A
mechanistic model based on brittle fracture
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The model is
based on the following simplifications:
1.
The workpiece material is an ideally brittle material;
2.
The material is removed by brittle fracture;
3.
The diamond abrasive particles are assumed to be rigid spheres of the same
size, and all the particles are assumed to take part in cutting during
each ultrasonic cycle.
To develop the
model, the first step is to study the interaction between the individual
diamond particle and the workpiece. Then, based on this, an expression for
the volume of material removed by a single diamond particle within a
single vibration cycle will be derived. Finally, the effect of all the
working diamond particle is aggregated to obtain the MRR of the process.
Because during
rotary ultrasonic machining the tool vibrates and rotates simultaneously,
the shape of the intersection crater between the workpiece and the diamond
particle can be approximately considered as a part of an ellipsoid. The
volume of this part of the ellipsoid, or, the indentation volume, can be
calculated by the following equation (see Figure 12).
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Figure 12
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(1)
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where, d is the
diameter of the spherical abrasive, d the depth
of maximum penetration, and L the distance moved by the particle during
penetration into the workpiece due to the rotary motion of the tool.
The volume of
the material removed by the diamond particle, W, is considered to be
proportional to V:
where the
constant of proportionality, k, is a function of the material properties
and the probability of causing fracture, etc.. The value of k is
experimentally determined.
The MRR is
given by the product of the volume of material removed by one particle, W,
the frequency of vibration, f, and the number of active diamond particles,
n. Hence, the MRR is given by the following equation:
Substituting
(1) into this equation, we get:
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(4) |
The model
is mechanistic in the sense that the parameter k can be determined from a
few experiments for a particular material and then used in prediction of
MRR over a wide range of process parameters. This is demonstrated for
magnesia stabilized zirconia, where very good predications are obtained
using an estimate of this single parameter. Figure 13 is the plot of the
experimental MRR for each experiment versus the corresponding nfp(1+L/d)(d/2-d/3)d2
value for the experiment. The slope of the least-squares straight line
gives the estimate of k. On the basis of this model, relations between MRR
and the controllable machining parameters are deduced. Some of these
relations are shown in Figure 14. These relationships agree well with the
experimental trends discussed in the previous section.
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Figure 13
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Figure 14 |