Fiber Orientation Tensor Research Articles

Smoothed particle hydrodynamics (SPH) modeling of fiber orientation in a 3D printing process

We present a numerical study of the fused deposition modeling 3D printing process of fiber-reinforced polymers by means of Smoothed Particle Hydrodynamics (SPH). For this purpose, a classical microstructure-based fiber suspension model coupled with a constitutive model for the suspending polymer is implemented within an SPH framework. The chosen model is reviewed, together with the details and specificities of its implementation in SPH. The results for several representative cases are then presented, mainly in terms of contours of fiber orientation tensor components and orientation distributions across the deposited layer thickness. The impact of the fiber concentration and its aspect ratio in a semi-concentrated regime and the effect of the ratio between extrusion and substrate velocities are investigated. Some insights into the link between the flow field and fiber orientation evolution within the printing head and as the material exits the nozzle are given. The main findings lie in the prediction of a skin/core structure in the deposited layer in which the skin regions exhibit a higher fiber alignment with respect to the core region. This effect is found to be enhanced by an increase in fiber concentration and to be sensitive to the substrate-to-extrusion velocity ratio. It is indeed enhanced in cases where the substrate velocity is low compared to the extrusion velocity and accompanied by a larger swelling of the deposit at the nozzle exit.

Prediction of Young’s Modulus for Injection Molded Long Fiber Reinforced Thermoplastics

In this article, the elastic properties of long-fiber injection-molded thermoplastics (LFTs) are investigated by micro-mechanical approaches including the Halpin-Tsai (HT) model and the Mori-Tanaka model based on Eshelby’s equivalent inclusion (EMT). In the modeling, the elastic properties are calculated by the fiber content, fiber length, and fiber orientation. Several closure approximations for the fourth-order fiber orientation tensor are evaluated by comparing the as-calculated elastic stiffness with that from the original experimental fourth-order tensor. An empirical model was developed to correct the fibers’ aspect ratio in the computation for the actual as-formed LFTs with fiber bundles under high fiber content. After the correction, the analytical predictions had good agreement with the experimental stiffness values from tensile tests on the LFTs. Our analysis shows that it is essential to incorporate the effect of the presence of fiber bundles to accurately predict the composite properties. This work involved the use of experimental values of fiber orientation and serves as the basis for computing part stiffness as a function of mold filling conditions. The work also explains why the modulus tends to level off with fiber concentration.

A novel pseudo-grain approach for the estimation of the elastic stress distributions within the matrix of short fiber-reinforced polymers

Abstract The aim of the work is to provide a new methodology for the calculation of the linear elastic stress distributions within the matrix of a short fiber-reinforced Representative Volume Element (RVE) with misaligned fibers, without relying on complex algorithms for the generation and solution of the microstructure. For this purpose, a Pseudo-Grain approach is adopted, the novelty of which lies in the computation of local stress distributions through the analytical combination of numerical solutions from unidirectional (UD) grains that can be easily and quickly solved through commercial codes. The new approach consists in two steps: i) the application of Voigt boundary conditions to Finite Element (FE) models (grains) with fully aligned fibers and ii) the mathematical combination of the computed Stress Cumulative Distribution Functions (S-CDF) on the basis of the second-order Fiber Orientation Tensor (FOT) of the microstructure of interest. The latter can be generally derived from process simulations or experimental analyses. The method is validated by comparing the resulting stress fields with those calculated from actual RVEs with different fiber volume fractions, fiber aspect ratios and Fiber Orientation Tensors, showing a very good agreement.

Stochastic reconstruction and microstructure modeling of SMC chopped fiber composites

To establish an integrated Processing-Microstructure-Property workflow for the prediction of material behaviors, this paper presents a new stochastic pseudo-3D microstructure reconstruction method for Sheet Molding Compounds (SMC) chopped fiber composites. The proposed method captures the bi-level microstructural features of SMC composites. At the higher level, a Voronoi diagram-based algorithm is developed to reconstruct the unique substructure features of SMC fiber tows. The geometry of Voronoi cells is adjusted by a Simulated Annealing (SA) algorithm to match the geometrical statistics of the real fiber tows. At the lower level, the algorithm assigns fiber orientation to each Voronoi cell (which represents a fiber tow). The fiber orientation angles are recovered from a statistical fiber orientation tensor. The proposed method is employed to establish a multi-layer pseudo-3D SMC Representative Volume Element (RVE) model for Finite Element Analysis (FEA) of SMC microstructure. This model enables the prediction of mechanical properties based on the material processing information (e.g. fiber orientation tensor obtained from compression molding simulation), and the microstructure information obtained from microscopic imaging for an SMC composite. The predicted properties are successfully validated by experimental tensile tests.

Rheology Effects on Predicted Fiber Orientation and Elastic Properties in Large Scale Polymer Composite Additive Manufacturing

Short fiber-reinforced polymers have recently been introduced to large-scale additive manufacturing to improve the mechanical performances of printed-parts. As the short fiber polymer composite is extruded and deposited on a moving platform, velocity gradients within the melt orientate the suspended fibers, and the final orientation directly affects material properties in the solidified extrudate. This paper numerically evaluates melt rheology effects on predicted fiber orientation and elastic properties of printed-composites in three steps. First, the steady-state isothermal axisymmetric nozzle melt flow is computed, which includes the prediction of die swell just outside the nozzle exit. Simulations are performed with ANSYS-Polyflow, where we consider the effect of various rheology models on the computed outcomes. Here, we include Newtonian, generalized Newtonian, and viscoelastic rheology models to represent the melt flow. Fiber orientation is computed using Advani–Tucker fiber orientation tensors. Finally, elastic properties in the extrudate are evaluated based from predicted fiber orientation distributions. Calculations show that the Phan–Thien–Tanner (PTT) model yields the lowest fiber principal alignment among considered rheology models. Furthermore, the cross section averaged elastic properties indicate a strong transversely isotropic behavior in these composites, where generalized Newtonian models yield higher principal Young’s modulus, while the viscoelastic fluid models result in higher shear moduli.

Multiscale finite element modeling of sheet molding compound (SMC) composite structure based on stochastic mesostructure reconstruction

Predicting the mechanical behavior of the chopped carbon fiber Sheet Molding Compound (SMC) due to spatial variations in local material properties is critical for the structural performance analysis but is computationally challenging. Such spatial variations are induced by the material flow in the compression molding process. In this work, a new multiscale SMC modeling framework and the associated computational techniques are developed to provide accurate and efficient predictions of SMC mechanical performance. The proposed multiscale modeling framework contains three modules. First, a stochastic algorithm for 3D chip-packing reconstruction is developed to efficiently generate the SMC mesoscale Representative Volume Element (RVE) model for Finite Element Analysis (FEA). A new fiber orientation tensor recovery function is embedded in the reconstruction algorithm to match reconstructions with the target characteristics of fiber orientation distribution. Second, a metamodeling module is established to improve the computational efficiency by creating the surrogates of mesoscale analyses. Third, the macroscale behaviors are predicted by an efficient multiscale model, in which the spatially varying material properties are obtained based on the local fiber orientation tensors. Our approach is validated through experiments at both meso- and macro-scales, such as tensile tests assisted by Digital Image Correlation (DIC) and mesostructure imaging.

Multi-scale modelling and analysis of strain rate dependent behaviour for long fibre reinforced thermoplastic based on X-ray computed tomography

This paper focuses on the multi-scale modelling method for strain rate dependent behaviour of long fibre reinforced thermoplastics based on the Mori–Tanaka scheme. In the present work, the anisotropic and nonlinear viscoelastic–viscoplastic behaviour of an injection moulded plate are experimentally characterised, and the sensitive range of strain rates is discussed. Fibre orientation tensors, content and fibre length are measured and analysed based on micro computed tomography images at different positions of the plate. Further, a multi-scale material model considering microstructure variables, constitutive models of two phases and failure criteria, is proposed. The accuracy of the model is validated by finite element simulations, including five strain rates and three loading directions. The comparison results indicate that the calculated stress–strain curves show a good consistency with the experimental data.

Prediction of Static and Dynamic Mechanical Properties of GFRTP Product through Injection Molding

This paper aims at evaluating the prediction accuracy of macroscopic static and dynamic mechanical properties, including damping properties, of glass fiber reinforced thermo plastics (GFRTP) obtained via computer simulations. The prediction procedure is based on the Eshelby-Mori-Tanaka approach using fiber orientation tensor. The prediction accuracy of mechanical properties depends on the fiber orientation tensor. In this paper, we compare that prediction accuracies obtained using measured fiber orientation tensor and using simulated fiber orientation tensor by resin flow analysis. Additionally, we propose a linearized damping model based on the Eshelby-Mori-Tanaka approach and modal strain energy method to predict damping property, as well as a method for evaluating the contribution of the elastic modulus in order to quantitatively comprehend the anisotropic characteristics of GFRTP.

Flow-induced anisotropic viscosity in short FRPs

BackgroundThe commonly used flow models for Fiber Reinforced Polymers (FRPs) often neglect the flow-induced 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 micromechanically-based constitutive law is proposed which considers the flow-induced anisotropic viscosity of the fiber suspension.MethodsThe introduced viscosity tensor can handle arbitrary anisotropy of the fluid-fiber 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.ResultsThe 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 Dinh-Armstrong constitutive law.ConclusionsThe introduced model is a possible way to describe the flow-induced anisotropic viscosity of a fluid-fiber suspension based on the mean field theory.

Effect of the initial fiber alignment on the mechanical properties for GMT composite materials

A glass-mat-reinforced thermoplastic (GMT) material is widely used in the automotive industry for components such as underbody shields, seat structures, front/rear bumper, and front-end modulus. Due to the higher residual length of the glass strands, GMT usually offers better mechanical properties than injection-molded fiber-reinforced thermoplastics. The GMT material is typically manufactured by compression molding (CM) of preimpregnated fibers–reinforced resin sheets called mat. Two types of mats, one with discontinuous random (RD) fibers and other with aligned continuous fibers, are considered in this study. A stack of such mats with different combinations is used to tailor the mechanical properties of the final part. During the CM, the fibers in the mat flow with the resin and change the alignment. In this study, we are presenting an approach to account for the initial condition, such as fiber length, orientation and concentration of the fibers in the mat, and process conditions used, to develop a material model for the finished part. First, a stack of mat with known fiber orientation, length, and concentration as initial conditions is simulated for CM to predict the fiber orientation in the finished part. Next, the material model for the finished parts is developed using a Mori–Tanaka homogenization approach. The fiber orientation in the finished part is mapped from the CM simulation. For the fiber concentration and fiber length distribution, we used an empirical approach. The cross section of the finished part is investigated under optical microscope, and the fiber length and concentration are estimated based on the microstructure and initial stacking of mats. The predicted fiber orientation tensor is verified with orientations measured using computerized tomography (CT) scan on actual parts. The material model is verified by comparing the predicted performance with the actual tensile and bending test results.

The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics

We present an algorithm for generating volume elements of short fiber reinforced plastic microstructures for prescribed fourth order fiber orientation tensor, fiber aspect ratio and solid volume fraction. The algorithm inserts fibers randomly into an existing microstructure, and removes the resulting overlap systematically based on a gradient descent method. In contrast to existing methods, large fiber aspect ratios (up to 150) and large volume fractions (60 vol% for isotropic orientation and aspect ratio of 33) can be reached. We study the effective linear elastic properties of the resulting microstructures, depending on fiber orientation, volume fraction as well as aspect ratio, and examine the size of a corresponding representative volume element.

Flow‐induced anisotropic viscosity in short fiber reinforced polymers

AbstractThe commonly used flow models for fiber reinforced polymers often neglect the flow induced mechanical anisotropy of the suspension. With an increasing fiber volume fraction, this plays, however, an important role. There are some models which count on this effect, they are, however, phenomenological and require a fitted model parameter. In this paper, a micromechanically based constitutive law is proposed which considers the flow induced anisotropic viscosity of the fiber suspension. The introduced viscosity tensor can handle arbitrary anisotropy of the fluid‐fiber mixture depending on the actual fiber orientation distribution. 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 by Jeffery's equation. A numerical implementation of the introduced model is applied to representative flow modes. The calculated stress values are analyzed in transient and stationary flow cases. They show a less pronounced anisotropic viscous behaviour in every investigated case compared to the results obtained by the use of the Dinh‐Armstrong constitutive law. The reason for the qualitative difference is that the presented model depends on the complete orientation information, while the other one is linear in the fourth‐order fiber orientation tensor. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thermal fiber orientation tensors for digital paper physics

We estimate the orientation of wood fibers in porous networks like paper, paperboard or fiberboard by computing digital thermal conductivity experiments on micro-computed tomography (μCT) images with artificial isotropic thermal conductivity parameters.The accuracy of mechanical and thermal constitutive models for porous wood fiber based materials crucially depends on knowing the wood fiber orientation. Unfortunately, due to the high porosity, the micro-heterogeneity of wood fibers, the high carbon content of organic materials and the unknown additives present in industrial paper, μCT-scans often exhibit low contrast and strong artifacts. Conventional image processing approaches encounter difficulties, as they rely upon convex fiber cross sections.We propose a solution by circumventing the segmentation of single wood fibers in μCT images, by performing thermal conductivity simulations on binarized wood fiber structures, where an artificial isotropic thermal conductivity is associated to the fibers and the pore space is considered as isolating. The local and global temperature fluxes are assembled into a fiber orientation tensor. This method overcomes the limitations of the mentioned local image processing approaches, as individual fibers need not be resolved and convergence for increasing resolution is a consequence of abstract mathematical theory.We use our novel method to analyze large three-dimensional μCT-scans and a synchrotron scan of a paperboard sample, serving as the starting point of an accurate micromechanical modeling of the effective anisotropic mechanical behavior of paper and paperboard. These results are crucial for calculating the mechanical strength of deep-drawn paperboard, which will be accomplished in a subsequent article.

Effects of extrudate swell and nozzle geometry on fiber orientation in Fused Filament Fabrication nozzle flow

The viability of Fused Filament Fabrication (FFF) as an Additive Manufacturing (AM) method is improved by including discrete carbon fibers in the polymer feedstock. The resulting carbon fiber polymer composite enhances mechanical and thermal properties, where improvements are directly related to the orientation of the carbon fibers within the extruded polymer composite. Fiber orientation in the polymer melt is affected by the flow field which is defined by the melt flow geometry including flow in the nozzle convergence and extrudate swell. This paper presents a computational approach for modeling the polymer melt flow and the associated effects of nozzle geometry and extrudate swell on fiber orientation in the extruded polymer. The finite element method is used to evaluate Stokes flow in the axisymmetric flow field of an FFF nozzle. Fiber orientation tensors are calculated throughout the fluid domain using the Orthotropic Fitted Closure and the Folgar-Tucker isotropic fiber interaction model. The radial and axial modulus of the extruded polymer is then calculated from fiber orientation tensors within the composite at the nozzle exit. Fiber orientation along the extrudate axis is shown to increase due to elongation flow within the nozzle convergence zone and decrease in the extrudate swell such that the modulus of elasticity of the extruded polymer is moderately affected by extrudate swell and substantially affected by the nozzle geometry.

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.

Simulation study for prediction of short Fiber Orientation reinforced thermoplastics

The purpose of this paper is the simulation image processing of thermoplastic reinforced with short fibers with an established code in Matlab for the determination of the sizes, orientation, and the ambiguity of the fibers. Which gives a general idea about what happens before crystallization in the mold.Generally, the simulation of short 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.

Coupling of Mold Flow Simulations with Two-Scale Structural Mechanical Simulations for Long Fiber Reinforced Thermoplastics

The entire simulation process for long fiber reinforced thermoplastics is examined to determine the effective mechanical properties which are influenced by the microstructural fiber orientation state. Therefore, flow and fiber orientation simulations are conducted and the obtained fiber orientation tensors are used in two-scale structural simulations. The fiber orientation distributions as well as the mechanical properties are compared with micro-computed tomography data and results from threepoint bending tests performed by dynamical mechanical analysis (DMA), respectively. The validated results show that prediction of the essential mechanical properties is possible with the applied combinated methods and that the knowledge of the fiber orientation and its gradients is of crucial importance for the entire simulation process.

Two-scale structural mechanical modeling of long fiber reinforced thermoplastics

Abstract The mechanical properties of long fiber reinforced thermoplastics (LFT), which highly depend on the fiber orientation induced through manufacturing on a direct LFT line, are predicted for compression molded rectangular plates. Therefore, three two-scale structural mechanical simulation schemes are applied and discussed: a two-step approach, the Mori-Tanaka scheme and the self-consistent method. Fiber orientation tensors based on measured micro computed tomography data of selected samples as well as on filling simulations are used for the determination of mechanical properties, as e.g. the storage modulus. The results have been compared with dynamic mechanical analysis measurements of tensile specimens. The influence of the initial strand position on the effective mechanical properties of the plate and the variation of those are examined.

An Approach for Predicting Fiber Orientation Distribution in Plastic Injection Molding of Composites

Aims: This paper presents an approach for predicting the fiber or ientation distribution in plastic injection composites. Methodology: By introducing fiber orientation tensor, the average distribution of fibers in the composite is acquired through injection molding simulation with Autodesk Moldflow Simulation Insight ver sion 2014 . Quantitative fiber orientation distribution along thickness at specific spot is obtained by image analysis on microscopic photos of samples. The proposed approach is demonstrated by a car instrument panel made from injection molding of thermopla stics with glass fibers. Tensile test samples are cut from the instrument panel with an angle of 0 o , 45 o , 90 o to the fiber orientation at sites where fibers are highly aligned for measuring tensile modulus. The fiber orientation distribution and its effect on the tensile modulus are investigated. Results: It is shown that the car instrument panel is heterogeneous in mechanical properties. The fibers are roughly distributed into three layers along the thickness with each layer having roughly

Numerical Simulation of Injection Molding using OpenFOAM

AbstractWe show how to use and extend OpenFOAM's incompressible two‐phase flow solvers for the simulation of injection molding with short fiber reinforced thermoplastics in a laminar flow regime. Second order fiber orientation tensors are computed using the Folgar‐Tucker equation (FTE) with quadratic closure. The FTE is coupled to the viscosity‐term of the Navier‐Stokes equations for the non‐Newtonian flow in a segregated manner. Phase dependent boundary conditions are implemented to simulate wall heat transfer, stickiness of the melt to the wall and to prevent air‐traps close to the wall. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)