Characterization and Finite Element Modeling of Microperforated Titanium Grade 2

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The mechanical properties of microperforated titanium sheets are studied in this work by experimental data and Finite Element Modeling.

Abstract

Hybrid Laminar Flow Control (HLFC) is a promising technology for reducing aircraft drag and, therefore, emissions and fuel consumption. The integration of HLFC systems within the small space of the wing leading edge, together with de-icing and high lift systems, is one of the main challenges of this technology. This challenge can be tackled by using microholes along the outer skin panels to control suction without the need for an internal chamber. However, microperforations modify the mechanical properties of titanium sheets, which bring new challenges in terms of wing manufacturability. These modified properties create uncertainty that must be investigated. The present paper studies the mechanical properties of micro-drilled titanium grade 2 sheets and their modeling using the Finite Element Method (FEM). First, an experimental campaign consisting of tensile and Nakajima tests is conducted. Then, an FEM model is developed to understand the role of the anisotropy in sheet formability. The anisotropy ratios are found by combination of Design of Experiments (DoE) and the Response Surface Method (RSM); these ratios are as follows: 1.050, 1.320, and 0.975 in the directions Y, Z, and XY, respectively. Some mechanical properties are affected by the presence of microholes, especially the elongation and formability that are significantly reduced. The reduction in elongation depends on the orientation: 20% in longitudinal, 17% in diagonal, and 31% in transversal.

1. Introduction

A new generation of large HLFC structures are being developed to reduce complexity by using microperforation along the outer skin to control suction without the need of internal chambering. Outer skins made of titanium alloys and variable microperforation patterns entail a new challenge in terms of forming. This must be investigated before deciding on the most suitable skin forming technology and alloy for these new HLFC structures, which will contribute to reduce 10% of both fuel consumption and pollutant emissions in future aircraft [1].

The process chain to manufacture HLFC components starts with the microperforation of metallic sheets, followed by a sheet metal forming process and, finally, cleaning tasks. The perforation is carried out by laser techniques that fulfill quality and high-rate production requirements [2,3]. On the other hand, several metal forming manufacturing processes are being considered to perform HLFC structures. Stretch Forming (SF) is one of the most promising technologies to manufacture HLFC components. SF consists of a rigid punch, called die, and a flat sheet clamped by gripping jaws at both edges. The sheet is forced to get in contact with the die while the tensile load is kept constant, which causes the sheet to stretch and bend over the die simultaneously [4]. Parts with various curve radii and that are wrinkle-free can be formed by this process at lower tooling costs in comparison with drawn stamping. The main disadvantages of SF are the limited ability to form sharp or re-entrant contours and the springback problem. To avoid these problems, different hot forming processes are nowadays commonly used. In the hot forming (HF) process, the metallic sheet is heated above recrystallization temperature to induce a soft and malleable state. Then, the sheet is positioned on a hot die while a hot punch goes down and shapes the component. The sheet is held under pressure and temperature for a time period and, finally, is released and cooled by natural convection. A soft and malleable state during large deformations reduces or avoids springback and cracking in small bend radii areas. On the other hand, there is elevated risk of oxidation in HF, and optimization of temperature exposure time, so using shielding gas and/or lubricants is needed. Other technologies such as laser forming or laser peen forming [5,6] can be considered, but they are currently in a lower TRL level. This paper focuses on the mechanical properties of micro-drilled and non-drilled titanium grade 2 sheets at room temperature.

Titanium is the ninth most abundant element in the earth’s crust (about 0.6%), and despite its relative scarcity, many industries have benefited from its favorable properties. High strength, low density, and excellent corrosion resistance are some of the main properties that make titanium an interesting material for a wide variety of applications, such as aerospace, energy, and chemical industries, or even uses in the biomedical field. Among all of them, titanium plays a key role in the aerospace industry, where weight plays a critical role in the design of more fuel-efficient aircraft. For instance, titanium alloys are employed for aero-engine applications or in the airframe systems where they can represent a moderate proportion of these components.

Pure titanium sheets, in their four distinct grades (1, 2, 3, and 4), show highly anisotropic mechanical behavior at room temperature due to the inherent crystallographic texture and the manufacturing process. It is crucial to use yield criteria that capture the fundamental characteristics of the plastic behavior during forming operations and under complex loading conditions [7]. This anisotropic behavior imposes severe restrictions on the form of yield conditions. The aim of this study is to characterize the planar anisotropic behavior of titanium sheets and to use a realistic yield criterion that can faithfully describe the material’s response and reproduce the experimental values. There are several models to describe the anisotropic plastic behavior and formability of sheet metals, and each of them uses different mathematical formulations. Among all the criteria, one of the most widely used is the model proposed by Hill in 1948 (called Hill48) [8], which was further improved by Hill90 [9]. Another relevant model, called Barlat89 [10], employs the transformation of the Cauchy stress or the BBC models (Banabic–Balan–Comsa) [11].

The Finite Element Method (FEM) has become a cornerstone in the field of mechanical analysis, offering a powerful computational tool to simulate and predict the behavior of materials and structures under various loading conditions [12,13,14]. Particularly in the aerospace industry [15], FEM is instrumental in assessing the performance of advanced materials like titanium alloys, which are prized for their strength, lightweight, and corrosion resistance. Microperforated titanium is a relatively new material that provides a novel solution for a variety of applications, from medical devices to aerospace components and sports equipment. The material is gaining interest among scientists, but there is not enough research to characterize and study the behavior of the material. This lack of research has resulted in a limited ability to fully comprehend its properties and potential applications. Hence, the aim of this work is to enhance comprehension and improve understanding of the material.

2. Materials and Methods

2.1. Tensile Testing

Tensile testing is a fundamental mechanical test widely used to determine the behavior of materials under axial forces. During this test, a specimen is subjected to a controlled tension until it fractures, allowing the measurement of material properties such as ultimate tensile strength, yield strength, elongation, and Young’s modulus. Initially, the sample experiences elastic elongation as it is pulled. With increasing stress, the specimen begins to undergo permanent deformation, known as plastic strain. The yield strength is the stress required to induce noticeable plastic deformation, typically defined at 0.2% strain. Once yielding begins, a significant dislocation motion occurs within the metal grains. It is expected that the presence of microperforations in the titanium specimens influences stresses, strains, and dislocations, which alters the mechanical properties.

The tensile tests were carried out with the aid of a Zwick 1484 (200 kN) testing machine (made by ZwickRoell GmbH & Co. KG, Ulm, Germany). The VideoXtens 1–120 was used for the strain measurement and the TestXpert III software (version 1.8) was employed to control the tests and collect data. The tests were performed at room temperature and the crosshead speed was set to 12 mm/min (which corresponds to the strain rate of 0.005 s⁻¹).

A sheet of titanium grade 2 with a thickness of 0.8 mm was microperforated and then cut into specimens by means of waterjet cutting. The geometry of the tested specimens is presented in Figure 1. They were then milled to ensure edge quality and to position microperforation exactly in the center of the specimen.

A microperforation squared pattern of 1 mm was introduced into some specimens to study how mechanical properties change in the presence of holes. These holes have a conical shape with a small diameter in one face (50.0 ± 3.0 µm) and a bigger diameter in the other (120.0 ± 6.0 µm). In all cases, microperforation pitch is aligned according to the rolling direction. Several microholes were observed by using an optical microscope and it was found that they are not exactly round, and the diameter varies from hole to hole. The authors assume that this fact does not affect the results.

Both perforated and non-perforated specimens were tested in different orientations: longitudinal (rolling direction), transversal, and diagonal. Moreover, all cases were repeated three times. In summary, 18 samples were tensile tested: 3 orientations × 2 materials × 3 repetitions. Perforated (PM) and base material (BM) are considered here as different materials.

2.2. Nakajima Testing

The Nakajima test is the primary method used to establish Forming Limit Curves (FLCs). In a Nakajima test series, several sheet metal specimens of different widths (see Figure 2) are deformed using a punch moved by a hydraulic press. The recorded data are used for creating accurate forming limit diagrams and understanding the material’s behavior under different forming conditions.

Erichsen 145/60 (made by Erichsen GmbH & Co. KG, Hemer, Germany) was used as a sheet testing device. The main properties of the sheet testing machine are summarized in

Table 1. Properties of Erichsen 145/60 universal sheet testing device.

Characteristic	Value
Nominal power	13.5 kW
Drawing force	up to 600 kN
Blanking force	up to 400 kN
Drawing speed	0–750 mm/min
Sheet thickness	0.2–6.0 mm

. Aramis 2M (made by GOM GmbH, Braunschweig, Germany) was used as a strain measuring device, and its properties are summarized in

Table 2. Properties of Aramis 2M optical measuring device.

Characteristic	Value
Camera resolution	1624 × 1236
Frame rate	up to 22 Hz
Strain measuring range	0.05–100%
Strain measuring accuracy	up to 0.02%
Measuring area	80 mm2–2.3 m2

. The Erichsen testing machine equipped with the Aramis 2M optical measuring device is shown in Figure 3.

Nakajima tests were performed according to ISO 12004 standards [16,17]. Specimen geometry affects the strain path; thus, different geometries must be manufactured to determine FLCs. At least five geometries for the description of a complete FLC are necessary, but a classical number of geometries is seven. The sheet of titanium grade 2 with a thickness of 0.8 mm was also microperforated and then cut into specimens, as presented in Figure 2.

Before the Nakajima tests, a speckle random pattern was introduced to the tested specimen by using white and black paints. Then, the specimen was placed in the Erichsen machine and deformed by the hemispherical punch until fracture. All specimens were in such a way that faces with bigger holes were in front of the punch. The blank holder force was set to 400 kN and the punch speed to 1 mm/s. PTFE foil and oil OKS 352 were used for lubrication purposes. Observation of deformation was performed by stereo cameras at a frame rate of 10 Hz, and strain calculation was carried out by comparison of stereo pictures. Eventually, the strain map was prepared and ready to be analyzed, as shown in Figure 4a. To define the values of major and minor strains determining the onset of necking, three linear sections are built perpendicular to the crack. Then, these sections were analyzed according to ISO 12004-2 (necking was compensated by performing curve fitting). An evaluation of section data according to ISO 12004 is presented in Figure 4b.

2.3. Finite Element Modeling

The authors also propose studying the material using a Nakajima test finite element model (FE model). Strain limits can be obtained for each mechanical behavior (equiaxial, plane strain, uniaxial tension, etc.) by deforming sheet metal blanks of different geometries with a hemispherical punch. The proposed model is a dynamic/explicit analysis which uses an explicit solution technique that integrates the equations of motion through time. This procedure is conditionally stable and has a robust contact functionality that easily solves even the most complex contact simulations.

The FE model of the Nakajima includes the main components such as the punch, blank holder, die, and sheet. The analysis was performed by modeling just a quarter of the model, taking advantage of the double symmetry of all Nakajima geometries (see Figure 2), which reduces the number of elements and saves time consumption. The following image shows the FE model for geometry G7.

The punch, blank, and holder have been modeled as discrete rigid components. The sheet is a 0.8 mm thick deformable shell (S4R elements) with variable geometry according to the shapes shown in Figure 2. The approximate size of the sheet elements is 2 mm. The material titanium grade 2 is anisotropic, so it is necessary to use a model that faithfully reproduces the mechanical properties. Considering that the material exhibits different yield in different directions, the Hill anisotropic yield criterion has been used [8]. This behavior is introduced in Abaqus through user-defined stress ratios ($R_{ij}$) that are applied in the quadratic Hill’s potential function.

Among the multiple yield criteria mentioned in the introduction, Hill48 is particularly well suited for modeling materials that exhibit anisotropic plasticity. This criterion was an extension of the isotropic von Mises yield criterion and can be presented as follows:

$$f\left(\sigma \right)=\sqrt{F{\left({\sigma }_{22}-{\sigma }_{33}\right)}^2+G{\left({\sigma }_{33}-{\sigma }_{11}\right)}^2+H{\left({\sigma }_{11}-{\sigma }_{22}\right)}^2+2L{{\sigma }_{23}}^2+2M{{\sigma }_{31}}^2+2N{{\sigma }_{12}}^2}$$

(1)

where F, G, H, L, M, and N are constants characterizing the anisotropy behavior of the material. Considering the model being studied in this work, it can be approximated as a plane strain case, and the Hill yield criterion is as follows:

$$f\left(\sigma \right)=\sqrt{\left(\mathrm{G}+H\right){{\sigma }_{11}}^2+\left(F+H\right){{\sigma }_{22}}^2-2H{\sigma }_{11}{\sigma }_{22}+2N{{\sigma }_{12}}^2}$$

(2)

The material constants (G, H, F, and N) can be expressed using stress ratios R_ij. The yield ratios are expressed relative to a reference yield stress (${\sigma }_0$), so if ${\sigma }_{ij}$ is applied as the only non-zero stress, the corresponding yield stress will be $R_{ij}{\sigma }_0$.

$$F={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{2}{{R_{22}}^2}}+{\displaystyle \frac{2}{{R_{33}}^2}}+{\displaystyle \frac{2}{{R_{11}}^2}}\right)$$

(3)

$$G={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{2}{{R_{33}}^2}}+{\displaystyle \frac{2}{{R_{11}}^2}}+{\displaystyle \frac{2}{{R_{22}}^2}}\right)$$

(4)

$$H={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{2}{{R_{11}}^2}}+{\displaystyle \frac{2}{{R_{22}}^2}}+{\displaystyle \frac{2}{{R_{33}}^2}}\right)$$

(5)

$$N={\displaystyle \frac{3}{{R_{12}}^2}}$$

(6)

In the case of planar anisotropy, the unknown ratios will be R₂₂, R₃₃, and R₁₂ since the rest has a common value of 1 (R₁₁ = R₂₃ = R₃₂ = 1). These yield ratios are the study variables which will be obtained by means of an RSM optimization process. The stress ratios are in turn related to the Lankford coefficients in the three angles from the rolling direction (0°, 45°, and 90°) which can be expressed as follows:

$$R_{22}=\sqrt{{\displaystyle \frac{r_{90}\left(r_0+1\right)}{r_0(r_{90}+1)}}}$$

(7)

$$R_{33}=\sqrt{{\displaystyle \frac{r_{90}\left(r_0+1\right)}{r_0+r_{90}}}}$$

(8)

$$R_{12}=\sqrt{{\displaystyle \frac{{3r}_{90}\left(r_0+1\right)}{{(r}_0+r_{90}\left)\right(2r_{45}+1)}}}$$

(9)

$$R_{11}=R_{13}=R_{23}$$

(10)

The model can be approached as a planar case of anisotropy because of the reduced thickness of the sheets [18]. Thus, the coefficients to be determined are R₂₂, R₃₃, and R₁₂ since R₁₁ = R₁₃ = R₃₂ = 1. It is necessary to define a local coordinate system associated with each element to correctly define the anisotropy.

The unknown yield ratios (R₂₂, R₃₃, R₁₂) will be obtained by means of RSM. The rest of the mechanical properties used in the FE model are provided by the tensile and Nakajima test mentioned above. The engineering curves from tensile tests were transformed to true stress–strain curves to be used in the model. Poisson’s ratio also has to be defined, however it is hard to define a reliable value for titanium grade 2 since anisotropy leads to small differences in both elastic and shear moduli. It was assumed a common Poisson’s ratio to all materials of a value of 0.37. Finally, it is also necessary to define a damage initiation criterion to indicate when the sheet starts to break. When the damage is produced, the analysis removes the elements from the mesh that have reached that criterion. In the proposed model, the forming limit diagram (FLD) is used as a damage initiation criterion.

The interaction between different components is conducted by means of a penalty contact with different friction coefficients: µ = 0.05 for punch–specimen contact and µ = 0.20 for die–specimen and blank holder–specimen contacts. Punch–specimen contact friction is lower due to the use of lubrication.

The boundary conditions applied to the model are as follows (Figure 5):

• Die movements are fully restricted in the reference point.
• The punch and blank holder can only move along the Y-axis.
• X-axis and Z-axis symmetry applied to the specimen.

Regarding the forces and displacements applied to the model, the punch stroke is 20 mm in the Y-axis direction. On the other hand, the holder force is not applied linearly but by means of a step function of 100 kN (one fourth of the total force due to symmetry). This variation of load throughout the analysis is modeled by amplitude curves defined as a mathematical function (see Figure 6).

2.4. Calibration Process

RSM is a group of statistical methods that uses DoE, polynomial functions, and the gradient descent method. Nowadays, RSM has been and continues to be used to implement multi-objective optimizations of processes both experimentally [19,20] and by models [21,22]. The first step is to define the variables that will be studied (inputs and outputs). In this case, the inputs are the anisotropic ratios (R₂₂, R₃₃, R₁₂) and the outputs are the errors of FE models’ results compared with Nakajima measurements. The range of the three inputs is 0.65–1.35. The flow diagram presented in Figure 7 summarizes the relationship between testing, FE modeling, and the calibration process.

In this case, there are three inputs (factors) and therefore the number of combinations is 3³ = 27 considering three values per factor. This kind of design is called a full three-level design (or 3 k) because it considers all possible combinations with three values (also called levels). There are seven geometries and three directions (except for G7) for each case; thus, there are 27·(6·3 + 1) = 513 models that must be launched. Note that geometry G7 has only one direction because of its round shape. There are several methods to reduce the number of combinations (such as Box–Behnken or Central Composite Designs [23]). However, such reduction is not needed in this case because the computational cost of Nakajima test models is relatively low: around 3 min using parallel computing with 12 cpus (Intel Xeon E5-2690v3, 192 GB RAM).

For each case, nineteen models are launched, each one with a different geometry (as shown in Figure 2) and different direction. When all these simulations are completed, an output for each case, geometry, and direction is calculated using the Root Mean Square Error (RMSE):

$$Y_{kjo}=RMS{E}_{kjo}=\sqrt{{\displaystyle \frac{1}{n_j}}·\sum _{i=1}^{n_j}{\left(Y_{joi}^{EXP}-Y_{kjoi}^{FEM}\right)}^2}$$

(11)

where

• Y_kjo is the output for geometry j and orientation o for case k. It quantifies how different the FE model punch is compared to the experimental punch;
• n_j is the number of measurements (time points) of Nakajima geometry j in each curve;
• $Y_{joi}^{EXP}$ is the experimental punch stroke of Nakajima geometry j and orientation o at time point i;
• $Y_{kjoi}^{FEM}$ is the FE model punch stroke of Nakajima geometry j and orientation o at time point i for case k.

Using all these inputs and outputs, it is possible to fit a low-degree polynomial. Quadratic functions are one of the most used functions to relate inputs and outputs:

$$Y=b_0+\sum _{i=1}^n{b}_i·x_i+\sum _{i=1}^n{b}_{ii}·x_i^2+\sum _{i=1}^{n-1}\sum _{j=i+1}^n{b}_{ij}·x_i·x_j+e$$

(12)

where the first summation is the linear component, the second is the quadratic component, the third is the product of the variables, and e is the error. The values of the coefficients b₀, b_i, b_ii, and b_ij must be calculated by the minimum squared method. Once the quadratic models are fitted, it is possible to find the optimum inputs (R₂₂, R₃₃, R₁₂) by using the steepest ascent method [24]. The objective of the optimization is to minimize the nineteen outputs at the same time. When a problem has multiple outputs, it is called the Multi-Response Surface Method (MRSM), which deals with conflict between responses. An optimal configuration for one output may diverge substantially from the optimal configuration for another output. Harrington [25] presented a compromise between responses by so-called desirability functions. In this case, there are nineteen outputs (one for each Nakajima geometry and orientation). Here, we use Harrington equations to optimize the anisotropy ratios to minimize the outputs defined in Equation (11).

3. Results and Discussion

3.1. Tensile Test Results

Microperforated specimens with longitudinal and transversal orientations show similar cracks, but specimens with a diagonal orientation look different. It can be concluded that cracks always move along the shortest path between two microholes (Figure 8).

A comparison of microperforated and non-microperforated tensile test results is shown in Figure 9. Based on these results, it can be concluded that microperforation significantly reduces the elongation and the elastic modulus. These reductions can be explained by the reduction in resistant area and the concentration of stresses caused by the holes. It is important to highlight that elongation reduction depends on the orientation: 20% in longitudinal, 17% in diagonal, and 31% in transversal.

Counterintuitively, yield stress and ultimate tensile strength increase slightly in microperforated material in the three orientations. Yield stress increases by 3–5% depending on orientation, and ultimate tensile strength near 3%. The most relevant attributes of stress–strain curves are summarized in

Table 3. Tensile properties of titanium grade 2.

	Orient.	Elastic Modulus (MPa)	UTS (MPa)	Yield Stress (MPa)	Elongation
BM	Long.	115,000	460 ± 4	330 ± 3	0.420 ± 0.010
	Trans.		439 ± 4	397 ± 7	0.435 ± 0.033
	Diag.		426 ± 2	365 ± 3	0.440 ± 0.003
PM	Long.	82,000	481 ± 3	342 ± 4	0.335 ± 0.005
	Trans.		461 ± 1	410 ± 1	0.300 ± 0.005
	Diag.		439 ± 3	373 ± 4	0.365 ± 0.005

. The explanation for these results is unclear. Only three repetitions per case were conducted, and the increments are small; thus, it could be a consequence of random variation rather than a real effect of the microholes.

Titanium grade 2 is anisotropic, and the introduction of microperforation does not change this fact. Figure 10 shows the average tensile test results for perforated material in the three orientations. These results indicate substantial differences between the flow curves, especially in terms of yield stress, ultimate tensile strength, and elongation. For example, the maximum tensile strength is reached in the longitudinal direction and the highest elongation in 45° specimens. These results are consistent with those obtained by Pham Quoc Tuan et al. [26].

3.2. Nakajima Test Results

Microperforated specimens oriented in longitudinal and transversal orientations have similar cracks, but specimens oriented in the diagonal orientation crack look different. Exactly as happened in tensile testing, cracks always move along the shortest path between two holes that correspond to longitudinal and transversal orientations.

The FLC points for microperforated samples are presented in

Table 4. FLC points for microperforated samples.

Geometry	Longitudinal		Transversal		Diagonal
	Minor Strain	Major Strain	Minor Strain	Major Strain	Minor Strain	Major Strain
G1	−0.164	0.287	−0.168	0.263	−0.230	0.348
G2	−0.127	0.254	−0.138	0.242	−0.178	0.305
G3	−0.090	0.211	−0.091	0.199	−0.115	0.240
G4	−0.059	0.177	−0.057	0.168	−0.068	0.188
G5	−0.052	0.157	−0.043	0.149	−0.041	0.163
G6	−0.028	0.132	−0.001	0.106	0.005	0.128
G7	0.087	0.106	0.087	0.106	0.087	0.106

, which were calculated as the mean of three repetitions for each geometry. Variability between repetitions is less than 3% in all cases. Figure 11 shows the FLCs for microperforated samples and compares them with ordinary titanium grade 2 obtained from the literature [27]. The microperforation significantly reduces the formability of titanium grade 2 sheets. The mayor strain in unperforated material at the plane strain condition is approximately 32%, and it reduces to 10–13% when the material is perforated. On the other hand, orientation does not play a significant role, only the formability of transversal specimens is slightly worse than other specimens. It seems like it is related to the fact that a crack always moves on the shortest way between two holes.

Another important piece of data obtained from the Nakajima test is the punch load–stroke curves (Figure 12). The punch–stroke curves for different orientations vary significantly, mainly at the end of the tests. At the beginning, the three orientations are remarkably similar for all geometries; however, after a certain stroke (near 15 mm), the longitudinal direction is appreciably stronger and requires more stroke to break. Diagonal and transversal orientations overlapped but transversal (green) always broke before diagonal (red). All these conclusions prove that anisotropy plays an important role in the microperforated material studied in this paper.

3.3. Finite Element Results

The mechanical properties of both base and perforated materials were calculated from tensile tests. Since the rolling direction does not play a significant role in the elastic regime, the elastic modulus was calculated as the average of nine tensile test results. In that way, the base material has an elastic modulus of 115,000 MPa and microperforated material of 82,000 MPa. The plastic behavior (true stress–strain curves) in the longitudinal orientation for both materials is shown in

Table 5. True stress–strain curves in longitudinal orientation.

Strain	Stress
	BM	PM
0.00	320.0	360.0
0.01	347.9	383.3
0.02	379.8	401.6
0.03	402.4	419.6
0.04	422.6	435.8
0.05	438.6	450.2
0.06	454.0	463.2
0.07	466.3	474.5
0.08	479.3	484.7
0.09	490.6	493.6
0.10	500.1	501.5
0.11	507.2	510.0
0.12	515.0	518.0
0.13	523.0	526.0
0.14	530.6	532.0
0.15	537.8
0.16	544.3

, discretized every 0.01 units.

FLC strains were obtained as the average of three repetitions for each geometry, which are shown in

Table 6. Forming Limit Curves for microperforated material.

Geometry	Strains
	Minor	Major
G1	−0.187	0.299
G2	−0.148	0.267
G3	−0.099	0.217
G4	−0.061	0.178
G5	−0.045	0.156
G6	−0.008	0.122
G7	0.087	0.106

in tabular form.

The anisotropy is introduced in the FE model by means of the previously introduced R_ij stress ratios. These ratios were unknown, and a DoE was implemented to find them. The full three-level factor design is summarized in

Table 7. Full three-level factor design matrix.

Case	Inputs			Case	Inputs
	R22	R33	R12		R22	R33	R12
1	0.65	0.65	0.65	15	1.00	1.00	1.35
2	0.65	0.65	1.00	16	1.00	1.35	0.65
3	0.65	0.65	1.35	17	1.00	1.35	1.00
4	0.65	1.00	0.65	18	1.00	1.35	1.35
5	0.65	1.00	1.00	19	1.35	0.65	0.65
6	0.65	1.00	1.35	20	1.35	0.65	1.00
7	0.65	1.35	0.65	21	1.35	0.65	1.35
8	0.65	1.35	1.00	22	1.35	1.00	0.65
9	0.65	1.35	1.35	23	1.35	1.00	1.00
10	1.00	0.65	0.65	24	1.35	1.00	1.35
11	1.00	0.65	1.00	25	1.35	1.35	0.65
12	1.00	0.65	1.35	26	1.35	1.35	1.00
13	1.00	1.00	0.65	27	1.35	1.35	1.35
14	1.00	1.00	1.00

.

The holder force and punch stroke were gathered for each FEM case. After, the outputs were calculated for each case as the difference between the FEM and experimental data (Equation (11)). Then, nineteen quadratic models were fitted to predict the differences between the FEM results, and the experimental data were fitted. The following equation is an example of these quadratic models:

$$Y_7=339.5+129.14·R_{22}-50.4·R_{22}^2-591.9·R_{33}-22.5·R_{22}·R_{33}+250.8·R_{33}^2-27.4·R_{12}-20.9·R_{33}·R_{12}+18.7·R_{12}^2$$

(13)

Finally, an optimization by gradient descent algorithm was launched. The objective of the optimization was to minimize all the distances between the Nakajima test results and the FE model results, ergo, to minimize all quadratic models at the same time. Usually, it is impossible to minimize all objectives together due to the conflicts between objectives, and the solution is generally a compromise. The optimum values that achieve this compromise solution are the anisotropy ratios summarized in the

Table 8. Anisotropic ratios for MP material.

Ratios	Symbol	Value
Anisotropy Y	R22	1.050
Anisotropy Z	R33	1.320
Anisotropy XY	R12	0.975

.

Finally, the experimental and finite element crack results can be compared visually. Figure 13 compares the results of geometry G4 in the longitudinal orientation after the Nakajima test is completed. Both results have similarities in terms of crack location and strains distribution. However, there are some differences due to the nature of the simulation. The FEM results present double symmetry because the model was a quarter of the whole test, while the experimental results present approximately symmetrical results.

4. Conclusions

The present study has described microperforated titanium grade 2 experimental testing and FE modeling. The non-perforated titanium grade 2 testing campaign consisted of nine tensile tests, while the microperforated testing campaign consisted of another nine tensile and several Nakajima tests. All of them were conducted at room temperature. In both cases, different orientations were tested: longitudinal (rolling direction), transversal, and diagonal. The main conclusions obtained from the testing campaign can be summarized as follows:

• Microperforation significantly reduces the formability and elongation of titanium grade 2 samples, between 17 and 31% depending on direction.
• Both yield stress and ultimate tensile strength increase slightly (less than 5%) with microperforations. It is not considered significant due to the small difference and the small number of repetitions.
• The orientation of specimens does not play a significant role in formability. Only the formability of transversal specimens is slightly worse than that of other specimens.
• Anisotropy has a relevant role in punch load–stroke curves.
• A crack always moves on the shortest path between two microholes regardless of the material’s anisotropy.

A dynamic explicit FE model has been built to characterize mechanical properties and anisotropy. The material characterization was carried out considering the results from the testing campaign. The most relevant conclusions obtained are listed in the following lines:

• Anisotropic ratios play a significant role in load–stroke curves.
• Hill anisotropic is not able to reproduce crossing plastic curves. However, the combination of Hill78 anisotropic and damage initiation criteria models accurately the behavior of the microperforated titanium grade 2.