Herein, we present a self-similar cavity expansion model that follows from the work of Cohen and Durban (2013b) to analyze the dynamic indentation response of elasto-plastic porous materials while accounting for the plastic strain gradient induced size eﬀect. The incorporation of the plastic strain gradient induced size eﬀect in the dynamic cavity expansion model for elasto-plastic porous materials is the key novelty of our model. The predictions of the cavity expansion model for the material hardness, for diﬀerent indentation depths and speeds, are compared against the available experimental results for OFHC copper, for strain rates varying from 10−4s−1to108s−1. We note that despite several simplifying assumptions, the predictions of our cavity expansion model show a reasonable agreement with the experimentally measured material hardness over a wide range of indentation depths and speeds. In addition, we have also carried out parametric analyses to elucidate the speciﬁc roles of indentation speed, size eﬀect and initial porosity, on the material hardness and cavitation ﬁelds that develop during the indentation process. In particular, our parametric analyses show that there exists a critical value of the indentation speed beyond which the contribution of inertial eﬀect becomes extremely important and the material hardness increases rapidly. While the inﬂuence of the initial porosity on the material hardness is found to increase with increasing indentation speed and decrease with increasing size eﬀect. Keywords: Dynamic indentation, Cavity expansion, size eﬀect, Ductile materials, Porous plasticity 1. Introduction Mok and Duﬀy (1964) published one of the ﬁrst experimental works on impact indentation of metallic materials. They dropped balls of hardened steel and aluminum of 12.7mm and 25.4mm diameter onto targets of pure lead and AA6061-T6, at a range of velocities (with a maximum of 6m/s) giving strain rates varying ∗Corresponding author. Tel. +34 916249904; Fax: +33 916249430. E-mail address:jarmarti@ing.uc3m.es Preprint submitted to Mechanics of Materials November 24, 2020 2 from quasi-static to ≈3·103s−1, and temperatures ranging from 295K to 750K. In these experiments, they measured the impact and rebound velocities, as well as the size of the crater formed in the target and the time of contact between the impactor and the target. Next, using a simple energy balance, equating the loss in kinetic energy of the impactor to the energy required to produce the indentation, Mok and Duﬀy (1964) determined the dynamic yield pressure (i.e. hardness) of the target for each test. At room temperature, in the case of pure lead, the dynamic yield pressure varied from 20MPa for quasi-static loading to 70MPa for the highest strain rate tested, and from 200to400MPa in the case of AA6061-T6. These experiments provided the ﬁrst evidence that material hardness changes with the impact velocity. Shortly after, Mok and Duﬀy (1965) performed additional tests on C1018 steel and AA 1100F. These experiments were performed at room temperature and for the same range of impact velocities as in Mok and Duﬀy (1964). These additional tests conﬁrmed the dependence of the material hardness on the strain rate, and showed that results of dynamic indentation experiments can be interpreted to provide a measure of the yield stress of the target material over a range of strain rates. Later, Tirupataiah and Sundararajan (1991) published what is probably the ﬁrst experimental work on very high velocity impact indentation of metallic materials. These authors launched very hard spherical tungsten carbide balls of 4.75mm diameter onto iron and copper targets at velocities ranging from 5m/s to180m/s. Similar to Mok and Duﬀy (1964, 1965), the dynamic hardness of the material was deﬁned as the resistance per unit area opposing the penetration of the ball, and it was calculated from an energy balance between the kinetic energy of the projectile and the energy required to form the resulting crater in the target, under the assumptionthatalltheenergyoftheimpactingball(exceptforthereboundenergy)wasconsumedintheplastic deformation of the target material. The experiments showed that material hardness generally increases under dynamic loading, reaching 800MPa for copper and 1600MPa for iron, and becoming up to three times higher than under quasi-static conditions for strain rates that were estimated to be as high as ≈104s−1. Tirupataiah and Sundararajan (1991) also brought out some limitations of the high velocity dynamic indentation tests, including the fact that the proper determination of the dynamic hardness requires the material to harden during the indentation process. Sundararajan and Tirupataiah (2006) performed additional experiments with copper, iron, steel, and AA7039 alloy samples, and showed that for the last three materials the dynamic hardness measured in the experiments displays a maximum for a critical strain, and then decreases due to the onset of plastic localization, which was attributed to the decrease in ﬂow stress due to the adiabatic heating of the material. Thirty-ﬁve years after the pioneering works of Mok and Duﬀy (1964, 1965), Subhash et al. (1999) developed anexperimentalsetuptoperformdynamicindentationtests, usingtheone-dimensionalelasticwavepropagation principle employed in Kolsky bar experiments, which allowed them to obtain time-resolved indentation depth 3 and load responses during the indentation process. Subhash et al. (1999) tested a variety of materials and the maximum strain rate attained in these experiments was ≈2·103s−1. Their experimental results showed that, for this high strain rate, the increase in dynamic hardness over quasi-static hardness depends on the material, ranging from 2%for aluminum to 30%for titanium. Later, Lu et al. (2003) developed an alternative dynamic indentation technique which enabled recording of the force and displacement of the indenter for test velocities ranging from 6m/s to35m/s. In these experiments, the measured indentation depth and load were used to estimate the strain rate sensitivity of OFHC copper, and the results obtained were shown to be in agreement with the data obtained with Kolsky bar experiments. More recently, Hassani et al. (2020) have developed an experimental setup to perform microparticle impact indentation to determine material hardness at strain rates beyond 106s−1(these are probably the ﬁrst measurements of hardness at such high strain rates). These authors employed laser ablation to impact ceramic alumina microparticles of 14µm diameter (as indenter) onto copper and iron targets, and obtained time-resolved measurements of the impact and rebound velocities together with post-impact measurements of the volume of the indent. Similarly to Tirupataiah and Sundararajan (1991), Hassani et al. (2020) calculated the dynamic hardness of the target material as the ratio of the plastic work to the indentation volume, and showed that, for strain rates greater than 106s−1, the hardness of the material increases up to 4−5GPa, i.e. it becomes one order of magnitude greater than under quasi-static loading conditions. Besides dynamic eﬀect, size eﬀect has also been shown to play an important role on the indentation response of a variety of materials, and it is generally observed that the hardness of the material increases as the size of the indent decreases. For instance, Poole et al. (1996) performed quasi-static micro-indentation experiments on annealed and work hardened copper specimens and showed a monotonic increase in the hardness for indentation depths below≈10µm. The experimental results were explained using a dislocation-based model which incorporated geometrically necessary dislocations due to the presence of strain gradients in the deformation zone around the indent. Later, McElhaney et al. (1998) obtained accurate measurements of the hardness of copper for indentation depths near 100nm. For such a small indentation depth, the material hardness was found to be above 2GPa, i.e. 4−5times higher than the value corresponding to the greater indentation depth (≈2000nm) tested. Moreover, the quasi-static nano-indentation experiments on copper performed by Lim and Chaudhri (1999) yielded the same trends obtained in the tests of Poole et al. (1996) and McElhaney et al. (1998), with greater hardness for small indentation depth. A number of experimental studies have also focused on the combined eﬀect of inertia and various size eﬀects on the material hardness. Trelewicz and Schuh (2008) were among the ﬁrst authors to perform high strain rate nano-indentation experiments. These authors tested nanocrystalline Ni-W samples with grain sizes ranging from 3to150nm. In these tests, the maximum penetration depth was 1µm and the characteristic strain rate 4 was estimated to be ≈103s−1. Trelewicz and Schuh (2008) found ≈10%increase in the material hardness under dynamic loading as compared to the quasi-static loading for grain sizes below 40nm. Using similar setup to Trelewicz and Schuh (2008), Arreguin-Zavala et al. (2013) performed nano-impact indentation experiments on nanostructured and ultraﬁne-grained Al-Si claddings at various temperatures ranging from 623K to 773K. The dynamic hardness was found to increase nonlinearly as the penetration depth decreases, from ≈1.5GPa for6µm to≈3.5GPa for 2.5µm. Later, Qin et al. (2019) determined the dynamic hardness of single crystal Al (110) at strain rates that were estimated to be ≈104s−1at the ﬁrst stages of the indentation process. The comparison of the high strain rate experimental results with quasi-static nano-indentation tests showed that the dynamic hardness of Al (110) single crystals is not only greater but is also more sensitive to size eﬀect. Mathematical modeling of the indentation process of metallic materials is in general based on the cavity expansionmodelfollowingthepioneeringworkofJohnson(1970), whoconsideredthattheindenterisencasedin a semi-cylindrical or hemispherical core (the cavity) subjected to hydrostatic pressure equal to the indentation pressure, and the core expands (i.e. material displaces radially) as the indenter penetrates the specimen. While outside the core, the stress and strain ﬁelds are considered to be radially symmetric and are the same as in an inﬁnite body containing a cylindrical or spherical cavity under pressure. Johnson (1970) obtained quasi- static solutions for elastic perfectly-plastic materials, and validated the theoretical results with experimental data obtained with diﬀerent wedge-shape indenters. Later, Fleck et al. (1992) extended the cavity expansion model of Johnson (1970) to elasto-plastic porous solids. These authors compared the theoretical results with ﬁnite element calculations, and showed that for nearly rate-independent porous materials, the cavity expansion model provides accurate qualitative predictions of the eﬀect of porosity, elastic modulus and indenter angle on the quasi-static indentation pressure. Mata et al. (2006) and Gao et al. (2006) revisited the analogy between indentation experiments and cavity expansion, and developed a quasi-static cavity expansion model for strain hardening elasto-plastic solids. These works led to closed-form solutions for the extension of the plastic zone that develops near the indenter (Mata et al., 2006) and demonstrated that for a given indenter geometry, indentation hardness depends on Young’s modulus, yield stress and strain-hardening exponent of the material (Gao et al., 2006). The plastic strain gradient induced size eﬀect is also incorporated in the quasi-static cavity expansion model by Gao (2006). In line with the experimental observations, Gao (2006) obtained the indentation size dependent hardness of the material under quasi-static loading conditions (smaller the indentation depth, greater the hardness of the material). Note that all the works discussed in the preceding paragraph are for the quasi-static indentation problems. This is despite the fact that numerous experimental works (as detailed in the earlier paragraphs) have high- lighted the eﬀects of dynamics and indentation size at high loading rates. Thus, here we present a dynamic cavity expansion model to analyze the high rate indentation response of elasto-plastic porous materials while 5 accounting for the plastic strain gradient induced size eﬀect. We note that dynamic cavity expansion models have been utilized to study the mechanisms of deep penetration and perforation in structural impact problems (e.g. Cohen and Durban (2013a)). However, their extension and application to estimate dynamic hardness of the material in which the inertial forces and size eﬀects are important has not been addressed before (to the best of our knowledge). The remainder of the paper is organized as follows. Section 2 describes the elasto- plastic constitutive framework used to model the material behaviour, in which yielding is described using the modiﬁed Gurson ﬂow potential while incorporating the size eﬀect following the work of Chen and Yuan (2002). The dynamic spherical cavity expansion model used to analyze dynamic indentation is developed in Section 3. Section 4 compares the results of the theoretical model with experimental results of Poole et al. (1996), McElhaney et al. (1998), Lim and Chaudhri (1999) and Hassani et al. (2020) for annealed OFHC copper, showing that the cavity expansion model developed herein very well captures the increase in material hardness with decreasing indentation depth and increasing strain rate. The results of parametric analyses elucidating the speciﬁc roles of indentation speed, size eﬀect and initial porosity, on the material hardness and cavitation ﬁelds that develop during the indentation process are also presented in Section 4. Finally, the main conclusions derived from this work are listed in Section 5. 2. Constitutive framework This section describes the elasto-plastic constitutive framework used to model the mechanical behavior of the porous material. Here, yielding is described by the heuristic modiﬁcation of the Gurson-Tvergaard ﬂow potential as proposed by Chen and Yuan (2002) in which the size eﬀect is included through by deﬁnition of the ﬂow strength of the matrix material. The rate of the deformation tensor, d, is taken to be the sum of an elastic part, de, and a plastic part, dp: d=de+dp(1) with the elastic part being related to the rate of the Cauchy stress tensor σby the following hypo-elastic law: Oσ=C:de(2) whereO ()denotes an objective derivative and Cis the fourth order isotropic elastic tensor given as: C=E 1 +νI/prime+E 3 (1−2ν)1⊗1 (3) 6 whereEis the Young’s modulus, νthe Poisson’s ratio, 1the unit second order tensor and I/primethe unit-deviatoric fourth order tensor. The plastic part of the rate of deformation tensor is written as: dp=˙λ∂Φ ∂σ(4) where ˙λis the rate of plastic multiplier and Φis the plastic ﬂow potential: Φ =/parenleftbiggσe ¯σ/parenrightbigg2 + 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 eﬀective stress, σe, and the hydrostatic stress, σh, are: σe=/radicalbigg3 2s:s, σh=1 3σ:1,s=σ−σh1 withsbeing the deviatoric part of the Cauchy stress tensor. Similarly to Chen and Yuan (2002), the ﬂow strength of the matrix material ¯σis deﬁned using a simple form of the generalized gradient plasticity model proposed by Aifantis (1984): ¯σ= ¯σhom(¯εp,T) +El/bardbl∇¯εp/bardbl (6) whereEistheYoung’smodulus(asmentionedbefore), listhelengthscaleparameterand /bardbl∇¯εp/bardbl= (∇¯εp·∇¯εp)1 2 stands for the Euclidean norm of ∇¯εp, with ¯εp=/integraltextt 0˙¯εp(τ)dτbeing the eﬀective plastic strain in the matrix material and ˙¯εpthe eﬀective plastic strain rate. The advantages of this simple gradient plasticity model are that it introduces a single length scale parameter and dismisses high order derivatives, which greatly facilitates the formulation of the dynamic cavity expansion model and the interpretation of results. Note that, for the sake of simplicity, we have considered that the parameter lis constant. Here ¯σhomis the ﬂow strength for homogeneous deformation that depends on the eﬀective plastic strain in the matrix material ¯εpand the temperature Tas follows: ¯σhom(¯εp,T) =σ0/parenleftbigg 1 +¯εp ε0/parenrightbiggn/parenleftBig 1 +bexp−c(T0−273)/parenleftBig exp−c(T−T0)−1/parenrightBig/parenrightBig (7) whereσ0is the initial yield stress of the matrix material, nis the strain hardening exponent, ε0is the reference strain, and b,candT0are parameters that determine the temperature sensitivity of the material. 7 The equality between the rates of macroscopic plastic work and matrix plastic work (Chen and Yuan, 2002): σ:dp= (1−f) ¯σ˙¯εp(8) provides a relation between the plastic part of the rate of deformation tensor and the eﬀective plastic strain rate in the matrix material. Next, assumingtheplasticincompressibilityofthematrixmaterial, theevolutionofthevoidvolumefraction is given as: ˙f= (1−f)dp:1 (9) Furthermore, assuming adiabatic conditions and considering that plastic work is the only source of heating, the temperature evolution is calculated as: ρCp˙T=χσ:dp(10) withρ,Cpandχbeing the current material density, the speciﬁc heat and the Taylor-Quinney coeﬃcient, respectively. The values of the parameters of the constitutive model used in the cavity expansion calculations presented in Section 4 are given in Table 1. The ﬂow strength for homogeneous deformation has been calibrated against the experimental results for annealed OFHC copper by Tanner and McDowell (1999). The elastic parameters, the initial material density and the speciﬁc heat also correspond to copper, and the standard values used in the literature are taken for the ﬂow potential parameters q1andq2, and the Taylor-Quinney factor χ. In the analysis performed in Section 4, parametric calculations are carried out for the values of the initial void volume fraction,f0and the length scale parameter, l. Note that, for the sake of simplicity, we have not considered the strain rate dependence of the ﬂow strength of the matrix material. While we are aware that this is a rather crude assumption for most metallic materials, it greatly facilitates the derivation of the theoretical model in Section 3 (rate-independent materials allow for self-similar cavity expansion solutions) and to interpret the results presented in Section 4 (isolating the eﬀect of inertia in the cavitation ﬁelds at high strain rates). Note that including the strain rate sensitivity of the material in the constitutive model also introduces a time scale into the cavity expansion problem, see dos 8 Symbol Property and units Value ρ0Initial density/parenleftbigkg/m3/parenrightbig8940 CpSpeciﬁc heat (J/kg K), Eq. (10) 384.6 EYoung modulus (GPa), Eq. (3) 116 νPoisson ratio, Eq. (3) 0.31 q1Material parameter, Eq. (5) 1.25 q2Material parameter, Eq. (5) 1.0 f0Initial void volume fraction, Eq. (5) 0.0,0.05,0.1 σ0Reference yield stress (MPa), Eq. (7) 55 nStrain hardening sensitivity parameter, Eq. (7) 0.32 lLength scale parameter (nm), Eq. (6) 0,10,20,30,50 ε0Reference strain, Eq. (7) 0.0026 bTemperature sensitivity parameter, Eq. (7) 10 cTemperature sensitivity parameter/parenleftbigK−1/parenrightbig, Eq. (7) 1.23×10−4 T0Reference temperature (K), Eq. (7) 293 χTaylor-Quinney coeﬃcient, Eq. (10) 0.9 Table 1: Values of the parameters of the constitutive model used in the cavity expansion calculations presented in Section 4. Santos et al. (2020). The Gurson porous plasticity model was derived based on several assumptions, including that the shape of thevoidremainssphericalduringthewholeloadingprocess(i.e. hightriaxialityloadings). However, theGurson model has been in general employed to simulate the response of porous materials subjected to indentation and penetration and has been shown to provide results that are more or less consistent with the experimental observations. For instance, Cohen and Durban (2013a) derived a cavity expansion model to calculate the penetration resistance of highly porous titanium plates, and the analytical results were in reasonable accord with experimental data reported in the literature, showing a decrease in the ballistic limit with increase in material porosity. Nevertheless, a future work will be to extend the analytical model presented in this paper to also consider other damage mechanisms that develop in the material during indentation such as, nucleation of secondary voids and shear softening and localization due to pre-existing voids. 3. Dynamic indentation model In this section, we extend the self-similar cavity expansion model developed by Cohen and Durban (2013b) for high velocity penetration problems to model dynamic indentation of rate-independent elasto-plastic porous solids exhibiting size eﬀect. The mechanical behavior of the material is described using the constitutive frame- work presented in Section 2. Following Johnson (1970), the process of conical indentation is idealized by the radial expansion of a 9 spherical cavity. The main assumption is that the indenter is encased in a rigid hemispherical core of radius, a, (the expanding cavity) which is subjected to a hydrostatic stress equal to the indentation pressure, p, see Fig. 1. This approach considers that neither pile-up nor sink-in of the free surface of the material occurs during the indentation. While this is a rough assumption, the works of Johnson (1970) and Fleck et al. (1992), among others, have shown that it is a reasonable approximation for a variety of metallic materials. Let us consider an Eulerian spherical coordinate system (r,θ,φ )with the origin located at the center of the hydrostatic core O, see Fig. 1. The indenter penetrates the solid at a constant velocity ˙h, such that a steady state indentation pressure is achieved whereby pdoes not change with time, thus causing the material to displace radially at velocity ˙r(i.e. ˙ris the radial velocity of a material point), see Fleck et al. (1992). The incompressibility of the material leads to the following relation (Johnson, 1970; Fleck et al., 1992): 2 ˙r=˙hatr=a(the cavity wall) (11) where ˙h= ˙atanβby geometry (see Fig. 1). Thus, the material velocity, ˙r, atr=arelates to the core expansion velocity, ˙a, by: ˙r ˙a=tanβ 2atr=a(the cavity wall) (12) Following Johnson (1970) and Fleck et al. (1992) and Danas et al. (2012), among others, in all the calcu- lations presented in this work the cone angle βis taken as 19.7◦in order to give the same displaced volume as the Vickers pyramid. Note that, for this cone angle, the material and the hydrostatic core (cavity) do not expand at the same speed, i.e. ˙r/negationslash= ˙a. In fact, for β < 63.43◦, the velocity of the cavity is greater than the velocity of the material. 10 Elastoplastic zone Elastic zoner=a r=ri r=rw rhIndenter Symmetry line Hydrost. coreO Figure 1: Illustration of the indentation problem. Following Durban and Masri (2004), we take the non-dimensional radial coordinate ξ=r/aas the only independentvariableoftheproblem(self-similarsteady-statesolution). Accordingly, thederivativewithrespect to the radial coordinate is: ∂() ∂r=∂() ∂ξ∂ξ ∂r= ()/prime1 a(13) where the prime superscript denotes diﬀerentiation with respect to ξ. In addition, the time derivative is: ˙() =∂() ∂ξ˙ξ= ()/prime˙a a(v−ξ) (14) withv=˙r/˙abeing the dimensionless radial velocity. Since ˙ais assumed to be constant (because ˙his assumed to be constant), the dimensionless acceleration is ˙v=¨r/˙a. Due to the spherical symmetry of the problem, the only active components of the Cauchy stress tensor are σrrandσθθ=σφφ. Hence, the linear momentum balance reduces to a single equation in the radial direction: ∂σrr ∂r+2 r(σrr−σθθ) =ρ¨r (15) which, employing equations (13) and (14), is rewritten as: Σ/prime rr+2 ξ(Σrr−Σθθ) =m2ρ ρ0(v−ξ)v/prime(16) where Σrr=σrr/EandΣθθ=σθθ/Eare the dimensionless radial and circumferential stresses, respectively, and 11 m=˙a/√ E/ρ0is the ratio between ˙aand the elastic wave velocity in a long rod/radicalbig E/ρ0, withρ0being the initial material density. Moreover, the only active components of the rate of deformation tensor are: drr=∂˙r ∂r(17) dθθ=dφφ=˙r r(18) The hypo-elastic law introduced in equation (2) can be rewritten as: de rr=1 E˙σrr−2ν E˙σθθ (19) de θθ=de φφ=(1−ν) E˙σθθ−ν E˙σrr (20) where in absence of material spin, the objective derivativeO ()(see equation (2)) has been replaced by a time derivative˙(). Furthermore, using the ﬂow potential (5), and the plastic work equivalence (8), the ﬂow rule given by equation (4) can be expressed as: dp rr= (1−f)˙¯εp˜Nrr (21) dp θθ=dp φφ= (1−f)˙¯εp˜Nθθ (22) 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(23) ˜Nθθ=1 2Σe¯Σ+q1q2fsinh/parenleftBig 3q2Σh 2¯Σ/parenrightBig /parenleftBig Σe¯Σ/parenrightBig2+q1q2fsinh/parenleftBig 3q2Σh 2¯Σ/parenrightBig 3Σh 2¯Σ(24) 12 where Σe=σe/E=(σθθ−σrr)/Eis the dimensionless eﬀective von Mises stress, ¯Σ = ¯σ/Eis the dimensionless matrix ﬂow strength and Σh=σh/Eis the dimensionless hydrostatic stress. Using the additive decomposition of the rate of deformation tensor, Eq. (1), together with equations (13)-(14) and (17)-(22), the following expressions are obtained: v/prime= (v−ξ)/bracketleftBig Σ/prime rr−2νΣ/prime θθ+ (1−f) (¯εp)/prime˜Nrr/bracketrightBig (25) v ξ= (v−ξ)/bracketleftBig −νΣ/prime rr+ (1−ν) Σ/prime θθ+ (1−f) (¯εp)/prime˜Nθθ/bracketrightBig (26) For a plastic loading, the relation between Σeand ¯Σis obtained when the ﬂow potential, Φ, given in Eq. (5) is zero: /parenleftbiggΣe ¯Σ/parenrightbigg2 + 2q1fcosh/parenleftbigg3q2Σh 2¯Σ/parenrightbigg −1−(q1f)2= 0 (27) For a spherically symmetric problem, using Eq. (13), the dimensionless ﬂow strength of the matrix material reads (see Eq. (6)): ¯Σ = ¯Σhom(¯εp,T) +l a(¯εp)/prime(28) where ¯Σhom=¯σhom/Eis the dimensionless ﬂow strength for homogeneous deformation given by (see Eq. (7)): ¯Σhom(¯εp,T) = Σ 0/parenleftbigg 1 +¯εp ε0/parenrightbiggn/parenleftBig 1 +bexp−c(T0−273)/parenleftBig exp−c(T−T0)−1/parenrightBig/parenrightBig (29) with Σ0=σ0/Ebeing the dimensionless initial yield stress of the matrix material. It is apparent from Eq. (28) 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(Danas et al., 2012), such that the size eﬀect decreases as the cavity radius increases (the cavitation ﬁelds becoming independent of lfora>>l). In addition, recalling the geometric relation h=atanβ, it is shown that increasing/decreasing the length scale parameter lhas the same eﬀect on the cavitation ﬁelds as decreasing/increasing the penetration depth h. The evolution of porosity given by expression (9) is rewritten using Eqs. (14) and (21)-(22), as: 13 f/prime= (1−f)2/parenleftBig˜Nrr+ 2˜Nθθ/parenrightBig (¯εp)/prime(30) Moreover, using the work equivalence principle, Eq. (8), and the time derivative introduced in Eq. (14), the energy balance given in Eq. (10) is rewritten as: ρCp ET/prime=χ(1−f)¯Σ (¯εp)/prime(31) Furthermore, the mass conservation principle in Eulerian description, and in the absence of material spin, reads: ˙ρ+ρ(drr+ 2dθθ) = 0 (32) Using equations (1), (9), (14) and (19)-(20), we obtain: ρ/prime ρ=−/bracketleftbigg 3(1−2ν)Σ/prime h+f/prime (1−f)/bracketrightbigg (33) Integrating equation (33) from a given coordinate ξto the elastic wave front ξw, wheref=f0,ρ=ρ0, and Σh= 0(see Fig. 1), a closed-form relation for the density ratio is obtained: ρ ρ0=(1−f) (1−f0)exp−3(1−2ν)Σh (34) In this model, Eqs. (16), (25)-(31) and (34) provide a system of three algebraic relations and six diﬀerential equations with nine unknowns/parenleftBig v,Σrr,Σθθ,¯Σ,¯Σhom,¯εp,f,T,ρ/parenrightBig . The problem is completed with the following boundary conditions: v=tanβ 2and Σrr=−Patξ= 1 (the cavity wall) (35) and v= 0 and Σ rr= Σθθ= 0 atξ=ξw(the elastic wave front) (36) whereP=p/Eis the dimensionless cavity pressure, see Fig. 1. Notice that the velocity boundary condition at 14 ξ= 1corresponds to equation (12). The integration is performed over the independent variable ξ, fromξwup toξ= 1(see Fig. 1). An iterative shooting method is used to enforce the velocity boundary condition at the cavity wall. The initial conditions, with an unknown constant, and the location of the elastic wave front, are determined employing the elastic solution derived by Durban and Masri (2004). Notethatconsideringthematerialstrainratesensitivityintheindentationmodelrequiresadditionaleﬀorts, in line with the recent paper of dos Santos et al. (2020), who developed a time-dependent cavity expansion model for size-independent elasto-viscoplastic materials modeled with von Mises plasticity, and performed ﬁnite element simulations to estimate the extend of the transient behavior that precedes the quasi-constant velocity expansion of the cavity. For rate-dependent materials it is essential to identify the loading time from which ˙abecomes quasi-constant because if the indentation process is much faster than the time required for the quasi-constant velocity cavity expansion to appear, then the constant velocity cavitation ﬁelds (hypothesis ˙a=constant, see below Eq. (14)) do not describe the actual underlying physical phenomena. Thus, the extension of the dynamic cavity expansion model presented in this paper to rate-dependent materials is left for a future work, in which ﬁnite element simulations should also be performed. 4. Results In this section we present the results obtained with the cavity expansion model developed in Section 3, for indentation velocities ˙hranging from 10−4m/s to≈400m/s. We compare the theoretical predictions with the experimentalresultsofPooleetal.(1996), McElhaneyetal.(1998), LimandChaudhri(1999)andHassanietal. (2020) to demonstrate the predictive capabilities of our analytical model. In addition, we perform parametric analyses to bring out the eﬀects that inertia, size eﬀect and initial porosity have on the material hardness and the elasto-plastic ﬁelds that develop near the indenter during the indentation process. We are aware that for the high indentation speeds investigated in this work it would have been pertinent to consider the material strain rate sensitivity, using a rate-dependent constitutive relation to describe the material ﬂow strength. However, as mentioned before, material viscosity introduces a time scale into the cavity expansion process which prevents using a steady-state self-similar approach to solve the indentation problem (the cavitation ﬁelds become time dependent). 4.1. Fully dense materials Figure 2 shows the evolution of the material hardness Hwith the indentation depth hpredicted by the theoretical model, and the experimental data of Poole et al. (1996), McElhaney et al. (1998) and Lim and 15 Chaudhri (1999) for annealed OFHC copper (using pyramidal indenters) under quasi-static loading conditions. The experimental results show that the hardness decreases nonlinearly with the indentation depth, displaying a concave-upwards shape and almost reaching an horizontal asymptote for large values of h. The hardness at h= 0.1µmisapproximatelyﬁvetimesgreaterthanthatat h= 10µm. Thedependenceof Hontheindentation depth shows the indentation size eﬀect. In the theoretical model predictions, the hardness corresponds to the hydrostatic core pressure p(see Fig. 1 and equation (35)), the indentation velocity is ˙h= 10−4m/s (quasi- static loading) and the initial porosity f0is set to 0(Poole et al. (1996); McElhaney et al. (1998); Lim and Chaudhri (1999) did not provide any measurement of the material porosity). The theoretical results are shown for diﬀerent values of the length scale parameter l= 0,10,20,30and50nm. Forl= 0, the cavity expansion model does not account for the size eﬀect, and the hardness is independent of the indentation depth. For l/negationslash= 0, the evolution of the hardness displays a concave-upwards shape (in qualitative agreement with the experiments), with increasing slope as the length scale parameter increases (the increase of lboosts the size eﬀect in the indentation hardness). Notice that, while the hardness for small indentation depth of 0.1µm is three times greater for l= 50nm than for l= 10nm, the value of Hbecomes practically independent of the length scale parameter for h= 10µm. Although not shown here, similar trends are obtained for all other indentation speeds considered in this paper. Moreover, note that the theoretical predictions for l= 20and 30nm show reasonable quantitative agreement with the experiments, for the whole range of indentation depths investigated, providing some validity to the cavity expansion model and the simple gradient plasticity model used in this work. 16 012345 0.1 1 10O 10 20 30 50 J L NExperiments: Poole et al. (1996) Experiments: McElhaney et al. (1998) Experiments: Lim and Chaudhri (1999)Material hardness, H (GPa)h=10-4 m/s. f 0=0Theoretical model: l=0 Theoretical model: l=10 nm Theoretical model: l=20 nm Theoretical model: l=30 nm Theoretical model: l=50 nm Indentation depth, h ( μm) Figure 2: Evolution of the material hardness Hwith the indentation depth hpredicted by the cavity expansion model for diﬀerent values of the length scale parameter l= 0,10,20,30and50nm. The hardness corresponds to the hydrostatic core pressure p, the indentation velocity is ˙h= 10−4m/s and the initial porosity is f0= 0. The experimental data of Poole et al. (1996), McElhaney et al. (1998) and Lim and Chaudhri (1999) for annealed OFHC copper (using pyramidal indenter) under quasi-static loading conditions are also plotted. The eﬀect of the indentation speed on the theoretical predictions of the material hardness is investigated in Fig. 3(a). We now ﬁx l= 30nm as this value of the length scale parameter provided good agreement with the experiments in Fig. 2. Speciﬁcally, Fig. 3(a) shows the evolution of the hardness, H, with the indentation depth,h, for six diﬀerent indentation speeds, ˙h=10−4,50,100,200,300and400m/s. We consider the same range of indentation depths shown in Fig. 2 (note that 0.1µm≤h≤10µm is a typical range of indentation depth investigated in the literature, e.g. Shu and Fleck (1998)). The greater indentation speeds are similar to the velocities used in the dynamic experiments performed by Sundararajan and Tirupataiah (2006) and Hassani et al. (2020), among others. For all indentation speeds, the H−hcurves display a concave-upwards shape, with decreasing hardness as the indentation depth increases (as in Fig. 2). A key outcome is that the increase in the material hardness with the indentation speed is signiﬁcant (only) for ˙h≥100m/s, which suggests that this is the critical indentation speed above which inertial eﬀects become important (at least for this cavity expansion model and the set of constitutive parameters considered). This conclusion is reinforced in Fig. 3(b), that displays the evolution of the material hardness, H, with the indentation speed, ˙h, at four values of the indentation depth, h= 0.1,0.2,1and3µm. It is shown that the material hardness is virtually insensitive to the indentation speed for ˙h≤100m/s. In addition, note that the increase in Hthat occurs for ˙h≥100m/s is more pronounced as the indentation depth increases. This behavior is physically sound: the 17 indentation depth increases the material volume displaced, and thus inertia eﬀects, which are responsible for the increase in material hardness at high indentation speeds (in the absence of strain rate sensitivity, the jump in hardness at≈100m/s is attributed to inertia). Recall that assuming that the material is rate-independent has the advantage of isolating the eﬀect of inertia in the cavitation ﬁelds at high strain rates. 01234567 0.1 1 100.0001 50 100 200 300 LMaterial hardness, H (GPa) Indentation depth, h ( μm)f 0=0h=10-4 m/s h=50 m/s h=100 m/s h=200 m/s h=300 m/s h=400 m/s. l=30 nm. . . . 3.7 107s-1 3.7 107s-1. .. (a) 01234567 10-50.0001 0.001 0.01 0.1 1 101102103B D F Hf0=0l=30 nmh=0.1 μm h=0.2 μm h=1μm h=3 μmMaterial hardness, H (GPa) Indentation speed, h (m/s). (b) Figure 3: Dynamic indentation model. The initial porosity, f0= 0and the length scale parameter, l= 30nm. (a) Evolution of the material hardness Hwith the indentation depth hfor six indentation speeds. ˙h=10−4,50,100,200,300and400m/s. (b) Evolution of the material hardness Hwith the indentation speed ˙hat four values of the indentation depth, h=0.1,0.2,1and 3µm. Figure 4 displays the evolution of the material hardness, H, with the eﬀective plastic strain rate, ˙¯εp, predicted by the theoretical model, and the comparison with the experimental data in Hassani et al. (2020) for OFHC copper. Note that these authors did not provide any experimental measurement of the material porosity. Therefore, as in Fig. 3, the dynamic cavity expansion calculations are performed for f0= 0while l= 30nm. The theoretical results are shown at three values of the indentation depth, h= 1,2and3µm. The experimental results for strain rates lower than 104s−1were taken by Hassani et al. (2020) from uniaxial stress data reported in Armstrong and Walley (2008), and converted to hardness with a Tabor factor of 3. The experimental results for strain rates greater than 106s−1were obtained by Hassani et al. (2020) via high velocity impact indentations with spherical micro-particles of average diameter of 14µm. The micro-particles were impacted onto copper targets at velocities near vi= 425±8m/s, leading to a maximum penetration depth of2.79µm. ThecharacteristicstrainrateoftheimpactindentationexperimentswascalculatedbyHassanietal. (2020) according to the relation: ˙¯εp=vi/φp, whereφpis the diameter of the impacting particle. In contrast, the model predictions consider the eﬀective plastic strain rate at the cavity wall, and a conical indenter (see Section 3). Nonetheless, there is a reasonable qualitative and quantitative agreement between the experimental data and the theoretical predictions, which provides additional validation for the analytical model developed 18 in this paper. For the low strain rates, the theoretical predictions for h= 1µm are closer to the experiments, however, at high strain rates, the best quantitative agreement with the tests is obtained for h= 3µm. A key outcome is that the theoretical model captures the increase by a factor of 10of the material hardness at high strain rates, making apparent the important role of inertia in high velocity indentation experiments, in agreement with the theoretical predictions previously shown in Fig. 3. Note that the cavity expansion results for h= 1,2and3µm intersect at 107s−1such that for strain rates <107s−1, the greater hardness is predicted for the smaller indentation depth, and vice versa for strain rates >107s−1. As mentioned before, increasing the indentation depth increases the material volume displaced, and thus increases the inertial eﬀects for a ﬁxed value of the high strain rate. Notice that the relation between the strain rate and the indentation speed predicted by the theoretical model is linear, and the smaller the indentation depth, the higher the strain rate, see Fig. 5. Hence, in order to reach a ﬁxed value of the high strain rate, the higher the indentation depth, the higher the indentation speed is required, which boosts the material hardness (the yellow markers in Figs. 3(a) and 5 correspond to a strain rate of 3.7·107s−1). On the other hand, Fig. 4 also shows that, while the increase in hardness predicted by the theoretical model at 107s−1seems to be too sharp, and a smoother evolution could have been expected (there is a sense of smooth increase of hardness in the experiments for ≈104s−1), the absence of experimental data for strain rates ranging from 104s−1to106s−1prevents drawing deﬁnite conclusions in this regard. Nevertheless, it is expected that including the strain rate sensitivity of the material in the cavity expansion model would provide a more gradual transition between the hardness obtained for 104s−1and107s−1, and also greater values of H, leading to the theoretical predictions being closer to the experiments performed for ˙¯εp>107s−1. Note that the strain rate sensitivity would lead to an increase in the material ﬂow strength with increasing strain rate, that shall trigger a gradual increase in the material hardness, particularly for strain rates below 107s−1(i.e. before inertia becomes the main contributor to material hardness). This point shall be addressed in future works, using an alternative method to the self-similar assumption used in this paper to solve the cavity expansion problem, see dos Santos et al. (2020). 19 0123456 10-50.001 0.1 101103105107109B K D F H Effective plastic strain rate, εp (s-1)Material hardness, H (GPa)Experiments: Armstrong and Walley (2008) Experiments: Hassani et al. (2020) - h=2.79 μm Theoretical model: h=1 μm Theoretical model: h=2 μm Theoretical model: h=3 μm f0=0l=30 nm . Figure 4: Evolution of the material hardness Hwith the eﬀective plastic strain rate ˙¯εppredicted by the theoretical model and comparison with the experimental data reported in Hassani et al. (2020) for OFHC copper. In the theoretical model predictions, the initial porosity, f0= 0and the length scale parameter, l= 30nm. Theoretical results are shown at three values of the indentation depth,h= 1,2and3µm. The experimental results for strain rates lower than 104s−1were taken by Hassani et al. (2020) from uniaxial stress data of Armstrong and Walley (2008), and converted to hardness with a Tabor factor of 3. The experimental results for strain rates greater than 106s−1were obtained by Hassani et al. (2020) performing high velocity indentation experiments of spherical micro-particles impacting onto OFHC copper targets. 10102103104105106107108109 10-50.0001 0.001 0.01 0.1 1 101102103B D F H Indentation speed, h (m/s)Effective plastic strain rate, εp (s-1). .h=0.1 μm h=0.2 μm h=1 μm h=3 μmf 0=0l=30 nmsee Fig. 3(a) 3.7 107s-1. Figure 5: Dynamic indentation model. Eﬀective plastic strain rate ˙¯εpversus indentation speed ˙hat four values of the indentation depth,h= 0.1,0.2,1and3µm. The initial porosity, f0= 0and the length scale parameter, l= 30nm. The inﬂuence of the indentation speed on the cavitation ﬁelds that develop near the indenter is investigated 20 in Fig. 6, which displays theoretical results for the evolution of the dimensionless eﬀective stress Σe, the dimensionless radial stress Σrr, the eﬀective plastic strain ¯εp, and the eﬀective plastic strain rate ˙¯εpwith the dimensionless radial coordinate ξ. The initial porosity, f0= 0, the indentation depth, h= 0.1µm and the length scale parameter, l= 30nm. Results are shown for ﬁve values of the indentation speed, ˙h=10−4,50, 100,200and300m/s. The eﬀective stress, the radial stress, the eﬀective plastic strain and the eﬀective plastic strain rate all decrease with the radial coordinate, displaying a concave-upwards shape, with the maximum values of Σe,Σrr,¯εpand ˙¯εplocated at ξ= 1(the cavity wall). Notice that the indentation speed hardly aﬀects the eﬀective stress Σeand the eﬀective plastic strain ¯εp(recall that the material is taken to be rate- independent). In contrast, the increase in ˙hboosts the radial stress and the eﬀective plastic strain rate near the cavity. Speciﬁcally, the radial stress at the cavity wall, which coincides with the material hardness (opposite sign), is 65%greater, in absolute value, for ˙h= 300m/s than for ˙h= 10−4m/s. Moreover, the strain rate at ξ= 1is six times greater for ˙h= 300m/s than for ˙h= 50m/s, which shows that at the cavity wall there is a linear relationship between the eﬀective plastic strain rate and the indentation speed, as anticipated in Fig. 5. Notice that the eﬀective plastic strain rate for ˙h= 10−4m/s is very low in comparison to the other indentation speeds (it is hardly noticeable in the graph), with a maximum value of ≈250s−1. 21 00.010.020.030.04 1 2 3 4 50.0001 50 100 200 300 Dimensionless radial coordinate, ξh=10-4 m/s h=50 m/s h=100 m/s h=200 m/s h=300 m/s. . . . .Dimensionless effective stress, Σ e f0=0l=30 nmh=0.1 μm (a) -0.04 -0.03 -0.01 0 1 2 3 4 50.0001 50 100 200 300Dimensionless radial stress, Σrr Dimensionless radial coordinate, ξh=10-4 m/s h=50 m/s h=100 m/s h=200 m/s h=300 m/s. . . . .f 0=0l=30 nmh=0.1 μm (b) 00.020.040.060.080.10.120.14 1 2 3 4 50.0001 50 100 200 300Effective plastic strain, ε p Dimensionless radial coordinate, ξh=10-4 m/s h=50 m/s h=100 m/s h=200 m/s h=300 m/s. . . . .f0=0l=30 nmh=0.1 μm (c) 02 1084 1086 1088 108 1 2 3 4 50.0001 50 100 200 300 Dimensionless radial coordinate, ξh=10-4 m/s h=50 m/s h=100 m/s h=200 m/s h=300 m/s. . . . .Effective plastic strain rate, ε p. f 0=0l=30 nmh=0.1 μm (d) Figure 6: Dynamic indentation model. Results for (a) dimensionless eﬀective stress Σe, (b) dimensionless radial stress Σrr, (c) eﬀective plastic strain ¯εpand (d) eﬀective plastic strain rate ˙¯εpversus dimensionless radial coordinate ξ. The initial porosity, f0= 0, the length scale parameter, l= 30nm and the indentation depth, h= 0.1µm. Results are shown for ﬁve values of the indentation speed, ˙h=10−4,50,100,200and300m/s. The inﬂuence of the indentation depth in the theoretical cavitation ﬁelds is investigated in Fig. 7, which shows the evolution of the dimensionless eﬀective stress Σe, the dimensionless radial stress Σrr, the eﬀective plastic strain ¯εp, and the eﬀective plastic strain rate ˙¯εpwith the dimensionless radial coordinate ξ, at four values ofh:0.1(the value taken in Fig. 6), 0.2,1and3µm. The results correspond to ˙h= 300m/s (the greatest value shown in Fig. 6), while f0= 0andl= 30nm. Figure 7(a) shows that the Σe−ξcurves shift downwards as the indentation depth increases, such that at the cavity wall the eﬀective stress at h= 0.1µm is≈11times greater than that at h= 3µm. In contrast, the accumulated eﬀective plastic strain is greater as the indentation depth increases (reaching ≈0.13in the case of h= 3µm), with slightly greater gradients of ¯εp ashincreases, see Fig. 7(c). Similar trends for the dependence of the eﬀective stress and the eﬀective plastic strain with the indentation depth were obtained from the quasi-static ﬁnite element calculations performed by 22 Danas et al. (2012) for fully-dense materials (see the discussion of Fig. 10 therein). Moreover, Fig. 7(b) shows that the radial stress decreases, in absolute value, as the indentation depth increases. At the cavity wall, where the radial stress equals the material hardness, the absolute value of Σrrath= 0.1µm is twice greater than that ath= 3µm. This is consistent with the results presented in Fig. 3(a). Note that, in absence of any size eﬀect, eﬀective stress, radial stress and eﬀective plastic strain are independent of the indentation depth, so that these results bring out the important role of the size eﬀect on the cavitation ﬁelds that develop during dynamic indentation. The relaxation of the material as the indentation proceeds is illustrated in Fig. 7(d), which shows that the eﬀective plastic strain rate decreases with the increasing indentation depth. At the cavity wall, ξ= 1, the value of ˙¯εpath= 0.1µm is approximately 20times greater than that at h= 3µm. Note that the eﬀective plastic strain rate depends on the indentation depth regardless of whether the size eﬀect is considered or not. 00.010.020.030.04 1 2 3 4 50.1 0.2 1 3 Dimensionless radial coordinate, ξDimensionless effective stress, Σ e f 0=0l=30 nmh=300 m/s.h=0.1 μm h=0.2 μm h=1 μm h=3 μm (a) -0.04 -0.03 -0.02 -0.01 0 1 2 3 4 50.1 0.2 1 3Dimensionless radial stress, Σrr Dimensionless radial coordinate, ξf 0=0l=30 nmh=300 m/s. h=0.1 μm h=0.2 μm h=1 μm h=3 μm (b) 00.020.040.060.080.10.120.14 1 2 3 4 50.1 0.2 1 3Effective plastic strain, ε p Dimensionless radial coordinate, ξf 0=0l=30 nmh=300 m/s.h=0.1 μm h=0.2 μm h=1 μm h=3 μm (c) 02 1084 1086 1088 108 1 2 3 4 50.1 0.2 1 3f 0=0l=30 nmh=300 m/s.h=0.1 μm h=0.2 μm h=1 μm h=3 μmEffective plastic strain rate, ε p. Dimensionless radial coordinate, ξ (d) Figure 7: Dynamic indentation model. Results for (a) dimensionless eﬀective stress Σe, (b) dimensionless radial stress Σrr, (c) eﬀective plastic strain ¯εpand (d) eﬀective plastic strain rate ˙¯εpversus dimensionless radial coordinate ξ. The initial porosity, f0= 0, the length scale parameter, l= 30nm and the indentation speed, ˙h= 300m/s. Results are shown at four values of the indentation depth, h=0.1,0.2,1and3µm. 23 4.2. Porous materials Figure 8 shows the predictions of the cavity expansion model for the evolution of the material hardness, H, with the indentation depth, h, for a ﬁxed length scale parameter, l= 30nm, and three values of the initial porosity,f0=0(fully dense material), 0.05and0.1. Similar range of initial void volume fractions has been considered by Fleck et al. (1992) to study quasi-static indentation of porous solids. Moreover, other authors, e.g. Czarnota et al. (2017), have also investigated initial void volume fractions as high as 0.1to bring out the role of porosity on the dynamic response of porous metallic materials subjected to impact loading conditions. Note that exploring such a wide spectrum of initial void volume fractions allows us to elucidate the role of porosity on the cavitation ﬁelds and material hardness, and to obtain trends for the interplay between size eﬀects and initial void volume fraction. In addition, sintered and additively manufactured metals may contain initial porosity ranging from 1%to10%(e.g. see Aboulkhair et al. (2014)), so that the range of void volume fractions investigated in this work is representative of some engineering materials, if not all. As shown in Fig. 8(a), for the indentation speed, ˙h= 10−4m/s, irrespective of the initial porosity, the H−hcurves show a concave-upwards shape (as in Figs. 2 and 3(a)). For ˙h= 10−4m/s, the hardness of the material only increases slightly with decreasing initial porosity. In contrast, the results in Fig. 8(b) show that for ˙h= 300m/s the inﬂuence of the initial porosity on the material hardness is signiﬁcant and dependent on the indentation depth, such that the diﬀerences between the H−hcurves obtained for diﬀerent values of f0increase as hincreases. In fact, the slope of the H−hcurves increases with the porosity, bringing out that f0slightly broadens the range of the indentation depths for which size eﬀect plays a role on the dynamic material hardness. 012345 0.1 1 100 0.05 0.1Material hardness, H (GPa) Indentation depth, h ( μm)f 0=0 f0=0.05 f0=0.1l=30 nmh=10-4 m/s. (a) 012345 0.1 1 10B D FMaterial hardness, H (GPa) Indentation depth, h ( μm)f 0=0 f0=0.05 f0=0.1l=30 nmh=300 m/s. (b) Figure 8: Dynamic indentation model. Evolution of the material hardness Hwith the indentation depth hfor three values of the initial porosity, f0=0(fully dense material), 0.05and0.1. The length scale parameter is l= 30nm, and the indentation speed is: (a)˙h= 10−4m/s and (b) ˙h= 300m/s. 24 Figure 9 illustrates the evolution of the material hardness, H, with the length scale parameter, l, for three values of the initial porosity, f0= 0,0.05and0.1. The indentation speed, ˙h= 300m/s. The material hardness increases non-linearly with the parameter l, displaying a concave-upwards shape. The comparison of the results ath= 0.1and1µm presented in Figs. 9(a) and 9(b), respectively, shows that the increase in hardness with the length scale parameter, i.e. with the size eﬀect, is more important for the lower indentation depth, in agreement with the results shown in Fig. 2. In addition, the comparison of Figs. 8 and 9 shows that increasing the length scalelhas the same eﬀect on the material hardness as decreasing the indentation depth, h(as anticipated in Section 3, see the discussion below equation (29)). Moreover, notice that the decrease in the hardness with an increase in the initial porosity is more important for small values of the parameter l, which illustrates that size eﬀect reduces the eﬀect of the porosity on the cavitation ﬁelds (this conclusion will be further discussed in Figs. 10 and 11). 0123456 1 10 100B D FMaterial hardness, H (GPa) Length scale parameter, l (nm)f0=0 f 0=0.05 f 0=0.1h=0.1 μmh=300 m/s. (a) 0123456 1 10 100B D FMaterial hardness, H (GPa) Length scale parameter, l (nm)f 0=0 f 0=0.05 f0=0.1h=1 μmh=300 m/s. (b) Figure 9: Dynamic indentation model. Evolution of the material hardness Hwith the length scale parameter lfor three values of the initial porosity, f0=0(fully dense material), 0.05and0.1. The indentation speed is ˙h= 300m/s, and the indentation depth is: (a)h= 0.1µm and (b)h= 1µm. The role of the initial porosity on the cavitation ﬁelds is illustrated in Fig. 10, which shows the evolution of the dimensionless eﬀective stress Σe, the dimensionless radial stress Σrr, the eﬀective plastic strain ¯εp, and the eﬀective plastic strain rate ˙¯εpwith the dimensionless radial coordinate ξ, for three values of f0:0(fully dense material), 0.05and0.1. The indentation speed, ˙h= 300m/s, the indentation depth, h= 0.1µm and the length scale parameter, l= 30nm. The inﬂuence of the initial material porosity on the cavitation ﬁelds is generally small, and the evolution of Σe,Σrr,¯εpand ˙¯εpwithξshows the same concave-upwards shape for the three values of f0considered. Speciﬁcally, the eﬀective stress and the eﬀective plastic strain are hardly aﬀected by the initial porosity, and the curves obtained for diﬀerent values of f0virtually overlap with each other, 25 see Figs. 10(a) and 10(c). Moreover, the radial stress, Fig. 10(b), slightly decreases with the initial porosity, especially for intermediate values of ξ. On the other hand, Fig. 10(d) shows that the ˙¯εp−ξcurves intersect for an intermediate value of ξ, such that the greatest eﬀective plastic strain rate corresponds to f0= 0.1near the cavity wall, and to f0= 0forξ>2.2. 00.010.020.030.04 1 2 3 4 5f=0 f=0.05 f=0.1 Dimensionless radial coordinate, ξDimensionless effective stress, Σ e h=0.1 μm l=30 nmh=300 m/s. f0=0 f0=0.05 f 0=0.1 (a) -0.04 -0.03 -0.02 -0.01 0 1 2 3 4 5f=0 f=0.05 f=0.1Dimensionless radial stress, Σrr Dimensionless radial coordinate, ξh=0.1 μm l=30 nmh=300 m/s. f 0=0 f 0=0.05 f0=0.1 (b) 00.020.040.060.080.10.12 1 2 3 4 5f=0 f=0.05 f=0.1Effective plastic strain, ε p Dimensionless radial coordinate, ξh=0.1 μm l=30 nmh=300 m/s. f0=0 f0=0.05 f 0=0.1 (c) 02 1084 1086 1088 1081 109 1 2 3 4 5f=0 f=0.05 f=0.1h=0.1 μm l=30 nmh=300 m/s. f 0=0 f 0=0.05 f 0=0.1Effective plastic strain rate, ε p. Dimensionless radial coordinate, ξ (d) Figure 10: Dynamic indentation model. Results for (a) dimensionless eﬀective stress Σe, (b) dimensionless radial stress Σrr, (c) eﬀective plastic strain ¯εpand (d) eﬀective plastic strain rate ˙¯εpversus dimensionless radial coordinate ξ, for three values of f0:0 (fully dense material), 0.05and0.1. The indentation speed, ˙h= 300m/s, the indentation depth, h= 0.1µm and the length scale parameter, l= 30nm. In comparison with Fig. 10, the results presented in Fig. 11 correspond to a greater indentation depth h= 1µm. The initial porosity, unlike in the case of h= 0.1µm, signiﬁcantly aﬀects the distributions of eﬀective stress Σe, radial stress Σrr, eﬀective plastic strain ¯εpand eﬀective plastic strain rate ˙¯εp. Speciﬁcally, the eﬀective stress at the cavity wall is greater for f0= 0.1, and it decreases faster as the initial porosity increases, see Fig. 11(a). In contrast, the greater value of Σrrobtained at the cavity wall for the fully dense 26 material is accompanied by a more sustained drop in the radial stress as the porosity decreases. Moreover, Fig. 11(c) shows that, while the eﬀective plastic strain at the cavity wall increases with the porosity, the plastic zone is smaller as f0increases, revealing that porosity favors localization of plastic deformation. Similarly, Fig. 11(d) shows that the eﬀective plastic strain rate is greater at the cavity wall, and decreases faster with the radial coordinate, as the initial porosity increases. Notice also that, unlike the results for the fully dense material, the ˙¯εp−ξcurves forf0= 0.05and0.1turn from concave-downwards to concave-upwards for an intermediate value of the radial coordinate, revealing the important eﬀect of porosity in the strain rate ﬁelds that develop near the indenter. The key outcome from Figs. 10 and 11 is that increasing the indentation depth, i.e. decreasing the size eﬀect, boosts the inﬂuence of the porosity on the cavitation ﬁelds. Although not shown here, the same overall trends are obtained for low indentation speed, ˙h= 10−4m/s, as well. 00.0020.0040.0060.008 1 2 3 4 5f=0 f=0.05 f=0.1 Dimensionless radial coordinate, ξDimensionless effective stress, Σ e h=1 μm l=30 nmh=300 m/s. f 0=0 f0=0.05 f0=0.1 (a) -0.02 -0.01 0 1 2 3 4 5f=0 f=0.05 f=0.1Dimensionless radial stress, Σrr Dimensionless radial coordinate, ξh=1 μm l=30 nmh=300 m/s.f0=0 f 0=0.05 f 0=0.1 (b) 00.050.10.150.2 1 2 3 4 5f=0 f=0.05 f=0.1Effective plastic strain, ε p Dimensionless radial coordinate, ξh=1 μm l=30 nmh=300 m/s. f 0=0 f0=0.05 f0=0.1 (c) 01 1082 108 1 2 3 4 5f=0 f=0.05 f=0.1h=1μm l=30 nmh=300 m/s.f 0=0 f 0=0.05 f 0=0.1 Dimensionless radial coordinate, ξEffective plastic strain rate, ε p. (d) Figure 11: Dynamic indentation model. Results for (a) dimensionless eﬀective stress Σe, (b) dimensionless radial stress Σrr, (c) eﬀective plastic strain ¯εpand (d) eﬀective plastic strain rate ˙¯εpversus dimensionless radial coordinate ξ, for three values of f0:0 (fully dense material), 0.05and0.1. The indentation speed, ˙h= 300m/s, the indentation depth, h= 1µm and the length scale parameter, l= 30nm. 27 5. Conclusions In this work we have developed a dynamic self-similar cavity expansion model to analyze the high rate indentation response of elasto-plastic porous materials while accounting for the plastic strain gradient induced size eﬀect. The main novelty of the model developed herein is the incorporation of the plastic strain gradient induced size eﬀect in the dynamic cavity expansion model for elasto-plastic porous materials. The predictions of the cavity expansion model for the material hardness, for indentation depths varying from 0.1µm to 10µm, and speeds yielding a variation in the strain rates from 10−4s−1to108s−1, are compared against the available experimental results on OFHC copper. We note that despite several simplifying assumptions, the predictions of our cavity expansion model show a reasonable agreement with the experimentally measured material hardness over such a wide range of indentation depths and strain rates. We have also carried out parametric analyses to elucidate the speciﬁc roles of inertia, size eﬀect and initial porosity on the material hardness and the cavitation ﬁelds that develop during the indentation process. The key conclusions of this work are as follows: •There exists a critical indentation speed ( ≈100m/s for the set of constitutive parameters considered) above which the inertial eﬀect becomes extremely important and the material hardness increases rapidly. •The rapid increase in the material hardness at very high strain rates becomes more pronounced as the indentation depth increases, highlighting the fact that the inertial eﬀect increases with increasing indentation depth. •Incorporation of the plastic strain gradient induced size eﬀect in the dynamic cavity expansion model leads to a decrease in the radial stress near the cavity wall with increasing indentation depth, while in the absence of any size eﬀect the radial stress will be independent of the indentation depth. •The eﬀect of initial porosity on the hardness of the material (i.e the hardness decreases with increasing initial porosity) increases with increasing indentation speed, and at suﬃciently high indentation speeds the eﬀect of initial porosity on the hardness of the material increases with increasing indentation depth. •The eﬀect of initial porosity on the hardness of the material decreases with increasing value of the length scale parameter, i.e. increasing size eﬀect. Acknowledgements TdS wishes to acknowledge the support of FAPERGS, Fundação de Amparo ã Pesquisa do Estado do Rio Grande do Sul, grant agreement 19/2551−0001054−0. 28 JAR-M acknowledges the ﬁnancial support obtained from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme. Project PURPOSE, grant agreement 758056.