Report 4 : Analysis of Shear Angles

Analytical Research on Cutting Mechanics of Metals

Abstract

(1) It is one of the essential ploblems to consider how the shear plane is determined in orthogonal machining.
In this report the process to determine a shear plane is explained by means of plastic dynamics.
The shear plane of orthogonal machining is treated as a plain strain problem, and the angle θ_f between the direction of principal stress and the rake surface of a tool is calculated mathematically from the normal stress and the shear stress on the rake surface of a tool.
The primary shear angle φ₁ is expressed as follows :
φ₁=45°-θ_f+α
(2) On the other hand, Mohr's stress circle and the diagram of a shear plane are shown for the orthogonal machining of lead.
From Mohr's circle of lead the following relation is obtained experimentally.
p/k=1.72.0
p : mean stress-p=1/3(σ_x+σ_y+σ_z)
k : shear yield stress of chip.
Lee-Shaffer assumes that one of the principal stress in the plastic zone of chip is zero.
i.e. σ_x=-p+k=0 ∴p/k=1
This assumption is not sufficient for the practical cutting process, and is considered as one of the reasons why Lee-Shaffer's formula of a shear angle shows the different value from the experimental shear angle.
(3) The stress-strain curves of material are seemed to be useful to investigate the relation between the sum of shear strains γ_t and the total mean value of shear stresses τ_t which are explained in Report 3.
Loizou and Sims have performed high speed compression tests for lead by using the cam plastometer under the conditions of various strains, strain rates and temperature.
From those results the following experimental fomula can be obtained, assuming that the temperature is constant at room temperature.
σ_ij=3.8·ε_i^0.23·ε_j^0.033
σ_ij : compression stress for ε_i and ε_j (kg/mm²)
ε_i : compression strain
ε_j : compression strain rate.
(4) The cutting tests of lead are performed at the room temperature and very slow cutting speed 50mm4, 000mm/min.
The mean temperature rises so little in those cutting tests that the effect of temperature on the shear yield stress of chip is negligible. According as the increase of rake angles ranging from 20° to 40°, γ_t decreases but τ_t increases a little.
The increase of τ_t is presumed that the strain rate increases according to the increase of rake angles, and hardend the chip.
As the results of calculation by the experimental formula of σ_ij, the mean strain rate γ_t showed the order of 10²10⁴ sec^-1.