Civil Engineering Reference
In-Depth Information
deformation of the chip and crack propagation, i.e., fracture. In conventional machining analysis, the
energy utilized for fracture is ignored [65], as it forms an insignificant proportion of the energy required
for plastic deformation. However, a recent investigation on brittle fracture by Gol'dshtein and Moslov
[23] highlights the role played by fracture energy in the formation of surface structures.
The crucial interpretation involves considering the workpiece as a hierarchy of scale structures. Griffith's
law characterizes ideal brittle fracture as follows: the growth of a crack results in a reduction in the system
potential energy, and this is balanced by the work of separation that forms the new surface [18, 40].
However, in the presence of a scale hierarchy the fracture energy forms a cascade, released from larger
scales to smaller ones, and finally to microscopic scales, where the new crack is formed. At each of the
smaller scales, the energy is modulated as it is passed from larger scales [50]. This energy cascade is
described using the following power law [23].
D
f
1
Gl
()
l
n
,
(1.2)
where
G
(
l
) is the elastic energy released on the
n
th scale,
l
, and
D
is the fractal dimension.
n
n
f
Premature Deformation in the Tool Path
In metal cutting, high stresses are generated at the tool tip, leading to fracture of the work material. In
addition, secondary compressive stresses are generated in the material just ahead of the tool tip. The
extent of material affected will depend on the cutting force and the area of stress distribution [26].
Depending on the magnitude of the compressive stress, the material in the tool path will be deformed,
elastically or plastically. The portions deforming elastically recover, but those deforming plastically do
not, i.e., there is differential recovery along the tool path. Consequently, there exists a varying deformation
pattern in the workpiece ahead of the cutting zone. This deformation can conceivably affect the cutting
characteristics (e.g., depth of cut), when the relative motion brings the tool to those points. This is yet
another possible cause for the existence of interrelated features in surface profiles, which can be char-
acterized by suitable fractal parameters.
Mathematical Implications: Fractal Models
The above subsections have explored the various possibilities for the manifestations of fractal structures
in surface errors, characterized by special long range correlations between various scales. An elegant
method of modeling the long-term dependence in data is by using fractal parameters [36]. The vehicle
we use for computation of these fractal parameters is the theory of wavelets, elucidated in the next section.
This also forms the mathematical prelude to the fractal-wavelet relationship, which forms the crux of
the analysis and synthesis using this technique.
Fractal-Wavelet Analysis
The objective of the fractal-wavelet analysis procedure is to take any experimental profile and abstract the
error information in terms of fractal parameters. This problem is ideally set up within the context of
multiresolution analysis. Here the experimental profile is studied at different resolutions, and at each distinct
resolution we obtain the approximation and detail. These operations are mathematically defined below.
Approximations
Let us denote a given product surface profile as a function
is the measurement coordinate.
In manufactured profiles, since the data is usually given in terms of height (from a specified datum) at
discrete points along the workpiece, this data sequence is indicated by
p
(
x
), where
x
A
k
()
p
A
0
. Here
implies the base
approximation of the profile
p
, and
k
is an integer denoting the
k
th data point. In general the data
A
k
()
p
sequence corresponding to an approximation at resolution level
m
is given by
[67]. Consider a
discrete filter
with the following impulse response:
h
k
(
1
x
()
,
(
xk
)
)
,
kZ
(1.3)
