Chapter 2
The Areal Field Parameters
François Blateyron
Abstract The vast majority of surface texture parameters are the field parameters.
The term field refers to the use of every data point measured in the evaluation area,
as opposed to feature parameters that only take into account specific points, lines
or areas. Field parameters allow the characterisation of surface heights, slopes,
complexity, wavelength content, etc. They are defined in the specification standard
ISO 25178 part 2. In this chapter the ISO areal field parameters will be presented
along with limited guidance on their use.
2.1 A Short History of Areal Parameters
The first areal surface texture measuring instruments were made available around
1987. Instrument manufacturers such as Zygo, Wyko and others started to provide
parameters calculated on the data. Early areal parameters were often simple
extrapolations of profile parameters (see Chap. 1 and Leach 2009 for a description
of the profile parameters) and were either named simply Ra or sRa, and sometimes
calculated using proprietary algorithms, leading to different parameter values on
different instruments.
The initial work by Stout et al. (1993a, b) leading to the ‘‘Blue Book’’ was
covered in Chap. 1. During the same period, ISO technical committee TC 57
introduced a new concept, called Geometrical Product Specification and Verification (GPS for short) in order to unify specification standards dealing with
dimensional analysis and surface texture. Then, in 1996 a new committee was
created, TC 213, to develop GPS specification standards. One of the first actions of
TC 213 was to entrust a group of researchers with the aim of developing the basis
F. Blateyron (&)
Digital Surf sarl, 16 rue Lavoisier, 25000 Besançon, France
e-mail: [email protected]
R. Leach (ed.), Characterisation of Areal Surface Texture,
DOI: 10.1007/978-3-642-36458-7_2, Springer-Verlag Berlin Heidelberg 2013
15
16
F. Blateyron
of areal surface texture parameters. The historical perspective of the SURFSTAND, which led to the so-called ‘‘Green Book’’ (Blunt and Jiang 2003) was
covered in Chap. 1.
The SURFSTAND results were presented to ISO in January 2002 and officially
transferred to TC 213, in order to start the standardisation process. In June 2002,
TC 213 voted for the creation of a new working group, which was assigned the
task of developing future international standards for areal surface texture. This
working group (known as WG 16) met for the first time in January 2003. At the
end of 2005, the ISO secretary allocated the number (ISO) 25178 to all areal
surface texture standards, thereby giving the standards their official birth. These
new standards were motivated by a shift that occurred in surface metrology: the
shift of surface measurement from profile to areal and contact to non-contact
(Jiang et al. 2007).
The work allocated to WG 16, which would be implemented in the ISO 25178
standards, contained two parts:
• to define the content of the areal surface texture standards, for specification and
verification; and
• to revise the existing profile standards to bring them into line with the new areal
standards.
The first part of ISO 25178, part 6 on classification of surface texture measuring
instruments, was published in January 2010 (ISO 25178 part 6 2010). This was
followed in June by a group of documents on instrument techniques: part 601 (ISO
25178 part 601 2010) and part 701 (ISO 25178 part 701 2010) on stylus profilometer and part 602 (ISO 25178 part 602 2010) on confocal chromatic instruments.
The main document defining areal parameters, ISO 25178 part 2, was published
in September 2012 (ISO 25178 part 2 2012). As highlighted in Chap. 1, more
documents will follow in the forthcoming years.
2.2 Naming, Filtering and Calculation Conventions
2.2.1 Naming
ISO 25178 part 2 defines symbols for surface texture parameters that have a prefix
that is the capital letters S or V followed by one or several small letters that form
the suffix. This suffix should not be written as a subscript but on the same line as
the prefix. However, the final version of ISO 25178 part 2 uses subscripts in
parameter names to comply with ISO, which states that symbols should only
contain one letter and optional subscripts.
The prefix S is used for the majority of parameters (for example, Sq, Sdr, Smr),
the alternative being volume parameters that start with the letter V (for example,
Vmp, Vvc).
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The Areal Field Parameters
17
2.2.2 Filtering Conditions
Profile parameters are named after the type of surface profile from which they are
calculated, for example R-parameters (Ra, Rsk, etc.) are calculated on the
roughness profile, W-parameters (Wa, Wsk, etc.) are calculated on the waviness
profile, while P-parameters are calculated on the primary profile (see ISO 4287
2000; Leach 2009). The type of surface is not taken into account with areal
parameters. The Sa parameter can be calculated on a primary surface or a filtered
surface—it will be called Sa in all cases. Therefore, it is important to provide the
filtering conditions together with the parameter value.
2.2.3 Sampling Area
Profile parameters are defined based either on a sampling length or the evaluation
length. If a parameter is defined on a sampling length, it is (by default) calculated
on each sampling length (ISO 4288 1996) and a mean value calculated (the default
number of sampling lengths is five). With surfaces and areal parameters, the
concepts of sampling and evaluation areas are still defined but the default is one
sampling area per evaluation area. This simply means that parameters are calculated on the measured surface without segmenting the surface into small sub-areas
that depend on the sampling length.
If the user wants to use more sampling areas in the evaluation of a surface, several
surfaces can be measured, either contiguous or separated, and a statistical evaluation
of the parameters calculated on each surface (mean, standard deviation, etc.).
2.2.4 Centred Heights
In surface texture parameter equations, the height function, z(x,y) must be centred.
This means that the mean height calculated on the definition area is already
subtracted from the heights. This leads to a simplified version of the parameter
equations as it is possible to express the equation as, for example
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ZZ
1
z2 ðx; yÞdxdy;
ð2:1Þ
Sq ¼
A
A
rather than
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ZZ
1
Sq ¼
½zðx; yÞ z2 dxdy
A
A
where z is the mean height of the surface on the definition area, A.
ð2:2Þ
18
F. Blateyron
In practice, the above simplification can lead to rounding errors because heights
are usually coded with integer numbers and the average height is, therefore,
rounded to the closest integer. Each value in the sum is slightly biased, leading to
rounding errors in the result, especially when high order powers are involved in the
calculation, as is the case with Ssk (see Sect. 2.4.2.1) or Sku (see Sect. 2.4.2.2). A
correct digitisation of heights with a small quantisation step makes it possible to
reduce the uncertainty due to rounding errors. In the rest of this chapter, heights
z(x,y) are understood to be already centred.
2.3 Continuous Against Discrete Definitions
In modern specification standards, parameter definitions are always given for the
continuous case, i.e. expressed with integrals, although in practice measured
profiles and surfaces are always sampled and digitised. The use of continuous
definitions ensures the correctness of the definition and does not imply any
numerical approximation. National Metrology Institutes (NMIs) can implement
the most accurate versions of the parameter definitions without taking into account
the need for calculation speed (Harris et al. 2012a). However, software engineers
providing commercial products will implement an algorithm that has appropriate
speed and accuracy.
A simple discrete implementation uses summations. For example, in the case of
the parameter Sa (see Sect. 2.4.1.2), on a surface sampled with ny lines of nx
points, the equation will be approximated by
1 Xny 1 Xnx 1
f
Sa ¼
jzðx; yÞj:
y¼0
x¼0
n x ny
ð2:3Þ
This simple implementation provides sound results providing that the data density
is sufficiently high.
Figure 2.1 shows a classical representation of a profile where data points are
joined by line segments. In 3D, data points are connected through a triangular
facet. However, when parameters are calculated, the approximation method
replaces integrals by summations and, therefore, corresponds more to the representation shown in Fig. 2.2.
Fig. 2.1 Sampled data points
represented as a series of line
segments
2
The Areal Field Parameters
19
Fig. 2.2 Integration using a
simple summation of heights
Fig. 2.3 Integration using
linear interpolation
Fig. 2.4 Integration using a
spline interpolation
This simple summation of heights approximation can be used when the density
of points is high enough, that is to say, for most amplitude parameters where zero
crossing is not involved.
Linear interpolation gives a lower value for the Sa parameter as shown in
Fig. 2.3, since the area enclosed between the profile and the horizontal axis is
smaller around the zero crossings compared to the simple summation (Fig. 2.2).
Figure 2.4 shows a spline interpolation (Unser 1999), which is usually closer to
the continuous profile (Harris et al. 2012a). In this case, the value of the Sa
parameter will be more accurate and will lie somewhere in between the values
calculated in the cases shown in Figs. 2.2 and 2.3. The drawback of spline
interpolation is that it generates overshoots around peaks and tends to give
excessive values for peak-to-valley parameters. However, certain varieties of
splines do not have overshoot compared to simple cardinal splines (Catmull and
Rom 1974).
Software implementations of surface texture parameters can be tested independently from instrument contributions by testing the calculation algorithms
against areal software measurement standards (Chap. 1; ISO 25178 part 71 2012;
ISO/CD 25178-72 2012; Harris et al. 2012b).
20
F. Blateyron
2.4 Height Parameters
The definitions of the height parameters are given in the following sections.
2.4.1 Mean Height of the Surface
2.4.1.1 Root Mean Square Height, Sq
The root mean square height or Sq parameter is defined as the root mean square
value of the surface departures, z(x,y), within the sampling area, A.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ZZ
1
Sq ¼
zðx; yÞdxdy
ð2:4Þ
A
A
2.4.1.2 Arithmetic Mean Height, Sa
The arithmetic mean height or Sa parameter is defined as the arithmetic mean of
the absolute value of the height within a sampling area,
ZZ
1
Sa ¼
ð2:5Þ
jzðx; yÞjdxdy:
A
A
The Sa and Sq parameters are strongly correlated to each other (Blunt and Jiang
2003). The Sq parameter has more statistical significance (it is the standard deviation) and often has a more physical grounding than Sa, for example, Sq is directly
related to surface energy and the way light is scattered from a surface (Leach 2009).
2.4.2 Skewness and Kurtosis
2.4.2.1 Skewness, Ssk
Skewness is the ratio of the mean of the height values cubed and the cube of Sq
within a sampling area,
ZZ
1 1
z3 ðx; yÞdxdy:
ð2:6Þ
Ssk ¼ 3
Sq A
A
This parameter can be positive, negative or zero, and is unit-less since it is normalised by Sq. The Ssk parameter describes the shape of the topography height
distribution. For a surface with a random (or Gaussian) height distribution that has
symmetrical topography, the skewness is zero. The skewness is derived from the
2
The Areal Field Parameters
21
amplitude distribution curve; it is the measure of the profile symmetry about the
mean line. This parameter cannot distinguish if the profile spikes are evenly distributed above or below the mean plane and is strongly influenced by isolated
peaks or isolated valleys. Skewness represents the degree of bias, either in the
upward or downward direction of an amplitude distribution curve. A symmetrical
profile gives an amplitude distribution curve that is symmetrical about the centre
line and an unsymmetrical profile results in a skewed curve. The direction of the
skew is dependent on whether the bulk of the material is above the mean line
(negative skew) or below the mean line (positive skew).
As an example, a porous, sintered or cast iron surface will have a large value of
skewness. A characteristic of a good bearing surface is that it should have a
negative skew, indicating the presence of comparatively few peaks that could wear
away quickly and relatively deep valleys to retain lubricant traces. A surface with a
positive skew is likely to have poor lubricant retention because of the lack of deep
valleys in which to retain lubricant traces. Surfaces with a positive skewness, such
as turned surfaces, have high spikes that protrude above the mean line. The Ssk
parameter correlates well with load carrying ability and porosity.
2.4.2.2 Kurtosis, Sku
The Sku parameter is a measure of the sharpness of the surface height distribution
and is the ratio of the mean of the fourth power of the height values and the fourth
power of Sq within the sampling area,
ZZ
1 1
z4 ðx; yÞdxdy:
ð2:7Þ
Sku ¼ 4
Sq A
A
Kurtosis is strictly positive and unit-less, and characterises the spread of the height
distribution. A surface with a Gaussian height distribution has a kurtosis value of
three. Unlike Ssk, use of this parameter not only detects whether the profile spikes
are evenly distributed but also provides a measure of the spikiness of the area. A
spiky surface will have a high kurtosis value and a bumpy surface will have a low
kurtosis value.
The Ssk and Sku parameters can be less mathematically stable than other
parameters since they use high order powers in their equations, leading to faster
error propagation.
2.4.3 Maximum Height of the Surface
The Sp parameter represents the maximum peak height, that is to say the height of
the highest point of the surface. The Sv parameter represents the maximum pit
height, i.e. the height of the lowest point of the surface. As heights are counted from
the mean plane and are signed, Sp is always positive and Sv is always negative.
22
F. Blateyron
The Sz parameter is the maximum height of the surface, i.e. is sum of the
absolute values of Sp and Sv,
Sz ¼ Sp þ jSvj ¼ Sp Sv:
ð2:8Þ
The maximum height parameters are to be used with caution as they are sensitive to isolated peaks and pits which may not be significant. However, Sz can be
pertinent on surfaces that have been filtered with a low-pass filter (S–F surfaces
with a large S nesting index or in other words, waviness surfaces—see Chap. 4) to
characterise the amplitude of waviness on the workpiece. Also, maximum height
parameters will succeed in finding unusual conditions such as a sharp spike or burr
on the surface that may be indicative of poor material or poor processing.
Alternative parameters that could be used as more robust versions of a maximum height parameter are the feature parameter S10z (see Chap. 3) and Sdc (see
Sect. 2.5.2.4).
2.5 Function Related Parameters
The definitions of the function related parameters are given in the following sections.
2.5.1 Height Distribution and Material Ratio Curve
The height distribution can be represented as a histogram of the surface heights that
quantifies the number of points on the surface that lie at a given height. The
material ratio curve is the cumulative curve of the distribution. The material ratio
curve is counted from the highest point on the surface (where the curve equals 0 %)
to its lowest point (where the curve reaches 100 %) (Fig. 2.5).
In the case of profiles, the material ratio is calculated using a cutting depth
c which is counted from the highest peak (ISO 4287 2000). This is not the most
0
5
5
4
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
µm
0
0.5
1
1.5
2
2.5 %
10
20
30
40
µm
Fig. 2.5 The height distribution (left) and material ratio curve (right)
50
60
70
80
90
100 %
2
The Areal Field Parameters
23
robust solution as it may be affected by outliers. In areal analysis, the value c is
counted on a surface from the mean plane, and this reference provides a more
robust definition for material ratio parameters.
2.5.2 Material Ratio Parameters
2.5.2.1 Areal Material Ratio, Smr
The areal material ratio is the ratio of the material at a specified height c to the
evaluation area expressed as a percentage (see Fig. 2.6). The heights are taken
from the reference plane. The Smr(c) function gives the material ratio p corresponding to a cutting height c given as a parameter.
2.5.2.2 Inverse Areal Material Ratio, Smc
The Smc(p) function evaluates the height value c corresponding to a material ratio
p given as a parameter (see Fig. 2.7).
2.5.2.3 Peak Extreme Height, Sxp
The Sxp parameter (see Fig. 2.8) is aimed at characterising the upper part of the
surface, from the mean plane to the highest peak without taking into account a
small percentage of the highest peaks that may not be significant,
Sxp ¼ Smcð2:5%Þ Smcð50 %Þ:
Fig. 2.6 Smr(c) is the
material ratio
p corresponding to a section
height c
0
5
4
3
2
1
0
-1
-2
-3
-4
-5
µm
10
20
30
40
50
ð2:9Þ
60
70
80
90 100 %
24
F. Blateyron
Fig. 2.7 Smc(p) is the height
section c corresponding to a
material ratio p
5
0
10
20
30
40
50
60
70
80
90 100 %
70
80
90
4
3
2
1
0
-1
-2
-3
-4
-5
µm
Fig. 2.8 Peak extreme
height Sxp defined as the
height difference between
two inverse material ratios at
2.5 and 50 %
5
0
10
20
30
40
50
60
100 %
4
3
2
1
0
-1
-2
-3
-4
-5
µm
The values 2.5 and 50 % are defined as the default values (ISO 25178-3 2011) and
can be set to other values depending on the application (they should be close to
these default values as this parameter is specifically defined for peak characterisation). For more general height differences, the Sdc parameter should be used
(see Sect. 2.5.2.4).
2.5.2.4 Surface Section Difference, Sdc
This parameter is not strictly defined in ISO 25178 part 2 (2012) but it is a simple
extension of the Rdc parameter that was part of ISO 4287 (2000) (and see Leach
2009)
2
The Areal Field Parameters
25
Sdc ¼ Smcð pÞ SmcðqÞ
ð2:10Þ
where p and q are two material ratios that can be chosen freely depending on the
application.
The Sdc parameter can be used to give the maximum height of the surface when
the extreme peaks and valleys are removed or a threshold is applied, for example,
with p = 2 % and q = 98 %.
2.5.3 Characterisation of Stratified Surfaces
ISO 13565 part 2 (1998) is based on the German standard DIN 4777 (1990) that
was the first to introduce functional parameters based on a graphical construction
on the Abbott-Firestone curve (Whitehouse 2011). These parameters, Rk, Rvk,
Rpk, Mr1 and Mr2, are extracted from a filtered surface using a robust filter
specially designed for stratified surfaces. The parameters Sk, Spk, Svk, Sr1 and Sr2
are the areal equivalent of the parameters defined in ISO 13565 part 2 when the
Abbott-Firestone curve is built from an areal surface (see Fig. 2.9). The surface
may be filtered prior to the calculation of these parameters, preferably using a
robust Gaussian filter (ISO/CD 16610 part 71 2011; Muralikrishnan and Raja
2009, and see Chap. 4).
ISO 13565 part 3 (2000) defines three further parameters that are extracted:
Spq, Smq and Svq. These parameters are calculated on the probability curve and
are specifically designed for the evaluation of plateau honed surfaces (Malburg and
Raja 1993).
Fig. 2.9 Graphical
construction of Sk parameters
Spk
Sk
Svk
0
10
Smr1
20
30
40
50
60
70
80
90 100 %
Smr2
26
Fig. 2.10 Void volume Vv
below a section height
defined by a material ratio mr
F. Blateyron
mr
0
10
20
30
40
50
60
70
80
90
100 %
Vv (mr)
2.5.4 Volume Parameters
2.5.4.1 Void Volume, Vv
The void volume or Vv(mr) parameter is the void volume calculated for a material
ratio mr. This parameter is calculated by integrating the volume enclosed above
the surface and below a horizontal cutting plane set at a height h = Smc(mr). This
can be expressed by the following
Z 100%
½Smcðmr Þ SmcðqÞdq
ð2:11Þ
VvðmrÞ ¼ k
mr
where k is a factor to convert the volume into the required unit, either [lm3], [lm3/
mm2] or [ml/m2]. Void volume can be represented on the Abbott-Firestone curve
as shown in Fig. 2.10.
For mr = 100 %, the void volume is zero. For mr = 0 %, the void volume is a
maximum (the cutting plane below the lowest point).
Void volume calculations are often useful to evaluate the surface texture of
mechanical components that are used in contact with other surfaces.
2.5.4.2 Material Volume, Vm
The material volume or Vm(mr) parameter is the material volume calculated for a
material ratio mr. The parameter is calculated by integrating the volume enclosed
2
The Areal Field Parameters
Fig. 2.11 Material volume
above the section height
defined by a material ratio mr
27
mr
0
10
20
30
40
50
60
70
80
90
100 %
Vm (mr)
below the surface and above a horizontal cutting plane set at a height h = Sdc(mr).
This parameter can be expressed by the following
Z mr
½SmcðqÞ SmcðmrÞdq:
ð2:12Þ
Vmðmr Þ ¼ k
0%
The material volume can be represented on the Abbott-Firestone curve as shown in
Fig. 2.11.
For mr = 100 %, the void volume is a maximum. For mr = 0 %, the void
volume is zero (the cutting plane above the highest point).
2.5.4.3 Peak Material Volume, Vmp
The peak material volume or Vmp parameter is the material volume calculated at a
fixed material ratio mr,
Vmp ¼ Vmðmr1Þ
ð2:13Þ
where mr1 = 10 % by default. The ratio mr1 may be changed for specific
applications and will always be specified together with the value of Vmp.
The Vmp parameter can be used for the same purpose as the Spk parameter, i.e.
to characterise the volume of material which is likely to be removed during running-in of a component.
28
F. Blateyron
2.5.4.4 Core Material Volume, Vmc
The core material volume or Vmc parameter is the difference between two material
volume values calculated at different heights
Vmc ¼ Vmðmr2Þ Vmðmr1Þ
ð2:14Þ
where mr2 = 80 % and mr1 = 10 % by default.
The Vmc parameter represents the part of the surface material which does not
interact with another surface in contact, and which does not play any role in
lubrication.
2.5.4.5 Core Void Volume, Vvc
The core void volume (the difference in void volume between the mr1 and mr2
material ratios) is given by
Vvc ¼ Vvðmr1Þ Vvðmr2Þ
ð2:15Þ
where mr2 = 80 % and mr1 = 10 % by default.
2.5.4.6 Dales Void Volume, Vvv
The dale volume at mr2 material ratio is given by
Vvv ¼ Vvðmr2Þ
ð2:16Þ
where mr2 = 80 % by default.
2.5.4.7 Examples of Volume Parameters
Figure 2.12 shows the four volume parameters Vmp, Vmc, Vvc and Vvv calculated
from two bearing ratio levels mr1 and mr2.
Volume parameters have shown good correlation with functional requirements in
several applications; see Waterworth (2006) for a thorough treatment. Volume
parameters have replaced the functional indices Sbi, Sci and Svi (Stout et al. 1993a, b)
as they have proved to be more stable while providing the same type of information
(see Sect. 2.9.1 and Jiang et al. 2000).
The Vvv parameter characterises the volume of fluid retention in the deepest
valleys of the surface. This parameter is not affected by wear processes applied on
the surface.
The Vmp parameter characterises the volume of material located on the highest
peaks of the surface which is removed during a wear process. On a used
mechanical component, after several hours of function, the highest peaks are cut
2
The Areal Field Parameters
Fig. 2.12 Definition of
volume parameters on the
bearing areal ratio curve
29
mr2
mr1
0
10
20
30
40
50
60
70
80
90
100 %
Vmp
Vvc
Vmc
Vvv
out or plastically deformed, and the corresponding particles of material are captured by the deepest valleys, so that the behaviour of the surface is more likely
described by Vmc and Vvc.
2.6 Hybrid Parameters
The hybrid parameters are defined in the following sections.
2.6.1 Root Mean Square Gradient, Sdq
The gradient of a surface point is defined for each axis x and y by oz=ox and oz=oy:
The implementation of these gradients on a sampled surface is given elsewhere
(Whitehouse 2011). The root mean square gradient is then calculated on the whole
surface with
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ZZ 2
1
oz
oz 2
dxdy
ð2:17Þ
þ
Sdq ¼
A
ox
oy
where A is the projected area of the surface. The Sdq parameter has a unit-less
positive value. Optionally, it can be expressed in [lm/lm] or [lm/mm], or even as
an angle by calculating the arctangent of Eq. (2.17). The Sdq parameter is useful
30
F. Blateyron
α
Fig. 2.13 Orientation b and
inclination a of a surface
facet
β
for assessing surfaces in sealing applications and for controlling surface cosmetic
appearance.
The Sdq parameter is also associated with two plots that represent the distribution of the horizontal and vertical angles a and b (and see Fig. 2.13)
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
oz 2 oz 2 A
a ¼ tan1 @
;
ð2:18Þ
þ
ox
oy
1
b ¼ tan
oz2
oy
oz2
ox
!
:
ð2:19Þ
The angle a characterises the steepest gradient in the vertical plane, and is given
as an angle between 0 and 90, 0 being a horizontal facet and 90 a vertical facet.
The angle b, when calculated on the whole surface, characterises the mean
orientation of the surface facets and is an evaluation of the texture direction. It is
given as an angle between 0 and 360, with 0 in the direction of the x axis,
counter clockwise.
2.6.2 Developed Interfacial Area Ratio, Sdr
The developed interfacial area of a surface is calculated by summing the local area
when following the surface curvature. It can be approximated by the mean area of
two triangles formed between four adjacent points.
Referring to Fig. 2.14, the area of a triangle is half the cross product of two
vectors,
2 The Areal Field Parameters
31
P11: z(x+1,y+1)
P11: z(x+1,y+1)
P01 : z(x,y+1)
P01: z(x,y+1)
P00: z(x,y)
P10: z(x+1,y)
P00: z(x,y)
P10: z(x+1,y)
Fig. 2.14 Area between four adjacent points calculated by the average of two triangulations
A00
2 ! ! ! ! 3
1
P00 P01 P00 P10 þ P11 P01 P11 P10 162
7
¼ 4 þ 5:
2 1 ! ! ! !
2 P10 P00 P10 P11 þ P01 P00 P01 P11 ð2:20Þ
As most surfaces are globally flat (the topography is seen only by expanding the
z axis), the developed area is usually slightly larger than the projected area—this is
why the Sdr parameter is expressed as the excess value above 100 %, thus
PP
Aij A
Sdr ¼
;
ð2:21Þ
A
where Aij is the mean area calculated at a point, and A is the projected area
calculated by the product of the lengths in x and y. Note that ISO 25178 part 2
(2012) defines the Sdr parameter for the continuous case, i.e. with integrals instead
of summations.
The Sdr parameter can be given as a unit-less positive number or as a percentage. It will usually produce a value of several percent (typically between 0 and
10 %). A perfectly flat and smooth surface would have Sdr = 0 %.
The Sdr parameter is used as a measure of the surface complexity, especially in
comparisons between several stages of processing on a surface, and it can provide
useful correlations in adhesion applications (Löberg et al. 2010; Barányi et al.
2011; Reizer and Pawlus 2011). The Sdr parameter is greatly influenced by the
sampling scheme (number of points and spacing in the x and y axes).
2.7 Spatial Parameters
The spatial parameters are defined in the following sections.
32
F. Blateyron
2.7.1 Autocorrelation Function
The autocorrelation function (ACF) evaluates the correlation of a part of an image
with respect to the whole image. The ACF is defined as a convolution of the
surface with itself, shifted by (sx, sy)
RR
zðx; yÞz x sx ; y sy dxdy
ACFðsx ; sy Þ ¼
:
ð2:22Þ
RR
zðx; yÞ2 dxdy
Fig. 2.15 Surface with PCB vias (left) and its autocorrelation plot (right)
Fig. 2.16 Abrasive surface with quartz grains (left) and its autocorrelation plot (right)
2 The Areal Field Parameters
33
Fig. 2.17 Surface of a DVD stamper (left) and its autocorrelation plot (right)
Fig. 2.18 Using autocorrelation for detecting surface patterns. When the image of the circuit
(left) is correlated with this pattern, it creates an image with correlation peaks at positions where
the pattern is found in the image (right)
The ACF corresponds to the autocovariance normalised by Sq2 [the denominator
in Eq. (2.22)]. The ACF produces a value between -1 and +1 for each point on the
surface. An ACF of +1 means a perfect correlation and zero means no correlation.
34
F. Blateyron
The maximum of the ACF is always at the centre (for a zero shift). Figures 2.15,
2.16, 2.17 show several examples of surface textures and their autocorrelation
plots.
The ACF is used to study periodicities on a surface, i.e. when a texture motif is
reproduced several times on the surface (see Fig. 2.18), or is used to assess the
isotropy of a surface (see also Sect. 2.7.3).
2.7.2 Autocorrelation Length, Sal
The autocorrelation length, Sal, is defined as the horizontal distance of the ACF(tx, ty)
which has the fastest decay to a specified value s, with 0 B s \ 1. The Sal parameter
is given by
Fig. 2.19 Autocorrelation peak with an applied threshold of 0.2 (white part above the threshold)
Fig. 2.20 Shortest radius
measured from the centre to
the contour of the thresholded
lobe on the autocorrelation
plot
2 The Areal Field Parameters
35
Sal ¼ min
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tx2 þ ty2 :
ð2:23Þ
Figure 2.19 shows the autocorrelation of a textured surface. The white part on the
central lobe is above the threshold s. A radius is calculated from the centre to the
perimeter of the lobe and the shortest radius is kept for Sal (see Fig. 2.20).
For all practical applications involving relatively smooth surfaces, the value for
s can be taken as 0.2 (ISO 25178 part 3 2012), although other values can be used
and will be subject to forthcoming areal specification standards. For an anisotropic
surface, Sal is in the direction perpendicular to the surface lay. A large value of Sal
denotes that that surface is dominated by low spatial frequency components, while
a small value for Sal denotes the opposite case.
The Sal parameter is a quantitative measure of the distance along the surface by
which a texture that is statistically different from that at the original location would
be found.
The contour of the central lobe is measured from the centre and the shortest
radius is identified. This radius gives the value of the Sal parameter.
2.7.3 Texture Aspect Ratio, Str
The texture aspect ratio parameter, Str is one of the most important parameters
when characterising a surface in an areal manner as it characterises the isotropy of
the surface.
The Sal parameter is calculated from the minimum radius on the central lobe of
the ACF. The Str parameter is calculated from the minimum, rmin, and maximum
radii, rmax (see Fig. 2.21), found under the same conditions, on the autocorrelation
plot after applying a threshold of 0.2
Fig. 2.21 Minimum and
maximum radii measured
on the central lobe of the
autocorrelation plot
r max
r min
36
F. Blateyron
Str ¼
rmin
rmax
ð2:24Þ
The Str parameter is unit-less and its values lies between 0 and 1. It can also be
expressed as a percentage between 0 and 100 %.
The Str parameter is an evaluation of the surface texture isotropy. If Str is close
to unity, then the surface is isotropic, i.e. it has the same properties regardless of
the direction. On an isotropic surface, it is possible to assess the surface texture
using a 2D (profile) surface texture measuring instrument. If Str is close to 0, then
the surface is anisotropic, i.e. it has a dominant texture direction. In this case, the
parameter Std will give the direction angle of the texture.
Fig. 2.22 Surface of artificial leather (left) and its Fourier spectrum (right)
Fig. 2.23 Radial integration
of frequency amplitudes of
the Fourier spectrum
2 The Areal Field Parameters
37
Fig. 2.24 Polar spectrum graph representing the texture directions
2.7.4 Texture Direction, Std
The texture direction parameter, Std, is assessed from the Fourier spectrum of the
surface. The Fourier spectrum gives the energy content of each spatial frequency
on the surface and is usually represented as a plot where amplitudes are coded with
a colour or grey level (see Fig. 2.22).
When moving from the centre to an edge of the spectrum in a given direction,
the spatial frequencies go from the lowest to the highest value. The frequency at
the centre corresponds to the continuous value in z (frequency of zero or infinite
wavelength). When the surface is centred, this offset is zero. Frequency amplitudes
along the radius at a given direction h can be integrated between two selected
spatial frequencies, fmin and fmax, in order to calculate a value A(h) that represents
the spatial frequency content in that direction (see Fig. 2.23).
By repeating this integration for all angles between 0 and 180, a polar spectrum
is obtained that can be represented with a semi-circular graph (see Fig. 2.24).
The maximum value of the graph shown by Fig. 2.24 is called the main texture
direction, or Std. The Std parameter is given in degrees between 0 and 180, and
should be considered as insignificant if the isotropy factor Str is below 0.6 and 0.8
(depending on the application). The definition of Std in ISO 25178 part 2 specifies
that the angle can be given from a reference angle s.
The Std parameter is a convenient parameter on surfaces showing scratches and
oriented texture (Schulz et al. 2010; McGarigal et al. 2009).
38
F. Blateyron
2.8 Areal Parameters from ASME B46.1
The US specification standard ASME B46.1 introduced areal parameters in its
1995 edition. The latest 2009 edition contains a set of areal parameters that is
similar to that in ISO 25178 part 2 (2012). ASME B46.1 (2010) defines the
following parameters identically to ISO 25178 part 2: Sa, Sq, Sp, Sv, Ssk, Sku, Sdq,
Str and Std. However, one parameter in ASME B46.1 is named differently: St
corresponding to Sz in the ISO standard.
Two other parameters, not defined in the ISO standard, are specific to the
ASME B46.1 standard:
SWt: peak to valley height of the waviness surface. This parameter can be emulated with an ISO parameter by calculating Sz on an S-L surface with the L nesting
index set at the same value as kc.
Sdq(h): directional root mean square slope. This parameter corresponds to Pdq (as
per ISO 4287 2000 and see Leach 2009) calculated on a profile extracted along the
h direction.
The 2002 edition of ASME B46.1 also defined SDa(h), which it is not in the
2009 edition.
2.9 Areal Parameters from Earlier Reference Documents
2.9.1 European Project Report EUR 15178 EN (1993)
The European project report EUR 15178 EN (Stout et al. 1993a, b) established a
list of areal parameters grouped into several families: amplitude parameters,
spatial parameters, hybrid parameters and functional parameters (see Table 2.1).
The parameters in Table 2.1 are still widely used in some industries and in
scientific publications, although updated and improved parameters have been
available for a long time (De Chiffre et al. 2000).
2.9.2 Basis for 3D Surface Texture Standards
‘‘SURFSTAND’’
The aim of the European project SURFSTAND was to improve areal parameters
defined in EUR 15178 EN (Stout et al. 1993a, b), investigate their correlation with
surface function, and to prepare the basis of an international specification standard
(now ISO 25178 part 2 2012). The parameter set developed in SURFSTAND is,
therefore, based on the ISO parameter set but with several modifications given
below.
Sdr
Stp
Smr
Svr
Sbi
SDq
Ssc
Sds
Str
Std
Sal
Sku
Sz
Ssk
Equivalent (see note 1)
Renamed (see note 6)
Renamed (see note 7)
Renamed (see note 7)
Specific
Equivalent (see note 1)
Different (see note 5)
Different (see note 3)
Equivalent
Equivalent (see note 4)
Equivalent
Equivalent (see note 1)
Different (see note 2)
Equivalent (see note 1)
Equivalent (see note 1)
Equivalent (see note 1)
Sa
Sq
Arithmetic mean deviation
Root-mean-square
deviation
Ten point height
Skewness of topography
height distribution
Kurtosis of topography
height distribution
Density of summits
Texture aspect ratio
Texture direction
Fastest decay
autocorrelation length
Root-mean-square slope
Arithmetic mean summit
curvature
Developed interfacial ratio
Surface bearing ratio
Material volume ratio
Void volume ratio
Surface bearing index
Compatibility
Table 2.1 Parameters from EUR 15178 EN (Stout et al. 1993a, b)
Symbol
Parameter name
Hybrid
Area & Volume
Area & Volume
Area & Volume
Functional
Hybrid
Hybrid
Spatial
Spatial
Spatial
Spatial
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
Family
(continued)
*
*
*
*
*
*
*
*
*
*
*
*
B’14*
2 The Areal Field Parameters
39
Core fluid retention index
Valley fluid retention index
Functional parameters from
DIN 4777
Sk, Spk, Svk, Sr1, Sr2
Parameter name
Specific
Specific
Equivalent (see note 8)
Compatibility
Functional
Functional
Functional
Family
ð2:25Þ
*
*
B’14*
In order to discriminate significant peaks and valleys, only one peak and one valley should be found per autocorrelation area, with side length equal to twice
the fastest autocorrelation decay Sal
Note 3 Sds corresponds to Spd in ISO 25178 part 2 but the discrimination method is different
Note 4 Std is defined in EUR 15178EN with the origin 0 on the y axis while in ISO 25178 part 2 it is defined with an origin s that can be set at any angle
Note 5 Ssc corresponds to Spc in ISO 25178 part 2 but the discrimination method is different
Note 6 Stp corresponds to Smr in ISO 25178 part 2
Note 7 Smr corresponds to Vm in ISO 25178 part 2 (should not be confused with the bearing ratio parameter that has the same name); Svr corresponds to Vv
in ISO 25178 part 2
Note 8 These parameters are extensions in 3D of the parameters defined in ISO 13565 part 2 (1996). The Sr1 and Sr2 parameters have been renamed Smr1
and Smr2 in ISO 25178 part 2. The standard also includes Spq, Svq and Smq from ISO 13565 part 3 (1996)
* These parameters are part of the so-called ‘‘Birmingham 14 parameters’’
P5 P5
zpi þ i¼1 jzvi j
:
Sz ¼ i¼1
5
Note 1 Parameter equations are all given for the discrete case, but their definitions are compatible with those of ISO 25178 part 2 (2012)
Note 2 This parameter is defined here from the five highest peaks and the five deepest valleys
Sci
Svi
Sk …
Table 2.1 (continued)
Symbol
40
F. Blateyron
2 The Areal Field Parameters
41
• Sp and Sv are now introduced as maximum surface peak height and maximum
surface valley depth.
• Sz is defined as the maximum height i.e. the sum of the absolute values of Sp and
Sv. The old ten-point height parameter Sz of the previous report is here renamed
as S10z.
• Ssc, Sds and S5z, which are parameters related to peaks, are now calculated from
peaks detected after a segmentation and Wolf pruning of 5 % of Sz (see Chap. 3).
The SURFSTAND report (published as a book, Blunt and Jiang 2003) also
introduces several new parameters given below.
• Sfd is the fractal dimension calculated from the volume-scale plot where the
volume is calculated between two morphological envelopes (see Chap. 6).
• Vmp and Vmc are introduced as material volume (see ISO 25178 part 2 2012),
respectively peak material volume and core material volume.
• Vvc and Vvv are introduced as void volume (see ISO 25178 part 2 2012),
respectively core void volume and valley void volume.
It is interesting to note that in the SURFSTAND report, the Sa parameter was
removed from the parameter list although it was part of the earlier work. The
authors of the study wanted to avoid encouraging people to use Sa in the same way
they use Ra on profiles, without really knowing if the parameter is the best correlated parameter for their needs. However, during the preparation of the ISO
25178 standard, some experts of WG 16 lobbied strongly in order to reintegrate Sa,
and it was finally added to the draft.
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