In order to meet the demand for the safe and lightweight design of passenger cars, Advanced High Strength Steel (AHSS) sheets have become the preferred materials for car body and some passive safety components. Compared with the cold rolled sheets for stamping and conventional high-strength low-alloy steels (HSLA), AHSS is not only excellent in the combination of strength and plastic, but also is more remarkable in local deformability as the result of their unique strengthening mechanism, like phase transformation, twinning and complex phase composition. Therefore, a series of state-of-the-art forming processes such as hydraulic forming, bending, rolling, torsion and hot stamping have further expanded the industrial applications of AHSS. Meanwhile, some structural components in passenger cars endure complex loading states during driving and especially under collision, more attention needs to be paid to the material constitutive relation, the material response, especially the resistance to the ductile fracture under different stress states such as compression, shear, and uniaxial/biaxial tension, necessitated by materials R&D and industrial applications.

On the other hand, Nakajima’s Forming Limit Diagram (FLD) test reveals that the arrest performance of sheet metal to the ductile fracture is significantly affected by plane stress loading conditions. As a result, Forming Limit Curve (FLC) defines the material ultimate deformation with respect to the different deformation paths. Through FLD test, Bai and Wierzbicki [1] proposed their analytical criterion models for predicting the stamping failure based on the plane stress triaxiality. However, until now it has still been of a hard job to plot a reproducible and analytical FLC curve through scattering major/minor strain points. Moreover, conventional Nakajima test can only provide three regular plane stress states, like tension, plane strain tension and bulging, and consequent FLC curve is significantly affected by the non-proportional deformation path, mold lubrication and other variables. More importantly, Nakajima test cannot provide shear, tension shear, which are extremely important loading conditions in automobile collision, especially for the components or structures made of light-weight high-strength materials.

Meanwhile, the Generalized Incremental Stress State Dependent Damage Model (GISSMO) in LS-DYNA^Ò is capable to predict the ultimate fracture behavior of the calibrated material in complex strain paths. For new materials, GISSMO needs real-tested tensile force-displacement curves of 5 - 8 types of samples with well-designed geometries, and their DIC frames during all loading periods until fracture to validate and calibrate the material parameters used in the model [2]. However, it was reported that the stress triaxiality confined in the fracture zone varies significantly, due to the loading alignment and the material local deformation [3] [4]. When using the theoretical geometry-related stress triaxiality (η) as the input to GISSMO, inestimable system deviation will be introduced.

Many studies [5] [6] have designed samples with typical theoretical triaxiality and obtained the force-displacement curve of the fracture zone through DIC technology to calibrate the accuracy of Finite Element Method. Then the whole process of loading the samples to fracture could be simulated to track the evolution history of physical characteristic parameters characterizing the toughness fracture, such as stress triaxiality, Lode angle, equivalent effect strain, etc. Because of the drastic changes, the strain integral averaging method was usually used to determine these characteristic parameters.

In this paper, the inconvenience of the application of FEM in industrial laboratory was avoided instead Moving Regression Algorithm was deployed upon the measured (ε₁, ε₂) to monitor the inherent relationship among the key parameters involved in the four variables Mohr-Coulomb model. That is, the transformation from the originally measured (ε₁, ε₂) into the explicit (η, ε ¯ ) required by M-C model has been well detailed. Also the locus of Mohr-Coulomb fracture curve with the transformed extended FLC curve inclusive of shear and tension-shear definition was also introduced to characterize the material resistance to the ductile fracture under various plane stress conditions.

Due to the requirement of thickness reduction and lightweight, AHSS sheets are usually delivered with thin thickness, so the material mainly experiences the plane stress state in service. Similar to the FLC representing various stamping deformation paths, the plane stress plasticity could be expressed by the principle true major strain ε₁ in loading direction and the true minor strain ε₂. Then, the strain ratio α is calculated by Equation (1).

α = Δ ε 2 / Δ ε 1 (1)

α in Equation (1) represents the slope of the curve of minor strain (ε₂) against major strain (ε₁) at a certain time. The regular FLD test shall consist of a group of samples, relative to three kinds of strain ratio: α = −0.5 is uniaxial tension, α = 0 is plane strain tension, and α = 1 is equibiaxial bulging. It should be noted that for the ideal specimen with special design, under the proportional loading or proportional strain path, the strain ratio can be simplified as the ratio of αε₂ to ε₁. However, the stress triaxiality usually changes with the deformation of the specimen under real conditions. Therefore, in the present study, the moving regression algorithm was applied to calculate the linear regression slope of each (ε₁, ε₂) scatter point to obtain the instantaneous α. Where N is the smooth bandwidth, which means N pairs of (ε₁, ε₂) scatter points neighbor to the event of η_i are to be included by using linear regression algorithm to obtain the final result, see Equations (2) and (3). The larger value of N causes the more smooth evolution curve of instantaneous η_i.

η i = Δ ε 2 Δ ε 1 ≃ ∑ j = i − N − 1 2 i + N − 1 2 ε 1 , j ε 2 , j − N ε ¯ 1 ε ¯ 2 ∑ j = i − N − 1 2 i + N − 1 2 ε 1 , j 2 − N ε ¯ 1 2           i = 1 , 2 , 3 , ⋯ , m (2)

ε ¯ 1 = ∑ j = i − N − 1 2 i + N − 1 2 ε 1 , j N ,       ε ¯ 2 = ∑ j = i − N − 1 2 i + N − 1 2 ε 2 , j N           i = 1 , 2 , 3 , ⋯ , m (3)

Since the z-component stress is neglected under the plane stress state, i.e. σ₃ = 0, the point (ε₁, ε₂) in real-tested strain space could be transformed into the point (σ₁, σ₂) as in the stress space [1], then the stress ratio β is as expressed in Equation (4).

β = σ 2 / σ 1 = 2 α + 1 2 + α (4)

Introducing the stress triaxiality η and Lode angle parameter θ ¯ , the (σ₁, σ₂, σ₃) in the Cartesian-coordinate stress space could also be transformed into corresponding points of ( σ ¯ , η, θ ¯ ) or ( ε ¯ , η, θ ¯ ) in the virtual spherical-coordinate stress or strain space respectively. All the variables can be determined through Equation (5) to Equation (8) [1] [7].

η = σ m σ ¯ = β + 1 3 β 2 − β + 1 (5)

θ ¯ = 1 − 2 π arccos ξ (6)

ξ = − 27 2 η ( η 2 − 1 3 ) (7)

d ε ¯ = 2 d ε 1 3 1 + α + α 2 (8)

Under the proportional loading of plane stress condition, the strain ratio α is constant, so that Equation (8) can be integrated and transformed into the Equation (9).

ε ¯ = 2 ε 1 3 1 + α + α 2 (9)

It should be noted that σ ¯ and ε ¯ are equivalent true stress and strain respectively in the spherical-coordinate space. So far, if the plane stress is applied to the specimen, the corresponding local plane stress triaxialityη, Lode angle parameter θ ¯ and equivalent true strain ε ¯ can be simultaneously calculated through the recorded major and minor true strains (ε₁,ε₂) on the local area of interest over all the loading duration. As a result, the characteristic value of ε ¯ f affected by the change of η implies the resistance performance of sheet metals to the ductile fracture, as experimentally studied in the paper.

On the other hand, Mohr Coulomb fracture model establishes the analytical relationship between loading triaxiality η and equivalent strain ε ¯ , which has been widely confirmed [7] [8] [9] [10]. Without considering anisotropy, the four-variable M-C model was described as Equation (11), here Hollomon flow hardening rule was included for simplification, see Equation (10):

σ ¯ = K ε ¯ n (10)

K is the strain strengthening coefficient obtained by the uniaxial tensile test (unit: MPa) and n is the strain hardening exponent.

c 2 1 + c 1 2 3 cos ( θ ¯ π 6 ) + c 1 [ η + 1 3 sin ( θ ¯ π 6 ) ] = { c θ s + ( c θ a x − c θ s ) ⋅ 3 2 − 3 [ sec ( θ ¯ π 6 ) − 1 ] } ⋅ K ε ¯ f n

c θ a x = { 1           θ ¯ ≥ 0 c θ c       θ ¯ < 0 (11)

Here, C₁ ³ 0 and C₂ > 0 are both material constants, in which C₁ is dimensionless and C₂ represents the stress (unit: MPa). According to different yield criteria, c θ s has different values. For example, when considering von Mises yield condition, c θ s = 1 , while c θ s = 3 / 2 is Tresca yield condition. Here for sheet samples loaded by uniaxial tension under plane stress states, c θ c = c θ s = 1 can be considered as a special case in general.

At present, the GISSMO model in LS-DYNA^Ò has been successfully used to optimize the cold forming process for AHSS and simulate the crashworthiness of passenger cars. As to validate new material parameters required by the GISSMO model through inverse identification procedure, a series of typical uniaxial-tension specimens with their theoretically given stress triaxiality η are well prepared and tested to obtain the load-displacement curve, including the pure shear, shear tension, uniaxial tension, plane strain tension and bulge [4] [11]. Since K, n, η and θ ¯ in Equation (11) all can be determined in advance, and the ultimate equivalent true strain ε ¯ f of the fracture zone can be measured by DIC during loading. In principle, it seems that only three or four samples are needed to obtain the four variables in Equation (11), as C₁, C₂, c θ s and c θ c , through the fitting optimization algorithm. However, the η value fluctuates significantly with the deformation [2] [3] [4], at the same time, η is strongly correlated with θ ¯ due to the plane stress state, indicated by Equation (6) and (7). Therefore, different sample types have great influences on the final curve-fitted parameters involved in the M-C model. To simplify the Equation (11), Coefficients a and b were defined as follows.

a = 1 3 cos ( θ ¯ π 6 ) (9)

b = η + 1 3 sin ( θ ¯ π 6 ) (10)

Under plane stress loading conditions, the coefficients a and b can be solely determined by the given stress triaxiality η as shown in Figure 1. The curve of the strain ratio α against η was also plotted in the figure to indicate the characteristic deformation path relative to the current η. All the values under typical

loading conditions were listed in Table 1.

As shown in Figure 1, when the plane stress triaxiality starts from the uniaxial tension, the coefficients a and b of the M-C model tend to be the same, covering the regular deep drawing strain paths according to the FLD test, like uniaxial tension, plane strain tension and bulge. Therefore, if choosing these three types of specimen, key parameters linked to the M-C model could hardly be well fitted, no matter how many specimens were involved in the testing.

In the meanwhile, the coefficients a and b are significantly different in the pure shear and shear-tension regions. As proposed in the present study, Therefore, the research shows that among the three or four samples, at least one pure shear or shear-tension sample, together with a uniformly deformed uniaxial tensile sample are required, in order to accurately fit and optimize the four variable parameters of M-C model, as to accurately predict the ultimate deformation performance of AHSS steel plate under different plane stress loading conditions.

Here in the present study, the approach to establish the M-C model was validated by comparing the reported results of 7075-T6 aluminum sheet in Lee’s research on its cold forming performance through GISSMO tools [8].

LE5105 electro-mechanical testing machine was used to record the tension force-displacement curve, simultaneously the evolution of strain on the specimen local area of interest was simultaneously recorded by 3D-DIC technology (GOM ARAMIS 5M). In general, a regular video extensometer could also be implemented to record the deformation of specimens during loading until breaking for further strain analyzing, illustrated in Figure 2. To meet the GISSMO requirements [2] [12] and concern the importance of the shear specimen mentioned above, at least three types of uniaxial loading specimens including 0˚pure shear (Figure 3(a)), 45˚shear-tension (Figure 3(b)), regular tensile (Figure 3(c)) were recommended in the present study.

It should be addressed that the alignment of the load chain and the centering of the specimen installment could be ensured by the help of laser level adopted. Therefore, each well designed specimen approximately experienced the deformation in the proportional strain path. See Figure 4, for all types of specimens as studied, the recorded true major and minor strain scattering points (ε₁, ε₂) of the breaking area are apparently linear from beginning to critical breaking. It is also shown that the Moving Regression Algorithm mentioned above could be reliable to monitor the instantaneous strain ratio α of the studied region at any given time.

The sampling frequency of GOM ARAMIS is 15 fps. Uniaxial loading is applied on the specimen by control the crosshead separation speed. For pure shear and tension-shear specimens, the total length of deformation area is only 6 mm, while the regular tensile specimen has the parallel length up to 70 mm. So the loading speed should be controlled about 2.5 mm/min for shear and shear-tension specimens and 25mm/min for large regular tensile specimen, i.e. about 0.04% - 0.05% change in strain every two frames of images, which is compatible to the best measurement capacity of the strain analyzer with its 0.005% accuracy during measurement.

Similar to the FEM calculation, the results of DIC strain measurement is also affected by the initial size of the virtual grid/gauge. In order to calibrate and regularize material parameters, GISSMO requires two series of fine measurement for small samples, like Figure 3(a) and Figure 3(b), prepared with two virtual grid sizes (0.5 mm and 1 mm). However, for such large specimens as Figure 3(c), it requires five series of data based on the virtual grid size including 0.5, 1, 2.5, 5 and 10 nm [2] [4]. In general, the smaller the grid size, the more accurate local deformation could be differentiated in detail, while more time consumption and more complex calculation will be performed. Here in the present study, 0.5 mm grid size was used to precisely locate the fracture zone and monitor the

change of the equivalent strain with its relative stress triaxiality as well.

The critical moment of the ductile fracture could be marked as the steep drop on the recorded force-displacement curve in the meanwhile the major and minor strain of the deformation area could be continuously recorded until cracking. Through the last several frame images, the fracture region with the ultimate deformation could be clearly identified. The necessary parameters required by the M-C model could be also evaluated, such as the triaxiality η, the equivalent fracture strain ε ¯ f . Repeating for all three types of specimens, key parameters to establish the M-C model, see Equation (11), of the sheet as received, including C₁, C₂, c θ s and c θ c = 1 (in the present work), could be solely determined, using non-linear curve fitting algorithm.

In this paper, a 1.4 mm-thick, ultra-high-strength and boron-contained steel sheet, 1800UHSS has been studied. The yield strength and the ultimate tensile strength in the rolling direction were 1350 MPa and 1850 MPa respectively, the tensile elongation at fracture was 7%, n value was 0.112, and K value was 2772 MPa. It should be noted that, according to tensile results, the material anisotropy was not very significant (the relative deviation of yield and tensile strength were both less than 5% in three directions 0˚, 45˚ and 90˚), which met the assumption of simplified M-C model.

The four variables Mohr-Coulomb models were introduced to study the ductile fracture of metal sheets under plane stress states. Initiating from the DIC recorded minor and major strains (ε₁, ε₂) which were distributed on the deformation area of uniaxial tension samples, several key parameters involved in the M-C model, including stress triaxiality η, Lode angle parameter θ ¯ and equivalent strain ε ¯ over the whole loading period could be determined by Moving Regression algorithm without the involvement of FEM into industrial application. Thereafter M-C model could be solely determined through three to four well-designed specimens which have quite different theoretical stress triaxiality.

It was shown that real-tested η values changed significantly during the loading period, deviating from their theoretical values relative to the geometry. Shear loading samples are of great importance to determine M-C Curve. In the present study, three types of samples including 0˚ pure shear, 45˚ shear-tension and regular tension were testified suitable to resolve the W-shaped M-C model, which has two apparent valleys in the shear zone η ∈ ( 0 , 1 / 3 ) and the uniaxial/biaxial tension zone η ∈ ( 1 / 3 , 2 / 3 ) respectively.

M-C curve with transformed extended-FLC curve covers shear, uniaxial tension, plane strain tension, biaxial tension/bulging loading states, effectively extending the conventional FLC into the field of shear loading, providing a guide to assess the material resistance against the ductile failures under various plane stress loading conditions.

The authors are grateful for the financial support received from the Scientific and Technical Funds provided by Central Research Institute, Baoshan Iron & Steel Co., Ltd. (Project Item Y20ECEE22F).

The authors declare no conflicts of interest regarding the publication of this paper.

Fang, J., Wang, B., Zhang, J.W. and Yang, X. (2022) Experimental Evaluation of Ductile Fracture of Sheet Metals under Plane Stress States. Materials Sciences and Applications, 13, 417-432. https://doi.org/10.4236/msa.2022.136024