Keywords

1 Introduction

Development of new constructions is associated among others with increasing demand for new engineering materials. These materials, in general, belong to the group of difficult-to-cut materials and are most commonly used in following types of industries: aerospace, maritime, petrochemical, and biomedical engineering. The increasing demand of these materials is justified through their unique properties, which include high strength at relatively low density, creep resistance, and resistance to fatigue failures.

Ezugwu in his work [1] suggested the division of difficult-to-cut materials into four main groups: titanium-based alloys, nickel-based alloys, ceramics, and ferrous alloys. Additional type of material (composite materials) has been added to this division which is presented on Fig. 1.

Fig. 1
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Division of difficult-to-cut materials

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2 The Characteristics of the Surface Layer Formed in the Cutting Process

Machining is an oriented and controlled process of cracking which is accompanied by progressive separation of the material in form of the chip with the use of wedge blade [2]. The cutting tool is a carrier of focused, external mechanical energy. The energy inside the material increases with the increase of plastic deformation. Crystal lattice is subjected to the increasing value of plastic deformation which alters properties of the workpiece.

During machining process, new stress state is generated which results in energy changes in the cutting zone. Stresses which are present in the material during external load and stresses present in the material in the absence of the influence of external forces can be distinguished [3]. Another division of stresses is based on the area of size in which they balance. According to Davidenkov’s classification, stresses can be divided into:

  • Macro-residual stresses (1st type of RS) appear in the area where their size is compared with the size of the workpiece. The existence of these stresses can be determined based on the displacement of diffraction lines in proper direction.

  • Micro-residual stresses (2nd type of RS) appear in the area where their size is compared with the size of the grain. These residual stresses can be further divided into oriented residual stresses which arise as a result of oriented plastic deformation, and they cause an angular shift of diffraction lines. Unoriented residual stresses cause diffraction line broadening.

  • Sub-residual stresses (3rd type of RS) appear in the area of the few atomic distances. The force and moment equilibrium exist in sufficiently large parts of a crystallite. These stresses are caused by numbers of deformations in crystal lattice (e.g., delamination and dislocations).

Stresses can be also distinguished based on the cause of their formation. Thereby, following types of stresses can be given: mechanical, thermal, and structural. Structural stresses arise from volumetric changes [4, 5]. They are generally caused by phase transitions and are determined by temperature and time. In case of increased specific volume, compressive stresses are present, otherwise tensile stresses occur. The uneven heating or cooling process of particular layers of the material leads to thermal stresses. After reaching a temperature below core temperature, compressive residual stresses are formed, while tensile stresses occur inside of the core. During machining process, mechanical stresses arise as a result of uneven deformation of the surface layer of the workpiece.

3 Impact of Cutting Conditions on Stress State in the Surface Layer of Inconel 718

Research was carried out on samples made out of nickel-based alloy—Inconel 718 through the wide range of its industrial applications. The main goal of the research was to determine the influence of peripheral milling on residual stresses state in the surface layer of the workpiece. During machining process, cutting forces were recorded by a Kistler three—component piezo dynamometer. The scheme of the workstation is presented on Fig. 2. The course of cutting forces was registered to correlate the obtained data with calculated values of residual stresses. After milling process, residual stresses were determined by X-ray diffractometer. Measurements were performed at the Institute for Sustainable Technologies—National Research Institute in Radom.

Fig. 2
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Scheme of the work station and dimensions of the test sample [6]

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Table 1 presents scheme of the research program, while Table 2 contains ranges of variable parameters during milling process of Inconel 718.

Table 1 Scheme of the research program
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Table 2 Variable cutting parameters used during milling process
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The values of calculated residual stresses are presented in Table 3. Residual stresses were identified in samples made out of nickel-based alloy—Inconel 718 after peripheral, down-cut milling. Research was carried out on a diffractometer (Brucker’s company), where σ r —stresses calculated in the parallel direction toward milling process, σ p —stresses calculated in the perpendicular direction toward milling process.

Table 3 Calculated stresses values generated in the surface layer of Inconel 718 [6]
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Carried out research confirms that a texturization of crystal lattice occurs in the surface layer during machining process in the direction of feed motion. Tensile stresses occurred in the surface layer of nickel-based alloys after milling.

The function of the research object was developed in order to determine the dependence between cutting parameters and residual stresses. Second degree of a polynomial function was created with interaction for three input parameters (v c , f, and a p ). Obtained results are shown in a graphical form in Figs. 3, 4, and 5.

Fig. 3
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Relationship between residual stresses and variable values of cutting parameters—feed and depth of cut (f, a p ) at constant cutting speed v c  = 62.1 [m/min] [6]

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Fig. 4
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Relationship between values of residual stresses and variable values of cutting parameters—cutting speed and depth of cut (v c , a p ) at constant feed f = 550 [mm/min] [6]

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Fig. 5
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Relationship between residual stresses and variable values of cutting parameters—cutting speed and feed (v c , f) at constant depth of cut a p  = 0.5 [mm] [6]

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4 Safety Factor in Terms of Construction Reliability

Heterogeneity of the structure of materials, dispersion of values of strength characteristics, and instability of exploitation conditions are factors which justify consideration of safety factor of construction. Based on the research developed by Loladze [7] and Betanelego [8], safety factor \( (\vartheta ) \) can be presented as followed:

$$ \vartheta = \frac{{X_{\sigma } }}{{X_{g} }} $$
(1)

where \( X_{\sigma } \)—determines the stress state in the material; \( X_{g} \)—determines the limit of ultimate strength of the material.

Strength characteristics of construction materials can be usually described by normal distribution. For a given quantile percentage (\( \gamma \)), the safety factor of the construction can be presented as followed:

$$ \vartheta = \frac{{1 - \omega_{\sigma } \cdot \gamma_{R} }}{{1 - \omega_{d} \cdot \gamma_{1 - R} }} $$
(2)

where \( \omega_{\sigma } = \frac{{D_{{X_{\sigma } }} }}{{X_{{\sigma_{s} }} }} \)—variation coefficient of stresses \( \omega_{d } = \frac{{D_{{X_{g} }} }}{{X_{{g_{s} }} }} \)—variation coefficient of the limit of ultimate strength of the material; \( \gamma_{R^{-}} \) R—quantile percentage of normalized normal distribution and \( \gamma_{1 - R} \) (1−p) quantile percentage of normalized normal distribution for determined level (p) of construction reliability (s); \( D_{{X_{\sigma } }} \)—standard deviation of stresses dispersion and \( D_{{X_{g} }} \)—standard deviation of the limit of ultimate strength of the material; \( X_{{\sigma_{s} }} \) average value of stresses and \( X_{{g_{s} }} \)—average value of the ultimate strength of the material.

The analysis of Eq. 2 indicates the safety factor of the construction that strongly depends on the dispersion of mechanical properties of the material and assumed/established level of construction reliability [9, 10]. To simplify (the model, the calculations), the authors assumed \( \omega = \omega_{\sigma } = \omega_{d} \). Figure 6 shows the change of safety factor \( \vartheta \).

Fig. 6
figure 6

Dependence between safety factor \( (\vartheta ) \); reliability level (R); and coefficient of variation \( \omega_{ } \) for two values of probability (p) of achieving assumed reliability (R); 1 p = 0.90; 2: p = 0.975

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The results obtained from carried out research show that residual stresses arise after machining process and are present on the surface layer of the workpiece. These stresses affect the behavior of the element and they have an impact on the construction reliability. Stresses arise in the material as a result of external load. Generated residual stresses can add up and cause number of undesired effects (e.g., delaminations, cracing, tribological wear). Therefore, residual stresses should be taken into account during determination of safety factor.

With the use of chi-square test χ2 authors demonstrated that in significance level equal to \( \alpha = 0.05 \), there is no reason to reject the hypothesis of the normal distribution of residual stress in the surface layer of Inconel 718 alloy. The dependence of safety factor takes the following form:

$$ \vartheta = \frac{{1 - \omega_{{\sigma_{w} }} \cdot \gamma_{R} }}{{1 - \omega_{d} \cdot \gamma_{1 - R} }} $$
(3)

where \( \omega_{{\sigma_{w} }} = \frac{{D_{{X_{\sigma } }} + D_{{X_{r} }} }}{{X_{{\sigma_{s} }} + X_{{\sigma_{r} }} }} \)—variation coefficient of combined stresses in a material, (X σ ) exploitation stresses, (X r ) residual stresses, \( D_{{X_{r} }} \)—standard deviation of residual stresses and \( X_{{\sigma_{r} }} \)—average value of residual stresses.

5 Conclusions

Multi-blade milling process of Inconel 718 alloy generates residual stresses in formed surface layer of the workpiece. The possibility of residual stresses summation must take into account during strength calculations when structures are subjected to external loads at the allowable level. Therefore, in order to ensure reliable operation of construction determination of the proper safety factor is necessary.