Introduction

The determination of mechanical properties from indentation tests is not direct because of the complex stress and strain fields that develop under the contact area. Nevertheless, the versatility and simplicity of the technique have inspired researchers to establish methodologies to solve the inverse analysis problem in indentation, i.e., the extraction of elastoplastic properties from indentation data. In addition, indentation testing is applicable over a wide range of length scales (from nm to mm), which enables characterization of mechanical behavior at different scales. Moreover, nanoindentation may be the only feasible mechanical characterization technique in some cases, such as in thin films,1 in mechanical property mapping at the microstructural scale,2 or for in situ testing of the matrix in fiber-reinforced polymer composites.3

The introduction of instrumented indentation entailed great advances in this field. In fact, the determination of hardness and elastic modulus based on the Oliver-Pharr method4,5 is now an industry standard.6 However, the determination of the plastic constitutive behavior still remains elusive. Some authors have approached the problem by developing analytical approximations. For sharp indentation, Johnson derived a set of equations based on the so-called expanding cavity model,7 which was originally proposed by Marsh.8 Tabor followed a different approach, based on the slip-line solution, and came up with its well-known expression that relates hardness, H, with yield stress, Y, by H≈3Y, for a rigid-perfectly plastic solid in sharp (self-similar) indentation.9 The analysis was later expanded for strain hardening materials by defining the relation \(H \approx 3\sigma_{r}\), where \(\sigma_{r}\) is the flow stress at a certain representative strain which is an average of the strain field beneath the indenter. Tabor established that the representative strain is around 0.08 for Vickers indenters. Similar analytical treatments were also developed for spherical indentation. Contrary to the case of self-similar sharp indenters, the strain field beneath the indenter changes with penetration for spherical indentation, and therefore the representative strain depends on the indenter depth. This means that a full stress-strain curve can be potentially extracted from spherical indentations performed at increasing depths.10

While analytical approaches offer simple and fast means of deriving mechanical properties from indentation, the complex stress and strain fields that develop under the indenter and the variety of existing material behaviors question the real applicability of these approaches. In addition, these methods do not take into account highly nonlinear phenomena like the piling up of material around the indenter. More recent studies have benefited from the advances in computational power and numerical methods like the finite element method (FEM) to devise more advanced inverse analysis methodologies. Some researchers have proposed semi-analytical or phenomenological expressions calibrated with FEM simulations that relate the metrics obtained in an instrumented indentation test with the elastoplastic properties of the material.11,12,13 In most of the cases, a power-law hardening material is assumed. Other researchers have used dimensional analysis to derive sets of dimensionless functions that relate the instrumented indentation metrics with the material properties,14,15 including substrate and coating properties in coated systems.16 Other studies in the literature employed iterative FEM approaches to match the simulation and experimental results by manually tuning the model parameters,17 or by making use of a “goodness of fit” parameter that evaluates the level of agreement of the predicted and measured results in a material property space.18

One of the challenges in these inverse analyses is the issue of uniqueness. Different material property sets can lead to the same indentation load-displacement curve. The hardening exponent is usually the most challenging to extract. Furthermore, the load-displacement curves are not very sensitive to changes in the material properties. This makes inverse analyses very susceptible to experimental errors and uncertainties in the measurement of the curves (thermal drifts, instrument compliance, zero-load point, etc.). To this end, some authors have suggested the use of additional information in the analysis, like for example the residual imprint,18 or the use of multiple sharp indenters.19,20 Other authors have taken advantage of the last advances in deep learning to propose inverse analyses based on neural networks that are trained with multifidelity data (2D and 3D FEM simulations and experimental results) to predict mechanical properties from the indentation load-displacement curve.21

The extraction of time-dependent mechanical constitutive behavior from instrumented indentation has also attracted the interest of researchers over the years. Most of the studies focused on the determination of the strain-rate sensitivity (SRS) of the hardness (H), assuming a material in which the flow stress (Y) is a power law function of strain rate, leading to a constant SRS value.22,23,24,25,26 Some researchers then made estimations of the strain-rate-dependent flow stress by using the SRS of the hardness and applying an appropriate constraint factor, usually 2.8–3 for metallic materials. In all these studies, \(\dot{h}/h\), where h is the indenter penetration, is considered the representative indentation strain rate, as shown by Cheng et al. for a power law creeping material by dimensional analysis.14

One of the main assumptions in all these studies is that the SRS of the hardness is equal to that of the flow stress. However, while this is valid for most metals in which the indentation is almost fully plastic, it is not the case for materials in which the elastic work of indentation is not negligible. Several authors have shown that for materials in which the elastic deformation cannot be neglected, which is the case for most polymers and hard ceramics, the SRS of the hardness is significantly lower than that of the flow stress.27,28,29,30,31 Nevertheless, Alkorta et al. and Elmustafa et al. developed methods to obtain the SRS of the flow stress from the SRS of the hardness.27,28 They found that the ratio of SRS of hardness to SRS of flow stress is approximately a unique function of the parameter \(H/E_{eff}\), where \(E_{eff}\) stands for the effective elastic modulus. Elmustafa et al. attributed the lower rate sensitivity of the hardness in materials with a high \(H/E_{eff}\) (above 0.01) to the fact that in this regime the constraint factor (H/Y) changes with the plasticity index (Y/Eeff) of the material.28

Most polymers are remarkably rate dependent and elastic materials (\(Y/E_{eff}\) between 0.01 and 0.1), thus the SRS of the hardness is also expected to be lower than that of the flow stress.30 Moreover, adding to their complex mechanical behavior is the fact that their yield stress is pressure sensitive, a feature that is shared by other amorphous solids like metallic and ceramic glasses. These types of materials can be classified as cohesive-frictional materials, which are characterized by an asymmetric plastic and fracture behavior (brittle in tension and ductile with large deformations in compression). Several authors have studied the inverse problem of indentation for pressure-sensitive materials, assuming the Mohr-Coulomb32 or the Drucker-Prager yield criteria,33,34,35,36 which are the two most popular material models to simulate plasticity in cohesive-frictional materials. For instance, Rodriguez et al. performed an extensive inverse analysis showing that pressure sensitivity increases the constraint factor found in indentation.35

Despite the fact that previous studies have devoted significant efforts to the extraction of mechanical properties in rate-sensitive materials, there are no studies up to date that aim at extracting mechanical properties from materials that combine both pressure and rate sensitivity, which is the case of polymers. The inverse analysis presented in this work aims at closing this gap.

Inverse Analysis Methodology

Dimensional Analysis for Frictional Materials

Dimensional analysis is a powerful tool for establishing relationships between the metrics obtained in an instrumented indentation test and the material parameters of a certain constitutive model. Cheng et al. carried out an extensive review of dimensional analysis of indentation, assuming sharp (self-similar) indenters and considering different types of material models.14 The work of Cheng et al. considered frictional materials for which the flow stress is a power law function of the strain rate, but assuming a rigid plastic material, an assumption that is not valid for many materials. Alkorta et al. later extended Cheng’s analysis to rate-dependent elastoplastic materials.27 For this, they assumed a rigid, self-similar indenter (defined by the included semi-angle \(\theta\)) with frictionless contact penetrating into a homogeneous and isotropic solid.27 The elastic properties were defined by the elastic modulus and Poisson’s ratio (combined in Eeff), while the yield stress was assumed to follow a power-law function of the strain rate with no hardening (\(Y = k\dot{\epsilon}^{m}\)), where m is defined as the SRS of the yield stress. Considering that the load P and contact area Ac in instrumented indentation are a function of the following variables:

$$ P = P\left( {k,m,E_{eff} ,\theta ,h,\dot{h}} \right) $$
(1)
$$ A_{c} = A_{c} \left( {k,m,E_{eff} ,\theta ,h,\dot{h}} \right) $$
(2)

The application of dimensional analysis results in these two dimensionless functions:

$$ P = k\left( {\frac{{\dot{h}}}{h}} \right)^{m} h^{2} {\Pi }_{P} \left( {m,\theta ,\frac{{k\left( {\dot{h}/h} \right)^{m} }}{{E_{eff} }}} \right) $$
(3)
$$ A_{c} = h^{2} {\Pi }_{{A_{c} }} \left( {m,\theta ,\frac{{k\left( {\dot{h}/h} \right)^{m} }}{{E_{eff} }}} \right) $$
(4)

which confirm that the magnitude \(\dot{h}/h\) can be still considered a representative strain rate of indentation for elastoplastic materials, because the hardness remains constant with depth if this magnitude is held constant throughout the indentation test. However, contrary to Cheng’s analysis, it considers the effect of the elastic properties on the indentation response. In any case, Cheng’s analysis for plastic rigid materials can be easily recovered if the ratio \(k\left( {\dot{h}/h} \right)^{m} /E_{eff}\) approaches zero, which is the case for many metals.

Extension to Cohesive-Frictional Materials Following a Drucker-Prager Yield Criterion

The methodology presented in this work is an extension of Alkorta’s analysis to cohesive-frictional materials, by introducing the effect of pressure sensitivity using a Drucker-Prager yield criterion. Cohesive-frictional materials present an asymmetric plastic behavior, having the compressive yield strength greater than the tensile yield strength. These materials generally fail in a brittle manner in tension while they exhibit large plastic deformation prior to fracture in compression. In addition, the yield strength is pressure-dependent. Granular-like materials used in civil and geotechnical engineering fall under this category: soils, rocks, and concrete. Furthermore, many amorphous solids like metallic and ceramic glasses and polymers also exhibit the characteristics of cohesive-frictional materials. The Drucker-Prager model is a common yield criterion used to simulate the mechanics of these materials. This model has proven effective in the extraction of mechanical properties from cohesive-frictional materials through instrumented indentation.35,36,37 The yield surface is defined by:

$$ F\left( {{\varvec{\sigma}},\beta } \right) = q - p\tan \beta - d = 0 $$
(5)

Where the parameters β and d represent the frictional angle and the cohesion strength, respectively. The friction angle is related to the pressure sensitivity of the material, while the cohesion is related to the yield stress. \({\varvec{\sigma}}\) is the Cauchy stress tensor, and q and p are the Mises equivalent stress and hydrostatic pressure stress, respectively. They are defined by:

$$ q = \sqrt {\frac{3}{2}\left( {{\varvec{\sigma}}^{^{\prime}} :{\varvec{\sigma}}^{^{\prime}} } \right)} , p = - \frac{1}{3}tr\left( {\varvec{\sigma}} \right) $$
(6)

Where \({\varvec{\sigma}}^{^{\prime}}\) is the deviatoric part of the stress tensor, defined as \({\varvec{\sigma}}^{^{\prime}} = {\varvec{\sigma}} + p{\varvec{I}}\). The cohesion strength of the material, d, can be defined as a function of the compressive yield stress and the frictional angle by:

$$ d = \sigma_{yc} \left( {1 - \frac{1}{3}\tan \beta } \right) $$
(7)

The Drucker-Prager model represents a cone in the meridional (p-q) plane. This means that the dependency of the Mises equivalent stress with hydrostatic stress follows a linear relation and the slope is defined by the frictional angle. If the frictional angle becomes zero (no pressure sensitivity), the classical Mises yield criterion is recovered.

The Drucker-Prager model thus needs the input of two parameters, the compressive yield stress and the frictional angle. Assuming the compressive yield stress to be a power-law function of the strain rate (\( \sigma = k\dot{\epsilon}^{m}\)) and that the frictional angle is rate independent, dimensional analysis (see supplementary material) provides the following expression for the hardness:

$$ H = k\left( {\frac{{\dot{h}}}{h}} \right)^{m} {\Pi }_{H} \left( {m,\theta ,\frac{{k\left( {\dot{h}/h} \right)^{m} }}{{E_{eff} }},\beta } \right) $$
(8)

Equation 8 shows that even assuming that the hardness (H) and the effective modulus (Eeff) can be determined from the indentation test, the constraint function \({\Pi }_{H}\) depends on three material properties, k, m, and β. A unique solution to the inverse analysis problem can only be found if one of the parameters is known. The inverse methodology developed in this work assumes that the pressure sensitivity, β, is known and focuses on the determination of m and k.

Numerical Analysis

FEM simulation was used to determine the constraint function \({\Pi }_{H}\) in Eq. 8. The proposed inverse analysis follows a two-step procedure. First, the strain-rate sensitivity of the flow stress, m is determined. Then, the constraint factor (H/Y) is computed from the values of m, β as a function of H/Eeff. The FEM simulations to support this methodology were carried out in the framework of the commercial FEM software package Abaqus. The Berkovich indenter was considered a rigid analytical solid with frictionless contact and it was modeled as an equivalent cone with an included semi-angle of 70.3°, which renders the same area-to-depth ratio. This is a common approach when modeling pyramidal indentation,14,35 because it reduces the problem to a 2D domain by assuming radial symmetry.

The indentation domain was defined as a square of side 300 hmax. The size was set large enough to avoid boundary effects. The optimum size was chosen from a parametric study using the worst-case scenario, which is represented by a very elastic material (Y/E > 0.01). In very plastic materials, the deformation is mostly confined beneath the indenter, so the size of the domain does not need to be as large. The domain was divided into two areas, as shown in Fig. 1. The area beneath the indenter, where the highest plastic strains are expected, was very finely meshed so that at least 40 elements were in contact with the indenter at the point of maximum displacement.35 The size of this area was 6 hmax. Outside the refined region, the element size gradually increased to reduce the computational time. For the mesh, 4- and 3-noded quadrilateral axisymmetric elements (CAX4 and CAX3) were selected. Reduced integration elements were chosen when the Explicit solver was used.

Fig. 1
figure 1

Schematics of the Abaqus FEM model used for simulation of conical indentation assuming axisymmetry.

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The FEM-based analysis was based on a series of parametric studies required to complete the described two-step procedure. The first set of simulations was designed to determine the strain-rate sensitivity master curve for a frictional material with no pressure sensitivity. For this, an isotropic linear elastic material with J2-plasticity and no hardening was implemented. Seven constant strain rate indentations at \(\dot{h}/h\) of 10−3, 10−2, 10−1, 100, 101, 102, and 103 s−1 were simulated for different material datasets covering a wide range of frictional properties and strain-rate sensitivities. In particular, the elastic modulus was set to 5, 30, and 70 GPa for a constant Poisson’s ratio of 0.3; m was varied between 0.02 and 0.2; and k was ranged from 100 to 500 GPa∙sm. The Abaqus Implicit solver was used, imposing a nonlinear, large deformation analysis.

For the second set of simulations, a Drucker-Prager yield criterion was implemented. The objective was to assess whether the implementation of pressure sensitivity had an influence on the strain-rate sensitivity master curve. The specifics of the model are described in “Extension To Cohesive-Frictional Materials Following a Drucker-Prager Yield Criterion” section. Perfect plasticity and a non-associated flow rule with dilation angle equal to zero were assumed. The cohesion, d, of the Drucker-Prager model was defined in terms of the compression yield stress, which was assumed to be a power-law function of the strain rate. For the parametric analysis, the elastic modulus was varied from 5 to 50 GPa with a constant Poisson’s ratio of 0.3, k was set to 200 GPasm, m was varied between 0.08 and 0.16, and two frictional angles of β were considered, 15° and 30°. Finally, a third set of simulations of constant strain rate indentation were carried out to determine the constraint function \({\Pi }_{H}\) in Eq. 8 as a function of frictional angle, β, and the SRS of the flow stress, m. For this, the elastic modulus was varied from 2 to 200 GPa with a constant Poisson’s ratio of 0.3; k was set to 200 GPasm; m was varied between 0 and 0.2; and four different frictional angles, β, of 0°, 10°, 20°, and 30°, were considered. The Abaqus Explicit solver was used in this case to avoid convergence problems, imposing nonlinear, large deformation analysis. An Arbitrary Lagrangian-Eulerian adaptive mesh option was used in the refined area beneath the indenter to reduce element distortion. Mass scaling was employed to reduce the computational time, setting up a Δttarget of 10−8 s.

Results and Discussion

Strain-Rate Sensitivity (SRS)

The first step in the inverse analysis methodology is the determination of the strain-rate sensitivity of the flow stress, m. This first part of the analysis was inspired by the work of Alkorta et al. and Elmustafa et al.27,28 They found that an approximately unique relationship exists between the ratio of SRS of hardness and flow stress and the parameter H/Eeff. This is crucial for the analysis because the SRS of the hardness is not equal to that of the flow stress when the elastic deformation in the indentation test cannot be neglected, which is the case for most polymers and ceramics.

Since the hardness H is not easily determined in cases where the indentation is affected by pile-up effects, it is useful at this point to use an alternative hardness definition, which will be referred to as the apparent hardness, Hap, following Alkorta et al.27 The apparent hardness can be defined as the indentation pressure assuming that there are no pile-up or sink-in effects. It has no physical meaning but it presents the advantage that, contrary to the real hardness, it can be simply computed from the area function of the indenter at the total indentation penetration. Since the FEM simulations use a conical indenter, the apparent hardness can be easily calculated as:

$$ H_{ap} = \frac{P}{A} = \frac{P}{{\pi h^{2} \tan^{2} \theta }} $$
(9)

while the strain-rate sensitivities of the hardness and apparent hardness can be computed from the simulations based on their definition using central differences:

$$ m_{H} = \frac{\partial \ln H}{{\partial \ln \left( {\dot{h}/h} \right)}}\,\,\,\,\,m_{{H_{ap} }} = \frac{{\partial \ln H_{ap} }}{{\partial \ln \left( {\dot{h}/h} \right)}} $$
(10)

Figure 2a presents the results of the parametric FEM analysis, together with literature data for frictional materials.27,28 Each point corresponds to one of the simulated materials in the parametric study. The results correlate well with the trends in the literature, confirming the validity of the simulations. The results show that the ratio of SRS of the hardness to flow stress, \(m_{H} /m\), line up in a single curve, meaning that it is a unique function of the parameter H/Eeff. Alkorta et al. showed that actually there is a slight dependency on m, but it can be considered negligible.27 The same occurs for the ratio of SRS of the apparent hardness to flow stress, \(m_{{H_{ap} }} /m\). All values line up in a unique curve when plotted as a function of Hap/Eeff. Of course, \(m_{H} /m\) and \(m_{{H_{ap} }} /m\) follow different curves due to the elastic contribution to deformation and they only merge in the limit where H/Eeff (Hap/Eeff)→0, which corresponds to the ideal rigid plastic material, in which the SRS of the hardness should be the same than that of the flow stress.

Fig. 2
figure 2

Ratio of SRS of the hardness (and apparent hardness) and the flow stress as a function of the dimensionless parameter H/Eeff. (a) Values obtained from an FEM-based parametric study using J2-plasticity, and comparison with literature results.27,28 (b) Values obtained from an FEM-based parametric study using Drucker-Prager plasticity, and compared with master curves obtained in (a) and literature.

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One important implication of the master curve in Fig. 2a is that more elastic materials exhibit less sensitivity to changes in the strain rate of indentation independently of the SRS of the flow stress. In other words, if two materials had the same m value, but one of them were much more elastic than the other (higher H/Eeff), the SRS of the hardness would be reduced in the case of the more elastic material. Furthermore, in the case of extremely elastic materials (H/Eeff >> 0.1), the ratio mH/m would approach zero, and no rate sensitivity could be measured in the hardness (even if the strain sensitivity of the flow stress m were high).

Figure 2b plots the results of the second set of simulations for cohesive frictional materials following the Drucker-Prager yield criterion. Figure 2b shows identical results to those obtained for frictional materials that follow the Mises yield criterion, independently of the frictional angle, β. This is a very important result because it corroborates that the curves plotted in Fig. 2 are universal and that the SRS of the flow stress can be directly determined from the SRS of the hardness (or the apparent hardness), provided that H/Eeff (or Hap/Eeff) is known, for any frictional or cohesive frictional material. It also has important practical implications because Hap/Eeff can be easily determined from indentation testing.

Constraint Factor

The second step in the inverse analysis is the determination of the constraint factor, i.e., the ratio between hardness and yield stress. The parametric FEM analysis carried out to determine the master curves of the constraint factor consisted of four sets of simulations, each one for a specific value of frictional angle (0°, 10°, 20°, and 30°). The master curves for apparent hardness and real hardness are presented in Figs. 3 and 4, respectively. All curves exhibit a similar trend. The constraint factor decreases with increasing H/Eeff, which is directly proportional to the plasticity index, Y/Eeff, and the ratio of elastic-to-total work of indentation, We/Wt. This result is in line with previous studies that have used dimensional analysis for inverse extraction of mechanical properties.14 Only at the limit of very plastic materials (H/Eeff < 10-2), does the constraint factor become nearly independent of the elastic deformation of the material through H/Eeff. In this case, the analysis of Cheng et al. is recovered (only for β=0) in which the constraint factor is a function of m and θ.14 Also, at this limit of very plastic materials, and in the case of β=0 and m=0, the well-known Tabor’s relation is recovered, i.e., H/Y ≈ 3. Furthermore, Figs. 3 and 4 indicate a decrease in the constraint factor with increasing the SRS of the flow stress and with decreasing the frictional angle. The latter is in line with the observations made by Rodriguez et al.35

Fig. 3
figure 3

Constraint factor (for apparent hardness) as a function of Hap/Eeff, strain-rate sensitivity of the flow stress, m, and friction angle, β.

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Fig. 4
figure 4

Constraint factor (for real hardness) as a function of H/Eeff, strain-rate sensitivity of the flow stress, m, and friction angle, β.

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These master curves also corroborate the decrease in the SRS of the hardness for very elastic materials since all the curves become closer for higher values of H/Eeff or Hap/Eeff, in line with the master curve of Fig. 2. At the limit of a fully elastic material (Sneddon’s solution), the material is likely to show no rate sensitivity in the hardness, even if the material has a strongly rate-dependent flow stress.

On certain occasions, it may be more convenient to employ the ratio of elastic-to-total work, We/Wt, instead of the parameter H/Eeff as dimensionless indentation metric. Cheng et al. showed that there exists a unique relation between these two dimensionless material metrics.14 This implies that either the elastic modulus or the hardness can be calculated if the other value is known, without the need of estimating the contact area. In the case of materials in which the yield stress is a power-law function of strain rate, Alkorta et al. showed that the relation between H/Eeff and We/Wt depends on the SRS of the flow stress, m, through the parameter crig, the pile-up parameter for quasi-rigid materials.27 Both dimensionless metrics have been compared to evaluate whether the friction angle or the exponent m have an effect. The curves are shown in Fig. 5. In line with the observations of Cheng et al.,14 it seems that the work ratio is a unique function of the parameter H/Eeff. A slight dependency with m is observed in the detail of the curve for real hardness, but as a first approximation it can be considered negligible. Taking the average value in the curve for real hardness will only induce an error of about 5%. On the other hand, the relation between work ratio and the parameter Hap/Eeff in the case of apparent hardness is reasonably unique. Therefore, the relations in Fig. 5 can be used to change between both dimensionless metrics, which gives more versatility to the inverse analysis methodology presented in this work.

Fig. 5
figure 5

Relationship between H/Eeff (and Hap/Eeff) factor and elastic-to-total work ratio We/Wt for all simulated cases (β = 0, 10, 20, and 30, and m = 0, 0.04, 0.08, 0.12, 0.16, and 0.2).

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Experimental Validation

Three materials with a well-characterized rate-dependent mechanical behavior were selected for validation of the inverse analysis methodology: PMMA, coarse-grain copper (as annealed state), and ultrafine grain (UFG) copper (produced by equal channel angular pressing). Specimens were ground and polished with 1-μm diamond paste to obtain a mirror-finish surface suitable for nanoindentation experiments. The nanoindentation tests were carried out on an instrumented nanoindenter NanoTest Alpha from Micro Materials Ltd (UK) equipped with a Berkovich indenter. A wide range of strain rates can be covered with this instrument: low strain rates with the conventional load-controlled set-up and high strain rates with the nano-impact set-up. A modified version of this instrument was used that incorporates a force sensor in contact with the sample to directly measure the applied impact force.38 Constant strain rate nanoindentation tests were carried out using an exponential loading scheme, up to a maximum load of 100 mN. Three strain rates (\(\dot{h}/h\)) were tested: 0.005, 0.05, and 0.5 s-1. A dwell period of 20 s was applied in all cases, followed by unloading at a constant unloading rate of 10 mN/s. Five repetitions were done per test. The nano-impact tests in the coarse-grain copper and UFG copper were performed with an impulse force of 5 mN and a 20-μm impulse distance. The nano-impact tests in the PMMA sample were performed with an impulse force of 10 mN and a 20-μm impulse distance. The impact conditions were selected in such a way that the resulting maximum displacements were close to the nanoindentation ones. Ten repetitions were carried out per test.

Figure 6 plots the apparent hardness and strain rate evolution with indentation depth in all cases. The indentation strain rates were constant for the nanoindentation tests, as expected, since a logarithmic loading scheme was imposed during the tests. During the nano-impact tests, however, the strain rate was maximum at first contact and gradually decreased with indentation depth, which is expected due to the nature of the test, in which the indenter is impacted against the specimen surface until it is arrested by the material response at maximum depth. The overall hardness results suggest that coarse-grain Cu shows the lowest strain-rate sensitivity, followed by UFG Cu and PMMA. However, it is instructive to follow the hardness evolution with indentation depth in each case. For instance, coarse-grain copper presents a significant indentation size effect, both in the case of the nanoindentation and the nano-impact results. This pronounced size effect is typical in coarse-grained metallic materials and is generally explained by the Nix-Gao model based on the concept of geometrically necessary dislocations.39 However, the indentation size effect is negligible in the UFG material because the severe plastic deformation process introduces a high dislocation density that reduces the characteristic internal length scale of the material.23,40 Similarly, PMMA does not exhibit an indentation size effect either in the nanoindentation results. The apparent indentation size effect observed in the nano-impact tests for UFG Cu and PMMA is not a real indentation size effect. It might be slightly influenced by uncertainties in the zero-load determination, which is necessary in the analysis of the nano-impact tests.38,41 However, most of it comes from the rate-dependent behavior of the materials and the decreasing indentation strain rate as the indenter penetrates the surface and gets arrested at maximum depth.

Fig. 6
figure 6

Evolution of apparent hardness and strain rate with indentation depth for the three tested materials. These curves were obtained by averaging the results of all test repeats: 5 in the case of nanoindentation and 10 in the case of nano-impact tests.

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Based on the curves of Figs. 6, 7 shows the resulting apparent hardness as a function of strain rate for the three materials. The apparent hardness of the quasistatic cases was calculated from the average of the values in Fig. 6 in the last 10% of the curve, to minimize indentation size effects, when present. The nano-impact values are represented as a cloud of points because the strain rate of indentation was not constant with indentation depth, as explained before. The solid symbol within each cloud of points represents the average value computed from the nano-impact tests. The strain-rate sensitivity of the apparent hardness was computed by central differences, and forward and backward differences for the first and last hardness points, respectively.

Fig. 7
figure 7

Apparent hardness as a function of strain rate obtained in the combined nanoindentation and nano-impact tests of PMMA, coarse-grain copper and ultrafine grain copper.

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Finally, Table I compiles the elastic modulus used in the inverse analysis for each material. An elastic modulus of 120 GPa (Eeff of 118.3 GPa) was assumed for Cu, independently of the indentation strain rate, which is reasonable for metallic materials. In the case of PMMA, this hypothesis does not hold, as the elastic modulus is expected to vary with strain rate. Therefore, the strain rate-dependent elastic moduli of PMMA were obtained from previous studies42,43,44 by applying a conversion factor of approximately 1.7, to compensate for the fact that the indentation modulus in polymers is typically higher than in macroscopic tests.45 A frictional angle of 20° was assumed for PMMA.46,47,48

Table I Reduced modulus values [GPa] and frictional angles [°] used as input in the inverse analysis
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Figure 8 plots the estimated flow stress as a function of strain rate that results from applying the inverse analysis based on the master curves of Figs. 2 and 3. The results of the analysis are compared with literature values for each material. The estimated flow stress in PMMA agrees reasonably well with the literature values. The results show a substantial rate-dependent behavior. This is typical in polymers and it is attributed to the characteristic times for activation of the different motions (rotations, chain alignment, etc.) of the molecular chains that constitute the polymer.

Fig. 8
figure 8

Estimations of flow stress from the inverse analysis for the three tested materials. Results are compared with the literature.46,49,50,51,52,53,54,55,56 Literature results for coarse-grain copper correspond to the flow stress at 15% deformation, and 10% in the case of ultrafine-grain copper.

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The estimated flow stress for the two metals is also close to literature values, but the deviations are higher, especially for coarse-grain copper. This is not surprising because coarse-grain copper is expected to display a high hardening rate, and the inverse analysis presented in this work is only valid for perfectly plastic materials. Moreover, the literature values were reported at a strain of 15%, while the representative strain for a Berkovich indenter should be closer to 8%,9 so it makes sense that the estimation is below the literature values. Despite these limitations, the variation of flow stress with strain rate agrees reasonably well with the literature. The flow stress remains nearly independent of strain rate up to strain rates around 103–104 s-1, for which the flow stress increases abruptly. This has been attributed to the increase in the dislocation accumulation rate when the strain rate reaches very high values, leading to a higher strain-rate sensitivity.52

The estimated flow stress for UFG copper correlates better to the literature, compared with the coarse-grain material. This is probably due to the fact that UFG copper shows negligible strain hardening, which better fits the assumptions made in the analysis. The UFG material displays a higher strain-rate sensitivity compared with the coarse-grain material. This is expected due to a change in deformation mechanisms when decreasing the grain size, from forest dislocation hardening to deformation controlled by the grain boundaries.56,57

Conclusion

Inverse analysis is often used for the extraction of elastoplastic properties in instrumented nanoindentation. In the case of strain rate and pressure-sensitive materials, which is the typical behavior of polymers, the studies found in the literature focused on one effect or the other, but there is no study considering the combined effect of pressure and strain rate. To close this gap, this work proposes an FEM-based inverse analysis methodology that accounts both for pressure sensitivity, through a Drucker-Prager yield criterion, and for strain-rate sensitivity, by assuming that the flow stress is a power-law function of the strain rate. The methodology does not consider strain hardening effects, which is reasonable for polymers, nor does it account for viscoelastic effects, which should be taken into account, when applicable, in the value of the strain-rate-dependent elastic moduli. With this limitation in mind, the analysis demonstrates that the ratio of SRS of the hardness to that of the flow stress is a nearly unique function of H/Eeff. Moreover, master curves are provided that consider the effect of strain-rate sensitivity and pressure sensitivity on the indentation constraint factor H/Y. Overall, the results allow the estimation of the flow stress as a function of strain rate for any rate-dependent non-hardening cohesive frictional material from indentation tests performed at varying indentation strain rates. The inverse analysis was validated by testing three materials (coarse-grain copper, ultrafine-grain copper and PMMA) with a well-known rate-dependent mechanical behavior. The predictions of the analysis agreed well with macroscale results from the literature, provided that indentation size effects are taken into account.