# Mathematical model of the relationship between the

2022-08-14
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Mathematical model of the relationship between grinding process and grinding surface residual stress

1 formation mechanism of grinding surface residual stress

influence of Plastic Bulge effect

during grinding, due to the large negative rake angle of the abrasive cutting edge, the plastic deformation in the deformation area is very serious, and a complex stress state will be formed in the area in front of the abrasive tip. On the surface just passed by the abrasive cutting edge, plastic shrinkage occurs along the surface direction, while tensile plastic deformation occurs in the vertical direction of the surface - this is the plastic bulge effect, resulting in residual tensile stress on the ground surface

influence of squeezing effect

in the process of cutting, there will be a force between the tool and the workpiece. The force perpendicular to the machined surface and the resulting friction force together produce a light squeezing effect on the machined surface. When the cutting edge is not sharp or the cutting conditions are bad, the influence of crowding effect is more obvious, and crowding effect will cause residual compressive stress on the surface of parts

influence of thermal stress

during grinding, the grinding surface layer produces thermal expansion under the action of grinding heat, and at this time, the substrate temperature is low, and the thermal expansion of the grinding surface layer is limited by the substrate, resulting in compressive stress. When the temperature of the surface layer exceeds the temperature allowed by the elastic deformation of the material, and the temperature of the surface layer decreases to be consistent with the temperature of the substrate, the surface layer produces residual tensile stress

cooling effect of grinding fluid

in the grinding process, due to the use of grinding fluid, the grinding surface layer will produce a cooling gradient in the cooling process, which is just opposite to the influence of thermal stress, and it can reduce the surface residual tensile stress caused by thermal stress

in the grinding process, in addition to the above factors affecting residual stress, there are secondary quenching of the surface layer and tempering of the surface layer

2 Establishment of mathematical model of residual stress on grinding surface

through the above analysis, it can be seen that the new types of plastic granulators that affect the grinding surface are constantly optimized and updated, and the main factors affecting the residual stress in the construction of national new raw material base can be summarized as follows: grinding force, grinding temperature and cooling of grinding fluid. Force and temperature are two kinds of grinding phenomena in the grinding process, which directly affect the residual stress; The influence of grinding fluid on residual stress is directly produced by the cooling process of the surface on the one hand, and indirectly produced by the influence on force and temperature on the other hand. This paper attempts to give a mathematical model reflecting the relationship between force, temperature, cooling performance of grinding fluid and surface residual stress through the mathematical processing of the test data of force and temperature and the test data of two-dimensional residual stress on the grinding surface. The mathematical model should include the above factors that affect the residual stress on the grinding surface, That is,

SRT = SF + Sr + SL

where: SRT - residual stress on grinding surface

SF - influence of grinding force

SR - influence of grinding temperature

SL - influence of cooling performance of grinding fluid

1) mathematical model of the relationship between grinding force and residual stress

first, analyze the relationship between residual stress and plastic deformation according to the model shown in Figure 1. Figure 1A shows the two springs in the free state, and figure 1b shows the state in which the two springs are placed between the rigid plates. According to the equilibrium conditions, it can be concluded that

n=k1k2 (L1-L2)/(k1+k2)

where: n - the internal force of the spring after the two springs are placed in the rigid plate

l1, L2 - the length of the two springs in the free state

k1, K2 - the elastic coefficient of the two springs

l1-l2 can be regarded as plastic deformation in the sense of this article. From the above formula, it can be concluded that the internal force is proportional to the plastic deformation, that is, the residual stress is proportional to the plastic deformation

Figure 1 residual stress and plastic deformation relationship model

Figure 2 is a simplified model of the relationship between stress s and strain E. It can be seen from the figure that

eb= (sb SS)/e1+ese 'a=eb/e

where: SS - yield limit of material

sb - stress under a certain grinding condition

e - elastic modulus of material

e1 - constant

Figure 2 stress σ According to figure 2, when the external force is released, the residual strain EB at point B is EP

ep=eb-e 'a= (sb SS)/e+es+sb/e

after the release of the slope OA. The above formula shows that the plastic deformation has a linear relationship with the force

based on the above analysis, it can be recognized that there is a linear relationship between residual stress and grinding force, and the relationship can be expressed as

sf=af+d1 (1)

where: A, D1 - coefficient

f - grinding force (tangential force can be used)

2) mathematical model of the relationship between grinding temperature and residual stress

Figure 3 intense market competition

residual stress generated by thermal stress can be analyzed in Figure 3. When the temperature in the grinding zone rises, the surface layer is heated and expanded to produce compressive stress σ， The stress increases linearly with the increase of temperature, and its value is approximately

s=a ′ edq

where: a '- linear expansion coefficient

e - elastic modulus of material

dq - temperature rise

when the grinding temperature continues to rise to QA, the thermal stress reaches the yield limit of the material. If the temperature rises again (QA → QB), the surface layer will produce plastic deformation, and the thermal stress value will stay at the yield limit of the material at different temperatures. After grinding the doors and windows, the temperature of the surface layer decreases, and the thermal stress decreases according to the original slope sb (along the BC curve) until it is consistent with the substrate temperature. At this time, the surface produces residual tensile stress. Its value is

sq=sd-sbsd=a ′ eqb

if sb is considered to be in a linear relationship with temperature, the mathematical model of the relationship between grinding temperature and residual stress can be obtained as

sq=a ′ eqb-b0qb+d2=bq+d2

B0, B, D2 in the formula

θ—— The highest temperature in the grinding area

3) mathematical model of the relationship between the cooling performance of grinding fluid and the residual stress on the grinding surface

the higher the temperature in the grinding area and the greater the cooling coefficient of grinding fluid, the greater the temperature difference of the lower surface layer, and the more the residual stress caused by thermal stress is reduced. The influence of the cooling performance of grinding fluid on the surface residual stress is related to the surface temperature. Therefore, the mathematical model of the relationship between grinding fluid and grinding surface residual stress is expressed as

sl=cqa+d3

where C, D3 - coefficient

α—— The mathematical model of the relationship between the grinding process and the residual stress on the grinding surface can be obtained from the comprehensive formulas (1), (2) and (3) of the cooling coefficient of grinding fluid

SRT = AF + BQ + CQA + d

in which a, B, C, D - reflect the grinding force, grinding temperature Coefficient of grinding fluid cooling performance affecting grinding surface residual stress

3 regression calculation and analysis of mathematical model

1) regression calculation of mathematical model

according to the mathematical model of residual stress expressed in formula (4), the following normal equations can be obtained by the least square method

in this paper, n=6, table 1 is the calculation table of a group of measured data (residual stress unit: MPa, the same below). Solve the normal equations and get

a=45.77 b=1.98 c=-6 × D=-1.47

from this, the fitting results are as follows

srt=45.77f+1.98q-6 × Q a-1.47 Table 1 regression calculation table

test piece fiqisiq2if2iqiqiqiaiq2ia2isifisiqisiqiaiqiai11 39515.. eighty-nine point four four × one thousand and seventy-one point one four ×.. twenty-one ×... 82326.456532.59.92 × one thousand and seventy-one point seven one × 109693.. seventeen ×.. 89360.. twelve point one × one thousand and seventy-nine point eight three × 109749.. seventy-one ×. 441.... twenty-four × one thousand and seventy-two point six two ×.. eighty-four ×... 64421.. forty-seven × one thousand and seventy-four point four three ×.. ninety-two ×... 77431.. forty-seven × one thousand and seventy-four point four three ×.. ninety-two × s9... 532338.. four hundred and twenty-six point seven one × ten thousand seven hundred and twenty-four point zero three ×.. sixty-six × one thousand and sixty-one point zero six × 106

2) analysis of calculation results of mathematical model

Table 2 is the error analysis table of calculation results. Where s*i is the calculated value, Si is the measured value, di=|s*i-si|. The maximum fitting error is 5.5%. Table 2 error analysis table

specimen s * isididi/s*i (%) 1434.53421.413.133.02510.87509.61.270.23436.394414.611.14362.21382.219.995.55460.89460.60.290.16462.66460.62.060.5

4 conclusion

this paper proposes a new method to study the residual stress of grinding surface, that is, using the force, temperature and grinding fluid that reflect the grinding process.The three comprehensive indexes of cooling coefficient are used to study the residual stress of grinding surface, The mathematical model is given. The mathematical model reflects the formation mechanism of residual surface stress on grinding surface, and this method is widely comparable with other single factor research methods. (end)

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