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A Review on Sheet Metal Forming Behavior in High-Strength Steels and the Use of Numerical Simulations

Luis Fernando Folle

^1,2

Tiago Nunes Lima

Matheus Passos Sarmento Santos

¹,

Bruna Callegari

Bruno Caetano dos Santos Silva

¹,

Luiz Gustavo Souza Zamorano

³ and

Rodrigo Santiago Coelho

^1,*

SENAI CIMATEC, Senai Institute of Innovation for Forming and Joining of Materials (CIMATEC ISI C&UM), Av. Orlando Gomes, 1845, Salvador 41650-010, Brazil

Technical and Industrial School of Santa Maria, Federal University of Santa Maria—UFSM-CTISM, Santa Maria 97105-900, Brazil

Ford Motor Company, Camaçari 42810-440, Brazil

Author to whom correspondence should be addressed.

Metals 2024, 14(12), 1428; https://doi.org/10.3390/met14121428

Submission received: 30 October 2024 / Revised: 2 December 2024 / Accepted: 5 December 2024 / Published: 13 December 2024

(This article belongs to the Special Issue Modeling, Simulation and Experimental Studies in Metal Forming)

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Abstract

High-strength steels such as Dual Phase (DP), Transformation-Induced Plasticity (TRIP), and Twinning-Induced Plasticity (TWIP) steels have gained importance in automotive applications due to the potential for weight reduction and increased performance in crash tests. However, as resistance increases, there is also an increase in springback due to residual stresses after the forming process. This is mainly because of the greater elastic region of these materials and other factors associated with strain hardening, such as the Bauschinger effect, that brings theory of kinematic hardening to mathematical modeling. This means that finite element software must consider these properties so that the simulation can accurately predict the behavior. Currently, this knowledge is still not widespread since it has never been used in conventional materials. Additionally, engineers and researchers use the Forming Limit Diagram (FLD) curve in their studies. However, it does not fully represent the actual failure limit of materials, especially in high-strength materials. Based on this, the Fracture Forming Limit Diagram (FFLD) curve has emerged, which proposes to resolve these limitations. Thus, this review aims to focus on how finite element methods consider all these factors in their modeling, especially when it comes to the responses of high-strength steels.

Keywords:

finite element simulation; ductile fracture; springback; high-strength steels; sheet metal forming

1. Introduction

Sheet metal forming processes are widely used in the manufacturing of structural vehicle components such as side members, pillar reinforcements, and floor panels. However, these processes pose challenges such as the occurrence of wrinkling, excessive thinning, premature failure, shear cracks, and springback. All these factors must be taken into account during the production of a component, while also meeting requirements for lightweight design, maintaining mechanical strength, and ensuring low production costs [1,2].

High-strength steels have helped to improve the performance of the automotive industry in terms of the characteristics required for the crash test, promoting greater safety in the event of collisions [3,4,5]. However, the increase in strength needs to relate to formability [6,7,8], where these steels must be able to transform into parts without significant increases in costs. In the case of sheet metal forming processes, some parameters, such as elongation, provide evidence of the forming capacity of the material, which is directly related to ductility, at least at one of the material’s demand levels, which is uniaxial stretching [9]. In general, as a metallic material increases its strength, the maximum elongation (or elongation at break) decreases, as in the case of adding carbon to steels. This characteristic is worrisome for sheet metal forming because, as there is less room for material deformation, less strain generates a quick nucleation of defects that can lead to premature rupture before the part is formed [10]. It is important to note that this analysis applies to cold forming, which comprises most of the sheet metal forming processes [11], because in the hot process other mechanisms are activated that change the response of the material, increasing elongation [12,13].

For the automobile industry, the relationship between ductility and mechanical resistance is particularly a problem, since more and more vehicles have requirements for reducing fuel consumption, which demands weight reduction but maintaining the same characteristics of structural strength [14,15,16]. This causes the ductility of steels to decrease, consequently promoting a reduction in their formability [17]. To minimize this problem, the metallurgical industries started to develop steels with microstructures that are able to meet both resistance and ductility requirements simultaneously [17,18,19]. Figure 1 shows how these steels behave in relation to strength and elongation. It is possible to see that there are steels in which, even with high resistance, there is a high elongation, differentiating from the effect found in conventional steels. This is particularly evident for the 2nd and 3rd generation steels shown in Figure 1.

In the literature it is possible to find several studies on advanced high-strength steel development [21,22,23], sheet metal forming processes, and the application of numerical simulations [24,25,26]. It was shown by several industrial examples that numerical simulations can predict technological problems and help to solve them [27,28]. However, studies developed in recent years show the effectiveness of using more complex models that can be applied to newly developed steels. There are some studies that concentrate on the simple shear testing of sheet metals and their application in sheet metal forming [29]. In Coppieters et al.’s [30] paper, they present an overview of the state-of-the-art methods to acquire the large strain flow curve of sheet metal.

The present review focuses on the sheet metal forming of advanced high-strength steels and the use of numerical simulations. From the emergence and application of steels with increasingly higher levels of mechanical strength, some complicated factors arise in the forming process [31]. This review aims to present some of these factors and what is being proposed by researchers to overcome these challenges using mainly numerical simulation combined with increasingly elaborate experimental tests.

In Section 2, we discuss one of the main problems in the sheet metal forming of high-strength steels, the springback phenomenon. In Section 3, we present the basic concepts and the state of the art in relation to the material and friction models and failure criteria used in sheet metal forming simulations. Finally, we discuss some works that show how these models and criteria adjustments can improve the simulation results in relation to springback.

2. Springback

Although formability is partially resolved by increasing the total elongation in a uniaxial test, the strength, and therefore the yield limit, remains high as well. This promotes an elastic response of the material with greater amplitude. This effect, known as springback, becomes more pronounced with a decrease in Young’s modulus and an increase in yield strength and mechanical resistance [32].

Springback is the phenomenon that occurs when a material is plastically deformed and presents a kind of “stress relief” that causes the material to move in the opposite direction to load application. For the case of bending a sheet metal at 90°, for example, after the force is removed, the material will tend to return to an earlier position, that is, with an angle greater than 90° [33], as shown in Figure 2. This phenomenon occurs because when a material is plastically deformed, it necessarily goes through elastic strains, which are basically linked to the interaction effect between the atoms of the elements (atomic bonding), and this is not undone upon plastic straining.

In the case of a tensile test, when the material is unloaded after a plastic strain (line AC of Figure 3), the stress–strain curve returns to zero with an inclined line with the same angle as the elastic strain line, which corresponds to the material’s Young’s modulus. Normally, materials that are of the same nature, for example, alloys of steel, aluminum, or titanium, have almost the same modulus of elasticity, regardless of the yield stress being lower or higher. Consequently, if the steel has greater mechanical resistance, it will necessarily present a higher springback, which corresponds to the BC line in Figure 3.

The springback induced by residual stresses in the material generates a geometric difference from what was designed. This is a problem because parts that should have a specific design present a dimensional distortion that hinders fitting and joining steps foreseen with other parts in the design. This phenomenon is particularly intensified in bending that takes place during deep drawing processes, shown in Figure 4a, because in V-bending, shown in Figure 4b, this effect is reduced.

In deep drawing, the sheet is initially bent around a radius and subsequently straightened, leading to the unbending of the material and generating distinct stress states across different regions of the sheet. During the initial bending, tensile stresses develop on the outer surface, while compressive stresses form on the inner surface. In the subsequent stage, as the material undergoes unbending, the stress states are reversed, resulting in tensile stresses on the inner surface and compressive stresses on the outer surface (Figure 5). This situation generates a residual stress that promotes a “torsion” in the part. This effect can be studied through a standard geometry as represented in Figure 6a, with the measurement of angles θ₁, θ₂ and of the radius ρ as shown in Figure 6b.

Springback has driven the development of technological alternatives to mitigate this effect, which often causes substantial changes in the geometry of the parts. One way to address this issue is by compensating the bending angles, allowing the part to achieve the desired geometry after springback. However, this approach requires the application of higher deformations, which can lead to material fracture, especially when forming high-strength steels. A more practical solution for stamping is the use of drawbeads, which effectively control material flow and enhance the formability of the part. These features introduce new stress states into the sheet being formed, thereby reducing the impact of springback [37,38].

3. Sheet Forming Simulation

3.1. Basic Concepts

Numerical simulation is a computational method that practically emerged with the appearance of the personal computer. Its most used form is the Finite Element Method (FEM) or Finite Element Analysis (FEA) which is a method widely used to solve engineering problems with mathematical models.

Numerical simulation, specifically by finite elements, is very similar to an analytical calculation, where the structure always starts from a real problem, for example, how much force a metallic beam can withstand when subjected to a certain load. For this, an idealized representation is made, where assumptions are made that facilitate mathematical calculation. Afterwards, the equations that describe the physical behavior of the model are listed, that is, the mathematical relationships known from the study of mechanics that translate to a given physical behavior. These equations are then solved with some hypothesis, for example, the equilibrium condition of the structure. With the mathematical manipulation of the equations, the unknown variables that can provide information about the behavior of the structure, such as displacements, strains, and stresses, are determined. From there, analysis of the results is made according to the expectations of the proposed model to verify its coherence with the real problem.

What was described above would be a straightforward case, where a calculation could be simply performed by hand, and it would be possible to have the required answer in a timely manner. However, if the structure becomes increasingly larger, so that the number of beams, or calculations, demands an excessively long time to be solved analytically, this project becomes unfeasible. In this context, numerical simulations emerged, where this time is spent by the computer to calculate the structure.

For a computer to “understand” this logic, finite element analyses emerged, where each part of the structure’s geometry is subdivided into several calculation points based on the boundary conditions, which are the conditions to which the structure is subjected and are known, such as fixed points, applied forces, geometry size, connection points between beams, etc. It is also important, for the numerical simulation, to know the properties of the materials to be calculated. The first simulations that were carried out in the past considered the linear elastic behavior of metallic materials, as they are the easiest to calculate. The linear elastic behavior occurs when the material is subjected to a force and generates a proportional deformation returning to the initial state when unloaded. This behavior is called the spring effect and Equation (1) describes this effect mathematically.

F = k \cdot d

(1)

where F is the applied force, d is the displacement caused in the material, and K is the constant of the spring effect.

For the case of a structure with several beams, for example, each beam will accommodate a vector direction of the applied force submitted to the same constant K. This generates several linear equations that are created for each beam. Thus, an “x” number of equations with “x” unknown variables are obtained. However, the computer is not able to solve systems of linear equations, thus, this is transformed into a matrix set as shown in Equation (2) where the matrix of the K coefficients is called the stiffness matrix.

\{\begin{matrix} f_{1} \\ f_{2} \\ \begin{matrix} f_{3} \\ \begin{matrix} ⋮ \\ f_{n} \end{matrix} \end{matrix} \end{matrix}\} = [\begin{matrix} \begin{matrix} \begin{matrix} K_{11} \\ K_{12} \end{matrix} \\ K_{13} \end{matrix} & \begin{matrix} \begin{matrix} K_{21} \\ K_{22} \end{matrix} \\ \dots \end{matrix} & \begin{matrix} \begin{matrix} K_{31} \\ K_{32} \end{matrix} \\ K_{33} \end{matrix} \\ ⋮ & ⋱ & ⋮ \\ K_{1 i} & \dots & K_{i j} \end{matrix}] \cdot \{\begin{matrix} d_{1} \\ d_{2} \\ \begin{matrix} d_{3} \\ \begin{matrix} ⋮ \\ d_{n} \end{matrix} \end{matrix} \end{matrix}\}

(2)

The stiff matrix is constructed from a geometry of a beam structure. It is possible to predict that as the structure becomes more and more complex, the greater the stiffness matrix will be. Each element of the matrix is a vector direction of the resultant force in each beam. These resulting forces are generated through the boundary conditions, which can be external forces or displacements applied to a beam and supports that are transferred between the beams until they travel through the entire structure. When the computer performs all the calculations, it is possible to know data on the internal forces of the structure, displacements, stress, and moments, among others and, with that, to be able to make decisions about reinforcements, reliefs, and beam formats that meet the design requirements.

However, if it is necessary to know what is happening in the middle of the beam, it is necessary to place one more calculation point which is called a mesh element and this will obviously generate more calculations and, therefore, more simulation time. Thus, the more mesh points the geometry has, the more confidence one has about the internal load transfers of the component. In terms of definition, the term “mesh” refers to the number of calculation points in a Finite Element Analysis (FEA). So, the more elements a geometry has, the more refined the mesh, and the fewer elements the geometry has, the coarser the mesh will be. The points where these lines are located are called “nodes”, which correspond to points of generation of results.

There is a relatively wide range of finite element mesh types, depending on the most suitable type of application. However, it is possible to classify the mesh types into two large groups, which are tetrahedral and hexahedral, for volume elements. Obviously, there are advantages and disadvantages for each one. The tetrahedral shape is more flexible in terms of complex geometries, where curved regions are better drawn by this type of shape, while the hexahedral shape is better represented in straight geometries. However, it is difficult, currently, to have shapes of pieces that are straight, without rounding or organic surfaces, but as compensation the hexahedral shape presents an advantage in relation to the tetrahedral shape, where, for the same spacing of elements, the hexahedral shape generates a mesh with a considerably smaller number of elements. It is important to note that computational time does not depend exclusively on the size and shape of the mesh, but this is one of its main determining factors. Another observation is that due to the smaller number of elements, the hexahedral mesh is more studied, so that there are subroutines in simulation software that adjust the shape of the hexahedral mesh for complex part geometry formats.

An important subject that arises when analyzing the finite element mesh is which mesh size generates good enough results for analysis purposes. It is highly recommended that the simulation starts with a coarse mesh so that the simulation can quickly present results on what is being studied and it is able to show if the data were inputted correctly. To obtain more assertive numerical results, the mesh may be more refined. The level of refinement that must be applied to a study is defined by a method commonly known as mesh convergence and should always be used at the beginning of a simulation.

Figure 7 shows a graph of the mesh density (number of nodes) by the normalized result for axial load applied. It is possible to see that from the fifth mesh there is no significant difference in the numerical results. This shows that using a certain number of elements in the mesh is enough to obtain a mesh convergence and solve the proposed problem. Figure 8 shows the error in the axial loading associated with the number of elements and element type. The mesh refinement can reduce significantly the error, but the computational time increases as a side effect from the insertion of more elements in the simulation.

During a simulation setup, besides the mesh convergency, it is necessary to define the boundary condition to which the piece or body is subjected. This condition is normally defined by a “support” restriction in finite element simulation software. The support (attachment), by default, will not deform, generating a point of stress concentration at the edge of the part that tends to infinity as the mesh is increasingly refined. This can be explained if we imagine that the calculations would have to meet a condition of zero tension and displacement at a mesh point in the attachment and a non-zero tension at the mesh point immediately outside the attachment and this will generate increasingly greater results as the mesh points become closer. In a real case, this fixation would be by welding, screws, glue, rivets, etc., where the fixation itself would also have a tension and displacement. Another observation is that the current finite element programs already have localized mesh refinements, which means that only the regions of interest will have a refined mesh, considerably reducing the number of total elements and, consequently, the simulation times.

It is perfectly acceptable for the question of why not simulate a completely real condition to arise. However, if the focus of analysis is not on the fixation region, any extra component will generate more simulation time, which will not contribute to the solution of the problem. Thus, the answer would simply be to ignore the stresses in the attachment and focus on the stresses in the hole.

Another issue would be to try to apply some deformation programmed in the support to try to generate results closer to reality. This would also be a problem in the simulation, since the software will basically calculate a system of equations and, mathematically, this system may not have a solution. In this way, the software will make an error and will not generate any results. In this case, the simplest solution, again, is to fix the part and disregard the stresses in that region.

Everything that has been mentioned so far serves a condition of elastic linear simulation, however, metallic materials exhibit another behavior when the deformations are permanent. If the focus of the simulation is to avoid the yield stress, the whole project will be performed so that the external forces do not generate permanent deformations, and the simulation will never overcome the linear region of the metals. However, if the aim is to analyze what happens to the metal during plastic deformations, the software will have to account for this effect in the calculations.

There are several types of material idealizations based on stress–strain curves: linear elastic or perfectly elastic behavior, elastic and perfectly plastic behavior, curves with linear hardening, and curves with non-linear hardening.

Finally, there is elasto-plastic behavior with non-linear hardening, where the vast majority of metals fit. This curve is the one that generates the longest simulation time since, as in the plastic region, there will be large permanent displacements of the nodes of the mesh, and the stiffness matrix will be drastically altered. This will mean that as the software applies an increment of displacement in the nodes, the stiffness matrix must be reformulated. Thus, the simulation time for non-linear conditions with large permanent deformations is greatly affected.

Everything mentioned above is valid for sheet simulation, and the basic difference in the finite element method for sheet stamping is the type of element that is normally used, called the shell type element. Figure 9 shows a representation of the shell element, where the nodes are positioned in the same central plane and the thickness information is given through integration points outside the node plane. The number of integration points is usually chosen by the software operator and is between three and seven points. Obviously, the greater the number of integration points, the longer the simulation will be, and the mesh convergence rule also applies to integration points in thickness.

There are a lot of programs that carry out finite element simulations. However, each one has its strongest area, since there are programming subroutines that help to improve both the simulation time and the results. Some examples can be mentioned, such as the programs specialized in CAD such as SolidWorks, Catia, Creo Parametric, and Inventor that are more dedicated to linear simulations of elastic and buckling in structures. For the forming area, Simufact and Forge software are more specifically dedicated to simulations with forging in general and LS-Dyna, AutoForm, and PamStamp software are more dedicated to sheet metal simulations. There are others that are more generalist, such as Ansys and ABAQUS, which aim to simulate a wider range of applications, including computational fluid dynamics. In this context, ABAQUS is widely used for scientific research because it has greater freedom to make changes in calculation methods, in constitutive equations, or in the generation of programming subroutines. Ansys is less suitable for these modifications, however, the user interface is more friendly.

3.2. Scientific Approach

Sheet metal forming and material plasticity should be evaluated with industrial applications in mind, making the use of macroscopic models more suitable. Given the significant disparity between the microstructural scale and a formed component, simulating at the microstructural level would generate an enormous amount of data, making it impractical to track all changes throughout a complete deformation process. To address this, microscopic information is incorporated into the macroscopic scale by introducing specific variables into the equations. The constitutive equations are formulated in two ways depending on the type of loading: scalar forms for uniaxial loading and tensorial forms for multiaxial loading. These equations are generally expressed as follows [40]:

\dot{ε} = (σ, Θ, x_{i}) \dot{x_{i}} = x_{i} (σ, Θ, x_{k})

(3)

where

\dot{ε}

represents strain rates,

σ

the stress,

Θ

the temperature, and

x_{i}

the state variables that macroscopically represent the microstructural state of the material.

When implementing a material model for simulation, specific formulations must be incorporated into the Finite Element (FE) code. These models are utilized by the simulation software through specialized calculation methods to obtain results for the variables involved in the forming process. In the work of Ablat and Qattawi (2016) [41], solution methods for sheet metal forming simulations were summarized into five categories: static explicit method, dynamic explicit method, static implicit incremental method, static implicit large-step method, and static implicit one-step method. However, variations may exist depending on the application, such as dynamic methods, which can sometimes also be applied implicitly.

Dynamic methods are frequently used for simulations involving sheet metal forming, as they are based on the body’s inertia and do not require static equilibrium. Explicit dynamic methods are not affected by divergence issues and typically require less memory and computational power. Implicit dynamic methods, on the other hand, use an iterative approach to solve equilibrium equations, offering unconditional stability and enabling the use of larger time increments [40,41].

These calculation methods are typically integrated into the commercial software mentioned above. During the simulation, the software automatically applies the appropriate calculation method based on the selected simulation module.

3.3. Fracture Criteria

The fracture criteria in the mechanical forming process are basically related to either ductile fracture or localized stress [42,43]. Materials that exhibit ductile behavior are those that deform plastically before breaking [44,45,46]. Due to the way the rupture is caused, it is important to study the topic [47]. Additionally, some studies have shown that the stress state has a great influence on sheet formability, where through-thickness shear stress on the sheet can improve sheet metal formability [48].

In terms of component performance in general, the disposal of a part encompasses a wider range of manufacturing defects, such as “scratched” surfaces, exaggerated thinning, wrinkles, “orange peel” effects, localized cracks, and edge cracks [42,49,50], among others [51]. For the analysis that will be performed here, only defects of premature ruptures will be considered, which makes the manufacture of a part made by sheet metal forming unfeasible. The other defects mentioned, in general, occur if the part is not broken, therefore, they will not be considered at this time.

In a uniaxial tensile test, it is possible to know the basic behavior of a material regarding the formation of the defect by ductile fracture. Figure 10 shows three types of deformation that occur in a uniaxial tensile test on a sheet. At the beginning of the plastic strain until the peak of the material’s strength, the strain is uniform, that is, distributed throughout the region that has the same geometry. At this point, the metal has practically the same strain in any region of the specimen of the same geometry. After ultimate tensile strength, the curve begins to fall slightly as a phenomenon of necking called “diffuse necking” begins to occur where the material has a decrease in width, which is in the central region of the specimen and differs from the rest. From that moment on, the rest of the material does not deform plastically, serving as an anchor for the deformations, that is, there will only be movement of material in the region of the strain. With the advance of the test displacement, a third type of deformation called “localized necking” appears [52], which occurs in the thickness of the material. At that moment, as the portion of the metal that is under movement is very small (practically only in thickness), failure or rupture is imminent and soon the fracture occurs. At this stage, the rest of the material, similarly to diffuse necking, will serve as an anchor and will no longer deform and therefore the failure is quickly reached. The localized necking is usually inclined in relation to the application of force in the test, reaching close to 55° of angulation. This is because this angle would have the highest shear stress, and this has already been proven mathematically.

As mentioned above, strains generate a material movement that will cause a localized rupture. This rupture is called “ductile fracture.” In microstructural terms, ductile fracture would be a “detachment” that occurs around inclusions of the material, which increase and coalesce, that is, joining together until a macroscopic-sized void promotes the rupture of the metal. Figure 11 shows three types of failure that can occur due to void nucleation, the first would be a fracture without diffuse necking, only localized, the second would be with diffuse and localized necking, and the third with only diffuse necking. Figure 12 shows these behaviors for three metals, Q195, HSLA350, and AL6061. Q195 is a structural carbon steel (a type of carbon steel). HSLA350 is a high-strength low-alloy steel. AL6061 is an aluminum alloy.

Materials that have some hardening mechanism usually present a type I fracture, as shown in Figure 11. Figure 13 shows the evolution of these voids from their formation to their junction (coalescence). Figure 13a shows a parallel type of coalescence and Figure 13b shows shear type coalescence.

The effect of localized and diffuse necking can also be seen in the thickness of the sheet. Figure 14 shows, for some metals, the types of fracture that can occur visibly through the thickness. Steel Plate Cold Commercial (SPCC) and Steel Plate Rolled Commercial (SPRC) materials are conventional carbon steels and in them the diffuse necking effect is visible. For the rest of the materials in Figure 14, which include steels DP, TRIP, TWIP, and hardened aluminum and magnesium alloys, only localized necking will occur.

What was mentioned earlier serves to describe how the sheet will break during strain. However, only the tensile test was shown, and this test does not represent the range of strains that can occur in a stamped part. For this, there is the Nakajima test, which aims to build a diagram called a Forming Limit Diagram (FLD). This curve or diagram is intended to provide a limit of strains that the part cannot exceed with the risk of breakage during forming. This curve is obtained through a test with several sizes of stainless steel specimens in which each sample will generate a point on the curve as shown in Figure 15.

The curve in Figure 15 is recommended for materials that exhibit diffuse necking behavior before fracture, because how it is obtained depends on a measurement that cannot get close enough to the failure. Figure 16 shows how strains are measured in the Nakajima test. As one can see, the measurable circles are those that have not been fractured (marked with an X) and that are relatively far from the fractured circles (marked with a dot), which are not entirely measured. This measurement methodology results in small errors, since it is not possible to measure the deformations in the failure region, and the FLD curve is produced from measurements of regions close to the fracture. Figure 17a shows an ideal measurement that would be in the zone of localized necking and Figure 17b shows where strains are generally measured in a Nakajima test.

Thus, the Forming Limit Diagram (FLD) is normally a curve that is lower than the ductile fracture zone represented by the Fracture Forming Limit Diagram (FFLD) curve. Several scientific studies have tried to obtain FFLD curves [59,60,61,62,63,64,65], as their development is still difficult and there are no standardized tests that can be applied to obtain this curve for different materials and conditions [66,67,68,69,70]. Some studies are more focused on the study and obtaining of these curves and others have already used them in some practical applications [71,72].

The FLD is a curve generated by tests that represent a stamping process. However, there may be situations in which there are deformations for which the FLD does not generate points, such as in the shear fracture region. In addition, the Nakajima test still faces the problem of not being able to reveal the deformations that occur close to failure. Thus, efforts have been made for several years to determine a failure model that has a mathematical formulation based on fracture mechanics and that can be calibrated by simple tests such as a tensile test.

Figure 18a shows three curves where the black and red curves are FLD curves obtained through calculations that are based on the hardening exponent and the blue curve is the FFLD curve developed through the Modified Mohr–Coulomb (MMC) theory. This theory defines a fracture criterion which considers that the fracture occurs when a combination of normal stresses and shear stress reaches a certain value [73,74,75]. The MMC curve is first obtained to plot the equivalent strain until fracture versus triaxial strain graph (Figure 18b). Triaxial stress is defined as the ratio of the mean stress (the three main stresses divided by 3) by the equivalent stress (von Mises stress). From the triaxial curve, the FLD is obtained. Thus, the FFLD curve can be added to software that simulates stamping processes and serves as a failure criterion, that is, any strain that goes beyond this curve will fracture.

In Li et al.’s work [76], a test for TRIP 690 steel was performed, where it was shown that the FFLD curve is most appropriate to predict the fracture in high-strength sheets, especially in the negative zones of lower deformation (second quadrant of the diagram). Figure 19 shows the result of the deformations in relation to the FFLD curve and Figure 20 shows the result of the experiment in relation to the simulation. It is possible to note that the conventional FLD curve will not predict rupture for deep-revenue deformations, which will be foreseen by the FFLD.

A study conducted by Chen et al. [73] evaluated the use of the MMC criterion to predict fracture during stamping. As shown in Figure 21, the simulation using the FLD curve was unable to predict fracture in the part radius, while the MMC theory successfully identified most regions susceptible to failure. However, the MMC model also predicted additional failure points around the part’s corner area, which do not occur during the stamping process, indicating that further calibration of the curve is necessary for industrial applications.

Thus, Hill was the first to create a mathematical model that takes into account the effects of material anisotropy, which is very common in metal sheets. Other researchers, based on the idea that a microscopic flaw could turn into a macroscopic flaw through the accumulation of deformation or, in other words, damage accumulation, proposed their models [71]. Some authors [77,78,79,80] developed several anisotropic yield functions by introducing more coefficients. Other works were developed based on the three invariants of the stress tensor [81,82,83,84]

In the last two decades, studies [61,85,86,87] have used in their models the Lode parameter or angle and the triaxial stress state to define in which situation the deformation is found. Figure 22 shows how the deformation can occur depending on the Lode parameter and the triaxial stress state, where Lode = −1 represents axisymmetric compression, Lode = 1 represents axisymmetric tension, and Lode = 0 is either plane strain or shear. The triaxiality value also reveals the type of stress to which the material is subjected, with −1/3 being uniaxial compression, 0 pure shear, 1/3 uniaxial tension, 1/√3, plane strain, and 2/3 biaxial tension. The surface in Figure 22 represents a mathematical model that can be calibrated using coefficients. These coefficients, in turn, can be obtained through tests.

A model that has been widely explored is the GTN, which was proposed by Gurson and Tvergaard and Needleman. However, the authors believe that the GTN model is too complicated to be applied in complex industries. However, this model was used in several works [89,90,91].

In commercial finite element software, these material definitions are sometimes not present in the selection list, making simulation subroutines necessary. The User Material (UMAT) and Vectorized User Material (VUMAT) subroutines, for example, are widely used in ABAQUS. These subroutines allow users to define custom material models and incorporate them into the simulation process. UMAT and VUMAT are user subroutines that allow users to implement their constitutive models in ABAQUS. The main difference between UMAT and VUMAT is that UMAT is for implicit analysis and VUMAT is for explicit analysis.

3.4. Material Definition Models

When a metallic material undergoes plastic strain, there is a characteristic response that can be modeled mathematically and serves to define this material.

The first model is the one that concerns the behavior in uniaxial deformation in the tensile test. In this test, with the results of true stress and strain, mathematical approximations of the curve are made, and the coefficients are added in the simulation software. Figure 23 shows how, from a tensile test, mathematical fit curves can be applied to the material. It is possible to see that the tensile data do not generate all the necessary information for larger deformation, and from there, the data are extrapolated. However, this can be a problem, because in sheet metal forming, there will be strains that will be greater than those given by the tensile test due to the demands to which a sheet may be subjected. From there, the real behavior of the material is unknown. To minimize this uncertainty, the bulge test is designed to provide more information about the curve. Figure 24 shows the tensile test data compared to the bulge test. It is possible to see that there is a greater amount of data when the mathematical approximation is complemented with data from the bulge test.

The main mathematical approximations that can be applied to a material are shown in Table 1, where it is possible to observe that as the material has a more complex behavior, the equation becomes larger. In each of these equations, the only variables are the “σ” stresses and the “ε” strains, the rest are fit coefficients that must be determined to be loaded into simulation software.

Another parameter that must be considered is anisotropy, which is applied practically only to metal sheets. The concept of anisotropy refers to the difference in mechanical properties for different directions on the Cartesian axis. In sheet forming, this occurs due to the lamination process that promotes a change in the shape of the grains, thus causing a difference in properties. However, as strains are the main properties analyzed in sheet metal forming, anisotropy is given by the relationship between strains in width and strains in thickness in a tensile test according to Equation (11) [99]. It is important to note that in sheet metal forming, there is always a concern about thickness, because when a sheet is deformed, the initial idea is that the thickness does not show any thinning, which ends up causing discarding of the part. Thickness thinning is represented in Equation (11) by a high strain, or a high numerical value in the denominator, giving a small number in the result. This means that the lower the anisotropy value, the more susceptible to premature thickness thinning the material will be. Another observation is that as thickness strains are difficult to measure due to the small dimensions they may have, the equation is rewritten, based on the volume constancy law, with strains in width and length.

r = \frac{ε_{w}}{ε_{t}} = \frac{l n (\frac{w_{f}}{w_{0}})}{l n (\frac{t_{f}}{t_{0}})} = \frac{l n (\frac{w_{0}}{w_{f}})}{l n (\frac{w_{f} \cdot l_{f}}{w_{0} \cdot l_{0}})}

(11)

where:

t₀ and t_f are the initial and final thicknesses, respectively.

w₀ and w_f are the initial and final widths, respectively.

l₀ and l_f are the initial and final lengths, respectively.

In Equation (11), the material will be isotropic when the value of “r” is close to 1. The material will tend to thin if the value is less than 1, where typical values are up to 0.8. Otherwise, the material will have a low tendency to thin when the anisotropy is greater than 1, where maximum values reach 2.2. The effect of changing properties is not only seen in the thickness but also in the plane of the sheet. Thus, it was agreed that the anisotropy should be measured in three directions in relation to the sheet rolling direction (0°, 45°, and 90°), as shown in Figure 25.

The anisotropy results will thus be represented through the values of r₀, r₄₅, and r₉₀. With these data, it is possible to calculate the mean or normal anisotropy and the planar anisotropy, according to Equations (12) and (13) [99], respectively. Normal anisotropy is just an average value for the three directions. Planar anisotropy, on the other hand, gives indications of the difference in the level of strains between directions, thus, the greater the difference between these values, the greater the strains in each direction. This causes the material to behave a little differently according to the geometry. A high planar anisotropy is the main source of the earing effect in deep drawing, which generates undulations, “ears”, along the edge of a stamped piece.

\bar{r} = \frac{r_{0 °} + 2 \times r_{45 °} + r_{90 °}}{4}

(12)

∆ r = \frac{r_{0 °} - 2 \times r_{45 °} + r_{90 °}}{2}

(13)

As seen so far, the anisotropy of the material generates differences in mechanical behavior, so these data should be entered as input material properties in sheet metal forming simulation software. This model is usually called Hill48, which comes from the author’s name (Rodney Hill) and the year of publication of the work (1948) [100].

What has been mentioned so far concerns the deformations that can be measured in relation to the rolling direction, but the theory behind this phenomenon is more complex. In a real part, there are stresses and deformations that are generated in the three Cartesian axes forming what is called a triaxial stress state where these stresses manifest themselves in the form of deformations, which are easier to measure. The ideal would be to measure the stresses directly, but in terms of experimental tests this is very difficult to evaluate. Thus, theories emerged that compare what occurs in a triaxial stress state with the uniaxial tensile test. The first theory developed was that of Tresca (1868) and was soon complemented by von Mises (1913), where, in these theories, the premise is that the material would enter into plastic deformation from an equivalent stress value calculated by an equation that accounts for the stresses in the other directions. This led to the definition of a yield surface as shown in the figure below.

However, these theories were formulated with the assumption that the material would behave the same regardless of the direction considered, i.e., isotropically, which does not occur in a sheet metal stamping situation. Therefore, in 1948, Rodney Hill updated this formulation considering the effects of anisotropy. In this theory, Hill introduced six anisotropy coefficients dependent on the material. These coefficients cannot be measured directly. However, they will be calculated using the values of “r” and the yield stress of the material in the three directions in relation to rolling. Hill’s material criterion causes a distortion in the yield surface as shown in the figure below. For r = 1 the yield surface coincides with von Mises, i.e., in the isotropic case [101].

Another widely used criterion was the one developed by Barlat in 1991, which has a lot of correspondence with aluminum alloys. This criterion also has coefficients that come from the calculation of the equivalent stress, but which can be calculated from the yield stresses of the material in the three directions in relation to the rolling and through the biaxial stress in the bulge test. However, in the Barlat criterion [102], there is an exponent that must be obtained through experimental adjustments, which makes its determination more complex. Even so, several studies have already been conducted in this sense showing that the coefficients that best fit are 6 for aluminum and 8 for steel.

In addition to the anisotropy coefficients in the three directions relative to rolling, there is biaxial anisotropy, where differences in properties are evaluated when a material is subjected to a balanced biaxial stress state. This property is obtained through a test called the “through-thickness disk compression test” where a plate disk is compressed to a certain strength and the deformations in two perpendicular directions are measured. Figure 26 shows disks that were compressed and Figure 27 shows the deformation graph in both directions. Figure 28 shows how the calculation of biaxial anisotropy is performed.

Another parameter that can be defined through a biaxial test is the biaxial yield stress. Figure 29 shows bulge tests for three materials compared to tensile tests, where it is possible to see that the results are very similar, but with small variations. The main difference is in the yield stress, and this must be measured to improve the material’s characterization. With the results of yield stress in three directions (σ₀, σ₄₅, σ₉₀), biaxial yield stress (σ_b), anisotropy in three directions (r₀, r₄₅, and r₉₀), and biaxial anisotropy (r_b) it is possible to use the model of material proposed by [20], called Barlat’s yield 2000.

There is yet another phenomenon that can occur in sheet metal forming processes, which is the Bauschinger effect, where the material under compression exhibits a different tensile behavior when one is applied in sequence to the other. Figure 5 already showed when this can occur in sheet metal forming, where the sheet is bent and unbent to pass through a radius, thus causing tensile and compressive stresses.

The Bauschinger effect, strictly speaking, generates a stress–strain curve under compression with less resistance than the curve under tension. Thus, material behavior is different depending on how strains are applied to a piece. Figure 30 shows how this phenomenon occurs in a tensile–strain curve, where the dashed line presents material behavior without the Bauschinger effect. A similar example of stress reversal occurs when material passes over a tool radius. During the first bend, there is tension on the outside of the sheet and a compression on the inside. At the exit of the radius, the state of stress on the thickness of the sheet is inverted and is directly influenced by the Bauschinger effect.

The mathematical modeling of the Bauschinger effect has already been performed by several authors [105,106,107], one of them is called “kinematic hardening” where some parameters can be obtained through a tensile–compression test. Figure 31 shows a typical tensile–compression test, where the sheet is lightly pressed so that there is no buckling, and the strains are measured through a Digital Image Correlation (DIC) camera.

The most used mathematical model to represent the Bauschinger effect is called Yoshida–Uemori kinematic hardening (YU model) [108,109], where 10 parameters (Y, B, C, Rsat, b, m, h, E0, EA, and ξ) are defined to represent the tension–compression curve with kinematic hardening. The first seven parameters can be set either by cross-curve coefficient fitting programs or manually by similar fitting methods. The last three parameters are also defined through a fitting curve but represent the decrease in Young’s modulus with plastic deformation. For this, a tensile test with loading–unloading–reloading is carried out to determine the Young’s modulus for each stage of unloading and recharging of the machine [110,111]. Figure 32 shows the curve generated from a test in which the specimen was initially elongated to a plastic strain of approximately 0.02. The procedure was then continued, creating cycles with increments in plastic strain. This test demonstrates that loading and unloading cycles alter the material’s behavior, affecting its mechanical properties in the pre-stretched condition. Specifically in this example, an increase in yield strength and a decrease in elastic modulus are observed.

The stress–strain curve with loading cycles presented in Figure 32 enables the derivation of another curve, shown in Figure 33, which was developed for a TRIP 1000 steel and provides a clearer visualization of the kinematic hardening effect on Young’s modulus. This effect, caused by the accumulation of plastic strain and the redistribution of internal stresses, can be mathematically fitted using Equation (14). This approach allows for the determination of the parameters E₀, E_A, and ξ, where E₀ and E_A represent the Young’s modulus for virgin and infinitely pre-strained materials, respectively, and ξ is a material constant related to the degradation of Young’s modulus.

E = E_{0} - (E_{0} - E_{A}) [1 - e x p (- ξ {\bar{ε}}_{p})]

(14)

Another mathematical model that describes the Bauschinger effect, more recent than the YU model, is the one proposed by Barlat et al. [112] and is called Homogeneous Anisotropic Hardening (HAH), where nine coefficients (a, q, k, k1, k2, k3, k4, and k5) must be determined so that the equations can describe the phenomenon.

The Bauschinger effect is a consequence of a phenomenon called kinematic hardening in metallic materials. Any cyclic deformation will inevitably generate an accumulation of plastic strain. This will cause the yield surface to shift in principal stress space. If the yield surface changes its size, we are dealing with isotropic hardening. The radius of the yield surface is described by a hardening function that approximates the uniaxial stress diagram in terms of stresses and strains. The hardening function can take different forms in true and engineering stress–strain coordinates. If the yield surface changes its position without changing its size, we are dealing with kinematic hardening. The distance over which the center of the yield surface shifts is represented by reverse stresses. There are at least four situations that can occur due to the phenomenon of cyclic loads: softening or hardening when the deformation is constant over time and softening and hardening when the stress is constant over time [113].

The kinematic hardening rule is important in terms of the need to capture the cyclic response of the material. In addition to the Yoshida–Uemori model, there are others that can be used, such as Chaboche or Armstrong–Frederick, and can be found in commercial software that uses the finite element method for simulation. Currently, almost all commercial software has one of these models [114].

3.5. Friction Models

So far, the material data that have been described refer to the behavior under stress loading generating a strain response. However, it is important to define what happens at the load application interface between two metals which is represented by the friction coefficient. This parameter no longer refers exclusively to the material that will be made into a piece but rather to the relationship between the sheet metal forming tools and the material of the part. Therefore, it is important not only to know its value but also to know which process variables can affect it. Typically, the coefficient of friction can vary with the contact pressure, the level of lubrication [115], the working temperature, the strains imposed, the finish and hardness of the materials involved, and the forming speed, among others. Figure 34 shows the possible variables that have a chance of influencing friction, where it is possible to observe that several factors are sources of variation in tribological properties. However, in practical terms, for a numerical simulation, it is important to quantify the most important factors.

Based on all these factors mentioned above, it is clear that accurate and efficient prediction of the formability and springback in metal forming requires enhanced friction modeling. In Lee et al.’s [117] study, an alternative to the simple Coulomb friction law, a multiscale friction model under a mixed lubrication condition, was numerically applied to consider realistic contact and friction behavior in the sheet metal forming process.

The first factor that has a big influence on friction is the shape and finish of the surfaces. Figure 35 shows work by Gantar et al. [27] where the surface type and hardness were varied in a pin-in-disk test. The idea of modifying the surface of tools is normally used to generate lubricant pockets from which, when the surface pressure increases, the lubricant is released, thus decreasing friction. This effect is also desired for large displacements between the tools and the workpiece to avoid wear.

The friction results of Figure 35 are shown in Figure 36 where it is possible to see that the surfaces that generate less friction are those that have lubricant storage cavities as in Figure 35a,b. Figure 35c does not show good results as the surface peaks are too high, causing a very punctual contact between the dies and the sheet and the lubricant not being able to act in the entire area.

The surfaces of materials in contact only change when there is wear, but there are factors that can change throughout the forming of a piece [119]. One of the process parameters that most influences friction is the contact pressure between the dies and the sheet. Figure 37 shows that contact pressure causes friction to decrease. This normally occurs because the surfaces of the tools store lubricants in the roughness valleys and, as the pressure increases, there is a local strain of the roughness peaks causing the lubricant to leak into the non-lubricated areas, thus reducing friction.

In addition to the contact pressure, other parameters that have a strong influence on friction are the sliding speed between the sheet and the matrix and the temperature at the piece/matrix interface due to friction itself and the level of strain. To illustrate this effect, Figure 38 and Figure 39 show how speed and temperature impact friction. It is possible to see that as the temperature increases, the friction also increases. Figure 38 presents the variation observed with a temperature change of 50 °C, a range achievable in cold forming processes. In Figure 39, the speed varies from 30 to 70 mm/s, showing that, at lower speeds, friction tends to be reduced, while an increase in speed leads to a corresponding rise in friction.

These data demonstrate that regardless of what is happening during a sheet metal forming process, friction will not be constant, and this is one of the main sources of error in numerical simulations. Therefore, Hora et al. [123] proposed an equation that joins the three main process variables that impact the friction coefficient.

μ = μ_{0} \cdot {(p_{0} - p)}^{- a} \cdot {(\frac{T - T_{0}}{T_{0}})}^{b} \cdot {(v_{0} - v)}^{- c}

(15)

where μ₀, p₀, T₀, and v₀ are the initial friction coefficient, the initial pressure, the initial temperature, and the initial velocity, respectively, and “a”, “b”, and “c” are the adjustment coefficients.

3.6. Simulation Results

As mentioned before, it is possible to determine a series of parameters that can better adjust the results of a simulation both in the definition of the material and in the material/tools interface. The results that will be shown in this section show how these adjustments can improve the simulation results in relation to springback.

Figure 6 shows a representation of how springback is studied using a standardized test. Figure 6a shows the geometry that is most used in scientific papers to study the springback and Figure 6b shows which measurements are taken to analyze the results of this test. In the specific case of the simulation analysis represented in Figures 42–44, a U-bending process was used, which is commonly applied during the optimization of numerical parameters. The blank material is aluminum, with dimensions of 0.81 mm × 35 mm × 350 mm, with a value of Young’s modulus of 71 GPa, a Poisson’s ratio of 0.33, a density of 2.7 × 10³ kg/m³, a friction coefficient of 0.162, and a blank holder force of 2.45 kN.

The first analysis that must be performed regarding the finite element simulation is the mesh convergence, described above. Figure 40 shows the result of a finite element simulation of the springback test where the mesh varies from an element size of 5 mm to 1.5 mm. In this figure, is possible to observe that the more refined the mesh, the closer the results converge. In this case it would be ideal to use a maximum element size of 1.5 mm.

Another property of numerical simulation is the “time step” which refers to the time interval or displacement in which the deformation and stress information of the elements will be updated, generating a new stiffness matrix. The “time step” works in the same way as the element size and the integration points in thickness, that is, the smaller it is, the more accurate the simulation will be. Figure 41 shows the level of time step that is required for results to converge.

So far, on all the parameters of the simulation that must be adjusted so that the results are not influenced by the numerical analysis itself have been commented on. However, several other sources of error can exist in the simulation that do not depend exclusively on the finite element method. The first source of error is the characterization of the sheet material. Figure 42 shows the error associated with a simulation when the material model is used without considering the Bauschinger effect. Two material models are compared with experimental results, the first model is isotropic hardening, that is, without the Bauschinger effect, and the second is the kinematic hardening, with the Bauschinger effect, where it is possible to see that when the material presents this behavior, the error when not using the material correctly becomes higher.

Another result shown in the work of Choi et al. [36] is the effect of the types of material models used to simulate the springback of two advanced high-strength steels (DP980 and TWIP980). Figure 43 and Figure 44 show the angles θ1, θ2 and wall radius ρ of the springback test for materials DP980 and TWIP980, respectively, where it is possible to see that the best results (those closest to the real one) are exactly the ones that have the best characterization, that is, they consider the Bauschinger effect and the variable modulus of elasticity. These results considered constant friction, which can also be a source of error.

The tested models are described below:

1—Swift flow curve + von Mises flow criterion + isotropic hardening (IH) + constant modulus of elasticity (E);

2—Swift flow curve + Hill (1948) (based on anisotropy − r) + IH + E;

3—Voce flow curve + Yld2000-2d + YU model + variable E;

4—Swift flow curve + Yld2000-2d + HAH model + variable E.

Lastly, as mentioned earlier, the coefficient of friction is also a source of variation in the sheet metal forming process, where finite element software normally considers it to be constant, adding error to the results. This was evaluated in the work of Wang et al. [118], where the friction was varied according to the contact pressure for the DP780 steel and later loaded in the simulation. Figure 45 shows that a variable friction generates a different springback in which the error in relation to the experimental test is 31.1% for constant friction and 7.4% in relation to variable friction. Thus, it is also important to evaluate how the friction coefficient behaves to obtain better results in the simulation.

However, there are other parameters beyond friction in the experiments that are crucial for proper calibration of the simulation. These include accurate experimental data on material properties, initial sheet thickness, adjusted tool curvature radius, forming speed (strain rate), and process temperature. For the validation of forming simulations, in addition to a visual geometric comparison between the stamped part and the simulated part, validation can be performed by measuring bending angles, especially when the simulation addresses springback. Another approach involves comparing the final thickness values obtained experimentally and in the simulation.

As discussed above, simulation can play an important role in evaluating sheet metal forming processes, but it also presents certain challenges for effective use and result interpretation. Table 2 presents crucial concepts that are essential for understanding the role of simulation, as widely discussed by several authors [38,40,41,112,113,114,124,125], outlining its advantages and disadvantages across various aspects.

4. Conclusions

As seen in this review, simulation is an indispensable tool for designing complex pieces like those used in the automotive industry, besides the steels that are also increasingly complex to suit the demands of low weight and high strength. Therefore, more and more new characterization tests of these materials will be necessary for the simulations to be successful, and this will have to be incorporated both by production plants and by stamping manufacturers. An example of this is that, in the past, a few parameters obtained by the tensile test and the forming limit curve were sufficient for the simulation to run well for common steels. However, for high-strength steels, tests such as tensile, cyclic tensile, tensile–compression, variable friction, bulge, and biaxial tests are already needed, all with a data acquisition system and automated loading control.

Within this context, the FLD curve is currently being replaced by the FFLD curve because the industry in general is increasingly demanding the use of ultra-high-resistance steels and these new materials require a more appropriate characterization. Therefore, finite element software that proposes to simulate and predict dedicated manufacturing processes such as the stamping of fine sheets is having to include these new modeling methods of materials so that the expected results are increasingly assertive and correlated with reality.

In the future, the triaxial curve will replace the FLD and even FFLD because, as in a tensile test, this curve compares tensions with deformations, not only deformations such as shown by an FLD and FFLD, which is more effective to characterize and predict the behavior of metallic materials.

Author Contributions

Conceptualization, investigation, methodology, visualization, writing—original draft, L.F.F.; investigation, methodology, supervision, writing—review and editing, T.N.L.; investigation, visualization, writing—original draft, M.P.S.S.; methodology, writing—review and editing, validation, B.C.; project administration, resources, B.C.d.S.S.; resources, supervision, L.G.S.Z.; methodology, project administration, resources, supervision, R.S.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by SENAI National Department, grant number 329784 and FORD MOTOR COMPANY OF BRASIL LTDA.

Data Availability Statement

Not applicable.

Conflicts of Interest

Luiz Gustavo Souza Zamorano was employed by the company Ford Motor Company. The authors declare that this study received funding from SENAI National Department and FORD MOTOR COMPANY OF BRASIL LTDA. The funders were not involved in the study design, collection, analysis, interpretation of data, the writing of this article or the decision to submit it for publication.

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Figure 1. Advanced high-strength steels developed for automotive applications [20]. Reproduced with permission from Elsevier, 2024.

Figure 2. Example of springback in sheet metal bent at 90° [34]. Reproduced with permission from Elsevier, 2024.

Figure 3. Effect of springback on high-strength steels [35]. Reproduced with permission from Elsevier, 2024.

Figure 4. Springback effect after (a) bending in deep drawing [36] and (b) V-bending. Reproduced with permission from Elsevier, 2024.

Figure 5. Influence of bending and straightening on residual stresses during deep drawing of metallic sheets.

Figure 6. (a) Standard geometry used for the study of springback; (b) measurements made on the part after bending [36]. Reproduced with permission from Elsevier, 2024.

Figure 7. Convergence analysis on the axial loading value (normalized) as a function of the density and the element type. Figure reproduced under Creative Commons Attribution 4.0 International License from [39].

Figure 8. The error in the axial loading estimation as a function of the CPU time, for different element types and numbers of elements. Figure reproduced under Creative Commons Attribution 4.0 International License from [39].

Figure 9. Shell element with integration points in the thickness.

Figure 10. Strain stages in a tensile test for a conventional material.

Figure 11. Types of localized failure that can occur through void nucleation: (a) Failure by localized shear plastic without necking, (b) Failure by localized shear plastic after necking and (c) Failure by void coalescence with obvious necking [53]. Reproduced with permission from John Wiley and Sons, 2024.

Figure 12. Examples of behavior under fracture of 3 metals [54]. Reproduced with permission from Elsevier, 2024.

Figure 13. Two types of mechanisms for void coalescence: (a) parallel connection between voids; (b) void shear connection [55]. Reproduced with permission from Elsevier, 2024.

Figure 14. Two failure mechanisms: necking for SPCC and SPRC and ductile fracture for other sheet metals. SPCC and SPRC are conventional carbon steels [56]. Reproduced with permission from Elsevier, 2024.

Figure 15. Forming Limit Diagram obtained by measuring the diffuse necking: (a) specimens, (b) fracture regions, and (c) plotted curve.

Figure 16. Visualization of the circles used in the Nakajima test to obtain the main strains [57]. Reproduced with permission from Elsevier, 2024.

Figure 17. Location of positions where deformations can be measured. (a) In the necking zone. (b) Out of the necking zone. Figure reproduced under Creative Commons Attribution 4.0 International License from [58].

Figure 18. Comparison between FLD (black and red) and FFLD (blue) curves: (a) in the space of major and minor principal strains; (b) in the space of stress triaxiality and equivalent strain to failure [76]. Reproduced with permission from Elsevier, 2024.

Figure 19. Global strain status, experimental FLC, and damage: (a) results at the integration points located on the negative surface; (b) results at the integration points located on the positive surface [76]. Reproduced with permission from Elsevier, 2024.

Figure 20. Comparison of the (a) experiment and (b) simulation [76]. Reproduced with permission from Elsevier, 2024.

Figure 21. Comparison between results obtained in a (a) simulation using the FLC curve and (b) a simulation using the MMC fracture criterion of an automotive front rail made from DP780 steel [73]. Reproduced with permission from SAE international, 2024.

Figure 22. Fracture strain versus stress triaxiality and Lode angle simulated. Figure reproduced under Creative Commons Attribution 3.0 International License from [88].

Figure 23. Mathematical adjustment curves for tensile testing. Figure reproduced under Creative Commons Attribution 4.0 International License from [92].

Figure 24. Comparison of tensile test data with Bulge test. Figure reproduced under Creative Commons Attribution 3.0 International License from [93].

Figure 25. Three main directions of anisotropy measurements in the sheet rolling Direction.

Figure 26. Compressed disk specimens of AA2090-T3 using different lubricants and at different thickness strains (ε_z) [80]. Reproduced with permission from Elsevier, 2024.

Figure 27. Strains measured in samples after the disk compression test [80]. Reproduced with permission from Elsevier, 2024.

Figure 28. Calculation of the biaxial anisotropy coefficient. Figure reproduced under Creative Commons Attribution 4.0 International License from [103].

Figure 29. Comparison of true stress–strain curves determined by uniaxial tension and Bulge test for 3 different steels [104]. Reproduced with permission from Elsevier, 2024.

Figure 30. Schematic representation of the Bauschinger effect.

Figure 31. Typical tensile–compression test [104]. Reproduced with permission from Elsevier, 2024.

Figure 32. Stress–strain curves of the tensile test under loading–unloading–reloading condition for determination of Young’s modulus of elasticity at different pre-strains [104]. Reproduced with permission from Elsevier, 2024.

Figure 33. Young’s modulus change with plastic strain.

Figure 34. Scheme of a tribological system in sheet forming. Figure reproduced under Creative Commons Attribution 4.0 International License from [116].

Figure 35. The 3D images and mean roughness Rz values of bionic structures (a) St1, (b) St2, (c) St3, (d) St4 (e) St5, and (f) flat reference surface [118]. Reproduced with permission from Elsevier, 2024.

Figure 36. Friction coefficient results obtained [118]. Reproduced with permission from Elsevier, 2024.

Figure 37. Coefficient of friction under different contact pressures [120]. Reproduced with permission from Elsevier, 2024.

Figure 38. Friction variation with working temperature: (a) Variation curves of the friction coefficient with time under different temperatures; (b) Experimental friction coefficients at different temperatures. Figure reproduced under Creative Commons Attribution 4.0 International License from [121].

Figure 39. Results of the coefficient of friction as a function of sliding speed and contact pressure. Figure reproduced under Creative Commons Attribution 4.0 International License from [122].

Figure 40. Comparison of springback for different mesh sizes in finite element simulation [38]. Reproduced with permission from Springer Nature, 2024.

Figure 41. Comparison of springback for different time steps in finite element simulation [38]. Reproduced with permission from Springer Nature, 2024.

Figure 42. The comparison of calculated springback between isotropic and kinematic hardening mode [38]. Reproduced with permission from Springer Nature, 2024.

Figure 43. Finite element simulation results for DP980 in a springback test: (a) θ₁, (b) θ₂, and (c) sidewall radius ρ [36]. Reproduced with permission from Elsevier, 2024.

Figure 44. Finite element simulation results for TWIP980 in a springback test: (a) θ₁, (b) θ₂, and (c) sidewall radius ρ [36]. Reproduced with permission from Elsevier, 2024.

Figure 45. Numerical simulation results for models with constant and variable friction [120]. Reproduced with permission from Elsevier, 2024.

Table 1. Mathematical models representing the most used strains.

Model	Equation
Hollomon [94]	$σ = K_{1} \cdot ε^{n}$	(4)
Ludwik [95]	$σ = K_{1} {+ K}_{2} \cdot ε^{n}$	(5)
Swift [96]	$σ = K_{1} \cdot {(ε + ε_{0})}^{n}$	(6)
Voce [97]	$σ = K_{1} + (K_{2} - K_{1}) \cdot e^{- m \cdot ε}$	(7)
Hockett/Sherby [98]	$σ = K_{1} + (K_{2} - K_{1}) \cdot e^{- m \cdot ε^{n}}$	(8)
Ghosh	$σ = K_{1} {+ K}_{2} \cdot {(K_{3} + ε)}^{n}$	(9)
Swift–Voce	$σ = K_{1} \cdot σ_{S w i f t} + (1 - K_{2}) \cdot σ_{V o c e}$	(10)

Table 2. Key aspects of numerical simulations: advantages and disadvantages.

Aspects	Advantages	Disadvantages
Cost and time	Significant cost reduction in trials and construction of prototypes.	High initial cost of software and hardware to run simulations.
	Shorter project development time (reduces lead time from design to production).	Considerable time to correctly set up and run complex simulations.
Predictability	Identification of process problems such as forming defects and springback.	Dependency on the quality of input data regarding material properties and process conditions.
	Enables obtaining a detailed analysis on stresses, strains, thickness variation.	Results obtained may be inaccurate if the numerical models applied cannot adequately represent real forming processes.
Flexibility	Facilitates testing of different process conditions such as materials, tools, and forming parameters.	Limited in analyzing process involving highly non-linear and complex phenomena.
	Enables process optimization without physical intervention.	Optimization obtained may not be feasible depending on technology restrictions.
Safety	Allows analysis of extreme conditions without risks to the operator or equipment.	Cannot fully replace experimental test for final validation.
Integration with design and project	Compatibility with other CAD/CAE software, facilitating adjustments and analysis.	Requires specialized training and knowledge to correctly interpret the results and adjust the design.

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MDPI and ACS Style

Folle, L.F.; Lima, T.N.; Santos, M.P.S.; Callegari, B.; Silva, B.C.d.S.; Zamorano, L.G.S.; Coelho, R.S. A Review on Sheet Metal Forming Behavior in High-Strength Steels and the Use of Numerical Simulations. Metals 2024, 14, 1428. https://doi.org/10.3390/met14121428

AMA Style

Folle LF, Lima TN, Santos MPS, Callegari B, Silva BCdS, Zamorano LGS, Coelho RS. A Review on Sheet Metal Forming Behavior in High-Strength Steels and the Use of Numerical Simulations. Metals. 2024; 14(12):1428. https://doi.org/10.3390/met14121428

Chicago/Turabian Style

Folle, Luis Fernando, Tiago Nunes Lima, Matheus Passos Sarmento Santos, Bruna Callegari, Bruno Caetano dos Santos Silva, Luiz Gustavo Souza Zamorano, and Rodrigo Santiago Coelho. 2024. "A Review on Sheet Metal Forming Behavior in High-Strength Steels and the Use of Numerical Simulations" Metals 14, no. 12: 1428. https://doi.org/10.3390/met14121428

APA Style

Folle, L. F., Lima, T. N., Santos, M. P. S., Callegari, B., Silva, B. C. d. S., Zamorano, L. G. S., & Coelho, R. S. (2024). A Review on Sheet Metal Forming Behavior in High-Strength Steels and the Use of Numerical Simulations. Metals, 14(12), 1428. https://doi.org/10.3390/met14121428

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