Abstract
The size effect in the small-scale indentation testing is studied for a functionally-graded material (FGM) whose shear elastic modulus varies according to the exponential law. Under the simplifying assumption of zero Poisson’s ratio, the asymptotic model of the indentation stiffness for an axisymmetric frictionless indenter is developed in the case when the contact radius is small compared to the inhomogeneity characteristic size. The so-called sample size effect is considered on the example of a simply supported FGM plate indented at the center of its top surface. A certain range of applicability of the first-order asymptotic models has been established by comparison with the approximate analytical solution available in the literature.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Abali BE, Völlmecke C, Woodward B, Kashtalyan M, Guz I, Müller WH (2014) Threedimensional elastic deformation of functionally graded isotropic plates under point loading. Composite Structures 118:367–376
Aizikovich SM, Alexandrov VM, Kalker JJ, Krenev LI, Trubchik IS (2002) Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties. International Journal of Solids and Structures 39(10):2745–2772
Ajzikovich SM, Aleksandrov VM (1986) Asymptotic solutions to contact problems of elasticity theory for half-space and half-plane inhomogeneous by depth (in Russ.). Izvestiya AN Armjanskoj SSR Mehanika 3:13–28
Albrecht HJ, Hannach T, Häse A, Juritza A, Müller K, Müller WH (2005) Nanoindentation: a suitable tool to determine local mechanical properties in microelectronic packages and materials? Archive of Applied Mechanics 74(11):728–738
Anstis GR, Chantikul P, Lawn BR, Marshall DB (1981) A critical evaluation of indentation techniques for measuring fracture toughness: I. Direct crack measurements. Journal of the American Ceramic Society 64(9):533–538
Argatov I (2010) Frictionless and adhesive nanoindentation: Asymptotic modeling of size effects. Mechanics of Materials 42(8):807–815
Argatov I, Sabina F (2017) A two-phase self-consistent model for the grid indentation testing of composite materials. International Journal of Engineering Science 121:52–59
Argatov I, Heß M, Popov VL (2018) The extension of the method of dimensionality reduction to layered elastic media. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 98(4):622–634
Borodich FM, Galanov BA (2008) Non-direct estimations of adhesive and elastic properties of materials by depth-sensing indentation. Proceedings: Mathematical, Physical and Engineering Sciences (The Royal Society) 464(2098):2759–2776
Borodich FM, Keer LM (2004) Contact problems and depth-sensing nanoindentation for frictionless and frictional boundary conditions. International Journal of Solids and Structures 41(9):2479–2499
Bulychev SI, Alekhin VP, Shorshorov MK, Ternovskij AP, Shnyrev GD (1975) Determination of young modulus by the hardness indentation diagram (in Russ. Zavodskaya Laboratoriya 41(9):1137–1140
Cheng L, Xia X, Yu W, Scriven LE, Gerberich WW (2000) Flat-punch indentation of viscoelastic material. Journal of Polymer Science Part B: Polymer Physics 38(1):10–22
Giannakopoulos AE, Suresh S (1997a) Indentation of solids with gradients in elastic properties: Part I. Point force. International Journal of Solids and Structures 34(19):2357–2392
Giannakopoulos AE, Suresh S (1997b) Indentation of solids with gradients in elastic properties: Part II. Axisymmetric indentors. International Journal of Solids and Structures 34(19):2393–2428
Gibson RF (2014) A review of recent research on nanoindentation of polymer composites and their constituents. Composites Science and Technology 105:51–65
Gouldstone A, Chollacoop N, Dao M, Li J, Minor AM, Shen YL (2007) Indentation across size scales and disciplines: Recent developments in experimentation and modeling. Acta Materialia 55(12):4015–4039
Gradshteyn IS, Ryzhik IM (1994) Tables of Integrals, Series and Products. Academic Press, New York
HeßM(2016) A simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials. International Journal of Engineering Science 104:20–33
Kashtalyan M (2004) Three-dimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of Mechanics - A/Solids 23(5):853–864
Koumi KE, Nelias D, Chaise T, Duval A (2014) Modeling of the contact between a rigid indenter and a heterogeneous viscoelastic material. Mechanics of Materials 77:28–42
Markworth AJ, Ramesh KS, Parks WP (1995) Modelling studies applied to functionally graded materials. Journal of Materials Science 30(9):2183–2193
Müller WH, Worrack H (2012) Analysis of nanoindentation experiments by means of rheological models. PAMM 12(1):293–294
Müller WH,Worrack H, Sterthaus J, Villain J,Wilden J, Juritza A (2009) How to extract continuum materials properties for (lead-free) solders from tensile tests and nanoindentation experiments. Microsystem Technologies 15(1):45–55
Müller WH, Worrack H, Sterthaus J, Wilden J (2009) Nanoindentation experiments at elevated temperatures for the determination of mechanical solder material properties. PAMM 9(1):719–720
Müller WH, Worrack H, Zapara M (2011) Analysis of nanoindentation experiments by means of atomic force microscopy. PAMM 11(1):413–414
Oliver WC, PharrGM(1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research 7(6):1564–1583
Plevako VP (1971) On the theory of elasticity of inhomogeneous media. Journal of Applied Mathematics and Mechanics 35(5):806–813
Suresh S, Giannakopoulos AE, Alcalá J (1997) Spherical indentation of compositionally graded materials: Theory and experiments. Acta Materialia 45(4):1307–1321
Villain J, Mueller WH, Haese A, Weippert C, Corradi U, Saeed U (2008) Determination of mechanical properties of small test volumes using nanoindentation - a critical review. In: 2008 2nd Electronics System-Integration Technology Conference, pp 155–162
Vorovich II, Aleksandrov VM, Babeshko VA (1974) Non-Classical Mixed Problems in the Theory of Elasticity (in Russ.). Nauka, Moscow
Weinberg K, Sterthaus J, Müller WH (2005) Determining material parameter of solder alloys by nanoindentation. PAMM 5(1):451–452
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Argatov, I. (2019). On the Size Effects in Indentation Testing of Elastic Functionally-graded Materials. In: Abali, B., Altenbach, H., dell'Isola, F., Eremeyev, V., Öchsner, A. (eds) New Achievements in Continuum Mechanics and Thermodynamics. Advanced Structured Materials, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-030-13307-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-13307-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-13306-1
Online ISBN: 978-3-030-13307-8
eBook Packages: EngineeringEngineering (R0)