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Nomenclature
ks = value of plasticity corresponding to yield strength
kf = value of plasticity corresponding to fracture strength
σy= yield strength of work material
ε = shear strain
σ0, n = parameters of hardening
a = undeformed chip thickness
a1 = chip thickness
ξ = a1/a = chip thickness coefficient
α = tool rake angle
Φ = imaginary shear angle
N = normal force on tool rake face
F = friction force on tool rake face
L = tool-chip contact length
σ = average normal stress/hydrostatical pressure
σn(x), τn(x) = normal and shear stress distribution on tool rake face
θ(x) = angle between tangent to α-slip line at current point M on tool-chip interface and X-axis
ϕ = angle between normal to the given contour and X-axis
Sf = true fracture strength of work material
Brief history

Timae proposed the model of chip formation with single shear plane
Zvorykin and Merchant applied minimum force and energy principles
Problems of model with single shear plane
Palmer and Oxley used cinema technique and found experimentally the behaviour of material particles in the primary deformation zone they used modified Henky and balance moment equations and concluded about acceptability of their model. The main problem in this approach is that the stress distribution on tool rake face is unknown and it is necessary to make an assumption about this distribution for solution of balance force equations
Lee and Shaffer were the first, who proposed the continuation of plastic deformation after primary shear.
SLIP LINE MODEL
Shear stress distribution
Split tool

distinction in experimental results is most probably caused by different ways for stress measurement.
Zorev and Bobrov used the split-tool method. (neglected the forces on the clearance face)
caused by built-up-edge formation and “agglutination” between two parts of the tool.
Gordon changed the design of the split-tool dynamometer, which can decrease the influence of forces on tool clearance face
Bagchi and Wright applied photo-elastic sapphire tool and got the same behaviour of experimental shear stress distribution for steels 1020 and 12L14.( the application of photo-elastic sapphire tool makes possible the reduction of errors caused by general split-tool method)

Slip line model & Tool-Chip Contact Length
Tresca Plastic flow criterion.(Assumed slip line field)
FABDEF- Primary deformation zone
AFG-Second central slip-line field
assumed that in general, there is no* shear stress on the tool-chip interface at tool edge i.e. point A
<FAG = <AGF = π /4 and so triangular AFG is isosceles
from the suggested geometry of slip-line field

…………..(1)