Use of combined elastic modulus in the analysis of depth-sensing indentation data
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It is shown that the substitution of reduced modulus for specimen modulus in the analysis equations for nanoindentation test data is valid. The methods of analysis use the slope of the unloading force–depth response which is assumed to be elastic. Because of this utilization of the slope or unloading stiffness, it makes no difference whether or not the deflection of the indenter is accommodated explicitly or transferred to that occurring within the specimen by artificially reducing the specimen modulus from its true value to lower value, the reduced modulus. Chaudhri1 has recently expressed some concern about the use of reduced elastic modulus in the contact equations commonly used in the analysis of nanoindentation test data. In this communication, the justification for this use is presented and shown to be admissible for this purpose. Following Johnson,2 we may define the quantity E* such that 1 共1 − 2兲 共1 − ⬘2兲 = + , E* E E⬘
(1)
whereupon the Hertz equation for elastic contact for a sphere of radius Ri and a flat surface of a specimen can be written: ␦3 =
冉 冊 3 4E*
2
P2 , Ri
(2)
which can be expressed as 3 d␦ . ␦= P 2 dP
(3)
The quantity E* is the effective or “combined” modulus of the system and allows us to treat the problem either as that occurring between a nonrigid sphere of modulus E⬘ and specimen of modulus E, or as a contact between a perfectly rigid sphere of radius Ri and a specimen of modulus E*. In both cases, ␦ is the load–point displacement or distance of mutual approach of distant points on indenter and the specimen. For the case of a rigid sphere, ␦ is equivalent to the depth of penetration beneath the specimen surface that would be achieved if the indenter were in fact rigid and the specimen modulus was given by E*. a)
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J. Mater. Res., Vol. 16, No. 11, Nov 2001 Downloaded: 17 Mar 2015
Although this procedure might satisfy the contact mechanics of the indentation (relating load–point displacement, radius of circle of contact, and load), it should be noted that, physically, the specimen material, viewed as a free body, experiences a contact with a “rigid” indenter of a larger radius R+ [see Eq. (4) below] that may be substantially different from the nominal indenter radius Ri. In terms of the radius of the contact circle a, the equivalent rigid indenter radius is given by R+ =
4a3E 3共1 − 2兲P
,
(4)
the value of which depends upon the load and where E is the actual modulus of the specimen. A comparison of the different radii is shown in Fig. 1. With reference to Fig. 1, the radius of circle of contact and load–point displacement are the same in each case, but the deformation experienced by the specimen is the profile of the surface shown by case 1 and not case 2. With respect to case 1, the specimen, viewed as a free body and with a modulus E, might just as well be indented by a perfectly rigid indenter of radius R+. The methods of analysis used in depth-sensing indentation experiments assume that general
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