Introduction

New promising materials and technologies for the production of gas turbine engines are increasingly being researched and implemented in recent years [1]. The heat-resistant Inconel 718 alloy, which is actively used in the petrochemical industry, is one of these materials. It is indispensable in the creation of nuclear reactors, aircraft, gas turbines, parts of rocket and aircraft engines (working blades), and space vehicles [2]. Today, 3D printing – a promising technology of additive production using methods of selective laser sintering or melting, is used for their production [3]. The different directions of individual element formation are one of its features for obtaining complex profiled elements.

It is extremely important to calculate the elastic parameters of the above-mentioned products to assess their strength [4]. As a rule, a crystallographic texture [5, 6] is formed in metal materials obtained under additive production conditions. This texture leads to anisotropy of the elastic parameters of semi-finished products and articles. To determine the influence of treatment characteristics on the resulting texture, appropriate methods for establishing elastic parameters are necessary. Since the modulus of elasticity of the material directly influences the calculation accuracy of the element stress state, then during the averaging of its values for all components of the element, significant errors may occur in the calculation of their safety factor.

The purpose of the research is to evaluate the moduli of elasticity and shear, Poisson’s ratio, as well as their anisotropy, determined by the elastic constants of the monocrystal and the texture characteristics using X-ray diffraction of Inconel 718 alloy samples (17–21 mass% Cr; 50–55 Ni; 2.8–3.3 Mo; 4.75–5.5 Nb; 0.65–1.15 Ti; ≤ 1 Co; ≤ 0.05 Ta; ≤ 0.06 B; ≤ 0.35 Mn; ≤ 0.35 Si; ≤ 0.015 P; ≤ 0.015 S; ≤ 0.08 C), obtained from powders by the selective laser sintering technology using 3D printing in the horizontal and vertical directions of samples construction and heat treated (post-treatment).

Research Methods and Materials

The alloy was used to produce high-loaded elements of the hot part of gas turbine engines operating at temperatures up to 700°C. The regularities of crystallographic texture formation were established on prismatic samples grown using the EOS M400 unit. The spherical granules (powders) were the starting material [6].

Samples printed in different directions in their original state (after printing) and after additional treatment were studied. In the horizontal direction, the central axis of the samples was placed parallel to the machine table (in the XY plane in the direction of the X axis), and in the vertical direction in the perpendicular direction (in the YZ plane in the direction of the Z axis). The texture of the samples was investigated by the X-ray method (see Fig. 1). Samples were previously chemically polished to a depth of 0.1 mm to remove the defective surface layer. radiation. Scanning was performed by a DRON-3M diffractometer in the filtered MoKα -radiation. Based on the results of the X-ray analysis, inverted polar figures (IPF) were built for different directions of the samples.

Fig. 1.
figure 1

Scheme of X-ray spectra taking off from the surfaces of samples built (a) in the XY plane and (b) in the Z axis direction.

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Results and Discussion

Modes of obtaining samples and their additional treatment (post-treatment) are given in Table 1.

Table 1 Modes of Obtaining and Additional Treatment of the Studied Specimens
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Since the polar density on the IPF is proportional to the volume fraction of crystals of the corresponding orientation, it was believed that Young’s modulus, shear modulus, and Poisson’s ratio of the studied polycrystalline samples in a certain direction were determined by the sum of specific contributions of the parameters of monocrystals of the corresponding orientation; the specific contribution is the normalized polar density \({P}_{hkl}^{N}\) on the IPF of the corresponding direction of the sample, i.e. the ratio of the polar density \({P}_{hkl}^{N}\) to the sum \(\sum_{hkl}{P}_{hkl}\) for all \({P}_{hkl}\) values on the corresponding IPF is:

$${P}_{hkl}^{N}=\frac{{P}_{hkl}}{\sum_{hkl}{P}_{hkl}}.$$
(1)

To calculate Young’s modulus, shear, and Poisson’s ratio, they were presented through Miller indices hkl, elastic compliance, and elastic moduli Cij of the alloy monocrystal [7,8,9].

Taking into account Eq. (1) for monocrystal elastic parameters, we received:

$$\frac{1}{{F}_{{\text{poly}}}}=\frac{{P}_{hkl}^{N}}{{F}_{hkl}},$$
(2)

where F is a corresponding characteristic (E, G, or v), numerical values of elastic moduli sources are given in Table 2.

Table 2 Elastic Moduli of Inconel 718 Alloy Monocrystal
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Since the Cij values differ somewhat, the averaged results were used in further calculations. The average values of the elastic yielding Sij of the Inconel 718 alloy monocrystal, obtained according to the known ratios [14] and the data in Table 2, are as follows: S11 = 0.008092321; S12 = –0.003143634; S44 = 0.009009009; \({S}_{0}={S}_{11}-{S}_{12}-\frac{1}{2}{S}_{44}=0.006731415.\) The normalized values of the polar density on IPF (Table 3) were also determined by Eq. (1).

Table 3 Calculated Values of Elastic Parameters
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A significant anisotropy of the studied properties was revealed. In particular, for samples obtained from powders using the PREP technology without additional post-printing processing (Nos. 1 and 2), the elastic characteristics calculated in the same directions (Nos. 1X, 2X, 1Z, and 2Z) differ: Young’s moduli by approx. 2.6–8%, shear moduli approx. by 0.3–8% and Poisson’s ratio by approx. 2–6%. For sample No. 1, calculated in the X direction, sample No. 2 in direction Z (i.e. Nos. 1X and 2Z), differ by approx. 6; 10 and 5%, respectively. At the same time, the elastic moduli of horizontal samples in the X direction prevail over those of vertical samples in the Z direction. Similar regularities were observed earlier [9]. For shear modulus and Poisson’s ratio, the trend is opposite.

After heat treatment (samples Nos. 3 and 4), elastic parameters calculated in the same directions (Nos. 3X, 4X, 3Z, and 4Z) differ: Young’s moduli by approx. 0.2–2.3%, shear moduli by approx. 0.9–7%, and Poisson’s ratio by approx. 1%, in particular, for sample No. 3, elastic parameters calculated in the X direction, and sample No. 4 calculated in the Z direction (that is, Nos. 3X and 4Z) differ by approx. 4, 4, and 2.5%, respectively.

Heat treatment against printing assists in the reduction of alloy anisotropy. With the complex application of hot isostatic pressing and heat treatment (samples Nos. 5 and 6), the difference between the studied parameters is no more than 1.5%. Samples from powders obtained by the VIGA technology (samples Nos. 7–10) have similar values. The established regularities agree with those given in [15, 16].

The experimental values of the elastic modulus of samples obtained by 3D printing in the horizontal and vertical directions are averaged over several measurements [16] and equal 202.2 and 179 GPa, respectively. Comparing the values calculated in the X direction of the horizontal sample and the Z direction of the vertical sample (samples Nos. 1 and 2) with the corresponding experimental values, it was revealed that the deviation from the experimental values does not exceed 6%. Thermal treatment after 3D printing promotes the increase of the modulus of elasticity, and combined with GIP, it is even more significant.

The average experimental values of the elasticity modulus of samples obtained under similar conditions [17] equal 304 and 310 GPa for horizontal and vertical directions, respectively. A comparison of the values calculated in the X and Z directions (samples Nos. 5, 6, 9, and 10) with the corresponding experimental ones [17] shows that the deviation does not exceed 17% and possibly is related to another mode of additional processing after printing. The shear modulus of the alloy wire is 77.2 GPa [18]. The maximum deviation of the calculated values (Table 3) from the given value is 10%. The determined values of Poisson’s ratio are in the range of 0.25–0.35, which is consistent with previously obtained results [19].

Conclusions

According to the results of the analysis of the inverted polar figures of the Inconel 718 alloy samples, obtained by the technology of selective laser sintering from powders by 3D printing and after post-printing processing in various combinations, the elastic and shear moduli, Poisson’s ratio and samples anisotropy in the horizontal and vertical directions of construction were theoretically estimated. For this, the monocrystal elastic constants and the texture characteristics, determined with the help of X-ray diffraction, were used. The anisotropy of the investigated properties was established. The elastic modulus of horizontal samples determined in the X direction is higher than that of vertical samples in the Z direction by 8–10%. The trend of shear modulus and Poisson’s ratio changes is opposite. The deviation of the calculated values of the elastic modulus of in the X direction of the horizontal sample and the Z direction of the vertical sample from the experimental values does not exceed 6%. Heat treatment of samples after 3D printing, as well as with hot isostatic pressing, promotes the increase of elastic and shear moduli, and Poisson’s ratio and the decrease of elastic and shear moduli. The calculated values of Poisson’s ratio of the alloy after 3D printing and additional processing were in the range of 0.31–0.33. It was established that the technology of powder production for 3D printing does not significantly change the anisotropy of the elastic parameters of the samples.

Author contributions. All authors have contributed equally to the work, authors also read and approved the final manuscript.

Conflict of interest. The authors declare that they have no potential conflict of interest about the study in this paper.

Funding. The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Disclosure statement. The authors have no relevant financial or non-financial interests to disclose.