Aerodynamic web forming: process simulation and material properties
- Simone Gramsch^{1},
- Axel Klar^{2},
- Günter Leugering^{3},
- Nicole Marheineke^{3}Email author,
- Christian Nessler^{2},
- Christoph Strohmeyer^{3} and
- Raimund Wegener^{1}
https://doi.org/10.1186/s13362-016-0034-4
© The Author(s) 2016
Received: 1 September 2016
Accepted: 18 November 2016
Published: 1 December 2016
Abstract
In this paper we present a chain of mathematical models that enables the numerical simulation of the airlay process and the investigation of the resulting nonwoven material by means of virtual tensile strength tests. The models range from a highly turbulent dilute fiber suspension flow to stochastic surrogates for fiber lay-down and web formation and further to Cosserat networks with effective material laws. Crucial is the consistent mathematical mapping between the parameters of the process and the material. We illustrate the applicability of the model chain for an industrial scenario, regarding data from computer tomography and experiments. By this proof of concept we show the feasibility of future simulation-based process design and material optimization which are long-term objectives in the technical textile industry.
Keywords
airlay process nonwoven material virtual tensile strength test fiber suspension flow fiber lay-down fiber networks model chain stochastic surrogates homogenization effective material lawsMSC
65Mxx 74Hxx 74Kxx 74Q15 76T201 Introduction
The aerodynamic web forming is a multi-scale two-phase problem whose monolithic handling and direct simulation based on a model of first principles are not possible due to its high complexity. So far, no simulation results exist in literature. In this paper, we establish a consistent, accurate and efficiently evaluable chain of mathematical models towards the simulation of the airlay process and furthermore the investigation of the material behavior. The models cover the dilute fiber suspension with elastic slender bodies in the turbulent flow, stochastic surrogates for the fiber lay-down and web formation as well as Cosserat networks with effective material laws for tensile strength tests. They are coupled by means of parameter identification. We illustrate the applicability of the model chain for an industrial set-up, regarding computer tomography data and tensile strength experiments of the airlay nonwoven materials.
The promising use of model hierarchies and model chains for the virtual production of filaments and nonwovens in the technical textile industry is topic in [2]. A model hierarchy for nonwoven manufacturing in the spunbond process was presented in [3], we adapt and transfer the ideas for the handling of the endless fibers in the respective entanglement and deposition regimes to the staple fibers in the airlay process at hand. The simulation of elastic fibers in the turbulent flow is performed on the works [4, 5], using an inextensible Kirchhoff beam model that is capable of large, geometrically nonlinear deformations and driven by a stochastic aerodynamic drag force. Presupposing a statistic turbulence model for the flow field, the turbulence impact on the fiber dynamics is described by a Gaussian white noise with a flow-dependent amplitude that carries the information of kinetic turbulent energy, dissipation rate, and correlation structure. Due to the huge amount of physical details such simulations are computationally extremely costly and practically limited to some hundreds of fibers. This motivates the introduction of a stochastic surrogate for the virtual web generation: a lay-down model describes the fiber position on the conveyor belt. Containing parameters that characterize the process, it is calibrated by means of a representative sample of dynamical fiber-flow simulations and allows for the fast and efficient computation of a web with millions of fibers. We refer to [6, 7] for lay-down models of endless fibers (2D/3D, isotropic/anisotropic, smooth/standard), to [8–10] for their analysis regarding ergodicity and existence results and to [11] for a comparison with computer tomography data for spunbond materials. Using the random topology generated by the lay-down model we design the elastic microstructure via Cosserat networks based on beams and trusses. Homogenization techniques allow for modeling effective material laws and investigating the tensile strength in dependence on characterizing net parameters. Stochastic fiber networks and non-periodic homogenization are a recent topic of research and were addressed in, e.g., [12–15]. For an homogenization approach on nonwoven materials see [16], this article also provides a remarkable survey over nonwoven microstructure models and studies in literature. Model- and simulation-based investigations of the tensile behavior and mechanical analysis of nonwoven materials can be found in, e.g., [16–18] and [19].
1.1 Industrial airlay process, reference scenario
Fiber properties in reference scenario
Property | Symbol | Unit | Bico fiber (PES/PET) | Solid fiber (PES) |
---|---|---|---|---|
Line density, titer | (ρA) | kg/m | 4.4⋅10^{−7} | 6.7⋅10^{−7} |
Density | ρ | kg/m^{3} | 1.325⋅10^{+3} | 1.38⋅10^{+3} |
Diameter | D | m | 2.1⋅10^{−5} | 2.5⋅10^{−5} |
Length (straight | crimped) | L | ℓ | m | 6.0∣5.1⋅10^{−2} | 6.0∣5.1⋅10^{−2} |
Crimp number | C | bow/m | 7⋅10^{+2} | 5⋅10^{+2} |
Elasticity modulus | E | N/m^{2} | 3⋅10^{+9} | 3⋅10^{+9} |
Shear modulus | G | N/m^{2} | 1.035⋅10^{+9} | 1.035⋅10^{+9} |
Bending stiffness | (EI) | Nm^{2} | 2.6⋅10^{−11} | 5.6⋅10^{−11} |
Tensile strength | S | N/(kg/m) | 3.3⋅10^{+5} | 3.0⋅10^{+5} |
Notation 1
Throughout this paper we typeset vector- and tensor-valued quantities in small and large boldfaced letters, respectively. Scalars are normal-typed, we especially indicate the scalar parameters specified for the industrial reference scenario (Section 1.1) by a Roman font. Sets are denoted by caligraphic letters. We use a tensor calculus with the dot operator ⋅ and the tensor product ⊗.
2 Process simulation
The core of the aerodynamic web forming is the dilute suspension behavior of flexible fibers in the turbulent flow. The random microstructure is essentially determined by the fibers’ deposition, i.e., the distribution of the fibers on the conveyor belt as well as their characteristic geometrical lay-down properties. The fiber-loaded turbulent flow is a multi-scale two-phase problem in a complex geometry. Direct numerical simulation based on the model of first principles as well as approaches like immersed boundary or fictitious boundary/domain are well investigated for fluid-structure problems, but their applicability is practically limited to laminar flows and a small number of suspended solids due to the required high computational demands (from the broad existing literature see, e.g., [20–26]). For flows with a high particle load kinetic modeling approaches have been established, leading to coupled Navier-Stokes Fokker-Planck systems (see the monographs [27, 28] and reference therein). However, these approaches do not cover flexible fibers with infinitely many degrees of freedom. In this work we follow [2, 5] and model a fiber asymptotically as elastic Kirchhoff beam that is capable of large, geometrically nonlinear deformations and driven by a stochastic aerodynamic drag force due to the surrounding turbulent flow field. The flow is specified by a statistic k-ϵ turbulence model. Because of the low load concentration in the airlay process we neglect fiber-fiber interactions as well as the fibers’ impact on the turbulent flow.
2.1 Elastic fiber dynamics in turbulent flows
Numerical treatment. Fiber-flow computations at industrial scale require a highly efficient numerical performance. We use the commercial CFD software^{1} ANSYS Fluent for the flow and the licensable research software^{2} FIDYST for the fiber simulations.
2.2 Aerodynamic web forming zone
The fiber suspension flow in the aerodynamic web forming zone is dilute. Therefore, we neglect fiber-fiber interactions as well as the fibers’ impact on the air flow. The fibers leave the rotating card cylinder continuously over time according to the machine’s mass rate (Figure 2). Due to their inertia they collide with the baffle pipe before they are suctioned onto the conveyor belt by the downwards directed turbulent air flow. The turbulent flow fluctuations cause the fibers to swirl and to form a random web. We are interested in the fibers’ distribution on the conveyor belt and their characteristic geometrical lay-down properties as starting point for generating the resulting nonwoven material by means of a stochastic surrogate lay-down model. Since the airlay process parameters are constant over time and all characteristic (statistic) properties are homogeneous in CD - apart from negligible boundary effects due to the plant edges - we can use the invariances when determining the transition probability of the process.
3 Virtual microstructure generation
3.1 Random fiber web topology
The nonwoven is the deposition image of the fibers. A striking characteristic in the microstructure is a ramp-like contour surface, see Figure 6 for a photo of a nonwoven sample. After the aerodynamic web forming the nonwoven material is thermobonded in a post-processing step. As result of heating the bicomponent fibers melt and glue the random individual fibers together to a solid fiber network which is then explored in material testing. Our strategy is to use the contour surface that results from the lay-down probability densities as basis for stochastic modeling the three-dimensional deposition image with crimped fibers. We identify the contact points of the fibers in the random web, specify the adhesive joints and generate the net topology by help of a graph where the adhesive joints are interpreted as nodes and the fibers as edges. The resulting network is equipped with constitutive relations in Section 3.2.
Superposing bicomponent and solid fibers according to their MD lay-down distributions, number ratio and calibrated model parameters results in a virtual fiber web. To obtain the thermobonded nonwoven to \(\mathcal {V}_{\mathrm{R}}\), the adhesive joints in the web are detected by help of a contact threshold \(a \geq0\) and the net topology is set up in terms of a graph. Details to the strategies and algorithms are given in the following subsections.
Parameter estimation and simulation. The parameters of the surrogate model (3.2a)-(3.2c) are calibrated to the airlay process, following the procedure for endless spunbond fibers in [3, 11]. Whereas \(A_{j}\), \(\sigma_{x,j}\), \(\sigma_{y,j}\) and \(\sigma_{z,j}\) depend on the specific fiber type (\(j=1,2\) for bicomponent and solid, respectively) and are estimated from a representative sample of dynamical fiber-flow simulations (cf. Section 2.2), the isotropy parameter B characterizes the total nonwoven produced. The required additional information about the full spatial orientation is taken from computer tomography data (cf. Figure 6). The respective image processing and analysis were performed and provided by the Fraunhofer ITWM, Department Image Processing with the software^{3} MAVI.
Detection of adhesive joints. The microstructure of the thermobonded nonwoven is the random web of the deposited fibers that is glued together by the bicomponent fibers. In terms of graph theory the random net topology can be represented by a graph \(\mathcal{G} = (\mathcal{N}, \mathcal{E})\) where the adhesive joints are considered as the nodes and the fiber curves as the edges. Here, \(\mathcal{N}\) and \(\mathcal{E}\) denote the index sets of nodes and edges, respectively.
3.2 Elastic fiber net
4 Investigation of effective nonwoven material behavior
4.1 Virtual tensile strength test
In the one-dimensional experimental tensile strength test a cuboidal material sample over the full fabric height H is glued with the upper and lower faces onto two parallel plates. The plates are pulled apart in direction of the nonwoven height while the reacting tensile force F is recorded as function of the strain ε. The strain is thereby determined from the actual sample height H as \(\varepsilon=(H-H^{\circ})/H^{\circ}\) with the referential height \(H^{\circ}\) corresponding to a pre-tensioning force \(F^{\circ}\), see Figure 10.
Effective force modeling based on fiber net behavior and RVE treatment. The effective force model \(N_{i}\) (4.1) is deduced from energetic investigations of the stress-strain relations for the ith RVE with the elastic fiber net (3.5a)-(3.5b) subjected to axial displacements - in analogy to the procedure presented in Section 3.2. Considering the RVE of referential height \(h_{i}\) (Figure 11), all fiber node points at the upper face are vertically shifted with magnitude u, i.e., \(\mathbf{r}^{\nu}=\mathbf {r}^{\mathbf{0}\nu}+u\mathbf{e}_{\mathbf{z}}\), \(\nu\in\mathcal{N}_{B,up}\), while the ones at the lower face are kept at the referential positions. At the lateral faces only motions on the face plane are allowed. Evaluating \(\varepsilon=u/h_{i}\), \(N_{i}=\sum_{\nu\in\mathcal{N}_{B,up}} \sum_{\mu\in\mathcal{E}(\nu)} \mathbf{n}^{\nu}_{\mu}\cdot\mathbf{e}_{\mathbf{z}}\) for various \(u\in\mathbb{R}_{0}^{+}\) yields then the effective force function.
4.2 Stress-strain results
5 Conclusion and outlook
In this paper, we established a consistent, efficiently evaluable chain of mathematical models that enabled the simulation of the airlay process and the investigation of the resulting material properties by virtual tensile strength tests. For the long-term industrial objective, the simulation-based process design towards the prediction and improvement of product properties, the mathematical mapping between the parameters of process and material is essential. We gave a proof of concept and showed the feasibility of a future optimization by applying our model chain to an industrial set-up. Proceeding from an airlay process simulation with a highly turbulent dilute fiber suspension flow, we used the process parameters (including fiber properties) and the numerically obtained deposition results to set up the stochastic surrogate model for the microstructure generation. Thereby, certain topological web parameters that characterize anisotropy, adhesive joints and height of the microstructure have to be identified from computer tomography data and calibrated by experiments (thermobonding effects). So far, the evaluation of the fiber distribution in height direction lacks from the image analysis of the computer tomography scans, but respective research work is in progress. The effective nonwoven material laws were deduced from the underlying fiber properties and simulation runs using energetic homogenization techniques. In the tensile strength tests the simulated and measured material behaviors match well for small strains but deviate for higher strains. This discrepancy might be due to plastic changes (rupture) that are not handled in the present elastic fiber net model. The break-up of adhesive joints could be certainly included but requires a deeper insight in the mechanism of thermobonding that was not analyzed in this paper.
At this point of research, however, we still face a difficulty: the model chain contains one parameter that was introduced for mathematical reasons, i.e., well-posedness of the elastic fiber net model, but turned out to strongly influence the tensile behavior. To gain understanding of its dependencies on the process parameters which will be necessary for future optimization issues, a sensitivity analysis in combination with a broader experimental study might be helpful.
www.itwm.fraunhofer.de. FIDYST is a software tool for fiber dynamics simulations, developed by the Fraunhofer ITWM, Germany. For details on the applicability spectrum, interfaces and algorithms we refer to [2]. We sketch here briefly the underlying relevant numerical schemes.
www.mavi-3d.de. MAVI (Modular algorithms for volume images) is a Fraunhofer software for image processing, analysis and visualization, for details we refer to [43, 44].
Declarations
Acknowledgements
The authors thank their industrial partners, Dr. Joachim Binnig and Christopher Schütt from AUTEFA Solutions as well as Olaf Döhring from IDEAL Automotive, for performing measurements, providing data and information about process and fabric and supporting the work with fruitful discussions. Moreover, the financial support of the German Bundesministerium für Bildung und Forschung, Project OPAL 05M13, is acknowledged.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Albrecht W, Fuchs H, Kittelmann W, editors. Nonwoven fabrics: raw materials, manufacture, applications, characteristics, testing processes. New York: Wiley; 2006. Google Scholar
- Wegener R, Marheineke N, Hietel D. Virtual production of filaments and fleece. In: Neunzert H, Prätzel-Wolters D, editors. Currents in industrial mathematics: from concepts to research to education. Berlin: Springer; 2015. p. 103-62. View ArticleGoogle Scholar
- Klar A, Marheineke N, Wegener R. Hierarchy of mathematical models for production processes of technical textiles. Z Angew Math Mech. 2009;89:941-61. MathSciNetView ArticleMATHGoogle Scholar
- Marheineke N, Wegener R. Fiber dynamics in turbulent flows: general modeling framework. SIAM J Appl Math. 2006;66(5):1703-26. MathSciNetView ArticleMATHGoogle Scholar
- Marheineke N, Wegener R. Modeling and application of a stochastic drag for fiber dynamics in turbulent flows. Int J Multiph Flow. 2011;37:136-48. View ArticleGoogle Scholar
- Götz T, Klar A, Marheineke N, Wegener R. A stochastic model and associated Fokker-Planck equation for the fiber lay-down process in nonwoven production processes. SIAM J Appl Math. 2007;67(6):1704-17. MathSciNetView ArticleMATHGoogle Scholar
- Klar A, Maringer J, Wegener R. A smooth 3D model for fiber lay-down in nonwoven production processes. Kinet Relat Models. 2012;5(1):57-112. MathSciNetView ArticleMATHGoogle Scholar
- Doulbeault J, Klar A, Mouhot C, Schmeiser C. Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes. Appl Math Res Express. 2013;2013:165-75. MathSciNetMATHGoogle Scholar
- Grothaus M, Klar A. Ergodicity and rate of convergence for a non-sectorial fiber lay-down process. SIAM J Math Anal. 2008;40(3):968-83. MathSciNetView ArticleMATHGoogle Scholar
- Kolb M, Savov M, Wübker A. (Non-)ergodicity of a degenerate diffusion modeling the fiber lay down process. SIAM J Math Anal. 2013;45(1):1-13. MathSciNetView ArticleMATHGoogle Scholar
- Grothaus M, Klar A, Maringer J, Stilgenbauer P, Wegener R. Application of a three-dimensional fiber lay-down model to non-woven production processes. J Math Ind 2014;4:4. MathSciNetView ArticleGoogle Scholar
- Briane M. Three models of nonperiodic fibrous materials obtained by homogenization. Modél Math Anal Numér. 1993;27(6):759-75. MathSciNetMATHGoogle Scholar
- Le Bris C. Some numerical approaches for weakly random homogenization. In: Kreiss G, Lötstedt P, Malqvist A, Neytcheva M, editors. Numerical mathematics and advanced applications 2009. Berlin: Springer; 2010. p. 29-45. View ArticleGoogle Scholar
- Lebée A, Sab K. Homogenization of a space frame as a thick plate: application of the bending-gradient theory to a beam lattice. Comput Struct. 2013;127:88-101. View ArticleGoogle Scholar
- Sigmund O. Materials with prescribed constitutive parameters: an inverse homogenization problem. Int J Solids Struct. 1994;31(17):2313-29. MathSciNetView ArticleMATHGoogle Scholar
- Raina A, Linder C. A homogenization approach for nonwoven materials based on fiber undulations and reorientation. J Mech Phys Solids. 2014;65:12-34. MathSciNetView ArticleGoogle Scholar
- Adanur S, Liao T. Fiber arrangement characteristics and their effects on nonwoven tensile behavior. Tex Res J. 1999;69(11):816-24. View ArticleGoogle Scholar
- Bais-Singh S, Goswami BC. Theoretical determination of the mechanical response of spun-bonded nonwovens. J Text Inst. 1995;186(2):271-88. View ArticleGoogle Scholar
- Farukh F, Demirci E, Sabuncuoglu B, Acar M, Pourdeyhimi B, Silberschmidt VV. Mechanical analysis of bi-component-fibre nonwovens: finite-element strategy. Composites, Part B, Eng. 2015;68:327-35. View ArticleGoogle Scholar
- Glowinski R, Pan TW, Hesla TI, Joseph DD, Périaux J. A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J Comput Phys. 2001;169:363-426. MathSciNetView ArticleMATHGoogle Scholar
- Hämäläinen J, Lindström SB, Hämäläinen T, Niskanen H. Papermaking fibre-suspension flow simulations at multiple scales. J Eng Math. 2011;71(1):55-79. View ArticleMATHGoogle Scholar
- Hu HH, Patanker NA, Zhu MY. Direct numerical simulation of fluid-solid systems using arbitrary Lagrangian-Eulerian technique. J Comput Phys. 2001;169:427-62. MathSciNetView ArticleGoogle Scholar
- Peskin CS. The immersed boundary method. Acta Numer. 2002;11:479-517. MathSciNetView ArticleMATHGoogle Scholar
- Stockie JM, Green SI. Simulating the motion of flexible pulp fibres using the immersed boundary method. J Comput Phys. 1998;147(1):147-65. View ArticleMATHGoogle Scholar
- Svenning E, Mark A, Edelvik F, Glatt E, Rief S, Wiegmann A, Martinsson L, Lai R, Fredlund M, Nyman U. Multiphase simulation of fiber suspension flows using immersed boundary methods. Nord Pulp Pap Res J. 2012;27(2):184-91. View ArticleGoogle Scholar
- Tornberg AK, Shelle MJ. Simulating the dynamics and interactions of flexible fibers in Stokes flow. J Comput Phys. 2004;196:8-40. MathSciNetView ArticleGoogle Scholar
- Barrett JW, Knezevic DJ, Süli E. Kinetic models of dilute polymers: analysis, approximation and computation. Prague: Nećas Center for Mathematical Modeling; 2009. Google Scholar
- Gidaspow D. Multiphase flow and fluidization: continuum and kinetic theory descriptions. San Diego: Academic Press; 1994. MATHGoogle Scholar
- Antman SS. Nonlinear problems of elasticity. New York: Springer; 2006. Google Scholar
- Baus F, Klar A, Marheineke N, Wegener R. Low-Mach-number - slenderness limit for elastic rods. 2015. arXiv:1507.03432.
- Lindner F, Marheineke N, Stroot H, Vibe A, Wegener R. Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting. Stoch Dyn. 2016. doi:https://doi.org/10.1142/S0219437175001622016. Google Scholar
- Schmeisser A, Wegener R, Hietel D, Hagen H. Smooth convolution-based distance functions. Graph Models. 2015;82:67-76. View ArticleGoogle Scholar
- Scott DW. Multivariate density estimation: theory, practice and visualization. New York: Wiley; 1992. View ArticleMATHGoogle Scholar
- Klar A, Maringer J, Wegener R. A 3D model for fiber lay-down in nonwoven production processes. Math Models Methods Appl Sci. 2012;22(9):1250020. MathSciNetView ArticleMATHGoogle Scholar
- Lagnese J, Leugering G, Schmidt E. Modeling, analysis and control of dynamic elastic multi-link structures. Boston: Springer; 1994. View ArticleMATHGoogle Scholar
- Hohe J, Becker W. Determination of the elasticity tensor of non-orthotropic celluar sandwich cores. Tech Mech. 1999;19(4):259-68. Google Scholar
- Munoz Romero J. Finite-element analysis of flexible mechanisms using the master-slave approach with emphasis on the modelling of joints [PhD thesis]. London: Imperial College; 2004. Google Scholar
- Bonilla LL, Götz T, Klar A, Marheineke N, Wegener R. Hydrodynamic limit for the Fokker-Planck equation describing fiber lay-down models. SIAM J Appl Math. 2007;68(3):648-65. MathSciNetView ArticleMATHGoogle Scholar
- Chang C, Gorissen B, Melchior S. Fast oriented bounding box optimization on the rotation group \(SO(3,\mathbb{R})\). ACM Trans Graph. 2011;30(5):122. View ArticleGoogle Scholar
- Ericson E. Real-time collision detection. London: CRC Press; 2004. Google Scholar
- Simo JC. A finite strain beam formulation. The three-dimensional dynamic problem - part I. Comput Methods Appl Mech Eng. 1985;49:55-70. View ArticleMATHGoogle Scholar
- Hill R. Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids. 1963;11(5):357-72. View ArticleMATHGoogle Scholar
- Ohser J, Schladitz K. 3D images of materials structures - processing and analysis. Weinheim: Wiley-VCH; 2009. View ArticleMATHGoogle Scholar
- Redenbach C, Rack A, Schladitz K, Wirjadi O, Godehardt M. Beyond imaging: on the quantitative analysis of tomographic volume data. Int J Mater Res. 2012;2:217-27. View ArticleGoogle Scholar