Dynamic surface shot peening is a continuous process of changing the state of the surface layer consisting of plastic deformation and other processes accompanying this phenomenon, the most important of which are phase transformations. Shot peening is characterised by the formation of compressive residual stresses. The quantities determining the state of compressive stresses are: the value of maximum stresses, the depth of their deposition and the distribution in the hardened surface layer. Research on the shot peening process is relatively more widely and more often used for the selection and optimisation of technological parameters and for forecasting the obtained states of stresses and deformation.
An X-ray diffractometer of the PSF-3M type from the Rigaku Company (Japan) was used to measure stresses on the surface of the samples, using a chromium lamp, using the sin2(
The test results (
The PSF-3M X-ray diffractometer was used to study the distribution of residual stresses in the surface layer with a thickness of 0.8 mm [2], [,3],[12].
A test sample with dimensions 40 mm × 40 mm is placed on the base of the instrument. The measurement zone on the sample is located in the central part of the sample. The diffractometer is equipped with two movable arms: one arm with a chromium X-ray lamp is used to set the angle of incidence
The X-ray method of measuring residual stresses is used for metals with a crystalline structure. Residual stresses are introduced into the surface layer of the object by heat treatment and shot peening, which causes displacements in the crystal lattice of the metal.
The stream of X-rays, after being reflected from the displaced crystal lattices, undergoes diffraction and is recorded in the transducer of the diffractometer [3], [12]. The image of the stream of X-ray waves after diffraction is shown in Fig. 1 [3]. For the subsequent angles of incidence
where
Hooke’s law relating the deformations
where
From the quoted relationships in Eqs (1) and (2), the dependence for the calculation of residual stresses in the X-ray method using the sin2
where m indicates the angular coefficient of a straight line approximating a set of points with the coordinates (sin2
An example of a test protocol [1] of the measurement of residual stresses using the X-ray diffractometer with an approximation line in coordinates (sin2
The printout (Fig. 2) shows:
the value of residual stresses
E = 210,000 MPa – Young’s modulus, and
as well as,
the approximation line in the coordinates: (sin2
In order to obtain the results of measuring residual stresses in the surface layer as a function of the distance
The tests of the shot peening process were carried out on samples that were heat treated (hardened and tempered at different temperatures to obtain different values of HRC hardness) and subjected to the pneumatic shot peening process [1]. The samples were divided into six groups as a function of hardness (HRC):
(1) HRC samples ε <25, 30>, (2) HRC samples ε <30, 35>, (3) HRC samples ε <35, 40>, (4) HRC samples ε <40, 45>, (5) HRC samples ε <45, 50> and (6) HRC samples ε (<50).
The samples were subjected to pneumatic shot peening with the following parameters:
four types of shot, marked with the numbers of the
1 – SW170 cast steel shot, granulation 0.4 mm, hardness 470 HV,
2 – SW330 cast steel shot, granulation 0.8 mm, hardness 470 HV,
2 – StD-03 steel shot, granulation 0.4 mm, hardness 640 HV, and
2 – Std-06 steel shot, granulation 0.6 mm, hardness 640 HV;
constant air pressure
shot peening time
100% shot peened surface coverage, and
the sample was rotating.
The shot peening process [8] was performed using the PEEN-IMP device, a patented station (patent PL204718), shown in Fig. 3.
The test results are summarised in Table 1. A total of 128 experiments were carried out. The independent variables are:
Measurements of stresses
LP | Distance from the surface d mm | Hardness HRC | Steel shot |
Stresses |
---|---|---|---|---|
1 | 0.000 | 35 | 1 | −506.37 |
2 | 0.079 | 35 | 1 | −465.59 |
3 | 0.127 | 35 | 1 | −390.06 |
4 | 0.194 | 35 | 1 | −397.08 |
5 | 0.241 | 35 | 1 | −244.17 |
6 | 0.299 | 35 | 1 | −209.32 |
7 | 0.399 | 35 | 1 | −232.61 |
8 | 0.456 | 35 | 1 | −172.24 |
… | ||||
25 | 0.000 | 30 | 2 | −448.40 |
26 | 0.046 | 30 | 2 | −540.25 |
27 | 0.106 | 30 | 2 | −571.50 |
28 | 0.150 | 30 | 2 | −547.69 |
29 | 0.210 | 30 | 2 | −527.38 |
30 | 0.258 | 30 | 2 | −514.13 |
31 | 0.315 | 30 | 2 | −427.26 |
32 | 0.340 | 30 | 2 | −354.94 |
33 | 0.396 | 30 | 2 | −305.50 |
… | ||||
70 | 0.000 | 45 | 3 | −682.34 |
71 | 0.051 | 45 | 3 | −602.13 |
72 | 0.111 | 45 | 3 | −615.84 |
73 | 0.191 | 45 | 3 | −552.10 |
74 | 0.206 | 45 | 3 | −395.75 |
75 | 0.227 | 45 | 3 | −259.79 |
76 | 0.340 | 45 | 3 | −249.39 |
77 | 0.393 | 45 | 3 | −293.09 |
78 | 0.419 | 45 | 3 | −195.22 |
… | ||||
120 | 0.000 | 50 | 4 | −643.16 |
121 | 0.116 | 50 | 4 | −737.91 |
122 | 0.166 | 50 | 4 | −745.57 |
123 | 0.220 | 50 | 4 | −509.45 |
124 | 0.284 | 50 | 4 | −394.01 |
125 | 0.347 | 50 | 4 | −432.17 |
126 | 0.401 | 50 | 4 | −321.01 |
127 | 0.462 | 50 | 4 | −254.92 |
128 | 0.565 | 50 | 4 | −261.30 |
where 0.000 indicates the free surface of the sample.
deposition depth of residual stresses
material hardness
the type of shot used marked x3 ε <1, 2, 3, 4>.
The dependent variable is the residual stress
The selected test measurements of stresses for the shot N = 1, 2, 3, 4 are summarised in Table 1.
The meddle in the stresses measured on the surface of the samples after heat treatment were:
hardness in the range of 25–30 HRC, stresses – б = –22 MPa;
hardness in the range of 35–40 HRC, stresses – б = –28 MPa; and
hardness in the range of 45–50 HRC, stresses – б = –145 MPa.
The test results (
The REGSTEP.EXE stepwise multiple regression program makes it possible to create many regression functions:
linear,
linear exponentials with the
second-degree non-linear, where the number of independent variables is one to four, and
the listed functions may have variables written in natural or logarithm form.
The form of the regression function is related to the number of experiments entered into the REGSTEP.EXE program. For three independent variables with interactions, the minimum number of experiments entered into the program is 10 [9]. The second-degree regression function created by the program for three independent variables with interactions is as follows:
where
The
After entering the data (Table 1) into the REGSTEP.EXE program, the coefficients of the regression function (Table 2) and their statistical evaluation were obtained using the Student’s
Coefficients of the regression function calculated by the program for the results of measurements.
Variable number | Regression coefficient | Std. Error of reg. coeff. | Computed T-value | |
---|---|---|---|---|
( |
1 | −0.845537E+02 | 303.47275 | −0.279 |
( |
7 | −0.865059E+01 | 1.82461 | −4.741 |
( |
6 | 0.259431E+02 | 52.86755 | 0.491 |
( |
2 | −0.109254E+03 | 26.07856 | −4.189 |
( |
10 | 0.798438E+02 | 14.47331 | 5.517 |
( |
9 | 0.160997E+01 | 0.35632 | 4.518 |
( |
3 | −0.114981E+03 | 60.96822 | −1.886 |
( |
5 | 0.138582e+02 | 7.15698 | 1.936 |
( |
8 | 0.493962E+03 | 288.88644 | 1.710 |
( |
0.168799E+04 |
For example, in Table 3, the results of the last (ninth) step of the statistical evaluation of the obtained regression function are presented.
Statistical evaluation of the obtained regression function using the statistical parameters
STEP 9 |
VARIABLE ENTERED..... 8 |
(FORCED VARIABLE) |
SUM OF SQUARES REDUCED IN THIS STEP.... 18743.844 |
PROPORTION REDUCED IN THIS STEP........ 0.006 |
CUMULATIVE SUM OF SQUARES REDUCED...... 2162765.250 |
CUMULATIVE PROPORTION REDUCED.......... 0.741 OF 2919265.000 |
FOR 9 VARIABLES ENTERED |
MULTIPLE CORRELATION COEFFICIENT... 0.861 |
(ADJUSTED FOR D.F.)........... 0.851 |
SQUARE MULTIPLE CORRELATION COEFFICIENT ⊠ 0.741 |
F-VALUE FOR ANALYSIS OF VARIANCE... 37.483 |
STANDARD ERROR OF ESTIMATE......... 80.069 |
(ADJUSTED FOR D.F.)........... 82.716 |
The REGSTEP.EXE program is based on IBM subprograms [7]. The more important components’in Table 3 have the following equivalents:
SQUARE MULTIPLE CORRELATION COEFFICIENT = 0.741 (R2 = 0.741 – multiple correlation coefficient)
F-VALUE FOR ANALYSIS OF VARIANCE... 37.483 (F Snedecor’s test = 37.483).
T-VALUE – Student’s t-test for regression function coefficients marked: 1, 2, 3, 5, 6, 7, 8, 9, 10
REGRESSION COEFFICIENT – coefficients of the regression function (1, 2, 3, 5, 6, 7, 8, 9, 10)
The regression function has the computational form:
The functional form in Eq. (5) allows the calculation of the values of residual stresses in the factor space determined for the values of the variables:
The polynomial regression function in Eq. (5) makes it possible to calculate the value of residual stresses into the material
Table 4 presents the results of the calculations employed for ascertaining the tabular distribution of residual stresses as a function of every 0.10-mm change in the
Values of residual stresses
Type of function - NATURAL -N- | ||
Values of Bi coefficients in the calculated function | ||
B0= 0.1688E+04, B1= −0.8455E+02, B2= −0.1093E+03, | ||
B3= −0.1150E+03 | ||
B12= 0.1386E+02, B13= 0.2594E+02, B23= −0.8651E+01, | ||
B11= 0.4940E+03, B22= 0.1610E+01, B33= 0.7984E+02 | ||
NUMBER OF POINTS N = 8 | ||
Values of X1, X2, X3 variables at set points | ||
X1 | X2 | X3 |
0.000 | 35.0000 | 1.0000 |
0.1000 | 35.0000 | 1.0000 |
0.2000 | 35.0000 | 1.0000 |
0.3000 | 35.0000 | 1.0000 |
0.4000 | 35.0000 | 1.0000 |
0.5000 | 35.0000 | 1.0000 |
0.6000 | 35.0000 | 1.0000 |
0.7000 | 35.0000 | 1.0000 |
0.8000 | 35.0000 | 1.0000 |
Xi depth and Yi function values at set points | ||
No. | Xi | Yi |
1 | 0.000 | −501.1677 |
2 | 0.1000 | −454.0123 |
3 | 0.2000 | −396.5508 |
4 | 0.3000 | −329.2101 |
5 | 0.4000 | −251.9901 |
6 | 0.5000 | −164.8909 |
7 | 0.6000 | −67.9124 |
8 | 0.7000 | 38.9453 |
9 | 0.8000 | 155.6822 |
The distribution of residual stresses shown in the tabular form (Table 4) is additionally presented in the analytical notation. The polynomial regression program REGPOLY.EXE developed on IBM subprograms was used to record the analytical tabular distribution of residual stresses (Table 4) [7]. The program for the tabular stress distribution generates successive polynomial regression functions of one variable, until the moment when the next added monomial does not improve the accuracy of the calculation of the value of the entire polynomial.
As a result of the calculations of the REGPOLY.EXE program, the second-degree polynomial describing the distribution of residual stresses in the surface layer was obtained. The second-degree function takes the following form:
The constants in the polynomial in Eq. (7) have the following values:
The notation of the residual stress distribution function is as follows:
The EXCEL Microsoft Office 2000 program was used to obtain a graphic form of the tabular distribution of residual stresses (Table 4). The stress distribution diagram is shown in Fig 4.
Curve 1 (Fig. 4) presents the values of the residual stresses
A total of 128 experiments involving the measurement of residual stresses performed on the C45 steel samples after heat treatment and shot peening [8] were used to determine the form of the regression function of three independent variables (depth extending into the material, hardness of the C45 steel (HRC) and shot in the shot peening process). The REGSTEP.EXE stepwise multiple regression program was used. The test samples were shot peened on both sides in a device for pneumatic shot peening [1]. An X-ray diffractometer was used to measure the stresses. The successive layers of material 0.04–0.06 mm thick were removed from the samples by electrochemical etching. The FUNVAL3.EXE function value program was used to obtain a tabular distribution of residual stresses into the material. The REGPOLY.EXE polynomial regression program was used to record the analytical distribution of residual stresses into the material. The results of the obtained calculations satisfactorily reproduce distributions of stresses in the surface layer, which is confirmed by the following statistical parameters: R2 (squared multiple correlation coefficient), Fisher’s Snedecor test and Student’s t-tests for regression function coefficients. The obtained regression function allows for the initial determination of shot peening parameters, and is recommended for obtaining the value of residual stresses in the surface layer.
The number of laborious experiments needed for determining the regression function of three independent variables can be reduced by applying the rules of the planned experiment [9].