The following points highlight the top five theories on mechanics of metal cutting in the industries. The theories are: 1. Ernst-Merchant Theory 2. Merchant Theory 3. Stabler Theory 4. Lee and Shaffer Theory.
1. Ernst-Merchant Theory:
According to this theory:
This is based on the assumptions:
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(i) That expenditure is minimum in the process, i.e., shear will take plane in a direction in which energy required for shearing is minimum.
(ii) That shear stress is maximum at the shear plane and it remains constant.
In the Merchant’s theory, it was seen that:
2. Merchant Theory:
Merchant has carried out lot of practical experiments on metal cutting and he found that the relationship 
does not hold good practically.
He then modified his theory by assuming that shear stress y along the shear plane varies linearly with normal stress, i.e. y = y0 + K σ, y0 is the value of y when normal stress σ = 0
He then derived 2 ɸ = C + α = β, where C = machining constant = arc cot k
Its value varies from 70° to 80° for various steels. i.e. it measures the dependence of shear strength on various steels.
3. Stabler Theory:
He modified the Ernst-Merchant equation as:
4. Lee and Shaffer Theory:
This theory is based on ideal theory of plasticity according to which shear occurs on a single plane.
This assumption when applied to cutting process, means that:
(i) Material ahead of tool behaves as ideal plastic mass,
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(ii) No hardening in chip occurs,
(iii) There exists a shear plane which separates the chip and work-piece.
Accordingly metal bounded by ABC in Fig. 22.25 is assumed to be rigid plastic, and to have been stressed throughout to its yield point. The chip above BC is assumed to be stress free and hence the slip lines meet BC at 45°. Shearing occurs along the entry slip line BA, which is a velocity discontinuity. Since the chip is assumed to be stress free, the normal stress at BC along the stress-free interface is zero. For these conditions it can be shown that β + ɸ – α = π/4 or 45°.
This relationship on graph of ɸ versus β – α will have intercept of ɸ/4 and slope of –1. This is also not accurate for most of the metals. To overcome its inadequacy, existence of a small built up edge of the tool was considered as shown in Fig. 22.26. Thus result was modified to take a factor θ to take into account the built up nose formation.
Even this relationship was not widely acceptable because of the presumptions like behaviour of material as ideal plastic. This built up edge actually is much greater in size and work hardening of metal also needs to be considered.
5. Christopherson, Oxley and Palmer’s Theory:
Christopherson, Palmer and Oxley made detailed study of mechanics of chip formation on the flow of grains in the material and found that the grains flow showed a streamlined pattern and that the boundary between elastic and plastic zones was not a straight line but a narrow wedge shaped zone. This theory is probably the most important recent development in the theory of metal cutting.