1 ELASTIC-PLASTIC CONTACT OF A ROUGH

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1 ELASTIC-PLASTIC CONTACT OF A ROUGH
ELASTIC-PLASTIC CONTACT OF A ROUGH SURFACE WITH WEIERSTRASS
PROFILE
Yan-Fei Gao and A. F. Bower
Division of Engineering, Brown University, Providence, RI 02912, USA
The classical approach to modeling contact between rough surfaces follows a model developed
by Greenwood and Williamson (1966), in which surface roughness is idealized as a collection of
discrete asperities, with a Gaussian height distribution, and (in the simplest model) identical
curvatures. Despite the widespread acceptance of this model, it has proved difficult to apply to
realistic surfaces, whose roughness resembles a self-affine fractal over a wide range of length
scales. For a perfectly fractal surface, the Greenwood-Williamson model breaks down
completely, because it is impossible to detect individual asperities on the surface, and to
determine their curvature. Several recent models of rough surface contact have therefore been
based on a fractal description of the surface geometry: the Weierstrass profile is one such
example. Ciavarella et al (2000) have recently reported a rigorous analysis of elastic contact of
the Weierstrass profile. Our goal is to extend their analysis to the more realistic case of an
elastic-perfectly plastic solid. To this end, we consider an isotropic, elastic – perfectly plastic
solid with a rough surface, which is characterized by the two-dimensional Weierstrass profile.
The solid is indented by a smooth frictionless cylinder. We attempt to compute quantities such
as the number of contact spots, their size (or more precisely, their size distribution); the true
contact pressure acting on the contact spots, and the total true area of contact. In addition, we
calculate critical conditions that will initiate plastic flow in the asperities, and determine the
extent of this plastic flow as a function of the applied loading, the material properties of the
deforming solid, and the surface roughness. An important conclusion of our study is that a
perfectly fractal description of surface roughness appears to lead to unphysical predictions of the
true contact size and number of contact spots, for both elastic and elastic-plastic solids. Possible
approaches to resolving these difficulties are discussed.
Keywords: self-affine fractal, contact mechanics, rough surface plasticity
1. Introduction
When two rough surfaces are pressed into contact, they meet only at the highest points on the
two surfaces. To predict the size of the resulting contact spots, the true contact pressure, and the
true area of contact is a venerable problem in contact mechanics, and is clearly of considerable
practical importance, since phenomena such as friction, wear, and contact fatigue are controlled
by the nature of asperity contacts. Nevertheless, a fully satisfactory solution remains elusive.
The essential character of surface roughness, and the implications on the nature of contact
between rough surfaces, were recognized in the pioneering work of Archard (1957). His
approach was to model a rough surface as a sphere, which has an array of smaller spherical
bumps on its surface. These spheres in turn have still smaller spheres on their surfaces. If this
geometric process is continued indefinitely, one constructs what would today be regarded as a
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surface with fractal geometry. Archard analyzed elastic contact between such a surface and a flat
indenter, and predicted the number of contact spots, the contact size, and contact pressure for
each successive scale of spheres. One important result of this calculation is that if the surface
roughness contains a finite number of scales of spheres, the true area of contact is proportional to
the load, thereby providing an elegant explanation for Amonton’s laws of friction. For the limit
of a perfectly fractal surface (with an infinite number of scales of spheres), one finds that the true
contact area consists of an infinite number of contact spots with zero size, which are subjected to
infinite contact pressure.
Archard’s model contains an excellent qualitative description of a rough surface, but cannot
easily be related to actual measurements of real surfaces. Greenwood and Williamson (1966)
therefore devised an alternative approach to characterize surface roughness, and to model contact
between two rough surfaces. In their original work, Greenwood and Williamson idealized a
rough surface as a collection of asperities with spherical tips of equal radius, but with a Gaussian
distribution of heights. They neglected interactions between neighboring asperities, and so were
able to relate the behavior of the surface to the force-displacement relation for an individual
asperity and the asperity height distribution function. This approach predicts a well-defined
number of total contact spots, contact size, contact pressure and contact area, which depend on
the curvature of the asperity tips, the asperity height distribution, and the elastic and plastic
properties of the two contacting surfaces.
The Greenwood-Williamson approach has been
refined and extended by a number of authors (e.g. Whitehouse & Archard 1970; Nayak 1971;
Bush et al 1975; McCool 1985; Greenwood 1984), and is widely used in many applications.
Since the early work of Greenwood and Williamson, a great deal of work has also been done to
characterize the statistical properties of surface roughness. Various approaches have been
developed, including through its autocorrelation function (Whitehouse & Archard 1970), power
spectral density (Sayles & Thomas 1978) or as a semi-deterministic fractal (Majumdar & Tien
1990; Majumdar & Bhushan 1990, 1991). There has been some debate as to the most accurate
description of realistic surfaces, but all models agree that surfaces have an approximately selfaffine fractal character over a wide range of length scales, and usually continue to exhibit fractal
behavior down to the limits of the resolution of standard surface profilometers. Indeed,
Majumdar and Bhushan (1990, 1991) show data that suggests that roughness spectra for hard
disk drives remain fractal down to wavelengths of a few hundred nanometers.
This behavior leads to some serious problems with Greenwood and Williamson’s approach,
because it is difficult to detect individual asperity tips on a surface, and even more difficult to
measure their curvature. Surface profiles are measured by sampling the surface at discrete
intervals. In two dimensions, asperity tips are identified by finding points that are higher than
their neighbors. Finding summits in three dimensions is more difficult, but conceptually similar.
The curvature of each asperity tip can be determined by fitting a curve (in two dimensions) or
surface (in three dimensions) through the points surrounding the peak. For a differentiable
surface, this procedure converges to identify a well-defined set of asperities as the sampling
interval is reduced. When this procedure is applied to experimentally measured surfaces,
however, the number of asperities and their curvature appear to increase without limit even as the
sampling interval is reduced to extremely small values. This, of course, is a consequence of the
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fractal nature of surface roughness. For a perfectly fractal surface, Greenwood and Williamson’s
procedure breaks down completely, since such a surface does not contain well-defined summits.
Many recent studies of rough surface contact have therefore abandoned the GreenwoodWilliamson description of surface geometry in favor of one that captures in some way the fractal
character of surface roughness. Various approaches have been proposed, including ad-hoc
procedures for adapting Greenwood and Williamson’s calculations to fractal surfaces (Majumdar
& Bhushan 1990, 1991), direct numerical simulations of experimentally measured profiles (e.g.
Sayles, 1996; Polonsky & Keer 2000a, 2000b; Borri-Brunetto et al 1999; Hyun et al 2004; Pei et
al 2005), or hybrid schemes which combine numerical simulations with Greenwood and
Williamson’s approach (see Mihailidis et al 2001 for a review). Attempts to adapt GreenwoodWilliamson to a fractal surface appear to suffer from the same difficulties as the original
Greenwood-Williamson analysis: there is no unambiguous way to count, or to characterize,
asperities on a fractal surface. Numerical simulations provide considerable insight into the
deformation fields associated with rough surface contact, but there is a limit to the spatial
resolution of any numerical scheme, which artificially truncates the roughness spectrum. This
again introduces problems very similar to those associated with the Greenwood-Williamson
model: for a fractal surface, predictions of the contact size, number of contacts, and true contact
area will not converge as the resolution of the numerical scheme is refined. Great care must be
taken to ensure that this truncation does not lead to spurious predictions (Borri- Brunetto et al
1999).
A significant step towards a satisfactory model of the elastic contact between rough fractal
surfaces was made recently by Ciavarella et al (2000) and by recent work of Persson (2001,
2005). The approximate approach of Ciavarella et al (2000) will be followed closely in our
work, and so will be described in more detail in Sections 2 and 3 of this paper. Briefly, they
chose to approximate surface roughness using the two-dimensional Weierstrass function given in
Eq. 1 below.
The Weierstrass function consists of a sum of sinusoidal profiles, with
progressively decreasing wavelengths Λ m = Λ0γ − m where m = 0,1,2K ∞ and Λ 0 and γ are
geometrical parameters. The amplitude of each term in the series varies with m as G = G0γ ( D− 2) m ,
and for 1 < D < 2 the function is a self-affine fractal with dimension D. It is difficult to analyze
contact of rough surfaces with realistic values of γ , but Ciavarella et al show that much progress
can be made by taking γ >> 1 . Under these conditions, the wavelengths in the Weierstrass
profile are widely separated, and each term can be analyzed independently of the rest. The
surface then resembles Archard’s early model, and can be treated in a similar way: instead of
attempting to identify actual summits on the surface, and to predict the true contact size, pressure
and area for these summits, Ciavarella et al compute the apparent contact size, pressure and total
area for each successive scale of roughness m. They formally take the limit m → ∞ , and recover
Archard’s original prediction that the true contact area consists of an infinite number of contact
spots, with zero size, infinite pressure, and vanishing total area (Ciavarella et al 2000; Ciavarella
& Demelio 2001). This is a purely formal mathematical result, of course: in practice phenomena
such as plastic flow or adhesion might alter these predictions, and in any event no real surface is
perfectly fractal. There are some applications, such as modeling friction, where the true contact
size and area are of great interest, and predicting these quantities remains an unresolved problem.
For modeling the compliance of the surface, or its electrical contact resistance, and possibly
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physical phenomena such as plastic flow, wear, and the initiation and growth of contact fatigue
cracks, the behavior of very short wavelength roughness is less important, since damage events
occur at sub- micron length scales rather than nanometer length-scales. One advantage of
Archard’s model, as extended by Civarella et al, is that the behavior of long-wavelength
roughness scales can be predicted in terms of measurable surface parameters without needing to
identify real asperities on the surface.
This work has provided an important step towards understanding contact between surfaces with a
fractal character. Their predictions leave several issues open, however. Their analysis shows
clearly that for elastic contact between perfectly fractal surfaces, the true contact area consists of
an infinite number of contact spots with vanishing size, which are subjected to infinite contact
pressure. The total true contact area also vanishes. This is clearly an unphysical prediction. Of
course, real surfaces differ from the idealized model in several respects, which include: (i) real
surfaces are not perfectly elastic; (ii) the surface roughness of a real surface cannot be perfectly
fractal – the spectrum must be truncated at some very short length-scale; (iv) adhesive forces
may act between real surfaces at very short length-scales; and (v) Ciavarella et al’s model is
restricted to surfaces with widely separated scales of roughness. Weierstrass profiles with large
values of real γ clearly do not accurately characterize real surface roughness, even if one accepts
that the Weierstrass representation with γ → 1 is a realistic model of a surface. The question is,
which of these effects plays the most important role in controlling the true contact area in real
solids? These are important issues for further research.
In this paper, our goal is to extend the calculations of Ciavarella et al to account for plastic
deformation. The problem to be solved is illustrated in Fig. 2. We consider an isotropic, elastic –
perfectly plastic solid with Young’s modulus E , Poisson’s ratio ν and yield stress σ Y . The
solid has a rough surface, whic h is characterized by the two-dimensional Weierstrass profile
given in Eq. 1. The solid is indented by a smooth cylinder with radius R, which is subjected to a
force P per unit length out of the plane. The goal of our analysis is to estimate quantities such as
the number of contact spots, their size (or more precisely, their size distribution); the true contact
pressure acting on the contact spots, and the total true area of contact. It is of particular interest
to calculate critical conditions that will initiate plastic flow in the asperities, and to determine the
extent of this plastic flow as a function of the applied loading, the material properties of the
deforming solid, and the surface roughness.
The remainder of this paper is organized as follows. In the next section, we give a more careful
statement of the problem to be solved, the assumptions underlying our calculations, and the
results that will be calculated. Section 3 gives a brief review of the indentation response of an
elastic-perfectly plastic solid with sinusoidal roughness, as the first step towards analyzing the
Weierstrass profile. In Section 4, we review the procedure proposed by Ciavarella et al to
analyze contact of the Weierstrass profile, and devise a simplified, approximate procedure that
can be used to derive analytical results for both elastic and plastic contacts. Section 5 presents
detailed results, and section 6 contains conclusions.
The principal conclusion of our calculation is that for a perfectly fractal elastic-plastic surface,
the true contact size consists of an infinite number of contact spots with zero size, which are
subjected to a finite contact pressure. This limiting contact pressure is approximately twice the
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material hardness, due to lateral interactions between neighboring contacts. In addition, the total
area of contact between the two surfaces is finite. Thus, plasticity resolves some, but not all, of
the singular behavior associated with elastic contact between ideally fractal surfaces.
Determining the true contact spot size, and the number of true contacts, remain open questions.
2
100
3
S(ω) (µm )
10
10-2
10
-4
10-6
10-2
10-1
ω (rad/µm)
(a)
100
(b)
Fig 1. (a) A typical rough surface topography of single crystal copper, measured by the atomic
force microscope (courtesy of B.C.P. Burke). (b) Fractal properties can be deduced from the
power spectrum. The straight line is by least squares fitting, giving rise to D ≈ 1.47 .
2. Rough Surface Model
Majumdar and Bhushan (1991) suggest that the two dimensional Weierstrass function
∞
h ( x ) = G0 ∑ γ
( D −2 )m
m =0
cos(2π xγ / Λ 0 )
m
(1)
with γ ≈ 1.5 and D ≈ 1.4 provides a good qualitative description of many practical surface
profiles. As a representative example, we show in Fig 1(a) a three dimensional profile z ( x , y ) of
a surface (in this case a copper single crystal) obtained with an atomic force microscope. To
compare the surface profile with the representation in Eq.(1), we take a two-dimensional trace
z ( x, y = 20µ m ) of the surface in the x direction. We then compute the Fourier transform zˆ(ω ) of
the two-dimensional trace, and plot the power spectral density S (ω ) in Fig. 1(b). The power
spectrum for the ideal Weierstrass profile is a series of Dirac delta functions, but Berry and
Lewis (1980) argue that the Weierstrass distribution can be used to approximate a continuous
power spectrum by ensuring that the integral of the continuous power spectral density matches
that of the Weierstrass function. The resulting continuous approximation is given by
G2Λ
S (ω ) = 0 0 (Λ 0ω )2 D−5
(2)
2 log(γ )
5
and would correspond to a straight line with slope 2D-5 on Fig 1(b).
The Weierstrass function is clearly not a fully accurate description of real surface roughness.
The actual power spectrum clearly appears to be better approximated by a continuous function
than a series of Dirac delta functions. In addition, roughness wavelengths in the range 50 µ m 1mm are strongly sensitive to processing and in general are not well characterized by even the
continuous approximation to the Weierstrass function. Similarly, the Weierstrass profile becomes
unphysical at very short wavelengths: as m → ∞ , the surface becomes needle- like, with
roughness amplitude much greater than its wavelength. Any departure from ideal behavior at
long wavelength (low m) will lead to quantitative, but not qualitative changes in our predictions
and could readily be incorporated in the computations. The departure from ideal behavior at
short wavelengths (large m) has a critical influence on the true size of the contact spots, and for
ideally elastic surfaces, on the total true area of contact and contact pressure. It will not,
however, have a significant influence on the behavior of lower scales of roughness.
Experiments suggest that γ ≈ 1.5 and D ≈ 1.4 in Eq. (1) give the best qualitative fit to real
surfaces. Under these conditions, roughness scales with successive values of m do not clearly
decouple, however, and analyzing the contact between such surfaces is difficult. Ciavarella et al
(2000) show that calculations are greatly simplified by approximating a fractal surface with a
larger value of γ > 5 . The surface then consists of a series of harmonic profiles, with widely
separated wavelength, each of which can be analyzed independently. Although the resulting
surface does not resemble any actual surface, it does capture the self-affine fractal character of
real surfaces, and so provides an invaluable qualitative picture of the behavior of two contacting
fractal surfaces.
We now consider an elastic perfectly-plastic solid, with Young’s modulus E , Poisson’s ratio ν
and yield stress σ Y , which has a surface roughness defined by Eq. (1). The solid is indented by a
smooth cylinder with radius R as illustrated in Fig. 2. The cylinder is loaded by a total force (per
unit out of plane length) P, to induce a contact pressure distribution p ( x) . Of course, the two
surfaces actually meet only at the tips of asperities. We shall find that for an elastic-perfectly
plastic solid with an ideal Weierstrass profile, the true contact consists of an infinite number of
contact spots of zero size, which are all subjected to the same, uniform contact pressure. This is
a purely formal result, however, and does not characterize the loading or deformation of the solid
in a useful manner. We obtain a better picture by regarding the loading and deformation to
consist of a series of separate scales, n = −1,0,1...∞ , as illustrated in Fig. 2. The lowest scale,
n = −1 corresponds to the nominal contact. Successive terms n = 0,1,2...∞ characterize the
behavior of the solid at length scales comparable to successive wavelengths in the Weierstrass
profile. The fields associated with each term interact weakly through the requirement that the
pressure at scale n must be statically equivalent to the pressure at scale n+1.
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Scale -1
P
R
p-1 (x)
a-1
Scale 0
p0
p0 (x)
2a0
λ0
Scale 1
p1
p 1 (x)
2a1
λ1
Figure 2: Idealized two-dimensional model of contact between a smooth cylinder and a surface
with Weierstrass profile with widely separated scales. Successive scales interact through the
condition that the nominal pressure acting on scale n is equal to the true pressure acting on scale
n-1.
The nominal contact area and pressure (scale n=-1) can be computed using results for smooth
contacts (see e.g. Johnson 1985). It is helpful to review these results briefly. The severity of the
nominal load is conveniently characterized by a dimensionless load parameter PE * / σ Y2 R . For
PE* / σ Y2 R < 3.2 , the contact is elastic, and the contact width and nominal contact pressure
distribution follow as
4 PR
a−1 =
π E*
(3)
4 mean
2
2
p−1 ( x) = p− 1 1 − x / a−1
π
where
p−mean
= P / 2 a−1 ≤ 1.4σ Y
(4)
1
7
is the average nominal contact pressure. For PE * / σ Y2 R > 40 , the contact is fully plastic, and
the contact width and stress distribution can be approximated using the rigid plastic slip-line field
solution, which gives
a−1 = P /(2 H 0σY )
(5)
p−1 ( x) = H0σ Y
where H 0 ≈ 2.9 is the ratio of the plane strain hardness of the solid to its tensile yield stress. In
the intermediate regime 3.2 < PE* / σY2 R < 40 , the contact width and pressure distribution
would need to be computed using numerical simulations. This will not be attempted here: we
shall focus instead on the limiting cases of perfectly elastic and fully plastic bulk contacts.
It is important to note that the nominal contact geometry introduces a natural and well defined
long wavelength cut-off for the roughness spectrum. Roughness wavelengths greater than a −1
clearly do not influence the contact pressure. The roughness component with wavelength closest
to a −1 should be regarded as part of the nominal contact geometry (its effect is equivalent to
modifying the shape of the indenter). The first component of the profile to influence the contact
pressure distribution has index m = 1 + log(Λ 0 / a−1 )/log γ , and has wavelength and amplitude
λ0 = a−1 / γ
g0 = G0 ( γΛ0 / a−1 )
(6)
Henceforth, we will characterize the surface roughness within the area of contact with a
truncated Weierstrass profile given by
( D −2)
∞
h ( x ) = g0 ∑ γ
( D −2) n
n= 0
cos(2π xγ / λ0 )
n
(7)
where g 0 and λ0 represent the amplitude and wavelength of the first roughness component that
has wavelength less than the nominal contact width. It will be important to keep in mind that
while γ , D, G0 and Λ 0 are properties of the surface geometry only, g 0 and λ0 depend also on
the nominal contact geometry as shown in Eq. (6). We shall return to this issue when presenting
our results in Section 5.
If we now examine a small region of the nominal contact, with length λ1 << dx << a−1 and
centered at some point x within the contact area, we observe a sinusoidal surface with
wavelength λ0 in contact with a nominally flat indenter. The indenter is subjected to a uniform
nominal contact pressure p0 = p−1 ( x) , which induces a contact width 2a0 and pressure p0 ( x) on
the zero-th roughness scale. We can then examine more closely a region with size
λ2 << dx << λ0 within one of these contacts. Here, we observe a sinusoidal surface with
wavelength λ1 , subjected to nominal contact pressure p1 = p0 ( x) , inducing a contact size a1 , and
contact pressure p1( x) . This process can evidently be continued indefinitely.
The goal of our analysis is to determine, for each roughness scale n, the total number of contact
spots N n , the apparent size of the contact spots a n , and the apparent contact pressure
distribution pn ( x ) . Calculations are complicated by the fact that the qua ntities of interest vary
8
with position on the surface. It is possible (but exceedingly cumbersome) to calculate this spatial
variation. In practice, it is preferable to quantify the variation by computing distribution
functions, which, for example, specify the fraction of asperity contacts with size between a n and
an + dan , or the fraction of the nominal contact area that is subjected to pressure between pn and
pn + dpn , for each scale n = 0...∞ .
The distribution function for the pressure plays a particularly important role in subsequent
calculations. For each scale of roughness, we define a dimensionless function qn ( p / σ Y ) such
that qn ( p / σ Y ) ( dp /σ Y ) is the fraction of the nominal contact (which has total width 2a−1 ) that
is subjected to a dimensionless pressure between pn / σ Y and ( pn + dpn ) / σ Y . As an example,
the distribution function q−1 ( p / σ Y ) characterizing the nominal contact can easily be computed.
The cumulative distribution Q ( p) (the proportion of the contact over which the pressure exceeds
p) can be computed from the contact pressure distribution as Q−1 ( p ) = x ( p ) / a−1 . The
distribution function q−1 ( p) then follows as q −1 ( p ) = − ∂Q−1 p . For example, if the nominal
contact is elastic, we have
Q−1 ( p) = x ( p) / a−1 = 1 − ( π p / 4 p−mean
)
1
2
(8)
The distribution function follows as
ξ (π / 4ξ −mean
)
dQ
1
q−1 (ξ ) = −σ Y
=
mean
dp
1 − (πξ / 4ξ −1 ) 2
2
(9)
where ξ = p σ Y , ξ −mean
= p−mean
/ σ Y . Similarly, for fully plastic contact, the distribution function
1
1
is evidently
q−1 (ξ ) = δ (ξ − H 0 )
(10)
where δ denotes the Dirac delta distribution and H 0 ≈ 2. 9 is the ratio of the hardness to the
tensile yield strength of the solid, as before. Similar distributions will characterize the pressure
distribution for each higher roughness scale. Although it is not possible to write down a simple
expression relating the contact pressure pn ( x ) to qn ( p) for n ≥ 0 , Ciavarella et al (2000) show
that this is not necessary. Their procedure for computing q n will be reviewed in detail in Section
4.
The distribution function qn ( p) provides a convenient description of the contact pressure acting
on the nth roughness scale, but is difficult to interpret physically. To provide a more direct
measure of the deformation of each roughness scale, we introduce the contact size distribution
function Sn ( a) defined such that
dn = Nn Sn ( a) da
(11)
with N n the total number of contact spots at the nth scale, gives the number of contacts with size
between a and a + da . Similarly, as a more direct measure of the loading applied to the
contacts in the nth scale, we define the mean contact pressure for an individual contact spot as
9
p mean =
a
1
p ( x ) dx
2 a −∫a
(12)
We then introduce the contact pressure distribution Pn ( p mean ) such that
dn = Nn Pn ( p mean )dp mean
(13)
gives the number of contacts in the nth scale that are subjected to mean pressure between p mean
and p mean + dp mean . Naturally, since there is a direct relationship between the contact size and
mean contact pressure for individual contact spots, there will be a similar direct relationship
between Sn ( a) and Pn ( p mean ) .
Although we shall formally compute expressions for Sn ( a) and Pn ( p mean ) for our model surface,
we readily concede that in view of the rather crude description of surface topography that forms
the basis for our calculations, the exact distribution functions may have little practical
significance. In many cases the average values of contact size and pressure for each scale of
contact are sufficient. These are formally related to Sn ( a) and Pn ( p mean ) through
∞
∞
an = ∫ S n ( a )a da
pnmean = ∫ Pn ( p) p dp
0
(14)
0
In practice, however, it is simpler to compute an and
distribution functions. The total contact area
An = Nn 2 an
and the nominal contact pressure
pnnom = P / An
are also of interest.
pnmean
directly, rather than via the
(15)
(16)
We shall calculate these quantities using the elegant procedure developed by Ciavarella et al
(2000) to model contact between elastic solids with a Weierstrass profile. A key assumption in
their work is that the behavior of a particular roughness scale n at a point on the surface can be
calculated from the solution to a sinusoidal surface indented by a flat, frictionless punch.
Successive roughness scales interact only through the requirement that the nominal pressure pn
that acts on scale n is equal to the true pressure pn−1 ( x ) for the preceding scale n-1. This
assumes that no interaction occurs between the deformation fields associated with successive
roughness scales.
For an elastic solid, this assumption can easily be justified using Saint Venant’s principle. For
elastic-plastic materials, no rigorous proof exists. We speculate, however, that provided the bulk
contact (at scale n-1) consists of a single, isolated contact area, with a size much smaller than any
characteristic structural dimension, the same decoupling can be applied for elastic-plastic solids.
As an extreme example, consider the limiting case of a rigid-perfectly plastic solid. The slip- line
field solutions for contact between a semi- infinite solid and a single indenter invariably show a
rigid region under the contact area (Hill 1950; also reproduced in Johnson 1985). Consequently,
the deformation associated with the first roughness scale in a Weierstrass surface (scale 0) may
be embedded within the rigid region associated with the nominal contact. The behavior of the
10
first scale of roughness is therefore equivalent to that of a sinusoidal surface, with rigid boundary
conditions at infinity. Our numerical solution to this problem shows that a rigid regio n forms
under each contact area. Consequently, the second roughness scale can similarly be embedded
within this region, and this process can be continued for all higher roughness scales. Since the
deformation field constructed in this manner forms a kinematically admissible collapse
mechanism, our solution should strictly be regarded as an upper bound to the collapse load, and a
lower bound to the contact size and contact area. We present some further support for our
calculations in the Appendix A, where the analytical calculations are compared with the
predictions of full- field finite element computations for up to three roughness scales. It is
important to recognize, however, that if the deformation field for the bulk contact (scale -1) is
not contained, a rigid region may not form below the bulk contact in the fully plastic limit. In
this case, the first scale of roughness is not contained, and this behavior will propagate upwards
throughout all subsequent scales of roughness. Such unconstrained flow is particularly likely to
occur in metal forming operations.
3. Indentation response of a sinusoidal surface
Clearly, establishing the response of a simple sinusoidal surface to indentation loading is a key
step towards solving the contact problem outlined in the preceding section. The indentation
response of an elastic-perfectly plastic solid with a harmonic roughness was analyzed in detail by
Gao et al (2005). Only a few of their results are needed here: these are reviewed briefly below.
We focus attention on a single roughness scale with wavelength
λn = λ0 / γ n
(17)
and amplitude
g n = g 0γ ( D −2) n
(18)
as illustrated in Fig. 2. The substrate is an elastic-perfectly plastic solid with Young’s modulus
E, Poisson’s ratio ν and yield stress σ Y . Gao et al (2005) show that the strength of the surface
is conveniently characterized by a material and geometric parameter given by
g E*
ψn = n
(19)
λnσ Y
with E * = E 1 − ν 2 . In the discussion to follow, the value of ψ n for the zero-th roughness scale
(n=0) plays a central role in characterizing the resistanc e of the rough surface to plastic flow. In
terms of this parameter, we have
g0 E*
ψ0 =
, ψ n = ψ 0γ (D −1)n
(20)
λ0 σ Y
Note that the resistance to plastic flow varies in inverse proportion to ψ : a rough surface with a
low yield stress has a high value of ψ , while a smooth, hard surface would have a low ψ . In
addition, note that ψ n increases with n, so we anticipate that roughness scales with low
wavelength will be elastic, but all scales beyond a critical value of n will deform plastically.
(
)
11
The solid is indented by a flat, rigid surface, under the action of a nominal pressure pn , as
illustrated in Fig. 2. The loading induces a contact width 2 an , pressure distribution pn ( x ) and
mean pressure pnmean given by functional relationships
p
g 
2 an / λn = A  n ,ψ n , n 
λn 
 σY
 x p
pn ( x)
g 
= Φ  , n ,ψ n , n 
σY
λn 
 λn σ Y
(21)
p
pnmean
g 
= Φ mean  n ,ψ n , n 
σY
λn 
 σY
Here, we have shown that formally that A, Φ and Φ mean depend on both ψ n and g n / λn . In fact,
if the surface remains elastic, exact calculations show that the solution depends only on ψ n and
is independent of g n / λn . If the surface deforms plastically, numerical simulations show that the
functions depend only weakly on g n / λn . It would not be difficult to include this variation in our
computations, but for simplicity we have chosen to neglect it.
0.5
Interaction between
neighboring asperities
0.4
0.3
0.2
a/λ
Fully-Plastic
Elastic-Plastic
0.1
10 -1
Elastic
100
10 1
ψ=E*g/λσY
Fig. 3: A map illustrating the regimes of behavior of an elastic-perfectly plastic solid with
sinusoidal surface when indented by a rigid flat punch.
Finally, we must also compute the pressure distribution function qn ( p / σ Y ) , which is defined so
that qn ( p / σY )( dp /σ Y ) is the fraction of the sinusoidal surface that is subjected to normalized
12
pressure between pn / σ Y and ( pn + dpn ) / σ Y . This distribution function can be determined from
the contact pressure, following the procedure given for the nominal contact in equations (8) and
(9). The result will be specified by the functional relationship
 p 
 p pn

qn 
(22)
 = q  , ,ψ n 
 σY 
 σY σY

We have calculated approximations to the functions A, Φ , Φ mean and q by fitting curves to the
numerical simulations reported in Gao et al (2005). To specify the results, it is necessary to
distinguish between four general regimes of behavior: (i) Elastic; (ii) Elastic-plastic; (iii) Fully
plastic, with no interaction between neighboring asperities and (iv) Fully plastic, with interaction
between neighboring asperities. The distinction between these regimes is discussed in detail in
Gao et al (2005), and will be summarized only briefly here. Elastic behavior is observed at low
values of applied load; while the elastic-plastic regime occurs for loads that slightly exceed the
elastic limit. For sufficiently large values of load, the contact pressure distribution becomes
approximately uniform. This condition defines fully plastic behavior. In the fully plastic
regime, if the contact spots are small compared with the wavelength of the roughness, the
asperity contacts are isolated and the contact pressure is equal to the plane strain hardness of the
solid (≈ 2.9σ Y ). If the contact width approaches the roughness wavelength, the plastic zones
under neighboring contact spots begin to interact. Under these conditions, the contact pressure
remains approximately uniform, but its magnitude increases with applied load, and approaches
values up to twice the material hardness as 2 an → λn . Gao et al (2005) show that these four
regimes of behavior can be conveniently displayed on a map with a / λ and ψ as parameters.
The map is shown in Fig. 3: the elastic limit is given approximately by the condition
ψ a / λ = 0.1 , while ψ a / λ = 1 defines the transition to fully plastic behavior. In the sections
below, we give expressions for the functions defined in Eqs (21) and (22) in each of the four
regimes.
3.1 Elastic Regime 0 < ξ < πψ sin 2 (π /10ψ )
In the elastic regime, A, Φ and Φ mean are given by the Westergaard solution (Westergaard 1939)
 2 −1 ξ
,
ξ < πψ
 sin
A(ξ ,ψ ) = π
πψ
(23)

1,
ξ ≥ πψ

1/2
 2ξ cos(πη )
sin 2 (π A /2) − sin 2 (πη )}
ξ < πψ
{
 2
Φ (η ,ξ ,ψ ) =  sin (π A /2)
(24)
ξ + πψ cos(2πη )
ξ > πψ

Φ
mean
πξ


−1
(ξ ,ψ ) =  2sin ( ξ /πψ )

ξ

13
ξ ≤ πψ
ξ ≥ πψ
(25)


2

2
q (η ,ξ ,ψ ) =   (πψ − ξ ) + η

0

{
η /π
}{2ξ (πψ − ξ +
( πψ − ξ )
2
+η
2
) }

−η 

1/2
η < 2 πψξ
2
(26)
η > 2 πψξ
Note that the elastic solution depends only on the normalized nominal pressure ξ = p / σY and
surface property ψ , and is independent of g n / λn . The critical load ξ = p / σ Y = πψ corresponds
to full contact ( 2 a / λ = 1 ). This suggests a second physical interpretation for the parameter ψ : a
surface with a low value for ψ can easily be flattened.
This solution is valid below the elastic limit: Gao et al (2005) suggest that a convenient (but
approximate) estimate for the yield point is given by the condition ψ a / λ < 0.1 , which requires
that ξ = pn / σ Y < πψ sin 2 (π /10ψ ) .
3.2 Elastic-Plastic regime πψ sin 2 (π /10ψ ) < ξ < 2 H 0 / ψ
The elastic-plastic regime of behavior is observed for applied loads that slightly exceed yield.
Under these conditions, the elastic and plastic strains under the indenter are comparable, and
exact expressions for A, Φ and Φ mean cannot be found. Detailed numerical results are given in
Gao et al (2005). For the problem at hand, however, we find that at most a single scale of
roughness is in the elastic-plastic regime, so it is not necessary to characterize this regime in
detail. Consequently, we have constructed simple functions for A, Φ and Φ mean by using a selfconsistent interpolation between the elastic and fully plastic regimes. The interpolation is
constructed to predict correctly: (i) the elastic limit, (ii) the critical load to transition to fully
plastic behavior; (iii) the contact pressure distribution at the elastic limit; and (iv) the contact
pressure at the fully plastic limit. In addition, we interpolate the contact size between the elastic
and fully plastic limits as follows
(ξ − ξ p ) A + (ξ − ξ e )0.9 A
A(ξ ,ψ ) =
(27)
(ξe − ξ p ) e (ξ p − ξ e )0.9 p
where
 π 
2H 0
ξ e = πψ sin 2 
ξp =
H 0 = 2.8

ψ
(28)
 10ψ 
Ae = 1/(5ψ )
Ap = 2 / ψ
define dimensionless values of the critical nominal pressure at yield; the critical nominal
pressure at full plasticity, and the normalized contact widths at yield and the fully plastic limit,
respectively. The contact pressure distribution is approximated as
1/2
2α (ξ ,ψ )cos(πη )
Φ (η ,ξ ,ψ ) =
sin 2 (π A /2) − sin 2 (πη )} + β (ξ ,ψ )
(29)
{
2
sin (π A /2)
where the scaling factors α and β are introduced to ensure that the contact pressure has the
correct average and maximum values. This requires
14
( Aξ
α (ξ ,ψ ) =
max
− ξ ) sin(π A /2)
2ξ − ξ max sin(π A /2)
(30)
2 A − sin(π A / 2 )
2 A − sin(π A /2)
where the maximum contact pressure is interpolated between its values at yield and full plasticity
as
( A − Ap ) ξ + H 0 ( A − A0 )
ξ max =
(31)
( A0 − Ap ) A ( Ap − A0 )
β (ξ ,ψ ) =
where
A0 =
(
Ae ( H 0 − ξe / Ae ) + Ap ξe / Ae − 2 πψξ e
)
H0 − 2 πψξ e
is a constant chosen to ensure that the peak contact pressure has the correct value
ξ max = 2 πψξe at the elastic limit ( A = Ae , ξ = ξ e ). The mean contact pressure follows as
Φ mean (ξ ,ψ ) = ξ / A
In addition, the contact pressure distribution function may be computed as
(η − β H 0 )/ π
q (η ,ξ ψ
, )=
1/2
{ω 2 + (η − β H ) 2} 2α ω + ω 2 + (η − β H ) 2 − (η − β H ) 2 
0
0
0


ω = α cot 2 (π A /2)
{ (
)
}
(32)
(33)
(34)
3.3 Fully plastic regime – non-interacting asperities 2 H 0 /ψ < ξ < 2H 0 / 3
In the fully plastic regime, the plastic strains under the indenter greatly exceed the elastic strains,
and the stress fields approach the rigid-plastic limit (given e.g. by slip line field solutions).
Here, it is necessary to distinguish between two distinct types of behavior. As long as the
contact spots are small (numerical results suggest 2 a / λ < 0.67 ), there is no interaction between
neighboring asperity contacts, and A, Π and Π mean can be estimated from slip- line field
solutions (or numerical simulations) for a flat punch indenting a rigid-plastic half-space as
A(ξ ) = ξ / H0
Φ(x ) = H 0
( x / a < 1)
(35)
Φ mean = H0
where H 0 ≈ 2.9 is the hardness of the material normalized by its tensile yield stress. Numerical
simulations and slip- line field solutions show that H 0 varies weakly with g / λ , but this
variation is insignificant for our present purposes. Note that in this regime the contact pressure
distribution is perfectly uniform, and the solution is independent of ψ . Finally, the pressure
distribution function follows as
q (η ,ξ ,ψ ) = A(ξ ,ψ )δ (η − H0 )
(36)
where δ ( x ) denotes the Dirac delta distribution.
15
3.4 Fully plastic regime – interacting asperities 2 H 0 / 3 < ξ
When the size of the contact spots becomes comparable to the wavelength, the plastic strain
fields associated with neighboring asperities begin to interact. Numerical simulations show that
in this regime, the contact pressure distribution remains approximately uniform, but the
magnitude of the pressure increases substantially, and in some cases reaches twice the hardness
of the material. This behavior has a dramatic influence on the behavior of fine roughness scales.
As in Section 3.2 we have devised approximate analytical approximations to numerical
simulations to simplify our calculations. It is simplest to specify the relationships between ξ ,
ψ , A and Φ mean parametrically as
Φ mean = H 0 + ( H 1(ψ ) − H 0 )( 3 A − 2 )
3
(37)
Approximation
Numerical
5
Mean contact pressure pmean/σY
Normalized contact pressure p/σY
ξ = AΦ mean
(38)
where H1(ψ ) is the limiting value of the normalized contact pressure p ( x ) / σ Y at full contact.
The following function gives an approximate fit to the results shown in Fig. 8 of Gao et al (2004)
H 1(ψ ) = 5.8 − 60/(16 +ψ 3 / 2 )
(39)
It is straightforward to solve these equations to determine the functions defined in Eqs (21). The
results involve the solution to a cubic equation, however, and so are too lengthy to be recorded
here. Finally, the contact pressure distribution function in the interacting asperity regime has the
same form as that given in Eq. (36).
ψ=50
ψ=10
4
ψ=3.5
3
ψ=2
2
ψ=1.0
1
0
0.2
0.4
0.6
0.8
1
Approximation
Numerical
5
ψ=50
ψ=10
4
ψ=3.5
3
ψ=2
2
ψ=1.0
1
0
0.2
0.4
0.6
0.8
1
Contact fraction 2a/λ
Contact fraction 2a/λ
Fig. 4: The relationship between nominal pressure and mean pressure and contact fraction for a
sinusoidal surface in contact with a rigid, flat surface. The symbols show the results of finite
element comp utations, while the solid lines show the approximate analytical expressions used in
our rough surface model.
16
Our approximate curves are compared with the results of finite element simulations in Fig. 4,
which shows a parametric plot of the nominal and mean contact pressures as a function of the
contact size as the load increases, for various values of ψ .
4. Contact mechanics for a general Weierstrass profile
We proceed to analyze contact of the Weierstrass profile. Our analysis follows exactly the work
of Ciavarella et al (2000) and so will be reviewed only briefly. We will first present a fully
general procedure for computing distribution functions for contact size and pressure for each
scale. Subsequently, we suggest a simple approximation that can be used to obtain closed- form
estimates for the number of contact spots, their size, and mean pressure for each scale of loading.
4.1 Results for general surface contact
Although it is difficult to compute the spatial contact pressure distribution for the Weierstrass
surface, it is straightforward to determine the probability distribution function qn ( p / σ Y ) defined
in Section 3. The calculation begins by computing the distribution function q−1 ( p / σY )( dp /σ Y ) ,
which specifies the proportion of the nominal contact area that is subjected to a pressure between
p σ Y and ( p + dp ) σ Y . The result is given in Eq. (9).
The distribution q0 ( p ) for the zero-th roughness scale may therefore be computed as follows.
Recall that we introduced the dimensionless function q, such that q ( p / σ Y , ξ ,ψ )( dp / σ Y )
specifies the proportio n of the sinusoidal surface that is subjected to pressure between p σ Y and
( p + dp ) σ Y when loaded by a normalized nominal pressure ξ . The zeroth roughness scale is
subjected to a distribution of nominal pressure, as quantified by q−1 ( p / σ Y ) . Summing over all
possible values of the contact pressure for the nominal contact, we find that
∞
q0 ( p / σY ) = ∫ q ( p / σY , ξ ,ψ 0 ) q−1 (ξ ) dξ
(40)
0
The upper limit on the integral here is purely formal, of course: the nominal contact pressure
does not exceed p−1 = 2 P / π a−1 , so q−1 (ξ ) = 0 for ξ > 2P / π a−1σ Y . This result can be applied
recursively, to see that
∞
qn ( p / σ Y ) = ∫ q ( p / σ Y , ξ ,ψ n ) qn−1 (ξ ) dξ
(41)
0
Once the hierarchy of distribution functions qn ( p / σ Y ) has been computed, various quantities of
interest can be deduced. For example, to calculate the total (apparent) area of contact for the nth
scale, note that the area of contact for one wavelength of the nth scale subjected to pressure p is
2 an = λn A ( p σ Y ,ψ n ) . In addition, there are 2 a−1 / λn such wavelengths within the nominal area
of contact. The total contact area therefore follows as
∞
An = 2a−1 ∫ A (ξ ,ψ n ) qn−1 (ξ )d ξ
0
17
(42)
The total number of contact spots at the n -th scale within the nominal contact is
A
N n = n−1
λn
whereupon the average size of the contacts at the nth scale follows as
A
A
2 an = n = λ n n
Nn
An−1
The average contact pressure can be found by a similar approach, with the result
pnmean =
(43)
(44)
∞
2a −1
Φ mean (ξ ,ψ n )qn −1 (ξ ) dξ
An−1 ∫
(45)
0
Finally, the nominal contact pressure for the nth scale follows as
mean
n
p
∞

P
mean 
=
= p−1 ∫ A ( ξ ,ψ n ) qn−1 (ξ ) d ξ 
An
0

−1
(46)
We can also find expressions for the contact size and mean pressure distribution functions Sn ( a)
and Pn ( p mean ) defined in Eqs. (11) and (13). To compute the distribution function for the contact
size, denote the pressure required to induce a contact size a is given by the functional
relationship p / σ Y = A−1(2a / λ ,ψ ) , where A−1 denotes the inverse of the function A defined in
equation (21). Given the distribution function for the contact pressure at (n − 1) -th scale, we can
easily compute the probability distribution function of the contact fraction for scale n as follows.
For a single contact, 2 an / λn = A ( pn−1 σ Y ,ψ n ) , so that let p n−1 / σ Y = A−1 ( 2an / λn ,ψ n ) . The
probability of finding a contact fraction in between 2 an / λn and 2(an + dan ) / λn is
S n (2a n λ n ) =
dA −1
q n−1 A −1 (2a n λn ,ψ n )
d (2a n λ n )
(
)
(47)
4.2 Simplified analysis – the uniform contact pressure assumption
The results listed in the preceding section can be simplified dramatically if the contact pressure
distribution for each scale in the contact (including the nominal contact area) is uniform. This is
an excellent approximation when each scale of contact is fully plastic. At lighter loads, the
nominal contact, and some long-wavelength roughness scales, will remain elastic, and for these
scales the distribution of contact pressure is not uniform. Surprisingly, however, we have found
that averaged quantities such as the total area of contact, the average contact size, and the
average contact pressure can be computed remarkably accurately even for elastic surfaces by
replacing this non-uniform contact pressure distribution with a statically equivalent, uniform
pressure.
Proceeding along these lines, therefore, we suppose that the entire nominal contact area is
subjected to the same contact pressure. For this approximation, the distribution function
q−1 ( p / σ Y ) , which specifies the fraction of the nominal contact area (scale -1) subjected to
pressure between p σ Y and ( p + dp ) σ Y , must therefore be given by
18
q−1 (ξ ) = δ (ξ − p−mean
/σY )
1
(48)
where δ ( x) denotes the Dirac delta distribution and p
= P / 2 a−1 . The distribution function
for the zero-th scale of roughness follows from Eq. (41) as
mean
−1
∞
q0 ( p / σ Y ) = ∫ q( p / σ Y , ξ ,ψ 0 )δ (ξ − p−mean
/ σ Y ) d ξ = q( p / σ Y , p−mean
/ σ Y ,ψ 0 )
1
1
(49)
0
Recall now that the function q ( p / σY , p−mean
/ σ Y , ψ 0 ) specifies the fraction of a sinusoidal surface
1
that is subjected to pressure between p σ Y and ( p + dp ) σ Y , when indented by a nominal
contact pressure p−mean
. If the pressure is approximated by a statically equivalent uniform
1
distribution, with magnitude p mean , it follows that
(
q ( p / σ Y , p−mean
/ σ Y ,ψ ) = A( p−mean
/ σY ,ψ )δ p / σ Y − Φ mean ( p−mean
/ σ Y ,ψ , )
1
1
1
)
(50)
where the functions A and Φ mean are defined in equations (21), and given specific forms in Eqs.
(23)--(39). Repeating this procedure for subsequent roughness scales gives the following result
for the distribution functions qn ( p)
{
}
qn ( p / σ Y ) = Π A( p j −1 / σY ,ψ j ) δ ( p / σ Y − Φ
n
mean
j =0
mean
( pn−1 / σY ,ψ n ) )
mean
Here, the mean contact pressure at each scale can be determined from
pnmean / σ Y = Φ mean ( pnmean
−1 / σ Y ,ψ n )
(51)
(52)
where the sequence is started with p−mean
given in Eq.(48).
1
Given pnmean for each scale, and the distribution functions qn ( p / σ Y ) , we can determine the
remaining quantities of interest using Eqs. (42)-(46). The total area of contact at the nth scale
follows as
∞
n
An = 2a−1 ∫ A (ξ ,ψ n ) qn −1 (ξ )d ξ = 2a −1 Π A( p mean
j −1 / σ Y , ψ j )
j= 0
0
(53)
where the function A is defined in Eq. (21). The number of contacts follows as
n −1
N n = 2( a−1 / λn ) Π A( p mean
j −1 / σ Y ,ψ j )
(54)
j= 0
and the average contact size is given by
2 an = λn A( pnmean
−1 / σ Y ,ψ n )
(55)
Finally, equation (46) provides an additional way to compute p
{
mean
n
}
as
−1
n
pnmean = p−mean
Π A( p jmean
1
−1 / σ Y ,ψ j )
j =0
(56)
from which we also recover the expected relationship between the mean pressure at successive
scales
mean
pnmean = pnmean
(57)
−1 A( pn −1 / σ Y , ψ n )
19
4.3 Approximate analytical expressions for average contact size, pressure, number of contacts,
and total contact area for elastic contact
If the contacts remain elastic, the procedure outlined in the preceding section can be used to
derive simple analytical expressions for the average contact size, average contact pressure.
Under these conditions we may approximate
2
ξ
2 ξ
A(ξ ,ψ ) = sin−1
≈
π
πψ π πψ
Φ
mean
(58)
πξ
π
(ξ ,ψ ) =
≈
πξψ
−1
2sin ( ξ /πψ ) 2
whereupon Eq. (52) reduces to
pnmean π pnmean
π
−1
=
πψ n =
σY
2 σY
2
This recursion relation is satisfied by
pnmean
−1
πψ 0 γ ( D−1) n
σY
2 −2− n
mean
2− n −1

pnmean  π 
( D −1) n  p−1
= 
γ D −1 )
πψ 0γ
(


σY
2
 σ Y πψ 0 
The average contact size may be estimated using Eq. (55) as
− 2− n
 p

π 
2 an = λn  
γ D−1 )
(


2
 σ Y πψ 0 
Similarly, the total area of contact at the nth scale follows as
2− n −1
mean
−1
2
2− ( n+1)
(60)
−(n +1)
(61)
1− 2
−n
(59)
2 −2
mean
− ( D −1)( n−1+ 2− n )  p

π 
−1
An = 2a−1  
γ


2
 σ Y πψ 0 
The total number of contacts at the nth scale can be estimated as
N n = An−1γ (n+1) (2a−1 )
For large n, these results may be approximated by their asymptotes
−( n +1)
(62)
(63)
2
pnmean  π 
=   πψ 0 γ ( D−1)n
σY
2
2 < an >= λn γ (1− D)
(64)
(65)
−2
mean
  π −2 P
−( D −1)n  p−1
π 
An = 2a−1   γ

= 
2
 σY πψ 0   2  σ Y πψ n
(66)
−2
mean
 π  D− ( D−2 ) n  p−1 
Nn =   γ


2
 σ Y πψ 0 
(67)
As expected, equation (66) shows that the total contact area at the nth scale (for large n) is
proportional to the contact load P and independent of the nominal contact size a −1 or the
indenter geometry (other than through its effect of setting the wavelength of the zeroth roughness
20
scale). This result also provides an opportunity to check the predictions obtained using the
constant pressure approximation (Sect 4.2) with the exact solution. Ciavarella et al. (2000) find
that An = 0.391P /(σ Y πψ n ) , giving a coefficient that is within 4% of the approximate value in
Eq. (66). A more detailed comparison of the approximate and exact calculations in section 5
shows similar agreement under all conditions.
Note also that Eq. (64) shows that the mean contact pressure acting on each roughness scale
becomes progressively more severe as n increases. Consequently, for an elastic-plastic solid, the
short wavelength roughness scales will always deform plastically. We also see that the formal
limit n → ∞ for a perfectly elastic solid consists of an infinite number of contact spots, with zero
size, which are subjected to infinite contact pressure. The total true contact area vanishes.
4.4 Deformation and loading for the most heavily loaded region at each scale (elastic contact)
To calculate the conditions necessary to initiate yield at each successive scale of roughness, or to
identify conditions that will just begin to bring a roughness scale into complete contact with the
indenter, it is of interest to determine the contact size and contact pressure for the most heavily
loaded asperities at each successive scale of roughness. We assume that all roughness scales, as
well as the nominal contact, remain elastic; and assume also that γ ( D−1) > 2 . Equation (3) shows
that the maximum contact pressure exerted by the nominal contact is
p−max
PE*
1
=
σY
π Rσ Y2
(68)
The most severely loaded asperities on the zeroth roughness scale are therefore subjected to a
nominal contact pressure p0 = p−max
1 . Equation (24) gives the maximum contact pressure for a
sinusoidal surface that is subjected to a normalized nominal pressure ξ = p / σY as
whence
max
0
p
the
maximum
p max (ξ ,ψ ) / σ Y = 2 πξψ
contact pressure on the
zeroth
roughness
(69)
scale is
/ σY = 2 π p ψ 0 / σY . Similarly, the maximum contact pressure on successive scales of
max
−1
roughness follows as pnmax / σY = 2 π pnmax
−1 ψ n / σ Y . The recursion relation is satisfied by
2− ( n+1)
max
2− n −1

pnmax
2− 2− n
( D −1) n  p−1
= ( 2)
γ D −1 )
πψ 0 γ
(70)
(


σY
 σ Y πψ 0 
Other than a numerical factor, this result is identical to eq (60), which estimates the average
contact pressure on each roughness scale. However, Eq. (70) does not assume isolated Hertztype contacts and so is valid for all values of ψ 0 , provided that no scale reaches full contact.
21
4.5 A graphical approach to calculating contact size, pressure and area for roughness scales in
the fully plastic regime
If the bulk contact is fully plastic (which requires
PE * / σ Y2 R > 40 ), and the surface roughness
is such that ψ 0 > 16 , then the procedure outlined in section 4.2 gives exact results, since all
roughness scales are in the fully plastic regime, so the contact pressure distribution at each scale
is uniform. In addition, under these conditions, the method can be illustrated using a simple
graphical procedure, originally suggested by Childs (1973, 1977) in a discussion of the
persistence of asperities under contact loading.
We begin by simplifying the behavior of the sinusoidal surface for the fully plastic regime.
Consider a single scale of roughness with wavelength λ , which is indented by a rigid platen.
The surface is subjected to a nominal contact pressure (force per wavelength) p , which induces
a contact size a. Each contact spot is subjected to a uniform contact pressure with magnitude
p mean . For values of ψ 0 > 16 , the relationship between the mean normalized contact pressure
and contact size for the fully plastic regimes given in Eqs. (35)-(39) can be approximated by
A< 2/3
 H0
p mean / σ Y = Φ mean = 
(71)
3
A > 2/3
 H0 + ( H1 (ψ → ∞ ) − H 0 )( 3 A − 2 )
where A = 2a / λ , H 0 ≈ 2.9 and H1(ψ → ∞ ) = 5.8 . In addition, the normalized nominal contact
Contact pressure p/σY
pressure follows as ξ = p / σ Y = AΦ mean . Both these functions are plotted in Fig. 5.
5
4
3
mean
pmean/σ Y
p1 /σ Y = p 2/σY
pmean
/σ Y = p 1/σY
0
mean
p -1 /σY = p0/σ Y
2
p/σY
2a2/λ
2a 1/λ
2a0/λ
1
0
0.2
0.4
0.6
0.8
1
Contact fraction 2a/λ
Fig. 5: A graphical technique for calculating the contact size and contact pressure for successive
scales of a rigid-perfectly plastic surface with Weierstrass profile. The solid lines show the mean
pressure p mean / σ Y and nominal pressure p / σ Y for a rigid plastic sinusoidal surface as a
function of contact fraction. Successive scales of roughness are related by the condition that
p n / σ Y = pnmean
−1 / σ Y .
22
Next, we apply the iterative scheme outlined in Section 4.2 for this simplified situation. Since
the nominal contact (scale -1) is in the fully plastic regime, the nominal contact pressure is
uniform, and has magnitude p−1 / σY = 2.9 . This implies that the nominal pressure exerted on
asperities in the zero-th roughness scale is p0 / σ Y = 2.9 . Fig. 5 can be used to deduce the
normalized contact size 2 a0 / λ0 corresponding to this nominal contact pressure; whence the true
contact pressure p0mean / σ Y can also be found. The same procedure may be used to progress to
higher scales of roughness: the condition pnmean / σ Y = p n+1 / σ Y determines the normalized contact
size an +1 / λn+1 at each successive roughness scale, whereupon pnmean
+1 / σ Y can be deduced. In
addition, the apparent total contact area at each scale follows as
n 2a
j
An = 2a−1∏
(72)
λ
j =0
j
In particular, the total contact area in the limit n → ∞ is
H0
P
Lim An = 2a−1
=
(73)
n →∞
H 1 (ψ → ∞ ) 5.8σ Y
Thus, we find that the contact area is again proportional to the normal load, but instead of
converging to zero as n → ∞ , the total contact area converges to a finite value.
A similar conclusion was reached by Willner (2004) using a more sophisticated series of
numerical simulations. His analyses did not account for interactions between neighboring
contact spots, however, and consequently gave a limiting area of Lim An = P H 0σ Y . More
n→ ∞
recently, Pei et al (2005) have conducted an extensive study of the indentation response of an
elastic-plastic solid with a nominally fractal surface roughness using three-dimensional finite
element computations. They found that the true area of contact is a function of the ratio of yield
stress to Young’s modulus for the solid: this behavior is most likely due to an artificial truncation
of the roughness spectrum resulting from the resolution of the finite element mesh. For
sufficiently low values of σ Y / E , however, they found that the true contact area was related to
the load by Atrue = P / 6σY , in excellent agreement with our approximate estimate.
The convergence of contact size and contact area to their limiting values are illustrated in Fig. 6.
Since the normalized contact size 2 an / λn → 1 as n → ∞ , while Lim An / 2a−1 = H 0 / H1 (ψ → ∞ ) ,
n →∞
we have plotted 2 an / λn − 1 and An /(2a−1 ) − H0 / H1(ψ → ∞) as functions of n. Evidently, both
contact size and total contact area converge to their asymptotic values exponentially: indeed, it is
straightforward to show that for sufficiently large n , the contact size and total contact area may
be approximated by
2 an / λn ≈ 1 − 0.1e −0.2n
(74)
An /(2a−1) ≈ H 0 (1 − 0.45e −0.2 n ) / H1 (ψ → ∞)
23
Difference from asymptote
Total contact area
An /a-1 -H0/H1 (ψ→∞ )
-1
10
10-2
10-3
0
Contact fraction
1-2an/λ n
5
10
15
20
Roughness scale n
Fig 6: Illustrating the convergence of total contact area and contact fraction towards the ir rigid
plastic asymptotes with roughness scale n.
These results can also be used to deduce the asymptotic behavior of fine-scale roughness on an
elastic-plastic solid. We saw in Section 4.3 that even if the bulk contact and the first few
roughness scales remain below yield, the contact pressure on an elastic surface increases for
progressively higher scales of roughness. Consequently, there will always be a critical scale of
roughness that reaches yield. By the same argument, there will always be a critical roughness
scale which reaches the fully plastic regime. For higher (shorter wavelength) scales of roughness
the graphical method illustrated in Fig 5 can again be used to deduce the contact pressure and
contact fraction. For sufficiently large n, the true contact area and contact fraction will again
approach their asymptotes as shown in Fig 6.
It is interesting to note the important role played by interactions between neighboring asperities
in determining the asymptotic behavior of fine scale roughness. Without this interaction, the
contact pressure on each scale of roughness is bounded by the material hardness (i.e. the limiting
pressure under an isolated indenter). For a rigid plastic contact, therefore, all roughness scales
(including n=0) must reach full contact an / λn = 1 to support the load. For a more general
elastic-plastic contact, the total true contact area and the true contact spot size would be
determined by the first roughness scale to reach the fully plastic limit, since all shorter
wavelength scales of roughness must be at full contact to support the contact pressure. The true
contact spot size could then be computed unambiguously. This, in fact, is the simple picture of
plastic contact between two rough surfaces proposed by Tabor (1951), although his focus was on
computing the total area of contact, and did not address the question of the contact spot size.
24
5. Results
We now discuss in detail the predictions of the calculations outlined in the preceding sections.
We suppose that the power spectral density S (ω ) has been measured for some (two dimensional)
surface of interest, as illustrated in Fig. 2. To extract the fractal parameters Λ 0 , G0 , D for the
surface, we fit a straight line through the high frequency portion of the spectrum, and find D by
noting that on a log-log plot the line has slope 2 D − 5 . The value of Λ 0 can be chosen
arbitrarily, as long as Λ 0 exceeds the expected width of the nominal contact. The value of G0
should then be chosen so that the point ( Λ 0 , G0 ) is consistent with the discrete approximation to
the PSD, which requires S (ω = 2π Λ 0 ) = G 02 Λ 0 2 log (γ ) . In addition, an arbitrary value may be
used for γ : Majumdar and Bhushan suggest that γ = 1.4 produces the most realistic looking
surfaces, but our calculations are strictly only valid for γ >> 1 , so choosing γ = 5 or greater is
more appropriate.
We now suppose that the surface is indented by a cylinder of radius R, which is subjected to a
load P per unit length out of plane, as illustrated in Fig. 1. It is convenient to characterize the
applied load, surface geometry and material properties using three dimensionless groups
 G0E * 
 γΛ02 
PE *
Σ=
Ψ =
Κ =
(75)


RσY2
 Λ 0σ Y 
 G0 R 
Here, Σ is a dimensionless load parameter, and is proportional to the mean nominal contact
pressure, through Eq. (4). The parameters Ψ and Κ together characterize the surface roughness
and its resistance to plastic flow.
As a preliminary step, we express several parameters of interest in terms of Σ , Ψ and Κ . If the
nominal contact is elastic ( Σ < 3.2 ) the contact size is a−1 / Λ0 = (2/ π ) γΣ /(ΨK ) , while for a
fully plastic nominal contact ( Σ > 40 ) a−1 / Λ0 = γΣ 2 /(2.8Ψ K ) . The wavelength of the zeroth
roughness scale (see Fig. 1) follows as λ0 = a−1 / γ , so we find that the plasticity index ψ 0
(defined in Eq. (19)) for the zeroth roughness scale is given by
Ψ D π Κ / 2Σ D−1 (elastic nominal contact Σ < 3.2)

ψ0 = 
(76)
D −1
 Ψ D ( 2.8Κ / Σ 2 )
(plastic nominal contact Σ > 40)

(
)
5.1 Conditions for inducing yield
Controlling plastic flow under contact loading is a central issue in designing failure resistant
surfaces for tribological applications. With this motivation, we begin by estimating the
conditions necessary to initiate yield in our idealized model of a surface with fractal roughness.
The general nature of the contact between a smooth cylinder and the Weierstrass surface is
illustrated in Fig. 2. As discussed in detail in section 2, we focus attention on each successive
scale of roughness under the contact in turn. Strictly, our calculations show that short
25
wavelength roughness scales will always deform plastically, so the surface does not have an
elastic limit. Instead, our goal is to calculate the number of roughness scales that remain elastic,
as a function of the load (parameterized by Σ ) and surface roughness and material properties,
parameterized by Ψ and Κ .
We assume for simplicity that the nominal contact remains elastic, which requires Σ ≤ 3.2
(Johnson, 1985). The most heavily loaded region of the zeroth scale of roughness occurs at the
center of the nominal contact, and is subjected to a normalized contact pressure
p0 / σ Y = p−max
1 / σ Y = Σ / π . The elastic solution for indentation of a sinusoidal surface (Section
3) shows that the zeroth roughness scale will behave in one of two ways, depending on the value
of ψ 0 . For ψ 0 < 0.2 , the zeroth roughness scale will remain elastic up to full contact, which
occurs at a critical load p0 / σ Y = Σ / π = πψ 0 . For ψ 0 > 0.2 , the zeroth scale of roughness will
reach yield before full contact. The critical nominal contact pressure at yield is given by
p0 / σ Y < πψ 0 sin 2 (π /10ψ 0 ) .
Nominal contact yields
10
-2
Sc ale
e las
-3
10
10
Scale 1 elastic
-1
tic
2 el a
sti c
10
Sca l
e3
Load factor Σ
10
All roughness
scales plastic
0
-2
Scale 0 elastic
10
-1
0
10
Normalized roughness ΨDΚD-1
10
1
Fig. 7: A map showing the behavior of successive roughness scales during indentation by a
smooth cylindrical indenter. Dashed lines show conditions necessary to bring each scale to full
contact; solid lines mark transitions from elastic to plastic behavior.
Similar arguments can be applied to each successive scale of roughness n = 1,2,3...∞ . The
maximum nominal contact pressure acting on each roughness scale can be computed from eq
max
2
(70); the criteria pnmax
−1 / σ Y = πψ n and pn −1 / σ Y < πψ n sin (π /10ψ n ) give the conditions for full
contact and yielding on the nth scale of roughness, respectively.
The results of these calculations are summarized in Fig 7, which shows the critical normalized
load Σ required to (i) just bring each roughness scale to full contact (shown as dashed lines), and
(ii) just cause yield in each roughness scale (shown as solid lines). The results depend (weakly)
on D and γ , and are shown for the particular case D = 1.5, γ = 5 .
26
At extremely low values of applied load, only the zeroth scale of roughness deforms elastically;
all shorter wavelength scales n = 1,2..∞ are plastic. As the load is increased, however, additional
roughness scales can transition to elastic behavior. This apparently contradictory behavior is
readily explained by recalling that the cont act size increases with load. While the contact is
small, all roughness scales with wavelength smaller than the contact size have large amplitude
compared with their wavelength, and therefore readily deform plastically. As the load is
increased, the nominal contact spreads, and therefore new roughness scales that come under the
influence of the indenter have a lower ratio of roughness amplitude to wavelength, and so may
remain elastic, and eventually reach full contact.
The parameter Ψ DΚ D −1 evidently plays the role of a plasticity index for the surface: a contact
with a low value of Ψ DΚ D −1 has a large number of roughness scales that deform elastically up to
the load where the nominal contact yields, while a contact with high Ψ DΚ D −1 will experience
extensive plastic flow even at modest loads.
0
Plastic zone depth dΨΚ/Λ0
10
ΨDΚD-1 = 1
-1
10
10-2
Ψ Κ
D
-3
10
D-1
= 0.2
10-4
ΨD ΚD-1 = 0.06
-5
10
-6
10
10
ΨD ΚD-1 = 0.02
-6
10
-5
10
-4
10
-3
10
-2
Load Factor Σ
10
-1
10
0
Fig 8: The depth of the plastically deformed layer induced by indenting a Weierstrass surface
with a rigid cylinder.
For practical purposes it is of particular interest to estimate the depth of the plastically deformed
region under the contact. As a rough guide, the depth of the plastically deformed region can be
taken to be one half the wavelength of the first roughness scale to reach yield, since longer
wavelengths remain below yield, while all shorter wavelengths will exceed yield. This
approximation gives d yield / Λ 0 ≈ a− 1 /(2Λ 0γ k +1) = Σ /(γ k ΨΚ π ) , where k is the number of
roughness scales that remain in the elastic regime, which may be estimated from Fig 7. The
normalized depth of the plastically deformed region as a function of the load parameter Σ is
shown in Fig 8, for various values of the plasticity index Ψ DΚ D −1 , for D = 1.5, γ = 5 . We
observe three general regimes of behavior: for large values of Ψ DΚ D −1 , only the zeroth
27
roughness scale remains elastic at light loads, so the depth of the deformed layer is proportional
to Σ . For Ψ DΚ D −1 > 0.25 , the zeroth roughness scale yields at large loads, which results in a
sudden increase in the depth of the deformed layer. For 0.08 < Ψ DΚ D−1 < 0.15 , the zeroth scale
remains elastic up to full contact, while all remaining scales yield – in this case there depth of the
deformed layer is proportional to Σ over the full range. Finally, for Ψ DΚ D −1 < 0.08 , the one or
more roughness scales transition to elastic behavior at sufficiently large loads. Under these
conditions the depth of the deformed layer is proportional to the load parameter Σ at light loads,
but is independent of Σ above a critical load.
0
ela
c
s ti
10-1
0
10-2
al e
-3
Sc
10
ic
elas t
-2
1
Scale
10
las tic
-1
e
Scale 2
10
All roughness
scales plastic
Scale 3 elas tic
Load factor p-1/σY
10
100
101
Normalized roughness ψ0
Fig. 9: A map showing the behavior of successive roughness scales under a uniform nominal
pressure p− 1 . Dashed lines show conditions necessary to bring each scale to full contact; solid
lines mark transitions from elastic to plastic behavior.
For comparison, we have also calculated the conditions necessary to induce yield in a rough
surface with truncated Weierstrass profile given in Eq. (7), subjected to a uniform nominal
pressure p− 1 . In this case, we characterize the surface roughness by ψ 0 , defined in Eq. (20), and
characterize the load using the dimensionless ratio p −1 / σY . The critical load required to initiate
yield in successive scales of roughness is shown as a function of ψ 0 in Fig. 9, for D = 1.5 and
γ = 5 . The results resemble the conditions necessary to initiate yield under a cylindrical
indenter, except that in this case the zero-th roughness scale has a fixed wavelength and
amplitude, and therefore roughness scales do not transition from plastic to elastic behavior. It is
interesting to note that the yield criterion for scales n=4 and above is essentially independent of
nominal load, and is determined primarily by the roughness amplitude of the zeroth roughness
scale and the elastic and plastic properties of the surface, characterized by ψ 0 .
28
n=0
n=1
Fully
plastic
transition
n=2
-1
10
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10
10-2
10-3
10 -3
10-2
n=18
10 -1
10 0
Fully
plastic
transition
n=1
n=2
n=3
10 -4
10
n=16
n=0
10 -2
10 -3
-4
n=11
Distribution function Qn(p/σ Y)
Distribution function Qn(p/σY)
10 0
101
n=4
n=6
10 -2
n=8
n=10
10 -1
n=12
100
101
Normalized pressure p/σY
Normalized pressure p/σ Y
(a)
(b)
Fig 10 Cumulative contact pressure distributions Qn ( p / σ Y ) which specify the proportion of
each scale of roughness over which the pressure exceeds p / σ Y . Results are shown for a
cylindrical indenter subjected to nominal load Σ = 0.001 and for surfaces with (a) ψ 0 = 0.0002 ,
(b) ψ 0 = 0.1 .
5.2 Contact pressure distribution functions
We proceed next to examine the contact pressure distributions in more detail. The pressure
acting on each scale of contact is completely characterized by the distribution functions qn (ξ )
specifying the fraction of the contact. For an elastic surface, qn (ξ ) are smooth functions (see
e.g. Fig. 5 of Ciavarella et al 2000). For plastic contacts, qn (ξ ) contains singularities (this is a
consequence of the fact that contact pressure under fully plastic contacts is uniform - see
Appendix B for a more detailed discussion). Consequently, it is more convenient to plot the
cumulative contact pressure distribution functions Qn ( p / σ Y ) , which specify the proportion of
each scale of contact over which the pressure exceeds p / σ Y . The cumulative contact pressure
distributions can be computed from the definition q n = −σ Y ∂Qn ∂p . Results are shown in Fig
10 for a dimensionless nominal pressure Σ = 0.001 and two values of surface roughness, with
γ = 5 and D = 1.5 . Fig 10(a) shows results for ψ 0 = 0.0002 , in which case a large number of
scales under the contact are elastic. Under these conditions, the pressure distributions for
roughness scales n = 0...9 , which deform elastically, tend towards a self-similar distribution.
(Ciavarella et al 2000). The most heavily lo aded region of the 10th roughness scale begins to
deform plastically, while the 11th reaches the fully plastic limit (which occurs when the contact
pressure reaches 2.9σ Y ), and begins to transition towards a uniform distribution. The pressure
acting on subsequent scales becomes increasingly uniform, and gradually increases towards the
limiting value 5.8σ Y . A contrasting result for a rougher surface with ψ 0 = 0.1 is shown in Fig.
10(b). In this case, only scales n = 0...1 are elastic; scale n = 2 exceeds yield, and a small
fraction of scale n = 3 reaches the fully plastic limit. Scale n = 4 is dominated by contacts in the
29
Plastic asymptote
10
10
10
0
-1
(a)
Contact fraction 2<an >/λn
(b)
Elastic asymptote
(c)
(d)
-2
Fully Plastic
10
-3
10
-4
10
7
10
5
No. contacts or pressure
Normalized contact size or area
fully plastic regime. The pressure acting on subsequent scales of roughness again becomes
progressively more uniform, and increases towards the limit 5.8σ Y . Pei et al (2005) have noted
the tendency of plasticity to drive contact size and pressure distributions to uniform distributions
in numerical simulations (they refer to plasticity as a `grand equalizer’). Our model suggests an
explanation for this phenomenon. The behavior of a sinusoidal surface is shown in Fig. 4, and is
approximated by Eqs (23)-(39).
Evidently, nominal pressures in the range
2 H 0 /ψ < p / σ Y < 2 H 0 / 3 acting on the sinusoid all induce the same true contact pressure. The
first roughness scale to reach fully plastic behavior therefore exerts a nearly uniform pressure on
the next scale of roughness.
Elastic
Elastic
asymptote
10
3
10
1
-1
10
Contact area An/2a-1
Elastic
asymptote
-3
-2
-1
0
1
2
3
4
5
10 10 10 10 10 10 10 10 10 10
Elastic asymptote
Plastic asymptote
Contact pressure P/AnσY
(a) (b) (c)
-3
(b)
(a)
6
No. contacts
Nn /N0
Plastic
asymptote
10
Plastic asymptote
(d)
(c) (d)
Fully Plastic
Elastic
-3
-2
-1
0
1
2
3
4
5
10 10 10 10 10 10 10 10 10 10
6
Roughness scale ψ n
Roughness scale ψ n
(a)
(b)
Fig 11: The variations of (a) normalized average contact size and total contact area, and (b) the
number of contact spots and normalized nominal contact pressure, for successive roughness
scales under a cylindrical indenter. The chain lines mark the transitions from elastic to elasticplastic behavior, and from elastic-plastic behavior to fully plastic behavior. Results are shown
for nominal pressure Σ = 0.001 and roughness values (a) ψ 0 = 0.0002 , (b) ψ 0 = 0.001 , (c)
ψ 0 = 0.01 , (d) ψ 0 = 0.1 .
5.3 Average contact size, contact pressure, total contact area, and number of contacts for each
scale
The overall behavior of the contacts at each scale may be computed using the distribution
functions given in the preceding section, and may also be estimated using the approximate
procedure outlined in Section 4.2. Figures 11 and 12 show the results of these calculations.
Figs. 11(a) and 12(a) show the contact fraction 2 an / λn and the ratio of true to nominal contact
area An / 2 a− 1 for successive roughness scales n under a contact; while Figs 11(b) and 12(b) show
the total number of contacts on each successive scale, normalized by the number of contacts in
the zeroth roughness scale, N n / N0 , and the dimensionless nominal contact pressure P / σ Y An on
30
each scale. All variables have been plotted as a function of ψ n = ψ 0γ ( D−1) n to illustrate their
asymptotic behavior clearly. Results are shown in Fig 11 for a nominal contact pressure
Σ = 0.001 and for several values of surface roughness (parameterized by ψ 0 = Ψ D
(
π Κ / 2Σ
)
D−1
);
while Fig 12 shows results for a fixed dimensionless surface roughness Ψ DΚ D−1 = 10− 3 and
progressively increasing load Σ . In each figure, the symbols show results obtained using the
approximate procedure outlined in Section 4.2, while the solid lines show the exact procedure
described in Section 4.1 and Appendix B. The approximate recursion scheme is clearly an
excellent approximation to the exact solution.
Fig. 11 shows clearly the influence of plastic flow on the behavior of successive scales of
roughness. For the conditions chosen, the first roughness scale (n=0) is always elastic. As
shown in Fig. 11(a), the contact size and contact area for subsequent elastic scales of roughness
approach the elastic asymptotes: 2 an / λn → γ (1−D ) and An / 2 a− 1 → Σ / π 5 / 2ψ n , respectively,
while the number of contact spots and nominal pressure approach N n / N0 → Σ γ (2− D)n / π 5 / 2ψ 0
and P / Anσ Y → π 3ψ n / 4 .
The transition from elastic to elastic-plastic behavior occurs at
ψ n ≈ 0.1γ ( D −1) (this approximation is strictly valid only if the contact size reaches its elastic
asymptote); while the transition to fully plastic behavior occurs near ψ n ≈ 2 . Only two or three
roughness scales lie in the elastic-plastic regime. For roughness scales that are loaded beyond the
fully plastic limit, the contact size and contact area approach the plastic asymptotes:
2 an / λn → 1 ; An / 2 a−1 → π Σ / 4 H1(∞ ) ≈ 0.076 Σ , while the number of contact spots and the
Contact fraction 2<an>/λn
0
10 (e)
-1
10
(d)
(c)
-2
10
(b)
-3
10
10
7
10
5
10
3
10
1
No. contacts or pressure
Normalized contact size or area
normalized nominal pressure approach N n / N0 → γ n and P / Anσ Y → H1 (∞ ) = 5.8 .
(a)
-4
10
No. contacts Nn/N0
-1
10
-5
10
Contact area An /2a- 1
-3
-2
-1
0
1
2
3
4
5
10 10 10 10 10 10 10 10 10 10
-3
10
-6
10
6
(e)
(d) (c)
(b)
(a)
Contact pressure P/AnσY
(e)
(d) (c)
-2
(b) (a)
-1
0
1
2
3
4
5
10 10 10 10 10 10 10 10 10
6
Roughness scale ψ n
Roughness scale ψ n
(a)
(b)
Fig 12 The variations of (a) normalized average contact size and total contact area, and (b) the
number of contact spots and normalized nominal contact pressure, for successive roughness
scales under a cylindrical indenter. Results are shown for constant normalized roughness
Ψ DΚ D−1 = 10− 3 and progressively increasing nominal pressure (a) Σ = 10−5 , (b) Σ = 10−4 , (c)
Σ = 10 −3 , (d) Σ = 10−2 (e) Σ = 0.03 .
31
The behavior of roughness scales under a surface with fixed roughness, which is indented by a
cylinder under progressively increasing load is illustrated in Fig. 12. Results are shown for
roughness Ψ DΚ D−1 = 10− 3 and five different progressively increasing loads. At the lightest load
( Σ = 10−5 ) only the zeroth scale of roughness is in the elastic regime; there are 2-3 scales in the
elastic-plastic regime, but the majority of roughness scales lie in the fully plastic regime and the
contact size, area, pressure and the number of contact spots quickly approach their fully plastic
asymptotes. As load is increased, additional roughness scales with a lower ratio of amplitude to
wavelength are brought within the contact, so the number of scales in the elastic regime increases
with load. At Σ = 0.03 , there are six roughness scales in the elastic regime: a sufficient number
to allow the contact size, area, pressure and the number of contact spots approach their elastic
asymptotes. Shorter wavelengths again transition to the fully plastic regime.
6. Discussion
We began this paper by pointing out that a fully satisfactory model of contact between rough
surfaces has not yet been developed, and must reluctantly conclude with the same statement.
The central issue is that measured surface geometries appear to be well characterized as a selfaffine fractal even at length-scales of a few hundred nanometers; calculations show, however,
that for ideally elastic, perfectly fractal surfaces one cannot obtain meaningful predictions for
any features of the true contact area. The situation is slightly better for elastic-plastic solids,
where the total contact area and contact pressure are well defined, but it remains impossible to
predict the actual number of contacts or their size.
The ambiguous nature of the true contact could be resolved in either of two ways: (i) No real
surface can be perfectly fractal, and provided that the surface geometry can be measured, and
characterized, at extremely short wavelengths where the fractal description breaks down,
standard techniques – including perhaps the Greenwood Williamson method – can be used to
model the contact. Consequently, a systematic study of surface geometry at nanometer length
scales would be of great interest. Available experimental studies provide strong evidence that
surfaces cease to be fractal at short length scales. Majumdar and Bhushan (1990) obtained bilinear power spectra for profiles of hard disk surfaces, suggesting that the fractal descriptio n is
accurate for roughness scales with wavelengths between about one hundred microns and a few
hundreds nanometers. For short wavelength roughness, the spectral density remains linear, but
with slope exceeding 3. Since 1 < D < 2 and 2D − 5 < 3 , thus, the surface retains its multi-scale
character (and has no well-defined surface curvature) but it is not fractal. Similar results were
obtained more recently by Bora et al (2004) for surfaces of MEMS devices. There is, of course,
no difficulty in applying Ciavarella’s approach to surfaces of this type, although the GreenwoodWilliamson model remains ill-posed, since the mean asperity curvature cannot be computed.
Preliminary calculations along these lines suggest that the contact spot size and area is well
defined for surfaces of this type. (ii) It is also possible that some physical process, which was not
included in our contact model, can regularize the ill posed nature of contact even for perfectly
fractal surfaces. In fact, the present work was motivated by the hope that plasticity would have
this effect. As we have shown, however, this is not the case, because of the stiffening caused by
32
lateral interactions between plastically deforming asperities. Some hope remains that this lateral
interaction may play a less significant role under sliding contact, or in situations (such as metal
forming) where the plastic region of the nominal contact is not contained. Behavior could also
be regularized by phenomena such as short-range adhesive forces acting on the surfaces; the
presence of a lubricant film; and time-dependent processes such as diffusion, which would tend
to quickly spread small contacts.
In our view, the most promising approach is to abandon attempts to characterize surface
roughness as a perfect fractal. Nevertheless it is clear that a fractal description of a surface has
several advantages over the conventional approach; and that some features of fractal behavior
must be retained in a complete analysis. In particular, the fractal description captures the
important fact that asperity contacts are spatially correlated. Since small asperities sit on top of
larger ones, contact spots tend to cluster together. At a distance, a cluster of small spots
resembles a single, larger contact – but these larger contacts also tend to cluster together. This
progression continues down to the nominal contact. In the Greenwood-Williamson description,
the focus is entirely on the smallest scale of contacts, which are assumed to be uniformly
distributed spatially, and to exert a nominally uniform pressure. In contrast, the fractal
description, together with the Ciavarella model, allows one to calculate the collective effect of
the contact clusters, without needing to describe the contacts themselves in detail. For many
applications, this is sufficient: one such example is our fractal based plasticity index Ψ DΚ D −1 ,
which determines the number of roughness scales that exceed yield, and the resulting depth of
the plastic zone as illustrated in Figs 7 and 8.
Our analytical calculations are also helpful to elucidate the behavior that one would expect to
observe in a numerical simulation of contact between rough, ideally fractal, surfaces. The most
sophisticated models of contact between rough surfaces rely on large-scale numerical
simulations. These have now reached an advanced stage, particularly for elastic solids, and it is
possible to compute contact pressure distributions and contact sizes for measured three
dimensional surface profiles in considerable detail. The calculations of Ciavarella et al (2000),
as well as those reported here, raise some questions regarding the interpretation of such
simulations.
It appears to be an inescapable conclusion that the true contact area for elastic contact between
two perfectly fractal surfaces consists of an infinite number of contact spots of zero size, which
are subjected to infinite contact pressure, and have a vanishing total area of contact. Numerical
simulations that predict contact sizes, contact pressures, or the true area of contact between two
elastic solids are therefore valid only if (i) the actual surface geometry ceases to be fractal at
some short wavelength (e.g. the power-spectral density could be truncated at some short
wavelength; or else decay sufficiently rapidly to give a differentiable autocorrelation function at
the origin); and (ii) the resolutions of both the surface profilometer used to measure the 3D
surface, as well as the numerical simulations, are sufficient to detect and to characterize surface
geometry at these short wavelengths. Real surfaces must meet the first criterion, but
measurements of actual surface roughness spectra suggest that real surfaces remain
approximately fractal down to wave lengths of a few hundred nanometers. It is extremely
difficult, therefore, to meet the second criterion, both when the surface is measured, and also in
the subsequent numerical simulations.
33
Numerical simulations of elastic-plastic contact are somewhat less sensitive to artificial
truncation of the roughness spectrum. In an elastic-plastic simulation, we would expect the
predicted contact pressure, and the total area of contact, to converge to well-defined limits as the
resolution is reduced, even for perfectly fractal surfaces. The total number of contacts, and their
size, however, would not converge for a perfectly fractal surface. Again, such simulations are
valid only if the surface roughness is not perfectly fractal, and if the simulations can properly
resolve and characterize surface topography at short wavelengths.
4
10
Contact pressure P/A nσYψ0
Plastic
asymptote
n=11
n=8
103
Elastic
asymptote
n=5
102
101
100
10 1
102
10 3
104
Normalized surface strength 1/ψ0
Fig. 13: The dimensionless true contact pressure for a truncated Weierstrass surface, as a
function of the dimensionless strength parameter 1/ψ 0 .
The qua litative effects of introducing an artificial short-wavelength cutoff into numerical
simulations of contact between fractal surfaces can be estimated using our simple model. As a
representative example, we consider a Weierstrass surface (Eq. (7)) containing only terms
n = 0,1,2...5 , which is indented by a perfectly flat, rigid indenter under a uniform nominal
pressure p− 1 . We restrict attention to modest nominal pressures, so that the true area of contact
is much less than the nominal contact area. Under these conditions, the true contact consists of a
well-defined set of contact spots. The average size of these spots, and the true contact pressure,
are essentially independent of nominal pressure, while the number of contact spots and the total
true contact area increase in proportion to the nominal pressure. The contact size and mean
contact pressure, as well as the relationship between true contact area and the number of contact
spots, is determined by the elastic and plastic properties of the surface and the surface roughness,
which may be characterized together by ψ 0 . The elastic map shown in Fig. 9 shows that for
ψ 0 < 0.006 , the surface deforms elastically: under these conditions the behavior of the contacts
can be estimated using the expressions given in Section 4.3, with n=5. For ψ 0 > 0.006 the
contact spots exceed yield. The effect of plastic flow is to increase the size and number of
contacts, to increase the true area of contact, and reduce the mean pressure. As ψ 0 increases, the
true contact progressively transitions to the limiting conditions for a rigid plastic surface: the
34
contact fraction for the 5th scale approaches a5 / λ5 → 1 , the true contact pressure approaches
p5 = 5.8σ Y and the ratio of true to nominal contact area approaches p −1 /5.8σY . The transition
with ψ 0 is gradual, however. To illustrate this transition from elastic to plastic behavior, we
have plotted the variation of the true contact pressure P Anσ Yψ 0 = Pλ0 An E * g 0 as a function
of 1 ψ 0 = σ Y λ 0 E * g 0 in Fig. 13 for a surface with D = 1.5 and γ = 5 . Results are shown for
surfaces truncated at scales n=5, n=8 and n=11. We have chosen to plot the results in this form
to compare our predictions with the detailed finite element simulations of indentation of a
nominally fractal surface reported by Pei et al (2005). In particular, their Fig 6 shows the
predicted true contact area as a function of the ratio of yield stress to elastic modulus σ Y / E .
The numerical results are in excellent qualitative agreement with our estimate. They observe a
clear transition from elastic behavior at high σ Y / E to plastic behavior at low σ Y / E ; moreover,
they predict P / An → 6σY in the limit σ Y / E → 0 , in excellent agreement with our estimate of
P / An → 5.8σ Y . In our calculations, the elastic-plastic transition is a direct consequence of
truncating the Weierstrass series, however. As the series is truncated at progressively shorter
wavelengths, the elastic regime disappears. It is probable, therefore, that the elastic-plastic
transition observed in numerical simulations is a consequence of a similar truncation introduced
by the finite element interpolation. These results illustrate the challenges associated with
numerical simulations of rough surface contact. The most satisfactory approach to avoiding
these difficulties would be to purposely truncate the roughness spectrum in a controlled manner,
and to ensure that the numerical computations can fully resolve the truncated spectrum.
Numerical predictions are likely to be sensitive to the details of the truncation, however.
Despite these concerns, numerical simulations of rough surface contacts can still provide useful
information. Under purely normal indentation, the behavior of long-wavelength scales of
roughness can be computed accurately even if the nature of the true contacts is not known.
Numerical simulatio ns that artificially truncate the roughness spectrum can therefore be
interpreted as modeling the behavior of roughness scales that exceed some critical cut-off. In
models of wear, and contact fatigue, the principal concern is to characterize the state of the solid
at microstructurally relevant length-scales. Predictions of the true contact area, the number of
contact spots, and the actual size of the contact spots are of less interest. For such applications
numerical simulations provide a useful way to compare surface roughness textures, even if the
details of the very finest scales of roughness are not captured accurately.
The situation for simulations of frictional sliding contact between rough surfaces is less clear. As
we have seen, under purely normal loading, the contact pressure acting on the nth scale of
roughness is determined fully by lower (longer wavelength) roughness scales, and so is not
sensitive to truncation. In contrast, the frictional forces acting on the nth scale of roughness are
determined by the behavior of higher (shorter wavelength) scales of roughness. Here, it seems
that artificial truncation of the roughness spectrum will lead to spurious results. A particularly
simple illustration of these difficulties is provided by an attempt to model frictional sliding
between two fractal, elastic, surfaces. A model that follows the usual assumption that shear
tractions are proportional to the true contact area arrives at the absurd conclusion that the
coefficient of friction vanishes.
35
The simple model we have developed here could be extended in several ways. For example, it
would be straightforward to extend the calculations described here to three dimensions: this
would require numerical simulation and analytical characterization of the indentation of an
elastic-plastic solid with a doubly sinusoidal surface; the remainder of the analysis would still
apply. In addition, it would not be difficult to extend our calculations to account for the scale
dependent nature of plastic flow: due to indentation size effects (Nix & Gao 1998) we might
expect short wavelength roughness to return to elastic behavior. It would be of great interest to
determine where this transition occurs, because then it would be possible to design surfaces that
remain elastic. Finally, the general procedure employed here is clearly not restricted to fractal
surfaces, and could be applied to any discrete approximation to a power spectral density,
provided that the discrete scales remain widely separated.
A more important issue, however, is the need to adopt a more realistic model of the surface
roughness. The Weierstrass profile gives a convenient way to characterize the multi-scale nature
of surface roughness, but is clearly not a realistic model of surface roughness. Furthermore, the
scale decoupling introduced by Ciavarella et al (2000), which forms the basis of our analysis,
strictly holds only in the limit γ → ∞ . A number of questions therefore need to be addressed.
Firstly, it is of interest to determine the range of γ for which the scale decoupling can be
applied. For elastic surfaces, this is straightforward. The scales will decouple as long as there are
sufficient contacts of scale n+1 within each contact of scale n. Recall that (for sufficiently large
n) the contact spot size for the nth scale can be estimated as 2 < an >= λn γ (1− D) . The number of
contact spots of scale n+1 within each contact of scale n follows as m = 2 < an > / λn+1 ≈ γ (2 −D ) .
Consequently, the critical value of γ to ensure m contact spots is γ ≥ m1/(2− D ) ≈ m 2 . Taking
m = 2 as an optimistic estimate requires γ ≥ 4 ; in practice, an estimate of γ ≥ 10 is more
reasonable. It is difficult to obtain similar analytical estimates for plastic contacts, so instead, we
have investigated this issue using finite element computations with a finite number of roughness
scales, some of which are described in Appendix A. Simulations with two and three scales of
roughness suggest that scales of roughness completely decouple for γ ≥ 10 . For smaller γ , we
find that a few asperities in the nth scale that are located near the peaks of scale n-1 may
coalesce, in contrast to the large γ approximation. This is because at light loads, the nth scale
contains only a few contacts, and insufficient lateral constraint develops to increase the contact
pressure for the finest scale. However, asperities on scale n that are located in the valleys of
scale n-1 are strongly constrained, and so exhibit behavior that is in good agreement with the
behavior of a surface with purely sinusoidal roughness. Consequently, the predictions of the
scale decoupling analysis continue to predict qualitatively the average behavior of each
roughness scale even for low values of γ . Nevertheless, further work is required to properly
characterize the behavior of random surfaces, for both elastic and plastic solids. Recent studies
by Persson (2001, 2005) are promising steps in this direction.
36
7. Conclusions
We have outlined an approximate model that is intended to estimate the behavior of a rough
elastic-perfectly plastic solid with (two-dimensional) self-affine fractal surface roughness
subjected to contact loading. The surface was idealized by a Weierstrass profile (Eq. (1)) with
widely separated scales of roughness ( γ >> 1 ). Under these conditions, each scale of roughness
can be regarded as a sinusoidal surface, indented by a flat surface. Successive scales are coupled
through the condition that the nominal pressure acting on scale n is equal to the true pressure
acting on scale n-1. On this basis, a recursive expression can be found for the contact pressure
distribution, the contact size, and the total area of contact for each scale of roughness, in terms of
the response of a sinusoidal surface (Ciavarella et al, 2000). In our calculations, the response of
an elastic-plastic solid with sinusoidal roughness was approximated by fitting curves to finite
element computations, as outlined in Section 3. We found that simple but highly accurate
estimates for the average contact fraction, pressure and area of contact on successive roughness
scales can be obtained by replacing the pressure distribution acting on each scale by a statically
equivalent uniform distribution.
We have used this approach to analyze the behavior of an elastic-perfectly plastic Weierstrass
surface when indented by a rigid cylinder. We found that the behavior of the solid is governed
by three dimensionless functions, defined in Eqs. (75). The parameter Σ quantifies the
magnitude of the applied load; while Ψ and Κ together characterize the surface roughness and
its resistance to plastic flow. The conditions necessary to induce plastic flow in successive
roughness scales under the cylinder are illustrated in Fig. 7. For light loads, the nominal contact
area is small, so roughness scales under the indenter have a large amplitude compared with their
wavelength, and consequently a low resistance to plastic flow. As a result, at light loads only the
zeroth scale of roughness deforms elastically; all other roughne ss scales are fully plastic. As the
applied load is increased, the nominal contact area spreads, and new roughness scales that come
into the contact area may remain elastic, and for sufficiently high loads will reach full contact.
The parameter Ψ DΚ D −1 was identified as a plasticity index for a surface with fractal roughness.
Surfaces with low values of Ψ DΚ D −1 have a large number of scales of roughness in the elastic
regime, and consequently plastic flow is restricted to a thin layer at the surface, Fig. 8. As
Ψ DΚ D −1 increases, the number of plastic roughness scales, and the corresponding depth of the
plastically deformed layer, increase.
We also calculated the average contact fraction, total contact area, number of contacts, and
average contact pressure for successive scales of roughness. The results are illustrated in Figs.
11 and 12. The behavior of the surface depends primarily on the dimensionless parameter ψ 0
defined in Eq. (20), which characterizes the resistance to plastic flow of the longest wavelength
scale of roughness under the nominal contact. This parameter can be computed in terms of Ψ
and Κ using Eqs. (76). For low values of ψ 0 , a large number of long-wavelength roughness
scales under the contact remain elastic: their behavior can be calculated using the closed- form
expressions given in Section 4.3. Short wavelength roughness scales always deform plastically,
and for large n approach the asymptotes given in Section 4.5. For an elastic-plastic solid with a
perfectly fractal surface, the true area of contact consists of an infinite number of contact spots of
37
zero size, with a true contact pressure lim n →∞ P / An = 5.8σ Y . The true contact area is bounded,
and proportional to the load.
The implications of these findings were discussed in detail in Section 6. Our calculations
suggest that idealizing surface roughness as a perfect fractal leads to unphysical predictions for
the contact size and number of contact spots. It is likely that these difficulties can only be
resolved by taking into account the departure of actual surfaces from perfectly fractal behavior at
very short wavelengths.
Acknowledgements
This work was performed as part of the General Motors Collaborative Research Laboratory on
Computational Materials Research at Brown University. The authors are grateful to M.
Ciavarella, R.W. Carpick and J.F. Molinari for helpful discussions and for providing pre-prints
of their recent publications.
38
Appendix A: Numerical simulations of plastic contact of a truncated Weierstrass profile
Mean contact pressure pm/σ Y
A central assumption that underlies the Ciavarella et al model is that successive scales of
roughness interact only through the requirement that the true contact pressure acting on the nth
scale must equal the nominal contact pressure exerted on the n+1th scale, and the deformation
fields otherwise fully decouple. Saint Venant’s principle can be used to justify this assumption
for elastic solids. For materials that deform plastically, however, there is no rigorous basis for
this approximation – and indeed there is good reason to believe that there are circumstances
where the deformation fields between successive roughness scales do not decouple. We have
therefore conduc ted finite element simulations of elastic-plastic contact between surfaces with
two or more scales of roughness, to investigate the validity of this assumption. A few
representative results of these simulations will be outlined here.
Approximation
Asperity 1
Asperity 2
Asperity 3
Asperity 4
Asperity 5
5
4
3
λ2
2
1
2
3
1
4
5
λ 1/2
0.25
0.5
0.75
Contact fraction 2a2/λ2
1
Fig 14: Results of a finite element simulation of the indentation response of an elastic-perfectly
plastic solid with two scales of surface roughness. The figure shows the variation of mean
pressure with contact fraction for each asperity contact on the finer scale as the load is
progressively increased. Our approximation to the behavior of a single roughness scale is shown
for comparison.
Fig 14 shows the behavior of an elastic-perfectly plastic solid with two roughness scales, which
is indented by a rigid flat indenter. Results are shown for γ =λ 1 / λ0 = 10
E σ Y = 200 ,
g 0 λ0 = 1100 , g1 λ1 = 1 4 0 and ν = 0.3 . These parameters correspond to ψ 0 = 2 , ψ 1 = 5 , and
D = 1.4 . We have computed the mean contact pressure as a function of contact size for each
asperity in the finer roughness scale, and compared the results with our approximation to the
behavior of a surface with a purely sinusoidal roughness. For small values of 2 a1 / λ1 the two
cases are indistinguishable. For values of 2 a1 / λ1 > 0.6 (where asperities on the finer scale begin
to interact), we find that the pressure on the two-scale surface rises somewhat more gradually
than that for a pure sinusoid, but the limiting behavior as 2a1 / λ1 → 1 is correctly modeled by a
single-scale surface.
39
The results of a second set of simulations are illustrated in Figs 15 and 16. Here, we compare
directly the behavior of surfaces with one, two and three scales of roughness, when indented by a
rigid flat surface. Results are shown for γ = 10 , E σ Y = 200 , ν = 0.3 , g 0 λ0 = 1100 ,
g1 λ1 = 1 4 0 , g 2 λ2 = 1 1 6 , which correspond to D = 1.4 , ψ 0 = 2.2 , ψ 1 = 5.5 , ψ 2 = 13.7 . All three
scales of roughness are therefore in the fully plastic regime for sufficiently high loads. Fig 15
shows the distribution of contact pressure for the three scales of roughness, for two values of
nominal load. These results clearly agree qualitatively with the predictions of our approximate
model. In particular, the computations demonstrate the progressive increase in contact fraction
and contact pressure on successive scales of roughness, and show also that asperity interactions
cause the average contact pressure on scales n=1 and 2 to exceed the material hardness.
Scale 0
Scale 1
Scale 2
4
3
2
1
0
0
0.1
0.2
0.3
Scale 0
Scale 1
Scale 2
5
Contact pressure p/σY
Contact pressure p/σY
5
0.4
4
3
2
1
0
0.5
0
Position x/λ0
0.1
0.2
0.3
0.4
0.5
Position x/λ0
(a)
(b)
Fig. 15: The contact pressure distribution predicted by finite element computations for truncated
Weierstrass surfaces with one, two and three scales of roughness that are indented by a rigid, flat
surface. Results are shown for two values of nominal pressure: (a) p −1 / σY = 0.624 (b)
p− 1 / σ Y = 2.10
A more quantitative comparison of our numerical and analytical predictions is shown in Fig 16.
Here, we have plotted the total true area of contact predicted by finite element computations for
each of the three roughness scales, and compared the results with our approximate analytical
expression. The analytical results clearly provide a good approximation to the finite element
computations.
40
Total contact area An/λ0
0.8
Finite element results
Approximation
0.7
0.6
0.5
A0
A1
A2
0.4
0.3
0.2
0.1
0
0
1
2
Nominal contact pressure p-1/σY
Fig 16: Comparison of the true area of contact predicted by finite element simulations with 1, 2
and 3 scales of roughness with the approximate analytical expressions predicted by assuming
each scale behaves as an isolated sinusoid.
41
Appendix B: Numerical computations of pressure distribution functions.
Elementary calculations show that the pressure distribution function for an elastic-plastic solid
will have the form
n
qn (ξ ) = qn (ξ ) + ∑ Q (jn )δ (ξ − ξ (j n) )
(B1)
j =0
where q n denotes a continuous function, and the sum consists of a series of direct Delta
distributions with magnitude Q(j n) , which occur at normalized pressures ξ (jn ) . Given the nth term
in the series, we seek to compute the next term. The general expression is given by
∞
qn (η ) = ∫ q (η ,ξ ,ψ n ) qn−1(ξ ) dξ
(B2)
0
This integral must be divided into separate contributions that correspond to the elastic regime,
elastic plastic regime, and fully plastic regime
qn (η ) =
e
ξ max
∫ q (η ,ξ ,ψ ) q
n −1
n
(ξ )d ξ +
e
ξ min
+
fp
ξ max
∫ q (η ,ξ ,ψ ) q
n
n −1
ep
ξ max
∫ q (η, ξ ,ψ ) q
n −1
n
(ξ ) dξ
ep
ξ min
(ξ )d ξ +
fp
(B3)
fpi
ξ max
∫ q (η, ξ ,ψ ) q
n
n −1
(ξ ) dξ
fpi
ξ min
ξ min
where the limits are
e
ξ min
= η 2 / 4πψ n
e
e
ξ max
= min(ξ nmax
−1 , ξn −1)
ep
ξ min
= max( p max −1 (η ), ξne−1 )
ep
−1
ξ max
= min(ξnmax
−1 , β (η ))
fp
ξ min
= max(ξ nmax
−1 , 2 H 0 / ψ n )
fp
ξ min
= max(ξ nmax
−1 , 2 H 0 /3)
fp
ξ max
= min(ξnmax
−1 ,2 H 0 /3)
(B4)
fpi
ξ max
= min(ξnmax
−1 , H 1(ψ n ))
where ξ ne denotes the critical value of dimensionless nominal pressure to induce yield in the nth
scale; ξnmax is the maximum dimensionless nominal pressure acting on the nth scale; p max −1
denotes the value of ξ that satisfies eq (31) with ξ max = η ; β − 1(η ) is the value of ξ that
satisfies the second of Eq. (30) with β = η . Next, recall that that
fp
fp
q (ξ ,η ) = A(ξ ,ψ )δ (η − H0 ) ξmin
< ξ < ξmax
(B5)
fpi
fpi
q (ξ ,η ) = A(ξ ,ψ )δ (η − Φ mean (ξ )) ξmin
< ξ < ξ max
Substitute into (B2) to see that
42
qn (η ) =
e
ξ max
∫ q (η ,ξ ,ψ ) q
n −1
n
(ξ )d ξ +
e
ξ min
+
fpi
ξ max
∫ A ( ξ ,ψ ) δ (η − Φ
mean
n
ep
ξmax
fp
ξ max
∫ q (η, ξ ,ψ ) q
n
n −1
ep
ξ min
(ξ ) dξ + δ (η − H0 ) ∫ A (ξ ,ψ n ) qn−1 (ξ ) dξ
fp
ξ min
n −1
(B6)
(ξ )) qn−1(ξ ) dξ + ∑ Q(jn−1) A(ξ jn−1, ψ n )δ (η − Φ mean (ξ jn−1,ψ n ))
j =0
fpi
ξ min
Hence
Q =
n
0
fp
ξ max
∫
Afp (ξ ,ψ n )qn−1(ξ ) dξ
fp
ξ min
Qnj = Qj( n−1−1) Afpi (ξ j(−n1−1) ,ψ n )
ξ nj = Φ mean (ξ nj−−11,ψ n )
qn (η ) =
(B7)
e
ξ max
∫ q (η ,ξ ,ψ ) q
n
n −1
(ξ )d ξ +
e
∫ q (η, ξ ,ψ ) q
n
ep
ξ min
+
ξ ep
max
ξ min
fpi
ξ max
∫ A (ξ ,ψ ) δ (η − Φ
mean
n
fpi
ξ min
43
(ξ )) qn−1(ξ )d ξ
n −1
(ξ ) dξ
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