where S is the sum of the square residuals and ri is given by the difference of the reaction force values f(δi) EXP obtained by the real experiment and f(δ i) FEM numerically calculated. N is the number of sampling points for the function f(δ) expressing the reaction forces values derived from a quasi-static indentation in function of the penetration depth δ. In this study, the maximum value for δ is 1 (mm). The benchmark study is performed taking into account N=10 and N=100. The influence of the number of sampling points is analyzed. The cost function Eq.(5) is optimized using the Modified Nelder-Mead Algorithm (MNMA) starting from an initial guess value G(0) (Fig.(1)). It is a direct-search optimization method for global minima determination. The description of its working principle is out of the scope of this work. Fig. 1 Inverse method scheme In order to evaluate the reliability of the methodology two different studies typologies have been conducted: a virtual benchmark and a real case. The implementation of a virtual benchmark study allowed the evaluation of accuracy in the parameters (c10 and c01) determination eliminating a possible error source due to real experimental data utilization. The benchmark “unknown” parameters have been determined by a numerical calculation of the reaction forces values at the interface of an analytical rigid surface (micro-indenter probe) and a deformable cubic specimen. Real experiment is performed using a dedicated micro-indentation system consisting of a linear actuator, a force transducer and an acquisition data card connected to a computer. The device has a maximum penetration resolution of 220 nm and is automatically driven at speed of 0.02 (mm/s) for a measured distance of 1 (mm) in a cubic silicon rubber specimen of 1000 ml volume. A flat cylindrical punch of 23
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