Stresses in a grinding wheel

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journal homepage: http://farplss.org

Volume 26, Number 2, 2018

225 Stresses in a grinding wheel

I. Belmas

ORCID 0000-0003-2112-0303

Dniprovsk State Technical University, Kamianske, Ukraine

G. Tantsura

ORCID 0000-0002-8672-1153

Dniprovsk State Technical University, Kamianske, Ukraine

S.Zaldya

Dniprovsk State Technical University, Kamianske, Ukraine

A.Gaponenko

Dniprovsk State Technical University, Kamianske, Ukraine

Article info

Received 21.03.2018

Accepted 30.04.2018

Dniprovsk State Technical University, Kamianske, Ukraine

2, Dniprostroevskaya Str, 51918, Dnipropetrovsk region, Kamenskoye, Ukraine

belmas09@meta.ua gannaivan71@gmail.com zaldyasveta@gmail.com

+38 (098) 541 28 12

Belmas, I., Tantsura, G.,Zaldya. S., Gaponenko, A. (2018). Stresses in a grinding wheel.

Fundamental and applied researches in practice of leading scientific schools, 26 (2), 225– 230.

The distribution of mechanical stresses in a part of a grinding wheel of limited size that can be compared with the size of grinding grains is investigated. The purpose of the paper is to establish a qualitative and quantitative dependency of stress distribution in the material of a grinding tool on a radial cutting force and mechanical properties of the material. The research was carried out by methods of linear theory of elasticity.

The material of the circle is considered as isotopic with averaged value of a shear modulus. The mean value of a shear modulus is calculated by the method offered by Feucht. The radial force is added at a point at a depth equal to the size of the grinding grain. The stressed state is deter-mined by methods of theory of elasticity, within the boundaries of Mindolin problem with the use of a stress function. A solution for a case of load of several grains located on one line is formulat-ed. It is established that the normal stress in the material of a grinding wheel near the grain de-pends on the load of the adjacent grain. Its extreme value exceeds the corresponding stress on a single grain by 20-25%. A gradient stress increases along a grain height in a case of increasing the distance between the grains. As Poisson coefficient increases, the influence of the number of load-ed grains on the value of normal stress decreases. This influence increases on the value of the tan-gential stress.

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226

research. Determining the latter will create the opportunity to optimize the technology of grinding and wear of grinding wheels.

Keywords: grinding wheel; cutting force; tangential stress; normal stress; mechanical parameters.

Introduction

Grinding takes up a significant place in metal processing. A grinding tool, usually made of grinding grains connected in a single structure by a jointing material. In the process of grinding, the cutting grains periodically come into contact with a processed part. The contact is accompanied by their periodic force interaction – the effect of the cutting force. The periodic load of grains by the cutting force causes a cyclically variable stress state of the material that holds abrasive grains. The interaction of grains with the material of the part during its processing leads to a wear of grains. The wear of cutting edges is particularly negative. The variable stress state of the material causes the surface layer of the grinding wheel to be damaged. The latter leads to the appearance of new cutting edges instead of worn ones. Slight adhesion between the contact surfaces of grains and jointing material significantly affects the indicated process.

The establishment of optimum grain loading conditions at which the time of wear of cutting edges of abrasive grains is equal to the appearance time of new edges on it due to the damage of the material that holds them is an actual scientific and technical problem. It includes the task of determining the stress state of the material that holds the abrasive grains, as the main factor which leads to the damage of the material during its cyclic loading.

A considerable amount of papers is devoted to the establishment of a stress state of grinding tools. Here are some of them. In the dissertation of Muzychka D.G. [1] the features of a shape change of a cutting surface of the grinding wheel, considering the temperature-force factors during the interaction of the tool with the part were investigated. Some aspects of the power interaction of abrasive grains and the material that holds them were investigated in the article by Ushakov A.N. [2]; in paper [3] the author applies external forces to the abrasive grain of a given form. The interaction surface of a grain with the jointing material is considered constant. In the article [4] the power parameters of a process of centerless grinding of roller bearings with intermittent grinding wheels were investigated. In paper [5], the grinding wheel is considered as an infinitely long cylinder loaded with evenly distributed normal and tangential force applied to a cylindrical surface

at an insignificant angle. In paper [6], the stress state of a grinding material in a case of grain load by concentrated force was investigated. In article [7] the stressed state of a porous material of a grinding wheel was investigated using the finite element method.

The above analysis indicated that in the study of a stress-strain state of the grinding wheel material the mechanism of redistribution of stresses in it was not investigated. This does not allow the use of known predicting methods of quantity of load cycles of abrasive grain jointing material in a grinding tool until it is damaged. In order to apply the predicting methods, it is necessary to consider the patterns of stress change during the load process and the endurance of the grinding tool material.

The purpose of the article is establishing the qualitative and quantitative dependency of stress distribution in the material of a grinding tool on the radial cutting force and the mechanical properties of the material.

Materials and Methods

The research was carried out by methods of linear theory of elasticity. The dimensions of the grinding wheel considerably exceed the dimensions of interaction surface with the part during its processing. The dimensions of the part are much larger than the size of grinding grains, as the main components that transfer the load to the jointing material.

Consider the distribution of stresses at the micro level in a part of the grinding wheel limited in size, which can be compared with the size of grinding grains. Note that the abrasive grains are bonded by another material. Together they form a composite material of chaotic structure. Abrasive grains are anisotropic. They are freely oriented in a grinding tool. The jointing material is isotropic. During the grinding process, a group of grains interact with the detail at a time. This allows to consider the material of the wheel as an isotropic material with averaged elasticity and to determine the average values of its stress-strain state during the part processing. The average value of the shear modulus by Feucht is:

1 1 2 2

G





G





G



...

,

where G1, G2, ...,

 

₁

,

₂

,...

– shear moduli and relative volume content of components in a composite material.

For a porous material, such as the one investigated in [7], the shear modulus of the volume part of pores (cavities in the material) should be considered to be zero. Consider a significant difference in a size of the grinding tool and the abrasive grains in it. Assume that the radius of the grinding wheel is boundless. One grain interacts with the processed part. The cutting force is applied to the grain normally to the surface of the grinding wheel. Cutting force is equal to one.

Bring the following computational model in correspondence with the physical model of a tool and a part interaction. The grinding tool takes the half-space

0   

z

. A part of a cubic shape with unit dimensions c3

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227

a cross-section

z



c

.Assume the values of force equal to one. Direct it to the surface normally

z



0

. In this formulation the problem is axially symmetric. In the theory

of elasticity, it is known as Mindolin's problem. It is solved using the following biharmonic stress function:





1





2



 





2



2

8

1

1 4 1 2

1 ln

8

1 P

cz

R

z

R

z

c

R















_



_







_









_



 



_

_



_

_

,

where μ – Poisson coefficient;

R

₁





z



c



2



r

2

;

R

2





z



c



2



r

2 ; r –radius-vector, from the point of force application to the current point.

The stresses are described by the following dependencies:

2

,

Rr

z



r













_



_









2

1 ,

z

r r









_









_















2

2 ,

Zz

z

 

z













_



_













2

1 ,

Rz

Zr

r

 

z













_



_









0,

R

 



r





z



Z





where

 

2 2

2

2 2 2 2

...

1 ...

...

.

r

z













With the use of the above, the stress state of the grinding wheel material was determined. Analysis of the results allows drawing the following conclusions. The stresses on the grain surface are distributed nonlinearly and descending along the grain height. Extreme values are proportional to Poisson coefficient. The maximum values of normal stress do not exceed 10% of the value of normal grain pressure on the jointing material. The maximum values of tangential stresses are 15%. Extreme values of tangential stresses are realized at a distance of about 60% from the surface of the grain deepening height into the grinding wheel material [6].

In the experiment [3] 60 grains were used in the grinding process with a placement density of 2.68 grains/mm2_{. Together they affect the stress state of the}

jointing material. Take into account that the extreme stresses in the material of grinding wheel occur on a surface of its interaction with a grain. Consider the load of several grinding grains. Assume that the grains are loaded with equal forces; points of application of forces are located in one plane. The distance between the grains is С.

Linear statement of the problem allows determining the stress state of the material of a grinding wheel as a sum of states. The first state is a stressed state, which is realized on the surface of the nearest grain. The second is a stressed state on the same surface, but caused by the load of a more distant grain. The third and the following – caused by the loads of even more distant grains. Thus, the stresses are determined by the following dependencies:



 







_

_





 





 1

2 2

1 ₁

2

1

8

1

c c С n

N

n c _{С n}

d

P

Rr

dr

dz

  

 





  

 _ _







_



 

_











,



 







_

_





 





 1

2 2

1 ₁

2

1

8

1

c c С n

N

n c _{С n}

d

P

B

dr

dz



  

 





  

 _ _





 



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Volume 26, Number 2, 2018

228 

























1

1 ,

2

N

n

c

Rz

  

 

С n



С n



















where N – the number of grinding grains, the influence of which is taken into account;







 























2 ₂

2 4

2 2

2

2 2

2 2 ₂

3

2 2

8

1

1 ₆

4

1 1 3

4 1 2

;

r

zc

_zcr

R

z

c

r

z

c

R

z

c

_R

_z

_c

_z

_c

R











_





_



















_

_

_

_

_

_









_

_





 

















2



 



2 2

2 1

4 1 2

1

2

8

1

1 ;

z

c

zc

R

z

c

R











_





_



  







 

















































 



2 2

3

1 1 2

2 2

2 3

2 2

5 2

2 2

8

1 1 4 1

1

4

8

1

2

4

1 1 2

1

6

4 .

z

c

R

c z

c

z

c

zc

z

c

R

z

R

z

c

z z

c

R









 













_





_





 















_

_



Using the obtained expressions, the stress state of the material of a grinding wheel is caused by the load on two adjacent grains. The distance between the grains C was chosen as part of the grain size. The results are shown in Figures 1-3. They show additional stresses (without

considering the stresses determined in [6]) in the jointing material on the boundary surface δ = 0 and at a distance from it at two values of the distance between the crystals C

for the case N = 2 and with a load on each grain of force

Р = 1.

a) μ = 0

b) μ = 0.5

Figure 1 – Distribution of relative normal stresses Rr0 in planes normal to a grain axis along its height z and in a radial

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229

The obtained graphs show that the normal radial stresses in the grinding wheel material near the grain depend on the load of the adjacent grain. Their extreme values exceed the stresses caused by the load of one grain by

20-25%. The stress gradient is greater along the grain height at a greater distance between the grains. As Poisson coefficient increases, the influence of increasing the number of loaded grains is smaller.

a) μ = 0 b) μ = 0.5

Figure 2 – Distribution of relative tangential stresses Br0 in planes normal to a grain axis along its height z and in a radial

direction δ for a material with different values of Poisson coefficient at different values of the distance between the abrasive grains С

According to the figure, the character of a stress dependency Br on the number of grains to which the force is applied is similar to the dependency of normal stresses. The

difference is the quantitative increase in maximum stresses. They increase up to 50%.

a) μ = 0 b) μ = 0.5

Figure 3 – Distribution of relative tangential stresses Rz0 in a plane parallel to the grain axis along its height z and in a

radial direction δ for a material with different values of Poisson coefficient at different values of the distance between the abrasive grains С

The tangential stresses in the material, like normal radial stresses near the grain, depend on a load of the adjacent grain. Their extreme values are close to the extreme stress values for the case of load on a single grain. The gradient of stresses along the grain height is greater at a greater distance between the grains. The character of influence of Poisson coefficient on the stress state, considering the presence of two grains, is opposite to the case of load on a

single individual grain. Extreme values of stress increase reach 100%.

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230

planes tangent to the surface of a processed part are increased by 10-12%, tangential stresses increase by a factor of two. The tangential stresses in planes normal to the surface of interaction with the grinding wheel and the processed part increase the most. This increase reaches 50%. The frequency of interaction between the grains and the part leads to a periodic change in the stress in the grinding wheel material from zero to maximum.

Conclusions. In terms of linear theory of elasticity and Mindolin problem, a solution was obtained for determining the stress state of the grinding wheel material, as an isotropic material, for the case of load by radial cutting force of a single grinding grain and several grains located on the same line. It is established that the normal radial stresses in the material of the grinding wheel near the grain depend on the loading of adjacent grains. Their extreme values exceed the stresses caused by the load on a single grain by 20-25%. Stress gradient of grains increases along the grain height in a

case of an increase in a distance between grains. As Poisson coefficient increases, the influence of increasing the number of loaded grains decreases.

An increase in the number of grains that simultaneously interact with a local processing area of a part leads to an increase in the maximum stresses in the material holding the abrasive grains. The tangential stresses in planes normal to the contact surface of the part and the abrasive wheel increase the most.

The character of stress change from the action of radial cutting force on a grinding wheel corresponds to the cycle, in which stress changes from zero to an extremum.

The direction of further investigation should consider determination of indices of durability of grinding wheel materials during the cycle, in which load changes from zero to an extremum. Using their basis, establish of optimum conditions of damage of a material that holds abrasive grains of a grinding wheel and the wear of their cutting edges.

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