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. 2025 Nov 5;15:38828. doi: 10.1038/s41598-025-23495-w

An improved NG-F model for capturing the three types of indentation size effect

Peina Wang 1, Yu Gao 2, Peihuan Wang 3,✉
PMCID: PMC12589521  PMID: 41193791

Abstract

Experimental studies indicate that three types of ISE are observed. However, developing a model to capture three types of ISE is one challenging task in materials science. In this paper, the relationship between k and h (k is the ratio of the plastic zone radius to the contact radius, and h is the indentation depth) is systematically analyzed. Then, an exponential decay function is modified to capture the relationship between k and h. The modified exponential decay function can accurately describe the nonlinear relationship between k and h. However, the modified exponential decay function still fails to capture the complex relationship between k and h. On this basis, the NG-F Model is improved with the modified exponential decay function. An analysis has been done to comprehensive comprehension the influence of the parameters on the improved NG-F model. The parameter analysis has been qualitatively proved that the improved NG-F model can capture three types of ISE. The competition between the relative rate of change of k3 and h is the mechanisms behind the improved NG-F model for capturing the three types of ISE. The improved NG-F model is validated with experimental results of three types of ISE. Surprisingly, the validation demonstrates that the improved NG-F model can successfully predict three types of ISE for different materials. However, the improved NG-F model’s predictive accuracy varies across materials, with relatively lower DC values observed for certain NR-ISE and C-ISE materials. This is because the dislocation mechanism underlying the improved NG-F model differs from the mechanisms responsible for these materials.

Keywords: Indentation size effect, Normal ISE, Transition of normal to reverse ISE, Complex ISE

Subject terms: Materials science, Nanoscale materials

Introduction

Hardness, a vital mechanical property of materials, is crucial for the design and application of micro–nano components, thin film coatings, and emerging materials in micro-nano scales1–8. Up to now, nanoindentation tests have evolved into a robust technique to determine material hardness at the micro-nano scale1,9–14. Nanoindentation tests have also shown that the measured hardness of materials is size-dependent, a phenomenon referred to as the indentation size effect (ISE)2,3,15,16. Meyer initially reported in 1908 12 that hardness exhibited a decline with increasing indentation depth, which is known as the normal ISE (N-ISE), as illustrated in Fig. 1. Since then, the N-ISE has also been observed in nanoindentation tests of different materials1,17–20. However, some nanoindentation experiments21–27 have shown that a hardening-softening transition occurred at shallow indentation depths, that is, the hardness-depth curve gradually transited from normal to reverse ISE (NR-ISE), as depicted in Fig. 1. Moreover, some nanoindentation tests have shown a complex ISE (C-ISE) in polycrystalline materials and nanostructured multilayers/substrate systems (see in Fig. 1)28,29, where observations indicate that hardness first decreases with increasing indentation depth, followed by an increase, a subsequent decrease, and finally convergence to a constant value.

Fig. 1.

Fig. 1

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Depiction of three types of ISE1.

Over the past few decades, numerous theories have emerged to elucidate the indentation size effects (ISEs) mentioned above30–32. Meyer proposed an empirical power-law to reflect the relationship between the applied force and the indentation diagonal12. Hays and Kendall33 identified that the ISE occurs when a critical load threshold is exceeded, initiating permanent deformation. Li and Bradt34 developed a proportional specimen resistance (PSR) model in which the indentation size is considered to exert a linearly increasing influence on the test specimen’s resistance to irreversible deformation. Gong et al.35 improved the PSR model by considering the effects of residual stresses. Bull et al.36 proposed an elastic-plastic model in which the elastic stress at the indentation periphery is a consequence of localized permanent deformation. Ruiz-Moreno et al.37 adopted an exponential function to characterize the nonlinear relation between the applied force and indentation depth. Hou and Jennet38 formulated a model that incorporates the effect of grain size and the pinning effect of dislocation defects on hardness. Nevertheless, the aforementioned ISE models are limited to describing the N-ISE.

One of the most widely cited models is the NG model introduced by Nix and Gao39. Rooted in the Taylor dislocation framework, it describes how the density of geometrically necessary dislocations (GNDs) varies with indentation depth. Subsequent nanoindentation studies, however, indicated that the original model tends to overpredict the hardness for very shallow indentations; this discrepancy has been attributed to the strong repulsive forces among GNDs40,41. Consequently, several researchers40,42,43 have improved the NG model by allowing the ratio between the plastic-zone size and the indentation depth to vary. Huang et al.44 further modified the model by imposing a maximum allowable GNDs density. Meanwhile, to more accurately describe the correlation between the size of the plastic zone and the indentation depth, Feng and Nix42 incorporated an exponential function into the NG model (NG-F model), while Haušild45 utilized a power function for the same purpose (NG-H model). Abu Al-Rub46 improved the NG model by introducing the Taylor hardening law, and Qiu et al.47 enhanced the NG model by incorporating friction stress unrelated to dislocation motion. Yuan and Chen48 refined the NG model by combining it with gradient plasticity and an intrinsic material length. In an effort to broaden the predictive scope of the NG model specifically for ISE at depths below 100 nm, Liu and his co-authors49 took into account the elastic deformation. However, a comparative study of NG model and improved NG models indicted that only NG-F and NG-H models can capture the NR-ISE of different materials, but these improved NG models cannot capture the C-ISE30,50.

The complex ISE (C-ISE) (see Fig. 1) is commonly observed in nanoindentation experiments of polycrystalline materials and nanostructured multilayers/substrate systems28,29. Jung and colleagues51 developed a modified MSG theory to predict the C-ISE in polycrystalline materials. The model incorporates the GB effect by assessing the GNDs density along the GBs. Nevertheless, the underlying mechanism of this model remains controversial, because it assumes that hardening and softening are mainly governed by the accumulation and transfer of dislocations across GBs. Voyiadjis et al.28,29 proposed a theoretical model that encompasses the impacts of temperature and strain rate on ISE by accounting for the GB effect; this model successfully describes the C-ISE in Cu bicrystals and Inconel 718. Xiao et al.52 introduced a theoretical model that considers GB resistance to plastic deformation to explain the C-ISE in bicrystals. Liu et al.53 developed a mechanistic model to provide a comprehensive understanding of the C-ISE in irradiated nanostructured multilayer/substrate systems. Their analysis examined several factors in depth, such as GNDs, irradiation hardening, and interface hardening. However, the ISE models proposed by Xiao et al.52 and Liu et al.53 are expressed as piecewise functions consisting of four segments; consequently, both their practical application and parameter determination are highly complex.

In this work, the NG-F model is firstly reviewed. The relationship between k and h (k is the ratio of the plastic zone radius to the contact radius, and h is the indentation depth) is systematically analyzed. Subsequently, the exponential decay function proposed by Feng and Nix42 is modified to characterize the correlation between k and h. On this basis, the NG-F model is further improved with the modified exponential decay function. To achieve a more comprehensive understanding, the effects of various parameters on the improved NG-F model have been thoroughly analyzed. The performance of the improved NG-F model is validated with experimental results of three types of ISE. In addition, the mechanisms behind the improved NG-F model for capturing the three types of ISE are clarified.

An improved NG-F model for capturing the three types of ISE

Establishment of the improved NG-F model

Based on a schematic representation of geometrically necessary dislocations (GNDs) underneath a conical indenter in Fig. 2, the following NG model is established by Nix and Gao to capture the N-ISE of crystalline materials39.

graphic file with name d33e483.gif 1

Fig. 2.

Fig. 2

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Schematic representation of geometrically necessary dislocations (GNDs) underneath a conical indenter used by the NG model.

where H and h are hardness and indentation depth, respectively. Inline graphic is the hardness for large value of h and Inline graphic. The Inline graphic is an intrinsic length and Inline graphic. The θ is the angle between the surface of the specimen plane and conical indenter, tan θ = b/s. The b is the magnitude of the Burgers vector or the height of each slip step, and s is the spacing between the GND loops, as shown in Fig. 2. The µ is the shear modulus, and α is the Taylor factor and within the range of 0.3 to 0.6 for face-centered cubic materials54. The ρs is the statistically stored dislocations (SSDs) density.

As reported by Feng and Nix42, the radius of the plastic zone beneath the indenter can be expressed as rp = ka. Here, the a represents the contact radius between the indenter and the indentation surface and the k denotes the ratio of the plastic zone radius rp = ka to the contact radius a, as shown in Fig. 2. Subsequently, the NG model is revised as follows.

graphic file with name d33e595.gif 2

Feng and Nix42 proposed an exponential decay function to describe the correlation between k and h, as presented below.

graphic file with name d33e613.gif 3

where r0 > 0 and h1 > 0 are material constants, which can be determined by fitting method42.

Experimental and simulated results qualitatively indicate that the k nonlinearly decreases as increasing h55–57. Moreover, Xiao et al.56 conducted a simulated indentation of stainless steel. It is found when h > 400 nm that the ratio k is a constant that is not equal to 1 for both un-irradiated and irradiated stainless steel. Therefore, an exponent n ≥ 0 is introduced into Eq. (3) to modulate the nonlinear relationship between k and h, and the parameter k∞ > 0 is introduced into Eq. (3) to reflect that the k is a constant that is not equal to 1 for h → +∞.

graphic file with name d33e684.gif 4

Figure 3 shows the k/k∞ versus h curves with different r0, h1 or n for Eq. (4). One can notice in Fig. 3 that as h increases, k/k∞ first approximates constant (Upper bound), then decreases, and finally tends to be constant (Lower bound). It can be found in Fig. 3(a) that as h1 decreases, the critical region of indentation depth where the k/k∞ versus h curves transition from the upper bound to the lower bound gradually increase. In Fig. 3(b), the r0 can only adjust the upper bound of the k/k∞ versus h curves. In Fig. 3(c), as n increases, the transition of the k/k∞ versus h curves from the upper bound to the lower bound become steeper and more localized.

Fig. 3.

Fig. 3

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Impact of the parameters (a) r0, (b) h1 and (c) n on Eq. (4).

To validate the predictability of the modified exponential decay function [Eq. (4)], the data of k versus h are assembled from un-irradiated FeCr alloy55, irradiated FeCr alloy55, irradiated 16MND5 steel56 and simulated material57. The parameters of Eqs. (3) and (4) are determined using the fitting method. These optimized parameters are given in Tables 1 and 2.

Table 1.

The optimized parameters of Eq. (3).

No. Material r 0 h1/nm
1 Un-irradiated FeCr alloy 224.60 25.34
2 Irradiated FeCr alloy 358.50 23.90
3 Irradiated 16MND5 steel 17,510 16.51
4 Simulated material 737.50 0.41
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Table 2.

The optimized parameters of Eq. (4).

No. Material r 0 h1/nm k∞ n
1 Un-irradiated FeCr alloy 10.86 99.87 0.9658 7.7170
2 Irradiated FeCr alloy 13.83 125.80 0.5893 4.1690
3 Irradiated 16MND5 steel 15.18 0.000298 7.8910 0.1015
4 Simulated material 1.22 164.40 0.2249 0.6846
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It can be observed from Fig. 4 that the exponential decay function [Eq. (3)] cannot capture the nonlinear relationship between k and h. Figure 4(a), (c) and 4(d) also show that the k-h curves predicted by the modified exponential decay function [Eq. (4)] agree well with the experimental data. Except for the irradiated FeCr alloy, the agreement between the experimental data and Eq. (4) is better than that with Eq. (3), as illustrated in Fig. 4. It must be acknowledged that fitting the five experimental data points of both the unirradiated and irradiated FeCr alloys with the four-parameter Eq. (4) carries a risk of overfitting, but this risk has been avoided for the following reasons.

Fig. 4.

Fig. 4

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Comparison between the experimental results comparison k ISE of (a) Un-irradiated FeCr alloy55, (b) Irradiated FeCr alloy55, (c) Irradiated 16MND5 steel56 and (d) Simulated material57 with predictions of Eq. (3) and Eq. (4).

  1. All optimized parameters presented in Tables 1 and 2 fall within the pre-specified range (r0 > 0, h1 > 0, n ≥ 0 and k∞ > 0). As demonstrated by the parameter analysis in Fig. 3, the variation trend of Eq. (4) exhibits a monotonically decreasing function, while the overfitting function usually shows fluctuations.

  2. Equation (4) outperforms Eq. (3) for three materials depicted in Fig. 4(a), (c) and (d), with supporting evidence from Table 3: the mean DC value is 0.9643 for Eq. (4) versus 0.8243 for Eq. (3). Notably, if overfitting were present, such improvement would tend to be material-specific rather than universal.

  3. Equation (4) is not arbitrarily constructed; instead, it extends the NG-F model through the introduction of two parameters: k∞ (to account for the fact that k converges to a constant as h approaches +∞) and n (to capture nonlinear decay behavior). This functional form is motivated by experimental and simulated observations in Fig. 4.

Table 3.

The DCs of the Eqs. (3) and (4).

No. Material Equation (3) Equation (4)
1 Un-irradiated FeCr alloy 0.6687 0.9794
2 Irradiated FeCr alloy 0.9731 0.9674
3 Irradiated 16MND5 steel 0.7457 0.9354
4 Simulated material 0.9098 0.9750
Mean DC 0.8243 0.9643
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It can be noticed that the experimental data of k-h exhibit a clear transition from descent to ascent at h ≈ 150 nm in Fig. 4(a), h ≈ 150 nm in Fig. 4(b), h ≈ 600 nm in Fig. 4(c) and h ≈ 150 nm in Fig. 4(d). Xiao et al.52 developed a dislocation-density-based model that relates k to indentation depth h in bicrystals while incorporating grain-boundary resistance. The predicted k-h curve for Fe-Si bicrystal (see in Fig. 5) also shows a descent-to-ascent transition. The dislocation-density-based model attributes this transition to the progressive accumulation of GNDs at grain boundaries. Once the local stress attains the critical threshold required for grain-boundary dislocation pile-up, the accumulated stress triggers dislocations to traverse the boundary, causing the plastic zone to expand rapidly into the underlying grain and, consequently, driving the k-h curve from descent to ascent. By comparing the parameter analysis of Eq. (4) (see in Fig. 3) with the Fig. 5, it becomes evident that the modified exponential decay function [Eq. (4)] cannot capture the complex relationship between k and h given in Fig. 552.

Fig. 5.

Fig. 5

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The complex relationship between k and h derived by Xiao et al.52 for Fe–Si bicrystal with GB effect.

The determination coefficient (DC) is utilized to quantitatively assess the accuracy of the Eqs. (3) and (4). When the experimental data and predicted data align perfectly, the highest DC is achieved. The calculation method of DC is referred to reference30. The DCs of Eqs. (3) and (4) are depicted in Table 3. The mean DC of Eqs. (4), 0.9643, is greater than that of Eqs. (3), 0.8243.

Although the modified exponential decay function [Eq. (4)] cannot capture the complex relationship between k and h, it can better describe the nonlinear characteristics between k and h than Eq. (3). Therefore, by substituting Eq. (4) into Eq. (2), the improved Nix-Gao-Feng model (Improved NG-F model) are got.

graphic file with name d33e1455.gif 5

where h2 = (k∞)−3h0.

Parameter analysis

In Eq. (5), the H0 only adjusts the value of the improved NG-F model without impacting it shape. Hence, Eq. (5) is normalized by H0 as follows.

graphic file with name d33e1495.gif 6

Figure 6 shows the impact of the r0, h1, h2 and n on the improved NG-F model [Eq. (6)]. In Fig. 6(a), the ISE curves of the improved NG-F model are seen to transition from the N-ISE to the C-ISE, and then to the NR-ISE as r0 increases. In Fig. 6(b) and (c), the improved NG-F model demonstrates that the NR-ISE becomes increasingly evident as h1 is reduced or h2 is elevated. In Fig. 6(d), as n is increased, the ISE curves of the improved NG-F model shift from the N-ISE to the NR-ISE, and then to the C-ISE. Meanwhile, the critical h for the NR-ISE exhibits a gradual decline. The C-ISE, which has been found in the nanoindentation of bicrystal FCC metals28, can be captured by the improved NG-F model, as depicted in Fig. 6. The parameter analysis has been qualitatively proved that the improved NG-F model can describe three types of ISE. Subsequently, the underlying mechanism of this finding is elucidated.

Fig. 6.

Fig. 6

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Impact of the (a) r0, (b) h2, (c) h1 and (d) n on the improved NG-F model [Eq. (6)].

Mechanism behind the improved NG-F model

By setting k1 = k / k∞ in Eq. (4), the improved NG-F model [Eq. (6)] can be simplified as follows.

graphic file with name d33e1625.gif 7

The derivative of Eq. (7) with respect to h is as follows.

graphic file with name d33e1639.gif 8

where (k13)′ and h′ are the derivative of k13 and h with respect to h, respectively. The (k13)′/ k13 and h′/h are the relative rate of change of k13 and h with respect to h, respectively. The (k13)′/ k13 is derived based on k1 = k / k∞ and Eq. (4) as follows.

graphic file with name d33e1721.gif 9

Since n, r0 and h1 > 0, (k13)′/ k13 < 0 and h′/h > 0 are determined by Eq. (8). Therefore, in Eq. (8), the condition [k13)′/ k13 + h′/h] > 0 leads to the N-ISE [(H/H0)2]′ > 0. Conversely, the reverse ISE [(H/H0)2]′ < 0 is obtained under the condition [k13)′/ k13 + h′/h] < 0. This indicates that the competition between (k13)′/ k13 and h′/h serves as the mechanism underlying the improved NG-F model for describing the three types of ISE.

Validation of the improved NG-F model

Experimental data collection

To examine the correctness and rationality of the improved NG-F model, three types of ISE data are collected from the literature, including nine indentation experiment of MgO single crystals, 16MND5 steel, High entropy alloy, ZrO2 ceramic, Cu single crystals, Y2O3-ZrO2 ceramic, Cu bicrystal, Inconel 718 and Fe-Si bicrystal. These data are documented in Table 4. To avoid overfitting and accurately estimate the model parameters, it is imperative to have an adequate amount of data. Consequently, the fitting method is utilized to calculate the parameters of the NG-F and improved NG-F models. The optimized parameters are presented in Tables 5 and 6.

Table 4.

Experimental data of ISE for various materials.

Type of ISE Number Material Data bulk Sources
N-ISE 1 MgO single crystals 41 Feng and Nix42
2 16MND5 steel 19 Renjo et al.49
3 High entropy alloy 29 Haušild45
NR-ISE 4 ZrO2 ceramic 59 Wang et al.21
5 Cu single crystals 64 Yang et al.22
6 Y2O3-ZrO2 ceramic 15 Luo et al.23
C-ISE 7 Cu bicrystal 147 Zhang et al.28
8 Inconel 718 120 Voyiadjis et al.29
9 Fe-Si bicrystal 57 Xiao et al.52
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Table 5.

The optimized parameters of the NG-F model for various materials.

No. Material H0/GPa h2/µm h1/µm r 0
1 MgO single crystals 9.2210 0.0981 0.0722 0.7202
2 16MND5 steel 1.8180 1.3320 0.5810 0.5214
3 High entropy alloy 0.4443 1.8540 0.2286 0.6206
4 ZrO2 ceramic 18.2200 0.3024 0.0330 7.5320
5 Cu single crystals 0.9818 0.8284 0.0072 41.7900
6 Y2O3-ZrO2 ceramic 10.0400 0.8521 0.1302 4.2910
7 Cu bicrystal 0.09854 31,640 50.3600 1.3930
8 Inconel 718 3.165 335.8 64.7500 2.1200
9 Fe-Si bicrystal 0.1134 245,800 122.6000 1.2410
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Table 6.

The optimized parameters of improved NG-F model for various materials.

No. Material H0/GPa h2/µm h1/µm r 0 n
1 MgO single crystals 9.2490 0.0940 0.0994 0.4858 1.3610
2 16MND5 steel 2.2860 0.2716 0.2346 0.0313 4.0310
3 High entropy alloy 0.4300 2.3400 2.28E-07 218.0000 0.1375
4 ZrO2 ceramic 17.4300 0.3763 0.0409 6.9900 1.1440
5 Cu single crystals 1.1780 0.5061 0.0132 14.2600 1.4010
6 Y2O3-ZrO2 ceramic 1.2130 248.8000 0.0020 30.2200 0.2550
7 Cu bicrystal 1.0330 75.1100 63.5700 0.5432 3.4550
8 Inconel 718 3.9890 85.2900 87.8400 1.5520 2.1740
9 Fe-Si bicrystal 0.1055 127300.0000 95.2900 0.5089 3.3540
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Model validation

By substituting the optimized parameters from Tables 5 and 6 into the NG-F and improved NG-F models, the predicted ISE curves can be obtained. The experimental results are compared with the predicted ISE curves derived from the NG-F and improved NG-F models, and these comparisons are represented in Figs. 7, 8 and 9. Figure 7 illustrates that both NG-F and improved NG-F models can precisely depict the experimental data of N-ISE. However, in Fig. 8, the NG-F and improved NG-F models can only describe the changing trends of the three experimental data of the NR-ISE, but cannot accurately depict their changing law. Furthermore, the NG-F and improved NG-F models also fail to describe the transformation of hardness from the hardening state to the softening state at relatively small indentation depths, as presented in Fig. 8. For the C-ISE in Fig. 9, although NG-F model can predict the experimental data of Inconel 718, it fails to describe the experimental data of Cu bicrystals and Fe-Si bicrystals. This is because the NG-F model cannot reflect the nonlinear characteristics between k and h [see in Fig. 4]. To our surprise, the improved NG-F model can accurately predict the experimental data of Cu bicrystals and Inconel 718 for the C-ISE, as illustrated in Fig. 9(a) and (b).

Fig. 7.

Fig. 7

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Comparison between the experimental results of (a) MgO single crystals, (b) 16MND5 steel and (c) high entropy alloy and the predicted curves of the NG-F and improved NG-F models for N-ISE.

Fig. 8.

Fig. 8

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Comparison between the experimental results of (a) ZrO2 ceramic, (b) Cu single crystals and (c) Y2O3-ZrO2 ceramic and the predicted curves of the NG-F and improved NG-F models for NR-ISE.

Fig. 9.

Fig. 9

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Comparison between the experimental results of (a) Cu bicrystal, (b) Inconel 718 and (c) Fe-Si bicrystal and the predicted curves of the NG-F and improved NG-F models for C-ISE.

The DCs of the NG-F and improved NG-F models are determined to quantitatively evaluate their accuracy for various materials, which are provided in Table 7. For materials exhibiting N-ISE, the improved NG-F model shows only marginal improvements over the original model. The DC values for MgO single crystals and 16MND5 steel are very close to 1, indicating high predictive accuracy for both models. The high entropy alloy also shows excellent predictive performance, with the improved model achieving a DC value of 0.9997 compared to 0.9995 for the original model. These results suggest that both models are highly effective for materials with N-ISE, but the improved NG-F model offers slight enhancements in predictive accuracy.

Table 7.

The DCs of the NG-F and improved NG-F models for different materials.

Type of ISE Number Material NG-F model Improved NG-F model
N-ISE 1 MgO single crystals 0.9968 0.9988
2 16MND5 steel 0.9975 0.9988
3 High entropy alloy 0.9995 0.9997
NR-ISE 4 ZrO2 ceramic 0.8017 0.8202
5 Cu single crystals 0.9589 0.9885
6 Y2O3-ZrO2 ceramic 0.7378 0.7890
C-ISE 7 Cu bicrystal 0.8739 0.9411
8 Inconel 718 0.8893 0.9352
9 Fe-Si bicrystal 0.7462 0.8916
Mean DC 0.8891 0.9292
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For materials exhibiting NR-ISE, the improved NG-F model shows only a slight enhancement in predictive accuracy, as reflected by its DC values. Specifically, for ZrO₂ ceramic and Y₂O₃-ZrO₂ ceramic, the DC values exhibit slight increases: from 0.8017 to 0.8202 for ZrO₂ ceramic, and from 0.7378 to 0.7890 for Y₂O₃-ZrO₂ ceramic, respectively. Similarly, a minor improvement is observed for Cu single crystals, with the DC value rising from 0.9589 to 0.9885.This slight degree of enhancement indicates that the modifications made to the NG-F model fail to effectively capture the mechanisms associated with NR-ISE. This is because the dislocation mechanism underlying the NG-F and improved NG-F model differs from the mechanisms responsible for NR-ISE. For instance, experimental analysis of ZrO₂ ceramic demonstrates that reverse ISE arises from the relaxation of surface stresses induced by indentation cracks under low loads21. For Cu single crystals, molecular dynamics simulations and experimental data reveal that the reverse ISE in nano-hardness is attributed to both the finite roundness of the indenter tip and the intrinsic property-transition behavior of the material22. Additionally, Luo et al. [23] reported that Y₂O₃-ZrO₂ ceramic undergoes a partial tetragonal-to-monoclinic transformation under indentation load, and this transformation is the cause of its reverse ISE23.

For materials with C-ISE, the improved NG-F model shows substantial improvements in predictive accuracy. The DC values for Cu bicrystal and Inconel 718 increase significantly from 0.8739 to 0.9411 and from 0.8893 to 0.9352, respectively. The Fe-Si bicrystal also exhibits a marked improvement, with the DC value rising from 0.7462 to 0.8916. These results highlight the effectiveness of the improved NG-F model in handling the intricate mechanical behaviors associated with C-ISE, where the NG-F model has struggled to provide accurate predictions.

Although the improved NG-F model significantly improves the DC of Fe-Si bicrystals from 0.7462 to 0.8916, this value remains lower than those of other C-ISE materials like Cu bicrystals (DC = 0.9411) and Inconel 718 (DC = 0.9352). A potential explanation for this limitation is that the improved NG-F model primarily optimizes the functional relationship between k and h, but does not fully incorporate the grain boundary resistance in Fe-Si bicrystals. This resistance induces a clear descent-to-ascent transition in the k-h curve (See in Fig. 5), which the modified exponential decay function employed in the improved NG-F model has yet to capture.

Furthermore, Table 7 shows that for the NG-F model, the DC values of Cu bicrystal are slightly lower than those of Inconel 718, whereas the opposite holds for the improved NG-F model. This reversed fitting performance ranking stems from the two models’ differential sensitivity to the curve nonlinearity of the two materials. The NG-F model’s accuracy relies on the extent to which Eq. (3)’s exponential decay of k matches actual behavior: Inconel 718 has a gentle nonlinear curve (k decreases gradually with h and plateaus at large h) that aligns with Eq. (3), yielding a higher DC (0.8893) than Cu bicrystal (0.8739); in contrast, Cu bicrystal’s pronounced nonlinearity (sharp k drop at shallow h, then plateau) deviates from Eq. (3), worsening fitting. The improved NG-F model, designed to capture steep nonlinearities via Eq. (4) (introducing parameters n and k∞), enhances DC more significantly for Cu bicrystal (n = 3.455, Table 6; DC: 0.8739→0.9411) than for Inconel 718 (n = 2.174, Table 6; DC: 0.8893→0.9352). This leads to reversed DC rankings in the improved NG-F model.

The mean DC values for the NG-F model and the improved NG-F model are 0.8891 and 0.9292, respectively. This indicates an overall improvement in predictive reliability when using the improved NG-F model to describe three types of ISE.

Conclusions

In this work, the relationship between k and h (k and h are the ratio of the plastic zone radius to the contact radius and indentation depth, respectively) is systematically analyzed. Then, an exponential decay function assumed by Feng and Nix42 is modified to capture the relationship between k and h. On this basis, the NG-F model is improved with the modified exponential decay function. To achieve a more comprehensive understanding, the effects of various parameters on the improved NG-F model have been thoroughly analyzed. The performance of the improved NG-F model is validated with experimental results of three types of ISE. In addition, the mechanisms behind the improved NG-F model for describing the three types of ISE are clarified. The key findings of this research are outlined below.

  1. The nonlinear relationship between k and indentation depth cannot be captured by the exponential decay function, but is accurately described by the modified exponential decay function. However, the modified exponential decay function still fails to capture the complex relationship between k and indentation depth derived by Xiao et al.52.

  2. The parameter analysis has been qualitatively proved that the improved NG-F model can capture the three types of ISE. The competition between the relative rate of change of k3 and h with h is the mechanisms behind the improved NG-F model for capturing the three types of ISE.

  3. Surprisingly, the validation demonstrates that the improved NG-F model can successfully predict three types of ISE for different materials, while the NG-F model is unable to predict the experimental results of C-ISE. However, the improved NG-F model’s predictive accuracy varies across materials, with relatively lower DC values observed for certain NR-ISE and C-ISE materials. This is because the dislocation mechanism underlying the improved NG-F model differs from the mechanisms responsible for these materials.

Author contributions

Peina Wang: Writing-original draft, Methodology, Investigation. Yu Gao: Data curation, Formal analysis. Peihuan Wang: Writing-review-editing, Supervision, Investigation, Conceptualization.

Data availability

Data are provided in the paper.

Declarations

Competing interests

The authors declare no competing interests.

Footnotes

Publisher’s note

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References

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