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Norbert Schwarzer
To cite this version:
Norbert Schwarzer. The extended Hertzian theory and its uses in analysing indentation experiments. Philosophical Magazine, Taylor & Francis, 2006, 86 (33-35), pp.5179-5197.
�10.1080/14786430600690507�. �hal-00513694�
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The extended Hertzian theory and its uses in analysing indentation experiments
Journal: Philosophical Magazine & Philosophical Magazine Letters Manuscript ID: TPHM-05-Oct-0455.R1
Journal Selection: Philosophical Magazine Date Submitted by the
Author: 15-Feb-2006
Complete List of Authors: Schwarzer, Norbert; SIO
Keywords: multilayers, coatings, indentation testing
Keywords (user supplied): method of images, concept of effective indenter, intrinsic stress
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The extended Hertzian theory and its uses in analysing indentation experiments
N. Schwarzer, Saxonian Institute of Surface Mechanics, Am Lauchberg 2, 04838 Eilenburg, Germany, Tel. ++493423 656639, Fax. ++493423 656666, E-Mail: n.schwarzer@esae.de
Abstract
Since the presentation of the complete elastic field of an extended Hertzian approach by the author (treating the problem of mechanical contact of two bodies with shape functions of symmetry of revolution of the type z(r)=r2/d0 + r4/d2 + r6/d4 + r8/d6 + …) quite a few publications have appeared showing some useful applications of this new theory. They are mainly dedicated to the problem of parameter identification in the case of nanoindentation experiments. In this paper these new analysing techniques are summarised and further applications concerning indentation testing are proposed.
Thus, the main topics of the paper are:
• purely elastic indentation experiments
o two spheres in contact revised (With the new theoretical results the limits of the classical Hertzian approach are investigated.)
o arbitrarily shaped indenters of symmetry of revolution and their resulting elastic field at each point within the either monolithic or layered sample material o the determination of the surface pressure distribution in the case of layered
materials
• elasto-plastic indentation experiments
o the concept of the effectively shaped indenter
o identification of Young’s modulus, hardness and yield strength for monolithic and layered materials
o about the determination of intrinsic stresses via indentation experiments an in principle feasibility study
o about pile-up and sink-in (Does the new mathematical tool also help in analysing and detecting indents where pile-up or sink-in has occurred?)
The new techniques are in parts demonstrated on real nanoindentation data. The results are discussed and - wherever possible – compared with the outcome of other analysing
procedures.
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Introduction
In 1881 Heinrich Hertz first published his theory of the linear elastic contact of two parabolic bodies [1, 2]. His solution, however, could not provide the complete elastic field within the two bodies but only at their surfaces and only for some stress components and the normal displacement. Since then many authors intended to extent Hertz´s work in order to complete his solution. Among them we find Huber, Fuchs, Hamilton and Goodman, Hanson and many others [3 - 11]. Hanson finally succeeded in presenting a very compact closed form solution of the Hertzian problem for the isotropic [10] and the transverse isotropic case [11]. He succeeded in applying a new mathematical method introduced by Fabrikant [12]. This has been 111 years after Hertz’ first publication about this topic. Later on numerous authors have extended contact problems for the homogeneous to the layered half spaces. A comprehensive discussion of the different methods used there has been given in [13]. The method we apply here is the so called method of image loads or image contacts [13]. Apart from other uses this method allowed for example the determination of the Young’s modulus of very thin coatings down to 4.3 nm [14], the optimisation of the load carrying capacity of thin film structures [15, 16], or the successful characterisation of functionally graded coatings by using differently sized spherical indenters [17, 18].
In 2002 Pharr [19] published a comprehensive work about the so called „concept of the effectively shaped indenter“ and its application onto nanoindentation experiments with sharp pointed indenter like Vickers and Berkovich. Here, the usually completely elastic upper part of the unloading curve is used for the parameter identification of these experiments. Due to the above mentioned restriction of the Hertzian theory to surface displacements of the parabolic shape of indenter wI and sample body wS
2
0
( ) ( )
S I
w r w r h r
+ = −d , (1)
(r is giving the distance to the contact centre and d0 is parameter depending on the radii of curvature of both indenter and sample-body) it has been impossible to present a complete solution of the elastic field of his effective indenter. So Pharr and his co-workers used a Sneddon approach (cf. [19]) instead. This approach, however, can provide only surface information. This is a mayor restriction because the knowledge about the complete elastic field of the effective indenter could yield important information about critical material parameters in the moment of beginning unloading. That is why Schwarzer 2004 extended the Hertzian theory [20] and evaluated the complete potentials necessary to obtain the elastic field for a governing contact equation of the type
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2 4 6 8
0 2 4 6
( ) ( )
S I
r r r r
w r w r h
d d d d
+ = − − − − . (2)
He also presented the procedures necessary to obtain the potential functions for even higher exponents n of rn. Thus, now normal and even tangential [21] load distributions of the form
2 2
0
0
( , ) N n
zz n
n
r c rσ a r
σ ϕ
=
=
∑
− (3)2 2
0
0
2 2
0
0
( , ) ( , )
N n
xz xn
n N
n
yz yn
n
r c r a r
r c r a r
τ
τ
τ ϕ
τ ϕ
=
=
= −
= −
∑
∑
(4)
with arbitrary constants c (and by following the instructions of the mathematical procedures for obtaining the complete potential functions as given in [20] and [21] even arbitrary high N) can be solved completely. This opens up a wide range of new analysing techniques in order to extract more information from indentation experiments [20-23]. Motivated by the structures of the new governing contact equation (2) as well as the old Hertzian one (either as (1) or in its original form (9)), this approach has been dubbed “extended Hertzian” by the author.
In [24] these new solutions have been used for the determination of the yield strength of very thin coatings of less than 100 nm thickness. Hereby the most interesting and also somewhat surprising fact is that not even a hardness could be measured because these coatings were simply too thin to do so.
The reader might ask why the author has restricted the approach to even exponents of r. The reason for this are some mathematical difficulties arising if one intends to find the complete solution for odd numbers, meaning r2k+1. But even though the author has not been able to find the complete solution for the whole body for general odd exponents he succeeded in deriving a complete surface solution of the type
2 3 4 5 6
0( ) 0 1 2 3 4 5 6
w r =q +q r+q r +q r +q r +q r +q r
and obtained for the corresponding normal surface stress distribution [20]
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0 0
6 5 3 2
6 5 3 5
4 2 2 4
4 6 1 3 5
2 2 4
2 4 6
2 4 6
7 2 0 2 4 6
2
( , ) ( )
18432 6750 675 (16 25 )
1024 (25 36 ) 225 (64 144 225 )
256 (225 400 576 )
1 128(225 900 1600 2304 )
28800
zz r zz r
a q a q a q q r
a q q r a q q r q r
a q q r q r
q q q r q r q r
H
a r
σ ϕ σ
π π
π
π
=
+ + +
+ + − + +
+ + +
= − + + +
− 2
2 4 2 2
1 3 5
2 2
225 (64 144 225 )
q q r q r ln a a r
a r r
π
+
+ + + −
−
. (5)
The constant q7 must be chosen such, that the integral of σzz0 over the whole contact zone gives the total pressure force p
∫ ∫
=
π
ϕ σ
2
0 0
0rdrd p
a
zz
Integration yields
( )
7 2 3 4 5 6
0 1 2 3 4 5 6
1680
3360 840 2240 630 1792 525 1536
q p
a q a q a q a q a q a q a q
π π π π
= − + + + + + + . (6)
It can be shown, that apart from the solution for the coefficient q1 (singularity at r=0) all other solutions can be made free of singularities and thus are in principle applicable within analysing procedures for the investigation of nanoindentation data. However, the additional degree of freedom one obtains by using odd exponents like r3 and r5 could always also be achieved by adding higher even exponents. As stated above for even exponents the complete field can be evaluated completely analytically, for odd exponents this is not possible that easily (By applying a mathematical trick introduced by Hanson [36] solutions can be found but the evaluations are more cumbersome). The same holds for fractional exponents.
Purely elastic indentation experiments
Two spheres in contact - revised
Despite the fame and frequent use of Hertz´s work often very little is known about its real content. So many authors have quoted his work and referred to it as the "Hertzian contact between two spheres". This, however, is not correct. From the mathematical point of view Hertz´s original papers are not treating the contact of two spheres, but that one of two ellipsoidal paraboloids with an parabolic behaviour of the type x*y, x2, y2 respectively r2. Because this knowledge effects principle parts of this paper, too, we want to elaborate some of the principle basics of Hertz´s original work. In the introduction part of both his original
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German papers (references [1] and [2]) Hertz in fact mentioned ellipsoids and gave their shape due to the expressions z1=A1*x2 + C*xy + B1*y2 and z2=A2*x2 + C*xy + B2*y2 (see page 157 of [1]). Defining wS(r) and wI(r) as the normal displacement of the sample-body and the indenter, respectively and denoting the overall approach with the letter h, this leads to the governing contact equation
2 2
( ) ( )
S I
w r +w r = −h Ax −By , (7)
which Hertz gave in the form
2 2
1 2 Ax By
ζ −ζ = −α − (p. 159 of [1]). (8)
We see, that Hertz used different definition of direction, so we have
1 2
( ) ; ( ) ;
S I
w r =ζ w r = −ζ h= α
Then, on page 167 of [1] Hertz introduces the problem of two spheres in contact by simplifying his “ellipsoidal results”. However, by doing so he came up with the following result for the “spherical contact”:
2
1 2 Ar
ζ −ζ = −α (9)
which is only a contact of two spheres in an approximated form, because both the shape of the original spheres and that one of the resulting sum of displacements in normal direction of the contact region in the case of a real spherical contact must be described by a function of the form:
2 4 6
2 2
3 5
( ) ...
2 8 16
r r r
z r R r R
R R R
= − = − − − − , (10)
with R giving the radius of curvature of either the sphere or the contact zone (absolute values might be added for convenience in order to define a proper origin). We should add here that in the case of a rigid sphere of radius RI being pressed into a half space, RI would also exactly be the radius of curvature of the deformed half space surface under the indenter. Thus, in order to solve the spherical contact problem less approximated an extension of the Hertzian theory starting with the governing equation (2) would be necessary.
Assuming sample and indenter being of spherical shape with the radii of curvature RS and RI, respectively the first four di must read:
1 1
0 2 3 3
1 1
4 5 5 6 7 7
1 1 1 1
2 ; 8 ;
1 1 128 1 1
16 ;
5
S I S I
S I S I
d d
R R R R
d d
R R R R
− −
− −
= + = +
= + = +
. (11)
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For bigger contact radii (see figure 1) the difference between the approximation of the Hertzian approach and the new extended one is quite notable. We see, that the approach (2) and the corresponding elastic field given in the papers [20] and [21] now provides the means to solve the problem of two spheres in contact (or even a sphere in a spherical cavity) in a more correct manner. Though still an approximation of the spherical contact, the main
advantage of the extended Hertzian approach in comparison to the known solutions of Maugis [25] and Schwarz [26] or the classic approach of Goodman and Keer [27] is, that a complete elastic field in the whole half space can be given.
One might argue here, that having a better approximation of the shape of the spherical contact area is of rather no use because of the limits of the linear elasticity being restricted to small strains and displacements only, but in the case of conforming cavities or even as a first
"linearised estimation" one could still find this extension useful. This means in the case of ball bearings or asperity contacts [28], [29] for example (conforming cavities) the resulting stress distributions are different from the Hertzian one even if one stays in the small deformation regime.
Arbitrarily shaped indenters of symmetry of revolution and their
resulting elastic field at each point within the either monolithic or layered sample material
Especially in the case of very low loads the real shape of a nanoindenter tip is very important.
Spherical, cone or similar simple approximations do not suffice because during the course of analysing the indentation experiment they lead to mechanical fields which can differ
significantly from the real ones. The extended Hertzian approach gives us the means of much better approximations of the indenter shapes in these cases. As an example we consider the surface stress in normal direction σzz in the case of a Berkovich-indentation into a thin DLC- coating on silicon (figure 2). We see a quite notable difference between the best
approximation of symmetry of revolution of the real indenter and the Hertzian one (“Hertzian surface stress” in figure 2). Due to the singularity in the contact centre the cone-
approximation results in completely unrealistic stress fields (for more information about this investigation of ultra thin coatings the reader is referred to [24]). It should be noted here, that the influence of the substrate (though the coating is quite thin) on the form of the surface pressure distribution is insignificant (below 1%) because the ratio of contact radius to coating thickness is ¼. In addition the differences between the Young’s moduli of coating and
substrate is relatively small (528GPa/165GPa) and it can be evaluated [30] using the
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procedure described in the next section that under those conditions any substrate-effect on the shape of the pressure distribution can well be neglected. Thus, the different pressure
distributions shown in figure 2 are completely dominated by geometrical and non-linear material effects and not due to any substrate influence.
The determination of the surface pressure distribution in the case of layered materials
The Hertzian stress distribution as for example shown in figure 3 gives the correct stress for a parabolic indenter or a spherical one in the case of small contact radii only in the case of homogeneous materials. In the case of layered materials the surface stress can differ
significantly from the Hertzian one. The question of how good the classical Hertzian approach with σzz ~ a2 −r2 does agree with the real pressure distribution in the case of spherical indentations into layered materials can not be answered in a simple and general manner but must be investigated for each case separately. Figures 3 and 4 show two opposite examples for a soft coating on a stiff substrate (fig. 3) and a hard coating on a much softer substrate (fig.
4). The evaluation of the resulting stress distribution is based on the principle approach (3).
From [20] one can extract the surface displacement w0 within the contact zone below the indenter resulting from this type of normal load stress in the case of monolithic half spaces
( ) ( )
( )
( )
( )
2 2 4 2 2 4
2 2
6 4 2 2 4 6
4
8 6 2 4 4 2 6 8
6
0 3 2 4 6
2 4 6
1 1
2 8 8 9
4 64
1 1
16 8 18 25
256
1 640 256 288 800 1225
16384
( ) 2 105 42 24 16 / 315
a r c a a r r
p c a a r a r r
E
c a a r a r a r r
w r
a c a c a c a
ν
− + + − +
− + + − +
+ + + −
= + + + , (12)
where we have set all cn>6=0 and c0=1 (p denotes the total normal load). Now we consider the governing contact equation of the extended Hertzian approach (2). While for the
homogeneous indenter equation (12) exactly gives the resulting displacement wI=w0, one has to combine the results from [20] (extended Hertzian approach) with the method of image loads [13] in order to obtain the displacement wS for the layered sample. So, the above
equation (2) has to be solved with respect to h and a by using (11) for the geometrical indenter parameters d0, d2, d4 and d6. This results in the following equation for homogeneous bodies (meaning both indenter and sample are monolithic)
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( ) ( )
( )
( )
2 2 4 2 2 4
2 2 2
6 4 2 2 4 6
4
8 6 2 4 4
6 2 6 8 2 4 6
3 2 4 6
0 2 4
2 4 6
1 1
2 8 8 9
4 64
1 1 1
16 8 18 25
256
640 256 288
1
16384 800 1225
2 105 42 24 16 / 315
S I
I S
a r c a a r r
p c a a r a r r
E E
a a r a r
c
a r r r r r
h d d d
a c a c a c a
ν ν
− + + − +
− + − + + − +
+ +
+ −
= − − − −
+ + +
8
6
r d (13) (ES, νS and EI, νI are denoting the Young’s modulus and Poisson’s ratio for the sample and the indenter, respectively) and a set of equations in the case where only the indenter is monolithic and the sample is of layered structure (here a single-layer-coating is assumed)
( )
2 4 6
2 4 6
2 4 6 1 2 1 2
3 5 7
2 4 6
315 64 16 8 5
( ,0, , , , , , , , , )
256 105 42 24 16
eff
S
p E c a c a c a
h w p a c c c E E d
a +c a +c a + c a ν ν
= +
+ + + , (14)
2 4 6 8
0 2 4 6 1 2 1 2
0 2 4 6
( ) ( , , , , , , , , , , )
i i i i
i S i
r r r r
h w r w p r a c c c E E d
d d d d ν ν
− − − − = + , (15)
with ri properly chosen r1, r2, r3, r4 within the range 0<r≤a (e.g. ri=(i-1)*a/4), Eeff given through
1 I2 eff
I
E E
ν
= − ,
E1, ν1, E2, ν2 and EI, νI are denoting the Young’s modulus and Poisson’s ratio for the sample layer, sample substrate and the indenter, respectively. We see, that due to (11-13) the
parameters describing the shape of the surfaces of the bodies in contact (di) and the
coefficients ci of the series of stresses of the form given due to (3) are related to each other.
However, this relation is of 4th order in the homogeneous (monolithic half space) case and has even transcendental character in the case of layered materials. Thus, it can not be given in a simple or closed form manner. The equation (11) and the system of equations (14) and (15) must be solved numerically in order to extract the contact parameters c2, c4, c6, h and a. The function wS( , , , , , , ,p r a c c c E E2 4 6 1 2, , , )ν ν1 2 d has to be evaluated according to the procedure described in the appendix A in [24]. This can be done applying an especially designed software package [30].
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Elasto-plastic indentation experiments
The concept of the effectively shaped indenter
The basic idea of this concept is the assumption of a plastically/elastically deformed zone beneath the penetrated indenter tip which builds up an “effective” indenter with an
“effectively” shaped tip acting onto the specimen. In this deformed zone residual stresses from the permanent deformation and indenter stresses from the applied load are present.
However according to the earlier work of Solomon [31] the unloading process should start completely elastically. So, when the indenter is repeatedly loaded and unloaded only the elastic indenter stresses give response to the effective indenter tip and can therefore be
separated by analysing the shape of the unloading curve applying a straight forward procedure [20, 30]. From the resulting pressure profile beneath the indenter tip the distribution of
stresses within the sample can be obtained considering all boundary conditions of the experimental set up correctly including even the layered character of the sample geometries [20 - 24].
A very nice introduction to this concept has been given by Woirgard and Dargenton [32] as well as Pharr and Bolshakov [19]. In addition there is a computer video animation available presenting the concept and its application in a comprehensive and figurative manner
(www.esae.de/downloads, contact: service@esae.de). Therefore and because of its
voluminous character, we restrain ourselves from presenting a more extensive elaboration of the method within this paper.
The benefit of the effectively shaped indenter concept is its applicability to any indentation unloading curve even if an – sufficiently recessive - inelastic deformation has occurred. The main limitation is given by the condition that the elastic indenter stresses dominate during the indentation and that the influence of the destructive deformation on the resulting stress field is small. This means concrete, that in the case of very brittle and soft materials like for example so called porous Xerogels, only relatively big spherical indenters can be used in order to obtain sufficiently high loads without to much damage of the material. In order to overcome this disadvantage the author has developed an extension of the method of the effectively shaped indenter for smaller tip radii [33].
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Identification of Young’s modulus, hardness and yield strength for monolithic and layered materials
Similar to the classical Oliver and Pharr method [34] the free geometrical indenter parameters d0, d2, d4 and d6 and the material parameters (Young's modulus and Poisson's ratio) have to be fitted to the unloading curve of the indentation data. While, for pure reversible and linear elastic indents this would result in an approximated (because we only have an approach of symmetry of revolution) description of the indenter shape, one obtains an effective indenter shape (an “elastic equivalent”) in the case of any inelastic behaviour. The fit procedure requires the repeated solution of the set of equations (14) and (15). It also results in the determination of the contact radius a in the moment of maximum load, which allows the evaluation of the hardness. Applying now the further results of [20] and combining them with the method of image loads [13] one can evaluate the complete stress field produced by the hypothetic effectively shaped indenter found by means of the fit procedure even in the case of layered materials. Here we are especially interested in the v. Mises stress field in the moment of beginning unloading. As the v. Mises stress sums up all shearing stresses within the material, its maximum, evaluated for the effectively shaped indenter in the moment of beginning unloading, might give us the yield strength of the investigated sample [22, 23].
This statement presumes of course that the contact is plastic at full load and that the v. Mises criterion had been satisfied. If the contact is elastic, then of course the v. Mises stresses will not reach the yield stress value of the material.
As an example we consider the case of a Berkovich-Indentation into a thin DLC coating of 95 nm deposited on a silicon substrate (see [24] for more information). Figure 5 shows the load displacement curve with a maximum load of about 50 µN. The solid line shows the equivalent indentation curve of the fitted “effective indenter”. How to determine these effective indenter fit parameters has been briefly described in the preceding section. Figure 2 presents the distribution of the surface stress in normal direction for this effective indenter in the moment of maximum penetration respectively beginning unloading. For comparison the stress
distribution produced by an ideal cone, a cone with a rounded tip (the best approximation of the real indenter shape) and the Hertzian stress distribution is given, too (all stress distribution are evaluated for the same over all loading force). How is the effective stress distribution to be understood? Within an ideal elastic material an ideal mathematical cone would produce a stress singularity in the centre of contact at r=0 (c.f. fig. 5). However, in practice indenters have rounded tips. Taking the shape of our ACCU-Tip approximated for symmetry of revolution and evaluating the resulting stress distribution in the case of an ideal elastic
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material, the singularity would disappear and the stress distribution would be flattened (details of the evaluation are given in appendix B of [24]). Assuming an yield strength of 31GPa (see below) one finds however, that such a stress distribution would exceed the elastic limit within the coating material (fig. 6). As one can see in figure 6 inelastic behaviour would occur below the indenter around the axis of indention at depth range from z=4nm to z=16nm. During this plastic deformation material would be moved from this area in mainly centrifugal direction.
This leads to a further flattening of the stress distribution as one can observe on the effective indenter stress (figure 2). Comparing this effective indenter stress with a Hertzian stress distribution for the same radius of contact and over-all-loading-force one finds however, that the effective indenter still appears to be “sharper” than an effective sphere would be. Driving the indenter further into the material would lead to a further decreased stress within the centre of the contact (figure 7a). In the case of a fully developed plastic region around the indenter the stress distribution would finally obtain the shape of a flat punch with rounded edges as proposed in [19]. Such a stress distribution is shown in figure 7b, which has been evaluated however for a different compound system and is here given only as an example for higher loads.
Using [30] we can now evaluate the von Mises stress of the "effective indenter" for our layered system in the moment of beginning unloading. By doing so, one finds the v. Mises stress distribution shown in figure 8 has a maximum value of 31.0 GPa. This value could be taken as an approximation of the yield strength in the case of rather low residual stresses. And in fact, by applying other means (c.f. [24]) one obtains values relatively close to this one, namely: (34 ± 5)GPa.
It has to be pointed out here, that the method only determines the resulting yield strength of the constrained case, because it can not separate residual stresses from the elastic indenter stresses during unloading. Thus, in those cases, where the residual stresses are quite high the measured yield strength can differ significantly from the one of the unstressed case. For example: In figure 9 it is shown how an biaxial intrinsic stress can influence the load carrying capacity with respect to plastic flow and thus would result in a yield strength result being significantly different from the yield strength of the corresponding stress free state. Within the section “About the determination of intrinsic stresses via indentation experiments” below it will be shown how the intrinsic stress can influence the measured “yield strength” and that – in principle - an additional piece of experimental information (meaning in addition to the pure normal loading like a combination of normal and mixed loading) is necessary in order to extract both, residual stresses and the “zero-stress yield strength”.
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An other important question arises concerning the problem of finding the correct “effective indenter shape” due to the above described fitting process. In other words: How unique is the solution and how do experimental uncertainties influence the determined parameters of the effective indenter and the subsequently evaluated elastic field? Well, as the new method must still be considered as non-standard and rather experimental no final statement can be made about the uncertainties. But taking into account the numerous measurements analysed by the author so far the following conclusions can be made:
1. The method works particularly well for all sorts of materials known for not producing high residual stresses during the loading state [20 - 24].
2. Concerning the uniqueness of the solution and the stability of the fitting process, it is advisable to start the fitting procedure on a lower (e.g. fitting only d0 and setting all other di=0) order and compare them with other methods (like the Oliver&Pharr method [34]) before subsequently moving on to fits of higher order (d2 – d6). In this manner the method must of course be considered more a mathematical tool to the classical analysing technique rather than a stand alone procedure. However, due to the possibility of evaluating the complete elastic indenter stress field it opens up wider range of features for parameter identification.
Finally, one should point out here, that in principle the method has the potential of completely separating the contributions of substrate and coating. However, as it has been demonstrated and discussed comprehensively elsewhere [17, 18] one must know the substrate parameters and the coating thickness, because in almost all cases an effective monolithic elastic half space could be found also fitting almost any unloading curve of a layered sample. Due to the lack of space the reader is referred to check this by using the software package [30], which has been made available as supplementary material in the archive of the Technical University of Chemnitz (web site: http://archiv.tu-chemnitz.de/pub/2006/0018).
About the determination of intrinsic stresses via indentation experiments
an in principle feasibility study
For the sake of practical reasons one would desire a method allowing the determination of the intrinsic stress and yield strength in “one go”, meaning that only one cycle of penetration should suffice in order to obtain a proper value for either the yield strength or other limiting mechanical parameters together with the intrinsic stress. Concerning the yield strength such a method is already at hand, namely the “concept of the effectively shaped indenter”.
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In order to determine the intrinsic stress, too, in [21] we have proposed the following measuring procedure:
1. The yield strength is determined using the method of the effectively shaped indenter as presented above. However, during unloading, the indenter is only drawn back to a distinct fraction of the maximum load p0. This load shall be called p1. The reader should note, that p1 must be chosen such, that on one hand there is enough unloading curve for the determination of the shape of the effective indenter and on the other hand the load is still big enough in order to avoid strong and dominant inelastic unloading effects like e.g. unloading fractures. In addition a p1 close to p0 also assures that the shape of the effective indenter only changes in an insignificant manner during unloading. We call the determined yield strength σMcrit.
2. Now a slowly oscillating tangential load component tx with increasing amplitude is added and the resulting lateral shift is measured. So we now have mixed load
conditions and approximately assume our - in step one determined - effectively shaped indenter acting with the combined load components p1 and tx onto the coating
substrate compound. Due to bonding effects additional field components might be necessary in order to obtain a proper model of the mixed load conditions under the indenter.
3. The slowly oscillating tangential loading with increasing amplitude is monitored with very high resolution (as well as the static normal load and displacement of course) until nonlinear behavior can be detected. Thus, the value of maximum tangential load tx=tcrit or maximum lateral displacement uσ is determined. Now we introduce the following assumption: The combined stresses add up to a mechanical stress field producing a maximum von Mises stress σM =σMcrit somewhere within the investigated coating material.
4. These values tcrit and uσ have now to be compared with those ones an unstressed material (σrrf = ) would require in order to reach its yield strength limit 0 σM =σMcrit. From this comparison one can deduce the value of σrrf residing in the coating. The governing formulae necessary to perform this are given in [21] appendix A (The equations necessary for the evaluation of the different – pure normal and mixed - elastic fields can be found in the appendix B of [21] and in [20]).
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This apparently simple method has apart from experimental problems1 one main theoretical draw back, namely the relatively complex theoretical apparatus necessary to model and analyze this experiment. The task of modeling a coated material “mixed loaded” with an
“effectively shaped indenter” requires a comprehensive and relatively unwieldy set of
formulae. Thus, a software package has been created providing the mathematical model in an easy to use manner [30]. It contains the solution for the above mentioned extended Hertzian approach for the case of mixed normal and tangential loading of the type:
2 2
0 0
( , ) N n n
x z n
y
r c rτ a r
τ ϕ
=
=
∑
− . (16)2 2
0
0
( , ) N n
zz n
n
r c rσ a r
σ ϕ
=
=
∑
− . (17)Applying the above described method of determining the intrinsic stresses a relation between the ratio of intrinsic stress σrrf to the critical von Mises stress σMcrit and the ratio of the
deviation of maximum lateral displacement of pre-stressed uσ and unstressed u0 case to u0 can be given. Evaluating this relation for a big variety of material combinations (mainly using the results of the experimental examples presented in [20 - 24]) a linear function can be extracted
0 0
*
f rr crit M
u u
const u
σ σ
σ
− = (18)
which gives the minimum accuracy the measurement must be performed with (see figure 10) in order to determine proper intrinsic stress values. Though no such measurement device seems to be at hand right now, the determination of intrinsic stresses should in principle be measurable with the next generation of nanoindenters as proposed for example by the companies Hysitron, MTS and ASMEC.
About pile-up and sink-in
It is well known, that pile-up or sink-in during nanoindentation result in over- or
underestimation of the contact zone and thus yields wrong values for Young's modulus, Hardness and Yield strength. The question is: Does the new mathematical tool of the extended Hertzian approach also help in analysing and detecting indents where pile-up or sink-in has occurred? The answer is NO, because independent additional information can not be gained
1 For one, there is no proper Nanoindenter with a sufficiently sensitive lateral force unit available yet. Though the author knows from some efforts, it still seems to be a long way until the “next generation of Nanoindenters”
can be used in order to tackle problems like those ones described here.
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from applying this method on the unloading curves of indentation data and thus without additional knowledge it can not be concluded whether or whether not effects like pile-up or sink-in have occurred. However, again, by combining the normal indentation with an additional tangential load component one might be able to obtain this necessary piece of information. We assume, that at maximum load a small tangential load component is added to the normal one. For very small tangential loads the indenter can be assumed to be bond to the sample surface and the following simple relation [35] between tangential load T and lateral displacement u would be valid (it was assumed, that no tilting moment is occurring):
[
1 2 1 2]
* I( , )I I S( , , , , )
T =a u E% ν +u E E% ν ν d , (19) E1, ν1, E2, ν2 and EI, νI are denoting the Young’s modulus and Poisson’s ratio for the sample layer, sample substrate and the indenter, respectively. The fact has been used, that in the case of a bonded punch the contact radius a can be factored out as follows u =a u*%. This way one would have a second measurement for the determination of the contact radius a and the material parameters being completely independent of the normal loading. With this additional information one could go into the fitting process described in the section "The concept of the effectively shaped indenter" and increase the number of extractable parameters. The fitting procedure would then consist of three basic equations, namely (14), (15) and (19) instead of just two. An experimental test of this new measurement technique is still awaited to be done.
Conclusions
The extended Hertzian approach provides a powerful mathematical tool for the analysis of nanoindentation data. Especially indentation measurements one can approximate by indenters with symmetry of revolution participate from the higher order of accuracy the measurement could be modelled with. So for example, within the elastic regime, the tip rounding of the indenter can be described with higher accuracy including the analytical evaluation of the complete elastic field. Additionally the real shape of spherical indenters and their resulting stress fields can be described even for relatively big radii of contact in comparison with the radius of the indenter. This also holds in the case of layered materials, where now the non Hertzian character of the surface stress distribution caused by the layered structure of the materials can be modelled in a more correct manner. In the elastic-plastic regime the extended Hertzian approach also provides the means for a higher order fit procedure within the
"concept of the effectively shaped indenter". From the theoretical point of view additional tangential load measurements should provide additional linear independent information allowing the extraction of more physical parameters like for example the intrinsic stress or the
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detection of pile-up and sink-in effects directly during the nanoindentation process without looking at the residual imprint.
References
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Berlin, November 1882, pp. 174-196
[3] M. T. Huber: "Zur Theorie der Berührung fester elastischer Körper", Annalen der Physik, Leipzig, Band 14, 1904, S. 153-163
[4] S. Fuchs: Phys. Zeitschrift vol. 14 (1913), 1282-
[5] W.B. Morton, L. J. Close: Phil. Mag. S. 6, Vol. 43, No. 254 (1922), 320-329
[6] R. F. Bishop, R. Hill, N. F. Mott: Proc. Phys. Soc. Vol. 57, 3, No. 321 (1945), 147-159 [7] I. N. Sneddon: Int. J. Engng Sci. Vol. 3 (1965), 47-57
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[9] W. T. Chen: Int. J. of Solids and Structure, Vol. 5, 1969, pp. 191-214
[10] M. T. Hanson, (1994) The elastic field for an upright or tilted sliding circular punch on a transversely isotropic half-space. Int. J. Solids Structures, Vol. 31, No. 4, 567 – 586 [11] M. T. Hanson, (1992) The elastic field for a spherical hertzian contact including
sliding friction for transverse isotropy. ASME J. of Tribology, Vol. 114, 606-611 [12] V. I. Fabrikant, (1989) Application of potential theory in mechanics: A Selection of
New Results. Kluver Academic Publishers, The Netherlands
[13] N. Schwarzer: ”Arbitrary load distribution on a layered half space”, ASME Journal of Tribology, Vol. 122, No. 4, October 2000, 672 – 681
[14] T. Chudoba, M. Griepentrog, A. Dück, D. Schneider, F. Richter, Young’s modulus measurements on ultra-thin coatings, J. Mater. Res., 19 (2004) 301
[15] O. Wändstrand, N. Schwarzer, T. Chudoba, Å. Kassman-Rudiphi: ”Load-carrying capacitity of Ni-plated media in spherical indentation: experimental and theoretical results”, Surface Engineering Vol. 18 (2002) No. 2, 98 – 104
[16] V. Linss, I. Hermann, N. Schwarzer, U. Kreissig, F. Richter: „Mechanical properties of thin films in the ternary triangle B-C-N“, Surf. Coat. Technol. 163-164 (2003) 220- 226 (ISSN 0257-8972)
[17] V. Linss, N. Schwarzer, T. Chudoba, M. Karniychuk, F. Richter: “Mechanical Properties of a Graded BCN Sputtered Coating with VaryingYoung's Modulus:
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