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Taihua Zhang,a) Yihui Feng, and Rong Yang State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China (Received 20 October 2011; accepted 3 April 2012)

Lee and Radok [J. Appl. Mech. 27, 438 (1960)] derived the solution for the indentation of a smooth rigid indenter on a linear viscoelastic half-space. They had pointed out that their solution was valid only for regimes where contact area did not decrease with time. In this article, a large number of finite element simulations and one typical experiment demonstrate that Lee-Radok solution is approximately valid for the case of reducing contact area. Based on this finding, three semiempirical methods, i.e., Step-Ramp method, Ramp-Ramp method and Sine-Sine method, are proposed for determination of shear creep compliance using the data of both loading and unloading segments. The reliability of these methods is acceptable within certain tolerance. I. INTRODUCTION

Instrumented indentation is an efficient and convenient tool for probing mechanical properties of viscoelastic materials, such as polymers and biomaterials. Due to the time-dependent behavior of viscoelastic materials, the widely used Oliver-Pharr1 method is not suitable here.2–8 Robust methods for characterization of mechanical properties of viscoelastic materials via instrumented indentation are therefore required. Over the past decade, a number of researchers9–17 have proposed methods for determining shear creep compliance and shear relaxation modulus from load-depth curves of indentation tests. A common limitation of these methods is that only the data of loading or holding segments, where contact area is nondecreasing, are used. Because they are based on the solution first obtained by Lee and Radok18 as following equations 4Cn F ðt Þ ¼ 1m

ðnþ1Þ=n

h

Zt

dhðnþ1Þ=n ðsÞ ds Gðt  sÞ ds

Poisson’s ratio; Cn is a constant related to indenter shape, n 5 1, C1 5 tan a/p for conical indenter, and a p isffiffiffithe  included half-angle [see Fig. 1(a)]; n 5 2, C2 ¼ 2 R 3 for spherical indenter, and R is the radius of spherical indenter [see Fig. 1(b)]. Lee and Radok18 pointed out their solution is valid only for cases that contact area does not decrease with time. Which means Lee-Radok solution is valid during loading and holding, but fails for unloading. Hunter,19 Graham,20 and Ting21 addressed the viscoelastic contact problem when contact area has a single maximum. They derived the solution for unloading. When contact area passes through the maximum, the solution is written as FðtÞ ¼

4Cn ðnþ1Þ=n

ðBn Þ

;

ð1aÞ

Zt J ðt  sÞ

dF ðsÞ ds ds

Gðt  sÞ



danþ1 ðsÞ ds; t > tm ds

;

ð2aÞ

0

0

1m ðtÞ ¼ 4Cn

ð1  mÞ

Zt1 ðtÞ

;

ð1bÞ

Zt Bn hðtÞ ¼ a ðtÞ 

0

whereG(t) and J(t) are the shear relaxation modulus and shear creep compliance, respectively; v is the time-independent

Jðt  sÞ

n

tm

Zs 

Gðs  gÞ t1 ðsÞ

@ @s

dan ðgÞ dgds; t > tm dg

; ð2bÞ

a)

Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2012.