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Research Papers

On the Applicability of Sneddon's Solution for Interpreting the Indentation of Nonlinear Elastic Biopolymers

[+] Author and Article Information
Man-Gong Zhang, Xi-Qiao Feng

AML,
Institute of Biomechanics
and Medical Engineering,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

Jinju Chen

School of Mechanical and System Engineering,
Newcastle University,
Newcastle Upon Tyne NE1 7RU, UK;
Arthritis Research UK (ARUK)
Tissue Engineering Centre,
Newcastle University,
Newcastle Upon Tyne NE1 7RU, UK

Yanping Cao

AML,
Institute of Biomechanics
and Medical Engineering,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: caoyanping@tsinghua.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 23, 2014; final manuscript received July 4, 2014; accepted manuscript posted July 9, 2014; published online July 16, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(9), 091011 (Jul 16, 2014) (7 pages) Paper No: JAM-14-1178; doi: 10.1115/1.4027973 History: Received April 23, 2014; Revised July 04, 2014; Accepted July 09, 2014

Indentation has been widely used to characterize the mechanical properties of biopolymers. Besides Hertzian solution, Sneddon's solution is frequently adopted to interpret the indentation data to deduce the elastic properties of biopolymers, e.g., elastic modulus. Sneddon's solution also forms the basis to develop viscoelastic contact models for determining the viscoelastic properties of materials from either conical or flat punch indentation responses. It is worth mentioning that the Sneddon's solution was originally proposed on the basis of linear elastic contact theory. However, in both conical and flat punch indentation of compliant materials, the indented solid may undergo finite deformation. In this case, the extent to which the Sneddon's solution is applicable so far has not been systematically investigated. In this paper, we use the combined theoretical, computational, and experimental efforts to investigate the indentation of hyperelastic compliant materials with a flat punch or a conical tip. The applicability of Sneddon's solutions is examined. Furthermore, we present new models to determine the elastic properties of nonlinear elastic biopolymers.

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Figures

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Fig. 1

Schematic plot of (a) flat punch indentation and (b) conical indentation

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Fig. 2

Finite element models of (a) conical indentation with θ=4π/9 and (b) flat punch indentation

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Fig. 5

Spherical indentation loading curves, Hertzian solution was used to evaluate the initial shear modulus of the indented material

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Fig. 6

Indentation loading curves for (a) flat punch indentation and (b) conical indentation. The Sneddon's solution was included with μ0 = 0.67 MPa.

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Fig. 3

Dependence of dimensionless functions ΠMRF, ΠMRC,θ=π/4, ΠMRC,θ=π/3, ΠMRC,θ=9π/4, ΠABF, ΠABC,θ=π/4, ΠABC,θ=π/3, and ΠABC,θ=4π/9 on the hyperelastic parameters, the ratio d/a and the tip angle θ for the Mooney–Rivlin and Arruda–Boyce models

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Fig. 4

(a) Spherical, flat punch, and conical indenters used in our experiments and (b) the ElectroForce® 3100 test instrument (Bose)

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Fig. 7

Reciprocals of condition numbers for the determination of λm for the Arruda–Boyce model under different values of d/a and θ

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