7 This paper investigates the steady-state elastoplastic ﬁelds induced by a pressurized cylindrical cavity 8 expanding dynamically in an anisotropic porous medium. For that task, we have developed a theoretical 9 model which: (i) incorporates into the formalism developed by Cohen and Durban (2013b) the eﬀect of 10 plastic anisotropy using the constitutive framework developed by Benzerga and Besson (2001) and (ii) uses 11 the artiﬁcal viscosity approach developed by Lew et al. (2001) to capture the shock waves that emerge at 12 high cavity expansion velocities. We have shown that while the development of the shock waves is hardly 13 aﬀected by the material anisotropy, the directionality of the plastic properties does have an eﬀect on the 14 elastoplastic ﬁelds that evolve near the cavity. The importance of this eﬀect is strongly dependent on the 15 cavity expansion velocity, the initial porosity and the strain hardening of the material. In addition, the 16 theoretical model has been used in conjunction with the Recht and Ipson (1963) formulas to assess the 17 ballistic performance of porous anisotropic targets against high velocity perforation. 18 Keywords: 19 Cylindrical cavity expansion, Dynamic loading, Shock waves, Porous materials, Anisotropic materials 20 1. Introduction 21 Spherical and cylindrical cavity expansion theories have been widely used over the last 50years to model 22 high velocity penetration and perforation problems in which the target exhibits ductile hole growth. The 23 cavity expansion models use the solution of the pressure for steady expansion of a cavity in an inﬁnite 24 medium as the contact pressure experienced by a projectile that penetrates a target. While the assumption 25 of spherical or cylindrical ﬂow ﬁelds in the target material during the penetration may be considered a 26 rough approximation to the actual ﬂow ﬁeld (Yarin et al., 1995; Roisman et al., 1997; Rubin et al., 2016), 27 cavity expansion theories have been widely accepted as a basic physical model underlying perforation and 28 penetration mechanics in metallic materials (Masri, 2016). According to Durban and Masri (2004) and 29 Cohen and Durban (2013a), key to the success of the cavity expansion theories is that they provide simple 30 ∗Corresponding author. E-mail address: jarmarti@ing.uc3m.es Preprint submitted to International Journal of Impact Engineering June 6, 2019 2 expressions for critical parameters like the resisting force, the ballistic limit and the penetration depth (e.g. 31 Forrestal and Warren (2008, 2009) and Cohen et al. (2010a)). 32 Spherical cavity expansion approximations have been primarily used to model deep penetration prob- 33 lems (Forrestal and Warren, 2009). Goodier (1965) employed the dynamic solution developed by Hopkins 34 (1960)todescribethepenetrationofsphericalimpactorsintometaltargetsmodelledaselastoplastic, incom- 35 pressible solids. Forrestal et al. (1988) developed a spherical cavity expansion model for rate-independent, 36 elastic-perfectly plastic materials that was validated against penetration depths obtained from tests in 37 which projectiles with diﬀerent geometries were impacted onto aluminum targets at velocities which ranged 38 from 0.4−1.4km/s. Forrestal et al. (1991) included the strain hardening of the material in the penetra- 39 tion equations developed by Forrestal et al. (1988), and showed the inﬂuence of material compressibility 40 in the predictions of the cavity expansion model. Warren and Forrestal (1998) incorporated the strain 41 rate sensitivity into the model of Forrestal et al. (1991), and concluded that it was necessary to take into 42 account the compressibility, the strain hardening, and the strain-rate sensitivity of the target material to 43 obtain good agreement with previously published depth of penetration data for striking velocities between 44 300and1200m/s. Durban and Masri (2004) developed a dynamic spherical cavity expansion model for 45 Drucker-Prager solids with a non-associated ﬂow rule. The model, which assumed a self-similar expansion 46 ﬁeld, showed that plastic pressure sensitivity causes an increase in the cavitation pressure. The self-similar 47 spherical cavity expansion approach was also used by Masri and Durban (2005) to develop penetration 48 equations for compressible von Mises solids. The penetration depth predictions obtained with the cavity 49 expansion model were in reasonable agreement with the experimental data reported by Forrestal et al. 50 (1988, 1991) for diﬀerent impact velocities and projectiles nose shapes. Cohen et al. (2010b) investigated 51 the elastoplastic ﬁelds induced by a pressurized spherical cavity expanding in von Mises solids at higher 52 velocities than those investigated by Masri and Durban (2005). They showed that when the speed of the 53 cavity is≈50%of the elastic wave velocity in a long rod, shock waves emerge in the material. Cohen and 54 Durban (2013b) found similar results for porous materials modelled with Gurson plasticity, and showed that 55 the increase of the material porosity favours the development of shock waves at lower cavitation velocities. 56 On the other hand, cylindrical cavity expansion approximations have been primarily used to model 57 perforation problems (Forrestal and Warren, 2009). Forrestal et al. (1990) obtained perforation equations 58 basedontheplanestrainexpansionofacylindricalcavitywhichprovidedresidualvelocityandballisticlimit 59 predictions in agreement with experimental data obtained from tests in which conical nose projectiles were 60 impacted onto 5083-H131 aluminium armour plates of various thickness. The analytical model considered 61 linear elastic, power law strain hardening plastic solids. The eﬀect of target inertia, that was neglected in 62 3 the perforation equations developed by Forrestal et al. (1990), was investigated by Forrestal and Warren 63 (2009) which showed that the eﬀect of inertia in the perforation process increases with the target thickness. 64 A dynamic cylindrical cavitation model valid for moderate expansion velocities was developed by Masri and 65 Durban (2006) for incompressible elastoplastic materials with a deﬁnite yield point. Shortly after, Masri 66 and Durban (2009) developed solutions for orthotropic materials, including the isotropic von Mises and 67 Tresca solids as special cases. The analysis was valid for any hardening response, including materials with 68 no deﬁnite yield point, and accounted for elastic compressibility in the linear elastic zone. Masri (2014) 69 also included the eﬀect of temperature in the cylindrical cavity expansion model, and showed that material 70 resistance to cavitation is weakened with adiabatic thermal softening. Very recently, Johnsen et al. (2017) 71 extended the cylindrical cavity expansion theory to include the Voce strain hardening rule. 72 Overall, manynumericalandtheoreticalstudiesaddressingtheperforationofisotropicmetallicmaterials 73 have been conducted over the last decades. However, as discussed by N’souglo et al. (2019), only few studies 74 considering anisotropy eﬀects on penetration problems have been reported in the literature (Grytten et al., 75 2009; Masri and Durban, 2009; Seidt et al., 2013; Mars et al., 2015; Koubaa et al., 2017). In this paper 76 we adapt the dynamic spherical cavity expansion model developed by Cohen et al. (2010b) and Cohen 77 and Durban (2013b) for isotropic solids to the case of a cylindrical cavity expanding under plane strain 78 conditions in anisotropic porous materials. Further, following the recent work of dos Santos et al. (2019), 79 we incorporate into the cylindrical cavity expansion model the artiﬁcial viscosity approach developed by 80 Lew et al. (2001) to capture the shock waves that emerge at high cavitation velocities. The respective 81 inﬂuence of anisotropy and porosity in the cavitation pressure and in the shock waves formation is exposed 82 and discussed. In addition, new results are presented regarding the stress and density ﬁelds that evolve 83 near the cavity for porous materials with varying degrees of mechanical anisotropy. Finally, the cylindrical 84 cavity expansion model is used in conjunction with the Recht and Ipson (1963) formulas to assess the 85 ballistic performance of porous anisotropic targets against high velocity perforation. The outline of the 86 paper is as follows. Section 2 presents the constitutive model developed by Benzerga and Besson (2001). 87 The cylindrical cavity expansion model for porous anisotropic solids is developed in Section 3. Theoretical 88 results for materials with diﬀerent degrees of mechanical anisotropy and initial void volume fractions are 89 presented in Section 4. We conclude in Section 5 with a summary of the main ﬁndings of this work. 90 2. Constitutive framework 91 The mechanical behaviour of the material is described using the elastic isotropic, plastic orthotropic 92 constitutive model developed by Benzerga and Besson (2001) for porous ductile metals. The model con- 93 4 sists of extending the constitutive framework developed by Gurson (1977) and Tvergaard (1981, 1982) for 94 spherical voids to the case of an orthotropic material. For that purpose, the mechanical behaviour of the 95 matrix material is described using Hill’s model (Hill, 1948). Only the main features of the Benzerga-Besson 96 model are presented in this section. For additional details, the reader is referred to the work of Benzerga 97 and Besson (2001) and the recent book of Cazacu et al. (2019). 98 99 In cylindrical coordinates (r,θ,z ), the ﬂow potential reads: 100 Φ =/parenleftBigg˜σe σyr/parenrightBigg2 + 2q1fcosh/parenleftBigg3q2σh ψσyr/parenrightBigg −1−(q1f)2≤0 (1) 101 whereq1andq2are material parameters, fis the void volume fraction, and σyris the matrix yield strength 102 in the radial direction (see equation (4)). Moreover, σh=1 3σ:1is the mean stress, with σbeing the 103 macroscopic Cauchy stress tensor and 1the unit second order tensor. The constant ψis deﬁned as: 104 ψ=/radicaltp/radicalvertex/radicalvertex/radicalbt4 5/parenleftBigg˜F0+˜G0+˜H0 ˜F0˜G0+˜G0˜H0+˜H0˜F0/parenrightBigg +6 5/parenleftBigg1 ˜L0+1 ˜M0+1 ˜N0/parenrightBigg (2) where ˜F0,˜G0,˜H0,˜L0,˜M0, and ˜N0are dimensionless anisotropy coeﬃcients. Moreover, ˜σeis the equivalent 105 stress expressed as: 106 ˜σe=/radicalBig ˜F0(σθθ−σzz)2+˜G0(σzz−σrr)2+˜H0(σrr−σθθ)2+ 2˜L0σ2 θz+ 2˜M0σ2 zr+ 2˜N0σ2 rθ (3) whereσrr,σθθandσzzare the radial, circumferential and axial normal stresses, and σrθ,σθzandσzrare 107 the corresponding shear stresses. Benzerga and Besson (2001) showed that if ˜F0=˜G0=˜H0=1 2and 108 ˜L0=˜M0=˜N0=3 2thenψ= 2, and equation (1) reduces to the ﬂow potential derived by Gurson (1977) 109 and Tvergaard (1981, 1982). Following the work of Benzerga and Besson (2001), it can be shown that 110 equation (1) reduces to the ﬂow potential derived by Hill (1948) provided that f= 0and the following 111 relations are fulﬁlled: 112 ˜F0=F0 G0+H0,˜G0=G0 G0+H0,˜H0=H0 G0+H0,˜L0=L0 G0+H0,˜M0=M0 G0+H0,˜N0=N0 G0+H0 where ˜F0,˜G0,˜H0,˜L0,˜M0, and ˜N0represent the deﬁnitions of the dimensionless anisotropy coeﬃcients in 113 equation (2). Moreover, F0,G0,H0,L0,M0,N0are the anisotropy coeﬃcients introduced by Hill (1948), 114 5 which are deﬁned as: 115 2F0=1 σ2 0θ+1 σ2 0z−1 σ2 0r, 2G0=1 σ2 0z+1 σ2 0r−1 σ2 0θ, 2H0=1 σ2 0r+1 σ2 0θ−1 σ2 0z 2L0=1 σ2 0θz, 2M0=1 σ2 0zr, 2N0=1 σ2 0rθ beingσ0r,σ0θandσ0zthe initial yield stresses of the matrix material along the radial, circumferential and 116 axial directions, respectively, and σ0rθ,σ0θzandσ0zrthe corresponding initial yield stresses in shear. 117 Basedonpreviousrelationships, boththeanisotropyparameter ψ, equation(2), andtheequivalentstress 118 ˜σe, equation (3), can be given in terms of the initial yield stresses of the matrix material. Consequently, it 119 is noticed that isotropy is recovered when σ0r=σ0θ=σ0z=√ 3σ0θz=√ 3σ0zr=√ 3σ0rθ. 120 The matrix yield strength along the radial direction σyr, which enters into equation (1), is described by 121 the following power-type relation: 122 σyr=σ0r/parenleftbigg 1 +¯εp ε0/parenrightbiggn (4) beingnthe strain hardening parameter and ε0a reference strain. As in Hill (1950), we assume that the 123 strain hardening evolves proportionally for any material direction, i.e. the ratio between the current yield 124 stresses, e.g.σyz σyr, remains constant through the deformation process. Recall that σyzandσyrare the 125 matrix yield strengths along the axial and radial directions, respectively. Moreover, ¯εp=/integraltextt 0˙¯εp(τ)dτis the 126 equivalent plastic strain, where ˙¯εpis the equivalent plastic strain rate in the matrix material. According to 127 Hill’s rationale (Hill, 1950), the equivalent plastic rate ˙¯εpis the work conjugate of ¯σy(see equation (10)), 128 which is deﬁned as: 129 ¯σy=σyr/radicaltp/radicalvertex/radicalvertex/radicalbt3 2/parenleftBigg1 ˜F0+˜G0+˜H0/parenrightBigg (5) Notethat, forthesakeofsimplicity, wehavenotconsideredthetemperatureandstrainratedependences 130 of the yield strength of the matrix material. This, in turn, facilitates to derive the theoretical model for 131 the cylindrical cavity expansion problem presented in section 3. 132 133 The rate of deformation tensor is taken to be the sum of an elastic part, de, and a plastic part, dp, as 134 follows: 135 6 d=de+dp(6) where the elastic part is related to the rate of the stress by the following hypo-elastic law: 136 Oσ=C:de=C: (d−dp) (7) withOσbeing an objective stress rate and Cbeing the isotropic elastic tensor given by: 137 C=E 1 +νI/prime+E 3 (1−2ν)1⊗1 (8) whereEis the Young’s modulus, νis the Poisson’s ratio, and I/primeis the unit deviatoric fourth order tensor. 138 139 The plastic part of the rate of deformation tensor follows the direction normal to the ﬂow potential: 140 dp=˙λ∂Φ ∂σ(9) where ˙λis the non-negative plastic ﬂow proportionality factor. 141 The plastic part of the rate of deformation tensor and the equivalent plastic strain rate in the matrix 142 material are related assuming that the rate of macroscopic plastic work is equal to the rate of equivalent 143 plastic work in the matrix material: 144 σ:dp= (1−f) ¯σy˙¯εp(10) Moreover, assuming the incompressibility of the matrix material, the evolution of the void volume 145 fraction is deﬁned as: 146 ˙f= (1−f)dp:1 (11) Note that void nucleation is not considered in the present analysis. Hence, if the initial void volume 147 fraction is null, the macroscopic material is fully dense. 148 The formulation of the constitutive model is completed with the loading/unloading Kuhn-Tucker con- 149 ditions: 150 ˙λ≥0; Φ≤0; ˙λΦ = 0 (12) 7 and the consistency condition: 151 ˙λ˙Φ = 0 (13) 2.1. Specialization of the constitutive equations 152 The constitutive equations presented above are specialized to describe four particular cases that will be 153 analysed in section 4, namely: isotropic fully dense material, transversely isotropic fully dense material, 154 isotropic porous material and transversely isotropic porous material. 155 156 •Isotropic fully dense material 157 The constitutive framework presented in section 2 reduces to standard von Mises plasticity. The 158 initial material porosity f0is zero. Moreover, the initial yield stress is the same for any direction, 159 which leads to: 160 ˜F0=˜G0=˜H0=1 2, ˜L0=˜M0=˜N0=3 2 161 •Transversely isotropic fully dense material 162 The constitutive framework presented in section 2 reduces to transversely isotropic Hill plasticity, 163 with mechanical properties that are symmetric about the z−axis. The initial material porosity f0is 164 zero. Moreover, the initial yield stress is the same for any direction in the r−θplane, which leads to: 165 ˜F0=˜G0, ˜N0=˜G0+ 2˜H0 The transverse isotropy about the z−axisleads to a constant Lankford coeﬃcient for uniaxial traction 166 alonganydirectioninthe r−θplane. Thisconstantcoeﬃcientmaybecalculatedconsideringauniaxial 167 traction along the r-direction: 168 R=dp θθ dp zz=˜H0 ˜G0(14) 8 The assumption of a transversely isotropic material, in addition to represent many practical appli- 169 cations (Hill, 1950), is particularly convenient from a modeling standpoint, since an axisymmetric 170 cylindrical cavitation model can be adopted when the material is isotropic in the r−θplane. 171 Moreover, since σ0r=σ0θ, the ratio between the initial yield stresses along the axial and radial 172 directions is: 173 K=σ0z σ0r=/radicaltp/radicalvertex/radicalvertex/radicalbt˜G0+˜H0 2˜G0(15) The combination of equations (14) and (15) provide the following relations between RandK: 174 R= 2K2−1 orK2=R+ 1 2 According to Hill (1948), Ris expected to be always positive, and therefore K≥/radicalBig 1 2≈0.7071. 175 Previous relations allow to express ˜F0,˜G0,˜H0and ˜N0as a function of a single scalar parameter, R 176 orK, as follows: 177 ˜F0=˜G0=1 2K2=1 R+ 1, ˜H0=2K2−1 2K2=R R+ 1, ˜N0=4K2−1 2K2=2R+ 1 R+ 1 Moreover, for sake of simplicity, we assume that ˜L0=˜M0=3 2(the isotropic value, see discussion 178 after equation (3)). This simpliﬁcation has also been assumed by Cazacu et al. (2019) in the absence 179 of suﬃcient experimental data. At this point, it is readily noticed that, for a transversely isotropic 180 material, the main features associated to its plastic anisotropy are represented by single scalar pa- 181 rameter,RorK. Note that if K= 1(orR= 1), then ˜F0=˜G0=˜H0=1 2and ˜N0=3 2, and the 182 material follows von Mises plasticity. 183 184 •Isotropic porous material 185 The constitutive framework presented in section 2 reduces to standard Gurson plasticity. The initial 186 material porosity f0is diﬀerent from zero. The isotropy of the material leads to: 187 ˜F0=˜G0=˜H0=1 2, ˜L0=˜M0=˜N0=3 2, ψ = 2 188 9 •Transversely isotropic porous material 189 The constitutive framework presented in section 2 reduces to transversely isotropic Benzerga-Besson 190 plasticity, with mechanical properties that are symmetric about the z−axis. The initial material 191 porosityf0is not zero. As in the case of the transversely isotropic fully dense material, the plastic 192 anisotropy parameters, expressed as a function of K, are: 193 ˜F0=˜G0=1 2K2, ˜H0=2K2−1 2K2, ˜N0=4K2−1 2K2 and the parameter ψ, expressed as a function of KorR, becomes: 194 ψ=/radicaltp/radicalvertex/radicalvertex/radicalbt4 52K2(2K2+ 1) 4K2−1+6 5/parenleftBigg2K2 4K2−1+4 3/parenrightBigg =/radicalBigg 4 5(R+ 2) (R+ 1) 2R+ 1+6 5/parenleftbiggR+ 1 2R+ 1+4 3/parenrightbigg where the value assigned to the parameters ˜L0and ˜M0was3 2(isotropic value). If K= 1(orR= 1), 195 then ˜F0=˜G0=˜H0=1 2,˜N0=3 2andψ= 2, and the material follows Gurson plasticity. 196 2.2. Material parameters 197 Table 1 shows the values of the initial density and the elastic constants, and the values of the parameters 198 corresponding to the ﬂow potential and the ﬂow strength of the matrix material, which have been taken 199 from Srivastava et al. (2014) and N’souglo and Rodríguez-Martínez (2018). In section 4 we carry out a 200 parametric analysis to investigate the eﬀect of initial porosity, strain hardening and plastic anisotropy in 201 the cavitation ﬁelds. In the parametric analysis diﬀerent values of f0,nandKare used, see Table 1. Note 202 that, while the initial radial yield stress is constant in the analysis developed in section 4, the initial axial 203 yield stress σ0zchanges with K, see equation (15). 204 3. Theoretical model 205 The theoretical model developed in this paper is based on the seminal contribution of Cohen and Durban 206 (2013b)whostudiedtheproblemofasphericalcavityexpandingdynamicallyinanisotropicporousmedium. 207 The main novelty of the present paper, which considers a cylindrical cavity expanding under plane strain 208 conditions, is to take into account the plastic anisotropy of the material. Another diﬀerence with the work 209 of Cohen and Durban (2013b) is that we use an artiﬁcial viscosity approach to capture the shock waves that 210 emergeathighcavitationvelocities. Theappearanceofshocks, whichforrateindependentmaterialsleadsto 211 discontinuities in the cavitation ﬁelds, causes diﬃculties in the integration of the governing equations using 212 10 Symbol Property and units Value ρ0 Initial density (kg/m3), Eq. (39) 7600 E Young’s modulus (GPa), Eq. (8) 70 ν Poisson’s ratio, Eq. (8) 0.3 q1 Material parameter, Eq. (1) 1.25 q2 Material parameter, Eq. (1) 1.0 σ0r Initial yield stress along the radial direction (MPa), Eq. (4) 300 n Strain hardening exponent, Eq. (4) 0.1, 0.3, 0.5 ε0 Reference strain, Eq. (4) 0.00429 K Anisotropy coeﬃcient, Eq. (15) 0.75, 1, 1.5, 2 f0 Initial void volume fraction 0, 0.001, 0.01, 0.1 Table 1: Material parameters. Data after Srivastava et al. (2014) and N’souglo and Rodríguez-Martínez (2018) . standard numerical methods. While Cohen and Durban (2013b) solved this problem imposing Hugoniot 213 conditions regarding mass and momentum conservation at the shock front, in this paper we use the artiﬁcial 214 viscosity approach developed, for instance, by Wilkins (1980) and Lew et al. (2001). This approach, which 215 leads to smooth solutions of the set of diﬀerential equations which govern the problem and stands out for 216 its simplicity, was applied –and validated with ﬁnite element simulations– in the recent paper of dos Santos 217 et al. (2019) to capture the emergence of shock waves in isotropic porous materials. We are aware that there 218 are more physically sound approaches to model shocks in porous materials; e.g. considering microinertia 219 eﬀects related to the rapid collapse of the voids (Czarnota et al., 2017). Nevertheless, analysing the role 220 of microinertia in the cavitation ﬁelds of elastoplastic porous materials is out of the scope of this research, 221 and it will be addressed in a future work. 222 3.1. Steady self-similar dynamic cylindrical cavity expansion 223 The problem is modelled as a pressurized cylindrical cavity of instantaneous radius aexpanding under 224 self-similar, steady-state conditions in an inﬁnite porous medium, see Fig. 1. Let (r,θ,z )stand for the 225 Eulerian coordinates of a material point with reference to a cylindrical system with the origin located at 226 the center of the cavity denoted by Oin Fig. 1. The axial direction zis under plane strain constraint. 227 Following Durban and Fleck (1997) and Durban and Masri (2004), we take the nondimensional radial 228 coordinate ξ=r aas the only independent variable of the problem. Accordingly, the derivative with respect 229 to the radial coordinate is: 230 d() dr=d() dξdξ dr= ()/prime1 a(16) 11 Initial distribution of porosityξ and stress-free turbed zone = f0 and ρ = ρ0 Elastic zoneElastoplastic zoneξw ξi ξ=1 Oξp Figure 1: Scheme of self-similar ﬁeld in dynamic expansion of a circular cylindrical cavity in an elastoplastic porous medium. The internal pressure pis applied at the cavity wall where the dimensionless radial coordinate ξ=r a= 1. The remote region, atξ>ξw, is a strain and stress-free undisturbed zone with f=f0andρ=ρ0, while behind the elastic wave front, at ξ=ξw, an elastic range develops. The transition between the elastic and elastoplastic regions is denoted by ξi, and the location of the plastic shock wave by ξp(a plastic shock wave emerges at high cavitation velocities). This ﬁgure is adapted from Durban and Masri (2004). where the prime superscript denotes diﬀerentiation with respect to ξ. Moreover, as shown by Durban and 231 Fleck (1997), the time derivative is: 232 ˙() =d() dξ˙ξ= ()/prime˙a a(v−ξ) (17) wherev=˙r ˙ais the dimensionless radial velocity. Since ˙ais constant (steady-state expansion), the dimen- 233 sionless acceleration reads ˙v=¨r ˙a. 234 235 We assume that the axes of orthotropy of the material are aligned with the radial, circumferential and 236 axial directions, i.e. the principal directions of stress and strain coincide. Therefore, under cylindrical 237 expansion, the only active components of the stress tensor are σrr,σθθandσzz. Similarly, the only active 238 components of the rate of deformation tensor dare: 239 drr=d˙r dr=˙a av/prime(18) dθθ=˙r r=˙a av ξ(19) 12 with 240 dzz= 0 (20) due to the plane strain condition imposed in the zdirection. Note that, to obtain expressions (18) and (19), 241 equations (16) and (17) have been employed. For the problem at hand, since the principal axes are ﬁxed, 242 i.e. no rotation is included, the rate of deformation tensor coincides with the strain rate tensor. Moreover, 243 the hypo-elastic law introduced in equation (7) can be rewritten as: 244 de rr=˙σrr E−ν E( ˙σθθ+ ˙σzz) (21) de θθ=˙σθθ E−ν E( ˙σrr+ ˙σzz) (22) de zz=˙σzz E−ν E( ˙σrr+ ˙σθθ) (23) Note that, in absence of material spin, the objective derivativeO ()in equation (7) has been replaced by 245 a time derivative˙(). 246 247 The ﬂow rule introduced in equation (9), using the ﬂow potential (1), the work conjugacy relation (10), 248 and relation (5), can be rewritten as: 249 dp rr= (1−f)˙¯εp˜Nrr (24) dp θθ= (1−f)˙¯εp˜Nθθ (25) dp zz= (1−f)˙¯εp˜Nzz (26) where 250 ˜Nrr=κ[˜G0(Σrr−Σzz)+˜H0(Σrr−Σθθ)] Σyr+q1q2f ψsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig /parenleftBig˜Σe Σyr/parenrightBig2+q1q2fsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig 3Σh ψΣyr(27) 13 ˜Nθθ=κ[˜F0(Σθθ−Σzz)+˜H0(Σθθ−Σrr)] Σyr+q1q2f ψsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig /parenleftBig˜Σe Σyr/parenrightBig2+q1q2fsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig 3Σh ψΣyr(28) ˜Nzz=κ[˜F0(Σzz−Σθθ)+˜G0(Σzz−Σrr)] Σyr+q1q2f ψsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig /parenleftBig˜Σe Σyr/parenrightBig2+q1q2fsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig 3Σh ψΣyr(29) withκ=/radicalbigg 3 2/parenleftBig 1 ˜F0+˜G0+˜H0/parenrightBig . Note that for an isotropic material κ= 1. In previous expressions Σrr=σrr E, 251 Σθθ=σθθ E, and Σzz=σzz Eare the dimensionless radial, circumferential, and axial elastoplastic stresses, 252 respectively. Moreover, Σh=σh E,˜Σe=˜σe EandΣyr=σyr Eare the dimensionless mean stress, equivalent 253 stress and yield strength of the matrix material along the radial direction, respectively. 254 255 Using the expressions for the strain rates given in (18)-(20), and the additive decomposition of the rate 256 of deformation tensor (6), the following three equations are obtained for the radial, circumferential and 257 axial directions: 258 v/prime= (v−ξ)/bracketleftBigg Σ/prime rr−ν(Σ/prime θθ+ Σ/prime zz) + (1−f)∂¯εp ∂ΣyrΣ/prime yr˜Nrr/bracketrightBigg (30) v ξ= (v−ξ)/bracketleftBigg Σ/prime θθ−ν(Σ/prime zz+ Σ/prime rr) + (1−f)∂¯εp ∂ΣyrΣ/prime yr˜Nθθ/bracketrightBigg (31) 0 = (v−ξ)/bracketleftBigg Σ/prime zz−ν(Σ/prime rr+ Σ/prime θθ) + (1−f)∂¯εp ∂ΣyrΣ/prime yr˜Nzz/bracketrightBigg (32) where the elastic deformation rates given in equations (21)-(23), and the plastic deformation rates given in 259 equations (24)-(26), have been employed. To obtain expressions (30)-(32), the time derivative (17) and the 260 relation 3Σh= Σrr+ Σθθ+ Σzzhave been used. The derivative∂¯εp ∂Σyr, which is obtained from equation (4), 261 is calculated as: 262 ∂¯εp ∂Σyr=ε0 nΣ0r/parenleftBiggΣyr Σ0r/parenrightBigg1 n−1 (33) being Σ0r=σ0r Ethe dimensionless initial yield stress of the matrix material along the radial direction. 263 264 Moreover, using equations (24)-(26), the expression for the evolution of the void volume fraction given 265 in (11) can be written as follows: 266 14 ˙f= (1−f)2˙¯εp/parenleftBig˜Nrr+˜Nθθ+˜Nzz/parenrightBig (34) Using the time derivative (17) and equations (27)-(29), previous expression is rewritten as: 267 f/prime=3κ ψ(1−f)2q1q2fsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig /parenleftBig˜Σe Σyr/parenrightBig2+q1q2fsinh/parenleftBig 3q2Σh ψΣyr/parenrightBig 3Σh ψΣyr∂¯εp ∂ΣyrΣ/prime yr (35) This equation shows, as noted by Benzerga and Besson (2001), that the porosity rate depends on the 268 anisotropy of the material through the constant ψ. Ifψ<2the porosity rate is greater than for an isotropic 269 material. On the other hand, if ψ>2the porosity rate decreases in comparison with an isotropic material. 270 Benzerga and Besson (2001) also noted that, since ψappears in the hyperbolic sine, any smallvariation of 271 ψhas signiﬁcant inﬂuence in the porosity rate. 272 273 Moreover, the ratio between current material density ρand initial material density ρ0can be obtained 274 from the balance of mass which, in Eulerian description, takes the form: 275 ˙ρ+ρtr (l) = 0 (36) wherelis the spatial velocity gradient. In absence of material spin l=d, and, using equations (17)-(20), 276 previous expression becomes: 277 (v−ξ)ρ/prime ρ+v/prime+v ξ= 0 (37) Inserting equations (30)-(32) into equation (37), the following expression is obtained: 278 ρ/prime ρ=−/bracketleftBigg 3(1−2ν)Σ/prime h+f/prime (1−f)/bracketrightBigg (38) where the deﬁnition for f/primegiven in (35) has been used. Following Cohen and Durban (2013b), integrating 279 equation (38) from a given coordinate ξto the elastic wave front ξw, wheref=f0,ρ=ρ0, and Σh= 0(see 280 Fig. 1), a closed-form relation for the density ratio is obtained: 281 ρ ρ0=/parenleftBigg1−f 1−f0/parenrightBigg exp [−3(1−2ν)Σh] (39) 282 Moreover, the balance of linear momentum along the radial direction, in Eulerian description, takes the 283 15 form: 284 ∂σt rr ∂r+σt rr−σt θθ r=ρ¨r (40) whereσt rrandσt θθare the totalradial and circumferential stresses given by the sum of the elastoplastic 285 stresses and the viscousstresses, i.e. σt rr=σrr+σv rrandσt θθ=σθθ+σv θθ. On the one hand, σrrandσθθare 286 related to the elastic and plastic deformation rates by the constitutive equations presented in section 2. On 287 the other hand, σv rrandσv θθare related to the total deformation rates by means of an artiﬁcial viscosity (Lew 288 et al., 2001) to be presented in the sequel. The artiﬁcial viscosity approach is only used in the deﬁnition 289 of the stresses that enter into the balance of linear momentum. Employing equations (16)-(17), the linear 290 momentum equation (40) can be rewritten as: 291 /parenleftBig Σt rr/parenrightBig/prime+1 ξ/parenleftBig Σt rr−Σt θθ/parenrightBig =m2ρ ρ0(v−ξ)v/prime(41) where Σt rr=σt rr EandΣt θθ=σt θθ Eare dimensionless total stresses, and m=˙a√ E/ρ0is the dimensionless 292 cavitation velocity deﬁned as the ratio between the cavity expansion velocity ˙aand the elastic wave velocity 293 in a long rod/radicalBig E/ρ0. 294 Following the development presented in the recent work of dos Santos et al. (2019), which is based on 295 previous works of Wilkins (1980) and Lew et al. (2001), the dimensionless viscous stresses, Σv rr=σv rr Eand 296 Σv θθ=σv θθ E, that enter into equation (41), are calculated as: 297 Σv rr=  −ηrrm|v/prime|ifv/prime<0 0 if v/prime≥0(42) and 298 Σv θθ=  −ηθθm|v|ifv<0 0 if v≥0(43) being 299 ηrr=c2 0(∆ξ)2ρ ρ0m|v/prime|+cL∆ξ/radicalBiggρ ρ0(44) ηθθ=c2 0/parenleftBigg∆ξ ξ/parenrightBigg2ρ ρ0m|v|+cL∆ξ ξ/radicalBiggρ ρ0(45) 16 the radial and circumferential artiﬁcial viscosity parameters, respectively. In expressions (44) and (45), 300 c0≈2andcL≈1are constant parameters, and ∆ξis a given increment along the dimensionless coordinate 301 ξ. The parameters c0andcLplay an important role in the artiﬁcial viscosity method. If the values of c0 302 andcLaretoosmall, the emergence of shock waves leads to discontinuities in the elastoplastic ﬁelds, and 303 no solution is reached. If the values of c0andcLare too large, the method provides unrealistic results, 304 overestimating the actual stresses in the material. Nevertheless, the values used in this paper, c0≈2 305 andcL≈1, which are commonly used in the literature, provide reasonable results. Note that theoretical 306 results using the artiﬁcial viscosity method were compared with ﬁnite element simulations in dos Santos 307 et al. (2019), and satisfactory agreement between both approaches was found for a wide range of cavitation 308 velocities. 309 310 In this formulation, equations (1), (30), (31), (32), (35), (41) and (42), provide a system of one algebraic 311 relation and six diﬀerential equations with derivatives of seven unknowns (Σrr,Σθθ,Σzz,Σyr,Σv rr,v,f), with 312 ∂¯εp ∂Σyr,ρandΣv θθbeing calculated from the algebraic relations (33), (39) and (43). The problem is completed 313 by the following boundary conditions: 314 v= 1,Σrr=−P,atξ= 1 (46) and 315 v= 0,Σrr= Σθθ= 0,atξ=ξw (47) whereP=p Eis the dimensionless applied pressure. 316 317 The integration is performed over the independent variable ξ, from the elastic wave front ξwup to the 318 cavity wall ξ= 1. An iterative shooting method is used to enforce the velocity boundary condition at the 319 cavity wall. The initial conditions, with an unknown constant, and the location of the elastic wave front are 320 determined adapting the elastic solution derived by Masri and Durban (2009) to the constitutive framework 321 used in this paper. 322 3.2. Elastic zone solution 323 The solution of Masri and Durban (2009) is based on the fact that, within the elastic zone, where 324 the plastic strain rates are zero, the dimensionless stresses are expected to be small |Σrr|,|Σθθ|,|Σzz|/lessmuch1. 325 Thus, equations (30), (31), (32), (37) and (41) can be solved analytically to obtain the following expressions 326 17 for the radial, circumferential and axial elasticstresses: 327 Σrr=−C√1−M2ξ2 ξ2−CM2 1−2νln/parenleftBigg1 +√1−M2ξ2 Mξ/parenrightBigg +B (48) Σθθ=C√1−M2ξ2 ξ2−CM2 1−2νln/parenleftBigg1 +√1−M2ξ2 Mξ/parenrightBigg +B (49) and 328 Σzz=−2νCM2 1−2νln/parenleftBigg1 +√1−M2ξ2 Mξ/parenrightBigg + 2νB (50) whereBandCare integration constants to be determined. The parameter M=˙a/CEcan be regarded as 329 the cavity expansion Mach number (Masri and Durban, 2009), being CE=/radicalBig (1−ν)E/(1+ν)(1−2ν)ρ0the elastic 330 dilatation wave speed. 331 Moreover, thecorrespondingmaterialdensity ρandvelocity varecalculatedbythefollowingexpressions: 332 ρ ρ0= exp (−3 (1−2ν) Σh) (51) and 333 v= 2 (1 +ν)C√1−M2ξ2 ξ(52) where the approximation exp (−(1 +ν) (Σθθ−Σrr))≈1−2C(1 +ν)√ 1−M2ξ2 ξ2has been used. 334 3.2.1. Determination of the integration constants 335 The elastic wave front is located at ξ=ξw, wherev= 0. Thus, equation (52) leads to: 336 ξw=1 M=CE ˙a(53) Knowing that at the elastic wave front Σrr= 0, previous expression is inserted into equation (48) to 337 obtain: 338 B= 0 (54) 339 Moreover, the constant Cis determined imposing ﬁeld continuity conditions at the (unknown) elastic / 340 18 elastoplastic interface ξ=ξi, where equation (1) is rewritten as: 341 /parenleftBigg˜Σe Σ0r/parenrightBigg2 + 2q1f0cosh/parenleftBigg3q2Σh ψΣ0r/parenrightBigg −1−(q1f0)2= 0 (55) Using equations (48)-(50), the mean stress Σhand the equivalent stress ˜Σeare calculated as: 342 3Σh=−2(1 +ν) (1−2ν)CM2ln 1 +/radicalBig 1−M2ξ2 i Mξi  (56) ˜Σe=/radicaltp/radicalvertex/radicalvertex/radicalbtC2˜F0/parenleftBiggA ξ2 i−M2ln/parenleftBigg1 +A Mξi/parenrightBigg/parenrightBigg2 +C2˜G0/parenleftBiggA ξ2 i+M2ln/parenleftBigg1 +A Mξi/parenrightBigg/parenrightBigg2 +C2˜H0/parenleftBigg −2A ξ2 i/parenrightBigg2 (57) whereA=/radicalBig 1−M2ξ2 i. Then, equations (56) and (57) are inserted into (55), and the resulting expression is 343 solved in terms of the constant C. Within the shooting method, the elastic / elastoplastic interface location 344 is iterated until the velocity boundary condition ( v= 1) is reached at the cavity wall. 345 4. Results 346 This section of the paper is split into two parts. Subsection 4.1 shows results for fully dense materials, 347 for a wide range of dimensionless cavitation velocities 0.04≤m≤0.6. For the material parameters of 348 Table 1, this range of values of mcorresponds to dimensional cavitation velocities ranging between 0.1214 349 and1.8209Km/s. Four diﬀerent values of the anisotropy coeﬃcient, K= 0.75,1,1.5and2, are used in the 350 calculations. Recall from equation (15) that the minimum possible value ofKis/radicalBig 1/2. The strain hardening 351 exponent is n= 0.3. Subsection 4.2 shows results for porous materials with two diﬀerent initial void volume 352 fractionsf0= 0.001and0.1. The cavitation velocities and anisotropy parameters investigated are the same 353 as in subsection 4.1. Two diﬀerent values of the strain hardening exponent nare considered: 0.1and0.5. 354 The results obtained for porous and fully dense materials are compared. 355 4.1. Fully dense materials 356 Fig. 2 shows the variation in dimensionless applied pressure Pwith dimensionless cavitation velocity m 357 for four diﬀerent values of the anisotropy parameter K:0.75,1(von Mises plasticity), 1.5and2. Recall that 358 for all the calculations reported in this subsection f0= 0andn= 0.3. For the four values of Kinvestigated 359 the cavitation velocity increases with the applied pressure, the P−mcurve featuring a concave-upward 360 shape. Fig. 2 reveals that the increase of the anisotropy parameter Kshifts downwards the P−mcurve. As 361 Kincreases, less pressure is required to reach a given cavitation velocity. In other words, for a ﬁxed value 362 of the initial radial yield stress, the cavitation pressure decreases as the initial axial yield stress increases, 363 19 see equation (15). Nevertheless, the inﬂuence of the anisotropy parameter Kin theP−mcurves is small, 364 and it decreases as mincreases. For m&0.3the results obtained for diﬀerent values of Kpractically lie 365 on top of each other. 366 00.10.20.30.40.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F HDimensionless applied pressure, P Dimensionless cavitation velocity, mFully dense material, f0=0 n=0.3 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 Figure 2: Variation in dimensionless applied pressure Pwith dimensionless cavitation velocity mfor four diﬀerent values of the anisotropy parameter K:0.75,1(von Mises plasticity), 1.5and2. Fully dense material: f0= 0. The strain hardening exponent is n= 0.3. 367 Fig. 3 illustrates the radial, circumferential, axial and equivalent stresses (Σrr,Σθθ,ΣzzandΣe), versus 368 thedimensionlessradialcoordinate ξ. Resultsareshownforfourdiﬀerentvaluesoftheanisotropyparameter 369 K:0.75,1(von Mises plasticity), 1.5and2. The cavitation velocity is m= 0.1. 370 Fig. 3(a) shows that Σrris negative at any value of the radial coordinate ξ. The radial stress decreases 371 monotonically with ξ. The kinkshown by the Σrr−ξcurves atξ≈8corresponds to the interface between 372 the elastic zone (on the right) and the elastoplastic zone (on the left). Note that, for this cavitation velocity, 373 the speciﬁc location of the elastic / elastoplastic interface, from now on denoted by ξi(see Fig. 1), slightly 374 depends on K, as it will be discussed later on (see Fig. 5). The inﬂuence of the anisotropy parameter K 375 in the radial stress is greater near the cavity, where Σrrincreases as Kdecreases, in agreement with the 376 results presented in Fig. 2 ( Σrr=−Pat the cavity wall). As we move away from the cavity, the Σrr−ξ 377 curves get closer to each other until they intersect at ξ≈6. For 6.ξ≤ξithe radial stress is slightly 378 greater as the anisotropy parameter Kincreases. In the elastic range, ξi≤ξ≤ξw, the inﬂuence of Kin 379 the radial stress is negligible. Moreover, the location of the elastic wave front ξw= 11.6is independent of 380 the plastic properties of the material. 381 Fig. 3(b) shows that the circumferential stress Σθθis compressive at any value of the radial coordinate 382 ξ. As we move away from the cavity, the circumferential stress ﬁrst increases rapidly, reaches a maximum, 383 20 and then decreases more slowly. The kink located in the Σθθ−ξcurves atξ≈8corresponds to the elastic / 384 elastoplastic interface (as in Fig. 3(a)). In the elastoplastic zone the circumferential stress is slightly greater 385 asKincreases. However, in the elastic zone, the inﬂuence of Kis negligible. Note that |Σθθ|<|Σrr|at 386 any radial coordinate. 387 Fig. 3(c) shows that the axial stress Σzzis also negative for any value of the radial coordinate ξ. 388 Similarly to the circumferential stress, the axial stress ﬁrst increases rapidly, reaches a maximum, and then 389 decreases more slowly. Near the cavity the axial stress is slightly greater as Kdecreases. As we move away 390 from the cavity, the Σzz−ξcurves get gradually closer and eventually intersect at ξ≈7. Fromξ≈7until 391 the elastic / elastoplastic interface, the axial stress is slightly greater as Kincreases. In the elastic zone, 392 the inﬂuence of Kis negligible. Note that |Σθθ|<|Σzz|<|Σrr|at any radial coordinate. 393 Fig. 3(d) shows that the equivalent stress Σedecreases monotonically with the radial coordinate ξ. Note 394 that the decrease of Σeis particularly fast near the cavity. In fact, according to the asymptotic analysis 395 developed by Durban and Masri (2004), Σeshould be unbounded at the cavity wall. However, this is not 396 the case for the calculations performed in this work. It seems that, due to the numerical solution, the 397 equivalent stress has a ﬁnite value at ξ= 1(dos Santos et al., 2019). We have checked that the speciﬁc 398 value of Σeforξ= 1slightly depends on the tolerance of the shooting method used to solve the problem 399 (see section 3). Near the cavity, the equivalent stress is slightly greater as Kdecreases. The diﬀerence 400 between the curves obtained for diﬀerent values of Kis gradually reduced with ξ. The four Σe−ξcurves 401 intersect at ξ≈6.5and their order is reversed until the the elastic / elastoplastic interface is reached. In 402 the elastic zone the inﬂuence of Kin the Σe−ξcurves is small. 403 404 Fig. 4 illustrates the radial, circumferential, axial and equivalent stresses (Σrr,Σθθ,ΣzzandΣe), versus 405 the dimensionless radial coordinate ξ. The diﬀerence with Fig. 3 is that the cavitation velocity has been 406 increased to m= 0.6. 407 Fig. 4(a) shows that the radial stress is negative and decreases with ξ. Unlike in the case of m= 0.1 408 illustrated in Fig. 3(a), for m= 0.6theΣrr−ξcurves feature a concave-downward shape. The sudden drop 409 of the radial stress indicates the emergence of a shock wave at ξp≈1.65. For this value of mthe locations 410 of the shock wave and the elastic / elastoplastic interface are very close to each other ξi≈ξp, as it will 411 be shown in Fig. 5. Note that the change in the radial stress induced by the shock, while abrupt, is not 412 a discontinuity, i.e. the shock has a ﬁnite width due to the artiﬁcial viscosity included in the model. The 413 inﬂuence of the anisotropy parameter Kin the radial stress proﬁles is very small. Similar qualitative results 414 are shown in Figs. 4(b) and 4(c) for the circumferential and axial stresses. Quantitatively, the radial stress 415 21 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 1 2 3 4 5 6 7 8 9 10 11 12B D F HFully dense material, f0=0 n=0.3m=0.1 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2Dimensionless radial stress, Σrr Dimensionless radial coordinate, ξintersectionelastic wave front, ξw elastic / elastoplastic interface, ξi (a) -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 1 2 3 4 5 6 7 8 9 10 11 12B D F HFully dense material, f0=0 n=0.3 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2Dimensionless circumferential stress, Σθθ Dimensionless radial coordinate, ξm=0.1 elastic wave front, ξw elastic / elastoplastic interface, ξi (b) -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 1 2 3 4 5 6 7 8 9 10 11 12B D F HDimensionless axial stress, Σzz Dimensionless radial coordinate, ξFully dense material, f0=0 n=0.3 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 elastic wave front, ξw elastic / elastoplastic interface, ξim=0.1 intersection (c) 00.010.020.030.040.050.06 1 2 3 4 5 6 7 8 9 10 11 12B D F HDimensionless equivalent stress, Σe Dimensionless radial coordinate, ξFully dense material, f0=0 n=0.3m=0.1 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 elastic wave front, ξwintersection elastic / elastoplastic interface, ξi (d) Figure 3: Dimensionless radial stress Σrr, circumferential stress Σθθ, axial stress Σzzand equivalent stress Σeversus dimen- sionless radial coordinate ξ. Fully dense material: f0= 0. Results are shown for four diﬀerent values of the anisotropy parameterK:0.75,1(von Mises plasticity), 1.5and2. The strain hardening exponent is n= 0.3. The cavitation velocity is m= 0.1. 22 is slightly greater than the axial stress, which is in turn slightly greater than the circumferential stress. Fig. 416 4(d) shows that the equivalent stress is a decreasing function of the radial coordinate. Note that, unlike 417 the radial, circumferential and axial stress proﬁles, the Σe−ξcurves display a concave-upward shape in 418 the elastoplastic zone. Note also that in the elastoplastic zone the values of Σeare one order of magnitude 419 smaller that the values Σrr,ΣθθandΣzz, i.e. the mean stress is one order of magnitude greater than the 420 equivalent stress. In addition, the inﬂuence of Kin the equivalent stress is more important than in the 421 radial, circumferential and axial stress proﬁles, particularly, near the cavity wall. 422 Note that all the stress ﬁelds shown in the paper correspond to the elastoplastic stresses. The viscous 423 stresseslackofphysicalmeaningandtheirroleinthetheoreticalmodelislimitedtofacilitatetheintegration 424 of the governing equations at high cavitation velocities. The reader is referred to Appendix A and the work 425 of dos Santos et al. (2019) to obtain additional insights into the respective contribution of elastoplastic and 426 viscous stresses into the total stresses that enter in equation (41). 427 428 Fig. 5 shows the variation in the locations of the elastic / elastoplastic interface ξiand the shock wave 429 ξpwith the dimensionless cavitation velocity m. The location of the shock wave is taken as the radial 430 coordinate for which v/prime≤−1.4. The condition v/prime≤−1.4is chosen arbitrarily. The value −1.4is chosen 431 because it was not possible to solve the problem without artiﬁcial viscosity for v/prime≤−1.4. On the other 432 hand, the location of the elastic / elastoplastic interface is taken as the minimum radial coordinate for 433 which the plastic strains are zero. The elastic / elastoplastic interface gets closer to the cavity as the 434 cavitation velocity increases. Note that the inﬂuence of the plastic anisotropy parameter Kin the location 435 of the elastic / elastoplastic interface decreases as mincreases. The location of the shock wave is virtually 436 independent of the anisotropy parameter K. The shocks emerge, for the four values of Kinvestigated, 437 form≥0.37. As the cavitation velocity increases ξpdecreases, i.e. the shock wave becomes closer to the 438 cavity. Note that the diﬀerence between ξiandξpis small for any value of mconsidered (diﬀerences in the 439 second decimal place), and becomes smaller as mincreases. This observation agrees with the theoretical 440 and numerical results reported by Cohen et al. (2010b) who showed for porous isotropic materials that 441 increasingmwill eventually lead to ξi=ξp. 442 4.2. Porous materials 443 Fig. 6 shows the variation in dimensionless applied pressure Pwith dimensionless cavitation velocity 444 mfor four diﬀerent values of the anisotropy parameter K:0.75,1(Gurson plasticity), 1.5and2. Diﬀerent 445 values of initial void volume fraction f0and strain hardening coeﬃcient nare considered. The results for 446 the fully dense material and K= 2are also included. 447 23 -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2B D F HFully dense material, f0=0 n=0.3m=0.6 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 Dimensionless radial coordinate, ξDimensionless radial stress, Σrr shock wave, ξp (a) -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2B D F HFully dense material, f0=0 n=0.3m=0.6 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2Dimensionless circumferential stress, Σθθ Dimensionless radial coordinate, ξshock wave, ξp (b) -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2B D F H Dimensionless radial coordinate, ξFully dense material, f0=0 n=0.3m=0.6Dimensionless axial stress, Σzz K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2shock wave, ξp (c) 00.010.020.030.040.050.06 1 1.2 1.4 1.6 1.8 2B D F H Dimensionless radial coordinate, ξDimensionless equivalent stress, Σe K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2Fully dense material, f0=0 n=0.3m=0.6 shock wave, ξp (d) Figure 4: Fully dense materials: f0= 0. Dimensionless radial stress Σrr, circumferential stress Σθθ, axial stress Σzzand equivalent stress Σeversus dimensionless radial coordinate ξ. Results are shown for four diﬀerent values of the anisotropy parameterK:0.75,1(von Mises plasticity), 1.5and2. The strain hardening exponent is n= 0.3. The cavitation velocity is m= 0.6. 123456789101112 123456789101112 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H J L N PShock wave location, ξ p Fully dense material, f0=0n=0.3Elastic / elastoplastic interface, ξ iK=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 Dimensionless cavitation velocity, mK=0.75 K=1 (Von Mises plasticity) K=1.5 K=2ξi ξp Figure 5: Fully dense materials: f0= 0. Variation in the locations of elastic / elastoplastic interface ξiand shock wave ξp with dimensionless cavitation velocity m. Results are shown for four diﬀerent values of the anisotropy parameter K:0.75,1 (von Mises plasticity), 1.5and2. The strain hardening exponent is n= 0.3. 24 Fig. 6(a) shows results for f0= 0.001andn= 0.1. TheP−mcurves obtained for diﬀerent values of 448 Kpractically lie on top of each other. Moreover, attending to the results for the fully dense material and 449 K= 2, it is inferred that the inﬂuence of this initial porosity in the cavitation pressure is negligible. In 450 Fig. 6(b) the initial porosity has been increased to 0.1. The results for diﬀerent values of Kalso show no 451 meaningful diﬀerences. On the other hand, this value of initial porosity leads to a drop in the cavitation 452 pressure which becomes increasingly important as mincreases. Note that this large value of initial porosity 453 f0= 0.1leads to a meaningful decrease of the material ﬂow stress. 454 Fig. 6(c) shows results for f0= 0.001andn= 0.5. The diﬀerence with Fig. 6(a) is that the strain 455 hardening exponent nhas been increased to 0.5. The increase of nshifts upwards the P−mcurves because 456 the strain hardening exponent increases the material ﬂow stress. In addition, unlike in the case of n= 0.1, 457 the parameter Kdoes inﬂuence the P−mcurves, especially for the lowest values of mconsidered. For 458 any given cavitation velocity, the cavitation pressure increases as Kdecreases. Moreover, for K= 2, the 459 diﬀerence between the results for f0= 0andf0= 0.001is imperceptible. In Fig. 6(d) the initial porosity 460 has been increased to 0.1, keeping the strain hardening exponent nequal to 0.5. The increase of porosity 461 fromf0= 0.001tof0= 0.1has shifted downwards the P−mcurves. 462 463 Fig. 7 shows the dimensionless equivalent stress Σeversus the dimensionless radial coordinate ξfor four 464 diﬀerent values of the anisotropy parameter K:0.75,1,1.5and2. Diﬀerent values of the cavitation velocity 465 mand the strain hardening coeﬃcient nare considered. The initial void volume fraction is f0= 0.001. The 466 results for the fully dense material and K= 2are also included. 467 Fig. 7(a) presents results for m= 0.1andn= 0.1. Near the cavity the ﬂow stress is slightly greater as 468 Kdecreases, i.e. as the axial yield stress decreases. At ξ≈6.8the four curves intersect, and their order is 469 reversed. The elastic / elastoplastic interface is located at greater values of ξasKincreases. In Fig. 7(b) 470 the cavitation velocity mhas been increased to 0.6. The emergence of a shock wave is illustrated by the 471 sudden drop of Σethat occurs at ξ≈1.65. The location of the shock wave practically coincides with the 472 elastic / elastoplastic interface, as it will be shown later on in Fig. 13(a). 473 Fig. 7(c) shows results for m= 0.1andn= 0.5. In comparison with Fig. 7(a) the strain hardening 474 coeﬃcient has been increased to 0.5. Note also that the scale of the y-axis has been increased one order of 475 magnitude. The increase of the strain hardening boosts the equivalent stress at the cavity wall. Note also 476 that, in the elastoplastic zone, the variation of Kinﬂuences the equivalent stress such that, as Kdecreases, 477 the equivalent stress is slightly greater. However, in the elastic zone, the impact of the anisotropy parameter 478 is imperceptible. Fig. 7(d) shows results for m= 0.6andn= 0.5, i.e. with respect to Fig. 7(c) the 479 25 00.10.20.30.40.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H JDimensionless applied pressure, P Dimensionless cavitation velocity, mK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) Porous material, f0=0.001n=0.1 (a) 00.10.20.30.40.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H JDimensionless applied pressure, P Dimensionless cavitation velocity, mK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) Porous material, f0=0.1n=0.1 (b) 00.10.20.30.40.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H JDimensionless applied pressure, P Dimensionless cavitation velocity, mPorous material, f0=0.001n=0.5K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) (c) 00.10.20.30.40.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H JDimensionless applied pressure, P Dimensionless cavitation velocity, mK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) Porous material, f0=0.1n=0.5 (d) Figure 6: Porous materials. Variation in dimensionless applied pressure Pwith dimensionless cavitation velocity mfor four diﬀerent values of the anisotropy parameter K:0.75,1(Gurson plasticity), 1.5and2. Diﬀerent values of initial void volume fractionf0and strain hardening coeﬃcient nare considered: (a) f0= 0.001andn= 0.1, (b)f0= 0.1andn= 0.1, (c) f0= 0.001andn= 0.5and (d)f0= 0.1andn= 0.5. The results for the fully dense material and K= 2are also included. 26 cavitation velocity has been increased to 0.6. The inﬂuence of the anisotropy parameter on the equivalent 480 stress in the elastoplastic zone is important. The shock wave, which is located at similar radial coordinate 481 than the elastic / elastoplastic interface (see Fig. 13(c)), is illustrated by the sudden drop of Σelocated at 482 ξ≈1.68. In the elastic zone, the inﬂuence of Kis negligible. 483 Note that in the four plots of Fig. 7 the curves of f0= 0andf0= 0.001forK= 2practically lie on 484 top of each other, which suggests that 0.1%porosity does not aﬀect signiﬁcantly the cavitation ﬁelds, in 485 agreement with the results presented in Figs. 6(a) and 6(c). Next, we analyse the cavitation ﬁelds for a 486 greater value of initial porosity. 487 00.0050.010.015 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Dimensionless radial coordinate, ξDimensionless equivalent stress, Σ e Porous material, f0=0.001n=0.1 m=0.1K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) intersectionelastic / elastoplastic interface, ξi elastic wave front, ξw (a) 00.0050.010.015 1 1.2 1.4 1.6 1.8 2B D F H JPorous material, f0=0.001n=0.1 m=0.6 Dimensionless radial coordinate, ξDimensionless equivalent stress, Σ e K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) shock wave, ξpelastic wave front, ξw (b) 00.050.10.15 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Dimensionless radial coordinate, ξDimensionless equivalent stress, Σ e Porous material, f0=0.001n=0.5 m=0.1K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) elastic / elastoplastic interface, ξielastic wave front, ξw (c) 00.050.10.15 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξDimensionless equivalent stress, ΣeK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)Porous material, f0=0.001n=0.5 m=0.6 elastic wave front, ξw shock wave, ξp (d) Figure 7: Porous materials: f0= 0.001. Dimensionless equivalent stress Σeversus dimensionless radial coordinate ξfor four diﬀerent values of the anisotropy parameter K:0.75,1(Gurson plasticity), 1.5and2. Diﬀerent values of cavitation velocity mand strain hardening coeﬃcient nare considered: (a) m= 0.1andn= 0.1, (b)m= 0.6andn= 0.1, (c)m= 0.1and n= 0.5and (d)m= 0.6andn= 0.5. The results for the fully dense material and K= 2are also included. 488 Fig. 8 shows the radial, circumferential, axial and equivalent stresses versus the dimensionless radial 489 coordinate. Calculations with four diﬀerent values of Kare presented. Both the strain hardening exponent 490 27 nand the cavitation velocity mare0.1. The initial void volume fraction is 10%. The results for the fully 491 dense material and K= 2are also presented. 492 The comparison between the fully dense material and the porous ones reveals that considering 10%of 493 initial porosity in the calculations has an important impact on the predicted stress ﬁelds. For instance, 494 Fig. 8(a) shows that the radial stress for the fully dense material is greater than for the porous materials 495 for any value of ξ. Note also that for the fully dense material the radial stress decreases smoothly with ξ. 496 However, the Σrr−ξcurves for the porous materials show a steep decline of the radial stress at ξ≈2.8. 497 Forξ <2.8the radial stress is greater as Kdecreases, and vice versa for ξ >2.8. Fig. 8(b) illustrates that 498 the circumferential stress is also signiﬁcantly greater for the fully dense material. Unlike in Fig. 8(a), the 499 Σθθ−ξcurve forK= 2is above the curves for K= 1.5,1and0.75for any radial coordinate. Fig. 8(c) 500 shows the proﬁles of axial stress, which are similar, both quantitatively and qualitatively, to the proﬁles of 501 radial stress. For the porous materials the Σzz−ξcurves also show a steep decline ofΣzzatξ≈2.8such 502 that forξ <2.8the axial stress is slightly greater as Kdecreases, and vice versa for ξ >2.8. The equivalent 503 stress proﬁles are shown in Fig. 8(d). Note that, for the sake of clarity, the scale of the y-axis has been 504 reduced in comparison with Figs. 8(a)-(b)-(c). Near the cavity, the equivalent stress is 2to4times smaller 505 than the radial, circumferential and axial stresses, i.e. the mean stress is greater than the equivalent stress. 506 For small values of the radial coordinate, the equivalent stress is smaller for the fully dense material than 507 for the porous materials. The Σe−ξcurve of the fully dense material intersects the Σe−ξcurves of the 508 porous materials at ξ≈2. The speciﬁc location of the intersection depends on the parameter K. For 509 greater values of ξ, the equivalent stress for the fully dense material is signiﬁcantly greater. Note also that, 510 while the equivalent stress decreases monotonically with the radial coordinate for the fully dense material, 511 this is not the case if f0= 0.1. For the porous materials the equivalent stress ﬁrst decreases, reaches a 512 minimum at ξ≈2.8(the radial coordinate corresponding to the steep decline of the stress in Figs. 8(a) 513 and 8(c)), and then slightly increases until a kink appears in the Σe−ξcurves. The minimum indicates 514 the incipient formation of a shock wave (a small increase in mwill lead to the emergence of a shock wave, 515 see Fig. 13(b)) and its location is practically independent of K. The kink corresponds to the elastic / 516 elastoplastic interface, and its location depends on the value of K, as further discussed in Fig. 13(b). To 517 the left of the kink, the equivalent stress is slightly greater as Kdecreases. On the contrary, in the elastic 518 zone, to the right of the kink, the equivalent stress in greater as Kincreases. 519 520 Fig. 9 illustrates the variation of the void volume fraction fand the dimensionless material density 521 ρ/ρ 0with the dimensionless radial coordinate ξ. Calculations with four diﬀerent values of Kare presented: 522 28 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Dimensionless radial coordinate, ξDimensionless radial stress, Σrr Porous material, f0=0.1n=0.1 m=0.1K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) intersection elastic wave front, ξw (a) -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Porous material, f0=0.1n=0.1 m=0.1 Dimensionless radial coordinate, ξDimensionless circumferential stress, Σ θθ K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) positive values of Σθθ (b) -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Dimensionless radial coordinate, ξPorous material, f0=0.1n=0.1 m=0.1K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)Dimensionless axial stress, Σzz intersection (c) 00.0020.0040.0060.0080.01 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Dimensionless radial coordinate, ξPorous material, f0=0.1n=0.1 m=0.1K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)Dimensionless equivalent stress, Σe elastic wave front, ξw elastic / elastoplastic interface, ξi (d) Figure 8: Porous materials: f0= 0.1. Dimensionless radial stress Σrr, circumferential stress Σθθ, axial stress Σzzand equivalent stress Σeversus dimensionless radial coordinate ξ. Results are shown for four diﬀerent values of the anisotropy parameterK:0.75,1(Gurson plasticity), 1.5and2. The strain hardening exponent is n= 0.1. The cavitation velocity is m= 0.1. The results for the fully dense material and K= 2are also included. 29 0.75,1(Gurson plasticity), 1.5and2. The corresponding values of ψare:2.052,2,2.18and2.466. Recall 523 that the porosity rate depends on the anisotropy through the constant ψ. As in Fig. 8 the strain hardening 524 exponentn, the cavitation velocity m, and the initial void volume fraction are 0.1. The results for the fully 525 dense material and K= 2are also included in the graph which shows the density proﬁles. 526 Fig. 9(a) shows that the f−ξcurves feature a sigmoidal shape. Near the cavity wall, where large 527 compressive stresses appear, the porosity is zero. The porosity takes positive values for a value of the 528 radial coordinate that depends on Ksuch that the greater the value of ψ, the smaller the value of ξfor 529 whichf > 0. Note that the f−ξcurves intersect at ξ≈2.86, which is the radial coordinate for which 530 the shape of the curves turns from concave-upward to concave-downward. For ξ < 2.86the porosity is 531 greater asψincreases, and vice versa for ξ >2.86. The porosity reaches the initial value f0= 0.1at the 532 elastic / elastoplastic interface. Fig. 9(b) shows that the dimensionless material density ρ/ρ 0decreases 533 with the radial coordinate. For the porous materials, the ρ/ρ 0−ξcurves feature a concave-downward shape 534 forξ <2.86, and a concave-upward shape for ξ >2.86. Note that for ξ <2.86the density is greater as 535 Kdecreases, and vice versa for ξ >2.86. Unlike in the case of the porous materials, for the fully dense 536 material and K= 2, theρ/ρ 0−ξcurve features a concave-upward shape from the cavity wall until the 537 elastic wave front. Near the cavity, for ξ <2.86, the density in the fully dense material is smaller than in 538 the porous materials. The plastic compressibility of the porous materials is responsible for the increased 539 density close to the cavity. However, at ξ≈2.86theρ/ρ 0−ξcurves for the porous and the fully dense 540 material intersect such that for ξ >2.86the density in the fully dense material is greater. These results 541 reveal that considering 10%of initial porosity in the calculations has an important impact on the predicted 542 density ﬁelds. We are aware that most metals and alloys display values of initial porosity smaller than 543 10%. However, considering a wide range of values of f0helps to determine the eﬀect of porosity in the 544 elastoplastic ﬁelds and in the formation of shock waves. 545 546 Fig. 10 illustrates the variation of the radial, circumferential, axial and equivalent stresses with the 547 dimensionless radial coordinate. Calculations with four diﬀerent values of Kare presented. The strain 548 hardening exponent is n= 0.1, and the initial void volume fraction is 10%. The results for the fully dense 549 material and K= 2are also included. The diﬀerence with the results presented in Fig. 8 is that the 550 cavitation velocity mhas been increased to 0.6. 551 Figs. 10(a)-(b)-(c) show that the proﬁles of radial, circumferential and axial stress for the porous 552 materials are very similar, qualitatively and quantitatively, and practically independent of the anisotropy 553 parameterK. Thevalueof Σrr,ΣθθandΣzzatthecavitywall is ≈−0.42. Aswemoveawayfromthecavity, 554 30 00.020.040.060.080.10.12 1 2 3 4 5 6 7 8 9 10 11 12B D F HPorous material, f0=0.1 n=0.1 m=0.1Void volume fraction, f Dimensionless radial coordinate, ξK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 00.020.040.060.080.10.12 1.5 2 2.5 3 3.5f 0 intersection zoom (a) 11.051.11.15 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Dimensionless radial coordinate, ξDimensionless material density, ρ/ρ0 Porous material, f0=0.1n=0.1 m=0.1 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) elastic wave front, ξwintersection (b) Figure 9: Porous materials: f0= 0.1. Void volume fraction fand dimensionless material density ρ/ρ0versus dimensionless radial coordinate ξ. Results are shown for four diﬀerent values of the anisotropy parameter K:0.75,1(Gurson plasticity), 1.5and2. The strain hardening exponent is n= 0.1. The cavitation velocity is m= 0.1. In the plot which shows the dimensionless material density ρ/ρ0, results for the fully dense material and K= 2are also included. the stresses decrease smoothly , featuring a concave-downward shape. At ξ≈1.54there is a sudden drop 555 of the stresses which indicates the emergence of a shock wave. The location of the shock wave practically 556 coincides with the location of the elastic / elastoplastic interface, i.e. ξp≈ξi, as it will shown later in Fig. 557 13(b). The comparison of the results for the porous materials with the fully dense material show that, in the 558 latter case, the radial, circumferential and axial stresses are signiﬁcantly higher. Note also that for the fully 559 dense material the drop of stress that indicates the shock wave occurs at a greater value of ξ. Moreover, 560 Fig. 10(d) shows that the evolution of the equivalent stress for the porous materials also depends very little 561 on the parameter K. Note that the equivalent stress is ≈50times smaller than the radial, circumferential 562 and axial stresses, i.e. the mean stress is ≈50times greater than the equivalent stress. Moreover, unlike 563 in the case of the fully dense material, the equivalent stress for the porous materials is not a decreasing 564 function of the radial coordinate. The development of the shock wave leads to a sudden decrease of Σe 565 that partially bounces back until reaching a local maximum that coincides with the elastic / elastoplastic 566 interface (i.e. ξp≈ξias mentioned before). Note that the equivalent stress for the fully dense material is 567 below the equivalent stress for the porous materials until, in the latter case, the shock wave emerges. 568 569 Fig. 11 shows the void volume fraction fand the dimensionless density ρ/ρ 0versus the dimensionless 570 radial coordinate ξ. Calculations with four diﬀerent values of Kare presented. The dimensionless density 571 ρ/ρ 0for the porous materials is compared with the fully dense material and K= 2. As in Fig. 10, the 572 strain hardening exponent nand the initial void volume fraction are 0.1. The diﬀerence with the results 573 presented in Fig. 9 is that the cavitation velocity mhas been increased to 0.6. 574 31 -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξDimensionless radial stress, Σrr Porous material, f0=0.1n=0.1 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)m=0.6 elastic wave front, ξwshock wave, ξp (a) -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)Porous material, f0=0.1n=0.1 m=0.6Dimensionless circumferential stress, Σθθ elastic wave front, ξwshock wave, ξp (b) -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξPorous material, f0=0.1n=0.1 m=0.6 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)Dimensionless axial stress, Σzz elastic wave front, ξwshock wave, ξp (c) 00.0020.0040.0060.0080.01 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξDimensionless equivalent stress, Σe Porous material, f0=0.1n=0.1 m=0.6 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)elastic wave front, ξwshock wave, ξpelastic / elastoplastic interface, ξi (d) Figure 10: Porous materials: f0= 0.1. Dimensionless radial stress Σrr, circumferential stress Σθθ, axial stress Σzzand equivalent stress Σeversus dimensionless radial coordinate ξ. Results are shown for four diﬀerent values of the anisotropy parameterK:0.75,1(Gurson plasticity), 1.5and2. The strain hardening exponent is n= 0.1. The cavitation velocity is m= 0.6. The results for the fully dense material and K= 2are also included. 32 Fig. 11(a) shows that from the cavity wall, until the shock wave location, the porosity is zero. Within 575 the shock the porosity increases rapidly and reaches the initial value f0= 0.1at the elastic / elastoplastic 576 interface. Within the elastic range, for ξ > ξi, the value of fis constant. Fig. 11(b) shows that, for 577 the porous materials, the inﬂuence of Kon the dimensionless density is negligible. The density decreases 578 smoothly as we move away from the cavity wall featuring a concave-downward shape until the shock wave 579 location is reached. The shock leads to a sudden drop of the density. Moreover, the comparison between 580 the results for f0= 0.1andf0= 0shows that, for the latter case, the density is 10%smaller near the cavity 581 and the shock wave is located at a greater value of ξ(as mentioned before). 582 00.020.040.060.080.10.12 1 1.2 1.4 1.6 1.8 2B D F HK=0.75 K=1 (Gurson plasticity) K=1.5 K=2Porous material, f0=0.1 n=0.1 m=0.6Void volume fraction, f Dimensionless radial coordinate, ξshock wave, ξpelastic / elastoplastic interface wave, ξi (a) 11.21.41.61.82 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)Porous material, f0=0.1n=0.1 m=0.6Dimensionless material density, ρ/ρ0 shock wave, ξp (porous materials) elastic wave front, ξwshock wave, ξp (fully dense material) (b) Figure 11: Porous materials: f0= 0.1. Void volume fraction fand dimensionless material density ρ/ρ0versus dimensionless radial coordinate ξ. Results are shown for four diﬀerent values of the anisotropy parameter K:0.75,1(Gurson plasticity), 1.5and2. The strain hardening exponent is n= 0.1. The cavitation velocity is m= 0.6. In the plot which shows the dimensionless material density ρ/ρ0, results for the fully dense material and K= 2are also included. 583 Fig. 12 shows, for porous materials, the dimensionless equivalent stress Σe, void volume fraction fand 584 dimensionless density ρ/ρ 0versus the dimensionless radial coordinate ξ. The initial void volume fraction 585 isf0= 0.1. As in previous graphs, calculations for K= 0.75,1,1.5and2are presented. Recall that the 586 corresponding values of ψare: 2.052,2,2.18and2.466. In Figs. 12(a)-(c)-(e) the cavitation velocity mis 587 0.1and in (b)-(d)-(f) is 0.6. The diﬀerence with the results presented in Figs. 8, 9, 10 and 11 is that the 588 strain hardening has been increased to n= 0.5. 589 Fig. 12(a) shows that the equivalent stress for m= 0.1decreases monotonically with ξ. Note that this 590 was not the case in Fig. 8(d) where the strain hardening exponent nwas0.1. Moreover, unlike in Fig. 8(d), 591 theΣe−ξcurves for the porous materials and the fully dense material are very similar. In addition, the 592 inﬂuence of plastic anisotropy in the Σe−ξcurves for the porous materials is small. In contrast, Fig. 12(b) 593 shows that the eﬀect of porosity and plastic anisotropy in the equivalent stress is important for m= 0.6. 594 33 For this cavitation velocity, the Σe−ξcurve for the fully dense material is below the curves for the porous 595 materials until the shock wave emerges for the latter at ξp≈1.55. Note that, unlike in Fig. 10(d), the 596 equivalent stress is monotonic in the shock region. Fig. 12(b) also shows that, for the porous materials and 597 ξ <ξp, the equivalent stress is smaller as Kincreases. The inﬂuence of the anisotropy parameter Kin the 598 Σe−ξcurves is greater in Fig. 12(b) than in Fig. 10(d) where the strain hardening exponent was 0.1. 599 Fig. 12(c) shows that, for m= 0.1, the porosity decreases nonlinearly as the cavity is approached. Note 600 that, at the cavity wall, while the porosity is small, it is not zero. Near the cavity the porosity is slightly 601 smaller as the parameter ψdecreases. The porosity proﬁles depicted in Fig. 12(d) for m= 0.6show that 602 f= 0ifξ <ξpandf= 0.1ifξ >ξp. Within the shock wave there is an abrupt variation of the porosity. 603 Fig. 12(e) shows the density proﬁles for m= 0.1. For the porous materials, the ratio ρ/ρ 0ﬁrst increases 604 withξnear the cavity wall, reaches a maximum and then decreases smoothly. The inﬂuence of Kin the 605 density proﬁles is small. For ξ <3.90the density of the porous materials is greater than the density of the 606 fully dense material. The opposite behaviour occurs for ξ >3.90. Fig. 12(f) shows that the density proﬁles 607 for the porous materials and m= 0.6are hardly dependent on the anisotropy parameter K. The density 608 for the fully dense material is ≈10%smaller than the density for the porous materials until the shock wave 609 emerges for the latter. Note that similar qualitative and quantitative results were shown for n= 0.1in Fig. 610 11(b). 611 612 The inﬂuence of initial porosity and strain hardening exponent on the locations of the elastic / elasto- 613 plastic interface and the shock wave is analysed in Fig. 13 which shows the variations of ξiandξpwith the 614 dimensionless cavitation velocity mfor: (a)n= 0.1andf0= 0.001, (b)n= 0.1andf0= 0.1, (c)n= 0.5 615 andf0= 0.001, (d)n= 0.5andf0= 0.1. 616 Fig. 13 shows that, irrespective of the values of f0andn, the elastic / elastoplastic interface and the 617 shock wave are located closer to the cavity as mincreases. In addition, we have that ξi≈ξpfor the 618 highest cavitation velocities investigated. Moreover, note that the anisotropy parameter Khardly aﬀects 619 the shock wave location. On the other hand, the elastic / elastoplastic interface is located further from 620 the cavity as Kincreases. The inﬂuence of Konξiis more important as mdecreases. These results ﬁnd 621 qualitative agreement with the data reported in Fig. 5 for fully dense materials. The comparison between 622 Figs. 13(a) and 13(b), and between Figs. 13(c) and 13(d), shows that the increase of the initial porosity 623 brings the elastic / elastoplastic interface closer to the cavity, boosts the inﬂuence of Konξi, and favours 624 the emergence of shock waves at lower cavitation velocities. Note that the increase of the strain hardening 625 exponentnhas the opposite eﬀect. 626 34 00.050.10.15 1 2 3 4 5 6 7 8 9 10 11 12B D F H J Dimensionless radial coordinate, ξK=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) Porous material, f0=0.1n=0.5 m=0.1Dimensionless equivalent stress, Σe elastic / elastoplastic interface, ξi (a) 00.050.10.15 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξPorous material, f0=0.1n=0.5 m=0.6 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)Dimensionless equivalent stress, Σe shock wave, ξp (porous materials)shock wave, ξp (fully dense material) (b) 00.020.040.060.080.10.12 1 2 3 4 5 6 7 8 9 10 11 12B D F HVoid volume fraction, f Dimensionless radial coordinate, ξPorous material, f0=0.1 n=0.5 m=0.1 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2elastic / elastoplastic interface, ξi 00.020.040.060.080.10.12 1 1.5 2 2.5 3zoom (c) 00.020.040.060.080.10.12 1 1.2 1.4 1.6 1.8 2B D F HVoid volume fraction, f Dimensionless radial coordinate, ξPorous material, f0=0.1 n=0.5 m=0.6 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2shock wave, ξpelastic / elastoplastic interface wave, ξi (d) 11.051.11.15 1 2 3 4 5 6 7 8 9 10 11 12B D F H JPorous material, f0=0.1 n=0.5 m=0.1K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0) Dimensionless radial coordinate, ξDimensionless material density, ρ/ρ 0 11.021.041.061.081.11.121.14 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4zoom (e) 11.21.41.61.82 1 1.2 1.4 1.6 1.8 2B D F H J Dimensionless radial coordinate, ξDimensionless material density, ρ/ρ0 Porous material, f0=0.1 n=0.5 m=0.6 K=0.75 K=1 (Gurson plasticity) K=1.5 K=2 K=2 (fully dense material f0=0)shock wave, ξp (porous materials)shock wave, ξp (fully dense material) (f) Figure 12: Porous materials: f0= 0.1. Dimensionless equivalent stress Σe, void volume fraction fand dimensionless material densityρ/ρ0versus dimensionless radial coordinate ξ. Results are shown for four diﬀerent values of the anisotropy parameter K:0.75,1(Gurson plasticity), 1.5and2. The strain hardening exponent is n= 0.5. In Figs. 12(a)-(c)-(e) the cavitation velocity ism= 0.1. In Figs. 12(b)-(d)-(f) the cavitation velocity is m= 0.6. In the plots which show the dimensionless equivalent stress Σeand the dimensionless material density ρ/ρ0, results for the fully dense material and K= 2are also included. 35 24681012 24681012 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H J L N PShock wave location, ξpElastic / elastoplastic interface, ξi Dimensionless cavitation velocity, mK=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2ξi ξp Porous material, f0=0.001n=0.1 (a) 24681012 24681012 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H J L N PK=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2ξi ξpShock wave location, ξpElastic / elastoplastic interface, ξi Dimensionless cavitation velocity, mPorous material, f0=0.1n=0.1 (b) 24681012 24681012 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H J L N PShock wave location, ξpElastic / elastoplastic interface, ξi Dimensionless cavitation velocity, mK=0.75 K=1 (Von Mises plasticity) K=1.5 K=2 K=0.75 K=1 (Von Mises plasticity) K=1.5 K=2ξi ξp Porous material, f0=0.001n=0.5 (c) 24681012 24681012 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7B D F H J L N PK=0.75 K=1 (Von Mises plasticity) K=1.5 K=2ξiShock wave location, ξpElastic / elastoplastic interface, ξi Dimensionless cavitation velocity, mK=0.75 K=1 (Von Mises plasticity) K=1.5 K=2ξp Porous material, f0=0.1n=0.5 (d) Figure 13: Porous materials. Variation in the locations of elastic / elastoplastic interface ξiand shock wave ξpwith dimen- sionless cavitation velocity m. Results are shown for four diﬀerent values of the anisotropy parameter K:0.75,1(Gurson plasticity), 1.5and2. Diﬀerent values of the strain hardening exponent and the initial void volume fraction are considered: (a)n= 0.1andf0= 0.001, (b)n= 0.1andf0= 0.1, (c)n= 0.5andf0= 0.001, (d)n= 0.5andf0= 0.1. 36 627 Finally, we apply the cylindrical cavity expansion model to assess the ballistic performance of porous 628 anisotropic targets against high velocity perforation. For that purpose, we obtain the Recht-Ipson curves 629 following the procedure reported by Cohen et al. (2010a). According to Recht and Ipson (1963) and Ipson 630 and Recht (1975), the residual velocity vrcan be calculated as: 631 vr=/radicalBig v2 0−v2 bwithvb=/radicaltp/radicalvertex/radicalvertex/radicalbt2πr2 ph mpσrs (58) wherev0andvbare the impact and ballistic limit velocities, rpandmpare the radius and mass of the 632 projectile,his the target thickness, and σrsis an eﬀective resisting stress. Equation (58) assumes that the 633 projectile impacting the target is rigid and has circular cross-section. In addition, equation (58) disregards 634 friction eﬀects between projectile and target, and does not account for target fragmentation and shear plug 635 formation. 636 We simulate the experiments of Rosenberg and Forrestal (1988) who launched conical nose projectiles 637 with massmp= 0.024 Kg, diameterdp= 7.1 mm, and apical angle 2ϕ≈36.73◦, against aluminum plates 638 withthickness h= 25.4 mm. Theimpactvelocityisrelatedtothecavityexpansionvelocityinthetheoretical 639 model byv0=˙a/tanϕ. The eﬀective resisting stress is taken to be the quasi-static cavitation pressure ps640 form= 0.04. Fig. 14 shows the variation of the residual velocity vrwith the impact velocity v0predicted 641 by the cylindrical cavity expansion model in conjunction with equation (58) for two diﬀerent values of the 642 anisotropy parameter K= 0.75and2, and two diﬀerent material porosities f0= 0(fully dense material) 643 and0.1. Results for two diﬀerent strain hardening exponents n= 0.1and0.5are shown in Figs. 14(a) and 644 (b), respectively. The residual velocity increases with the impact velocity, non-linearly close to the ballistic 645 limit, and quasi-linearly for greater impact velocities. The residual velocity is greater for K= 2than for 646 K= 0.75, for both strain hardening exponents considered, which shows the inﬂuence of anisotropy on the 647 ballistic performance of the target. Note that the inﬂuence of Kin the results increases with the value of n. 648 These observations are consistent with the cavitation pressure results for m= 0.04reported in Fig. 6, which 649 showed that Pwas greater for K= 0.75than forK= 2, with increasing diﬀerences as nincreases. Note 650 also that the porosity increases the residual velocity, and thus decreases the energy absorption capacity 651 of the target. On the other hand, the comparison between Figs. 14(a) and (b) shows that the strain 652 hardening exponent decreases the residual velocity and increases the ballistic limit. The speciﬁc values of 653 the ballistic limit for the calculations presented in Fig. 14 are given in Table 2. The theoretical Recht-Ipson 654 curves are also compared with the experimental data reported by Rosenberg and Forrestal (1988). Despite 655 the constitutive model used in this paper was not calibrated to describe the mechanical behavior of the 656 37 aluminum targets tested in Rosenberg and Forrestal (1988), the theoretical results for n= 0.1are in good 657 agreement with the experiments. 658 00.511.52 0 0.5 1 1.5 2 2.5 3B D F H JResidual velocity, Vr (Km/s) Impact velocity, V 0 (Km/s)K=0.75 K=2 K=2 (fully dense material f0=0) Experimental data: Rosenberg and Forrestal (1988) - Conical projectile Experimental data: Rosenberg and Forrestal (1988) - Arrowhead projectile zoomPorous material, f0=0.1n=0.1 00.51 0 0.5 (a) 00.511.52 0 0.5 1 1.5 2 2.5 3f0=0.1 - n=0.5 - K=0.75 f0=0.1 - n=0.5 - K=2 f0=0 - n=0.5 - K=0.75 H JResidual velocity, Vr (Km/s) Impact velocity, V 0 (Km/s)zoomPorous material, f0=0.1n=0.5K=0.75 K=2 K=2 (fully dense material f0=0) Experimental data: Rosenberg and Forrestal (1988) - Conical projectile Experimental data: Rosenberg and Forrestal (1988) - Arrowhead projectile 01 0.5 1 (b) Figure 14: Porous materials: f0= 0.1. Residual velocity vrversus impact velocity v0(Recht-Ipson curves). Results are shown for two diﬀerent values of the anisotropy parameter K:0.75and2. Two values of the strain hardening exponent are considered: (a) n= 0.1and (b)n= 0.5. Results for the fully dense material and K= 2are also included. The theoretical results are compared with the experimental data reported by Rosenberg and Forrestal (1988). Anisotropy coeﬃcient, KInitial porosity, f0Strain hardening exponent nBallistic limit, vb(Km/s) 0.75 0 .1 0 .1 0 .3303 2 0 .1 0 .1 0 .3035 2 0 0 .1 0 .3374 0.75 0 .1 0 .5 0 .6449 2 0 .1 0 .5 0 .5240 2 0 0 .5 0 .5630 Table 2: Ballistic limit velocities vbfor diﬀerent values of the anisotropy coeﬃcient K, the initial porosity f0and the strain hardening exponent n. These results are obtained using equation (58), in conjunction with the cylindrical cavity expansion model. The impact velocity is calculated using the cavity expansion velocity in the theoretical model v0=˙a/tanϕ. The eﬀective resisting stress σrsis taken as the quasi-static cavitation pressure psin the theoetical model for m= 0.04. 5. Summary and concluding remarks 659 In this paper we have developed a theoretical model to study the self-similar elastoplastic ﬁelds induced 660 by a cylindrical cavity expanding dynamically in a rate-independent porous medium. The main novelties of 661 the model with respect to the work of Cohen and Durban (2013a) are that it takes into account the plastic 662 38 anisotropy of the material, and it uses viscous damping to capture the shock waves that emerge at high 663 cavitation velocities. The viscous damping is introduced using the artiﬁcial viscosity approach proposed 664 by Lew et al. (2001). The plastic anisotropy is included using the constitutive framework developed by 665 Benzerga and Besson (2001) to model the porous medium. We have investigated materials with several 666 degrees of anisotropy, K= 0.75,1.5and2, and diﬀerent values of initial porosity, f0= 0.001,0.01and 667 0.1. The special cases for which the material is fully dense, f0= 0, and/or isotropic, K= 1, have also 668 been considered. Diﬀerent strain hardening exponents have been used in the calculations: n= 0.1,0.3 669 and0.5. In addition, a wide range of cavitation velocities has been investigated 0.04≤m≤0.6, and the 670 inﬂuence of material porosity and anisotropy on the emergence and development of shock waves has been 671 analysed. Finally, the cylindrical cavity expansion model has been used to calculate the Recht-Ipson curves 672 and assess the ballistic performance of porous anisotropic targets against high velocity perforation. The 673 main outcomes of the research are summarized below: 674 •The cavitation pressure Pincreases as the parameter Kdecreases, i.e. as the axial yield stress 675 decreases. The eﬀect of anisotropy on the cavitation pressure is greater for low cavitation velocities 676 and it becomes more important as the material strain hardening nincreases. The strain hardening 677 also boosts the cavitation pressure. 678 •The formation and location of shock waves is practically independent of the anisotropy parameter 679 K. In contrast, the initial porosity is shown to promote the emergence of shocks for lower cavitation 680 velocities. On the contrary, the increase of the material strain hardening delays the shocks formation. 681 Irrespective of the values of K,f0andn, as the cavitation velocity increases, the shock forms closer 682 to the cavity. 683 •The elastic / elastoplastic interface is closer to the cavity wall as the cavitation velocity mincreases. 684 At high cavitation velocities, the interface and the shock wave are located at similar radial coordinate. 685 The decrease of the anisotropy parameter Kand the increase of the initial porosity f0also move the 686 elastic / elastoplastic interface closer to the cavity. The eﬀect of anisotropy is particularly important 687 at low cavitation velocities. On the contrary, the increase of the strain hardening parameter nreduces 688 the eﬀect of Kon the interface location, which is situated further from the cavity as nincreases. 689 •The radial, circumferential and axial stresses, Σrr,ΣθθandΣzz, are of the same order of magnitude 690 than the equivalent stress Σefor low cavitation velocities. However, the mean stress Σhincreases sig- 691 niﬁcantlywith msuchthat, forthegreatestcavitationvelocitytested m= 0.6, withintheelastoplastic 692 zone, Σhcan be up to ﬁfty times greater than Σefor low values of the strain hardening exponent. 693 39 While Σrr,ΣθθandΣzzare greater for the fully dense material than for the porous materials, the 694 opposite behaviour is observed for the equivalent stress Σenear the cavity wall. 695 •The increase of the cavitation velocity mdecreases the inﬂuence of the plastic anisotropy in the radial, 696 circumferential and axial stresses. On the contrary, the anisotropy parameter Kplays a meaningful 697 roleintheequivalentstressforanyvalueof mconsidered. Thespeciﬁcinﬂuenceof Kontheequivalent 698 stress depends on the radial coordinate ξ, the strain hardening nand the initial porosity f0. 699 •The porosity fdecreases as the cavity is approached. For low cavitation velocities the evolution of 700 fwith the radial coordinate is smooth, generally being the porosity in the material greater as ψ 701 increases. The increase of the strain hardening slows down the porosity decrease. For high cavitation 702 velocities, the variation of porosity is localized within the shock wave such that for ξ <ξpthe porosity 703 is zero and for ξ >ξpthe porosity is f0. 704 •The material density, near the cavity wall, is greater for the porous materials than for the fully 705 dense. This behavior is attributed to the plastic compressibility of the porous materials. Far from the 706 cavity, the opposite behaviour is observed. For the porous materials, the inﬂuence of the anisotropy 707 parameterKin the density ﬁelds decreases as the cavitation velocity increases. For low values of m, 708 near the cavity, the density is smaller as Kincreases. Far from the cavity, the trend is reversed. 709 •The Recht-Ipson curves show that the residual velocity is greater for K= 2than forK= 0.75, 710 which revels the inﬂuence of anisotropy on the ballistic performance of metallic targets. For n= 0.1, 711 decreasing Kfrom 2to0.75increases the ballistic limit ≈8%. Forn= 0.5, the increase of the 712 ballistic limit velocity is signiﬁcantly greater ≈23%. Moreover, increasing the initial porosity f0713 from 0(fully dense material) to 0.1increases the residual velocity, and thus decreases the energy 714 absorption capacity of the target. The increase of the strain hardening exponent nfrom 0.1to0.5 715 has the opposite eﬀect. 716 Acknowledgements 717 The research leading to these results has received funding from the European Research Council (ERC) 718 under the European Union’s Horizon 2020 research and innovation programme. Project PURPOSE, grant 719 agreement 758056. 720 40 Appendix A. The respective contribution of elastoplastic and viscous stresses to total stresses 721 Fig. A.15showsthetotal, elastoplasticandviscousradialstresses (Σt rr,Σrr,Σv rr)versusthedimensionless 722 radial coordinate ξ, form= 0.60,f0= 0.1,K= 1, and two values of the strain hardening exponent, n= 0.1 723 andn= 0.5. Note that the elastoplastic radial stress Σrrwas denoted simply as radial stress in Figs. 3, 724 4, 8 and 10. For ξ <ξpthe contribution of the viscous radial stress to the total radial stress is very small 725 and the ratioΣv rr/Σrris less than 0.001. The emergence of the shock leads to a sharp increase of Σv rr, such 726 that the ratioΣv rr/Σrrreaches values greater than 1. Note that the viscous stress increases the width of the 727 shock wave. The viscous stress quickly drops to zero for values of the radial coordinate greater than ξp. 728 The observed behavior is readily understood relying on equation (42) which shows that gradients of velocity 729 (which is the case within the shock wave) lead to large values of Σv rr. 730 -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2T E V Dimensionless radial coordinate, ξDimensionless total, elastoplastic and viscous radial stress, Σ rr, Σ rr and Σ rr Porous material, f0=0.1n=0.1 m=0.6 K=1 (Gurson plasticity) Σrr Σ rr Σrr t v elastic wave front, ξwshock wave, ξp t v-0.5 -0.4 -0.3 -0.2 -0.1 0 1.52 1.56zoom Σrrv (a) -0.5 -0.4 -0.3 -0.2 -0.1 0 1 1.2 1.4 1.6 1.8 2T E V Dimensionless radial coordinate, ξDimensionless total, elastoplastic and viscous radial stress, Σ rr, Σ rr and Σ rr t vPorous material, f0=0.1n=0.5 m=0.6 K=1 (Gurson plasticity) Σrr Σ rr Σrr elastic wave front, ξwt vshock wave, ξp -0.5 -0.4 -0.3 -0.2 -0.1 0 1.54 1.56 1.58zoom Σrrv (b) Figure A.15: Porous materials: f0= 0.1. Dimensionless total radial stress Σt rr, elastoplastic radial stress Σrr(denoted simply as dimensionless radial stress in Figs. 3, 4, 8 and 10) and viscous radial stress Σv rrversus dimensionless radial coordinate ξ. The anisotropy parameter is K= 1(Gurson plasticity) and the cavitation velocity is m= 0.60. Two strain hardening exponents are considered: (a) n= 0.1and (b)n= 0.5.