This paper presents an investigation on the formation of adiabatic shear bands in metallic sheets subjected to dynamic shear-compression loading under plane-stress conditions. For that purpose, we have developed a theoretical model based on perturbation analysis, and have performed nite element calculations. The material behavior has been modeled with von Mises plasticity, and the evolution of the yield stress has been considered dependent on plastic strain, plastic strain rate, and temperature. The theoretical model extends the linear stability analysis for simple-shear formulated by Molinari (1997) to shear-compression loading. The numerical simulations have been performed in ABAQUS/Explicit (2016) using a unit-cell model based on the work of Rodr guez-Mart nez et al. (2015) in which the shear band formation is favored introducing geometric and material imperfections. Moreover, we have developed a calibration procedure for the stability analysis that enables to make qualitative and quantitative comparisons with the nite element calculations. Both stability analysis predictions and nite element results display the same overall trends, and show that any small negative triaxiality has a profound stabilizing e ect on the material behavior delaying, or even preventing, the formation of a shear band. This key outcome has been substantiated for a wide range of strain rates and for materials with di erent strain hardening coecients, strain rate sensitivities, and thermal softening behaviors. Keywords: Shear-compression loading, Adiabatic shear bands, Inertia, Linear stability analysis, Finite element calculations 1. Introduction The formation of shear localization bands generally precedes ductile failure of metals and alloys subjected to shear-dominated dynamic loading scenarios, e.g. during machining (Molinari et al., 2002; Cai and Dai, 2014), high-speed penetration (Borvik et al., 2001) and high-velocity metalworking operations (Chen et al., Corresponding author. Tel. +34 916248064; Fax: +33 916249430. E-mail address:gvadillo@ing.uc3m.es Preprint submitted to Elsevier June 27, 2020 2 2003). The interested reader can nd a wealth of experimental evidence and additional examples in which shear bands lead to catastrophic failure of structures in Chapter 2 of the book edited by Bai and Dodd's (Bai and Dodd, 2012). Molinari and Clifton (1987) divided shear localization bands into two types: those in which thermal softening plays a negligible role in their formation and those in which thermal softening plays a primary role. In the rst case, the shear bands, which are often referred to as isothermal shear bands , are formed as a result of microstructural softening mechanisms such as dynamic recrystallization (Rittel et al., 2008; Osovski et al., 2012, 2013) or deformation texture evolution (Kuroda and Tvergaard, 2007; Kuroda and Needleman, 2019). The second type of shear localization bands correspond to the so-called adiabatic shear bands , which are formed when the thermal softening caused by the local increase of temperature under adiabatic conditions of deformation overcomes the strain hardening of the material (Zener and Hollomon, 1944). While both types of shear bands, isothermal and adiabatic , have been extensively studied over the past 80 years, our interest in this paper is limited to those of the latter type. The experiments of Hartley et al. (1987) and Marchand and Du y (1988) marked a turning point in the research on adiabatic shear bands. These authors twisted thin-walled tubes made of AISI 1018 cold rolled steel, AISI 1020 hot rolled steel and HY-100 structural steel at shear strain rates ranging from 1000 s1to 5000 s1 using a torsional Kolsky bar and they measured, for the rst time, both local strains and local temperatures during the specimen deformation. The experimental evidence enabled to identify three stages in the initiation and growth of an adiabatic shear band: in the rst step, the strain in the specimen was homogeneous; in the second step, after the maximum stress was reached, the strain started to localize with a progressive decrease in the width of the region over which localization was occurring; and in the third stage, the strain localized further into an ever narrower shear band, leading to an important increase in the material temperature and a sharp drop in the ow stress, with the subsequent fracture of the specimen. Du y and co-workers (Cho et al., 1990; Du y, 1991; Du y and Chi, 1992; Liao and Du y, 1998) published additional experimental results for two tempers of AISI 4340 VAR steel of varying hardness and Ti6Al4V titanium alloy. All these papers concluded that the mechanisms involved in the development of the adiabatic shear band are cumulative, i.e. the high temperature band becomes progressively narrower, and ever higher strain rates and temperatures are attained in this narrowing band, which accelerates the localization process. This autocatalytic behavior , as it was called by Molinari and Clifton (1987), has also been observed by other authors over the years (e.g. Wang et al. (1988); Zhou et al. (1996b,a); Bonnet-Lebouvier and Klepaczko (2002)), and has motivated that the adiabatic shear bands were accepted as a thermomechanical (material) instability. Bai (1982) published what is probably the rst paper in the open literature in which the formation of 3 adiabatic shear bands is modeled using linear stability theory. The technique consists of evaluating the rate of growth of an in nitesimal spatio-temporal perturbation which is added to the fundamental solution of the problem at di erent loading times, i.e. for di erent background strains, to determine whether shear localization can occur: if there is any perturbation mode that grows with time, the fundamental solution is unstable, and a shear band can be formed. The assumption is that the perturbation growth represents the rst stages of the shear localization process. Bai (1982) demonstrated that for isotropic materials with yield stress dependent on strain, strain rate and temperature, the condition for the shear instability to occur, i.e. for the initiation of the shear band, is that the thermal softening caused by plastic work overcomes the work hardening of the material, in agreement with the classical explanation of Zener and Hollomon (1944). Batra and Chen (2001) demonstrated that for adiabatic simple-shearing deformations, the critical strain obtained from the instability criterion of Bai (1982) equals that given by the maximum stress condition derived from the Consid ere (1885) criterion. Moreover, Molinari (1988) used the linear stability analysis to calculate the critical strain leading to the onset of shear localization in AISI 1018 cold rolled steel, and found very good agreement with the experimental results reported by Hartley et al. (1987) for a shear strain rate of 1000 s1. Molinari (1988) also performed a qualitative evaluation of the dynamical and heat conduction e ects on the perturbation growth, and showed that the stabilizing e ects of inertia and thermal conductivity on small and large perturbation modes, respectively, promote the speed of growth of intermediate modes. The perturbation mode that grows the fastest is referred to as the critical perturbation mode , and the corresponding perturbation growth as the critical perturbation grow rate. Molinari (1997) showed for AISI 1018 steel and titanium materials, modeled as isotropic solids with yield stress described by a thermo-viscoplastic power-law constitutive equation, that the critical perturbation grow rate evolves with the strain of the fundamental solution such that it rst increases, reaches a maximum, and then decreases. It was assumed that the background strain associated to the largest critical perturbation grow rate corresponds to the strain for which shear bands form in the experiments of Nesterenko et al. (1995b,a), in which thick-walled cylinders were collapsed radially at strain rates of the order of 10000 s1. Good agreement between theoretical predictions and experimental results was found for both steel and titanium materials. In these experiments, in which the material is subjected to pure-shear deformations, the shear bands emanate from the inner boundary of the cylinder, with spiral trajectories and regular spacing. Molinari (1997) also showed that the average bands spacing in these tests nds good agreement with the predictions of the stability analysis for the wavelength associated to the perturbation mode corresponding to the largest critical perturbation grow rate. Following the work of Molinari (1997), Daridon et al. (2004) and Batra and Wei (2006) used the stability analysis to explore the constitutive sensitivity of the average shear bands spacing in the dynamic radial collapse 4 of thick-walled cylinders, and showed that the spacing increases with the thermal conductivity and the strain rate sensitivity, and decreases with the strain hardening of the material. The e ect of material behavior on the shear bands spacing was further con rmed in the experiments performed by Xue et al. (2002, 2004) and Lovinger et al. (2011, 2015) with various metallic materials including Ti6Al4V, AISI 304L steel and Mg-AM50. Other recent works that have used the linear stability theory to investigate the formation of shear bands in metallic materials are the papers of Rodr guez-Mart nez et al. (2015) and Long ere (2018), who analyzed the respective roles of thermal softening and dynamic recrystallization in the initiation of shear bands in Ti6Al4V, and the article of Gurrutxaga-Lerma (2018), who studied the conditions that dislocation generation and motion must ful ll to promote the development of adiabatic shear bands in crystalline metals. In addition, Arriaga et al. (2015, 2016) recast the stability analysis as a generalized eigenvalue problem that can be implemented in a nite element code to determine the onset of shear localization, damage and fracture, at the element level, in general multidimensional applications. Numerical results on a 1D problem representative of a dynamic torsion experiment showed that the eigenvalue analysis recovers the instability point predicted by the classical analytical criterion of Bai (1982). The eigenvalue analysis was later used in the nite element simulation of a series of 2D problems, including the Kaltho experiment, bringing out the capacity of this approach to detect the formation and propagation of cracks, even in complex situations like crack branching (Arriaga and Waisman, 2017, 2018a,b). Today, almost 40 years after Bai's paper (Bai, 1982), it is widely accepted that the stability analysis is a very useful tool for identifying the essential material features that control the formation of adiabatic shear bands, and that it provides results which are in qualitative and quantitative agreement with the experimental evidence. However, all the analytical stability analyses cited in this introduction are concerned only with one-dimensional simple-shear deformations and, to the authors' knowledge, the application of the linear perturbation theory to investigate analytically the formation of shear bands under two-dimensional shear-dominated loadings is still missing. This is precisely the goal of this paper, in which the linear stability analysis for simple-shear deformations developed by Molinari (1997) is extended to consider dynamic shear-compression loadings under plane-stress conditions. The reason to investigate such stress state is given by the dynamic shear experiments in sheet materials performed by Peirs et al. (2012) and Xu et al. (2017, 2018, 2019), among others, who showed that, despite the e orts to improve the specimens design to obtain stress homogeneity and zero triaxiality in the gauge, undesired normal stresses frequently arise in the shear zone of the samples during the tests. In this regard, the linear stability analysis developed in this paper shows that any small negative triaxiality has a profound stabilizing e ect on the material behavior delaying, or even preventing, the formation of a shear band. The 5 theoretical predictions have been compared with nite element calculations performed in ABAQUS/Explicit (2016) for von Mises materials with di erent strain hardening coecients, strain rate sensitivities, and thermal softening behaviors. A key outcome of this investigation is that both stability analysis predictions and nite element results display the same overall trends, and the agreement is also quantitative for some of the strain rates and material behaviors investigated. 2. Constitutive framework The constitutive model follows von Mises plasticity. The strain rate tensor dis decomposed as the sum of an elasticdeand a plastic part dp: d=de+dp(1) The relation between the elastic strain rate and the stress rate is given by the following hypo-elastic law: _=C:de(2) where _is an objective derivative of the Cauchy stress tensor and Cis the tensor of isotropic elastic moduli given by: C= 2G~I+K1 1 (3) whereGis the elastic shear modulus, Kis the bulk modulus, 1is the second-order unit tensor and ~Iis the fourth-order deviatoric unit tensor. Moreover, the plastic part of the strain rate tensor is given by the following associated ow rule: dp= _@f @(4) 6 where _is the rate of the plastic multiplier and f= Yis the yield condition, with  =q 3 2s:sbeing the equivalent stress, where s=h:1andh=1 3:1. Moreover, Yis the yield stress of the material de ned as: Y("p;_"p;) =0 Y+K("p)n_"p _"refm ref (5) where "p=Rt 0_"p( )d is the equivalent plastic strain, _"p=q 2 3dp:dpis the equivalent plastic strain rate and is the material temperature. The parameters 0 YandKare the initial yield stress and the plastic modulus, respectively. Moreover n,mandare constants which represent the strain, strain rate and temperature sensitivities of the material. The parameters _"refandrefare the reference strain rate and temperature, respectively. The values of the material constants used in the constitutive model are taken from Vaz-Romero et al. (2015), and correspond to the AISI 430 steel, see Table 1. Unless otherwise noted, this set of parameters is used in the stability analysis calculations and nite element simulations presented in this paper. Symbol Property and units Value o Initial density (kg =m3), Eq. (21) 7740 Cp Speci c heat (J =kg K), Eq. (21) 460 k Thermal conductivity (W =m K), Eq. (21) 26.1 Thermal expansion coecient (K1) 0.00001 K Bulk modulus (GPa), Eq. (3) 196 G Shear modulus (GPa), Eq. (3) 75 0 Y Initial yield stress (MPa), Eq. (5) 175.67 K Strain hardening modulus (MPa), Eq. (5) 530.13 n Strain hardening exponent, Eq. (5) 0.167 _"ref Reference strain rate (s1), Eq. (5) 0.01 m Strain rate sensitivity exponent, Eq. (5) 0.0118 ref Reference temperature (K), Eq. (5) 300  Temperature sensitivity exponent, Eq. (5) -0.51 Taylor-Quinney coecient, Eq. (21) 0.9 Table 1: Conventional material constants and parameters used in the constitutive model for AISI 430 steel. Data after Vaz-Romero et al. (2015). 7 3. Theoretical model We develop a linear stability analysis for the theoretical modelling of shear instabilities in von Mises ma- terials subjected to shear-compression loading. A rectangular sheet of initial thickness b0and edges of initial lengthL0 xandL0 y, withL0 x>>L0 y>>b0, is considered for the analysis. Note that the superscript ()0is used to denote quantities at time t= 0. As illustrated in Fig. 1, the sheet is subjected to prescribed velocities on opposed sides, VyandVx. It is assumed that this loading condition is always satis ed. Moreover, the elastic deformations are neglected and from now on  "= "pwill be named equivalent strain and _"=_"pequivalent strain rate. Note that ( X;Y;Z ) and (X0;Y0;Z0) are the Lagrangian Cartesian frames associated to the applied velocity eld and the shear plane, respectively, and ( x;y;z ) and (z0;y0;z0) are their Eulerian counterparts. Note also thatZ=Z0andz=z0, being zero the stresses in this direction (plane-stress condition). The shear plane, which contains the x0axis, is determined by the condition Vx0jy0=0= 0. The orientation of the shear plane with respect to the xaxis is de ned by the angle . Note that this angle varies with the loading time t. Note also that, while the imposed velocity eld is expressed as a function of the Lagrangian coordinates, the problem will be formulated in the ( x0;y0;z0) frame. The kinematics of the problem in the ( x;y;z ) and (x0;y0;z0) frames is shown in Appendix A. Figure 1: Scheme of a rectangular sheet of initial thickness b0and edges of initial length L0 xandL0 ysubjected to an imposed velocity eld given by Vx= _"0 xX+ 2 _"0 xyYandVy=_"0 yY. Note that ( X;Y;Z ) and (X0;Y0;Z0) are the Lagrangian Cartesian frames associated to the applied velocity eld and the shear plane, respectively, and ( x;y;z ) and (x0;y0;z0) are their Eulerian counterparts. Note also that Z=Z0andz=z0, being zero the stresses in this direction (plane-stress condition). The shear plane is determined by the condition Vx0jy0=0= 0. The orientation of the shear plane with respect to the xaxis is de ned by the angle . This angle varies with the loading time. 8 3.1. Governing equations The unknowns of the problem are: velocities ( Vx0,Vy0), strain rates ( dx0x0,dy0y0,dz0z0,dx0y0), thickness (b), stresses ( x0x0,y0y0,x0y0), equivalent stress ( ), yield stress ( Y), equivalent strain ( "), equivalent strain rate ( _"), rate of plastic multiplier ( _ ) and temperature ( ). The fundamental equations governing the loading process are presented below: Rate of deformation tensor dx0x0=@Vx0 @x0= 0 (6) dy0y0=@Vy0 @y0(7) dx0y0=1 2@Vx0 @y0+@Vy0 @x0 (8) where the explicit expressions of Vx0andVy0as a function of x0andy0are given in Appendix A.2, see equation A.14. Incompressibility dx0x0+dy0y0+dz0z0= 0 (9) Thickness dz0z0=_b b(10) Previous expression is obtained from the kinematics of the problem described in Appendix A. Equivalent strain rate and equivalent strain 9 _"=r 2 3 d2 x0x0+d2 y0y0+d2 z0z0+ 2d2 x0y0 (11) "=Zt 0_"( )d (12) Yield condition f= Y= 0 (13) where, as reported in Section 2,  andYare the equivalent stress and the yield stress, respectively. Equivalent stress =q 2 x0x0+2 y0y0x0x0y0y0+ 32 x0y0 (14) Yield stress Y(";_";) =0 Y+K(")n_" _"refm ref (15) Previous expression is obtained replacing  "pand _"p, by "and _", in equation (5). Flow rule dx0x0= _2x0x0y0y0 2= 0 (16) 10 dy0y0= _2y0y0x0x0 2(17) dx0y0= _3x0y0 2(18) Balance of linear momentum @ @x0(bx0x0) +@ @y0 bx0y0 =bDVx0 Dt(19) @ @x0 bx0y0 +@ @y0 by0y0 =bDVy0 Dt(20) The momentum balance equation in the out-of-plane direction has been disregarded. Energy c_=k@2 @x02+@2 @y02 +k b@ @x0@b @x0+@ @y0@b @y0 + _" (21) wherec=Cp,kand are the heat capacity per unit volume, the thermal conductivity and the Taylor- Quinney coecient, respectively. Recall that the material is considered incompressible, and thus =0. Note that in previous expression we have assumed no heat ow at the specimen's boundaries and neglected the contribution of thermoelastic e ects. The set of equations (6)-(21) is to be solved under the following initial and boundary conditions expressed in the (X;Y;Z ) frame: 11 Vx(X;Y; 0) = _"0 xX+ 2 _"0 xyY; Vy(X;Y; 0) =_"0 yY; ij(X;Y; 0) =0 Y "(X;Y; 0) = 0;  (X;Y; 0) =0; b (X;Y; 0) =b0(22) Vx(X;L0 y 2;t) = _"0 xX_"0 xyL0 y; Vx(L0 x 2;Y;t) =_"0 xL0 x 2+ 2 _"0 xyY Vy(X;L0 y 2;t) =_"0 yL0 y 2; Vy(L0 x 2;Y;t) =_"0 yY @ @X X=L0x=2=@ @Y Y=L0y=2= 0(23) where we take 0=refandb0= 2 mm. The constants _ "0 x0, _"0 y0 and _"0 xy>0 de ne the values of the initial strain rates in the sheet. In this paper we limit our attention to shear-dominated loading conditions such that _"0 xy>>_"0 xand _"0 xy>>_"0 y, and assume that the sheet does not buckle due to the load applied in the Ydirection. Note that if _ "0 x= _"0 y= 0, we retrieve the simple-shear problem studied, for instance, by Wright (1995), Molinari (1997), Daridon et al. (2004) and Gurrutxaga-Lerma (2018). The applied loading is selected through the input value of the initial equivalent strain rate _"0, and the initial strain rates _ "0 xand _"0 y. Moreover, the value of _ "0 xywhich enters into the initial and boundary conditions is obtained from the following expression: _"0 xy=r 3 4(_"0)2( _"0x)2 _"0y2_"0x_"0y (24) which is derived from the de nition of the equivalent strain rate at t= 0 using the incompressibility condition. 3.2. Linear stability analysis Let us denote by S(x0;y0;t) the fundamental solution of the problem. The solution at t=t1is: S1= V1 x0;V1 y0;d1 x0x0;d1 y0y0;d1 z0z0;d1 x0y0;"1;_"1;_1;b1;1;1 x0x0;1 y0y0;1 x0y0;1;1 YT(25) All the unknowns of the problem at time t1can be readily obtained by integration of the governing equations stated in Section 3.1, satisfying the initial and boundary conditions given in (22) and (23). Note that the 12 superscript ()1is used to denote quantities evaluated at t1. Note also that this fundamental solution considers large strains and deformations, as can be seen in the kinematics of the problem detailed in Appendix A. Next, a small perturbation Sgiven by: S(y0;t)t1=S1eiy0e(tt1)(26) is imposed over the fundamental solution, where: S1= Vx0;Vy0;dx0x0;dy0y0;dz0z0;dx0y0;";_";_;b;; x0x0;y0y0;x0y0;;YT(27) is the perturbation amplitude, is the wavenumber and is the growth rate of the perturbation at time t1. Note that t1is the time at which the perturbation is inserted. Note also that will be referred to as instantaneous instability index everywhere in this paper. Moreover, while in this paper we have assumed the classical frozen coecients theory and the exponential form of the time dependency of the perturbation for the sake of simplicity, Xavier et al. (2020) have recently developed a promising method to relax these assumptions. Furthermore, we seek only for shear instability modes such that the perturbation is imposed in the Eulerian coordinate y0. For the current shear-compression problem, we assume that no other instabilities like necking can develop (necking requires thickness reduction). On the other hand, note that there are several papers in the literature in which speci c stability analyses have been formulated to study the formation of dynamic necks in tensile dominated loadings, e.g. Fressengeas and Molinari (1985, 1994), Guduru and Freund (2002), Mercier and Molinari (2003, 2004) and N'souglo et al. (2020). In future works, in order to investigate the formation of both shear and necking instabilities, the stability analysis should be extended to consider shear-tensile loadings. The solution of the problem is: S=S1+S (28) withjSj S1 . By substituting equation (28) into equations (6)-(21), and retaining only rst-order terms in 13 the increments S1, linearized equations are obtained. In order to scale the problem and bring out the key variables which control the loading process, we introduce the following nondimensional groups: ^Vx0=2 L0yVx0 _"1; ^Vy0=2 L0yVy0 _"1; ^dx0x0=dx0x0 _"1; ^dy0y0=dy0y0 _"1; ^dz0z0=dz0z0 _"1; ^dx0y0=dx0y0 _"1 ^"= ";^_"=_" _"1; ^_=_ _"1; ^b=b b0; ^= 0 ^x0x0=x0x0 0 Y; ^y0y0=y0y0 0 Y; ^x0y0=x0y0 0 Y; ^= 0 Y; ^Y=Y 0 Y Likewise, the nondimensional variables de ning the perturbation are: ^= _"1; ^=L0 y 2 that yield to the following linearized equations. Rate of deformation tensor ^dx0x0= 0 (29) ^dy0y0i^^Vy0= 0 (30) ^dx0y0i^ 2^Vx0= 0 (31) Incompressibility 14 ^dx0x0+^dy0y0+^dz0z0= 0 (32) Thickness  ^d1 z0z0^ ^b+^b1^dz0z0= 0 (33) Equivalent strain rate and equivalent strain ^_"2 3 ^d1 x0x0^dx0x0+^d1 y0y0^dy0y0+^d1 z0z0^dz0z0+ 2^d1 x0y0^dx0y0 = 0 (34) ^_"^^"= 0 (35) Yield condition ^^Y= 0 (36) Equivalent stress ^1^+^1 y0y02^1 x0x0 2^x0x0+^1 x0x02^1 y0y0 2^y0y03^1 x0y0^x0y0= 0 (37) Yield stress 15 ^Y^H^"^S^_"^P^= 0 (38) where ^H=1 0 Y@Y @" t1,^S=_"1 0 Y@Y @_" t1and ^P=0 0 Y@Y @ t1. Flow rule ^1^dx0x0+^d1 x0x0^^_1^x0x0+^_1^y0y0 22^1 x0x0^1 y0y0 2^_= 0 (39) ^1^dy0y0+^d1 y0y0^^_1^y0y0+^_1^x0x0 22^1 y0y0^1 x0x0 2^_= 0 (40) ^1^dx0y0+^d1 x0y0^3^_1 2^x0y03^1 x0y0 2^_= 0 (41) Balance of linear momentum i^ ^b1^x0y0+ ^1 x0y0^b ^^b1 ^L2^Vx0= 0 (42) i^ ^b1^y0y0+ ^1 y0y0^b ^^b1 ^L2^Vy0= 0 (43) where ^L1=_"1L0 y 2r 0 Yis the inertial resistance to motion. Energy 16  ^^c+^2^k ^ ^1^_" ^= 0 (44) where ^c=c0 0 Yand^k=k2 L0y20 0 Y_"1. A non-trivial solution for S1can only be obtained if the determinant of the matrix of coecients of the system of linear algebraic equations (29)-(44) is equal to zero. Application of this condition leads to a fourth-degree polynomial in ^ : B4(S1)^4+B3(S1;^)^3+B2(S1;^)^2+B1(S1;^)^+B0(S1;^) = 0 (45) where the coecients Bi, withi= 0;:::;4, are not shown explicitly for the sake of brevity. Equation (45) has four roots in ^ . The requisite for unstable growth of the perturbation is given by the real part of the instantaneous instability index ^ to be positive. The perturbation growth represents the rst stage of the shear localization process, i.e. if the perturbation grows a shear band can be formed. The root of equation (45) with the greatest positive real part is the physical solution of the problem and it is considered for the analysis (Dudzinski and Molinari, 1991). Equation (45) shows that the dimensionless instantaneous instability index ^  depends on the fundamental solution S1(i.e. on the loading time tand thus on the equivalent strain in the material, see Fig. 3 in Section 5) and the dimensionless wavenumber ^. For a given value of the loading time t, the stabilizing e ect of inertia on short wavenumbers, and of strain rate sensitivity and thermal conductivity on large wavenumbers, promotes the growth of intermediate modes (see Section 5). The mode with the greatest value of ^is referred to as the critical instantaneous wavenumber ^c ^, and the corresponding grow rate of the perturbation is referred to as the critical instantaneous grow rate ^ c. The value of ^ c, which evolves with the loading time (i.e. with the equivalent strain in the material, see Fig. 5 in Section 5), is generally used as an indicator of the material stability in shear band problems: the greater the critical instantaneous grow rate, the more prone the material is to develop shear localization (see also the works of Molinari (1997) and Rodr guez-Mart nez et al. (2015)). An alternative indicator of the stability of the solution was proposed by Fressengeas and Molinari (1994) who de ned a cumulative instability index, ^I=Rt 0^_"1dt, which tracks the history of the instantaneous grow rate of all the growing modes. The cumulative instability index has been 17 used by several authors to predict the formation of necks at high strain rates (see Petit et al. (2005), El Ma  et al. (2014) and N'souglo et al. (2020)). To calculate ^I, we introduce the perturbation at di erent times and sum the dimensionless instantaneous instability index ^ obtained for each loading time. The mode with the greatest value of the cumulative instability index ^Iis called the critical cumulative wavenumber ^c ^I, and the corresponding cumulative index is referred to as the critical cumulative grow rate ^Ic. Like the critical instantaneous grow rate, ^Icalso evolves with the loading time, and thus with the equivalent strain in the material (see Fig. 5 in Section 5). Results of the stability analysis for both dimensionless instantaneous and cumulative instability indexes are shown in Section 5. Moreover, the calibration of the linear stability analysis and the comparison between stability analysis predictions and numerical simulations for the shear bands formation are presented in Section 7. 4. Finite element model This section describes the 2D nite element model developed in ABAQUS/Explicit (2016) to simulate shear localization under shear-compression loading. Accordingly to the theoretical model, the problem consists of a long thin sheet extended in the Xdirection and subjected to plane-stress conditions, see Fig. 2. Material points are referred to using a Cartesian coordinate system with positions in the reference con guration denoted as (X;Y;Z ). The origin of coordinates is located in the center of mass of the specimen. The initial and boundary conditions are the same used in the theoretical model (see Fig. 1 and equations (22) and (23) in Section 3.1). Following the work of Molinari and Clifton (1987), a sinusoidal spatial imperfection has been introduced in the nite element model to promote early shear localization: Lx=L0 x 1 + cos2Y L0y (46) whereLxis the perturbed length of the specimen and is the amplitude of the imperfection. Consistent with the notation used in the theoretical model presented in Section 3, L0 xandL0 yare the unperturbed initial length and initial width of the sample, respectively. We have chosen =L0 x 100andL0 x= 23:5L0 y, withL0 y= 0:508 mm as in Rodr guez-Mart nez et al. (2015). The dimensions of the nite element model are based on the (typical) slot of the Shear-Compression Specimen (Vural et al., 2011). Notice that the value of L0 yis similar to the average spacing between shear bands in the experimental and analytical results of Nesterenko et al. (1995b) 18 and Molinari (1997). Moreover, in order to break the fundamental solution and favor shear localization, the initial yield stress of the material 0 Y(see Table 1) has been perturbed in a narrow strip of width , extended in the Xdirection and located in the center of the specimen (see Fig. 2). Four elements on each side of the Xaxis have been perturbed (i.e. = 0:1 mm). The number of perturbed elements has been chosen arbitrarily. The results reported in Appendix B.1 show the e ect of the number of elements perturbed in the shear band formation (i.e. the e ect of ). The initial yield stress of each perturbed element is a random number generated using the Marsaglia polar method according to a normal distribution function. The mean of this distribution is =0 YCuand the standard deviation is s=0 YCv, whereCudetermines the average perturbed yield stress and Cvthe maximum dispersion of the generated random values of perturbed yield stress. The nite element calculations have been performed for Cu= 0:975 andCv= 0:01. The speci c in uence of Cuin the shear band formation is shown in Appendix B.2. 19 Figure 2: Finite element model. Mesh and boundary conditions. (For interpretation of the