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. Aramis 2M (made by GOM GmbH, Braunschweig, Germany) was used as a strain measuring device, and its properties are summarized in
Table 2. 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 (
) 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:
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:
The material constants (
G,
H,
F, and
N) can be expressed using stress ratios
Rij. The yield ratios are expressed relative to a reference yield stress (
), so if
is applied as the only non-zero stress, the corresponding yield stress will be
.
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:
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 3
3 = 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):
where
Ykjo 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;
nj is the number of measurements (time points) of Nakajima geometry j in each curve;
is the experimental punch stroke of Nakajima geometry j and orientation o at time point i;
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:
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,
bi,
bii, and
bij 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).