Identification of LLDPE Constitutive Material Model for Energy Absorption in Impact Applications

Copyright: c © 2021 by the authors. Submitted to Polymers for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/ 4.0/). 1 New Technologies Research Centre, University of West Bohemia, 301 00 Plzeň, Czech Republic; hyncik@ntc.zcu.cz (L.H.); spicka@ntc.zcu.cz (J. Š.); tomasz@ntc.zcu.cz (T. B.); cimrman3@ntc.zcu.cz (R. C.) 2 Faculty of Applied Sciences, University of West Bohemia, 301 00 Plzeň, Czech Republic; kochovap@kme.zcu.cz (P.K.); kottner@kme.zcu.cz (R.K); kanaksan@kme.zcu.cz (S.K.) 3 MECAS ESI s.r.o., Brojova 2113, 326 00 Plzeň, Czech Republic; Miloslav.Pasek@esi-group.com (M.P.) * Correspondence: hyncik@ntc.zcu.cz Abstract: Current industrial trends bring new challenges in energy absorbing systems. Polymer 1


Introduction
Thin-layered polymer materials are traditionally used for packaging goods to 20 protect them during transportation. Therefore the major desired properties relate to 21 thickness, density (which relates to weight), strengths, elongation, puncture resistance 22 and stretching level, see Table 1. On the other hand, preliminary experimental tests show 23 also a good performance of such materials in energy absorption. 24 Current trends in the automotive industry regarding future mobility bring new 25 challenges for energy-absorbing safety systems. Non-traditional seating configurations 26 in autonomous vehicles and complex crash scenarios including multi-directional loading 27 are to be considered hand in hand with the advanced materials for energy absorption. 28 As virtual prototyping plays an important role in the design of new products 36 nowadays, the paper aims to identify the linear low-density polyethylene (LLDPE) 37 material parameters for both static and dynamic loading, to implement them into a 38 constitutive material model and to verify the material model by numerical simulations 39 representing the experiments. As the static tests are represented by quasi-static loading 40 conditions, the dynamic tests represent the scenario close to the one schematically 41 described in Figure 1.   LLDPE films have been identified as the most promising material in cases, where 43 the impact loading is assumed, because of their higher average peak force and the 44 energy to peak force when compared to LDPE [3]. LLDPE is a linear polyethylene with a 45 significant number of short branches (see Figure 2) commonly made by copolymerization 46 of ethylene and another longer olefin, which is incorporated to improve properties such 47 as tensile strength or resistance to harsh environments. The structure of LLDPE leads to 48 its heterogeneous non-linear behavior. LLDPE is very flexible, elongates under stress, absorbs a high level of impact 50 energy and thus is suitable to make thin and ultra-thin films [5][6][7][8]. The mechanical 51 properties of polyethylene depend on its complex structure [9], which leads to a non- while the yield strain decreases with the increasing strain rate [9]. The double yield 64 point is also mentioned by Plaza et al. (1997). The relation between the yield stress, 65 the temperature and the strain rate can be described by constitutive laws [5,6,9,17]. 66 The temperature-dependent mechanical properties of thin-layered materials are also 67 described by Luyt et al. (2021). By comparison among LDPE, LLDPE and HDPE, LLDPE 68 showed greater rate sensitivity than the other two materials under both static and 69 dynamic regions of a compression test [9]. 70 The typical stress-strain relation as well as the strain-rate dependence are sketched II., see Figure 3. The third region III. is the stiffening leading to the material rupture.

76
LLDPE has anisotropic behavior due to its chain structure. The chain structure 77 creates the anisotropy in 2 perpendicular directions, called the machine direction (MD) 78 and the transversal direction (TD). The local preferential orientation of chains in LLDPE 79 affects the tensile strength in MD and TD [11]. In the direction of the main chain 80 orientation, mostly the MD, LLDPE are stiffer than in the perpendicular direction, mostly 81 the TD [3,5,6,15]. The tensile stress-strain relations in MD and TD play an important role 82 during the biaxial deformation of the impact test [3].

84
The material parameter identification is applied to the LLDPE thin foil produced 85 by Tic (2020). Table 1 summarizes its parameters presented by the producer. The unilateral quasi-static loading test of the material sample was executed using 88 the testing machine 574LE2 TestResources. From the material roll (see Figure 4) provided 89 by the producer, the testing samples of length l 0 = 5 mm and width w = 10 mm were 90 extracted, see Figure 5 (a), where the left and right yellow sides are fixed to the testing 91 machine yaws. The thickness of the sample was h = 12 µm. The samples were fixed in 92 the testing machine jaws (see Figure 5 (b)) and stretched in 2 major orthotropic directions 93 (MD and TD). MD is in the direction, in which the material is winded up on the roll, 94 whist TD is perpendicular to MD, see Figure 4.

86
The constant Young modulus E was also identified as the slope of the initial elastic 106 region as Fulfilling the aim of this study, the quasi-static tests were reproduced by numerical where the shear angle is calculated based on the deformed sample angle ψ as where L is the side of the square sample, d is the displacement in direction D3 and F 3 is 126 the force recorded in direction D3. Therefore the shear stress can be calculated as 127 τ = Q Lh (7) and the shear stress resultant is The thickness of the material is h = 12 µm as defined by the producer [22]. The     The acceleration was measured using a uniaxial piezoelectric accelerometer Kistler tric accelerometer cannot measure a free-fall gravity acceleration [26]. The experimental 172 acceleration curve decreases to minus g just after releasing and reaches the equilibrium 173 0 g during the free fall, so the experimental acceleration curve needs to be adjusted to be 174 comparable to the simulation results.

175
As the measured displacement was limited by the range of the laser measuring to check the correctness of the calculations and to identify the energy absorption. Here total energies, the energy absorption was calculated as the energy loss where  values. According to Figure 3, the stiffness of region I. and the yield stress were optimized.

203
The MD and TD curves in region I. were updated as by multiplying by dimensionless coefficients k y , 1 Coefficient k y is the yield stress multiplier.

209
The optimization process was run in a loop controlled by a MATLAB script updating 210 the constitutive material curves in MD and TD according to Equations (12) and (13). The 211 cost function in the optimization measured the relative acceleration error E a defined as noindent where a e (t) is the time-dependent acceleration signal measured from by the CFC 1000 filter [25]. Figure 10 shows the simulation setup for optimization where d e (t) is the time-dependent displacement signal obtained by the double integra-

234
The following figures and tables summarize the results from the quasi-static tests 235 as well as the identification of LLDPE parameters under dynamic loading.    the curves for each direction were averaged as shown in Figure 12. It can be seen that   By detailed analysis of the measured data in Figure 12    using Equation (2) by averaging the slopes of the elastic regions of all curves, see Table 6.  the constitutive material model, see Figure 21. A single-element numerical simulation to 270 reproduce the stretching was run. Figure 13 shows perfect fit to the experimental curves.

Dynamic loading 272
The acceleration decrease interval from 0 g to minus g approximately within the 273 first 32 ms was used as the approximated parabolic acceleration ramp (see Figure 14)  Such process led to a perfect fit in both measured and calculated displacements 286 (both shown in Figures 15 -19) identifying also the real impact velocity (see Table 7).

287
The only exception was scenario 1509, where the displacement measurement failed. So 288 the impact velocity was estimated to fit the remaining part of the displacement curve.  Table 7) used for the constitutive 293 material model. Several approaches to optimize the strain-rate dependent constitutive material 296 curves were used and finally, the same stiffening ratio in MD and TD was proposed.

297
The optimization process controlled by a MATLAB script involved running a series 298 of simulations for updating the constitutive material model curves. The quasi-static 299 response was taken as the initial guess for the optimization.
300 Table 8 shows the coefficients coming from the optimization process. Table 8 also 301 shows the number of iterations leading to the optimized constitutive material curves as 302 well as the errors from the cost function calculated by Equation (14) and the error in the 303 displacement calculated by Equation (15). The original experimental curves are in red dashed lines, the updated target curves

336
The quasi-static experiments were performed in 2 perpendicular directions sup-  were supposed for developing the dynamic constitutive material model in MD and TD.

358
The optimized multipliers as well as optimization process errors are stated in Table   359 8.

378
The paper contributes to the field of virtual testing by developing the material 379 model and identifying its constitutive parameters. The target material was LLDPE, a 380 material traditionally used for packaging goods to protect them during transportation.

381
The paper proves a high energy absorption of the material suitable for impact protection, 382 also due by its low weight. Both quasi-static and dynamic responses of the material were 383 considered in the constitutive material model.    The following abbreviations are used in this manuscript: