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Research Papers

# Conical Indentation of a Viscoelastic Sphere

[+] Author and Article Information

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

## Abstract

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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## References

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## Figures

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

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

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

Fig. 4

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

Fig. 5

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

Fig. 6

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

## Errata

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