A. Simulation results

After simulation, the relationship between the calculated hardness, the ratio of σ_y/E, and indentation depth h for different material properties is depicted in Fig. 3(a). Obviously, it can be seen that as the indentation depth decreased and the strain hardening exponent increased, the ratios of H/σ_y increase accordingly. In terms of Fig. 3(b), the ratios of h _c/h _m increase with the indentation depth and decreasing strain hardening exponent, however, it is not distinct with the increasing σ_y/E. Besides, P _m/σ_y increases with the increasing indentation depth and strain hardening exponent, see Fig. 3(c). According to Eq. (1), the fitting loading curvature C is also plotted as shown in Fig. 3(d). With the increasing indentation depth h, C is decreasing a little, and it increases with the increasing strain hardening exponent. Still, no clear trend with σ_y/E can be found.

FIG. 3.

The relation between (a) H/σ_y; (b) h_c/h_m; (c) P_m/σ_y; (d) loading curvature C; versus σ_y/E at different indentation depth.

In a certain indentation depth, the H/σ_y is a function of σ_y/E and strain hardening exponent. To find a clear relation between hardness and material properties, a model is adopted as

(8)$${H \over {{\sigma _{\rm{y}}}}} = \kappa {\left( {1 + {\varepsilon _{\rm{r}}}{E \over {{\sigma _{\rm{y}}}}}} \right)^n}\quad,$$

where κ is the coefficient and ε_r is the representative strain. Although the representative strain defined by Dao et al. only works for a limited property range,^{Reference Cao and Huber25} the representative strain in this paper was chosen as 0.033 as given in Ref. Reference Dao, Chollacoop, Van Vliet, Venkatesh and Suresh2, due to the material properties selected as shown in Table I. By using Eq. (8), the fitting results of κ are shown in Table II. It can be seen that as the indentation depth increases, the coefficient decreases. From this result, it is hard for us to get a uniform model to characterize the material properties. So, the indentation depth (size effect) should also be considered simultaneously at this level.

TABLE II.

The fitting results by using Eq. (8).

B. The relation between microhardness and material properties

According to Cheng and Cheng,^{Reference Cheng and Cheng19} the relationship between microhardness, material properties (E, n, σ_y, ν), and simulation parameters (indenter angle θ) at certain indentation depth h, where the friction effect was not considered, can be written as

(9)$${H \over {{\sigma _{\rm{y}}}}} = {\Pi _h}\left( {{{{\sigma _{\rm{y}}}} \over E},\nu ,n,\theta } \right)\quad.$$

If the indentation depth size effect is considered, Eq. (9) can be rewritten as

(10)$$H = H({\kappa _0},E,n,{\sigma _{\rm{y}}},\nu ,h,{h^ * })\quad.$$

In general, the material hardness is related to the material flow stress σ_flow as^{Reference Gouldstone, Chollacoop, Dao, Li, Minor and Shen1,Reference Dao, Chollacoop, Van Vliet, Venkatesh and Suresh2,Reference Cheng and Cheng5,Reference Cheng and Cheng19}

(11)$${H_0} = {\kappa _0}{\sigma _{{\rm{flow}}}}\quad,$$

where H ₀ is the hardness of the material at macroscale, corresponding to the measured hardness at the greatest depth tested,^{Reference Swadener, George and Pharr26} and σ_flow is the uniaxial tensile flow stress^{Reference Huang, Zhang, Hwang, Nix, Pharr and Feng8} or representative stress. κ₀ is a constraint factor depending on indenter angle and material properties, which increases with E/σ_y (or 1/ε_y) and approaches a constant (∼3) when E/tanθ(1−ν²)σ_y > 20.^{Reference Chen and Vlassak27,Reference Johnson28} The constraint factor κ₀ is 2.75 referring to the study by Cheng and Cheng.^{Reference Cheng and Cheng18}

According to the study by Huang et al.,^{Reference Huang, Zhang, Hwang, Nix, Pharr and Feng8} the flow stress is determined from the uniaxial tension stress–strain curve, which is related to the plastic strain ε^p as

(12)$${\sigma _{{\rm{flow}}}} = {\sigma _{{\rm{ref}}}}f({\varepsilon ^{\rm{p}}})\quad,$$

where σ_ref is a reference stress and f(ε^p) is a nondimensional function, which can be written in a different form according to the material properties adopted.^{Reference Cheng and Cheng5,Reference Huang, Zhang, Hwang, Nix, Pharr and Feng8,Reference Wei and Hutchinson17}

Combining Eqs. (11) and (12), the relationship between hardness and flow stress can be rewritten as

(13)$${H_0} = {\kappa _0}{\sigma _{{\rm{ref}}}}f({\varepsilon ^{\rm{p}}})\quad.$$

Here, the reference stress is taken as σ_ref = σ_y, and the function of plastic strain from Huang et al.^{Reference Huang, Zhang, Hwang, Nix, Pharr and Feng8} is

(14)$$f({\varepsilon ^{\rm{p}}}) = {\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon ^{\rm{p}}}} \right)^n} = {\left( {1 + {{{\varepsilon ^{\rm{p}}}} \over {{\varepsilon ^{\rm{y}}}}}} \right)^n}\quad.$$

Using Eq. (14) and σ_ref = σ_y in Eq. (13), it can be rewritten as

(15)$${H_0} = {\kappa _0}{\sigma _{{\rm{flow}}}} = {\kappa _0}{\sigma _{{\rm{ref}}}}f({\varepsilon ^{\rm{p}}}) = {\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon ^{\rm{p}}}} \right)^n}\quad.$$

If ε^p is chosen as a representative strain ε_r, the relationship can be obtained, which agrees with Eq. (8):

(16)$${H_0} = {\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)^n}\quad.$$

From the previous research studies on size effect,^{Reference Nix and Gao7,Reference Huang, Zhang, Hwang, Nix, Pharr and Feng8,Reference Durst, Backes and Göken13} the material hardness at microscale is

(17a)$${{{H_n}} \over {{H_0}}} = \sqrt {1 + {{{h^*}} \over h}}\quad ,$$

(17b)$${h^*} = {{81} \over {2{f^3}}}b{\alpha ^2}\;{\tan ^2}\theta {\left( {{\mu \over {{H_0}}}} \right)^2}\quad.$$

In the above equations, H ₀ is the macroscopic hardness, relating to the density of statistically stored dislocations (SSDs) according to Nix/Gao model and h ^∗ is a characteristic length for the material.^{Reference Nix and Gao7} If the hardness H ₀ in Eqs. (16) and (17a) is the same, and using Eq. (16) in Eq. (17a), one can find

(18)$$H = H({\kappa _0},E,n,{\sigma _{\rm{y}}},\nu ,h,{h^ * }) = {H_0}\sqrt {1 + {{{h^*}} \over h}} = {\kappa _0}{\sigma _{\rm{r}}}\sqrt {1 + {{{h^*}} \over h}} = {\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)^n}\sqrt {1 + {{{h^*}} \over h}}\quad .$$

It is the relationship between hardness and material properties with considering size effect at a certain Poisson’s ratio. In the proposed model as in Eq. (18), h ^∗ relates to the material properties, definitely not a constant for all materials. It is related to the intrinsic characteristic length associating with size effect, as shown in Eq. (17b). If shear modulus, $\mu = {E \over {2(1 + \nu )}}$ and macroscopic hardness ${H_0} = {\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)^n}$ with representative strain ε_r = 0.033 as in the study by Dao et al.^{Reference Dao, Chollacoop, Van Vliet, Venkatesh and Suresh2} are substituted in Eq. (18), the model is determined as

(19)$$H = {\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)^n}\sqrt {1 + {{81} \over {2{f^3}h}}b{\alpha ^2}\;{{\tan }^2}\theta {{\left( {{\mu \over {{H_0}}}} \right)}^2}} = {\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)^n}\sqrt {1 + {{81} \over {2{f^3}h}}b{\alpha ^2}\;{{\tan }^2}\theta {{\left( {{E \over {2(1 + \nu ){\kappa _0}{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}}}} \right)}^2}\quad}.$$

For a Berkovich indenter, θ = 70.3°, Burgers vector b is at quality of 10⁻⁴ μm, and empirical coefficient is α = 0.1–0.5. So, the part of $hr = {{81} \over {2{f^3}}}b{\alpha ^2}{\tan ^2}\theta$ should be at the order of magnitude of 10⁻⁴–10⁻². From the proposed model, a better fitting result is obtained with the constraint factor as κ₀ = 2.90016, and hr = 0.000613247, which meets the requirement of magnitude order, with correlation coefficient (R): 0.9839. The proposed model reflects a good relationship between material hardness and their properties.

By the process of Nix and Gao, H ₀ can be obtained by the linear fitting H ² versus h ⁻¹ by ${H^2} = H_0^2\left( {1 + {{{h^*}} \over h}} \right)$.^{Reference Nix and Gao7} With the simulation results, the relationship between the fitting H ₀ and predicted H ₀ is plotted in Fig. 4, where predicted hardness ${H_0} = 3{\sigma _{\rm{y}}}{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)^n}$, in which the constraint factor is set as 3.^{Reference Cheng and Cheng5,Reference Tabor20} We can find that the predicted H ₀ is a bit greater than experimental results. The reason can be explained by the previous fitting result κ₀ = 2.90016, which is less than the traditional one used.^{Reference Cheng and Cheng5,Reference Tabor20}

FIG. 4.

The relationship between the experiment H ₀ and predicted H ₀.

In microscale, the hardness of solids is determined by many factors, such as material elastic–plastic properties (E, σ_y, n), indenter geometry (half apex angle, indenter shape), friction factor, etc.^{Reference Cheng and Cheng5,Reference Cheng and Cheng18,Reference Johnson28} To find a proper and precise relation between hardness and material properties, the forward problem has been focused on since long.^{Reference Cheng and Cheng5,Reference Cheng and Cheng18,Reference Tabor20,Reference Johnson28} Tabor proposed that the hardness is 3 times of the flow stress at 8% representative strain.^{Reference Gouldstone, Chollacoop, Dao, Li, Minor and Shen1} The result of Johnson for a large indent in an elastic–perfectly plastic solid is $H = {2 \over 3}{\sigma _{\rm{y}}}(1 + \ln (\tan \beta /3{\varepsilon _{\rm{y}}}))$, which is in agreement with the limit for large indents for the solid with low strain hardening.^{Reference Wei and Hutchinson17,Reference Johnson28} Following Tabor's model, Cheng and Cheng set up a model between hardness and representative stress (the yield stress at 10% of strain) with a different yield strain, since with great representative strain, the effect of strain hardening exponent vanished.^{Reference Cheng and Cheng19} As used in the concept, the material hardness is independent to indentation depth.^{Reference Cheng and Cheng5} This previous model was not considered size effect at microscale, which cannot obtain better results in this study. In regard to the model of Eq. (18) with size effect, the part of $\sqrt {1 + {{{h^*}} \over h}}$ is combined with material properties. When the indentation depth is large enough, the part of $\sqrt {1 + {{{h^*}} \over h}}$ will be close to 1, and then the hardness will not relate to size effect and just depends on material properties. Besides, according to Cao et al.,^{Reference Cao, Xue, Chen and Raabe15} he defined a new hardness as H ₀ⁿ, which is the nominal hardness at the macroscale^{Reference Cao, Xue, Chen and Raabe15} or is the hardness in the limit of infinite depth.^{Reference Nix and Gao7} It was H _n = 4.4σ_r, in which the constraint factor κ₀ was set as 4.4, different from the general result κ₀ = 3.^{Reference Ogasawara, Chiba and Chen14,Reference Chen and Vlassak27} According to the proposed model, $H = {H_0}\sqrt {1 + {{{h^*}} \over h}} = {\kappa _0}{\sigma _{\rm{r}}}\sqrt {1 + {{{h^*}} \over h}}\,$, at an infinite size or indentation depth h ₀ⁿ, the material hardness is given as $H_0^n = {H_0}\sqrt {1 + {{{h^*}} \over {h_0^n}}} = {\kappa _0}{\sigma _{\rm{r}}}\sqrt {1 + {{{h^*}} \over {h_0^n}}}\,$. Combining the part of ${\kappa _0}\sqrt {1 + {{{h^*}} \over {h_0^n}}}\,$, it is greater than 3 for the nominal hardness of the material at an infinite size, or at the limitation of size effect, which is similar to the results of Cao et al.^{Reference Cao, Xue, Chen and Raabe15} For h ₀ⁿ, it may be related to the limited depth of size effect, which is needed for further investigation.

Accordingly, by using Eqs. (1) and (6) the maximum load is determined as

(20)$${P_{\rm{m}}} = {P_{\rm{m}}}({\kappa _0},E,n,{\sigma _{\rm{y}}},\nu ,h,{h^ * }) = HA = kh_{\rm{c}}^2H = kh_{\rm{c}}^2{\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {\varepsilon _{\rm{r}}}{E \over {{\sigma _{\rm{y}}}}}} \right)^n}\sqrt {1 + {{{h^*}} \over h}}\quad .$$

As given in Eq. (6), the hardness H is the ratio between maximum load P _m and contact area A: $H = {{{P_{\rm{m}}}} \over A} = {{{P_{\rm{m}}}} \over {kh_{\rm{c}}^2}} = {{{P_{\rm{m}}}} \over {\pi {{\tan }^2}\theta h_{\rm{c}}^2}}$.

C. The relationship between loading curvature and material properties

Similarly, loading curvature C, which depends on elastic and plastic material properties, as well as indenter geometry,^{Reference Gouldstone, Chollacoop, Dao, Li, Minor and Shen1} is the ratio of load and indenter depth h by Eq. (1). During indentation process, C is a constant, which can be obtained from the fitting process as shown in Fig. 5. Also, it can be determined at peak depth h _m as $C = {C_{\rm{m}}} = {\left. {{P \over {{h^2}}}} \right|_{h = h{\rm{m}}}} = {{{P_{\rm{m}}}} \over {h_{\rm{m}}^2}}$, which is similar to the definition as hardness H. One relates to the contact depth h _c, and the other associates to the maximum depth h _m. From Eq. (6), it can be found that C _mh _m² = Hkh _c². So, loading curvature C can be determined as $C = {C_{\rm{m}}} = kH{\left( {{{{h_{\rm{c}}}} \over {{h_{\rm{m}}}}}} \right)^2}$. From the obtained results as shown in Fig. 3(d), the relationship between C and $H{\left( {{{{h_{\rm{c}}}} \over {{h_{\rm{m}}}}}} \right)^2}$ is fitted, a good result for coefficient k is obtained as 24.399–24.673, which is close to the value of k = πtan²θ = 24.56, with correlation coefficient (R): 0.9997 (see Fig. 6).

FIG. 5.

Examples of fitting loading curvature C with n = 0.3 and h = 1 (red lines are the fitting lines and scattered symbols are the original data).

FIG. 6.

The relation between loading curvature C and H(h _c/h _m)².

Accordingly, combining Eqs. (18), (20), and (1), the loading curvature C can be rewritten as

(21)$$C = kH{\left( {{{{h_{\rm{c}}}} \over {{h_{\rm{m}}}}}} \right)^2} = k{\kappa _0}{\sigma _{\rm{r}}}{\left( {{{{h_{\rm{c}}}} \over {{h_{\rm{m}}}}}} \right)^2}\sqrt {1 + {{{h^*}} \over {{h_{\rm{m}}}}}} = k{\kappa _0}{\sigma _{\rm{y}}}{\left( {1 + {\varepsilon _{\rm{r}}}{E \over {{\sigma _{\rm{y}}}}}} \right)^n}{\left( {{{{h_{\rm{c}}}} \over {{h_{\rm{m}}}}}} \right)^2}\sqrt {1 + {{{h^*}} \over {{h_{\rm{m}}}}}}\quad ,$$

in which C depends on elastic and plastic material properties, as well as indenter geometry.^{Reference Gouldstone, Chollacoop, Dao, Li, Minor and Shen1} From Eq. (21), loading curvature C is related to the material properties with ${\kappa _0}{\sigma _{\rm{r}}}\sqrt {1 + {{{h^*}} \over {{h_{\rm{m}}}}}}$, i.e., the macronominal hardness, H ₀, indenter geometry k = πtan²θ, and certain ratio between contact depth and maximum depth $\left( {{{{h_{\rm{c}}}} \over {{h_{\rm{m}}}}}} \right)$, which vary slightly with increasing indentation depth.

D. Characteristic length h ^∗

For size effect, the hardness relates to the dislocation in the material during deformation as GND density (ρ_G) which responds to the ‘additional’ hardness, and the original SSDs density (ρ_S) and the size effect of the material can be rewritten as

(22a)$${{{H_{\rm{n}}}} \over {{H_0}}} = \sqrt {1 + {{{h^*}} \over h}} = \sqrt {1 + {{{\rho _{\rm{G}}}} \over {{\rho _{\rm{S}}}}}}\quad ,$$

(22b)$${h^*} = {{{\rho _{\rm{G}}}} \over {{\rho _{\rm{S}}}}}h\quad.$$

in which h ^∗ is not a constant for a given material and indenter geometry. Rather it depends on the SSD density through H ₀.^{Reference Nix and Gao7} And GND density (ρ_G) and SSD density (ρ_S) are

(23a)$${\rho _{\rm{G}}} = \bar r{{{\eta ^p}} \over b} = {3 \over {2{f^3}\;{{\tan }^2}\theta bh}}\quad,$$

(23b)$${\rho _{\rm{S}}} = {[{\sigma _{{\rm{ref}}}}f({\varepsilon ^{\rm{p}}})/(M\mu \alpha b)]^2}\quad,$$

in which a coefficient f is added into GND density [ρ_G as Eq. (23b)], when the profile of indenter shape is not considered,^{Reference Durst, Backes and Göken13,Reference Swadener, George and Pharr26} and M is the Taylor factor which acts as an isotropic interpretation of the crystalline anisotropy at the continuum level, M = 3 for FCC metals as well as for BCC metals on {1 1 0} slip planes. The relation between GND density (ρ_G) and SSD density (ρ_S) is

(24)$${{{\rho _{\rm{G}}}} \over {{\rho _{\rm{S}}}}} = {{{3 \over {2{f^3}{{\tan }^2}\theta bh}}} \over {{{[{\sigma _{{\rm{ref}}}}f({\varepsilon ^{\rm{p}}})/(M\mu \alpha b)]}^2}}} = {{3{{(M\mu \alpha )}^2}b} \over {2{f^3}{{\tan }^2}\theta h{{[{\sigma _{{\rm{ref}}}}f({\varepsilon ^{\rm{p}}})]}^2}}}\quad.$$

And h ^∗ can be rewritten as

(25)$${h^ * } = {3 \over {2{f^3}}}{{{{(M\alpha \mu )}^2}b} \over {{{({\sigma _{{\rm{ref}}}}f({\varepsilon ^{\rm{p}}})\tan \theta )}^2}}}\quad.$$

In Nix and Gao,^{Reference Nix and Gao7} the hardness also was determined as

(26)$${H_0} = {\kappa _0}M\alpha \mu b\sqrt {{\rho _{\rm{S}}}}\quad .$$

In Nix/Gao model, the SSD density (ρ_S) was ${\rho _{\rm{s}}} \approx {1 \over {L_{\rm{s}}^2}}$, where L _s is the mean spacing between SSDs.^{Reference Nix and Gao7} However, for the case of materials strengthened by dispersoids or precipitations, the flow stress in the absence of gradients might be expressed in terms of the mean spacing between particles, L _p, ${\rho _{\rm{s}}} \approx {1 \over {{{(\alpha {L_{\rm{p}}})}^2}}}$ according to^{Reference Nix and Gao7}

(27a)$${\rho _{\rm{s}}} \approx {1 \over {L_{\rm{s}}^2}}\quad,$$

(27b)$${\rho _{\rm{s}}} \approx {1 \over {{{(\alpha {L_{\rm{p}}})}^2}}}\quad.$$

Referring to Eqs. (16) and (26), the SSD density (ρ_S) can be rewritten as

(27c)$${\rho _{\rm{S}}} = {({H_0}/{\kappa _0}M\alpha \mu b)^2} = {\left( {{{{\kappa _0}{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}} \mathord{\left/ {\vphantom {{{\kappa _0}{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}} {{\kappa _0}M\alpha \mu b}}} \right. \kern-\nulldelimiterspace} {{\kappa _0}M\alpha \mu b}}} \right)^2} = \left( {{{{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}} \mathord{\left/ {\vphantom {{{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}} {M\alpha \mu {b^2}}}} \right. \kern-\nulldelimiterspace} {M\alpha \mu {b^2}}}} \right)\quad.$$

According to Eq. (27c), SSD density (ρ_S) is related to the material properties, i.e., the macronominal hardness, H ₀. When the material is selected, the SSD density mainly depends on the deformation degree in the material. With the increasing plastic strain, the SSD density (ρ_S) increases accordingly. Additionally, the loading curvature is related to dislocation density as $C = k{\kappa _0}M\alpha \mu b\sqrt {{\rho _S}} {\left( {{{{h_{\rm{c}}}} \over {{h_{\rm{m}}}}}} \right)^2}$.

Moreover, if GND density (ρ_G) is taken as given in Ref. Reference Durst, Backes and Göken13${\rho _{\rm{G}}} = \bar r{{{\eta ^p}} \over b} = {3 \over {2{f^3}{{\tan }^2}\theta bh}}$, in which a coefficient f was used for the plastic deformation scope under indenter tip.^{Reference Durst, Backes and Göken13} Accordingly, the characteristic length h ^∗ is determined as

(28)$${h^*} = {{{3 \over {2{f^3}{{\tan }^2}\theta bh}}} \over {{\rho _{\rm{S}}}}}h = {3 \over {2b{f^3}{\rho _{\rm{S}}}{{\tan }^2}\theta }}\quad.$$

Substituting SSD density (ρ_S) in Eq. (28), h ^∗ can be rewritten as

(29a)$$h^* = \left\{ {\matrix{ {{{3L^2 _{\rm{S}} } \over {2bf^3 \tan ^2 \theta }}} \cr {{{3\alpha ^2 L^2 _{\rm{P}} } \over {2bf^3 \tan ^2 \theta }}} \cr } } \right.\quad \matrix{ {{\rm{for}}\;{\rm{pure}}\;{\rm{metal}}} \cr {{\rm{for}}\;{\rm{strengthened}}\;{\rm{materials}}} \cr }\quad ,$$

or

(29b)$${h^*} = {3 \over {2b{f^3}{\rho _{\rm{S}}}{{\tan }^2}\theta }} = {3 \over {2b{f^3}{{\left( {{{{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}} \mathord{\left/ {\vphantom {{{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}} {M\alpha \mu b}}} \right. \kern-\nulldelimiterspace} {M\alpha \mu b}}} \right)}^2}{{\tan }^2}\theta }} = {{3{{(M\alpha \mu )}^2}b} \over {2{f^3}{{\left( {{\sigma _{\rm{y}}}{{\left( {1 + {E \over {{\sigma _{\rm{y}}}}}{\varepsilon _{\rm{r}}}} \right)}^n}} \right)}^2}{{\tan }^2}\theta }}\quad.$$

The characteristic length h ^∗ only depends on the material properties, especially the average distance between dislocations or dispersoids or precipitations. With the increasing SSD density or dispersoids or precipitations, the distance between them is reduced accordingly. This effect strengthens the material with small characteristic length h ^∗. Furthermore, the value of characteristic length h ^∗ is a measurement for size effect. When the ratio of h ^∗/h is small enough at great indentation depth, the material hardness will become constant, while it is sensitive to indentation depth when the ratio of h ^∗/h is a non-missing part.