Forces, Power, and Stresses in Machining Paul H. Cohen, The Pennsylvania State University

Introduction

THE MODELING AND ANALYSIS of chip formation has been a continuing exercise over the past century. The metal cutting process is a unique and complex production process distinguished by:

• Large shear strains, usually of the order of 2 to 5 (Ref 1) • Exceptionally high shear strain rates, typically from 103 to 105 s-1 with local variations as high as 107 s-1

(Ref 2, 3) • The rubbing of the tool flank over a freshly cut surface that is chemically clean and active • Many process and tooling parameters with a wide range of settings that can drastically alter the cutting

process • A large number of metallurgical parameters in the workpiece that can influence its response to the

cutting tool

These factors and others make the modeling of metal cutting a difficult task that continues to evolve over time. The models and the discussion presented in this article will attempt to explain the basic concepts of the many complex factors that influence the forces, power, and stresses in machining.

Forces and Energy in Orthogonal Machining

Although most production machining processes are oblique (that is, having three component forces), models of the orthogonal (that is, two force) machining of metals are useful for understanding the basic mechanics of machining and can be extended for modeling of the production processes.

Forces. The classical thin zone mechanics was developed for materials that yield continuous chips with a planar shear process coupled with the following assumptions (Ref 4, 5):

• The tool tip is sharp, and no rubbing occurs between the tool and the workpiece • Plane strain conditions prevail (that is, no side spread occurs) • The stresses on the shear plane are uniformly distributed • The resultant force, R, on the chip is equal, opposite, and colinear to the force R' at the tool/chip

interface (Fig. 1)

feed (in unit distance) per tooth, ft, is specified because too large a cut with any tooth may damage the tooth and because different cutters may contain varying numbers of teeth. The feed rate is related to the feed per tooth by:

fm = fr · T · N (Eq 33)

and the material removal rate is specified by:

MRR = Uncut area · Feed rate MRR = (d · w)fm MRR = dwfrTN

(Eq 34)

The power required is therefore Ps · MRR.

Factors Affecting Specific Power. The specific power is a material-related property that can be significantly altered by cutting parameters, prior strain hardening of the workpiece material, and the material and geometry of the cutting tool.

The cutting velocity has a significant effect on the specific power and/or energy because the coefficient of friction on the rake face is speed dependent. Increasing speeds decrease the friction (up to a point), thus decreasing the specific power through the frictional component of specific power (Fig. 7).

Fig. 7 Influence of speed, tool geometry, and prior strain hardening on the specific energy of brass

The cutting tool material will also change the specific power due to changes in friction. For example, under similar conditions, the coefficient of friction and therefore the frictional component of the specific power will be lower for carbide tools than for high-speed steel tools. Therefore, the specific power will be higher, resulting in higher cutting forces. The tool geometry will also play a role in determining the specific power. In particular, the back rake angle, ,

Power in Turning. As with the plate, the total power required in a turning operation can be calculated as P = Ps · MRR. However, the material removal rate must be redefined for turning. Consider the turning operation illustrated in Fig. 6(a), in which a billet of diameter D is turned with depth of cut d to diameter D1. The billet is rotated at N revolutions per minute, while the tool is fed at fr units (millimeters or inches) per revolution, which can be set directly on the machine. Recommended cutting speeds (in meters or feet per minute) are generally available from handbooks and can be converted to rotational speed where V = DN. Suggested feeds are also available.

Fig. 6 Setups for turning (a), drilling (b), and milling (c) operations

The material removal rate has been defined as the uncut area multiplied by the rate at which the material is removed perpendicular to the area. For turning, the area removed is an annular ring of outside diameter D and inside diameter D1.

Thus, the uncut area is (D2 - )/4. The rate at which the tool is fed, fm (in unit distance per minute), is fr · N. Therefore, the material removal rate for turning is:

(Eq 30)

and the total cutting power is Ps · MRR, which can then be adjusted for machine tool efficiency.

The specific power and the material removal rate can also be used to estimate the main cutting force, Fc. The total horsepower can be written as P = Ps · MRR or as the product of the main cutting force multiplied by the velocity, as described for the plate in the section "Energy of Chip Formation" in this article. Equating these two equivalent expressions yields:

(Eq 31)

Power in Drilling. In Fig. 6(b), a drill of diameter D is rotated at N revolutions per minute and fed at fr (unit distance per revolution). The uncut area is the area covered by the drill, or D2/4, while the feed rate perpendicular to the area is fm = fr · N. As with turning, recommended velocities, V, and feeds, fr, are available in handbooks and the literature. The cutting velocity is again related to the rotational velocity by V = DN. The material removal rate is D2/4 · fm, and the power can be calculated as:

P = Ps( D2/4 · fr · N) (Eq 32)

Thus, the calculation of the material removal rate and the power is quite analogous to the case of turning.

Power in Milling. For the milling operation illustrated in Fig. 6(c), the cutter has diameter D with T teeth. The cutter is rotated at N revolutions per minute. Unlike turning and drilling, the table (tool) is indexed at a feed rate of fm (millimeters or inches per minute), which is set directly on the machine. However, the feed rate is not given in handbooks. Instead, the

Thus, the shear stress over the entire face can be conveniently expressed as:

(Eq 26)

To determine the friction force, F, on the rake face, the shear stress given in Eq 26 must be integrated over the area of contact. This yields:

(Eq 27)

where s is the length of seizure (that is, - f).

Although such models are useful in understanding the process, it is difficult to determine the lengths associated with seizure and sliding. Typically, a tool is ground to restrict the total length of contact and the length of seizure determined by the flow lines, as detailed previously.

From this simple orthogonal model, it is clear that the shear strength of the chip material, the relative amount of seizure, and other parameters will significantly alter the machining forces. Increasing shear strengths and lengths of seizure (for constant contact length) will increase the cutting forces. Therefore, the use of tool materials with less propensity for chip seizure or the use of lubricants that decrease s will lower the cutting forces accordingly.

Power Consumption in Production Processes

Although one may wish to describe the energy per unit volume needed to form the chip, machine tools are typically rated in terms of power. Unit (or specific) power values can be calculated by dividing the power input to the process, FcV, by the volumetric rate at which material is removed and then dividing this quantity by 33,000 to convert to horsepower. The specific power, Ps, is a measure of the difficulty involved in machining a particular material and can be used to estimate the total cutting power, P. Typical specific horsepower values are given in Table 1.

The specific power is the power required to remove a unit volume per unit time. Therefore, the specific and total powers are related as follows:

P = Ps · MRR (Eq 28)

where MRR is the material removal rate, or volume of material removed per unit time. The material removal rate can be computed as the uncut area multiplied by the rate at which the tool is moved perpendicular to the uncut area. As previously determined for a plate, the material removal rate is the uncut area, t · w, multiplied by the velocity of the tool, V. Thus, the cutting parameters and machine tool kinematics define the material removal rate. There are many standard sources for specific power values for a variety of materials.

Unfortunately, machine tools are not completely efficient. Losses due to component wear, friction, and other sources prevent some power from reaching the tool. Therefore, the gross power, Pg, needed by the motor can be defined as:

(Eq 29)

where is the efficiency of the machine.

Fig. 5 Model of stress distribution on tool during cutting. Source: Ref 14

The normal stress on the rake face is defined by:

f = max(x/ )n (Eq 23)

where max is the maximum normal stress at the tool tip (or x = ), is the total length of contact of the chip on the tool, x is the distance from the point at which the chip leaves the tool to the point of interest, and n is the exponent. The normal force can be obtained by integrating the normal stress over the area of contact on the tool face:

(Eq 24)

The shear stress is more complicated to evaluate because the behavior of the chip material as it passes over the tool varies along the rake face. The region of seizure close to the tool tip must be modeled differently from the region of sliding (Coulombic) friction. Over the region of seizure ( f x ), the shear stress has a constant value, , because the chip material shears internally, as illustrated by the flow lines in Fig. 4.

Over the sliding region, the shear and normal stresses are related by:

= f = max(x/ )n

(Eq 25)

Fig. 4 Flow lines in a chip

This concept of seizure is quite different from the standard notions of sliding friction. Because of the high interface temperatures and pressures, the material adjacent to the tool surface is almost stationary, and relative shearing takes place in the chip.

As originally developed by Zorev (Ref 14) and consistent with the empirical results presented, the stresses on the rake face are inherently nonlinear, as shown in Fig. 5. The normal stress is assumed to take on a maximum value, max, at the tip of the tool; the stress then decreases as a power function of the distance from the tool tip to the point at which the chip leaves the tool (Fig. 5). The shear stress is constant in the region of seizure and then decreases as a power function to the point at which the chip leaves the tool.

(Eq 21)

Analogously, the normal stress on the rake face can be written as:

(Eq 22)

These models have been found to be useful approximations of the behavior of the chip as it slides over the tool. However, there is a large body of experimental evidence to suggest that the stresses are not uniformly distributed on the rake face.

Nonuniform Stress Distributions on the Rake Face. The body of experimental evidence indicating the nonuniformity of the stresses on the rake face is extensive, using a wide variety of experimental techniques and observations. Perhaps the simplest observation supporting this conclusion is the transfer of workpiece material to the tool as observed by the unaided eye, light microscope, or scanning electron microscope (Ref 10). The transfer of this material does not occur over the entire contact area, but near the tip of the tool. Experiments utilizing photography through transparent sapphire tools (Ref 11), photoelastic tools (Ref 12), quick-stop devices to observe metal flow in the chip (Ref 13), and other techniques have revealed the nonuniformity of the stresses.

Quick-stop devices that separate the tool from the chip freeze the flow pattern of the material in the chip. Such studies have revealed two major regions on the rake face with respect to flow. When polished and etched, it is clear from the flow lines that the material near the tool tip is seized by the tool. This can be shown by the flow lines in Fig. 4, which run parallel to the tool face. The rest of the contact area exhibits sliding contact.

Fig. 3 Shear stress variation with Brinell hardness for ferrous and nonferrous metals. Source: Ref 9

Stress Distributions on the Rake Face. The nature of the tool/chip interface and the distribution of the shear and normal stresses are critical in understanding the cutting process and the performance of cutting tools. The high stresses, coupled with the high temperatures and large strains in the chip adjacent to the tool face, make the secondary shearing process difficult to model.

Uniform Stresses on the Rake Face. The classical analysis of the forces and stresses on the rake face assumes that Coulombic sliding friction is present and that the stresses are uniformly distributed. Therefore, the coefficient of sliding friction is simply the frictional force, F, divided by the normal force, N, acting on the rake face. Thus:

(Eq 19)

The coefficient of friction is velocity dependent, with increasing speeds yielding lower friction.

The area of contact on the tool/chip interface is the product of the width of cut, w, and the length of sliding contact, , as illustrated in Fig. 1. Thus, the area of sliding contact on the rake face is:

Af = w · (Eq 20)

and the shear stress at the interface can be calculated as:

(Eq 17)

and the normal stress can be computed similarly as:

(Eq 18)

Therefore, the stresses rely only on measured cutting forces (Fc and Ft), the geometry of the cut (t and w), and the deformation geometry ( ).

The shear stress, s, takes on a constant value for a particular material. Figure 3 and Table 1 provide typical values for a variety of metals.

Table 1 Shear stresses and specific horsepowers of selected engineering materials

Material Shear stress, psi

Specific horsepower, hp/in.3/min

Hardness, HB

Magnesium 28,000 0.17 . . . 1100 aluminum alloy 16,700 . . . . . . 6061-T6 aluminum alloy 35,722 0.35 . . . 2024-T4 aluminum alloy 50,000 0.46 . . . Copper 44,850 0.78 . . . 60-40 brass 47,000 65-35 brass 50,000 70-30 brass 56,940 0.59 . . .

0.58 150-175 AISI 1020 steel 61,500 0.67 176-200

AISI 1112 steel 63,500 0.5 150-175 Type 304 stainless steel 105,000 1.1-1.9 . . . Titanium 173,500 1.9 . . .

plane. Experimental results have shown that the material is elastically deformed at distances sufficiently far from the tool tip. As the material approaches the tool, the compressive stresses will begin to plastically deform the workpiece material as shown in Fig. 2. Behind the tool tip, the stresses will be tensile.

Fig. 2 Stresses in the workpiece

The distance of the elastic-plastic boundary from the tool tip will depend on tooling parameters, cutting parameters, and workpiece material properties. In particular, the amount of prior strain hardening and the ability of the workpiece material to work harden will alter the magnitude of the stresses in the workpiece and will affect the placement of the elastic-plastic boundary. Materials with little prior strain hardening will extend their boundaries farther out from the tool tip.

Stresses on the Shear Plane. Consistent with the assumptions in the section "Forces" in this article, the shear plane is generally modeled to have uniform distributions of both shear and normal forces over its entire area. The shear area, As, is the area of cut (Ac = t · w) inclined at the shear angle . Thus, the shear area is:

(Eq 16)

The shear stress on the shear plane can then be calculated as follows:

The shear energy per unit volume can be calculated by substituting the energy per unit time necessary to shear the material in place of the total energy per unit time in Eq 8. Thus:

(Eq 11)

where Vs is the shear velocity (where Vs = V cos /cos( - ), as defined in the article "Mechanics of Chip Formation" in this Section; see the discussion of shear angle measurement during orthogonal machining). The shear energy per unit volume is the largest of the four components, typically representing more than 75% of the total.

The friction energy per unit volume is consumed as the chip slides on the rake face of the tool. This component is very sensitive to cutting velocity and can be written as:

(Eq 12)

where Vc is the velocity of the chip as it flows over the tool (Vc = V sin /cos( - ), as defined in Eq 4 of the article "Mechanics of Chip Formation" in this Section).

The kinetic (momentum) energy per unit volume required to accelerate the chip is generally neglected but takes on increasing importance with very high speed machining. It can be written as:

(Eq 13)

where Fm is the momentum force = V2tw sin , where is the density of the material being cut and is the shear strain.

Additional energy is required to produce a new uncut surface. The surface energy per unit volume needed to create this new surface can be written as:

(Eq 14)

where T is the surface energy of the material being cut. This component is also generally neglected.

Therefore, for most machining applications, the specific energy can be accurately estimated as:

u us + uf (Eq 15)

except at high speeds (above 900 to 1200 m/min, or 3000 to 4000 sfm) for which the kinetic specific energy should be included. Specific energies can be used to calculate the power per unit volume per unit time (specific horsepower) and are readily accessible for most engineering materials. They are a good measure of the difficulty involved in machining a particular material.

Stress Distributions in Metal Cutting

High shear and normal stresses occur both in the primary shear plane and on the rake face of the tool. This region of friction or secondary shear is critical in understanding the process mechanics and the wear of cutting tools.

Stresses in the Workpiece. As discussed in the article "Mechanics of Chip Formation" in this section, the fundamental mechanism of chip formation requires prior work hardening before the workpiece material reaches the shear

stages within a scanning electron microscope, is typically computed by using a ratio of the depth of cut, t, to chip thickness, tc, as follows:

(Eq 1)

The shear strain necessary to shear the work material at this angle, , is:

= tan ( - ) + cos (Eq 2)

Analyzing the primary shearing process in Fig. 1, the shear and normal forces on the shear plane can be written as functions of the measured horizontal and vertical (dynamometer) forces and shear angle, as follows:

Fs = Fc cos - Ft sin (Eq 3)

Fn = Fc sin + Ft cos (Eq 4)

Similarly, the forces on the rake face can be written as functions of the same measured force components and the tool back rake angle as:

F = Fc sin + Ft cos (Eq 5)

N = Fc cos - Ft sin (Eq 6)

The resultant force, R, which acts on the chip and is shown in Fig. 1(b), can be written as the vector sum of the measured forces, the forces acting on the shear plane, or the forces acting on the rake face of the tool. Therefore:

R = ( + )1/2 (Eq 7)

R = ( + )1/2 (Eq 8)

R = (F2 + N2)1/2 (Eq 9)

Energy of Chip Formation. During the cut, the total energy per unit time (or power) can be calculated simply as the product of the primary cutting force, Fc, and the velocity of cut, V. However, because many parameters can be varied in the cutting process that change the total energy consumed, this energy value is typically normalized by dividing by the rate at which material is removed. The material removal rate is calculated by multiplying the area being cut (t · w for the case of the plate of width w shown in Fig. 1a), by the velocity perpendicular to that area at which the material is removed (V in this case). Thus, the energy per unit time, or specific energy, u, can be calculated as:

(Eq 10)

The specific energy can be partitioned into four components (Ref 6, 7):

• Shear energy per unit volume, us • Friction energy per unit volume, uf • Kinetic (momentum) energy per unit volume, um • Surface energy per unit volume, ua

Fig. 1 The geometry (a) and forces (b) in orthogonal cutting

The modeling of the orthogonal cutting process defines two regions of deformation (primary and secondary), each described by its own set of orthogonal forces, as shown in Fig. 1(b). Because these force components cannot be directly measured (except for the forces on the rake face of the tool when = 0°), a dynamometer must be used to measure the primary (horizontal) cutting force, Fc, and the tangential (vertical) force, Ft. Thus, the measured forces can be resolved onto the shear plane through the shear angle, , and onto the rake face through the back rake angle, .

The shear angle, , is the angle the primary shear plane makes with respect to the horizontal motion of the tool. Although it is possible to observe and measure this angle in special experiments by using high-speed photography or the machining

will influence the friction and therefore the specific power. Larger back rake angles will result in lower power and/or energy consumed per unit volume per unit time, provided the tool maintains its integrity. Figure 7 illustates this in the machining of brass. In addition, cutting tools (inserts) are available in a variety of nose radii. The radius of the tool has been shown to change the specific power and cutting forces significantly with larger radii yielding higher specific power.

Research has shown that ductile materials such as aluminum, brass, or copper will behave quite differently during machining, depending on the amount of strain hardening from prior processing. These materials have a great propensity for seizing the tool when in an annealed state. Cutting forces and specific powers are greater than for the same materials with significantly more work hardening. Thus, the processing history of a metal is an important determining factor for the specific power requirements.

References

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