Prediction of the bilinear stress-strain curve of engineering material by nanoindentation test
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1 Prediction of the bilinear stress-strain curve of engineering material by nanoindentation test T.S. Yang, T.H. Fang, C.T. Kawn, G.L. Ke, S.Y. Chang Institute of Mechanical & Electro-Mechanical Engineering, National Formosa University 6, Wunhau Rd., Yunlin, 63, Taiwan Tel.: + [886] Fax: + [886] tsyang@nfu.edu.tw Department of mechanical engineering, Nan Kai University of Technology Abstract Instrumented indentation is widely used to probe the elastic and plastic properties of engineering materials. FEM has been widely used for numerical simulation of indentation tests on bulk and film material in order to analyze its deformation response. This study proposed an improved technique to determine the stress-strain curve of bulk material. The finite element method (FEM) in conjunction with an abductive network is used to predict the stress-strain relationship of bilinear elastic-plastic material from the nanoindentation test s force-displacement curve. Keywords: Finite element method, abductive network, nanoindentation, stress-strain curve. Introduction Finite element method (FEM) has been widely used for numerical simulation of indentation tests on bulk material in order to analyze its deformation response and investigate the influence of indenter geometry, friction and material elastic and plastic properties. Pelletier et. al [] have investigated the influence of material bilinear elastic-plastic behaviour model for numerical simulation of nanoindentation testing of various bulk metals. The indenter and the specimen were treated as a revolution body in order to have threedimensional situation. The numerical simulation results of loads verse displacement compare reasonable well to experimental results of nanoindentation tests of pure metals as Fe, Ni, Ti and Cu. Pelletier [] used a comprehensive parametric study of 8 cases was conducted. They defined two dimensionless equations which link the parameters extracted from the experimental load displacement curve with material parameters, such as Young s modulus, yield stress and tangent modulus. The load displacement curves simulated by FEM and then experimental curves obtained by nanoindentation testing using a Berkovich tip. In order to verify the FEM simulation results of the mechanical parameters such as Young s modulus, yield stress and tangent modulus, the experimental data are compared with the results of the current simulation. The abductive network was then applied to synthesize the data sets obtained from the numerical simulation. After employing the predictive model can provide valuable references in prediction of the mechanical parameters after nanoindentation tests.. Finite element indentation analysis Figure (a) shows the FE-model of nanoindentation process simulated in DEFORM-D. An axisymmetric cone with half-included angle of 7.3 in which the conical indenter has the same area function as a Berkovich tip was used in this study. The indenter is constrained to move only in the thickness-direction of the bulk metal. Nanoindentation is performed under a precisely continuous measurement of the force (load) and the displacement (depth) during the loading and unloading process (Fig. (b)). Figure (c) presents the stress-strain curve with bilinear constitutive law of a bulk material. Note that E is Young s modulus; E T is tangent -33
2 modulus; Y is yielding strength. During the analyses, the conical indenter is assumed to be rigid. The bulk metal is assumed to be elastic-linear work-hardening. Because of symmetry only a half of the conical indenter and bulk material were taken for simulation. The nodes along the axis of rotation can move only along the y-axis and all the nodes on the bottom of the mesh are fixed. The meshes near the indenter needed to be very fine to be able to describe the deformation and stress gradient below indenter with sufficient accuracy. The interface between the indenter and the specimen was assumed to be frictionless. In this nanoindentation case, the bulk metal has a nm thickness and a, nm width. The model adopts - node tetrahedral elements for the bulk metal in two-dimensional simulation. The proper size of the meshes for the FE models is determined through convergence studies. The bulk metal is meshed about 787 elements and 956 nodes in this study. The indentation process is simulated both during loading and unloading. Fig. (a) FE-model of (b) load and the depth curve (c) stress-strain curve indentation process. To verify the FEM simulation results obtained by DEFORM-D software for the load and unload curve of bulk metal during the nanoindentation process, experimental data obtained by Pelletier et al. [] are compared with simulation results obtained by this study. The experimental conditions in Pelletier et al. [] are adopted as the simulation input parameters. Figures (a) and (b) show the comparison between the current simulation and the results of Pelletier et al. [] for loading and unloading curves of Fe and Ni metals. The results predicted by present simulation are in good agreement with FE simulation by Pelletier et al. [], but the experimental data show a stiffer response to the FE simulation results. This is attributed to the radius of tip caused by the wear of indenter []. Therefore, the DEFORM- D can effectively capture the loading and unloading curves in nanoindentation process. 3 Fe nm compared curve Fe Experimental curve Pelletier simulated curve 6 Ni nm com pared curve Ni Experimental curve Pelletier simulated curve Pelletier's FEM simulation : E = 5 G P a Y o = M P a E t= 6 G P a Pelletier's FEM simulation : E = G P a Y o = 5 3 M P a E t= 9 G P a Fig. (a) Comparison between the current (b) Comparison between the current simulation and results of Pelletier et. al [] simulation and results of Pelletier et. al [] for loading and unloading curve for loading and unloading curve (bulk material Fe and h = nm). (bulk material Ni and h = nm). -3
3 The effect of the nonlinear material behavior on the indentation response is examined for a bulk metal. The bilinear constitutive law of the bulk material is used by present simulation. To investigate the effects of process and material parameters of bulk metal, such as Young's modulus E, tangent modulus E T and yielding strength Y, on the loading and unloading curve of bulk metal. Numerical analysis was performed for each change in these values. Figures 3(a) and (b) show the effect of the bulk metal Young s modulus E on the loading and unloading curve of nanoindentaion process. Large values of film Young s modulus result in high values of indentation load for the same indentation depth. The slope of unloading curve increases as the bulk metal Young s modulus increases. Figures (a) and (b) show the effect of the yielding stress Y on the loading and unloading curve. Large values of yielding stress result in high values of the loading force for the same indentation depth. The slopes of unloading curves are almost the same as the yielding stress increases. Figures 5(a) and (b) show the effect of the bulk metal tangent modulus E T on the loading and unloading curve. Large values of tangent modulus result in high values of the loading force for the same indentation depth. The slopes of unloading curves are almost the same as the tangent modulus increases..3.3 Elastic modulus 7 Elastic modulus 56 Elastic modulus 7 Elastic modulus 56 Elastic modulus 3 Elastic modulus Fig. 3 (a) effect of the Young s modulus E (b) effect of the Young s modulus E Yield stress Yield stress 35 Yield stress 6 Yield stress Yield stress 35 Yield stress Fig. (a) effect of the yielding stress Y (b) effect of the yielding stress Y.8 E t 3.6 Et.8 E t E t 3.6 Et.8 E t 6 8 Fig. 5 (a) effect of the tangent modulus E T (b) effect of the tangent modulus E T -35
4 3. Abductive network indentation models Abductive networks have been used over the last decade in a wide range of engineering applications. The main advantage of abductive network is the ability to learn complex trends and generalize the behavior from a collection of discrete data sets. In the abductive network, a complex system can be decomposed into smaller, simpler subsystems grouped into several layers using polynomial functional nodes. The polynomial network proposed by Ivakhnenko [3] is a group method of data handing techniques. Theses nodes evaluate the limited number of inputs by a polynomial function and generate an output to serve as an input to subsequent nodes of the next layer. It consists of sigma (summation) units in the hidden layer and pi (product) units in the output layer. For more detailed explanation of these polynomial functional nodes, please refer to the paper of Ivakhnenko [3]. To build a complete abductive network, the first requirement is to train the database. The information given by the input and output parameters must be sufficient. A predictive square error criterion is then used to automatically determine an optimal structure. The yielding stress is varied between -6 MPa, whereas the other material s properties were selected by varying the Young s modulus, the tangent modulus and the indentation depth in the ranges of 7-3 GPa, 3.6- GPa and - nm, respectively. There are four variables of material properties, each of which was set at three levels. Therefore, 8 (3*3*3*3) combinations of parameters of material s property are constituted totally. The load displacement curves obtained by FEM software DEFORM-D. The curves corresponding to the loading part have been fitted using a polynomial law, P=Ah +Bh; and the unloading part can be mathematically described with a polynomial law, P=Ch +Dh+F. The coefficients of loading-unloading curves are investigated for different material parameters such as Young s modulus E, yield stress Y and tangent modulus E T, by finite element analysis. Base on the training database regarding the parameters of material s property and coefficient of polynomial law such as Y, E, E T, A, B, C, D, F and h, the abductive networks with a criterion of minimum square error can be developed for predicting the E, Y and E T under a suitable range of parameters such as A, B, C, D, F and h. The networks are built for prediction of the tangent modulus and yielding stress. Notably, The Young's modulus can be determined by indentation test. Figures 6(a) and (b) show the compasions between the FEM simulations and abductive networks predictions for Ti and Fe materials. The abductive networks predictions model agree well with the FEM simulations (DEFORM-D) results..6 Ti nm com pared curve Fe nm compared curve Ti Experimental curve Fe Experimental curve. AIM predict curve 3 AIM predict curve E = 3 G P a Y o = 6M P a E t= 3 6 G P a E = 5 G P a Y o = 6 M P a E t= 6 G P a Fig. 6(a) Comparison between FEM and abductive networks (AIM) prediction for Ti (b) Comparison between FEM and Abductive networks (AIM) prediction for Fe.. Prediction of stress-strain behavior by experimental data The loading-displacement curves can be obtained by nanoindentation tests. The aim of this study is the stress-strain behavior of a local region of a material can be directly extracted by the comparison of load-displacement behavior between the abductive network prediction -36
5 and an indentation test. The prediction model of abductive network can provide valuable references conveniently in prediction of the mechanical parameters such as E, Y and E T after nanoindentation tests. Figures 7(a) and (b) show the stress-strain behavior is directly extracted by the comparison of load-displacement behavior for Cu and Ti materials. The FEM simulation (DEFORM-D) and abductive networks (AIM) are adopted to predict the mechanical parameters (Y and E T ). The Young s modulus E can be obtained directly by nanoindentation test. The value of E is 98 GPa and GPa for Cu and Ti, repectively. The predictive values of Y and E T are 3 MPa and 5GPa by FEM simulation for Cu material. Moreover, the predictive values of Y and E T are 3 MPa and 6GPa by abductive networks for Cu material. The predictive values of Y and E T are 5 MPa and 3GPa by FEM simulation for Ti material. Moreover, the predictive values of Y and E T are MPa and 6GPa by abductive networks for Ti material. Cu material Nanotest curve (experiment) FEM simulation (DEFORM-D) Abductive networks prediction FEM simulation: E = 98 GPa, Y = 3 Mpa, Et = 5 GPa Abductive networks predition: E = 98 GPa, Y = 3 Mpa, Et = 6 GPa Ti material Nanotest curve (experiment) FEM simulation (DEFORM-D) Abductive networks prediction FEM simulation: E = GPa, Y = 5 Mpa, Et = 3 GPa Abductive networks predition: E = GPa, Y = Mpa, Et = 6 GPa Indentation Depth h (nm) Fig. 7(a) The prediction of E, Y and E T of Cu material. (b) The prediction of E, Y and E T of Ti material. 5. Conclusion This study applies the finite element method (FEM) in conjunction with an abductive network to predict stress-strain curve of bulk metal during the nanoindentation process. To verify the prediction of FEM simulation for loading and unloading process, the experimental data are compared with the results of current simulation. Loading-unloading curve are investigated for different material parameters, such as Young s modulus, yielding stress and tangent modulus of nanoidentation process, by finite element analysis. The abductive network is then utilized to synthesize the data sets obtained from numerical simulations, and the prediction model is established for predicting loading-displacement curve under a wide range of process parameters. Finally, the stress-strain behavior of a local region of a material is directly extracted by the comparison of load-displacement behavior between the abductive network prediction and an indentation test. 6. Acknowledgements The authors wish to thank the support from National Science Council under grants NSC- 97--E-5-6-MY. References. H. Pelletier, J. Krier, A. Cornet, P. Mille. Thin Solid Films., 379, pp H. Pelletier. Tribology International. 6, 39, pp A.G. Ivakhnenko. IEEE Trans. Syst. 97,, pp
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