Accepted Manuscript Numerical investigation, experimental validation and macroscopic yield criterion of Al5056 honeycombs under mixed shear-compression loading R. Tounsi, E. Markiewicz, B. Zouari, F. Chaari, G. Haugou PII: DOI: Reference: S0734-743X(16)31101-0 10.1016/j.ijimpeng.2017.05.001 IE 2916 To appear in: International Journal of Impact Engineering Received date: Revised date: Accepted date: 30 December 2016 28 April 2017 1 May 2017 Please cite this article as: R. Tounsi, E. Markiewicz, B. Zouari, F. Chaari, G. Haugou, Numerical investigation, experimental validation and macroscopic yield criterion of Al5056 honeycombs under mixed shear-compression loading, International Journal of Impact Engineering (2017), doi: 10.1016/j.ijimpeng.2017.05.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Highlights • Yield behaviour of Al5056 aluminium alloy honeycombs is investigated IP T under mixed shear-compression. • Numerical simulations allow to overcome a limitation of the experimental measurements. US CR • Numerical and experimental investigations allow to investigate the normal and shear behaviours separately. • A macroscopic yield criterion expressed as a function of the impact AC CE PT ED M AN velocity, the loading angle Ψ and the in-plane orientation angle β. 1 ACCEPTED MANUSCRIPT IP T Numerical investigation, experimental validation and macroscopic yield criterion of Al5056 honeycombs under mixed shear-compression loading R. Tounsia , E. Markiewiczb,∗, B. Zouaric , F. Chaarib , G. Haugoub a University of Technology Belfort-Montbliard (UTBM), 25200 Montbliard, France of Valenciennes (UVHC), LAMIH, 59313 Valenciennes, France c National Engineering School of Sfax (ENIS), LA2MP, B.P. W3038, Sfax, Tunisia US CR b University Abstract Numerical simulations of honeycomb behaviour under mixed shear- AN compression loading are performed to overcome a limitation of the experimental measurements and to investigate the normal and the shear honeycomb behaviours separately. A detailed FE model allowing to simulate the mixed M shear-compression honeycomb behaviour is presented. A validation between numerical and experimental results in terms of crushing responses and collapse mechanisms allows to dissociate the normal and shear forces components. They PT ED are used to identify the parameters of a macroscopic yield criterion expressed as a function of the impact velocity, the loading angle and the in-plane orientation angle. A well known dynamic enhancement phenomenon is confirmed by this macroscopic yield criterion. However, as a new result, this dynamic enhancement is reversed when the loading angle reaches a critical value. An CE analysis of the collapse mechanisms is carried out under both quasi-static and dynamic loading conditions in order to explain this inversion. AC Keywords: Honeycomb, FE model, experimental validation, macroscopic yield criterion, ∗ Corresponding author Email addresses: tounsi.f@gmail.com (R. Tounsi), eric.markiewicz@univ-valenciennes.fr (E. Markiewicz), bzouari@yahoo.com (B. Zouari), fahmi.chaari@univ-valenciennes.fr (F. Chaari), gregory.haugou@univ-valenciennes.fr (G. Haugou) Preprint submitted to International Journal of Impact Engineering May 2, 2017 ACCEPTED MANUSCRIPT mixed shear-compression 1 1. Introduction Cellular materials are increasingly used in the transportation industry due 3 to their high strength/weight ratio which contributes to the development of 4 environmentally friendly vehicles. Among this class of materials, aluminium 5 alloy honeycombs have an outstanding capability for absorbing energy. Several 6 studies reported by [1], [2], [3], [4], [5], [6], [7], [8] and [9] have investigated the 7 quasi-static and dynamic behaviours of honeycombs under uni-axial or bi-axial 8 compression loadings ( out-of-plane or in-plane loadings). US CR IP T 2 A limited number of studies reported by [10],[11] [12, 13], [14, 15], [16], [17], 10 [18, 19, 20] and [21] have investigated honeycomb behaviour under more realistic 11 working conditions that occur in crash events where shear and compression 12 loadings are mixed. In this case, two angles are defined : the loading angle 13 ψ is the angle between load direction and out-of-plane direction and the in- 14 plane orientation angle β is the angle between shear load direction and ribbon 15 direction in the cell plane. M AN 9 Aluminium alloy honeycombs are considered as materials for 17 different applications including lightweight structures combining a 18 high stiffness and energy absorption capabilities under dynamic 19 loading. The state of the art of all experimental studies and numerical 20 studies on honeycombs allows concluding that a macroscopic yield 21 criterion expressed as function of the impact velocity, the loading 22 angle and the in-plane orientation angle is required to provide 23 a constitutive description of the yield behaviour of honeycomb AC CE PT ED 16 24 structures. 25 For that, Mohr and Doyoyo[11] suggested a linear fit for the crushing 26 envelope obtained from their honeycomb specimens tested with only one in-plane 27 orientation angle β= 90◦ . Hong et Al. [12, 13] developed a quadratic yield 28 criterion that gives a good description of the macroscopic crush behaviour of 3 ACCEPTED MANUSCRIPT honeycomb specimens under quasi- static and dynamic loading conditions with 30 different in-plane orientation angles (β= 0◦ ; β= 30◦ and β= 90◦ ). However, 31 they investigated the impact velocity for only one loading angle ψ= 15◦ . An 32 elliptical yield envelope is found for both the quasi-static and dynamic loading 33 cases by Hou et Al. [15] using the Levenberg-Marquardt Algorithm (LMA). 34 Their macroscopic yield criterion takes into account the loading angle ψ and the 35 impact velocity but without any influence reported on the in-plane orientation 36 angle for β= 0◦ and β= 90◦ . However, a significant effect of this in-plane 37 orientation angle was reported by Zhou et Al. [16] on the experimental yield 38 surface of Nomex honeycombs. US CR IP T 29 In this paper we propose to investigate the combined effects of the in- 40 plane orientation angle, the loading angle and the impact velocity on the 41 macroscopic yield criterion of an Al5056 honeycomb. The paper is organized as 42 follows. Section 2 presents the experimental set-up and limitations. Section 43 3 is dedicated to numerical simulations used to perform virtual crushing 44 tests. Comparative studies between numerical and experimental results are 45 proposed in section 4. In particular, a validation is performed in terms of 46 both crushing responses and collapse mechanisms under both quasi-static and 47 dynamic loadings. Section 5 presents separately the numerical shear and normal 48 honeycomb behaviours based on the validated numerical model. The shear 49 and normal crushing responses are used in section 6 in order to present the 50 macroscopic yield criterion as function of ψ, β and the impact velocity. CE PT ED M AN 39 2. Experimental set-up and measurement limitations 52 2.1. Specimens and experimental set-up AC 51 53 Al5056-N-6.0-1/4-0.003 aluminium alloy honeycomb specimens are 54 considered. The relative density (the ratio of the honeycomb density 55 and the base material density) is ρ∗ = 3 %. The cell wall width is D = 3.67 56 mm, the single cell wall thickness is t = 76 µm , the cell angle is α = 120◦ 57 and the cell size is d = 6.35 mm. The specimen contains 39 full cells on the 4 ACCEPTED MANUSCRIPT honeycomb cross-section. The specimen dimensions are 44 x 41 x 25 mm in 59 the directions of X, Y and Z respectively. X and Y directions are the in-plane 60 directions. Z direction is the out-of-plane direction (figure 1). Under mixed 61 shear-compression loading, the loading angle ψ is defined by the angle between 62 the out-of-plane direction and the load direction. The in-plane orientation 63 angle β is defined by the angle between the ribbon direction and the shear load 64 direction (figure 1). Five loading angles are considered for the experimental 65 study. ψ = 0◦ corresponds to uni-axial compression loading and ψ = 15◦ / ψ = 66 30◦ / ψ = 45◦ / ψ = 60◦ correspond to mixed shear-compression loadings. For 67 every loading angle ψ, the in-plane orientation angle β is varied. Four in-plane 68 orientation angles (β = 0◦ / 30◦ / 60◦ / 90◦ ) are considered. PT ED M AN US CR IP T 58 Figure 1: The honeycomb specimen geometry, cell parameters, loading angle and in-plane orientation angle. [20] The experimental program is divided into 2 parts : The specimens are 70 crushed with an impact velocity of 15 m/s for the dynamic experiments and with 71 a loading speed of 1 mm/min for the quasi-static experiments. A mixed shear- AC CE 69 72 compression loading device is introduced in a Nylon SHPB set-up to perform the 73 dynamic experiments and is adapted to a universal tensile/compression machine 74 to perform the quasi-static ones (figure 2). 75 As detailed in Tounsi et Al. [19] a high strength steel sleeve of 10 76 mm thick with a Teflon sleeve of 5 mm thick put inside are used in 5 ACCEPTED MANUSCRIPT 77 the experiments in order to limit the expansion phenomenon and to 78 ensure a good alignment during tests (figures 2 and 3). In the following, analyses are focused on the initial peak force FP eak and the 80 average crushing force FAverage (plateau) that characterise a typical response 81 of honeycomb under impact loadings. Quasi-static experiments IP T 79 Dynamic experiments using SHPB AN US CR SPECIMEN Figure 2: The quasi-static and dynamic experimental set-up.[20] 2.2. Experimental set-up limitations M 82 In order to investigate in detail the mixed shear-compression behaviour of 84 honeycombs and to develop a macroscopic yield criterion taking into account 85 the loading angle ψ, the in-plane orientation angle β and the impact velocity, 86 the normal and shear responses should be dissociated. However, the device 87 used to load the specimens (figure 3) leads to a transverse component force 88 FY which can not be measured experimentally. Indeed, the SHPB set-up only 89 provides the axial component force (FZ ) of the crushing responses of aluminium CE PT ED 83 honeycomb under mixed shear-compression loadings. The relationship between 91 the forces obtained by the experimental set-up and the normal and shear forces 92 on honeycomb specimen are presented by figure 3 and by the following equations AC 90 93 : FX ≈ 0 (1) FY = FN sin(ψ) − FS cos(ψ) (2) FZ = FN cos(ψ) + FS sin(ψ) (3) 94 95 6 ACCEPTED MANUSCRIPT Y Specimen Teflon sleeve Z Ψ Output Bar FZ Steel sleeve Short cylindrical bars with one bevel end US CR FN Aluminium support IP T FY FS Input Bar Figure 3: Scheme of the force components under mixed loading. So, the normal and the shear force components respectively FN and FS , 97 which are required for the macroscopic yield criterion, can not be calculated 98 directly using equations 2 and 3. AN 96 One way to overcome this limitation is to simulate the experimental tests 100 in order to have access to local force components. That is why, numerical 101 simulations are carried out taking the effect of the loading angle ψ and the 102 in-plane orientation angle β into consideration. In addition, an analysis of the 103 crushing responses and the collapse mechanisms is realised. 104 3. General description of the FE numerical model PT ED For obvious CPU-Time cost, only the honeycomb specimen placed between CE 105 M 99 the two beveled bars is modelled with a refined mesh. The experimental input 107 and output velocities are used in the numerical simulations and are applied to 108 the two beveled bars. The numerical model as well as the boundary conditions 109 are shown in figure 4. The FEM commercial code Abaqus/explicit is used for 110 this numerical study. AC 106 7 US CR IP T ACCEPTED MANUSCRIPT Figure 4: Scheme of the numerical model and the boundary conditions under mixed loading. 111 3.1. The FE numerical model and materials The material constitutive model parameters are determined using an inverse 113 approach of identification proposed by Markiewicz et Al. [22]. This approach 114 uses the optimization methods and allows one to determine the parameters 115 of constitutive models by correlating the results of the numerical simulation 116 with the global variables obtained by the experimental crushing tests of thin- 117 walled components. The honeycomb cell wall material is an aluminium alloy 118 (Al5056-O) well known to be rate insensitive and it is assumed to exhibit a 119 bilinear elasticplastic response. That is why only the quasi-static uni-axial 120 crushing response is used in the inverse approach. The inverse approach 121 solutions allow to identify a material yield stress (σy = 0.38 GPa) and 122 a hardening modulus (Et = 0.5 GPa) which are in accordance with 123 the published parameters reported in the literature for this alloy by CE PT ED M AN 112 Hou et Al. [15]. As an example, figure 5 (a) under uni-axial (ψ = 0◦ ) 125 and (b) mixed shear-compression (ψ = 30◦ / β = 60◦ ) illustrate the 126 sensitivity of the finite-element model to the yield stress parameter AC 124 127 for a fixed hardening modulus on the numerical crushing responses. 128 Other material parameters such as density (ρ = 2640 kg/m3), Young’s modulus 129 (E = 70 GPa) and Poisson’s ratio (ν = 0.35) are taken from literature. 8 ACCEPTED MANUSCRIPT 20 20 FZ _ψ = 0°_EXP 14 FZ _ψ = 30° β = 60°_EXP 18 FZ _ψ = 30° β = 60°_NUM_ σy =280 MPa FZ _ψ = 0°_NUM_ σS =380 MPa 16 FZ _ψ = 30° β = 60°_NUM_ σy =330 MPa FZ _ψ = 0°_NUM_ σS =480 MPa 14 FZ _ψ = 30° β = 60°_NUM_ σy =380 MPa 12 10 8 FZ _ψ = 30° β = 60°_NUM_ σy =480 MPa 10 8 6 6 4 4 2 FZ _ψ = 30° β = 60°_NUM_ σy =430 MPa 12 2 0 0 2 4 6 8 10 0 2 4 6 8 10 US CR 0 IP T 16 FZ _ψ = 0°_NUM_ σS =280 MPa FORCE (kN) FORCE (kN) 18 Disp (mm) Disp (mm) (a) ψ = 0◦ (b) ψ = 30◦ / β = 60◦ Figure 5: Sensitivity of the finite-element responses to the yield stress parameter for a fixed hardening modulus : (a) Uni-axial ψ = 0◦ and (b) Mixed shear-compression ψ = 30◦ / β = 60◦ . The two short beveled bars are made of Teflon presented by a linear material 131 with elastic properties (Density ρ = 2200 kg/m3 , Young’s modulus E = 1.5 GPa 132 and Poisson’s ratio ν = 0.46). The material properties are presented in table 1. AN 130 M Table 1: Aluminium honeycomb specimen and bars material properties Aluminium honeycomb Teflon parts 2640 70 0.35 0 0.38 0.5 2200 1.5 0.46 - 3 3.2. FEM: Mesh and boundary conditions CE 133 PT ED Density ρ (kg/m ) Young’s Modulus E (GPa) Poisson’s Ratio ν Plastic Poisson’s Ratio νp Yield Stress σy (GPa) Hardening modulus Et (GPa) All the honeycomb cells are meshed using four-node-doubly curved thin shell 135 elements with a reduced integration scheme (finite membrane strains, active AC 134 136 stiffness hourglass control (S4R) and 5 integration points through cell wall 137 thickness). A sensitivity analysis on the mesh size was carried out in order 138 to prevent mesh effects on the numerical results. This analysis has led to choose 139 an element size of 0.25 mm corresponding to a discretisation of the complete 140 model geometry with 232,600 elements (figure 6). 9 US CR Figure 6: FE Model: Mesh. IP T ACCEPTED MANUSCRIPT A general contact algorithm was used for the whole model with a Coulomb 142 friction coefficient equal to 0.3 [15]. At the interfaces between the specimen and 143 the inclined surfaces of the beveled parts, surface-to-surface contact algorithm 144 with a rough contact method is applied. The double thickness wall in the real 145 honeycomb is composed of two single thickness walls bonded by a thin adhesive 146 layer. During simulations, these double walls are represented by single shells 147 with double thickness assuming that the bonding is not delaminated [11]. All 148 degrees of freedom of nodes of the two short beveled bars are fixed except the 149 velocity in Z direction in order to ensure the good alignment corresponding to 150 the experimental set-up. The impact velocities for the numerical simulation 151 are taken equal to the input velocity and the output velocity obtained in the 152 experimental study on respectively the input and output short beveled bars 153 (figure 4). 154 elements with reduced integration (C3D8R) with an element size of 3 mm (figure 155 6). PT ED M AN 141 AC CE The short beveled bars are meshed with 8-node bilinear brick 10 US CR IP T ACCEPTED MANUSCRIPT Figure 7: The in-plane orientation angle β on the numerical simulations. Four configurations are presented in figure 7 that illustrates the position and 157 the orientation of the specimen to obtain the four in-plane orientation angles β 158 = 0◦ , β = 30◦ , β = 60◦ and β = 90◦ for every loading angle ψ such as performed 159 in the experiments. 160 4. Numerical results and experimental validation PT ED M AN 156 161 The aim of this section is to validate the numerical model developed 162 previously by comparison with the experimental results presented in the 163 previous published works by Tounsi et Al. [19] under uni-axial compression 164 and mixed shear-compression loadings. The comparison between experimental and numerical crushing responses in the Z direction (Fz) and collapse 166 mechanisms in both cases of loadings are presented. This comparison study 167 is carried out in terms of the initial peak force and the average crushing force AC CE 165 168 as well as the deforming patterns modes. 169 4.1. The crushing responses under quasi-static loading 170 In order to perform the quasi-static numerical tests, the numerical model 171 is used by fixing one of the beveled bars. As the constitutive material is rate 11 ACCEPTED MANUSCRIPT insensitive, a loading velocity of 1 m/s is applied on the other beveled bar. This 173 loading velocity is chosen after simulations with various loading velocity (0.1, 174 0.5 and 1 m/s). For this loading velocity, the ratio of the kinetic energy to the 175 strain energy is very small (about 10−4 ) and inertia effects are insignificant. IP T 172 Figure 8 presents a comparison between the experimental and numerical 177 crushing responses under uni-axial compression loading and for one 178 configuration of the mixed shear-compression loading. A good agreement is 179 observed between the numerical curves and the experimental ones. US CR 176 16 FZ _ Ψ = 0° _QS_ NUM 14 FZ _ Ψ = 0° _QS_ EXP 12 AN Force (kN) 10 8 6 4 0 0 M 2 2 4 6 8 10 12 Disp (mm) PT ED (a) Uni-axial compression Ψ = 0 ° 16 14 FZ _ Ψ = 30° / β = 0° _QS_ NUM FZ _ Ψ = 30° / β = 0° _QS_ EXP 12 Force (kN) 10 8 6 4 AC CE 2 0 0 2 4 6 8 10 12 Disp (mm) (b) Mixed shear-compression Ψ = 30 °/β = 0° Figure 8: Comparison between the experimental and numerical crushing responses under quasi-static (a) uni-axial compression ψ = 0◦ and (b) mixed shear-compression ψ = 30◦ and β= 0◦ . 12 ACCEPTED MANUSCRIPT In figures 9, the numerical and experimental results (with standard 180 deviation) under quasi-static loading conditions are presented. 182 agreement between experimental and numerical results at the initial peak force 183 and the average crushing force is provided for all configurations under quasi- 184 static loading with an acceptable difference. Average Average F _EXP_QS_β = 30° Average F _NUM_QS_β = 30° Peak F _EXP_QS_β = 0° Average F _NUM_QS_β = 0° 16 16 14 14 12 FORCE (kN) FORCE (kN) 12 10 8 6 30 45 (a) β = 0◦ Average Peak F _EXP_QS_β = 60° Average F _NUM_QS_β = 60° 0 15 M 30 15 30 45 (b) β = 30◦ Average Peak F _EXP_QS_β = 90° Average F _NUM_QS_β = 90° F _EXP_QS_β = 90° Peak F _NUM_QS_β = 90° 12 10 8 6 4 45 60 0 15 30 45 Loading angle (deg) (c) β = 60◦ (d) β = 90◦ AC Figure 9: Numerical and experimental initial peak and average crushing forces under quasistatic mixed shear-compression loading for (a) β = 0◦ (b) β = 30◦ (c) β = 60◦ and (d) β = 90◦ . 185 60 Loading angle (deg) 14 Loading angle (deg) CE 0 16 FORCE (kN) FORCE (kN) PT ED 14 4 60 F _EXP_QS_β = 60° Peak F _NUM_QS_β = 60° 16 6 AN 15 Loading angle (deg) 8 8 4 0 10 10 6 4 12 Peak F _EXP_QS_β = 30° Peak F _NUM_QS_β = 30° US CR F _EXP_QS_β = 0° Peak F _NUM_QS_β = 0° A good IP T 181 4.2. The crushing responses under dynamic loading 186 For the numerical simulations under dynamic loading, the use of the input 187 and the output velocities obtained by the experiments Tounsi et Al. [19] 188 provided a crushing displacement of 11 mm. 13 60 ACCEPTED MANUSCRIPT 189 Figure 10 presents the crushing responses under dynamic uni-axial 190 compression loading and dynamic mixed shear-compression loading (ψ = 30 191 ◦ 192 numerical responses. IP T and β= 60◦ ). A good agreement is observed between the experimental and 20 18 FZ _ Ψ = 0° _ EXP 16 Force (kN) 12 10 8 6 4 2 0 0 2 4 6 8 DISP (mm) US CR FZ _ Ψ = 0° _ NUM 14 10 (a) Uni-axial compression Ψ = 0 ° 20 AN 18 FZ _ Ψ = 30° and β = 60° _ EXP 16 FZ _ Ψ = 30° and β = 60° _ NUM Force (kN) 14 12 10 8 4 2 0 0 M 6 2 4 6 8 10 DISP (mm) PT ED (b) Mixed shear-compression Ψ = 30 °/β = 60° CE Figure 10: Comparison between the experimental and numerical crushing responses under dynamic (a) uni-axial compression ψ = 0◦ and (b) mixed shear-compression ψ = 30 ◦ and β= 60◦ . To summarise the experimental and numerical results for the different 194 loading angles with various in-plane orientation angles, figure 11 presents the 195 initial peak and average crushing forces for all configurations of loading under AC 193 196 dynamic condition. 14 ACCEPTED MANUSCRIPT Average F _EXP_DYN_β = 0° Peak F _NUM_DYN_β = 0° Average 18 16 16 14 14 12 10 8 12 10 8 6 4 4 US CR 6 Peak F _EXP_DYN_β = 30° Peak F _NUM_DYN_β = 30° IP T 18 2 2 0 15 30 45 60 Loading angle (deg) Average F _EXP_DYN_β = 60° Average F _NUM_DYN_β = 60° 20 0 15 30 45 (b) β = 30◦ Peak F _EXP_DYN_β = 60° Peak F _NUM_DYN_β = 60° 18 Average F _EXP_DYN_β = 90° Average F _NUM_DYN_β = 90° 20 Peak F _EXP_DYN_β = 90° Peak F _NUM_DYN_β = 90° 18 16 16 14 FORCE (kN) AN 14 12 10 8 12 10 8 6 4 2 0 15 30 M 6 45 4 2 60 Loading angle (deg) 0 15 30 45 Loading angle (deg) (d) β = 90◦ PT ED (c) β = 60◦ Figure 11: Numerical and experimental initial peak and average crushing forces under dynamic mixed shear-compression loading for (a) β = 0◦ (b) β = 30◦ (c) β = 60◦ and (d) β = 90◦ . Under dynamic conditions, the uni-axial and mixed shear-compression 198 numerical responses show a good correlation with experimental results for the CE 197 199 loading angle. Nevertheless, when the loading angle increases ψ ≥ 45◦ , the 200 initial peak force is overestimated in the numerical responses. The average 201 crushing force shows a good correlation between the numerical results and the AC 60 Loading angle (deg) (a) β = 0◦ FORCE (kN) F _EXP_DYN_β = 30° Average F _NUM_DYN_β = 30° 20 FORCE (kN) FORCE (kN) Peak F _EXP_DYN_β = 0° Average F _NUM_DYN_β = 0° 20 202 experimental ones. The error becomes more pronounced for ψ = 60◦ . This 203 could be explained by the complexity of the experimental collapse mechanisms 204 when the loading angle increases. 205 Based on the comparison between the experimental and numerical results 206 under quasi-static and dynamic mixed loading, a good correlation is reported 15 60 ACCEPTED MANUSCRIPT in term of initial peak and average crushing forces. The numerical model was 208 less successful on the initial peak force as well as the average crushing force 209 when the loading increases ψ ≥ 45◦ . So, to explain this difference, a numerical 210 investigation of the collapse mechanisms including the deforming pattern modes 211 is performed. 212 4.3. The collapse mechanisms 213 4.3.1. Numerical deforming pattern modes and cell in-plane orientation effects 214 The collapse mechanisms including the global deforming pattern modes of 215 the specimens and the local collapse mechanisms at cells level are investigated 216 under quasi-static and dynamic loading conditions. 217 description of the numerical collapse mechanisms and the effect of the in-plane 218 orientation angle are presented. An experimental validation is realised focusing 219 on the deforming pattern modes. US CR AN 220 IP T 207 In this section, a full Three deforming pattern modes are identified numerically (figure 12) such [19] and Hou et Al. [14]. as observed experimentally by Tounsi et Al. 222 The numerical simulations illustrate definitions and reasons of the occurrence 223 of each mode, including the analysis of folding mechanisms using cross-section 224 views which are presented in previous experimental works reported by Tounsi 225 et Al.[19]. AC CE PT ED M 221 16 Horizontal Plastic Hinge Lines Inclined Plastic Hinge Lines Inclined Plastic Hinge Lines IP T ACCEPTED MANUSCRIPT Compatibility zones Compatibility zones US CR Inclined Plastic Hinge Lines MODE I MODE II and β = (b) Mode II (ψ = 45◦ and β = 0◦ ) Horizontal Plastic Hinge Lines Compatibility zones M Inclined Plastic Hinge Lines 90◦ ) AN (a) Mode I (ψ = 15◦ MODE III PT ED (c) Mode III (ψ = 30◦ and β = 90◦ ) Figure 12: The three deforming pattern modes I, II and III under dynamic mixed shearcompression loading : numerical results. 226 For the experiments under dynamic loading, the collapse mechanisms are 227 observed through the high speed video at the side of the specimen. Indeed, the experimental local collapse mechanisms (inside the specimen) are not accessible. 229 The numerical results can overcome this experimental limitation and be helpful 230 to analyse the collapse mechanisms in detail at the cell level. AC CE 228 231 Figure 13 presents the mode I. The shear load direction is perpendicular to 232 the double wall thickness (β = 90◦ ). This leads to the folding of the double 233 wall thickness without any shearing. Thus, the formed plastic hinge lines are 234 horizontal. The progressive folding system is localised at only one side of the 235 specimen (top or bottom). 17 ACCEPTED MANUSCRIPT Mode I Double wall thickness Horizontal plastic hinge lines US CR Shear load IP T β = 90° Simple wall thickness AN Figure 13: The deforming pattern mode I at the cell level for β = 90◦ . Figure 14 presents the mode II. The shear load direction is parallel to the 237 double wall thickness (β = 0◦ ). This leads to the folding of double and single wall 238 thickness with shearing. Thus, all the formed plastic hinge lines are inclined. 239 The progressive folding system is localised at the both sides of the specimen 240 that allows the rotation of the cell axis simultaneously with the folding process. PT ED M 236 Mode II Both sides of the specimen β = 0° AC CE Double wall thickness Inclined plastic hinge lines Shear load Simple wall thickness Figure 14: The deforming pattern mode II at the cell level for β = 0◦ . 18 ACCEPTED MANUSCRIPT Figure 15 presents the mode III. The shear load direction is perpendicular 242 to the single wall thickness (β = 30◦ ). This leads to the folding of single wall 243 thickness without shearing and the double wall thickness with shearing. Thus, 244 the formed plastic hinge lines are horizontal (single wall thickness) and inclined 245 (double wall thickness). The progressive folding system is localised at the both 246 sides of the specimen that allows to the global rotation of the specimen. US CR IP T 241 Mode III Both sides of the specimen β = 30° Double wall thickness AN Horizontal + inclined plastic hinge lines Shear load M Simple wall thickness 247 248 PT ED Figure 15: The deforming pattern mode III at the cell level for β = 30◦ . 4.3.2. Numerical and experimental deforming pattern modes In this section, a comparative study is achieved between the experimental and numerical collapse mechanisms. Under both quasi-static and dynamic 250 mixed shear-compression loading, the deforming pattern modes are compared CE 249 251 taking into account the coupled effect of the loading angle and the in-plane 252 orientation angle. AC 253 254 The three identified deforming pattern modes are verified both numerically and experimentally and shown in figure 16. 19 ACCEPTED MANUSCRIPT Experimental Numerical US CR IP T Ψ = 15° / β = 30° Mode I AN Ψ = 30° / β = 0° PT ED Ψ = 45° / β = 60° M Mode II Mode III CE Figure 16: Good correlation between the experimental and numerical deforming pattern modes. 255 to the numerical ones under quasi-static and dynamic loadings, respectively. AC 256 Tables 2 and 3 present the experimental deforming pattern modes compared 20 ACCEPTED MANUSCRIPT Table 2: The deforming patterns modes under quasi-static mixed shear-compression loading: numerical and experimental results 0◦ 15◦ 30◦ 45◦ 60◦ β = 60◦ NUM EXP β = 90◦ NUM EXP MI MII MII MII MII MI MI MI MIII MIII MI MI MIII MIII MIII MI MI MIII MIII MIII MI MII MII MII MII MI MI MII MIII MIII MI MI MII MIII MIII MI MIII MI MI MI IP T = = = = = β = 30◦ NUM EXP US CR ψ ψ ψ ψ ψ β = 0◦ NUM EXP Table 3: The deforming pattern modes under dynamic mixed shear-compression loading: numerical and experimental results 0◦ 15◦ 30◦ 45◦ 60◦ β = 60◦ NUM EXP β = 90◦ NUM EXP MI MII MII MII MII MI MIII MIII MIII MIII MI MII MIII MIII MIII MI MIII MIII MIII MIII MI MI MII MII MII MI MI MII MIII MIII AN = = = = = β = 30◦ NUM EXP MI MI MI MIII MIII MI MI MI MI MIII M ψ ψ ψ ψ ψ β = 0◦ NUM EXP For β = 0◦ , the deforming pattern mode “ mode II ” is the dominant 258 mode numerically and experimentally for all loading angle under quasi-static 259 and dynamic loading. For β = 30◦ and β = 60◦ , the three deforming pattern 260 modes “ mode I ”, “ mode II ” and “ mode III ” are observed. When the 261 loading angle ψ increases the mode “ mode III ” becomes more dominant. For 262 β = 90◦ , a difference on the deforming modes is reported between numerical 263 and experimental collapse mechanisms. 264 4.3.3. Numerical CE PT ED 257 and experimental local collapse mechanisms It’s difficult to observe the local collapse mechanisms for all configurations. 266 So a set of configurations is chosen to realise a section cut of the experimental 267 crushed specimens under quasi-static loading. 268 achieved between the experimental local collapse mechanisms and the simulated 269 ones. AC 265 21 Therefore, a comparison is ACCEPTED MANUSCRIPT The comparison is performed for the following loading configurations, firstly 271 for the uni-axial loading (ψ = 0◦ ), secondly for seven configurations of the mixed 272 shear-compression loading : (ψ = 15◦ / β = 30◦ , ψ = 30◦ / β = 0◦ , ψ = 30◦ / β 273 = 90◦ , ψ = 45◦ / β = 0◦ , ψ = 45◦ / β = 60◦ , ψ = 60◦ / β = 0◦ and ψ = 60◦ / β 274 = 90◦ ) such as presented in figure 17. Experimental US CR Numerical Ψ = 15° / β = 30° Ψ = 30° / β = 0° PT ED Ψ = 45° / β = 60° M Ψ = 45° / β = 0° AN Ψ = 30° / β = 90° IP T 270 AC CE Ψ = 60° / β = 0° Ψ = 60° / β = 90° Figure 17: The comparison between the experimental and numerical local collapse mechanisms under quasi-static mixed shear-compression loading. 275 Figure 17 shows that the numerical model is able to reproduce the observed 276 experimental collapse mechanisms. A good correlation is observed not only for 277 the deforming pattern modes but also for the local collapse mechanisms at the 22 ACCEPTED MANUSCRIPT cell level under quasi-static loading except for some cases of mixed loading (ψ 279 = 30◦ / β = 90◦ ). This difference could be explained by the side effect and the 280 boundary conditions. This phenomenon is observed experimentally under both 281 uni-axial and mixed loadings such as shown in figure 18. It suggests that the 282 distribution of collapse mechanisms is different from the side to the middle of 283 the crushed specimen. Moreover, the side effect becomes more significant when 284 the loading angle ψ increases. US CR IP T 278 Ψ = 0° Section cut view M Ψ = 30° AN Side view Section cut view PT ED Side view Ψ = 45° Section cut view Side view AC CE Top view Figure 18: The side effect on the collapse mechanisms under mixed shear-compression loading. 285 As a conclusion, a good correlation between experimental and numerical 286 results is observed on the crushing responses. The finite element model allows 287 to reproduce faithfully the experimental collapse mechanisms with a good 23 ACCEPTED MANUSCRIPT agreement in the deforming pattern modes. The validated finite element model 289 is used in the next section to separate the normal and the shear behaviours under 290 mixed shear-compression loading that will be used to determine the macroscopic 291 yield criterion. 292 5. The normal and shear behaviours IP T 288 The finite element model developed and validated previously is used to get 294 access to the tangential force component FY . Therefore, the normal and shear 295 honeycomb behaviours under mixed shear-compression loading are presented 296 separately. 297 calculated in order to identify the parameters of a macroscopic yield criterion 298 for the aluminium honeycomb under mixed shear-compression. US CR 293 AN In addition, the average crushing normal and shear forces are Based on the numerical simulations, all the raw data of force-displacement 300 curves (the axial force component FZ and the tangential force component FY ) 301 for all configurations (ψ and β) are presented in appendix 7. Therefore, the 302 normal and the shear forces in the frame of the specimen (figure 3) are calculated 303 using the following equations : PT ED FN = FZ cos(ψ) + FY sin(ψ) (4) FS = FZ sin(ψ) − FY cos(ψ) (5) AC CE 304 M 299 24 ACCEPTED MANUSCRIPT 16 FN _ Ψ = 45° and β = 0° _NUM 14 FN _ Ψ = 45° and β = 30° _NUM FN _ Ψ = 45° and β = 60° _NUM 12 FN _ Ψ = 45° and β = 90° _NUM IP T Force (kN) 10 8 6 4 0 0 2 4 6 Disp (mm) US CR 2 8 10 12 AN Figure 19: The numerical crushing force components under dynamic mixed shearcompression loading for ψ = 45◦ with various β: FN normal force component. 16 FS _ Ψ = 45° and β = 0° _NUM 14 FS _ Ψ = 45° and β = 30° _NUM FS _ Ψ = 45° and β = 60° _NUM 12 FS _ Ψ = 45° and β = 90° _NUM M Force (kN) 10 8 6 PT ED 4 2 0 0 2 4 6 8 10 12 Disp (mm) CE Figure 20: The numerical crushing force components under dynamic mixed shearcompression loading for ψ = 45◦ with various β: FS shear force component. The influence of the in-plane orientation angle is fairly significant in the 306 tangential force component FY . As a result, a significant effect is reported AC 305 307 in the normal force component FN and the shear force component FS . For a 308 loading angle ψ = 45◦ as an example, the in-plane orientation angle effect is 309 more pronounced in the shear force FS than in the normal force FN such as 310 illustrated by figures 19 and 20. 311 Now, to study the combined effect of the three parameters: the in-plane 25 ACCEPTED MANUSCRIPT orientation angle, the loading angle and the impact velocity for all loading 313 configurations, a macroscopic yield criterion is suggested. It is based on the 314 normal and shear crushing forces. The average crushing normal and shear forces 315 are presented in the figure 21. and they are calculated by the following equations 316 : Z 1 Crmax − Crpeak Z FSAverage = 18 FN (Cr)dCr Crpeak (6) Crmax FS (Cr)dCr (7) Crpeak FN _ Ψ = 30° and β = 0° _NUM 16 FS _ Ψ = 30° and β = 0° _NUM AN 14 12 FN AVERAGE 10 8 6 4 M Force (kN) Crmax 1 = Crmax − Crpeak US CR FNAverage IP T 312 FS AVERAGE 2 0 1 2 PT ED 0 CrPEAK 3 Disp (mm) 4 5 6 CrMAX Figure 21: The numerical average crushing force components under mixed shear-compression loading for ψ = 30◦ and β= 0◦ : the normal and shear average crushing forces. 317 Finally, based on the numerical results for all loading configurations, the numerical simulations allow to determine the average crushing normal and shear 319 forces. Thus, the identification of the macroscopic yield criterion parameters 320 may be performed. AC CE 318 321 6. Macroscopic Yield Criterion 322 The aluminium honeycomb behaviour under mixed shear-compression 323 loading is characterized by the normal crush strength σ and the shear crush 324 strength τ defined by: 26 ACCEPTED MANUSCRIPT 326 FNAverage SSpecimen (8) τ= FSAverage SSpecimen (9) and where SSpecimen represents the cross specimen section area. IP T 325 σ= Under mixed shear-compression quasi-static loading, equation 10 reported 328 by Hong et Al. [12] defines the macroscopic yield criterion of the aluminium 329 honeycomb taking into account the in-plane orientation angle β ( US CR 327 τ 2 σ 2 ) + (A cos2 (β) + B sin2 (β))( ) =1 σQS σQS (10) where σQS is the crush strength under quasi-static uni-axial compression loading 331 and A and B are the material constants. Based on the fitted strength contours, 332 the values of A and B parameters could be determined by the non linear 333 least squares fits method with Levenberg-Marquardt Algorithm (LMA) for 334 the specimens with β= 0◦ and β= 90◦ , respectively. Therefore, the material 335 constants identified are A = 8.62 with R2 = 0.972 and B = 22.44 with R2 = 336 0.991. PT ED M AN 330 For a validation of the proposed yield criterion, the normalized strength 338 contour for β= 30◦ and β= 60◦ are determined. The proposed yield criterion 339 provides a very good description of the quasi-static honeycomb behaviour. 340 Based on equation 10, figure 22 presents the macroscopic yield criterion for 341 the Al5056-N-6-1/4-0.003 aluminium honeycomb under quasi-static loading. AC CE 337 27 ACCEPTED MANUSCRIPT ] US CR IP T Normalized normal crush strength [ Normalized shear strength Figure 22: The macroscopic yield criterion under quasi-static mixed shear-compression loading. Under dynamic loading, the equation 10 presents the quadratic yield 343 criterion generalised to be valid at different impact velocities. It is defined 344 by the following equation reported by Hong et Al. [13] : σ τ )2 + (Ad (VImp ) cos2 (β) + Bd (VImp ) sin2 (β))( )2 = 1 σDY N (VImp ) σDY N (VImp ) (11) PT ED ( M AN 342 345 where σDY N is the crush strength under dynamic uni-axial compression loading. 346 The material constants Ad (VImp ) and Bd (VImp ) at the impact velocity VImp = 347 15 m/s are obtained from the normal and shear crush strengths for β= 0◦ and 348 β= 90◦ . Such as under quasi-static conditions, the non linear least squares fits 349 method with Levenberg-Marquardt Algorithm (LMA) of the numerical results suggests the following material constants identified as Ad = 10.04 with R2 = 351 0.972 and Bd = 28.73 with R2 = 0.915. CE 350 AC 352 353 Based on equation 11, figure 23 presents the macroscopic yield criterion for the Al5056-N-6-1/4-0.003 aluminium honeycomb under dynamic loading. 28 ACCEPTED MANUSCRIPT ] US CR IP T Normalized normal crush strength [ Normalized shear strength Figure 23: The macroscopic yield criterion under dynamic mixed shear-compression loading. The symbols in the two figures 22 and 23 represent the normalized normal 355 crush and shear strengths (quasi-static and dynamic, respectively). The lines 356 represent the macroscopic yield envelope based on the non linear least squares 357 fits method of the numerical results. M AN 354 A significant effect of the in-plane orientation angle on the macroscopic yield 359 criterion from β= 0◦ to β= 90◦ is confirmed. The most important effect is 360 obtained for β= 0◦ . This corresponds to the shearing of the double 361 wall thickness that requires a large quantity of energy. 362 analyse the impact velocity effect, the superposition of the macroscopic yield 363 criterion under quasi-static and dynamic loading requires that the dynamic 364 macroscopic yield criterion must be normalised by the crush strength under 365 quasi-static uni-axial compression loading such as defined by the following 366 equation: Finally, to AC CE PT ED 358 ( τ 2 σDY N 2 σ 2 ) + (Ad (VImp ) cos2 (β) + Bd (VImp ) sin2 (β))( ) =( ) σQS σQS σQS (12) 367 Thus, equations 10 and 12 leads to present the macroscopic yield criterion 368 as a function of the loading angle ψ, the in-plane orientation angle β and the 29 ACCEPTED MANUSCRIPT impact velocity VImp (figure 24). β = 30° β = 60° US CR β = 90° IP T β = 0° Normalized normal crush strength 369 Normalized shear strength AN Figure 24: The macroscopic yield criterion under mixed shear-compression loading as function of ψ, β and VImp . A dynamic enhancement phenomenon observed by Tounsi et Al. [20] 371 is confirmed by the macroscopic yield criterion up to a critical loading angle 372 ψcritical determined through the solution of the equations system (10 and 12) 373 and given by : PT ED M 370 ψcritical = arctan( 374 where τcritical = σcritical CE 375 s τcritical ) σcritical R2 − 1 KDY N − KQS R2 R= σDY N σQS (13) (14) (15) 376 KDY N = Ad (VImp ) cos2 (β) + Bd (VImp ) sin2 (β) (16) KQS = A cos2 (β) + B sin2 (β) (17) AC 377 378 For ψ > ψcritical , the quasi-static crushing responses become higher than the 379 dynamic ones. This phenomenon is explained by the difference of the collapse 380 mechanisms. An analysis of the collapse mechanisms is carried out under both 30 ACCEPTED MANUSCRIPT quasi-static and dynamic loading conditions in order to explain the negative 382 dynamic enhancement rate [20]. The collapse mechanisms and a schematic of 383 quasi-static and dynamic collapse mechanisms is presented by figure 25. The 384 folds number is higher under quasi-static loading than under dynamic one. The 385 main reason is due to inertia effects which promote the global rotation of the 386 cell axis rather than the formation of folds. As a consequence, the crushing 387 under quasi-static loading requires more quantity of energy than under dynamic 388 loading. This explains the negative dynamic enhancement observed for ψ > 389 ψcritical . Quasi-static (section cut view) US CR IP T 381 (Ψ = 60°) Quasi-static 2f Zoom view (Ψ = 60°) f : fold 4f AN 6f Dynamic (top view) Crushing Dynamic M 2f (a) Collapse mechanisms views 2f 2f (b) Scheme of collapse mechanisms PT ED Figure 25: Comparison between quasi-static and dynamic collapse mechanisms for ψ = 60◦ .[20] 390 The proposed macroscopic criterion allows to describe the mixed 391 shear-compression honeycomb behaviour taking into consideration 392 not only of the loading angle ψ but also of the in-plane orientation 393 angle β and the impact velocity. Next steps of this work will be to rewrite the proposed yield criterion in a suitable form for a 395 ’honeycombs’ material model, such as the Deshpande and Fleck model 396 [23]. AC CE 394 397 7. Conclusion 398 The mixed shear-compression experiments are reproduced using detailed FE 399 simulations by modelling the specimen placed between two beveled bars and 400 using the experimental input and output velocities. 31 ACCEPTED MANUSCRIPT A comparison between the numerical results and the experimental ones are 402 carried out on the initial peak force and on the average crushing force. A good 403 correlation is observed under quasi-static and dynamic mixed shear-compression 404 loadings. The collapse mechanisms are investigated numerically and the three 405 deforming pattern modes experimentally observed are also identified. Numerical 406 results suggest that the combined effect of the in-plane orientation angle and 407 the loading angle has an influence on the deforming pattern modes. Good 408 correlations between experimental and numerical are reported in terms of global 409 mechanisms (deforming patterns modes) and local collapse mechanisms (at the 410 cell level). US CR IP T 401 Finally, using the validated numerical model, the normal and the shear 412 behaviours are separated to determine the parameters of a macroscopic yield 413 criterion. A significant effect of the in-plane orientation angle is highlighted on 414 the mixed shear-compression behaviour by the macroscopic yield criterion. The 415 superposition of quasi-static and dynamic macroscopic yield criterion confirms 416 the dynamic enhancement up to a critical loading angle ψcritical depending of 417 the in-plane orientation angle β and the impact velocity. It decreases for the 418 loading angle 0◦ ≤ ψ < ψcritical and a negative enhancement is observed for 419 ψ > ψcritical . This phenomenon is attributed to the an unexpected combined 420 effect of the loading angle, the in-plane orientation angle and the impact velocity 421 which affects the collapse mechanisms. 422 Acknowledgements CE PT ED M AN 411 423 This research is conducted through collaboration between the University of 424 Valenciennes and the National Engineering School of Sfax. This collaboration is jointly financed by the National Centre of Scientific Research and the General 426 Direction of Scientific Research in Tunisia. The present research work has also 427 been supported by the International Campus on Safety and Intermodality in 428 Transportation, the Nord-Pas-de-Calais region, the European Community, the 429 Regional Delegation for Research and Technology, the Agence Universitaire de AC 425 32 ACCEPTED MANUSCRIPT la Francophonie and by the Ministry of Higher Education and Research. The 431 authors gratefully acknowledge the support of these institutions. 432 References 434 [1] Gibson, L., Ashby, M., 1997. Cellular Material: Structure and Properties, 2nd Edition. Cambridge University Press, Cambridge, UK. US CR 433 IP T 430 435 [2] Papka, S., Kyriakides, S., Oct. 1999. In-plane biaxial crushing of 436 honeycombs: Part II: analysis. International Journal of Solids and 437 Structures 36 (29), 4397–4423. [3] Yang, M.-Y., Huang, J.-S., Nov. 2005. Elastic buckling of regular hexagonal 439 honeycombs with plateau borders under biaxial compression. Composite 440 Structures 71 (2), 229–237. AN 438 [4] Hu, L.L. and Yu, T.X., May. 2010. Dynamic crushing strength of hexagonal 442 honeycombs. International Journal of Impact Engineering 37 (5), 467–474. 443 [5] Wilbert, A., Jang, W.-Y., Kyriakides, S., Floccari, J.F., 2011. Buckling and 444 progressive crushing of laterally loaded honeycomb. International Journal 445 of Solids and Structures 48, 803-816. 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International Journal of Solids and Structures 48 (5), 474 687–697. 475 PT ED 471 [15] Hou, B., Pattofatto, S., Li, Y., Zhao, H., Mar. 2011. Impact behavior of honeycombs under combined shear-compression. part II: analysis. 477 International Journal of Solids and Structures 48 (5), 698–705. CE 476 AC 478 [16] Zhou, Z., Wang, Z., Zhao, L., Shu, X., Jun. 2012. Experimental 479 investigation on the yield behavior of nomex honeycombs under combined 480 shear-compression. Latin American Journal of Solids and Structures 9 (4), 481 515. 34 ACCEPTED MANUSCRIPT 482 [17] Jimnez, F. L., Triantafyllidis, N., 2013. Buckling of rectangular and 483 hexagonal honeycomb under combined axial compression and transverse 484 shear. International Journal of Solids and Structures 50, 3934-3946. [18] Tounsi, R., Zouari, B., Chaari, F., Haugou, G., Markiewicz, E., 486 Dammak, F., Dec. 2013. Reduced numerical model to investigate the 487 dynamic behaviour of honeycombs under mixed shear-compression loading. 488 International Journal of Thin-Walled Structures 73, 290–301. US CR IP T 485 [19] Tounsi, R., Markiewicz, E., Haugou, G., Chaari, F., Zouari, B., Feb. 2016. 490 Dynamic behaviour of honeycombs under mixed shear-compression loading 491 : Experiments and analysis of combined effects of loading angle and cells 492 in-plane orientation. International Journal of Solids and Structures 80, 501– 493 511. AN 489 [20] Tounsi, R., Markiewicz, E., Haugou, G., Chaari, F., Zouari, B., Mar. 2016. 495 Combined effects of the in-plane orientation angle and the loading angle on 496 the dynamic enhancement of honeycombs under mixed shear-compression 497 loading. European Physical Journal Special Topics, 1–10. PT ED M 494 498 [21] Ashab, AMS., Ruan, D., Lu, G., Wong, Y.C., May. 2016. Quasi-static 499 and dynamic experiments of aluminum honeycombs under combined 500 compression-shear loading. Materials and Design 97, 183–194. [22] Markiewicz, E., Ducrocq, P., Drazetic, P. 1998. An inverse approach to 502 determine the constitutive model parameters from axial crushing of thin- CE 501 503 walled square tubes. International Journal of Impact Engineering , 21 (6), 504 433-449. AC 505 506 [23] Deshpande, V.S., Fleck, N.A., 2000. Isotropic constitutive 507 models for metallic foams. Journal of the Mechanics and Physics 508 of Solids, 48, 1253-1283. 35 ACCEPTED MANUSCRIPT 509 Appendix 510 Raw data for quasi-static loading conditions FZ _ Ψ = 15° and β = 0° _NUM_QS FY _ Ψ = 15° and β = 0° _NUM_QS FZ _ Ψ = 15° and β = 30° _NUM_QS FY _ Ψ = 15° and β = 60° _NUM_QS Force (kN) FZ _ Ψ = 15° and β = 90° _NUM_QS 10 FY _ Ψ = 15° and β = 30° _NUM_QS 8 FZ _ Ψ = 15° and β = 60° _NUM_QS 12 8 6 FY _ Ψ = 15° and β = 90° _NUM_QS 6 4 4 2 2 0 0 0 2 4 6 8 10 12 0 Disp (mm) 10 FZ _ Ψ = 30° and β = 0° _NUM_QS Force (kN) FZ _ Ψ = 30° and β = 90° _NUM_QS 8 6 2 0 2 4 6 8 Disp (mm) M 4 0 10 6 0 PT ED CE 4 FY _ Ψ = 30° and β = 90° _NUM_QS 2 4 6 8 10 12 10 FY _ Ψ = 45° and β = 0° _NUM_QS FY _ Ψ = 45° and β = 30° _NUM_QS 8 FY _ Ψ = 45° and β = 60° _NUM_QS Force (kN) Force (kN) 6 FY _ Ψ = 30° and β = 60° _NUM_QS (d) FY , ψ = 30◦ FZ _ Ψ = 45° and β = 90° _NUM_QS 8 FY _ Ψ = 30° and β = 30° _NUM_QS Disp (mm) FZ _ Ψ = 45° and β = 60° _NUM_QS 10 12 0 12 FZ _ Ψ = 45° and β = 30° _NUM_QS 12 10 2 FZ _ Ψ = 45° and β = 0° _NUM_QS 14 8 4 (c) FZ , ψ = 30◦ 16 6 FY _ Ψ = 30° and β = 0° _NUM_QS 8 Force (kN) FZ _ Ψ = 30° and β = 60° _NUM_QS AN FZ _ Ψ = 30° and β = 30° _NUM_QS 10 4 (b) FY , ψ = 15◦ 16 12 2 Disp (mm) (a) FZ , ψ = 15◦ 14 US CR 14 Force (kN) IP T 10 16 FY _ Ψ = 45° and β = 90° _NUM_QS 6 4 2 2 0 AC 0 0 2 4 6 8 10 12 0 4 6 8 10 12 Disp (mm) (e) FZ , ψ = 45◦ (f) FY , ψ = 45◦ 10 16 FZ _ Ψ = 60° and β = 0° _NUM_QS 14 FY _ Ψ = 60° and β = 0° _NUM_QS FZ _ Ψ = 60° and β = 30° _NUM_QS FY _ Ψ = 60° and β = 60° _NUM_QS Force (kN) FZ _ Ψ = 60° and β = 90° _NUM_QS 10 FY _ Ψ = 60° and β = 30° _NUM_QS 8 FZ _ Ψ = 60° and β = 60° _NUM_QS 12 Force (kN) 2 Disp (mm) 8 6 FY _ Ψ = 60° and β = 90° _NUM_QS 6 4 4 2 2 0 0 2 4 6 8 Disp (mm) (g) FZ , ψ = 60◦ 10 36 12 0 0 2 4 6 8 10 12 Disp (mm) (h) FY , ψ = 60◦ Figure 26: Numerical axial and tangential forces under quasi-static mixed shear-compression loadings. ACCEPTED MANUSCRIPT Raw data for dynamic loading conditions 10 20 FZ _ Ψ = 15° and β = 0° _NUM FY _ Ψ = 15° and β = 60° _NUM 7 FZ _ Ψ = 15° and β = 90° _NUM Force (kN) Force (kN) FY _ Ψ = 15° and β = 30° _NUM 8 FZ _ Ψ = 15° and β = 60° _NUM 14 FY _ Ψ = 15° and β = 0° _NUM 9 FZ _ Ψ = 15° and β = 30° _NUM 16 IP T 18 12 10 8 FY _ Ψ = 15° and β = 90° _NUM 6 5 4 6 3 4 2 1 2 0 0 0 2 4 6 8 0 10 (a) FZ , ψ = 15◦ 10 FZ _ Ψ = 30° and β = 0° _NUM 12 Force (kN) 8 10 12 10 12 10 12 FY _ Ψ = 30° and β = 90° _NUM 6 AN 10 5 4 3 2 1 0 0 2 4 6 Disp (mm) 10 0 20 10 8 6 4 2 CE 0 0 2 4 6 FY _ Ψ = 45° and β = 30° _NUM 8 FY _ Ψ = 45° and β = 60° _NUM 7 FZ _ Ψ = 45° and β = 90° _NUM 12 8 FY _ Ψ = 45° and β = 0° _NUM 9 FZ _ Ψ = 45° and β = 60° _NUM 14 6 10 FZ _ Ψ = 45° and β = 30° _NUM 16 4 (d) FY , ψ = 30◦ FZ _ Ψ = 45° and β = 0° _NUM 18 2 Disp (mm) PT ED (c) FZ , ψ = 30◦ 8 M 0 Force (kN) Force (kN) 12 2 Force (kN) 10 FY _ Ψ = 30° and β = 60° _NUM 7 FZ _ Ψ = 30° and β = 90° _NUM 4 FY _ Ψ = 45° and β = 90° _NUM 6 5 4 3 2 1 0 8 10 0 2 4 Disp (mm) 6 8 Disp (mm) (e) FZ , ψ = 45◦ (f) FY , ψ = 45◦ 10 20 FZ _ Ψ = 60° and β = 0° _NUM 18 FY _ Ψ = 60° and β = 30° _NUM 8 FZ _ Ψ = 60° and β = 60° _NUM 14 FY _ Ψ = 60° and β = 0° _NUM 9 FZ _ Ψ = 60° and β = 30° _NUM 16 FY _ Ψ = 60° and β = 60° _NUM 7 FZ _ Ψ = 60° and β = 90° _NUM Force (kN) Force (kN) 8 FY _ Ψ = 30° and β = 30° _NUM 8 FZ _ Ψ = 30° and β = 60° _NUM 6 AC 6 FY _ Ψ = 30° and β = 0° _NUM 9 FZ _ Ψ = 30° and β = 30° _NUM 14 4 (b) FY , ψ = 15◦ 20 16 2 Disp (mm) Disp (mm) 18 US CR 511 12 10 8 FY _ Ψ = 60° and β = 90° _NUM 6 5 4 6 3 4 2 1 2 0 0 0 2 4 6 8 0 10 Disp (mm) (g) FZ , ψ = 60◦ 2 4 6 8 Disp (mm) 37 (h) FY , ψ = 60◦ Figure 27: Numerical axial and tangential forces under dynamic mixed shear-compression loadings.