Introduction

Microindentation techniques have been widely applied to solid materials, including rock-forming minerals, to characterize their mechanical properties (e.g. Evans and Goetze 1979; Darot et al. 1985; Whitney et al. 2007). The results have been used to understand both ductile and brittle behavior in the Earth’s crust and mantle (e.g. Watts and Zhong 2000; deMartin et al. 2004; Kumamoto et al. 2017). Microhardness (\(H_{{\text{v}}} :{\text{GPa}})\) and fracture toughness (\(K_{{{\text{ifr}}}} :{\text{MPa}}\;{\text{m}}^{1/2} )\) are extracted from an indentation impression size and cracks that initiate from the impression center using the respective relationships:

$$H_{{\text{v}}} = A \cdot P/a^{2} ,$$
(1)

and

$$K_{{{\text{ifr}}}} = B \cdot \left( {E/H_{{\text{v}}} } \right)^{0.5} \cdot {(}P{/}c^{1.5} {),}$$
(2)

where \(P\) is the load (N), \(a\) is an impression dimension (µm), such as half the diagonal length of a pyramidal impression, \(E\) is Young’s modulus (GPa), c is the crack length (m) measured from the contact impression center (Fig. 1), and \(A\) and \(B\) are calibration constants that depend on the type of indentation test (e.g., Anstis et al. 1981; Miyoshi et al. 1985). In this study, we focus on hardness (\(H\)) and fracture toughness (\(K_{{{\text{IC}}}}\)) measured from Vickers indention tests, which we refer to specifically as \(H_{{\text{v}}}\) and \(K_{{{\text{ifr}}}}\), respectively. Mechanical characterizations using the above relationships often assume constant \(H_{{\text{v}}}\) and \(K_{{{\text{ifr}}}}\) values at different values of \(P\); however, changes in \(H_{{\text{v}}}\) with \(P\), especially at small loads, have been identified in a number of materials including metals, ceramics, and glass (e.g. Manika and Maniks 2006). The effect of load on hardness is often referred to as the indentation size effect (ISE), which is considered to manifest itself as changes in \(H\) with indentation depth (h) (e.g., Pharr et al. 2010). Recent developments in nanoindentation techniques allow loads and displacements to be monitored at a fine scale. The results are often compared to a power law relationship of the form \(H \propto h^{ - 0.5}\), which is predicted from a mechanistic model based on the concept of geometrically necessary dislocations (GND) (Nix and Gao 1998), although some studies demonstrate difficulties in explaining the observed ISE with a power law relationship (e.g., Pharr et al. 2010; Manika and Maniks 2006).

Fig. 1
figure 1

Schematic representation of indentation experiments. P: load; 2a: diagonal length of an indentation impression; h: indentation depth; 2c: length of a half-penny-type crack

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Another size effect derives from the grain size of the materials analyzed. A typical grain-size effect is described by \(H \propto d^{ - 0.5}\), which is known as the Hall–Petch relationship (Hall 1951; Petch 1953). This relationship predicts larger hardness values for finer-grained materials, which drives material scientists to explore techniques to reduce the grain size of materials. In Earth science, only a few studies have considered the Hall–Petch effect on the mechanical properties of rocks (White et al. 1980; Druiventak et al. 2011; Hansen et al. 2019), while no studies have confirmed the Hall–Petch relationship in rock-forming minerals. The grain-size effect is characterized by the same or a similar power law as the ISE, which implies a common source (e.g., Jung et al. 2013; Li et al. 2016). However, along with the ISE, the dependence of \(H\) on the inverse square of grain size has been questioned in recent studies (e.g., Dunstan and Bushby 2014; Li et al. 2016).

Kumamoto et al. (2017) reported a relationship of the form \(H \propto h^{ - x}\) with x values of 0.07–0.09 for \(h\) values of 0.03–0.94 μm in nanoindentation tests on olivine single crystals. They compiled previous experimental results on the low-temperature plasticity of olivine using this power law, where \(H\) is related to the yield stress and \(h\) is grain size in the case of polycrystalline materials. The size exponent (x) they obtained is much lower than the typical value of 0.5, and they considered power-law size effects for a size range up to ~ 1 mm, which is much larger than the commonly observed ISE size range, at less than a few tens of µm (e.g., McElhaney et al. 1998; Manika and Maniks 2006).

In this study, we prepared Fe-free and Fe-bearing olivine single crystals with surfaces displaying different crystallographic planes along with high-density polycrystalline material with different average grain sizes ranging from 170 to 890 nm. We used these materials for Vickers indentation tests with different loads to explore the effects of grain size, indentation size (or load), composition, and crystallographic orientation on the values of \(H_{{\text{v}}}\) and \(K_{{{\text{ifr}}}}\) in olivine.

Experimental

Materials

The olivine single crystals used for this study include a synthetic Fe-free crystal (manufactured by Oxide Corp.) and Fe-bearing (~ Fo91) gem-quality olivine from San Carlos, New Mexico. Both crystals were free from cracks and pores according to observations in optical and scanning electron microscopes (SEM) (Fig. 2).

Fig. 2
figure 2

Scanning electron micrographs of indent impressions after applying a load of 0.98 N on an Fe-free olivine single crystal. Each image corresponds to the impression on a sample surface parallel to a different crystallographic plane: a (100), b (010), c (001). Crystallographic axes are indicated in the images

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For polycrystalline olivine, we used synthetic samples from our laboratory (Koizumi et al. 2010). Hardness and fracture toughness are sensitive to pre-existing cracks and pores. Grain boundary cracks form easily during sample cooling due to interfacial stress produced by anisotropic shrinkage of grains. Such interfacial stress-induced cracking can be suppressed by reducing grain size (Evans 1978). Synthesized olivine aggregates usually require a grain size of < 5 μm to prevent such cracking (Cooper 1990; Koizumi et al. 2010). We first prepared initial mineral powders of Fe-free and Fe-bearing (Mg1.8Fe0.2SiO4) olivine with a particle size of ~ 100 nm and < 5 vol% enstatite through solid-state reaction of nanoscale powders of Mg(OH)2 and colloidal SiO2, with or without ferric nitrate at 1000 °C. The subsequent calcined powders were formed by slip casting using a porous alumina block covered with a 0.2 μm membrane filter into a φ10 × 10 mm cylindrical powder green body (Koizumi et al. 2016). In this study, we used the spark plasma sintering (SPS) technique (Guignard et al. 2011), an alternative to conventional vacuum sintering, to minimize olivine grain growth during the sintering. The green body was fitted into a graphite die with a 10-mm inner diameter in an SPS machine (SPS-110N; Sinterland, Inc., Nagaoka, Japan). Under vacuum conditions (~ 0.4 Pa), the powder was heated to 1150 °C at a heating rate of 10ºC/min and then held for 20 min under a compressive load of 50 MPa. After the SPS treatment, the applied pressure was reduced to below 13 MPa and the specimens were subsequently annealed at 1000 °C for 10 min to relieve the residual strain. Through this procedure, we obtained disc-shaped (10-mm diameter and 3-mm thickness) specimens of 170-nm grain size. Some of the specimens were subsequently annealed at a temperature of 100 °C above the sintering temperature with different durations to obtain various grain-sized materials through static grain growth. Ultimately, we were able to prepare Fe-free aggregates with six different average grain sizes ranging from 170 to 890 nm and Fe-bearing aggregates with grain sizes of 260 and 360 nm. All the aggregates exhibit common microstructural characteristics: (1) near-zero porosity, (2) equigranular grains, and (3) equiaxed grain shapes (Fig. 3). The water content in the aggregates is below the detection limit of Fourier transform infrared (FT-IR) spectrometry (Fei et al. 2016).

Fig. 3
figure 3

Scanning electron micrographs of indent impressions after applying a load of 0.98 N on Fe-free polycrystalline olivine with various grain sizes. The lower images show the sample microstructures. The samples in the lower figures were thermally etched to enhance grain boundary visibility. a 170-nm grain-size aggregate. A bulk sample with a thickness of 1 mm is also shown in the lower figure. b 410-nm grain-size aggregate. c 890-nm grain size aggregate

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The indentation surfaces were prepared by grinding with a 30-μm grit-coated diamond lapping film and given a final polish with a 0.5-μm diamond paste. The surfaces of the single crystal samples were parallel to either the (100), (010), or (001) planes of olivine. These orientations were verified by electron backscatter diffraction with SEM. Mechanical polishing can introduce extra defects (e.g., dislocations and/or an amorphous layer), which are concentrated at the sample surfaces and can affect the indentation results. To check for such polishing effects, we also prepared samples that were additionally annealed at 1000–1100 °C for 30 min after polishing to remove any surface defects. We analyzed the surfaces of samples (i.e., 10 μm × 10 μm areas) with and without thermal treatments using atomic force microscopy (SPM-9700HT; Shimadzu Japan). The heat treatment had the effect of smoothing the grain surfaces and creating grooves in the polished surface along grain boundaries, which helped to establish equilibrium dihedral angles between olivine grain boundaries and the surface (Maruyama and Hiraga 2017). As a consequence, both samples exhibit a similar roughness (Ra) of 1.8 ± 0.2 nm.

Indentation methods

Vickers indentation tests were conducted at eight different loads in the range 0.10–9.81 N for a constant period of 15 s using a standard microhardness tester (MVK-E Akashi, Japan). We checked the consistency of the tests using a beryllium copper standard block. We typically produced ten impressions with the same load in the same sample. The single crystals of olivine were oriented with their crystallographic axes (i.e., [100], [010] and [001]) aligned with the directions of the pyramidal indentation tip and its corners. The offset in alignment between the crystal axes and the indenter dimensions was < 1° and < 4° for Fe-free and Fe-bearing crystals, respectively, so that crack propagation was expected to occur parallel to either the (100), (010), or (001) olivine planes. The sample surfaces were observed after the tests by backscattered electron imaging in a field emission gun SEM (JSM-7001F, JEOL), which allowed the dimensions of the indentation impressions and their associated cracks to be measured at a high resolution (i.e., ± 10 nm).

We determined Vickers hardness (\(H_{{\text{v}}}\)) using Eq. (1) with a value of \(A\) of 463.6 (e.g. Li et al. 1989), and measured values of \(P\) and \(a\). We measured the lengths of the two diagonals (= \(2a\)) as shown in Fig. 1 and used their arithmetic means to obtain a representative value of \(a\) for each impression. The impression depths (\(h\)) were calculated from these \(a\) values and the relationship \(a = 3.5 \times h\), which is valid for the geometry of the indentation tip used in this study.

We determined Vickers fracture toughness (\(K_{{{\text{ifr}}}}\)) using Eq. (2) with a \(B\) coefficient value of 0.018 (Miyoshi et al. 1985), experimental \(P\) values, and \(c\) and \(E\) values determined as follows. We measured \(2c\) for the two sets of orthogonal cracks as shown in Fig. 1 and used their arithmetic means to obtain a representative c value for each impression in the polycrystalline samples. For the single crystals, the \(2c\) of each orthogonal crack was measured to obtain a c value representing the length of a crack parallel to each olivine crystallographic plane. We used \(E\) based on the isotropic Young’s modulus given by \(E = 9KG/\left( {3K + G} \right)\), where \(K\) (= 128.6 and 129.5 GPa for Fe-free and Fe-bearing olivine, respectively) and \(G\) (= 81.1 GPa, and 77.6 GPa for Fe-free and Fe-bearing olivine, respectively) are the bulk and the shear modulus, respectively (Kumazawa and Anderson 1969; Webb 1989).

Results

Observations

We were able to impose indent impressions comparable to the structure shown in Fig. 1 on most of the samples (Figs. 2, 3). The smallest and largest impressions analyzed have \(2a\) values corresponding to 3.3 and 49.1 µm, respectively (Tables 1, 2). The indentation depths (h) exceeded the grain size of all the polycrystalline materials except for indentation on the 890-nm-sized sample at the lowest load (i.e., 0.1 N) (Table 2).

Table 1 Results of indentation tests on olivine single crystals
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Table 2 Results of indentation tests on olivine polycrystals
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Cracks were generated from the impression center and propagated parallel to the impression diagonals (Figs. 2, 3). The crack widths are commonly continuous from the exterior to the interior of the impressions, indicating the formation of half-penny cracks (Cook and Pharr 1990). The lengths of the two major orthogonal cracks are essentially equal (< 2% difference) in the polycrystalline samples (Fig. 3), while they differ by up to 30% in the single crystals (Fig. 2, Table 1). Occasionally, the cracks branch off at their tips (Fig. 2c) and secondary cracks emanate from the middle of impression edges in the single crystals (Fig. 2b). We measured the lengths of only the major cracks to obtain toughness values. The cracks formed in the single crystals are very sharp and straight (Fig. 4a), while those in the polycrystalline samples are more irregular (Fig. 4b). Finer-scale examinations of the cracks in the polycrystalline sample revealed that crack propagation occurred in both inter- and intra-granular regions at a ratio of approximately 1:5 (Fig. 4b). The cracks tend to kink when they intersect different grains and grain boundaries.

Fig. 4
figure 4

Scanning electron micrographs of half-penny cracks formed in Fe-free olivine at a load of 1.96 N. The inserts (lower left) show the locations of the crack tips in the larger images. a A single crystal with a surface parallel to (001) and horizontal and vertical axes parallel to [100] and [010], respectively. b A polycrystal with a grain size of 170 nm. This sample was thermally etched prior to indentation

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We failed to obtain \(2a\) and \(2c\) values from all the impressions due to irregular cracking, which was significant at larger loads. Tests with fewer than 10 analyzed impressions listed in Tables 1, 2 are a result of such failures. In theory, an impression center should bisect the total length of a half-penny crack (Fig. 1). The distances of the impression center to the two crack ends differ by < 3% in the polycrystalline samples, but we occasionally observed large differences of > 30% in the single crystals, which were excluded from the toughness calculations in this study. Some indentations, especially at the lower loads, were not associated with the formation of cracks. Most of the blank entries for crack length (2c) in Table 1 correspond to such a case.

Microhardness

The average values of \(H_{{\text{v}}}\) calculated from the sizes of impressions at the same loads (Eq. (1)) are listed in Tables 1, 2 and are plotted against load in Fig. 5a, b. We focus here on the average values, referred to as \(H_{{{\text{v}}\_{\text{sing}}}}\) and \(H_{{{\text{v}}\_{\text{poly}}}}\) for the single crystal and polycrystalline samples, respectively. The minimum and maximum \(H_{{\text{v}}}\) values are also listed in the tables and are connected by vertical bars in most of the figures. Tests at larger loads exhibited smaller \(H_{{\text{v}}}\) values, but this trend ceased at loads of > 4 N, above which the values are nearly constant for all types of samples.

Fig. 5
figure 5

Load (P) versus Vickers hardness (\(H_{{\text{v}}}\)). a Results for single crystals. The \(H_{{\text{v}}}\) values obtained from indent impressions on different crystallographic planes are shown by different symbols. Blue: Fe-free; green: Fe-bearing. b Results for polycrystals. Different colors and symbols indicate grain size and the presence of Fe. The 170-nm grain-size samples denoted a1 and a2 correspond to the thermally treated samples (a1: 1000 °C for 30 min; a2: 1100 °C for 30 min) prior to the indentation tests

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We did not observe a systematic difference among \(H_{{{\text{v}}\_{\text{sing}}}}\) values of different crystallographic surfaces (Fig. 5a). At large loads, the Fe-bearing single crystal has an \(H_{{\text{v}}}\) of ~ 8 GPa, which is the smallest value we obtained in this study and is ~ 10% smaller than that of the Fe-free crystal. The difference between the Fe-free and Fe-bearing crystals becomes smaller with decreasing load.

The finest-grained Fe-free polycrystalline sample (= 170 nm) has an \(H_{{\text{v}}}\) of ~ 17 GPa at the lowest load (= 0.1 N), which is the largest value we obtained in this study (Fig. 5b). The samples given an additional heat treatment to remove possible surface damage layers have essentially the same \(H_{{{\text{v}}\_{\text{poly}}}}\) values at similar loads and grain sizes. Despite the small amount of data for Fe-bearing samples, the Fe-bearing sample with a 360-nm grain size has the lowest \(H_{{{\text{v}}\_{\text{poly}}}}\) among the polycrystalline samples, which might reflect an Fe-weakening effect, as seen in the single crystals.

To better view any exponential characteristics of \(H_{{\text{v}}}\) as a function of size, the same \(H_{{\text{v}}}\) values presented in Fig. 5 were replotted against h (Fig. 6a, b) and d (Fig. 7) in log space. Previously reported values are also shown for comparison. Overall, each \(H_{{\text{v}}}\) data set from the same samples (plus the same crystallographic surfaces in case of the single crystals) can be well fit by a single line at \(h\) of < 4 µm, which indicates a relationship of \(H_{{\text{v}}} \propto h^{ - x} { }\) with an \(x\) value of ~ 0.1. The dependence on orientation and Fe content of the exponent \(x\), which corresponds to the slope in the figure, is not clear (Fig. 6a), although the value exhibits grain-size sensitivity (Fig. 6b). A larger \(x\) value of ~ 0.2 best describes the \(h\) dependence of \(H_{{{\text{v}}\_{\text{poly}}}}\) of the finest-grained sample, while the smaller value of ~ 0.1 best describes the coarser-grained materials. The smaller exponent value is essentially identical to that found for \(H_{{{\text{v}}\_{\text{sing}}}}\) (Fig. 6a). The \(H_{{\text{v}}} - h^{ - x}\) relationship identified for h of < 4 µm clearly underestimates \(H_{{\text{v}}} { }\) at h of > 4 µm for both single crystal and polycrystalline samples, which agrees with our finding that \(H_{{\text{v}}}\) is load independent at > 4 N in Fig. 5.

Fig. 6
figure 6

Data for \(H_{{\text{v}}}\) dependence on indentation depth (\(h\)), plotted in log space for a single crystals and b polycrystals. Previously reported values are included, some of which have been converted to \(H_{{\text{v}}}\) values using calibration coefficients for different indentation methods (see text). The representative depth exponents corresponding to the slopes are shown in the figures

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Fig. 7
figure 7

Data for \(H_{{\text{v}}}\) dependence on grain size (\(d\)). Single crystal data are shown in the right-hand column for comparison. Previously reported values are also included. H17: Hassanzadeh-Tabrizi (2017); KF10: Kharaziha and Fathi (2010); FK09: Fathi and Kharaziha (2009). The curves correspond to Eq. (9) with representative indentation depths for each load (see text)

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Due to the limited grain-size range of the Fe-bearing samples, we analyzed the effect of grain size on \(H_{{\text{v}}}\) for Fe-free olivine only (Fig. 7). Each \(H_{{\text{v}}}\) data set at the same load can be well fit with a single line, which indicates that \(H_{{{\text{v}}\_{\text{poly}}}} \propto d^{ - x}\) (Fig. 7). As we will discuss later, we interpret \(H_{{{\text{v}}\_{\text{poly}}}}\) to be a function of \(d\) in a different manner, corresponding to the dotted curves in the figure. However, for the sake of simplicity, here we discuss the \(H_{{{\text{v}}\_{\text{poly}}}}\) data in terms of their best-fit \(x\) values, which correspond to the slopes in the figure. The \(x\) value appears to be load sensitive. The largest \(x\) value of ~ 0.25 describes the grain-size dependence of \(H_{{{\text{v}}\_{\text{poly}}}}\) at the smallest load, while the exponent decreases to ~ 0.05 at larger loads. If we extrapolate the \(H_{{{\text{v}}\_{\text{poly}}}}\)—\(d\) relationship of each load to larger grain sizes, most of the resulting \(H_{{{\text{v}}\_{\text{poly}}}}\) values are close to the mean values of \(H_{{{\text{v}}\_{\text{sing}}}}\) for the three different surfaces at the same loads as at d of ~ 1 µm (see the right column in Fig. 7).

We have included previous results from nanoindentation and Vickers microindentation tests on San Carlos olivine in Fig. 6a (Evans and Goetze 1979; Kranjc et al. 2016; Kumamoto et al. 2017). To compare the hardness obtained from different types of indentation tests, the hardness obtained in the previous nanoindentation tests conducted with a Berkovich-type tip were multiplied by a factor of 0.906 (ISO 14577). The previous results are all reasonably consistent with our results at a given h (Fig. 6a). As well as the absolute values, the changes in \(H{ }\) with \(h\) reported by Kumamoto et al. (2017), who fit their data with exponent values of 0.07–0.09, are also consistent.

The previously reported \(H_{{{\text{v}}\_{\text{poly}}}}\) values for Fe-free olivine (Fathi and Kharaziha 2009; Kharaziha and Fathi 2010; Hassanzadeh-Tabrizi 2017) are all slightly lower than our results at a given grain size and \(h\) (Figs. 6b, 7), which can be attributed to porosities of ≥ 1 vol% in their samples.

Fracture toughness

A number of fracture toughness studies exhibit \(K_{{{\text{ifr}}}}\)—\(c\) relationships that depend on the crack morphologies at the indentation contacts (Cook and Pharr 1990). The relationship of Eq. (2) is applicable for a half-penny crack with a length much greater than the impression size (i.e., \(\left( {c - a} \right) \gg a\) in Fig. 1) (e.g., Evans and Charles 1976; Niihara et al. 1982; Cook and Pharr 1990), which effectively corresponds to the geometry of the majority of the cracks observed in this study (Figs. 2, 3, 4). Thus, the measured crack length was analyzed based on \(c^{1.5} /a\) against \(P^{0.5}\), which are expected to be linearly correlated (Eqs. (1), (2)) (Fig. 8a). The data from the single crystals were combined according to the fracture planes. For example, cracks parallel to the (100) plane were formed during indention on (010) and (001) olivine surfaces, as in Fig. 2b, c. As a consequence, we obtained two \(c^{1.5} /a\) data points for each \(P\) (Fig. 8a), except in the case of indentation failures (Table 1). The two \(c^{1.5} /a\) data from the same \(P\) do not differ by more than 30% (Fig. 8a). Overall, each \(c^{1.5} /a\)—\(P^{0.5}\) data set for a certain grain size and a certain fracture plane in the case of the polycrystals and single crystals, respectively, is well explained by a linear relationship corresponding to the fitted lines in the figure. The polycrystals tend to exhibit larger \(c^{1.5} /a\) values relative to the single crystals at a given \(P\). No clear differences between Fe-free and Fe-bearing samples are observed.

Fig. 8
figure 8

Fracture toughness (\(K_{{{\text{ifr}}}}\)) analyses. a Crack lengths to the power of 1.5 normalized by the impression diagonal lengths (a) versus the inverse square of the load (\(P^{0.5}\)) (Eq. (2)). The best-fit lines for each data set of the same grain size and the same crystallographic surface (plane) that satisfies \(P \propto c^{1.5} /a\) are included. b The \(K_{{{\text{ifr}}}}\) values calculated from the data collected at loads of 4.9 and 9.8 N are plotted against grain size for polycrystals (left) and against fracture planes for single crystals (right). The fracture planes are assumed to be parallel to the olivine crystallographic planes. Values for San Carlos olivine single crystals from Swain and Atkinson (1978) and Darot et al. (1985) are included

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The large crack-to-indent ratio needed to assume \(\left( {c - a} \right) \gg a\) is known to be valid for \(c/a \ge 2.5\) (Niihara et al. 1982; Niihara 1983; Ponton and Rawlings 1989), which applies to all the cracks formed at loads of \(\ge\) 0.98 N in our tests (Tables 1, 2). Thus, we used the data for \(c\), \(a\), and \(H_{{\text{v}}}\) at loads of 4.90 and 9.81 N, where \(H_{{\text{v}}}\) is free from ISE, to obtain representative \(K_{{{\text{ifr}}}}\) values based on Eq. (2). The calculated \(K_{{{\text{ifr}}}}\) values were plotted against grain size and crystallographic fracture planes for polycrystalline and single crystal samples, respectively (Fig. 8b). As expected from the \(c^{1.5} /a\)—\(P^{0.5}\) plots (Fig. 8a), the \(K_{{{\text{ifr}}}}\) value increases weakly from 0.8 to 1.1 MPa m1/2 with increasing grain size. For the single crystals, we see fracture plane-dependent \(K_{{{\text{ifr}}}}\) values ranging from 0.5 to 0.9 MPa·m1/2 with relatively large variations (± 0.2 MPa m1/2). \(K_{{{\text{ifr}}}}\) values from previous Vickers indentation tests on San Carlos olivine (Swain and Atkinson 1978; Darot et al. 1985) are included in the figure. These studies reported only fracture surface energies (\(\gamma\)) that were obtained from \(K_{{{\text{ifr}}}}\) values; thus, we converted the \(\gamma\) values of three crystallographic planes to \(K_{{{\text{ifr}}}}\) values using the same relationship used in the earlier analyses (Atkinson 1984):

$$2\gamma = K_{{{\text{ifr}}}}^{2} \left( {1 - v^{2} } \right)/E,$$
(3)

where \(v\) is Poisson’s ratio, which is 0.24 and 0.25 for Fe-free and Fe-bearing olivine, respectively (Kumazama and Anderson 1969; Webb 1989). The \(K_{{{\text{ifr}}}}\) values from our study and the previous studies are essentially identical (Fig. 8b). The effect of Fe on \(K_{{{\text{ifr}}}}\) values seems to be very minor.

Discussion

Indentation size effect on hardness

We observed hardness to depend on load (or indentation depth) and grain size (Figs. 6, 7). The larger ISE for the finer-grained materials indicates a strong coupling between ISE and the grain-size effect (Fig. 6b). We first consider ISE in the single crystals where the grain-size effect is not involved. The ISE is usually interpreted based on \(H/H^{0} \propto h^{ - x}\) with a threshold depth (\(h^{0}\)) above which \(H\) reaches a constant value of \(H^{0}\) (Nix and Gao 1998). Thus, we expect the hardness of the single crystal (\(H_{{{\text{sing}}}}\)) to be a function of \(h\) according to:

$${\log}H_{{{\text{sing}}}} \left( h \right) = {\log}H_{{{\text{sing}}}}^{0} - m_{{{\text{sing}}}}^{{{\text{ISE}}}} \cdot {\log}\left( {h/h^{0} } \right),$$
(4)

where \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) is the depth exponent and \(H_{{{\text{sing}}}}^{0}\) is the ISE-free \(H_{{{\text{sing}}}}\) value at \(h > h^{0}\). This relationship is expected to hold at \(h \le h^{0}\). As seen in Fig. 6a, the overall characteristics of olivine \(H_{{{\text{v}}\_{\text{sing}}}}\) values are well represented by Eq. (4) with values for \(H_{{{\text{sing}}}}^{0}\) of 9 and 8 GPa for Fe-free and Fe-bearing olivine, respectively, \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) of ~ 0.1, and \(h^{0}\) of ~ 4 µm.

Only a few studies have examined both nanohardness and microhardness in the same materials (e.g., Manika and Maniks 2006; Al-Rub 2007). These two studies showed that both types of hardness are comparable at the same indentation depth and that they both change with \(h\) in the same manner, i.e., they exhibit the same ISE. The comparison of our results with previous studies on single-crystal olivine leads to the same conclusion (Fig. 6a). Estimation of the value of \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) should be more accurate in studies using nanoindentation due to its ability to detect continuous changes in \(H\) with depth (Kumamoto et al. 2017). At the same time, because the technique is specialized for depth sensing at the submicron level, it is not suitable for investigating \(h^{0}\), which is typically on the order of a micron. As summarized by Manika and Maniks (2006), the \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) value varies from 0.1 to 0.3 depending on the material, with \(h^{0}\) values of ~ 1 µm for alkali halides and 10–30 µm for metals. Overall, the nature of the ISE in olivine single crystals follows that of other solid materials. Our estimated \(h^{0}\) value of 4 µm indicates that all previous indentation studies have measured the \(H_{{{\text{sing}}}}\) of olivine within its ISE regime (Fig. 6a).

Grain-size effect on hardness

Here we examine the effect of grain size on \(H_{{\text{v}}}\) where \(H_{{\text{v}}}\) is free from ISE (i.e., \(H_{{\text{v}}}\) at \(h > h^{0}\)). We refer to \(H_{{{\text{v}}\_{\text{poly}}}}\) from the ISE-free regime as \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\) hereafter. \(H_{{{\text{poly}}}}\) is commonly analyzed by viewing \(H_{{{\text{poly}}}}\) versus \(d^{ - 1/2}\) under the assumption of a Hall–Petch relationship (Hall 1951; Petch 1953); however, we keep the grain size exponent as a variable by referring to \(m_{{{\text{poly}}}}^{{{\text{HP}}\_0}}\), which is expected to describe the grain-size dependence of \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\) (Dunstan and Bushby 2014; Li et al. 2016). The value of \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\) decreases with increasing \(d\) (Fig. 6b). The \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\) values of the coarser-grained samples are essentially identical to \(H_{{{\text{v}}\_{\text{sing}}}}^{0}\) (Fig. 7), which suggests the presence of a threshold grain size (\(d^{0}\)) above which \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\) does not change with \(d\). Although we were not able to analyze the \(d\) dependence of \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\) at \(d\) values of > 890 nm in this study, \(d^{0}\) was estimated as ~ 1 μm (Fig. 7). Overall, we can consider the grain-size effect in a similar manner to ISE, which leads us to expect \(H_{{{\text{poly}}}}^{0}\) as a function of \(d\) with a similar form to Eq. (4) as

$${\log}H_{{{\text{poly}}}}^{0} \left( d \right) = {\log}H_{{{\text{sing}}}}^{0} - m_{{{\text{poly}}}}^{{{\text{HP}}\_0}} \cdot {\log}\left( {d/d^{0} } \right).$$
(5)

We plotted \(d/d^{0}\) versus \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\) by setting \(d^{0}\) to 1 μm in log space (Fig. 9). As seen in the figure, the observed \({ }H_{{{\text{v}}\_{\text{poly}}}}^{0}\) values are well represented by Eq. (5), with the best-fitting \(m_{{{\text{poly}}}}^{{{\text{HP}}\_0}}\) values in the range 0.06–0.09. The \(m_{{{\text{poly}}}}^{{{\text{HP}}\_0}}\) values are comparable to those found for \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\); however, they are much smaller than the exponent value in the Hall–Petch relationship. We will discuss this point later.

Fig. 9
figure 9

Data for \(H_{{{\text{poly}}}}^{0}\) dependence on grain size (\(d\)) The grain size is normalized by the threshold grain size (\(d^{0}\)) of 1 µm. The \(H_{{{\text{poly}}}}^{0}\) values correspond to the hardness measured at loads of 4.9 and 9.8 N. The slopes correspond to \(m_{{{\text{poly}}}}^{{{\text{HP}}\_0}}\) in Eq. (5)

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Source of size effects on hardness

The source of ISE has been a matter of debate for decades (Dunstan and Bushby 2014), with proposed explanations including (1) friction between the indenter tip and the sample surface (e.g., Li et al. 1993), (2) the presence of a surface damage layer due to polishing (e.g., Becher 1974), (3) limited dislocation movement near the sample surface (e.g., De Guzman et al. 1993), and (iv) dislocation–dislocation interactions (Elmustafa and Stone 2002).

Friction can be significantly affected by the state of the surface; however, \(H_{{\text{v}}}\) of our samples with and without annealing are essentially identical (Fig. 5b). The grain-size dependent ISE is difficult to explain by friction alone. A power law of essentially the same size explains both ISE and the grain-size effect (i.e., \(m_{{{\text{sing}}}}^{{{\text{ISE}}}} \approx m_{{{\text{poly}}}}^{{{\text{HP}}\_0}}\)), which is consistent with the idea that they have a common source (Jung et al., 2013; Li et al. 2016).

Equation (4) predicts \(H_{{{\text{sing}}}}\) values of ~ 20 and ~ 18 GPa for Fe-free and Fe-bearing olivine, respectively, at h of 5 × 10–3 μm, which is equivalent to the olivine lattice constant. Such \(H_{{{\text{sing}}}}\) values are ~ 1.5 times the theoretical strength estimated as \(G/2\pi\). Manika and Maniks (2006) extrapolated the \({\log}H_{{{\text{sing}}}} - {\log}h\) curves for various materials using an h value equal to their lattice constants, and found that such \(H_{{{\text{sing}}}}\) values compared well to the theoretical strength within a factor of < 2. These observations might indicate deformation by a dislocation-free process at the very surface region of the materials where pre-existing dislocations are absent and/or limited in movement. Kumamoto et al. (2017) explained their obtained very large \(H_{{{\text{sing}}}}\) (~ 30 GPa) at a very small load by the similar physical process. The behavior of grain boundaries can be compared to such surfaces, which explains the common power law for the grain size effect and ISE.

A sample volume extending to ~ 10 times the indentation depth is considered to contribute to \(H_{{\text{v}}}\) (Yang and Vehoff 2007), indicating that our measurements reflect the properties of both intra- and inter-granular regions in the polycrystalline samples, even at the minimum depth and largest grain size. The ISE has often been interpreted based on GND theory (Nix and Gao 1998), which predicts a depth exponent of 0.5. The grain-size effect, namely the Hall–Petch effect is often described by a grain size exponent of 0.5, which is usually explained by dislocation pile-ups against a grain boundary. The GND theory has been extended to include grain-size effects with the relation \(H/H^{0} \propto \sqrt {1 + \frac{E}{d} + \frac{F}{h}}\), where \(E\) and \(F\) are material constants (Jung et al. 2013). However, this relationship does not explain features in our observations such as the larger grain-size dependence of \(H_{{\text{v}}}\) at smaller \(h\) (Fig. 6b). Krell (1998) measured \(H_{{\text{v}}}\) of polycrystalline alumina with 0.5- and 3-µm grain sizes and 3 mol% Y2O3-doped zirconia with 0.28- and 0.46-µm grain sizes at loads of 3–100 N. Manika and Maniks (2006) examined \(H_{{\text{v}}}\) of polycrystalline copper with 1-, 3-, and 5-µm grain sizes at indentation depths of 0.7–8 µm. These studies showed that finer-grained materials tend to exhibit smaller or little ISE. In contrast to the individual effects of grain size and ISE, the universal effect of their coupling has not been identified yet.

Combined indentation and grain-size effects on hardness

Although little is known about \(H_{{\text{v}}}\) when the two size effects are involved, we found a strong interplay between ISE and grain-size effects in olivine (Fig. 6b). Here we attempt to formulate their combined effect on olivine \(H_{{\text{v}}}\) with an appreciation of all the assumptions and derivations that we made to quantify both ISE and grain-size effects above. We can summarize our observations of changes in \(H_{{\text{v}}}\) with \(h\) as shown in Fig. 10, where \(H_{{{\text{v}}\_{\text{poly}}}}^{0}\)≈\(H_{{{\text{v}}\_{\text{sing}}}}^{0}\) at \(d\) of ≥ \(d^{0}\) with a common \(h_{0}\) for both single and polycrystals. Thus, we can expand Eq. (4) to include the case of a polycrystal as

$${\log}H = {\log}H^{0} - m^{{{\text{ISE}}}} \cdot {\log}\left( {h/h^{0} } \right).$$
(6)
Fig. 10
figure 10

Schematic of the \({\log}h\)–\({\log}H\) relationship identified in this study. \(h^{0}\): threshold indentation depth;\(H_{{{\text{sing}}}}\) and \(H_{{{\text{poly}}}}\) are \(H\) values of single- and poly-crystals, respectively; \(H_{{{\text{sing}}}}^{0}\) and \(H_{{{\text{poly}}}}^{0}\) are ISE-free \(H_{{{\text{sing}}}}\) and \(H_{{{\text{poly}}}}\), respectively. The slopes of \(H\) at \(h < h^{0}\) correspond to \(m^{{{\text{ISE}}}}\) (Eq. (7))

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The \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) value does not change with the presence of Fe; thus, the larger ISE for the finer-grained materials can be interpreted by the \(m^{{{\text{ISE}}}}\) dependence on \(H^{0} /H_{{{\text{sing}}}}^{0}\). Thus,

$${\log}m^{{{\text{ISE}}}} = {\log}m_{{{\text{sing}}}}^{{{\text{ISE}}}} + x \cdot {\log}\left( {H^{0} /H_{{{\text{sing}}}}^{0} } \right),$$
(7)

where

$$m^{{{\text{ISE}}}} /m_{{{\text{sing}}}}^{{{\text{ISE}}}} = \left( {d/d^{0} } \right)^{{ - m_{{{\text{poly}}}}^{{{\text{HP}}\_0}} \cdot x}} ,$$
(8)

based on Eqs. (5) and (7). We explored the value of \(x\) by plotting \(m^{{{\text{ISE}}}} /m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) versus \(d/d^{0}\) in log space and setting \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) = \(m_{{{\text{poly}}}}^{{{\text{HP}}\_0}}\) = 0.09 (Fig. 11). All \(m^{{{\text{ISE}}}}\) values used in the plots correspond to the best-fit values that satisfy Eqs. (5) and (6) for each grain size (i.e., the slopes in Fig. 6b). As shown in Fig. 11, an \(x\) value of 5.5 reasonably explains the \(m^{{{\text{ISE}}}}\) values of different grain sizes. Thus, combining Eqs. (4), (5), and (7), we obtain

$$\frac{H}{{H_{{{\text{sing}}}}^{0} }} = \left\{ {\frac{{d^{0} }}{d} \cdot \left( {\frac{{h^{0} }}{h}} \right)^{{\sqrt {d^{0} /d} }} } \right\}^{0.09} .$$
(9)
Fig. 11
figure 11

Analysis of \(m^{{{\text{ISE}}}}\) dependence on \(d\) (Eq. (8)). Both values are normalized by \(m_{{{\text{sing}}}}^{{{\text{ISE}}}}\) (= 0.09) and \(d^{0}\) (= 1 µm), respectively. The slope of the dotted line corresponds to \(- m_{{{\text{poly}}}}^{{{\text{HP}}\_0}} \cdot x\) (see text for details), which gives a \(x\) value of 5.5 by setting \(m_{{{\text{poly}}}}^{{{\text{HP}}\_0}}\) to 0.09

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We checked our observed \(H_{{\text{v}}}\) against simulated values calculated by Eq. (9), which are plotted in Figs. 7 and 12. The \(H\) curves predicted by substituting \(H_{{{\text{sing}}}}^{0}\) (= 9 GPa for Fe-free olivine), \(d^{0}\) of 1 µm, \(h^{0}\) of 4 µm, and representative \(h\) values for each load into Eq. (9) are shown in Fig. 7. Overall, the curves reproduce the systematic changes in \(H_{{{\text{v}}\_{\text{poly}}}}\) with grain size as well as their absolute values. The predicted \(H_{{{\text{v}}\_{\text{poly}}}}\) values at a grain size of 1 µm (= \({ }d^{0}\)) correspond reasonably well to the average values of the observed \(H_{{{\text{v}}\_{\text{sing}}}}^{0}\) of three different directions at any load.

Fig. 12
figure 12

Comparison of the observed \(H_{{\text{v}}}\) values normalized by \(H_{{{\text{v}}\_{\text{sing}}}}^{0}\) with predicted values based on Eq. (9). The single crystal data are plotted by setting \(d^{0} /d\) = 1. The data from ISE-free regions are plotted by setting \(h^{0} /h\) = 1. The values used for other parameters in Eq. (9) are shown. Black symbols: Fe-free samples; grey symbols: Fe-bearing samples. Closed and open circles correspond to our data from polycrystals and single crystals, respectively. Other symbols represent single crystal data from previous studies

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As explained above, the grain-size effect is often described by \(H \propto d^{ - x}\), where x is 0.5 for the classic Hall–Petch relationship, while Eq. (9) represents the exponent as a manifestation of the strong interplay between ISE and the grain-size effect. We are usually limited to measuring \(H\) of materials with a grain size range of less than an order of magnitude. The predicted curves shown in Fig. 7 indicate that these values can be reasonably approximated by \(H/H^{0}\)\(\propto d^{ - x}\) over such a grain size range. We can predict the large x value of 0.5 at small \(h\) (or load) and grain size. Dunstan and Bushby (2014) showed that a plot of \(H\) versus \(d^{ - 0.5}\) could mislead researchers to conclude a Hall–Petch relationship for materials. Instead, they showed a number of ways to describe the observed grain-size dependence of \(H\) other than by a Hall–Petch-type relationship. Our conclusion supports recent criticisms of Hall–Petch relationships (Dunstan and Bushby 2014).

We compared the observed \(H_{{\text{v}}}\) values, which are normalized by \(H_{{{\text{v}}\_{\text{sing}}}}^{0}\) (= 9 and 8 GPa for Fe-free and Fe-bearing olivine, respectively), with their predicted values based on Eq. (9) in Fig. 12. Equation (9) was derived for \(H_{{\text{v}}}\) of polycrystalline olivine in the ISE regime; however, we can incorporate the data for single crystals and the ISE-free regime by setting \(d^{0} /d\) and \(h^{0} /h\) to 1, respectively. Along with our data, the \(H_{{{\text{v}}\_{\text{sing}}}}\) values from previous studies on San Carlos olivine are also shown in the figure for comparison (Evans and Goetze 1979; Kranjc et al. 2016; Kumamoto et al. 2017). As seen in the figure, the \(H_{{\text{v}}}\) values of polycrystalline Fe-free olivine are well reproduced over their entire range, which is partially due to the derivation of Eq. (9) relying mainly on data for that material. In addition, the equation explains other \(H_{{\text{v}}}\) values for different Fe contents and single crystals fairly well.

Size effects on fracture toughness

Commonly, toughness decreases with decreasing grain size, while single crystals often exhibit lower toughness than polycrystals (e.g., Townsend and Field 1990). As a consequence, toughness peaks at a moderate grain size. We observed such size-dependent toughness, where the toughness increases weakly with increasing grain size, while the single crystals exhibit lower toughness values that depend on orientation (Fig. 8). Finer-grained materials should provide more locations at which grain boundaries can open up relative to coarser materials during crack propagation, which is well supported by our observations (Fig. 4b). As a consequence, the work needed to create new surfaces is expected to be lower by the amount of grain boundary energy in such cases. The single crystals were well aligned for cracks to propagate parallel to specific low-index crystallographic planes (Fig. 4a), which should have lower surface energies than surfaces of high-index crystallographic planes. Atomistic simulations of the surfaces in Fe-free olivine estimate the surface energy (\(\gamma\)) of different crystallographic planes as \(\gamma_{010} :\gamma_{001} : \gamma_{100}\) = 1.0: 1.7: 2.1 (Watson et al. 1997), while our obtained \(K_{{{\text{ifr}}}}\) values from the single crystal predict \(\gamma_{010} :\gamma_{001} : \gamma_{100}\) = 1.0: 1.4: 1.7 (Eq. (3)). These small differences can be attributed to the different atmospheres involved, as the simulation considered the surfaces to be exposed to a vacuum. We conclude that the observed \(K_{{{\text{ifr}}}}\) values are primarily determined by the olivine interfacial energies.