Theory Of Metal Cutting- Theory Of Chip Formation

LECTURE-08 THEORY OF METAL CUTTING - Theory of Chip Formation

NIKHIL R. DHAR, Ph. D. DEPARTMENT OF INDUSTRIAL & PRODUCTION ENGINEERING BUET

Chip Reduction Coefficient (ξ)

Chip reduction coefficient (ξ) is defined as the ratio of chip thickness (a2) to the uncut chip thickness (a1). This factor, ξ, is an index of the degree of deformation involved in chip formation process during which the thickness of layer increases and the length shrinks. In the USA, the inverse of ξ is denoted by rc and is known as cutting ratio. The following Figure shows the formation of flat chips under orthogonal cutting conditions. From the geometry of the following Figure.

a1 a2 A

B

Chip

β O

γo

C

Workpiec e

Tool

a 2 AC OA cos(β − γ 0 ) cosβ cosγ 0 + sinβ sinγ 0 ξ= = = = − − − −[1] a 1 AB OA sinβ sinβ Department of Industrial & Production Engineering

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Shear Angle (β) From Equation [1]

cosβ cosγ + sinβ sinγ cosγ 0 0 0 + sinγ ξ= = 0 sinβ tanβ cosγ 0 tanβ = ξ − sinγ o  cosγ  o  Shear angle β = tan − 1  ξ − sinγ  o 

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Condition for maximum chip reduction coefficient (ξ) from Equation [1]

dξ d  cos(β − γ 0 )  = 0 or  =0 dβ dβ  sinβ 

[

]

sinβ − sin(β − γ ) − cos(β − γ )cosβ 0 0 =0 2 sin β π cos(β − γ ) cosβ + sin(β − γ ) sinβ = 0 = cos 0 0 2 π cos(β − γ + β) = cos 0 2 1π  ∴ β =  + γ  Shear angle 0 22 Department of Industrial & Production Engineering

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Velocity Relationships The following Figure shows the velocity relation in metal cutting. As the tool advances, the metal gets cut and chip is formed. The chip glides over the rake surface of the tool. With the advancement of the tool, the shear plane also moves. There are three velocities of interest in the cutting process which include:

VC

= velocity of the tool relative to the workpiece. It is called cutting velocity

Vf

β

= velocity of the chip (over the tool rake) relative to the tool. It is called chip flow velocity

V s=

velocity of displacement of formation of the newly cut chip elements, relative to the workpiece along the shear plane. It is called velocity of shear

Vf

Vc

Vs Vc

γo -β

90o -γo

Vf

Workpiece

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γo

Chip

γo

β

Vs

90o -β+γo

Tool

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According to principles of kinematics, these three velocities, i.e. their vectors must form a closed velocity diagram. The vector sum of the cutting velocity, Vc, and the shear velocity, Vs, is equal to chip velocity, Vf. Thus,

V =V +V f c s V V V s c = = f sin(90o − γ ) sin 90o − (β − γ  sinβ o  o  V sinβ V sinβ c V =V = = c f c ξ sin 900 − (β − γ ) cos(β − γ o )  o  V or, c = ξ Vf

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γo -β

90o -γo Vf

Vc β

Vs

γo 90o -β+γo

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Kronenberg derived an interesting relation for chip reduction coefficient (ξ) which is of considerable physical significance. Considering the motion of any chip particle as shown in the following Figure to which principles of momentum change are applied:

dv F = −m dt dθ 2 N = mω r = mv dt F dv μ= =− N v dθ

Vc

π ( − γ0 ) 2

N

F

Vf

γo

dv − = μ dθ v Department of Industrial & Production Engineering

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As the velocity changes from Vc to Vf, hence

π ( -γo ) V f dv 2 = ∫ πdθ ∫ − π ( −γ ) v 0 2 V Vc c V  π    f − ln = μ − γ  Vf γo V  o 2   F  c N   π    μ − γ V 2 0  c =e  V f This equation demonstrates that the chip reduction coefficient and chip   flow velocity is dependant on the frictional aspects at the interface as π  μ − γ  well as the orthogonal rake angle (γ0). If γ0 is increased, chip reduction 2  0  coefficient decreases. ξ=e  0

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Shear Strain (ε) The value of the shear strain (ε) is an indication of the amount of deformation that the metal undergoes during the process of chip formation. The shear strain that occurs along the shear plane can be estimated by examining the following Figure. The shear strain can be expressed as follows:

ε=

AC AD + CD AD CD = = + = cot β + tan(β − γ ) - -[1] o BD BD BD BD Shear plane

Magnitude of strained material

β γo Workpiece

A

Chip=parallel shear plates

Tool

A

β Plate thickness

C

a

D C

β-γo

γo

B

B b

c

Shear strain during chip formation (a) chip formation depicted as a series of parallel sliding relative to each other (b) one of the plates isolated to illustrate the definition of shear strain based on this parallel plate model (c) shear strain triangle Department of Industrial & Production Engineering

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From equation [1]

cos γ

o ε = cot β + tan(β − γ ) = − −[2] o sin β. cos (β - γ ) o From velocity relationship V cos γ s = o − − − [3] V cos (β - γ ) c o From equation [2] and equation [3] V s Shear strain ε= V sin β c

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