Residual Stress Analysis of TC4/Inconel718 Functionally Graded Material Produced by Laser Additive Manufacturing Based on Progressive Activation Element Method

Residual Stress Analysis of TC4/Inconel718 Functionally Graded Material Produced by Laser Additive Manufacturing Based on Progressive Activation Element Method | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Residual Stress Analysis of TC4/Inconel718 Functionally Graded Material Produced by Laser Additive Manufacturing Based on Progressive Activation Element Method Hongjian Zhao, Chi Gao, Zihao Wang, Quanyi Wang, Changsheng Liu, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2497853/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 30 Sep, 2023 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted 5 You are reading this latest preprint version Abstract With the continuous development of preparation technology, laser additive manufacturing (LAM) has become one of the effective ways to manufacture functionally graded materials due to its unique layer-by-layer stacking technology. However, the repeated and repeated rapid heating and cooling processes in the manufacturing process will generate large residual stress inside the structure, resulting in the destruction of the structure. In this paper, based on a new finite element method called progressive activation element method (PAE), a thermomechanical coupling model for simulating the process of LAM is established, and the influence of laser power and composition ratio of transition layers on the residual stress of the overall structure is discussed. The results show that there is a positive correlation between the laser power and the residual stress. The PAE method is compared with the traditional “Model Change” method, and it is found that the PAE method has advantages in computational efficiency, especially when calculating the residual stress of functionally graded materials, the efficiency can be improved by about 1650%. When the TC4/Inconel718 functionally graded material is prepared experimentally, the optimal composition ratio of the transition layers is 8:2. This paper provides reference for the understanding and reasonable suppression of residual stress of functionally graded materials in LAM. laser additive manufacturing functionally graded material residual stress progressive activation element Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1 Introduction Due to the rapid development of aerospace industry, the material selection of aerospace rockets and airplanes has become the focus of scholars[ 1 – 5 ]. Functionally graded materials (FGM) were first proposed by Japanese scholars M. Niino and T. Hrai[ 6 ]. They could combine two or more materials with different properties through advanced manufacturing processes, so that the internal materials of the structure change in a gradient and there is no obvious layered interface inside. It is the existence of this characteristic that makes the FGM composed of partial materials have high specific strength, good oxidation resistance and low density. For example, TC4/Inconel718 superalloys are often used in aircraft engines that need to work in extreme environments. In recent years, the rapid development of laser additive manufacturing (LAM) provides a new technical support for the fabrication of FGMs. LAM technology, with its unique manufacturing advantages of layer-by-layer stacking, has shown unparalleled huge advantages in manufacturing FGMs with special geometric shapes, expensive processing costs, and difficult to prepare by traditional methods. The existence of residual stress is a factor that must be considered regardless of the preparation of FGM by any method. In the process of LAM, due to the high temperature required to melt the metal, a significant thermal gradient will be generated around the molten pool[ 7 ]. Residual stresses are mainly caused by plasticity resulting from shrinkage of the welded area or solid-state phase transformations present in the material[ 8 ]. Another source of residual stress is the difference in thermal expansion coefficients between different phases of the same material. Residual stresses can also be the result of chemical composition gradients within workpieces of the same material[ 9 ]. High residual stress can cause severe damage to the structures[ 10 ], such as deformation[ 11 ], loss of geometric tolerance[ 12 , 13 ], delamination during deposition, poor fracture resistance[ 14 ], etc. However, FGMs possess a systematically varied microstructure and/or composition resulting in gradual changes in their mechanical, physical and/or chemical properties across the geometry[ 15 ], resulting in the thermal and mechanical problems are very complicated. Therefore, it is difficult to detect the residual stress produced by LAM experimentally. S.G. Jeong et al[ 16 ] used the direct energy deposition method to prepare the samples, and used the neutron diffraction method to detect the residual stress generated during the preparation process, which was in good agreement with the finite element method simulation results. Their study has demonstrated that using a substrate with a softer material reduces residual stress in additively manufactured structures. E.R. Denlinger et al[ 17 ] conducted experimental observations on the laser deposition construction process of Inconel625 alloys and TC4 alloys, and studied the effect of interlayer residence time on the structural residual stress, and found that different residence times had a significant effect on the accumulation of deformation, and shorter residence times can significantly lower residual stress. J.V. Gordon et al[ 18 ] studied the fatigue crack growth phenomenon of 304L stainless steel produced by wire and arc additive manufacturing and correlated it with residual stress and microstructure. It was found that the retained compressive residual stress had a positive effect on the structure. Jing et al[ 19 ] presented a new method for controlling residual stress in laser directed energy deposition additive manufacturing products, and results showed that weaker constraints on the substrate can greatly reduce residual stress in additively manufactured products. A.S. Wu et al[ 20 ] used experimental methods (digital image correlation combined with build plate removal and sectioning combined with neutron diffraction) to investigate the effects of laser scanning mode, power, scanning speed and build direction on residual stress in powder bed fusion additive manufacturing. A parametric study was carried out to gain a better understanding of the factors affecting the macroscopic residual stress. Lu et al[ 21 ] studied the yield temperature related to thermal deformation and mechanical restraint, discussed the effect of such factors on controlling residual stress, and believed that increasing the yield temperature had a positive effect on controlling the accumulation of residual stress. The scholars mentioned above have done a lot of excellent works, but most of them focus on using experimental methods to detect and suppress residual stress. There are relatively few literatures on residual stress analysis of LAMed structures, and most use traditional “Model Change” method. In this paper, based on the relatively novel progressive activation element (PAE) method, the main factors of the additive manufacturing process, the laser power and the material composition ratio of the transition layers, are simulated and analyzed, and the simulation results are compared with the experimental results to verify the correctness of the simulation results. This paper could provide useful guidance for understanding, controlling and rationally suppressing residual stress in experiments. 2 Basic Theory 2.1 Characterization of physical parameters of transition layers materials Each transition layer is mixed with TC4 alloy and Inconel718 high-temperature alloy in different proportions. The mechanical properties of the heterogeneous material of the transition layers could be deduced with composite mechanics. Knowing the density and volume fraction of each component, the density of the mixed material is $$\rho ={\rho _1}{\varphi _1}+{\rho _2}{\varphi _2}$$ 10 where ρ 1 and ρ 2 are the density of each component, and φ 1 and φ 2 are the volume fraction of each component. The Halpain-Tsai model is usually used to calculate the Young’s modulus of heterogeneous materials. Following is the calculation formula: $$E=\frac{{{\varphi _1}{E_1}\left( {G+{E_2}} \right)+{\varphi _2}{E_2}\left( {G+{E_1}} \right)}}{{{\varphi _1}\left( {G+{E_2}} \right)+{\varphi _2}\left( {G+{E_1}} \right)}}$$ 11 where E 1 and E 2 are the Young’s modulus of each component; G is the interface shear modulus of the material, which is obtained by Voigt formula: $$G={\mu _1}\frac{{{E_1}}}{{2\left( {1+{\mu _1}} \right)}}+{\mu _{\text{2}}}\frac{{{E_2}}}{{2\left( {1+{\mu _2}} \right)}}$$ 12 where µ 1 and µ 2 are the Poisson’s ratios of the two materials. The following is the formula for calculating the yield strength σ S of the heterogeneous material: $${\sigma _{\text{S}}}={\varphi _1}{\sigma _{{\text{S1}}}}+{\varphi _2}{\sigma _{{\text{S2}}}}$$ 13 where σ S1 and σ S2 are the yield strength of the two materials respectively. The Lichtenecker empirical formula is widely used by many researchers to calculate the thermal conductivity λ of the heterogeneous material. The following is the Lichtenecker empirical formula: $$\lg \lambda ={\varphi _1}\lg {\lambda _1}+{\varphi _2}\lg {\lambda _2}$$ 14 where λ 1 and λ 2 are the thermal conductivity of the two materials. The empirical formula is put forward by Turner, which is for the thermal expansion coefficient of composite materials as follows: The Neumann-Kopp rule gives the calculation formula of specific heat capacity as follows: $$c={W_1}{c_1}+{W_2}{c_2}$$ 16 Where c 1 and c 2 are the specific heat capacities of the two materials; W 1 and W 2 are the mass fractions of the two materials. The physical parameters of TC4 and Inconel718 are shown in Table 1 and Table 2 [ 22 , 23 ]. Table 1 The physical parameters of TC4 Temp Density Young's Modulus Poisson's Ratio Specific Heat Capacity Thermal Conductivity Thermal expansion coefficient ℃ kg/m3 GPa J/(kg·K) W/(m·K) 1/K 75 4420.73 116.05 0.32 549.67 5.14 85 4420.14 115.79 0.32 551.69 5.27 8.90E-06 100 4418.95 115.27 0.32 555.57 5.52 8.93E-06 200 4412.91 112.66 0.32 572.58 6.71 9.08E-06 300 4405.42 109.50 0.32 589.62 7.99 9.26E-06 500 4391.03 103.68 0.33 616.11 10.09 9.59E-06 800 4367.18 94.57 0.33 652.10 13.03 1.01E-05 1000 4350.76 88.60 0.34 675.07 14.83 1.04E-05 1200 4333.59 82.57 0.34 699.68 16.59 1.07E-05 1500 4309.28 74.27 0.35 872.09 19.52 1.08E-05 Table 2 The physical parameters of Inconel718 Temp Density Young's Modulus Poisson's Ratio Specific Heat Capacity Thermal Conductivity Thermal expansion coefficient ℃ kg/m3 GPa J/(kg·K) W/(m·K) 1/K 75 8266.85 207.23 0.31 425.09 11.57 85 8265.23 206.94 0.31 426.34 11.65 1.31E-05 100 8261.97 206.37 0.31 428.76 11.80 1.31E-05 200 8245.39 203.45 0.31 439.60 12.56 1.34E-05 300 8224.93 199.88 0.31 450.90 13.47 1.36E-05 500 8185.88 193.09 0.32 469.54 15.10 1.40E-05 800 8121.74 182.00 0.32 497.06 17.58 1.47E-05 1000 8077.92 174.45 0.33 515.23 19.18 1.51E-05 1200 8032.37 166.60 0.33 534.09 20.76 1.56E-05 1500 7943.32 150.52 0.34 708.30 23.29 1.71E-05 2.2 Heat source model In the process of LAM, a suitable heat source model is necessary to study the thermo-mechanical coupling models. Three common heat source models are used widely, including concentrated heat source model, plane heat source model and volumetric heat source model. The situation is that the laser power is low and the molten pool depth is small. The plane heat source is the best, including plane Gaussian heat source model and double elliptical heat source model. X.L. Luo et al [ 24 ] studied the influence of different heat source models by simulating the single-layer single-pass laser melting forming, concluded that the calculation result of the double-ellipsoid heat source model is better than the calculation result of the Gaussian heat source model. The expression of the double-ellipsoid heat source model is $$q\left( {x,y,z} \right)=\left\{ {\begin{array}{*{20}{c}} {\frac{{6\sqrt 3 AP}}{{\pi \sqrt \pi {a_r}bc}}\exp \left\{ { - 3\left[ {\left( {\frac{{{x^2}}}{{{a_r}^{2}}}} \right)+\left( {\frac{{{y^2}}}{{{b^2}}}} \right)+\left( {\frac{{{z^2}}}{{{c^2}}}} \right)} \right]} \right\},\left( {x \leqslant 0} \right)} \\ {\frac{{6\sqrt 3 AP}}{{\pi \sqrt \pi {a_f}bc}}\exp \left\{ { - 3\left[ {\left( {\frac{{{x^2}}}{{{a_f}^{2}}}} \right)+\left( {\frac{{{y^2}}}{{{b^2}}}} \right)+\left( {\frac{{{z^2}}}{{{c^2}}}} \right)} \right]} \right\},\left( {x \geqslant 0} \right)} \end{array}} \right.$$ 1 where a r and a f are the first half radius and the second half radius of the double-ellipsoid on the x-axis; b is the radius length on the y-axis of the double-ellipsoid; c is the laser penetration depth; A is the absorption rate of the metal powder to the laser; P is the laser power. 2.3 Heat conduction equation It is a nonlinear transient heat conduction problem and the temperature of the process of LAM changes rapidly and drastically. The governing equation is listed as below: $$c\rho \frac{{\partial T}}{{\partial t}}=\left[ {{k_x}\frac{\partial }{{\partial x}}\left( {\frac{{\partial T}}{{\partial x}}} \right)+{k_y}\frac{\partial }{{\partial y}}\left( {\frac{{\partial T}}{{\partial y}}} \right)+{k_z}\frac{\partial }{{\partial z}}\left( {\frac{{\partial T}}{{\partial z}}} \right)} \right]+q\left( {x,y,z} \right)$$ 2 Where k x , k y , and k z are the heat transfer coefficient in the x , y , and z directions respectively; c and ρ are the specific heat capacity and density of the material; q ( x , y , z ) is the internal heat source of the solution domain; T is the temperature; t is the time of heat transfer. Since the physical properties of materials change with time, formula (2) is extended to anisotropic gradient structure, and the heat conduction differential equation is $${\nabla ^\lambda }\left[ \mathbf{K} \right]\nabla T+q\left( {x,y,z} \right)=\frac{\partial }{{\partial t}}\left( {\rho cT} \right)$$ 3 Where \(\nabla\) is the gradient operator of space coordinates; the superscript λ represents transpose; [ K ] represents the matrix of heat transfer coefficient. $$\nabla =\left( {\frac{\partial }{{\partial x}},\frac{\partial }{{\partial y}},\frac{\partial }{{\partial z}}} \right)$$ 4 The initial condition and boundary condition are set to [22] : $$T\left( {x,y,z,0} \right)={T_0}$$ 5 $$k\frac{{\partial T}}{{\partial n}} - q+{h_c}\left( {T - {T_0}} \right)+{\sigma _b}{\varepsilon _b}\left( {{T^4} - T_{0}^{4}} \right)=0,\;\;\;\;\;\;\left( {x,y,z} \right) \in {S_n}$$ 6 Where T 0 is the ambient temperature; n is the size of a normal vector on the model surface; h c is the natural convection heat transfer coefficient; σ b is the Stefan-Boltzmann constant; ε b is the emissivity of the material; S n is the model surface. The stress-strain equation of the thermal-mechanical model is $$\varepsilon ={D^{ - 1}}\sigma +{\varepsilon ^{{\text{th}}}}{\text{+}}{\varepsilon ^p}$$ 7 where ε is the total strain vector; D is the elastic matrix; σ is the stress vector; ε th is the thermal strain vector, \({\varepsilon ^p}\) is the plastic strain. The thermal strain is calculated according to the temperature field distribution and the thermal expansion coefficient of the material: $${\varepsilon ^{{\text{th}}}}=\int_{{{T_{{\text{ref}}}}}}^{T} {\alpha \left( T \right){\text{d}}T}$$ 8 where T ref is the ambient temperature; α is the thermal expansion coefficient. According to the residual stress characteristics of LAM and the basic theory of plastic mechanics, the Mises yield criterion is adopted, and the equivalent stress calculation formula is $${\sigma _i}=\sqrt {\frac{1}{2}\left[ {{{\left( {{\sigma _1} - {\sigma _2}} \right)}^2}+{{\left( {{\sigma _2} - {\sigma _3}} \right)}^{\text{2}}}{\text{+}}{{\left( {{\sigma _{\text{3}}} - {\sigma _1}} \right)}^2}} \right]}$$ 9 3 Finite Element Models 3.1 Progressive activation element Different with “Model Change” method, progressive activation element does not need to set a large number of steps and interactions to realize the movement of heat source, which can save us a lot of energy and time. Python is used to implement the loading and movement of the double ellipsoid heat source, and realize the deposition process of the FGM powder. Multiple attempts were taken to achieve parameter settings that were nearly identical to the experimental procedure, such as the laser power, scanning speed, etc. Finally, calculate the compiled .inp file. 3.2 Models building In order to save computation time and improve efficiency, axisymmetric models are adopted. The whole model is shown in Fig. 1 , the lower part of the model is the substrate, which is made of Inconel718 alloy, and the upper part of the model is the laser cladding layers, which are made of TC4/Inconel718 FGM. The material of the cladding layers is changed every two layers, the bottom is made by Inconel718 alloy, and the top is made by TC4 alloy. The material composition of each layer is shown in Table 3 . The stress changes and distribution of the cladding layers should be concerned. So, the equal and dense meshes are set there, and sparse and large meshes are set with the substrate plate. After many attempts, considering both calculation time and result accuracy, the mesh size of the cladding layers should be set to \(0.3{\text{3}} \times 0.33 \times 0.25\) mm, and the size of each cladding layer in the model is set to \({\text{60}} \times {\text{1}} \times {\text{0}}{\text{.5}}\) mm, the size of the substrate plate is set to \({\text{70}} \times {\text{10}} \times {\text{5}}\) mm. At this size, each cladding layer has 1080 meshes, and the entire model has a total of 58464 meshes. Table 3 Powder mixing ratio from layer 1 to layer 10 Bottom-up layer number Powder mixing ratio 1–2 100%Inconel718 3–4 80%Inconel718,20%TC4 5–6 50%Inconel718,50%TC4 7–8 20%Inconel718,80%TC4 9–10 100%TC4 3.3 Models verification After processing of LAM, set 300s for the natural cooling, and the obtained stress distribution in the scanning direction is shown in Fig. 2 . The thermal and mechanical problems presented by FGMs are very complicated because the gradient layers are heterogeneous both microscopically and macroscopically. Because of the large temperature gradients and high-frequency heating-cooling cycles during the LAM process, it is difficult to monitor the temperature and thermal stress in real time during experiments[ 25 ]. In order to verify the correctness of this model, a TC4 alloy model of the same size is established with the same method, and the result graph calculated by the commonly used “Model Change” method is provided, showing the difference between the two methods. After that, the residual stress of the sample was detected by the experimental method, and the comparison chart between the stress value of the two methods and the experimental value was obtained. Laser ultrasonic technology is used to detect the residual stress of the structure during all the experiments. Laser ultrasonic has shown strong potential due to its advantages, such as non-contact, high-precision, non-destructive, strong anti-interference ability, high efficiency and portability. In previous research, a lot of work has been done on the measurement of residual stress with laser ultrasonic. The result of residual stress measurement of TC4 titanium alloy was published[ 26 ]. Almost the same parameters and experimental conditions as the experiment are adopted in this paper. The whole experiment procedure consists of two parts, preparation of the experiment samples with LAM, and measurement of the residual stress with laser ultrasonic technology. The Laser melting deposition system was used in the experiment (MetLas3.0, Liaoning Yutong laser application technology Engineering Co., Ltd). The powder materials used for deposition were TC4 alloy powder produced by AVIC Maite powder metallurgy technology (Beijing) Co., Ltd. The powder is spherical. The particle size is 140 ~ 200µm. The laser ultrasonic experiment system is developed and applied in residual stress measurement. The Nd:YAG laser setup (Dawa-100, Beamtech Optronics Ltd., Beijing, China) is used to generate the ultrasonic waves. A laser pulse is sent by the Nd: YAG laser setup, and focused by the cylindrical lens as a line source (0.6 mm width, 20 mm length) to generate the ultrasonic surface waves. The laser Doppler vibrometer (Sdptop LV-S01, Sunny Optical Technology Ltd., Suzhou, China) is used to detect the ultrasonic vibration information. The experimental setup is shown in Fig. 4 (a). The ultrasonic system was used to detect residual stress of TC4 sample. In order to improve the intensity of the detection signal received by laser Doppler vibrometer, mechanical polishing of the surface was set. It is assumed that mechanical polishing does not alter the stress state. The polishing process is carefully controlled to ensure that there is no excessive external force. Normally, the thickness of the polishing layer is less than one tenth of the surface wave incident depth. The porosity of the samples is measured by Archimedean method, and the average value is 0.16%. The research focuses on the measurement of macro residual stress, and the porosity of the samples is low, so the influence of porosity on the residual stress is neglected. As shown in Fig. 4 (a)(b), M 0 represents the measuring point and all the measuring points are lay out on the measuring line l 0 (blue line). N 1 and N 2 represent the positions of detection and excitation laser, respectively. l 1 , l 2 and l 3 represent the auxiliary lines (black line). It is noteworthy that the stress at the measuring point is actually the average value within the propagation distance of the surface wave. Obviously, the stress magnitudes obtained by the two methods are similar to the experimental results, and the result obtained by PAE method is more accurate than the one by Model Change method. Therefore, the correctness of these two methods have been verified (see Fig. 5 ). 4 Results In the process of LAM, drastic change in temperature is the main cause of the residual stress. It is important to observe the temperature changes currently to analyze the residual stress conveniently later. As the processing progresses, the temperature of the heat source of the LAM increases (see Fig. 6 ). Residual stress is our focus in simulating LAM. The power of the laser heat source is one of the most important reasons for the residual stress. Therefore, simulation models for different powers are established to verify the distribution law of residual stress under different machining powers, as shown in the Fig. 7 . Before that, we set up several reference points to detect the residual stress around. As mentioned above, two layers of each material were printed, and the middle common nodes of the two cladding layers was used as the reference points. It can be seen from the Fig. 7 that with the increase of laser power, the residual stress gradually increases, and the two are positively correlated. However, due to the influence of the material gradient in the stacking direction, the influence of different cladding layers is also different. The laser power received at the bottom layer of the structure (100% Inconel718) has a greater influence, while the upper layer of the structure, especially the layers 7 and 8(20% Inconel718 and 80% TC4), the influence of the laser power is smaller. The residual stress distribution of the overall structure presents an “inverted bowl shape”, the residual stress on both sides is small, and a large residual stress is generated in the middle part of the structure. In actual engineering, we should pay more attention to the middle part of the structure, and the fracture phenomenon of a large number of additively manufactured specimens occurs mostly in the middle of the structure. Compared with isotropic materials, the biggest feature of this paper is the material anisotropy of the structure in the stacking direction. Due to the differences in the physical and mechanical properties of materials, materials with different mixing ratios have different resistance to residual stress. In order to facilitate the experimental preparation of functionally graded materials, simulation models with different transition ratios are designed, which are divided into four types: 6:4, 7:3, 8:2 and 9:1. The LAM simulation model designed in this paper has a total of 10 cladding layers, of which the 1st and 2nd layers are 100% Inconel718 material, the 9th and 10th layers are 100% TC4 material, and the 5th and 6th layers are 50% TC4 and 50%Inconel718 material mixing, these three parts always remain unchanged, the 3rd, 4th, 7th and 8th layers are designed as transition layers, taking the transition ratio of 8:2 as an example, that is, the 3rd and 4th layers are 80%Inconel718 mixed with 20% TC4 material, the 7th and 8th layers are mixed with 20% Inconel718 and 80% TC4 material. As mentioned above, the residual stress mainly takes a large value in the middle part of the structure. In order to facilitate the detection of residual stress, a total of 15 reference points is averagely set on three lines, which are called "left line", “middle line” and “right line” (see Fig. 8 (d)). These three lines are taken at the quarter points of the scanning direction of the structure, that is, the two adjacent lines are separated by 45 meshes (1.5cm). Figure 8 (a), (b) and (c) show the residual stress distribution on these three lines. Due to the influence of the gradient, the variation of the residual stress does not have an obvious regularity like the laser power. In general, when the composition ratio of the transition layer is 6:4 and 9:1, the residual stress of the structure changes most obviously, which is obviously not conducive to the stability of the structure. When the composition ratio of the transition layer is 7:3 and 8:2, the change of residual stress is relatively moderate, and when the two are compared, the composition ratio of 8:2 has a smoother residual stress transition, which is not easy to cause the buckling of the structure. and breakage issues. Therefore, when using the LAM method to prepare the TC4/Inconel718 functionally graded material, it is more in line with the actual needs to use the transition layer ratio of 8:2. In the end, in order to reflect the difference between the PAE method and traditional “Model Change” method, the calculation time of the above models are listed, as shown in Table 4 . Compared with the “Model Change” method, the PAE method has relatively high computational efficiency and can save a lot of computational time. When calculating anisotropic materials, “Model Change” method will consume a lot of time for stress analysis due to the change of the material composition of the transition layers. This advantage of PAE will be more obvious when calculating gradient materials. When calculating isotropic materials (such as TC4 alloy), the calculation efficiency can be increased by 995%, and when calculating anisotropic materials, the higher the laser power, the more obvious the efficiency improvement of the PAE method; the calculation time of combining these models, the PAE method can probably improve by about 1650% than the “Model Change” method. It is worth stating that the CPU used in this paper is Intel® Core™ i5-10600KF CPU @ 4.10GHz, and the utilization rate is kept above 90%. The memories are two Essencore DDR4-3200 8GB and two Crucial Technology DDR4-2666 8GB. Table 4 Calculation time of above models Model Calculation time Efficiency PAE Model Change Isotropic Pure TC4 3h15mins 35h44mins 995% Anisotropy 400W 3h30mins 71h18mins 1937% 500W 4h15mins 92h36mins 2078% 600W 5h3mins 108h36mins 2050% 6:4 4h47mins 113h7mins 2265% 7:3 4h10mins 98h25mins 2262% 8:2 4h15mins 94h36mins 2126% 9:1 4h39mins 116h14mins 2400% 5 Conclusions When the TC4/Inconel718 functionally graded material is prepared by the laser additive manufacturing method, the size of the laser power will greatly affect the residual stress of the specimen after processing, and the two are positively correlated. The residual stress of the specimen is reasonably controlled to meet the requirements of industrial use. Due to the different composition of materials in the stacking direction, the residual stress exhibits irregular changes. When the excessive ratio of 6:4 and 9:1 is used, the interlayer change of residual stress is too large, which is easy to cause damage to the structure. TC4/Inconel718 functionally graded material was prepared with a composition ratio of 8:2 for the transition layers. Declarations Acknowledgement This study is supported by the National Natural Science Foundation of China Project (Grant No. 51771051), the Natural Science Foundation of Liaoning Province Project (Grant No. 2021-MS-102) and the Fundamental Research Funds for the Central Universities (Grant No. N2105021). Author Statement Hongjian Zhao(first author): Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing-Original Draft Chi Gao: Formal analysis, Writing-Original Draft Zihao Wang: Software, Formal analysis Quanyi Wang: Validation Changsheng Liu: Resources, Supervision, Funding acquisition Yu Zhan (Corresponding Author): Visualization, Resources, Writing-Review & Editing, Funding acquisition Conflict of Interest The work presented here was performed in collaboration among all authors. Hongjian Zhao designed, analyzed, and wrote the paper. Chi Gao, Zihao Wang, Quanyi Wang, Changsheng Liu and Yu Zhan provided and analyzed the experimental data. All authors contributed to and approved the manuscript. References Tascopglu E, Karabulut YY, Kaynak Y (2020) Correction to: Influence of heat treatment temperature on the microstructural, mechanical, and wear behavior of 316L stainless steel fabricated by laser powder bed additive manufacturing. Int J Adv Manuf Technol 107:1957. https://doi.org/10.1007/s00170-020-05115-1 Guo S, Chen M et al (2023) 3D printed hierarchically porous zero-valent copper for efficient pollutant degradation through peroxymonosulfate activation. Sep Purif Technol 305:122437 Tang HB, Huang HJ et al (2021) Multi-Scale modelling of structure-property relationship in additively manufactured metallic materials. Int J Mech Sci 194:106185 Hong X, Xiao G, Zhang Y et al (2021) Research on gradient additive manufacturing of ultra-large hot forging die based on automatic wire arc additive manufacturing technology. Int J Adv Manuf Technol 116:2243–2254. https://doi.org/10.1007/s00170-021-07424-5 Du D, Wang L et al (2022) Promoting the densification and grain refinement with assistance of static magnetic field in laser powder bed fusion. Int J Mach Tools Manuf 183:103965 Niino M, Hirai T, Watanabe R (1987) Functionally Gradient Materials - Toward Super Heat Resistant Materials for Spacecraft. J Compos Mater 13(6):257–264. doi.org/10.6089/jscm.13.257 Arcam U (2011) Manual, Arcam AB, Denlinger ER, Michaleris P (2016) Effect of stress relaxation on distortion in additive manufacturing process modeling. Addit Manuf 12:51–59. doi.org/10.1016/j.addma.2016.06.011 Fergani O, Berto F, Welo T et al (2017) Analytical modelling of residual stress in additive manufacturing. Fatigue Fract Eng M 40(6):971–978. doi.org/10.1111/ffe.12560 Mercelis P, Kruth JP (2006) Residual stresses in selective laser sintering and selective laser melting. Rapid Prototyp J 12(5):254–265 Prabhakar P, Sames WJ et al (2015) Computational modeling of residual stress formation during the electron beam melting process for inconel 718. Addit Manuf 7:83–91 Kempen K, Thijs L, Vrancken B et al (2013) Producing crack-free, high density M2 Hss parts by selective laser melting: preheating the baseplate. In: proceedings of the 24th international solid freeform fabrication symposium. Austin(TX): Laboratory for freeform fabrication, p.131-9 Zhao XM, Lin X, Chen J et al (2009) The effect of hot isostatic pressing on crack healing, microstructure, mechanical properties of Rene88DT superalloy prepared by laser solid forming. Mat Sci Eng A-Struct 504(1–2):129–134. doi.org/10.1016/j.msea.2008.12.024 Grilli N, Tarleton E, Cocks ACF (2021) Coupling a discrete twin model with cohesive elements to understand twin-induced fracture. Int J Fract 227:173–192. doi.org/10.1007/s10704-020-00504-9 Dacner CEJ, Achintha M, Salter CJ et al (2012) Residual stress distribution in a functionally graded alumina-silicon carbide material. Scripta mater 67:281–284. doi.org/10.1016/j.scriptamat.2012.05.002 Jepng SG, Ahn SY, Kim ES et al (2022) Effect of substrate yield strength and grain size on the residual stress of direct energy deposition additive manufacturing measured by neutron diffraction. Mat Sci Eng A-Struct 851:143632. doi.org/10.1016/j.msea.2022.143632 Denlinger ER, Heigel JC, Michaleris P et al (2015) Effect of inter-layer dwell time on distortion and residual stress in additive manufacturing of titanium and nickel alloys. J Mater Process Tech 215:123–131. doi.org/10.1016/j.jmatprotec.2014.07.030 Gordon JV, Haden CV, Nied HF et al (2018) Fatigue crack growth anisotropy, texture and residual stress in austenitic steel made by wire and arc additive manufacturing. Mat Sci Eng A-Struct 724:431–438. doi.org/10.1016/j.msea.2018.03.075 Jing H, Ge P, Zhang Z et al (2022) Numerical Studies of the Effects of the Substrate Structure on the Residual Stress in Laser Directed Energy Additive Manufacturing of Thin-Walled Products, Metals, 12:462, doi.org/10.3390/met12030462 Wu AS, Brown DW, Kumar M et al (2014) An Experimental Investigation into Additive Manufacturing-Induced Residual Stresses in 316L Stainless Steel. Metall Mater Trans A 45:6260–6270. doi.org/10.1007/s11661-014-2549-x Lu XF, Cervera M, Chiumenti M et al (2021) Residual stresses Control in Additive Manufacturing. J Manuf Mater Process 5:138. doi.org/10.3390/jmmp5040138 Saunders N, Li X, Miodownik AP et al An integrated approach to the calculation of materials properties for Ti-alloys. Ti-2003: Proc 10th World Conference on Titanium, July 13–18, Hamburg, Germany Guo ZL, Saunders N, Miodownik AP et al (2007) Quantification of high temperature strength of nickel-based superalloys. MSF, pp 546–549. org/10.4028/www.scientific.net/msf.546-549.1319 Luo XL, Liu MH, Li ZH et al (2021) Butong reyuan moxing dui xuanqu jiguang ronghua 18Ni300 wenduchang jisuan jieguo de yingxiang[The Influence of Different Heat Source Models on the Calculation Results of the Laser Melting 18Ni300 Temperature Field]. Zhongguo Jiguang 48(14):52–62. doi.org/10.3788/CJL202148.1402005 Grilli N, Hu D, Yushu D et al (2022) Crystal plasticity model of residual stress in additive manufacturing using the element elimination and reactivation method. Comput Mech 69:825–845. doi.org/10.1007/s00466-021-02116-z Zhan Y, Liu C, Zhang JJ et al (2019) Measurement of residual stress in laser additive manufacturing TC4 titanium alloy with the laser ultrasonic technique. Mat Sci Eng A-Struct 762:138093. doi.org/10.1016/j.msea.2019.138093 Cite Share Download PDF Status: Published Journal Publication published 30 Sep, 2023 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted Editorial decision: Major Revisions Needed 28 May, 2023 Reviewers agreed at journal 26 Jan, 2023 Reviewers invited by journal 24 Jan, 2023 Editor assigned by journal 23 Jan, 2023 First submitted to journal 19 Jan, 2023 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-2497853","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":170353182,"identity":"9d80d1bc-a088-4bcf-8d95-599b2de4e3ce","order_by":0,"name":"Hongjian Zhao","email":"","orcid":"","institution":"","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Hongjian","middleName":"","lastName":"Zhao","suffix":""},{"id":170353183,"identity":"45fb4e49-c732-4a10-9300-26f9cad9bed6","order_by":1,"name":"Chi Gao","email":"","orcid":"","institution":"","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Chi","middleName":"","lastName":"Gao","suffix":""},{"id":170353184,"identity":"f09bed7b-18eb-4c0b-944d-0e92c39a4961","order_by":2,"name":"Zihao Wang","email":"","orcid":"","institution":"","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Zihao","middleName":"","lastName":"Wang","suffix":""},{"id":170353185,"identity":"b77fcb19-7589-4aec-9c17-5d58d6b74fde","order_by":3,"name":"Quanyi Wang","email":"","orcid":"","institution":"","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Quanyi","middleName":"","lastName":"Wang","suffix":""},{"id":170353186,"identity":"301097c3-78c1-435d-ba88-1fe5b0abc20d","order_by":4,"name":"Changsheng Liu","email":"","orcid":"","institution":"","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Changsheng","middleName":"","lastName":"Liu","suffix":""},{"id":170353187,"identity":"0b721d8b-364c-4794-8b83-8e488cf01aa8","order_by":5,"name":"yu zhan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVRIiWNgGAWjYDACCTBpw8DAzNwAFUogSksaUAsjaVoOAzGxWuRnNz97zNt2Ppq/nbGBuaDiMAM/e44Bw88duLUwzjlmbszbdjt3xmGglhlnDjNI9rwxYOw9g1sLs0SCmTRISwNIC2/bYQaDGzkGzIxtuLWwSaR/A2o5lzsfrOXfYQZ7Qlp4JHJAthzI3QDW0gC0RYKAFgmJnDLJOeeSczcCtRzmOZbOI3HmWcHBXjxa5Gekb5N4U2aXO+/84YOPeWqs5fjbkzc++IlHCwgw8UAZB0AuhTHwAsYfhFSMglEwCkbByAYA0pVMZB+kSSEAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-1723-2612","institution":"NORTHEASTEN","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"yu","middleName":"","lastName":"zhan","suffix":""}],"badges":[],"createdAt":"2023-01-20 05:41:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2497853/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2497853/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00170-023-12348-3","type":"published","date":"2023-09-30T15:01:21+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":32136859,"identity":"0afe8ddd-eed9-42e6-99ed-aeec8b0f237d","added_by":"auto","created_at":"2023-01-27 21:10:07","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":339904,"visible":true,"origin":"","legend":"\u003cp\u003eModel and mesh diagram, the laser processed along the x direction from point A to point B.\u003c/p\u003e","description":"","filename":"Figure1.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/b5562740aff4bb7122add36e.png"},{"id":32136863,"identity":"d75624d9-ef1b-4a34-b165-c46f27a9d0bd","added_by":"auto","created_at":"2023-01-27 21:10:07","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":102886,"visible":true,"origin":"","legend":"\u003cp\u003eTC4/Inconel718 alloy distribution in scanning direction\u003c/p\u003e","description":"","filename":"Figure2.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/8df33d269d8ff0b3fe735c98.png"},{"id":32136857,"identity":"fb211291-5900-44e3-bf39-ffb92bea2f30","added_by":"auto","created_at":"2023-01-27 21:10:07","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":196331,"visible":true,"origin":"","legend":"\u003cp\u003eTC4 alloy stress distribution in scanning direction for both methods: (a) is Model Change; (b) is progressive activation element.\u003c/p\u003e","description":"","filename":"Figure3.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/87b77eff50d57a0c75b98e3f.png"},{"id":32136862,"identity":"3aac18d6-3ef1-45ad-b689-14391563d66e","added_by":"auto","created_at":"2023-01-27 21:10:07","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":541865,"visible":true,"origin":"","legend":"\u003cp\u003e(a) (a) Laser ultrasonic system, (b) Experimental sample and layout of measuring line as measuring the velocity \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eM\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e represents the measuring point, \u003cem\u003el\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e represents the measuring line, \u003cem\u003el\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003el\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e and \u003cem\u003el\u003c/em\u003e\u003csub\u003e3 \u003c/sub\u003erepresent the auxiliary lines, \u003cem\u003eN\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eN\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003eare the positions of the laser.\u003c/p\u003e","description":"","filename":"Figure4.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/c439cff8a604a5a57afdb016.png"},{"id":32137319,"identity":"403cd216-1954-4d84-adc3-f6d7ac654775","added_by":"auto","created_at":"2023-01-27 21:18:07","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":16009,"visible":true,"origin":"","legend":"\u003cp\u003eComparison results between experiments and finite element analysis.\u003c/p\u003e","description":"","filename":"Figure5.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/4f6fd5cf1f7a978380e0ea53.png"},{"id":32136860,"identity":"15d0e8b1-9843-4114-a5ad-8e318df57cd7","added_by":"auto","created_at":"2023-01-27 21:10:07","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":54797,"visible":true,"origin":"","legend":"\u003cp\u003eIsothermal fields at different times\u003c/p\u003e","description":"","filename":"Figure6.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/0cdc3e0347f3d51c96bf39ae.png"},{"id":32137320,"identity":"15db339f-1f1a-494e-ab8f-a92d049398f0","added_by":"auto","created_at":"2023-01-27 21:18:07","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":39871,"visible":true,"origin":"","legend":"\u003cp\u003eThe distribution of residual stress in different layers: (a) layers 1 and 2; (b) layers 3 and 4; (c) layers 5 and 6; (d) layers 7 and 8; (e) layers 9 and 10.\u003c/p\u003e","description":"","filename":"Figure7.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/6472e3bcc0e073d976e6d15f.png"},{"id":32136864,"identity":"58796965-54a1-43ed-bf95-e5c855dffe6e","added_by":"auto","created_at":"2023-01-27 21:10:07","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":78064,"visible":true,"origin":"","legend":"\u003cp\u003e(d) The distribution of residual stress in different composition ratios of transition layers: (a) left line; (b) middle line; (c) right line; (d) the distribution of reference points\u003c/p\u003e","description":"","filename":"Figure8.png","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/a3e991d82221933f72eecb74.png"},{"id":43974608,"identity":"f4215e52-e4a6-4a0d-bf7b-f79bdefcf0a8","added_by":"auto","created_at":"2023-10-02 15:08:56","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1731321,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2497853/v1/c0ae32e0-e9f5-41b0-86fd-32516ca84bec.pdf"}],"financialInterests":"","formattedTitle":"Residual Stress Analysis of TC4/Inconel718 Functionally Graded Material Produced by Laser Additive Manufacturing Based on Progressive Activation Element Method","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eDue to the rapid development of aerospace industry, the material selection of aerospace rockets and airplanes has become the focus of scholars[\u003cspan additionalcitationids=\"CR2 CR3 CR4\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Functionally graded materials (FGM) were first proposed by Japanese scholars M. Niino and T. Hrai[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. They could combine two or more materials with different properties through advanced manufacturing processes, so that the internal materials of the structure change in a gradient and there is no obvious layered interface inside. It is the existence of this characteristic that makes the FGM composed of partial materials have high specific strength, good oxidation resistance and low density. For example, TC4/Inconel718 superalloys are often used in aircraft engines that need to work in extreme environments. In recent years, the rapid development of laser additive manufacturing (LAM) provides a new technical support for the fabrication of FGMs. LAM technology, with its unique manufacturing advantages of layer-by-layer stacking, has shown unparalleled huge advantages in manufacturing FGMs with special geometric shapes, expensive processing costs, and difficult to prepare by traditional methods.\u003c/p\u003e \u003cp\u003eThe existence of residual stress is a factor that must be considered regardless of the preparation of FGM by any method. In the process of LAM, due to the high temperature required to melt the metal, a significant thermal gradient will be generated around the molten pool[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. Residual stresses are mainly caused by plasticity resulting from shrinkage of the welded area or solid-state phase transformations present in the material[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. Another source of residual stress is the difference in thermal expansion coefficients between different phases of the same material. Residual stresses can also be the result of chemical composition gradients within workpieces of the same material[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. High residual stress can cause severe damage to the structures[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], such as deformation[\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], loss of geometric tolerance[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], delamination during deposition, poor fracture resistance[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], etc. However, FGMs possess a systematically varied microstructure and/or composition resulting in gradual changes in their mechanical, physical and/or chemical properties across the geometry[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], resulting in the thermal and mechanical problems are very complicated. Therefore, it is difficult to detect the residual stress produced by LAM experimentally.\u003c/p\u003e \u003cp\u003eS.G. Jeong et al[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] used the direct energy deposition method to prepare the samples, and used the neutron diffraction method to detect the residual stress generated during the preparation process, which was in good agreement with the finite element method simulation results. Their study has demonstrated that using a substrate with a softer material reduces residual stress in additively manufactured structures. E.R. Denlinger et al[\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] conducted experimental observations on the laser deposition construction process of Inconel625 alloys and TC4 alloys, and studied the effect of interlayer residence time on the structural residual stress, and found that different residence times had a significant effect on the accumulation of deformation, and shorter residence times can significantly lower residual stress. J.V. Gordon et al[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] studied the fatigue crack growth phenomenon of 304L stainless steel produced by wire and arc additive manufacturing and correlated it with residual stress and microstructure. It was found that the retained compressive residual stress had a positive effect on the structure. Jing et al[\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] presented a new method for controlling residual stress in laser directed energy deposition additive manufacturing products, and results showed that weaker constraints on the substrate can greatly reduce residual stress in additively manufactured products. A.S. Wu et al[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] used experimental methods (digital image correlation combined with build plate removal and sectioning combined with neutron diffraction) to investigate the effects of laser scanning mode, power, scanning speed and build direction on residual stress in powder bed fusion additive manufacturing. A parametric study was carried out to gain a better understanding of the factors affecting the macroscopic residual stress. Lu et al[\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] studied the yield temperature related to thermal deformation and mechanical restraint, discussed the effect of such factors on controlling residual stress, and believed that increasing the yield temperature had a positive effect on controlling the accumulation of residual stress.\u003c/p\u003e \u003cp\u003eThe scholars mentioned above have done a lot of excellent works, but most of them focus on using experimental methods to detect and suppress residual stress. There are relatively few literatures on residual stress analysis of LAMed structures, and most use traditional \u0026ldquo;Model Change\u0026rdquo; method. In this paper, based on the relatively novel progressive activation element (PAE) method, the main factors of the additive manufacturing process, the laser power and the material composition ratio of the transition layers, are simulated and analyzed, and the simulation results are compared with the experimental results to verify the correctness of the simulation results. This paper could provide useful guidance for understanding, controlling and rationally suppressing residual stress in experiments.\u003c/p\u003e"},{"header":"2 Basic Theory","content":"\u003cdiv class=\"Section2\" id=\"Sec3\"\u003e\n \u003ch2\u003e2.1 Characterization of physical parameters of transition layers materials\u003c/h2\u003e\n \u003cp\u003eEach transition layer is mixed with TC4 alloy and Inconel718 high-temperature alloy in different proportions. The mechanical properties of the heterogeneous material of the transition layers could be deduced with composite mechanics.\u003c/p\u003e\n \u003cp\u003eKnowing the density and volume fraction of each component, the density of the mixed material is\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ1\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\rho ={\\rho _1}{\\varphi _1}+{\\rho _2}{\\varphi _2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026rho;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026rho;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the density of each component, and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026phi;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026phi;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the volume fraction of each component.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eThe Halpain-Tsai model is usually used to calculate the Young\u0026rsquo;s modulus of heterogeneous materials. Following is the calculation formula:\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ2\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$$E=\\frac{{{\\varphi _1}{E_1}\\left( {G+{E_2}} \\right)+{\\varphi _2}{E_2}\\left( {G+{E_1}} \\right)}}{{{\\varphi _1}\\left( {G+{E_2}} \\right)+{\\varphi _2}\\left( {G+{E_1}} \\right)}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eE\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eE\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the Young\u0026rsquo;s modulus of each component;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eG\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the interface shear modulus of the material, which is obtained by Voigt formula:\u003c/span\u003e\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ3\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$$G={\\mu _1}\\frac{{{E_1}}}{{2\\left( {1+{\\mu _1}} \\right)}}+{\\mu _{\\text{2}}}\\frac{{{E_2}}}{{2\\left( {1+{\\mu _2}} \\right)}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026micro;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026micro;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the Poisson\u0026rsquo;s ratios of the two materials.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eThe following is the formula for calculating the yield strength \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e of the heterogeneous material:\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ4\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e$${\\sigma _{\\text{S}}}={\\varphi _1}{\\sigma _{{\\text{S1}}}}+{\\varphi _2}{\\sigma _{{\\text{S2}}}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026sigma;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003eS1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026sigma;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003eS2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the yield strength of the two materials respectively.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eThe Lichtenecker empirical formula is widely used by many researchers to calculate the thermal conductivity \u0026lambda; of the heterogeneous material. The following is the Lichtenecker empirical formula:\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ5\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e$$\\lg \\lambda ={\\varphi _1}\\lg {\\lambda _1}+{\\varphi _2}\\lg {\\lambda _2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026lambda;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026lambda;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the thermal conductivity of the two materials.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eThe empirical formula is put forward by Turner, which is for the thermal expansion coefficient of composite materials as follows:\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ6\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\u003cimg src=\"data:image/png;base64,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\"\u003e\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eThe Neumann-Kopp rule gives the calculation formula of specific heat capacity as follows:\u003c/span\u003e\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ7\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e$$c={W_1}{c_1}+{W_2}{c_2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eWhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ec\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ec\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the specific heat capacities of the two materials;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eW\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e1\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eW\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e2\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the mass fractions of the two materials.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eThe physical parameters of TC4 and Inconel718 are shown in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e[\u003cspan class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e].\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tab1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eThe physical parameters of TC4\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTemp\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDensity\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eYoung\u0026apos;s Modulus\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003ePoisson\u0026apos;s Ratio\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSpecific Heat Capacity\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eThermal Conductivity\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eThermal expansion coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e℃\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ekg/m3\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eGPa\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eJ/(kg\u0026middot;K)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eW/(m\u0026middot;K)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e1/K\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e75\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4420.73\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e116.05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e549.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4420.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e115.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e551.69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.90E-06\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4418.95\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e115.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e555.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.93E-06\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e200\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4412.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e112.66\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e572.58\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.08E-06\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e300\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4405.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e109.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e589.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e7.99\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.26E-06\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4391.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e103.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e616.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e10.09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.59E-06\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e800\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4367.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e94.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e652.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e13.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.01E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4350.76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e88.60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e675.07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e14.83\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.04E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1200\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4333.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e82.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e699.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e16.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.07E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4309.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e74.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e872.09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e19.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.08E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tab2\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eThe physical parameters of Inconel718\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTemp\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDensity\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eYoung\u0026apos;s Modulus\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003ePoisson\u0026apos;s Ratio\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSpecific Heat Capacity\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eThermal Conductivity\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eThermal expansion coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e℃\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ekg/m3\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eGPa\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eJ/(kg\u0026middot;K)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eW/(m\u0026middot;K)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e1/K\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e75\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8266.85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e207.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e425.09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e11.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8265.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e206.94\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e426.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e11.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.31E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8261.97\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e206.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e428.76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e11.80\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.31E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e200\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8245.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e203.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e439.60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e12.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.34E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e300\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8224.93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e199.88\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e450.90\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e13.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.36E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8185.88\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e193.09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e469.54\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e15.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.40E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e800\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8121.74\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e182.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e497.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e17.58\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.47E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8077.92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e174.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e515.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e19.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.51E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1200\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8032.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e166.60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e534.09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e20.76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.56E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e7943.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e150.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e708.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e23.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.71E-05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec4\"\u003e\n \u003ch2\u003e2.2 Heat source model\u003c/h2\u003e\n \u003cp\u003eIn the process of LAM, a suitable heat source model is necessary to study the thermo-mechanical coupling models. Three common heat source models are used widely, including concentrated heat source model, plane heat source model and volumetric heat source model. The situation is that the laser power is low and the molten pool depth is small. The plane heat source is the best, including plane Gaussian heat source model and double elliptical heat source model. X.L. Luo et al [\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e] studied the influence of different heat source models by simulating the single-layer single-pass laser melting forming, concluded that the calculation result of the double-ellipsoid heat source model is better than the calculation result of the Gaussian heat source model. The expression of the double-ellipsoid heat source model is\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ8\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e$$q\\left( {x,y,z} \\right)=\\left\\{ {\\begin{array}{*{20}{c}} {\\frac{{6\\sqrt 3 AP}}{{\\pi \\sqrt \\pi {a_r}bc}}\\exp \\left\\{ { - 3\\left[ {\\left( {\\frac{{{x^2}}}{{{a_r}^{2}}}} \\right)+\\left( {\\frac{{{y^2}}}{{{b^2}}}} \\right)+\\left( {\\frac{{{z^2}}}{{{c^2}}}} \\right)} \\right]} \\right\\},\\left( {x \\leqslant 0} \\right)} \\\\ {\\frac{{6\\sqrt 3 AP}}{{\\pi \\sqrt \\pi {a_f}bc}}\\exp \\left\\{ { - 3\\left[ {\\left( {\\frac{{{x^2}}}{{{a_f}^{2}}}} \\right)+\\left( {\\frac{{{y^2}}}{{{b^2}}}} \\right)+\\left( {\\frac{{{z^2}}}{{{c^2}}}} \\right)} \\right]} \\right\\},\\left( {x \\geqslant 0} \\right)} \\end{array}} \\right.$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ea\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ea\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the first half radius and the second half radius of the double-ellipsoid on the x-axis;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eb\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the radius length on the y-axis of the double-ellipsoid;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ec\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the laser penetration depth;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eA\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the absorption rate of the metal powder to the laser;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eP\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the laser power.\u003c/span\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec5\"\u003e\n \u003ch2\u003e2.3 Heat conduction equation\u003c/h2\u003e\n \u003cp\u003eIt is a nonlinear transient heat conduction problem and the temperature of the process of LAM changes rapidly and drastically. The governing equation is listed as below:\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ9\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e$$c\\rho \\frac{{\\partial T}}{{\\partial t}}=\\left[ {{k_x}\\frac{\\partial }{{\\partial x}}\\left( {\\frac{{\\partial T}}{{\\partial x}}} \\right)+{k_y}\\frac{\\partial }{{\\partial y}}\\left( {\\frac{{\\partial T}}{{\\partial y}}} \\right)+{k_z}\\frac{\\partial }{{\\partial z}}\\left( {\\frac{{\\partial T}}{{\\partial z}}} \\right)} \\right]+q\\left( {x,y,z} \\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eWhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ek\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e,\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ek\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e, and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ek\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003ez\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the heat transfer coefficient in the\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ex\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e,\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ey\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e, and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ez\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;directions respectively;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ec\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026rho;\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;are the specific heat capacity and density of the material;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eq\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e(\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ex\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e,\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ey\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e,\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003ez\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e) is the internal heat source of the solution domain;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eT\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the temperature;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003et\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the time of heat transfer. Since the physical properties of materials change with time, formula (2) is extended to anisotropic gradient structure, and the heat conduction differential equation is\u003c/span\u003e\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ10\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e$${\\nabla ^\\lambda }\\left[ \\mathbf{K} \\right]\\nabla T+q\\left( {x,y,z} \\right)=\\frac{\\partial }{{\\partial t}}\\left( {\\rho cT} \\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eWhere\u0026nbsp;\u003c/span\u003e\u003cspan class=\"InlineEquation\" style=\"text-align: inherit;\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\nabla\\)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the gradient operator of space coordinates; the superscript\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026lambda;\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;represents transpose; [\u003c/span\u003e\u003cstrong style=\"text-align: inherit;\"\u003eK\u003c/strong\u003e\u003cspan style=\"text-align: inherit;\"\u003e] represents the matrix of heat transfer coefficient.\u003c/span\u003e\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ11\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e$$\\nabla =\\left( {\\frac{\\partial }{{\\partial x}},\\frac{\\partial }{{\\partial y}},\\frac{\\partial }{{\\partial z}}} \\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eThe initial condition and boundary condition are set to\u003c/span\u003e\u003csup style=\"text-align: inherit;\"\u003e[22]\u003c/sup\u003e\u003cspan style=\"text-align: inherit;\"\u003e:\u003c/span\u003e\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ12\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e$$T\\left( {x,y,z,0} \\right)={T_0}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv class=\"Equation\" id=\"Equ13\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e$$k\\frac{{\\partial T}}{{\\partial n}} - q+{h_c}\\left( {T - {T_0}} \\right)+{\\sigma _b}{\\varepsilon _b}\\left( {{T^4} - T_{0}^{4}} \\right)=0,\\;\\;\\;\\;\\;\\;\\left( {x,y,z} \\right) \\in {S_n}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eWhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eT\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e0\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the ambient temperature;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003en\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the size of a normal vector on the model surface;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eh\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the natural convection heat transfer coefficient;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026sigma;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the Stefan-Boltzmann constant;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026epsilon;\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the emissivity of the material;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eS\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the model surface.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eThe stress-strain equation of the thermal-mechanical model is\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ14\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e$$\\varepsilon ={D^{ - 1}}\\sigma +{\\varepsilon ^{{\\text{th}}}}{\\text{+}}{\\varepsilon ^p}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026epsilon;\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the total strain vector;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eD\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the elastic matrix;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026sigma;\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the stress vector;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026epsilon;\u003c/em\u003e\u003csup style=\"text-align: inherit;\"\u003eth\u003c/sup\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the thermal strain vector,\u0026nbsp;\u003c/span\u003e\u003cspan class=\"InlineEquation\" style=\"text-align: inherit;\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varepsilon ^p}\\)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the plastic strain.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eThe thermal strain is calculated according to the temperature field distribution and the thermal expansion coefficient of the material:\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ15\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e$${\\varepsilon ^{{\\text{th}}}}=\\int_{{{T_{{\\text{ref}}}}}}^{T} {\\alpha \\left( T \\right){\\text{d}}T}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003ewhere\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003eT\u003c/em\u003e\u003csub style=\"text-align: inherit;\"\u003eref\u003c/sub\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the ambient temperature;\u0026nbsp;\u003c/span\u003e\u003cem style=\"text-align: inherit;\"\u003e\u0026alpha;\u003c/em\u003e\u003cspan style=\"text-align: inherit;\"\u003e\u0026nbsp;is the thermal expansion coefficient.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eAccording to the residual stress characteristics of LAM and the basic theory of plastic mechanics, the Mises yield criterion is adopted, and the equivalent stress calculation formula is\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ16\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e$${\\sigma _i}=\\sqrt {\\frac{1}{2}\\left[ {{{\\left( {{\\sigma _1} - {\\sigma _2}} \\right)}^2}+{{\\left( {{\\sigma _2} - {\\sigma _3}} \\right)}^{\\text{2}}}{\\text{+}}{{\\left( {{\\sigma _{\\text{3}}} - {\\sigma _1}} \\right)}^2}} \\right]}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\n \u003c/div\u003e\n\u003c/div\u003e"},{"header":"3 Finite Element Models","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Progressive activation element\u003c/h2\u003e \u003cp\u003eDifferent with \u0026ldquo;Model Change\u0026rdquo; method, progressive activation element does not need to set a large number of steps and interactions to realize the movement of heat source, which can save us a lot of energy and time. Python is used to implement the loading and movement of the double ellipsoid heat source, and realize the deposition process of the FGM powder. Multiple attempts were taken to achieve parameter settings that were nearly identical to the experimental procedure, such as the laser power, scanning speed, etc. Finally, calculate the compiled .inp file.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Models building\u003c/h2\u003e \u003cp\u003eIn order to save computation time and improve efficiency, axisymmetric models are adopted. The whole model is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the lower part of the model is the substrate, which is made of Inconel718 alloy, and the upper part of the model is the laser cladding layers, which are made of TC4/Inconel718 FGM. The material of the cladding layers is changed every two layers, the bottom is made by Inconel718 alloy, and the top is made by TC4 alloy. The material composition of each layer is shown in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The stress changes and distribution of the cladding layers should be concerned. So, the equal and dense meshes are set there, and sparse and large meshes are set with the substrate plate. After many attempts, considering both calculation time and result accuracy, the mesh size of the cladding layers should be set to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(0.3{\\text{3}} \\times 0.33 \\times 0.25\\)\u003c/span\u003e\u003c/span\u003emm, and the size of each cladding layer in the model is set to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{60}} \\times {\\text{1}} \\times {\\text{0}}{\\text{.5}}\\)\u003c/span\u003e\u003c/span\u003emm, the size of the substrate plate is set to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{70}} \\times {\\text{10}} \\times {\\text{5}}\\)\u003c/span\u003e\u003c/span\u003emm. At this size, each cladding layer has 1080 meshes, and the entire model has a total of 58464 meshes.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePowder mixing ratio from layer 1 to layer 10\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBottom-up layer number\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePowder mixing ratio\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u0026ndash;2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e100%Inconel718\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u0026ndash;4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e80%Inconel718,20%TC4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u0026ndash;6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e50%Inconel718,50%TC4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u0026ndash;8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e20%Inconel718,80%TC4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e100%TC4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Models verification\u003c/h2\u003e \u003cp\u003eAfter processing of LAM, set 300s for the natural cooling, and the obtained stress distribution in the scanning direction is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe thermal and mechanical problems presented by FGMs are very complicated because the gradient layers are heterogeneous both microscopically and macroscopically. Because of the large temperature gradients and high-frequency heating-cooling cycles during the LAM process, it is difficult to monitor the temperature and thermal stress in real time during experiments[\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. In order to verify the correctness of this model, a TC4 alloy model of the same size is established with the same method, and the result graph calculated by the commonly used \u0026ldquo;Model Change\u0026rdquo; method is provided, showing the difference between the two methods. After that, the residual stress of the sample was detected by the experimental method, and the comparison chart between the stress value of the two methods and the experimental value was obtained.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eLaser ultrasonic technology is used to detect the residual stress of the structure during all the experiments. Laser ultrasonic has shown strong potential due to its advantages, such as non-contact, high-precision, non-destructive, strong anti-interference ability, high efficiency and portability. In previous research, a lot of work has been done on the measurement of residual stress with laser ultrasonic. The result of residual stress measurement of TC4 titanium alloy was published[\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. Almost the same parameters and experimental conditions as the experiment are adopted in this paper.\u003c/p\u003e \u003cp\u003eThe whole experiment procedure consists of two parts, preparation of the experiment samples with LAM, and measurement of the residual stress with laser ultrasonic technology. The Laser melting deposition system was used in the experiment (MetLas3.0, Liaoning Yutong laser application technology Engineering Co., Ltd). The powder materials used for deposition were TC4 alloy powder produced by AVIC Maite powder metallurgy technology (Beijing) Co., Ltd. The powder is spherical. The particle size is 140\u0026thinsp;~\u0026thinsp;200\u0026micro;m. The laser ultrasonic experiment system is developed and applied in residual stress measurement. The Nd:YAG laser setup (Dawa-100, Beamtech Optronics Ltd., Beijing, China) is used to generate the ultrasonic waves. A laser pulse is sent by the Nd: YAG laser setup, and focused by the cylindrical lens as a line source (0.6 mm width, 20 mm length) to generate the ultrasonic surface waves. The laser Doppler vibrometer (Sdptop LV-S01, Sunny Optical Technology Ltd., Suzhou, China) is used to detect the ultrasonic vibration information. The experimental setup is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(a).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe ultrasonic system was used to detect residual stress of TC4 sample. In order to improve the intensity of the detection signal received by laser Doppler vibrometer, mechanical polishing of the surface was set. It is assumed that mechanical polishing does not alter the stress state. The polishing process is carefully controlled to ensure that there is no excessive external force. Normally, the thickness of the polishing layer is less than one tenth of the surface wave incident depth. The porosity of the samples is measured by Archimedean method, and the average value is 0.16%. The research focuses on the measurement of macro residual stress, and the porosity of the samples is low, so the influence of porosity on the residual stress is neglected. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(a)(b), \u003cem\u003eM\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e represents the measuring point and all the measuring points are lay out on the measuring line \u003cem\u003el\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e (blue line). \u003cem\u003eN\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eN\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e represent the positions of detection and excitation laser, respectively. \u003cem\u003el\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003el\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e and \u003cem\u003el\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e represent the auxiliary lines (black line). It is noteworthy that the stress at the measuring point is actually the average value within the propagation distance of the surface wave.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eObviously, the stress magnitudes obtained by the two methods are similar to the experimental results, and the result obtained by PAE method is more accurate than the one by Model Change method. Therefore, the correctness of these two methods have been verified (see Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Results","content":"\u003cp\u003eIn the process of LAM, drastic change in temperature is the main cause of the residual stress. It is important to observe the temperature changes currently to analyze the residual stress conveniently later. As the processing progresses, the temperature of the heat source of the LAM increases (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e).\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eResidual stress is our focus in simulating LAM. The power of the laser heat source is one of the most important reasons for the residual stress. Therefore, simulation models for different powers are established to verify the distribution law of residual stress under different machining powers, as shown in the Fig.\u0026nbsp;\u003c/span\u003e\u003cspan class=\"InternalRef\" style=\"text-align: inherit;\"\u003e7\u003c/span\u003e\u003cspan style=\"text-align: inherit;\"\u003e. Before that, we set up several reference points to detect the residual stress around. As mentioned above, two layers of each material were printed, and the middle common nodes of the two cladding layers was used as the reference points. It can be seen from the Fig.\u0026nbsp;\u003c/span\u003e\u003cspan class=\"InternalRef\" style=\"text-align: inherit;\"\u003e7\u003c/span\u003e\u003cspan style=\"text-align: inherit;\"\u003e that with the increase of laser power, the residual stress gradually increases, and the two are positively correlated. However, due to the influence of the material gradient in the stacking direction, the influence of different cladding layers is also different. The laser power received at the bottom layer of the structure (100% Inconel718) has a greater influence, while the upper layer of the structure, especially the layers 7 and 8(20% Inconel718 and 80% TC4), the influence of the laser power is smaller. The residual stress distribution of the overall structure presents an \u0026ldquo;inverted bowl shape\u0026rdquo;, the residual stress on both sides is small, and a large residual stress is generated in the middle part of the structure. In actual engineering, we should pay more attention to the middle part of the structure, and the fracture phenomenon of a large number of additively manufactured specimens occurs mostly in the middle of the structure.\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eCompared with isotropic materials, the biggest feature of this paper is the material anisotropy of the structure in the stacking direction. Due to the differences in the physical and mechanical properties of materials, materials with different mixing ratios have different resistance to residual stress. In order to facilitate the experimental preparation of functionally graded materials, simulation models with different transition ratios are designed, which are divided into four types: 6:4, 7:3, 8:2 and 9:1. The LAM simulation model designed in this paper has a total of 10 cladding layers, of which the 1st and 2nd layers are 100% Inconel718 material, the 9th and 10th layers are 100% TC4 material, and the 5th and 6th layers are 50% TC4 and 50%Inconel718 material mixing, these three parts always remain unchanged, the 3rd, 4th, 7th and 8th layers are designed as transition layers, taking the transition ratio of 8:2 as an example, that is, the 3rd and 4th layers are 80%Inconel718 mixed with 20% TC4 material, the 7th and 8th layers are mixed with 20% Inconel718 and 80% TC4 material. As mentioned above, the residual stress mainly takes a large value in the middle part of the structure. In order to facilitate the detection of residual stress, a total of 15 reference points is averagely set on three lines, which are called \u0026quot;left line\u0026quot;, \u0026ldquo;middle line\u0026rdquo; and \u0026ldquo;right line\u0026rdquo; (see Fig.\u0026nbsp;\u003c/span\u003e\u003cspan class=\"InternalRef\" style=\"text-align: inherit;\"\u003e8\u003c/span\u003e\u003cspan style=\"text-align: inherit;\"\u003e(d)). These three lines are taken at the quarter points of the scanning direction of the structure, that is, the two adjacent lines are separated by 45 meshes (1.5cm). Figure\u0026nbsp;\u003c/span\u003e\u003cspan class=\"InternalRef\" style=\"text-align: inherit;\"\u003e8\u003c/span\u003e\u003cspan style=\"text-align: inherit;\"\u003e(a), (b) and (c) show the residual stress distribution on these three lines. Due to the influence of the gradient, the variation of the residual stress does not have an obvious regularity like the laser power. In general, when the composition ratio of the transition layer is 6:4 and 9:1, the residual stress of the structure changes most obviously, which is obviously not conducive to the stability of the structure. When the composition ratio of the transition layer is 7:3 and 8:2, the change of residual stress is relatively moderate, and when the two are compared, the composition ratio of 8:2 has a smoother residual stress transition, which is not easy to cause the buckling of the structure. and breakage issues. Therefore, when using the LAM method to prepare the TC4/Inconel718 functionally graded material, it is more in line with the actual needs to use the transition layer ratio of 8:2.\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: inherit;\"\u003eIn the end, in order to reflect the difference between the PAE method and traditional \u0026ldquo;Model Change\u0026rdquo; method, the calculation time of the above models are listed, as shown in Table\u0026nbsp;\u003c/span\u003e\u003cspan class=\"InternalRef\" style=\"text-align: inherit;\"\u003e4\u003c/span\u003e\u003cspan style=\"text-align: inherit;\"\u003e. Compared with the \u0026ldquo;Model Change\u0026rdquo; method, the PAE method has relatively high computational efficiency and can save a lot of computational time. When calculating anisotropic materials, \u0026ldquo;Model Change\u0026rdquo; method will consume a lot of time for stress analysis due to the change of the material composition of the transition layers. This advantage of PAE will be more obvious when calculating gradient materials. When calculating isotropic materials (such as TC4 alloy), the calculation efficiency can be increased by 995%, and when calculating anisotropic materials, the higher the laser power, the more obvious the efficiency improvement of the PAE method; the calculation time of combining these models, the PAE method can probably improve by about 1650% than the \u0026ldquo;Model Change\u0026rdquo; method. It is worth stating that the CPU used in this paper is Intel\u0026reg; Core\u0026trade; i5-10600KF CPU @ 4.10GHz, and the utilization rate is kept above 90%. The memories are two Essencore DDR4-3200 8GB and two Crucial Technology DDR4-2666 8GB.\u003c/span\u003e\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tab4\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eCalculation time of above models\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"5\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eModel\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eCalculation time\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eEfficiency\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePAE\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eModel Change\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eIsotropic\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePure TC4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3h15mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e35h44mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e995%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eAnisotropy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e400W\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3h30mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e71h18mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1937%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e500W\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4h15mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e92h36mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2078%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e600W\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5h3mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e108h36mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2050%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6:4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4h47mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e113h7mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2265%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7:3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4h10mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e98h25mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2262%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8:2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4h15mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e94h36mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2126%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9:1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4h39mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e116h14mins\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2400%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003col\u003e\n \u003cli\u003eWhen the TC4/Inconel718 functionally graded material is prepared by the laser additive manufacturing method, the size of the laser power will greatly affect the residual stress of the specimen after processing, and the two are positively correlated. The residual stress of the specimen is reasonably controlled to meet the requirements of industrial use.\u003c/li\u003e\n \u003cli\u003eDue to the different composition of materials in the stacking direction, the residual stress exhibits irregular changes. When the excessive ratio of 6:4 and 9:1 is used, the interlayer change of residual stress is too large, which is easy to cause damage to the structure. TC4/Inconel718 functionally graded material was prepared with a composition ratio of 8:2 for the transition layers.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAcknowledgement\u003c/h2\u003e\n\u003cp\u003eThis study is supported by the National Natural Science Foundation of China Project (Grant No.\u0026nbsp;51771051), the Natural Science Foundation of Liaoning Province Project (Grant No.\u0026nbsp;2021-MS-102)\u0026nbsp;and\u0026nbsp;the Fundamental Research Funds for the Central Universities (Grant No.\u0026nbsp;N2105021).\u003c/p\u003e\n\u003ch2\u003eAuthor Statement\u003c/h2\u003e\n\u003cp\u003eHongjian Zhao(first author): Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing-Original Draft\u003c/p\u003e\n\u003cp\u003eChi Gao: Formal analysis, Writing-Original Draft\u003c/p\u003e\n\u003cp\u003eZihao Wang: Software, Formal analysis\u003c/p\u003e\n\u003cp\u003eQuanyi Wang: Validation\u003c/p\u003e\n\u003cp\u003eChangsheng Liu: Resources, Supervision, Funding acquisition\u003c/p\u003e\n\u003cp\u003eYu Zhan (Corresponding Author): Visualization, Resources, Writing-Review \u0026amp; Editing, Funding acquisition\u003c/p\u003e\n\u003ch2\u003eConflict of Interest\u003c/h2\u003e\n\u003cp\u003eThe work presented here was performed in collaboration among all authors. Hongjian Zhao designed, analyzed, and wrote the paper. Chi Gao, Zihao Wang, Quanyi Wang, Changsheng Liu and Yu Zhan provided and analyzed the experimental data. All authors contributed to and approved the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003e\u003cspan\u003eTascopglu E, Karabulut YY, Kaynak Y (2020) Correction to: Influence of heat treatment temperature on the microstructural, mechanical, and wear behavior of 316L stainless steel fabricated by laser powder bed additive manufacturing. Int J Adv Manuf Technol 107:1957. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1007/s00170-020-05115-1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eGuo S, Chen M et al (2023) 3D printed hierarchically porous zero-valent copper for efficient pollutant degradation through peroxymonosulfate activation. Sep Purif Technol 305:122437\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eTang HB, Huang HJ et al (2021) Multi-Scale modelling of structure-property relationship in additively manufactured metallic materials. Int J Mech Sci 194:106185\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eHong X, Xiao G, Zhang Y et al (2021) Research on gradient additive manufacturing of ultra-large hot forging die based on automatic wire arc additive manufacturing technology. Int J Adv Manuf Technol 116:2243\u0026ndash;2254. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1007/s00170-021-07424-5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eDu D, Wang L et al (2022) Promoting the densification and grain refinement with assistance of static magnetic field in laser powder bed fusion. Int J Mach Tools Manuf 183:103965\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eNiino M, Hirai T, Watanabe R (1987) Functionally Gradient Materials - Toward Super Heat Resistant Materials for Spacecraft. J Compos Mater 13(6):257\u0026ndash;264. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.6089/jscm.13.257\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eArcam U (2011) Manual, Arcam AB,\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eDenlinger ER, Michaleris P (2016) Effect of stress relaxation on distortion in additive manufacturing process modeling. Addit Manuf 12:51\u0026ndash;59. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1016/j.addma.2016.06.011\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eFergani O, Berto F, Welo T et al (2017) Analytical modelling of residual stress in additive manufacturing. Fatigue Fract Eng M 40(6):971\u0026ndash;978. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1111/ffe.12560\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eMercelis P, Kruth JP (2006) Residual stresses in selective laser sintering and selective laser melting. Rapid Prototyp J 12(5):254\u0026ndash;265\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003ePrabhakar P, Sames WJ et al (2015) Computational modeling of residual stress formation during the electron beam melting process for inconel 718. Addit Manuf 7:83\u0026ndash;91\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eKempen K, Thijs L, Vrancken B et al (2013) Producing crack-free, high density M2 Hss parts by selective laser melting: preheating the baseplate. In: proceedings of the 24th international solid freeform fabrication symposium. Austin(TX): Laboratory for freeform fabrication, p.131-9\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eZhao XM, Lin X, Chen J et al (2009) The effect of hot isostatic pressing on crack healing, microstructure, mechanical properties of Rene88DT superalloy prepared by laser solid forming. Mat Sci Eng A-Struct 504(1\u0026ndash;2):129\u0026ndash;134. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1016/j.msea.2008.12.024\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eGrilli N, Tarleton E, Cocks ACF (2021) Coupling a discrete twin model with cohesive elements to understand twin-induced fracture. Int J Fract 227:173\u0026ndash;192. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1007/s10704-020-00504-9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eDacner CEJ, Achintha M, Salter CJ et al (2012) Residual stress distribution in a functionally graded alumina-silicon carbide material. Scripta mater 67:281\u0026ndash;284. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1016/j.scriptamat.2012.05.002\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eJepng SG, Ahn SY, Kim ES et al (2022) Effect of substrate yield strength and grain size on the residual stress of direct energy deposition additive manufacturing measured by neutron diffraction. Mat Sci Eng A-Struct 851:143632. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1016/j.msea.2022.143632\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eDenlinger ER, Heigel JC, Michaleris P et al (2015) Effect of inter-layer dwell time on distortion and residual stress in additive manufacturing of titanium and nickel alloys. J Mater Process Tech 215:123\u0026ndash;131. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1016/j.jmatprotec.2014.07.030\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eGordon JV, Haden CV, Nied HF et al (2018) Fatigue crack growth anisotropy, texture and residual stress in austenitic steel made by wire and arc additive manufacturing. Mat Sci Eng A-Struct 724:431\u0026ndash;438. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1016/j.msea.2018.03.075\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eJing H, Ge P, Zhang Z et al (2022) Numerical Studies of the Effects of the Substrate Structure on the Residual Stress in Laser Directed Energy Additive Manufacturing of Thin-Walled Products, Metals, 12:462, \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.3390/met12030462\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eWu AS, Brown DW, Kumar M et al (2014) An Experimental Investigation into Additive Manufacturing-Induced Residual Stresses in 316L Stainless Steel. Metall Mater Trans A 45:6260\u0026ndash;6270. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1007/s11661-014-2549-x\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eLu XF, Cervera M, Chiumenti M et al (2021) Residual stresses Control in Additive Manufacturing. J Manuf Mater Process 5:138. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.3390/jmmp5040138\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eSaunders N, Li X, Miodownik AP et al An integrated approach to the calculation of materials properties for Ti-alloys. Ti-2003: Proc 10th World Conference on Titanium, July 13\u0026ndash;18, Hamburg, Germany\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eGuo ZL, Saunders N, Miodownik AP et al (2007) Quantification of high temperature strength of nickel-based superalloys. MSF, pp 546\u0026ndash;549. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003eorg/10.4028/www.scientific.net/msf.546-549.1319\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eLuo XL, Liu MH, Li ZH et al (2021) Butong reyuan moxing dui xuanqu jiguang ronghua 18Ni300 wenduchang jisuan jieguo de yingxiang[The Influence of Different Heat Source Models on the Calculation Results of the Laser Melting 18Ni300 Temperature Field]. Zhongguo Jiguang 48(14):52\u0026ndash;62. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.3788/CJL202148.1402005\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eGrilli N, Hu D, Yushu D et al (2022) Crystal plasticity model of residual stress in additive manufacturing using the element elimination and reactivation method. Comput Mech 69:825\u0026ndash;845. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1007/s00466-021-02116-z\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eZhan Y, Liu C, Zhang JJ et al (2019) Measurement of residual stress in laser additive manufacturing TC4 titanium alloy with the laser ultrasonic technique. Mat Sci Eng A-Struct 762:138093. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003edoi.org/10.1016/j.msea.2019.138093\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"laser additive manufacturing, functionally graded material, residual stress, progressive activation element","lastPublishedDoi":"10.21203/rs.3.rs-2497853/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2497853/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWith the continuous development of preparation technology, laser additive manufacturing (LAM) has become one of the effective ways to manufacture functionally graded materials due to its unique layer-by-layer stacking technology. However, the repeated and repeated rapid heating and cooling processes in the manufacturing process will generate large residual stress inside the structure, resulting in the destruction of the structure. In this paper, based on a new finite element method called progressive activation element method (PAE), a thermomechanical coupling model for simulating the process of LAM is established, and the influence of laser power and composition ratio of transition layers on the residual stress of the overall structure is discussed. The results show that there is a positive correlation between the laser power and the residual stress. The PAE method is compared with the traditional \u0026ldquo;Model Change\u0026rdquo; method, and it is found that the PAE method has advantages in computational efficiency, especially when calculating the residual stress of functionally graded materials, the efficiency can be improved by about 1650%. When the TC4/Inconel718 functionally graded material is prepared experimentally, the optimal composition ratio of the transition layers is 8:2. This paper provides reference for the understanding and reasonable suppression of residual stress of functionally graded materials in LAM.\u003c/p\u003e","manuscriptTitle":"Residual Stress Analysis of TC4/Inconel718 Functionally Graded Material Produced by Laser Additive Manufacturing Based on Progressive Activation Element Method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-01-27 21:10:02","doi":"10.21203/rs.3.rs-2497853/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major Revisions Needed","date":"2023-05-28T09:57:50+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2023-01-26T07:31:21+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2023-01-24T13:44:58+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2023-01-23T13:07:46+00:00","index":"","fulltext":""},{"type":"submitted","content":"The International Journal of Advanced Manufacturing Technology","date":"2023-01-20T00:41:03+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"8524939a-5782-4e58-8fbb-e61ca858eea1","owner":[],"postedDate":"January 27th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2023-10-02T15:05:53+00:00","versionOfRecord":{"articleIdentity":"rs-2497853","link":"https://doi.org/10.1007/s00170-023-12348-3","journal":{"identity":"the-international-journal-of-advanced-manufacturing-technology","isVorOnly":false,"title":"The International Journal of Advanced Manufacturing Technology"},"publishedOn":"2023-09-30 15:01:21","publishedOnDateReadable":"September 30th, 2023"},"versionCreatedAt":"2023-01-27 21:10:02","video":"","vorDoi":"10.1007/s00170-023-12348-3","vorDoiUrl":"https://doi.org/10.1007/s00170-023-12348-3","workflowStages":[]},"version":"v1","identity":"rs-2497853","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2497853","identity":"rs-2497853","version":["v1"]},"buildId":"rHA-KDH7Qsr4HCuvH75dn","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}