Quantitative Evaluation of Stress-Strain Curves by Spherical-Tip Nanoindentation with Variable Radius Method

Abstract

We propose a novel evaluation method to obtain precise stress-strain curves by spherical-tip nanoindentation with continuous multiple loading technique. The small-sized spherical-tip indenters have an imperfect shape and it causes miss-fitting with tensile test result. We adopted the variable radius method, where we defined the precise indenter radius as the function of contact depth to calculate the stress-strain response precisely. We calculated the representative stress and representative strain by combining Hollomon's law, Hill's model, proper strain model, and the variable radius and compared with the tensile test result. It is found that stress-strain curves obtained by using the variable radius method agree well with the tensile test result rather than using nominal radius. These results clearly indicate the effectiveness of our method to evaluate the stress-strain response by using the small-sized spherical-tip indenter.

Modelling and simulation of the hardness profile and its effect on the stress‐strain behaviour of punched electrical steel sheets

“…Regarding the determination of the true stress‐strain curve, Tabor's approach of representative strain and stress can be applied for spherical instrumented indentation tests. The mean pressure $${{p}_{{\rm m}}}$$ in the indentation region in the fully plastic regime is assumed to be proportional to the representative stress $${{\sigma }_{{\rm r}}}$$ [4, 13, 27, 28]: $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr {\sigma }_{{\rm r}}={{{p}_{{\rm m}}}\over{\psi }}={{L}\over{\psi \pi {a}^{2}}}\hfill\cr}}$$$ …”

“…With increasing indentation depth, the material can sink in or pile up at the edge of the indenter, which results in a change of the contact area between specimen and indenter, Figure 2 [27, 28]. The amount of sink‐in or pile‐up depends on the strain‐hardening exponent $${n}$$ of the investigated material.…”

“…The amount of sink‐in or pile‐up depends on the strain‐hardening exponent $${n}$$ of the investigated material. A dimensionless correction parameter has been developed to consider the impact on the contact radius $${a}$$ [27, 28, 30]: $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr a=\sqrt{{c}^{2}{a}^{{\rm { {^\prime}}}2}}=\sqrt{{{5}\over{2}}{{2-n}\over{4+n}}\left(2R{h}_{{\rm c}}-{h}_{{\rm c}}^{2}\right)}\hfill\cr}}$$$ …”

“…The amount of sink-in or pile-up depends on the strain-hardening exponent n of the investigated material. A dimensionless correction parameter has been developed to consider the impact on the contact radius a [27,28,30]:…”

“…Often the two‐parametric Hollomon function is applied to the plastic regime. The elastic regime with stresses below the yield strength $${{\sigma }_{y}}$$ is described by Hooke's law [13–15, 28, 29, 31]: $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr \sigma \left(\varepsilon{} \right)=\left\{\begin{matrix}E\varepsilon{}, \ \sigma \char60 {\sigma }_{{\rm y}} \\ K{\varepsilon{} }^{n},\ \sigma \ge {\sigma }_{{\rm y}} \end{matrix} \right.\hfill\cr}}$$$ …”