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TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 63/67 pp393-399 Volume 13, Number S1, October 2008
Comparative Study of Reverse Algorithms via Artificial Neural Networks Based on Simulated Indentation Tests*
Somsak Swaddiwudhipong**, Edy Harsono, Liu Zishun†
Department of Civil Engineering, National University of Singapore, Singapore 119260; † Institute of High Performance Computing, Singapore 138632
Abstract: The advances in the instrumented indentation equipments and the need to assess the properties
of materials of small volume such as those constitute the micro-electro-mechanical devices, micro-electronic
packages, and thin films have propelled the interest in material characterization via indentation tests. The
load-displacement curves and their characteristics, namely, the curvature of the loading path, C, and the ra-
tio of the remaining and total work done, WR / WT, can be conveniently obtained from finite element simula-
tions for various elasto-plastic material properties. The paper reports the comparative study on two reverse
neural networks algorithms involving several combinations of databases established from the results ob-
tained from simulated indentation tests. The performance of each set of results is analyzed and the most
appropriate algorithm identified and reported. The approach with the selected neural networks model has
great potential in practical applications on the characterization of a small volume of materials.
Key words: artificial neural networks; finite element simulation; friction; least square support vector
machines; material characterization; indentation tests
Introduction
The urgent needs for material characterization on small volumes, such as thin films, have motivated the re-search works in this area of study. Traditional testing methods are not suitable because a “bulk” quantity of the materials may not be available and the mechanical properties of materials at small scale are usually differ-ent from those of bulk materials. It is not only the hardness of the indented material[1] that is of interest to material scientists. Indentation tests can also be used to calculate other mechanical properties of materials such
as Young’s modulus, E, yield strength, Y, and strain hardening exponent, n. Methods and algorithms to ex-tract these mechanical properties from indentation curves of elasto-plastic material obeying power law strain hardening have been one of the main focus re-search interest on instrumented indentation. The ex-traction of elasto-plastic material properties (E, Y, n) from the indentation curves has been considered by many researchers[2-8]. Dimensional analysis has been used to propose the universal functions which relate characteristics of indentation curves and parameters of material properties[9,10]. By adopting these functions, forward and reverse analyses have been proposed to extract material properties from indentation curves.
Closed form solutions for indentation curve are not readily available due to the complexity of contact problem as well as highly nonlinearity of materials in-volved in indentation tests. Artificial neural network
Received: 2008-05-30
* Supported by the Singapore Ministry of Education’s ACRF Tier 1Funds (No. R-264-000-186-112)
** To whom correspondence should be addressed. E-mail: [email protected]; Tel: 65-65162173
Tsinghua Science and Technology, October 2008, 13(S1): 393-399 394
(ANN) and support vector machine (SVM) are gaining popularity in the field of engineering as their capa-bilities to mimic the method of computation adopted by human brain. They do not require complicated mathematical model as required in conventional meth-ods. Neural network models have been widely applied to solve any reverse analyses involving complicated functions as those of indentation tests. These models have been used to characterize the mechanical proper-ties of materials from indentation curves by several researchers[11-15].
Comparative study on application of neural network models in material characterization using simulated dual indentation tests has been carried out in the pre-sent study. Simulated indentation tests encompassing most of metallic material properties were conducted via extensive 2D finite element analysis. The predicted elasto-plastic material properties obtained from the neural network models are in good agreement with the expected prescribed values.
1 Finite Element Analysis
Finite element simulations of indentation tests have been carried out for 640 different combinations of elasto-plastic properties for three different conical in-denter tips of 50°, 60°, and 70.3°. The conical indenter is modelled as a rigid body while the target material is treated as a deformable body using more than 10 000 axisymmetric quadrilateral elements in each model. A finer mesh is adopted near the contact region and the mesh is gradually coarser further away from the con-tact region. The convergence of the mesh is verified through extensive convergence studies and the mesh used in the analysis is also tested to ensure that the re-sults are insensitive to far-field effect. For ease of ref-erence, conical indenters with half-angles of 50°, 60°, and 70.3° are designated as C50, C60, and C70, respec-tively. A typical finite element model is illustrated in Fig. 1.
In this study, the response of a wide practical range of elasto-plastic materials obeying power law strain-hardening under different conical indenter tips has been simulated. Figure 2 shows a typical load-displacement curve obtained from indentation simulation.
Rigid cone indenter
50°, 60°, 70.3°Indented materialO
A B
C
Fig. 1 Finite element mesh for conical indentation
Fig. 2 A typical load-displacement of sharp indentation curve
2 Characteristics of Indentation Curves
The reverse analysis based on results from just one in-denter is not sufficient to provide a unique set of solu-tions for E*/Y and n and the P/h curves from at least dual indenters of different geometries are required[8,16]. From dimensional analysis[9,10], it can be shown that the relationships between load-displacement parame-ters and the elasto-plastic material properties are given by
1,* ,C E n
Y Y (1)
2,* ,R
T
W E nW Y
(2)
1, 1
13, 12
21, 2
* ,* ,
* ,
E nC EY n
EC YnY
(3)
Somsak Swaddiwudhipong et al Comparative Study of Reverse Algorithms via Artificial Neural ... 395
where indicates the type and geometry of the in-denters considered, C is the curvature of the loading curve, WT and WU are the areas under the loading and unloading curves. Indentation parameters adopted for dual indenters are indentified for each of the indenter tips denoted by the subscript 1 and 2.
Extensive finite simulations covering the values of E*/Y from 10 to 1500 and n from 0.025 to 0.6 have
been carried out. The surfaces denoted by the functions 1 and 2 shown in Figs. 3-5 illustrate the varia-
tions of C/Y and WR/WT with respect to E*/Y and n for different indenter tips. The second subscript indicates the angle of indenter tip used.
The surfaces describing the ratio of C/Y for different combinations of two indenter tips are presented in Figs. 6-8.
(a) (b)
Fig. 3 Variation of (a) C/Y ( 15) and (b) WR /WT ( 25) based on conical indenter of 50°
(a) (b) Fig. 4 Variation of (a) C/Y ( 16) and (b) WR /WT ( 26) based on conical indenter of 60°
(a) (b) Fig. 5 Variation of (a) C/Y ( 17) and (b) WR /WT ( 27) based on conical indenter of 70.3°
Tsinghua Science and Technology, October 2008, 13(S1): 393-399 396
Fig. 6 Variation of C50 /C60 ( 3,65)
Fig. 7 Variation of C70 /C60 ( 3,76)
Fig. 8 Variation of C70 /C50 ( 3,75)
3 Construction of Neural Network Models
In the present study, artificial neural network (ANN) and support vector machine (SVM) models have been established to carry out mapping of the characteristics of
the indentation curves back to the material properties. Neural network can be established without knowing how the output functionally depends on the input. The surfaces described by functions, 1, 2, and 3 as shown in Figs. 3-8 are used to serve as the training and validating data sets for ANN and SVM models. More details on the establishment of neural network models are given in Refs. [14] and [15].
Two reverse analysis algorithms based on ANN and SVM models have been constructed to allow recovery of elasto-plastic material properties from indentation curves. Flowcharts illustrating the ANN and the SVMs models are shown in Figs. 9 and 10.
Fig. 9 Flowchart for ANN
Fig. 10 Flowchart for SVMs
4 Numerical Results
Simulated indentation tests based on four material properties reported in earlier study[8] were conducted for different conical indenter tip angles. The quantities C/Y and WR /WT have been identified from indentation curves and their values depicted in Table 1.
Indentation parameters obtained from four material properties are fed into the tuned neural network and sup-port vector machine models. The material properties predicted by the proposed reverse analysis models are shown in Tables 2 and 3 for ANN and SVMs.
Somsak Swaddiwudhipong et al Comparative Study of Reverse Algorithms via Artificial Neural ... 397
Table 1 Indentation parameters obtained from forward analyses
Material properties Indentation parameters from forward analyses C50 C60 C70 Materials
E* / GPa Y / MPa n C/Y WR /WT C/Y WR /WT C/Y WR /WT
AL6061 70.2 284 0.080 21.22 0.958 44.27 0.939 101.40 0.906 AL7075 73.4 500 0.122 21.78 0.926 41.65 0.896 91.53 0.848
Steel 194.3 500 0.100 25.10 0.969 50.65 0.956 117.62 0.933 Iron 170.8 300 0.250 44.36 0.960 84.03 0.945 185.17 0.922
Table 2 Comparison identified material properties obtained from ANN algorithm
Identified elastic-plastic material properties for ANN algorithm
E* (% Error)
Y (% Error)
n (% Error)
Materials
C50-C60 C60-C70 C50-C70 C50-C60 C60-C70 C50-C70 C50-C60 C60-C70 C50-C70
AL6061 70.16 ( 0.06)
70.21 (0.01)
70.37 (0.24)
283.8 ( 0.07)
284.0 (0.00)
284.0 (0.00)
0.0791 ( 1.13)
0.0799 ( 0.13)
0.0790 ( 1.25)
AL7075 74.42 (1.39)
73.54 (0.19)
73.04 ( 0.49)
500.3 (0.06)
499.5 ( 0.10)
500.7 (0.14)
0.1292 (5.90)
0.1227 (0.57)
0.1270 (4.10)
Steel 201.32 (3.61)
194.74 (0.23)
197.45 (1.62)
499.4 ( 0.12)
499.4 ( 0.12)
499.4 ( 0.12)
0.1034 (3.40)
0.0999 ( 0.10)
0.1039 (3.90)
Iron 172.35 (0.91)
171.28 (0.28)
171.67 (0.51)
300.0 (0.00)
300.4 (0.13)
299.7 ( 0.10)
0.2525 (1.00)
0.2498 ( 0.08)
0.2506 (0.24)
Table 3 Comparison identified material properties obtained from SVM algorithm
Identified elastic-plastic material properties for SVM algorithm E*
(% Error) Y
(% Error) n
(% Error) Materials
C50-C60 C60-C70 C50-C70 C50-C60 C60-C70 C50-C70 C50-C60 C60-C70 C50-C70
AL6061 68.01
( 3.12) 70.03
( 0.24) 70.52 (0.46)
265.7 ( 6.44)
284.0 (0.00)
283.9 ( 0.04)
0.0801 (0.13)
0.0812 (1.50)
0.0791 ( 1.13)
AL7075 76.84 (4.69)
73.65 (0.34)
73.97 (0.78)
499.5 ( 0.10)
499.5 ( 0.10)
500.1 (0.02)
0.1279 (4.84)
0.1217 ( 0.25)
0.1292 (5.90)
Steel 193.62 ( 0.35)
195.11 (0.42)
203.26 (4.61)
499.5 ( 0.10)
499.5 ( 0.10)
499.6 ( 0.08)
0.1046 (4.60)
0.0991 ( 0.90)
0.1048 (4.80)
Iron 173.53 (1.60)
170.79 ( 0.01)
172.84 (1.19)
300.3 (0.10)
300.3 (0.10)
299.9 (-0.03)
0.2512 (0.48)
0.2503 (0.12)
0.2512 (0.48)
It can be observed that the proposed reverse analysis algorithms based on database of several combinations of conical indenter tips predict the elasto-plastic mate-rial properties accurately. The predicted material prop-erties based on ANN are marginally better than those of SVMs. However, the SVMs are more structured and less computational effort is required when compared to ANN.
The best predicted material properties are obtained from reverse analysis algorithms derived based on datasets of conical indenter tips of 60° and 70.3°. The tuned neural network models created from datasets of
sharper indenter tip of 50° with either of 60° or 70.3° half-angle, produce results with somewhat more deviations than the former combination especially for the values of n. Simulated indentation tests carried out by using the sharpest indenter tip (50°) introduce more friction affecting the indentation curves. It is observed that it is harder to achieve smooth indentation curves for the sharpest indenter tip of 50° in this study. Ele-ments near the contact region also suffer greater distor-tion for the same indentation depth. Indentation pa-rameters obtained from the sharpest indenter tip are usually less accurate than those of 60° and 70.3°.
Tsinghua Science and Technology, October 2008, 13(S1): 393-399 398
5 Sensitivity Study
The sensitivity of the predicted elasto-plastic material properties to the perturbations of the input variables has been investigated in detail. Based on the Monte Carlo method, all inputs for three combinations conical indenter tip datasets are perturbed simultaneously. The perturbation values of 1%, 1%, 0.5%, and 0.5% are adopted for 1 2 1/ , / , /R TC Y C Y W W , and 2/R TW W ,
respectively. The sensitivity of all combinations of datasets has been carried out. Only combination of datasets from conical indenter tips of 60° and 70.3° for each algorithm is reported herein. Figures 11-13 show the sensitivity of the proposed reverse analysis algo-rithms for material characterization via instrumented indentation tests. It can be observed that E* is more sensitive to the perturbation of the characteristic values of indentation curves while the value of Y and n are
(a) SVMs (b) ANN algorithm
Fig. 11 Sensitivity of E* for the proposed neural networks
(a) SVMs (b) ANN algorithm
Fig. 12 Sensitivity of Y for the proposed neural networks
(a) SVMs (b) ANN algorithm
Fig. 13 Sensitivity of n for the proposed neural networks
Somsak Swaddiwudhipong et al Comparative Study of Reverse Algorithms via Artificial Neural … 399
better behaved. For both algorithms, the largest devia-tions of the values of E* occur in the region of high values of E*/Y and n close to 0, implying materials with a short elastic regime and very flat plastic behav-ior, which are less common practically. Material prop-erties obtained from ANN algorithm are more sensitive to the perturbation compared to those from SVMs.
6 Conclusions
Neural network models based on dual indenters con-cept are constructed for material characterization via instrumented indentation tests. From the present com-parative study, it can be concluded that the prediction of elasto-plastic material properties is more accurate if combination of datasets of conical indenter tips of 60° and 70.3°, where elements near these indenter tips are less distorted and less friction involved, are adopted. The results obtained from ANN model are more accu-rate compared to those from SVMs. However, SVMs require less computationally efforts to calibrate the network as compared to ANN models. Generally speaking, these two proposed algorithms are able to predict the mechanical properties of materials rather accurately. The proposed neural network models can be extended to other applications in engineering and natural systems.
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