In nanoindentation tests14reduced modulus of elasticity ({E}^{*}) can be obtained as

$${E}^{*}=frac{1}{2beta }Sfrac{sqrt{pi }}{sqrt{A}}$$

(1)

or (A) represents the projected contact area, and (beta ) denotes a constant which depends on the geometry of the indenter ((beta =1.034) for a Berkovich indenter15). Generally, the contact stiffness (S) is taken from the elastic unloading, which is the initial part of the unloading curve, while it can alternatively be measured during the loading part of an indentation test using the CSM technique16. The reduced modulus of elasticity ({E}^{*}) can be expressed as follows:

$$frac{1}{{E}^{*}}=frac{(1-{nu }_{s}^{2})}{{E}_{s}}+frac{ (1-{nu }_{i}^{2})}{{E}_{i}}$$

(2)

or (E) and (v) represent the modulus of elasticity and the Poisson’s ratio corresponding to the diamond indenter I and sample (s), respectively. As a geometrically similar indenter, the ideal projected contact area (A({h}_{c})) of the Berkovich point can be expressed as follows:

$$A({h}_{c})=24.50times {{h}_{c}}^{2}$$

(3)

or ({h}_{c}) is the contact depth. Combine eq. (1) and (3), we obtain:

$$S=2beta {E}^{*}sqrt{frac{24.5}{pi }}times {h}_{c}$$

(4)

In dimensional analysis, the contact depth can be obtained as17

$${h}_{c}=h{prod }_{beta }left(frac{Y}{E},v,n,theta right)$$

(5)

or ({prod }_{beta }={h}_{c}/h). Besides, (Y), (E), (v), (not)and (theta ) are respectively the elastic limit, the modulus of elasticity, the Poisson’s ratio, the exponent of work hardening and the geometry of the indenter. The linear dependence between ({h}_{c}) and (h) has been verified by finite element calculations for a wide range of values ​​of (Y), (E)and (not) 18. The linear relationship between ({h}_{c}) and (h) was also verified by Li et al16Zhu et al.19 and Gao et al20.21.

Therefore, eq. 4 can be written

$$S=2beta {E}^{*}sqrt{frac{24.5}{pi }}{prod }_{beta }left(frac{Y}{E},v, n,theta right)times h$$

(6)

Considering the zero point shift in practice with the equations. (1), (3) and ({H}_{IT}=P/A)or (P) represents the load applied to the tip, the errors of modulus of elasticity and indentation hardness can be derived as

$$frac{{Delta E_{IT} }}{{E_{IT} }} = frac{{E_{IT}^{^{prime}} – E_{IT} }}{{E_{ IL} }} = frac{{sqrt A – sqrt {A^{{prime }} } }}{{sqrt {A^{prime}} – frac{{left( {1 – nu_{i}^{2} } right)}}{{E_{i} }} times frac{sqrt pi times S}{{2beta }}}} = frac{{ sqrt A – sqrt {A^{{prime }} }}{{sqrt {A^{prime}} – 0.00074 times S}};{text{and}}$$

(7)

$$frac{{Delta H_{{IT}} }}{{H_{{IT}} }} = frac{{H_{{IT}}^{{prime }} – H_{{IT} } }}{{H_{{IT}} }} = frac{{A – A^{{prime }} }}{{A^{{prime }} }} = frac{{h_{c } ^{2} – h_{c}^{{‘2}} }}{{h_{c}^{{{prime }2}} }} = – frac{{frac{{Delta h_ {c} }}{{h_{c} }}left( {frac{{Delta h_{c} }}{{h_{c} }} + 2} right)}}{{left( {frac{{Delta h_{c} }}{{h_{c} }}+1} right)^{2} }},~;{text{where}};;Delta h_{c} = h_{c}^{{prime }} – h_{c}$$

(8)

Primed symbols denote parameters after the zero point shift.

Contact stiffness S of the Berkovich indenter on homogeneous materials should be a linear function with respect to the depth of indentation , implying that this linear function can be a criterion for finding the effective zero point with the modified Berkovich indenter. However, it should be noted that the initial contact of Berkovich’s practical tip is elastic due to the rounding of the tip. The roughness of the surface causes the zero point to shift, and the rounding of the tip magnifies the difference between the first touch and the actual zero point. In addition to the problems mentioned above, the loss of contact tapping in the CSM method22 would inevitably fail regular zero point determination. Therefore, the normal method of determining the zero point and even the current linear relationship between S and h (Eq. 6) cannot be applied at the onset of increasing contact stiffness. The present study performed a linear fit to the contact stiffness values ​​from the elastic-plastic stable contact point. The different definitions of the zero point of these methods are illustrated schematically in Fig. 1. The normal method generally estimates that the zero point is deeper than the actual first contact, resulting in a larger contact area than the actual one. In contrast, the present method estimates that the zero point is shallower than the actual first contact, but the zero point is the ideal sharp point.

The indentation size effect (ISE), which is a decrease in hardness with increasing indentation depth, is often observed for various nanoindentation hardness materials. The Nix-Gao model23based on the concept of geometrically necessary dislocations formed around the indentation indentation, has been widely accepted and explains the ISE as

$$H={H}_{0}sqrt{1+frac{{h}^{*}}{h}}$$

(9)

or ({H}_{0}) and ({h}^{*}) represent the equivalent bulk hardness corresponding to infinite depth and the length which characterizes the depth dependence of hardness, respectively. Although recent literature recommends that the CSM method should not be used for the study of ISE due to the uncertainty of hardness data at shallow depth22exploring the effect of zero point adjustment on ISE is considered valuable in understanding ISE.