The key parameter to calculate hardness is the indentation area. Rather than imaging the hardness impression, the Oliver-Pharr method allows us to obtain the contact area based on the area function. The basic assumption of the method is that the contact periphery sinks in, which does not account for the pile-up around the contact impression in many elastic-plastic materials (Oliver and Pharr, 2004). When pile-up occurs, the predicted contact area by the method is less than the actual area, consequently overestimating the hardness values.

It has been reported that the ratio of the elastic modulus to yield strength ( E / σ y ) and the work hardening coefficient (n) are the fundamental material properties affecting the pile-up behavior (Bolshakov and Pharr, 1998; McElhaney et al., 1998). In general, the pile-up is greatest in materials with large E / σ y and small n. In addition, the pile-up can be more prominent during indenting the thin coating/substrate systems (Pharr and Oliver, 1992), like the structure of ion-irradiated materials, i.e., the thin damaged layers on the underlying unirradiated soft substrate. Hardie et al. investigated the role of pile-up effect on the mechanical property evaluation of self-irradiated Fe12%Cr alloys (Hardie et al., 2015), and found that the pile-up behavior between unirradiated and irradiated materials was significantly different using a cube-corner indenter, producing large errors in the irradiation hardening evaluation based on the area function. They attributed the pile-up to the restricted growth of the plastic zone in the damaged layer and the direct evaluation using the uncorrected indentation data that involve the combined effects of both irradiation hardening and irradiation-induced indentation pile-up.

Hence, the differences in the pile-up effects between the unirradiated and irradiated materials need to be considered when evaluating the irradiation strengthening, to obtain the actual hardness or modulus changes of irradiated materials.

The limited ion penetration depth and the inhomogeneous dose profile are inevitable for ion irradiation. As could be conveniently estimated using the Monte-Carlo codes such as The Stopping and Range of Ions in Matter (SRIM) (Ziegler et al., 2010), the dose and dose rate dramatically vary with irradiation depth, and the maximum dose is, in general, many times or even orders greater than that in the platform region. On the other hand, the plastic zone is usually much greater than the indentation depth (Johnson, 1970; Kiener et al., 2006) [this review focuses on the metals and alloys, and this fact is also applicable for ceramics (Hockey et al., 1978; Lankford and Davidson, 1979)]. Thus, a wide dose depth range is sampled within the plastic zone under the sample surface during indentation, likely involving the unirradiated soft substrate. Such damage gradient effects and soft substrate effects strongly challenge the data analyses.

The surface nanoindentation (see Figures 2A–C) is the most widely applied method to evaluate nanohardness (Gao et al., 2019; Cui et al., 2017; Zhang et al., 2016), due to the simple sample preparation procedure. But the DGE, SSE, ISE, and surface effects are inevitable, as shown in Figure 2B. Regarding the SSE, Hosemann et al. proposed a “simple law of mixture” (Hosemann et al., 2012; Kiener et al., 2012), in which the plastic zone is semi-spherically assumed with a radius of five times to indentation depth for simplicity, to calculate the “fictive” hardness of irradiated materials. By calculating the volume fraction of the unirradiated materials sampled in the total sampling plastic zone, the measured hardness data at the specific indentation depth can be converted to the “fictive hardness” as if the plastic zone is performed exactly as the entire irradiation zone, as illustrated with the blue dotted line in Figure 2C. This simplified approach only gives a semi-quantitative estimate for the true hardness of irradiated materials. Compared with the “simple law of mixture,” the precise plastic zone of the indention on irradiated materials can be obtained by transmission electron microscope (TEM) observation and finite element simulation, as reported in (Kiener et al., 2006; Dolph et al., 2016; Saleh et al., 2016), and thus the more accurate “fictive hardness” can be calculated.

In contrast to the surface indentation, the cross-sectional nanoindentation (see Figures 2D–F) may “deconvolute” the indentation hardness to a specific depth (actually a depth range) to allow the correlation between the hardness and dose, as shown in Figure 2F. Thus, it could ignore the irradiation-induced surface contamination effects (although additional cross-sectional surface is made), and eliminate the DGE and SSE, as illustrated with Figure 2E. Over the past decades, the cross-sectional nanoindentation has gained increasing popularity for ion-irradiated materials (Hosemann et al., 2009; M. Rice and Stoller, 1997; Snead et al., 1992). However, the preparation of cross-sectional samples is tedious and difficult, especially to obtain a sharp and clean edge. Moreover, it needs to note that the lateral size of the plastic zone should not be ignored either, which could also cover hundreds of nanometers to blur the corresponding dose of certain indents. Therefore, the cross-sectional method usually contains quite few numbers of indents across the damage depth especially for the shallow irradiation.

The size-dependent behavior can be observed in indentation tests. At small scales, the apparent hardness changes with the indentation sizes, bringing challenges for the actual hardness evaluation, so-called ISE. Over the past decades, two types of effects have been reported (Bückle, 1959; Upit and Varchenya, 1973; Ma and Clarke, 1995; Sangwal, 2000; Pharr et al., 2010; Renjo et al., 2014). The normal ISE, i.e., the hardness increases at small indentation depths, is the more frequently seen effect. The reverse type of ISE, where the hardness increases with the increasing indentation sizes, has also been observed (Sangwal, 2000), but is usually ascribed to testing artifacts (Pharr et al., 2010). Here we focus on the frequently observed normal ISE, called ISE for simplicity in the following.

Several models have been built to understand the ISE, of which the most widely accepted one is the Nix-Gao model (Nix and Gao, 1998). The model separates the effects caused by geometrically necessary dislocations (GNDs) (Ashby, 1970) and statistically stored dislocations (SSDs), to describe the depth dependence of hardness for crystalline, as shown in Eq. 1. The linear relationship between the square of nanoindentation hardness (H ² ) and the reciprocal of indentation depth (1/h) has been widely used to fit the experimental data. H H 0 = 1 + h ∗ h (1) where H is the hardness of a given depth h; H ₀ is the hardness at infinite depth, i.e., macroscopic hardness, as described with the following Eq. 2; h* is a characteristic length that depends on the materials (the shear modulus   μ and H ₀ ) and the shape of the indenter, described with Eq. 3. H 0 = 3 3 α μ b ρ s (2) h ∗ = 81 2 b α 2 ⁡ tan 2 ⁡ θ ( μ H 0 ) 2 (3) where α is a constant; b is the length of the Burgers vector; ρ s is the SSD density; and θ is the angle between the surface of the indenter and the plane of the surface.

Nonetheless, such a simple linear relationship may not hold for irradiated materials. For example, the bi-linear behavior has been reported in the irradiated ferritic alloys, stainless steels, and F82H steels, and the underlying mechanism is believed to be related to the SSE (Kasada et al., 2011; Yabuuchi et al., 2014; Takayama et al., 2013). Moreover, the ISE could be weaker in the irradiated materials. For example, Jin et al. found that the ISE of ion irradiated Mo became weaker with increasing irradiation doses, as shown in Figure 3A (Jin et al., 2018). They believed that the increased density of SSDs induced by ion irradiation weakens the role of the depth-dependent GNDs, and so the ISE is less pronounced for irradiated materials. Similarly, the weaker ISE was discovered in the proton-irradiated single crystal copper after correcting the nanohardness data, based on the "simple law of mixture" (Hosemann et al., 2012). Other than the conventional room temperature tests, Prasitthipayong et al. also found that the ISE-dependent length h* decreased with the increasing temperatures for both unirradiated and irradiated 800H steels (Prasitthipayong et al., 2018b), which can be explained by the decreasing GND density at elevated temperatures, as shown in Figure 3B.

In addition to the irradiated materials, poor fitting also occurs when indentation depth is below 100 nm. To address this issue, Huang et al. modified the Nix-Gao model by introducing the concept of “maximum allowable GND density ( ρ G max ),” i.e., the upper limit of GND density, and found that the model agreed reasonably with the nanohardness data even below 100 nm of MgO and iridium (Huang et al., 2006). The feasibility of the model on ion-irradiated materials has recently been supported by Yang et al. (2020), when fitting the nanohardness data effectively in different ferritic steels. However, the modified model based on the ρ G max also faces some challenges such as the more complex parameters.

The yield strength of ion irradiated materials could be evaluated in nanoindentation tests using either the Berkovich indenter, through a so-called “three-step approach” (Lupinacci et al., 2015; Hosemann, 2018) to convert the nanohardness to the yield strength, or the spherical indenter to directly obtain the stress-strain curves (Herbert et al., 2001; Basu et al., 2006; Kalidindi and Pathak, 2008; Pathak and Kalidindi, 2015).

Compared with nanoindentation, the micro-compression technique is more direct to evaluate the irradiation strengthening (Yu et al., 2010; Lupinacci et al., 2014; Shin et al., 2014; Yano et al., 2017). The accuracy is affected by the geometric factors of pillars, such as the taper of pillars, fillet radius (the curvature at the bottom of pillars connecting to the base), and the aspect ratio of the pillars (the height/diameter ratio). Using two-dimensional (2D) and three-dimensional (3D) finite element modeling, Zhang et al. recommend minimal taper, fillet radius of 0.2–0.5, and pillars aspect ratios of 2-3 to provide sufficient testing accuracy (Zhang et al., 2006). These suggestions have been adopted to fabricate defined micro-pillars in unirradiated and ion-irradiated materials (Jin et al., 2016; Reichardt et al., 2017; Heo et al., 2018; Fan et al., 2019; Paccou et al., 2019).

The key challenge to correlate the small-scale yield strength values with those at the macro-scales is also the size effects. In the micro-size regime, the “smaller is stronger” phenomenon has been discovered for abundant metallic materials, as expressed with a generic form (Eq. 9) (Dehm, 2009; Greer and De Hosson, 2011). Some researchers have established dislocation-based models to explain the scaling behavior, such as the dislocation-starvation (Greer et al., 2005) and the single-ended dislocation source model (Parthasarathy et al., 2007). σ = A D m (9) where σ is the compressive flow strength; D refers to the pillar diameter; m is the size-effect exponent; and A is a constant.

When more grains are sampled with increasing sample sizes, the strength may increase toward the bulk strength due to the “grain boundary strengthening.” Combining the two conditions in the micro- and meso-scales leads to the black curve in Figure 4A (Hosemann et al., 2015; Hosemann, 2018), which indicates the size-effect behavior of the materials with low defect density. The size effect of irradiated materials (red curve) is significantly different from that of the unirradiated materials, i.e., the weaker size effects and the missing region “B” (Figure 4A), as well as the smaller specimen dimension to arrive at the size-independent strength (Figure 4B) (Kiener et al., 2011a). These differences may be related to the interaction between dislocations and irradiation induced defects, but the underlying mechanism and quantitative relationships of size effects remain unclear.

The most direct quantification method of strength and critical resolved shear stress (CRSS) for ion-irradiated materials could be the micro-tensile tests (Vo et al., 2017; Ajantiwalay et al., 2019). For example, using this method, Vo et al. (2017) found significant increase in the yield strength in 304SS after irradiation. The yield strength and CRSS values were in good agreements with the macroscopic tensile test results of neutron-irradiated stainless steels in the literature. Ajantiwalay et al. (2019) also discovered that the yield strength and CRSS of 304SS matches well with the macro-tensile test values in the literature. These findings demonstrated that the reliable CRSS values can be estimated locally using the micro-tensile tests.

Ajantiwalay et al. have investigated the applicability of the proposed size scaling behavior [Figure 4A (Hosemann et al., 2015)] for unirradiated 304SS based on micro-tensile tests, as illustrated in Figure 5 (Ajantiwalay et al., 2019). The yield strength values follow the overall trend of the proposed curve. Further studies are required for the irradiated materials to verify the feasibility of the proposed size scaling behavior.

Over the past decades, some works have combined or compared different small-scale mechanical property tests to study the irradiation strengthening (Hosemann et al., 2008; Vo et al., 2015; Weaver et al., 2017b; Prasitthipayong et al., 2018a).

Hosemann et al. found that the yield strength values of unirradiated and irradiated stainless steels using micro-compression tests were in relatively good agreements with nanoindentation data using the “three-step” method with the Busby’s pre-factor (Hosemann et al., 2008). Prasitthipayong et al. also found that the yield strength values of irradiated 800H and T91 converted from nanohardness are comparable with the micro-compression measurements (Prasitthipayong et al., 2018a). These results demonstrate the consistency of the yield strength values obtained from different small-scale tests. However, some studies reported the discrepancies on the irradiation strengthening using different evaluation methods (Moschetti et al., 2020; Sadeghilaridjani et al., 2020). For example, Sadeghilaridjani et al. found that the irradiation hardening effects evaluated by micro-compression (28%) were more significant than nanoindentation (13%) for HfTaTiVZr (Sadeghilaridjani et al., 2020).

Weaver et al. conducted a direct comparison between spherical nanoindentation, micro-compression, and micro-tensile tests, by converting the indentation strain-stress curves to uniaxial responses, as shown in Figure 6 (Weaver et al., 2017b). The main finding was that the work hardening behaviors of these techniques were basically alike for unirradiated materials (Figures 6A,B) while strong disagreements existed for the irradiated materials (Figures 6C,D). They believed that the little to no hardening behaviors of the irradiated pillars or tensile bars may be explained by the dislocation channeling mechanism.

Indentation technique is a simple, rapid, and inexpensive method to evaluate the fracture toughness. Based on the Lawn, Evans, and Marshall (LEM) model and its modified expressions proposed in the 1980s (Lawn et al., 1980; Anstis et al., 1981; Laugier, 1987), the fracture toughness (K _Ic ) can be measured with the crack lengths (c) for the Vickers indentation cracks, as shown in Eq. 10. The geometry-dependent constant α in Eq. 10 must be revised for nanoindentation tests because the indenters are standard Berkovich and cube-corner rather than Vickers, such as 0.036 for cube-corner indenters while 0.016 for Vickers indenters (Harding et al., 1994; Cuadrado et al., 2012). K I c = α ( E H ) 1 / 2 ( P c 3 / 2 ) (10) where E is the elastic modulus; H is the hardness; P is the peak indentation load; and α refers to an empirical constant which depends on the indenter geometry and the crack morphology.

Many researchers have used the nanoindentation to evaluate the fracture toughness of ion-irradiated SiC (Park et al., 2002; Yang et al., 2015; Leide et al., 2021), and obtained similar results that the radiation damage seems to improve the fracture toughness. Yang et al. explained such unusual phenomenon by the compressive stress and the deflection, pinning, and branching of cracks induced by irradiation defects (Yang et al., 2015). By means of high-resolution electron backscatter diffraction (HR-EBSD) and Raman spectroscopy, Leide et al. concluded that no crack in irradiated SiC is an artificial consequence of compressive residual stress, caused by constrained radiation swelling (Leide et al., 2021).

It is regrettable that the true fracture toughness values of ion-irradiated materials are difficult to assess using the classical indentation-based method and the method cannot be utilized on semi-brittle and ductile materials because of the high cracking thresholds.

The fracture toughness values could be calculated based on a simple correlation between the critical load at failure (P _c ), and the pillar radius (R), as shown in Eq. 11 (Sebastiani et al., 2015a). For the pillar splitting method, there is no need to image and measure the indentation crack lengths accurately; the residual stress can be eliminated by making the pillar diameter approximately equal to its length; the substrate effects are also minimized. K c = γ ⋅ P c R 3 / 2 (11) where the dimensionless coefficient γ is calibrated with the cohesive finite element simulation, not related to the LEM coefficient α in Eq. 10.

The micro-pillar splitting method has been successfully performed on various materials, such as thin ceramic films and coating (Sebastiani et al., 2015a; Sebastiani et al., 2015b; Bolelli et al., 2019), silicon (Jaya et al., 2015; Ghidelli et al., 2017; Lauener et al., 2018), battery (Mughal et al., 2016; Mughal et al., 2017; Wang et al., 2018a), and composite materials (Liu et al., 2017). In addition, this method could be applied at high temperatures on silicon and ceramic hard coatings (Lauener et al., 2018; Best et al., 2019). Nevertheless, it is impractical for most metals because their high toughness prevents splitting.

Researchers have reported that the fracture toughness values vary significantly with the different testing methods. For example, Jaya et al. reviewed that the fracture toughness of pure single crystal silicon scattered from 0.7 to 2.1 MPa m^1/2 due to the different sample dimensions, sample geometries, and preparation techniques (Jaya et al., 2015).

However, the consistency of fracture toughness evaluated using different methods has also been found. For example, the average fracture toughness values of silicon, deduced by four sample geometries, i.e., single-cantilever, double-cantilever, clamped beams, and pillars, are nearly constants (∼0.80 MPa m^1/2) (Jaya et al., 2015). Besides, the fracture toughness values of CrN coating show great agreements between the single-cantilever bending and pillar splitting methods, while the values obtained by double-cantilever and pillar splitting method show reasonable agreements (within 25%) (Sebastiani et al., 2015b).

Some researchers believed that the discrepancies on the fracture toughness values may be caused by the systematic errors in calibration procedures (Sebastiani et al., 2015b), and the multiple microstructural complexities in materials cannot be captured using some testing methods (Jaya et al., 2015).

The early indentation creep experiments can be dated back to at least the 1960s (Mulhearn and Tabor, 1960). With the development of the load and depth sensing indentation techniques, nanoindentation creep tests have been used to investigate the creep response of materials at small scales. The basic parameters of indentation creep experiments are stress σ and strain rate ε ˙ , as shown in Eq. 13. The projected contact area (A) could be calculated by A = r × h 2 for simplicity, in which r is 24.5 for the Berkovich indenter. The displacement rate ( h ˙ ) is measured by h ˙ = d h / d t . h = h 0 + a t b + c t is one of the empirical equations to fit the h-t curves (Choi et al., 2012). σ = F A   ε ˙ = h ˙ h (13) σ is taken as the applied load (F) divided by the projected contact area (A); while ε ˙ is taken as the displacement rate ( h ˙ ) divided by the instantaneous indent displacement (h).

The conventional uniaxial creep power-law equation is suitable for the stress σ and steady state strain rate ε ˙ data of the steady indentation creep, and the key parameter, creep stress exponent (n), can be yielded using Eq. 14. The creep stress exponent (n) is not only a valuable indicator for creep mechanism, but also corresponds closely to the maximum total elongation (k) during tension creep tests. Frost and Ashby, who integrated the numerical results of Burke and Nix (1975), proposed an empirical correlation between n and k (Eq. 15) (Lund and Nix, 1976), in which k ≈ 2-3. ε ˙ = C σ n = B σ n ⁡ exp ( − Q R T ) (14) where C is a fitting parameter associated with temperatures and materials; B is a material constant; Q refers to the activation energy; R is the universal gas constant; and T is the absolute temperatures. E l o n g a t i o n ( % ) = k n − 1 × 100 % (15)

Using the ex-situ nanoindentation creep tests, some works have studied the evolution of creep parameters (n , Q , k , etc.) for various ion-irradiated materials (Huang et al., 2014; Liu et al., 2020; Zhu et al., 2020). Huang et al. evaluated the changes of creep properties after high-temperature irradiation (up to 600°C) (Huang et al., 2014), while Zhu et al. compared the creep resistance of different materials at room-temperature irradiation (Zhu et al., 2020). Zhu et al. also found that the creep plasticity degrades with increasing ion dose for all samples and thought that the irradiation defects perform as the barriers to hinder the dislocation gilding, consequently degrading the creep plasticity. The nanoindentation creep tests show the capability to qualitatively compare the creep properties by comparing the obtained creep parameters (n , k) of different materials. However, the “true” creep parameters from conventional tensile creep tests can hardly be comparable quantitatively with those from the nanoindentation creep tests (Zhu et al., 2020).

In recent years, simple models for indentation creep have been developed to correlate the indentation creep parameters with those obtained in uniaxial creep tests (Ginder et al., 2018; Xiao and Yu, 2020b), but the applicability of the models on ion-irradiated materials has not be proven. Further efforts need to be made to obtain reliable creep parameters for ion-irradiated materials.

The stress states of micro-pillars uniaxial creep tests are similar to those of conventional creep tests, making it more reliable to assess the creep properties. Hence, the micro-pillars uniaxial creep tests have emerged on the study of small-scale creep behaviors at room temperature or high temperature (Wang et al., 2010; Choi et al., 2013; Mayer et al., 2015; Kim et al., 2016).

Based on the conventional power-law equation (Eq. 14), the important creep parameters n of micro-pillars uniaxial creep can be extracted by fitting the log ε ˙ − log ⁡ σ data. The steady state strain rate ε ˙ can be measured from the ε -t curves and Garofalo’s mathematical fitted equation, ε c r e e p = ε 0 + α ( 1 − e − r t ) + ω t , which was originally suggested for conventional tensile creep analysis, can be adapted for the micro-pillars creep curves (Choi et al., 2013; Kim et al., 2016).

An in-situ micro-pillars compression creep apparatus was developed by Özerinç et al., which consists of an accelerator and a custom designed spring loaded device, and allows them to perform in-situ IIC experiments during heavy ion irradiation (Özerinç et al., 2014). They later upgraded the apparatus for elevated temperatures (Özerinç et al., 2016), and verified the apparatus suitable for accurate creep properties evaluation due to the thermal and mechanical stability.

The real time observation of IIC can be achieved by an in-situ ion irradiation-TEM at Sandia National Laboratories (Hattar et al., 2014). The TEM observation ensures that the samples align with the loading apparatus, enables direct verification for the measured strain values (Dillon et al., 2017), and allows for the IIC compliance (B) measurements, as shown with Eq. 16 (Jawaharram et al., 2020). B = 1 σ 0 d ε d ϕ (16) where σ 0 is the applied stress; ϕ refers to the ion dose (dpa); and d ε / d ϕ is the normalized creep rate.

Many reports have confirmed that the in-situ experiments are capable of quantifying the IIC for various materials, with the pillar geometry or three-point beam geometry (Dillon et al., 2017; Jawaharram et al., 2018; Jawaharram et al., 2020). During the in-situ TEM observation on the ion-irradiated nanocrystalline high entropy alloys (HEAs), Jawaharram et al. found that the IIC compliance values of the HEAs are similar to the values of nanocrystalline Cu alloys at the same irradiation conditions in the literature. The results suggested that the point defect concentrations are comparable for the two alloys and indicated that the recombination is not significantly enhanced in HEAs (Jawaharram et al., 2020).

Jawaharram et al. have investigated the size effects on IIC of different materials, using the in-situ ion irradiation TEM observation (Jawaharram et al., 2018; Jawaharram et al., 2020). Different size effects were discovered between single crystalline Ag pillars and nanocrystalline HEA. For single-crystal Ag, the creep compliance increases parabolically with the increasing pillar sizes, as a result of the competition between dislocation loops and surfaces acting as sinks (Jawaharram et al., 2018). But the IIC compliance scaled inversely with grain sizes of nanocrystalline HEA in the sink-limited regime because the creep behaviors are controlled by the grain boundary mechanism (Jawaharram et al., 2020).

The different hypothesized regimes of IIC responses as a function of grain sizes are illustrated in Figure 9 (Jawaharram et al., 2020). Specifically, in the nanocrystalline regime, the grain boundary mechanism controls the creep behaviors and the IIC compliance decreases with increasing grain sizes. When the grain size is above the critical sizes, in which the loop nucleation occurs, the dislocation-based mechanism begins to dominate the creep process and the IIC compliance increases with increasing grain sizes.