Relationships between initial unloading slope, contact depth, and mechanical properties for conical indentation in linea
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Che-Min Cheng Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China (Received 18 December 2004; accepted 26 January 2005)
Using analytical and finite element modeling, we studied conical indentation in linear viscoelastic solids with either displacement or load as the independent variable. We examine the relationships between initial unloading slope, contact depth, and viscoelastic properties for various loading conditions such as constant displacement rate, constant loading rate, and constant indentation strain rate. We then discuss whether the Oliver–Pharr method for determining contact depth, originally proposed for indentation in elastic and elastic-plastic solids, is applicable to indentation in viscoelastic solids. We conclude with a few comments about two commonly used experimental procedures for indentation measurements in viscoelastic solids: the “hold-at-peak-load” technique and the constant indentation strain-rate method.
I. INTRODUCTION
Instrumented indentation is playing an increasing role in the study of small-scale mechanical behavior of “soft” matters, such as polymers, composites, biomaterials, and food products. Many of these materials exhibit viscoelastic behavior, especially at elevated temperatures. Modeling of indentation into viscoelastic solids thus forms the basis for analyzing indentation experiments in these materials. Theoretical studies of contacting linear viscoelastic bodies became active since the mid 1950s by the work of Lee,1 Radok,2 Lee and Radok,3 Hunter,4 Gramham,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.10–15 In this paper, we examine, through analytical and finite element modeling, the relationship between initial unloading slope, contact depth, and viscoelastic properties. We then discuss whether the commonly used Oliver–Pharr method16,17 is applicable to indentation in viscoelastic
solids. We conclude with a critique of two commonly used experimental procedures, the “hold-at-peak-load” technique and the constant indentation strain-rate method, for indentation measurements in viscoelastic solids. II. ANALYTICAL RESULTS A. Conical indentation in linear viscoelastic solids
We consider a rigid, smooth, and frictionless conical indenter with half-angle indenting a viscoelastic solid that can be described by the following constitutive relationships18,19 between deviatoric stress and strain, sij and dij, and between dilatational stress and strain, ii and ⑀ii, t
兰
⭸dij共兲 d ⭸
t
⭸⑀ii共兲 d ⭸
sij共t兲 = 2 G共t − 兲 0
兰
ii共t兲 = 3 K共t − 兲 0
,
(1)
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.2005.0141 1046
J. Mater. Res., Vol. 20, No. 4, Apr 20
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