0
Research Papers

Identification of Plastic Properties From Conical Indentation Using a Bayesian-Type Statistical Approach

[+] Author and Article Information
Yupeng Zhang

Department of Materials Science and
Engineering,
Texas A&M University,
College Station, TX 77843

Jeffrey D. Hart

Professor
Department of Statistics,
Texas A&M University,
College Station, TX 77843

Alan Needleman

Professor
Fellow ASME
Department of Materials Science
and Engineering,
Texas A&M University,
College Station, TX 77843

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 23, 2018; final manuscript received August 28, 2018; published online October 1, 2018. Editor: Yonggang Huang.

J. Appl. Mech 86(1), 011002 (Oct 01, 2018) (9 pages) Paper No: JAM-18-1487; doi: 10.1115/1.4041352 History: Received August 23, 2018; Revised August 28, 2018

The plastic properties that characterize the uniaxial stress–strain response of a plastically isotropic material are not uniquely related to the indentation force versus indentation depth response. We consider results for three sets of plastic material properties that give rise to essentially identical curves of indentation force versus indentation depth in conical indentation. The corresponding surface profiles after unloading are also calculated. These computed results are regarded as the “experimental” data. A simplified Bayesian-type statistical approach is used to identify the values of flow strength and strain hardening exponent for each of the three sets of material parameters. The effect of fluctuations (“noise”) superposed on the “experimental” data is also considered. We build the database for the Bayesian-type analysis using finite element calculations for a relatively coarse set of parameter values and use interpolation to refine the database. A good estimate of the uniaxial stress–strain response is obtained for each material both in the absence of fluctuations and in the presence of sufficiently small fluctuations. Since the indentation force versus indentation depth response for the three materials is nearly identical, the predicted uniaxial stress–strain response obtained using only surface profile data differs little from what is obtained using both indentation force versus indentation depth and surface profile data. The sensitivity of the representation of the predicted uniaxial stress–strain response to fluctuations increases with increasing strain hardening. We also explore the sensitivity of the predictions to the degree of database refinement.

FIGURES IN THIS ARTICLE
<>
Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Cheng, Y. T. , and Cheng, C. M. , 1999, “ Can Stress-Strain Relationships Be Obtained From Indentation Curves Using Conical and Pyramidal Indenters?,” J. Mater. Res., 14(9), pp. 3493–3496. [CrossRef]
Alkorta, J. , Martinez-Esnaola, J. M. , and Sevillano, J. G. , 2005, “ Absence of One-to-One Correspondence Between Elastoplastic Properties and Sharp-Indentation Load-Penetration Data,” J. Mater. Res., 20(2), pp. 432–437. [CrossRef]
Chen, X. , Ogasawara, N. , Zhao, M. , and Chiba, N. , 2007, “ On the Uniqueness of Measuring Elastoplastic Properties From Indentation: The Indistinguishable Mystical Materials,” J. Mech. Phys. Solids, 55(8), pp. 1618–1660. [CrossRef]
Dao, M. , Chollacoop, N. V. , Van Vliet, K. J. , Venkatesh, T. A. , and Suresh, S. , 2001, “ Computational Modeling of the Forward and Reverse Problems in Instrumented Sharp Indentation,” Acta Mater., 49(19), pp. 3899–3918. [CrossRef]
Ogasawara, N. , Chiba, N. , and Chen, X. , 2005, “ Representative Strain of Indentation Analysis,” J. Mater. Res., 20(8), pp. 2225–2234. [CrossRef]
Cheng, Y. T. , and Cheng, C. M. , 1998, “ Scaling Approach to Conical Indentation in Elastic-Plastic Solids With Work Hardening,” J. Appl. Phys., 84(3), pp. 1284–1291. [CrossRef]
Huber, N. , and Tsakmakis, C. , 1999, “ Determination of Constitutive Properties From Spherical Indentation Data Using Neural Networks—Part I: The Case of Pure Kinematic Hardening in Plasticity Laws,” J. Mech. Phys. Solids, 47, pp. 1569–1588. [CrossRef]
Huber, N. , and Tsakmakis, C. , 1999, “ Determination of Constitutive Properties From Spherical Indentation Data Using Neural Networks—Part II: Plasticity With Nonlinear Isotropic and Kinematic Hardening,” J. Mech. Phys. Solids, 47, pp. 1589–1607. [CrossRef]
Huber, N. , and Tyulyukovskiy, E. , 2004, “ A New Loading History for Identification of Viscoplastic Properties by Spherical Indentation,” J. Mater. Res., 19(1), pp. 101–113. [CrossRef]
Tyulyukovskiy, E. , and Huber, N. , 2006, “ Identification of Viscoplastic Material Parameters From Spherical Indentation Data—Part I: Neural Networks,” J. Mater. Res., 21(3), pp. 664–676. [CrossRef]
Klötzer, D. , Ullner, C. , Tyulyukovskiy, E. , and Huber, N. , 2006, “ Identification of Viscoplastic Material Parameters From Spherical Indentation Data—Part II: Experimental Validation of the Method,” J. Mater. Res., 21 (3), pp. 677–684. [CrossRef]
Wang, M. , Wu, J. , Zhan, X. , Guo, R. , Hui, Y. , and Fan, H. , 2016, “ On the Determination of the Anisotropic Plasticity of Metal Materials by Using Instrumented Indentation,” Mater. Des., 111, pp. 98–107. [CrossRef]
Wang, M. , Wu, J. , Hui, Y. , Zhang, Z. , Zhan, X. , and Guo, R. , 2017, “ Identification of Elastic-Plastic Properties of Metal Materials by Using the Residual Imprint of Spherical Indentation,” Mater. Sci. Eng.: A, 679, pp. 143–154. [CrossRef]
Mostafavi, M. , Collins, D. M. , Cai, B. , Bradley, R. , Atwood, R. C. , Reinhard, C. , Jiang, X. , Galano, M. , Lee, P. D. , and Marrow, T. J. , 2015, “ Yield Behavior beneath Hardness Indentations in Ductile Metals, Measured by Three-Dimensional Computed X-Ray Tomography and Digital Volume Correlation,” Acta Mater., 82, pp. 468–482. [CrossRef]
Mostafavi, M. , Bradley, R. , Armstrong, D. E. J. , and Marrow, T. J. , 2016, “ Quantifying Yield Behaviour in Metals by X-Ray Nanotomography,” Sci. Reports, 6, p. 34346. [CrossRef]
Babuska, I. , Sawlan, Z. , Scavino, M. , Szabó, B. , and Tempone, R. , 2016, “ Bayesian Inference and Model Comparison for Metallic Fatigue Data,” Comput. Methods Appl. Mech. Eng., 304, pp. 171–196. [CrossRef]
Rovinelli, A. , Sangid, M. D. , Proudhon, H. , Guilhem, Y. , Lebensohn, R. A. , and Ludwig, W. , 2018, “ Predicting the 3D Fatigue Crack Growth Rate of Small Cracks Using Multimodal Data Via Bayesian Networks: In-Situ Experiments and Crystal Plasticity Simulations,” J. Mech. Phys. Solids, 115, pp. 208–229. [CrossRef]
Madireddy, S. , Sista, B. , and Vemaganti, K. , 2015, “ A Bayesian Approach to Selecting Hyperelastic Constitutive Models of Soft Tissue,” Comput. Methods Appl. Mech. Eng., 291, pp. 102–122. [CrossRef]
Asaadi, E. , and Heyns, P. S. , 2017, “ A Computational Framework for Bayesian Inference in Plasticity Models Characterisation,” Comput. Methods Appl. Mech. Eng., 321, pp. 455–481. [CrossRef]
Rappel, H. , Beex, L. A. , and Bordas, S. P. , 2018, “ Bayesian Inference to Identify Parameters in Viscoelasticity,” Mech. Time-Depend. Mater., 22(2), pp. 221–258. [CrossRef]
Worthen, J. , Stadler, G. , Petra, N. , Gurnis, M. , and Ghattas, O. , 2014, “ Towards Adjoint-Based Inversion for Rheological Parameters in Nonlinear Viscous Mantle Flow,” Phys. Earth Planet. Inter., 234, pp. 23–34. [CrossRef]
Prudencio, E. E. , Bauman, P. T. , Williams, S. V. , Faghihi, D. , Ravi-Chandar, K. , and Oden, J. T. , 2013, “ A Dynamic Data Driven Application System for Real-Time Monitoring of Stochastic Damage,” Procedia Comp. Sci., 18, pp. 2056–2065. [CrossRef]
Prudencio, E. E. , Bauman, P. T. , Faghihi, D. , Ravi-Chandar, K. , and Oden, J. T. , 2015, “ A Computational Framework for Dynamic Data-Driven Material Damage Control, Based on Bayesian Inference and Model Selection,” Int. J. Numer. Methods Eng., 102(3–4), pp. 379–403. [CrossRef]
Vigliotti, A. , Csányi, G. , and Deshpande, V. S. , 2018, “ Bayesian Inference of the Spatial Distributions of Material Properties,” J. Mech. Phys. Solids, 118, pp. 74–97. [CrossRef]
Fernandez-Zelaia, P. , Joseph, V. R. , Kalidindi, S. R. , and Melkote, S. N. , 2018, “ Estimating Mechanical Properties From Spherical Indentation Using Bayesian Approaches,” Mater. Des., 147, pp. 92–105. [CrossRef]
Needleman, A. , Tvergaard, V. , and Van der Giessen, E. , 2015, “ Indentation of Elastically Soft and Plastically Compressible Solids,” Acta Mech. Sin., 31(4), pp. 473–480. [CrossRef]
Hoff, P. D. , 2009, A First Course in Bayesian Statistical Methods, Springer Science & Business Media, New York, pp. 67–87.
Matlab, 2016, MATLAB Release 2016a, Function Normrnd, The MathWorks, Natick, MA.

Figures

Grahic Jump Location
Fig. 1

Sketch of the indentation configuration analyzed

Grahic Jump Location
Fig. 2

Normalized indentation force F/(Ehref2) versus normalized indentation depth h/href during loading and unloading for three input materials. The symbols are the points used as the “experimental” values in the statistical calculations (finput). ◻:m1, Δ:m2, ∇:m3. The parameters for the three input materials are given in Table 1.

Grahic Jump Location
Fig. 3

Uniaxial stress–strain curves for three input materials. The parameters for the three input materials are available in Table 1.

Grahic Jump Location
Fig. 4

Normalized surface profiles near the indenter after unloading for the three input materials. The symbols are the points used as the “experimental” values in the statistical calculations (sinput). ◻:m1, Δ:m2, ∇:m3. The parameters for the three input materials are given in Table 1.

Grahic Jump Location
Fig. 5

Uniaxial stress–strain curves showing the effect of the choice of database: G0 (dashed lines); G2 (dash dot lines); and Gf (dash dot dot lines). The stress–strain curves are for the predicted combinations of Y/E and N with the highest posterior probability for each database grid. The solid lines show the uniaxial stress–strain curves for the three input materials in Table 1. For material m1, only the dash line does not overlap with the solid line. For material m2, dash dot line and dash dot dot line overlap.

Grahic Jump Location
Fig. 6

Uniaxial stress–strain curves comparing the predictions using only indentation force versus indentation depth data (dash dot lines) and both indentation force versus indentation depth and surface profile data (dashed lines). The stress–strain curves are for the predicted combinations of Y/E and N with the highest posterior probability using database grid G2. The solid lines show the uniaxial stress–strain curves for the three input materials in Table 1. For material m1, the solid line and the dashed line overlap.

Grahic Jump Location
Fig. 7

Posterior probability distribution p for various combinations of Y/E and N for the material m1 using database grids (a) G2 and (b) Gf

Grahic Jump Location
Fig. 8

Uniaxial stress–strain curves showing the effect of the values of the noise measures ηs and ηf in Eq. (23): ηs = ηf = 0.001 (dashed lines); ηs = 0.001 and ηf = 0.01 (dash dot dot lines); and ηs = ηf = 0.01 (dash dot lines). The stress–strain curves are for the average predicted combinations of Y/E and N with highest posterior probability using database grid G2. The solid lines show uniaxial stress–strain curves for the three input materials in Table 1. For material m1, only the dash dot line does not overlap with the solid line. For materials m2 and m3, the dashed line and the dash dot dot line overlap.

Grahic Jump Location
Fig. 9

One realization of posterior probability distribution p for various combinations of Y/E and N with values of the noise measures ηs = ηf = 0.001 using database grid G2: (a) for material m1 and (b) for material m2

Grahic Jump Location
Fig. 10

One realization of posterior probability distribution p for various combinations of Y/E and N with values of the noise measures ηs = ηf = 0.01 using database grid G2. The crosses mark the third and fourth highest probability values: (a) for material m1 and (b) for material m2.

Grahic Jump Location
Fig. 11

Uniaxial stress–strain curves obtained using the predicted values of Y/E, N that correspond to: the highest posterior probability value (dashed lines); the third highest posterior probability value (dash dot lines); and the fourth highest posterior probability value (dash dot dot lines). For materials m1 and m2, the third highest and fourth highest posterior probability values are marked by an x in Fig. 10 where ηs = ηf = 0.01. Database grid G2 was used. The solid lines show the stress–strain curves for the three input materials in Table 1. For material m1, the dash dot dot line overlaps with the solid line. For material m2, the dashed line and dash dot line overlap.

Grahic Jump Location
Fig. 12

The variation of the average of predicted posterior probability with the value of surface profile data noise measure ηs for material m3 using database grid G1 (red lines) and database grid G2 (blue lines). The solid lines show the average of the highest posterior probability and the dashed lines show the average of the second highest posterior probability. The bars show the corresponding standard deviations.

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In