Effect of unloading strain rate on the elastic modulus of a viscoelastic solid determined by nanoindentation
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Michael V. Swain Biomaterials Unit, Department of Oral Sciences, School of Dentistry, University of Otago, Duniden, New Zealand; Biomaterials Science Research Unit, Faculty of Dentistry, University of Sydney, United Dental Hospital, Surry Hills NSW 2010, Australia; and School of Aerospace, Mechanical & Mechatronic Engineering, University of Sydney, Sydney NSW 2006, Australia (Received 27 October 2005; accepted 5 December 2005)
The elastic modulus of an amorphous polymer was investigated by nanoindentation using combinations of ten total penetration depths and three constant unload rates. This experimental design provided a range of unloading strain rates coexisting with a range of depths. The elastic modulus of the material was found to correlate strongly with the unloading strain rate, whereas its correlation with the indentation depth was statistically nonsignificant. Thus, the increase of elastic modulus that occurred with decreasing depth at each constant unload rate was merely due to the increasing unloading strain rate associated with the decreasing depth. The true depth dependence of a rate-dependent material can therefore be studied only by maintaining a constant unloading strain rate across the entire depth range. The implications of these results to the continuous stiffness measurement technique are considered.
I. INTRODUCTION
Depth-sensing indentation or nanoindentation has become a commonly accepted technique for measuring the surface mechanical properties of a material.1 It was developed originally for a material the mechanical response of which is independent of time scales used in indenting the specimen. On the basis of the Sneddon’s solution2 to the contact problem of an elastic half space loaded with an axis-symmetric punch, the technique derives the reduced modulus as1 Er =
公 2
S
公A
,
(1)
where A is the projected area of the surface in contact with the indenter at a peak indentation load, S is the contact stiffness due to unloading from the peak load, and  is a constant associated with the geometry of the indenter. Er is related to the Poisson’s ratio and the elastic modulus E of the material through the relationship 1/Er ⳱ (1 − 2)/E + (1 − i2)/Ei, where i and Ei are the
a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2006.0087 708
J. Mater. Res., Vol. 21, No. 3, Mar 2006
Poisson’s ratio and elastic modulus of the indenter. Nanoindentation determines the contact stiffness as the slope of the load (F) versus displacement (h) curve at the onset of unloading. The area of contact is determined as a function of the depth of contact hc, based on the knowledge of the indenter tip geometry. The depth of contact is deduced using the following equation by Oliver and Pharr1,3: hc = ht − ⑀
Ft , S
(2)
where ht and Ft are the total displacement and load at the onset of unloading, and ⑀ is a constant associated with the indenter tip geometry. For both conical and spherical indenters, the constant may be approximated to be 0.75.3 Based on the f
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