Indentation is a simple and one of the oldest small-scale test methods for characterizing the mechanical response of materials. Recently, there has been a growing interest in dynamic indentation due to its potential to characterize the mechanical response of small volume of materials at high strain-rates. Herein, we focus on understanding the synergistic e ects of materials' inherent strain-rate sensitivity and inertia on the scaling of dynamic hardness with indentation strain-rate. Speci cally, we analyze the dynamic indentation response of ductile materials over a wide range of indentation velocities, utilizing both nite element calculations and an analytical cavity expansion model. The materials are assumed to fol- low isotropic elastic-viscoplastic constitutive relations, with the viscoplastic part described by either an overstress or a simple power-law model. Our results show that below a crit- ical indentation strain-rate, the scaling of dynamic hardness with indentation strain-rate is the same as the viscoplastic constitutive description. Therefore, at these strain-rates, dynamic hardness can e ectively characterize a material's strain-rate sensitivity, provided its viscoplastic constitutive description is known beforehand. However, above the critical indentation strain-rate, the dynamic hardness increases rapidly with indentation strain-rate. This phenomenon indicates an apparent strain-rate sensitivity that exceeds the expected re- sponse of the viscoplastic constitutive description. Moreover, above the critical indentation strain-rate, the indentation depth acts as a natural length-scale, with higher hardness ob- served at greater depths due to increased inertial e ects. In other words, above the critical indentation strain-rate, dynamic hardness cannot be taken as an intrinsic material property. Keywords: Dynamics; Indentation and hardness; Elastic-viscoplastic material; Constitutive behaviour; Finite elements; Dynamic cavity expansion Corresponding author: ankit.sri@tamu.edu (A. Srivastava) Preprint submitted to Journal of the Mechanics and Physics of Solids August 9, 2023 1. Introduction Characterizing the strain-rate dependent mechanical response of materials at high strain- rates is essential for a wide range of applications, including automotive and aerospace en- gineering, as well as for ballistic and extraterrestrial impact events (Meyers, 1994; Ramesh, 2008). Such a characterization enables enhanced safety, performance optimization, and better material selection. The range of strain-rates of interest depends on the speci c ap- plication. For instance, high-speed machining typically involves strain-rates ranging from 102to 104s1, whereas impact loading can result in strain-rates ranging from 103to 106s1 (Meyers, 1994). In hypervelocity extraterrestrial impact events, the strain-rates can reach as high as 108s1(Ramesh, 2008). Experimental techniques for characterizing the strain- rate dependent mechanical response of materials at high strain-rates include split Hopkinson pressure bars (Hopkinson, 1914; Kolsky, 1949), Taylor impact (Taylor, 1946), plate impact (Gray, 2000), and dynamic indentation (Shore, 1907; Subhash et al., 1999), to name a few. The dynamic indentation test is particularly appealing since it requires small volume of materials and can be used as a high-throughput test method (Liu et al., 2021; Schmalbach et al., 2021). There are two approaches to conducting dynamic indentation tests. The rst approach involves impact experiments using an indenter of known geometry without directly measuring the indentation force and depth (Shore, 1907; Batson, 1918; Tabor, 1948; Davis and Hunter, 1960; Davies, 1949; Mok and Du y, 1965; Tirupataiah and Sundararajan, 1991; Yoshioka and Yoshioka, 1995). One of the simplest methods based on this approach is conducting drop impact tests. In these tests, the rebound height or velocity change of an impacting indenter is related to the initial drop height, the impact velocity, and the volume of deformed material in the hardness impression to arrive at a \rebound hardness" (Shore, 1907). Using laser ablation techniques these tests have also been extended to micro- and nano-scales, wherein small hard particles are propelled to impact the specimen, resulting in strain-rates as high as 108s1(Hassani et al., 2020). The second approach involves directly measuring the time-resolved indentation force (Crook, 1952; Nobre et al., 1997) or directly measuring both time-resolved indentation force and depth (Koeppel and Subhash, 1997; Subhash et al., 1999; Lu et al., 2003) during the dynamic indentation process. In Koeppel and Subhash (1997) and Subhash et al. (1999), the dynamic indentation test involved propelling a striker bar using a gas gun, which accelerated an incident bar with an indenter on the other end to indent the sample. The indentation force was then measured using a high- frequency piezoelectric load cell placed behind the specimen, and the indentation depth was deduced from a strain gauge mounted on the incident bar. Subsequently, Lu et al. (2003) utilized optical interferometry to more accurately measure the indentation depth. Similar concepts have been extended to characterize high strain-rate indentation force and depth response of materials at small scales using nanoindentation systems and testing techniques (Sudharshan Phani and Oliver, 2017; Guillonneau et al., 2018; Zehnder et al., 2018; Qin et al., 2019; Merle et al., 2020; Rueda-Ruiz et al., 2020; Phani et al., 2023; Hackett et al., 2023). One way of characterizing the strain-rate dependent mechanical response of materials using dynamic indentation tests is to determine their strain-rate sensitivity exponent from 2 the variation of measured hardness with (a measure of) imposed strain-rate (Subhash et al., 1999; Lu et al., 2003; Sudharshan Phani and Oliver, 2017; Calle et al., 2018). However, me- chanical tests do not directly measure material properties but rather the structural response, which depends on the material, specimen geometry and loading conditions. In fact, relating measured response quantities to material properties can be complex and may not always be unique, especially in indentation tests (Chen et al., 2007). Additionally, de nition of several material parameters depends on the constitutive description of the material. This raises a fundamental question: Does the scaling of dynamic hardness with indentation strain-rate follow the same scaling as the strain-rate dependent (i.e., viscoplastic) constitutive descrip- tion of materials? Furthermore, it has been shown that even for a material that follows strain-rate independent constitutive description, there exists a critical value of the indenta- tion loading-rate above which the inertial e ects become extremely important, leading to a rapid increase in material hardness (dos Santos et al., 2021). This raises a second question: How does the interplay of materials' strain-rate dependence and the e ects of inertia af- fect the scaling of dynamic hardness with indentation strain-rate? These two questions are extremely important since, to the best of the authors' knowledge, all prior works assumed a simple power-law dependence of dynamic hardness on the indentation strain-rate. Addi- tionally, none of the prior works considered the potential in uence of inertia on strain-rate dependent dynamic hardness measurements. To answer these questions, in this work, we extensively analyze dynamic indentation response of ductile materials when indented using a rigid conical indenter. The materials are assumed to follow isotropic elastic-viscoplastic constitutive relations, with the viscoplas- tic part described by either an overstress power-law (Cowper and Symonds, 1957) or the widely used simple power-law (Hutchinson and Neale, 1977; Needleman, 2018). We solve the dynamic indentation problem computationally using both nite element calculations and an analytical cavity expansion model. The computations are carried out for a wide range of indentation velocities. Our results show that there exists a critical indentation strain- rate below which the scaling of dynamic hardness with indentation strain-rate is the same as the viscoplastic constitutive description. Therefore, a material's strain-rate sensitivity exponent can be characterized from dynamic indentation results for a known constitutive description. However, for both viscoplastic constitutive descriptions, at any given indenta- tion strain-rate above the critical value, the hardness depends on the indentation depth and increases rapidly compared to what is expected based on the constitutive description. In other words, at suciently high indentation strain-rates, the indentation depth emerges as a natural length-scale that needs to be considered when interpreting the results of dynamic indentation tests. To the best of the authors' knowledge, this paper presents the rst-ever demonstration that inertia e ects make dynamic hardness size-dependent. This nding emphasizes a limitation in using dynamic indentation tests to characterize the strain-rate sensitivity or strain-rate-dependent ow strength of materials at high strain-rates. 3 Figure 1: A schematic representation of the three-dimensional specimen being indented with a rigid conical indenter and the nite element mesh of the specimen together with a zoomed view of the same in the center of the top surface of the specimen. The dimensions of the specimen are lx,lyandlz,_h0is the constant velocity imposed on the conical indenter, and 90 is the cone angle. 2. Finite Element Method A series of nite element calculations of dynamic indentation of a 3D specimen with a rigid conical indenter were performed, as illustrated in Fig. 1. The specimen dimensions were set to lx=ly=lz=50 mm. The indenter had a tip radius of rt=0.1 mm and cone angle of 90 with =19.7, consistent with the work of Lu et al. (2003). The nite element mesh (Fig. 1) of the specimen consisted of 140 ;220 twenty-node brick elements, with a very ne uniform mesh in a 5 52 mm3region surrounding the actual indented volume. The following displacement boundary conditions were imposed on the analyzed region: Thexandydisplacements of the specimen faces normal to the xandyaxes, respectively, and thezdisplacement of the bottom face of the specimen normal to the zaxis in the reference con guration were constrained. The velocity loading condition imposed on the specimen (which is initially at rest) follows the pro le of the indenter penetrating the specimen and is as follows: The node at the center of the top surface of the specimen, initially in contact with the indenter tip, was given a Heaviside step velocity of _h0along the negative zaxis. As deformation continued and the indenter tip penetrated the specimen, additional nodes on the top surface came into contact with the indenter. These nodes were also assigned the samezvelocity, but their xandyvelocities were constrained, mimicking a perfect adhesion of the material to the indenter. The assumption of perfect adhesion between the material and the indenter is based on the work of Johnson (1970). Speci cally, Johnson (1970) demonstrated that assuming the material perfectly adheres to the indenter in the nite element method results in a deformation pattern consistent with experimental evidence, i.e., the deformation occurs beneath the indenter, and the displacement elds are approximately radial. It has also been shown that for indentation using a conical indenter, perfect adhesion or frictionless conditions 4 between the material and the indenter have no signi cant e ect on the resulting indentation force-depth response (Bhattacharya and Nix, 1988). The perfect adhesion condition has been utilized in a number of other works, see Fleck et al. (1992) and Danas et al. (2012), among others. All the nite element calculations were carried out using our in-house data-parallelized nite element code (Osovski et al., 2015; N'souglo et al., 2018; Liu et al., 2019, 2020; Zheng et al., 2020). The nite element code is based on the dynamic principle of virtual work using a nite deformation Lagrangian convected coordinate formulation. Integration of internal force contributions was performed using eight-point Gaussian integration for each twenty- node element, and twenty-seven-point Gaussian integration was used for the element mass matrix. Moreover, lumped masses were used so that the mass matrix is diagonal. The discretized equations were integrated using the explicit Newmark -method (Belytschko et al., 1976), and the constitutive behavior was updated based on the rate tangent modulus method (Peirce et al., 1984). 2.1. Constitutive Model The material was modeled as elastic-viscoplastic, with yielding described by the von Mises criterion. Herein, the rate of deformation tensor dis represented as the sum of an elastic partdeand a plastic part dp, i.e.,d=de+dp. The elastic part is related to the Jaumann rate of the Cauchy stress tensor as ^=C:de(1) with^() denoting the objective derivative and Cbeing the fourth rank isotropic elastic tensor C=E 1 +I0+E 3 (12)1 1 (2) whereEis the Young's modulus, is the Poisson's ratio, I0is the unit-deviatoric fourth rank tensor, and 1is the unit second rank tensor. The plastic part of the rate of deformation tensor is given as dp=_@ @(3) where _is a non-negative plastic multiplier rate and  = Y160 (4) is the von Mises ow potential, where  =q 3 2s:sis the von Mises e ective stress with s=1 3(:1)1, and Y=0 1 +"p "0n 1 +_"p _"0m G(T) (5) 5 is the yield stress, where _"p=q 2 3dp:dpis the equivalent plastic strain-rate,  "p=Rt 0_"p( )d is the accumulated equivalent plastic strain, and Tis the current temperature; while 0is the reference yield strength, nis the strain hardening exponent, "0is a reference strain, m is the strain-rate sensitivity (or hardening) exponent, and _ "0is a reference stain rate. The dependence of Yon_"pin Eq. (5) is referred to as the overstress power-law (Cowper and Symonds, 1957). The temperature dependence of Yin Eq. (5) is taken as G(T) = 1 +bexp (c[T0273]) [exp (c[TT0])1] (6) wherebandcare constitutive parameters and T0is a reference temperature. Assuming adiabatic deformation process (no heat ux), and considering that plastic work is the only source of heating, the temperature evolution is given as _T= _"p Cp(7) where is the Taylor-Quinney coecient, Cpis the speci c heat, and =0=det(F) is the current material density, with 0being the initial material density and Fthe deformation gradient tensor. Finally, from the equivalence of macroscopic plastic work and microscopic plastic dissi- pation i.e.,:dp= _"p, we get _=_"pin Eq. (3). 2.2. Constitutive Model Calibration The constitutive model described in Section 2.1 contains various parameters that need to be determined for a given material. In this work, we focused on OFHC copper as a model material since it has been extensively characterized using a variety of techniques. Additionally, this choice is consistent with the dynamic indentation experiments in Lu et al. (2003). As such the elastic constants were taken to be E= 116 GPa and = 0:31, the initial density was set to 0= 8940 kg/m3, and the speci c heat was set to Cp= 384:6 J/kg-K. The Taylor-Quinney coecient was assumed to be = 0:9. The values of the remaining parameters, i.e., those in Eqs. (5) and (6), were determined using the experimental results of mechanical testing of OFHC copper found in the literature (Tong et al., 1992; Tanner and McDowell, 1999; Rittel et al., 2002). In these works, the quasi-static mechanical testing was performed using servo-hydraulic machines, while dynamic mechanical testing was conducted using a Kolsky bar and pressure-shear plate impact tests. These tests quanti ed the material's stress-strain response (Tong et al., 1992) and the relationship between ow stress and the imposed strain-rate (Tong et al., 1992; Rittel et al., 2002) and test temperature (Tanner and McDowell, 1999), as illustrated in Fig. 2. From these experimental results, the values of the parameters in Eq. (5) were found to be 0= 302 MPa, n= 0:21,"0= 0:44, m= 0:35, and _"0= 2:2104s1, and those in Eq. (6) were found to be b= 5:71 and c= 3:52104withT0= 293 K. Note that 293 K is also the initial temperature in all calculations, meaning T=T0att= 0. The comparison of the experimental and predicted mechanical response of the material using these values of the constitutive parameters is 6 (a) (b) (c) Figure 2: Comparison of the mechanical response of an OFHC copper obtained experimentally (symbols) and predicted (solid line) using the calibrated constitutive model with overstress power-law description for the strain-rate dependence. (a) Quasi-static true stress ( ) - e ective plastic strain ( "p) response. (b) The dependence of the normalized ow stress ( = 0) on the imposed nominal e ective strain-rate ( _"). (c) The dependence of = 0on temperature ( T). The experimental data in (a) are from Tanner and McDowell (1999). The experimental data in (b), square and diamond symbols, are from Rittel et al. (2002) and Tong et al. (1992), respectively. The experimental data in (c) are from Tanner and McDowell (1999). In (b) and (c),0= 302 MPa. The horizontal axis in (b) is on the log scale. shown in Fig. 2. We note that the dynamic indentation test results of Lu et al. (2003) are not utilized for the calibration of the constitutive model. Instead, in this paper, these dynamic indentation test results are solely employed to validate the nite element predictions. 7 3. Finite Element Results (a) (b) Figure 3: (a) Comparison of the experimentally (Exp) obtained and calculated (Cal) indentation force ( F) - depth (h) response for two values of the imposed indentation velocity ( _h0). (b) The zoomed-in view of the plot in (a) showing the comparison of the experimental and calculated F-hresponse at lower indentation depths. The nite element calculations were carried out using the calibrated constitutive model with overstress power-law description for the strain-rate dependence. The experimental data are taken from Lu et al. (2003). A comparison of the indentation force ( F) - depth ( h) response for two values of the imposed indentation velocity ( _h0) predicted using the fully calibrated overstress power law constitutive model and the experimental results of Lu et al. (2003) is shown in Fig. 3. An excellent quantitative agreement between the predicted and experimentally measured Fh response for the full range of the values of his noted from Figs. 3(a) and (b). As discussed in Section 2.2, we calibrated the constitutive model using the results of quasi-static uniaxial mechanical tests and dynamic mechanical tests involving Kolsky bar and pressure-shear plate impact tests (Tong et al., 1992; Tanner and McDowell, 1999; Rittel et al., 2002), and not the dynamic indentation test results of Lu et al. (2003). This is because even though the indentation response of a material using a conical indenter may not be unique (Chen et al., 2007), meaning a wide range of constitutive parameter values can lead to the same indentation response, the indentation response of a fully calibrated constitutive model is unique. The experimental work of Lu et al. (2003) represents a signi cant milestone in the area of dynamic indentation testing and is one of the few works that reported full indentation force - depth response. The observed quantitative agreement between our predictions and their experimental results provides robust validation of our computational approach and instills con dence in the numerical predictions presented hereafter. The indentation Fhresponse of the material for a xed value of the imposed _h0is 8 used to estimate the material's hardness ( H) and the nominal indentation strain-rate ( _"i) as H=F(h) A(h); _"i=_h0 h(8) whereA(h) is the cross sectional area of the conical indenter in contact with the material at an indentation depth h. The de nition of _"ifollows from the work of Subhash et al. (1999). Next, to answer the question of whether the scaling of Hwith _"ifollows the same scaling as the strain-rate dependent constitutive description of the material, we also repeated all the dynamic indentation nite element calculations with a simple power-law description of the strain-rate dependence in Eq. (5). The simple power-law based description of the strain-rate dependence is given as (Hutchinson and Neale, 1977; Needleman, 2018) Y=0 1 +"p "0n_"p _"0m G(T) (9) The values of the constitutive parameters for the simple power-law in Eq. (9) are also taken to be the same as for the overstress power-law in Eq. (5) and reported in Section 2.2. For both descriptions of the strain-rate dependence of the material, nite element calcula- tions of dynamic indentation are carried out for twenty two values of _h0ranging from 0 :3 m/s to 158 m/s. (a) (b) Figure 4: Variation of hardness ( H) with indentation depth ( h) for four values of the imposed indentation velocity ( _h0) obtained from the nite element calculations. (a) For calculations using the overstress power- law description for the strain-rate dependence. (b) For calculations using the simple power-law description for the strain-rate dependence. Figure 4 shows the evolution of Has a function of hfor four values of _h0. The results obtained with overstress and simple power-law descriptions of strain-rate dependence of the 9 material are shown in sub gures (a) and (b), respectively. As shown in the gure, at the start of the indentation process, the value of His unbounded because A(h) is in nitesimally small. Additionally, since the specimen is initially at rest and the imposed indentation loading condition is a Heaviside step velocity, the value of _"iis extremely large in the beginning, and the sudden acceleration of the specimen leads to oscillations in the Hh curves forh<0:2. These oscillations do decay gradually with increasing h. As the value of hincreases, the value of A(h) also increases while that of _"idecreases, resulting in a rapid decrease in the value of H. Forh > 0:2, the value of Htends to decrease gradually with a continued increase in h. This is because the value of _"iis not constant for a xed value of the imposed _h0, and it depends on the value of h, see Eq. (8). Thus, even for a constant _h0, the value of Hdepends on the value of h. TheHhcurves for all values of _h0are qualitatively similar for both viscoplastic constitutive models. For both models, an increase in the value of _h0for a given value of hresults in an increase in the value of Hdue to the associated strain-rate hardening and inertia. However, at any given value of hand for a xed value of _h0, the overstress power-law model exhibits a greater value of Hcompared to the simple power-law model. This is due to the fact that the ow strength of the material with the same set of constitutive parameters, but governed by the overstress power-law model, is greater than that of the material governed by the simple power-law model for the same imposed strain-rate, see Eqs. (5) and (9). (a) (b) Figure 5: Variation of hardness ( H) with nominal indentation strain-rate ( _"i) obtained from the nite element calculations for the values of the imposed indentation velocity ( _h0) ranging from 0 :3 m=s to 158 m=s and for the values of the indentation depth ( h) ranging from 0 :25 mm to 1 mm. (a) For calculations using the overstress power-law description for the strain-rate dependence. (b) For calculations using the simple power-law description for the strain-rate dependence. The horizontal axis in both (a) and (b) are on the log scale. The evolution of Has a function of _"ifor the values of _h0ranging from 0 :3 m/s to 158 10 m/s is shown in Fig. 5. Here as well, the results obtained with overstress and simple power- law models are shown in sub gures (a) and (b), respectively. In Fig. 5, the evolution of H with _"iis plotted for h2[0:25;1] mm with di erent colors for all values of _h0considered in this work. Additionally, the variation of the value of Hwith _"iis also shown for h= 0:25 mm and 1 mm to illustrate the bounds. The range of values of _"ishown in Fig. 5 spans from relatively low values that are consistent with the works of Subhash et al. (1999) and Lu et al. (2003) to high values that are consistent with the works of Tirupataiah and Sundararajan (1991) and Hassani et al. (2020). However, these prior works did not address the two fundamental questions posed in this work: Does the scaling of dynamic hardness with indentation strain-rate follow the same scaling as the strain-rate dependent constitutive description of materials? and how does the interplay of the materials' strain-rate dependence and the e ects of inertia a ect the scaling of dynamic hardness with indentation strain-rate? It is also worth noting that Tirupataiah and Sundararajan (1991) hypothesized that the inertial resistance of the target material should be negligible compared to its resistance to plastic ow for the interpretation of dy- namic hardness measurements to be considered valid. Answering the two aforementioned fundamental questions will provide veri cation and rationale for this three-decades-old hy- pothesis. While Lu et al. (2003) performed nite element analyses of dynamic indentation, their focus was solely on comparing the experimentally and computationally obtained vari- ations of a measure of ow strength of the material with a measure of strain-rate. Both the ow strength and strain rate in their work were empirically deduced from the dynamic indentation results. Thus, the existing literature has left these two critical questions largely unexplored, and addressing them is a central focus of the results presented hereafter. As shown in Fig. 5, both overstress and simple power-law models produce qualitatively similarH_"iresponses, but for a xed value of _"i, the overstress power-law model yields higher values of Hcompared to the simple power-law model. This is because the ow strength of the material governed by the overstress power-law model is greater than that of the material governed by the simple power-law model for the same imposed strain-rate, as noted earlier. Nonetheless, while the value of Hremains nite for the overstress power-law model at all values of _"i, see Fig. 5(a), the value of Hfor the simple power-law model tends to zero, see Fig. 5(b), as the value of _"idecreases. This is due to the fact that the ow strength of the material governed by the overstress power-law model tends to a reference value as the imposed strain-rate tends to zero while that of the material governed by the simple power-law model tends to zero, see Eqs. (5) and (9). The results in Fig. 5 demonstrate that, for both viscoplastic constitutive models and indentation depths in the range of 0 :25 to 1 mm, the value of His independent of hand increases with an increasing value of _"ifor_"i<5104s1. This nding implies that the value ofHfor_"i<5104s1is solely determined by the nominal strain-rate, and as such, is a material property directly linked to the strain-rate dependent ow strength of the material. In contrast, for _"i>5104s1(referred to hereafter as the critical indentation strain-rate), the evolution of Hwith _"ibecomes dependent on the value of hand increases rather rapidly with further increases in _"i. Importantly, for a given value of _"i>5104 s1, the dependence of Honhis such that the value of Hincreases with increasing value 11 ofh, indicating that it can no longer be treated as a material property. The h-dependent evolution of Habove a critical indentation strain-rate is due to inertial e ects since the constitutive models do not have any explicit length-scale. Nonetheless, this behavior is physically sound because for a given indentation strain-rate, a greater depth of indentation requires more material to be displaced, resulting in an increase in the inertial e ect and the hardness of the material. This is the rst-ever demonstration that inertia e ects make dynamic hardness measurements size-dependent, even in the absence of any length-scale in the constitutive model. The value of the critical indentation strain-rate, 5104s1, identi ed for hvalues ranging from 0 :25 to 1 mm for both viscoplastic constitutive models, above which inertial e ects become increasingly signi cant, is not likely to be universal and will depend on various other parameters. These parameters will include both material and geometrical factors, such as material density and indenter shape. (a) (b) Figure 6: Graphical representation of (a) Eq. (10) corresponding to the overstress power-law model and (b) Eq. (11) corresponding to the simple power-law model showing the dependence of hardness ( H) on the nominal indentation strain-rate ( _"i) at two values of the indentation depth ( h) and for the range of indentation velocity ( _h0) considered. The normalization parameters Ho rand _"o rin (a) are tting constants, whileHs rand _"s rin (b) are the values of the hardness and indentation strain-rate, respectively, corresponding to the lowest value of the indentation velocity considered herein. The parameter min (a) and (b) is the slope of the linear portion of the curve. In Fig. 6, we attempt to quantify the scaling of Hwith _"iand check to what extent it follows the same scaling as the strain-rate dependent constitutive description of the material. To this end, we assume that the increase in Hwith increasing _"iis a result of the strain-rate sensitivity of the material as described by the constitutive models in Eqs. (5) and (9). To quantify this relationship, we derive analogous expressions for the dependence of Hon_"ias follows 12 logH Ho r1 =mlog_"i _"o r (10) for the overstress power-law description for the strain-rate dependence and logH Hs r =mlog_"i _"s r (11) for the simple power-law description of the strain-rate dependence of the material. Figures 6(a) and (b) show the plots of Eqs. (10) and (11), respectively. The parameters Ho r= 1:17 GPa, _"o r= 3:17105s1, andm= 0:35 in Eq. (10) are identi ed by tting the equation to the Hversus _"idata for _"i<104s1obtained from the nite element calculations using overstress power-law model. In Eq. (11), for the simple power-law model, the values of the parameters Hs r= 0:125 GPa, _"s r= 3102s1, andm= 0:35. The values of Hs rand _"s rare simply the values of Hand _"iath= 1 mm for _h0= 0:3 m/s , which is the lowest value of the indentation velocity considered in this work, while the value of mis identi ed by tting the equation to the Hversus _"idata for _"i<104s1obtained from the nite element calculations using simple power-law model. Of particular importance is the fact that the slopes ( m) of the linear portion of the log-log plots in Figs. 6(a) and (b) are the same as the strain-rate sensitivity exponents used in the nite element calculations only if the hardness vs. indentation strain-rate data are analyzed using a relation consistent with the underlying viscoplastic constitutive description of the material. In other words, if the data in Fig. 6(a) were analyzed as per Eq. (11), the resultant slope is not the same as the strain- rate sensitivity exponents used in the nite element calculations. Furthermore, analyzing the data beyond the linear portion of the log-log plots in Figs. 6(a) and (b) will yield a value ofmthat is greater than the actual strain-rate sensitivity exponent of the material. Beyond the linear portion, the scaling of Hwith _"ino longer follows the same trend as expected based on materials' strain-rate dependent constitutive description, con rming that dynamic hardness is no longer an intrinsic material property. The increase in dynamic hardness beyond the strain-rate hardening of the material is attributed to the development of large hydrostatic stresses induced by inertia in the specimen as the indentation velocity increases. Contour plots of the variation of hydrostatic stress around the indenter at an indentation depth of h= 1 mm for two values of the imposed indentation velocity, _h0= 0:96 m/s and 100 m/s, are shown in Fig. 7. In the gure, the results are shown for both overstress and simple power-law models. As can be seen in the gure, both constitutive models yield similar qualitative results, showing the formation of a hemispherical core below the indenter tip with large hydrostatic stress. The di erence in results obtained for the two constitutive models is quantitative in nature. Speci cally, the hydrostatic stress in the hemispherical core is lower for the simple power-law model compared to the overstress power-law model, because the ow strength of the former is less than that of the latter at the same imposed strain-rate. However, as the indentation velocity increases 13 Figure 7: Contour plots showing the variation of hydrostatic stress around the indenter at an indentation depth ofh= 1 mm for two values of the imposed indentation velocity, _h0= 0:96 m/s and 100 m/s (corresponding to _"i= 960 s1and 105s1), obtained from nite element calculations using both overstress power-law (OSP) and simple power-law (SP) descriptions for the strain-rate dependence. from _h0= 0:96 m/s to 100 m/s, the inertial e ects dominate, and the increase in hydrostatic stress is signi cantly greater than the expected increase due to the strain-rate-dependent enhancement of the material's ow strength. Nonetheless, the increase in hydrostatic stress is relatively similar for both models as the indentation velocity increases. For example, at a depth of 1 mm below the indenter tip (i.e., 2 mm below the specimen surface), the increase in hydrostatic stress is 510 MPa and 582 MPa for the overstress and simple power-law models, respectively. Similarly, at a depth of 3 mm below the indenter tip, the increase in hydrostatic stress is 238 MPa and 301 MPa for the overstress and simple power-law models, respectively. 4. Analytical Model The dynamic cavity expansion model utilized in this paper adapts the viscoplastic for- mulation of dos Santos et al. (2020) to the constitutive framework introduced in Section 2.1, and the process of conical indentation is idealized as the radial expansion of a spherical cavity into an in nite medium (Johnson, 1970; Studman et al., 1977; Fleck et al., 1992; Gao et al., 2006; Mata et al., 2006). The cavity expansion model assumes that the indenter is encased in a rigid hemispherical core of radius a(the expanding cavity) that is subject to 14 Figure 8: A schematic illustrating the idealization of the indentation process using a conical indenter as the radial expansion of a spherical cavity. a hydrostatic stress equal to the indentation pressure p, which corresponds to the specimen hardness, as shown in Fig. 8. Note that the assumption of a hydrostatic core developing below the indenter tip is consistent with the nite element results presented in Fig. 7. We consider an Eulerian spherical coordinate system ( r;; ) with the origin located at the center of the hydrostatic core. The indenter penetrates the solid at a constant velocity _h, such that a steady-state indentation pressure is achieved, causing the material to displace radially at velocity _ r. We assume that all eld variables depend on the normalized radial coordinate =r=a(Durban and Masri, 2004; Masri and Durban, 2005; Cohen and Durban, 2013b), which allows us to transform the eld equations (i.e., balance of linear momentum, constitutive relations, heat equation, and conservation of mass) to obtain the following set of expressions: 0 rr+2 (rr) =M2 0(V)V0(12) V0= (V) 0 rr20 ("p)0 (13) V = (V) 0 rr+ (1) 0 +1 2("p)0 (14) ("p)0=a _a(V)_"02 4rr 0 1 +"p "0n G(T)13 51 m (15) 15 Cp ET0=  ("p)0(16)  0= exp [(12)(rr+ 2)] (17) which have been normalized using the following dimensionless groups: rr=rr E;  = E;  0=0 E;  = E P=p E;M=_aq E 0;V=_r _a with the prime superscript denoting di erentiation with respect to . The Eq. (15) is obtained with the overstress power-law description of the strain-rate dependence, see Eq. (5), so that considering the simple power-law in the cavity expansion model requires using Eq. (9) to compute ( "p)0. Moreover, the strain-rate sensitivity of the material makes the dynamic cavity expansion model to be time-dependent, so that it is required to specify the time tat which the corresponding cavitation eld is evaluated (dos Santos et al., 2020). For a constant-velocity expansion, the current hydrostatic core radius is calculated as a=a0+ _at, wherea0is the hydrostatic core radius at t= 0 (hydrostatic core radius at a reference time). The set of Eqs. (12)-(17) is solved using an iterative shooting method under the following boundary conditions V=tan 2and rr=P;at= 1 V= 0;rr= = 0;at=w to obtain the values of six unknowns ( V;rr;;"p;T;), and the calculated radial stress at the cavity boundary provides the specimen hardness. The reader is referred to dos Santos et al. (2020, 2021), Cohen et al. (2010) and Cohen and Durban (2013a) for more details on the derivation of the dynamic cavity expansion model. 5. Analytical Model Results and Discussion The analytical model predictions for the variation of hardness ( H) with indentation strain-rate ( _"i) using the two viscoplastic constitutive descriptions, overstress and simple power-law, are shown in Figs. 9(a) and (b), respectively. The results are obtained for two values of the indentation depth, h= 1 mm and 0 :25 mm, and for a wide range of values of the imposed indentation velocity ( _h0). To facilitate direct comparison with the analytical model predictions, the nite element results shown in Fig. 5 are also reproduced in Fig. 9. As the overstress power-law constitutive model was calibrated for OFHC copper, limited 16 (a) (b) Figure 9: Comparison of the variation of hardness ( H) with indentation strain-rate ( _"i) obtained from the nite element (FE) calculations and the analytical (Theory) model at two values of the indentation depth (h) and for a wide range of values of the imposed indentation velocity ( _h0). (a) For overstress power- law description for the strain-rate dependence. (b) For simple power-law description for the strain-rate dependence. The horizontal axis in both (a) and (b) are on the log scale. The experimental (Exp) results with square symbols in (a) are the uniaxial stress data of Rittel et al. (2002) converted to Hwith a Tabor factor of 3 and the strain-rate is the nominal uniaxial strain-rate, while those with circles are the indentation force - depth data of Lu et al. (2003) converted to Hand _"ias per Eq. (8). available experimental data for this material from the works of Rittel et al. (2002) and Lu et al. (2003) are also included in Fig. 9(a) for direct comparison. As can be seen in Fig. 9, the analytical model and nite element results show the same dependence of Hon_"ifor both viscoplastic constitutive models. The analytical model predicts, like the nite element calculations, that at any given value of _"i<5104s1 (referred to as critical strain-rate in Section 3) the hardness is independent of the indentation depth. Similarly, for any given value of _"i>5104s1, the analytical model predicts, in agreement with the nite element results, that the value of Hdepends on the indentation depth, with higher values obtained at greater depths, and increases rather rapidly with further increase in _"i. The limited experimental data for OFHC copper in Fig. 9(a) further con rms the rapid increase in the value of Hwith increasing _"iabove a critical value of approximately 5 104 s1, as predicted by both the nite element calculations and the analytical model. However, the lack of hardness measurements at di erent indentation depths for a xed value of the indentation strain rate does not capture the dependence of material hardness on indentation depth within these limited experimental results. The exact value of the critical indentation strain rate,5104s1, obtained from the nite element calculations and analytical model, as well as observed in the experiments for OFHC copper subjected to dynamic indentation 17 (a) (b) Figure 10: Comparison of the variation of hydrostatic stress ( H) with indentation strain-rate ( _"i) obtained from the nite element (FE) calculations and the analytical (Theory) model at a depth of 1 mm below the indenter tip for an indentation depth of h= 1 mm and for a wide range of values of the imposed indentation velocity ( _h0). (a) For overstress power-law description for the strain-rate dependence. (b) For simple power-law description for the strain-rate dependence. The horizontal axis in both (a) and (b) are on the log scale. using a rigid conical indenter of xed geometry, is likely to be in uenced by both material and geometrical factors. The rapid increase in material hardness (greater than what is expected based on the strain-rate dependent constitutive description) above a critical indentation strain-rate is due to the inertia-induced rapid increase in the hydrostatic stress ( H). The rapid increase inHwith _"iabove a critical value of _"iis predicted by both the analytical model and the nite element calculations for both constitutive models, as shown in Fig. 10. For values of_"ibelow the critical value, inertia e ects are negligible, and the increase in hydrostatic stress is proportional to the strain-rate dependent increase in the material's ow strength. However, for values of _"iabove the critical value, inertia e ects dominate, and the increase in hydrostatic stress is signi cantly greater than the expected increase due to the strain-rate dependent increase in the material's ow strength. Moreover, for _"igreater than the critical value, the relative contributions of inertia and strain-rate hardening to the development of the hydrostatic stress eld are similar for both overstress and simple power-law models (for the range of indentation velocities considered in this work). A similar rapid increase in the value of Hwith increasing _"iabove a critical value of _"i has been observed, even for materials that follow a strain-rate independent constitutive de- scription (dos Santos et al., 2021). For strain-rate independent materials, the rapid increase inHabove a critical value of _"iwas also attributed to the inertia-induced rapid increase in hydrostatic stress. This phenomenon, where inertia-induced hydrostatic stress becomes in- 18 creasingly signi cant above a certain loading-rate, is not unique to dynamic indentation and has been observed in various other dynamic problems. Examples of such problems include the formation of multiple necks in metal rings and shells subjected to rapid radial expansion (Rodr guez-Mart nez et al., 2013, 2017), the competition between necking and fracture in porous ductile materials under dynamic loading (Zheng et al., 2020), and the collapse of voids in porous ductile materials subjected to shock loading (Lovinger et al., 2021), to name a few. Although the analytical dynamic cavity expansion model and nite element calculation results are qualitatively consistent in Figs. 9 and 10, there are quantitative di erences. Speci cally, the nite element calculations consistently yield higher values of H(andH) compared to the analytical model. The reasons for this quantitative discrepancy are not entirely clear but may be attributed to the simpli cations made in the analytical dynamic cavity expansion model, such as the assumption of a spherically symmetric mechanical eld and the absence of deviatoric stress within the hemispherical core representing the expand- ing cavity. Consequently, the simple analytical model cannot fully replace the necessity of nite element calculations when evaluating a material's response under dynamic inden- tation. Nevertheless, our results demonstrate that the simple analytical model accurately captures the fundamental physics of the dynamic indentation process and provides physi- cally sound trends and results. Therefore, it can be employed in future works for promptly interpreting experimental results, identifying the underlying viscoplastic constitutive model of the material, and narrowing down the regions in the parameter space that merit further exploration. In summary, the results presented here clearly demonstrate that the dependence of a material's hardness on indentation strain-rate below a critical value can be used to charac- terize the material's strain-rate sensitivity, provided the hardness vs. indentation strain-rate data are analyzed using a relation consistent with the underlying viscoplastic constitutive description of the material. Furthermore, the results also show that above the critical in- dentation strain-rate, the measured hardness is no longer characteristic of the material; instead, it is determined by the rapid increase in hydrostatic stress induced by inertia. As a consequence, any analysis of the material's hardness dependence on indentation strain-rate above the critical value will result in an apparent strain-rate sensitivity exponent that is greater than the inherent strain-rate sensitivity of the material. Moreover, above the critical indentation strain-rate, the indentation depth emerges as a natural length-scale that a ects the measured hardness. This emergence of indentation depth as a length-scale is attributed to the inertial e ects since the constitutive description of the material does not have an explicit length-scale. The signi cance of this length-scale is physically sound, as a deeper indentation requires more material displacement, leading to a stronger inertial e ect and yielding higher hardness values. These ndings were not only obtained from the results of extensive nite element analyses but also using a simple analytical dynamic cavity expansion model. Both the nite element method and the analytical model employed in this study rely on well-established theories and numerical tools, further ensuring the validity of these ndings. Lastly, to the best of the authors' knowledge, this is the rst-ever demonstration that inertia e ects make dynamic hardness size-dependent and emphasizes the limitations 19 of using high-rate indentation tests to characterize the strain-rate sensitivity or strain-rate dependent ow strength of materials at high strain rates. 6. Concluding Remarks This work focuses on two fundamental questions related to the dynamic indentation response of ductile materials: (i) Does the scaling of dynamic hardness with indentation strain-rate follow the same scaling as the strain-rate dependent constitutive description of materials? (ii) How does the interplay of materials' strain-rate dependence and the e ects of inertia a ect the scaling of dynamic hardness with indentation strain-rate? To answer these questions, we extensively analyzed the dynamic indentation response of ductile materials when indented using a rigid conical indenter over a wide range of indentation velocities. The materials are assumed to follow isotropic elastic-viscoplastic constitutive relations, with the viscoplastic part described by either an overstress or a simple power-law model. The dynamic indentation problem is solved computationally using both the nite element method and an analytical cavity expansion model. The predictions of the overstress power-law constitutive model were also validated using limited available experimental results for OFHC copper. The key conclusions are as follows: Below a critical indentation strain-rate, the scaling of dynamic hardness with indenta- tion strain-rate follows the constitutive model for both overstress and simple power-law models. Thus, allowing for the characterization of the material's strain-rate sensitivity, provided the hardness vs. indentation strain-rate data are analyzed using a relation consistent with the underlying constitutive description of the material. Above the critical indentation strain-rate, the dynamic hardness increases rather rapidly with increasing indentation strain-rate for both overstress and simple power-law mod- els. This indicates that above the critical indentation strain-rate, dynamic hardness is no longer an intrinsic material property. The rapid increase in dynamic hardness with indentation strain-rate above the critical indentation strain-rate is due to the inertia-induced rapid increase in the hydrostatic stress in the specimen. Despite the absence of an explicit length-scale in materials' constitutive description, above the critical indentation strain-rate, the indentation depth emerges as a length- scale, resulting in higher dynamic hardness at greater depths. This is due to the increase in inertial e ects resulting from the larger volume of material that needs to be displaced for deeper indentations. In this work, we have addressed two fundamental questions associated with the dynamic indentation response of ductile materials. These two questions are extremely important since, to the best of the authors' knowledge, all prior works not only analyzed the dependence of dynamic hardness on indentation strain-rate assuming a simple power-law scaling between 20 the two but also neglected any potential in uence of inertia. Additionally, to the best of the authors' knowledge, this paper presents the rst-ever demonstration that inertia e ects make dynamic hardness size-dependent and emphasizes the limitations in using high-rate indentation tests to characterize the strain-rate sensitivity or strain-rate-dependent ow strength of materials at high strain-rates. All these conclusions were not only drawn from the results of extensive nite element analyses but were also veri ed using a simple analytical dynamic cavity expansion model. Both the nite element method and the analytical model employed in this study rely on well-established theories and numerical tools, further ensuring the validity of these conclu- sions. Moreover, the analytical dynamic cavity expansion model presented in this work e ectively captures the fundamental physics of the dynamic indentation process and yields trends consistent with the full eld nite element results. Thus, it can be utilized in future works for quick interpretation of experimental results, identi cation of the material's consti- tutive model, and narrowing down the regions in the parameter space that require further investigation. Although in this work, we focused on the dynamic indentation of ductile materials char- acterized by two viscoplastic constitutive descriptions and used a rigid conical indenter of xed geometry, the conclusions drawn here are likely applicable to a wide range of circum- stances. However, we do note that some of the quantitative results, such as the value of the critical indentation strain-rate, will depend on both material and geometric parameters, such as material density and indenter geometry. Furthermore, investigating the dynamic indentation response of materials that follow pressure-dependent constitutive responses will be of great interest. This analysis will enable us to examine the e ect of plastic compress- ibility on dynamic hardness measurements, particularly at higher indentation strain-rates where hydrostatic stress controls the dynamic hardness measurements. We hope that our work will inspire further research in these areas. Acknowledgments The nite element calculations reported on were carried out using high performance research computing resources provided by Texas A&M University. Funding This work was supported by the European Union's Horizon 2020 Programme (Excellent Science, Marie-Sklodowska-Curie Actions) under REA grant agreement 777896 (Project QUANTIFY); FAPERGS (Foundation for the Support of Research in the State of Rio Grande do Sul, Brazil), grant agreement 21/2551-0000744-3; and the U.S. National Science Foundation grant CMMI - 1944496. Con ict of interest The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to in uence the work reported in this paper. 21 Author contributions Zahra Ghasemi : Formal analysis; Investigation; Methodology; Software; Validation; Visualization; Writing - original draft. Tiago dos Santos : Formal analysis; Investigation; Funding acquisition; Methodology; Software; Validation; Visualization; Writing - review & editing. Jos e A. Rodr guez-Mart nez : Conceptualization; Formal analysis; Inves- tigation; Funding acquisition; Methodology; Project administration; Supervision; Writing - original draft. Ankit Srivastava : Conceptualization; Data curation; Formal analysis; Funding acquisition; Methodology; Project administration; Supervision; Writing - original draft.