Indentation of elastically anisotropic half-spaces by cones and parabolae of revolution

Swadener, J. G.; Pharr, G. M.

Affiliations

¹Metals and Ceramics Division, Oak Ridge National Laboratory, PO Box 2008, MS-6116, Oak Ridge, Tennessee 37831-61 16, USA

²Department of Materials Science and Engineering, The University of Tennessee, Knoxville, Tennessee 37996-2200, USA

Abstract

Indentation of ceramic materials with smooth indenters such as parabolae of revolution and spheres can be conducted in the elastic regime to relatively high loads. Ceramic single crystals thus provide excellent calibration media for load- and depth-sensing indentation testing; however, they are generally anisotropic and a complete elastic analysis is cumbersome. This study presents a simplified procedure for the determination of the stiffness of contact for the indentation of an anisotropic half-space by a rigid frictionless parabola of revolution which, to first order, approximates spherical indentation. Using a similar approach, a new procedure is developed for analysing conical indentation of anisotropic elastic media. For both indenter shapes, the contact is found to be elliptical, and equations are determined for the size, shape and orientation of the ellipse and the indentation modulus.

Links

DOI: 10.1080/01418610108214314

WoS: 000167052500014

History

Received: 1999-09-22

Accepted: 2000-05-15

Cited Works (1)

Year	Entry
1992	Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res. 1992;7(6):1564-1583.

Cited By (30)

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2011	Oyen ML. Mechanics of indentation. In: Oyen ML, ed. Handbook of Nanoindentation With Biological Applications. Singapore: Pan Stanford Publishing Pte. Ltd; 2011:123-152.
2010	Reisinger AG, Pahr DH, Zysset PK. Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods. Biomech Model Mechanobiol. October 2010;9(5):499-510.
2011	Reisinger AG, Pahr DH, Zysset PK. Elastic anisotropy of bone lamellae as a function of fibril orientation pattern. Biomech Model Mechanobiol. February 2011;10(1):67-77.
2010	Wolfram U, Wilke H-J, Zysset PK. Rehydration of vertebral trabecular bone: influences on its anisotropy, its stiffness and the indentation work with a view to age, gender and vertebral level. Bone. February 2010;46(2):348-354.
2016	Mirzaali MJ, Schwiedrzik JJ, Thaiwichai S, Best JP, Michler J, Zysset PK, Wolfram U. Mechanical properties of cortical bone and their relationships with age, gender, composition and microindentation properties in the elderly. Bone. December 2016;93:196-211.
2012	Spiesz EM, Roschger P, Zysset PK. Influence of mineralization and microporosity on tissue elasticity: experimental and numerical investigation on mineralized turkey leg tendons. Calcif Tiss Int. 2012;90(4):319-329.
2009	Lukes J, Otahal S. Nanoindentation of porcine bone lamellae. Comput Methods Biomech Biomed Eng. 2009;12(suppl 1):177-178.
2016	Schwiedrzik J, Gross T, Bina M, Pretterklieber M, Zysset P, Pahr D. Experimental validation of a nonlinear μFE model based on cohesive-frictional plasticity for trabecular bone. Int J Num Meth Biomed Eng. April 2016;32(4):e02739.
2003	Hengsberger S, Enstroem J, Peyrin F, Zysset P. How is the indentation modulus of bone tissue related to its macroscopic elastic response? a validation study. J Biomech. October 2003;36(10):1503-1509.
2009	Franzoso G, Zysset PK. Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation. J Biomech Eng. February 2009;131(2):021001.
2010	Carnelli D, Gastaldi D, Sassi V, Contro R, Ortiz C, Vena P. A finite element model for direction-dependent mechanical response to nanoindentation of cortical bone allowing for anisotropic post-yield behavior of the tissue. J Biomech Eng. August 2010;132(8):081008.
2001	Swadener JG, Rho J, Pharr GM. Effects of anisotropy on elastic moduli measured by nanoindentation in human tibial cortical bone. J Biomed Mater Res. October 2001;57(1):108-112.
2011	Reisinger AG, Pahr DH, Zysset PK. Principal stiffness orientation and degree of anisotropy of human osteons based on nanoindentation in three distinct planes. J Mech Behav Biomed Mater. November 2011;4(8):2113-2127.
2012	Spiesz EM, Roschger P, Zysset PK. Elastic anisotropy of uniaxial mineralized collagen fibers measured using two-directional indentation: effects of hydration state and indentation depth. J Mech Behav Biomed Mater. August 2012;12:20-28.
2019	Asgari M, Abi-Rafeh J, Hendy GN, Pasini D. Material anisotropy and elasticity of cortical and trabecular bone in the adult mouse femur via AFM indentation. J Mech Behav Biomed Mater. May 2019;93:81-92.
2002	Fan Z, Swadener JG, Rho JY, Roy ME, Pharr GM. Anisotropic properties of human tibial cortical bone as measured by nanoindentation. J Orthop Res. July 2002;20(4):806-810.
2005	Gupta HS, Schratter S, Tesch W, Roschger P, Berzlanovich A, Schoeberl T, Klaushofer K, Fratzl P. Two different correlations between nanoindentation modulus and mineral content in the bone–cartilage interface. J Struct Biol. February 2005;149(2):138-148.
2009	Zysset PK. Indentation of bone tissue: a short review. Osteoporos Int. June 2009;20(6):1049-1055.
2017	Wahlquist JA. Zonal Articular Cartilage Possesses Complex Mechanical Behavior Spanning Multiple Length Scales: Dependence on Chemical Heterogeneity, Anisotropy, and Microstructure [PhD thesis]. Boulder, CO: University of Colorado; 2017.
2011	Donaldson FE. On Incorporating Bone Microstructure in Macro-Finite-Element Models [PhD thesis]. Edinburgh, UK: University of Edinburgh; March 2011.
2002	Hengsberger S. Mechanical Characterization of Bone from the Tissue Down to the Lamellar Level by Means of Nanoindentation [PhD thesis]. Lausanne, Switzerland: École Polytechnique Fédérale de Lausanne; 2002.
2006	Garcia D. Elastic Plastic Damage Laws for Cortical Bone [PhD thesis]. Lausanne, Switzerland: École Polytechnique Fédérale de Lausanne; 2006.
2014	Carretta R. Post-Yield Mechanics and Material Composition of Single Trabeculae: A Combined Experimental and Modelling Approach [PhD thesis]. Swiss Federal Institute of Technology Zürich; 2014.
2019	Groetsch A. Micro- and Nanomechanics of Mineralised Collagen Fibre Elasto-Plasticity [PhD thesis]. Ebinburgh, Scotland: Heriot-Watt University; November 2019.
2010	Ortega JA. Microporomechanical Modeling of Shale [PhD thesis]. Cambridge, MA: Massachusetts Institute of Technology; February 2010.
2011	Wu Z. Measures of Bone Quality and Their Relationship to Bone Mechanical Properties [PhD thesis]. University of Notre Dame; December 2011.
2011	Reisinger AG. Modeling and Validation of Multiscale Lamellar Bone Elasticity [PhD thesis]. Vienna University of Technology; November 2011.
2011	Spiesz EM. Experimental and Computational Micromechanics of Mineralized Tendon and Bone [PhD thesis]. Vienna University of Technology; October 2011.
2014	Schwiedrzik JJ. Experimental, Theoretical and Numerical Investigation of the Nonlinear Micromechanical Properties of Bone [PhD thesis]. Universität Bern; 2014.
2011	Wolfram U. Mechanical Multiscale Characterisation of Vertebral Trabecular Bone for the Prediction of Vertebral Fracture Risk [PhD thesis]. University of Ulm; 2011.