IDENTIFICATION OF MATERIAL MODELS IN HARD SYSTEM OF NANOCOATINGS USING METAMODEL M. Kopernik, A. Stanisławczyk, J. Kusiak, M. Pietrzyk Akademia Górniczo-Hutnicza, Mickiewicza 30, 30 – 059 Kraków, Poland
[email protected],
[email protected] Keywords: inverse analysis, metamodel, artificial neural network (ANN)
1. INTRODUCTION Hard systems of nanocoatings deposited using PVD (physical vapour deposition) are used in various applications. Numerical models of deformation of these coatings are needed for aiding design of new applications of these coatings. Correct determination of nanomaterial parameters is crucial for accuracy of simulations. FEM and ANN (Koker et al., 2007) are often used to obtain parameters of models for various materials. The objective of the present work is identification of material parameters of nanocoatings in hard system using both mentioned methods (Kopernik et al., 2007). The inverse analysis is performed using a metamodel (Kusiak et al., 2005). 2. MODEL OF NANOINDENTATION TEST Experimental nanoindentation test is performed using a Nano Test System. In the present work deformed 840 nm thick and 2600 nm wide specimen is a system of 3 hard nanocoatings. Two coatings are deposited periodically, respectively coating 1 (elastic, 400 nm thick) is repeated two times and coating 2 (elastoplastic, 40 nm thick) is a single interlayer. Indentation test, which is depth controlled, supplies force versus indentation depth data. Diamond (Young modulus E = 1141 GPa, Poisson ratio ν = 0.07), pyramid, deformable indenter (radius R = 150 nm, pyramid angle α = 70.32º) penetrates into specimen at a depth of 100 nm. Application of the inverse analysis to interpretation of the test results is the objective of the project. Inverse algorithm proposed in (Szeliga et al., 2006) is used. Due to very high computing costs, the
concept of the metamodel (Kusiak et al., 2005) is applied in optimization. ANN was used as metamodel. FORGE 2 FEM code is used as direct problem model. Axisymmetric 2D FEM solution is performed. The friction coefficient µ is assumed 0. The following material model is identified: (1) σ = Kε n where: σ – flow stress, ε - effective strain, K, n –parameters, which are optimization variables. Young modulus E is the third variable. Since testing of the approach is the main objective, the experimental data were generated by the FEM code. Two cases are considered. Two sets were assumed as the real material parameters: I) n = 0.125, E = 380 GPa, K = 290 MPa and II) n = 0.175, E = 400 GPa, K = 270 MPa. 120 simulations were performed to supply data for training ANN, for the following parameters: a) K = 50, 60, 100, 110, 300, 310 MPa; b) n = 0.1, 0.15, 0.2, 0.25; c) E = 330, 350, 370, 390, 410 GPa. 3. METAMODEL AND RESULTS Approximation of FEM output data is done using MLP - type of ANN with architecture 42-1 and logistic functions of transfer in the first and second network layers, and linear function of activation in output network layer. Optimization variables in the inverse analysis (E, K, n) and strain are the input data and the load is the output. 90 training sets of data for n, K and E were used as an input. Each set is composed of 20 values of force versus displacement data. The network was tested for n = 0.15, E = 370 GPa, K = 100 MPa and the results are shown in
Figure 1. Mean square error for test sets is equal to 40 µN2 what confirms good predictive capability of the network as the metamodel. Inverse analysis was performed next. The goal function is the mean square root error between experimental data and network output:
φ (n, E, K ) =
1 N Fexp ( i) − FANN (n, E, K , di )) 2 ( ∑ N i=1
where: Fexp – experimental force, generated by FEM for the assumed real material parameters, FANN - force predicted by ANN, di – displacement, N - umber of sampling points. Genetic algorithm is used as optimization algorithm. The results for both cases I and II are presented in Figure 2. Evaluated minimum is found at n = 0.11, E = 384 GPa, K = 280 MPa for case I and n = 0.17, E = 397 GPa, K = 331 MPa for case II. The goal function is respectively φ = 34 µN2 and φ = 21 µN2. Figure 3 shows results of simulation of strain distribution in the sample, using the material parameters determined from the inverse analysis for the case I.
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Fig. 3. Strain distribution in the 2D axisymmetric sample for material parameters determined from the inverse analysis for the case I. 6. CONCLUSIONS Presented approach is useful technique for optimization in problems characterized by high goal function computing costs. The algorithm allows to decrease the number of timeconsuming FEM calculations. Presented problem of optimization of flow stress parameters for nanocoatings proved efficiency of the method. Good predictive capability of the trained ANN was confirmed. Low value of the goal function was obtained also in the inverse analysis, but the minimum is weak and problem of uniqueness of the solution exists.
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displacement [nm] Fig. 1. Results of the network test set for workhardening curve (n = 0.15, E = 370 GPa, K = 100 MPa).
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output of FEM simulation for case I ANN output for case I output of FEM simulation for case II ANN output for case II
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displacement [nm] Fig. 2. Results of inverse analysis for the case I and the case II.
Koker, R., Altincock, N. and Demir, A., (2007): Neural network based prediction of mechanical properties of particulate reinforced metal matrix composites using various training algorithms, Materials and Design, 28, 616–627. Kopernik, M., Szeliga, D., Stanisławczyk, A., Pietrzyk, M. and (2007): Review of application of numerical methods to indentification material model of hard nanocoating, IXth Conf. COMPLAS, (submitted). Kusiak., J., śmudzki, A., and DanielewskaTułecka, A. (2005): Optimization of materials processing using a hybrid technique based on artificial neural networks, Arch. Metall. Mater., 50, 609620. Szeliga D., Gawąd J., Pietrzyk M., Inverse Analysis for Identification of Rheological and Friction Models in Metal Forming, Comp. Meth. Appl. Mech. Eng., 195, 2006, 6778-6798.
