A Material Model for the Orthotropic and Viscous Behavior of Separators in Lithium-Ion Batteries Under High Mechanical Loads

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Article A Material Model for the Orthotropic and Viscous Behavior of Separators in Lithium-Ion Batteries under High Mechanical Loads

Marian Bulla 1,* , Stefan Kolling 2 and Elham Sahraei 3,4

1 Altair Engineering GmbH, Josef-Lammerting-Allee 10, 50933 Cologne, Germany 2 Institute of Mechanics and Materials, Technische Hochschule Mittelhessen, Wiesenstr. 14, 35390 Giessen, Germany; [email protected] 3 Electric Vehicle Safety Lab (EVSL), Temple University, Philadelphia, PA 19122, USA; [email protected] 4 Impact and Crashworthiness Lab, Massachusetts Institute of Technology, Cambridge, MA 02139, USA * Correspondence: [email protected]; Tel.: +49-160-96367668

Abstract: The present study is focused on the development of a material model where the orthotropic- visco-elastic and orthotropic-visco-plastic mechanical behavior of a polymeric material is considered. The increasing need to reduce the climate-damaging exhaust gases in the automotive industry leads to an increasing usage of electric powered drive systems using Lithium-ion (Li-ion) batteries. For the safety and crashworthiness investigations, a deeper understanding of the mechanical behavior under high and dynamic loads is needed. In order to prevent internal short circuits and thermal runaways within a Li-ion battery, the separator plays a crucial role. Based on results of material tests, a novel material model for ﬁnite element analysis (FEA) is developed using the explicit solver Altair Radioss. Based on this model, the visco-elastic-orthotropic, as well as the visco-plastic-orthotropic, Citation: Bulla, M.; Kolling, S.; behavior until failure can be modeled. Finally, a FE simulation model of the separator material Sahraei, E. A Material Model for the is performed, using the results of different tensile tests conducted at three different velocities, Orthotropic and Viscous Behavior of · −1 · −1 · −1 Separators in Lithium-Ion Batteries 0.1 mm s , 1.0 mm s and 10.0 mm s and different orientations of the specimen. The purpose under High Mechanical Loads. is to predict the anisotropic, rate-dependent stiffness behavior of separator materials in order to Energies 2021, 14, 4585. https:// improve FE simulations of the mechanical behavior of batteries and therefore reduce the development doi.org/10.3390/en14154585 time of electrically powered vehicles and consumer goods. The present novel material model in combination with a well-suited failure criterion, which considers the different states of stress and Academic Editor: Carlos Miguel anisotropic-visco-dependent failure limits, can be applied for crashworthiness FE analysis. The Costa model succeeded in predicting anisotropic, visco-elastic orthotropic and visco-plastic orthotropic stiffness behavior up to failure. Received: 29 June 2021 Accepted: 23 July 2021 Keywords: polyethylene separator; visco-elasticity; visco-plasticity; elasto-plasticity; orthotropy; Published: 29 July 2021 material model; ﬁnite element model; safety; crashworthiness

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional afﬁl- 1. Introduction iations. Increasing efficiency, defined as energy consumption relative to the distance travelled, is considered a core development goal of the automotive industry. Customer demands for vehicles with larger mobility range in combination with legal requirements for climate protection by limiting CO emissions are the motivation for increasing the efficiency of electric Copyright: © 2021 by the authors. 2 Licensee MDPI, Basel, Switzerland. vehicles. In addition to the ecological motivation, economic factors are a major incentive for This article is an open access article vehicle manufacturers to focus on the reduction in the development time of their vehicles by distributed under the terms and using high fidelity computational simulation tools such as finite element analysis. Product conditions of the Creative Commons development to raise battery performance of portable electronic devices such as laptops or Attribution (CC BY) license (https:// mobile phones also benefits from an increased accuracy of FEA results [1,2]. To achieve creativecommons.org/licenses/by/ this goal, more precise FE models are required to describe the mechanical behavior of all 4.0/). involved materials under high mechanical loads. Recent developments, especially in the

Energies 2021, 14, 4585. https://doi.org/10.3390/en14154585 https://www.mdpi.com/journal/energies Energies 2021, 14, 4585 2 of 17

automotive industry, with focus on the electrification require precise prediction of how Li-ion cells respond to high mechanical load cases, due to misuse or crash events. Here, understanding the properties of the polymeric separator layer, maintaining a physical barrier between anodes and cathodes plays an important role. Therefore, knowing its mechanical behavior, especially in the nonlinear domain, is essential. In the past, the determination of material properties for plastics was mainly limited to the determination of elastic parameters and yield strength. These characteristic values were determined almost exclusively under quasi-static loading at one defined orientation. Typical characteristic values for FE modeling of plastics are available, for example, in the Computer Aided Material Preselection by Uniform Standards (CAMPUS) database [3]. Since the largest part of energy in crash processes is dissipated by plastic deformation, the material models, used for crashworthiness analysis, must also be able to describe the plastic material behavior accurately. The determination of the plastic material behavior is difficult for many plastics, since engineering strains with more than 100% value can occur under tensile loading. In addition, it is possible to have inhomogeneous distribution of strains in such plastic parts, which is also referred to as strain localization [4]. In the tensile test, this strain localization is noticeable by a constriction of the specimen. Standard strain measurement methods, such as strain gauges or extensometers, cannot consider these localizations. Therefore, the displacement at the clamping of the specimen is often used for strain measurement in the entire specimen. This determination of the strain distribution over the free gripping length of the specimen is also proposed in the standard for determining the tensile properties of plastics, ISO EN 527 [5]. However, this procedure can lead to considerable errors in the strain measurement as local strain overshoots are not detected. A reliable determination of the stress-strain curve, which represents the physical material behavior, which is necessary as a basis for the development of material models, is thus not readily available of such materials [6,7]. A further difficulty arises from the widely varying material behavior of different plastics [8]. The behavior of polymers is affected by many factors, e.g., moisture, temperature and others. In addition, polymers are sensitive to the rate of loading. An increase in the strain rate often results in a decrease in the ductility of the polymer. In contrast, the modulus and the yield or tensile strength increase with increasing loading rate [9]. Many authors have examined the influence of temperature and strain rate on the material behavior of polymers. Walley and Field [10] performed tests at room temperature over strain rates ranging from 10−2 s−1 to 104 s−1. Carnella [11] provided rate-dependent tests at high strain rates under tension and compression. Therefore, several problems arise in the development of material models for the numerical representation of thermoplastic components under dynamic and high load cases. Neither in commercial simulation programs nor in the literature do material models exist that have been developed specifically for the crash simulation of thermoplastic components. For this reason, material models developed for metallic materials are often used in industrial applications to illustrate the dynamic material behavior of thermoplastics. This includes, above all, the von Mises plasticity models. However, this material model is not suitable for mapping the mechanical behavior of thermoplastics and, as shown in [12] and this leads to incorrect results. The reason for this lies in two aspects, which are important for a precise mechanical response prediction: (1) viscose effects in the elastic domain and (2) anisotropic material hardening in the plastic domain. Deeper investigation of anisotropic behavior is performed in Pfeiffer et al. [13]. In this work, the focus is on the modeling of the separator material used in Li-ion batteries using time explicit method of nonlinear dynamic FEM. Figure1 shows the multiscale view of a Panasonic 18650AF cell used in electric cars as well as in notebooks and other consumer goods showing the cell cross section and the multilayered structure inside that, including electrodes and the separator. Within this work, a novel material model is developed, which considers the visco-elastic-orthotropic as well as the visco-plastic-orthotropic (strain rate dependent) behavior for modeling of the separator. Energies 2021, 14, x FOR PEER REVIEW 3 of 18 Energies 2021, 14, 4585 3 of 17

model is developed, which considers the visco-elastic-orthotropic as well as the visco- plastic-orthotropicIn addition to a (strain suited rate failure dependent) model [behavior14,15], the for materialmodeling stiffens, of the separator. and the damage and fractureIn of addition a high to dynamic a suited mechanical failure model load [14,15 can], bethe predicted.material stiffens, In Bulla and et the al. damage [16], a failure and fracture of a high dynamic mechanical load can be predicted. In Bulla et al. [16], a model is proposed, which considers the state of stress as well as the strain-rate effect based failure model is proposed, which considers the state of stress as well as the strain-rate on the tests of a PE separator. effect based on the tests of a PE separator.

FigureFigure 1. Panasonic 1. Panasonic CGR18650AF CGR18650AF Li-Ion Li-Ion cell: cell: ( a(a)) realreal photophoto and and (b (b) )schematic schematic cut-view, cut-view, (c) Finite-Element (c) Finite-Element-mesh-mesh using an using an 1 × 1×1 ×1 1× mm 1 mm representative representative volume, volume, ( (dd)) detailed detailed view view of the of the3d hexahedron 3d hexahedron elements elements with an with average an element average size element of 0.01 size of 0.01 ×× 0.01 ×× 0.010.01 mm mm showing showing the theseparator separator layer layer in white. in white.

2. Separator Materials and Methods 2. Separator Materials and Methods The investigated commercial polymeric separator was provided to us from a battery manufacturingThe investigated company. commercial The exact polymeric chemical composition separator wasof the provided materials to used us fromin these a battery manufacturingsamples was not company. specified. TheThe exactseparators chemical had the composition porosity of 36–46% of the materials[17]. used in these samplesIn wasthis study, not specified. the mechanical The separators properties hadof the the specimen porosity were of 36–46%studied by [17 conducting]. tensileIn this tests study, on small the strips mechanical cut out propertiesin 2 perpendicular of the specimenspatial directions were studied corresponding by conducting to tensilemachined tests direction on small (MD) strips and cut transversal out in 2 perpendicular direction (TD) of spatial the separator directions samples. corresponding Figure to machined2 shows directionthe experimental (MD) and setup transversal which was direction used for (TD) performing of the separator the tensile samples. tests. EachFigure 2 showsspecimen the experimentalwith dimensions setup of 12 × which 60 × 0.025 was mm used was forglued performing at each end theto metal tensile specimen tests. Each specimenholders which with dimensions led to a free oftest 12 length× 60 of× 360.025 mm. mm The was specimens glued atwere each marked end to with metal a gray specimen holderspattern which to enable led todigital a free image test length correlation of 36 mm.(DIC), The as specimensshown in Figure were marked3a,b. The with 3D a gray patternARAMIS to enable system digital from imageGOM correlation(GOM GmbH, (DIC), Schmitzstraße as shown in 2, Figure 381223a,b. Braunschweig, The 3D ARAMIS Germany) was used for analysis of the strain field, shown in Figure 3c,d. Each test was system from GOM (GOM GmbH, Schmitzstraße 2, 38122 Braunschweig, Germany) was repeated 5 times to ensure repeatability. Three different tension velocities were used for used for analysis of−1 the strain−1 field, shown− in1 Figure3c,d. Each test was repeated 5 times−1 to testing: 0.1 mm·s , 1 mm·s and 10 mm·s corresponding to strain rates 0.002778 s , −1 ensure0.02778 repeatability. s−1, and 0.2778 Three s−1. different tension velocities were used for testing: 0.1 mm·s , 1 mm·s−1 and 10 mm·s−1 corresponding to strain rates 0.002778 s−1, 0.02778 s−1, and 0.2778 s−1. Figure3 shows one specimen before and after testing. The recording frequency for the force and displacement measurements ranged from 2 Hz for the slowest tension velocity up to 40 Hz for the fastest ones. The measured elongation in MD direction is in the range of 10–19 mm, which results in 27–53% total strain. Figure4 shows the engineering stress vs. engineering strain curves for the three tested velocities in MD. The Young’s moduli are computed using the initial slope. Within this work, the statistical R-squared value (coefficient of determination) is used for the fitted regression line, starting with the first 2 measuring points. When the adjusted R-squared value drops below a defined threshold, we use the previous fit line as

the Young’s modulus. After failure, the specimen retained almost all its extension, which indicates a low relaxation. In conclusion, the material exhibits high plasticity and low elasticity behavior. Localization of strain occurs close to the center and proceeds to one of the ﬁxed ends, where ﬁnal rupture occurs. Energies 2021, 14, 4585 4 of 17 Energies 2021, 14, x FOR PEER REVIEW 4 of 18

Energies 2021Figure, 14, x FOR 2. Experimental PEER REVIEW setup for the tension tests on the PE isolator specimen. 5 of 18 Figure 2. Experimental setup for the tension tests on the PE isolator specimen. Figure 3 shows one specimen before and after testing. The recording frequency for the force and displacement measurements ranged from 2 Hz for the slowest tension velocity up to 40 Hz for the fastest ones.

FigureFigure 3. Set-up 3. ofSet-up prepared specimens of prepared in MD before specimens the test (a) in and MD at the before end of the the test, testafter failure (a) and (b). Specimen at the using end of the test, after DIC measurement at the beginning of the test (c) and at the end of the test, briefly before failure (d), showing the inhomogeneousfailure (b strain). Specimen field distribution using withing DIC measurementthe specimen. at the beginning of the test (c) and at the end of the test, brieﬂy before failure (d), showing the inhomogeneous strain ﬁeld distribution withing the specimen. The measured elongation in MD direction is in the range of 10–19 mm, which results in 27–53% total strain. Figure 4 shows the engineering stress vs. engineering strain curves for the three tested velocities in MD. The Young’s moduli are computed using the initial slope. Within this work, the statistical R-squared value (coefficient of determination) is used for the fitted regression line, starting with the first 2 measuring points. When the adjusted R-squared value drops below a defined threshold, we use the previous fit line as the Young’s modulus. After failure, the specimen retained almost all its extension, which indicates a low relaxation. In conclusion, the material exhibits high plasticity and low elasticity behavior. Localization of strain occurs close to the center and proceeds to one of the fixed ends, where final rupture occurs.

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Figure 4. Tested engineering stress vs. strain curves in machine direction (MD = 0°), at three different ◦ FigurevelocitiesFigure 4.4. TestedTested with the engineeringengineering corresponding stress stress vs.Young’s vs. strainstrain moduli curves curves in (dashedin machine machine lines). direction direction (MD (MD = = 0 0°),), atat threethree differentdifferent velocitiesvelocities withwith thethe correspondingcorresponding Young’s Young’s moduli moduli (dashed (dashed lines). lines). The measured elongation in TD direction was in the range of 40–53 mm, which The measured elongation in TD direction was in the range of 40–53 mm, which resultsThe in measured 111–147% elongation total strain. in Figure TD direction 5 show wass the in engineering the range of stress 40–53 vs. mm, strain which curves results for results in 111–147% total strain. Figure 5 shows the engineering stress vs. strain curves for inall 111–147% three tested total velocities strain. Figure in TD.5 showsThe Young’ the engineerings moduli are stress computed vs. strain using curves the for initial all three slope all three tested velocities in TD. The Young’s moduli are computed using the initial slope testedas shown velocities in Figure in TD. 6, Thewhich Young’s shows moduli a cutout are (indicated computed with using the the red-dotted initial slope rectangle as shown in as shown in Figure 6, which shows a cutout (indicated with the red-dotted rectangle in inFigure Figure 5).6 , which shows a cutout (indicated with the red-dotted rectangle in Figure5). Figure 5).

FigureFigure 5.5. TestedTested engineeringengineering stressstress vs.vs. strainstrain curvescurves inin transversetransverse directiondirection (TD(TD == 9090°),◦), atat threethree differentFigure 5. velocitiesTested engineering with the stresscorresponding vs. strain Young’scurves in moduli transverse (dashed direction lines) (TD (dotted = 90°), rectangle at three differentdifferent velocities velocities with with the the corresponding corresponding Young’s Young’s moduli moduli (dashed (dashed lines) (dotted lines) rectangle(dotted indicatesrectangle indicates the zoomed area). theindicates zoomed the area). zoomed area).

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Figure 6. Detailed view of engineering stress vs. strain curves in TD direction (TD = 90°), 90◦), at three different velocities with the corresponding Young’sYoung’s modulimoduli (dashed(dashed lines).lines).

Table1 1 shows shows thethe estimatedestimated Young’sYoung’s modulimoduli overover thethe nominalnominal strainstrain ratesrates (based(based onon the usedused tensiontension velocities). velocities). The The values values for thefor diagonalthe diagonal direction direction DD (=45 DD◦) were(= 45°) obtained were obtainedfrom Sahraei from [17 Sahraei]. [17].

Table 1. Young’s modulimoduli dependentdependent onon orientationorientation andand engineeringengineering strainstrain rate.rate.

𝜺. −1 −1 . 𝜺 −1 −1 .𝜺 −1 −1 OrientationOrientation (Degree)(Degree) ε1𝟏= = 0.002778 0.002778 (s (s) ) ε2=𝟐 0.02778= 0.02778 (s (s) ) ε3𝟑= 0.2778= 0.2778 (s (s) ) 0 1300 1300 MPa MPa 1400 1400 MPa MPa 1500 1500 MPa MPa 45 800 800 MPa MPa 900 900 MPa MPa 950 950 MPa MPa 90 300 300 MPa MPa 400 400 MPa MPa 500 500 MPa MPa

3. Modeling Modeling and and Results 3.1. Viscosity Viscosity in Elastic Region Hereinafter the denotation of a matrix is deﬁneddefined as [∎], and of a vector as {∎}. . The viscosity effect is easily treated by explicit solvers because of the usage the equation of motion and solving for unknownunknown values without the need of inverting the entire model: .. . [M] xn + [C] xn + [K]{xn} = {Fext(tn)}, (1) M𝑥 + 𝐶𝑥 + 𝐾𝑥 = 𝐹(𝑡), (1) where: where:[M ] = Mass matrix (kg), [C𝑀] = = Damping Mass matrix matrix (kg), (kg ·m·s−1), 𝐶 −1 [K] = = StiffnessDamping matrix matrix (N (kg·m·s·m−1), ), −1 𝐾.. = Stiffness matrix (N·m ), −2 xn = Acceleration vector (m·s ), 𝑥. = Acceleration vector (m·s−2), · −1 xn = Velocity vector (m s−1 ), 𝑥 = Velocity vector (m·s ), {xn} = Displacement vector (m), 𝑥 = Displacement vector (m), {Fext(tn)} = External force (N). 𝐹(𝑡) = External force (N).

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Due to the usage of the small strain formulation, where the stress increment is inte- grated at each time step and considering a stiffness that depends on the local strain rate in individual time steps, the integration schema—which solves the equation of motion in the time domain—leads to high oscillations of the total strain rates. Figure7a shows consider- Energies 2021, 14, x FORable PEER strain REVIEW rate oscillations within the specimen during loading in the elastic domain. To 9 of 19

reduce these oscillations, several approaches are used by commercial solvers [18–20].

(a) (b) Figure 7. Visualization of strain rates within a tensile test specimen in the elastic domain. Unfiltered (a) and filtered (b) Figure 7. Visualization of strain rates within a tensile test specimen in the elastic domain. Unfiltered (a) and filtered (b) using using the exponential moving average filter. the exponential moving average filter.

The approach3.2. Plasticity chosen here is to apply an exponential moving average filter [21] to the strain rate as a timeIn the series novel filter developed that removes visco-elastic-plasti high frequencyc oscillations:model, the true-stress vs. true-plastic- strain curves must be derived from the tested engineering-stress vs. engineering-strain . average . average . curves. For theεn calculation= αEMA of· εthen−1 plasticity+ (1 − cuαEMArves) in·εn the, MD direction, the approach (2) based on Swift–Voce interpolation, described by Bulla [16], is used. However, the TD behavior where: is showing a much higher plasticity range beyond 100% strain. Figure 8 shows one . average −1 εn =representative Filtered actual engineering strain rate (sstress), vs. engineering strain curve close to 120% total strain . average −1 εn−1 =before Filtered the previousmaterial strainfractures rate (green (s ), curve). For usage in an elasto-plastic material model, . −1 εn = Actualthe parametrization (not filtered) strain for rate the (s engineering), stress vs. engineering strain curve must be αEMA =modified Exponential further. moving The filtertrue stress value (vs.−). true strain curve (Figure 8: red curve) is derived and then the elastic portion of the strain is subtracted from the total strain, using the linear With the αEMA parameter it is possible to control the response of the filter. If αEMA = 0, no filtering isstrain applied.–stress Values relation close corresponding to 1 apply a strong to the low-pass strain filter.rate, Figuredependent7b shows Young’s the modulus result of applying(Figure the 6: straindashed rate lines). filtering. It leads to a smoother strain rate distribution and more realistic behaviorDue to withinthe need the for FE values model. beyond The strain the curve, rate filtering derived plays from antesting, important it is necessary to role in explicitextrapolate time integration. the true stress vs. true plastic strain curve. Based on the work of G’Sell et al. [22] who tested such high ductile polymer materials, the parameter identification is 3.2. Plasticitycarried out using the modified equation: In the novel developed visco-elastic-plastic model, the true-stress vs. true-plastic- 𝜎 =𝐴𝐵1−exp−𝐶𝜀. 1 𝐷 . 𝜀 𝐹𝜀. , (3) strain curves must be derived from the tested engineering-stress vs. engineering-strain curves. Forwhere: the calculation of the plasticity curves in the MD direction, the approach based on Swift–Voce𝐴 = initial interpolation, yield stress described parameter by (MPa); Bulla [ 16], is used. However, the TD 𝐵 = hardening coefficient (MPa); 𝐶 = hardening plastic strain coefficient (−) 𝐷 = 2nd hardening plastic strain coefficient (−); 𝐹 = 3rd hardening plastic strain coefficient (−);

𝜀. = equivalent plastic strain (−). Table 2 shows the identified parameter used for the hardening curve, based on G’Sell equation.

Energies 2021, 14, x FOR PEER REVIEW 9 of 18

on Swift–Voce interpolation, described by Bulla [16], is used. However, the TD behavior is showing a much higher plasticity range beyond 100% strain. Figure 8 shows one representative engineering stress vs. engineering strain curve close to 120% total strain before the material fractures (green curve). For usage in an elasto-plastic material model, the parametrization for the engineering stress vs. engineering strain curve must be modified further. The true stress vs. true strain curve (Figure 8: red curve) is derived and then the elastic portion of the strain is subtracted from the total strain, using the linear strain–stress relation corresponding to the strain rate, dependent Young’s modulus (Figure 6: dashed lines). Due to the need for values beyond the curve, derived from testing, it is necessary to extrapolate the true stress vs. true plastic strain curve. Based on the work of G’Sell et al. [22] who tested such high ductile polymer materials, the parameter identification is carried out using the modified equation:

𝜎 =𝐴+𝐵1−exp−𝐶𝜀. 1 + 𝐷 . 𝜀 +𝐹𝜀. , (3)

where: 𝐴 = initial yield stress parameter (MPa); 𝐵 = hardening coefficient (MPa); 𝐶 = hardening plastic strain coefficient (−) 𝐷 = 2nd hardening plastic strain coefficient (−); 𝐹 = 3rd hardening plastic strain coefficient (−); Energies 14 2021, , 4585 𝜀. = equivalent plastic strain (−). 8 of 17 Table 2 shows the identified parameter used for the hardening curve, based on G’Sell equation. behavior is showing a much higher plasticity range beyond 100% strain. Figure8 shows Table 2. Parameter list obtained for the G’Sell hardening curve. one representative engineering stress vs. engineering strain curve close to 120% total strain beforeA (MPa) the material fractures B (MPa) (green curve). C For(−) usage in anD ( elasto-plastic−) F material(−) model,0.005826 the parametrization for0.00422 the engineering stress77.76 vs. engineering0.09992 strain curve2.803 must be modiﬁed further. The true stress vs. true strain curve (Figure8: red curve) is derived andFigure then the 9 shows elastic very portion good of agreement the strain isbetween subtracted the true from stress the total vs. true strain, plastic using strain the curveslinear strain–stressobtained from relation real test corresponding data (black curve) to the and strain the rate,calculated dependent hardening Young’s curve modulus based on(Figure G’Sell’s6: dashed equation. lines).

Figure 8. Preparation ofof thethe TD TD test test curve curve from from engineering engineering stress stress vs. vs. engineering engineering strain stra (greenin (green curve), curve), to true to true stress stress vs. true vs. straintrue strain (red curve)(red curve) to the to true the stresstrue stre vs.ss true vs. plastictrue plastic strain strain (black (black curve). curve).

Due to the need for values beyond the curve, derived from testing, it is necessary to extrapolate the true stress vs. true plastic strain curve. Based on the work of G’Sell et al. [22] who tested such high ductile polymer materials, the parameter identiﬁcation is carried out using the modiﬁed equation:

2 σyield = A + B· 1 − exp −C·εpl. · 1 + D·εpl. + F·εpl. , (3)

where: A = initial yield stress parameter (MPa); B = hardening coefficient (MPa); C = hardening plastic strain coefficient (−) D = 2nd hardening plastic strain coefficient (−); F = 3rd hardening plastic strain coefficient (−); εpl. = equivalent plastic strain (−). Table2 shows the identified parameter used for the hardening curve, based on G’Sell equation.

Table 2. Parameter list obtained for the G’Sell hardening curve.

A (MPa) B (MPa) C (−)D(−)F(−) 0.005826 0.00422 77.76 0.09992 2.803 Energies 2021, 14, 4585 9 of 17 Energies 2021, 14, x FOR PEER REVIEW 10 of 18

WithinFigure9 the shows further very FEA good analysis, agreement the G’Sel betweenl-based the hardening true stress curve vs. true will plastic be used strain for Energies 2021, 14, x FOR PEER REVIEWcurves obtained from real test data (black curve) and the calculated hardening curve10 based of 18 the TD behavior. Figure 10 shows parametrized hardening curves resulting from test data foron G’Sell’sdifferent equation. strain rates.

Within the further FEA analysis, the G’Sell-based hardening curve will be used for the TD behavior. Figure 10 shows parametrized hardening curves resulting from test data for different strain rates.

Figure 9. Hardening curve from real test (black curve) vs. calculated curve based on G’Sell’s equation.

Within the further FEA analysis, the G’Sell-based hardening curve will be used for the TD behavior. Figure 10 shows parametrized hardening curves resulting from test data for Figure 9. Hardeningdifferent curve from strain real rates. test (black curve) vs. calculated curve based on G’Sell’s equation.

Figure 10. Fitted parametrized hardening curves resulting from test data for different strain rates in TD.

3.3. Stability Investigation In the explicit FEA, the numerical stability plays an important role, since the Figure 10. Fitted parametrized numerical methodhardening is curv curvesunstablees resulting if the fromtimestep test data is above forfor different a critical strainstrain timestep ratesrates inin [23,24].TD.TD. Another instability results from the material behavior. This material instability is investigated further3.3. Stability since Investigation the tested separator material shows a large strain in the TD direction. InIn therealthe explicit testsexplicit with FEA, FEA, metallic the the numerical numericalmaterials, stability stabthe playsilityengineering anplays important an stress important role, vs. sinceengineering role, the numericalsince strain the curvenumericalmethod shows is unstablemethod monotonic is if unstable the increase timestep if the in is timestepstress above with a criticalis above the timestepincrease a critical in [23 timestep strain,,24]. Another starting [23,24]. instability withAnother the slopeinstabilityresults of from the results theYoung’s material from modulus. the behavior. material When This behavior. materialapproaching instabilityThis plasticmaterial is investigateddeformation, instability further isthe investigated stress since still the increases,furthertested separator since but the with materialtested decreasing separator shows slope, a material large until strain showsa maximum in the a large TD stress direction. strain value in the is TD reached. direction. Then, the measuredIn real stress tests startswith tometallic decrease. materials, At this thpoe intengineering of maximal stress stress, vs. localization engineering occurs, strain wherecurve showsa certain monotonic area continues increase plastic in stress deformation with the without increase an in increase strain, starting in tension with force. the slope of the Young’s modulus. When approaching plastic deformation, the stress still increases, but with decreasing slope, until a maximum stress value is reached. Then, the measured stress starts to decrease. At this point of maximal stress, localization occurs, where a certain area continues plastic deformation without an increase in tension force.

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In real tests with metallic materials, the engineering stress vs. engineering strain curve shows monotonic increase in stress with the increase in strain, starting with the slope of the Young’s modulus. When approaching plastic deformation, the stress still increases, but with decreasing slope, until a maximum stress value is reached. Then, the measured stress starts to decrease. At this point of maximal stress, localization occurs, where a certain area continues plastic deformation without an increase in tension force. This important point is so called the necking point. For metals, the material instability is reached, and the material starts to damage and soften until final separation and failure will occur. In the MD, the specimens reach their terminal strain shortly after the maximum stress is reached. With a well calibrated failure criterium, the finite elements, where this point is reached, will be deleted from further calculation. In the TD of the tested material, the strain increases significantly after reaching the maximum stress value, without fracture. Therefore, the stability will be investigated more thoroughly. Inspired by the Drucker stability condition [25], which is widely used in hyper-elastic material models [26], the slope of the stress vs. strain curve will be analyzed. Using the true strain:

Z L 1 dL L1 L0 + ∆L ∆L εtrue = = ln = ln = ln 1 + = ln 1 + εeng (4) L0 L L0 L0 L0 where:

εeng = engineering strain (−); εtrue = true strain (−); L0 = initial length (mm); L1 = ﬁnal length (mm); L = measured Length (mm); ∆L = relative displacement (mm) of the parallel length and the true stress: · · F F L F L εtrue σtrue = = = = σeng·e = σeng· 1 + εeng , (5) S S·L S0·L0 where:

σtrue = true stress (MPa); F = measured force (N); 2 S0 = initial cross section (mm ); S = actual cross section (mm2) of the parallel length. The assumption of constant volume is made due to the lack of measurement of volume change, because of the small thickness of the separator material. The extension of this work, including consideration of the volume change during the test, is the topic of ongoing work and will be the subject of future investigation. Assuming a stable behavior of a material with a positive slope in the engineering stress–strain relationship and isochoric behavior in plastic region, the limit is reached where the slope becomes zero:

dσeng = 0 (6) dεeng

This indicates the necking point. Proceeding with the slope in the true stress vs. true strain domain, used in the elasto-plastic material models:

2 dσ dσeng· 1 + εeng + σeng·dεeng 1 + εeng ·dσeng 1 + εeng ·σeng·dεeng true = = + 1 , (7) dεtrue ·dεeng dεeng dεeng (1+εeng) Energies 2021, 14, 4585 11 of 17

we obtain as a condition for the necking point:

dσtrue = σtrue (8) dεtrue Figure 11 shows the true stress vs. true plastic strain and the ﬁrst derivative in the same chart. According to Equation (8), the material instability occurs when the 1st derivative of the true stress vs. true strain curve equals the true stress value, which is the ﬁrst crossing Energies 2021, 14, x FOR PEER REVIEWpoint. This corresponds to the maximum stress in the engineering stress vs.12 engineering of 18 Energies 2021, 14, x FOR PEER REVIEW 12 of 18 strain diagram.

FigureFigureFigure 11. 11.Extrapolated, 11. Extrapolated, Extrapolated, ﬁtted fitted fitted hardeninghardening hardening curve curve curve in in TDin TD TD (green (green(green curve) curve) and and andits its1st its 1st derivative, 1st derivative, derivative, indicating indicating indicating the the material material the material stability stability stability range (red curve). rangerange (red (red curve). curve).

FromFrom this this point, point, the the the first ﬁrst first derivative derivative derivative is is belois below below wthe the the true true true stress stress stress vs. vs. vs. true truetrue plastic plasticplastic strain strainstrain curve. Thiscurve.curve. indicates This This indicates anindicates unstable an anunstable materialunstable materi behavior.material albehavior. behavior. Proceeding Proceeding Proceeding further further withfurther thewith with elongation the the up elongation up to a strain value of approximately 0.6, the derivative crosses again the true toelongation a strain valueup to a of strain approximately value of approximately 0.6, the derivative 0.6, the derivative crosses againcrosses the again true the stress true vs. true stress vs. true strain curve and continues above this curve. This indicates a stabilization of strainstress vs. curve true andstrain continues curve and above continues this above curve. this This curve. indicates This indicates a stabilization a stabilization of the ofmaterial thethe material material due due to topotential potential orientation orientation of ofthe the molecular molecular chains chains in inthe the loading loading direction. direction. due to potentialA similar effect orientation is observed of the in real molecular tests in TD chains direction. in the Figure loading 12 sows direction. two examples A similar similar effect effect is isobserved observed in real in real tests tests in TD in direction. TD direction. Figure Figure 12 sows 12 two sows examples two examples in inTD, TD, showing showing the the localization localization through through the the rest rest of ofthe the specimen specimen (a) (a) or ora localization a localization in ina a in TD, showing the localization through the rest of the specimen (a) or a localization in a certaincertain area area with with a seconda second localization localization occu occurringrring while while proceeding proceeding with with the the tension tension test test certain(b).(b). area with a second localization occurring while proceeding with the tension test (b).

Figure 12. Test results TD showing a continuous localization starting close to the center of the FigureFigure 12. TestTest results results TD TD showing showing a continuous a continuous localization localization starting starting close to close the center to the of center the of the specimen and proceeding towards one end (a) and a second localization started at the other end specimenspecimen and and proceeding proceeding towards towards one one end end(a) and (a) anda second a second localization localization started startedat the other at the end other end duringduring the the loading loading phase phase (b). ( b). during the loading phase (b). 3.4.3.4. Material Material Modeling Modeling In Ina recenta recent work, work, a nonlineara nonlinear visco-elasti visco-elastic-orthotropicc-orthotropic and and visc visco-plastic-orthotropico-plastic-orthotropic materialmaterial model model was was developed, developed, based based on on the the investigation investigation of ofthe the PE PE material material used used as as separatorseparator in inLi-Ion Li-Ion batteries. batteries. In Inrecent recent publications publications and and commercial commercial FEA FEA products products and and solvers,solvers, there there exists exists several several material material models models to tomodel model the the behavior behavior of ofpolymers polymers [27–29]. [27–29].

Energies 2021, 14, 4585 12 of 17

3.4. Material Modeling In a recent work, a nonlinear visco-elastic-orthotropic and visco-plastic-orthotropic material model was developed, based on the investigation of the PE material used as separator in Li-Ion batteries. In recent publications and commercial FEA products and solvers, there exists several material models to model the behavior of polymers [27–29]. Within the nonlinear and crashworthiness FEA models, used by the automotive and other industries, the by far most used material model is based on elasto-visco-plasticity, using the Von Mises plasticity formulation [30]. This is due to the fact that the user can easily obtain the necessary values to ﬁt the material model by using results from various performed tensile tests. Another very important advantage of this model is based on the efﬁciency and performance in huge FE models, which lead to its wide usage in industrial environments [9]. Therefore, the Von Mises plasticity model was chosen as a basis and enhancements were implemented to handle the visco-elastic and orthotropic behavior of the PE separator. Material modeling starts with the description of the isotropic, elastic behavior, which is based on Hooke’s law: {σ} = [D]{ε}, (9) where: {σ} = 2nd order stress tensor (MPa); [D] = 4th order elasticity tensor (MPa); {ε} = 2nd order strain tensor (−). The elasticity tensor [E] may be written in Voigt’s notation as 6 × 6 matrix

 1 − ν ν ν 0 0 0   1 − ν ν 0 0 0    E  1 − ν 0 0 0  [D] =  1−2ν , (10) (1 + ν)(1 − 2ν)  2 0 0   1−2ν   Symmetry 2 0  1−2ν 2 where: E = Young’s modulus (MPa); ν = Poisson’s ratio (−). Now, the Young’s modulus E will be replaced by the ﬁltered, orthotropic, strain- . rate dependent Young’s modulus Eε interpolated nonlinearly between the three tested directions: MD, TD, DD (see Table1). For the nonlinear interpolation, between the three orthotropic directions, the cosine- interpolation function is used, as described by Bulla [16]. With the assumption of no sudden changes in the orthotropy of neighboring elements, the same method is applied to the Von Mises plastic potential: p f (J2, k) = 3J2 − k = 0, (11)

where: f = Flow potential (MPa); J2 = deviatoric stress tensor (MPa); k = yield stress obtained from the uniaxial tension test (MPa). We obtain the 2nd deviatoric stress invariant, using Einstein’s notation:

1 J = S S , (12) 2 2 ij ij Energies 2021, 14, 4585 13 of 17

with the additive decomposition of the stress tensor, the deviatoric stress is computed using:

1 S = σ − I δ , (13) ij ij 3 1 ij Energies 2021, 14, x FOR PEER REVIEW 14 of 18 where:

Sij = deviatoric stress tensor (MPa); σij = Cauchy stress tensor (MPa); 𝛿 = Kronecker delta (−). δij = Kronecker delta (−). With: With: 𝐼 =𝜎, I1 = σkk, (14) where: I𝐼1 = = 1st1st invariantinvariant onon thethe stressstress tensortensor (MPa).(MPa). Using a similar approach for viscosity and and orthotropy, orthotropy, as used in the elastic region, the yieldyield stress stressk isk dependentis dependent on the on ﬁltered the filtered strain ratestrain and rate orthotropy and orthotropy direction measureddirection measuredin the real test.in the With real the test. assumption With the assumpti of no suddenon of changes no sudden in the changes orthotropy, in the the orthotropy, associated theﬂow associated rule is used flow within rule is this used new within material this model. new material model.

3.5. FE-Model FE-Model In presentpresent workwork the the new new developed developed material materi modelal model is used is used and validatedand validated by the by tested the testedresults. results. The orthotropic The orthotropic visco-elastic visco-elastic properties, properties, obtained obtained from test from results test (seeresults Table (see1) Tableare used, 1) are to used, model to the model orthotropic-visco-elasticity. the orthotropic-visco-elasticity. During theDuring loading the loading phase, aphase, precise a precisemodeling modeling of the elastic of the properties elastic properties is important is important to accurately to predictaccurately the predict stress response the stress of responsethe entire of model the entire since model all elements since all are elements participating are participating in this mechanical in this mechanical region. region. Figure 1313 shows shows the the three three visco-plastic visco-plastic hardening hardening curves curves for the for MD the direction MD direction obtained obtainedby the test by carried the test outcarried during out thisduring study this and study the and DD the direction DD direction result result curves curves taken taken from fromSahraei Sahraei [17]. The [17]. engineering The engineering stress vs.stress engineering vs. engineering strain curves strain in curves MD direction in MD direction are ﬁtted areto the fitted Swift to andthe VoceSwift equations and Voce [ 31equations] and extrapolated [31] and extrapolated to 100 % strain, to 100 using % anstrain, interpolation using an interpolationfunction as described function in as Bulla described [16]. Figure in Bulla 14 shows[16]. Figure the FE 14 results shows of thethe materialFE results behavior of the materialin MD direction. behavior Itin canMD be direction. seen that It can after be the seen ﬁrst that localization after the first occurs, localization the localized occurs, area the localizedremains at area the sameremains group at ofthe elements. same group These elementsof elements. are furtherThese extendingelements whereasare further the extendingneighboring whereas elements the remainneighboring at the elements strain level, remain they at had the before strain the level, localization they had started. before theFor localization these material started. curves For (shown these material in Figure curv 13es, left(shown side) in the Figure. slope 13, of left the side) true the stress slope vs. oftrue the plastic true stress strain vs. curve true isplastic decreasing, strain curve which is means,decreasing, the material which means, will not the become material stable will notback become again. stable back again.

Figure 13. StrainStrain raterate dependent,dependent, orthotropic, orthotropic, hardening hardening true true stress stress vs. vs. true true plastic plastic strain strain curves curves ﬁtted fitted to test to test results results for MD for (0MD◦) direction(0°) direction (left ( sideleft )side and) derivedand derived from from tests tests provided provided in DD in (45 DD◦) (45°) direction direction (right (right side ).side).

Figure 15 shows the loading direction in the TD direction. As it can be seen in the hardening curve (Figure 13 right side), the slope decreases until ~11% plastic strain and then increases again, which results in stabilization of the material.

Energies 2021, 14, x FOR PEER REVIEW 15 of 18

EnergiesEnergies2021 2021, ,14 14,, 4585 x FOR PEER REVIEW 1514 of 1718

Figure 14. FE results showing the loading phases in the MD direction at initial stage (a), short before the first localization occurs (b) and further development of the total elongation (c).

For materials with such hardening curves, the material becomes unstable as soon as the 1st derivative decreases and undergoes the hardening true-stress-strain curve. Then, the first localization can occur (Figure 15b) and become stable again. During further loading (Figure 15c) the material stiffness increases, and the first derivative raises above FigureFigure 14.14.FE FE resultsresults showingshowingthe the thetrue-stress-strain loadingloading phasesphases in inhardening thetheMD MD directiondirection curve.at atTheinitial initial materialstage stage( (a abecomes),), shortshort beforebefore stable the the again ﬁrst first localization localization and localized occursoccurs (b(b)) and and further further development developmentstrain occurs of of the the totalin total another elongation elongation area ( c( c).(Figure). 15d) during further loading of the structure. This numerical behavior, which can be simulated only with a sufficient complex material model,FigureFor is materialsalso 15 observed shows with the insuch loadingreal hardening tests direction (Figure curves, 12). in the Figurethe TD material direction.16 shows becomes the As corresponding it unstable can be seen as soon inforce the as vs.hardeningthe displacement 1st derivative curve (Figuresimulation decreases 13 right andresult undergoes side), curves. the slope Thethe hardeninginfluence decreases oftrue-stress-strain until the ~11% strain plastic rate curve.dependent strain Then, and Young’sthenthe first increases moduli localization again, (see Table whichcan occur1) results is clearly (Figure in stabilization visible 15b) (Figureand ofbecome the16, material.right stable side). again. During further loading (Figure 15c) the material stiffness increases, and the first derivative raises above the true-stress-strain hardening curve. The material becomes stable again and localized strain occurs in another area (Figure 15d) during further loading of the structure. This numerical behavior, which can be simulated only with a sufficient complex material model, is also observed in real tests (Figure 12). Figure 16 shows the corresponding force vs. displacement simulation result curves. The influence of the strain rate dependent Young’s moduli (see Table 1) is clearly visible (Figure 16, right side).

Figure 15. FE results showing the loading phases in in the the TD TD direction at at initial initial stage stage ( (aa),), after after first ﬁrst localization localization occurs occurs ( (bb),), thenthen proceeding proceeding through the specimen (c) and shortly before failurefailure ((dd).).

For materials with such hardening curves, the material becomes unstable as soon as the 1st derivative decreases and undergoes the hardening true-stress-strain curve. Then, the first localization can occur (Figure 15b) and become stable again. During further loading (Figure 15c) the material stiffness increases, and the first derivative raises above the Figure 15. FE results showingtrue-stress-strain the loading phases hardening in the TD curve. direction The at material initial stage becomes (a), after stable first again localization and localized occurs (b strain), then proceeding through theoccurs specimen in another (c) and areashortly (Figure before 15 failured) during (d). further loading of the structure. This numerical behavior, which can be simulated only with a sufficient complex material model, is also observed in real tests (Figure 12). Figure 16 shows the corresponding force vs. displacement simulation result curves. The influence of the strain rate dependent Young’s moduli (see Table1) is clearly visible (Figure 16, right side). The simulations were conducted with the explicit FEA solver Radioss (version 2021) by Altair Engineering Inc. (1820 E. Big Beaver Rd., Troy MI 48083, USA), on a Windows64 computer using four Intel i7-6820 CPUs at 2.7 GHz with 32GB RAM. The material model was developed in FORTRAN and linked to the FEA solver as a dynamic linked library (dll) compiled using the freeware MinGW-W64 project with a gcc 7.2.0 compiler [32]. The developed validation models were meshed using 4-node shell elements with a mesh size of 1 × 1 mm and a defined thickness of 0.025 mm.

Energies 2021, 14, 4585 15 of 17 Energies 2021, 14, x FOR PEER REVIEW 16 of 18

FigureFigure 16. 16.Force Force vs. vs. displacement displacement curves curves fromfrom simulationssimulations for different different velocities. velocities. (a ()a load) load in in TD, TD, (b ()b load) load in inMD. MD.

The simulations were conducted with the explicit FEA solver Radioss (version 2021) 4. Discussion by Altair Engineering Inc. (1820 E. Big Beaver Rd., Troy MI 48083, USA), on a Windows64 computerThe objective using four of theIntel presented i7-6820 CPUs work at was 2.7 theGHz development with 32GB RAM. of a ﬁniteThe material element model material model,was developed for nonlinear, in FORTRAN structural and CAE linked simulation to the FEA and solver predicting as a dynamic the mechanical linked library behavior of(dll) a PE compiled separator using material the freeware under MinGW-W64 dynamic and project high with mechanical a gcc 7.2.0 load, compiler used for[32]. the The pre- dictiondeveloped of stiffness validation and models failure were in Li-ion meshed batteries. using 4-node Nevertheless, shell elements our focus with soa mesh far is size set on crashworthinessof 1 × 1 mm and analysisa defined where thickness batteries of 0.025 undergo mm. very complicated state of deformation where bending plays an important role. Therefore, we set our focus on tensile loading until fracture4. Discussion for the separators. BasedThe objective on the real of the test presented results, performed work was th withe development different velocities of a finite and element different material orienta- tionsmodel, of thefor specimen,nonlinear, astructural novel material CAE simulati modelon was and developed. predicting It the considers mechanical the visco-elasticbehavior orthotropic,of a PE separator visco-plastic material orthotropic under dynamic behavior and under high high mechanical mechanical load, loadings. used for The the de- velopedprediction FE of material stiffness model and failure accounts in Li-ion for the batteries. strain-rate Nevertheless, effect in elastic our focus loading so far phase is set and differenton crashworthiness material orientations, analysis aswhere well asbatteri the differentes undergo elasto-plastic-orthotropic very complicated state behavior of indeformation the plastic materialwhere bending domain, plays which an important was measured role. Therefore, in the real we tests. set our Within focus theon tensile material model,loading the until Von fracture Mises potential for the separators. was used, which is widely used in the industry and suitable for modelingBased on elasto-plastic the real test materials.results, perform For materialed with behaviordifferent velocities characterization and different the only necessaryorientations information of the specimen, was the resultsa novel frommate realrial testsmodel in was the threedeveloped. directions It considers (MD, DD, the TD), usingvisco-elastic a simple orthotropic, tension test vi undersco-plastic different orthotropic loading velocities.behavior under Microscopic high mechanical investigation ofloadings. the battery The separatordeveloped after FE material high mechanical model acco loadingunts for along the strain-rate the MD and effect TD in directions elastic areloading published phaseby and Zhang different [33 ].material Once the orientatio numericalns, as material well as the reaches different instability, elasto-plastic- a further considerationorthotropic behavior of damage in the propagation plastic material and domain, failure must which be was considered. measured in the real tests. WithinCompared the material to existing model, orthotropic the Von Mises material pote models,ntial was as used, developed which is by widely Hill or used Barlat in [the34,35 ], theindustry modeling and is muchsuitable easier for due modeling to the fact, elasto-plastic that the orthogonal materials. directions For material are obtained behavior directly bycharacterization using the real the tests only as inputnecessary for this information model. It was is also the results possible from to applyreal tests the in Hill the or three Barlat yield surface if further studies show this need.

Energies 2021, 14, 4585 16 of 17

5. Conclusions and Outlook Within this work, our novel material model and the developed FE model succeeded in predicting the test for all three directions (MD, DD, TD) with different velocities in the linear elastic as well as the plastic domain within a range of 5%. This lies within the scatter range of the tested specimen. For predicting the damage during this dynamic and high mechanical load, a suited failure model should be used in addition to the material model, as proposed by Bulla [16]. This should be useful in the design process of batteries and applications that contain batteries and will serve as an important new computational tool for assessing the safety of lithium-ion batteries against high mechanical loading and crashworthiness. The material model will be implemented in the commercial FE solver Altair Radioss and will thus be available for practical application in e-mobility simulation in near future. For porous polymers such as separators, pore orientation and connectivity affect their effective mechanical properties and performance in a battery, too [36]. However, this is the topic of further investigation and the usage of an extended model.

Author Contributions: Conceptualization, M.B., S.K. and E.S.; Formal analysis, S.K.; Methodology, M.B. and E.S.; Resources, S.K. and E.S.; Software, M.B.; Supervision, S.K. and E.S.; Validation, M.B., S.K. and E.S.; Visualization, M.B.; Writing—original draft, M.B. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Acknowledgments: The authors would like to thank to Boston Powers for providing the isolator materials. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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