Determining the instantaneous modulus of viscoelastic solids using instrumented indentation measurements

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Wangyang Ni Brown University, Providence, Rhode Island 02912

Che-Min Cheng Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China (Received 13 June 2005; accepted 2 August 2005)

Instrumented indentation is often used in the study of small-scale mechanical behavior of “soft” matters that exhibit viscoelastic behavior. A number of techniques have recently been proposed to obtain the viscoelastic properties from indentation load–displacement curves. In this study, we examine the relationships between initial unloading slope, contact depth, and the instantaneous elastic modulus for instrumented indentation in linear viscoelastic solids using either conical or spherical indenters. In particular, we study the effects of “hold-at-the-peak-load” and “hold-at-the-maximum-displacement” on initial unloading slopes and contact depths. We then discuss the applicability of the Oliver–Pharr method (Refs. 29, 30) for determining contact depth that was originally proposed for indentation in elastic and elastic-plastic solids and recently modified by Ngan et al. (Refs. 20–23) for viscoelastic solids. The results of this study should help facilitate the analysis of instrumented indentation measurements in linear viscoelastic solids.


Instrumented indentation is becoming a powerful tool for the study of small-scale mechanical behavior of “soft” matters, such as polymers, composites, biomaterials, and food products. Since many of these materials exhibit viscoelastic behavior, modeling of indentation in viscoelastic solids is essential. Theoretical studies of indentation in linear viscoelastic bodies can be traced back to the mid 1950s by the work of Lee,1 Radok,2 Lee and Radok,3 Hunter,4 Graham,5,6 Yang,7 and Ting.8,9 In recent years, a number of authors have extended the early work to the analysis of indentation measurements in viscoelastic solids.10–27 One of the widely used methods is to obtain the elastic modulus from the initial unloading stiffness or slope (Fig. 1), S ⳱ (dF/dh)m, of the unloading curve at the maximum indenter displacement hm28–30 a)

Address all correspondence to this author. e-mail: [email protected] This author was an editor of this journal during the review and decision stage. For the JMR policy on review and publication of manuscripts authored by editors, please refer to http://www.mrs. org/publications/jmr/policy.html. DOI: 10.1557/JMR.2005.0389 J. Mater. Res., Vol. 20, No. 11, Nov 2005

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dF 4G a= = dh h=hm 1 − ␯


公␲共1 − ␯2兲




where G is the shear modulus, E ⳱ 2G(1 + ␯) is Young’s modulus, ␯ is Poisson’s ratio, a is the contact radius, and A ⳱ ␲a2 is the contact area. Equation (1) can be derived from the theory for elastic contacts between flat surfaces and spheres,31 flat punches,31 and conical punches.32 More generally, Sneddon has derived expressions for load, displacement, and contact depth for elastic contacts between a rigid, axisymmetric punch with an arbitrary smo