Bieler, S.

Dehm, G.

Heußen, D.

Kang, S.

Ramachandramoorthy, R.

Weinberg, K.

English

MPG.PuRe

Received: 24 June 2021 Accepted: 31 August 2021DOI: 10.1002/pamm.202100155Investigation of copper lattice structures using a Split HopkinsonPressure BarSören Bieler1,∗, Sung Gyu Kang2, Daniel Heußen3, Rajaprakash Ramachandramoorthy2, GerhardDehm2, and Kerstin Weinberg11 Lehrstuhl für Festkörpermechanik, Universität Siegen, Paul-Bonatz-Straße 9-11, 57068 Siegen, Germany2 Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-Straße 1, 40237 Düsseldorf, Germany3 Fraunhofer-Institut für Lasertechnik ILT, Steinbachstr. 15, 52074 Aachen, GermanyThe paper deals with experiments on 3D printed lattices in a Split Hopkinson Pressure Bar. An energy-based evaluation ofthe measured wave signals enables us to compare the damping properties of two different copper lattice structures.© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.1 IntroductionThe ability to model a wide variety of structures through 3D printing processes is a relatively new field, which is highlyimportant for many disciplines. The response of such structures on mechanical or thermal loadings is therefore of great interest.This paper presents experiments on high strain-rate loading of printed copper lattices performed with a Split HopkinsonPressure Bar (SHPB).Typically, the SHPB is used to determine elastic and plastic material properties, and a cylindrical probe made of homoge-nous material works as a specimen. The conventional evaluation of the SHPB experiment presumes a uniform state of stressin the specimen. Therefore samples with a micro-structure are only suitable to a limited extent for determining materialproperties since the waves do not move through the specimen’s body over a homogeneous cross-section.With the aim to understand the energy absorption of 3D-printed structures under impact, we chose here an energy-basedevaluation of the measured signals. Our SHPB device uses aluminum material for the striker, the incident, and the transmissionbar. The bars have a diameter of 20 mm and a length of 1800 mm. Strain gauges are applied in the center of each bar, whichallows us to record the propagating pulses. Further details on our setup can be found in [1, 2].2 Copper lattice structuresWe investigate two types of 3D-printed lattice structures. They have an octet-truss and a F2CCZ lattice as unit-cells; bothshow different nodal connectivity. The truss diameter and the length of the unit cell are 200 µm and 1 mm, respectively. Eachlattice structure has 5× 5× 5 unit-cell repetitions. For the fabrication of specimens, a system with infra-red laser source wasused. The material was pure Cu-ETP powder with diameter of 16-63 µm. Laser power, scan speed, and layer height are 600W, 800 mm/s, and 30 µm, respectively. We used a contour scanning strategy, and the beam compensation was 70 µm. Thefabricated lattices are cube-shaped with a length of 5 mm, as shown in Fig. 1.Fig. 1: Two copper lattice structures: octet truss (left) and F2CCZ truss (right)3 Energy of specimen deformation in the SHPBThe elastic energy of a pulse in a bar can be determined asW =∫VwedV = Ac t0∫ ǫ0σ(ǫ̃) dǫ̃ with t0 = 2lStrikerc(1)∗ Corresponding author: e-mail soeren.bieler@uni-siegen.de, phone +49 (0271) 740-4644, fax +49 (0271) 740-12225This is an open access article under the terms of the Creative Commons Attribution License, which permits use,distribution and reproduction in any medium, provided the original work is properly cited.PAMM · Proc. Appl. Math. Mech. 2021;21:1 e202100155. www.gamm-proceedings.com 1 of 3https://doi.org/10.1002/pamm.202100155 © 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.2 of 3 Section 4: Structural mechanicswhere V , A and c are the volume, the cross section area and the elastic wave speed of the bar, respectively. The durationof the pulse is t0 and determined by the length of the striker in a SHPB. Assuming a rectangular strain pulse and an elasticbehavior of the bar, σ = E ǫ, we obtain for a constant Young’s modulus E after integration: W (ǫ) = 1/2EAc t0ǫ2. In theSHPB experiment three waves are measured: the incident (I), the reflected (R) and the transmitted strain pulse (T ). For eachof these pulses corresponding energy can be calculated:WI =12E Ac t0 ǫI2 WR =12E Ac t0 ǫR2 WT =12E Ac t0 ǫT2 (2)For a system of two bars without a specimen, the energy balance holds that: WI = WR + WT . When a specimen issandwiched between the bars, an energy difference is measured. This energy difference can be calculated accordingly, ∆W =WI −WR −WT . We assume the specimen to be small and in stress equilibrium [3], i. e.,ǫT = ǫI + ǫR (3)and after evaluation of these expressions, we obtain the energy difference:∆W = E Ac t0 (ǫI + ǫR) (ǫI − ǫT ) . (4)This energy is stored in the specimen and can considered to be the damping of the propagating wave. Depending on thespecific material used, this energy can be converted into elastic, viscoelastic or elastic-plastic deformation.0 1 2 3 4 5 6 710-4-4-3-2-1012310-40 1 2 3 4 5 6 710-4-4-3-2-101210-4Fig. 2: Wave signals measured from strain gauges of the SHPB with a striker length of 300 mm and a velocity of 3.3 m/s4 Results and conclusionFrom the measured incident, reflected, and transmitted signals of the SHPB, we calculated the relative energy difference forthe two different copper lattice structures as∆WWI(F2CCZ) = 0.218 and∆WWI(octet-truss) = 0.229 . (5)The stored energy of two lattice structures differs because of their different deformation mechanisms. The octet-truss latticehas higher nodal connectivity than the F2CCZ and so the deformation is mostly stretch-dominated [4]. Apart from that, theF2CCZ lattice has relatively low nodal connectivity, and the nodal bending or shear lowers the compressive strength of thestructure. Therefore, an octet-truss structure requires more energy to deform than a F2CCZ structure. Our SHPB resultsconfirm this general observation.This short study demonstrates that specimens with internal structure can be characterized and that the energy-based SHPBapproach can be applied to 3D architectured material to evaluate the energy or shock absorption efficiency.Acknowledgements Open access funding enabled and organized by Projekt DEAL.References[1] Weinberg, K.; Khosravani, M.;Thimm, B.; Reppel, T.; Bogunia, L.; Aghayan, S. and Nötzel, R. Hopkinson bar experiments as a methodto determine impact properties of brittle and ductile materials. GAMM-Mitteilungen – Surveys for Applied Mathematics and Mechanics,41(2):e201800008, 2018.© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH. www.gamm-proceedings.comPAMM · Proc. Appl. Math. Mech. 21:1 (2021) 3 of 3[2] Khosravani, M.; Silani,M.; Weinberg, K..: An investigation of the mechanical properties of materials at very high rates of loading,Theoretical and Applied Fracture Mechanics 93, 302–310, 2018[3] Chen, W.; Song, B.: Split Hopkinson (Kolsky) bar: design, testing and applications. Springer Science & Business Media,2010.[4] Maconachie,T. et al.: SLM lattice structures: Properties, performance, applications, and challenges, Materials and Design 183 (2019)108137.www.gamm-proceedings.com © 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.

Investigation of copper lattice structures using a Split Hopkinson Pressure Bar

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Investigation of copper lattice structures using a Split Hopkinson Pressure Bar

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