Fiber Orientation Tensor Research Articles

Homogenisation of the Local Thermal Conductivity in Injection-Moulded Short Fibre Reinforced Composites.

This paper deals with predicting the effective thermal conductivity (ETC) of injection-moulded short fibre reinforced polymers (SFRPs) using two different homogenisation schemes: a scheme based on the dielectric theory for pseudo-oriented inclusions and a two-step homogenisation model based on the mean-field homogenisation approach. In both cases, the fibre orientation tensor (FOT) obtained from Autodesk Moldflow® simulation was used. The Moldflow FOT predictions were validated via structure tensor analysis of micro-computed X-ray tomography (micro-CT) scans of the part. In the dielectric-wise approach, the orientation of fibres was originally defined by a scalar parameter, which is related to the diagonal components of the FOT. In the two-step homogenisation approach, an interpolative model based on the Mori–Tanaka theory is used in the first step for calculating the ETC for the ideal case of unidirectional fibre alignment, followed by a second step in which orientation averaging based on the FOT inside each element is applied. The ETC was calculated using both schemes for the specific case of uniform fibre orientation distribution and at three different locations with non-identical FOTs of an injection-moulded SFRP part. The results are compared with each other and evaluated against the direct numerical simulation for the uniform fibre orientation and experimental measurements for the injection-moulded SFRP. This shows that while the two-step homogenisation can predict the ETC in the full range of orientations between the perfectly aligned and uniformly distributed fibres, the dielectric-wise approach is only capable of modelling the ETC when distributions are close to the two extreme ends of the orientation spectrum.

Fiber orientation distributions based on planar fiber orientation tensors of fourth order

Fiber orientation tensors represent averaged measures of fiber orientations inside a microstructure. Although, orientation-dependent material models are commonly used to describe the mechanical properties of representative microstructure, the influence of changing or differing microstructure on the material response is rarely investigated systematically for directional measures which are more precise than second-order fiber orientation tensors. For the special case of planar orientation distributions, a set of admissible fiber orientation tensors of fourth-order is known. Fiber orientation distributions reconstructed from given orientation tensors are of interest both for numerical averaging schemes in material models and visualization of the directional information itself. Focusing on the special case of planar orientations, this paper draws the geometric picture of fiber orientation distribution functions reconstructed from fourth-order fiber orientation tensors. The developed methodology can be adopted to study the dependence of material models on planar fourth-order fiber orientation tensors. Within the set of admissible fiber orientation tensors, a subset of distinct tensors is identified. Advantages and disadvantages of the description of planar orientation states in two- or three-dimensional tensor frameworks are demonstrated. Reconstruction of fiber orientation distributions is performed by truncated Fourier series and additionally by deploying a maximum entropy method. The combination of the set of admissible and distinct fiber orientation tensors and reconstruction methods leads to the variety of reconstructed fiber orientation distributions. This variety is visualized by arrangements of polar plots within the parameter space of fiber orientation tensors. This visualization shows the influence of averaged orientation measures on reconstructed orientation distributions and can be used to study any simulation method or quantity which is defined as a function of planar fourth-order fiber orientation tensors.

On the dependence of orientation averaging mean field homogenization on planar fourth-order fiber orientation tensors

A comprehensive study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffnesses predicted by orientation averaging mean field homogenization is given. Fiber orientation states of sheet molding compound (SMC) are identified to be in most cases approximately planar. In the planar case, all possible fourth-order fiber orientation tensors are given by a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors. This set defines a three-dimensional body and forms the basis for a comprehensive study on the influence of a fiber orientation distribution in terms of a fourth-order tensor on homogenized stiffnesses. The methodology of this study is the main contribution of this work and can be adopted to analyze the orientation dependence of any quantity which is a function of a planar fourth-order fiber orientation tensor. At specific points inside the set of planar fiber orientation tensors, effective stiffnesses are calculated with selected mean field homogenization schemes. These schemes are based on orientation averaging of transversely isotropic elasticity tensors following Advani and Tucker (1987), which is explicitly recast as linear invariant composition in the fiber orientation tensors of second and fourth order of Kanatani third kind. A maximum entropy reconstruction of a fiber orientation distribution function based on leading fiber orientation tensors, enables a new numerical formulation of the Advani and Tucker average for the special planar case. Polar plots of Young’s modulus and generalized bulk modulus obtained by selected homogenization schemes are arranged on two-dimensional slices within the body of admissible fiber orientation tensors, visualizing the influence of the orientation tensor on the stiffness tensor. The orientation-dependence of the generalized bulk modulus differs significantly between selected homogenizations. Restrictions on the effective anisotropic material response caused by orthotropy of closure approximations are discussed.

Analysis of Fiber Orientation and a Simulation of Damage Fracture Prediction Using an X-ray Phase Imaging Method for Discontinuous Carbon Fiber Random Laminated Molded Products

In chopped carbon fiber tape reinforced thermoplastic（CTT）material, the anisotropic region of fiber orientation after press molding often causes the degradation of mechanical properties and damage and fracture. This study explores means of obtaining the fiber shape and fiber orientation tensors from molded products and then predicting the CTT mechanical properties and damage fracture locations. Extensive fiber shape data obtained by X-ray CT image processing are analyzed using a simple method to create a fiber waviness model.The fiber orientation tensor of the molded product is obtained through X-ray phase imaging. The material properties of a finite element model with CTT are calculated and mapped from the fiber waviness model and orientation tensor, and damage fracture prediction is simulated. Results showed that the fracture positions of both the test and simulation agreed in specimens having large anisotropic regions. However, the specimens with small anisotropic regions did not match. Therefore, the images of strain distribution of the test piece under a tensile load were compared by cross-correlation. Results showed a large correlation between the tensile test and simulation results, thus confirmed the effectiveness of the simulation.

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.

Development and verification of mechanical characteristics of a composite material made of a thermoplastic matrix and short glass fibers

The paper presents the results of computer modeling and prediction of the mechanical properties of composite materials with a polycarbonate matrix filled with short glass inclusions. At the micro-level, the influence of the volume of inclusions on the mechanical properties of the designed composite based on polycarbonate matrix is studied in the DIGIMAT (France) program. It was found that with a ratio of the sizes of inclusions in the range of 468÷60, the particles have a needle shape, and the material with such inclusions has a higher stress limit and elastic modulus than with a shape coefficient less than 50. The components of the fiber orientation tensor were also determined, at which the values of computer modeling are in good agreement with experimental data. The influence of the size of the finite element grid on the characteristics of the composite at the macro level was studied, and recommendations were given for choosing the size of the face of the finite element. The adequacy of computer models was confirmed by the results of field tests. The paper presents the results of testing flat samples made by injection molding technology. Mechanical tests were carried out for three variants of samples made of composite material based on a polycarbonate matrix with 10 %, 20 % and 30 % inclusions. The discrepancy between the experimental and computer results for samples with 10 %, 20 % content of short chopped fibers is explained by the influence of technological factors on the properties of the material at the macro-level. The conducted research allowed us to develop a computer modeling technique used at the stage of development of polymer composites based on thermoplastic matrices with short glass inclusions

Multi-scale analysis on the tensile properties of UHPC considering fiber orientation

The effect of fiber orientation distribution on the multi-scale tensile properties of ultra-high performance concrete (UHPC) was investigated. Firstly, the orientation tensor description used to characterize and calculate the orientation properties of non-aligned short fiber composites were analyzed. The tensile tests of UHPC with four different fiber contents were carried out. On the basis of analyzing the characteristics of the tensile fracture, the fiber orientation characteristics near and far from the fracture surface were investigated by the image processing. Then, the orientation distribution function (ODF) of non-aligned inclusions was reconstructed by using the hybrid closure approximation method. The feasibility of the method for UHPC was verified by comparing the reconstructed ODF with the experimental ODF. Finally, the elastic properties of UHPC considering the fiber orientation were calculated by the two-step mean field homogenization method. On this basis, combined with Puck failure criterion, the tensile strength of UHPC with different fiber content and orientation tensor was predicted.

Artificial intelligence approach for increasing the fidelity of the second order fibre orientation tensor for use in finite element analysis

A more efficient integrated computational approach for the design of injection and compression moulded composites which maps the results of manufacturing simulations as input to structural simulations is proposed. This methodology overcomes the limitations caused by insufficient detail provided by the second order fibre orientation tensor (FOT) output from manufacturing simulations which are needed in advanced structural finite element models that incorporate damage and failure. An artificial neural network (ANN) framework is proposed to predict a fibre orientation distribution function (ODF) from the FOT. The ANN contains the equation for calculating the FOT from the ODF such that the ODF is optimized to reproduce the input FOT. It is shown that the proposed framework is nearly as accurate as using experimentally measured ODF for predicting the stress strain response and failure of a compression moulded composite.

Asymptotic fiber orientation states of the quadratically closed Folgar–Tucker equation and a subsequent closure improvement

Anisotropic fiber-reinforced composites are used in lightweight construction, which is of great industrial relevance. During mold filling of fiber suspensions, the microstructural evolution of the local fiber arrangement and orientation distribution is determined by the local velocity gradient. Based on the Folgar–Tucker equation, which describes the evolution of the second-order fiber orientation tensor in terms of the velocity gradient, the present study addresses selected states of deformation rates that can locally occur in complex flow fields. For such homogeneous flows, exact solutions for the asymptotic fiber orientation states are derived and discussed based on the quadratic closure. In contrast to the existing literature, the derived exact solutions take into account the fiber-fiber interaction. The analysis of the asymptotic solutions relying upon the common quadratic closure shows disadvantages with respect to the predicted material symmetry, namely, the anisotropy is overestimated for strong fiber-fiber interaction. This motivates us to suggest a novel normalized fully symmetric quadratic closure. Two versions of this new closure are derived regarding the prediction of anisotropic properties and the fiber orientation evolution. The fiber orientation states determined with the new closure approach show an improved prediction of anisotropy in both effective viscous and elastic composite behaviors. In addition, the symmetrized quadratic closure has a simple structure that reduces the effort in numerical implementation compared to more elaborated closure schemes.

Micromechanical Modeling of Anisotropy and Strain Rate Dependence of Short-Fiber-Reinforced Thermoplastics

The anisotropy and strain rate dependence of the mechanical response of short-fiber-reinforced thermoplastics was studied using a straightforward micromechanical finite element analysis of representative volume elements (RVEs). RVEs are created based on the fiber orientation tensor, which quantifies the processing-induced fiber orientation distribution. The matrix is described by a strain rate-dependent constitutive model (the Eindhoven glassy polymer (EGP) model), which accurately captures the intrinsic response of amorphous polymers. The micromechanical results indicate that the influence of strain rate and that of the loading direction on the yield stress are multiplicatively decouplable, which confirms previous experimental observations. Moreover, it is demonstrated that the yield stress, to a good approximation, can be directly linked to the fiber orientation in the direction of loading. This leads to a new relation that uniquely links the rate dependence of the yield stress to the fiber orientation in loading direction.

Theoretical approximation of hydrodynamic and fiber-fiber interaction forces for macroscopic simulations of polymer flow process with fiber orientation tensors

Flow processes of discontinuous fiber reinforced polymers (FRPs) are the essence of several polymer-based manufacturing processes. FRPs show a transient chemo-thermomechanical matrix behavior and fiber-induced anisotropic physical properties. Therefore, they are one of the most complex materials used in volume production. The general flow behavior is influenced by fibers and their interactions with the matrix and other fibers. The consideration of individual fibers is numerically not capable for process simulation of FRP parts. Therefore, orientation tensors are used in macroscopic simulations, leading to a loss of information about the fiber network. Within this work, novel approximation schemes are presented to determine hydrodynamic and fiber-fiber contact forces with information provided by the second order fiber orientation tensor. Approximation of these forces can henceforth facilitate fiber breakage modeling in macroscopic process simulations. The results are verified by numerical simulations with individual fibers of different orientation states and lengths, showing good agreement with the verification results.

Numerical investigation on the influence of fiber orientation mapping procedure to the mechanical response of short-fiber reinforced composites using Moldflow, Digimat and Ansys software

Within this paper, a numerical study was performed, aiming to assess the influence of fiber orientation on the mechanical properties of tensile specimens, by using the finite element method. The specimen, designed according to ISO-527-2, is type 1BA and consists of a reinforced polyamide with 30% glass fiber (PA66 GF30). The specimens were extracted from 2 mm thickness plates, for which the fiber orientation tensor was determined using the dedicated Moldflow software for injection molding. The composite material was modelled using Digimat mean-field homogenization tool. The mechanical evaluation was performed in Ansys finite element software by coupling it with Digimat and thus considering the anisotropic behavior of the material. Different mapping procedures and mesh sizes for the Moldflow model and Ansys model were considered.

Additive manufacturing of compliance optimized variable stiffness composites through short fiber alignment along curvilinear paths

Abstract Short fiber-reinforcement has been used to improve the structural performance of extrusion based 3D printed polymer materials. The converging nozzle preferentially orients the short fibers along the printing direction. By locally steering the print paths to orient the fiber directions, the benefit of the anisotropic fiber reinforcement can be optimized. In this study, a methodology is developed to design and manufacture 3D printed short fiber-reinforced composite materials with optimized fiber or print directions. The standard open-hole tension sample geometry was employed as a test case. The least squares & continuity constraints (LSC) method was used to design the locally varying fiber orientation of each ply/layer to optimize for minimum compliance. An algorithm was developed to convert optimum discrete fiber angles to continuous paths. Material properties were estimated based the measured the fiber orientation tensor in conjunction with the Halpin–Tsai homogenization method. Strain measurements on open-hole tensile specimens by digital image correlation (DIC) were in good agreement with finite element model predictions. Despite some defects at path drop offs, LSC optimized samples showed improved stiffness and strength over zero degree orientation samples. In addition, samples prepared with print paths following streamlines for potential flow over a cylinder yielded higher strength and stiffness.

Experimental and computational analysis of structure-property relationship in carbon fiber reinforced polymer composites fabricated by selective laser sintering

A fundamental understanding of structure-property relationship of carbon fiber reinforced polymer (CFRP) composites fabricated by selective laser sintering (SLS) is essential to provide guidance to improve the mechanical properties in this promising additive manufacturing process. In the present study, we propose a new computational framework to evaluate the mechanical behavior of SLSed CFRP based on the representative volume element (RVE) model. Firstly, a new voxel-based fiber packing algorithm is proposed with the three-dimensional fiber orientation tensor to describe the fiber distribution. The porosity generated in the manufacturing process is also considered by adding spherical voids in the matrix region. In order to remove the artificial stress concentration induced by the voxel-based algorithm, a python script is developed to regenerate the geometries of different phases with conforming mesh. Meanwhile, the constitutive models of different phases in SLSed CFRP, i.e., matrix, fiber and interface, are established according to the material characteristics. Next, the proposed computational framework of SLSed CFRP is validated by the X-ray computed tomography (XCT) measurements and tensile tests of a representative SLSed carbon fiber/polyamide 12 (CF/PA12) composite. The effects of fiber volume fraction and fiber orientation distribution on the mechanical behavior of SLSed CF/PA12 composite are further quantitatively explored and ranked with regards to their influence on stiffness and failure strength.

Prediction of injection molding parameters for symmetric spur gear.

Polymer gears pose major advantages, like noiseless operation, resistive against corrosion, low weight, ability to damp vibrations, ease of manufacturability, and ability to operate without lubrication like in printers, household appliances, etc. In order to enhance mechanical properties of gear materials, various reinforcing materials are added such as glass and carbon fibers. The orientation of these fibers and distribution are critical parameters at the microstructural level for polymer reinforced with short fibers, which defines the strength and life of gears. The geometric accuracy and precision of molded gears are improved by the injection molding technique. The fiber orientation prediction is a new and novel aspect for high performance and life, as these injection-molded gears have complex patterns of fiber orientation. This also affects material properties such as elastic modulus, strength, and gear geometrical dimensional properties shrinkage and warpage. In this present work, an attempt is made to develop 3D symmetric spur gear tooth geometry using Autodesk Fusion 360. The injection molding parameters such as fiber orientation tensor, volumetric shrinkage at ejection, weld lines, deflection, and confidence in filling are studied for modeled gear having symmetric teeth profile for unreinforced and 20%, 30%, 40%, 50%, and 60% glass fiber-reinforced nylon 6/6 (PA66) by using Autodesk Moldflow Adviser 2017. The result obtained from mold simulation tool indicates that fiber orientation tensor for varying glass fiber contents was close to unity. The volumetric shrinkage considerably reduced from unreinforced PA66 to glass-reinforced PA66.Graphical abstract.

Effective elastic modulus of foamed long glass fiber reinforced polypropylene: Experiment and computation

The cross-section of an injection-molded plate of foamed long glass fiber reinforced polypropylene was analyzed using scanning electron microscopy. The distribution of the glass fiber orientations and the microcellular structure in the thickness direction were also studied. A multilayer representative volume element was constructed based on the fiber orientation tensors and the cell distribution. Nested and two-step homogenization methods based on the Mori–Tanaka and Voigt models were used to homogenize each layer of the representative volume element. Finally, classic laminate theory was used to obtain the effective elastic modulus of the material. The computed elastic moduli of the single-layer and multilayer representative volume element models with different loading directions predicted by the homogenization and finite element methods were compared with the experimental results. We found that the constructed multilayer representative volume element model can predict the elastic moduli of the foamed glass fiber reinforced polypropylene effectively and that the predicted results were accurate and stable.

CFD Analysis of Turbulent Fibre Suspension Flow

A new model for turbulent fibre suspension flow is proposed by introducing a model for the fibre orientation distribution function (ODF). The coupling between suspended fibres and the fluid momentum is then introduced through the second and fourth order fibre orientation tensors, respectively. From the modelled ODF, a method to construct explicit expressions for the components of the orientation tensors as functions of the flow field is derived. The implementation of the method provides a fibre model that includes the anisotropic detail of the stresses introduced due to presence of the fibres, while being significantly cheaper than solving the transport of the ODF and computing the orientation tensors from numerical integration in each iteration. The model was validated and trimmed using experimental data from flow over a backwards facing step. The model was then further validated with experimental data from a turbulent fibre suspension channel flow. Simulations were also carried out using a Bingham viscoplastic fluid model for comparison. The ODF model and the Bingham model performed reasonably well for the turbulent flow areas, and the latter model showed to be slightly better given the parameter settings tested in the present study. The ODF model may have good potential, but more rigorous study is needed to fully evaluate the model.

Computational micromechanics model based failure criteria for chopped carbon fiber sheet molding compound composites

Chopped carbon fiber sheet molding compound has a great potential in lightweight automotive, marine, and aerospace applications. One of the most challenging tasks is to predict the failure strength of the material due to its anisotropy and heterogeneity, as well as complex stress states in real-world working conditions. In this paper, a novel constitutive model of carbon fiber chip is proposed to capture the pre- and post-failure behaviors under different loading modes. On this basis, we propose a new computational micromechanics model, which is calibrated and validated by uniaxial tensile, compressive, and in-plane shear experiments. Furthermore, a set of microstructures representative volume element (RVE) models under complex loading conditions are reconstructed to understand the relationship between the microstructure characteristics and the failure envelopes. Finally, several modified versions of classical failure criteria are proposed for anisotropic materials with consideration of the fiber orientation tensor. The modified Tsai-Wu failure criterion, which shows the best accuracy among all failure criteria, is highlighted in the comparative study.

Selective Laser Sintering of PA6: Effect of Powder Recoating on Fibre Orientation

In Selective Laser Sintering, fibres are strongly orientated during the powder recoating process. This effect leads to an additional increase of anisotropy in the final printed parts. This study investigates the influence of process parameter variation on the mechanical properties and the fibre orientation. A full factorial design of experiment was created to evaluate the processing parameters of recoating speed, layer thickness and laser power on the part’s modulus of elasticity. Based on the mechanical testing, computed tomography was applied to selected samples to investigate the process-induced fibre microstructure, and calculate the fibre orientation tensors. The results show increasing part stiffness in the deposition direction, with decreasing layer thickness and increasing laser power, while the recoating speed only shows little effect on the mechanical performance. This complies with computed tomography imaging results, which show an increase in fibre orientation with smaller layer thickness. With thinner layers, and hence smaller shear gaps, shear stresses induced by the roller during recoating increase significantly, leading to excessive fibre reorientation and alignment. The high level of fibre alignment implies an increase of strength and stiffness in the recoating direction. In addition, thinner layer thickness under constant laser energy density results in improved melting behaviour, and thus improved fibre consolidation, consequently further increasing the mechanical properties. Meanwhile, the parameters of recoating speed and laser power do not have a significant impact on fibre orientation within their applicable process windows.

Integrated Prediction of Mechanical Behavior for the Non-Aligned Fiber Composites With Experimental Validation

Abstract This paper aims to present an integrated multi-scale method for predicting the anisotropic and nonlinear elasto-plastic behavior of short glass fiber-reinforced polymer (GFRP) materials typically produced by injection molding. The proposed method combines injection molding and microstructure together, with considering the nonaligned fibers and their corresponding anisotropy, to semi-analytically estimate the local effective mechanical properties at every point of GFRP. Micro-computed tomography measurement and injection molding simulation are used to obtain the fiber orientation tensor. The two-step mean-field homogenization method is applied to calculate the mechanical behaviors of the PA66GF30 GFRP with distributed-orientation fibers based on the fiber orientation tensor. Reverse engineering is used to obtain the optimized parameters of J2-plasticity and Tsai-Hill three-dimensional transversely isotropic stain-based failure criterion. Moreover, the integral mapping method can complete the transformation of the fiber orientation tensor from injection simulation to structure simulation model. The proposed integrated approach with the optimized parameters is verified by predicting the ring samples’ behavior from injection plates. The results from this investigation are expected to provide some design guidelines for GFRP composites.