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 The ductility of metals and alloys is limited by the onset of macroscopic strain localization, as it marks the end of uniform deformation and acts as a precursor to failure. Two distinct mechanisms have been identi ed that can lead to macroscopic strain localization in porous ductile materials (Teko glu et al., 2015). The rst mechanism of macroscopic strain localization involves strain softening due to mi- crostructural changes, thermal e ects and/or damage evolution resulting in local degradation in material's load carrying capacity. The local degradation in the material's load carrying capacity, causes localized deformation in a thin band with smoothly varying deformation pattern outside the  Email address: gvadillo@ing.uc3m.es (Guadalupe Vadillo) Preprint submitted to International Journal of Plasticity March 23, 2020 band (Fressengeas and Molinari, 1985; Tvergaard, 1981; Pardoen and Hutchinson, 2000; Mercier and Molinari, 2003; Aretz, 2007; Pineau et al., 2016). This results in a weak discontinuity along the thin band at the micro-scale and a discontinuity in the displacement eld at the macro-scale. In porous ductile materials, damage (void growth) induced softening is the dominant mechanism that triggers the onset of macroscopic localization (Tvergaard, 1982), resulting in void sheet fracture. The width of the thin band engul ng the voids in this scenario has a length scale comparable or greater than the mean spacing between the voids. Under these circumstances the onset of macroscopic localization is strongly dependent on material properties, initial porosity, orientation of the band undergoing localized deformation and the imposed stress state (Needleman and Rice, 1978; Yamamoto, 1978). The second mechanism of macroscopic strain localization in porous ductile materials involves void coalescence. Void coalescence is a local instability phenomenon where the interaction between neighboring voids plays a critical role. The onset of void coalescence is associated with a sudden concentration of plastic strain in the ligaments linking neighboring voids (Koplik and Needleman, 1988). Thus, following the onset of void coalescence, the kinematics of the void enlargement signi cantly di ers from that of void growth prior to this instability mode. For void coalescence induced macroscopic localization the width of the micro-scale localization band is much more narrower because the deformation is limited to the ligaments between the neighbouring voids. The two aforementioned mechanisms of macroscopic localization in porous ductile materials, void growth induced softening and void coalescence, are distinct and have been identi ed in Teko glu et al. (2015); Guo and Wong (2018). In particular, for a strain rate independent material, Teko glu et al. (2015) showed that depending on the value of the imposed stress triaxiality a clear separation exists between the two mechanisms of macroscopic localization. This raises a fundamental question: how does the material's strain rate sensitivity a ect the two mechanisms of macroscopic localization or ductile failure as a function of the imposed stress triaxiality? The strain rate sensitivity is an important material parameter and it greatly a ects the onset of localized deformation and damage evolution (Marciniak et al., 1973; Hutchinson and Neale, 1977; Ghosh, 1977; Hutchinson et al., 1978; Taya and Seidel, 1981; Budiansky et al., 1982; Cocks and Ashby, 1982a,b; Pan, 1983a; Pan et al., 1983; Yoon and Taya, 1984; Duva, 1986; Nemat-Nasser et al., 1986; Ortiz and Molinari, 1992; Czarnota et al., 2006; Vadillo et al., 2012; Srivastava and Needleman, 2013; Agoras and Ponte Casta~ neda, 2014; Osovski et al., 2015; Wang et al., 2018). This in turn a ects the performance, safety, reliability and manufacturability of engineering components and structures; for example, the crash worthiness of automobiles, the blast resistance of ships and airplane cargo holds, and the manufacturability of sheet metal components. In general, increasing the strain rate sensitivity of a ductile material can delay the onset of localized plastic deformation. Furthermore, it has also been shown that an increase in the material's strain rate sensitivity can slow down the evolution of porosity and delay the onset of void coalescence in a porous ductile material. In this work we focus on analyzing the in uence of materials strain rate sensitivity on the two mechanisms of macroscopic localization or ductile failure as a function of the imposed stress triaxiality. Several analytical and computational studies have been carried out in past to analyze the onset of localized deformation (Marciniak and Kuczy nski, 1967; Rudnicki and Rice, 1975; Rice, 1977; Saje et al., 1982; Pan, 1983b; Mear and Hutchinson, 1985; Nahshon and Hutchinson, 2008). The micro-mechanical approach that constitute the basis of these works involves analyzing an 2 in nite sheet or block with a defect band inclined at an angle with respect to the imposed loading direction. Alternatively, the onset of localization in a porous material can also be analyzed via unit cell model calculations. In the unit cell model calculations, a periodic distribution of voids in the material is assumed, that allows us to model a single void in the material and impose proportional loading throughout the deformation history (Koplik and Needleman, 1988; Scheyvaerts et al., 2011; Fritzen et al., 2012; Srivastava and Needleman, 2013; Br unig et al., 2013; Dunand and Mohr, 2014; Teko glu et al., 2015; Liu et al., 2016; Torki et al., 2017; Guo and Wong, 2018; Luo and Gao, 2018). Thus, unit cell model calculations not only provide the macroscopic response of the material but also provide the details of the void growth and coalescence, as a function of the imposed constant stress triaxiality values. In this work, three dimensional nite element calculations of unit cells have been carried out to model macroscopic strain localization due to damage induced softening, void growth and/or coalescence, in an in nite block containing a periodic distribution of initially spherical voids in a band. Note that we have assumed a state of the material where either the voids are initially present or have already nucleated (uniformly for the sake of simplicity) in a narrow region in the material. The unit cell modeled consists of a central block with a single void in its center within two semi-in nite void-free blocks. The matrix material of the unit cell is considered to follow a strain rate dependent elastic perfectly plastic ow response. Limited calculations have also been carried out to explore the combined e ect of material's strain rate sensitivity and strain hardening. The unit cell calculations are carried out for one initial void volume fraction, a range of strain rate sensitivity parameter, a range of imposed stress triaxiality, and three initial orientations of the voided band. In the calculations, onset of macroscopic localization is de ned as the point when the ratio of the rate of deformation in the band and outside the band reaches a critical level, while the onset of void coalescence is de ned when the ratio of the maximum to the minimum e ective plastic strain rate at the surface of the void reaches a critical value following the work of Teko glu et al. (2015); Guo and Wong (2018). The remainder of this paper are organized as follows. The problem formulation and numerical method are detailed in Section 2. The numerical results and the discussion of the key results are presented in Section 3. The main conclusions of this work are summarized in Section 4. 2. Problem Formulation and Numerical Method Three dimensional nite element calculations are carried out to model plastic ow localization and void coalescence in an in nite block containing a periodic distribution of initially spherical voids in a band as shown in Fig. 1 (top left). The con guration analyzed here is numerically similar to the works of Tvergaard (1989); Barsoum and Faleskog (2011); Teko glu et al. (2015) and Guo and Wong (2018). 2.1. Unit Cell Model As shown in Fig. 1, for the periodic distribution of the voids in the voided band, a unit cell can be de ned as a central block with a single void in its center within two void-free semi-in nite blocks. The initial volume of the central block is taken to be Vcentral block =D10D20D30=D3 0 and the initial volume fraction of the spherical void with radius R0isf0= 4R3 0=3D3 0. In the nite element calculations, the semi-in nite blocks along x1direction are assumed to be of nite 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 orientation angle, u, of the voided band in the initial state, the orientation angle of the band at any deformation angle progress with the deformation of the outer blocks as: tan( ) =e(E1E3)tan ( u) (1) _ =1 2sin (2 ) _E1_E3 (2) where,E1; E2; E3, are the macroscopic logarithmic principal strains given as: 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 Li being the current lengths of the unit cell along xiaxes. The current lengths of the unit cell follow, 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- rithmic principal strains are: _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 cell needs to be analyzed. The boundary conditions uABba 2 =uabcd 2=udcCD 2 = 0 anduA0B0b0a0 2 = ua0b0c0d0 2 =ud0c0C0D0 2 =U2are imposed on the external faces of the unit cell normal to x2axis. Also, we assume that the external faces of the upper and lower blocks of the unit cell (see Fig. 1) that are initially straight and normal to the x1andx3axes remain normal and straight throughout the deformation history. This results in boundary conditions, uDD0C0C 1 = 0 anduAA0B0B 1 =U1, and uBB0b0b 3 =ucc0C0C 3 = 0 anduAA0a0a 3 =udd0D0D 3 =U3. Next, following Tvergaard (1989), boundary conditions applied on the external faces of the band in the unit cell that are initially straight and normal to the x3axis are, uaa0d0d 1ubb0c0c 1=L30 eE3tan ( )tan ( u) uaa0d0d 3ubb0c0c 3=L30 eE31 (6) where,L30=D30=D0cos ( u). The voided part of the cell is weaker than the outer blocks. This is re ected by additional rigid body displacements among the outer blocks with values 2 1and 2 3. The values  1and 3vanish if the cell has no void. By symmetry, there is no relative displacement of the outer blocks inx2direction. The corresponding relative displacement eld in the upper block outside the voided cell (evaluated at point ain Fig. 1 and relative to the center of the void) can then be given as: 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 outside the band plus the strain associated with the rigid body displacements  1and  3: _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 to the band that consequently rotates with the band. The components of the strain rate _Eijare however expressed with respect to x1,x2andx3axes. His the current thickness of the band (in x0 1direction), being H0=D10cos u, with _H=_Eb 11H. The _Eb eqand _Eeqare given as: _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 . The macroscopic true principal stresses  1, 2and  3, or the average reaction forces per unit area at the deformed cell boundary, are de ned in terms of the Cauchy stress components, ii, as: 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: T=h eq;L=2213 13(10) with  hand  ebeing the hydrostatic and equivalent macroscopic stresses, respectively, and are given as: 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, with the range of possible values: 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 the Lode parameter L, while the strain states are characterized by the equivalent strain in the band,Eb eq, and the macroscopic equivalent strain, Eeq, in the upper/lower faces of the outer blocks. Finite element unit cell calculations are carried out for prescribed constant stress triaxiality, T, and Lode parameter, L, values using the commercial nite element code ABAQUS/Standard (2017). The periodic boundary conditions on the faces of the unit cell are implemented as multi-point 6 constraints in ABAQUS/Standard (2017), while the boundary conditions required to prescribe constant values of TandLare implemented in ABAQUS/Standard (2017) using an MPC user de ned subroutine. More details of the boundary conditions to prescribe constant values of Tand Lare given in Appendix A. The nite element calculations use C3D20R elements from the in-built element library of ABAQUS/Standard (2017), and the number of elements in the nite element mesh varies from 9 ;672 to 18;316 for initial band angle varying from u= 0to 39, respectively. A cropped region of a unit cell with u= 39,f0= 0:001 andD0= 1mm with nite element mesh in the voided region is shown in Fig. 1. 2.2. Constitutive Relation The matrix material of the unit cell is modeled using nite strain J2 ow theory. In the nite element calculations, updated Lagrangian formulation is used to account for large deforma- tions. For hypo elastic-plastic materials, the macroscopic rate of plastic deformation tensor, _Ep, is assumed to be related to the macroscopic stress rate by: _=C: _E_Ep (13) where, Cis the linear isotropic elastic tensor de ned by the fourth order tensorial relation: C= 2GIdev+K1 1 (14) withIdevbeing the unit deviatoric fourth order tensor having the form, (Idev)ijkl= 1=2 (ikjl+iljk)1=3ijkl (15) whileG=E=(2(1 +)) andK=E=(3(12)) are elastic constants. Assuming Mises plasticity, the yield function and the associated ow rule are given as: =  eq( _"p) _Ep=_@ @_(16) where, follows: =E" " "0 0(1 + (_"p _"0)m)">" 0(17) In Eq. (17), 0represents a reference yield stress, mis the strain rate sensitivity parameter of the material, _ "0is the reference strain rate, Eis the Young's modulus, "0=0=E, and plastic equivalent strain rate, _ "p=q 2 3_Ep:_Ep. 7 3. Results and Discussion In a porous ductile material, onset of macroscopic localization due to the softening induced by void nucleation and growth, or due to void coalescence resulting from plastic strain localization in the ligaments between the voids, marks the end of uniform deformation. For a strain rate independent porous ductile material, Teko glu et al. (2015) showed that depending on the value of the imposed stress triaxiality a clear separation exists between the two modes of macroscopic localization. This raises a fundamental question: how does the material's strain rate sensitivity a ect the two modes of localization or ductile failure as a function of the imposed stress triaxiality? In this section, we present results obtained using the numerical method detailed under Section 2 to address this question. 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 in its center within two void-free semi-in nite blocks, di ers from that of Teko glu et al. (2015). Teko glu et al. (2015) only considered the inclined part of the unit cell within the band, Fig. 1, and modeled the void-free semi-in nite block (along x1) through complex boundary conditions. On the other hand, here we assume that (see Section 2) if the length of the void-free block along x1is suciently greater than the characteristic length, D0, of the unit cell, it can mimic the semi- in nite block. Thus, it is warranted to validate the predictions of our numerical method against the available results of Teko glu et al. (2015). To this end, we use a strain rate independent ow response, as in Teko glu et al. (2015): =E" ""0 0("p "0)N">" 0(18) instead of Eq. (17). The calculations use, E= 167GPa ,0= 418MPa ,"0= 0:0025,= 0:3 andN= 0:1. The evolution of the normalized macroscopic stress along x1, 1=0, with the strain 8 in the band, Eb 11, forT= 3:0,L=1, u= 0andf0= 0:001 predicted using the numerical method detailed under Section 2 are compared with the results of Teko glu et al. (2015) in Fig. 2(a). Similarly, the evolution of  1=0with the macroscopic equivalent strain in the outer blocks, Eeq, forT= 1:5 and 3:0, and the same values of L, uandf0predicted using the current method are compared with the results of Teko glu et al. (2015) in Fig. 2(b). The results (of the current work) presented in Fig. 2, are for a unit cell whose length along x1direction is six times the characteristic length, D0, of the problem i.e. L10= 6D0. The strikingly good correlation between the predictions using our method and the results of Teko glu et al. (2015) in Fig. 2, clearly shows that the assumption of L10= 6D0numerically represent the unit cell consisting of a central block with a single void in its center within two void-free semi-in nite blocks. After validating our numerical method, we now focus on analyzing the in uence of material's strain rate sensitivity on the onset of macroscopic localization due to void growth induced soft- ening and/or void coalescence. All the results presented here onwards are from the nite element calculations utilizing strain rate dependent ow response de ned in Eq. (17). The full set of con- stitutive parameters used to de ne material (in the band as well as in the outer block) in the calculations are: E= 167GPa ,= 0:3,0= 418MPa ,"0= 0:0025, and _"0= 1s1, while the strain rate sensitivity parameter, m, is varied from 0 to 0 :25. For each strain rate sensitivity parameter value, unit cells with one initial void volume fraction, f0= 0:001, but three initial orientations of the voided band, u= 0, 20and 39are considered. The choice of the initial orientation of the voided band, u, is based on the results of Teko glu et al. (2015) for strain rate independent material. Finally, for each strain rate sensitivity parameter, m, value and ini- tial orientation of the voided band, u, calculations are carried out for prescribed constant stress triaxiality,T, values in the range, 0 :75T3 and one Lode parameter, L=1. 3.1. Strain Localization, Void Growth and Void Coalescence The evolution of the equivalent strain inside the band, Eb eq, with the macroscopic equivalent strain outside the band, Eeq, in unit cells with initial orientation of the voided band, u= 0and 39are shown in Fig. 3. As shown in Figs. 3, during initial stages of deformation Eb eqis same as Eeq, but at later stages of deformation, strain localizes in the band, and thereafter Eb eqincreases rapidly while Eeqincreases slightly in the material outside the band. The onset of localization is assumed to occur when the ratio between the rate of macroscopic equivalent strain in the band and outside the band reach the value of 5. The onset of void coalescence is assumed to occur when the ratio of the maximum to the minimum equivalent plastic strain rate at the void surface rst exceeds the threshold of 20, as in Teko glu et al. (2015). For T= 1:25, the onset of coalescence occurs soon after the onset of localization. But, for T= 2:75, the onset of coalescence occurs at a much greater value of Eb eqfollowing the onset of localization. A comparison of the results shown in Figs. 3(a) and (b), and Figs. 3(c) and (d), show that increasing the strain rate sensitivity of the material delays the onset of both, the localization and coalescence. However, the relative stabilizing e ect of material's strain rate sensitivity is greater at the onset of localization than at 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 of localization and ( Eb eq)m=0:25=(Eb eq)m=02:8 at the onset of void coalescence. Also a comparison of the results shown in Figs. 3(a) and (c), and Figs. 3(b) and (d), show that there is an increase in the relative stabilizing e ect of material's strain rate sensitivity with the increasing value of the imposed stress triaxiality. 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 band,Eeq, forT= 1:25, is shown for two unit cells with initial orientation of the voided band u= 0and u= 39in Figs. 4(a) and (c), respectively. The same is shown for T= 2:5, for two unit cells with u= 0and u= 39in Figs. 4(b) and (d), respectively. To quantify the in uence of the strain rate sensitivity parameter on the evolution of the porosity, results are shown in Fig. 4 form= 0;0:05, and 0:1. As shown in Fig. 4, for both the values of Tand u, a higher value of mshifts thef=f 0versusEeqcurves to the right i.e. the stabilizing e ect of the material's strain rate sensitivity slows down the evolution of porosity. For T= 2:5, the value of f=f 0at the onset of localization is less than the value of f=f 0at the onset of void coalescence for both the values 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 less than the value of f=f 0at the onset of void coalescence for u= 0form> 0. However, for T= 1:25, the value of f=f 0at the onset of localization is same as the value of f=f 0at the onset of void coalescence for u= 39for all three values of m. The e ect of the material's strain rate sensitivity on deformation in the band and distribution of equivalent plastic strain on the void surface is highlighted in Fig. 5. In Fig. 5, results are shown for two strain rate sensitivity parameters, m= 0 and 0:25, for a unit cell with the initial orientation of the band, u= 20and subjected to an imposed stress triaxiality, T= 1:0. As can be seen in Fig. 5, the edges, adandbc, that were initially straight and normal to the x3axis, undergo deformation following the periodicity of the initial voids in the band. In addition, the results also show that at the same equivalent strain levels in the band, Eb eq, the maximum value 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. The lower levels of "pon the void surface for m= 0:25 compared to m= 0 is consistent with the e ect ofmon the evolution of f=f 0shown in Fig. 4. 3.2. E ect of Strain Rate Sensitivity on Critical Porosity 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 at the onset of strain localization and void coalescence. The critical value of porosity at the onset of void coalescence is widely used as an input parameter to simulate ductile fracture due to void nucleation, growth and coalescence (Besson et al., 2003; Srivastava et al., 2014; Cheng et al., 2014). 12 The variation of the normalized critical value of porosity, fc=f0, at the onset of localization and void coalescence with material's strain rate sensitivity parameter, m, in two unit cells with initial orientation of the voided band, u= 0and 39, for two values of imposed stress triaxiality, 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 of localization and void coalescence are roughly the same for m< 0:05. However, for greater value ofmthe value of fc=f0at the onset of localization is less than the value of fc=f0at the onset of void coalescence. For T= 1:25 and u= 39, the values of fc=f0at the onset of localization and void coalescence are roughly the same for all values of m. On the other hand, for T= 2:5, the values of fc=f0at the onset of localization is less than the values of fc=f0at the onset of void coalescence for all values of m, for both u= 0and 39. Also, forT= 2:5 the di erence between the values of fc=f0at the onset of localization and void coalescence increases with increasing value ofm. The results presented in Fig. 6 show that at low imposed stress triaxiality values, the extent of void growth induced softening is not sucient to trigger macroscopic localization, thus localization does not occur until the onset of void coalescence. On the other hand, the void growth induced softening is sucient to trigger macroscopic localization at high imposed stress triaxiality values and the macroscopic localization occurs much before the void coalescence. The increase in the value of fcwith increasing value of mis because an increase in the value of mresults in increased strain rate hardening thus requiring an increase in damage induced softening to trigger any localized deformation in the voided band or within the ligament between the neighboring voids to trigger void coalescence. 3.3. E ect of Strain Rate Sensitivity on Critical Strain 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 imposed stress triaxiality a ect the onset of strain localization and void coalescence. Following this, we analyze the variation of critical macroscopic equivalent strain outside the band, Ec eq, and critical equivalent strain inside the band, Ecb eq, with strain rate sensitivity parameter, m, shown in 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 Fig. 9. As shown in Fig. 7, the value of Ec eqincreases with increasing value of strain rate sensitivity parameter, m, for all three initial orientation of the voided band, u= 0, 20and 39, and for both the imposed stress triaxiality values, T= 1:5 and 2:75. As also shown in the gure, for all three uvalues and for T= 1:5, the value of Ec eqis same at the onset of localization and void coalescence. For T= 2:75,Ec eqvalues at void coalescence are slightly greater than the values ofEc eqat the onset of localization. This is consistent with the observation that at low imposed stress triaxiality values, the extent of void growth induced softening is not sucient to trigger macroscopic localization, whereas at high imposed stress triaxiality values the void growth induced softening is sucient to trigger macroscopic localization. As also shown in Fig. 7, the e ect of initial band orientation on the value of Ec eqdepends on both, the values of Tandm. For example, the minimum value of Ec eqat the onset of localization corresponds to u= 20for T= 1:5 while it corresponds to u= 39forT= 2:75. Similarly, the value of the critical equivalent strain in the band, Ecb eq, at the onset of localiza- tion/coalescence increases with increasing value of mfor all three uand for both T, Fig. 8. In addition, the variation of Ecb eqwithmforT= 1:5, Fig. 8(a), roughly follows the same trend as shown in Fig. 7(a) for Ec eq. But forT= 2:75, the value of Ecb eqat the onset of localization is less than the value of Ecb eqat the onset of void coalescence for all the values of mand u, Fig. 8(b). ForT= 2:75, the initial band orientation that gives lowest values of Ecb eqis u= 0, and the orientation that gives largest values of Ecb eqis u= 20. This is in contrast to the dependence of Ec eqon uas shown in Fig. 7(b). This is because at high imposed stress triaxiality values, the onset of macroscopic localization occurs due to void growth induced softening that signi cantly a ects the deformation in the band. Figures 9(a) and (b) compare the variation of Ecb eqat the onset of localization and coalescence, with imposed stress triaxiality values, T, form= 0:05 and 0:25, respectively. As shown in 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 increasing value of the imposed T. But for suciently large values of T, with increasing T, the decrease in Ecb eqat void coalescence is rather gradual as compared to the decrease in Ecb eq at localization. This results in a clear separation between the values of Ecb eqat localization and coalescence for T2, with the increasing di erence between Ecb eqat localization and coalescence with increasing value of T. 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 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 of localization, ( Ecb eq)Coa=(Ecb eq)Loc, withTandmin Figs. 10 and 11, respectively, for two initial orientation of the voided band, u. As shown in Fig. 10, for all mand for both the values of u, the ratio ( Ecb eq)Coa=(Ecb eq)Locis approximately one for T < 2 and forT2, the ratio (Ecb eq)Coa=(Ecb eq)Locincreases with increasing value of T. This is because both ( Ecb eq)Coaand (Ecb eq)Loc decrease with increasing Tbut forT2, (Ecb eq)Locdecreases more rapidly with increasing Tthan (Ecb eq)Coa. Furthermore, as shown in Fig. 11, for T > 2, the value of ( Ecb eq)Coa=(Ecb eq)Locdecreases with increasing mfor both the values of u. The decrease in the value of ( Ecb eq)Coa=(Ecb eq)Locwith increasingmbecomes more pronounced with increasing value of the imposed T. Note, that both (Ecb eq)Coaand (Ecb eq)Locincrease with increasing m, thus a decrease in the value of ( Ecb eq)Coa=(Ecb eq)Loc suggests that ( Ecb eq)Locincreases more rapidly with increasing mfor higher stress triaxiality values. To summarize, the relative stabilizing e ect induced by strain rate sensitivity of the material not only increases with the increasing value of the imposed stress triaxiality, but also the relative stabilizing e ect of the strain rate sensitivity is greater at the onset of strain localization than at the onset of void coalescence. 3.4. E ect of Strain Hardening In this work, we have focused on answering the question: How does the materials strain rate sensitivity a ect the two mechanisms of macroscopic localization or ductile failure as a function of the imposed stress triaxiality? To answer this question, we have carried out a series of unit cell calculations, where the matrix material of the unit cell is considered to follow a strain rate dependent elastic perfectly plastic ow response. So the question arises as to what extent strain hardening in the material a ects our results. A complete study of the combined e ect of material's strain rate sensitivity, strain hardening and imposed stress triaxiality is beyond the scope of this work. However, our limited unit cell calculations presented in this section show that incorporating both material's strain rate sensitivity and strain hardening does not a ect the qualitative e ect 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. The results of our limited unit cell calculations to explore the combined e ect of material's strain rate sensitivity, strain hardening and stress triaxiality on the two mechanisms of macroscopic localization is shown in Fig. 12. The combined e ect of material's strain rate sensitivity and strain hardening is explored using the ow response given as, =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 E= 167GPa ,0= 418MPa ,"0= 0:0025, _"0= 1s1,N= 0 and 0:1, andm= 0:05 and 0:25 are used in Eq. (19). The results presented in Fig. 12 show that incorporating strain hardening in to the ow response of the material does not a ect the qualitative e ect of material's strain rate sensitivity and stress triaxiality on the onset of localization and void coalescence reported in this work. For example, irrespective of the value of the strain hardening exponent, N: (i) for low stress triaxiality value, T= 0:75, the value of Eb eqat the onset of localization and void coalescence are the same, (ii) the value of Eb eqat the onset of localization and void coalescence increases with the increasing value of strain rate sensitivity parameter, m, for both the values of T, (iii) for high stress triaxiality value, T= 2:5, the value of Eb eqat the onset of localization is less than the value ofEb eqat the onset of void coalescence, and (iv) the di erence between the values of Eb eqat the onset of localization and void coalescence decreases with increasing value of m. 17 4. Summary and Concluding Remarks Strain localization and void coalescence (resulting in the formation of micro-cracks) in a porous ductile material mark the end of uniform deformation and limits the ductility of the material. In this work the focus is con ned to analyzing the in uence of material's strain rate sensitivity on the onset of strain localization and void coalescence 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 localization and coalescence in an in nite block containing a periodic distribution of initially spherical voids in a band. The unit cell modeled consists of a central block with a single void in its center within two semi-in nite void-free blocks. The matrix material of the unit cell is considered to follow a strain rate dependent elastic perfectly plastic ow response. Limited calculations have also been carried out to explore the combined e ect of material's strain rate sensitivity and strain hardening. The unit cell calculations are carried out for one initial void volume fraction ( f0= 0:001), a range of strain rate sensitivity parameter (varying in the range 0m0:25), a range of imposed stress triaxiality (varying in the range 0 :75T3) with one Lode parameter ( L=1), and three initial orientations of the voided band ( u= 0, 20and 39). In the calculations, onset of localization is de ned as the point in the deformation history when the ratio of the rate of deformation in the band and outside the band reaches a critical level, while the onset of coalescence is de ned when the ratio of the maximum to the minimum e ective plastic strain rate at the surface of the void reaches a critical value following the work of Teko glu et al. (2015). Our results show that both the critical void volume fraction and strain, at the onset of localization and coalescence, are strongly in uenced by the material's strain rate sensitivity and the imposed stress triaxiality. The main conclusions of this work are as follows: 1. In uence of material's strain rate sensitivity, m, and imposed stress triaxiality, T, on the critical void volume fraction, fc, at the onset of strain localization and void coalescence: (a) For low Tvalues,fcis roughly the same at the onset of localization and coalescence, whereas for greater values of T,fcat the onset of coalescence is greater than fcat the onset of localization. (b) For all values of T, the value of fcat localization/coalescence increases with increasing m. 2. In uence of material's strain rate sensitivity, m, and imposed stress triaxiality, T, on the critical equivalent strain in the band, Ecb eq, at the onset of strain localization and void coa- lescence: (a) For low Tvalues,T < 2,Ecb eqis same at the onset of localization and coalescence, whereas for T2,Ecb eqat the onset of coalescence is greater than Ecb eqat the onset of localization. (b) For all values of m, the value of Ecb eqat the onset of localization/coalescence decreases with increasing T. But forT2, the value of Ecb eqat localization decreases more rapidly with increasing Tcompared to the value of Ecb eqat coalescence. (c) For all values of T, the value of Ecb eqat the onset of localization/coalescence increases with increasing value of m. But forT2, the value of Ecb eqat localization increases more with increasing value of mcompared to the value of Ecb eqat coalescence. 18 (d) The concluding points, 2(b) and (c), clearly show that the relative stabilizing e ect of mincreases with the increasing value of T, and the relative stabilizing e ect of mis greater at the onset of localization than at the onset of coalescence. Our limited results aimed at exploring the combined e ect of material's strain rate sensitivity, strain hardening and imposed stress triaxiality on the onset of localization and void coalescence show that incorporating strain hardening into the ow response of the material does not a ect the aforementioned qualitative e ect of material's strain rate sensitivity and stress triaxiality on the onset of localization and void coalescence. In this work, we have focused on one Lode parameter value,L=1. The value of the Lode parameter may also a ect the onset of localization and void coalescence especially at low values of stress triaxiality. The numerical method presented in this work can be used for future research to explore the combined e ect of material's strain rate sensitivity and strain hardening, and imposed stress triaxiality and Lode parameter on the onset of localization and void coalescence. Furthermore, our results motivate the need to incorporate the e ect of material's strain rate sensitivity in the porosity and stress triaxiality dependent ow potential (or constitutive response) of the material. Acknowledgements The nancial support provided by the European Union's Horizon2020 Programme (Excellent Science, Marie-Sklodowska - Curie Actions, H2020 - MSCA - RISE - 2017) under REA grant agreement 777896 (Project QUANTIFY) are gratefully acknowledged. Shmuel Osovski would also like to acknowledge the nancial support of the Pazy foundation (Young researchers award grant no. 1176). We are thankful to Dr. J.A. Rodr guez-Mart nez of University Carlos III of Madrid for numerous fruitful discussions. The authors also acknowledge to Katarzyna Kowalczyk-Gajewska from Institute of Fundamental Technological Research, Polish Academy of Sciences, for helpful discussions. 19 Appendix A. Procedure for the Multipoint constraint The procedure to prescribe proportional loading i.e. xed ratio of principal stresses, R=  2=1 andQ=  3=1to maintain constant macroscopic stress triaxiality, T, and Lode parameter, L, throughout the deformation history, follows the work of Vadillo et al. (2016). Since the macroscopic true stresses ( 1, 2, 3) and the macroscopic strain rates ( _E1,_E2,_E3) are equal to the volume average values in a cell (Hill, 1967), the total rate of deformation work in the whole cell _Wconsidering jointly upper, lower and central voided block, can be written as: _W=V1_E1+V2_E2+V3_E3 (A.1) whereVis the total volume of the RVE. De ningP1=V1,P2=V2andP3=V3as generalized forces and work rate conjugate quantities to _E1,_E2and _E3, respectively, the expression of _Wbecomes: _W=P1_E1+P2_E2+P3_E3 (A.2) in whichP1,P2andP3must satisfy: P2 P1=R;P3 P1=Q (A.3) The ratiosRandQcan be written in terms of the imposed stress triaxiality, T, and Lode param- eter,L: Q=3Tp L2+ 33L 3Tp L2+ 3 + 3L;R=3Tp L2+ 3 + 2L 3Tp L2+ 3 + 3L(A.4) Next, with the transformation: (A.5) 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: N=0 @A11A12A13 A21A22A23 A31A32A331 A; with the elements: 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: _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 prescribed as stress uniaxial: 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 and considering the relations given in Eq. (A.5), the three relations: (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: (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. 21