30-04-2014, 12:36 PM
Cutting Mechanics and Analytical Modeling
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Questions and Answers on Machining Modeling
Prior to the description of the most important modeling methods and their features,
it would be helpful to introduce some questions that may come to mind of those
who want to use modeling, and attempt to give answers. Although some answers
are already given in the previous chapter, a more elaborated approach is presented
in this section. The questions raised apply to all kinds of modeling; the answers
mostly concern FEM, without excluding all the other methods. In the next chapter
some more questions and answers, this time solely for FEM, will be presented.
A first question would be: what is modeling and what is simulation? A model
can be defined as an abstract system which is equivalent to the real system with
respect to key properties and characteristics, and is used for investigations,
calculations, explanation of demonstration purposes, which would otherwise be
too expensive or not possible. A model permits general statements about elements,
structure and behavior of a section of reality. Simulation is an imitation of a
dynamic process in a model in order to obtain knowledge which can be transferred
to reality. Both definitions, for model and simulation, are quoted from Ref. [1]; the
former is from Brockhaus while the latter from VDI Guideline 3663.
Orthogonal and Oblique Cutting
The chip flow in all wedged-tool machining processes can be described, in theory,
in a common way by two different cutting schemes termed orthogonal cutting and
oblique cutting, depicted in Figs. 2.1 and 2.2 respectively. In orthogonal cutting
the cutting edge of the tool is perpendicular to the direction of relative workpiece-
cutting tool motion and also to the side face of the workpiece. From the relative
movement of workpiece and cutting tool, a layer of material in the form of chip is
removed. In order to continue removing material at a second stage, the tool is
taken back to its starting position and fed downwards by the amount f, the feed of
the process. Perpendicular to f, d is the depth of cut, which is smaller than or equal
to the width of the tool edge. The surface along which the chip flows is the rake
face of the tool. The angle between the rake face and a line perpendicular to the
machined surface is called rake angle c. The face of the tool that is near the
machined surface of the workpiece is the flank face. The angle between the flank
face of the tool and the workpiece is called clearance angle a.
Cutting Mechanics and Analytical Modeling
The history of research pertaining to metal cutting is well documented by Finnie
[6] who pinpoints the work of Cocquilhat [7] in 1851 as the first research in the
area of measuring the work required to remove a given material volume by
drilling. However, the first work on chip formation by Time [8] in 1870 presented
the results obtained when observing cutting. In this publication it was argued that
the chip is created by shearing ahead of the tool. Astakhov claims that this is one
of the first publication that a shear plane theory is suggested [9], probably the first
being the one by Usachev in 1883 [10]. It is also shown that there is no contra-
diction between Time and Tresca [11]; Tresca argued that the chip in metal cutting
is produced by compression ahead of the tool. Zvorykin [12] was the first to
provide physical explanation for this model; his work resulted to an equation
predicting the shear angle. In 1881, Mallock [13] also identified the shearing
mechanism in chip formation and emphasized the importance of friction in the
tool-chip interface. However, it was the work of Ernst and Merchant [14] in 1941
that made the shear plane model popular; most of the fundamental works on metal
cutting mechanics reference this paper and many analytical models of orthogonal
cutting still use the relations derived from this work. In the following paragraphs
some key points of analytical modeling and advances in mechanics of cutting will
be discussed.
Slip-Line Field Models
Stress analysis in a plane strain loaded material indicates that at any point there are
two orthogonal directions that the shear stresses are reaching a maximum, but
these directions can vary from point to point. A line, which generally speaking is
curved, tangential along its length to the maximum shear stress is called a slip-line;
a complete set of slip-lines in a plastic region forms a slip-line field. The slip-line
field theory must follow rules that allow the construction of a slip-line field for a
particular case. First of all, the boundary between a part of a material that is
plastically loaded and another that has not yielded is a slip-line.
Shear Zone Models
The next step in analytical modeling was to enhance some features that were
neglected or simplified in previous models but play an important role in metal
cutting. Most shear plane models assume that shear stress on the shear plane is
uniform, no strain hardening is considered and that friction along the cutting tool-
chip interface is characterized by a constant friction coefficient; this last
assumption is in contradiction with experimental data. If it is assumed that
deformation takes place in a narrow band centered on the shear plane, more
general material assumptions can be used. The effects of yield stress varying with
strain and sometimes with strain rate and temperature were considered and
simplification of the equilibrium and flow was achieved. Pioneering work in this
area is associated with the work of Oxley. Based on experimental data, where the
plastic flow patterns are observed, it is assumed that the shear zone thickness is
about one tenth of the shear zone length.