The onset of macroscopic strain localization limits the ductility of many ductile materials. For porous ductile materials, two distinct mechanisms of macroscopic localization have been identi- ed: void growth induced softening and void coalescence. In this work we focus on analyzing the in uence of materials strain rate sensitivity (SRS) on the two mechanisms of macroscopic localization or ductile failure as a function of the imposed stress triaxiality. To this end, three dimensional nite element calculations of unit cells have been carried out to model void growth and coalescence in an in nite block containing a periodic distribution of initially spherical voids in a band. The matrix material of the unit cell is considered to follow a strain rate dependent elastic perfectly plastic ow response. The unit cell calculations are carried out for a range of SRS parameter, imposed stress triaxiality and initial orientations of the voided band. Our results show that both the critical porosity and strain at the onset of localization and coalescence are strongly in uenced by the SRS parameter and the imposed stress triaxiality values. Furthermore, the relative e ect of the SRS parameter is found to increases with the increasing value of the imposed stress triaxiality. Keywords: Ductility (A), Fracture Mechanisms (A), Elastic-viscoplastic Material (B), Porous Material (B), Finite Elements (C) 1. Introduction 1 The ductility of metals and alloys is limited by the onset of macroscopic strain localization, 2 as it marks the end of uniform deformation and acts as a precursor to failure. Two distinct 3 mechanisms have been identi ed that can lead to macroscopic strain localization in porous ductile 4 materials (Teko glu et al., 2015). 5 The rst mechanism of macroscopic strain localization involves strain softening due to mi- 6 crostructural changes, thermal e ects and/or damage evolution resulting in local degradation in 7 material's load carrying capacity. The local degradation in the material's load carrying capacity, 8 causes localized deformation in a thin band with smoothly varying deformation pattern outside the 9  Email address: gvadillo@ing.uc3m.es (Guadalupe Vadillo) Preprint submitted to International Journal of Plasticity December 28, 2019 band (Fressengeas and Molinari, 1985; Tvergaard, 1981; Pardoen and Hutchinson, 2000; Mercier 10 and Molinari, 2003; Aretz, 2007; Pineau et al., 2016). This results in a weak discontinuity along the 11 thin band at the micro-scale and a discontinuity in the displacement eld at the macro-scale. In 12 porous ductile materials, damage (void growth) induced softening is the dominant mechanism that 13 triggers the onset of macroscopic localization (Tvergaard, 1982), resulting in void sheet fracture. 14 The width of the thin band engul ng the voids in this scenario has a length scale comparable 15 or greater than the mean spacing between the voids. Under these circumstances the onset of 16 macroscopic localization is strongly dependent on material properties, initial porosity, orientation 17 of the band undergoing localized deformation and the imposed stress state (Needleman and Rice, 18 1978; Yamamoto, 1978). 19 The second mechanism of macroscopic strain localization in porous ductile materials involves 20 void coalescence. Void coalescence is a local instability phenomenon where the interaction between 21 neighboring voids plays a critical role. The onset of void coalescence is associated with a sudden 22 concentration of plastic strain in the ligaments linking neighboring voids (Koplik and Needleman, 23 1988). Thus, following the onset of void coalescence, the kinematics of the void enlargement 24 signi cantly di ers from that of void growth prior to this instability mode. For void coalescence 25 induced macroscopic localization the width of the micro-scale localization band is much more 26 narrower because the deformation is limited to the ligaments between the neighbouring voids. 27 The two aforementioned mechanisms of macroscopic localization in porous ductile materials, 28 void growth induced softening and void coalescence, are distinct and have been identi ed in 29 Teko glu et al. (2015); Guo and Wong (2018). In particular, for a strain rate independent material, 30 Teko glu et al. (2015) showed that depending on the value of the imposed stress triaxiality a 31 clear separation exists between the two mechanisms of macroscopic localization. This raises a 32 fundamental question: how does the material's strain rate sensitivity a ect the two mechanisms 33 of macroscopic localization or ductile failure as a function of the imposed stress triaxiality? 34 The strain rate sensitivity is an important material parameter and it greatly a ects the onset of 35 localized deformation and damage evolution (Marciniak et al., 1973; Hutchinson and Neale, 1977; 36 Ghosh, 1977; Hutchinson et al., 1978; Taya and Seidel, 1981; Budiansky et al., 1982; Cocks and 37 Ashby, 1982a,b; Pan, 1983a; Pan et al., 1983; Yoon and Taya, 1984; Duva, 1986; Nemat-Nasser 38 et al., 1986; Ortiz and Molinari, 1992; Czarnota et al., 2006; Vadillo et al., 2012; Srivastava and 39 Needleman, 2013; Agoras and Ponte Casta~ neda, 2014; Osovski et al., 2015; Wang et al., 2018). This 40 in turn a ects the performance, safety, reliability and manufacturability of engineering components 41 and structures; for example, the crash worthiness of automobiles, the blast resistance of ships and 42 airplane cargo holds, and the manufacturability of sheet metal components. In general, increasing 43 the strain rate sensitivity of a ductile material can delay the onset of localized plastic deformation. 44 Furthermore, it has also been shown that an increase in the material's strain rate sensitivity can 45 slow down the evolution of porosity and delay the onset of void coalescence in a porous ductile 46 material. In this work we focus on analyzing the in uence of materials strain rate sensitivity on 47 the two mechanisms of macroscopic localization or ductile failure as a function of the imposed 48 stress triaxiality. 49 Several analytical and computational studies have been carried out in past to analyze the 50 onset of localized deformation (Marciniak and Kuczy nski, 1967; Rudnicki and Rice, 1975; Rice, 51 1977; Saje et al., 1982; Pan, 1983b; Mear and Hutchinson, 1985; Nahshon and Hutchinson, 2008). 52 The micro-mechanical approach that constitute the basis of these works involves analyzing an 53 2 in nite sheet or block with a defect band inclined at an angle with respect to the imposed loading 54 direction. Alternatively, the onset of localization in a porous material can also be analyzed via unit 55 cell model calculations. In the unit cell model calculations, a periodic distribution of voids in the 56 material is assumed, that allows us to model a single void in the material and impose proportional 57 loading throughout the deformation history (Koplik and Needleman, 1988; Scheyvaerts et al., 2011; 58 Fritzen et al., 2012; Srivastava and Needleman, 2013; Br unig et al., 2013; Dunand and Mohr, 2014; 59 Teko glu et al., 2015; Liu et al., 2016; Torki et al., 2017; Guo and Wong, 2018; Luo and Gao, 2018). 60 Thus, unit cell model calculations not only provide the macroscopic response of the material but 61 also provide the details of the void growth and coalescence, as a function of the imposed constant 62 stress triaxiality values. 63 In this work, three dimensional nite element calculations of unit cells have been carried out 64 to model macroscopic strain localization due to damage induced softening, void growth and/or 65 coalescence, in an in nite block containing a periodic distribution of initially spherical voids in 66 a band. Note that we have assumed a state of the material where either the voids are initially 67 present or have already nucleated (uniformly for the sake of simplicity) in a narrow region in the 68 material. The unit cell modeled consists of a central block with a single void in its center within 69 two semi-in nite void-free blocks. The matrix material of the unit cell is considered to follow a 70 strain rate dependent elastic perfectly plastic ow response. Limited calculations have also been 71 carried out to explore the combined e ect of material's strain rate sensitivity and strain hardening. 72 The unit cell calculations are carried out for one initial void volume fraction, a range of strain rate 73 sensitivity parameter, a range of imposed stress triaxiality, and three initial orientations of the 74 voided band. In the calculations, onset of macroscopic localization is de ned as the point when 75 the ratio of the rate of deformation in the band and outside the band reaches a critical level, while 76 the onset of void coalescence is de ned when the ratio of the maximum to the minimum e ective 77 plastic strain rate at the surface of the void reaches a critical value following the work of Teko glu 78 et al. (2015); Guo and Wong (2018). 79 The remainder of this paper are organized as follows. The problem formulation and numerical 80 method are detailed in Section 2. The numerical results and the discussion of the key results are 81 presented in Section 3. The main conclusions of this work are summarized in Section 4. 82 2. Problem Formulation and Numerical Method 83 Three dimensional nite element calculations are carried out to model plastic ow localization 84 and void coalescence in an in nite block containing a periodic distribution of initially spherical 85 voids in a band as shown in Fig. 1 (top left). The con guration analyzed here is numerically 86 similar to the works of Tvergaard (1989); Barsoum and Faleskog (2011); Teko glu et al. (2015) and 87 Guo and Wong (2018). 88 2.1. Unit Cell Model 89 As shown in Fig. 1, for the periodic distribution of the voids in the voided band, a unit cell 90 can be de ned as a central block with a single void in its center within two void-free semi-in nite 91 blocks. The initial volume of the central block is taken to be Vcentral block =D10D20D30=D3 092 and the initial volume fraction of the spherical void with radius R0isf0= 4R3 0=3D3 0. In the 93 nite element calculations, the semi-in nite blocks along x1direction are assumed to be of nite 94 3 Figure 1: Three dimensional representation of an in nite block containing a periodic distribution of initially spherical voids in a band initially inclined at an angle uwith respect to the x2-x3plane (top left). The two dimensional sections of the block along x1-x3(top center) and x1-x2(top right) planes are also shown in the gure to highlight the periodicity of the voids. Owing to the periodic distribution of the voids, the voided band can be represented as a central block with a single void in its center within two void-free semi-in nite blocks (bottom center). The initial dimensions of the central block are D10,D20, andD30alongx1,x2andx3axes, respectively (bottom left). A zoomed in view of the nite element model of 1 =2 unit cell with initial void volume fraction, f0= 0:001 and u= 39(bottom right). length but long enough to numerically mimic semi-in nite blocks. For a given loading state and 95 orientation angle, u, of the voided band in the initial state, the orientation angle of the band at 96 any deformation angle progress with the deformation of the outer blocks as: 97 tan( ) =e(E1E3)tan ( u) (1) _ =1 2sin (2 ) _E1_E3 (2) where,E1; E2; E3, are the macroscopic logarithmic principal strains given as: 98 E1= ln (1); E 2= ln (2); E 3= ln (3) (3) 4 In Eq. (3), 1=L1=L10,2=L2=L20and3=L3=L30, withLi0being the initial, and Li99 being the current lengths of the unit cell along xiaxes. The current lengths of the unit cell follow, 100 L1=L10+U1;L2=2 =L20=2 +U2;L3=L30+U3 (4) where,Ui, are taken at point A0in Fig. 1. Following Eq. (3), the rates of macroscopic loga- 101 rithmic principal strains are: 102 _E1=_1 1;_E2=_2 2;_E3=_3 3(5) The unit cell shown in Fig. 1 (bottom) is symmetric along x2axis, so that only 1 =2 of the unit 103 cell needs to be analyzed. The boundary conditions uABba 2 =uabcd 2=udcCD 2 = 0 anduA0B0b0a0 2 = 104 ua0b0c0d0 2 =ud0c0C0D0 2 =U2are imposed on the external faces of the unit cell normal to x2axis. Also, 105 we assume that the external faces of the upper and lower blocks of the unit cell (see Fig. 1) that 106 are initially straight and normal to the x1andx3axes remain normal and straight throughout 107 the deformation history. This results in boundary conditions, uDD0C0C 1 = 0 anduAA0B0B 1 =U1, and 108 uBB0b0b 3 =ucc0C0C 3 = 0 anduAA0a0a 3 =udd0D0D 3 =U3. Next, following Tvergaard (1989), boundary 109 conditions applied on the external faces of the band in the unit cell that are initially straight and 110 normal to the x3axis are, 111 uaa0d0d 1ubb0c0c 1=L30 eE3tan ( )tan ( u) uaa0d0d 3ubb0c0c 3=L30 eE31 (6) where,L30=D30=D0cos ( u). 112 The voided part of the cell is weaker than the outer blocks. This is re ected by additional 113 rigid body displacements among the outer blocks with values 2 1and 2 3. The values  1and 114 3vanish if the cell has no void. By symmetry, there is no relative displacement of the outer 115 blocks inx2direction. The corresponding relative displacement eld in the upper block outside 116 the voided cell (evaluated at point ain Fig. 1 and relative to the center of the void) can then be 117 given as: 118 u1= (11)x1+  1 (7) u2= (21)x2 u3= (31)x3+  3 The strain rate components in the band, _Eb ij, are the result of the sum of the uniform strain 119 outside the band plus the strain associated with the rigid body displacements  1and  3: 120 _Eb 11= cos2( )_E1+ sin2( )_E3+2_1cos ( )2_3sin ( ) H 5 _Eb 22=_E2 (8) _Eb 33= sin2( )_E1+ cos2( )_E3 _Eb 13= sin (2 ) _E1_E3 +2_3cos ( ) + 2 _1sin ( ) H Here, the components of _Eb ijare de ned with respect to the local axes ( x0 1; x0 2; x0 3) attached 121 to the band that consequently rotates with the band. 122 The components of the strain rate _Eijare however expressed with respect to x1,x2andx3axes. 123 His the current thickness of the band (in x0 1direction), being H0=D10cos u, with _H=_Eb 11H. 124 The _Eb eqand _Eeqare given as: 125 _Eb eq=r 2 3_Eb ij_Eb ij;_Eeq=r 2 3 _E2 1+_E2 2+_E2 3 (9) and,Eb eq=Rt 0_Eb eqd andEeq=Rt 0_Eeqd . 126 The macroscopic true principal stresses  1, 2and  3, or the average reaction forces per unit 127 area at the deformed cell boundary, are de ned in terms of the Cauchy stress components, ii, as: 128 i=1 LjLkZLj 0ZLk 0[ii]xi=Lidxjdxkwithi;j;k = 1;2;3 The macroscopic stress triaxiality, T, and the Lode parameter, Lare de ned as: 129 T=h eq;L=2213 13(10) with  hand  ebeing the hydrostatic and equivalent macroscopic stresses, respectively, and 130 are given as: 131 h=1+  2+  3 3;eq=1p 2 (12)2+ ( 13)2+ ( 23)21=2(11) Relations, Eq. (10) and Eq. (11) allows to express  1, 2, 3as functions of  eq,TandL, 132 with the range of possible values: 133 0e1;1T1;1L1 (12) In this work, the stress states are characterized by the values of the stress triaxiality, T, and 134 the Lode parameter L, while the strain states are characterized by the equivalent strain in the 135 band,Eb eq, and the macroscopic equivalent strain, Eeq, in the upper/lower faces of the outer blocks. 136 Finite element unit cell calculations are carried out for prescribed constant stress triaxiality, T, and 137 Lode parameter, L, values using the commercial nite element code ABAQUS/Standard (2017). 138 The periodic boundary conditions on the faces of the unit cell are implemented as multi-point 139 6 constraints in ABAQUS/Standard (2017), while the boundary conditions required to prescribe 140 constant values of TandLare implemented in ABAQUS/Standard (2017) using an MPC user 141 de ned subroutine. More details of the boundary conditions to prescribe constant values of Tand 142 Lare given in Appendix A. The nite element calculations use C3D20R elements from the in-built 143 element library of ABAQUS/Standard (2017), and the number of elements in the nite element 144 mesh varies from 9 ;672 to 18;316 for initial band angle varying from u= 0to 39, respectively. 145 A cropped region of a unit cell with u= 39,f0= 0:001 andD0= 1mm with nite element 146 mesh in the voided region is shown in Fig. 1. 147 2.2. Constitutive Relation 148 The matrix material of the unit cell is modeled using nite strain J2 ow theory. In the 149 nite element calculations, updated Lagrangian formulation is used to account for large deforma- 150 tions. For hypo elastic-plastic materials, the macroscopic rate of plastic deformation tensor, _Ep, 151 is assumed to be related to the macroscopic stress rate by: 152 _=C: _E_Ep (13) where, Cis the linear isotropic elastic tensor de ned by the fourth order tensorial relation: 153 C= 2GIdev+K1 1 (14) withIdevbeing the unit deviatoric fourth order tensor having the form, 154 (Idev)ijkl= 1=2 (ikjl+iljk)1=3ijkl (15) whileG=E=(2(1 +)) andK=E=(3(12)) are elastic constants. 155 Assuming Mises plasticity, the yield function and the associated ow rule are given as: 156 =  eq( _"p) _Ep=_@ @_(16) where, follows: 157 =E" " "0 0(1 + (_"p _"0)m)">" 0(17) In Eq. (17), 0represents a reference yield stress, mis the strain rate sensitivity parameter 158 of the material, _ "0is the reference strain rate, Eis the Young's modulus, "0=0=E, and plastic 159 equivalent strain rate, _ "p=q 2 3_Ep:_Ep. 160 7 3. Results and Discussion 161 In a porous ductile material, onset of macroscopic localization due to the softening induced by 162 void nucleation and growth, or due to void coalescence resulting from plastic strain localization 163 in the ligaments between the voids, marks the end of uniform deformation. For a strain rate 164 independent porous ductile material, Teko glu et al. (2015) showed that depending on the value 165 of the imposed stress triaxiality a clear separation exists between the two modes of macroscopic 166 localization. This raises a fundamental question: how does the material's strain rate sensitivity 167 a ect the two modes of localization or ductile failure as a function of the imposed stress triaxiality? 168 In this section, we present results obtained using the numerical method detailed under Section 2 169 to address this question. 170 E11bΣ1/σ0 0 0.05 0.1 0.15 0.2012345Current work Teko lu et al. (2015 ) (a) EeqΣ1/σ0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35012345 Current work , T=1.5 Teko lu et al. (2015 ), T=1.5 Current work , T=3.0 Teko lu et al. (2015 ), T=3.0 (b) Figure 2: Comparison of the evolution of (a) normalized  1=0withEb 11forT= 3:0 and that of (b)  1=0with EeqforT= 1:5 andT= 3:0, predicted in the current work with the results of Teko glu et al. (2015). In both (a) and (b),L=1, u= 0,f0= 0:001 andN= 0:1. Our numerical method to model the unit cell that includes a central block with a single void 171 in its center within two void-free semi-in nite blocks, di ers from that of Teko glu et al. (2015). 172 Teko glu et al. (2015) only considered the inclined part of the unit cell within the band, Fig. 1, 173 and modeled the void-free semi-in nite block (along x1) through complex boundary conditions. 174 On the other hand, here we assume that (see Section 2) if the length of the void-free block along 175 x1is suciently greater than the characteristic length, D0, of the unit cell, it can mimic the semi- 176 in nite block. Thus, it is warranted to validate the predictions of our numerical method against 177 the available results of Teko glu et al. (2015). To this end, we use a strain rate independent ow 178 response, as in Teko glu et al. (2015): 179 =E" ""0 0("p "0)N">" 0(18) instead of Eq. (17). The calculations use, E= 167GPa ,0= 418MPa ,"0= 0:0025,= 0:3 180 andN= 0:1. The evolution of the normalized macroscopic stress along x1, 1=0, with the strain 181 8 in the band, Eb 11, forT= 3:0,L=1, u= 0andf0= 0:001 predicted using the numerical 182 method detailed under Section 2 are compared with the results of Teko glu et al. (2015) in Fig. 2(a). 183 Similarly, the evolution of  1=0with the macroscopic equivalent strain in the outer blocks, Eeq, 184 forT= 1:5 and 3:0, and the same values of L, uandf0predicted using the current method 185 are compared with the results of Teko glu et al. (2015) in Fig. 2(b). The results (of the current 186 work) presented in Fig. 2, are for a unit cell whose length along x1direction is six times the 187 characteristic length, D0, of the problem i.e. L10= 6D0. The strikingly good correlation between 188 the predictions using our method and the results of Teko glu et al. (2015) in Fig. 2, clearly shows 189 that the assumption of L10= 6D0numerically represent the unit cell consisting of a central block 190 with a single void in its center within two void-free semi-in nite blocks. 191 After validating our numerical method, we now focus on analyzing the in uence of material's 192 strain rate sensitivity on the onset of macroscopic localization due to void growth induced soft- 193 ening and/or void coalescence. All the results presented here onwards are from the nite element 194 calculations utilizing strain rate dependent ow response de ned in Eq. (17). The full set of con- 195 stitutive parameters used to de ne material (in the band as well as in the outer block) in the 196 calculations are: E= 167GPa ,= 0:3,0= 418MPa ,"0= 0:0025, and _"0= 1s1, while 197 the strain rate sensitivity parameter, m, is varied from 0 to 0 :25. For each strain rate sensitivity 198 parameter value, unit cells with one initial void volume fraction, f0= 0:001, but three initial 199 orientations of the voided band, u= 0, 20and 39are considered. The choice of the initial 200 orientation of the voided band, u, is based on the results of Teko glu et al. (2015) for strain 201 rate independent material. Finally, for each strain rate sensitivity parameter, m, value and ini- 202 tial orientation of the voided band, u, calculations are carried out for prescribed constant stress 203 triaxiality,T, values in the range, 0 :75T3 and one Lode parameter, L=1. 204 3.1. Strain Localization, Void Growth and Void Coalescence 205 The evolution of the equivalent strain inside the band, Eb eq, with the macroscopic equivalent 206 strain outside the band, Eeq, in unit cells with initial orientation of the voided band, u= 0and 207 39are shown in Fig. 3. As shown in Figs. 3, during initial stages of deformation Eb eqis same as 208 Eeq, but at later stages of deformation, strain localizes in the band, and thereafter Eb eqincreases 209 rapidly while Eeqincreases slightly in the material outside the band. The onset of localization is 210 assumed to occur when the ratio between the rate of macroscopic equivalent strain in the band 211 and outside the band reach the value of 5. The onset of void coalescence is assumed to occur when 212 the ratio of the maximum to the minimum equivalent plastic strain rate at the void surface rst 213 exceeds the threshold of 20, as in Teko glu et al. (2015). For T= 1:25, the onset of coalescence 214 occurs soon after the onset of localization. But, for T= 2:75, the onset of coalescence occurs 215 at a much greater value of Eb eqfollowing the onset of localization. A comparison of the results 216 shown in Figs. 3(a) and (b), and Figs. 3(c) and (d), show that increasing the strain rate sensitivity 217 of the material delays the onset of both, the localization and coalescence. However, the relative 218 stabilizing e ect of material's strain rate sensitivity is greater at the onset of localization than at 219 the onset of void coalescence. For example, for T= 2:75, (Eb eq)m=0:25=(Eb eq)m=05:3 at the onset 220 of localization and ( Eb eq)m=0:25=(Eb eq)m=02:8 at the onset of void coalescence. Also a comparison 221 of the results shown in Figs. 3(a) and (c), and Figs. 3(b) and (d), show that there is an increase in 222 the relative stabilizing e ect of material's strain rate sensitivity with the increasing value of the 223 imposed stress triaxiality. 224 9 EeqEeqb 0 0.2 0.4 0.6 0.800.20.40.60.8 LocalizationCoalescence(a) EeqEeqb 0 0.02 0.04 0.06 0.08 0.100.040.080.120.160.2 LocalizationCoalescence (b) EeqEeqb 0 0.2 0.4 0.6 0.800.20.40.60.8 LocalizationCoalescence (c) EeqEeqb 0 0.02 0.04 0.06 0.08 0.100.040.080.120.160.2 LocalizationCoalescence (d) Figure 3: Evolution of equivalent strain in the band, Eb eq, with macroscopic equivalent strain outside the band, Eeq, for (a) imposed stress triaxiality, T= 1:25 and strain rate sensitivity parameter, m= 0, (b)T= 2:75 and m= 0, (c)T= 1:25 andm= 0:25, and (d) T= 2:75 andm= 0:25. The initial orientation of the voided band is u= 0for the cases shown in (a) and (c), and u= 39for the cases shown in (b) and (d). In the gures the value ofEb eqat the onset of localization and void coalescence are marked with dashed horizontal lines. The evolution of normalized porosity, f=f 0, with macroscopic equivalent strain outside the 225 band,Eeq, forT= 1:25, is shown for two unit cells with initial orientation of the voided band 226 u= 0and u= 39in Figs. 4(a) and (c), respectively. The same is shown for T= 2:5, for two 227 unit cells with u= 0and u= 39in Figs. 4(b) and (d), respectively. To quantify the in uence 228 of the strain rate sensitivity parameter on the evolution of the porosity, results are shown in Fig. 4 229 form= 0;0:05, and 0:1. As shown in Fig. 4, for both the values of Tand u, a higher value of 230 mshifts thef=f 0versusEeqcurves to the right i.e. the stabilizing e ect of the material's strain 231 rate sensitivity slows down the evolution of porosity. For T= 2:5, the value of f=f 0at the onset 232 of localization is less than the value of f=f 0at the onset of void coalescence for both the values 233 10 Eeqf/f0 0 0.1 0.2 0.3 0.4 0.5 0.60306090120150m=0 m=0.05 m=0.1(a) Eeqf/f0 0 0.02 0.04 0.06 0.08 0.1020406080 m=0 m=0.05 m=0.1 (b) Eeqf/f0 0 0.1 0.2 0.3 0.4 0.5 0.60306090120150m=0 m=0.05 m=0.1 (c) Eeqf/f0 0 0.02 0.04 0.06 0.08 0.1020406080 m=0 m=0.05 m=0.1 (d) Figure 4: Evolution of the normalized porosity, f=f 0, withEeqfor (a) u= 0andT= 1:25, (b) u= 0and T= 2:5, (c) u= 39andT= 1:25, and (d) u= 39andT= 2:5. The open circles in the gure mark the onset of localization while the closed triangles mark the onset of void coalescence. of uand all three values of m. ForT= 1:25, the value of f=f 0at the onset of localization is 234 less than the value of f=f 0at the onset of void coalescence for u= 0form> 0. However, for 235 T= 1:25, the value of f=f 0at the onset of localization is same as the value of f=f 0at the onset 236 of void coalescence for u= 39for all three values of m. 237 The e ect of the material's strain rate sensitivity on deformation in the band and distribution 238 of equivalent plastic strain on the void surface is highlighted in Fig. 5. In Fig. 5, results are 239 shown for two strain rate sensitivity parameters, m= 0 and 0:25, for a unit cell with the initial 240 orientation of the band, u= 20and subjected to an imposed stress triaxiality, T= 1:0. As 241 can be seen in Fig. 5, the edges, adandbc, that were initially straight and normal to the x3axis, 242 undergo deformation following the periodicity of the initial voids in the band. In addition, the 243 results also show that at the same equivalent strain levels in the band, Eb eq, the maximum value 244 11 Figure 5: Deformed shape of the band and the distribution of the equivalent plastic strain, "p, in the band with initial orientation u= 20under imposed stress triaxiality T= 1:0 for two strain rate sensitivity parameters, m= 0 and 0:25 at two equivalent strain levels in the band, Eb eq= 0:18 and 0:36. of equivalent plastic strain, "p, on the void surface for m= 0:25 is much less than that for m= 0. 245 The lower levels of "pon the void surface for m= 0:25 compared to m= 0 is consistent with the 246 e ect ofmon the evolution of f=f 0shown in Fig. 4. 247 3.2. E ect of Strain Rate Sensitivity on Critical Porosity 248 mfc / f0 0 0.05 0.1 0.15 0.2 0.25050100150200250300350 0o (Localization ) 0o (Coalescence ) 39o (Localization ) 39o (Coalescence ) (a) mfc / f0 0 0.05 0.1 0.15 0.2 0.25050100150200250300350 0o (Localization ) 0o (Coalescence ) 39o (Localization ) 39o (Coalescence ) (b) Figure 6: Variation of the normalized critical value of porosity, fc=f0, at the onset of localization and void coales- cence with material's strain rate sensitivity parameter, m, for (a)T= 1:25 and (b) T= 2:5. We now analyze the e ect of material's strain rate sensitivity, m, on critical values of porosity 249 at the onset of strain localization and void coalescence. The critical value of porosity at the onset 250 of void coalescence is widely used as an input parameter to simulate ductile fracture due to void 251 nucleation, growth and coalescence (Besson et al., 2003; Srivastava et al., 2014; Cheng et al., 2014). 252 12 The variation of the normalized critical value of porosity, fc=f0, at the onset of localization 253 and void coalescence with material's strain rate sensitivity parameter, m, in two unit cells with 254 initial orientation of the voided band, u= 0and 39, for two values of imposed stress triaxiality, 255 T= 1:25 and 2:5, are shown in Fig. 6. For T= 1:25 and u= 0, the values of fc=f0at the onset 256 of localization and void coalescence are roughly the same for m< 0:05. However, for greater value 257 ofmthe value of fc=f0at the onset of localization is less than the value of fc=f0at the onset 258 of void coalescence. For T= 1:25 and u= 39, the values of fc=f0at the onset of localization 259 and void coalescence are roughly the same for all values of m. On the other hand, for T= 2:5, 260 the values of fc=f0at the onset of localization is less than the values of fc=f0at the onset of void 261 coalescence for all values of m, for both u= 0and 39. Also, forT= 2:5 the di erence between 262 the values of fc=f0at the onset of localization and void coalescence increases with increasing value 263 ofm. The results presented in Fig. 6 show that at low imposed stress triaxiality values, the 264 extent of void growth induced softening is not sucient to trigger macroscopic localization, thus 265 localization does not occur until the onset of void coalescence. On the other hand, the void growth 266 induced softening is sucient to trigger macroscopic localization at high imposed stress triaxiality 267 values and the macroscopic localization occurs much before the void coalescence. The increase 268 in the value of fcwith increasing value of mis because an increase in the value of mresults in 269 increased strain rate hardening thus requiring an increase in damage induced softening to trigger 270 any localized deformation in the voided band or within the ligament between the neighboring 271 voids to trigger void coalescence. 272 3.3. E ect of Strain Rate Sensitivity on Critical Strain 273 mEeqc 0 0.05 0.1 0.15 0.2 0.2500.10.20.30.40.5 0o (Localization ) 0o (Coalescence ) 20o (Localization ) 20o (Coalescence ) 39o (Localization ) 39o (Coalescence ) (a) mEeqc 0 0.05 0.1 0.15 0.2 0.2500.020.040.060.080.10o (Localization ) 0o (Coalescence ) 20o (Localization ) 20o (Coalescence ) 39o (Localization ) 39o (Coalescence ) (b) Figure 7: Variation of the critical macroscopic equivalent strain, Ec eq, outside the band at the onset of localiza- tion/coalescence with material's strain rate sensitivity parameter, m, for (a)T= 1:5 and (b)T= 2:75. The results presented thus far clearly show that both the material's strain rate sensitivity and 274 imposed stress triaxiality a ect the onset of strain localization and void coalescence. Following 275 this, we analyze the variation of critical macroscopic equivalent strain outside the band, Ec eq, and 276 critical equivalent strain inside the band, Ecb eq, with strain rate sensitivity parameter, m, shown in 277 13 mEeqcb 0 0.05 0.1 0.15 0.2 0.250.10.20.30.40.50.6 0o (Localization ) 0o (Coalescence ) 20o (Localization ) 20o (Coalescence ) 39o (Localization ) 39o (Coalescence )(a) mEeqcb 0 0.05 0.1 0.15 0.2 0.2500.050.10.150.20.25 0o (Localization ) 0o (Coalescence ) 20o (Localization ) 20o (Coalescence ) 39o (Localization ) 39o (Coalescence ) (b) Figure 8: Variation of the critical equivalent strain, Ecb eq, inside the band at the onset of localization/coalescence with material's strain rate sensitivity parameter, m, for (a)T= 1:5 and (b)T= 2:75. Figs. 7 and 8, respectively, and the variation of Ecb eqwith imposed stress triaxiality, T, shown in 278 Fig. 9. 279 As shown in Fig. 7, the value of Ec eqincreases with increasing value of strain rate sensitivity 280 parameter, m, for all three initial orientation of the voided band, u= 0, 20and 39, and 281 for both the imposed stress triaxiality values, T= 1:5 and 2:75. As also shown in the gure, 282 for all three uvalues and for T= 1:5, the value of Ec eqis same at the onset of localization 283 and void coalescence. For T= 2:75,Ec eqvalues at void coalescence are slightly greater than the 284 values ofEc eqat the onset of localization. This is consistent with the observation that at low 285 imposed stress triaxiality values, the extent of void growth induced softening is not sucient to 286 trigger macroscopic localization, whereas at high imposed stress triaxiality values the void growth 287 induced softening is sucient to trigger macroscopic localization. As also shown in Fig. 7, the 288 e ect of initial band orientation on the value of Ec eqdepends on both, the values of Tandm. 289 For example, the minimum value of Ec eqat the onset of localization corresponds to u= 20for 290 T= 1:5 while it corresponds to u= 39forT= 2:75. 291 Similarly, the value of the critical equivalent strain in the band, Ecb eq, at the onset of localiza- 292 tion/coalescence increases with increasing value of mfor all three uand for both T, Fig. 8. In 293 addition, the variation of Ecb eqwithmforT= 1:5, Fig. 8(a), roughly follows the same trend as 294 shown in Fig. 7(a) for Ec eq. But forT= 2:75, the value of Ecb eqat the onset of localization is less 295 than the value of Ecb eqat the onset of void coalescence for all the values of mand u, Fig. 8(b). 296 ForT= 2:75, the initial band orientation that gives lowest values of Ecb eqis u= 0, and the 297 orientation that gives largest values of Ecb eqis u= 20. This is in contrast to the dependence of 298 Ec eqon uas shown in Fig. 7(b). This is because at high imposed stress triaxiality values, the 299 onset of macroscopic localization occurs due to void growth induced softening that signi cantly 300 a ects the deformation in the band. 301 Figures 9(a) and (b) compare the variation of Ecb eqat the onset of localization and coalescence, 302 with imposed stress triaxiality values, T, form= 0:05 and 0:25, respectively. As shown in 303 14 TEeqcb 1.5 1.75 2 2.25 2.5 2.75 300.050.10.150.20.250.3 0o (Localization ) 0o (Coalescence ) 20o (Localization ) 20o (Coalescence ) 39o (Localization ) 39o (Coalescence )(a) TEeqcb 1.5 1.75 2 2.25 2.5 2.75 300.10.20.30.40.50.6 0o (Localization ) 0o (Coalescence ) 20o (Localization ) 20o (Coalescence ) 39o (Localization ) 39o (Coalescence ) (b) Figure 9: Variation of the critical equivalent strain, Ecb eq, inside the band at the onset of localization/coalescence with imposed stress triaxiality, T, for (a)m= 0:05 and (b) m= 0:25. 4 the gure, for both the values of m, the value of Ecb eqat localization/coalescence decreases with 304 increasing value of the imposed T. But for suciently large values of T, with increasing T, 305 the decrease in Ecb eqat void coalescence is rather gradual as compared to the decrease in Ecb eq306 at localization. This results in a clear separation between the values of Ecb eqat localization and 307 coalescence for T2, with the increasing di erence between Ecb eqat localization and coalescence 308 with increasing value of T. 309 T(Eeqcb)Coa / (Eeqcb)Loc 0.5 1 1.5 2 2.5 300.511.522.533.54m=0 m=0.05 m=0.1 m=0.25 (a) T(Eeqcb)Coa / (Eeqcb)Loc 0.5 1 1.5 2 2.5 300.511.522.533.54m=0 m=0.05 m=0.1 m=0.25 (b) Figure 10: Variation in the ratio ( Ecb eq)coa=(Ecb eq)Locwith imposed stress triaxiality values, T, for initial orientation of the voided band, (a) u= 0and (b) u= 20. We now analyze the variation of the critical equivalent strain in the band at the onset of 310 15 m(Eeqcb)Coa / (Eeqcb)Loc 0 0.05 0.1 0.15 0.2 0.2500.511.522.533.54 T=1.0 T=1.5 T=2.0 T=2.5 T=2.75(a) m(Eeqcb)Coa / (Eeqcb)Loc 0 0.05 0.1 0.15 0.2 0.2500.511.522.533.54 T=1.0 T=1.5 T=2.0 T=2.5 T=2.75 (b) Figure 11: Variation in the ratio ( Ecb eq)coa=(Ecb eq)Locwith material's strain rate sensitivity parameter, m, for initial orientation of the voided band, (a) u= 0and (b) u= 20. coalescence normalized with the value of the critical equivalent strain in the band at the onset 311 of localization, ( Ecb eq)Coa=(Ecb eq)Loc, withTandmin Figs. 10 and 11, respectively, for two initial 312 orientation of the voided band, u. As shown in Fig. 10, for all mand for both the values 313 of u, the ratio ( Ecb eq)Coa=(Ecb eq)Locis approximately one for T < 2 and forT2, the ratio 314 (Ecb eq)Coa=(Ecb eq)Locincreases with increasing value of T. This is because both ( Ecb eq)Coaand (Ecb eq)Loc315 decrease with increasing Tbut forT2, (Ecb eq)Locdecreases more rapidly with increasing Tthan 316 (Ecb eq)Coa. Furthermore, as shown in Fig. 11, for T > 2, the value of ( Ecb eq)Coa=(Ecb eq)Locdecreases 317 with increasing mfor both the values of u. The decrease in the value of ( Ecb eq)Coa=(Ecb eq)Locwith 318 increasingmbecomes more pronounced with increasing value of the imposed T. Note, that both 319 (Ecb eq)Coaand (Ecb eq)Locincrease with increasing m, thus a decrease in the value of ( Ecb eq)Coa=(Ecb eq)Loc320 suggests that ( Ecb eq)Locincreases more rapidly with increasing mfor higher stress triaxiality values. 321 To summarize, the relative stabilizing e ect induced by strain rate sensitivity of the material not 322 only increases with the increasing value of the imposed stress triaxiality, but also the relative 323 stabilizing e ect of the strain rate sensitivity is greater at the onset of strain localization than at 324 the onset of void coalescence. 325 3.4. E ect of Strain Hardening 326 In this work, we have focused on answering the question: How does the materials strain rate 327 sensitivity a ect the two mechanisms of macroscopic localization or ductile failure as a function 328 of the imposed stress triaxiality? To answer this question, we have carried out a series of unit 329 cell calculations, where the matrix material of the unit cell is considered to follow a strain rate 330 dependent elastic perfectly plastic ow response. So the question arises as to what extent strain 331 hardening in the material a ects our results. A complete study of the combined e ect of material's 332 strain rate sensitivity, strain hardening and imposed stress triaxiality is beyond the scope of this 333 work. However, our limited unit cell calculations presented in this section show that incorporating 334 both material's strain rate sensitivity and strain hardening does not a ect the qualitative e ect 335 16 EeqEbeq 0 0.4 0.8 1.2 1.6 200.40.81.21.62 m=0.05, N=0.1 m=0.05, N=0m=0.25, N=0.1 m=0.25, N=0(a) EeqEbeq 0 0.04 0.08 0.12 0.16 0.200.060.120.180.240.3 m=0.05, N=0.1 m=0.05, N=0m=0.25, N=0.1m=0.25, N=0 (b) Figure 12: A comparison of the evolution of equivalent strain in the band, Eb eq, with macroscopic equivalent strain outside the band, Eeq, in strain hardening and non strain hardening materials for (a) imposed stress triaxiality, T= 0:75 and (b) T= 2:5. The initial orientation of the voided band is u= 0for all the cases shown. In the gures the open circles mark the onset of localization while the closed triangles mark the onset of void coalescence. of material's strain rate sensitivity and stress triaxiality reported in this work. 336 The results of our limited unit cell calculations to explore the combined e ect of material's 337 strain rate sensitivity, strain hardening and stress triaxiality on the two mechanisms of macroscopic 338 localization is shown in Fig. 12. The combined e ect of material's strain rate sensitivity and strain 339 hardening is explored using the ow response given as, 340 =E" " "0 0("p "0)N(1 + (_"p _"0)m)">" 0(19) instead of Eq. (17). For the calculations corresponding to results in Fig. 12, the values of 341 E= 167GPa ,0= 418MPa ,"0= 0:0025, _"0= 1s1,N= 0 and 0:1, andm= 0:05 and 0:25 342 are used in Eq. (19). The results presented in Fig. 12 show that incorporating strain hardening 343 in to the ow response of the material does not a ect the qualitative e ect of material's strain 344 rate sensitivity and stress triaxiality on the onset of localization and void coalescence reported in 345 this work. For example, irrespective of the value of the strain hardening exponent, N: (i) for low 346 stress triaxiality value, T= 0:75, the value of Eb eqat the onset of localization and void coalescence 347 are the same, (ii) the value of Eb eqat the onset of localization and void coalescence increases with 348 the increasing value of strain rate sensitivity parameter, m, for both the values of T, (iii) for high 349 stress triaxiality value, T= 2:5, the value of Eb eqat the onset of localization is less than the value 350 ofEb eqat the onset of void coalescence, and (iv) the di erence between the values of Eb eqat the 351 onset of localization and void coalescence decreases with increasing value of m. 352 17 4. Summary and Concluding Remarks 353 Strain localization and void coalescence (resulting in the formation of micro-cracks) in a porous 354 ductile material mark the end of uniform deformation and limits the ductility of the material. In 355 this work the focus is con ned to analyzing the in uence of material's strain rate sensitivity on 356 the onset of strain localization and void coalescence as a function of the imposed stress triaxiality. 357 To this end, three dimensional nite element calculations of unit cells have been carried out 358 to model localization and coalescence in an in nite block containing a periodic distribution of 359 initially spherical voids in a band. The unit cell modeled consists of a central block with a single 360 void in its center within two semi-in nite void-free blocks. The matrix material of the unit cell 361 is considered to follow a strain rate dependent elastic perfectly plastic ow response. Limited 362 calculations have also been carried out to explore the combined e ect of material's strain rate 363 sensitivity and strain hardening. The unit cell calculations are carried out for one initial void 364 volume fraction ( f0= 0:001), a range of strain rate sensitivity parameter (varying in the range 365 0m0:25), a range of imposed stress triaxiality (varying in the range 0 :75T3) with one 366 Lode parameter ( L=1), and three initial orientations of the voided band ( u= 0, 20and 367 39). In the calculations, onset of localization is de ned as the point in the deformation history 368 when the ratio of the rate of deformation in the band and outside the band reaches a critical level, 369 while the onset of coalescence is de ned when the ratio of the maximum to the minimum e ective 370 plastic strain rate at the surface of the void reaches a critical value following the work of Teko glu 371 et al. (2015). Our results show that both the critical void volume fraction and strain, at the onset 372 of localization and coalescence, are strongly in uenced by the material's strain rate sensitivity and 373 the imposed stress triaxiality. 374 The main conclusions of this work are as follows: 375 1. In uence of material's strain rate sensitivity, m, and imposed stress triaxiality, T, on the 376 critical void volume fraction, fc, at the onset of strain localization and void coalescence: 377 (a) For low Tvalues,fcis roughly the same at the onset of localization and coalescence, 378 whereas for greater values of T,fcat the onset of coalescence is greater than fcat the 379 onset of localization. 380 (b) For all values of T, the value of fcat localization/coalescence increases with increasing 381 m. 382 2. In uence of material's strain rate sensitivity, m, and imposed stress triaxiality, T, on the 383 critical equivalent strain in the band, Ecb eq, at the onset of strain localization and void coa- 384 lescence: 385 (a) For low Tvalues,T < 2,Ecb eqis same at the onset of localization and coalescence, 386 whereas for T2,Ecb eqat the onset of coalescence is greater than Ecb eqat the onset of 387 localization. 388 (b) For all values of m, the value of Ecb eqat the onset of localization/coalescence decreases 389 with increasing T. But forT2, the value of Ecb eqat localization decreases more 390 rapidly with increasing Tcompared to the value of Ecb eqat coalescence. 391 (c) For all values of T, the value of Ecb eqat the onset of localization/coalescence increases 392 with increasing value of m. But forT2, the value of Ecb eqat localization increases 393 more with increasing value of mcompared to the value of Ecb eqat coalescence. 394 18 (d) The concluding points, 2(b) and (c), clearly show that the relative stabilizing e ect of 395 mincreases with the increasing value of T, and the relative stabilizing e ect of mis 396 greater at the onset of localization than at the onset of coalescence. 397 Our limited results aimed at exploring the combined e ect of material's strain rate sensitivity, 398 strain hardening and imposed stress triaxiality on the onset of localization and void coalescence 399 show that incorporating strain hardening into the ow response of the material does not a ect the 400 aforementioned qualitative e ect of material's strain rate sensitivity and stress triaxiality on the 401 onset of localization and void coalescence. In this work, we have focused on one Lode parameter 402 value,L=1. The value of the Lode parameter may also a ect the onset of localization and 403 void coalescence especially at low values of stress triaxiality. The numerical method presented in 404 this work can be used for future research to explore the combined e ect of material's strain rate 405 sensitivity and strain hardening, and imposed stress triaxiality and Lode parameter on the onset 406 of localization and void coalescence. Furthermore, our results motivate the need to incorporate 407 the e ect of material's strain rate sensitivity in the porosity and stress triaxiality dependent ow 408 potential (or constitutive response) of the material. 409 Acknowledgements 410 The nancial support provided by the European Union's Horizon2020 Programme (Excellent 411 Science, Marie-Sklodowska - Curie Actions, H2020 - MSCA - RISE - 2017) under REA grant 412 agreement 777896 (Project QUANTIFY) are gratefully acknowledged. Shmuel Osovski would also 413 like to acknowledge the nancial support of the Pazy foundation (Young researchers award grant 414 no. 1176). We are thankful to Dr. J.A. Rodr guez-Mart nez of University Carlos III of Madrid for 415 numerous fruitful discussions. The authors also acknowledge to Katarzyna Kowalczyk-Gajewska 416 from Institute of Fundamental Technological Research, Polish Academy of Sciences, for helpful 417 discussions. 418 19 Appendix A. Procedure for the Multipoint constraint 419 The procedure to prescribe proportional loading i.e. xed ratio of principal stresses, R=  2=1 420 andQ=  3=1to maintain constant macroscopic stress triaxiality, T, and Lode parameter, L, 421 throughout the deformation history, follows the work of Vadillo et al. (2016). 422 Since the macroscopic true stresses ( 1, 2, 3) and the macroscopic strain rates ( _E1,_E2,_E3) 423 are equal to the volume average values in a cell (Hill, 1967), the total rate of deformation work in 424 the whole cell _Wconsidering jointly upper, lower and central voided block, can be written as: 425 _W=V1_E1+V2_E2+V3_E3 (A.1) whereVis the total volume of the RVE. 426 De ningP1=V1,P2=V2andP3=V3as generalized forces and work rate conjugate 427 quantities to _E1,_E2and _E3, respectively, the expression of _Wbecomes: 428 _W=P1_E1+P2_E2+P3_E3 (A.2) in whichP1,P2andP3must satisfy: 429 P2 P1=R;P3 P1=Q (A.3) The ratiosRandQcan be written in terms of the imposed stress triaxiality, T, and Lode param- 430 eter,L: 431 Q=3Tp L2+ 33L 3Tp L2+ 3 + 3L;R=3Tp L2+ 3 + 2L 3Tp L2+ 3 + 3L(A.4) Next, with the transformation: 432 (A.5) 433 0 @_E(I) _E(II) _E(III)1 A=N0 @_E1 _E2 _E31 A;0 @P(I) P(II) P(III)1 A=N0 @P1 P2 P31 A where Nis an orthonormal N1=NT unsymmetric matrix: 434 N=0 @A11A12A13 A21A22A23 A31A32A331 A; with the elements: 435 A11=1p 1 +R2+Q2;A12=Rp 1 +R2+Q2;A13=Qp 1 +R2+Q2 A21=Rp 1 +R2;A22=1p 1 +R2;A23= 0: (A.6) A31=Qp (1 +R2) (1 +R2+Q2);A32=RQp (1 +R2) (1 +R2+Q2) A33=(1 +R2)p (1 +R2) (1 +R2+Q2) 20 Now, _Wcan be expressed in terms of the transformed rates of deformation and forces, as: 436 _W=P(I)_E(I)+P(II)_E(II)+P(III)_E(III) (A.7) If in the transformed coordinate system, the imposed incremental boundary conditions are 437 prescribed as stress uniaxial: 438 P(II)= 0;P(III)= 0; _E(I)=_EI (A.8) the total rate of deformation work has the form _W=P(I)_EI, that follows, in the initial system 439 and considering the relations given in Eq. (A.5), the three relations: 440 (1)P(II)= 0!A21P1+A22P2+A23P3= 0 (A.9) (2)P(III)= 0!A31P1+A32P2+A33P3= 0 (3) _E(I)=_EI!A11_E1+A12_E2+A13_E3=_EI or in a similar manner, if values of R,Qand _EIare prescribed on the unit cell, this leads to: 441 (1) ( 2=1) =R (A.10) (2) ( 3=1) =Q (3) _E1+R_E2+Q_E3 =_EIp 1 +R2+Q2 This procedure was implemented into ABAQUS/Standard (2017) as MPC user subroutine. 442 21