25 To assess the effect of stress state and strain r ate on damage and fracture of a commercial 26 DP1000 steel with a very fine microstructure, an extensive series of tests were performed. Using 27 finite element simulations, eight different sample geometries, including a dogbone, a centr al 28 hole, a shear and several notched samples, were designed to achieve both proportional and non - 29 proportional stress states using conventional test benches. Tested at quasi -static, intermediate 30 and, dynamic deformation rates, in total 175 tests were perfor med. Local strain fields were 31 obtained by digital image correlation. A correction procedure was worked out to eliminate the 32 influence of thermal softening. After testing, scanning electron microscopy was employed to 33 analyse the fracture surfaces. Tests and fractography allowed to draw systematic conclusions 34 on the response of the DP1000 steel. A two -stage strain rate sensitivity of strength is found with 35 a gradually increasing slope at low strain rates and a much steeper rise at high strain rates, which 36 is further amplified at higher triaxiality stress states. The experimentally derived fracture loci 37 revealed a dominant, detrimental impact of the stress triaxiality that is most pronounced at 38 intermediate strain rates. A remarkable, non -monotonic evolution of the fracture strain with 39 strain rate is observed: the highest values were obtained at intermediate rates. Scanning electron 40 microscopy images of the fracture surfaces indicate a void -assisted ductile fracture, though with 41 the occurrence of brittle feature s triggered at dynamic strain rates. Fracture morphology and 42 dimple features are heavily dependent on s tress state, strain rate and loading path. 43 Keywords 44 Ductile fracture, impact testing, multiaxial stress state, ductility, strain rate behaviour, dual- 45 phase steel, fracture mechanisms 46 47 48 Article Highlights 49 1. Design of test campaign imposing proportional and non -proportional stress paths 50 2. Extensive test dataset on strain rate dependent plasticity and fracture of DP1000 51 3. DP1000 shows a positive, two -stage strain rate sensitivity of strength 52 4. Remarkable, non -monotonic evolution of fracture strain of DP1000 with strain rate 53 5. Pronounced stress state influence on DP1000 hardening, ductility and failure modes 54 55 56 1 1. Introduction 57 The advent of advanced high strength steels (AHSS) has succeeded in reconciling the often 58 conflicting demands of enhanced safety and reduced fuel consumption in the automo tive 59 industry. Among AHSS, dual -phase (DP) steels have a great potential for vehicle body design 60 [ULSAB, 2016 ], particularly, for crash applications due to their excellent combination of 61 strength, initial work hardening and ductility. 62 The mechanical properties of DP steels are the result of a multiphase microstructure consisting 63 of hard martensite islands, or in some cases , bainite with traces of retaine d austenite, spatially 64 dispersed in a soft ferritic matrix. The ferritic phase lies at the origin of the excellent ductility, 65 whereas the high strength is mainly provided by the martensite. However, the presence of 66 different microstructural constituents an d their inevitable interaction, introduce another level of 67 complexity to damage and fracture of multiphase steels [ Lian, 2015 ]. The fairly straightforward 68 sequence of plastic deformation, severe strain localization and ductile fracture observed for 69 common single -phase ductile steels may no longer be applicable. Premature failure of 70 multiphase steels during forming processes or in service can restrict the applicability of 71 multiphase steels and should therefore be accounted for. Differences in mechanical resp onse 72 under dynamic loading, as compared to quasi -static loading, substantiate the need to explore 73 the intrinsic dynamic nature of the damage and fracture phenomena in multiphase automotive 74 steels. 75 The effect of strain rate on the mechanical properties of m etal alloys, including DP steels, has 76 been extensively studied [ Chen et al., 2011; Singh et al., 2011; Kim et al., 2013; Qin et al., 77 2013; Anderson et al., 2014; Rahman et al., 2016, Erice et al., 2018 ]. Several studies reported 78 a positive effect of strain rate on the streng th and strain hardening of dual -phase steels [ Beynon 79 et al., 2005; Oliver et al., 2007; Yu et al., 2009; Chen et al., 2011; Singh et al., 2011; Qin et 80 al., 2013; Kim et al., 2013; Anderson et al., 2014; Cao et al., 2015; Rahman et al., 2 016; Erice 81 2 et al., 2018 ]. However, significantly different strain rate dependencies of the flow stress were 82 observed in DP steels with strength levels ranging from 600 to 1000 MPa attributed to changing 83 martensite phase fractions [ Kim et al., 2013; Wang et al., 2013; Rahman et al. 2016, Das et al., 84 2017; Xu et al., 2019 ]. The strain rate effect on ductility, however, led to contradictory 85 conclusions [ Yu et al., 2009; Wang et al., 2013; Alturk et al., 2017 ]. The studies demonstrate 86 the uniqueness of the micr ostructure -dynamic property relations in multiphase high strength 87 steels and accentuate the need for in -depth characterization in light of their reliable utilization 88 in various sectors. 89 Extrinsic factors such as imposed strain, stress state, temperature a nd strain rate affect damage 90 and fracture in DP steels next to the material morphological features. Concerning the stress 91 state, the stress triaxiality and Lode angle are the critical factors that govern ductile fracture in 92 metals [ Bao and Wierzbicki, 2004; Bai and Wierzbicki, 2008 ]. Indeed, depending on the stress 93 state imparted on the material, the dominant fracture mode can alternate between void -assisted 94 tensile or shear failure and a combined shear -tensile failure. A wide variety of damage 95 mechanis ms can thus be unlocked by introducing various stress states. These states can be 96 achieved by imposing different loads or load combinations on well-designed samples [ Bao and 97 Wierzbicki, 2004; Mohr and Doyoyo, 2004; Mohr and Henn, 2007; Roth and Mohr 2014 ]. 98 However, very often, the reliability of the damage and fracture assessment is detrimentally 99 affected by the complex design of purpose -developed test benches and the critical 100 manufacturing of the complicated samples geometries. More recently, specimen geom etries 101 have been designed imposing specific stress states on metal samples using universal tensile 102 machines [ Bao and Wierzbicki; 2004; Mohr and Doyoyo, 2004; Mohr and Henn, 2007; 103 Driemer et al., 2010; Peirs et al., 2012; Roth and Mohr, 2014 ]. Damage and fr acture 104 investigations on DP steels [ Steinbrunner et al., 1988; Mazinani and Poole, 2007; 105 Kadkhodapour et al., 2011; Landron et al., 2013; Ghadbeigi et al., 2013; Anderson et al., 2014; 106 3 Erice et al., 2018; Chiyatan and Uthaisangsuk, 2020 ] revealed a negativ e impact of stress 107 triaxiality on plastic strain to failure while the strain rate yields a positive response in some steel 108 grades. Certain studies considered, next to stress triaxiality, also the Lode angle and represented 109 the stress state dependency on fr acture initiation and failure strain by a three -dimensional locus 110 [Bai and Wierzbicki, 2008; Lian et al., 2013; Roth and Mohr, 2014; Wu et al., 2017; Erice at 111 al., 2018; Roth and Mohr, 2018 ] using finite element simulations. A ductile mode of failure 112 was g enerally observed in DP steels. Seemingly divergent void nucleation mec hanisms 113 revealed the complexity, making a clear understanding and prediction of the damage behaviour 114 complicated. Upon critical review of studies concerning ductile fracture, it is evid ent that the 115 vast majority of material specimens undergoes a continuous variation in loading and/or strain 116 path during the deformation process. The stress and strain path changes result in a non - 117 straightforward assessment of material damage and fracture, i ncluding the fracture locus. 118 Moreover, non -proportional loading paths limit the direct correlation between stress state and 119 observed microstructural damage mechanisms. Hence, the establishment of proportional 120 loading paths in experimental design is crucial for the unambiguous analysis of damage and 121 fracture. 122 The present study aims at an in -depth characterisation of the plasticity, damage and fracture of 123 a commercial DP1000 steel with an ultra -fine grain microstructure. To trigger the multitude of 124 damage mec hanisms typically met in crash energy absorption components, an extensive test 125 campaign was designed aiming at a wide range of stress states and strain rates. Eight carefully 126 designed sample geometries were tested in uniaxial test setups. Deliberately, bot h proportional 127 and non -proportional stress paths were imposed. Indeed, although proportional loading states 128 facilitate systematic analysis and interpretation of the test result, in real car components, 129 proportional loading states are quite exceptional. All samples were tested at several, 7 in total, 130 quasi -static, intermediate and high strain rate tests using a conventional tensile test machine and 131 4 a split Hopkinson bar tensile setup. Including repetitions and tests performed for a temperature 132 correction, 17 5 tests were carried out. 133 Several advanced measurement techniques were used during all tests, including static and 134 dynamic 3D digital i mage correlation to obtain full -field strain fields. Additional temperature 135 measurements allowed to accurately determine the Taylor -Quinney coefficient, i.e., the fraction 136 of plastic work converted into heat. An experimental strategy was devised to uncouple the effect 137 of strain rate and temperature on the plastic response of the studied steel. To identify and 138 distinguish the various damage and fracture modes prevalent in the DP1000 steel, fracture 139 surfaces of the samples were analysed using scanning electron microscopy (SEM). 140 Finite element simulations were used for the specimen geometry optimization and to support 141 test anal ysis. The desired stress states are attained by well -conceived, easy -to-manufacture 142 sample geometries tested using widely available experimental setups and measurement 143 techniques. 144 2. Material and Methods 145 2.1 Material 146 A commercial dual -phase steel (DP1000) in the form of a rolled sheet with a thickness of 1.5 147 mm is used in this investigation. The microstructure consists of a mixture of ferrite grains (55%) 148 and martensite islands (45%). The average grain size of the ferrite is less than 2μm. 149 2.2 Experimental methods 150 2.2.1 Test setups, measurements and data processing 151 The static and intermediate strain rate tests are performed on an Instron material testing machine 152 (Model 5569). The deformation rate is controlled by the speed of the cross -head. A calibrated 153 load cell is used to measure the force. 154 5 An optical full -field surface deformation measurement technique based on digital image 155 correlation (DIC) using the commercial MatchID software is adopted to monitor the evolution 156 of the sample surface strain during testing. To this purpose, prior to testing, a speckle pattern i s 157 applied to the sample which is recorded during deformation by two cameras positioned under 158 an angle (30 -40°). High -resolution images are obtained using 5Mpxl F -033B Allied Vision 159 Stingray cameras together with appropriate LED lighting. The setup is calib rated to accurately 160 quantify 3D, i.e., both in -plane and out of plane, displacements. 161 The split Hopkinson bar tensile (SHBT) setup of DyMaLab at Ghent University is used for the 162 high strain rate tests. In the setup, the sample is positioned between two lo ng bars, i.e., the input 163 and output bar. A tensile loading wave is generated by the direct impact of an impactor on the 164 free end of the input bar. The incident loading wave propagates towards the sample and is partly 165 reflected and partly transmitted. By co ntrolling the velocity of the impactor, the strain rate in 166 the sample can be adjusted. Strain gauge measurements on the bars of the waves allow to 167 calculate the force, relative velocity and displacement imposed on the sample/bar interfaces 168 [Verleysen et al ., 2008 ]. Also for the dynamic tests, 3D DIC is used for full -field strain 169 measurements. The applied speckle pattern is recorded during the tests by two high speed 170 Photron Mini AX200 cameras. Depending on the test, framerat es between 50,000 and 100, 000 171 fps are used. The sample surface is illuminated using a high -intensity Dedocool light set. 172 To ensure the statistical relevance of the results, for each test condition, three test repetitions 173 are performed. For both static and dynamic DIC processing, subset, s tep and strain window 174 sizes are adopted based on the average speckle size (25 -55 μm) and resolution (5 -10 μm/pixel) . 175 The following strain formulations are used in the result section. For uniaxial tensile tests, often 176 the nominal (or engineering) strain 𝜀𝑛𝑜𝑚is defined as: 177 ℰ𝑛𝑜𝑚 =𝛥𝑙 𝑙0 (1) 178 6 Where 𝛥𝑙 is the elongation of a gauge section with an initial length 𝑙0. The plasti c strain 𝜀𝑝 can 179 then be derived as follows: 180 ℰ𝑝= ℰ𝑛𝑜𝑚 −𝜎 𝐸 (2) 181 Where 𝜎 and 𝐸 are respectively the nominal stress and Young’s modulus. Correspondingly, the 182 true plastic strain can be estimated by substituting the nominal stress and strain with its true 183 values. Logarithmic or true Euler -Almansi strain from DIC is given by: 184 ℰ𝑙𝑛= ln𝑉= ln√𝐹𝐹𝑇 (3) 185 In which 𝑉2=𝐹𝐹𝑇 is the left stretch tensor and 𝐹, the deformation gradient. Strain components 186 ℰ𝑖𝑗 (𝑖,𝑗=1,2,3) can be decomposed into an elastic strain ℰ𝑖𝑗𝑒 and a plastic strain ℰ𝑖𝑗𝑝, that is, 187 ℰ𝑖𝑗= ℰ𝑖𝑗𝑒+ℰ𝑖𝑗𝑝. Adopting classical J2 - (or Mises) plasticity, the equivalent plastic strain ℰ𝑒𝑞𝑝, 188 (or PEEQ) defined as work -conjugate of the equivalent str ess, can be calculated from the 189 deviatoric plastic strain components 𝑒𝑖𝑗𝑝 : 190 ℰ𝑒𝑞𝑝= ∫𝜀̇𝑒𝑞𝑑𝑡𝑡 0= ∫√2 3𝑒̇𝑖𝑗𝑝𝑒̇𝑖𝑗𝑝 𝑑𝑡𝑡 0 (4) 191 In the principal strain space, based on the ass umption of conservation of volume in the plastic 192 regime, i.e., ℰ𝐼̇𝑝+ℰ𝐼𝐼̇𝑝+ℰ𝐼𝐼𝐼̇𝑝=0, eq. 4 reduces to: 193 ℰ𝑒𝑞𝑝=∫√4 3((ℰ𝐼̇𝑝)2+(ℰ𝐼̇𝑝ℰ𝐼𝐼̇𝑝)+(ℰ𝐼𝐼̇𝑝)2)𝑡 0 𝑑𝑡 (5) 194 Where ℰ𝐼= ln(1+𝜕𝑢𝑥 𝜕𝑋) and ℰ𝐼𝐼= ln(1+𝜕𝑢𝑦 𝜕𝑌) are the principal logarithmic strains 195 measured directly on the specimen surface at the necking section. X and Y are the reference 196 coordinates along the principal axes, and 𝑢𝑥 and 𝑢𝑦 are the corresponding displacements. 197 The temperature evolution in the centre of the sample ’s gauge section is monitored using a 198 Pt100 thermocouple with an exposed tip. The sensor used in conjunction with a PID controller 199 7 unit and a data acquisition system results in reliable temperature measurem ents at a frequency 200 of 500 kHz. 201 202 Fig 1. DyMaLab static (left) and dynamic (right) tensile test setups (Inset: Speckled notched R4.5 specimen) 203 The test setups are shown in Fig.1. The specimen family designed for the characterization of 204 plasticity, damage and failure of DP1000 is presented in Fig. 2. Details on the sample geometry 205 optimisation procedure, including design criteria, are discussed in section 2.2.3. The red dots 206 indicate the gauge used in DIC processing to calculate the relative displacements and velocities 207 reported in the result section. For the uniaxial tensile sample, the distance between the red dots 208 is used as gauge length 𝑙0 to calculate the average nominal strain 𝜀𝑛𝑜𝑚 and strain rate. 209 The nominal or engineering stress 𝜎𝑛𝑜𝑚 is calculated as the ratio of the measured force to the 210 initial cross -section in the central section of the sample, except for the shear sample for w hich 211 the section of the cross -section of the shear region is used. For the dogbone sample, the true 212 values of the stress 𝜎𝑡𝑟 are obtained using: 213 𝜎𝑡𝑟= 𝜎𝑛𝑜𝑚(1+ℰ𝑛𝑜𝑚) (6) 214 True stress for the notched tensile and shear samples is defined as: 215 𝜎𝑡𝑟= 𝐹 𝐴 (7) 216 F is the applied force and A is the instantaneous area of the specimen minimum cross -section. 217 The a rea of the notched samples is estimated as the produ ct of the in -plane width reduction and 218 8 out of plane thickness reduction calculated from the post -processing of DIC results. 219 Symmetrical out of plane displacement about the specimen mid -thickness plane is assumed to 220 derive the variations in the sample thick ness throughout the deformation. For the shear sample, 221 the area is defined as the product of length and thickness of the shear region considering the 222 condition of pure shear within this zone. 223 From the engineering stress -strain curves, the yield strength ( YS) is calculated as the offset 224 stress at 0.2 % strain as per EN ISO 6892 -1:2019 [International Organization for 225 Standardization , 2019 ]. The ultimate tensile strength (UTS) is the highest nominal stress 226 reached. The fracture surfaces of the tested specimens are investigated using a n FEI Quanta 227 450f SEM. 228 229 230 Fig 2. Purpose -designed specimens for plasticity, damage and fracture characterization of DP1000 steel at static, 231 intermediate and high strain rates. Dimensions are given in mm. 232 9 2.2.2 Tem perature correction procedure 233 During testing, a major fraction of plastic work is converted into heat resulting in a temperature 234 rise in the sample. The tests at the lowest speed are quasi -isothermal, however, as will be shown 235 in section 3.1, in the SHTB t ests adiabatic conditions prevail. To isolate the effect of 236 temperature and unmask the true strain rate sensitivity of the material, a correction procedure 237 is introduced. To this purpose, tensile tests have been carried out at a low, isothermal strain rate 238 of 0.0001 s-1 at -40°C, room temperature and 100 °C, see Fig. 3a. In Fig. 3b, the ratios of the 239 stress at a certain temperature divided by the reference, room temperature stress is represented 240 for different values of true plastic strain before necking. These data points are used to fit a 241 temperature correction function 𝑓(𝑇) [Novokshanov et al., 2015; Liu et al., 2020 ]: 242 𝑓(𝑇)=𝑐1𝑇.𝑒𝑥𝑝(−(𝑐2𝑇.𝑇))+𝑐3𝑇 (8) 243 244 Fig 3. (a) True stress -plastic strain curves of tensile specimen at -40, 25 and 100 °C and (b) temperature dependence 245 function f(T) with 𝑐1𝑇, 𝑐2𝑇 and 𝑐3𝑇 taking the values of 0.620146, 0.004954 and 0.858314 respectively. 246 The function f(T) and measured temperatures are used to convert the stress 𝜎𝑡𝑟 measured at a 247 specific temperature T into the isothermal value 𝜎𝑡𝑟𝑖𝑠𝑜 according to: 248 𝜎𝑡𝑟𝑖𝑠𝑜= 𝜎𝑡𝑟𝑓(𝑇) ⁄ (9) 249 250 251 10 2.2.3 Design of experimental matrix for plasticity and fracture characterization 252 To identify the plastic properties at different strain rates, a conventional dogbone sample is 253 used. Aiming at a systematic analysis of the effect of stress state on the damage and fracture 254 properties of the DP1000 steel, 7 sample geometries are designed to be tested in conventional 255 test setups. The geometrical optimization is performed using a finite element based parametric 256 analysis procedure. A Levenberg -Marquardt algorithm is employed in Python programming 257 language to minimize a cost function and, as such, to attain t he solution of the optimization 258 problem. SHBT tests require the use of small samples. Identical samples and boundary 259 conditions are employed for static, intermediate and dynamic tests to avoid discrepancies and 260 to ensure that that differences between the r esults can be fully assigned to the deformation rate. 261 For damage studies, the stress state in a material is usually characterized by stress triaxiality η 262 and Lode angle parameter 𝜃. Stress triaxiality is defined as the ratio of hydrostatic pressure (σ m) 263 and von Mises equivalent stress ( 𝜎): 264 𝜂= 𝜎𝑚 𝜎 (10) 265 The Lode angle parameter is related to the normalized third deviatoric stress invariant and is 266 expres sed as: 267 𝜃=1−2 𝜋cos−1{3√3 2 𝐽3 𝐽232⁄ } (11) 268 Where J 2 and J 3 are the second and third deviatoric stress invariants. A detailed description of 269 the formulation and the physical meaning of these parameters is given in [ Bai and Wierzbicki, 270 2008; Lian et al., 2013 ]. For the sample geometry optimization, it is assumed th at the damage 271 initiation, as a precursor to the failure, occurs at the location of maximum plastic equivalent 272 strain (PEEQ). This location is denoted as the critical location and the element(s) encompassing 273 this lo cation is defined as the critical element( s). The main objectives of the optimization 274 process are the following: 275 11 1) The critical element(s) should yield the desired stress state during plastic deformation 276 over damage initiation till fracture. 277 2) The critical elements should remain at the same location . 278 The finite element simulations are performed using Abaqus/explicit. An isotropic von Mises 279 plasticity model is incorporated since a quasi -isotropic mechanical behavio ur is reported for 280 similar dual -phase (DP) steels [ Lian, 2015; Roth and Mohr, 2018; San tos et al., 2019 ]. The 281 hardening behaviour is characterized by a uniaxial tensile test under quasi -static conditions on 282 a smooth dogbone specimen dete rmined as per EN ISO 6892 -1:2019 [International 283 Organization for Standardization , 2019 ]. A combined Swift -Voce law is adopted to describe 284 the DP1000 flow curve at room temperature for a large strain range. 285 𝜎𝑡𝑟(ℰ𝑝)= 𝛼.𝐴(ℰ0+ℰ𝑝)𝑛+(1−𝛼).[𝑘0+𝑄(1−𝑒𝑥𝑝(−𝛽𝑣ℰ𝑝))] (12) 286 𝜎 and ℰ𝑝 are the true stress and true plastic strain, respectively. A, ℰ0, n, k 0, Q, β v and α are the 287 material parameters. The first term of the equation refers to the Swift l aw, the second to the 288 Voce law. 𝛼 is a weighting parameter. The resultant flow curve is shown in Fig. 4. For all 289 specimens, the finite element analysis is performed up to a maximum PEEQ value of around 290 0.6, reported as a realistic value for the f racture strain for several dual -phase steels [ Lian et al., 291 2013; Tasan et al., 2014 ]. Where possible, geometrical and loading symmetries are exploited: 292 one-eighth is modelled of the notched and unnotched tensile samples and one half of the in - 293 plane shear sample. Regular solid elements with reduced integration points (C3D8R) are 294 applied. In the cr itical region, an element size of 0.1 mm x 0.1 mm x 0.1 mm is chosen for all 295 specimen geometries since this size is similar to the length of a detectable micro -crack in the 296 tests. Additionally, a convergence analysis shows that further reducing the mesh si ze does not 297 result in significant changes in local stress and strain, as was also found in [ Lian et al., 2013 ]. 298 12 299 Fig 4. Experimental and extrapolated flow curve according to a combined Swift -Voce law (Eq. 12 ). The material 300 constants are: A = 1305 MPa, ℰ0= 2.25 e -14, n = 0.0768, k 0 = 776.28 MPa, Q = 265.49 MPa, β v = 74.34 and α = 301 0.5. 302 a) Sample geometry for identification of plasticity behaviour 303 The plasticity behaviour of DP1000 steel at different strain rates is characterized by uniaxial 304 tensile tests on smooth dogbone specimens in accorda nce with DIN EN ISO 26203 -1:2018 305 [International Organization for Standardization , 2018 ]. The dogbone specimen has a gauge 306 length of 6 mm and a width of 3mm. The short gauge length is selected to accelerate the 307 establishme nt of force equilibrium under dynamic tensile testing. Moreover, it allows to achieve 308 the high strain rates targeted in the present study. 309 310 311 13 Fig 5. (a) Distribution of equivalent plastic strain, stress triaxiality and Lode angle parameter in one -fourth of the 312 dogbone sample at a deformation resulting in a maximum equivalent strain of 0.62 and (b) loadin g paths in the 313 critical element. (Best viewed in colour) 314 Fig. 5a displays the distribution of PEEQ, η and 𝜃 at a maximum equivalent plastic strain of 315 0.62. The loading paths, i.e., the evolution of η and 𝜃 with respect to PEEQ, in the critical 316 element are shown in Fig. 5b. The critical location is concentrated around the centre of the 317 gauge section. Because of early necking, the loading path rapidly devia tes from the initial 318 uniaxial stress state characterised by η = 0.33 and 𝜃 = 1, to one by η = 0.36 and 𝜃 = 0.96 . 319 b) Sample geometries for fracture identification under non -proportional loading 320 Three notched specimens are derived from the dogbone spec imen geometry presented in the 321 previous section. As such, similar loading and measurement techniques can be adopted 322 throughout the experimental program. The notch radius and section width at the notch are found 323 to control the strain gradients in -plane and through the thickness of the sample and are therefore 324 selected as the optimisation parameters. Aiming at stress triaxialities between uniaxial tension 325 and plane strain, finally the three notched dogbone specimens, namely R4.5, R1.8 and R0.4, 326 presented in Fig. 2 are obtained. For the three specimens, during the entire plastic deformation, 327 the maximum PEEQ and stress triaxiality are obtained in a limited region around the centre of 328 the gauge section, see Fig. 6 a -c. As such, the optimization target concerning the critical location 329 is satisfied. The loading histories in the critical central element of the notched R4.5, R1.8 and 330 R0.4 specimens, represented in Fig. 6d show that both η and 𝜃 diverge continuously from their 331 initial values at the onset of yielding to values beyond plane strain. 332 333 14 334 335 336 Fig 6. Distribution of the equivalent plastic strain, stress triaxiality and Lode angle parameter in (a) one -fourth of 337 the notched R4.5, (b) R1.8, and (c) R0.4 samples at a PEEQ in the central critical elem ent of approximately 0.6. 338 Loading paths at the critical element (d) of the R4.5, R1.8 and R0.4 geometries. (Best viewed in colour) 339 c) Sample geometries for fracture identification under proportional loading 340 The geometries presented in the previous section give stress states corresponding to those in 341 crumple zones of cars. However, to investigate the fundamental damage and fracture 342 mechanisms in the DP1000 steel and to determine reliable damage and fracture loci, stable 343 stress states are highly preferred. Th erefore, four proportional specimen geometries are 344 designed to cover a wide range of stress states: a central hole, two plane strain and an in -plane 345 shear specimen. 346 15 347 348 349 350 351 Fig 7. Distribution of the equivalent plastic strain, stress triaxiality and Lode angle parameter in the (a) shear, (b) 352 one-fourth of the central hole, (c) PS2.5 and (d) PS4 samples at a PEEQ of approximately 0.6 in the central critical 353 element. (e) Loading paths at the critical element of the shear, central hole, PS2.5 and PS4 geo metries. (Best 354 viewed in colour) 355 16 For the central hole specimen, hole diameter and ligament width are the critical parameters 356 controlling the stress state. The parametric analysis shows that larger hole sizes tend to shift the 357 critical zone from the hole edge to the cent re of the ligament, whereas smaller holes introduce 358 larger mechanical gradients. The ligament width additionally controls the through -thickness 359 gradients. The optimization process results in a hole diameter of 2mm and a ligament width of 360 1mm. For these dimensions, the critical zone is stable and located in the middle over the 361 thickness at the hole, see Fig. 7b. During stable plastic deformation, the stress state in the critical 362 element features values of the stress triaxiality between 0.32 a nd 0.33 and a Lode angle of 363 approx. 0.91, see Fig. 7 e. Thus, values close to the values for uniaxial tension (η=0.33 and 𝜃 364 ~1) are obtained and maintained throughout loading. 365 Two plane strain samples are added to the matrix, namely PS2.5 and PS4, to feat ure a 366 generalized plane strain tension state under high stress triaxialities. Both samples have the same 367 notch radius, namely 1mm, whereas the widths are modified to achieve different η values. The 368 PS2.5 specimen, as shown in Fig. 7c and e, results in a pr oportional stress state (η~0.6 and 𝜃 369 ~0.0004) in the centre of the gauge section during plastic deformation. A stable loading path is 370 also established in the centre of the PS4 specimen, see Fig. 7d and e, however, under higher 371 stress triaxiality, i.e., η~ 0.69. 372 The shear specimen is optimized to achieve a pure shear stress state characterized by η=0 and 373 𝜃 =0 starting from the shear specimen geometry proposed in [ Peirs et al., 2012 ]. The key 374 attributes of the geometry are the length of the shear zone, the radius of the notches which 375 delimit the shear region and the eccentricity of the notches. After optimization, a proportional 376 shear stress state with η~0.0008 and 𝜃 =0.0002 is obtained in the critical shear deformation 377 region in and around the centre, see Fig. 7a and e, throughout the plastic deformation. Moreover, 378 a homogenous strain distribution is observed along the shear zone with negligible thinning of 379 17 the shear region. Experimental determination of the true material response up to large strains is 380 thus possible. 381 3. Results and Analyses 382 In section 3.1 and 3.2, the results are presented of the test campaign using the samples shown 383 in Fig. 2. In sections 3.3 and 3.4, relevant data extracted from the test series are gathered and 384 analysed focussing on the i nfluence of strain rate and stress state. 385 3.1. Uniaxial tensile tests 386 Tensile tests are carried out on the dogbone specimens at strain rates ranging from 0.0001 s-1 to 387 1080 s-1 averaged over the gauge length during the plastic deformation. Quasi -static ( 0.0001 - 388 0.01 s-1), intermediate (1.05 s-1) and dynamic (350 -1080s-1) loading regimes are covered. Fig. 389 8a presents representative engineering stress -strain curves. The DP1000 steel exhibits 390 continuous yielding behaviour and pronounced strain hardening at al l strain rates. 391 392 Fig 8. (a) Engineering stress -strain curves of uniaxial tensile specimens tested at quasi -static, intermediate and 393 dynamic strain rates and (b) time evolution of temperature and plastic strain in the centre of a specimen tested at 394 800 s-1. 395 Temperature measurements performed in the centre of the samples show that isothermal 396 conditions only exist at the lowest strain rate of 0.0001 s-1. When the test speed is increased, a 397 measurable, though low for the static tests, the temperature increase is observed. A t 800 s-1, a 398 18 temperature as high as 150˚C just before failure, i.e., at a logarithmic strain of ~0.6, is captured , 399 see Fig. 8b . Using the temperature correction procedure described in section 2.2.2, the 400 isothermal stress -strain curves at stra in rates of 0.01 and 500 s-1, together with the reference 401 curve at 0.0001 s-1, are presented in Fig. 9a. It is clear that sample heating induces significant 402 stress softening which becomes more pronounced at higher strain rates. 403 Under adiabatic conditions, the temperature increase in the sample as a function of the strain is 404 given by [ Taylor and Quinney, 1934 ]. 405 ∆𝑇= 𝛽 𝜌.𝐶𝑝∫𝜎ℰ 0 𝑡𝑟.𝑑ℰ𝑝 (13) 406 Where ρ is the density of the material (8050 kg/m3), C p the thermal capacity (466 J/kg/k) and 407 β the fraction of mechanical energy transformed into heat or Taylor -Quinney coefficient. At 408 lower, non -adiabatic strain rates, heat is partially dissipated to th e environment resulting in a 409 lower temperature rise than the one obtained by Eq. 13. In Fig. 9b, the 𝛽-values at necking 410 calculated from Eq. 13 are presented as a function of the logarithm of the strain rate. The 411 following Boltzmann -type equation is fitte d to the experimental data to determine the transition 412 curve: 413 𝛽𝑓𝑖𝑡= 𝐴1−𝐴2 1+𝑒𝑥𝑝 ((log ℰ̇−𝑥0) ∆)+𝐴2 (14) 414 Where 𝐴1=-1.87442x10 -16, 𝐴2=0.88, 𝑥0=-1 and ∆=0.66667. From a strain rate of approximately 415 10 s-1, 𝛽 is found to saturate around a value of 0.88, i.e., from that strain rate on adiabatic 416 conditions apply. 417 19 418 Fig 9. (a) Measured and corrected isothermal stress -strain curves for DP1000 steel at strain rates of 0.01 and 500 419 s-1 together with the reference curve at 0.0001 s-1, and the temperature evolution in the centre of the sample, (b) 𝛽- 420 value as a function of the logarithm of the strain r ate together with the transition curve given by Eq. 1 4. 421 3.2. Fracture tensile tests 422 Figs. 10 a -g give representative engineering stress –displacement curves for the tensile tests on 423 the purpose -designed fracture specimens performed at seven different deformation speeds. 424 425 426 20 427 428 Fig 10. Engineering stress -displacement plots for tests on a) central hole, b) notched R4.5, c) notched R1.8, d) 429 notched R0.4, e) shear, f) PS2.5 and g) PS4 samples at 7 test velocities. The velocity mentioned in the legend 430 refers to the average relative v elocity measured across the red dots using DIC. The inserted contour plot represents 431 the logarithmic axial strain distribution before fracture for all samples except the shear sample for which the 432 logarithmic shear strain is given. 433 3.2.1. Central hole ten sion 434 The stress -displacement curves feature a stress maximum before reaching fracture, see Fig. 10a. 435 Continuous yielding and significant strain hardening are observed at all the loading speeds. A 436 positive effect of the strain rate on strength levels is cle arly observed, even though thermal 437 softening effects at the higher strain rates mask the full effect of strain rate. The logarithmic 438 axial surface strain field plot obtained before fracture, see insert in Fig. 10a, indicates that strain 439 localization takes place at the intersection of the hole edge and the transverse mid -plane of the 440 specimen. Moreover, at each side of the hole, a pair of inclined localization bands is generated 441 when the hole is stretched. Thus, strong strain gradients develop in the ligamen ts along the 442 transverse direction and through the sample thickness beyond the point of local instability. 443 21 Fracture initiates at the hole and propagates towards the outer edge until final rupture. 444 Increasing strain rates initially improve the displacement a t failure, however, at dynamic strain 445 rates the fracture displacement decreases. Additionally, in the post -necking regime , a more 446 gradual decrease in the stress level is observed in the static tests as compared to the intermediate 447 and dynamic tests. This i s partly attributed to a more localised strain distribution at higher strain 448 rates and thermal softening. 449 3.2.2 Notched dogbone tension 450 The engineering stress -displacement curves of the three notched samples, i.e., R4.5, R1.8 and 451 R0.4, are presented in Fi g. 10b, c and d respectively. Initial yield and maximum values of the 452 stress increase as a function of deformation speed. Axial surface strain fields of all the notched 453 specimens clearly indicate strong strain localisation within a narrow zone around the c entral 454 cross -section area after maximum force. Thermal softening becomes prominent in this localized 455 deformation zone, especially at the higher strain rates. This effect coupled with damage 456 contributes to a more rapid decline in stress in the post -necking region at dynamic strain rates. 457 Increasing deformation speeds, however, result in a non -monotonic response on the 458 displacement at maximum force and the final displacement. For all the notched samples, these 459 displacements increase with the strain rate in th e static and intermediate zone. At dynamic rates, 460 a descent in the displacement values is observed, even to values below the reference. The 461 logarithmic surface strain fields inserted in Figs. 10, show that axial strain localization shifts 462 from the cent re (Fig. 10b) to the outer edge (Fig. 10d) of the specimen with increase in stress 463 triaxiality. This may seem contradictory to the optimisation criterion that imposed a maximum 464 value of the plastic equivalent strain in the centre of the samples. However, it sh ould be noted 465 that PEEQ values in the centre differ significantly from axial strain values, especially at the 466 sample surface. In all the notched samples, fracture initiates in the centre. In the R4.5 sample, 467 the crack propagates from the centre along the transverse axis. However, for the R1.8 and R0.4 468 22 samples, the centre crack is rapidly followed by cracks originating at the outer edge, see section 469 3.5. 470 3.2.3 Plane strain tension 471 Representative stress -displacement curves of the two plane strain specimens are given in Fig. 472 10f and g. The plane strain experiments show a significant strain rate sensitivity: a gradual 473 increase in yield and ulti mate stress is observed in the static and intermediate range. However, 474 although the stress is not yet corrected for thermal softening, a sharp increase is obtained going 475 from intermediate to dynamic rates. After the maximum force is reached, in the strain 476 localization phase, the stress decreases. Partly due to thermal softening, a more rapid decrease 477 is observed at intermediate and dynamic rates. However, as opposed to the samples discussed 478 in sections 3.2.1 and 3.2.2, no sudden shift to a more rapid declin e towards fracture is observed, 479 except for the intermediate range. Again, a non -monotonic trend is seen for the strain rate 480 influence on the failure displacement: in the static and intermediate regime, the failure 481 elongation increases with the imposed defo rmation speed, at higher rates it decreases. Axia l 482 strain field contours (figs.10 f and g) depict clear strain gradients along the specimen width with 483 maximum values at the edges. Here again, the final fracture initiates in the centre of the sample, 484 i.e., w here the highest value of the equivalent plastic strain is reached there. However, 485 subsequently , multiple cracks initiate along the transverse direction with increasing plastic 486 deformation, all of which eventually coalesce leading to failure. 487 3.2.4 In -plane shear 488 Representative stress -displacement shear curves are shown in Fig. 10e. In contrast to the other 489 samples, the shear samples exhibit strain hardening up to very large deformations. A minor, 490 though positive, effect of the strain rate on the stress l evels is seen under static testing velocities. 491 Strain rate sensitivity further increases in the intermediate and dynamic strain rates. In all tests, 492 23 the stress continuously increases up to a sudden failure. A sharp drop in stress is visible during 493 static t ests, whereas a more progressive decline is observed for all intermediate and high strain 494 rate shear experiments. Considering global ductility, fracture elongation experiences an 495 appreciable gain from static to intermediate strain rate tests. High strain r ate conditions again 496 reduce the final elongation. From th e shear strain fields in Fig. 10 e, strain localization within a 497 narrow band around the centre is discernible. Although in the centre the highest plastic 498 equivalent strain is obtained, since the devel opment of damage is affected by both plastic strain 499 and stress state, fracture might occur in the region at the notch edges where a positive stress 500 triaxiality is obtained. 501 3.3 Overview test results 502 To provide a comprehensive overview o f DP1000 plasticity , damage and fracture, the following 503 sections present key properties derived from the 175 experiments considered in the study. In 504 section 3.3.1 the focus lies on strength properties, in section 3.3.2 deformation values are 505 comprehensibly presented. 506 3.3.1 Strength levels 507 Fig. 11 presents the yield strength and isothermal ultimate tensile strength and strain hardening 508 rate obtained for the dogbone samples, normalized to the corresponding values of the reference 509 test, as a function of the logarithm of the str ain rate. The strain hardening rate at ultimate tensile 510 strength ( 𝜃𝑢), is calculated by: 511 𝜃𝑢= 𝜎𝑡𝑟𝑢𝑖𝑠𝑜−𝜎𝑡𝑟𝑦 ℰ𝑒𝑞𝑢−ℰ𝑒𝑞𝑦 (15) 512 𝜎𝑡𝑟𝑦 and 𝜎𝑡𝑟𝑢𝑖𝑠𝑜are the true yield and isothermal true tensile strength; ℰ𝑒𝑞𝑦 and ℰ𝑒𝑞𝑢 are the 513 equivalent s trains at yield and maximum force obtained by DIC. The introduced strain 514 hardening rate is used to identify the general hardening response and does not fully represent 515 24 the hardening definition devised in plasticity models. An approximate curve that follows the 516 experimental strength values is determined using the following function: 517 𝜎𝑓𝑖𝑡𝑛𝑜𝑚=𝐴1+𝐴2.𝑒𝑥𝑝 (logℰ̇ 𝐴3)+𝐴4.(logℰ̇)2 (16) 518 𝐴1= (1.044293, 1.036886), 𝐴2= (0.000 63, 0.000974), 𝐴3= (0.578374, 0.711387) and 𝐴4= (- 519 0.00292, -0.00245) are the fitted parameters. The first and second quantity within the brackets 520 is related to the normalised yield and tensile strength, respectively. Both 𝜎𝑡𝑟𝑦 and 𝜎𝑡𝑟𝑢𝑖𝑠𝑜 increase 521 with strain rate, however, the increment in 𝜎𝑡𝑟𝑦 is more prominent than in 𝜎𝑡𝑟𝑢𝑖𝑠𝑜. The strain 522 hardening rates in Fig. 11 indicate a depreciating hardening capacity at higher deformation 523 speeds. Although at the onset of yielding higher initial strain hardening rates are experienced 524 by the material with the increase in strain rate, the strain rate hardening capacity diminishes at 525 higher plastic deformations. 526 527 Fig 11. Evolution of normalized yield strength ( 𝜎𝑡𝑟𝑦), isothermal ultimate tensile strength ( 𝜎𝑡𝑟𝑢𝑖𝑠𝑜) and strain 528 hardening rate as a function of the logarithm of the strain rate. 529 Isothermal engineering stress -displacement plots for the lowest quasi -static, intermediate and 530 middle dynamic deformati on rates are presented in figs.12a, b and c, respectively. From the 531 reference series (Fig. 12a), it is clear that the maximum stress increases considerably with stress 532 triaxiality. However, the strength enhancement occurs at the expense of global ductility . The 533 shear tests exhibit the lowest strength. The absence of diffuse and localized necking phenomena 534 25 results in an extended uniform plastic deformation range in these tests. The plane strain tensile 535 tests, on the other end, show the highest strength and l owest fracture strain. 536 537 538 Fig 12. Combined engineering stress –displacement curves of damage/fracture tensile tests at different test velocity 539 ranges: a) lowest quasi -static, b) intermediate and c) medium dynamic 540 Similar observations are made for the intermediate and high strain rate tests. However, the stress 541 drop after the severe strain localisation is less abrupt compared to the static tests. 542 To elucidate the relationship between strain rate and stress state, t he evolution of 𝜎𝑡𝑟𝑦 and 𝜎𝑡𝑟𝑢𝑖𝑠𝑜 543 across the entire range of imposed stress states and strain rates is shown in Fig.13. A positive 544 strain rate sensitivity is apparent for all stress states. The largest increase in strength levels is 545 evident at th e transition from intermediate to dynamic rates. The gain in strength is significantly 546 more pronounced for the samples with higher stress triaxialities. It has to be noted that the 547 deformation rate has an influence on the stress state, especially at higher levels of deformation. 548 26 However, for the material considered here, even at UTS, the strain is relatively low, so also the 549 effect on the stress state at the onset of necking will be limited. 550 551 Fig 13. Effect of strain rate and stress state on the yield and tensile strength converted to isothermal values. For 552 each stress state, the imposed deformation rate increases from left to right; the corresponding test velocities are 553 indicated along the bars. 554 Values of 𝜃𝑢 at the reference deformation rate for all tes ted samples are given in Fig. 14a. An 555 increase in the strain hardening rate is obtained with increasing stress triaxiality. To uncouple 556 the influence of strain rate of f the stress state, Fig. 14b presents the isothermal strain hardening 557 ability normalized to its values under quasi -static reference state, for each geometry. Strain rate 558 clearly has a negative impact on the hardening capacity of the DP1000 steel, particularly under 559 dynamic loading. This conclusion applies to all the tests conducted in this study. However, an 560 increase in stress triaxiality retards and reduces the strain rate induced loss in strain hardening 561 rate. The most pronounced degradation in hardening rate is observed for the shear stress state, 562 the least pronounced for the highest tria xiality plane strain samples. 563 27 564 Fig 14. (a) Strain hardening rate at the lowest, reference deformation rate for all sample geometries and (b) 565 normalized hardening rate as a function of test velocity. 566 3.3.2 Uniform elongation and fracture loci 567 As opp osed to the strength values, increasing the strain rate results in a non -monotonic 568 evolution of the deformation capacity of the DP1000 steel. Indeed, as can be seen in Fig. 15, 569 which represents the uniform strain and total strain as a function of strain ra te for the dogbone 570 specimen, the elongation values increase as a function of strain rate in the static and 571 intermediate strain rate regime, though a significant reduction is observed at dynamic strain 572 rates. The experimental data is captured by a growth -decay function: 573 ℰ𝑓𝑖𝑡=𝑧0+𝑏.logℰ̇+𝑐.(logℰ̇)2+𝑑.𝑒𝑥𝑝 (−𝑒𝑥𝑝 (logℰ̇−𝑘𝑡 𝑤)) (17) 574 Where 𝑧0= (0.549568, 0.779497), 𝑏= (-0.0966, -0.1111), 𝑐= (-0.00924, -0.01127), 𝑑= (-0.8014, 575 -0.86843), 𝑘𝑡= (1.61302, 1.193892) and 𝑤= (2.38104, 2.381365) are the estimated parameter 576 sets. For each term, the first and the second value within the parentheses correspond to the 577 uniform and total strain, respectively. The relatively higher enhancement in total el ongation 578 compared to uniform elongation at static and intermediate strain rates indicates that much of 579 the gain in ductility occurs in the post -necking phase. 580 28 581 Fig 15. Evolution of uniform and total strain as a function of the logarithm of the strain rate. 582 Fracture elongation reflects only the global effect of strain rate and stress state on material 583 ductility. Since in all samples the strain fields are highly h eterogeneous and different volumes 584 of the material are involved in the deformation process, local ductility values are paramount to 585 assess the material response. To this end, DIC local equivalent plastic strains, calculated using 586 Eq.5, are further consider ed. With the local ductility values, a 3D material fracture locus is 587 established in the space of equivalent plastic strain to fracture ( 𝜀𝑒𝑞𝑝𝑓), stress triaxiality and Lode 588 angle parameter. To establish the relationship between material ductility and s tress state for the 589 DP1000 steel, only the proportional loading conditions are utilized. The test data are used to 590 determine the parameters of the symmetric formulation of the 3D fracture locus defined in the 591 extended modified Bai Wierzbicki model [ Lian et al., 2013; Wu et al., 2017 ]: 592 𝜀𝑒𝑞𝑝𝑓(𝜂,𝜃)=[𝐷1.𝑒𝑥𝑝(−𝐷2𝜂)−𝐷3.𝑒𝑥𝑝(−𝐷4𝜂)].𝜃2+𝐷3.𝑒𝑥𝑝(−𝐷4𝜂) (18) 593 𝐷1, 𝐷2, 𝐷3 and 𝐷4 are the material fracture parameters. A trust region -based surface fitting 594 optimization algorithm is employed in Matlab to determine these material parameters. For the 595 parameter optimization, the shear, the central hole, the two plane strain and the R4.5 tensile 596 tests are used. The R4.5 test is added to improve the quality of the fitting process. Indeed, 597 despite the initial non -proportionality, the R4.5 specimen demonstrates a constant stress state 598 in the later stage of plastic deformation. The thus obtain ed fracture locus at quasi -static strain 599 29 rates for DP1000 steel is depicted in Fig.16. The corresponding plane stress path on the fracture 600 locus is depicted using a red dotted line. 601 602 Fig 16. Fracture locus of DP1000 steel for proportional loading obtaine d with D 1=4.11224, D 2=3.85212, 603 D3=2.01749, D 4=1.68049. Black spheres denote the experimental data. Fracture locus for plane stress proportional 604 loading is illustrated b y red dotted lines. (Best viewed in colour) 605 A detrimental impact of stress triaxiality o n 𝜀𝑒𝑞𝑝𝑓 is evident. With a 𝜀𝑒𝑞𝑝𝑓 of 1.989, the shear 606 sample demonstrates that fracture initiation and rupture is effectively delayed at low 607 triaxialities. At the highest stress triaxiality, plane strain condition, on the contrary, a reduction 608 of the 𝜀𝑒𝑞𝑝𝑓 value of 66 % is observed. 609 Fracture loci at intermediate and high strain rates are computed in the same way as for the static 610 tests. Two different views of the three failure loci are p resented in Fig. 17. The colour scale in 611 the figure represents the equivalent strain level under different conditions. The ductility 612 deterioration with increasing stress triaxiality is comparable across different strain rate 613 intervals. However, the strain rate dependency on the fracture strain is not monotonic. Indeed, 614 as is clear from Fig. 17, from quasi -static to intermediate strain rates, a rise in 𝜀𝑒𝑞𝑝𝑓 is observed 615 for all combinations of stress triaxiality and Lode angle parameter considered. A fu rther rise in 616 strain rate results in an adverse effect on local ductility. 617 30 618 Fig 17. Failure loci (two views) at various strain rate intervals. Experimental data sets are denoted using the 619 corresponding legends in the figure. Red dotted lines denote the plane stress proportional loading under each strain 620 rate interval. (Best viewed in colour) 621 As is also the case for the hardening rate, see Fig. 14b, the most significant differences between 622 the fracture strains in different strain rate regimes are perceived at low stress triaxialities. 623 The strain rate effect on ℰ𝑒𝑞𝑝𝑓 is visualiz ed for different stress triaxialities in Fig.18. The figure 624 further endorses the statement regarding the non -monotonic evolution of the fracture strain with 625 strain rate. The following modified, non -linear, extreme value function is devised to replicate 626 the 3D surface: 627 ℰ𝑒𝑞𝑝𝑓𝑓𝑖𝑡=𝑆1+𝑆2𝑙𝑜𝑔𝑣 +𝑆3(𝑙𝑜𝑔𝑣 )2+𝑆4exp (−𝑒𝑥𝑝 {𝑆5−𝜂 𝑆6})+𝑆7𝑒𝑥𝑝 (−𝑒𝑥𝑝 {𝑆8−𝑙𝑜𝑔𝑣 𝑆9}) (19) 628 The optimized function parameters are obtained using the Levenberg Marquardt iteration 629 algorithm. 𝑆1= 0.7283, 𝑆2= 0.01797, 𝑆3= -0.0006618, 𝑆4= 4.05, 𝑆5= -0.02496, 𝑆6= -0.4968, 𝑆7= 630 -3.856, 𝑆8=2.514 and 𝑆9=1.811 are the values obtained after the regression fit. For all the 631 considered stress states, ℰ𝑒𝑞𝑝𝑓 initially increases with loading velocity from quasi -static up to 632 intermediate strain rates. Thereafter dynamic loading induces an earlier fracture. Furthermore, 633 the detrimental nature of the dynamic loading on the local ductility in DP1000 steel is very 634 evident in Fig. 18. 635 31 636 Fig 18. Effect of strain rate and stress triaxiality on equivalent plastic strain to fracture of DP1000 steel. Individual 637 dots represent data points from experiments. (Best viewed in colour) 638 Equivalent plastic strains for the shear tests measured at the instant of maximum force are 639 shown in Fig 1 9a for all test velocities. Similarly, for all the other samples, the equivalent plastic 640 strain at the onset of instability is compared across various test speeds, see Fig 19b. 641 642 Fig 19. (a) Equivalent plastic strain levels for shear tests across various test velocities measured at the instant of 643 maximum force. Corresponding test velocities are specified along with the bar plots. (b) Effect of test velocity on 644 the equivalent plastic strain at necking for all the tensile samples 645 A non -monotonic evolution of the plastic strain with a rising deformation rate applies in this 646 case as well. Here again, the equivalent plastic strain experiences an increase in its value from 647 quasi -static to intermediate strain rates. A sharp drop is evident thereafter, upon entering the 648 32 dynamic loading regim e. It can thus be established that strain rate is highly influential from the 649 early stage of plastic deformation until fracture for DP1000 steel. An i ncrease in stress 650 triaxiality limits the development of equivalent plastic strain in the material thereby triggering 651 an earlier onset of instability. Similar to the observations in Fig 16, the shear sample exhibits 652 the highest strain before localization/instability throughout the strain rate testing range while 653 the plane strain sample shows the lowest. It can additionally be seen that the gain in equivalent 654 plastic strain from quasi -static to intermediate strain rate is negatively affected by stress 655 triaxiality, see Fig 19b. Shear samples gain the most in deformation capacity, plane strain 656 samples the least. Lo ading path changes introduce a sharper drop in the strain at the onset of 657 instability at dynamic loading rates, particularly for the R1.8 and R0.4 samples. Comparing the 658 equivalent strain values at the onset of instability and at fracture, it is perceived that, in all the 659 fracture tests under all the tested conditions, post instability strain localization events contribute 660 significantly to failure strain. 661 The evolution of ℰ𝑒𝑞𝑝𝑓 with respect to initial stress state parameters ( 𝜂 and 𝜃 ) is plotted acr oss 662 all the loading speeds to establish an overview o f the impact of stress state parameters on 663 fracture, see Fig 20a and 20b. 664 665 Fig 20. a) Stress triaxiality and b) Lode parameter dependencies on the fracture strain across various strain rates. 666 33 Besides the dominating influence of stress triaxiality on ℰ𝑒𝑞𝑝𝑓, the effect of the Lode parameter 667 on the fracture strain becomes more pronounced with a rising deformation rate. The effect is 668 most apparent for the shear samples, demonstrated by the high est strain rate influence on the 669 fracture strain. 670 3.4 Fractographic analysis 671 Post mortem, the fracture surfaces of selected samples are observed by SEM. For each sample 672 series, three samples, i.e., one tested at static, one at intermediate and one at hig h strain rate, 673 are selected for this analysis. Figs. 21 a -g present the fractographs at two magnifications, i.e., 674 one showing the entire fracture surface and one with more details at the critical location, i.e., 675 where the fracture is assumed to initiate. 676 The dependence of the ℰ𝑒𝑞𝑝𝑓 on the strain rate, as presented in section 3.3.2, is reflected in the 677 images of the entire fracture surfaces. Indeed, except for the shear samples, the area reduction, 678 especially thinning, is most pronounced for the samples tested at intermediate strain rate, 679 followed by the static and then the dynamic samples. Dimples with average sizes ranging from 680 1 to 5 micromet re depending on the test condition are observed on all surfaces. A process of 681 void nucleation, growth and coale scence is clearly the primary contributor to macroscopic 682 failure. Moreover, in the centre of the dogbone, notched and plane strain samples, and at the 683 hole edge of the central hole sample, mainly equiaxed dimples are found typical for a tension 684 failure whi ch supports the assumption that fracture initiates in the critical zone. Away from the 685 fracture initiation location, elongated dimples are observed typical ly for tensile tearing or shear 686 in the inclined lips formed in certain samples towards the outer samp le edges. 687 Fractographs of the dogbone samples are depicted in Fig. 21a. At static and intermediate strain 688 rates, mainly small dimples are found. However, also some larger ones with secondary dimples 689 are observed. In line with ℰ𝑒𝑞𝑝𝑓, the deepest dimples are seen at intermediate rates, fewer and the 690 34 shallowest ones at dynamic rates. Moreover, at dynamic rates, macroscopic shear lips constitute 691 almost the entire fracture surface. In the lips, elongated dimples are seen. The p resence of quasi 692 cleavage facets is additionally detected, as indi cated by the ellipses in Fig. 21a. Quasi -cleavage 693 is favoured by high strain rates and triaxial stress states. The latter might occur at the tip of the 694 propagating macro crack. 695 Similar frac ture features are found in the central hole samples, see Fig. 21b. Apart from a few 696 enlarged dimples, the critical zone is mainly covered by homogenously distributed primary 697 dimples. 698 699 700 701 35 702 703 704 705 Fig 21. SEM fractographs of (a) dogbone (b) centra l hole, (c) R4.5, (d) R1.8, (e) R0.4, (f) PS2.5 and (g) shear 706 specimens at two magnifications. For each sample series, the left, middle and right images correspond with a static, 707 intermediate and dynamic test, respectively. Orange ellipses display quasi -cleavage facets. 708 36 At all the strain rates, fracture starts at the hole edge, propagates along the ligament and finally 709 a shear lip is formed at the outer edge completing the macroscopic failure. At intermediate strain 710 rates, micro -cracks are recognized along with the dimples. The cracks combined with the fine, 711 uniformly distributed deep dimples result in the highest fracture strain at this rate. The most 712 shallow dimples are again detected at dynamic strain rates, however, as opposed to the dogbone 713 samples, no quasi -cleavage zones are observed. 714 The fracture surfaces of the notched R4.5, R1.8 and R0.4 specimens, presented in figs.21c, d 715 and e respectively, show that an increased stress triaxiality gives rise to a more pronounced 716 heterogeneity in the size, shape and distribution of the dimples. The smallest dimples are found 717 for the R4.5 samples, followed by the R1.8 and R0.4 samples. Higher stress triaxialities 718 accelerate void growth at the expense of void nucleation. At the same time, a decreasing Lode 719 angle parameter induces a change in dimple sh ape. Intermediate testing speeds again result in 720 deeper dimples and a higher dimple density. So, both an enhanced nucleation and growth of 721 voids, gives rise to the higher fracture strain compared to the static samples. At all the strain 722 rates, the higher t he stress triaxiality, the lower the sample thinning. At dynamic strain rates, 723 shear lips constitute a major fraction of the fracture surface. The shear lips together with 724 sporadic quasi -cleavage facets hinder local thinning. Non -proportional loading condi tions 725 introduce mixed tensile -shear mode of fracture in these samples which is evident from the 726 aspect ratio and orientation of dimples in the critical zone. However, the presence of special 727 features such as serpentine glide on the dimple walls, particular ly for the samples with the 728 highest stress triaxialities, see figs.21d and e, suggests the presence of significant plasticity in 729 DP1000 even under such unfavourable circumstances. Secondary to the central critical zone, 730 high dimple and void densities are i dentified close to the notch root. Such an observation 731 indicates that void growth and coalescence events can occur sequentially at the specimen cent re 732 and its periphery (near the root of the notch). The micro -cracks formed after the coalescence 733 37 process at these locations thus propagate in opposing directions. These cracks eventually 734 conjoin to effectuate macroscopic failure in R1.8 and R0.4. 735 Fig. 21f presents the fracture features of PS2.5 plane strain samples. As opposed to all other 736 samples, for the PS2.5 samples, the smallest dimples are found at static rates, the largest at 737 dynamic rates. Especially at intermediate rates, the deviation in dimple sizes becomes larger 738 with many small dimples in between large dimples containing secondary dimples. At this ra te, 739 the dimples are also deeper, indicating a more sustained void growth before coalescence. 740 Similar to the notched tensile samples, dimple and void fractions are dispersed along the 741 transverse axis following the occurrence of maximum density at the cent re. Hence, the 742 subsequent void coalescence process will initiate cracks at multiple locations along the 743 transverse mid plane. The a malgamation of these cracks thereafter results in failure. For the 744 dynamic and static samples, the generated dimples are shallo wer. Quasi -cleavage is also 745 observed in the dynamic plane strain tests, as indicated by ellipses, which further reduces the 746 deformation capacity. Similar features are also found in PS4 samples . 747 The SEM fractographs of the shear fracture tests are displaye d in Fig. 21g. As opposed to all 748 other samples, the smallest dimples, even the smallest of all tests considered, are found for the 749 static samples. Void initiation is quasi -immediately followed by coalescence, explaining the 750 abrupt drop in force (Fig s.10 e and 12 a). Deeper and thus more visible dimples can be seen at 751 the intermediate and high strain rates. The void growth lying at its origin results in a more 752 gradual decline in load -carrying capacity at these rates (Figs. 12b and c ). 753 For the samples dominated by tensile loading, both dimple size and shape evolve in a non - 754 monotonic way as a function of strain rate. Indeed, for each tensile geometry, the smallest, 755 though deepest, and the most equiaxed dimples are found at intermediate st rain rates. 756 Consequently, not only void initiation, though also void growth is favoured at intermediate 757 rates. Compared to static rate samples, shallow dimples with slightly higher sizes are found at 758 38 dynamic rates. Hence, the lower fracture strains obtaine d at dynamic rates are mainly explained 759 by a suppressed void growth. The shear samples deviate from the tensile samples. The smallest 760 dimple sizes are found at static rates, the largest at intermediate rates. 761 Dimple size is clearly linked with stress tria xiality. For all the strain rates, in the tensile - 762 dominated samples, the smallest dimples are found for the central hole and dogbone samples, 763 followed by the R4.5, R1.8, R0.4 samples and then the plane strain samples. 764 The shape of the dimples is dominated by the Lode angle. Quasi -equiaxed dimples are obtained 765 for the central hole, R4.5 and dogbone samples. The more the Lode angle parameter evolves to 766 0, the more the dimples evolve to elliptical shapes. For the shear samples, strongly elongated 767 dimples are observed for all samples. Strain rate further favours deviations from the equiaxed 768 dimple shape, however, again in a non -monotonic way: for all the tensile dominated samples, 769 the highest differences between the elliptical dimples axes are found at dynamic rates, the 770 lowest at intermediate rates. For the shear samples, the highest difference is found for the 771 dynamic samples, followed by the intermediate and static samples. 772 4. Discussion 773 The aforementioned results provide a comprehensive insight into the stress state and strain rate 774 dependent plasticity and ductile fracture of the studied DP1000 steel. Irrespective of the test 775 condition, the material exhibits a continuous yielding behaviour with excellent deformation 776 values, in addition to excellent strength lev els. Both are attributed to the ultra -fine grain 777 microstructure. Higher deformation rates induce higher dislocation speeds which require higher 778 stress levels to overcome the critical resolved shear stress. As a result, a positive effect of strain 779 rate on t he flow stress is obtained. Especially, at dynamic deformation rates a steeper rise of 780 flow stress with strain rate is observed, resulting in a two -stage dependency of strength on strain 781 rate in agreement with [ Beynon et al., 2005; Oliver et al., 2007; Yu and Gao, 2009; Kim et al., 782 2013; Cao et al., 2015 ]. The significant strain hardening exhibited by the DP1000 could be due 783 39 to the unlocking of dislocations which are introduced into the ferrite during material processing 784 by stresses induced by the formation of martensite as described in [Cao et al., 2015 ]. 785 Adiabatic heating of the material at high strain rates often counteracts the effect of strain rate. 786 Indeed, an increased temperature decreases the dislocation resistance and induces material 787 softening. He nce, the outcome of dynamic tests is affected by both strain rate and temperature. 788 In the present work, a procedure is adopted to separate the temperature and strain rate effects, 789 thus allowing to reveal the real strain rate sensitivity of the material. Th e procedure relies on 790 temperature measurements performed during all tests. Furthermore, the temperature profiles 791 allowed to calculate the fraction of plastic work contributing to a temperature rise of the 792 material. Under dynamic testing conditions, this fr action, i.e., the Taylor -Quinney coefficient, 793 saturates at a value of 0.88 which is in line with values mentioned in [ Taylor and Quinney, 794 1934; Zaera et al., 2013; Roth and Mohr, 2014; Lian, 2015 ]. It is also worth mentioning that 795 the temperature correctio n approach employed in this work is suitable for the vast majority of 796 bcc steels. 797 For metals, generally, the gain in strength with rising strain rates occurs at the expense of 798 ductility in the material. However, in the present study, a peculiar, non -monotonic dependence 799 of the deformation capacity on the strain rate is observed in all test series and confirmed by the 800 fracture surface analysis. Stable plastic and fracture deformation are found to increase with 801 strain rate in the quasi -static and intermediate strain rate regime. The highest deformation values 802 are found at intermediate strain rates, the lowest at dynamic rates. This finding challenges the 803 generally assumed decrease of fracture strain with increasing strain r ate adopted in the 804 formulation of widely used material damage models. It also reflects the complexity associated 805 with the investigated material. The non -monotonic evolution of the deformation might be 806 attributed to the local temperature rise due to deforma tion localization at the ferrite -martensite 807 interfaces which further improves dislocation mobility and deformation capacity. Localized 808 40 heating, although not yet fully adiabatic at static and intermediate rates, compensates the 809 increased, strain rate induce d deformation moti on resistance to such an extent, resulting in 810 improved deformation values. At dynamic strain rates, the temperature increase, even though 811 much more significant, no longer compensates for the increased resistance. The accompanying 812 heat gen eration in the quasi -static and intermediate tests causes a temperature rise which 813 accelerates the dislocation mobility in the ferrite. The assumed critical dependence on 814 temperature might be further supported by the fact that DP steels are known to be pro ne to 815 dynamic strain aging (DSA) phenomena, which get activated at specific temperature ranges. 816 For DP1000 steel, the DSA effect is delayed at high strain rates [ Chandran et al., 2019 ]. As 817 such, it can be speculated that the temperature rise does not yet t rigger the DSA phenomenon 818 at intermediate loading speeds. However, at dynamic strain rates, the significantly higher 819 temperature rise in the samples pushes the internal temperature closer to the DSA activation 820 temperature thus locking further dislocation m ovement in the ferrite. In this case, martensite is 821 forced to accommodate the strain incompatibilities which favours fracture at ferrite -marten site 822 grain boundaries or in the martensite grains [ Steinbrunner et al., 1988; Mazinani and Poole, 823 2007; Kadkhoda pour et al., 2011; Landron et al., 2013; Ghadbeigi et al., 2013 ]. Moreover, 824 significant strain localization in the martensite might result in dislocation clustering and 825 entanglement which then enforces fragmentation in martensite laths with cleavage facets [Das 826 et al., 2017; Xu et al., 2019 ]. This effect assists early void formation at dynamic strain rates 827 which substantially deteriorates both strain hardening and local ductility [ Steinbrunner et al., 828 1988; Mazinani and Poole, 2007; Kadkhodapour et al., 201 1]. Analysis of the failure loci for 829 the different strain rate regimes confirms this judgement. A non -monotonic influence of strain 830 rate (Figs.17 and 18) on local ductility is thus established for the DP1000 steel. For quasi -static 831 up to intermediate strai n rates, increasing mobile dislocations could alter the dislocation 832 configuration near the grain boundaries. Under this circumstance, dislocations start to pile up. 833 41 This results in an enhanced density of dislocations at the ferrite -martensite interface. A 834 continuous increase in the dislocation density with increasing strain rate, motivated additionally 835 by the gradually increasing internal temperature, encourages the generation of dislocation cells 836 in the ferrite grains. The increasing difference in the plas tic deformation energy between the 837 ferrite matrix and the martensite/ferrite interface could activate multiple slip systems in some 838 ferrite grains . A more uniform energy distribution prevalent in such a case could actively delay 839 the initiation and propagat ion of micro voids. Thus, an improvement in fracture elongation 840 could be ascertained in these strain rate regimes. On the other hand, an accelerated generation 841 of dislocation tangles at high strain rates does not leave room for an efficient plastic deforma tion 842 energy transfer and thereby makes the material susceptible to early instabilities. It has to be 843 mentioned that the aforementioned statements are first -order postulates that could explain the 844 complex experimental observations made in this work. More ad vanced microstructural 845 characterization is needed to fully reveal the root causes of the phenomena revealed in this 846 research. 847 Next to rising deformation speeds, increasing the stress triaxiality proves to be harmful to the 848 material ductility. Multiaxial s tress states promote accelerated void growth at interfaces close 849 to the numerous martensite grains [ Ghadbeigi et al., 2013; Tasan et al., 2014; Das et al., 2017; 850 Santos et al., 2019 ] which could further induce the formation of shear slip bands along 851 marten site thereby constraining further deformation. Moreover, superior strain hardening rates 852 under higher stress triaxiality prompt strain incompatibilities and promote faster void 853 nucleation. The detrimental effect of stress triaxiality on the failure strain is observed over the 854 entire range of strain rates . The fracture surfaces of the tensile -dominated samples show highly 855 plastically deformed zones that failed perpendicular to the loading direction and shear lips 856 aligned with inclined, localised deformation bands. All fracture surfaces are mainly covered by 857 dimples indicating a ductile fracture mode governed by void initiation, growth and coalescence. 858 42 Similar findings are reported in [ Liu et al., 2019 ]. The dimple size is linked with the initiation 859 of voids, which is clearly favoured at intermediate rates. An increase in stress triaxiality 860 accelerates the void growth and restricts further void nucleation thus lowering local ductility. 861 Intensification of localised shear bands, resulting in shear lips and quasi -cleavage facets under 862 dynamic loading rates further attenuate the deformation capacity. Indeed, a significantly larger 863 portion of the fracture surface is occupied by shear lips at dynamic deformation rates. The 864 highly localised strain in these shear region s restricts further thinning of the sample. 865 The obtained 3D fracture loci suggest competing influences of stress triaxiality and Lode angle 866 that control the course of the plane stress fracture envelope (red dotted lines in Figs. 16 and 17) 867 between pure sh ear and tensile -dominated modes. An increase in stress triaxiality increases the 868 tensile stress acting on the plane of maximum shear. The detrimental effect of the normal stress 869 results in a higher ductility for pure shear than for uniaxial tension. The Lo de angle parameter, 870 on the contrary, points in the opposite direction. Indeed, at lower Lode angle parameters, the 871 probability of shear localization is higher. While following the trajectory of the red dotted lines 872 (in Figs. 16 and 17), i.e. the plane stre ss path, from pure shear to tension, we have 873 (𝜕𝜀𝑒𝑞𝑝𝑓𝜕𝜂⁄ )𝑑𝜂<0 and (𝜕𝜀𝑒𝑞𝑝𝑓𝜕𝜂⁄ )𝑑𝜂>0 thus suggesting the competition between the two 874 stress state parameters. However, from the fracture envelopes in Fig. 20a , it can be concluded 875 that the stress triaxiality effect is stronger than the Lode angle effect for DP1000 steel. The 876 observation of the lowest strength exhibited by the shear stress state together with the high 877 sensitivity of the fracture strain o n strain rate for samples with l ow Lode angle parameter 878 suggest potential secondary Lode angle effects. 879 Loading histories, in terms of 𝜂 and 𝜃, additionally play an important role in the occurrence of 880 certain failure modes in DP1000 steel. Fracture experiments with both proportional and more 881 complex loading paths allowed to derive the path dependence of ductile fracture. Besides 882 establishing the relationships between imposed stress states and the underlying damage 883 43 mechanisms, proportional loading conditions ensure that a unique, material intr insic fracture 884 locus is obtained which enhances the reliability of the calibration of phenomenological ductile 885 fracture models. Non -proportional loading paths, on the other hand, are a more realistic 886 representation of loading paths occurring in structural components. Fracture assessment under 887 non-proportional stress states for the DP1000 reveals the presence of mixed modes of failure 888 arising from increased strain heterogeneity in the microstructure. Associated fracture surfaces 889 consist of non -equiaxed dimpl es with traces of serpentine glides on the dimple walls. Increasing 890 stress triaxialities give rise to higher strain hardening capability, however, at the expense of 891 ductility. However, this rise in hardening capacity is comparatively diminished under non - 892 proportional loading paths. Similar conclusions are made based on numerical simulations and 893 analysis [Benzerga et al., 2012; Basu and Benzerga, 2015 ]. Loading path changes further 894 provoke instability prematurely at dynamic strain rates. 895 Irrespective of the stress state, lower strain hardening capability together with reduced 896 deformation capacity under dynamic loading diminishes the energy absorption potential of 897 DP1000 steel under impact loading conditions. 898 5. Concluding remarks 899 With the ai m of elucidating the effect of strain rate on plasticity, damage and fracture properties 900 of DP1000 steel, an extensive series of tests were carried out. A wide range of explicitly defined 901 stress states and deformation rates were considered. The following c onclusions were drawn 902 from the research: 903 1. Excellent strength and deformation capacity are exhibited in all tests by the DP1000 904 steel by virtue of the ultra -fine grain size prevalent in the material. 905 2. The mechanical response of DP1000 is sensitive to both the stress state and strain rate. 906 A positive strain rate effect on the material strength is apparent across the entire strain 907 44 rate range and for all the stress states. A steeper rise in flow stress at dynamic loading 908 rates revealed a two -stage relation in st rain rate sensitivity. The difference in strain rate 909 sensitivity between the two regimes is amplified by the stress triaxiality. 910 3. A temperature rise in the samples tested at high strain rates induces a coupled strain 911 rate-temperature response. A correction procedure is applied to isolate thermal softening 912 allowing to distinguish between real strain rate and temperature effects. 913 4. Comparison of measured and calculated temperature rises suggests that adiabatic 914 conditions can be assumed from a strain rate of ar ound 10 s-1 with a Taylor -Quinney 915 coefficient of 0.88. 916 5. Fracture experiments with proportional loading histories facilitate the determination of 917 reliable material intrinsic fracture loci for DP1000 steel and its dependency on strain 918 rate. Next to clearly sh owing the influence of stress triaxiality and Lode angle on the 919 fracture strain at different strain rates, the loci can be used in ductile fracture models. 920 6. A rising stress triaxiality has a detrimental impact on the material’s global and local 921 ductility. I ndependent of the Lode angle, a monotonically decreasing fracture strain with 922 respect to stress triaxiality is observed at all the loading rates. However, a competing 923 effect of stress state parameters on material toughness was also evident. Dominant stress 924 triaxiality effect is established throughout the tested range for DP1000 steel. The 925 influence of the Lode parameter is eminent with rising strain rate. 926 7. Fracture experiments at varying loading speeds exposed a non -monotonic strain rate 927 dependency on strai n to failure. The low Lode angle zone displays elevated sensitivity 928 to this strain rate induced fluctuations in fracture strain. 929 8. Loading path dependence on the material behaviour is perceptible particularly 930 concerning material hardening, ductility and the resultant fracture modes. Non - 931 45 proportional loading histories result in reduced strain hardenability, earlier onset of 932 instability and give rise to a mixed tensile/shear mode of fracture. 933 9. Ductile fracture governed by void initiation, growth and coalescenc e was noticed in all 934 fracture experiments. From shear to plane strain tension, the dominant failure mode 935 transitioned from shear failure to tensile dominated fracture with shear lips. High stress 936 triaxialities resulted in early failure as a result of accel erated void growth and 937 coalescence process. Intensification of localized shear regions and quasi -cleavage facets 938 under dynamic loading rates further attenuated the dimple fracture and in turn the 939 deformation capacity. 940 10. Finite element simulations were used t o optimise the sample geometries aiming at 941 specific stress states interesting for damage characterization. Although the outcome of 942 the specimen design is dependent on the material behaviour, the proposed research 943 methodology to identify the damage behaviou r of metals, and more specifically how 944 damage is affected by stress state, stress path and strain rate, on a non -ambiguous basis, 945 is widely applicable. Indeed, the proposed samples are easy -to-manufacture, and 946 required test setups and measurement technique s are widely available. 947 Acknowledgement 948 The authors gratefully acknowledge the European Commission Research Fund for Coal and 949 Steel (RFCS) for the funding of project “Toolkit for the design of damage tolerant 950 microstructures” [grant number : 709711] which supports the investigations. 951