Refined model of elastic nanoindentation of a half-space by the blunted Berkovich indenter accounting for tangential displacements on the contact surface

Abstract

Elastic contact between a non-ideal Berkovich indenter and a half-space is investigated. The derived mathematical model of the contact allows for tangential displacements of the boundary points of the half-space. The tip of the blunted indenter is simulated as a smooth surface. The boundary element method is implemented in the model for numerical simulation of nanoindentation. The relative deviation function is introduced and calculated to quantify the influence of the tangential displacements on the load–displacement curves. A simple expression is derived for the impact of the tangential displacements on the values of the reduced Young’s modulus determined due to nanoindentation studies. The refined model was successfully applied to simulate the experimental load–displacement curves gained by elastic nanoindentations of flat LiF and KCl samples. Such values of the indenter bluntness (the varying parameter) were found that the simulated load–displacement curves coincided with those of the experimental data at displacements higher than 7.5 nm. The model neglecting tangential displacements gives slightly differing values for the parameter of the indenter bluntness.

Appendices

Appendix A

Accordingly to [8] the nonlinear boundary integral equation of the contact problem accounting for tangential displacements is:

$$ \mu v^{ - } (M) + \lambda \mathop{\iint}\limits_{\Upomega } {\frac{{v^{ + } (N)}}{{R_{MN} }}\hbox{d}S_{N} = h - f\left( {x_{1} + u(M),\;x_{2} + w\left( M \right)} \right),\quad M(x_{1} ,\,x_{2} ),N(\xi ,\,\eta )} \in \Upomega . $$

(13)

Here u(M) and w(M) are the tangential displacements in point M in the directions Ox₁ and Ox₂ respectively, v ⁺(M) is the contact pressure. We assume that these displacements are small compared to the dimensions of the contact region. In this case the function $f(x_{1} + u(M),\,x_{2} + w(M))$ from Eq. 13 can be approximated by:

$$ f(x_{1} + u(M),x_{2} + w(M)) = \frac{{(x_{1} + u(M))^{2} + (x_{2} + w(M))^{2} }}{{2R(x_{1} + u(M),x_{2} + w(M))}} \approx f(M) + \frac{{x_{1} \cdot u(M) + x_{2} \cdot w(M)}}{R(M)}, $$

where f(M) is defined in (5). We denote further R(M) as R.

The tangential displacements induced by the surface of the bluntness can be introduced as [24]:

$$ u(M) = - \lambda \varepsilon \mathop{\iint}\limits_{\Upomega } {\frac{{v^{ + } (N) \cdot (x_{1} - \xi )}}{{R_{MN}^{2} }}\hbox{d}S_{N} ,}\quad w(M) = - \lambda \varepsilon \mathop{\iint}\limits_{\Upomega } {\frac{{v^{ + } (N) \cdot (x_{2} - \eta )}}{{R_{MN}^{2} }}\hbox{d}S_{N} .} $$

Therefore,

$$ f\left( {x_{1} + u(M),x_{2} + w(M)} \right) \approx f(M) - \frac{\lambda \varepsilon }{R}\mathop{\iint}\limits_{\Upomega } {\frac{{x_{1}^{2} + x_{2}^{2} - x_{1} \xi - x_{2} \eta }}{{R_{MN}^{2} }}v^{ + } (N)\hbox{d}S_{N} }. $$

and Eq (13) allowing for the tangential displacements can be written as

$$ \begin{gathered} \mu v^{ - } (M) + \lambda \mathop{\iint}\limits_{\Upomega } {\frac{1}{{R_{MN} }}\left[ {1 - \frac{\varepsilon }{R} \cdot \frac{{x_{1}^{2} + x_{2}^{2} - (x_{1} \xi + x_{2} \eta )}}{{R_{MN} }}} \right]v^{ + } (N)\hbox{d}S_{N} = h - f(M),} \hfill \\ \,M(x_{1} ,x_{2} ),\;N(\xi ,\eta ) \in \Upomega . \hfill \\ \end{gathered} $$

(14)

In the present formulation of the contact problem the magnitude of the radius R depends on the position of the point M.

Appendix B

Let us formulate the problem (2) in a dimensionless coordinate system $x_{\text{i}} = \sqrt {2hd} x_{{0{\text{i}}}}$ and also introduce the dimensionless unknown function $U(x_{01} ,x_{02} ) = \lambda \cdot \sqrt {{{2d} \mathord{\left/ {\vphantom {{2d} h}} \right. \kern-\nulldelimiterspace} h}} \cdot v(x_{1} ,x_{2} ).$ We then write the NBIEs (2) in the following dimensionless form:

$$ \begin{gathered} U^{ - } (M_{0} ) + \mathop{\iint}\limits_{{\Upomega_{0} }} {\frac{1}{{R_{{M_{0} N_{0} }} }}\left[ {1 - \sqrt {\frac{2h}{d}} \cdot \frac{\varepsilon }{{s(x_{01} ,x{}_{02})}} \cdot \frac{{x_{01}^{2} + x_{02}^{2} - (x_{01} \xi_{0} + x_{02} \eta_{0} )}}{{R_{{M_{0} N_{0} }}^{{}} }}} \right]} \cdot U^{ + } (N_{0} )\hbox{d}S_{{N_{0} }} \hfill \\ = 1 - \frac{{x_{01}^{2} + x_{02}^{2} }}{{s(x_{01} ,x{}_{02})}},\quad \hfill \\ \mathop{\iint}\limits_{{\Upomega_{0} }} {U^{ + } (N_{0} )}{\text{d}}S_{{N_{0} }} = P_{0} (h),\quad M_{0} ,N_{0} \in \Upomega_{0} , \hfill \\ \end{gathered} $$

(15)

where M₀ and N₀ are the points on the plane x₀₃ = 0 with the coordinates (x₀₁, x₀₂) and $(\xi_{0} ,\eta_{0} )$ respectively. P₀(h) is the dimensionless compression force. The square $\Upomega_{0} = \left\{ {M_{0}{:}\;\left| {x_{01} } \right| \le 6,\;\left| {x_{02} } \right| \le 6} \right\}$ is an image of the square Ω after changing variables. The value 6 is chosen in order that the square Ω₀ includes the contact region $S_{0} = \left\{ {M_{0} :\;U(M_{0} ) \ge 0} \right\} \subseteq \Upomega_{0} .$ Similarly, the dimensionless tangential displacements are given by:

$$ u_{0} (M_{0} ) = - \varepsilon \sqrt {\frac{h}{2d}} \cdot \mathop{\iint}\limits_{{\Upomega_{0} }} {\frac{{U^{ + } (N_{0} ) \cdot (x_{01} - \xi_{0} )}}{{R_{{M_{0} N_{0} }}^{2} }}\hbox{d}S_{{N_{0} }} ,}\quad w_{0} (M_{0} ) = - \varepsilon \sqrt {\frac{h}{2d}} \cdot \mathop{\iint}\limits_{{\Upomega_{0} }} {\frac{{U^{ + } (N_{0} ) \cdot (x_{02} - \eta_{0} )}}{{R_{{M_{0} N_{0} }}^{2} }}\hbox{d}S_{{N_{0} }} .} $$

(16)

The final formula for constructing the P(h) diagram for the approach of the indenter and the sample can be obtained by substituting the second equation in (15) into the second equation in (2)

$$ P = \frac{{\sqrt {2d} }}{\lambda } \cdot P_{0} (h) \cdot h^{3/2} . $$

(17)

The collocation method is used for the discretization of (15). The integrals are approximated by the rectangle formula. To solve the discretized equation the generalized Newton’s method is applied.

The region Ω₀ is divided into 1024 equal pieces (i.e. 1089 nodes). Numerical simulations show that the generalized Newton’s method converges to the exact solution of the discretized form of the first equation in (15) usually after five iterations at the discrepancy for each node of 1 × 10⁻⁵.

Before starting the numerical simulations, the solution of the NBIEs (15) was compared with that one of the Hertzian problem. For this purpose we assume $s(x_{1} ,x_{2} ) = {\text{const}} = 1$ and υ_s = 0.5. In this case:

$$ P = \frac{{\sqrt {2R} }}{\lambda } \cdot P_{0} \left( h \right) \cdot h^{3/2} .$$

holds for the compressive force P (see (17)). The solution of the dimensionless NBIEs (15) yields $P_{0} (h) = {\text{const}} = 0.3 \pm 2 \cdot 10^{ - 4} .$ Let us compare with the Hertzian problem [17, 24]:

$$ P = \frac{2\sqrt 2 }{3\pi } \cdot \frac{{\sqrt {2R} }}{\lambda} \cdot h^{3/2} , $$

where $\frac{2\sqrt 2 }{3\pi } \approx 0.30011.$