*Article* **Influence of Relative Humidity and Oxygen Concentration on Corrosion Behaviour of Copper in H2S-Containing Liquid Petroleum Gas**

**Xianqiang Li <sup>1</sup> , Yuan Lu <sup>2</sup> , Qiang Wei <sup>2</sup> , Hu Wang 1,\* and Juan Xie 1,\***


**\*** Correspondence: senty78@hotmail.com (H.W.); jennyx99@126.com (J.X.)

**Abstract:** In this paper, the influences of relative humidity (*RH*) and concentration of O<sup>2</sup> on copper corrosion in H2S-containing LPG (liquid petroleum gas) were studied. The corrosion products obtained in different environments were also analysed by scanning electron microscopy (SEM), energy dispersive spectrometry (EDS), grazing incidence X-ray diffraction (GIXRD), X-ray photoelectron spectroscopy (XPS) and Fourier transform infrared spectroscopy (FTIR). In H2S-containing LPG, *RH* has pronounced influence on the corrosion grade of copper. The variation in the critical point (*CP*) with the *RH* of LPG is a linear relationship. The presence of O<sup>2</sup> in dry H2S has limited influence on the corrosion of copper. In the presence of different *RH*s, the *CP* always follows a negative exponential function with O<sup>2</sup> concentration. The analysis of different corrosion products implies different corrosion behaviours and mechanisms, which are dependent on the presence or absence of water vapour. The corrosion mechanisms obtained in four different environments were also proposed.

**Keywords:** copper corrosion; liquefied petroleum gas; H2S corrosion; SEM; XPS

**1. Introduction**

Natural gas is widely used in industry and our daily life. As an important existence of natural gas, liquefied petroleum gas (LPG) is more commonly utilized for its accessibility in transportation. In the exploitation and processing of LPG, sulphur removal is inevitable for the sake of alleviating corrosion attack by sulphide existing in produced LPG [1–3]. Various forms of sulphide, such as hydrogen sulphide, mercaptan and carbonyl sulphide, may lead to corrosion of copper components in production and storage facilities. Although the sulphur removal process can eradicate most of the sulphide to some extent, the residual sulphide, in trace amounts, can also be harmful to the corrosion of copper components. Of those common active sulphides, H2S is the most harmful to copper, even it is presented in low concentration. Although previous studies have revealed the behaviour of H2S on copper corrosion in LPG, the influence of other environmental factors, such as the presence of O<sup>2</sup> and relative humidity (*RH*), also contribute to corrosion attack.

Recently, the corrosion behaviour of copper in the presence of various sulphides has been extensively studied [4–7]. Echeverria investigated the copper corrosion in SO<sup>2</sup> through atomic force microscopy (AFM). The results showed that the microscopic topography and roughness of the copper surface changed after several weeks in a polluted atmosphere containing SO<sup>2</sup> [8]. Majtás discovered that low concentrations of H2S can corrode copper parts, resulting in electrical failure of electronic equipment. Additionally, the adsorbed water can promote corrosion attack [9]. Zhu proved that the corrosion rate of copper in an SO<sup>2</sup> environment first increases and then gradually decreases with exposure time. Conversely, the corrosion rate of copper in H2S increases slowly at first and then sharply declines [10]. Araban studied the corrosion behaviour of copper in different rural atmospheres. The results showed that corrosion product of Cu2O formed preferentially, in which

**Citation:** Li, X.; Lu, Y.; Wei, Q.; Wang, H.; Xie, J. Influence of Relative Humidity and Oxygen Concentration on Corrosion Behaviour of Copper in H2S-Containing Liquid Petroleum Gas. *Metals* **2022**, *12*, 2015. https:// doi.org/10.3390/met12122015

Academic Editor: David M. Bastidas

Received: 12 October 2022 Accepted: 17 November 2022 Published: 24 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

relative humidity and ammonium sulphate had remarkable influence on the corrosion behaviour [11]. Monzó found that sulphide has an obvious influence on corrosion between the boundary and the centre of a copper sheet. Additionally, elemental sulphur is more corrosive than ethanethiol. The corrosion products of elemental sulphur are in the form of nodule particles, and the ethanethiol is in the form of a uniform film [12]. García found that, at low concentrations of elemental sulphur (5 ppm), mercaptans can significantly promote corrosion. At high concentrations of elemental sulphur (25 ppm), mercaptans inhibit the corrosion of elemental sulphur. Disulphide has an obvious inhibition effect on the corrosion of elemental sulphur [13]. Studies on the corrosion behaviour of copper in the outdoor natural atmosphere have also been reported [14–20]. Kong proved that the uneven corrosion of copper in the atmosphere of Turpan is caused by the dry–wet cycle and the cold–heat cycle [21]. Lopesino believed that the corrosion of copper is more serious when closer to the coast, and the degree of patina coverage depends on the concentration of chloride in the atmosphere [22]. Yan confirmed that the corrosion rate of copper in the atmosphere with sodium chloride is almost 30000 times higher than that in the blank atmosphere [23]. Some other studies focused on the influence of *RH* on the corrosion behaviour of copper [24–26]. Odnevall believed that in the rural atmosphere containing ammonium sulphate, the *RH* of the gas had a great influence on the corrosion behaviour of copper [12]. Sharma proposed that regarding copper in H2S with low relative humidity, the Cu2O layer resulted by air has a good protective effect on H2S. It almost has no protection under high *RH* [27]. Wu proved that the *RH* of the chloride-containing atmosphere is a key factor affecting the corrosion behaviour of copper wires [28]. The corrosion behaviour of chloride on copper has also been extensively studied [29]. Chen proposed that the non-uniform growth of corrosion products on the copper surface in chloride-containing sulphide aqueous solutions resulted in a potential difference between the "thick film" and the "thin film", and this small potential difference accelerated the occurrence of corrosion [30]. Lu believed that chloride ions in the marine atmosphere of Nansha are the key factors to accelerate the corrosion of copper, and the corrosion products are Cu2O and Cu2Cl(OH)<sup>3</sup> [31]. Schindelholz believed that sodium chloride is favourable for the formation of NaOH-rich diffusion regions, and copper preferentially forms Cu2O and Cu(OH)<sup>2</sup> [32]. There are also few reports on the electrochemical study of corrosion products on copper surfaces [33]. Tran found that the growth of corrosion product films of copper exposed to H2S-containing subsurface gas has three successive stages: the first stage is a linear growth rate in thin layers (less than 15 nm). In the second stage, the oxidation rate is limited by the diffusion of copper(I) ions through the thicker corrosion layer. The third stage is linear growth [34]. Fiaud believes that both hydrogen sulphide concentration and relative humidity can promote the growth of oxide and sulphide corrosive substances. The growth mechanism of Cu2O is an electrochemical mechanism, and the growth mechanism of Cu2S is a mixed chemical and electrochemical mechanism [35]. Some other reports aimed at the corrosion behaviour of copper regarding other aspects, for example, application of theoretical calculations to copper corrosion [5–7,24], the influence of various organic acids on copper corrosion [36,37], the influence of changes in magnetic field on copper corrosion [38] and corrosion behaviour of copper by oxygen plasma [39].

Although some behaviours of copper corrosion in H2S have been studied, there is still some insufficiency. It is necessary to investigate the corrosion behaviour and mechanism of copper in LPG containing H2S at different conditions, including the presence of different *RH*s and O<sup>2</sup> concentrations. In this paper, the influence of *RH* and O<sup>2</sup> on the corrosion behaviour of copper in H2S-containing LPG was studied, the corrosion products on the surface of copper sheets were characterized and analysed and the corresponding corrosion mechanism of H2S on copper was proposed.

#### **2. Experimental Methods**

## *2.1. Materials*

Copper sheets used in corrosion experiments were purchased from Fushun Keruisi Instrument Co., Ltd. Fushun Liaoning Province, China, which strictly follows the requirement of the ASTM standard [40]. The size of cuboid copper sheet is 75 mm × 12.5 mm × 3 mm, with the purity higher than 99.9%. Copper powder (analytical grade, Chengdu Kelon Chemical Co., Ltd. Chengdu, China) was used in X-ray photoelectron spectroscopy (XPS) and FTIR (Fourier transform infrared spectrometry). The purity of powder is 99.5% and the average particle size is 23 µm. The components of LPG are listed in Table 1. H2S and O<sup>2</sup> gas used in the experiments were purchased from Zhengrong Gas company (Chengdu, China). The purity of H2S and O<sup>2</sup> is 99.9%.

**Table 1.** The components of LPG.


## *2.2. Copper Corrosion Tests*

Copper corrosion tests were carried out according to the ASTM standard [40]. The copper sheet was first abrased with 65 µm silicon carbide sandpaper. Then, it was washed with isooctane. The copper surface was polished by 105 µm silicon carbide particles, which were operated with the assistance of isooctane-soaked degreasing cotton. The prepared copper sheets were suspended into the cylinder (the special closed container for corrosion test) in three parallel experiments. Then, high-purity N<sup>2</sup> was used to remove the air inside the cylinder, by ventilating N<sup>2</sup> into the cylinder to substitute the air. Then, to control the mass flow of LPG and H2S through a flowmeter (FMA5400A, Omega, San Antonio, TX, USA), they were injected into the cylinder and the gas was mixed evenly. Subsequently, the valve was fastened. Finally, the cylinder with mixture of LPG and H2S was vertically immersed in a water bath at a constant temperature (40 ± 0.5 ◦C) for 60 ± 5 min. After the experiment time was over, the liquid and gas in the cylinder were discharged. The copper sheets were taken out and compared with the standard colour plate [40]. Next, the corrosion grades of the copper sheets were evaluated. The details of the grade evaluation are shown in Table 2. The standard stipulates that if the corrosion level of a copper sheet reaches 2a and above, it is regarded as unqualified in corrosion.

**Table 2.** The grading table of copper corrosion standard swatches [40].


## *2.3. Analysis of Corrosion Products*

To facilitate the characterization of corrosion products, the concentrations of H2S and O<sup>2</sup> were increased to 50 ppm and 10 ppm in copper corrosion tests. The copper sheets were used in corrosion tests for characterizations such as scanning electron microscopy (SEM), energy dispersive spectrometry (EDS) and grazing incidence X-ray diffraction (GIXRD). Samples of corrosion products for XPS and FTIR were prepared with powder, which could provide better results than copper sheets. The copper powder samples were applied by placing 0.1 g of copper powder in a glass sample bottle. Then, the glass sample bottle was suspended in a cylinder to conduct a copper corrosion experiment.

SEM (Model EVO MA15, ZEISS, Jena, Germany) was used to observe the morphology of the corrosion products on the copper surface. EDS (Model X-MaxN, OXFORD INSTRUMENTS Company, Abingdon, UK) was used to analyse the elemental composition of the corrosion products on the copper sheet surface. XPS (Nexsa type, Thermo Scientific, Waltham, MA, USA) was used to analyse the elemental composition and valence distribution of the corrosion products on the copper powder surface. The original XPS image was fitted with Casa XPS software. GIXRD (SmartLab 9 kw, Rigaku, Tokyo, Japan) was used to analyse the phase composition of corrosion products on the copper sheet surface. FTIR (INVENIO R, Bruker Optik GmbH, Bremen, Germany) was carried out to test the infrared spectrum of the corrosion products on the copper powder surface.

#### **3. Results and Discussion**

## *3.1. Influence of Humidity on Copper Corrosion in H2S-Containing LPG*

The corrosion behaviour of copper in H2S-containing LPG is very sensitive to the gas humidity. The higher the gas humidity, the more easily copper is corroded by H2S. Figure 1a shows the variation in the corrosion grade of copper in H2S-containing LPG at different gas humidities. In LPG containing 3 ppm H2S, the corrosion grade of copper gradually intensifies with the increase in gas humidity. At 0–30% *RH*, there is no significant corrosion on the copper surface (corrosion grade is 1a). At 50–100% *RH*, the surface of the copper sheet begins to corrode (at 50% *RH*, the copper corrosion grade is 2a). The degree of corrosion varies with humidity. The copper corrosion grade reaches 2e at 100% *RH*. Meanwhile, the variation in the copper corrosion grade in LPG without H2S in the presence of different gas humidities is used as comparison, in which the copper surface does not corrode at all humidities (corrosion grade is 1a). *Metals* **2022**, *12*, 2015 5 of 21

**Figure 1.** (**a**) Variation in copper corrosion grade in H2S-containing LPG with humidity; (**b**) the influence of H2S concentration and *RH* on the corrosion grade of copper in LPG. **Figure 1.** (**a**) Variation in copper corrosion grade in H2S-containing LPG with humidity; (**b**) the influence of H2S concentration and *RH* on the corrosion grade of copper in LPG.

The influence of *RH* on the *CP* is shown in Figure 2. The results of the *CP* at different *RH*s show a linear relationship. The fitted data obey Equation (1) as follows: = −0.021 +3.77 , <sup>0</sup> = 0.96 (1) where *CP* is the critical point of copper corrosion, corresponding to the lowest H2S concentration for reaching the corrosion grade of 2a. *RH* is the relative humidity for the corrosion test. *R* is the coefficient of determination. The results imply that with the in-The critical point (*CP*) of copper corrosion in H2S-containing LPG decreases with the increase in the relative humidity of LPG. The *CP* is defined as the lowest H2S concentration to reach the corrosion grade of 2a at specific environmental conditions. Figure 1b shows the influence of H2S concentration and *RH* on the corrosion grade of copper in LPG. It can be seen that the *CP* gradually decreases with the increase in gas humidity. From 0% *RH* to 100% *RH*, the *CP* decreases from 3.8 ppm H2S to 1.7 ppm H2S. It indicates that higher humidity is more beneficial to the corrosion process. In higher humidity, the thin film of

crease in *RH*, the thickness of the water film formed at the copper surface increases accordingly. The thicker water film is more favourable for H2S dissolution. Consequently,

**Figure 2.** The influence of *RH* on *CP* of copper corrosion in H2S-containing LPG.

*3.2. Synergistic Effect of Oxygen and Humidity on Copper Corrosion in H2S-Containing LPG*

A small amount of O<sup>2</sup> has limited influence on the corrosion of copper in dry LPG (0% *RH*). Figure 3a shows the variation in copper corrosion grade with O<sup>2</sup> content in LPG (0% *RH*) containing H2S. As can be seen from Figure 3a, in pure LPG (without H2S), the corrosion grade of copper does not change with the increase in O<sup>2</sup> content (corrosion water at the copper surface forms more easily, which provides an electrolyte environment for H2S dissolution and electrochemical corrosion. With the increase in *RH*, the thickness of the water film at the interface increases, which will provide a better condition for electrochemical corrosion. Therefore, the corrosion attack is more severe at higher *RH* and the *CP* would be lowered with the increase in *RH*. **Figure 1.** (**a**) Variation in copper corrosion grade in H2S-containing LPG with humidity; (**b**) the influence of H2S concentration and *RH* on the corrosion grade of copper in LPG.

*Metals* **2022**, *12*, 2015 5 of 21

The influence of *RH* on the *CP* is shown in Figure 2. The results of the *CP* at different *RH*s show a linear relationship. The fitted data obey Equation (1) as follows: The influence of *RH* on the *CP* is shown in Figure 2. The results of the *CP* at different *RH*s show a linear relationship. The fitted data obey Equation (1) as follows:

$$CP = -0.021 \ RH + 3.77 \ , \ R\_0 = 0.96$$

where *CP* is the critical point of copper corrosion, corresponding to the lowest H2S concentration for reaching the corrosion grade of 2a. *RH* is the relative humidity for the corrosion test. *R* is the coefficient of determination. The results imply that with the increase in *RH*, the thickness of the water film formed at the copper surface increases accordingly. The thicker water film is more favourable for H2S dissolution. Consequently, the electrochemical corrosion process is enhanced. where *CP* is the critical point of copper corrosion, corresponding to the lowest H2S concentration for reaching the corrosion grade of 2a. *RH* is the relative humidity for the corrosion test. *R* is the coefficient of determination. The results imply that with the increase in *RH*, the thickness of the water film formed at the copper surface increases accordingly. The thicker water film is more favourable for H2S dissolution. Consequently, the electrochemical corrosion process is enhanced.

**Figure 2.** The influence of *RH* on *CP* of copper corrosion in H2S-containing LPG. **Figure 2.** The influence of *RH* on *CP* of copper corrosion in H2S-containing LPG.

#### *3.2. Synergistic Effect of Oxygen and Humidity on Copper Corrosion in H2S-Containing LPG 3.2. Synergistic Effect of Oxygen and Humidity on Copper Corrosion in H2S-Containing LPG*

A small amount of O<sup>2</sup> has limited influence on the corrosion of copper in dry LPG (0% *RH*). Figure 3a shows the variation in copper corrosion grade with O<sup>2</sup> content in LPG (0% *RH*) containing H2S. As can be seen from Figure 3a, in pure LPG (without H2S), the corrosion grade of copper does not change with the increase in O<sup>2</sup> content (corrosion A small amount of O<sup>2</sup> has limited influence on the corrosion of copper in dry LPG (0% *RH*). Figure 3a shows the variation in copper corrosion grade with O<sup>2</sup> content in LPG (0% *RH*) containing H2S. As can be seen from Figure 3a, in pure LPG (without H2S), the corrosion grade of copper does not change with the increase in O<sup>2</sup> content (corrosion grade is 1a). No apparent corrosion happened at the copper surface at such condition. In LPG containing trace H2S (1 ppm), slight corrosion on the copper surface appears with the increase in O<sup>2</sup> content. Among them, there is no apparent corrosion on the copper surface from 0 to 5 ppm O2, and the corrosion grade of copper sheets in 10 ppm O<sup>2</sup> begins to rise to 1b, which is also below 2a.

to rise to 1b, which is also below 2a.

**Figure 3.** Influence of oxygen concentration on copper corrosion grade in H2S-containing LPG at (**a**) 0% *RH*, (**b**) 100% *RH*. **Figure 3.** Influence of oxygen concentration on copper corrosion grade in H2S-containing LPG at (**a**) 0% *RH*, (**b**) 100% *RH*.

grade is 1a). No apparent corrosion happened at the copper surface at such condition. In LPG containing trace H2S (1 ppm), slight corrosion on the copper surface appears with the increase in O<sup>2</sup> content. Among them, there is no apparent corrosion on the copper surface from 0 to 5 ppm O2, and the corrosion grade of copper sheets in 10 ppm O<sup>2</sup> begins

A small amount of O<sup>2</sup> has a pronounced effect on the corrosion of copper in wet LPG (100% *RH*). Figure 3b is the variation in copper corrosion grade in H2S-containing LPG (100% *RH*) with O<sup>2</sup> concentration. It demonstrates that in the absence of H2S, the copper corrosion grade can hardly be changed with the increase in O<sup>2</sup> content. The copper surface does not corrode in 0–5 ppm O2, displaying a corrosion grade of 1a. In the presence of 10 ppm O2, the copper corrosion grade is slightly promoted to 1b. In the presence of a trace amount of H2S (1 ppm) at 100% *RH*, the degree of corrosion is sharply intensified with the increase in O<sup>2</sup> content. An amount of 0.5 ppm of O<sup>2</sup> can lead to unqualified copper corrosion (grade 2a). When the O<sup>2</sup> content increases to 1 ppm, the corrosion grade rapidly climbs to grade 2d. When the content of O<sup>2</sup> continues to increase, the corrosion grade is stabilized at grade 2d. A small amount of O<sup>2</sup> has a pronounced effect on the corrosion of copper in wet LPG (100% *RH*). Figure 3b is the variation in copper corrosion grade in H2S-containing LPG (100% *RH*) with O<sup>2</sup> concentration. It demonstrates that in the absence of H2S, the copper corrosion grade can hardly be changed with the increase in O<sup>2</sup> content. The copper surface does not corrode in 0–5 ppm O2, displaying a corrosion grade of 1a. In the presence of 10 ppm O2, the copper corrosion grade is slightly promoted to 1b. In the presence of a trace amount of H2S (1 ppm) at 100% *RH*, the degree of corrosion is sharply intensified with the increase in O<sup>2</sup> content. An amount of 0.5 ppm of O<sup>2</sup> can lead to unqualified copper corrosion (grade 2a). When the O<sup>2</sup> content increases to 1 ppm, the corrosion grade rapidly climbs to grade 2d. When the content of O<sup>2</sup> continues to increase, the corrosion grade is stabilized at grade 2d.

Figure 3 reveals that the coexistence of gas humidity and O<sup>2</sup> has a notable synergistic effect on the corrosion of copper in LPG in the presence of H2S. Compared with pure LPG (0% *RH*, 0 ppm O2), the copper in LPG containing wet H2S and O<sup>2</sup> is more easily cor-Figure 3 reveals that the coexistence of gas humidity and O<sup>2</sup> has a notable synergistic effect on the corrosion of copper in LPG in the presence of H2S. Compared with pure LPG (0% *RH*, 0 ppm O2), the copper in LPG containing wet H2S and O<sup>2</sup> is more easily corroded.

roded. In order to further study the synergistic effect of oxygen and humidity on the *CP* of copper corrosion in H2S-containing LPG, the corrosion behaviour of copper in H2S-containing LPG in the presence of different oxygen concentrations was studied at 0% *RH*, 30% *RH*, 50% *RH*, 80% *RH* and 100% *RH*, respectively. The results are shown in Figure 4. It can be seen that at every *RH* condition, the *CP* gradually decreases with the increase in O<sup>2</sup> concentration. At the same oxygen concentration, *CP* gradually declines with In order to further study the synergistic effect of oxygen and humidity on the *CP* of copper corrosion in H2S-containing LPG, the corrosion behaviour of copper in H2Scontaining LPG in the presence of different oxygen concentrations was studied at 0% *RH*, 30% *RH*, 50% *RH*, 80% *RH* and 100% *RH*, respectively. The results are shown in Figure 4. It can be seen that at every *RH* condition, the *CP* gradually decreases with the increase in O<sup>2</sup> concentration. At the same oxygen concentration, *CP* gradually declines with the increase in the gas humidity. *Metals* **2022**, *12*, 2015 7 of 21

**Figure 4.** Influence of oxygen content on *CP* at (**a**) 0% *RH*, (**b**) 30% *RH*, (**c**) 50% *RH*, (**d**) 80% *RH*, (**e**)

More precise behaviour can be illustrated by interpreting the relationship between the *CP* and O2 concentration at different *RH*s, as is shown in Figure 5. The discussion is

**Figure 4.** *Cont*.

100% *RH*.

carried out at different *RH*s.

**Figure 4.** Influence of oxygen content on *CP* at (**a**) 0% *RH*, (**b**) 30% *RH*, (**c**) 50% *RH*, (**d**) 80% *RH*, (**e**) 100% *RH*. **Figure 4.** Influence of oxygen content on *CP* at (**a**) 0% *RH*, (**b**) 30% *RH*, (**c**) 50% *RH*, (**d**) 80% *RH*, (**e**) 100% *RH*.

More precise behaviour can be illustrated by interpreting the relationship between the *CP* and O2 concentration at different *RH*s, as is shown in Figure 5. The discussion is carried out at different *RH*s. More precise behaviour can be illustrated by interpreting the relationship between the *CP* and O<sup>2</sup> concentration at different *RH*s, as is shown in Figure 5. The discussion is carried out at different *RH*s. *Metals* **2022**, *12*, 2015 8 of 21

**Figure 5.** Fitted data of the *CP* in different LPG environments. water film on the copper surface makes the corrosion process different. According to the **Figure 5.** Fitted data of the *CP* in different LPG environments.

30% = 0.720

50% = 0.720

80% = 0.793

100% = 1.123

(a) In the absence of water (0% *RH*), *CP* follows a linear relationship with O<sup>2</sup> concentra-

where *CP*x% is the critical point of copper corrosion in H2S-containing LPG (x% *RH*), corresponding to the minimum H2S concentration for corrosion grade 2a. *C*<sup>0</sup> is the oxygen concentration of LPG in the copper corrosion test. *R* is the coefficient of determination. (b) In the presence of water (30% *RH*), *CP* follows a negative exponential function with

(c) In the presence of water (50% *RH*), *CP* follows a negative exponential function with

(d) In the presence of water (80% *RH*), *CP* follows a negative exponential function with

(e) In the presence of water (100% *RH*), *CP* follows a negative exponential function with

In the absence of H2O, the contribution of O<sup>2</sup> to copper corrosion is relatively even, which is consistent with a previous report [27]. However, the presence of H2O makes copper corrosion more sensitive to O<sup>2</sup> even at low O<sup>2</sup> concentration. The formation of a

, <sup>1</sup> = 0.98 (2)

−3.82 <sup>0</sup> + 2.585 , <sup>2</sup> = 0.99 (3)

−3.82 <sup>0</sup> + 1.685 , <sup>3</sup> = 0.99 (4)

−4.74 <sup>0</sup> + 1.305 , <sup>4</sup> = 0.99 (5)

−10.20 <sup>0</sup> + 0.564 , <sup>5</sup> = 0.98 (6)

0% = 3.81 −0.978 <sup>0</sup>

the O<sup>2</sup> concentration. The fitted data obey Equation (3) as follows:

the O<sup>2</sup> concentration. The fitted data obey Equation (4) as follows:

the O<sup>2</sup> concentration. The fitted data obey Equation (5) as follows:

the O<sup>2</sup> concentration. The fitted data obey Equation (6) as follows:

(a) In the absence of water (0% *RH*), *CP* follows a linear relationship with O<sup>2</sup> concentration. The fitted data obey Equation (2) as follows:

$$CP\_{0\%} = 3.81 - 0.978 \text{ } \text{C}\_0 \text{ } \text{ } R\_1 = 0.98 \tag{2}$$

where *CP*x% is the critical point of copper corrosion in H2S-containing LPG (x% *RH*), corresponding to the minimum H2S concentration for corrosion grade 2a. *C*<sup>0</sup> is the oxygen concentration of LPG in the copper corrosion test. *R* is the coefficient of determination.

(b) In the presence of water (30% *RH*), *CP* follows a negative exponential function with the O<sup>2</sup> concentration. The fitted data obey Equation (3) as follows:

$$\text{CP}\_{30\%} = 0.720 \, e^{-3.82 \, \text{C}\_0} + 2.585 \, \text{ }, \quad \text{R}\_2 = 0.99 \tag{3}$$

(c) In the presence of water (50% *RH*), *CP* follows a negative exponential function with the O<sup>2</sup> concentration. The fitted data obey Equation (4) as follows:

$$\text{CP}\_{50\%} = 0.720 \, e^{-3.82 \, \text{C}\_0} + 1.685 \, \text{ } , \quad \text{R}\_3 = 0.99 \tag{4}$$

(d) In the presence of water (80% *RH*), *CP* follows a negative exponential function with the O<sup>2</sup> concentration. The fitted data obey Equation (5) as follows:

$$\text{CP}\_{80\%} = 0.793 \, e^{-4.74 \, \text{C}\_0} + 1.305 \, \text{ }, \quad \text{R}\_4 = 0.99 \, \text{} \tag{5}$$

(e) In the presence of water (100% *RH*), *CP* follows a negative exponential function with the O<sup>2</sup> concentration. The fitted data obey Equation (6) as follows:

$$CP\_{100\%} = 1.123 \, e^{-10.20 \, \text{C}\_0} + 0.564 \quad \text{ } \quad R\_5 = 0.98 \tag{6}$$

In the absence of H2O, the contribution of O<sup>2</sup> to copper corrosion is relatively even, which is consistent with a previous report [27]. However, the presence of H2O makes copper corrosion more sensitive to O<sup>2</sup> even at low O<sup>2</sup> concentration. The formation of a water film on the copper surface makes the corrosion process different. According to the Arrhenius Equation, Equations (7) and (8), in kinetics, when the temperature of the reaction system is constant, the rate constant of a specific chemical reaction is related to the activation energy of the reaction. The lower the activation energy of the reaction, the faster the reaction rate is. In the presence of H2O, the activation energy in the reaction system decreases (Equation (9)). Compared with the reaction system without H2O, the reaction rate constant (*k*) is larger, so the reaction rate is faster. This explains why the *CP* at 0% *RH* is higher than the *CP* in the presence of H2O.

$$k\_{(0)} = A \ e^{(-E\_{a(0)}/RT)} \tag{7}$$

$$k\_{\text{(c)}} = A \, e^{\left(-E\_{\text{a(c)}}/RT\right)} \tag{8}$$

from Equations (7) and (8):

$$E\_{a(c)} = E\_{a(0)} - RT \ln \left( k\_{\text{(c)}} / k\_{\text{(0)}} \right) \tag{9}$$

where *k*(0) is the rate constant of the reaction, *k*(c) is the rate constant of the reaction after adding the catalyst, *<sup>E</sup>*a(0) is the activation energy of the reaction (kJ·mol−<sup>1</sup> ), *E*a(c) is the activation energy of the reaction after adding the catalyst (kJ·mol−<sup>1</sup> ), *A* is the pre-exponential factor, *<sup>e</sup>* is the natural base (2.718), *<sup>R</sup>* is the gas constant (8.314 J·mol−<sup>1</sup> ·K −1 ) and *T* is the thermodynamic temperature (*K*). The presence of the water film, which acts as a catalyst in the system at the interface, changes the kinetics of the corrosion process.

The variation in the *CP* with O<sup>2</sup> at different *RH*s can also be explained by Equations (7)–(9). It is well known that the presence of a water film at the interface can reduce the activation of the reaction, which can increase the number of activated molecules in the reaction system by increasing the number of effective collisions. Thus, it significantly accelerates the reaction rate. A higher *RH* in the reaction system means a thicker water vapour film on the copper surface, which implies greater effectiveness in promoting the corrosion process. When the reaction concentration is constant, the thicker water film can generate more activated molecules in the reaction system, the reaction rate constant (*k*) is larger and more effective collisions are generated per unit time to form more Cu2S and Cu2O.

## *3.3. Corrosion Mechanism of Copper in Different H2S-Containing LPG Environments* 3.3.1. Surface Morphologies after Corrosion

In different environments, LPG with H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + O<sup>2</sup> + H2O, the microscopic morphologies of the corrosion products on the copper surface sheet are shown in Figure 6. In the absence of H2S, the copper surface displays an uncorroded appearance with grooves of abrasion. When corroded in H2S-containing LPG, the copper surface is evenly covered with a thick corrosion product film. The corrosion products are in the shape of a regular hexagon with sharp edges and corners. In LPG (100% *RH*) containing H2S + H2O, the copper surface is evenly covered with a thick layer of corrosion product film. The corrosion products are spherical and accumulate at the grooves of scratches, indicating that the nucleation and growth of corrosion tend to preferentially happen at grooves of scratches [41]. A similar phenomenon also appears in other environments. In H2S + O2, the amount of corrosion products is significantly reduced. The corrosion products are sporadically distributed on the copper surface. The white particles of corrosion products are irregular in shape and size. It can be seen from the morphology that the general corrosion at this condition is significantly reduced, which is consistent with the previous experimental results. The corrosion attack happens at localized active sites, not on the whole surface. In H2S + H2O (100% *RH*) + O2, it exhibits a thick corrosion product film on the copper surface. Some irregular white corrosion products attach on the film surface. *Metals* **2022**, *12*, 2015 10 of 21

**H2S+H2O H2S+H2O Figure 6.** *Cont*.

**H2S+O2 H2S+O2**

**H2S+H2O+O2 2000**☓ **H2S+H2O+O2**

**2000**☓

**2000**☓

H2O, H2S + O2 and H2S + O2 + H2O.

**Figure 6.** SEM images of corrosion products of copper sheets after corrosion tests in H2S, H2S +

**5000**☓

**5000**☓

**5000**☓

**2000**☓

**Blank Blank** 

**H2S H2S** 

**2000**☓ **5000**☓

**5000**☓

**Figure 6.** SEM images of corrosion products of copper sheets after corrosion tests in H2S, H2S + H2O, H2S + O2 and H2S + O2 + H2O. **Figure 6.** SEM images of corrosion products of copper sheets after corrosion tests in H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + O<sup>2</sup> + H2O.

## 3.3.2. EDS Analysis of Corrosion Products

EDS was used to analyse the elemental information of corrosion products. Figure 7 and Table 3 manifest the elemental content of copper corrosion products at four medium conditions (H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + H2O + O2). It can be seen from the results that the corrosion products of LPG in H2S mainly contain S and Cu, indicating that the corrosion products are only composed of copper sulphides. The corrosion products of LPG containing H2S + H2O mainly contain S, Cu and O, implying that they are mainly composed of copper sulphides and oxides. It is also possible that oxides were generated by exposure of the sample to air. The corrosion products in H2S+O<sup>2</sup> mainly contain S, Cu and O. It shows that the content of S is much higher than that of O, indicating that the corrosion products are mainly composed of a large amount of copper sulphides and a small amount of copper oxides. The corrosion products in H2S + H2O + O<sup>2</sup> mainly contain S, Cu and O elements, meaning that the corrosion products are mainly composed of copper oxides and sulphides. The content of O is much higher than that of S, implying that the existence of H2O and O<sup>2</sup> is more favourable for the growth of copper oxide. In the presence of H2O (100% *RH*), it is more favourable to form a water film on the copper surface, which in turn leads to the dissolution and diffusion of oxygen. The electrochemical corrosion happens with the cathode process of oxygen depolarization reaction. Additionally, the dissolution of H2S in water film leads to the emergence of H<sup>+</sup> . The hydrogen depolarization reaction sion effects.

Additionally, the dissolution of H2S in water film leads to the emergence of H<sup>+</sup>

*Metals* **2022**, *12*, 2015 11 of 21

EDS was used to analyse the elemental information of corrosion products. Figure 7 and Table 3 manifest the elemental content of copper corrosion products at four medium conditions (H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + H2O + O2). It can be seen from the results that the corrosion products of LPG in H2S mainly contain S and Cu, indicating that the corrosion products are only composed of copper sulphides. The corrosion products of LPG containing H2S + H2O mainly contain S, Cu and O, implying that they are mainly composed of copper sulphides and oxides. It is also possible that oxides were generated by exposure of the sample to air. The corrosion products in H2S+O<sup>2</sup> mainly contain S, Cu and O. It shows that the content of S is much higher than that of O, indicating that the corrosion products are mainly composed of a large amount of copper sulphides and a small amount of copper oxides. The corrosion products in H2S + H2O + O<sup>2</sup> mainly contain S, Cu and O elements, meaning that the corrosion products are mainly composed of copper oxides and sulphides. The content of O is much higher than that of S, implying that the existence of H2O and O<sup>2</sup> is more favourable for the growth of copper oxide. In the presence of H2O (100% *RH*), it is more favourable to form a water film on the copper

3.3.2. EDS Analysis of Corrosion Products

as a cathodic process also happens. The two cathodic processes occur at the same time in H2S + O<sup>2</sup> + H2O, which induces the synergistic corrosion effects. drogen depolarization reaction as a cathodic process also happens. The two cathodic processes occur at the same time in H2S + O<sup>2</sup> + H2O, which induces the synergistic corro-

. The hy-

**Figure 7.** EDS images of corrosion products in different LPG environments. **Figure 7.** EDS images of corrosion products in different LPG environments.



3.3.3. GIXRD Analysis of Corrosion Products

In order to further reveal the corrosion mechanism of copper sheets in different LPG environments, GIXRD was used to analyse the corroded copper sheets in the presence of H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + H2O + O2, respectively, and the incident angle of GIXRD was 0.7◦ . The results in Figure 8 show that the spectrum of corroded copper in H2S-containing LPG is mainly the diffraction peaks of Cu and Cu2S, in which Cu2S preferentially grows on the (−536) crystal plane. The spectrum of corroded copper in LPG containing H2S+H2O is mainly the diffraction peaks of Cu and Cu2S, in which Cu2S preferentially grows on the (−232) crystal plane. The expected spectrum of Cu2O cannot be obtained, which has been proven by EDS. This is due to a too little amount of corrosion products. The spectrum of corrosion products in LPG containing H2S+O<sup>2</sup> is similar to the diffraction peaks of corrosion products at H2S conditions, mainly Cu and Cu2S diffraction peaks. Similarly, Cu2S grows preferentially on the (−536) crystal plane. The spectrum of corrosion products in LPG containing H2S+O2+H2O is mainly Cu, Cu2S and Cu2O. The diffraction peak of Cu2O is obviously stronger than that of Cu2S, indicating that

the conditions are more preferable to the growth of Cu2O [34,35,42]. In addition, in this circumstance Cu2S preferentially grows along the (034) crystal plane, which is different from other environmental conditions. ditions are more preferable to the growth of Cu2O [34,35,42]. In addition, in this circumstance Cu2S preferentially grows along the (034) crystal plane, which is different from other environmental conditions.

*Metals* **2022**, *12*, 2015 12 of 21

3.3.3. GIXRD Analysis of Corrosion Products

**Element**

**Table 3.** The elemental content of corrosion products at four different corrosion conditions.

O - - 5.18 16.61 2.85 9.20 8.40 25.99 S 9.94 17.95 8.66 13.85 14.98 24.10 3.50 5.40 Cu 90.06 82.05 86.15 69.53 82.17 66.70 88.09 68.60 Total 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00

In order to further reveal the corrosion mechanism of copper sheets in different LPG environments, GIXRD was used to analyse the corroded copper sheets in the presence of H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + H2O + O2, respectively, and the incident angle of GIXRD was 0.7°. The results in Figure 8 show that the spectrum of corroded copper in H2S-containing LPG is mainly the diffraction peaks of Cu and Cu2S, in which Cu2S preferentially grows on the (−536) crystal plane. The spectrum of corroded copper in LPG containing H2S+H2O is mainly the diffraction peaks of Cu and Cu2S, in which Cu2S preferentially grows on the (−232) crystal plane. The expected spectrum of Cu2O cannot be obtained, which has been proven by EDS. This is due to a too little amount of corrosion products. The spectrum of corrosion products in LPG containing H2S+O<sup>2</sup> is similar to the diffraction peaks of corrosion products at H2S conditions, mainly Cu and Cu2S diffraction peaks. Similarly, Cu2S grows preferentially on the (−536) crystal plane. The spectrum of corrosion products in LPG containing H2S+O2+H2O is mainly Cu, Cu2S and Cu2O. The diffraction peak of Cu2O is obviously stronger than that of Cu2S, indicating that the con-

**H2S H2S + H2O H2S + O<sup>2</sup> H2S + O<sup>2</sup> + H2O** *wt***% Atomic %** *wt***% Atomic %** *wt***% Atomic %** *wt***% Atomic %**

**Figure 8.** GIXRD patterns of corrosion products at different conditions. **Figure 8.** GIXRD patterns of corrosion products at different conditions.

3.3.4. XPS Analysis of Corrosion Products

XPS was also applied to analyse the valence states of corrosion products of copper sheets in different LPG environments, including H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + H2O + O2. The results are shown in Figure 9. Figure 9a is a comparison diagram of the Cu 2p spectrum of the corrosion products in different LPG environments. In the Cu 2p spectrum of the corrosion products in H2S, the peaks at 932.75 eV and 945 eV correspond to the characteristic peak of Cu2S and the satellite peak of Cu<sup>+</sup> at the 2p3/2 orbital, respectively. The peaks at 932.77 eV, 934.10 eV and 943.00 eV in Cu 2p spectrum of corrosion products in H2S+H2O correspond to the characteristic peaks of Cu2S, CuO and the satellite peaks of Cu2+ at the 2p3/2 orbital, respectively. The peaks at 932.70 eV, 934.27 eV and 943.00 eV in the Cu 2p spectrum of corrosion products in H2S+O<sup>2</sup> correspond to the characteristic peaks of Cu2S, CuO and the satellite of Cu2+ at the 2p3/2 orbit, respectively. The peaks at 932.80 eV, 932.51 eV and 945.00 eV in the Cu 2p spectrum of the corrosion products with H2S + H2O + O<sup>2</sup> correspond to the characteristic peaks of Cu2S, Cu2O and the satellite of Cu<sup>+</sup> at the 2p3/2 orbital, respectively, which are consistent with the XRD results. Figure 9b is the analysis result of the high-resolution S 2p spectrum. As seen in Figure 9b, in different LPG environments, the XPS signal of S is weak and exists in the form of Cu2S.

3.3.4. XPS Analysis of Corrosion Products

form of Cu2S.

XPS was also applied to analyse the valence states of corrosion products of copper sheets in different LPG environments, including H2S, H2S + H2O, H2S + O<sup>2</sup> and H2S + H2O + O2. The results are shown in Figure 9. Figure 9a is a comparison diagram of the Cu 2p spectrum of the corrosion products in different LPG environments. In the Cu 2p spectrum of the corrosion products in H2S, the peaks at 932.75 eV and 945 eV correspond to the characteristic peak of Cu2S and the satellite peak of Cu<sup>+</sup> at the 2p3/2 orbital, respectively. The peaks at 932.77 eV, 934.10 eV and 943.00 eV in Cu 2p spectrum of corrosion products in H2S+H2O correspond to the characteristic peaks of Cu2S, CuO and the satellite peaks of Cu2+ at the 2p3/2 orbital, respectively. The peaks at 932.70 eV, 934.27 eV and 943.00 eV in the Cu 2p spectrum of corrosion products in H2S+O<sup>2</sup> correspond to the characteristic peaks of Cu2S, CuO and the satellite of Cu2+ at the 2p3/2 orbit, respectively. The peaks at 932.80 eV, 932.51 eV and 945.00 eV in the Cu 2p spectrum of the corrosion products with H2S + H2O + O<sup>2</sup> correspond to the characteristic peaks of Cu2S, Cu2O and the satellite of Cu<sup>+</sup> at the 2p3/2 orbital, respectively, which are consistent with the XRD results. Figure 9b is the analysis result of the high-resolution S 2p spectrum. As seen in Figure 9b, in different LPG environments, the XPS signal of S is weak and exists in the

**Figure 9.** XPS images of corrosion products in different LPG environments: (**a**) Cu 2p data and fits; (**b**) S 2p data and fits; (**c**) the content ratio of each phase in Cu 2p spectrum. **Figure 9.** XPS images of corrosion products in different LPG environments: (**a**) Cu 2p data and fits; (**b**) S 2p data and fits; (**c**) the content ratio of each phase in Cu 2p spectrum.

Figure 9c shows the comparative analysis results of the content of Cu2S, Cu2O and CuO in the Cu 2p spectrum in different LPG environments. The detailed information of each phase is shown in Table 4. It can be seen from Figure 9c that Cu2S exists in all four conditions, and the proportion is the highest in pure H2S. With the addition of H2O and O2, the content of Cu2S decreases and the content of CuO increases gradually. In the Figure 9c shows the comparative analysis results of the content of Cu2S, Cu2O and CuO in the Cu 2p spectrum in different LPG environments. The detailed information of each phase is shown in Table 4. It can be seen from Figure 9c that Cu2S exists in all four conditions, and the proportion is the highest in pure H2S. With the addition of H2O and O2, the content of Cu2S decreases and the content of CuO increases gradually. In the presence of H2O + O2, Cu2S content begins to rise again, and Cu2O appears in large quantities, which agrees with the previous EDS and XRD results.


**Table 4.** The content ratio of each phase in the corrosion products in the Cu 2p spectrum.

#### 3.3.5. FTIR Analysis of Corrosion Products

FTIR was used to analyse the corrosion products in different LPG environments. The results are shown in Figure 10. It compares the FTIR spectra of copper powder and copper powder with four different corrosion products. The full FTIR spectra of all corrosion products are similar to the blank. The absorption peak at 3743 cm−<sup>1</sup> represents the stretching vibration absorption peak of free-state O-H in the impurities on the copper powder surface. Compared with the blank, the weakening of the peak intensity is caused by the accumulation of a thick layer of corrosion products on the copper powder surface. Compared with blank, the absorption peak of 3446 cm−<sup>1</sup> is a composite peak, which is composed of a stretching vibration absorption peak of the associative O-H and Cu(I)-S on the copper powder surface. The phenomenon of increased peak intensity is due to the presence of a large amount of Cu2S (3470 cm−<sup>1</sup> ) after corrosion. The absorption peaks of 2923 cm−<sup>1</sup> and 2854 cm−<sup>1</sup> represent the stretching vibration absorption peaks of the C-H bond in the methylene group. The phenomenon of peak intensity enhancement is caused by the residual LPG in the corrosion products. In the FTIR spectrum of the bank, there are many unknown absorption peaks around 1500 cm−<sup>1</sup> , which should be caused by impurities of the copper powder surface. Figure 10b is an FTIR comparison of 920– 1230 cm−<sup>1</sup> . Compared with the blank, the absorption peak at 1160 cm−<sup>1</sup> is significantly enhanced, which can be attributed to the stretching vibration of Cu(I)-S (1145 cm−<sup>1</sup> ). The reason why the peak intensity at this position is similar to the blank can be ascribed to little corrosion products on the copper powder surface in H2S + O2. Compared with the blank, the peak position of 1081 cm−<sup>1</sup> appears blue-shifted, which is caused by the vibration of a large amount of Cu(I)-S (1115 cm−<sup>1</sup> ). This phenomenon is more pronounced in conditions with more corrosion products. Figure 10c shows the FTIR comparison between 400 cm−<sup>1</sup> and 800 cm−<sup>1</sup> for the four corrosion products and the blank. Compared with the blank, the apparent enhancement of the absorption peak at 713 cm−<sup>1</sup> is due to the in-plane rocking vibration of C-H in the methylene group of the residual LPG adsorbed in the corrosion product. As for the weak absorption peak at 619 cm−<sup>1</sup> in H2S and H2S + H2O, it can be regarded as the stretching vibration peak of Cu(I)-S (618 cm−<sup>1</sup> ). In H2S + O<sup>2</sup> and H2S + H2O + O2, a strong absorption characteristic peak at 619 cm−<sup>1</sup> can be regarded as the composite absorption peak of Cu2S and Cu2O (the absorption peak positions of Cu(I)-S (618 cm−<sup>1</sup> ) and Cu(I)-O (614 cm−<sup>1</sup> ) are very similar). The strong absorption peak at 499 cm−<sup>1</sup> is due to the stretching vibration of Cu=O [42–45]. *Metals* **2022**, *12*, 2015 15 of 21

**Figure 10.** FTIR images of corrosion products on copper powder in different LPG environments: (**a**) FTIR full spectrum, (**b**) FTIR images of 920~1230 cm−1 , (**c**) FTIR images of 445~800 cm−1 . **Figure 10.** FTIR images of corrosion products on copper powder in different LPG environments: (**a**) FTIR full spectrum, (**b**) FTIR images of 920~1230 cm−<sup>1</sup> , (**c**) FTIR images of 445~800 cm−<sup>1</sup> .

The results of the analysis on the differences in the absorption peaks in the FTIR reveal that Cu2S will be formed on the copper surface in the four conditions. The presence of H2O promotes the production of CuO. The results of the analysis on the differences in the absorption peaks in the FTIR reveal that Cu2S will be formed on the copper surface in the four conditions. The presence of H2O promotes the production of CuO.

The corrosion mechanism of copper in LPG only containing H2S is a chemical corrosion process. The corrosion process is carried out according to the reaction Equations (10) and (11). Figure 11 is a schematic diagram of the corrosion steps in H2S-containing LPG. In the first step, H2S gas in the LPG is adsorbed on the copper surface. Then, H2S reacts with Cu atoms at the surface to generate H<sup>2</sup> and Cu2S. The whole corrosion process is a chemical process. Electrochemical corrosion cannot occur due to the absence of the

**Figure 11.** Schematic diagram of corrosion steps of copper sheet in H2S-containing LPG. (**a**) step1.

2() ⇌ 2() (10)

2() + 2 ⇌ 2 + <sup>2</sup> (11)

as follows.

electrolyte.

(**b**) step2.

3.4.1. Corrosion Mechanism in H2S

*3.4. Corrosion Mechanism of Copper in Different LPG Environments*

#### *3.4. Corrosion Mechanism of Copper in Different LPG Environments 3.4. Corrosion Mechanism of Copper in Different LPG Environments* Based on the results of copper corrosion tests and corrosion product analysis, the

FTIR full spectrum, (**b**) FTIR images of 920~1230 cm−1

of H2O promotes the production of CuO.

*Metals* **2022**, *12*, 2015 15 of 21

Based on the results of copper corrosion tests and corrosion product analysis, the copper corrosion mechanisms of copper at different LPG environments can be proposed as follows. copper corrosion mechanisms of copper at different LPG environments can be proposed as follows. 3.4.1. Corrosion Mechanism in H2S

**Figure 10.** FTIR images of corrosion products on copper powder in different LPG environments: (**a**)

The results of the analysis on the differences in the absorption peaks in the FTIR reveal that Cu2S will be formed on the copper surface in the four conditions. The presence

#### 3.4.1. Corrosion Mechanism in H2S The corrosion mechanism of copper in LPG only containing H2S is a chemical cor-

The corrosion mechanism of copper in LPG only containing H2S is a chemical corrosion process. The corrosion process is carried out according to the reaction Equations (10) and (11). Figure 11 is a schematic diagram of the corrosion steps in H2S-containing LPG. In the first step, H2S gas in the LPG is adsorbed on the copper surface. Then, H2S reacts with Cu atoms at the surface to generate H<sup>2</sup> and Cu2S. The whole corrosion process is a chemical process. Electrochemical corrosion cannot occur due to the absence of the electrolyte. rosion process. The corrosion process is carried out according to the reaction Equations (10) and (11). Figure 11 is a schematic diagram of the corrosion steps in H2S-containing LPG. In the first step, H2S gas in the LPG is adsorbed on the copper surface. Then, H2S reacts with Cu atoms at the surface to generate H<sup>2</sup> and Cu2S. The whole corrosion process is a chemical process. Electrochemical corrosion cannot occur due to the absence of the electrolyte.

$$\rm H\_2S\_{(g)} \rightleftharpoons H\_2S\_{(ads)}\tag{10}$$

, (**c**) FTIR images of 445~800 cm−1

.

$$\rm H\_2S\_{(ads)} + 2Cu \rightleftharpoons Cu\_2S + H\_2 \tag{11}$$

**Figure 11.** Schematic diagram of corrosion steps of copper sheet in H2S-containing LPG. (**a**) step1. (**b**) step2. **Figure 11.** Schematic diagram of corrosion steps of copper sheet in H2S-containing LPG. (**a**) step1. (**b**) step2.

## 3.4.2. Corrosion Mechanism in H2S+H2O

The corrosion mechanism of copper in LPG containing H2S + H2O is an electrochemical corrosion process. The corrosion process is carried out according to the reaction Equations (12)–(16). Figure 12 shows the schematic corrosion steps in LPG with H2S + H2O. The process of electrochemical corrosion can be explained by Figure 12a–c. Firstly, H2O exists in the form of a water film on the copper surface. H2S in LPG dissolves into the water film and further hydrolyses to form a great amount of HS−, H3O− (hydronium ions) and a small amount of S2−. Furthermore, the process of the anodic reaction is losing electrons of a Cu atom to form Cu2S with S2−, as expressed in Equation (15). The cathodic reaction can be undertaken as Equation (16), which is the traditional hydrogen depolarization reaction. At this time, the massive consumption of S2<sup>−</sup> further promotes the dissolution and hydrolysis of H2S in LPG, which continuously accelerates the corrosion of copper and results in a great amount of Cu2S precipitation. However, when the water film is insufficient (low humidity), it is difficult to form an effective electrochemical corrosion environment. In this circumstance, the chemical process may be the main reason for corrosion. This is consistent with the results of XRD, XPS and FTIR. A small amount of CuO in XPS and FTIR is likely due to the contamination of oxidization in air.

$$\rm H\_2S\_{(g)} \rightleftharpoons H\_2S\_{(l)}\tag{12}$$

H2S(l) +H2O HS<sup>−</sup> + H3O <sup>+</sup> (13)

$$\text{HS}^- + \text{H}\_2\text{O} \rightleftharpoons \text{S}^{2-} + \text{H}\_3\text{O}^+ \tag{14}$$

$$\text{2Cu} + \text{S}^{2-} \rightleftharpoons \text{Cu}\_2\text{S} + \text{2e}^- \tag{15}$$

+

<sup>−</sup> + 3

2− + 3

2() ⇌ 2() (12)

, H3O<sup>−</sup>

, as expressed in Equation (15). The cathodic re-

. Furthermore, the process of the anodic reaction is losing elec-

(hydronium ions)

(13)

further promotes the dissolu-

<sup>+</sup> (14)

$$2\text{H}\_3\text{O}^+ + 2\text{e}^- \rightleftharpoons 2\text{H}\_2\text{O} + \text{H}\_2\tag{16}$$

*Metals* **2022**, *12*, 2015 16 of 21

water film and further hydrolyses to form a great amount of HS<sup>−</sup>

tion reaction. At this time, the massive consumption of S2<sup>−</sup>

The corrosion mechanism of copper in LPG containing H2S + H2O is an electrochemical corrosion process. The corrosion process is carried out according to the reaction Equations (12)–(16). Figure 12 shows the schematic corrosion steps in LPG with H2S + H2O. The process of electrochemical corrosion can be explained by Figure 12a–c. Firstly, H2O exists in the form of a water film on the copper surface. H2S in LPG dissolves into the

action can be undertaken as Equation (16), which is the traditional hydrogen depolariza-

tion and hydrolysis of H2S in LPG, which continuously accelerates the corrosion of copper and results in a great amount of Cu2S precipitation. However, when the water film is insufficient (low humidity), it is difficult to form an effective electrochemical corrosion environment. In this circumstance, the chemical process may be the main reason for corrosion. This is consistent with the results of XRD, XPS and FTIR. A small amount of

CuO in XPS and FTIR is likely due to the contamination of oxidization in air.

2()+2 ⇌ H

<sup>−</sup> + 2 ⇌

3.4.2. Corrosion Mechanism in H2S+H2O

trons of a Cu atom to form Cu2S with S2<sup>−</sup>

and a small amount of S2<sup>−</sup>

**Figure 12.** Schematic diagram of corrosion steps of copper sheet in LPG with H2S+H2O. (**a**) step1. (**b**) step2. (**c**) step3. **Figure 12.** Schematic diagram of corrosion steps of copper sheet in LPG with H2S+H2O. (**a**) step1. (**b**) step2. (**c**) step3.

## 3.4.3. Corrosion Mechanism in H2S + O<sup>2</sup>

The corrosion mechanism of copper in LPG with H2S + O<sup>2</sup> is a chemical corrosion process. The corrosion process is carried out according to Equations (10), (11) and (17)–(19). Figure 13 displays a schematic diagram of the corrosion process in LPG containing H2S + O2. In the first step, H2S and O<sup>2</sup> molecules in LPG adsorb on the copper surface. Then, the chemical reaction of copper with H2S and O<sup>2</sup> molecules generate Cu2S, H<sup>2</sup> and Cu2O, respectively. Part of Cu2O is further oxidized to CuO by O2. The corrosion mechanism shows that the presence of only O<sup>2</sup> will not significantly promote the corrosion grade of the copper sheet, which is consistent with the experimental results. In this condition, the peak of CuO does not appear in the XRD results. However, a small amount of CuO was detected in XPS, indicating that the content of CuO is quite low in corrosion products. With the FTIR spectrum, the amount of Cu2O is significantly more than CuO, which implies Cu2O is more stable than CuO.

$$\mathcal{O}\_{2(g)} \rightleftharpoons \mathcal{O}\_{2(ads)}\tag{17}$$

$$\text{O}\_{2(ads)} + 4\text{Cu} \rightleftharpoons 2\text{Cu}\_2\text{O} \tag{18}$$

$$\text{O}\_{2(ads)} + 2\text{Cu}\_2\text{O} \rightleftharpoons 4\text{CuO} \tag{19}$$

**Figure 13.** Schematic diagram of corrosion steps of copper sheet in LPG with H2S + O2.(**a**) step1. (**b**) step2. **Figure 13.** Schematic diagram of corrosion steps of copper sheet in LPG with H2S + O<sup>2</sup> . (**a**) step1. (**b**) step2.

The corrosion mechanism of copper in LPG with H2S + O<sup>2</sup> is a chemical corrosion process. The corrosion process is carried out according to Equations (10), (11) and (17)– (19). Figure 13 displays a schematic diagram of the corrosion process in LPG containing H2S + O2. In the first step, H2S and O<sup>2</sup> molecules in LPG adsorb on the copper surface. Then, the chemical reaction of copper with H2S and O<sup>2</sup> molecules generate Cu2S, H<sup>2</sup> and Cu2O, respectively. Part of Cu2O is further oxidized to CuO by O2. The corrosion mechanism shows that the presence of only O<sup>2</sup> will not significantly promote the corrosion grade of the copper sheet, which is consistent with the experimental results. In this condition, the peak of CuO does not appear in the XRD results. However, a small amount of CuO was detected in XPS, indicating that the content of CuO is quite low in corrosion products. With the FTIR spectrum, the amount of Cu2O is significantly more than CuO,

2() ⇌ 2() (17)

2() +4Cu ⇌ 22O (18)

2() + 22O ⇌ 4 (19)

3.4.4. Corrosion Mechanism in H2S + H2O + O<sup>2</sup>

3.4.3. Corrosion Mechanism in H2S + O<sup>2</sup>

which implies Cu2O is more stable than CuO.

3.4.4. Corrosion Mechanism in H2S + H2O + O<sup>2</sup> The corrosion mechanism of copper in LPG containing H2S + H2O + O<sup>2</sup> is an electrochemical corrosion process. The corrosion process is carried out according to the Equations (12)–(16) and (20)–(22). The electrochemical corrosion process can be illustrated by the schematic diagram in Figure 14. Firstly, H2O in LPG exists in the form of a water film on the copper surface. Two electrochemical reactions happen at the interface. The dissolution of H2S in the water film leads to the electrochemical reaction in Equations (12)–(16). Meanwhile, the dissolution and diffusion of O<sup>2</sup> also cause another electrochemical corrosion reaction, as in Equations (20)–(22), which is the oxygen depolarization process. After H2S and O<sup>2</sup> molecules in LPG dissolve into the water film to form H2S(l) and O2(l), H2S(l) is further hydrolysed into a large amount of HS- , H3O<sup>+</sup> and a small amount of S2- . Cu loses electrons to form Cu2S and Cu2O with S2- and OH- , according to the anodic reaction of Equations (15) and (22) [46]. H3O<sup>+</sup> and O2(l) obtain electrons from copper to form H2, H2O and OH- , which are regarded as cathode reactions (Equations (16) and (21)) [34,35,47]. Because the H3O<sup>+</sup> in the water is consumed in large quantities, it is conducive to the ionization of water to move to the right to generate a large amount of OH- , which increases the pH of the water film. A large amount of OHis not only conducive to the dissolution of H2S in LPG in the water film, improving the solubility of H2S, but is also The corrosion mechanism of copper in LPG containing H2S + H2O + O<sup>2</sup> is an electrochemical corrosion process. The corrosion process is carried out according to the Equations (12)–(16) and (20)–(22). The electrochemical corrosion process can be illustrated by the schematic diagram in Figure 14. Firstly, H2O in LPG exists in the form of a water film on the copper surface. Two electrochemical reactions happen at the interface. The dissolution of H2S in the water film leads to the electrochemical reaction in Equations (12)–(16). Meanwhile, the dissolution and diffusion of O<sup>2</sup> also cause another electrochemical corrosion reaction, as in Equations (20)–(22), which is the oxygen depolarization process. After H2S and O<sup>2</sup> molecules in LPG dissolve into the water film to form H2S(l) and O2(l), H2S(l) is further hydrolysed into a large amount of HS−, H3O<sup>+</sup> and a small amount of S2-. Cu loses electrons to form Cu2S and Cu2O with S2- and OH−, according to the anodic reaction of Equations (15) and (22) [46]. H3O<sup>+</sup> and O2(l) obtain electrons from copper to form H2, H2O and OH−, which are regarded as cathode reactions (Equations (16) and (21)) [34,35,47]. Because the H3O<sup>+</sup> in the water is consumed in large quantities, it is conducive to the ionization of water to move to the right to generate a large amount of OH−, which increases the pH of the water film. A large amount of OH− is not only conducive to the dissolution of H2S in LPG in the water film, improving the solubility of H2S, but is also conducive to the continuous hydrolysis of H2S(l) and HS<sup>−</sup> in the water film to form a large amount of HS<sup>−</sup> and S2−, which promotes the formation of lots of Cu2S precipitates. Thereby, the solubility of H2S in the water film is further increased to promote the corrosion of copper by H2S. Figure 14a–c are schematic diagrams of corrosion steps in LPG containing H2S + H2O + O2. Combined with the previous theoretical analysis and Figure 14, it is shown that when both O<sup>2</sup> and H2O exist in LPG, the corrosion grade of copper will be significantly increased. A large amount of Cu2S and Cu2O will be generated, which is consistent with the previous experimental results of the synergistic effect of humidity and O2. Moreover, Cu2O is thermodynamically more stable than Cu2S (the standard free energies of formation at room temperature for Cu2O and Cu2S are −34.98 and −20.6 *kcal/mole*, respectively [27]), indicating that Cu2O is preferentially formed at the same conditions. Therefore, the amounts of corrosion products in Cu2O are more than Cu2S. This conclusion is consistent with the results of XRD and EDS. In addition, a large amount of Cu2O in XPS and FTIR also proves it.

$$\mathcal{O}\_{2(g)} \rightleftharpoons \mathcal{O}\_{2(l)}\tag{20}$$

$$\text{2O}\_2\text{(l)} + 2\text{H}\_2\text{O} + 4\text{e}^- \rightleftharpoons 4\text{OH}^- \tag{21}$$

$$2\text{Cu} + 2\text{OH}^- \rightleftharpoons \text{Cu}\_2\text{O} + \text{H}\_2\text{O} + 2\text{e}^-\tag{22}$$

conducive to the continuous hydrolysis of H2S(l) and HS-

amount of Cu2O in XPS and FTIR also proves it.

amount of HS-and S2<sup>−</sup>

in the water film to form a large

(21)

(22)

, which promotes the formation of lots of Cu2S precipitates. There-

<sup>−</sup> ⇌ 4 O

<sup>−</sup> ⇌ 2 +2 + 2

−

−

2() ⇌ 2() (20)

by, the solubility of H2S in the water film is further increased to promote the corrosion of copper by H2S. Figure 14a–c are schematic diagrams of corrosion steps in LPG containing H2S + H2O + O2. Combined with the previous theoretical analysis and Figure 14, it is shown that when both O<sup>2</sup> and H2O exist in LPG, the corrosion grade of copper will be significantly increased. A large amount of Cu2S and Cu2O will be generated, which is consistent with the previous experimental results of the synergistic effect of humidity and O2. Moreover, Cu2O is thermodynamically more stable than Cu2S (the standard free energies of formation at room temperature for Cu2O and Cu2S are −34.98 and −20.6 *kcal/mole*, respectively [27]), indicating that Cu2O is preferentially formed at the same conditions. Therefore, the amounts of corrosion products in Cu2O are more than Cu2S. This conclusion is consistent with the results of XRD and EDS. In addition, a large

<sup>2</sup> () + 22 + 4

2Cu +2O

**Figure 14.** Schematic diagram of corrosion steps of copper sheet in LPG with H2S + H2O + O2. (**a**) step1. (**b**) step2. (**c**) step3. **Figure 14.** Schematic diagram of corrosion steps of copper sheet in LPG with H2S + H2O + O<sup>2</sup> . (**a**) step1. (**b**) step2. (**c**) step3.

#### **4. Conclusions**

**4. Conclusions** In this paper, the influence of humidity and O<sup>2</sup> on copper corrosion in H2S-containing LPG was studied. The corrosion products were characterized and ana-In this paper, the influence of humidity and O<sup>2</sup> on copper corrosion in H2S-containing LPG was studied. The corrosion products were characterized and analysed to reveal the corrosion mechanism. The following conclusions were obtained:


react with copper directly at the interface. The corrosion mechanism of copper in LPG containing H2S + H2O and H2S + H2O + O<sup>2</sup> is an electrochemical corrosion process. In H2S + H2O, the corrosion proceeds with an anodic reaction of copper oxidization and a cathodic reaction of traditional hydrogen depolarization. In H2S + H2O + O2, two different electrochemical reactions happen: one is the same as in H2S + H2O, and the other electrochemical reaction displays as the corrosion of O<sup>2</sup> in neutral medium, in which the cathodic process is oxygen depolarization.

**Author Contributions:** Conceptualization, H.W. and J.X.; methodology, H.W.; validation, H.W., X.L. and J.X.; formal analysis, Q.W.; investigation, Y.L.; writing—original draft preparation, X.L.; writing review and editing, X.L.; supervision, H.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Nissar Ahmed <sup>1</sup> , Imad Barsoum 1,2,\* and Rashid K. Abu Al-Rub 1,2**


**Abstract:** The layer-by-layer process of additive manufacturing (AM) is known to give rise to high thermal gradients in the built body resulting in the accumulation of high residual stresses. In the current study, a numerical investigation is conducted on the effect of residual stresses on the mechanical properties of IN718 triply periodic minimal surface (TPMS) lattices fabricated using the selective laser melting (SLM) process for different relative densities. The AM simulation of four different sheet- and ligament-based TPMS topologies, namely, Schwarz Primitive, Schoen Gyroid, Schoen IWP-S, and IWP-L, are performed using a sequentially coupled thermomechanical finite element model to evaluate the thermal histories and residual stress evolution throughout the SLM process. The finite element results are utilized to obtain the effective mechanical properties, such as elastic modulus, yield strength, and specific energy absorption (SEA), of the TPMS lattices while accounting for the residual stress field arising from the SLM process. The mechanical properties are correlated to relative density using the Gibson–Ashby power laws and reveal that the effect of the residual stresses on the elastic modulus of the as-built TPMS samples can be significant, especially for the Schwarz Primitive and Schoen-IWP-L TPMS topologies, when compared to the results without accounting for residual stresses. However, the effect of the residual stresses is less significant on yield strength and SEA of the TPMS samples. The work demonstrates a methodology for numerical simulations of the SLM process to quantify the influence of inherited residual stresses on the effective mechanical properties of complex TPMS topologies.

**Keywords:** residual stresses; additive manufacturing; selective laser melting (SLM); finite element modeling (FEM); triply periodic minimal surface (TPMS)

Received: 29 June 2022

Accepted: 8 August 2022 Published: 12 August 2022

**Citation:** Ahmed, N.; Barsoum, I.; Abu Al-Rub, R.K. Numerical Investigation on the Effect of Residual Stresses on the Effective Mechanical Properties of 3D-Printed TPMS Lattices. *Metals* **2022**, *12*, 1344. https://doi.org/10.3390/met12081344 Academic Editor: Zhanyong Zhao

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

Over the past couple of decades, advancements in additive manufacturing (AM) have enabled the fabrication of complex lattice structures including the triply periodic minimal surface (TPMS) [1,2], but these structures are difficult to produce with conventional manufacturing methods due to the associated high cost and low efficiency [3]. TPMS structures are three-dimensional open-celled structures composed of one or more repeating unit cells in an orderly pattern [4]. These structures have a high specific strength and stiffness, and a high surface-area-to-volume ratio, which are features desired in many mechanical applications [5,6]. However, due to the sequential layer-by-layer nature of the selective laser melting (SLM) process, part of the heat is absorbed during melting of the powder layer and part of the heat is conducted to the already solidified layers below and through convection to the surroundings. This cyclic nature of thermal loads leads to a transient change in temperature. The steep thermal gradients induce a strain mismatch between the newly formed layers and pre-solidified layers beneath during heating and cooling cycles, resulting in the accumulation of residual stresses [7]. The inherited residual stresses in SLM are known to affect the dimensional accuracies and shape of the parts, contributing to crack

97

formation and delamination of layers. Additionally, residual stress drastically affects the fatigue life and the loading capacity of the parts [8,9]. Hence, quantifying residual stresses arising from the SLM process is becoming more critical. Depending on the magnitude of the residual stresses and their effect, they can be classified as type I, II, or III residual stresses [10]. Type I residual stresses act over larger length scales, e.g., macroscale, due to nonuniform heating and cooling rates throughout the AM process, which can result in nonnegligible deformations. Type II stresses that act at the microstructural level, e.g., grain size level, can occur due to phase transformation in the material at the microscale, while type III stresses exist over atomic scales caused by dislocation and point defects. The focus of the current study is macroscopic residual stresses of type I. As the SLM process involves many process parameters, experimentally evaluating residual stresses is time-consuming [11,12], and hence, numerical analysis based on the finite element method (FEM) is an efficient way simulate the thermomechanical behavior and quantify the residual stresses during the SLM process. The most widely used thermomechanical approach is the indirect sequentially coupled analysis that is performed in two stages. First, a transient temperature field of the built part is simulated, and the temperature results are then used to perform the mechanical analysis. This method is computationally less expensive compared to fully coupled analysis in which the temperature and displacement degrees of freedom are solved simultaneously and updated for each time increment [13]. An indirect coupling approach has been successfully used to investigate the AM process to predict residual stress formations and geometric deformations to good accuracies [14–19].

Numerical methods based on finite element analysis (FEA) have been extensively used alongside experiments to investigate the mechanical properties of TPMS structures fabricated from the AM process [20–24]. Al-Rub et al. [25] used FEA to investigate the effective anisotropic elastic and plastic properties along with the deformation mechanism under different load combinations of Schoen's IWP TPMS and proposed a cost-effective finite-element simulation framework for the IWP structural system. Al-Ketan et al. [26] studied the topology–property relationship of lattice structures based on the TPMS IWP minimal surfaces, and FEA was used to investigate the stiffness and strength of different relative densities of the TPMS structure along with experimental investigations. The work showed that the sheet-based IWP lattice structure exhibited high structural efficiency compared to other strut-based and skeletal-based lattice structures. Yang et al. [27] used a numerical approach to quantify the influence of various geometric factors such as surface thickness, sample size, and number of surface periods, etc., on the overall structural response of gyroid structures. Lee et al. [28] investigated the mechanical properties and deformation behavior under several different stress states of the Schwarz Primitive unit cell under periodic boundary conditions. Zheng et al. [29] using the finite volume method found that the mechanical properties are highly dependent on topological architectures of TPMS structures. Abueidda et al. [30] in their work predicted and compared the electrical/thermal conductivities, elastic properties, and anisotropies of different TPMS foams. Maskery et al. [31] used FEA to predict the stress distribution in polymer-based TPMSs under compressive loads with good agreement with experiments, and the investigation also indicated that the cell geometry played a key role in determining the deformation mechanism, failure modes, and the associated mechanical properties.

In general, FEA has been effectively used to investigate the elastic and plastic properties of TPMS structures; however, to the best of the authors knowledge, numerical thermomechanical analysis to quantify the effect of residual stresses inherited from the SLM process on the effective mechanical properties of TPMS structures is a topic not addressed in the literature. Hence, in this work, an FEA simulation scheme is proposed to evaluate the residual stresses from the SLM process and to quantify its influence on the effective mechanical properties of TPMS structures. Four different TPMS topologies are considered: Schwarz sheet-based Primitive, Schoen sheet-based Gyroid, Schoen sheet-based I-WP, and ligament-based I-WP (henceforth referred to P-S, G-S, IWP-S, and IWP-L, respectively). Inconel 718 (IN718), which is a commonly used grade for metallic 3D printing, is considered

as the base material, and the effective mechanical properties from the thermomechanical simulations of the AM-built TPMS lattices are compared with their reference counterparts, i.e., lattices without residual stresses.

#### **2. Modeling Framework**

Figure 1 shows a schematic representation of the AM process with governing thermal conditions. Assuming a homogenous medium with isotropic thermal properties, e.g., no spatial change in thermal conductivity of the material, the transient temperature distribution *T* (*x*, *y*, *z*, *t*) throughout the workpiece during the SLM process is governed by the following three-dimensional thermal transient conduction equation:

$$k(T)\ \nabla^2 T + Q = \rho \ C\_p(T) \ \frac{\partial T}{\partial t} \tag{1}$$

where *C<sup>p</sup>* is the temperature-dependent specific heat capacity, *ρ* is the density, *T* is the temperature, *t* is the time, *Q* is the internal heat generation rate, and *k* is the temperaturedependent thermal conductivity of the isotropic material. The initial and final (e.g., cooling stage) thermal condition for temperature distribution throughout the powder bed is given by

$$T(\mathbf{x}, y, z, t = 0) = T\_0, \ T(\mathbf{x}, y, z, t = \infty) = T\_0 \tag{2}$$

where *T*<sup>0</sup> = 22 ◦C corresponds to room temperature. The base of the substrate is preheated and subjected to a temperature of 100 ◦C during the build stage and is brought back to room temperature during the cooling phase. The boundary conditions for all other surfaces are assumed to be under heat loss to the surrounding gas by free air convection with a heat transfer coefficient, *<sup>h</sup>* = 1 <sup>×</sup> <sup>10</sup>−<sup>5</sup> W/mm<sup>2</sup> ◦C.

**Figure 1.** Representation of the AM process with thermal conditions.

The layer-by-layer formation during the SLM process gives rise to large thermal gradients in parts, which, in turn, leads to the accumulation of residual stress and strain during the solidification phase. The related stress and strain in the parts are associated with the following equation:

$$\{\sigma\} = [D]\{\varepsilon^{\ell}\}\tag{3}$$

where {*σ*} is the stress vector, [*D*] is the elasticity stiffness matrix, and {*ε e* } is the elastic strain vector. Using a simplified elastic–plastic hardening model, {*ε e* } can be expressed as:

$$\{\varepsilon^{\varepsilon}\} = \{\varepsilon\} - \{\varepsilon^{p}\} - \{\varepsilon^{t}\} \tag{4}$$

where *ε* is the total strain vector, {*ε p* } is the plastic strain vector, and {*ε t* } is the thermal strain vector. Using Equation (4) in Equation (3) may be written as follows:

$$\{\varepsilon\} = [D]^{-1}\{\sigma\} + \{\varepsilon^p\} + \{\varepsilon^t\} \tag{5}$$

For isotropic material, the above stress–strain relationship can be elaborated in Cartesian co-ordinates as follows [32]:

$$\varepsilon\_{\mathbf{x}} = \frac{1}{E} \left[ \sigma\_{\mathbf{x}} - \left[ v(\sigma\_{\mathbf{y}} + \sigma\_{\mathbf{z}}) \right] + \varepsilon\_{\mathbf{x}}^{p} + \varepsilon^{t} \right. \tag{6}$$

$$
\varepsilon\_{\mathcal{Y}} = \frac{1}{E} \left[ \sigma\_{\mathcal{Y}} - v(\sigma\_{\mathcal{X}} + \sigma\_{\mathcal{z}}) \right] + \varepsilon\_{\mathcal{Y}}^p + \varepsilon^t \tag{7}
$$

$$\varepsilon\_{z} = \frac{1}{E} \left[ \sigma\_{z} - v \left( \sigma\_{x} + \sigma\_{y} \right) \right] + \varepsilon\_{z}^{p} + \varepsilon^{t} \tag{8}$$

$$\gamma\_{\mathbf{x}\mathbf{y}} = \frac{\tau\_{\mathbf{x}\mathbf{y}}}{\mathbf{2G}} + \gamma\_{\mathbf{x}\mathbf{y}}^{p}, \quad \gamma\_{\mathbf{x}\mathbf{z}} = \frac{\tau\_{\mathbf{x}\mathbf{z}}}{\mathbf{2G}} + \gamma\_{\mathbf{x}\mathbf{z}}^{p}, \quad \gamma\_{\mathbf{y}\mathbf{z}} = \frac{\tau\_{\mathbf{y}\mathbf{z}}}{\mathbf{2G}} + \gamma\_{\mathbf{y}\mathbf{z}}^{p} \tag{9}$$

where *E* is the elastic modulus; *G* is the shear modulus; *ν* is Poisson's ratio; *τxy*, *τxz*, and *τyz* are the shear stress components; *γxy*, *γxz*, and *γyz* are the corresponding shear strain components. The thermal strain component arising due to volume change caused by temperature variations can be expressed as follows:

$$
\varepsilon^t = \mathfrak{a}\_\ell \Delta T = \mathfrak{a}\_\mathfrak{e} \left( T - T\_{ref} \right) \tag{10}
$$

where *α<sup>e</sup>* is the coefficient of thermal expansion, *T* is the instantaneous temperature, and *Tref* is the reference temperature with respect to time at *t* = 0. The deviator stresses according to the Prandtl–Reuss equation of plasticity can be represented as follows:

$$\frac{d\varepsilon\_x^p}{\sigma'\_x} = \frac{d\varepsilon\_y^p}{\sigma'\_y} = \frac{d\varepsilon\_z^p}{\sigma'\_z} = \frac{d\gamma\_{xy}^p}{\tau\_{xy}} = \frac{d\gamma\_{yz}^p}{\tau\_{yz}} = \frac{d\gamma\_{zx}^p}{\tau\_{zx}} = d\lambda \tag{11}$$

$$
\sigma'\_{\text{x}} = \sigma\_{\text{x}} - \sigma\_{\text{m}} \qquad \sigma'\_{\text{y}} = \sigma\_{\text{y}} - \sigma\_{\text{m}} \qquad \quad \quad \sigma'\_{\text{z}} = \sigma\_{\text{z}} - \sigma\_{\text{m}} \tag{12}
$$

where *σ* 0 *<sup>x</sup>*, *σ* 0 *<sup>y</sup>*, and *σ* 0 *<sup>z</sup>* are the deviator stresses in the *x*, *y,* and *z* directions, respectively; *dλ* is the instant positive constant of proportionality. Σ*<sup>m</sup>* refers to the mean stress, which can be evaluated as follows:

$$
\sigma\_m = \frac{\sigma\_x + \sigma\_y + \sigma\_z}{3} \tag{13}
$$

Substituting the values of *ε p <sup>x</sup>*, *ε p <sup>y</sup>*, *ε p z ,* and *ε t* in Equations (6)–(9), the resultant equations can be stated as below in Equations (14)–(17):

$$
\varepsilon\_{\mathbf{x}} = \frac{1}{E} \left[ \sigma\_{\mathbf{x}} - v (\sigma\_{\mathbf{y}} + \sigma\_{\mathbf{z}}) \right] + \int \sigma'\_{\mathbf{x}} d\lambda + \mathfrak{a}\_{\mathbf{t}} \Delta T \tag{14}
$$

$$\varepsilon\_{y} = \frac{1}{E} \left[ \sigma\_{y} - v(\sigma\_{x} + \sigma\_{z}) \right] + \int \sigma'\_{y} d\lambda + \mathfrak{a}\_{\ell} \Delta T \tag{15}$$

$$\varepsilon\_{z} = \frac{1}{E} \left[ \sigma\_{zy} - v \left( \sigma\_{x} + \sigma\_{y} \right) \right] + \int \sigma'\_{z} d\lambda + a\_{\varepsilon} \Delta T \tag{16}$$

$$\gamma\_{xy} = \frac{\tau\_{xy}}{2\mathcal{G}} + \int \ \tau\_{xy} d\lambda \ \ \gamma\_{xz} = \frac{\tau\_{xz}}{2\mathcal{G}} + \int \ \tau\_{xz} d\lambda \ \ \gamma\_{yz} = \frac{\tau\_{yz}}{2\mathcal{G}} + \int \ \tau\_{yz} d\lambda \tag{17}$$

Finally, the Von Mises stress *σ<sup>m</sup>* is computed as follows:

$$
\sigma\_{\mathfrak{m}} = \sqrt{\frac{1}{2} \left[ \left( \sigma\_{\mathfrak{x}} - \sigma\_{\mathfrak{y}} \right)^2 + \left( \sigma\_{\mathfrak{y}} - \sigma\_{\mathfrak{z}} \right)^2 + \left( \sigma\_{\mathfrak{z}} - \sigma\_{\mathfrak{x}} \right)^2 + 6 \left( \tau\_{\mathfrak{x}\mathfrak{y}}^2 + \tau\_{\mathfrak{y}\mathfrak{z}}^2 + \tau\_{\mathfrak{z}\mathfrak{x}}^2 \right) \right]} \tag{18}
$$

where *σx*, *σy*, and *σ<sup>z</sup>* are the *x*, *y,* and *z* components of stress, respectively.

## *2.1. Material*

Temperature-dependent physical and thermal properties of Inconel 718 (IN718) including density *ρ*, specific heat *Cp*, and thermal conductivity *k* are used as input properties to perform the transient thermal analysis. For the transient mechanical analysis, the

temperature-dependent elastic modulus *E*, yield strength *σy*, Poisson's ratio *ν*, plastic tangent modulus *ET,* and coefficient of thermal expansion *α* are used.

The stress–strain curves of IN718 at various temperatures are shown Figure 2a, pertaining to stress normalized with the yield strength (*σy*0) at room temperature (*T*0). The evolution of the temperature-dependent thermal, physical, and mechanical properties is shown in Figure 2b,c, respectively, where the properties are normalized with their corresponding value at room temperature, and the temperature is normalized with the melting temperature value of IN718 (e.g., *T<sup>m</sup>* = 1260 ◦C). The corresponding properties at room temperature for IN718 are provided in Table 1.

**Figure 2.** Normalized temperature-dependent properties for IN718: (**a**) stress–strain curves, (**b**) mechanical properties, and (**c**) thermal and physical properties.


**Table 1.** IN718 physical, mechanical, and thermal material properties at room temperature (*T*<sup>0</sup> ) [33].

#### *2.2. Design of Triply Periodic Minimum Surfaces (TPMS)*

In TPMS structures, the mean curvature of a minimal surface is zero at every point and periodic in the three perpendicular directions. Table 2 provides the level-set approximations for the TPMS surfaces used in this work (P-S for Schwartz Primitive, G-S for Schoen Gyroid, and IWP for Schoen I-wrapped package) for generating the lattices in terms of local Cartesian coordinates, *x*, *y*, and *z*, and a specified level-set constant, *c.* A detailed design process for creating TPMS topologies was covered in the work by Al-Ketan and Abu Al-Rub [4]. Furthermore, Al-Ketan and Abu Al-Rub [34] developed a free software called MSLattice for generating TPMS-based lattices of either sheet-network type or ligamentnetwork type. In sheet-network TPMS lattices, the level-set parameter *c* is set to zero such that the minimal surface splits the 3D space into two domains of equal volumes, and the relative density is varied through the thickening of the minimal surface, whereas, in ligament-network TPMS lattices, the level-set parameter *c* is used to control the proportion volumes of the two domains split by the minimal surface such that the smaller volume is solidified (e.g., for *c* = 0, the relative density is 50%). For this study, TPMS lattice structures shown in Figure 3 are investigated, where P-S, G-S, and IWP-S are sheet-based TPMS structures, while IWP-L is a ligament-based TPMS lattice. Three relative densities (*ρ* = *ρ*/*ρs*) of values 10%, 20%, and 30% of the mentioned TPMS structure are considered for this study, where *ρ* is the apparent density of the lattice structure and *ρ<sup>s</sup>* is the density of its base solid material. The 3 × 3 × 3 cell configuration is selected for this study based on the yield stress convergence studies performed using different cell configurations (e.g., 3 × 3 × 3, 4 × 4 × 4, 5 × 5 × 5, and 6 × 6 × 6) considered for the P-S (*ρ* = 20%) lattice structure, with the method of evaluating yield strength as described in Section 2.4. As shown in Figure 4, the normalized yield stress value of different cell configurations is compared with that calculated from the equation by Lee et al. [28], which uses a unit cell P-S lattice structure with periodic boundary conditions and assumes the base material to be an isotropic solid with elastic modulus *E<sup>s</sup> =* 200 GPa, yield strength *σ<sup>s</sup>* = 400 MPa, and Poisson's ratio *ν<sup>s</sup>* = 0.3. Although it is observed that a higher cell configuration of 6 × 6 × 6 gives a value closer to matching the results by Lee et al. [28], the time required to simulate such an AM build configuration will be very large and with only a marginal gain in accuracy. Hence, to reduce the overall computational time, the 3 × 3 × 3 cell configuration is used for simulating residual stresses of the TPMS structures as well as their effective mechanical properties, which is within a 2% agreement with the results in [28]. The TPMS lattices are designed using the MSLattice, Al-Ketan and Abu Al-Rub (Abu Dhabi, UAE) simulation tool [34].

**Table 2.** Level-set equation for P-S, G-S, and IWP [4], where *x*, *y*, and *z* are Cartesian coordinates, and *c* is the level-set constant that controls the thickness of TPMS surfaces.


**Figure 3.** TPMS unit cell topologies pertaining to, e.g., relative density of *ρ* = 10, with unit cell dimensions of 10 mm × 10 mm × 10 mm.

**Figure 4.** Convergence study of normalized yield strength values for different cell configurations of P-S (*ρ* = 20%) compared with calculated value from Lee et al. [28].

#### *2.3. Residual Stress Simulation*

A nonlinear transient thermal analysis of the layer-by-layer buildup of the process is first performed to obtain the temperature histories during the melting and solidification stages. This is followed by a transient elastic–plastic mechanical analysis, where the transient temperature profile from the transient thermal analysis is used as input in the mechanical model. Ansys-WB [33] provides the multiphysics capability for performing such sequentially coupled thermomechanical analysis. The internal residual stress formation occurs during the cyclic heating and cooling due to the layer-by-layer nature of the SLM process and continually evolves during the process. The residual stress profile throughout the part is attained after the final cooling phase to room temperature (e.g., 22 ◦C). The baseplate removal option available in the simulation tool [33] is performed and the final residual stress state of the build is used as the initial stress profile for evaluating the effective mechanical properties of the TPMS samples.

The processing parameters used for performing the TPMS AM build simulation with IN718 material are as follows: the laser scan speed is 1200 mm/s, the powder layer thickness is 40 µm, and the preheat temperature of the IN718 base plate is set to 100 ◦C and switches to room temperature during the cooling phase after completion of the build, where build supports are not considered for this study. The simulation is performed on the four TPMS topologies considering three different relative densities, e.g., *ρ* = 10%, 20%, and 30%. The AM simulation tool performs a thermal analysis followed by a quasi-static elastic–plastic mechanical analysis. The thermal analysis records the temperature histories of elements activated layer-wise using the element birth and death technique until completion of the build and cooling to room temperature; this layer-wise activation is illustrated in Figure 5, e.g., the first element layer (highlighted in purple) is activated for deposition while the rest of the element layers above represent the material to be deposited and are to be activated in subsequent stages. The thermal histories are then used as input for mechanical analysis

to evaluate the stresses by layer-wise activation in a similar pattern as in the thermal analysis, and the stress state at the cooling of the build is calculated. As modeling every physical layer of deposition is computationally expensive and impractical, a simplified lumped-layer approach is used in which activation of each finite element layer accounts for the deposition of several layers of actual metal powder at once [33]. Incorporating this approach, the TPMS lattice is modeled with hexahedral elements (e.g., HEX8) with an element layer height of 0.4 mm as per the size range (i.e., 10~20 times the powder layer) [33]; thus, for this work, a total build height of 30 mm will have 75 element activation layers corresponding to 750 physical layers of actual metal powder, where the base plate (50 mm × 50 mm × 10 mm) uses a coarser mesh size of 5 mm. The meshed models for the four TPMS lattice structures are shown in Figure 6, pertaining to relative density *ρ* = 10%, with a total number of elements in the range of 250,000 to 600,000 for different relative densities of the TPMS lattice.

**Figure 5.** The illustration shows the activation of the first layer of elements (highlighted in purple color) for deposition; the rest of the layers above are in the deactivated state representing the material to be deposited.

**Figure 6.** 3 × 3 × 3 cell configuration FEA models used for AM buildup simulation on an IN718 baseplate: (**a**) P-S (**b**) G-S (**c**) IWP-S, and (**d**) IWP-L.

The various stages of build formation during the thermal analysis are consolidated and shown in Figure 7a–f pertaining to the P-S lattice structure (*ρ* = 10%) through the built stage, e.g., build height *zbh*. The thermal analysis terminates after the cooling phase when the part cools to room temperature (22 ◦C). Similarly, layer-by-layer simulation is performed during transient structural analysis using temperature histories, and the final vector principal stress on the nodes is grouped and exported to be used as an input as pre-stress to evaluate the mechanical properties during tensile and compression loading simulation.

**Figure 7.** Various stages of 3 × 3 × 3 AM buildup simulation for P-S (*ρ* =10%) for build height: (**a**) *zbh* = 5 mm, (**b**) *zbh* = 10 mm, (**c**) *zbh* = 15 mm, (**d**) *zbh* = 20 mm, (**e**) *zbh* = 25 mm, and (**f**) *zbh* = 30 mm at completion of build.

## *2.4. Mechanical Simulation*

The final stress state evaluated from the AM simulation is imported as the initial residual stresses field for investigating the mechanical properties of the corresponding TPMS lattices under tensile and compressive loading. The mechanical properties of the TPMS are determined considering uniaxial tensile and compression loading in the build direction (*z*-direction). The finite element model uses a tetrahedral element with a mesh size of 0.25 mm from mesh convergence studies performed for estimating the yield strength of the configuration of the P-S (*ρ* = 20%) lattice structure, and a bilinear elastic-plastic material model with linear plastic hardening is used for all mechanical simulations. Figure 8 shows the representation of the tensile and compression loading case and symmetric boundary conditions in the *z*-direction applied on the base plane for a P-S lattice structure. In the mechanical simulation, all six components of the stress tensor are mapped onto the nodes of the FE model as initial conditions. A displacement load in the *z*-direction is applied to the top surface of the lattice, with a magnitude of ∆ = ±3 mm in tension or compression, respectively, which corresponds to a total normal average strain of 10%. This is applied

via a remote reference point coupled to the top built surface [33]. The reaction force at the remote node is monitored to obtain the uniaxial stress–strain response during the loading of the TPMS lattices, where the uniaxial stress is defined as the ratio between the reaction force and the area of the TPMS lattice (30 mm × 30 mm). The yield strength is determined from the stress–strain response through the 0.2% offset strain method, whereas the elastic modulus in tension and compression is evaluated from the slope of the stress–strain curves in the elastic region. The uniaxial tensile and compression loading simulations are also performed incorporating the effect of the full-field residual stress profile imported from the AM simulation (c.f., Figure 10) and the results are compared with the case without residual stresses as the initial condition for all the TPMS lattices and relative densities.

**Figure 8.** FE model of P-S pertaining to *ρ* = 10% with uniaxial loading and symmetric BC.

#### **3. Results and Discussion**

To verify the modeling framework proposed herein and the residual stress results, the current simulation approach of layer-wise activation described in Section 2.3 is compared with a case of experimentally measured residual stress by An et al. [35] on a curved thin-walled plate, for which the details are presented in Appendix A.

For the TPMS lattice structures and for their three relative densities, the evolution of stresses for the total build height of the samples is shown in Figure 9. The stress is calculated by determining the reaction force of the constrained base nodes in the build direction (*z*-direction) over its projected base area (30 mm × 30 mm). This provides a measure of the level of the stress in the build direction, which, as can be seen in Figure 9, is compressive in nature during the build-up stage, with an increase in magnitude as the relative density increases. The stress level during the build-up stage is observed to be higher in G-S and IWP-S lattice structures as compared to P-S and IWP-L for the same densities. In Figure 10, which shows the Von Mises residual stress contours for all TPMS lattices at relative density *ρ* =10%, the color contours are unified and used within the same limits for comparison purposes. As evident, the residual stress profiles differ for each TPMS, and hence, its effect on the effective mechanical properties will vary for each topology. It is also interesting to note that the residual stresses are nonuniformly distributed through the printing *z*-direction. The predicted stress values are higher in magnitudes than the initial yield stress of the material (e.g., 648 MPa). This is expected as the finite element model used in the current study does not consider the convective heat transfers in the molten pool; hence, computed peak temperatures and temperature gradients are slightly overestimated and consequently higher residual stress values are determined [36].

**Figure 9.** Stress evolution in build direction during the build height (30 mm) of TPMS for various relative densities: (**a**) PS, (**b**) G-S, (**c**) IWP-S, and (**d**) IWP-L.

**Figure 10.** Von Mises residual stress field in the TPMS lattices build of *ρ* = 10%: (**a**) P-S, (**b**) G-S, (**c**) IWP-S, and (**d**) IWP-L.

#### *3.1. Stress–Strain Curves*

The uniaxial tensile and compression stress–strain curves for all four TPMS lattices and the three relative densities considered are shown in Figures 11–13, both for the cases with and without the incorporation of residual stress effects from the AM process. As can be seen for P-S in Figures 11a, 12a and 13a, the effect of the residual stress on both stiffness and onset of yielding is notable, especially under compression loading. This is attributed to the high level of residual stresses present in the P-S topology, as shown in Figure 10a, as compared to the G-S and IWP-S in Figure 10b,c, respectively. A similar trend in the influence of the residual stresses on the mechanical response can be observed in Figures 12d and 13d for IWP-L, which is also attributed to the level of residual stresses associated with the IWP-L topology, as shown in Figure 10d. For all the sheet-based TPMS topologies, e.g., P-S, G-S, and IWP-S, subjected to compression loading, e.g., Figure 11a–c, softening is observed beyond the point of onset of yielding, especially for *ρ* =10%, which is associated with localized deformation and buckling of the TPMS walls. This trend diminishes as *ρ* increases; however, the difference in mechanical response between the compression and tension loading persists for all *ρ*. At large deformations, e.g., strain values larger than 2%, it is observed that the mechanical response of all the TPMS topologies and for all relative densities is not affected by the presence of residual stress, neither for compression nor tension loading.

**Figure 11.** The stress–strain curve for tension/compression loading for *ρ* = 10% with and without residual stress: (**a**) P-S, (**b**) G-S, (**c**) IWP-S, and (**d**) IWP-L.

**Figure 12.** *Cont*.

**Figure 12.** The stress–strain curve for tension/compression loading for *ρ* = 20% with and without residual stress: (**a**) P-S, (**b**) G-S, (**c**) IWP-S, and (**d**) IWP-L.

**Figure 13.** The stress–strain curve for tension/compression loading for *ρ* = 30% with and without residual stress: (**a**) P-S, (**b**) G-S, (**c**) IWP-S, and (**d**) IWP-L.

#### *3.2. Effective Elastic Modulus and Yield Strength*

The elastic modulus and yield strength of the different TPMS lattices for the three different densities considered are extracted from the stress–strain curves, for both tension and compression loading, with and without the residual stress effects. The Gibson–Ashby power law is used to correlate Young's modulus *E* and yield strength *σ* to relative densities *ρ* of TPMS structures and is given by the following equations [37];

$$E/E\_0 = \mathcal{C}\_1 \left(\overline{\rho}\right)^n \tag{19}$$

$$
\sigma/\sigma\_0 = \mathbb{C}\_2 \left(\overline{\rho}\right)^m \tag{20}
$$

where *E*<sup>0</sup> and *σ*<sup>0</sup> correspond to the Young's modulus and yield strength of the solid base material at room temperature (e.g., see Table 1), while *E* and *σ* are the elastic modulus and yield strength of the lattice at a relative density *ρ*, *C*<sup>1</sup> and *C*<sup>2</sup> are the scaling coefficients, and *n* and *m* are the scaling exponents of the fitting curves.

The normalized Young's modulus (*E/E*0) and normalized yield strength (*σ/σ*0) values for different relative densities are shown in Figures 14 and 15, respectively. The elastic modulus and yield strength of the lattice structures increase with the increase in relative densities for all TPMS structures, as expected. Considering the effect of residual stresses, e.g., Figure 14, the TPMS accounting for residual stresses shows a lower elastic modulus in the build *z*-direction compared with the reference TPMS without residual stresses. The effect of residual stress is most prominent on the sheet-based TPMS P-S in Figure 14a and the ligament-based IWP-L in Figure 14d. There is also a notable difference in stiffness under compression and tension loading. However, as seen in Figure 15, there is an insignificant effect of the residual stress on the yield strength in the build *z*-direction. Figures 16 and 17 show the amount of reduction (%) in Young's modulus due to the effect of residual stress in tensile and compressive loading, respectively. In tensile loading, the amount of reduction in elastic modulus due to residual stress is observed to be more prominent (>25%) in IWP-L for all relative densities, and for P-S structures, a reduction of more than 15% is observed at higher relative density, and the percentage reduction in elasticity was observed to be least in the case of IWP-S. For the compressive loading case, the reduction in elastic modulus is less prominent for IWP-L compared to in tensile loading and a very nominal reduction in elastic modulus is found in the case of G-S and IWP-S lattices due to residual stress. However, the P-S structure showed almost the same amount of reduction in elastic modulus compared to the tensile loading case. The power law coefficient and exponents of the scaling laws in Equations (19) and (20) for elastic modulus and yield strength for the four TPMS topologies are given in Table 3. The effect of the residual stress can be realized from the values of the fitting constant and exponent *C*<sup>1</sup> and *n*, respectively, on the elastic modulus, and similarly for the effect of indicating distinctly different values, especially for the IWP-L topology. The nature of elastic deformation behaviors of the TPMS structures under tensile/compression loading can be best explained using the exponent value *n* of the power law in Equation (19) [38]. From the *n* exponent values of the respective TPMS in Table 3, in uniaxial tensile and compression loading, the sheet-based P-S, G-S, and IWP-S structures show a mixed mode of deformation (e.g., 1 < *n* < 2), although stretching may be more pronounced, while the ligament-based IWP-L lattice shows a predominantly bending behavior (*n* ≥ 2), indicating that deformation behavior is strongly related to the topology of the TPMS. These observations agree well with the behavior reported in the literature on TPMSs [26,39]. In addition, it can be observed that due to predominantly bending behavior, the IWP-L shows less mechanical stiffness and strength compared to the sheet-based TPMS of the same relative densities as indicated by the respective values of magnitude in Figures 14 and 15.


**Table 3.** Scaling law parameters, normalized young's modulus, and yield strength.

**Figure 14.** Effect of residual stresses on the normalized Young's modulus (*E/E*<sup>0</sup> ) with respect to relative density *ρ*: (**a**) P-S, (**b**) G-S, (**c**) IWP-S, and (**d**) IWP-L.

**Figure 15.** Effect of residual stresses on the normalized yield strength (*σ/σ*<sup>0</sup> ) with respect to relative density *ρ*: (**a**) P-S, (**b**) G-S, (**c**) IWP-S, and (**d**) IWP-L.

**Figure 16.** The reduction in Young's modulus in percentages for tensile loading for all TPMSs and their relative densities.

**Figure 17.** The reduction in Young's modulus in percentages for compressive loading for all TPMSs and their relative densities.

#### *3.3. SEA Simulation*

The specific energy absorption (SEA) value is defined as the energy absorbed by a material under uniaxial compression, normalized by its density, and involves large deformation up to the point of densification of the lattice structure. The objective here is to investigate the effect of the residual stress on the SEA capacity. The SEA is expressed as

$$\text{SEA} = \frac{1}{\overline{\rho}} \int\_0^{\varepsilon\_D} \sigma \, d\varepsilon \tag{21}$$

where *σ* is the compressive stress, *ε* is the compressive strain, and *ε<sup>D</sup>* is the densification strain. The full-scale FEA models of the two TPMS configurations shown in Figure 18 are subjected to compressive loads to investigate the SEA value with and without accounting for residual stresses for different relative densities. The lower rigid plate is fixed, whereas the upper rigid plate is given a prescribed total displacement of ∆ = 15 mm, corresponding to a total compressive strain of 50% in the build *z*-direction. A general contact interaction is defined between the upper and lower rigid plates, and the TPMS lattice with a frictional coefficient of 0.1 and the self-contact between surfaces of the TPMS are assigned a friction coefficient of 0.2.

**Figure 18.** Equivalent plastic strain (PEEQ) contour plots at 50% deformation P-S with: (**a**) *ρ* = 10% and (**b**) *ρ* = 20%.

Figure 18 shows the equivalent plastic strain (PEEQ) for the P-S lattice for *ρ* = 10% and 20%. The reaction force on the upper rigid plate is obtained from which the effective compressive stress over the lattice is determined for both the cases with and without residual stresses accounted for. As shown in Figure 19, there is a change in elastic modulus as observed in previous sections due to residual stress; however, beyond the yield region, the stress–strain response reveals that the effect of the residual stresses is insignificant on the SEA capacity of the lattice structure, and is found to be same as compared to the reference TPMS, e.g., without residual stresses.

**Figure 19.** Stress–strain compression response for P-S lattice of *ρ* = 10% and 20%, with and without the effect of residuals tresses.

#### **4. Conclusions**

The effective mechanical properties such as elastic modulus, yield strength, and SEA of four TPMS structures with varying relative densities (*ρ* = 10%, 20% and 30%) are investigated considering the influence of residual stress evaluated from the SLM process through sequentially coupled thermomechanical simulations. When incorporating the residual stresses arising from the SLM process, the effective elastic modulus of all TPMS lattices in the build direction is lower in comparison with the reference TPMS in which the residual stresses are not accounted for. The decrease in stiffness is more prominent in sheet-based P-S and ligament-based IWP-L due to the higher accumulation of residual stress, and a stiffness reduction of more than 25% is observed in IWP-L. In addition, the fitting constant and exponent *C*<sup>1</sup> and *n*, respectively, in the Gibson–Ashby power law show distinctly different values, especially for the IWP-L topology, indicating the influence of residual stress on the elastic modulus of the TPMS. The sheet-based TPMS (P-S, G-S, and IWP-S) with a relative density of 10% show softening behavior beyond the yield region in compression loading, while for higher densities of all TPMSs, the tensile and compression behaviors show consistent hardening. The influence of residual stresses on yield strength and SEA is insignificant in the build direction for all TPMSs, indicating that residual stresses have little effect beyond the yield region in both compression and tension loading.

The findings with regard to the variation in stiffness due to residual stress inherited from the SLM process are significant as stiffness is known to considerably affect the vibration and fatigue properties of parts and hence needs to be accounted for in any design process. Further investigations for the anisotropic behavior of the TPMS structures considering the loading in other directions, the effect of various scanning strategies and post-heat treatment procedures on residual stress mitigations in TPMS lattices, and its influence on mechanical properties may be undertaken.

**Author Contributions:** Conceptualization, I.B. and R.K.A.A.-R.; methodology, N.A., I.B. and R.K.A.A.-R.; software, N.A.; validation, N.A.; formal analysis, N.A.; investigation, N.A.; resources, I.B. and R.K.A.A.-R.; data curation, N.A.; writing—original draft preparation, N.A.; writing—review and editing, I.B. and R.K.A.A.-R.; visualization, N.A.; supervision, I.B. and R.K.A.A.-R.; project administration, I.B. and R.K.A.A.-R.; funding acquisition, I.B. and R.K.A.A.-R. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors acknowledge the financial support provided by Khalifa University under Award No. RCII-2019-003.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Verification of Residual Stresses**

For verification of residual stress values obtained through the simulations in the current study, experimentally measured values from the published literature [35] are used. The referenced work uses the neutron diffraction method to measure the inherited residual stress in a curved thin-walled plate fabricated through a metallic LPBF process. Figure A1a shows the geometric dimensions of the build made of Inconel 625 (IN625) material grade deposited on a stainless-steel base plate. The example case selected for comparison is deposited in a vertical orientation with a powder layer thickness of 30 µm. The simulation model uses an element layer height of 0.3 mm as per the recommended size range mentioned in Section 2.3, and a coarse mesh size of 2 mm is used for the baseplate. The location of the residual stress measurement path (e.g., T1) is at mid-height of the build indicated by the red dotted line shown in Figure A1a. Figure A1b depicts the comparison of stress component *σ<sup>z</sup>* in the build direction along path T1, with markedly good agreement between the simulation results and the experimental measurements.

**Figure A1.** Sample used for comparison with current simulation study: (**a**) geometric dimension of the curve plate and stress measurement location (T1) indicated by red dotted line; (**b**) comparison of simulated residual stress component *σz* with experimental measurements from literature [35].

## **References**

