1. Introduction
Over the past decades, the automotive industry has continuously strived to reduce the product development time in order to increase their competitiveness and profit. By the use of numerical modelling with the Finite Element method (FEM), it is now possible to perform virtual testing of components with high accuracy, thus reducing the amount of costly experiments and prototyping. However, with extended use of computer aided engineering in the product development process comes new challenges and demands on the numerical FE-modelling. One of the challenges of numerical FE-modelling is how to accurately describe damage generation in terms of micro-cracks, sharp notches and voids. Damage markedly affects sheet formability and part performance of high strength metallic sheets, such as Advanced High Strength Steels (AHSS) or high strength aluminium alloys. Most manufacturing defects in such materials are related to the edge cracking phenomenon, that is, the generation of small cracks at sheared (damaged) edges during stretching or bending. Regarding part performance, crash resistance involves the generalized cracking (damage) prior to final fracture. High strength metallic sheets are widely used for automotive lightweight constructions. Thus, FE-analysis of sheet metal forming and crash modelling requires numerical models that accurately handle damage, that is, the formation and propagation of cracks.
Shear cutting of high strength sheet metals is such process where the material separation involves the formation of cracks, and inherited damage from the cut affects the formability of the part. Publications by Konieczny and Henderson [
1], Dykeman et al. [
2], Thomas [
3] and Sigvant et al. [
4] show that the shear cutting process degrades the stretch-flangeability of the material when comparing formability of sheared edges to laser-cut or polished edges. This statement is further supported through work by Mori et al. [
5], Mori et al. [
6], Gläsner et al. [
7] and Saengkhiao et al. [
8], who present warming and smoothing techniques that improve the cut edge quality, thus improving the sheet formability. Microscopical investigations of sheared cut edges performed by Wu et al. [
9] and Yoon et al. [
10] also show the formation of micro-cracks along the cut edge surface, which are considered to mainly cause the limited formability of the cut edge due to edge-cracking.
Casellas et al. [
11] showed that the Essential Work of Fracture (EWF), a mechanical property determined by fracture mechanics that is equivalent to the conventional Elastic Plastic Fracture Mechanics (EPFM) toughness value
and measures crack propagation resistance, effectively describes stretch-flangeability and edge cracking resistance of AHSS. Such work pointed out that crack-related phenomena in sheet metal forming can be addressed by fracture mechanics concepts. Frómeta et al. [
12] proposed to use the EWF to rank crashworthiness in AHSS, because in crash tests the generation of damage (cracks) and the further crack propagation is related to the overall crash energy absorption. The work shows excellent correlation between crack propagation resistance, that is, fracture toughness, and crashworthiness for different AHSS grades. Recently, Frómeta et al. [
13] proposed to use fracture mechanics parameters related to the EWF to understand cracking resistance of AHSS and describe the damage that limits the formability of AHSS. The above publications related to the fracture toughness and cracking resistance of AHSS suggest that damage during forming and part performance should be addressed considering the crack nucleation and propagation. This directly implies that numerical modelling of shear cutting and forming of AHSS also should be capable to capture and follow the crack nucleation and propagation.
Numerical modelling of the shear cutting process is a challenging forming operation to simulate, as shear cutting involves both large material deformation, shear failure and duplex crack initiation. In addition, the numerical failure model should be able to handle a large range of stress states. The complex failure assessment is conventionally managed by strain-driven ductile failure models with stress state dependency. This was done by Hambli and Potiron [
14], who implemented a Lemaitre damage model presented by Lemaitre and Chaboche [
15] for an axisymmetrical shear cutting model of 1060 steel for process optimisation purposes. Hambli and Potiron [
14] defined the crack initiation and propagation through loss of element stiffness when a critical damage value was obtained in the element. Similarly, Thipprakmas et al. [
16] utilised an axisymmetrical model along with a Rice and Tracey [
17] fracture criterion to analyse fine-blanking of medium steel S45C. In order to avoid numerical divergence due to distorted elements, a re-meshing algorithm adjusted the elements of the shear affected zone. To ensure a correct crack path, Bacha et al. [
18] developed a two-step model for trimming of aluminium consisting of re-meshing and arbitrary Lagrangian formulation of the shearing process until crack initiation, from where the plastic equivalent strain acted as a damage variable for the following crack propagation. Work that has received considerable attention was presented by Wang et al. [
19] and consisted of a three dimensional punching and forming model where the modified Mohr–Coulomb model by Bai and Wierzbicki [
20,
21] defined the material failure. Wang et al. [
19] could predict edge cracking of high strength steel during hole expansion with inherent edge damage from shear cutting, using element deletion to simulate material failure. The works mentioned have in common that they utilise strain-driven ductile failure models or limiting failure strain values for simulating the fracture process through element deletion or stiffness reduction. The failure strain values are mainly determined from experimental tensile testing. This makes them suitable for the global failure analysis of undamaged materials, such as for damage initiation in crash analysis. Applying strain-driven ductile failure models in crack initiation analysis will, according to Anderson [
22], cause mesh size dependency and often need fine tuning to fit the experimental data. This procedure contrasts with fracture mechanical methodologies, such as the J-integral, where energy measures define the crack opening and propagation. Fracture mechanical approaches generally focus on the analysis of crack tip conditions and are nearly independent of the size of the crack tip. However, as stated by Anderson [
22], incorporating fracture mechanical approaches for FE-frameworks often requires cumbersome re-meshing, which makes fracture mechanical techniques seldom used in commercial fracture analysis.
It is clear that the intended area of usage differs between strain-driven ductile failure models and fracture mechanics, but for the numerical analysis of the shear cutting process there is an apparent need for both a failure assessment of undamaged material and crack tip analysis. To investigate the applicability of using only strain-driven ductile failure models for processes involving cracks, such as shear cutting, the authors of this article perform numerical modelling of cracked high strength sheet steel without taking notch effects or crack-tip singularities into account. The current work focuses on well-known fracture mechanical lab tests in order to ensure controlled crack initiation and propagation. Results from the laboratory tests were compared with corresponding simulation results using the Generalised Incremental Stress State Dependent Damage Model (GISSMO) as a ductile failure model in order to investigate how cracks limit the modelling accuracy. The laboratory tests experience Mode I crack opening with a minimum of uncertain details such as friction, dynamics or joints. The simplicity of the validation cases makes them suitable when evaluating the numerical modelling as every assumption made for the model can cause deviation from the actual load case. The material investigated is a complex phase steel with an ultimate tensile strength of 1000 MPa approximately and high ductility (named as CP1000HD from now). It is an AHSS grade with 1.5 mm thickness, commonly used for automotive crash components due to its high strain hardening rate and ductility.
2. Introduction to Damage- and Failure Modelling
In this work, the Generalised Incremental Stress State Dependent Damage Model (GISSMO) was used for damage and failure modelling. GISSMO was developed by Neukamm et al. [
23] and extended to include lode angle dependency by Basaran et al. [
24]. Further description of the GISSMO model was presented by Andrade et al. [
25]. GISSMO was implemented for the FE-software LS-Dyna and is a phenomenological damage and failure model that uses stress state dependent equivalent plastic strain as failure criteria. It is uncoupled from the plasticity model, thus compatible with several of the available material models in the LS-Dyna library. Ever since the implementation of GISSMO in LS-Dyna has its ability to generate high accuracy results been on display in several publications. Omer et al. [
26] used GISSMO for the modelling of crush members and Chen et al. [
27] evaluated the fracture predictability of an AHSS GISSMO model through a number of validation crash simulation. Similarly, Pérez Caro et al. [
28] could detect forming damages of Alloy 718 through the use of GISSMO. Considering its well documented ability to model failure, it is made clear that GISSMO is a useful tool for detecting where and when material failure occurs for certain purposes.
The damage accumulation rule is defined by Neukamm et al. [
23] according to Equation (
1), where
denotes the incremental damage value,
is the stress state dependent failure strain value,
is the accumulated plastic strain rate and
n is the damage exponent.
When the accumulated damage value reaches
material failure occurs. If wanted by the user, material failure can be preceded by material degradation through coupling stresses to the damage level. The damage/stress coupling is displayed in Equation (
2), where it is shown how the stress is coupled through the accumulated damage value
D and the damage value at material instability
. The fading exponent
m controls the non-linearity and the exponential damage coupling.
The material instability value
F controls when damage/stress coupling is initiated and accumulates in a similar way as the damage value
D, as shown in Equation (
3). An accumulated instability value of
defines the point of diffuse necking and the damage/stress coupling is initiated.
The instability strain
can either be a fixed value or a stress state dependent curve. Material instability or point of diffuse necking is a complex phenomenon and several instability criteria have been developed over the years. A commonly used instability criteria worth mentioning is the Considére criterion, defined as the strain at maximum engineering stress with a corresponding plastic strain equal to the work hardening exponent, presented by Considére [
29]. The damage and failure implementation provides the option to use arbitrary curves and functions to describe the material instability, denoted
ECRIT. Consequently, it is up to the user to determine how to define the material instability criteria that suit the particular material of interest.
Not seldom is the stress state dependent failure strain and material instability obtained through parameter calibration, where parameters such as , m, n and are fitted to give a correlation between simulation and corresponding test results.
Material failure in finite element analysis is generally manifested through element erosion when the failure strain is obtained for the element. However, the actual effective strain value of an element is affected by the element length, especially for geometries where stress/strain localization occurs. The element length dependency of the effective strain gives a mesh dependency during failure modelling that needs to be accounted for. The mesh dependency is treated through mesh regularisation, where a factor defined by the user scales the failure strain value. The user can input a curve defining scale factor versus element size in order to cover the mesh dependency over a range of element lengths.
The recent development of damage and failure modelling includes anisotropic hardening and failure behaviour. This development is driven by the urge of predicting failure in complex geometries consisting of anisotropic materials such as high strength steel, aluminium and composites. Such work was presented by Park et al. [
30], which modified the Lou–Huh ductile fracture criterion based on the Hill’s 48 anisotropic yield function, thus incorporating the anisotropic effect of the material in the failure strain. Furthermore, Basak et al. [
31] showed that the formability and failure of deep-drawn anisotropic materials can be predicted using anisotropic material models along with MK-FLD and GISSMO. The extension of GISSMO presented by Koch et al. [
32] also meant a damage and failure model with similar capabilities as its precursor, but with the possibility of including individual damage and failure characteristics depending on the deformation direction. Anisotropic behaviour of the CP1000HD was not considered in this work because it is expected to have a low impact on the modelling results. First, the specimens are thin (1.5 mm) and, secondly, the crack imitation and propagation occur locally under pure tensile loading. In complex tensile states situations, such as punching or shear, anisotropy effects should be considered.
A detailed description of the damage and failure model and element erosion techniques for LS-Dyna is found in the LS-Dyna user’s manual by the Livermore Software Technology [
33] and the theory manual by Hallquist [
34].
4. Results
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16 show the simulation results for triaxiality tests, VDA 238-100 and notched and cracked DENT specimens. Together with the simulation results, corresponding experimental results are included for comparison.
Since the CP1000HD damage and failure model was calibrated for matching the force-displacement results from the R3.75 tensile testing, the correlating results between simulation and experiment in
Figure 12a are expected. However, for the remaining tensile tests, a similar correlation is found, as shown in
Figure 12b–d.
Figure 13 shows the simulation results for the VDA 238-100 model along with the four sets of experimental results. As seen in
Figure 13, there is a clear correlation between the FE-model results and the experimental results both in terms of force-displacement response and for predicting the bending punch stroke until failure.
Figure 14 shows the observed cracks from experiments compared to the crack initiation predicted by simulation, defined as deleted elements.
The comparison between experimental and simulation results for notched DENT specimens are shown in
Figure 15. The comparison in
Figure 15 shows that the modelling of the notched DENT specimen correlates well with the experimental results until the final part of the fracture process. The points of crack initiation are marked in
Figure 15 and
Figure 16, where (•) and (∘) identifies the crack initiation at the specimen surface from FE-models and experiments, respectively. In tensile testing of DENT specimens, it is well reported that the crack initiates at the centre of the specimen, where stress triaxialtiy is higher than the outer free-surfaces of the specimen, as stated by Frometa et al. [
40]. This implies that crack initiation, in the specimen centre, is experimentally difficult to assess as the experimental DIC procedure could only capture data from the specimen surface. Thus, many works report crack initiation when it reaches the specimen’s surface. However, modelling can provide this crack initiation point and give relevant information for cracking prediction. The asterisk mark (∗) in
Figure 15 and
Figure 16 denotes the first point of crack initiation predicted by numerical models, occurring at the centre of the specimen. As stated, crack initiation in the notched DENT specimen centre was not possible to detect experimentally, which is why the validation of the numerically obtained crack initiation of the specimen centre could not be done. Comparing points of crack initiation at the specimen’s surface between numerical models and experiments show a decent correlation. Meanwhile, it is shown in
Figure 16 that the numerical modelling of cracked DENT specimens could not accurately reproduce the force displacement results from experiments. The results show an over-prediction of displacement at surface crack initiation with 15–35%, where the largest deviation occurs for the shortest ligament lengths. Considering the poor correlation between the modelling and experimental results of cracked DENT specimens, it can be stated that the current damage and failure model could not accurately predict the fracture behaviour of specimens containing cracks. The results in
Figure 16 show that inaccurate crack initiation from modelling produces overly high forces for the remaining displacement. This trend applies for the entire range of ligament lengths.
Comparing the equivalent strain field of the DENT ligament for both cracked and notched DENT specimen right before crack initiation is a supplementary indication of the modelling accuracy.
Figure 17 shows the equivalent strain field for notched DENT specimen with a ligament length of 6 mm. Similarly,
Figure 18 shows the equivalent strain field for notched DENT specimen with a ligament length of 5.087 mm. A similar set of images is presented in
Figure 19 for notched DENT specimen with ligament length of 14 mm, and
Figure 20 for cracked DENT specimen with ligament length of 13.213 mm.
5. Discussion
The good correlation between experiments and simulation for VDA 238-100 and notched DENT in
Section 4 shows that the damage and failure model accurately predicts crack initiation where undamaged sheet steel of CP1000HD grade is subjected to tensile and bending deformation. The CP1000HD damage and failure model presented in this article is therefore useful for predicting failure through thinning and necking in, for instance, forming applications as drawing or deep drawing. From
Figure 15, it is also apparent that the damage and failure model is capable of reproducing the fracture evolution process of the notched DENT specimen with decent accuracy, almost until the final fracture of the specimen.
However, the study of cracked DENT specimens in
Figure 16 shows poor simulation accuracy in defining the crack initiation. The numerical models of the cracked DENT tests all experience over-predicted displacement for crack initiation compared to experiments, with an average of 28% over-prediction, which is why the following fracture evolution process will inherit inaccurate force levels. Numerical modelling of crack initiation and propagation generally imposes stringent requirements on the FE-discretisation. The crack-tip needs to be sufficiently resolved and structured in order to account for the crack-tip singularity, and the crack propagation path is inevitably controlled by the mesh, as stated by Anderson [
22]. Neither the mesh size nor the mesh structure of the numerical model of the cracked DENT specimens were sufficient to reproduce the crack initiation. The combination of damage and failure modelling and FE-discretisation used in this article causes a plastic process before crack initiation and the crack-tip singularity is not accurately treated. This inaccuracy is accentuated when comparing the ligament strain field for cracked DENT specimens shown in
Figure 18 and
Figure 20, where a more extensive strain field is obtained at crack initiation in a simulation compared to the experimental results. On the contrary,
Figure 17 and
Figure 19 show good ligament strain field accuracy for notched DENT specimens. An even finer mesh than used in this article would cause crack initiation at lower displacements as the crack-tip elements would reach failure strain faster, and thus generate an improved correlation between the simulation and the test for the cracked DENT specimens. However, smaller element lengths (
mm) would generate premature failure for modelling of the triaxiality tensile specimens as well as for the VDA 238-100 model since the damage and failure model is calibrated towards an element length of
mm. Through mesh regularisation of the damage and failure model, a finer mesh would generate similar results as a coarser mesh, with the drawback that cracked DENT modelling results would again diverge from experiments. Calibration of the failure surface to fit the separate case of cracked DENT specimens would solve the over-predicted displacement at crack initiation for this particular problem, but would mean that the generality of the material characterisation method used in this article is lost. Failure surface calibration to specific problems would also infer inconvenience when using the CP1000HD damage and failure model on geometries where cracks emerge. This case is, for instance, found in the modelling of shear cutting, where the shearing process is followed by the opening of cracks preceding the fracture process, as shown by Dalloz et al. [
43], Wang and Wierzbicki [
44] and Zhang et al. [
45] in interrupted punching tests.