical results of mean axial velocity and turbulent shear stress along the lateral direction were validated by comparing with the experimental ones. The results show that most fibers are aligned with the flow direction as they go downstream, and fibers are more aligned with the flow direction within the region near the jet core. The fibers with high aspect ratio tend much easier to align with the flow direction, and the fiber orientation distribution is not sensitive to fiber aspect ratio when fiber aspect ratio is larger than 5. Fiber density has no obvious influence on the fiber orientation distribution and fiber orientation tensor. The randomizing effect of turbulence is insignificant in the regions near outside and jet core, and becomes significant in the region between outside and jet core. The randomizing effect increases with the increasing of the distance from the jet exit. Different components of fiber orientation tensor show a similar distribution pattern.

Irreversible thermodynamics based constitutive theory for fiber suspended polymeric liquids

A rheological model for a short fiber suspension in polymeric liquid is proposed according to irreversible thermodynamics of viscoelastic deformation of polymers with the effect of fiber orientation taken into account. In this model, the evolution of elastic deformation tensor, namely, reversible Finger strain tensor, is governed by not only the reversible Finger strain tensor but also the fiber orientation tensor in a manner of a positive entropy production. The final form of stress tensor combines the viscoelasticity of polymeric matrix of the Leonov model and the fiber contribution of the Dinh–Armstrong model. The Folgar–Tucker model is employed for fiber orientation kinematics along with an invariant-based optimal fitting closure approximation. The rheological behavior of the model is comprehensively studied using steady and transient shear flows.

Fibre orientation distribution in turbulent fibre suspensions flowing through an axisymmetric contraction

AbstractThe Reynolds averaged Navier–Stokes equation was solved numerically with the Reynolds stress model to get the mean fluid velocity and the turbulent kinetic energy in turbulent fibre suspensions flowing through an axisymmetric contraction. The fluctuating fluid velocity was represented as a Fourier series with random coefficients. Then the slender‐body theory was used to predict the fibre orientation distribution and orientation tensor. Some numerical results are compared with the experimental ones in the turbulent fibre suspensions flowing through a contraction with a rectangular cross‐section. The results show that the fibres with high aspect ratio tend to align its principal axis with the flow direction much easier. High contraction ratio makes the fibre alignment with the flow direction much easier. The contraction ratio has a strong effect on the fibre orientation distribution. Only a small part of the fibre is aligned with the flow direction in the inlet region, while most fibres are aligned with the flow direction when they approach to exit. The fibres are aligned with the flow direction rapidly in the inlet region, after that the fibre orientations change little in the most of the downstream region. The fibres with high aspect ratio are aligned with the flow direction faster when they enter the contraction. The randomising effect of the turbulence becomes significant in the downstream region because of the high turbulent intensity.

Modeling and Simulation of Fiber Orientation in Injection Molding of Polymer Composites

We review the fundamental modeling and numerical simulation for a prediction of fiber orientation during injection molding process of polymer composite. In general, the simulation of fiber orientation involves coupled analysis of flow, temperature, moving free surface, and fiber kinematics. For the governing equation of the flow, Hele‐Shaw flow model along with the generalized Newtonian constitutive model has been widely used. The kinematics of a group of fibers is described in terms of the second‐order fiber orientation tensor. Folgar‐Tucker model and recent fiber kinematics models such as a slow orientation model are discussed. Also various closure approximations are reviewed. Therefore, the coupled numerical methods are needed due to the above complex problems. We review several well‐established methods such as a finite‐element/finite‐different hybrid scheme for Hele‐Shaw flow model and a finite element method for a general three‐dimensional flow model.

Multiaxial fatigue life assessment for reinforced polymers

The multiaxial fatigue behaviour of PBT–PET GF30 (polybutylene terephthalate–polyethylene terephthalate with 30% mass short glass fibres) and PA66 GF35 (Polyamide 66 with 35% mass short glass fibres) is studied under constant amplitude force loading. Standard flat specimens are used to establish the influence of microstructure on the tension fatigue behaviour. Tubular specimens are used to study the influence of loading path on the tension–torsion fatigue behaviour. Finally an experimental device is developed to test dedicated flat samples in order to study the influence of microstructure on the pure shear fatigue behaviour. All fatigue tests are performed at two load ratios: −1 and 0, 1. Experimental results are analysed using a Through Process Modelling (TPM) based on three major steps. The first one is based on the simulation of the whole process in order to get the fibre orientation tensor at each point of the part. The second one uses the orientation tensor to perform a two-step homogenization procedure in order to estimate local effective properties and compute the stress–strain response as a function of local anisotropy. Mechanical fields thus obtained are used as input data for the application of a fatigue criterion in the third step. Results show that energetic fatigue criterion is an excellent compromise to get good fatigue life assessment for only one experimental input fatigue data.

Microstructures of fiber suspensions in complex geometry

AbstractEvolutions of molecular conformation and fiber orientation in fiber suspensions are investigated by collocated finite volume method on unstructured triangular meshes. FENE-P (Finite Extensible Nonlinear Elastic Dumbbell model with Peterlin’s approximation) model which is microstructure-based is chosen to describe the polymeric matrix and TIF (transversely isotropic fluid) model is used to calculate the fiber contribution in order to realize the coupling of flow and fiber orientation. Microstructures of molecule and fiber are obtained by analyzing the information of molecular conformation tensor and second-order fiber orientation tensor respectively. Two numerical examples are considered, namely, a planar contraction flow and a planar flow past a confined cylinder. Present results are hoping to give more insight into the microscopic details of complex flows and thus be more helpful for industrial application.

Orientation distribution and rheological properties of fiber suspensions flowing through curved expansion and rotating ducts

In dilute fiber suspensions, the Jeffery equation is employed to predict the fiber orientation distribution through a curved expansion and rotating duct. The results show that the fiber flips more quickly in the inlet region, especially in the region close to the concave wall. In the central region and the downstream region, the fiber orientation distributions are more uniform. In semi-dilute fiber suspension, the fiber interactions can be represented by a statistical model developed by Folgar & Tucker. Then a deduced Fokker-Planck equation can be used to describe the fiber orientation distribution probability. In the present work, the Fokker-Planck equation is directly solved by FVM. Then with the probability, the fiber orientation tensors are obtained. Furthermore, the fiber extra stresses are gained with the Batchelor model. The results show that the shear stress and normal stress difference are concentrated around the inlet close to the concave wall regions. In the central regions, these properties are less obvious. And in the downstream, these properties are negligible.

Rheological properties of fiber suspensions flowing through a curved expansion duct

A numerical method based on finite volume method for solving Fokker-Planck equation on a unit sphere was first developed to simulate numerically the fiber orientation distribution of fiber suspensions flowing through a curved expansion duct. The momentum equation with fiber extra stress was used to explore the rheological properties of the fiber suspensions. The second-order fiber orientation tensors in present simulation are compared with the previous numerical results for validating the model and computational code. The results showed that the pressure increases from inlet to outlet, and changes along the lateral direction nonmonotonically except at outlet. The extra shear stress and normal stress difference decrease from inlet to outlet, and changes along the lateral direction nonmonotonically. The effect of fiber aspect ratio on the extra shear stress and normal stress difference is negligible near the wall region. The extra shear stress and normal stress difference increase with the fiber aspect ratio near the centerline. The lateral distribution curves of extra shear stress and normal stress difference become flat with increasing the distance from the inlet.

Study on Fiber Weight Fraction Effect on Tensile Modulus of Polystyrene (PS) Composites Reinforced with Short Glass Fiber (SGF) Based on Their Fiber Orientation

In this study, the tensile modulus of injection molded SGF/PS composites is investigated. The experimental and simulation results show, increasing of tensile modulus of these composites, is inconsistent with increasing of fiber content in high fiber weight fraction. This inconstancy is more distinguishable for fiber weight fractions greater than 40%wt of GF. To explain away this observation, the fiber orientation tensor of molded part is simulated. The behavior of a11, the first orientation tensor component in filling flow direction, through the thickness of the part is obtained for different SGF/PS compositions. The consequent curves can explain the unexpected behavior and support this idea that the effects of fiber orientation should be considered in calculation of SFG composite properties.

Elastic Properties of Short-fiber Polymer Composites, Derivation and Demonstration of Analytical Forms for Expectation and Variance from Orientation Tensors

Existing methods for predicting elastic properties of short-fiber polymer composites from fiber orientation tensors are based on the orientation average of a transversely isotropic stiffness tensor. These evaluations focus solely on average properties and have yet to include a quantitative measure of property variation. Recognizing the statistical nature of fiber orientations within the composite commonly defined through the fiber orientation distribution function, analytical expressions are developed here to predict both expectation and variance of the material stiffness tensor from a fiber orientation distribution function. The fiber orientation distribution function is expanded through the Laplace series of complex spherical harmonics and results demonstrate that material stiffness tensor expectation is a function of orientation tensors up through fourth-order and the corresponding variance requires orientation tensors up through eighth-order. Numerical simulations obtained with the method of Monte-Carlo for sample sets generated from statistically independent unidirectional samples belonging to the fiber orientation distribution function from the accept—reject generation algorithm are shown to agree with the analytic expressions for material expectation and variance.

Effects of Some Injection Molding Process Parameters on Fiber Orientation Tensor of Short Glass Fiber Polystyrene Composites (SGF/PS)

Calculation of fiber orientation distribution is the reference standard for determining the numerous physical and mechanical properties of injection molded short fiber reinforced thermoplastics. On the other hand, most of the properties of these composites depend on their fiber orientation pattern, which can be represented by tensorial notation, aij. This work focuses on numerical calculation of a fiber orientation tensor of an injection molded short glass fiber polystyrene (SGF-PS) composite part, shaped as a rectangular plate. In this study, the effects of some parameters of the injection molding process including injection flow rate, mold wall temprature, packing pressure and also fiber content on the changes of fiber orientations of this specimen are investigated. Our studies are concentrated on determining the changes of the first orientation tensor component in the flow direction due to different injection parameters. In addition, the changes on tensile modulus of the injected part are simulated in every condition. The simulations show how these molding parameters can influence both fiber orientation and tensile modulus of injection molded parts. At the first step, in order to be confident about the software results, they are validated by experimental data in two cases.

The effect of fibre orientation closure approximations on mechanical property predictions

Abstract Current approaches used to simulate the flow of short-fibre suspensions employ fibre orientation tensors where the dependence upon higher-order orientation tensors is alleviated through the use of a closure. Unfortunately, the effect that a given closure has on material property predictions has received little attention in the literature. Accordingly this investigation demonstrates that current objective fourth-order closures assume an orthotropic form, whereas existing sixth-order closures are shown to provide a material representation with fewer symmetries than orthotropic. Numerical calculations demonstrate the existence of shear–extensional and shear–shear coupling in simple homogeneous flow and for a center-gated disk. Although these components are minor in comparison to principal components of the stiffness tensor, these components can be obtained via existing sixth-order closures.

Effects of tensor closure models and 3-D orientation on the stability of fiber suspensions in a channel flow

Three different kinds of closure model of fiber orientation tensors were applied to simulate numerically the hydrodynamic stability of fiber suspensions in a channel flow. The effects of closure models and three-dimensional (3-D) orientation distribution of fibers on the results of stability analysis were examined. It is found that the relationship of the behavior in hydrodynamic stability and the parameter of the fiber given by all the three models are the same. However, the attenuation of flow instability is most distinct using 3-D hybrid model because the orientation of the fiber departures from the flow direction, and least apparent using its 2-D counterpart for that the fibers show a tendency towards alignment with the flow direction in this case.

Global Stiffness of a SMC Panel Considering Process Induced Fiber Orientation

A material model, that translates into a stiffness matrix, the second order fiber orientation tensor, described by Advani and Tucker, and the stiffness matrix of a composite with aligned ellipsoidal inclusions, has been implemented in a FE programme and validated. The stiffness of a SMC panel with known state of fiber orientation is calculated using FEM. The influence of process induced fiber orientation is analysed. The fiber orientation for a realistic charge pattern for the panel has been obtained through mould filling simulation in a separate project. It is found that the fiber orientation has a rather small impact on the global stiffness. Only 0.8% lower stiffness compared to isotropic material model is obtained taking into account the fiber orientation distribution. The main reason for the low impact of the process induced fiber orientation is that the charge is symmetrically placed in the mould leading to a symmetric fiber orientation distribution.

Remodelling of continuously distributed collagen fibres in soft connective tissues

Extracellular matrix remodelling plays an essential role in tissue engineering of load-bearing structures. The goal of this study is to model changes in collagen fibre content and orientation in soft connective tissues due to mechanical stimuli. A theory is presented describing the mechanical condition within the tissue and accounting for the effects of collagen fibre alignment and changes in fibre content. A fibre orientation tensor is defined to represent the continuous distribution of collagen fibre directions. A constitutive model is introduced to relate the fibre configuration to the macroscopic stress within the material. The constitutive model is extended with a structural parameter, the fibre volume fraction, to account for the amount of fibres present within the material. It is hypothesised that collagen fibre reorientation is induced by macroscopic deformations and the amount of collagen fibres is assumed to increase with the mean fibre stretch. The capabilities of the model are demonstrated by considering remodelling within a biaxially stretched cube. The model is then applied to analyse remodelling within a closed stented aortic heart valve. The computed preferred fibre orientation runs from commissure to commissure and resembles the fibre directions in the native aortic valve.