In this paper, we have studied the hypervelocity expansion of a spherical cavity in an inﬁnite medium modeled with the extension of the porous plasticity criterion of Gurson [23] developed by Chen and Yuan [7] to account for plastic strain gradient induced size eﬀects. Following the self-similar, steady-state solution derived by Cohen and Durban [9] for size-independent porous materials, we have computed the critical cavity expansion velocity which leads to the emergence of plastic shock waves for a wide range of initial void volume fractions and diﬀerent values of the length scale parameter that controls the eﬀect of size. We have shown that size eﬀects hinder the emergence of plastic shock waves, so that as the length scale parameter increases, the expansion velocity required for the plastic shock to be formed increases. In addition, while porosity favors the formation of plastic shocks, as shown by Cohen and Durban [9], our results indicate that the eﬀect of initial void volume fraction on plastic shock wave formation decreases for size-dependent materials. Keywords: Dynamic cavity expansion, Porous plasticity, Size eﬀects, Plastic shock waves 1. Introduction High-velocity impact of small particles onto mechanical structures has been an important concern in many engineer- ing applications in the past decades [15, 51]. Speciﬁcally in aerospace industry, the degradation of engine components and protection shields due to high-velocity collision with small par- ticles and debris has been intensively investigated in order to understand and avoid associated structural failure [11, 6]. In the speciﬁc scenario of the high-velocity impact of small hard particles onto metallic materials, two main features are ex- pectedtoplaymajorroles: (i)sizeeﬀectsduetosmall-sizedin- dentations [37, 32, 36]; and (ii) shock wave propagation caused by high velocity impact [10, 9]. From a experimental point of view, Mok and Duﬀy [33] were the ﬁrst authors to show that the results of dynamic inden- tation tests can be interpreted to provide a measure of the yield stress of metals over a range of strains taking into ac- count the inﬂuence of strain-rate. Their work paved the way to subsequent studies using macro [45, 43, 28] and micro/nano [46, 5, 21, 38] tests to assess the mechanics of high strain- rate indentation and penetration of metallic materials. In this context, Hassani et al. [24] have recently developed an experi- mental setup to perform micro-particle impact indentation to determine material hardness at strain-rates beyond 106s−1. These authors employed laser ablation to impact ceramic alu- mina micro-particles of 14µmdiameter onto copper and iron targets, and obtained time-resolved measurements of the im- pact and rebound velocities together with post-impact mea- surements of the size and volume of the indent. On the modeling side, after the works of Hopkins [26] and Goodier [22], cavity expansion theories have been widely ap- plied and considered as a simple and accurate tool to model the mechanics of indentation [27, 18, 20] and penetration ∗Corresponding author Email address: tiago.santos@ufsm.br (T. dos Santos)[19, 16, 31] of metallic targets. In a context in which porous metallic materials are increasingly used in aerospace and air- craft industries (e.g. aircraft structures manufactured with 3D printed metallic materials [44, 50]), special attention shall be given to the hypervelocity cavity expansion model devel- oped by Cohen and Durban [9] for elastoplastic porous duc- tile materials. They showed that material porosity favors the development of plastic shock waves and that the level of dis- continuities induced by the plastic shock increases with the void volume fraction. These ﬁndings were also evidenced by dos Santos et al. [40] using ﬁnite element simulations. More- over, Cleja-Tigoiu et al. [8] developed a rate-dependent cavity model to address hypervelocity penetration in porous concrete structures. Recently, the model of Cohen and Durban [9] has been extended by dos Santos et al. [41] in an eﬀort to account for plastic strain gradient induced size eﬀects on the dynamic micro-indentation of porous materials. As a subsequent study, this work uses the formulation of dos Santos et al. [41] to eluci- date the eﬀects of size and material porosity on the formation of plastic shocks emerging during hypervelocity cavitation of porous solids. 2. Constitutive framework The constitutive framework is the same used in the recent paper of dos Santos et al. [41]. The rate of deformation tensor is considered to be the sum of an elastic part, de, and a plastic part,dp: d=de+dp(1) The elastic part is related to the rate of the Cauchy stress tensorσby the following hypo-elastic law: Oσ=C:de(2) whereO ()denotes an objective derivative and Cis the fourth- order isotropic elastic tensor: Preprint submitted to Mechanics Research Communications March 23, 2021 2 C=E 1 +νI/prime+E 3 (1−2ν)1⊗1 (3) whereEandνare the Young’s modulus and the Poisson’s ratio, respectively, 1is the unit second-order tensor and I/primeis the unit deviatoric fourth-order tensor. The plastic part of the rate of deformation tensor is: dp=˙λ∂Φ ∂σ(4) where ˙λis a non-negative plastic multiplier and Φis the plastic ﬂow potential [23, 48, 49]: Φ =/parenleftBigσe ¯σ/parenrightBig2 + 2q1fcosh/parenleftbigg3q2σh 2¯σ/parenrightbigg −1−(q1f)2≤0(5) whereq1andq2are material parameters and fis the void volume fraction (see equation (9)). The eﬀective von Mises stressσeand the hydrostatic stress σhare: σe=/radicalbigg 3 2s:s, σh=1 3σ:1,s=σ−σh1 withsbeing the deviatoric part of the Cauchy stress tensor. Similarly to Chen and Yuan [7], the ﬂow strength of the matrix material ¯σis deﬁned using a simple form of the generalized gradient plasticity model developed by Aifantis and co-workers [2, 3, 4, 47]: ¯σ= ¯σhom+lE/bardbl∇¯εp/bardbl (6) wherelis the length scale parameter and /bardbl∇¯εp/bardbl= (∇¯εp·∇¯εp)1 2, with ¯εp=/integraltextt 0˙¯εp(τ)dτbeing the eﬀective plastic strain in the matrix material and ˙¯εpthe eﬀective plastic strain rate. To be emphasized that the constitutive relation adopted in Eq. (6) intends to mimic plastic strain gradient induced size eﬀects, which arise in polycrystalline metals. Therefore, the length scale parameter lmay be related to microstructure aspects, suchasthematerialgrainsizeandtheaccumulationof geometrically necessary dislocations [17, 36]. Moreover, ¯σhom in Eq. (6) is the ﬂow strength for homogeneous deformation: ¯σhom=σ0/parenleftbigg 1 +¯εp ε0/parenrightbiggn (7) whereσ0is the initial yield stress of the matrix material, nis the strain hardening exponent and ε0is the reference strain. The equality between the rates of macroscopic plastic work and matrix plastic work [7]: σ:dp= (1−f) ¯σ˙¯εp(8) provides a relation between the plastic part of the rate of de- formation tensor and the eﬀective plastic strain rate in the matrix material. Moreover, the evolution of the void volume fraction is: ˙f= (1−f)dp:1 (9) The values of the parameters of the constitutive model, which are based on [42, 40], are given in Table 1.Symbol Property and units Value ρ0Initial density (kg/m3) 7600 EYoung’s modulus (GPa), Eq. (3) 70 νPoisson’s ratio, Eq. (3) 0.3 q1Material parameter, Eq. (5) 1.25 q2Material parameter, Eq. (5) 1.0 σ0Initial yield stress (MPa), Eq. (7) 300 nStrain hardening exponent, Eq. (7) 0.1 ε0Reference strain, Eq. (7) 0.00429 Table 1: Constitutive model parameters [42, 40]. 3. Dynamic spherical cavity expansion model Dynamic spherical cavity expansion theories have been widely accepted as a basic physical model underlying indenta- tion and penetration mechanics in metallic materials [18, 29]. The problem studied is that of a pressurized spherical cavity of instantaneous radius aexpanding under self-similar, steady- state conditions in an inﬁnite medium described with the con- stitutive framework of Section 2. Following Masri and Durban [30] and Durban and Masri [16], the cavity expansion model is formulated using an Eulerian spherical system of coordinates (r,θ,φ )with origin located at the center of the cavity denoted byOin Fig. 1. The only independent variable is the non- dimensional radial coordinate ξ=r/a, so that the derivative with respect to ris: ∂() ∂r=∂() ∂ξ∂ξ ∂r= ()/prime1 a(10) where the prime superscript denotes diﬀerentiation with re- spect toξ, and the time derivative is: ˙() =∂() ∂ξ˙ξ= ()/prime˙a a(v−ξ) (11) wherev= ˙r/˙ais the dimensionless radial velocity. Initial distribution of porosityξ and stress-free turbed zone = f0 and ρ = ρ0 Elastic zoneElastoplastic zoneξw ξi ξ=1 Oξp Figure 1: Illustration of the self-similar, steady-state cavity expansion ﬁeld. The internal pressure pis applied at the cavity wall where ξ= 1. The remote region ξ > ξ wis a strain and stress-free undisturbed zone withf=f0andρ=ρ0. The elastic wave front is denoted by ξ=ξw, the transition between the elastic and elastoplastic regions by ξi, and the shock wave location by ξp. This ﬁgure is adapted from Durban and Masri [16]. Owing to the spherical symmetry of the problem, the only non-zero components of the Cauchy stress tensor are normal stresses along the radial, σrr, circumferential and azimuthal, 3 σθθ=σφφ, directions. Therefore, the Eulerian version of the balance of linear momentum reduces to: ∂σrr ∂r+2 r(σrr−σθθ) =ρ¨r (12) whereρis the current material density. EmployingEqs. (10)and(11), previousexpressionbecomes: (Σrr)/prime+2 ξ(Σrr−Σθθ) =m2ρ ρ0(v−ξ)v/prime(13) where Σrr=σrr EandΣθθ=σθθ Eare dimensionless stresses, andm=˙a√ E/ρ0is the dimensionless cavity expansion velocity, which is assumed to be constant in this steady-state problem. Moreover, the non-zero components of the strain rate tensor are: drr=∂˙r ∂r=˙a av/prime(14) dθθ=dφφ=˙r r=˙a av ξ(15) where Eqs. (10) and (11) have been employed. The hypo- elastic law introduced in equation (2) can be rewritten as: de rr=˙σrr E−2ν˙σθθ E(16) de θθ=de φφ=(1−ν) ˙σθθ E−ν˙σrr E(17) Noticethat, inabsenceofmaterialspin, theobjectivederiva- tiveO (), see equation (2), has been replaced by a simple time derivative˙(). Furthermore, consideringtheﬂowrule(4), theﬂowpotential (5) and relation (8), the active plastic deformation rates are: dp rr= (1−f)˙¯εP˜Nrr (18) dp θθ=dp φφ= (1−f)˙¯εP˜Nθθ (19) with ˜Nrr=1 2−2Σe¯Σ+q1q2fsinh/parenleftBig 3q2Σh 2¯Σ/parenrightBig /parenleftBig Σe¯Σ/parenrightBig2 +q1q2fsinh/parenleftBig 3q2Σh 2¯Σ/parenrightBig 3Σh 2¯Σ(20) ˜Nθθ=1 2Σe¯Σ+q1q2fsinh/parenleftBig 3q2Σh 2¯Σ/parenrightBig /parenleftBig Σe¯Σ/parenrightBig2 +q1q2fsinh/parenleftBig 3q2Σh 2¯Σ/parenrightBig 3Σh 2¯Σ(21) where Σe=σe E=σθθ−σrr EisthedimensionlessvonMisesstress, ¯Σ =¯σ Eis the dimensionless matrix ﬂow strength and Σh=σh E is the dimensionless hydrostatic stress. Moreover, using equations (1), (14), (15), (18) and (19), the following expressions are obtained: v/prime= (v−ξ)/bracketleftbig Σ/prime rr−2νΣ/prime θθ+ (1−f) (¯εp)/prime˜Nrr/bracketrightbig (22) v ξ= (v−ξ)/bracketleftbig −νΣ/prime rr+ (1−ν) Σ/prime θθ+ (1−f) (¯εp)/prime˜Nθθ/bracketrightbig (23) For a spherically symmetric problem, using Eq. (10), the ﬂow strength of the matrix material given in Eq. (6) becomes: ¯Σ = ¯Σhom+l a(¯εp)/prime(24)where ¯Σhom=¯σhom Eis the dimensionless ﬂow strength for ho- mogeneous deformation given by (see Eq. (7)): ¯Σhom= Σ 0/parenleftbigg 1 +¯εp ε0/parenrightbiggn (25) with Σ0=σ0 Ebeing the dimensionless initial yield stress of the matrix material. Equation (24) brings out that the eﬀect of plastic strain gradients in the cavitation ﬁelds depends on the ratio between the length scale parameter land the current cavity radius a, so that for a/greatermuchlthe cavitation ﬁelds become size-independent (see Danas et al. [14]). Moreover,usingEqs. (11),(18)and(19),thematerialporos- ity evolution, Eq. (9), becomes: f/prime= (1−f)2/parenleftbig˜Nrr+ 2˜Nθθ/parenrightbig (¯εp)/prime(26) For a plastic loading, the relation between Σeand¯Σis given when the ﬂow potential, Eq. (5), is Φ = 0: /parenleftbiggΣe ¯Σ/parenrightbigg2 + 2q1fcosh/parenleftbigg3q2Σh 2¯Σ/parenrightbigg −1−(q1f)2= 0 (27) Diﬀerentiating Eq. (27) with respect to ξ, using Eq. (26) and the relations 3Σh= Σrr+ 2ΣθθandΣe= Σθθ−Σrr, the following expression is obtained: χ1Σ/prime rr+χ2Σ/prime θθ+χ3¯Σ/prime+χ4(1−f)2/parenleftbig˜Nrr+ 2˜Nθθ/parenrightbig (¯εp)/prime= 0 (28) with χ1=1 2q1q2fsinh/parenleftbigg3q2Σh 2¯Σ/parenrightbigg −Σe ¯Σ(29) χ2=q1q2fsinh/parenleftbigg3q2Σh 2¯Σ/parenrightbigg +Σe ¯Σ(30) χ3=−q1fsinh/parenleftbigg3q2Σh 2¯Σ/parenrightbigg3q2Σh 2¯Σ−Σ2 e ¯Σ2(31) χ4=q1/bracketleftbigg cosh/parenleftbigg3q2Σh 2¯Σ/parenrightbigg −q1f/bracketrightbigg ¯Σ (32) The material porosity is obtained solving Eq. (27) for the physical branch of f(see Cohen and Durban [9]): f=1 q1 cosh/parenleftbigg3q2 2Σh Σy/parenrightbigg −/radicalBigg sinh2/parenleftbigg3q2 2Σh Σy/parenrightbigg +/parenleftbiggΣe Σy/parenrightbigg2  (33) Furthermore, the principle of mass conservation, in Eulerian description, yields: ˙ρ+ρ(drr+ 2dθθ) = 0 (34) Integration of previous expression, from a given coordinate ξ, to the elastic wave front ξw, using relations (1), (9), (11), (16) and (17), leads to: ρ ρ0=/parenleftbigg1−f 1−f0/parenrightbigg exp [−3(1−2ν)Σh] (35) whereρ0is the initial material density (see Table 1). Note that, in order to obtain previous expression, the conditions f=f0,ρ=ρ0andΣh= 0have been considered at ξ=ξw (see Fig. 1). 4 In this model, Eqs. (13), (22), (23), (24), and (28), pro- vide a system of diﬀerential equations with derivatives of ﬁve unknowns/parenleftbig v,Σrr,Σθθ,¯Σ,¯εp/parenrightbig , with ¯Σhom,fandρbeing calcu- lated from the algebraic relations (25), (33) and (35), respec- tively. The problem is completed with the following boundary conditions: v= 1 and Σ rr=−P,atξ= 1 (36) and v= 0,Σrr= Σθθ= 0,atξ=ξw (37) whereP=p Eis the dimensionless applied pressure. The in- tegration is performed over the independent variable ξ, from ξwup toξ= 1(see Fig. 1). An iterative shooting method is used to enforce the velocity boundary condition, while the stress boundary condition is used to calculate the correspond- ing applied pressure. Moreover, we consider a shock to be formed when the fol- lowing function: ∆ =m2ρ ρ0(v−ξ)2−(1−f)∂¯εp ∂¯Σ˜Nθθχ2−(1−ν)χ5 −(1−2ν) (1 +ν)χ5+χ6(38) with χ5=χ3+χ4(1−f)2/parenleftbig˜Nrr+ 2˜Nθθ/parenrightbig∂¯εp ∂¯Σ(39) χ6= (1−f)∂¯εp ∂¯Σ/braceleftbig/bracketleftbig (1−ν)˜Nrr+ 4ν˜Nθθ/bracketrightbig χ1+˜Nθθχ2/bracerightbig (40) becomes zero, i.e. ∆ = 0is the shock wave condition which serves to identify both the plastic shock and the elastic wave front (see Fig. 1). Hereinafter, ∆will be called shock wave function, and the dimensionless velocity for which the plastic shock wave emerges will be referred to as critical dimensionless velocitymc. Note that expression (38) was derived by Cohen and Durban [9] for size and rate independent porous materials. Thus, forl= 0(i.e. ¯Σ = ¯Σhom, see Eq. (24)) the derivative ∂¯εp/∂¯Σcan be calculated analytically using Eq. (25). In con- trast, forl>0, it has to be approximated numerically. 4. Results Figure 2 shows the critical dimensionless velocity mcversus the dimensionless length scale parameter l/a, for diﬀerent ini- tialporosities f0= 0,0.01,0.05and0.1. Exploringsuchawide spectrum of initial void volume fractions helps to elucidate the role of porosity in the emergence of plastic shock waves, providing deﬁnite trends for the interplay between size eﬀects and initial void volume fraction. In addition, note that it is relatively common that sintered and additively manufactured metals display levels of initial porosity between 1%and10% (e.g. see Aboulkhair et al. [1] and Heidari et al. [25]), so that the range of void volume fractions investigated in this work can be representative of actual metallic materials. Moreover, the range of values investigated for the dimensionless length scale parameter 0≤l/a≤0.002is similar to that considered by dos Santos et al. [41]. The critical dimensionless velocity increases with the normalized length scale, bringing out that size eﬀects hinder the emergence of plastic shocks. To the au- thors’ knowledge, this is an original outcome of this paper. The explanation is that the plastic strain gradients introduced by equation (24) have a regularizing eﬀect that prevents the development of singularities (we will further elaborate on this issue with the results presented in Fig. 5). On the other hand,the increase of initial porosity shifts the mc−l/acurves down, showing that porosity favors the formation of plastic shock waves (as demonstrated by Cohen and Durban [9]). Neverthe- less, notice that the diﬀerence between the results obtained for diﬀerent values of f0decreases with l/a, revealing that the in- ﬂuence of porosity on plastic shock wave formation is reduced as the eﬀect of size increases. In fact, we have checked that for l/a≥0.004the diﬀerence in the critical dimensionless velocity obtained for the fully dense material and the three initial void volume fractions considered is below 0.01. To the best of our knowledge, this is also an original contribution of this paper. 00.20.40.60.81 0 0.0005 0.001 0.0015 0.002Critical dimensionless velocity, mc Dimensionless length scale, l/aFully dense material: f 0=0 Initial void volume fraction: f 0=0.01Initial void volume fraction: f 0=0.05 Initial void volume fraction: f 0=0.1 Δ=0 Figure 2: Evolution of the critical dimensionless velocity mcfor which the shock wave emerges ( ∆ = 0) versus the dimensionless length scale l/a forf0= 0,0.01,0.05and0.1. Figure 3 shows the evolution of the critical dimensionless velocitymcwith the initial porosity f0, for diﬀerent values of the dimensionless length scale parameter l/a= 0,0.0004, 0.001and0.002. Consistent with the results pictured in Fig. 2, the porosity favors the emergence of plastic shocks, and in- creasingf0boosts the inﬂuence of the dimensionless length scale parameter on the critical velocity leading to shocks for- mation. For l/a= 0and0.0004themc−f0curves display a concave-upwards shape, so that the critical dimensionless ve- locity decreases nonlinearly with the initial porosity, being the slope of the mc−f0curves smaller as the initial porosity in- creases. In contrast, for l/a= 0.002the decrease of the critical dimensionless velocity with the porosity is mild and roughly linear. To be highlighted that, for the rate independent ma- terial employed in this work, the only feature regularizing and delaying the shock wave formation is the plastic strain induced size eﬀects. However, for rate dependent porous metals, vis- cous [34, 39] and micro-inertia induced pore size eﬀects [12, 13] are also expected to attenuate plastic shock waves. Figure 4 shows the evolution of the shock wave function ∆ with the dimensionless radial coordinate ξfor diﬀerent values of the dimensionless length scale l/a= 0,0.0004,0.001and 0.002. Figure 4a displays results for the fully dense material f0= 0andm= 0.55. This value of mis slightly lower than mc forf0= 0andl/a= 0(size-independent material). The shape of the ∆−ξcurves is similar for all the dimensionless length scale parameters. The function ∆ﬁrst increases, reaches a maximum, drops rapidly and then increases again. The ﬁrst maximum corresponds to the elastic/elastoplastic interface ξi. The zoomed area in the right upper part of the plot shows that it is only for the size-independent material for which the shock wave emergence condition ∆→0is met (the plastic shock is located at the elastic/elastoplastic interface for the material withl/a= 0). For the size-dependent materials, l/a>0, the 5 00.20.40.60.81 0 0.02 0.04 0.06 0.08 0.1Critical dimensionless velocity, mc Initial void volume fraction, f 0l/a=0 l/a=0.0004l/a=0.001 l/a=0.002 Δ=0 Figure 3: Evolution of the critical dimensionless velocity mcfor which the shock wave emerges ( ∆ = 0) versus the initial void volume fraction f0forl/a= 0,0.0004,0.001and0.002. shock condition ∆ = 0is not met, and the ﬁrst maximum of the curve ∆−ξis smoother (i.e. the curvature is smaller). The second maximum corresponds to the elastic wave front, that yields a discontinuity in the cavitation ﬁelds so that the condition ∆ = 0is met for all values of the parameter l/a tested (see Cohen et al. [10]). Figure 4b presents the ∆−ξ curves forf0= 0.05andm= 0.17. This value of mtends tomcforf0= 0.05andl/a= 0, so that the maximum of the ∆−ξcurve for l/a= 0corresponds to the emerging plastic shock. Note that, unlike what occurs with the fully dense material (see Fig. 4a), the plastic shock does not emerge at the elastic/elastoplastic interface (which corresponds to the kink of the ∆−ξcurve located at ξ≈3.1), but it is located closer to the cavity wall, within the elastoplastic zone (see Fig. 1). Note also that the maximum of the ∆−ξcurves corresponding to l/a>0doesnotmeetthecondition ∆ = 0, showingthathigher values ofmare required for the plastic shock to emerge if size eﬀects are accounted for (as anticipated in Fig. 2). Moreover, note that the elastic wave front ξeis located at ξ>4and thus it is not shown in the graph (recall that at the elastic wave front the condition ∆ = 0is met irrespective of the value of l/a). Figure 5 displays the evolution of the dimensionless plastic strain gradient (¯εp)/primewith the dimensionless radial coordinate ξforl/a= 0,0.0004,0.001and0.002(the same values of the dimensionless length scale parameter considered in Fig. 4). Results are shown for f0= 0,0.01,0.05and0.1in subﬁgures (a), (b), (c) and (d), respectively. The cavity expansion ve- locity, for the four void volume fractions investigated, tends to the critical dimensionless velocity mcof the size-independent material (i.e. for l/a>0a larger value of mis required for the plastic shock to be formed). The value of (¯εp)/primeﬁrst de- creases as we move away from the cavity wall, reaches a local minimum, then increases until a maximum is attained for an intermediate value of ξ(corresponding to the plastic shock on- set in the case of l/a= 0), and then decreases again, eventually dropping to zero right after the elastic/elastoplastic interface (which is identiﬁed in the plots as ξi). The comparison of the results for diﬀerent void volume fractions shows that the value of(¯εp)/primeis generally smaller as the initial void volume fraction increases (notice that the scale of the y-axis of subﬁgures (a), (b), (c) and (d) is diﬀerent). This is most likely because the value ofmcorresponding to subﬁgures (a), (b), (c) and (d) decreases as the porosity increases (since porosity promotes -1-0.8-0.6-0.4-0.200.2 1 2 3 4B D F HShock wave function, Δ Dimensionless radial coordinate, ξFully dense material: f 0=0 zoomm=0.55 l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002ξwξp=ξi -0.2-0.15-0.1-0.0500.05 1.6 1.7 1.8(a) -1-0.8-0.6-0.4-0.200.2 1 2 3 4B D F HShock wave function, Δ Dimensionless radial coordinate, ξInitial void volume fraction: f 0=0.05 l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002m=0.17 zoom -0.2-0.15-0.1-0.0500.05 1.5 2 2.5 ξp ξi (b) Figure 4: Shock wave function ∆versus dimensionless radial coordinate ξ for diﬀerent values of the dimensionless length scale l/a= 0,0.0004,0.001 and0.002. (a)Fullydensematerial f0= 0anddimensionlessvelocity m= 0.55. (b) Initial void volume fraction f0= 0.05and dimensionless velocity m= 0.17. Theyellowlinecorrespondstotheshockwavecondition ∆ = 0. Recall that ξidenotes the elastic/elastoplastic interface and ξpthe plastic shock location for l/a= 0. plastic shock formation). On the other hand, for a given value off0, the only signiﬁcant diﬀerence between the results ob- tained for diﬀerent dimensionless length scale parameters is that the maximum of (¯εp)/primethat appears for an intermediate value ofξis lower as l/aincreases, bringing to light that size eﬀects smooth the plastic strain gradients thus preventing dis- continuities in the plastic strain ﬁeld (i.e. hindering plastic shocks formation). These ﬁndings are in line with the results reported by Molinari and Ravichandran [35], who investigated plasticstraingradienteﬀectsonplasticshocksformationinpe- riodic laminates and showed that, for a ﬁxed stress amplitude, increasing the length scale parameter reduces the maximum plastic strain rate and increases the shock width. The length scale used in Molinari and Ravichandran [35] was related to the geometry of the layers and material characteristics of the constituents. Figure 6 shows the dimensionless von Mises stress Σever- sus the dimensionless radial coordinate ξ, for the same dimen- sionless length scale parameters, initial void volume fractions and cavity expansion velocities considered in Fig. 5. Notice that, while the plastic strain gradients near the cavity showed little dependence on the length scale parameter, see Fig. 5, the value of l/ahas a signiﬁcant inﬂuence on the von Mises stress ﬁelds that develop near the cavity, so that both the von Mises stress and the gradients of the von Mises stress increase asl/aincreases. For the fully dense material, Fig. 6a, the elastic/elastoplastic interface, that in the case of l/a= 0de- termines the location of the plastic shock (recall that for the 6 05 1061 1071.5 107 1 2 3 4B D F H Dimensionless radial coordinate, ξDimensionless plastic strain gradient, ( ε p)' l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002zoom ξw Fully dense material: f 0=0 m=0.55 ξp=ξi 05 1061 1071.5 107 1.6 1.7 1.8 (a) 02 1064 1066 1068 1061 107 1 2 3 4B D F H Dimensionless radial coordinate, ξInitial void volume fraction: f 0=0.01 m=0.27 l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002zoom ξp ξi Dimensionless plastic strain gradient, ( ε p)' 02 1064 1066 1068 1061 107 2.3 2.4 2.5 2.6 (b) 02 1064 1066 1068 106 1 2 3 4B D F H Dimensionless radial coordinate, ξInitial void volume fraction: f 0=0.05 m=0.17 l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002zoom ξp ξi Dimensionless plastic strain gradient, ( ε p)' 02 1064 1066 1068 106 2.05 2.1 2.15 2.2 2.25 2.3 2.35 (c) 01 1062 1063 1064 106 1 2 3 4B D F H Dimensionless radial coordinate, ξInitial void volume fraction: f 0=0.1 m=0.13 ξp ξi Dimensionless plastic strain gradient, ( ε p)' l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002 (d) Figure 5: Dimensionless plastic strain gradient (¯εp)/primeversus dimensionless radial coordinate ξfor diﬀerent values of the dimensionless length scale l/a= 0,0.0004,0.001and0.002. (a) Fully dense material f0= 0and dimensionless velocity m= 0.55. (b) Initial void volume fraction f0= 0.01and dimensionless velocity m= 0.27. (c) Initial void volume fraction f0= 0.05and dimensionless velocity m= 0.17. (d) Initial void volume fractionf0= 0.1and dimensionless velocity m= 0.13. Recall that ξi denotestheelastic/elastoplasticinterfaceand ξptheplasticshocklocation forl/a= 0.size-independent material the plastic shock emergence condi- tion∆→0has been met), leads to a sudden drop in the value ofΣewhich is more pronounced as the length scale parameter decreases. Incontrast, fortheporousmaterials, Figs. 6(b)-(c)- (d), the elastic/elastoplastic interface ξicorresponds to a kink in the Σe−ξcurves, and the plastic shock ξp, that has emerged forl/a= 0within the elastoplastic zone, is represented by a bump, i.e. a brief drop of Σe, which partially bounces back immediately after, and it is followed by a plateau that even- tually extends until the elastic/elastoplastic interface. Notice that the bump is ﬂatten with the increase of the length scale parameter, showing that size eﬀects in porous materials pre- vent the formation of discontinuities in the von Mises stress ﬁelds within the elastoplastic zone, and thus the formation of plastic shocks (as mentioned before). Figure 7 shows the evolution of the void volume fraction f with the dimensionless radial coordinate ξ, for the same di- mensionless length scale parameters considered in Fig. 6, and three diﬀerent values of the initial porosity f0= 0.01,0.05 and0.1. The values of the cavity expansion velocity are the same employed in Figs. 6 (b)-(c)-(d), namely m= 0.27,0.17 and0.13in Figs. 7(a)-(b)-(c), respectively. The f−ξcurves display a sigmoidal shape so that the porosity is zero near the cavity (the pores have closed), and f=f0forξ>ξi(no plasticdeformation). Thevariationofthevoidvolumefraction occurs within an intermediate range of values of ξ, near the ra- dial coordinate corresponding to the plastic shock location for l/a= 0(recall that for the values of mconsidered the plastic shock condition ∆→0has been met for the size-independent material). Notice that the slope of the f−ξcurves increases as the dimensionless length scale parameter decreases, so that f/prime→∞for the size independent material, showing that size eﬀects prevent the formation of discontinuities in the porosity ﬁeld. 5. Concluding remarks The extension of the hypervelocity cavity expansion model of Cohen and Durban [9] to size-dependent materials devel- oped by dos Santos et al. [41] has been used to determine the eﬀect of size, induced by plastic strain gradient, on the for- mation of plastic shocks in porous materials. In particular, we haveshownthatthecriticalcavityexpansionvelocityforwhich plasticshocksemergeincreaseswiththelengthscaleparameter that controls the eﬀect of size, and decreases with the mate- rial porosity, with the inﬂuence of porosity decreasing as the eﬀect of size increases. In fact, investigation of the cavitation ﬁelds has shown that size eﬀects act as a regularizing factor that tempers the gradients of plastic strain, thus hindering the formation of discontinuities in the ﬁeld variables within the elastoplastic zone that develops near the cavity. Ultimately, the results presented in this paper shall be used to predict plastic shocks formation in hypervelocity nano-indentations. However, the validation of the theoretical predictions shown herein will be pending until relevant experimental results be- come available. Acknowledgements TdS wishes to acknowledge the support of FAPERGS, Fun- dação de Amparo à Pesquisa do Estado do Rio Grande do Sul, grant agreement 19/2551−0001054−0. JAR-M acknowledges the ﬁnancial support obtained from the European Research Council (ERC) under the European 7 00.010.02 1 2 3 4B D F H Dimensionless radial coordinate, ξDimensionless von Mises stress, Σe l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002m=0.55Fully dense material: f 0=0 zoom ξp=ξi 0.0020.0040.0060.008 1.6 1.7 1.8 (a) 00.010.02 1 2 3 4B D F H Dimensionless radial coordinate, ξDimensionless von Mises stress, ΣeInitial void volume fraction: f 0=0.01 m=0.27l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002zoom ξi ξp 0.0030.0040.005 2.3 2.4 2.5 2.6 (b) 00.010.02 1 2 3 4B D F H Dimensionless radial coordinate, ξDimensionless von Mises stress, Σe l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002Initial void volume fraction: f 0=0.05 m=0.17ξi ξp zoom 0.00250.005 1.8 2 2.2 2.4 2.6 (c) 00.010.02 1 2 3 4B D F H Dimensionless radial coordinate, ξDimensionless von Mises stress, Σe l/a=0 l/a=0.0004 l/a=0.001 l/a=0.002Initial void volume fraction: f 0=0.1 m=0.13zoom ξi ξp 0.00250.005 1.6 2 2.4 (d) Figure 6: Dimensionless von Mises stress Σeversus dimensionless radial coordinateξfor diﬀerent values of the dimensionless length scale l/a= 0, 0.0004,0.001and0.002. (a) Fully dense material f0= 0and dimen- sionless velocity m= 0.55. (b) Initial void volume fraction f0= 0.01 and dimensionless velocity m= 0.27. (c) Initial void volume fraction f0= 0.05and dimensionless velocity m= 0.17. (d) Initial void volume fractionf0= 0.1and dimensionless velocity m= 0.13. Recall that ξide- notes the elastic/elastoplastic interface and ξpthe plastic shock location forl/a= 0. 00.0020.0040.0060.0080.010.012 1 2 3 4B D F H Dimensionless radial coordinate, ξl/a=0 l/a=0.0004 l/a=0.001 l/a=0.002Initial void volume fraction: f 0=0.01 m=0.27ξi ξp Void volume fraction, f 00.0020.0040.0060.0080.010.012 2.3 2.4 2.5 2.6zoom(a) 00.010.020.030.040.050.06 1 2 3 4B D F H Dimensionless radial coordinate, ξVoid volume fraction, fl/a=0 l/a=0.0004 l/a=0.001 l/a=0.002Initial void volume fraction: f 0=0.05 m=0.17ξi ξp 00.010.020.030.040.050.06 1.8 2 2.2 2.4 2.6 (b) 00.020.040.060.080.10.12 1 2 3 4B D F H Dimensionless radial coordinate, ξVoid volume fraction, fl/a=0 l/a=0.0004 l/a=0.001 l/a=0.002Initial void volume fraction: f 0=0.1 m=0.13ξi ξp 00.020.040.060.080.10.12 1.6 2 2.4 (c) Figure 7: Void volume fraction fversus dimensionless radial coordinate ξfor diﬀerent values of the dimensionless length scale l/a= 0,0.0004, 0.001and0.002. (a) Initial void volume fraction f0= 0.01and dimen- sionless velocity m= 0.27. (b) Initial void volume fraction f0= 0.05 and dimensionless velocity m= 0.17. (c) Initial void volume fraction f0= 0.1and dimensionless velocity m= 0.13. Recall that ξidenotes the elastic/elastoplastic interface and ξpthe plastic shock location for l/a= 0.