Studies on Finite Element Analysis in Hydroforming of Nimonic 90 Sheet

Abstract

The primary goal of this study was to investigate the formability of Nimonic 90 sheet which performs well at high temperatures and pressures, making it ideal for applications in the aerospace, processing, and manufacturing industries. In this present study, finite element analysis (FEA) and optimization of process parameters for formability of Nimonic 90 in sheet hydroforming were investigated. The material’s mechanical properties were obtained by uniaxial tensile tests as per the standard ASTM E8/E8M. The sheet hydroforming process was first simulated to obtain maximum pressure (53.46 MPa) using the FEA and was then validated using an experiment. The maximum pressure obtained was 50.5 MPa in experimentation. Since fully experimental or simulation designs are impractical, the Box–Behnken design (BBD) was used to investigate various process parameters. Formability was measured by the forming limit diagram (FLD) and maximum deformation achieved without failure. Analysis of variance (ANOVA) results also revealed that pressure and thickness were the most effective parameters for achieving maximum deformation without failure. Response surface methodology (RSM) optimizer was used to predict optimized process parameter to achieve maximized response (deformation) without failure. Experimental validation was carried out for the optimized parameters. The percentage of error between experimental and simulation results for maximum deformation was less than 5%. The findings revealed that all the aspects in the presented regression model and FEM simulation were effective on response values.

1. Introduction

In most the industries, including the automotive and aerospace sectors, hydroforming was used to manufacture components which are challenging in metal forming [1]. It produces structurally strong components with complex geometry quickly, efficiently, and cost-effectively [1,2,3]. The benefits of hydroformed parts include improved strength to weight ratios, weight savings from section designs that are more effective, fewer parts, and reduced costs associated with tooling development, better dimensional stability and reproducibility of produced components, and subassemblies [4]. Since there are no welding joints, the parts produced in hydroforming can absorb more crash energy. This means that vehicles are more crashworthy, which translates into improved safety for vehicle occupants in the event of a crash [2]. All complex geometries of automotive components, such as rear axle subframe, a front axle, twin elbow exhaust manifold, fuel tank and roofs for luxury class cars, can be obtained through hydroforming [2,5]. Hydroforming processes are eco-friendly, as they reduce the amount of scrap, emit less noise pollution, and protect the environment, as forming is carried out only by the liquid medium [2]. Tube hydroforming and sheet hydroforming are the two primary divisions of the hydroforming process [4]. Sheet hydroforming is a near net shape manufacturing process, which means the parts it produces are very close to the final specified geometry and require very little rework [1]. This process will result in a reduction in the number of production steps and components in an assembly. This would reduce dimensional variations and make assembly easier [3]. The sheet hydroforming process, as shown in Figure 1, uses high pressure fluid for deformation of a blank (sheet) into a desired shape with die.

The most common material instabilities in sheet metals are wrinkling and tearing. Parameters of the sheet hydroforming process must be adequately modified to produce the desired results without wrinkling and tearing. The forming process parameters of sheet hydroforming includes pressure, blank holder force, sheet thickness, etc. Although most hydroforming operations are kept under 1000 bar, pressure intensification systems on high pressure hydroforming equipment can reach pressures ranging from 1000 to 4000 bar. Exceeding 4000 bar is possible, but it reduces the equipment’s service life while drastically increasing its complexity [6,7]. Blank holding force (BHF) will depend on the magnitude of the fluid pressure and the area of the blank in contact with the blank holder. It should be noted that in sheet hydroforming, the area of the blank in contact with the blank holder continuously decreases, and so, proper BHF is required to avoid wrinkling and rupture [8]. Sheet thickness influences formability and forming limits [9].

Improved formability in hydroforming is primarily caused by more evenly distributed strain, which results in less thinning at the corners [10]. In all forming operations using sheet metal as an input material, it is critical to understand the conditions that cause necking (instability of material) or fracture. Such limits can be represented as a forming limit diagram (FLD) shown in Figure 2, which plots the curve of major and minor strain coordinates [11]. The strain in the direction of the maximum strain is defined as the major strain. The strain perpendicular to the major strain is known as the minor strain. The major strain is always positive and is plotted vertically, while the minor strain is plotted horizontally [12]. The combinations of major and minor strains lying below the forming limit curve (FLC) define a safe operating region and failure is represented by the region above the FLC. FLD offers a useful summary. Formability helps to quickly identify key areas that need additional investigation, especially for early feasibility studies.

As the experimental procedure for the metal forming process is costly and time-consuming, the finite element method (FEM) has the advantage of lowering production costs by predicting part defects such as spring-back, rupture, wrinkling, buckling, and shape errors, as well as optimizing process parameters [13].

Design of experiments (DOE) approaches were used to maximize response variables in the presence of multiple factors. DOE is the process of using geometric concepts to statistical sampling in order to produce desired outputs. The DOE’s primary goal is to obtain the desired response with the fewest possible trials because conducting fewer experiments results in a reduction in the cost and time needed to carry out the experiments [14].

Nickel alloys are long-lasting materials known for their ability to operate at extremely high temperatures for extended periods of time. Nickel-based superalloys with outstanding high-temperature tensile strength, improved oxidation resistance, weldability, fatigue resistance, corrosion resistance, and long-term structural stability were used in high-temperature parts of aeroengines and industrial gas turbines [15]. Single crystal superalloys based on nickel have outstanding high-temperature mechanical properties and are commonly employed as turbine blade materials in current aviation gas turbine engines [16]. Although strong, nickel alloys are also relatively ductile, allowing them to be formed using a variety of different processes, although at higher pressures than other metals [17]. Nimonic 90, an nickel alloy, is an ideal material to use in aircraft parts, exhaust nozzles, and gas turbine components where the pressure and heat are extreme [18]. Nimonic 90 has high strength at high temperature levels, and it is highly resistant to scaling, oxidation, heat, and corrosion [19].

Existing studies lack in hydroforming Nimonic 90 sheet. The aim of this study is to propose an FEA model for formability analysis in the hydroforming of Nimonic 90 sheets. The following objectives will help to achieve this goal:

• Derivation of FEA model for sheet hydroforming;
• Validation of FEA result with experimentation;
• Evaluation of forming limit diagram;
• Determination of optimum process parameters for hydroforming of Nimonic 90 sheet;
• Discussion of the FEA model’s accuracy.

In this present study, first, mechanical properties of Nimonic 90 sheet were obtained by uniaxial tensile test as per the standard ASTM E8/E8M. Secondly, finite element method (FEM) simulation of the process was run to obtain the maximum pressure and blank holder force and was compared to experimental results. Thirdly, Box–Behnken design (BBD) of response surface methodology (RSM) was used to design the experiments by using lower and higher levels of variable parameters. Fourthly, FEM simulations were carried out as per the design of experiments (DOE). Fifthly, the impact of process factors (Pressure, Blank Holder Force, and Sheet Thickness) during the hydroforming of Nimonic 90 sheets was analyzed using RSM. Sixth, RSM optimizer was used to predict the optimized process parameter to achieve maximized response (deformation) without failure (crack or wrinkling). Lastly, a validation experiment was conducted, and the findings were discussed.

2. Material and Methodology

In this section, details of material properties, computer-aided design (CAD) modeling, FEM simulation, experimentation DOE, and optimization are discussed.

2.1. Material and Its Properties

In this study, the material used was Nimonic 90. The material was tested in the AUM Meta Lab, Mumbai, India, to obtain the chemical composition that is illustrated in

Table 1. Composition of Nimonic 90.

Ni	Cr	Co	Ti	Al	Fe	Mn	Si	Cu	C	P	S
Bal	18.65	16.7	2.37	1.38	0.98	0.67	0.34	0.085	0.082	0.008	0.007

.

Uniaxial tensile tests were carried out as per the standard ASTM E8/E8M as shown in Figure 3. The specimens were cut using electrical discharge machining, as shown in Figure 4.

The mechanical properties of the Nimonic 90 sheet as determined from the tensile test are given in

Table 2. Mechanical Properties of Nimonic 90.

Material Properties	Value
Yield stress (0.2%)	587 MPa
Young’s Modulus	105.95 GPa
Poisson’s Ratio	0.28
Ultimate Tensile Stress	1271 MPa
Strain Hardening Exponent at n value	0.30398
Strength Coefficient at n value	1555.36974

. True stress (σ_t) and True strain (ε_t) [20] were calculated using Equation (1).

σ_t = σ_e (1 + ε_e), and ε_t = ln(1 + ε_e),

(1)

where σ_e is Engineering stress and ε_e is Engineering strain.

The multilinear points of the stress–strain curve can be obtained using Ramberg–Osgood equation [21]. Where $\epsilon$ is strain, E is Young’s modulus, σ is stress, σ_y is yield strength, and n is strain-hardening coefficient

$$\epsilon =\frac{\mathsf{\sigma }}{\mathrm{E}}+0.002{\left(\frac{\mathsf{\sigma }}{\mathsf{\sigma }\mathrm{y} }\right)}^{\left(1/n\right)}$$

(2)

This Ramberg–Osgood equation shown in Equation (2) was used to approximate the non-linear relationship between strain and stress.

2.2. CAD Modeling and Finite Element Simulation

2.2.1. CAD Modeling

In this study, Autodesk Fusion 360 software was used to model components as shell elements for FEA simulations, as shown in Figure 5 Shell elements were used in FEA to achieve better results, because they allowed modelling of narrow features with fewer mesh components [22], and computational time was reduced. As per the approximation model shown in Figure 6, thin shell approximation was applied in this model as the ratio between thickness and length (h/L Chart) was within 0.3 [23].

Thickness of all surface components were assigned in FEA simulation. Dimensions of all components used in CAD modeling are shown in

Table 3. Dimensions of Die and Sheet.

Component	Dimension (mm)
Top Die	100 × 100
Bottom Die	100 × 100
Sheet	65 × 65
Sheet Thickness	0.8, 1 and 1.2
Die Cavity	38 × 38 × 20

. The die corner radius should be greater than four times the material thickness [24].

2.2.2. Development of Finite Element Model

To input the properties of Nimonic 90, two regions, elastic and plastic regions, were considered. The elastic properties were assigned as listed in Table 2 and true stress and true strain of the plastic region of the material were given as per Figure 7.

Meshing was carried out using Ansys Mechanical. The elements used for meshing were quadrilateral elements as they produced far smoother surfaces than triangular elements, as triangular elements frequently produced visible anomalies on the surface [25]. All the free elements were set to quad type and the element size was 2 mm. Some of the critical areas such as the blank surface, fillets, and corner surfaces were defined with smaller element sizes of 1 mm using the face sizing option. Figure 8a–c shows the FEA mesh model. Once the mesh was generated, it was exported in STL format to LS Dyna. In this work, mesh refinement for convergence study was performed. Simulations were run for various mesh refinement stages.

2.3. Development of Experiment Model

To validate the outcomes of numerical simulations and identify the optimum process parameters, experimental work was carried out using a 100-ton hydraulic press, as shown in Figure 9, and 1000 bar pressure pump.

The dies shown in Figure 10 and Figure 11 were made of P20 tool steel material which has a high degree of resistance to the deformation [26].The top plate and the bottom plate were attached to the respective dies with M8 bolts, which allowed the die to be clamped with the hydraulic press. The dimension of die and sheet used are given in Table 3. The specimen, Nimonic 90, was cut as per the dimension shown in Figure 12 using a laser cutting machine.

The validation experiment was conducted for maximum pressure and for the optimum process parameters that were obtained using an RSM optimizer.

Since the finite element simulation was validated using the experimental model and it was in the permissible limit, the same was proceeded with in the design of experiments approach.

2.4. Design of Experiments

The Box–Behnken design (BBD) of RSM was used in this study to analyze the regression model and determine the effects of variable parameters on the outputs. In the present study, three processing parameters including pressure (Pr), blank holder force (BHF), and thickness (T) of sheet were considered and their effects on the deformation without failure were investigated using RSM. The experiments were designed in accordance with the BBD by using lower and higher levels of variable parameters, procuring 15 experiments to run using Minitab software.

Table 4. BBD- Design of Experiments.

Std Order	Run Order	Pt Type	Blocks	Pressure (MPa)	BHF (kN)	Thickness (mm)
14	1	0	1	40	180	1
4	2	2	1	50	220	1
12	3	2	1	40	220	1.2
1	4	2	1	30	140	1
3	5	2	1	30	220	1
15	6	0	1	40	180	1
6	7	2	1	50	180	0.8
8	8	2	1	50	180	1.2
9	9	2	1	40	140	0.8
10	10	2	1	40	220	0.8
11	11	2	1	40	140	1.2
13	12	0	1	40	180	1
7	13	2	1	30	180	1.2
2	14	2	1	50	140	1
5	15	2	1	30	180	0.8

illustrates the conditions under which the simulations were performed.

The primary goal of RSM was to achieve an optimal response through a series of designed experiments. In most cases, the RSM regression model was a quadratic full equation [13] as Equation (3), where y is the response variable. Additionally, α₀, α₁, α₂, and α₃ are constant, linear, quadratic, and interaction coefficients, respectively. Additionally, x_i and x_j are the independent variables and E is the statistical error. The effectiveness of the regression model was then assessed using R² as Equation (3), which can be obtained from ANOVA.

$$y=\mathsf{\alpha }0+{\displaystyle \sum }_{i=0}^n\mathsf{\alpha }1 \mathrm{xi}+{\displaystyle \sum }_{i=0}^n\mathsf{\alpha }0{ \mathrm{xi}}^2+{\displaystyle \sum }_{i=0}^n{\displaystyle \sum }_{i=0}^n\mathsf{\alpha }3 \mathrm{xixj}+E$$

(3)

$$R^2=1-\frac{S_r}{S_t}$$

(4)

3. Results and Discussion

Figure 13 shows the analysis result of maximum pressure for failure. The simulation result for maximum pressure was validated using experimentation. Figure 14 represents the validation result of maximum pressure for failure during hydroforming.

Table 5. Maximum Pressure obtained by FEA Simulation and Experiment.

	Maximum Pressure (MPa)	Percentage Error (%)
FEA Simulation	53.46	5.53
Experimentation	50.5

represents the maximum pressure obtained during hydroforming in FEA simulation and experimentation. Then, the finite element simulation process was carried out under different conditions according to the design of experiments (Table 4), and the response variable, i.e., deformation (De), was obtained, as shown in

Table 6. Deformation results obtained from FEA Simulation.

	Inputs			Output
Run Order	Pr (MPa)	BHF (kN)	T (mm)	De (mm)
1	40	180	1	7.88
2	50	220	1	9.62
3	40	220	1.2	6.65
4	30	140	1	6.19
5	30	220	1	6.16
6	40	180	1	7.88
7	50	180	0.8	11.9
8	50	180	1.2	8.11
9	40	140	0.8	9.7
10	40	220	0.8	9.66
11	40	140	1.2	6.68
12	40	180	1	7.88
13	30	180	1.2	5.27
14	50	140	1	9.65
15	30	180	0.8	7.52

. Forming limit diagram obtained using LSdyna, as shown in Figure 15, depicts the major and minor strain of Nimonic 90 in hydroforming. Strain combinations over the FLC will result in fracture, whereas those below the wrinkling limit line will result in wrinkles. For a fixed minor strain, a larger gap between the FLC and wrinkling limit lines signifies more potential for forming [27]. In Figure 15, the gap between the FLC and wrinkling limit line was more, the Nimonic 90 sheet was more suitable for forming.

The ANOVA analysis yielded regression models for estimating the value of deformation (De) for Nimonic 90 as Equation (5).

De = 8.52 + 0.3483 × Pr − 0.00044 × BHF − 14.59 × T + 0.000262 × Pr × Pr − 0.000001 × BHF × BHF + 7.344 × T × T − 0.000000 × Pr × BHF − 0.1925 × Pr × T + 0.00031 × BHF × T

(5)

During this study, a confirmatory experiment was carried out to validate the optimized RSM solutions. Figure 16 shows the deformed Nimonic 90 sheet.

Table 7. Confirmatory test results.

Process Parameter			Predicted	Experimental	Percentage Error (%)
Pressure (MPa)	BHF (kN)	T (mm)	Deformation (mm)	Deformation (mm)
42.32	144.04	0.8	10.199	9.72	4.92

compares the predicted and experimental results of the response (deformation) in the formability of Nimonic 90. The table shows that the error percentage between predicted and experimental results was less than 5%.

ANOVA (analysis of variance) generally helps to understand the influence of independent input parameters on dependent output parameter(s). ANOVA helps the users to prove cause and effect relationships in various forms such as R² value, pareto chart, p-values, etc. Here, the Pareto chart shown in Figure 17 depicts the standardized effects of input parameters on output parameter (i.e., deformation). The R² value for the present regression model of deformation regarding formability of Nimonic 90 was greater than 95%, indicating the authenticity of model [17]. Furthermore, according to

Table 8. ANOVA results for each coefficient for deformation.

Coefficient of Regression Model	p-Value
Model	0.000
Linear	0.000
Pr	0.000
BHF	0.378
T	0.000
Square	0.000
Pr × Pr	0.337
BHF × BHF	0.962
T × T	0.000
2-Way Interaction	0.000
Pr × BHF	1.000
Pr × T	0.000
BHF × T	0.920

, the p value for most of the input terms were less than 0.05, implying that these terms had a significant influence on the value of deformation in the sheet [28].

Figure 18 shows the optimized process parameter achieved in RSM optimizer. In experimental validation, the error percentage between experimental and simulation was less than 10%. This indicates that the proposed simulation model is capable of making accurate predictions [28,29]. Figure 19 shows the mesh convergence study.

4. Conclusions

In this study, a finite element simulation of formability of Nimonic 90 in sheet hydroforming for investigating required pressure, blank holder force, and thickness reductions was conducted. The main conclusions from this study can be concisely summarized as follows.

• The Nimonic 90 sheet tested showed good formability. The formability was higher in the plane strain and biaxial tension condition compared to the tension–compression condition;
• The fluid pressure in sheet hydroforming caused the sheet to stretch in the flange area, forcing strains above the wrinkling limit curve in the forming limit diagram (FLD);
• Since the FLD indicated no failure zone, these process parameter values were acceptable;
• As p value in the two-way interaction between pressure and thickness was less than 0.05, it was vital in achieving maximum deformation;
• Based on finite element analysis and verified experiments, BHF of 144.04 kN, pressure of 42.32 MPa, and sheet thickness of 0.8 mm were the key parameters to prevent wrinkling under the forming state for achieving the maximum deformation;
• The statistical results revealed that the models proposed in analysis had a high accuracy to estimate the optimum pressure, blank holder force, and thickness for achieving maximum deformation in formability of Nimonic 90 sheets in sheet hydroforming;
• The results demonstrated that the most effective parameters on deformation were pressure and thickness;
• The proposed FEA model is capable of accurate predictions, as the error percentage between the experiment and simulation was less than 5%

Future research will include unconventional optimization techniques to predict the optimum process parameters for better formability and applying the same methodology for different super alloys and different shapes that are more complex. An optimization code will be utilized to create the multiple response optimization for hydroforming complex automotive parts.