# Chip Thickness Ratio: Derivation, Calculation, Relation ɸ & α

In this article, you will learn what is chip thickness ratio, how to derive it, how to calculate it with a formula as well as will learn on the relation between chip thickness ratio, shear angle, and rack angle.

So without wasting time let's get started,

## What is Chip Thickness Ratio?

Chip thickness ratio is the ratio of the thickness of metal before cutting to the thickness of metal after cutting.

During cutting, the cutting age of the tool is positioned a certain distance below the original this corresponds to the thickness of the chip.

Prior to chip formation t as the chip is formed along the shear plane its thickness increase tc. (After cutting chip Thickness)

So the ratio t/tc is called the chip thickness ratio.

It is represented by r.

So,

r = t/tc

Where,

t = uncut chip thickness
tc= cut chip thickness

## Derivation of Chip Thickness Ratio

The chip thickness ratio can be derived from the volume of the chip.

Let assume,

l = length of the chip before cutting
b = width of the chip before cutting
t = thickness of chip before cutting

lc = length of the chip after cutting
bc = width of chip after cutting
tc = thickness of chip after cutting

As we know,

The volume of the chip before cutting is equal to the volume of the chip after cutting.

So,

l × b × t = lb × bc × tc

Generally, the width of the chip is to be taken as negligible.

So,

b = bc

Then,

l × t = lb × tc

t/tc = lb/l

So,

Chip thickness ratio,

r =  t/tc = lb/l

## Chip Reduction Ratio or Coefficient

It is defined as the ratio of chip thickness after cutting to the ratio of chip thickness before cutting.

The chip reduction ratio or coefficient is always inversely proportional to the chip thickness ratio.

It is represented by E.

Now, we will know the relation between chip thickness ratio, shear angle, and rack angle but before knowing its relation first we need to know what is shear and rack angle.

## What is Shear Plane Angle?

The metal at the front of the tool is compressed, causing shear stress, which is maximized along a plane called the shear plane.

The angle made by the shear plane with the direction of movement of the tool is called the shear plane angle.

It is denoted by ɸ.

The chip flow velocity is affected by the shear plane angle.

## What is Rack Angle?

It is the angle between the rack face and normal to the machine direction.

It is denoted by α.

## Relation Between Chip Thickness Ratio, Shear Angle and Rack Angle

See in the figure chip produced during the machining process.

Where,

ɸ = Shear Angle
α = Rack Angle
t = uncut chip thickness
tc= cut chip thickness

Now if we make separate angle then,

See in triangle ABC,

As we know that,

Sinɸ = Perpendicular/Hypotenuse

So,

Sinɸ = t/AB

t  = AB×Sinɸ

Now, See in triangle ABD,

As we know that the sum of the total triangle angle is equal to 180°.

Hence,

x + 90 + (90 - ɸ)  + α = 180°

x = 180 - 90 - (90 - ɸ) - α

x  = ɸ - α

So as we know,

Cos(ɸ - α) = Base/Hypotenuse

Cos(ɸ - α) = tc/AB

tc = AB × Cos(ɸ - α)

So, as we know that,

r = t/tc

r = AB×Sinɸ/AB × Cos(ɸ - α)

r = Sinɸ/Cos(ɸ - α)

r = 1/E

Where,

E = Chip reduction ratio or coefficient

So,

As we know the chip thickness ratio,

r = Sinɸ/Cos(ɸ - α)

Then

r × Cos(ɸ - α)=  Sinɸ

Now open Cos(ɸ - α) by Trigonometry formula then,

r × ( cosɸ × cosα + sinɸ × sinα) = sinɸ

r × cosɸ × cosα = (1 - r × sinα)sinɸ

cosɸ/sinɸ = (1 - r × sinα)/r cosα

tanɸ = r × cosα/ (1 - r × sinα)

## Calculation of Chip Thickness Ratio and Shear Angle

### Question

In a machining operation of orthogonal cutting the cutting tool rack angle is 10° and chip thickness before cutting is 0.50 mm and after cutting is 1.125 mm. Calculate the chip thickness ratio and shear plane angle?

### Solution

Given Data,

α = 10°
t = 0.50mm
tc = 1.125mm

As we know that,

Chip thickness ratio,

r =  t/tc

r  = 0.50 / 1.125

= 0.44mm

So chip thickness ratio will be 0.44mm.

Now,

As we know,

tanɸ = r × cosα/ (1 - r × sinα)

tanɸ = 0.44 × cos10/(1-0.44 × sin10)

tanɸ = 0.4691

Hence,

Shear Ange,

ɸ = 25.13°

That is it, thanks for reading this article. If you have any questions about “chip thickness ratio” you can ask me in the comments. If you like “chip thickness ratio” then you can share it with your friends.

## FAQ Related to Chip Thickness Ratio

### Why is the chip thickness ratio is always less than 1?

The chip thickness after cutting is always greater than before cutting, so the chip ratio is always less than 1.

### Will chip reduction coefficient be greater than 1 always Why?

As we know that chip reduction coefficient is inversely proportional to the chip thickness ratio and the chip thickness ratio is always less than 1 that's why the chip reduction coefficient is always greater than 1.

### Why is the chip thickness ratio important?

The chip thickness ratio is an important parameter for machining analysis as it indicates the deformation of the chip across its thickness.