Description of Residual Stress Distribution in the Surface Layer After Heat Treatment and Shot Peening

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(ψ) method. The 40 × 40 mm amples, having a hardness of 30–50 HRC, were made of C45 steel. Four types of shot were used to harden the surface layer of the samples: cast steel shot SW170 with a diameter of 0.4 mm and SW330 with a diameter of 0.8 m and a hardness of 470 HV; and steel shot StD-03 with a diameter of 0.4 mm and StD-06 with a diameter of 0.6 mm and a hardness of 640 HV. The shot peening time of the sample was 120sec. The air pressure during peening was 0.45MPa. The depth of the hardened layer was in the order of 0.8 mm. The subsequent layers of the samples with the thickness of 0.04–0.06 mm were removed by an electrochemical method. The number of removed layers was in the range of 10–12. For the analytical description of the measurements of the residual stress distribution tests after heat treatment and shot peening, the multiple step regression program REGSTEP.EXE¹ was used (using the results the work of Mońka and Janowski [1]). The independent variables in the stress tests were: the depth of the hardened layer deposition d, the hardness of the HRC sample and the type of shot marked with successive natural numbers N (1, 2, 3, 4). Residual stresses σ MPa measured by X-ray diffraction were the dependent variable.

The test results (d, HRC, N, σ) were entered into the stepwise multiple regression program REGSTEP.EXE. In total, 128 experiments were carried out as part of the present study. The studies in the factor space (d ε <0.0–0.8> mm, HRC ε <25–50>, N ε <1, 2, 3, 4>) were not performed according to the rules of the planned experiment [9], but were random. The residual stress distribution function σ, derived from the REGSTEP.EXE program, is a function of three second-degree independent variables with interactions. The obtained function σ is used to obtain the tabular distribution of stresses into the surface layer using the FUNVAL3.EXE² program. The REGPOLY.EXE³ polynomial regression program was used for analytical recording of the tabular distribution of residual stresses. The EXCEL Microsoft Office 2000 program was used to graphically present the distribution of residual stresses in the form of a polynomial.

The REGSTEP.EXE stepwise multiple regression program makes it possible to create many regression functions:

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:

4 y=c0+c1⋅x1+c2⋅x2+c3⋅x3+c5⋅x1⋅x2+c6⋅x1⋅x3+c7⋅x2⋅x3++c8⋅x12+c9⋅x22+c10⋅x32

where c₀, c₁, c₂, c₃, c₄, c₅, c₆, c₇, c₈, c₉ and c₁₀ are coefficients calculated by the program.

The c₄ coefficient determines the y independent variable, has the value c₄ = 1 and is not printed by the program.

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 t-test (T-VALUE). Regression coefficients are written in exponential format (E).

For example, in Table 3, the results of the last (ninth) step of the statistical evaluation of the obtained regression function are presented.

The REGSTEP.EXE program is based on IBM subprograms [7]. The more important components’in Table 3 have the following equivalents:

The regression function has the computational form:

5 y=0.168799E+04+−0.845537E+02⋅x1+−0.109254E+03⋅x2+−0.114981E+03⋅x3+0.138582e+02⋅x1⋅x2+0.259431E+02⋅x1⋅x3+−0.865059E+01⋅x2⋅x3++0.493962E+03⋅x12+0.160997E+01⋅x22+0.798438E+02⋅x32}

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:

6 x1(depthd∈<0.0−0.8>mmx2(hardnessHRC)∈<25−50>x3(shotN)∈<1,2,3,4> }

The polynomial regression function in Eq. (5) makes it possible to calculate the value of residual stresses into the material σ for the assumed parameters of the factor space in Eq. (6) [5], [10], [11]. For practical purposes, the universal FUNVAL3.EXE program was developed, which, for the assumed values of the coefficients of the regression function and the values of three independent variables (x₁, x₂, x₃), calculates the value of the function in Eq. (5).

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 X₁ depth extending into the surface layer.

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:

7 y=a0+a1⋅x+a2⋅x2

The constants in the polynomial in Eq. (7) have the following values:

a0=−0.50159×103,a1=0.42643×103and a2=0.49396×103.

The notation of the residual stress distribution function is as follows:

8 σ=−0.50159⋅103+0.42643⋅103⋅x+0.49396⋅103⋅x2[MPa]

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.

Figure 4:

Curve 1 (Fig. 4) presents the values of the residual stresses σ MPa measured by the sin² Θ method using a diffractometer. Curve 2 represents the average value obtained by the REGSTEP.EXE stepwise multiple regression program. The presented curve 2 with the fixed parameters of the shot peening process allows an estimation of the distribution of residual stresses in the hardened surface layer.

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].