Multiscale modeling of unsaturated flow in dual-scale fiber preforms of liquid composite molding III: reactive flows

The woven, stitched or braided fabrics used in liquid composite molding (LCM) display partial saturation behind moving flow-front in an LCM mold which is caused by delayed impregnation of fiber tows. In this part 3 of the present series of three papers, a novel multiscale approach proposed in parts 1 and 2 [1] and [2] is adapted for modeling the unsaturated flow observed in the dual-scale fabrics of LCM under non-isothermal, reactive conditions. The volume-averaged species or resin cure equation, in conjunction with volume-averaged mass, momentum and energy (temperature) equations, is employed to model the reactive resin flow in the inter-tow (gap) and intra-tow (tow) regions with coupling expressed through several sink and source terms in the governing equations. A coarse global-mesh is used to solve the global (gap) flow over the entire domain, and a fine local mesh in form of the unit-cell of periodic fabrics is employed to solve the local (tow) flows. The multiscale algorithm based on the hierarchical computational grids is then extended to solve the dual-scale flow under reactive conditions. The simulation is compared with a two-color experiment and a previously published two-layer model. Significant differences between the temperatures and cures of the gap and tow regions of the dual-scale porous medium are observed. The ratio of pore volumes in the tow and gap regions, the effective thermal conductivity in the tows, and the reaction rate are identified as the important parameters for temperature and cure distributions in the gap and tow regions.     

Multiscale modeling of unsaturated flow in dual-scale fiber preforms of liquid composite molding I: Isothermal flows

Mold-filling simulation of unsaturated flows in LCM is important for optimizing mold design quickly and cost-effectively in the virtual space. For the first time, a true multiscale approach is developed for simulating the unsaturated flow under isothermal conditions in the dual-scale fiber-mats of RTM. To solve the coupled macro-micro equation-set, a coarse global mesh is used to solve the global flow equations over the entire domain while fine local meshes in form of the periodic unit-cells of fabrics are employed to solve the local tow-impregnation process. A multiscale algorithm based on hierarchical computational grids has been proposed to simulate the unsaturated flow in the dual-scale fiber mats under isothermal conditions. The predictions are compared with measurements for a 1-D flow experiment which indicates that the proposed approach can be used to simulate the unsaturated flow accurately through dual-scale fiber mats in LCM without the use of any fitting parameters.     

Modelling microscopic flow in woven fabric reinforcements and its application in dual-scale resin infusion modelling

A physical unit cell impregnation model is proposed for the micro-scale flow in plain woven reinforcements. The modelling results show a characteristic relationship between tow impregnation speed, the surrounding local macro-scale resin pressure and the tow saturation within the unit cell. This relationship has been formulated into a mathematical algorithm which can be directly incorporated into a continuum dual-scale model to predict the ‘sink’ term. The results using the dual-scale model show a sharp resin front in inter-tow-pore spaces and a partially saturated front region in intra-tow-pore spaces. This demonstrates that the impregnation of fibre tows lags behind the resin front in the macro pore spaces. The modelling results are in agreement with two reported experimental observations. It has been shown that the unsaturated region at the flow front could increase or have a fixed length under different circumstances. These differences are due to the variation in tow impregnation speed (or the time required for the tow to become fully impregnated), the weave architecture and the nesting and packing of plies. The modelling results have also demonstrated the drooping of the inlet pressure when flow is carried out under constant injection rates. The implementation of the algorithm into a dual-scale model shows coherence with a single-scale unsaturated model, but demonstrates an advantage in flexibility, precision and convenience in application.     

Characterising and modelling variability of tow orientation in engineering fabrics and textile composites

Variability of tow orientation is unavoidable for biaxial engineering fabrics and their composites. Since the mechanical behaviour of these materials is strongly dependent on the fibre direction, variability should be considered and modelled as exactly as possible for more realistic estimation of their forming and infusion behaviour and their final composite mechanical properties. In this study, a numerical code, ‘VariFab’, has been written to model realistic full-field variability of the tow directions across flat sheets of biaxial engineering fabrics and woven textile composites. The algorithm is based on pin-jointed net kinematics and can produce a mesh of arbitrary perimeter shape, suitable for subsequent computational analysis such as finite element forming simulations. While the shear angle in each element is varied, the side-length of all unit cells within the mesh is constant. This simplification ensures that spurious tensile stresses are not generated during deformation of the mesh during forming simulations. Variability is controlled using six parameters that can take on arbitrary values within certain ranges, allowing flexibility in mesh generation. The distribution of tow angles within a pre-consolidated glass–polypropylene composite and self-reinforced polypropylene and glass fabrics has been characterised over various length scales. Reproduction of the same statistical variability of tow orientation as in these experiments is successfully achieved by combining the VariFab code with a simple genetic algorithm.     

Flow of non-Newtonian liquid polymers through deformed composites reinforcements

The flow of non-Newtonian liquid polymers through fibrous reinforcements is a phenomenon which is often encountered during polymer composites manufacturing. In a previous work, we have proposed from a multiscale theoretical approach a method to model this phenomenon when the polymer can be regarded as a generalised Newtonian fluid [Orgéas et al. J. Non-Newtonian Fluid Mech. 2007; 145]. In this paper, the capability of the method is tested with power-law fluids flowing through deformed plain weave fabrics. For that purpose, the flow problem is firstly analysed at the mesoscale from numerical simulations performed on representative elementary volumes of the fabrics. The influences of both the current deformation of the fabrics and the fluid rheology on the macroscopic flow law are emphasised. Secondly, it is shown that the proposed method allows a nice fit of numerical results.     

双尺度纤维织物二维非饱和流动的数值模拟与实验

液体模塑成型工艺(LCM)中非饱和流动的填充模拟对于在虚拟空间中快速、高效地优化工艺参数具有重要意义。采用了一种模拟双尺度纤维织物在等温条件下非饱和流动的双尺度计算模型,通过引入沉浸函数求解宏观-微观流动控制方程组,同时考虑了在微观浸渍中毛细压力的影响,在有限元/控制体积算法中实现了对非饱和流动的数值模拟。随后对三向缝合纤维织物进行了二维径向填充实验,将实验结果与数值模拟的预测值对比。结果表明,该计算模型可以较精确地模拟双尺度纤维织物中的非饱和流动。在此计算模型的基础上,讨论了流体黏度、注射流量及纤维束孔隙率对非饱和填充浸润的影响。结果表明,不同流体黏度、注射流量及纤维束孔隙率对纤维织物填充过程中非饱和区域长度、入口压力曲线及填充时间影响不同。研究结果可以对合理预测纤维织物的浸润及树脂填充过程中入口压力提供指导。     

The impact of air evacuation on the impregnation time of Out-of-Autoclave prepregs

Most Out-of-Autoclave prepregs (OoA) are only partially impregnated with resin. Their impregnation completes during the cure cycle, solely driven by the difference between atmospheric and vacuum pressure. Increased part length leads to an impregnation time gradient caused by the transient air flow inside the fibrous medium. In this work, a novel numerical approach capable of predicting the local impregnation time of a fibrous domain with resin, at isothermal conditions, under the influence of transient air flow, is proposed (delayed air evacuation). Sensitivity studies prove the robustness of the numerical scheme, for a large range of flow time-scales. The same approach is used to predict the local impregnation time of a commercial OoA prepreg tow, for a wide range of part lengths. It is demonstrated that for manufacturing long parts OoA, accurately capturing the influence of the air pressure on the local impregnation state of the tow, is important for quantifying the risk for residual tow porosity.     

恒压下树脂在双尺度多孔介质中的流动特性

基于复合材料液态模塑(LCM)工艺过程中存在半饱和区域的实验现象以及对预制体双尺度效应的逐步认识, 一些学者提出用沉浸模型来研究双尺度多孔介质的不饱和流动。通过体积均匀化方法描述了双尺度多孔介质复合材料液态模塑工艺模型的特征, 得到含有沉浸项的双尺度多孔介质的质量守恒方程, 并采用有限元法对方程进行数值求解, 通过具体算例计算了考虑双尺度效应时恒压树脂注射下不同时段的压力分布状态, 得到树脂在填充过程中流动前沿半饱和区域从出现到消失的过程, 采用不同注射压力进行模拟并比较。结果表明, 与单尺度多孔介质模型不同, 双尺度多孔介质模型更能反映实际树脂填充过程中出现的半饱和区域现象。     

Stress analysis of multi-phase and multi-layer plain weave composite structure using global/local approach

Detailed stress analyses of multi-phase and multi-layer (MPML) composite structures are computationally challenging due to the complexities of the microstructure. In this study, an effective bottom-up global/local analysis strategy is employed to determine local stresses in the MPML plain weave composite structures. On the basis of the finite element analysis, the procedure is carried out sequentially from the homogenized composite structure of the macro-scale to the parameterized detailed fiber tow model of the micro-scale. The bridge between two scales is realized by mapping the global analysis result as the boundary conditions of the local tow model and hence the influence of the global model refinement on the computing accuracy of local stress is particularly addressed. To verify the computing results of such a bottom-up global/local analysis, we use a refined finite element mesh of the MPML structure whose solution is considered as the standard of comparison. The proposed approach is finally applied to the MPML plain weave composite panel.     

The Sink Term to Different Kinds of Fibrous Mat During the Unsaturated Filling Process

The fibrous pre-form of resin transfer molding is a dual-scale porous medium with two distinct scales of pores, i.e., pores in intra- and inter-tow, which produce an unsaturated infiltration phenomenon during filling. A sink term representing the delayed flow rate from the inter-tow gap into the intra-tow one is introduced to establish governing equations. This study mainly analyzes the sink term by tow saturation during the microscopic flow. First, fiber-tow permeability is calculated by FLOTRAN of ANSYS, Second, periodic unit cells are built according to different structures, and the concrete expression of the sink term is indirectly obtained through the numerical simulation and date fitting of tow saturation under different pressure and viscosity conditions. Results indicate that: the FLOTRAN module can be used to calculate the permeability of fiber tow in two directions; Moreover, the filling time and infiltration process for diverse unit cells with the same volume fraction are different; under the same injection condition, different unit cells have different parameters for the sink term.     

Inter-ply stitching optimisation of highly drapeable multi-ply preforms

An efficient finite element model has been developed in Abaqus/Explicit to solve highly non-linear fabric forming problems, using a non-orthogonal constitutive relation and membrane elements to model bi-axial fabrics. 1D cable-spring elements have been defined to model localised inter-ply stitch-bonds, introduced to facilitate automated handling of multi-ply preforms. Forming simulation results indicate that stitch placement cannot be optimised intuitively to avoid forming defects. A genetic algorithm has been developed to optimise the stitch pattern, minimising shear deformation in multi-ply stitched preforms. The quality of the shear angle distribution has been assessed using a maximum value criterion (MAXVC) and a Weibull distribution quantile criterion (WBLQC). Both criteria are suitable for local stitch optimisation, producing acceptable solutions towards the global optimum. The convergence rate is higher for MAXVC, while WBLQC is more effective for finding a solution closer to the global optimum. The derived solutions show that optimised patterns of through-thickness stitches can improve the formability of multi-ply preforms compared with an unstitched reference case, as strain re-distribution homogenises the shear angles in each ply.     

Influence of fabrics’ design parameters on the morphology and 3D permeability tensor of quasi-unidirectional non-crimp fabrics

This research addresses the effects of quasi-UD non-crimp fabric (NCF) design parameters on the fabric architecture and on the permeability tensor. These fabrics are designed for the Liquid Resin Infusion (LRI) of large and thick composite parts. Three fabrics’ parameters intended to bring a flow enhancement to the NCF are investigated: the stitch spacing, the stitch pattern and the weft tow lineal weight. Image analysis is undertaken to characterize the morphology of non-crimp fabric composite. A new continuous permeability measurement method based on compressive tests is proposed to relate the permeability of the quasi-UD NCF to the design parameters during the infusion process. The latter are proven to influence significantly both the fabric architecture and the permeability tensor coefficients.     

Modeling of heat transfer and unsaturated flow in woven fiber reinforcements during direct injection-pultrusion process of thermoplastic composites

This paper provides a methodology for the modeling of heat transfer and polymer flow during direct thermoplastic injection pultrusion process. Pultrusion was initially developed with thermosets which have low viscosity. But the impregnation becomes a critical point with thermoplastics which exhibit higher viscosity. There are very few reported works on direct thermoplastic impregnation with injection within the die. In addition, the rare studies have not adequately addressed the issue of unsaturated flow in woven fiber reinforcements. The solution proposed here, models the polymer flow through dual-scale porous media. A heat transfer model is coupled to a flow model enriched with a sink term. Specific changes of variables are made so as to model the steady state solution of unsaturation along a continuous process. The sink term, added to the continuity equation, represents the absorption rate of polymer by the bundles. Data were measured on a pultrusion line and micrographs confirmed the modeling strategy with an unsaturated flow approach. The flow modeling coupled to heat transfer of the thermoplastic pultrusion process aims at determining the saturation evolution through the die so as to manufacture pultruded profiles with the lowest residual porosity.     

Modelling the effect of material properties and process parameters on tow impregnation in out-of-autoclave prepregs

Out-of-autoclave prepregs based on woven fabrics initially consist of dry tows and resin-rich areas. The tows allow air evacuation in the initial stages of processing and are subsequently infiltrated by surrounding matrix. The following study analyzes the relationship between material properties, process parameters and tow impregnation for three OOA prepregs. First, a representative model for tow impregnation is developed. Then, the model parameters are determined and the model predictions are correlated to impregnation data measured by X-ray microtomography. Finally, the model is used in a parametric study to investigate the effect of fibre architecture, cure cycle temperature and resin initial degree of cure on tow impregnation rate and to predict the possible occurrence of flow-induced micro-voids.     

Mesh superposition-based multiscale stress analysis of composites using homogenization theory and re-localization technique considering fiber location variation

In this article, the accuracy of multiscale stress analysis of heterogeneous materials (unidirectional fiber-reinforced composite) was improved by considering microscopic geometrical variation using the mesh superposition method. When analyzing the stress distribution in a composite with the finite element method considering the variation of the fiber location, updating the mesh significantly is necessary; however, generating an appropriate mesh for a large geometrical variation is difficult. Therefore, we focused on the mesh superposition method, which can easily generate a numerical FE model, even if the internal structure is complex, because the matrix and inclusion can be expressed by the global mesh and local mesh independently. However, in the original mesh superposition method, the analysis accuracy may be degraded owing to the mesh overlap conditions. Therefore, an improved method was applied to the homogenization theory-based analysis. In this article, the effectiveness of the proposed approach was discussed by comparing the numerical results of this method with those of conventional mesh superposition method and standard finite element method. From the numerical results, accuracy improvement by the proposed approach for the multiscale stochastic stress analysis is confirmed.     

Governing equations for unsaturated flow through woven fiber mats. Part 1. Isothermal flows

Correct modeling of resin flow in liquid composite molding (LCM) processes is important for accurate simulation of the mold-filling process. Recent experiments indicate that the physics of resin flow in woven (also stitched or braided) fiber mats is very different from the flow in random fiber mats. The dual length-scale porous media created by the former leads to the formation of a sink term in the equation of continuity; such an equation in combination with the Darcy's law successfully replicate the drooping inlet pressure history, and the region of partial saturation behind the flow-front, for the woven mats. In this paper, the mathematically rigorous volume averaging method is adapted to derive the averaged form of mass and momentum balance equations for unsaturated flow in LCM. The two phases used in the volume averaging method are the dense bundle of fibers called tows, and the surrounding gap present in the woven fiber mats. Averaging the mass balance equation yields a macroscopic equation of continuity which is similar to the conventional continuity equation for a single-phase flow except for a negative sink term on the right-hand side of the equation. This sink term is due to the delayed impregnation of fiber tows and is equal to the rate of liquid absorbed per unit volume. Similar averaging of the momentum balance equation is accomplished for the dual-scale porous medium. During the averaging process, the dynamic interaction of the gap flow with the tow walls is lumped together as the drag force. A representation theorem and dimensional analysis are used to replace this drag force with a linear function of an average of the relative velocity of the gap fluid with respect to the tow matrix for both the isotropic and anisotropic media. Averaging of the shear stress term of the Navier–Stokes equation gives rise to a new quantity named the interfacial kinetic effects tensor which includes the effects of liquid absorption by the tows, and the presence of slip velocity on their surface. Though the gradient of the tensor contributes a finite force in the final momentum balance equation, a scaling analysis leads to its rejection in the fibrous dual-scale porous medium if the permeability of flow through the gaps is small. For such a porous medium, the momentum equation reduces to the Darcy's law for single-phase flow.     

A new computational model for entire pulse tube refrigerators: Model description and numerical validation

In present paper, a new modeling approach for the performance of pulse tube refrigerator is proposed. The new approach combines one-dimensional and two-dimensional models (1-D and 2-DCC model) together, and can be used to simulate the fluid flow and heat transfer processes of the basic type, orifice type and double-inlet type pulse tube refrigerators (PTRs). With the present model, the complicated fluid flow and heat transfer characteristics in the PTR system can be efficiently depicted and the computational time can be greatly reduced. Then based on the approach, the distribution characteristics of the flow and temperature fields of the three types of PTR are numerically analyzed. The complicated fluid flow and heat transfer phenomena in PTR, such as DC flow, velocity and temperature annular effects are presented vividly. The numerical results show that the 1-D and 2-DCC model is reliable and practical, which can be used to explore the physical mechanism of the thermodynamic processes of the PTR system and optimize the design of the PTR system and its components.     

Simulation of void formation in liquid composite molding processes

Air entrapment within and between fiber tows during preform permeation in liquid composite molding (LCM) processes leads to undesirable quality in the resulting composite material with defects such as discontinuous material properties, failure zones, and visual flaws. Essential to designing processing conditions for void-free filling is the development of an accurate prediction of local air entrapment locations as the resin permeates the preform. To this end, the study presents a numerical simulation of the infiltrating dual-scale resin flow through the actual architecture of plain weave fibrous preforms accounting for the capillary effects within the fiber bundles. The numerical simulations consider two-dimensional cross sections and full three-dimensional representations of the preform to investigate the relative size and location of entrapped voids for a wide range of flow, preform geometry, and resin material properties. Based on the studies, a generalized paradigm is presented for predicting the void content as a function of the Capillary and Reynolds numbers governing the materials and processing. Optimum conditions for minimizing air entrapment during processing are also presented and discussed.     

Numerical prediction of in-plane permeability for multilayer woven fabrics with manufacture-induced deformation

A unit cell based Computational Fluid Dynamics model is presented for predicting permeability of multilayer fabric structures. In Liquid Composites Moulding processes, fabric lay-ups undergo significant manufacture-induced deformation, combining compression, shear, and inter-layer nesting. Starting from the configuration of un-deformed fabric, the deformation is simulated geometrically by accounting for self-imposed kinematic constraints of interweaving yarns. The geometrical modelling approach is implemented in the open-source software TexGen. The permeability tensor is retrieved from flow analysis in ANSYS/CFX, based on TexGen voxel models. Using only measured geometrical data for un-deformed fabrics, deformed plain weave fabric and twill weave fabric lay-ups were modelled and their permeability tensors predicted. Comparison with experimental data demonstrates the generally good accuracy of predictions derived from the proposed numerical method.     

RTM工艺中单向纤维排列中的缺陷迁移的数值模拟

Decreasing the amount of residual voids during the resin infiltration into fibrous porous media is an important aspect in manufacturing high performance composite materials.In order to better understand void transports and flow behaviors in filling process,which affects immediately the final void content,a finite-element scheme for transient simulations of the void migration in a transverse flow through the uniaxial micro-structured fibrous media is developed in this work.A volume-of-fluid (VOF) method has been incorporated in the Eulerian frame to capture the free surface of the resin flow.The implementation of periodic boundary condition to the vertical direction avoids unwanted wall effect.The void migration in a dual-scale fiber tow model was investigated.The voids are observed to be transported through the inter-tow region as well as entrapped into fiber tow.It is that the motion of void lagged behind macro flow front which implies that the adequate resin bleeding after mold filling is crucial to remove the entrapped air.