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Experimental and numerical characterization method for forming behavior of thermoplastics reinforced with woven fabrics
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Procedia Manufacturing 29 (2019) 443–449 Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia
18th International Conference on Sheet Metal, SHEMET 2019 18th International Conference on Sheet Metal, SHEMET 2019
Experimental and numerical characterization method for forming Experimental and numerical characterization method for forming behavior of thermoplastics reinforced with fabrics Manufacturing Engineering Society International Conference 2017,woven MESIC 2017, 28-30 June behavior of thermoplastics reinforced with woven fabrics 2017, Vigo (Pontevedra), Spain Bernd-Arno Behrens, Alexander Chugreev, Hendrik Wester* Bernd-Arno Behrens, Alexander Chugreev, Hendrik Wester*
Leibniz Universität Hannover - IFUM, An der Universität 2, in 30583Industry Garbsen, Germany Costing models for capacity optimization 4.0: Trade-off Leibniz Universität Hannover - IFUM, An der Universität 2, 30583 Garbsen, Germany between used capacity and operational efficiency
Abstract Abstract A. Santanaa, P. Afonsoa,*, A. Zaninb, R. Wernkeb The automotive and aviation industry has to achieve significant weight reduction in order to fulfil legal obligations. This leads to The automotive aviation industry hasa University to achievecombinations weight reductionPortugal in order to fulfil legal leads to an increasing useand of new materials or new material like Guimarães, fibre-reinforced plastics (FRP) as obligations. they provideThis a high ofsignificant Minho, 4800-058 b of lowcombinations an increasing use of new or new material like fibre-reinforced plastics (FRP) as they provide a high lightweight potential due materials to the combination density and high tensile strength. pre-impregnated sheets with a Unochapecó, 89809-000 Chapecó, SC, Brazil Meanwhile lightweight potential due to the combination of low density and high tensile strength. pre-impregnated with a thermoplastic matrix reinforced with woven carbon fibres are commercially available.Meanwhile This has led in a significant sheets cost reduction thermoplastic reinforced woven carbon commercially available.properties, This has led in a significant cost reduction and hence, thematrix FRP have becomewith affordable for largefibres scaleare production. The material in particular the forming and and hence, the FRP become affordable large production.metal The material properties, theTherefore, forming and failure behaviour of have the FRP, differ strongly for from thatscale of conventional materials like steel in or particular aluminium. new failure of the FRP, differ strongly from that of conventional metal materials or aluminium. new Abstract materialbehaviour characterisation techniques, investigation methods as well as numerical modelslike aresteel required. The mainTherefore, focus of this material techniques, investigation methods wellfor as numerical models are required. main focus of this paper liescharacterisation on the development of a non-orthogonal material as model the FRP, its implementation in aThe commercial FE-software paper the development of a non-orthogonal materialprocesses model for the implementation inSince a commercial FE-software as welllies as use ofof a combined experimental-numerical procedure forFRP, material characterisation. the properties of these Under theonconcept "Industry 4.0", production will be its pushed to be increasingly interconnected, as well as are on strongly the use on oftemperature aacombined experimental-numerical procedure for material characterisation. Sincecapacity the properties these materials dependent, thenecessarily, forming process of reinforced thermoplastics typically carried out at of elevated information based real time basis and, much more efficient. In this is context, optimization materials are strongly temperature dependent, forming of reinforced thermoplastics is typically carried outmodel at elevated temperatures. Thus, temperature sensitivity hasthe to be takenprocess into account duringalso experimental testing as well as in the goes beyond the traditional aim of capacity maximization, contributing for organization’s profitability and value. temperatures. temperature sensitivity has toout be taken on intoanaccount during experimental testing asoptimization well asFor in this the model approach.lean TheThus, model parameterisation is carried based iterative numerical optimization procedure. purpose, Indeed, management and continuous improvement approaches suggest capacity instead the of approach. The model parameterisation is carriedby outmeans basedof ondigital an iterative numerical optimization procedure. For thismodel purpose, experimentally obtained results are investigated image correlation and linked with the numerical in the maximization. The study of capacity optimization and costing models is an important research topic that deserves experimentally obtained results are investigated by means of digital image correlation and linked with the numerical model in combination with an automated optimization process. contributions from both the practical and theoretical perspectives. This paper presents and discusses a mathematical combination with an automated optimization process. model for capacity management based on different costing models (ABC and TDABC). A generic model has been © 2018 The Authors. Published by Elsevier B.V. © 2019 Theand Authors. by Elsevier B.V. developed it wasPublished used to analyze idleBY-NC-ND capacity and to design strategies towards the maximization of organization’s © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is The an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) value. trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity This is an openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection Selection and and peer-review under under responsibility responsibility of of the the organizing organizing committee committee of of SHEMET SHEMET 2019. 2019. optimization might hide under operational inefficiency. Selection and peer-review responsibility of the organizing committee of SHEMET 2019. © 2017 The Authors. Published by Elsevier B.V. Keywords: Fabric-Reinforced Thermoplastics; FE-Modelling; Material Characterisation Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference Keywords: Fabric-Reinforced Thermoplastics; FE-Modelling; Material Characterisation 2017. Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency
1. Introduction
* Corresponding author. E-mail address: [email protected] * The Corresponding author. cost of idle capacity is a fundamental information for companies and their management of extreme importance E-mail address: [email protected] in modern production systems. In general, it is defined as unused capacity or production potential and can be measured 2351-9789 © 2018 The Authors. Published by Elsevier B.V. in several ways: tons ofunder production, available hours of manufacturing, etc. The management of the idle capacity This is an open access the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) 2351-9789 © 2018 Thearticle Authors. Published by Elsevier B.V. * Paulo Afonso. Tel.: +351 253 510 761; fax: +351 253 604 741 Selection under responsibility of the organizing committee of SHEMET 2019. This is an and openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) E-mail address: [email protected] Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under of the scientificbycommittee the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2019responsibility The Authors. Published Elsevier of B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 10.1016/j.promfg.2019.02.160
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1. Introduction New materials, like woven composites, offer a combination of low density and high specific tensile strength. Thus, they provide a high light-weight potential and are increasingly used. However, the high strength is only offered in the fibre direction. This strong anisotropic behaviour differs from commonly used metal materials. Due to their woven fibre arrangement, organic sheets offer a higher overall stiffness as compared with UD-laminates. Furthermore, the formability is increased by yarn rotation and stretching of the undulated fibres [1]. The use of thermoplastics as a matrix material is gaining more importance particularly in the automotive industry, since it can be formed to a complex geometry under elevated temperatures [2; 3]. The used material, organic sheets, consist of a thermoplastic PA6 matrix and woven glass fibres as reinforcement. The overall properties are a superimposition of matrix properties and fibre properties [4]. This leads to a strong anisotropic behaviour, where the highest strength is observed in the fibre direction. Therefore, the shearing of the fibres at the cross points is the main in plane forming mode of woven fabrics [5]. During the deformation, the angle between the fibres changes and once it reaches a critical value, referred to as locking angle, the shear stress undergoes a substantial increase due to shear locking. In the molten state, the shear resistance can be assumed to be negligible until reaching the specific locking angle of the woven fabric [6]. In contrast to the glass fibres, the forming behaviour of the thermoplastic matrix material is strongly temperature dependent [2]. Above the melting temperature, the stiffness and strength are significantly reduced. However, in case of a matrix solidification during the forming phase, the influence of the matrix on the overall forming behaviour increases significantly. The influence of temperature evolution has been investigated in [7] by means of the draping behaviour of a double dome geometry. The authors have concluded that the temperature evolution over the entire process including the heating, transfer and forming phase has a decisive impact on the precise prediction of the draping process. The bending properties of reinforced thermoplastic sheets were measured in [8] by means of cantilever test, which were performed in an environmental chamber. In order to characterise the main in plane forming mode, the shear deformation, the picture frame or bias extension test can be applied. Both methods are discussed in [6] and [9] and lead to similar results. The material properties of the glass fibres are analysed by means of uniaxial tensile tests, whereby the loading is introduced in fibre direction. This contribution deals with experimental investigations to determine the mechanical properties of organic sheets. Considering of the temperature dependency the tests have been performed in a wide temperature range. Based on the experimental findings, a numerical model has been developed and implemented into the commercial FESoftware Abaqus CAE 6.14. The model parameterisation has been subsequently carried out by a numerical optimisation process. 2. Material characterisation of organic sheets In the scope of this work organic sheet with a PA6 matrix reinforced with woven glass fibres (2/2 twill) and a thickness of 1 mm was investigated. Since the considered thermoplastic PA6 is strongly temperature dependent the tests were carried out inside an environmental chamber at temperatures between RT and 240 °C and a test speed of 1 mms-1. In order to take the effect of fibre direction into account tensile tests in fibre direction as well as bias extension tests have been performed. The experimental tests have been carried out at LMT, ENS Paris-Saclay and for each parameter combination up to 5 repetitions have been conducted. During the bias extension test the load is introduced with an initial angle of 45° to the fibres, which leads to pure shear in the middle of the sample. A detailed explanation of the bias extension test is given in [5]. During the tests the displacements have been captured by means of an optical measurement system with a frame rate of 4 pictures/mm. The ratio of height (H) to width (W) was chosen to 2.33 and thus higher than 2.0 in order to achieve more homogeneous shear deformation in the bias extension tests like proposed in [5]. Furthermore, heating tests have been carried out in order to ensure a homogeneous sample temperature. The used test setup as well as the fibre orientations for the different tests are shown in Fig. 2. In Fig. 2b, c measured force displacement curves for uniaxial tensile test and bias extension tests are presented for different testing temperatures at a test speed of 1 mm/s. The comparison between tensile test (b) and bias extension test (c) shows the strong influence of the fibre direction. During tensile testing, the introduced load is borne by the fibres. The observed maximum forces
Bernd-Arno Behrens et al. / Procedia Manufacturing 29 (2019) 443–449 Bernd-Arno Behrens / Procedia Manufacturing 00 (2018) 000–000
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are in general higher as compared to the bias extension test and only small strains exist before fracture. The temperature dependency is also small. a
c
b
Sample
6000
10000
5000
8000
4000
6000 RT 180 °C 220 °C
4000 2000 0
Bias extension test, fibre direction
Forec F [N]
Tensile test, 90 fibre direction
12000
Forec F [N]
Upper die
Lower die
0
1 2 3 Displacement u [mm]
RT 215 °C
185 °C 220 °C
3000 2000
1000 4
0
0
10 20 Displacement u [mm]
30
Fig. 1. (a) Experimental test setup for uniaxial tensile test and bias extension test with corresponding fibre directions; (b) Experimental obtained force-displacements curves for various temperatures at a test speed of 1 mm/s: uniaxial tensile tests with 90° fibre direction; (c) bias extension tests with ± 45° fibre direction.
In contrast, the results of bias extension test demonstrate a more complex material behaviour. Due to the loading direction the mechanical behaviour is a superimposition of matrix and fibre properties. A high temperature dependency can be seen due to the increased influence of the thermoplastic matrix material. The initial fibre orientation in the bias extension test is ±45°. However, the shear deformation leads to fibre rotation with increasing displacement and the fibre get more and more aligned in loading direction. Friction between the fibre bundles themselves as well as between the fibre bundles and the matrix influence the movement. If the specific lock angle is reached, further deformation leads to a sharp increase of force as well as out of plane wrinkling due to compaction of fibre bundles. For a detailed material description the change of shear angle with increasing displacement is of great interest. The current shear angle 𝛾𝛾 has been calculated based on optical measurement data by means of digital image correlation with following equation (1) [10]: 𝜋𝜋 1 + 𝜀𝜀𝑥𝑥 (1) 𝛾𝛾 = − 2 ∗ arctan ( ) 2
1 + 𝜀𝜀𝑦𝑦 1 𝐻𝐻 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾) = [( − 1) 𝐹𝐹𝑎𝑎𝑎𝑎 (cos − sin ) − 𝑊𝑊 𝐹𝐹𝑆𝑆𝑆𝑆 ( ) cos ] (2𝐻𝐻 − 3𝑊𝑊) cos 𝛾𝛾 𝑊𝑊 2 2 2 2
(2)
where εx and εy are the engineering strains. Furthermore, the material description requires the actual shear stress which depends on the shear angle. This dependency leads to a high non-linear behaviour. In this work a constant test speed of 1mms-1 has been investigated. Therefore, the matrix viscosity is assumed to be rate independent. Following the basic research of [11] the normalised shear force 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾) per unit length can be calculated based on the geometric properties of the bias extension sample with equation (2), whereby H and W are the specimen height and width, respectively. The current value of 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾) depends on 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾⁄2), hence the shear force needs to be calculated in an iterative procedure by means of a Matlab code. The shear angle is taken into account by correlating the measured axial force 𝐹𝐹𝑎𝑎𝑎𝑎 with the optical measured shear angle. The shear angle dependent shear stress τ12 can be derived by referring the shear force on the specimen thickness. The Fig. 2 presents exemplary calculated shear stress - shear angle curves for the temperature of 185 °C, 200 °C and 215 °C and a test speed of 1 mm/s. The shear angles distribution calculated by means of digital image correlation (DIC) show some variation at the sides. This is caused by loose fibre ends, which are not sheared but captured by the DIC and lead to failures in this region. Therefore, during the investigations, three regions of interest (ROI) have been chosen on the sample surface and a representative mean shear angle has been calculated and subsequently used for further calculation steps (Fig. 2a). The nonlinear dependency between shear stress and shear angle can be described by means of a polynomial function. In order to obtain numerical stability a continuous description of the temperature dependent material behaviour is necessary.
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a
b
Figure 2 (a) Shear angle obtained by means of DIC at a displacement of 18 mm for a temperature of 215 °C ; (b) shear stress - shear angle curve for the temperatures of 185 °C, 200 °C and 215 °C.
Shear angle – shear stress relationship
12
=
1 𝛾𝛾
+
2 𝛾𝛾
+
𝛾𝛾 +
𝛾𝛾 2 + 𝛾𝛾
)∗
Temperature influence
Shear stress [MPa]
Therefore, the polynomial function has been extended by an multiplicative term, considering an exponential relationship, whereby the coefficients p1 – p5 are material specific constants and T 0 is a scaling factor for the temperature dependency (Fig. 3a). The Fig. 3b presents the predicted shear stress surface as well as the experimental optained shear angle – shear stress curves, marked with black dots. The correlation coefficient is determined to R2=0.99, thus the introduced model provides a good description of the material behaviour under consideration of elevated temperatures. a b 80 60 40 20 0
190
200
210
0
0.2
0.4
0.6
0.8
1.0
Fig. 3. (a) Polynomial function for the description of the nonlinear relationship between shear stress and shear angle under consideration of temperature dependency; (b) Comparison between experimental data and the model predicted shear stress.
3. Numerical model and material data optimisation For numerical description of organic sheets deformation, a non-orthogonal hypo-elastic material model has been established and implemented in the FE-Software Abaqus CAE 6.14 by means of a user defined routine. Hypo-elastic material models are widely used in FE codes to describe the material behaviour at large strains [12]. Due to strong changes of the fibre direction during the forming process the shear stiffness needs to be described as a function of the shear angle. Taking into account the works [1; 13] the current fibre direction during deformation can be calculated by means of fixed body rotation matrix and deformation gradient. A schematic representation of fibre rotation is given in Fig. 4. The deformation caused change of fibre direction is considered by transforming the obtained strain increment from the global orthonormal frame (g) to the fibre frames (f). The stress and strain increments for both fibre directions (f1, f2) are calculated within the fibre frames based on the updated stiffness matrix and subsequently transferred back to Cauchy stress. The update of the stress increments is carried out by using the Hughes and Winget formulation [14]. 1 1 and 𝑓𝑓2𝑥𝑥 The shear stress increment is calculated in the non-orthogonal frame defined by the two fibre directions 𝑓𝑓1𝑥𝑥 based on the change of the angle 𝜓𝜓 and the updated nonlinear shear modulus. Finally, the shear stress increment is also transferred back to the global orthonormal frame. The stress tensor is updated by a superimposition of the calculated stress increments of the different frames. The superimposition is carried out in the global orthonormal frame. As an initial approach only tensile modulus in fibre direction and shear modulus depending on shear angle have been considered. The effect of poisson’s ratio has been neglected.
Bernd-Arno Behrens et al. / Procedia Manufacturing 29 (2019) 443–449 Bernd-Arno Behrens / Procedia Manufacturing 00 (2018) 000–000 Initial state
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Deformed state
=initial 1st fiber direction = initial 2nd fiber direction =initial orthonormal material frame
= deformed 1st fiber direction = deformd 2nd fiber direction =deformed orthonormal material frame
Fig. 4. Schematic representation of fibre orientation during deformation of woven fabrics.
Due to the superimposition of matrix and fibre properties, the nonlinear relationship between shear angle and shear stress as well as the calculation of the shear angle based on DIC-data, the parameterisation of the numerical material model is challenging. Therefore, an iterative numerical optimisation process has been developed. The process itself is controlled by a Python file, which modifies the user defined input parameters, the base points p1-p5 inside the Abaqus job file, submit the job as well as perform post processing operations on the results file. The shear stress – shear angle behaviour is described by the base points. To modify the material behaviour in fibre direction, the experimental determined elastic modulus 𝐸𝐸11 in fibre direction is multiplied by a scaling factor A. is introduced, which is multiplied by the elastic modulus in fibre direction. The sum of squared errors of the experimental measured and numerical calculated axial forces has been chosen as target figure. The optimisation is carried out by means of multi objective genetic algorithm (MOGA), which can be used efficiently for multi objective optimisation problems. In order to meet the shape of the force – displacement curve the deviation at characteristic displacements as well as the total deviation have been chosen as objectives. For a first approach, the optimisation process has been performed for bias extension test at a temperature of 215 °C and a test speed of 1 mms-1. b
c
Shear angle γ= -2θ=1.05 rad
d
γ ≈ 1.02 rad
ROI
Shear angle γ [rad] 1.1
ROI
1000 800
600
0.83
400
0.55
200
0.28
0.0
Experiment First fit Optimised material data
1200
γ ≈ 1.06 rad
2θ=0.52 rad
1600 1400
Axial force [N]
a
0
x y
0
5
10 15 Displacement [mm]
20
Figure 5 Comparison of manually measured shear angle (a), the shear angle distribution obtained by (b) DIC and by (c) numerical simulation at a displacement of 18 mm; (d) force – displacement curves based on experimental data and numerical simulation at a temperature of 215 °C.
The determination of the shear angle evolution by means of DIC analysis as well as the numerical model itself and the numerical optimisation process are validated by a comparison of numerical calculated data with experimental data. Fig. 5 shows a comparison between manually measured angle between the fibre direction (Fig. 5a), the shear angle obtained by optical measurement and DIC analysis (Fig. 5b) as well as the numerical calculated shear angle at the ROI position in the middle of the sample (Fig. 5c) for a bias extension tests at 215 °C and a displacement of 18 mm.
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The numerical results, presented in Fig. 5c, are calculated based on the optimised material data. An overview over the initial material parameters, determined based on the experimental data and the finally used optimized data is given in table 1. Table 1 Overview over initial and optimised material data for the numerical simulation p1
p2
p3
p4
p5
A
E11 [GPa]
Initial data
97.00
-524.00
1332.00
-1485.00
609.00
1.00
8000
Optimised data
106.89
-513.22
1343.39
-1610.50
637.09
1.19
8000
The determined shear angles show a good qualitative agreement. A comparison of experimental measured and numerical calculated force displacement curves is presented in Fig. 5d. The numerical calculated curves are based on the initial determined material data as well as on the material data which has been obtained after the numerical optimisation. The optimised material data shows a good agreement with the experimental measured curve. Furthermore, it can be seen that the optimisation process leads to a strong increase in the quality of the numerical results. 4. Conclusion In this research the complex material behaviour of organic sheets, which consist of a thermoplastic matrix with woven glass fibres, has been characterised by means of uniaxial tensile tests and bias extension tests under consideration of a wide temperature range. The results show a strong dependency on fibre direction, shear angle and temperature. A continuous model approach, which takes the change of fibre direction into account, has been implemented by means of a user subroutine. The model has been parametrized under consideration of the nonlinear relationship between shear angle and shear modulus. The change of the shear angle has been determined by means of digital image correlation in an incremental procedure based on the optical measured displacement data. The material model has been parameterised by means of an iterative optimisation process. The comparison of experimental measured and numerical calculated shear angle distribution showed a good qualitative agreement. Furthermore, the optimisation of the material data by means of a genetic optimisation algorithm leads to estimation of force displacement curves. Acknowledgements This work is funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) through the International Research Training Group 1627 “Virtual Materials and Structures and their Validation”. References
[1] P. Xue,, X. Peng, J. Cao, A non-orthogonal constitutive model for characterizing woven composites, Composites Part A: Applied Science and Manufacturing, 34 (2) (2003) 183–193. [2] B.-A. Behrens, S. Hübner, A. Neumann, Forming Sheets of Metal and Fibre-reinforced Plastics to Hybrid Parts in One Deep Drawing Process, Procedia Engineering, 81 (2014) 1608–1613. [3] B.-A. Behrens, A. Raatz, S. Hübner, C. Bonk, F. Bohne, C. Bruns, M. Micke-Camuz, Automated Stamp Forming of Continuous Fiber Reinforced Thermoplastics for Complex Shell Geometries, Procedia CIRP, 66 (2017) 113–118. [4] B.-A. Behrens, C. Bonk, N. Grbic, M. Vucetic, Numerical analysis of a deep drawing process with additional force transmission for an extension of the process limits, IOP Conference Series: Materials Science and Engineering, 179 (2017) 12006. [5] M. Machado, Modelling & simulation of continuous fibrereinforced thermoplastic-matrix composites, Dissertation, Universität Linz (2015). [6] I. Taha, Y. Abdin, S. Ebeid, Comparison of picture frame and Bias-Extension tests for the characterization of shear behaviour in natural fibre woven fabrics, Fibers and Polymers, 14 (2) (2013) 338–344. [7] P. Harrison, R. Gomes, N. Curado-Correia, Press forming a 0/90 cross-ply advanced thermoplastic composite using the double-dome benchmark geometry, Composites Part A: Applied Science and Manufacturing, 54 (2013) 56–69. [8] B. Liang, N. Hamila, M. Peillon, P. Boisse, Analysis of thermoplastic prepreg bending stiffness during manufacturing and of its influence on wrinkling simulations, Composites Part A: Applied Science and Manufacturing, 67 (2014) 111–122. [9] W. Lee, J. Padvoiskis, J. Cao, E. de Luycker, P. Boisse, F. Morestin, J. Chen, J. Sherwood, Bias-extension of woven composite fabrics.
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International Journal of Material Forming, 1 (S1) (2008) 895–898. [10] S.V. Lomov, B. Boisse, E. de Luycker, F. Morestin, K. Vanclooster, D. Vandepitte, I. Verpoest, A. Willems, Full-field strain measurements in textile deformability studies, Composites Part A: Applied Science and Manufacturing, 39 (8) (2008) 1232–1244. [11] F. Härtel, P. Harrison, Evaluation of normalisation methods for uniaxial bias extension tests on engineering fabrics, Composites Part A: Applied Science and Manufacturing, 67 (2014) 61–69. [12] M.A. Khan, T. Mabrouki, E. Vidal-Sallé, P. Boisse, Numerical and experimental analyses of woven composite reinforcement forming using a hypoelastic behaviour, Application to the double dome benchmark. Journal of Materials Processing Technology, 210 (2) (2010) 378–388. [13] X.Q. Peng, J. Cao, A continuum mechanics-based non-orthogonal constitutive model for woven composite fabrics, Composites Part A: Applied Science and Manufacturing, 36 (6) (2005) 859–874. [14] T.J.R. Hughes, J. Winget, Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis, International Journal for Numerical Methods in Engineering, 15 (12) (1980) 1862–1867.
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18th International Conference on Sheet Metal, SHEMET 2019 18th International Conference on Sheet Metal, SHEMET 2019
Experimental and numerical characterization method for forming Experimental and numerical characterization method for forming behavior of thermoplastics reinforced with fabrics Manufacturing Engineering Society International Conference 2017,woven MESIC 2017, 28-30 June behavior of thermoplastics reinforced with woven fabrics 2017, Vigo (Pontevedra), Spain Bernd-Arno Behrens, Alexander Chugreev, Hendrik Wester* Bernd-Arno Behrens, Alexander Chugreev, Hendrik Wester*
Leibniz Universität Hannover - IFUM, An der Universität 2, in 30583Industry Garbsen, Germany Costing models for capacity optimization 4.0: Trade-off Leibniz Universität Hannover - IFUM, An der Universität 2, 30583 Garbsen, Germany between used capacity and operational efficiency
Abstract Abstract A. Santanaa, P. Afonsoa,*, A. Zaninb, R. Wernkeb The automotive and aviation industry has to achieve significant weight reduction in order to fulfil legal obligations. This leads to The automotive aviation industry hasa University to achievecombinations weight reductionPortugal in order to fulfil legal leads to an increasing useand of new materials or new material like Guimarães, fibre-reinforced plastics (FRP) as obligations. they provideThis a high ofsignificant Minho, 4800-058 b of lowcombinations an increasing use of new or new material like fibre-reinforced plastics (FRP) as they provide a high lightweight potential due materials to the combination density and high tensile strength. pre-impregnated sheets with a Unochapecó, 89809-000 Chapecó, SC, Brazil Meanwhile lightweight potential due to the combination of low density and high tensile strength. pre-impregnated with a thermoplastic matrix reinforced with woven carbon fibres are commercially available.Meanwhile This has led in a significant sheets cost reduction thermoplastic reinforced woven carbon commercially available.properties, This has led in a significant cost reduction and hence, thematrix FRP have becomewith affordable for largefibres scaleare production. The material in particular the forming and and hence, the FRP become affordable large production.metal The material properties, theTherefore, forming and failure behaviour of have the FRP, differ strongly for from thatscale of conventional materials like steel in or particular aluminium. new failure of the FRP, differ strongly from that of conventional metal materials or aluminium. new Abstract materialbehaviour characterisation techniques, investigation methods as well as numerical modelslike aresteel required. The mainTherefore, focus of this material techniques, investigation methods wellfor as numerical models are required. main focus of this paper liescharacterisation on the development of a non-orthogonal material as model the FRP, its implementation in aThe commercial FE-software paper the development of a non-orthogonal materialprocesses model for the implementation inSince a commercial FE-software as welllies as use ofof a combined experimental-numerical procedure forFRP, material characterisation. the properties of these Under theonconcept "Industry 4.0", production will be its pushed to be increasingly interconnected, as well as are on strongly the use on oftemperature aacombined experimental-numerical procedure for material characterisation. Sincecapacity the properties these materials dependent, thenecessarily, forming process of reinforced thermoplastics typically carried out at of elevated information based real time basis and, much more efficient. In this is context, optimization materials are strongly temperature dependent, forming of reinforced thermoplastics is typically carried outmodel at elevated temperatures. Thus, temperature sensitivity hasthe to be takenprocess into account duringalso experimental testing as well as in the goes beyond the traditional aim of capacity maximization, contributing for organization’s profitability and value. temperatures. temperature sensitivity has toout be taken on intoanaccount during experimental testing asoptimization well asFor in this the model approach.lean TheThus, model parameterisation is carried based iterative numerical optimization procedure. purpose, Indeed, management and continuous improvement approaches suggest capacity instead the of approach. The model parameterisation is carriedby outmeans basedof ondigital an iterative numerical optimization procedure. For thismodel purpose, experimentally obtained results are investigated image correlation and linked with the numerical in the maximization. The study of capacity optimization and costing models is an important research topic that deserves experimentally obtained results are investigated by means of digital image correlation and linked with the numerical model in combination with an automated optimization process. contributions from both the practical and theoretical perspectives. This paper presents and discusses a mathematical combination with an automated optimization process. model for capacity management based on different costing models (ABC and TDABC). A generic model has been © 2018 The Authors. Published by Elsevier B.V. © 2019 Theand Authors. by Elsevier B.V. developed it wasPublished used to analyze idleBY-NC-ND capacity and to design strategies towards the maximization of organization’s © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is The an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) value. trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity This is an openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection Selection and and peer-review under under responsibility responsibility of of the the organizing organizing committee committee of of SHEMET SHEMET 2019. 2019. optimization might hide under operational inefficiency. Selection and peer-review responsibility of the organizing committee of SHEMET 2019. © 2017 The Authors. Published by Elsevier B.V. Keywords: Fabric-Reinforced Thermoplastics; FE-Modelling; Material Characterisation Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference Keywords: Fabric-Reinforced Thermoplastics; FE-Modelling; Material Characterisation 2017. Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency
1. Introduction
* Corresponding author. E-mail address: [email protected] * The Corresponding author. cost of idle capacity is a fundamental information for companies and their management of extreme importance E-mail address: [email protected] in modern production systems. In general, it is defined as unused capacity or production potential and can be measured 2351-9789 © 2018 The Authors. Published by Elsevier B.V. in several ways: tons ofunder production, available hours of manufacturing, etc. The management of the idle capacity This is an open access the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) 2351-9789 © 2018 Thearticle Authors. Published by Elsevier B.V. * Paulo Afonso. Tel.: +351 253 510 761; fax: +351 253 604 741 Selection under responsibility of the organizing committee of SHEMET 2019. This is an and openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) E-mail address: [email protected] Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under of the scientificbycommittee the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2019responsibility The Authors. Published Elsevier of B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 10.1016/j.promfg.2019.02.160
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1. Introduction New materials, like woven composites, offer a combination of low density and high specific tensile strength. Thus, they provide a high light-weight potential and are increasingly used. However, the high strength is only offered in the fibre direction. This strong anisotropic behaviour differs from commonly used metal materials. Due to their woven fibre arrangement, organic sheets offer a higher overall stiffness as compared with UD-laminates. Furthermore, the formability is increased by yarn rotation and stretching of the undulated fibres [1]. The use of thermoplastics as a matrix material is gaining more importance particularly in the automotive industry, since it can be formed to a complex geometry under elevated temperatures [2; 3]. The used material, organic sheets, consist of a thermoplastic PA6 matrix and woven glass fibres as reinforcement. The overall properties are a superimposition of matrix properties and fibre properties [4]. This leads to a strong anisotropic behaviour, where the highest strength is observed in the fibre direction. Therefore, the shearing of the fibres at the cross points is the main in plane forming mode of woven fabrics [5]. During the deformation, the angle between the fibres changes and once it reaches a critical value, referred to as locking angle, the shear stress undergoes a substantial increase due to shear locking. In the molten state, the shear resistance can be assumed to be negligible until reaching the specific locking angle of the woven fabric [6]. In contrast to the glass fibres, the forming behaviour of the thermoplastic matrix material is strongly temperature dependent [2]. Above the melting temperature, the stiffness and strength are significantly reduced. However, in case of a matrix solidification during the forming phase, the influence of the matrix on the overall forming behaviour increases significantly. The influence of temperature evolution has been investigated in [7] by means of the draping behaviour of a double dome geometry. The authors have concluded that the temperature evolution over the entire process including the heating, transfer and forming phase has a decisive impact on the precise prediction of the draping process. The bending properties of reinforced thermoplastic sheets were measured in [8] by means of cantilever test, which were performed in an environmental chamber. In order to characterise the main in plane forming mode, the shear deformation, the picture frame or bias extension test can be applied. Both methods are discussed in [6] and [9] and lead to similar results. The material properties of the glass fibres are analysed by means of uniaxial tensile tests, whereby the loading is introduced in fibre direction. This contribution deals with experimental investigations to determine the mechanical properties of organic sheets. Considering of the temperature dependency the tests have been performed in a wide temperature range. Based on the experimental findings, a numerical model has been developed and implemented into the commercial FESoftware Abaqus CAE 6.14. The model parameterisation has been subsequently carried out by a numerical optimisation process. 2. Material characterisation of organic sheets In the scope of this work organic sheet with a PA6 matrix reinforced with woven glass fibres (2/2 twill) and a thickness of 1 mm was investigated. Since the considered thermoplastic PA6 is strongly temperature dependent the tests were carried out inside an environmental chamber at temperatures between RT and 240 °C and a test speed of 1 mms-1. In order to take the effect of fibre direction into account tensile tests in fibre direction as well as bias extension tests have been performed. The experimental tests have been carried out at LMT, ENS Paris-Saclay and for each parameter combination up to 5 repetitions have been conducted. During the bias extension test the load is introduced with an initial angle of 45° to the fibres, which leads to pure shear in the middle of the sample. A detailed explanation of the bias extension test is given in [5]. During the tests the displacements have been captured by means of an optical measurement system with a frame rate of 4 pictures/mm. The ratio of height (H) to width (W) was chosen to 2.33 and thus higher than 2.0 in order to achieve more homogeneous shear deformation in the bias extension tests like proposed in [5]. Furthermore, heating tests have been carried out in order to ensure a homogeneous sample temperature. The used test setup as well as the fibre orientations for the different tests are shown in Fig. 2. In Fig. 2b, c measured force displacement curves for uniaxial tensile test and bias extension tests are presented for different testing temperatures at a test speed of 1 mm/s. The comparison between tensile test (b) and bias extension test (c) shows the strong influence of the fibre direction. During tensile testing, the introduced load is borne by the fibres. The observed maximum forces
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are in general higher as compared to the bias extension test and only small strains exist before fracture. The temperature dependency is also small. a
c
b
Sample
6000
10000
5000
8000
4000
6000 RT 180 °C 220 °C
4000 2000 0
Bias extension test, fibre direction
Forec F [N]
Tensile test, 90 fibre direction
12000
Forec F [N]
Upper die
Lower die
0
1 2 3 Displacement u [mm]
RT 215 °C
185 °C 220 °C
3000 2000
1000 4
0
0
10 20 Displacement u [mm]
30
Fig. 1. (a) Experimental test setup for uniaxial tensile test and bias extension test with corresponding fibre directions; (b) Experimental obtained force-displacements curves for various temperatures at a test speed of 1 mm/s: uniaxial tensile tests with 90° fibre direction; (c) bias extension tests with ± 45° fibre direction.
In contrast, the results of bias extension test demonstrate a more complex material behaviour. Due to the loading direction the mechanical behaviour is a superimposition of matrix and fibre properties. A high temperature dependency can be seen due to the increased influence of the thermoplastic matrix material. The initial fibre orientation in the bias extension test is ±45°. However, the shear deformation leads to fibre rotation with increasing displacement and the fibre get more and more aligned in loading direction. Friction between the fibre bundles themselves as well as between the fibre bundles and the matrix influence the movement. If the specific lock angle is reached, further deformation leads to a sharp increase of force as well as out of plane wrinkling due to compaction of fibre bundles. For a detailed material description the change of shear angle with increasing displacement is of great interest. The current shear angle 𝛾𝛾 has been calculated based on optical measurement data by means of digital image correlation with following equation (1) [10]: 𝜋𝜋 1 + 𝜀𝜀𝑥𝑥 (1) 𝛾𝛾 = − 2 ∗ arctan ( ) 2
1 + 𝜀𝜀𝑦𝑦 1 𝐻𝐻 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾) = [( − 1) 𝐹𝐹𝑎𝑎𝑎𝑎 (cos − sin ) − 𝑊𝑊 𝐹𝐹𝑆𝑆𝑆𝑆 ( ) cos ] (2𝐻𝐻 − 3𝑊𝑊) cos 𝛾𝛾 𝑊𝑊 2 2 2 2
(2)
where εx and εy are the engineering strains. Furthermore, the material description requires the actual shear stress which depends on the shear angle. This dependency leads to a high non-linear behaviour. In this work a constant test speed of 1mms-1 has been investigated. Therefore, the matrix viscosity is assumed to be rate independent. Following the basic research of [11] the normalised shear force 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾) per unit length can be calculated based on the geometric properties of the bias extension sample with equation (2), whereby H and W are the specimen height and width, respectively. The current value of 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾) depends on 𝐹𝐹𝑆𝑆𝑆𝑆 (𝛾𝛾⁄2), hence the shear force needs to be calculated in an iterative procedure by means of a Matlab code. The shear angle is taken into account by correlating the measured axial force 𝐹𝐹𝑎𝑎𝑎𝑎 with the optical measured shear angle. The shear angle dependent shear stress τ12 can be derived by referring the shear force on the specimen thickness. The Fig. 2 presents exemplary calculated shear stress - shear angle curves for the temperature of 185 °C, 200 °C and 215 °C and a test speed of 1 mm/s. The shear angles distribution calculated by means of digital image correlation (DIC) show some variation at the sides. This is caused by loose fibre ends, which are not sheared but captured by the DIC and lead to failures in this region. Therefore, during the investigations, three regions of interest (ROI) have been chosen on the sample surface and a representative mean shear angle has been calculated and subsequently used for further calculation steps (Fig. 2a). The nonlinear dependency between shear stress and shear angle can be described by means of a polynomial function. In order to obtain numerical stability a continuous description of the temperature dependent material behaviour is necessary.
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a
b
Figure 2 (a) Shear angle obtained by means of DIC at a displacement of 18 mm for a temperature of 215 °C ; (b) shear stress - shear angle curve for the temperatures of 185 °C, 200 °C and 215 °C.
Shear angle – shear stress relationship
12
=
1 𝛾𝛾
+
2 𝛾𝛾
+
𝛾𝛾 +
𝛾𝛾 2 + 𝛾𝛾
)∗
Temperature influence
Shear stress [MPa]
Therefore, the polynomial function has been extended by an multiplicative term, considering an exponential relationship, whereby the coefficients p1 – p5 are material specific constants and T 0 is a scaling factor for the temperature dependency (Fig. 3a). The Fig. 3b presents the predicted shear stress surface as well as the experimental optained shear angle – shear stress curves, marked with black dots. The correlation coefficient is determined to R2=0.99, thus the introduced model provides a good description of the material behaviour under consideration of elevated temperatures. a b 80 60 40 20 0
190
200
210
0
0.2
0.4
0.6
0.8
1.0
Fig. 3. (a) Polynomial function for the description of the nonlinear relationship between shear stress and shear angle under consideration of temperature dependency; (b) Comparison between experimental data and the model predicted shear stress.
3. Numerical model and material data optimisation For numerical description of organic sheets deformation, a non-orthogonal hypo-elastic material model has been established and implemented in the FE-Software Abaqus CAE 6.14 by means of a user defined routine. Hypo-elastic material models are widely used in FE codes to describe the material behaviour at large strains [12]. Due to strong changes of the fibre direction during the forming process the shear stiffness needs to be described as a function of the shear angle. Taking into account the works [1; 13] the current fibre direction during deformation can be calculated by means of fixed body rotation matrix and deformation gradient. A schematic representation of fibre rotation is given in Fig. 4. The deformation caused change of fibre direction is considered by transforming the obtained strain increment from the global orthonormal frame (g) to the fibre frames (f). The stress and strain increments for both fibre directions (f1, f2) are calculated within the fibre frames based on the updated stiffness matrix and subsequently transferred back to Cauchy stress. The update of the stress increments is carried out by using the Hughes and Winget formulation [14]. 1 1 and 𝑓𝑓2𝑥𝑥 The shear stress increment is calculated in the non-orthogonal frame defined by the two fibre directions 𝑓𝑓1𝑥𝑥 based on the change of the angle 𝜓𝜓 and the updated nonlinear shear modulus. Finally, the shear stress increment is also transferred back to the global orthonormal frame. The stress tensor is updated by a superimposition of the calculated stress increments of the different frames. The superimposition is carried out in the global orthonormal frame. As an initial approach only tensile modulus in fibre direction and shear modulus depending on shear angle have been considered. The effect of poisson’s ratio has been neglected.
Bernd-Arno Behrens et al. / Procedia Manufacturing 29 (2019) 443–449 Bernd-Arno Behrens / Procedia Manufacturing 00 (2018) 000–000 Initial state
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Deformed state
=initial 1st fiber direction = initial 2nd fiber direction =initial orthonormal material frame
= deformed 1st fiber direction = deformd 2nd fiber direction =deformed orthonormal material frame
Fig. 4. Schematic representation of fibre orientation during deformation of woven fabrics.
Due to the superimposition of matrix and fibre properties, the nonlinear relationship between shear angle and shear stress as well as the calculation of the shear angle based on DIC-data, the parameterisation of the numerical material model is challenging. Therefore, an iterative numerical optimisation process has been developed. The process itself is controlled by a Python file, which modifies the user defined input parameters, the base points p1-p5 inside the Abaqus job file, submit the job as well as perform post processing operations on the results file. The shear stress – shear angle behaviour is described by the base points. To modify the material behaviour in fibre direction, the experimental determined elastic modulus 𝐸𝐸11 in fibre direction is multiplied by a scaling factor A. is introduced, which is multiplied by the elastic modulus in fibre direction. The sum of squared errors of the experimental measured and numerical calculated axial forces has been chosen as target figure. The optimisation is carried out by means of multi objective genetic algorithm (MOGA), which can be used efficiently for multi objective optimisation problems. In order to meet the shape of the force – displacement curve the deviation at characteristic displacements as well as the total deviation have been chosen as objectives. For a first approach, the optimisation process has been performed for bias extension test at a temperature of 215 °C and a test speed of 1 mms-1. b
c
Shear angle γ= -2θ=1.05 rad
d
γ ≈ 1.02 rad
ROI
Shear angle γ [rad] 1.1
ROI
1000 800
600
0.83
400
0.55
200
0.28
0.0
Experiment First fit Optimised material data
1200
γ ≈ 1.06 rad
2θ=0.52 rad
1600 1400
Axial force [N]
a
0
x y
0
5
10 15 Displacement [mm]
20
Figure 5 Comparison of manually measured shear angle (a), the shear angle distribution obtained by (b) DIC and by (c) numerical simulation at a displacement of 18 mm; (d) force – displacement curves based on experimental data and numerical simulation at a temperature of 215 °C.
The determination of the shear angle evolution by means of DIC analysis as well as the numerical model itself and the numerical optimisation process are validated by a comparison of numerical calculated data with experimental data. Fig. 5 shows a comparison between manually measured angle between the fibre direction (Fig. 5a), the shear angle obtained by optical measurement and DIC analysis (Fig. 5b) as well as the numerical calculated shear angle at the ROI position in the middle of the sample (Fig. 5c) for a bias extension tests at 215 °C and a displacement of 18 mm.
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The numerical results, presented in Fig. 5c, are calculated based on the optimised material data. An overview over the initial material parameters, determined based on the experimental data and the finally used optimized data is given in table 1. Table 1 Overview over initial and optimised material data for the numerical simulation p1
p2
p3
p4
p5
A
E11 [GPa]
Initial data
97.00
-524.00
1332.00
-1485.00
609.00
1.00
8000
Optimised data
106.89
-513.22
1343.39
-1610.50
637.09
1.19
8000
The determined shear angles show a good qualitative agreement. A comparison of experimental measured and numerical calculated force displacement curves is presented in Fig. 5d. The numerical calculated curves are based on the initial determined material data as well as on the material data which has been obtained after the numerical optimisation. The optimised material data shows a good agreement with the experimental measured curve. Furthermore, it can be seen that the optimisation process leads to a strong increase in the quality of the numerical results. 4. Conclusion In this research the complex material behaviour of organic sheets, which consist of a thermoplastic matrix with woven glass fibres, has been characterised by means of uniaxial tensile tests and bias extension tests under consideration of a wide temperature range. The results show a strong dependency on fibre direction, shear angle and temperature. A continuous model approach, which takes the change of fibre direction into account, has been implemented by means of a user subroutine. The model has been parametrized under consideration of the nonlinear relationship between shear angle and shear modulus. The change of the shear angle has been determined by means of digital image correlation in an incremental procedure based on the optical measured displacement data. The material model has been parameterised by means of an iterative optimisation process. The comparison of experimental measured and numerical calculated shear angle distribution showed a good qualitative agreement. Furthermore, the optimisation of the material data by means of a genetic optimisation algorithm leads to estimation of force displacement curves. Acknowledgements This work is funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) through the International Research Training Group 1627 “Virtual Materials and Structures and their Validation”. References
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