Simulation of sub-micron indentation tests with spherical and Berkovich indenters
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The present work is concerned with the methods of simulation of data obtained from depth-sensing submicron indentation testing. Details of analysis methods for both spherical and Berkovich indenters using multiple or single unload points are presented followed by a detailed treatment of a method for simulating an experimental load–displacement response where the material properties such as elastic modulus and hardness are given as inputs. A comparison between simulated and experimental data is given.
I. INTRODUCTION
Indentation testing on the submicron scale enables the mechanical properties of thin films and very small volumes of materials to be measured at modest cost and effort. Usually the principal goal of such testing is to extract elastic modulus and hardness of the specimen from experimental readings of indenter load and depth of penetration. The methods of determining the area of contact from depth measurements, and hence hardness, and extraction of modulus from the unloading response are founded upon the elastic equations of contact of Hertz.1,2 In the present work, we will refer to the two most commonly used methods as the (i) multiple-point unload method3,4 and the (ii) single-point unload method.5 The methods rely on the analysis of the shape of the elastic unloading curve following elastic-plastic contact between an indenter and a specimen. In the present work, we show how these two methods of analysis may be both applied to spherical and Berkovich indenters and how they both may be used to generate a simulated force– displacement response using material properties such as elastic modulus and hardness as inputs.
II. ANALYSIS OF LOAD-DISPLACEMENT DATA A. Spherical indenter
depth of penetration beneath the original specimen free surface is ht at full load Pt. When the load is removed, assuming no reverse plasticity, the unloading is elastic and at complete unload, there is a residual impression of depth hr. It is assumed that the chordal radius of the residual impression is the same as the radius of the circle of contact at full load. During unloading, the contact is elastic through a distance he ⳱ ht − hr according to P=
4E* 1 Ⲑ 2 3 Ⲑ 2 , R he 3
(1)
where R is the combined radius of the indenter and the residual impression in the specimen surface and E* is the combined modulus of the indenter and specimen given by 1 1 1 = − , R Ri Rr
(2)
1 − 2 1 − ⬘ 2 1 = + , E* E E⬘
(3)
and
1. The single-point unload method
Indentation tests with a spherical indenter are characterized by an initial elastic deformation followed by an elastic-plastic response. With reference to Fig. 1, the
a)
e-mail: [email protected] J. Mater. Res., Vol. 16, No. 7, Jul 2001
where the superscripted terms refer to the properties of the indenter. In terms of the radius of the circle of contact, Eq. (1) is P=
4 E*ahe . 3
© 2001 Materials Research Society
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A.C. Fischer-Cripps: Simulation of sub-micron indentation tests with spherical and Berkovich indenters
FIG. 1. (a) Geometry of loading a pre
