Research Papers

Conical Indentation of a Viscoelastic Sphere

[+] Author and Article Information
J. K. Phadikar

Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19716

T. A. Bogetti

U.S. Army Research Laboratory,
Aberdeen Proving Ground, MD 21001

V. N. Kaliakin

Department of Civil and Environmental Engineering,
University of Delaware,
Newark, DE 19716

A. M. Karlsson

Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19716;
Fenn College of Engineering,
Cleveland State University,
Cleveland, OH 44115-2214
e-mail: a.karlsson@csuohio.edu

“Small” is relative to the indentation depth; that is, the indentation depth affects the overall behavior of the sphere and cannot be considered to be local around the indentation.

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received January 10, 2013; final manuscript received April 21, 2013; published online June 10, 2013. Assoc. Editor: Georges Cailletaud.

J. Eng. Mater. Technol 135(4), 041001 (Jun 10, 2013) (5 pages) Paper No: MATS-13-1008; doi: 10.1115/1.4024395 History: Received January 10, 2013; Revised April 21, 2013

Instrumented indentation is commonly used for determining mechanical properties of a range of materials, including viscoelastic materials. However, most—if not all—studies are limited to a flat substrate being indented by various shaped indenters (e.g., conical or spherical). This work investigates the possibility of extending instrumented indentation to nonflat viscoelastic substrates. In particular, conical indentation of a sphere is investigated where a semi-analytical approach based on “the method of functional equations” has been developed to obtain the force–displacement relationship. To verify the accuracy of the proposed methodology selected numerical experiments have been performed and good agreement was obtained. Since it takes significantly less time to obtain force–displacement relationships using the proposed method compared to conducting full finite element simulations, the proposed method is an efficient substitute of the finite element method in determining material properties of viscoelatic spherical particles using indentation testing.

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Johnson, K. L., 1987, Contact Mechanics, Cambridge University Press, Cambridge, UK.
Cheng, Y. T., and Cheng, C. M., 2004, “Scaling, Dimensional Analysis, and Indentation Measurements,” Mater. Sci. Eng. R, 44, pp. 91–149. [CrossRef]
Kucharski, S., and Mróz, Z., 2007, “Identification of Yield Stress and Plastic Hardening Parameters From a Spherical Indentation Test,” Int. J. Mech. Sci., 49, pp. 1238–1250. [CrossRef]
Yan, J., Karlsson, A. M., and Chen, X., 2007, “Determining Plastic Properties of a Material With Residual Stress by Using Conical Indentation,” Int. J. Solids Struct., 44, pp. 3720–3737. [CrossRef]
Phadikar, J. K., Bogetti, T. A., and Karlsson, A. M., 2012, “On Establishing Elastic–Plastic Properties of a Sphere by Indentation Testing,” Int. J. Solids Struct., 49, pp. 1961–1972. [CrossRef]
Kristiansen, H., Shen, Y., and Liu, J., 2001, “Characterisation of Mechanical Properties of Metal-Coated Polymer Spheres for Anisotropic Conductive Adhesive,” First International IEEE Conference on Polymers and Adhesives in Microelectronics and Photonics (POLYTRONIC 2001), Potsdam, Germany, October 21–24, pp. 344–348. [CrossRef]
Kwon, W. S., and Paik, K. W., 2006, “Experimental Analysis of Mechanical and Electrical Characteristics of Metal-Coated Conductive Spheres for Anisotropic Conductive Adhesives (ACAs) Interconnection,” IEEE Trans. Compon. Packag. Technol., 29, pp. 528–534. [CrossRef]
Misawa, H., Koshioka, M., Sasaki, K., Kitamura, N., and Masuhara, H., 1991, “Three Dimensional Optical Trapping and Laser Ablation of a Single Polymer Latex Particle in Water,” J. Appl. Phys., 70, pp. 3829–3836. [CrossRef]
Tamai, H., Hasegawa, M., and Suzawa, T., 1989, “Surface Characterization of Hydrophilic Functional Polymer Latex Particles,” J. Appl. Polym. Sci., 38, pp. 403–412. [CrossRef]
Dao, M., Lim, C., and Suresh, S., 2003, “Mechanics of the Human Red Blood Cell Deformed by Optical Tweezers,” J. Mech. Phys. Solids, 51, pp. 2259–2280. [CrossRef]
Cheng, L., Xia, X., Scriven, L. E., and Gerberich, W. W., 2005, “Spherical-Tip Indentation of Viscoelastic Material,” Mech. Mater., 37, pp. 213–226. [CrossRef]
Vandamme, M., and Ulm, F. J., 2006, “Viscoelastic Solutions for Conical Indentation,” Int. J. Solids Struct., 43, pp. 3142–3165. [CrossRef]
Francius, G., Hemmerlé, J., Ball, V., Lavalle, P., Picart, C., and Voegel, J. C., 2007, “Stiffening of Soft Polyelectrolyte Architectures by Multilayer Capping Evidenced by Viscoelastic Analysis of AFM Indentation Measurements,” J. Phys. Chem. C., 111, pp. 8299–8306. [CrossRef]
Cheng, Y. T., and Cheng, C. M., 2005, “General Relationship Between Contact Stiffness, Contact Depth, and Mechanical Properties for Indentation in Linear Viscoelastic Solids Using Axisymmetric Indenters of Arbitrary Profiles,” Appl. Phys. Lett., 87, p. 111914. [CrossRef]
Zhou, Z., and Lu, H., 2010, “On the Measurements of Viscoelastic Functions of a Sphere by Nanoindentation,” Mech. Time-Depend. Mater., 14, pp. 1–24. [CrossRef]
Lee, E. H., and Radok, J. R. M., 1960, “The Contact Problem for Viscoelastic Bodies,” ASME J. Appl. Mech., 27, pp. 438–444. [CrossRef]
Buckingham, E., 1914, “On Physically Similar Systems, Illustrations of the Use of Dimensional Equations,” Phys. Rev., 4, pp. 345–376. [CrossRef]
Haddad, Y., 1995, Viscoelasticity of Engineering Materials, Springer, New York.
Dassault Systèmes., 2009, abaqus Theory Manual version 6.9-2.
matlab R2011a, 2010, Natick, The MathWorks, Inc., Natick, MA.


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Fig. 1

(a) Force–displacement relationship of a typical indentation experiment and (b) conical indentation of a sphere resting on a rigid and flat surface

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Fig. 2

Assumed constitutive behavior of the viscoelastic material and the loading function: (a) standard three-element solid model for deviatoric behavior, (b) spring element for spherical (volumetric) behavior, and (c) triangular loading

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Fig. 3

Finite element model used in abaqus, including enlargement of the refined mesh (plotted at the same scale) at the top of the sphere (conical indentation) and the bottom of the sphere (contact with the rigid surface) for the present indentation problem

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Fig. 4

The force–displacement relationships for the elastic indentation problem, for selected Poisson's ratios as obtained from geometrically nonlinear finite element analysis

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Fig. 5

Normalized force–displacement relationships for the elastic indentation problem, for four selected Poisson's ratios

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Fig. 6

Comparison of force–displacement curves obtained using the proposed semi-analytical approach and abaqus for four selected loading times, T



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