Ratcheting Simulation of Additively Manufactured Aluminum 4043 Samples through Finite Element Analysis

Abstract

This study presents a finite element-based ratcheting assessment of additively manufactured aluminum 4043 samples undergoing asymmetric loading cycles. The Chaboche material model in ANSYS was utilized and the effects of mesh and element type were examined. Different element numbers were used in a thorough convergence study to obtain independent meshing structures. The coefficients of this model were defined through stress–strain hysteresis loops determined from the strain-controlled tests. The backstress evolution and the corresponding yield surface translation in the deviatoric stress space were discussed as three different mesh elements of linear brick, quadratic tetrahedron, and quadratic brick were adopted. The magnitude of backstress was affected as different element types were employed. The first-order brick elements resulted in the highest backstress increments, while the lowest backstresses were determined when quadratic brick elements were taken. Backstress increments are positioned in an intermediate level with the use of quadratic tetrahedron elements. The choice of the element type, shape, and number influenced material ratcheting response over the loading process. The use of quadratic brick elements elevated the simulated ratcheting curves. The quadratic tetrahedron and linear brick elements, however, suppressed ratcheting level as compared with those of experimental data. The closeness of the simulated ratcheting results to those of the measured values was found to be highly dependent on these finite element variables.

1. Introduction

Additive manufacturing (AM) is a fabrication technology used to build three-dimensional complex components through a layer-by-layer process. Additive manufacturing techniques have empowered industries including the aerospace, automotive, and medical industries to create delicate load-bearing parts with a high accuracy through a controlled fabrication time and process [1,2,3]. AM techniques have been broadly developed for different metallic alloys of steel, titanium, aluminum, nickel, and superalloys [4,5].

In recent years, the additive manufacturing industry has offered versatile techniques to fabricate load-bearing machinery parts undergoing in-service dynamic loads. Metallic AM components, through their layer-by-layer stacking structure and formation of porosities between debonding layers, are noticeably prone to fatigue damage and failure as subjected to stress cycles. In addition to the inhomogeneity of formed layers, porosity, and dendritic microstructure, asymmetric applied stress cycles further promote progressive plastic strain and ratcheting in AM components [6,7]. A tomography analysis on the AM AlSi10Mg samples fabricated by laser powder bed fusion (LPBF) revealed larger fatigue cracks on the parallel plane to the build direction as compared to the perpendicular plane [8]. A recent review on the metallurgical aspects on the laser additive manufacturing process revealed that the use of the hot isostatic pressed (HIP) post-processing treatment led to the closure of pores between layers, improving tensile properties comparable to those of thermo-mechanically treated samples [9]. Hot isostatic pressing also showed a greater ability to reduce subsurface crack growth for the laser-melted specimens in the high-cycle fatigue regime as opposed to the shot peening post-processing treatment. While the literature has widely addressed mechanical properties of AM samples under well-defined specimen-built and post-treated surface conditions, process interruption has been inevitable and infeasible. Richter et al. [10] and Pouille et al. [11] tested AM samples fabricated by the laser powder bed fusion (LPBF) method to study the effect of process interruption on the fatigue response of AM materials. Process interruption decreased the tensile and fatigue properties of the AM samples.

The ratcheting response of conventionally manufactured materials was broadly studied over the last century [12]. The open literature however lacks a pertinent ratcheting data bank and analysis for AM materials. The metallic AM samples undergoing asymmetric loading cycles experienced fatigue and ratcheting phenomena in aluminum alloys [13,14]. Wang et al. [15] tested different AM Al 4043 samples taken from the bottom, middle and upper portions of the 3D printed plates. They found that the prepared samples possessed different microstructural features with dendritic grains growing in different directions resulting in various mechanical properties and ratcheting responses. An investigation of the ratcheting of Ti–6Al–4V cellular solid with different structures was conducted by Zhao et al. [16]. With further strut rearrangements, they were able to control fatigue performance and ratcheting rates of AM samples. At different stress levels, Wu et al. [17] examined the effects of ratcheting on Ti–6Al–4V samples made through the selective laser melting (SLM) technique. For stress levels corresponding to 70% of fatigue life, the ratcheting strain rate for AM titanium samples increased gradually. Beyond this stress level, the ratcheting strain rate increased to a greater extent. Servatan et al. [18] predicted the ratcheting response of AM Al 4043 through the Ahmadzadeh–Varvani (A–V) kinematic hardening rule [19] combined with an isotropic hardening description of Lee–Zaverl [20] for various stress levels. The predicted ratcheting results by the combined hardening framework [18] closely agreed with the experimentally measured data.

In addition to analytical approaches for ratcheting assessment of AM samples, the literature holds a limited number of studies to simulate cyclic behavior through FE analysis. An overview of recent progresses of ratcheting response of materials through experimental investigation and finite element analysis was comprehended by Chen et al. [21]. Safety design, protocol, and assessment of engineering structures against ratcheting and related boundaries have been examined by Chen and Chen [22] through experimental and FEM research. They simulated the ratcheting response of pressurized pipes made of Z2CND18.12 N austenitic stainless steel through ANSYS software (2022 R1) by applying the modified Ohno-Wang model. Zhang et al. [23] simulated the fatigue response of AM AlSi10Mg samples within high-cycle and very-high-cycle regimes by means of the crystal plasticity finite element method. They found that samples with defects have a much shorter fatigue life than samples with no defects. In their studies, aluminum specimens with 0° angles performed better under fatigue cycles than those specimens with 90° angles. To predict the fatigue life of 3-D printed IN718 specimens, Prithivirajan and Sangid [24] encountered the effect of manufacturing pores in crystal plasticity FEM simulations. Their method offered a safe-life design criterion for the additively manufactured aircraft components assessed for their fatigue crack initiation lives. To study the influence of pores on the fatigue response of materials, fatigue damage metrics were also incorporated to include plastic strain accumulation and dissipative energy, along with refinement of pores near the meshing elements [25]. Using the dislocation-density-based crystal plasticity theory, Pal and Stucker [25] developed an FEA model framework to assess microstructural evolution during ultrasonic consolidation in aluminum foils. A quasistatic formulation of large plastic deformation, a material model based on non-local dislocation density, and process boundary conditions were employed in their simulation modeling. The simulated average grain diameter was reduced from 13 $\mathsf{\mu }\mathrm{m}$ prior to processing to 1.2 $\mathsf{\mu }\mathrm{m}$ after processing which was found in a close agreement with the experimental results. Ghosh et al. [26] simulated the ratcheting of AM titanium samples through the finite element method in order to gain a better understanding of the correlation between micro- and macro-scale deformation behavior. Servatan et al. [27] predicted the ratcheting response of AM samples employing the A-V hardening rule. They further simulated the ratcheting–fatigue damage interaction of SS304L and AlSi10Mg samples undergoing asymmetric loading cycles through the FE analysis. They employed a model to partition overall damage into ratcheting damage and fatigue damage enabling a robust design of additively manufactured samples against progressive deformation and failure. The hardening framework and a finite element method (FEM) [28] were employed to evaluate the backstress evolution, the yield surface translation, and the ratcheting response of various AM Al 4043, SS316L, and Ti–6Al–4V samples.

The present study intends to investigate the impact of element type, shape, and number on the ratcheting response of AM aluminum 4043 samples. The choice of AM aluminum samples in this study is mainly attributed to the high strength, fracture toughness, specific modulus, corrosion resistance, fatigue resistance plus the applicability of this material in the aerospace industry and medical equipment [29,30]. Ratcheting simulation of Al 4043 was conducted through finite element Ansys software [31] and through a nonlinear material model developed by Chaboche [32]. The backstress evolution and the corresponding yield surface translation for different element types of linear brick, quadratic tetrahedron, and quadratic brick were discussed as Al4 034 were tested at various uniaxial asymmetric stress cycles.

2. Mathematical Formulation

2.1. Elastic and Plastic Strain Components

The cyclic plasticity constitutive model consists of the von Mises yielding criterion, plastic flow rule, and a kinematic hardening rule. According to the von Mises criterion, the onset of yielding demarks the elastic and plastic domains and is defined as:

$$f\left(\mathrm{s}. \overline{a}. {\sigma }_y\right)=\sqrt{\frac{3}{2}\left(s-\overline{a}\right).\left(s-\overline{a}\right)}-{\sigma }_y{}^0$$

(1)

In Equation (1), ${\sigma }_y{}^0$ is the initial yield stress. Backstress is denoted by ā and determines the extent of the loading state outside the elastic limit. Term s is the deviatoric stress tensor and is expressed as:

$$s=\sigma -\frac{1}{3}\left(\sigma .I\right)I$$

(2)

Terms $\sigma$, $I$ are the applied stress and unit tensor, respectively.

The combination of elastic strain, ${\epsilon }^e$, and plastic strain, ${\epsilon }^p$, results in the total elastic–plastic strain as:

$$\epsilon ={\epsilon }^e+{\epsilon }^p$$

(3)

The elastic strain is defined by Hooke’s law as:

$${\epsilon }^e=\frac{\sigma }{2G}-\frac{\vartheta }{E}\left(\sigma .I\right)I$$

(4)

where G and E in Equation (4) are shear and elastic moduli, respectively. Term $\vartheta$ is Poisson’s ratio. A flow rule was used to express relation between stress–strain within the plastic domain:

$$d{\epsilon }^p=\frac{1}{H_p}⟨ds.n⟩n$$

(5)

The unit normal vector on the yield surface is given as:

$$n=\frac{\left(s-\overline{a}\right)}{\left|\left(s-\overline{a}\right)\right|}$$

(6)

2.2. The Chaboche Kinematic Hardening Rule

Chaboche [32] developed a kinematic hardening model to govern the incremental evolution of backstress along with the yield surface translation in the deviatoric stress space for applied loads within the plastic domain. He proposed integrating backstress increments $d\overline{\alpha }=\sum _{i=1}^{M}d{\overline{\alpha }}_i$ (i = 1,2,3) as loading progressed beyond the elastic domain. The Chaboche nonlinear hardening rule is expressed as:

$$d\overline{\alpha }={\sum }_{i=1}^{M}d{\overline{\alpha }}_i$$

(7a)

where

$$d{\overline{\alpha }}_i=\frac{2}{3}{C}_{i}d{\overline{\epsilon }}_P-{\gamma }_i^{\text{'}}{\overline{\alpha }}_{i}dp$$

(7b)

Terms $C_{1-3}$ and ${\gamma }_{1-3}^{\prime }$ are material-dependent coefficients specified from strain-controlled cyclic tests. Coefficients $C_1$ and $C_3$ were obtained from the slope yielding in the linear part of the strain-based hysteresis loop. Backstress components ${\overline{\alpha }}_i$ (i = 1,2,3) corresponded to the loading paths of each proceeding cycle. Through using the measured uniaxial ratcheting data over a number of cycles within the first-stage parameter, ${\gamma }_1^{\prime }$is obtained, and a proper value is selected for parameter ${\gamma }_3^{\prime }$ to maintain the steady rate of the ratcheting strain. Terms $C_2$ and ${\gamma }_2^{\prime }$ are acquired through a trial and error procedure. The increment in the equivalent plastic strain, $dp$, is defined as:

$$dp=\sqrt{\frac{2}{3}d{\overline{\epsilon }}_P.d{\overline{\epsilon }}_P}$$

(8)

where $d{\overline{\epsilon }}_P$ represents the increment in plastic strain.

2.3. Method of Solution

The ratcheting of additively manufactured samples made of Al 4043 alloy was studied through the finite element approach. Different solid elements of linear brick elements with 8 nodes, the quadratic tetrahedron with 10 nodes, and the brick elements with 20 nodes were taken to analyze plastic deformation through the Chaboche material model over the asymmetric loading cycles. To study the plastic accumulation over loading cycles, the Chaboche hardening rule was employed. The coefficients of the Chaboche hardening model were calibrated through the hysteresis loops constructed based on the strain-controlled tests. The geometry of AM samples was modeled in ANSYS software. Material properties and Chaboche coefficients of AM Al 4043 were determined as input values to simulate the ratcheting response. For each element type, meshing convergence was studied, and the required results were obtained. Figure 1 presents a stepwise flowchart for the ratcheting simulation of AM Al 4043 samples through use of the Chaboche hardening framework.

3. Results and Discussion

A finite element model was developed to study the ratcheting of additively manufactured samples made of Al 4043 alloy through a metal 3D printing process based on the cold metal transfer (CMT) welding technique. Different solid elements, shapes, and numbers were taken to analyze plastic deformation through the Chaboche material model over the asymmetric loading cycles. The coefficients of the Chaboche hardening model were calibrated through hysteresis loops constructed based on the strain-controlled tests.

3.1. Materials and Ratcheting Data

To simulate the ratcheting strain of AM Al 4043 samples, ratcheting data at various stress levels were taken from a research paper published by Wang et al. [15]. They employed a Fronius cold metal transfer (CMT) welder and an ABB IRB 4600 industrial robot to print Al 4043 plates. After the first layer of aluminum was formed, the welding gun rose 4.5 mm and the printing process was continued along the previous layer. After each layer was printed, there was a 120 s cooling interval between layers. The Al 4043 samples were then cut from the 3D-fabricated aluminum plate. Figure 2 represents the stress–strain curves of AM aluminum alloy examined in this study.

A bidirectional testing system, IBTC-300, was used to perform cyclic tests on Al 4043 samples at 0.01 mm/s and at different stress levels. Four applied stress levels of ${\sigma }_m\pm {\sigma }_a=$ 70 ± 55, 75 ± 55, 65 ± 65, and 75 ± 60 MPa were taken to simulate the ratcheting of AM Al 4043 samples.

3.2. Chaboche Coefficients

The coefficients of the Chaboche hardening rule were determined from stress–strain hysteresis loops obtained from strain-controlled testing results. A proper set of material coefficients were acquired as the simulated and measured hysteresis loops overlap closely with nearly the same shape and size. Figure 3 presents the comparison of three different sets of Chaboche coefficients for the Al 4043 sample tested under constant strain amplitude of ±1%. The deviation of the simulated and the measured data in Figure 3 is attributed to the values taken as material coefficients of $C_{1-3}$ and ${\gamma }_{1-3}^{\prime }$. The wider loop in Figure 3 was obtained as $C_1$ increased from 280 to 318. In constant to $C_1$, as coefficients $C_2$ and ${\gamma }_1^{\prime }$ reduced, narrower loops were achieved. For Al 4043 alloy, Chaboche coefficients of $C_{1-3}$ = 280, 16, and 5 GPa and ${\gamma }_{1-3}^{\prime }$ = 4500, 800, and 0 resulted in a tight overlap of the experimental and simulated loops. The material properties and coefficients for AM Al 4043 alloy were tabulated in

Table 1. Coefficients of the Chaboche hardening rule.

Material	${\mathit{\sigma }}_{\mathit{m}}\pm {\mathit{\sigma }}_{\mathit{a}} \left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$	Chaboche Coefficients
	75 ± 55	$C_{1 }\left(\mathrm{GPa}\right)=280$	${\gamma }_1^{\prime }=4500$
Al 4043	70 ± 55	$C_{2 }\left(\mathrm{GPa}\right)=16$	${\gamma }_2^{\prime }=800$
	65 ± 65	$C_{3 }\left(\mathrm{GPa}\right)=5$	${\gamma }_3^{\prime }=0$
	75 ± 60

.

3.3. The Finite Element Method

Through the use of Ansys 2022 R [31], a finite element model was developed to numerically assess the ratcheting response of AM Al 4043 samples. Figure 4 represents the flat shape geometry of aluminum samples with thickness of a 1 mm. Test sample movements in x, y, and z directions were limited by fixing one end of the sample. Sinusoidal loading cycles were applied with frequency of 0.01 mm/s from another specimen end. Through the developed FE model, different element types were examined as shown in Figure 4. The solid elements in this figure were created through using the sweep mesh method. Using the sweep method, a source surface was meshed, and it was then swept through a body to generate volume meshing. Three meshing structures of the linear brick elements with 8 nodes, the quadratic tetrahedron with 10 nodes, and the brick elements with 20 nodes were designed. Figure 4 presents mesh structures for the brick solid and the tetrahedron solid elements.

The convergence of displacement for each element type was studied to determine the independent nature of the result from the number of elements. Figure 5a presents the convergence curve for the linear brick element. As the number of elements increased, there was no substantial change in the result and the convergence was achieved at 119 elements. The quadratic tetrahedron element in Figure 5b shows a constant displacement when the number of taken elements exceeded 190. In this figure, as the number of elements was increased from 192 elements to 500 elements, the difference in displacement was found to be lower than 0.14%. The trend of convergence the quadratic brick elements in Figure 5c revealed that there was no significant change in the result above 120 elements and the model converged at this point. A sufficient number of elements led to an accurate estimation of the maximum and minimum strain values at the mid-section (gauge length) of samples.

Figure 6 exhibits the contour plot presentation of total strain for the AM Al 4043 sample under cyclic loading of 75 ± 60 MPa. The maximum total strain of 0.040996 was achieved at the specimen gauge length with the smallest area. The minimum strain of 0.000314 was obtained in the locations farther than critical area. In a distance far enough from this section, the contour lines of the total strain stay constant. As the elements corners are nodal locations, the Gaussian integration points are positioned within the element. Contour plots are generated from FE results by extrapolating stress and strain values from Gaussian integration points to nodal locations. This extrapolation process is performed using the shape functions.

Table 2. Different mesh structures information [33].

Element Type	Ansys Software Code	Integration Points	Pictorial [33]
8-Node Brick	SOLID185	$2\times 2\times 2$
10-Node Tetrahedron	SOLID187	4
20-Node Brick	SOLD186	$2\times 2\times 2$

presents the details of element type, shape, and nodal features employed in this study.

3.4. Backstress Increments and Yield Surface Evolution

In the deviatoric stress space, the backstress increments represent the magnitude and direction of the yield surface translation over the loading process beyond the elastic limit. According to Chaboche postulation, the backstress increment was achieved through integration of backstress increments, $d\overline{\alpha }={\sum }_{i=1}^{M}d{\overline{\alpha }}_i$. Figure 7 represents the yield surface translation for the AM aluminum sample for three different element types. The initial yield surface of the Al 4043 sample is depicted by a solid line. This figure also presents the stress–strain curve of additively manufactured Al 4043 through using right vertical and upper horizontal axes. When the applied load increases beyond the elastic limit, the yield surface translates (dashed-line surfaces) into the deviatoric stress space, intercepting the stress–strain curve. Beyond the elastic limit, the initial yield surface of Al 4043 with the origin of O was shifted to center $O^{*}$. The translated yield surface is shown with dashed lines intercepting the stress–strain curve at point $P^{*}$. The evolution of the yield surface by means of different element types resulted in the same direction of backstress increment $d\overline{\alpha }$with various magnitudes as shown in Figure 7. The lower amount of backstress increment resulted in a smaller vector of $\overline{O{O}_1^{*}}$ achieved through the second-order brick elements, while the linear brick elements possessed a larger translation from $\overline{O{O}_3^{*}}$. By surpassing the yield limit, the origin of the yield surface with the second-order tetrahedron element was shifted to $O_2^{*}$ and was placed between the translated centers for two other element types.

The mean values of backstress increments determined from the peak and valley of each loading cycle were plotted for different element types at various applied stress levels in Figure 8. The yield surface translation was governed by the Chaboche hardening rule through the backstress increments as the number of cycles progressed. Figure 8 shows that as the number of cycles increases, the average backstress moderately declines and results in a steady-state condition at longer cycles. The average backstress through using the second-order brick elements possessed a lower amount as compared to other element types. The highest backstress increment was associated with the first-order brick elements, which manifested the larger translation of the yield surface in Figure 7. The backstress curve of the quadratic tetrahedron element collapsed between the two other element curves consistent with the vector $\overline{O{O}_2^{*}}$ and the corresponding translated yield surface.

3.5. Simulated Ratcheting Results

The ratcheting response of AM Al 4043 samples under different stress levels was simulated through the finite element method and different element types, shapes, and nodes. The Chaboche kinematic hardening rule as a material model in Ansys software was employed to evaluate the ratcheting response of Al 4043 samples with three different element types. Figure 9 illustrates the simulated ratcheting strain curves for aluminum samples tested at 75 ± 55 MPa, 70 ± 55 MPa, 65 ± 65 MPa and 75 ± 60 MPa. In each loading condition, the simulated ratcheting curves through using quadratic brick elements resulted in larger magnitudes, while the ratcheting level declined, respectively, for the quadratic tetrahedron and the linear brick elements. The simulated curves based on different element types revealed that the ratcheting amount was increased by increasing the stress levels. The ratcheting magnitude was increased by as high as 12% in Figure 9b as compared with the simulated ratcheting curve in Figure 9a as the stress amplitude was increased from 70 MPa to 75 MPa. The simulated ratcheting curves closely agreed with the experimental values. The deviation of simulated ratcheting curves from those of measured values stayed between 2% and 12%.

Figure 10 represents the simulated stress–strain hysteresis loops for AM Al 4043 samples at various stress levels of 75 ± 55 MPa, 70 ± 55 MPa, 65 ± 65 MPa and 75 ± 60 MPa. Hysteresis loops for the 1st and 250th cycles are, respectively, simulated through FE analysis and the use of three different element types. Figure 10a,c,e,g present the first-order simulated stress–strain hysteresis loops for the first stress cycle applied with different stress levels. These figures present simulated hysteresis loops constructed based on different element types and those obtained experimentally. In the first cycle, the simulated loops based on the quadratic tetrahedron, linear brick, and quadratic brick elements resulted in a small difference as compared with the measured loops. As the number of cycles increased, samples with the linear brick elements tested at 75 ± 55 MPa (see Figure 10b) and 70 ± 55 MPa (see Figure 10d) closely agreed with the measured loops. At these stress levels, samples meshed with the quadratic tetrahedron as well as quadratic brick elements resulted in simulated loops with larger deviations, respectively. At stress levels of 65 ± 65 MPa (see Figure 10f) and 75 ± 60 MPa (see Figure 10h), the simulated loops by the second-order brick elements demonstrated a better agreement with the experimental loops at the 250th cycle. The evolution in hysteresis loops during 250 stress cycles revealed that the loops obtained from the quadratic brick element progressed more than the simulated loops with the quadratic tetrahedron and linear brick elements. In Figure 10, the stress–strain hysteresis loops for the first cycle were found to be almost identical in shape and width as compared with the corresponding measured loops. The decrease in the width of hysteresis loops over the number of cycles manifested the cyclic hardening response in AM Al 4043, leading to a drop in the ratcheting rate of aluminum samples. The minimum deviation between simulated and measured hysteresis loops in the 250th cycle is related to the linear brick element in Figure 10b with 3% and the maximum deviation is as high as 12%, as shown in Figure 10d for quadratic brick elements.

4. Conclusions

The finite element approach is a more affordable and reliable way to examine the efficiency of employing AM materials in various industries under different loading conditions. A major cause of catastrophic failure in the AM industry is the progression of plastic deformation over asymmetric loading cycles. Material damage can be assessed efficiently and reliably using the finite element method and the use of a prominent hardening model. The literature holds a limited number of studies to simulate the ratcheting response of AM materials through FE analysis. The choice of different element types greatly influenced the simulated ratcheting of AM Al 4043 samples by means of finite element analysis at different stress levels. Three meshing elements of the linear brick elements with 8 nodes, the quadratic tetrahedron with 10 nodes, and the brick elements with 20 nodes were employed to study the effect of meshing structure on the cyclic behavior of AM materials. The Chaboche hardening rule was employed to translate the yield surface of materials through an increment in backstress. A proper set of coefficients for the Chaboche model was obtained through strain-controlled tests and constructed stress–strain hysteresis loops. Any deviation from the Chaboche coefficients resulted in considerable differences between the simulated and experimental loops. The increment in backstress controlled yield surface translation with different magnitudes, as different element types of the linear brick, quadratic tetrahedron, and quadratic brick were employed. The simulation based on quadratic brick elements resulted in the smallest yield surface translation, while the linear brick elements led to the highest yield surface translation. Through use of quadratic brick elements, the simulated ratcheting curve shifted to a higher level, while the linear bricks and the quadratic tetrahedrons elements resulted in lower simulated ratcheting curves. The simulated stress–strain hysteresis loops for various types of elements presented a small difference in the first cycle, while quadratic brick elements progressed more during 250 stress cycles. The simulated ratcheting curves with different element types demonstrated a deviation ranging between 3% and 12% as compared with the experimentally measured values. The minimum deviation was related to linear brick elements, while the quadratic brick element results in maximum deviation.