What is Shear Plane Angle in machining and its formula derivation

**What is Shear Plane Angle in Tool Machining?**

It is defined as the plane angle at which the chip is start separating from the workpiece during the metal cutting operation.

With the help shear angle you can calculate cutting force , friction force , shear force, surface smoothness and efficiency of metal removal process

**Derivation of Shear angle in machining calculation **

Geometry of Orthogonal chip formation |

**Assumption**

Let shear plane angle is denoted by `\phi `

Let Rake angle is denoted by `\alpha `

Let thickness of chip before cutting is `t_{1} `

Let thickness of chip after cutting is `t_{2} `

Let chip thickness ratio (chip thickness before cutting /chip thickness after cutting) =`r_{c} `

from the figure we can see right angled triangle ABC

= `\frac{BC}{AB} = sin \phi `

= `AB = \frac{BC}{sin \phi } = \frac{ t_{1} }{sin \phi} ` ........(1)

From right angled triangel ABD,

`\frac{BD}{AB} = sin(90- \phi - \alpha ) = cos( \phi - \alpha )`

` \frac{ t_{2}}{AB} =cos( \phi - \alpha )` .......(2)

From equation (1) and (2) , we get

`\frac{ t_{1} }{sin \phi } = \frac{ t_{2} }{cos( \phi - \alpha )} `

put the value of `t_{1}` and t2 in chip thickness raito

= `r_{c} = \frac{t_{1}}{t_{2}} = \frac{sin \phi }{cos( \phi - \alpha )} = \frac{sin \phi}{cos \phi cos \alpha +sin \phi sin \alpha}`

= `\frac{r_{c}cos \phi cos \alpha}{sin \phi } + \frac{r_{c}sin \phi sin \alpha}{sin \phi} =1`

= `\frac{r_{c}cos \alpha}{tan\phi } + r_{c} sin \alpha =1`

= `\frac{r_{c}cos \alpha}{tan\phi } =1- r_{c} sin \alpha`

= `tan \phi = \frac{ r_{c} cos \alpha }{1- r_{c} sin \alpha } `

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