Uncoupled ductile fracture criterion considering secondary void band behaviors for failure prediction in sheet metal forming

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Uncoupled ductile fracture criterion considering secondary void band behaviors for failure prediction in sheet metal forming Quach Hung , Jin-Jae Kim , Duc-Toan Nguyen , Young-Suk Kim PII: DOI: Reference:

S0020-7403(19)32319-7 https://doi.org/10.1016/j.ijmecsci.2019.105297 MS 105297

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

27 June 2019 8 October 2019 31 October 2019

Please cite this article as: Quach Hung , Jin-Jae Kim , Duc-Toan Nguyen , Young-Suk Kim , Uncoupled ductile fracture criterion considering secondary void band behaviors for failure prediction in sheet metal forming, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105297

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Highlights 

 

A new phenomenological ductile fracture criterion considering micro mechanisms of void nucleation, void growth, and evolution of void coalescence is presented in this paper. The secondary voids band and rotation of voids effect are considered in the new ductile fracture criterion. The new ductile fracture criterion can be utilized for predicting initial fracture in sheet metal forming with lager range of stress triaxiality.

1

Uncoupled ductile fracture criterion considering secondary void band behaviors for failure prediction in sheet metal forming

Quach Hung1, Jin-Jae Kim1, Duc-Toan Nguyen2, Young-Suk Kim3,# 1

Graduate School of Mechanical Engineering, Kyungpook National University, Korea

2

School of Mechanical Engineering, Hanoi University of Science and Technology, Vietnam

3

School of Mechanical Engineering, Kyungpook National University, Korea

#

Corresponding Author / E-mail: [email protected]

Abstract: A new phenomenological ductile fracture criterion that is proposed. The proposed model is associated with the micro mechanisms of void nucleation, void growth, and evolution of void coalescence. The secondary voids band and rotation of voids effect are considered in the new ductile fracture criterion. A series of upsetting test results of aluminum 2024-T351 and TRIP RA-K40/70 steel are used to construct and compare the accuracy of fracture locus proposed by new ductile fracture criterion, Modified Mohr-Coulomb criterion and extend Lou-Huh criterion. The fracture locus constructed using the proposed criterion is close to the experimental data points over a wide stress state range from uniaxial compression to balanced biaxial tension. Then, a series of upsetting tests and square cup drawing tests are conducted with Al6014-T4 to evaluate the accuracy of the proposed criterion. All results indicate that the proposed ductile fracture criterion can be utilized for predicting initial fracture in sheet metal forming. KEYWORDS: Al2024, TRIP RA-K40/70, Al6014-T4, Stress triaxiality, Ductile fracture criteria.

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1. Introduction Ductile fracture behavior is well-recognized as an important property of metals that leads to cracking in metal parts. Microscopically, such a failure can be ascribed to the micro mechanisms of void nucleation, void growth, and void coalescence under moderate and high-stress triaxiality or shear band movement under low-stress triaxiality. Macroscopically, failure is associated with the progressive degradation of metals, and under large plastic deformation, the stiffness and strength of a metallic material decrease. Numerous fracture criteria have been developed over several decades. In view of the interaction between theoretical models and material responses, these fracture models can be classified into the following main groups: uncoupled models, which neglect the effects of damage on the yield surface of materials; coupled models, which incorporate damage accumulation into the constitutive equations; and continuum damage mechanics (CDM) models, in which damage is based on the internal constitutive variables. One of the first schemes for analyzing microscopic mechanisms was proposed by McClintock et al. [1], who investigated the evolution of an isolated cylindrical void in a ductile elastoplastic matrix to expose the role of micro-voids. McClintock et al. [2] proposed a cumulative damage model, which was dependent on the transverse principal stresses, equivalent stress, and strain-hardening exponent. Based on a study of the model of a finite sphere containing an isolated spherical void in a rigid perfectly plastic matrix, Gurson [3] proposed an approximate yield criterion and flow rules for porous ductile materials, highlighting the role of hydrostatic stress in plastic yield and void growth. Tvergaard and Needleman [4] proposed a modified Gurson damage model called the GTN model by incorporating additional parameters (q1, q2, q3) into the original Gurson criterion. This model considered damage accumulation induced by nucleation, growth, and void coalescence. Xue [5] modified the GTN model by incorporating the void shear effect into it. In Xue’s model, the void shear effect introduced the dependency of damage evolution on the third stress invariant. For CDM models, Lemaitre proposed a damage model [6] that was developed from the thermodynamics of continuum damage mechanics. However, the Lemaitre model is limited in its applicability to shear-dominated and complex forming processes. Cao et al. [7] modified the Lemaitre model by coupling it with the Lode-dependent parameter to overcome the abovementioned limitation. In uncoupled models, damage accumulation is formulated empirically or semi-empirically. It is postulated that fracture occurs at a material point in a body when the fracture scalar function 3

reaches a critical value as a “damage indicator”. From tensile test results obtained under hydrostatic pressure with observation of workability the maximum deformation before failure, Cockcroft and Latham [8] (C-L model) proposed that fracture is controlled by the maximum principal tensile stress integrated over the plastic strain. Brozzo et al. [9] modified the C-L model to incorporate an explicit dependence on hydrostatic stress for predicting the formability limit of sheet metal. This criterion was successfully applied to predict the failure of bulk metals subjected to extrusion and drawing. Rice and Tracey [10] (R-T model) studied the growth of a single spherical void in an infinite solid subjected to remote normal stresses and proposed a semiempirical model that considers stress triaxiality. In the R-T model, hydrostatic stress strongly depends on the equivalent plastic strain to fracture. Johnson and Cook [11] proposed a ductile fracture model to predict the failure of materials by considering strain rate and temperature conditions. The central idea of these uncoupled ductile fracture criteria originated from the simple effect of maximum principal stress and hydrostatic stress on the equivalent plastic strain to fracture, so these fracture criteria are suitable for describing the failure of a material under tension. Consequently, these models are limited in terms of predicting material failure in the lowstress triaxiality state or in a complicated loading mode. Bao and Wierzbicki [12] performed several types of tests on Al2024-T351 specimens, including uniaxial tension tests of notched specimens, dog-bone specimens, central hole specimen, in-plane shear specimen, and tensile tube, and compressive upsetting test of cylindrical bars. The results showed the shapes of the fracture locus corresponding to various stress states with the stress triaxiality ranging from -1/3 to 1.0. Bao and Wierzbicki introduced that ductile fracture would not occur when the stress triaxiality is lower than cut-off value of stress triaxiality ηcut-off =-1/3. Based on experimental results, Bao and Wierzbicki constructed an empirical ductile fracture model considering effect of the stress triaxiality parameter on fracture strain with good accuracy. Nevertheless, this experimental model indicated positive dependence of plastic fracture strain on stress triaxiality, which conflicted with the generally accepted concept that high-stress triaxiality accelerates void growth while reducing the plastic fracture strain of metals. This conflict originated because the authors neglected the effect of the Lode angle parameter in the ductile fracture model. Bai and Wierzbicki [13] introduced a yield criterion containing a term of stress triaxiality and the Lode angle parameter and proposed the Bai and Wierzbicki ductile fracture model. Thereafter, Bai and Wierzbicki [14] modified the Mohr-Coulomb criterion to describe ductile fracture of metal. The 4

modified Mohr-Coulomb (MMC) fracture locus was found to be a good approximation of the experimental data points of Al 2024-T351. The MMC criterion was successfully utilized in numerical analysis to predict ductile fracture in metal forming processes. Li et al. [15, 16] applied the MMC criterion to predict shear-induced fracture in the forming of TRIP 690, a thin sheet manufactured from Al 6061-T6. Beese et al. [17] partially coupled the MMC criterion with the anisotropic Hill48 yield function of Al 6061-T6. Besides, Mohr and Marcadet [18] proposed the Hosford-Coulomb model by substituting the maximum principal shear stress term (or Tresca equivalent stress) in the Mohr-Coulomb criterion with the Hosford equivalent stress considering the anisotropic behaviors of materials. Moreover, from the viewpoint of the micro mechanisms of void nucleation, void growth, and void coalescence, Lou et al. [19] proposed the Lou–Huh model to predict ductile failure of advanced high-strength steel DP980. Similar to Bao’s model, this model assumed that ductile fracture does not occur when stress triaxiality is lower than -1/3. However, this assumption conflicted with the experimental results obtained by Khan and Liu [20]. Khan and Liu conducted biaxial compression experiments by setting stress triaxiality as 0.496. In addition, Lou et al. [21] introduced the value C into the extended Lou–Huh model (extended L–H) fracture model to consider a changeable cutoff value of stress triaxiality. Nevertheless, the value of C must be determined by another test of uniaxial tension with round bar specimens under hydrostatic pressure. Next, Lou et al. [22] introduced a constant parameter into the extended L–H model to compensate for effect of the Lode parameter on the torsion of voids in anisotropic materials. More recently, Hu et al. [23] proposed the Hu–Chen uncoupled fracture model based on discussions that fracture occurs owing to tension and shearing; Mu et al. [24] proposed another phenomenological uncoupled ductile fracture criterion by considering two different void deformation modes of isotropic materials. Wen and Mahmoud [25, 26] employed the maximum shear stress ratio for developing three bounding curves fracture model under both monotonic loading and reverse loading conditions. The Wen and Mahmoud model, which could be simplified as the maximum shear criterion or Rice and Tracey criterion, was successful in capturing block shear failures in bolted connection. It is noted that a non-linear damage accumulation rule proposed by Wen and Mahmoud [27] under reverse loading condition showed a good performance in predicting failure in structural components. This development of damage rule is essential to investigate the failure of structural components that usually works under cyclic loading condition. On the other hand, in specific deformation processes such as extrusion, 5

drawing, hub-hole expanding or rolling the loading force direction does not significantly change during forming process. Moreover, almost fracture calibrate testing is performed under monotonic loading condition to identify material parameters of ductile fracture model. So, in this study, all fracture tests are investigated under monotonic loading condition. And, a damage evolution rule is considered to verify the ductile fracture criterion performance under nonproportional loading condition. Although there are great efforts on modeling and predicting onset of ductile fracture behaviors, the previous described fracture models still keep some limitations such as purely empirical(B-W model), considering only one of fracture mechanism (maximum shear stress model, Ko-Huh model) or lack of discussion about micro mechanism of material components(MMC model). Additionally, the recent discussion about secondary void band and rotation of void should be examined. In the present study, an uncoupled ductile fracture model is proposed for predicting the onset of ductile fracture for isotropic materials under monotonic loading conditions. The proposed ductile fracture criterion considered ductile fracture behaviors under both states of two macro fracture mechanisms and micro mechanism of void. On the macroscopic view, the proposed ductile fracture model introduces influence of maximum shear and maximum tension mechanism on failure. On the microscopic view, void nucleation is controlled by equivalent plastic strain, void growth is reviewed and presented as function of both stress triaxiality and Lode parameter in the proposed model, and the effect of secondary void band and rotation of void in coalescence stage is introduced by function of Lode parameter. Revised data of Al2024T351 and TRIP RA-K40/70 steel are used to verify the accuracy of proposed fracture model. Several upsetting tests are performed with Al6014-T4 specimens to calibrate and determine material constant of the proposed fracture criterion. Then, the proposed criterion was successfully applied to predict the initial fracture behavior of Al6014-T4 in a cup drawing test. All comparison results show that the proposed ductile fracture criterion can accurately predict the initial fracture of material in sheet metal forming. 2. Proposed ductile fracture criterion 2.1 Characterization of stress states The hydrostatic stress of isotropic material can be formulated in terms of the stress tensor [σ] as follows:

6

1 1 𝜎 = 𝑡𝑟([𝜎]) = (𝜎 + 𝜎 + 𝜎 ) 3 3

(1)

1 𝜎̅ = √ [(𝜎 − 𝜎 ) + (𝜎 − 𝜎 ) + (𝜎 − 𝜎 ) ] 2 𝜂 = σ ⁄σ ̅ = (σ + σ + σ )⁄(3σ ̅)

(2) (3)

⁄ (4) 27 𝑟 = [ (σ − σ )(σ − σ )(σ − σ )] 2 Where σ1, σ2, and σ3 are the principal stresses, it is assumed that σ1 ≥ σ2 ≥ σ3, 𝜎̅ is the equivalent

stress, η is the stress triaxiality parameter, r is the third invariant of a stress state in the space of principal stress. The Lode angle is defined as the angle between the projection of the stress vector OP and the projection of the maximum principal axis on the π-plane, as shown in Fig. 1a in the (σ1, σ2, σ3) space. The Lode angle θ can be calculated via the third invariant as follows: 𝑟 ξ = cos(3θ) = ( ) ̅ σ 𝜃̅ = 1 −

(5)

6𝜃 2 = 1 − 𝑎𝑟𝑐𝑐𝑜𝑠(ξ) 𝜋 𝜋

(6)

Where 𝜃̅ is the Lode angle parameter. Lode [28] used another formulation to calculate the Lode parameter, which is as follows: µ=

2σ − σ − σ σ −σ

− 1 ≤ 𝜇 ≤ 1, 𝑤𝑖𝑡ℎ σ ≥ σ ≥ σ

(7)

The relationship between the Lode parameter µ and the Lode angle θ is given as follows: (8) √3(µ + 1) 3−µ Three principal stresses as functions of the stress triaxiality η, the Lode parameter μ, and the tan(𝜃) =

equivalent stress are expressed as follows: σ =σ + σ =σ + σ =σ −

(3 − µ)σ ̅ 3√µ + 3 2µσ ̅ 3√µ + 3 (3 + µ)σ ̅ 3√µ + 3

= (η + = (η + = (η −

(3 − µ) 3√µ + 3 2µ

(9)

)σ ̅

3√µ + 3 (3 + µ) 3√µ + 3

(10)

)σ ̅

(11)

)σ ̅

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For plane stress conditions, the stress space is reduced from three-dimensional (3D) to twodimensional, which results in a functional relationship between the Lode angle parameter and stress triaxiality, as shown in Fig. 1b. The normalized maximum shear stress can be calculated easily as follows: τ ̅ σ

=

σ −σ 1 = 2σ ̅ √µ + 3

(12)

(a)

(b)

Fig. 1 (a) Lode angle in space of principal stress, and (b) relationship between Lode parameter and stress triaxiality for plane stress condition. 2.2 Microscopic analyses of ductile fracture The mechanical properties of materials observed at the macroscopic scale correspond to the phenomena occurring at multi length-lower scales. According to the microscopic viewpoint, metals and alloys are usually damaged owing to void nucleation, void growth, and micro void coalescence. These mechanisms must be analyzed carefully and described with suitable models. According to a number of previous fracture models (see appendix A), each micro mechanism just

plays a single role in describing the effect of stress triaxiality and Lode parameter on metal failure behaviors, while SEM images show more complicated behavior of ductile fracture. Besides, the ductile fracture criterion should consider both scales of fracture mechanisms to explain accuracy onset of fracture. Therefore, this subsection briefly reviews the microscopic mechanisms of ductile fracture under influence of both tension and shear fracture mechanism. 8

Moreover, the presence of secondary void band and rotation of void at coalescence stage as shown in report of Weck and Wilkinson [32] and Achouri et al. [36] should be considered on constructing fracture model. 2.2.1. Void nucleation The occurrence of void nucleation accompanied by plastic deformation plays a very important role in the fracture process and in determining the mechanical responses of the material. Voids nucleate at large particles, and the resulting local stress fields cause plastic deformation. However, the voids created during this stage are small, and therefore, they do not have a visible influence on the macroscopic behavior of the material. In many studies on the modeling of void nucleation, two types of models are employed: strain-based nucleation criterion, and energy and stress-based nucleation criterion. Based on critical stress, Goods and Brown [31] proposed an energy criterion to describe nucleation in terms of matrix particle de-cohesion. Lee and Mear [32] employed the stress concentration factor at the interface and inside the particle to describe void nucleation. Gurson [3] proposed a strain-controlled nucleation model to show that the rate of void nucleation is a function of the equivalent plastic strain as shown in Eq. (13). Additionally, Needleman and Rice [28] developed a phenomenological-nucleation model, in which void nucleation is governed by the equivalent plastic strain of the matrix and hydrostatic stress. 𝑓̇

= A𝜀̅ ̇ + 𝐵𝜎̅ ̇

Where A and B are two non-constant parameters, 𝑓̇

(13) is the rate of change in void volume

due to void nucleation. Briefly, void nucleation is presented as a function of the equivalent plastic strain as follows: D

= D(𝜀̅ )

(14)

2.2.2. Void growth After nucleation, voids grow under the effects of plastic deformation and hydrostatic stress, and eventually, they are linked together. Voids nucleate over a large range of plastic strain, and void growth occurs simultaneously, resulting in the formation of voids of different shapes and sizes. Rice and Tracey [33] described void growth with mathematical models elucidating the effect of mean stress. Moreover, void growth depends on the loading conditions and the microstructure of the material. Most empirical fracture models indicate a strong effect of the maximum principal stress, which represents the loading condition, on the equivalent plastic strain to failure. 9

Furthermore, Thomason [34] showed that for low-stress triaxiality cases, ductile fracture occurs because of large shape-changing and relatively small volume-changing under void growth process. Nahshon and Hutchinson [35] demonstrated that shear and normal localization at lowstress triaxiality can be predicted by considering the effect of voids. Li et al. [36], Khan and Liu [20] showed by means of scanning electron microscopy (SEM) the fracture surface of a material subjected to upsetting, on which multiple elongated voids could be observed in negative stress triaxiality, as shown in Fig. 2.

Fig. 2. SEM image of shear fracture surface with elongated voids at low and negative stress triaxiality (Li et al. [36]). It is proved that the Lode parameter, which represents the maximum shear stress term, strongly influences void shape changes during void growth. Consequently, a mathematical model of void growth should consider both stress triaxiality and the Lode parameter in term of the maximum tensile stress and shear stress, as follows: D 2.2.3. Coalescence of voids

= D(

σ τ , ) ̅ ̅ σ σ

(15)

The final stage of failure of a material is void coalescence, which is thought to be caused by the localization of plastic flow among enlarged voids. Therefore, coalescence mechanisms are related to the macroscopic fracture surface. Weck and Wilkinson [29] performed experiments 0.1mm in thickness tensile specimens with multiple laser-drilled holes of diameter 10 µm as the model materials, and the progress of void coalescence under tension was observed with SEM. According to Weck and Wilkinson, void coalescence progresses according to two modes: necking of matrix particles between neighbors of voids, that is the internal necking mode; and reduction of particle spacing under relatively low-stress triaxiality, that is the shear localization mode. The internal necking mode is caused by the maximum principal stress, while the shear 10

localization mode is caused by shear-linking of voids along the direction of the maximum shear stress. When internal necking occurs on a large surface of a specimen, shear localization occurs along the thickness direction of the specimen. According to the fracture model proposed by Brown and Embury [37], void coalescence occurs when shear band is at 45°. By contrast, according to the McClintock [2] fracture model, void coalescence takes place in a localized shear band where the shear-linking of voids occurs under the plane strain condition. Moreover, remarkably, when void coalescence occurs, the material inside the primary band of void localization undergoes accelerated void growth and nucleation well known as secondary void band. Figure 3 shows the distribution of secondary voids in the SEM images obtained by Weck and Wilkinson [29]. These images show that in a region with shear-dominated stress, voids can be enlarged by a factor of two or three, resulting in the formation of microcracks. In a comparison of the SEM of fracture surfaces of smooth specimens and specimens with various types of notches, Bron and Besson [38] showed that the occurrence of void coalescence is associated with the creation of smaller dimples inside the first void bands. Furthermore, Pardoen et al. [39] recommended that void distribution strongly influences the onset of void coalescence, while its effect can be neglected in the micro void growth process. Achouri et al. [30] showed detailed SEM observations of voids development in a shear stress state region obtained with a shear test on high-strength steel. These authors observed that matrix-particles debonding happened at interface then void grew and finally cracked. The void is rotated in these shear stress states before fracture happens. Therefore, the effects of secondary voids and voids rotation should be considered in the development of a ductile fracture criterion. As a result, void coalescence is described with a function that considers the influence of maximum shear stress as follows: D

= D(

11

τ 𝜎̅

)

(16)

Fig. 3. SEM images of secondary voids between laser holes at (a) 90° and (b) 45° with respect to the tensile axis; (c) magnified view of the ligament between two holes from an array at 45° (Weck and

Wilkinson [29] SEM images). 2.3. Proposed ductile fracture criterion 2.3.1 Proposed ductile fracture criterion Compared to the complicated coupled model and its modified forms, simple uncoupled phenomenological models are preferred for industrial applications because of their simplicity and less material constants to be calibrated using experimental data points. Uncoupled ductile fracture criteria are based on observation of extensive experiments, analytical studies, math simplicity, numerical results or some combination of them. A number of uncoupled ductile fracture models, which are initially utilized in bulk forming processes, have been proposed relatively simple crack formation in the past. On the macroscopic view, maximum tension mechanism and shearing mechanism are two main fracture mechanism factors leading to failure of material. On maximum tension mechanism approach, Cockcroft and Latham [8] introduced that fracture is governed by the maximum principal stress integrated over the plastic strain path. Oh et al [40] introduced equivalent stress into the C-L model to examine the workability of al 2024-T351 in extrusion and SAE 1144 cold-drawn steel in drawing as below: ̅

∫

𝜎 dε̅ = 𝐶 𝜎̅

(17)

Where 𝐶 is the material constant. Ko et al [41] combined the effect of both maximum principal stress and the stress triaxiality to describe the ductile fracture behavior of metal. The Ko-Huh criterion was successfully utilized to predict the hub-hole expanding process of high strength steel SAPH440, CT440, and FB590. It is noted that the maximum principal stress easily 12

transformed into the function of stress triaxiality as shown in section 2.1, these previous criteria can examine the influence of stress triaxiality in high-stress triaxiality scope. On the maximum tension mechanism approach, stress triaxiality is negative and Lode parameter that is presented by shear mechanism dominates the failure behavior of material. The simple stress-based criterion- maximum shear stress criterion, which introduced that failure occurs when the maximum shear stress reaches a prescribed value as shown in Eq. (18), fitted well with the test in low and negative stress triaxiality range for Al2024-T351 compared with six fracture models as reported by Wierzbicki et al [42]. 𝜎 −𝜎 (18) 2 Moreover, Vallellano et al [43] found that the maximum shear stress criterion fitted the measured 𝜏

=

biaxial strain failure data of Al2024-T3 sheet under stretch much better than other criteria. Lou et al proposed Lou-Huh model by combining maximum shear stress term and stress triaxiality under discussion of micro mechanism of material to predict the onset of fracture for DP980 steel. Appendix A summarizes various uncoupled models proposed in the literature, accounting for triaxiality and Lode parameter. In this study, a new uncoupled ductile fracture criterion considered both assumptions of two macroscopic fracture mechanisms and micro-mechanism of void nucleation, growth and evolution of void coalescence is proposed as follow: 𝜀̅ =

𝐶

(19)

𝜏 𝜏 𝜎 ( 𝜎̅ + 𝜎̅ ) *(3 + √3𝐶 ) 𝜎̅ − 𝐶 +

Where C1, C2, C3 material-dependent parameters. Following Eq. (19), on the macroscopic, the proposed fracture criterion considered the maximum principal stress and maximum shear stress under term of ( , ) respectively. By substituting Eq. (9) and Eq. (12) into Eq. (19), ̅

̅

maximum principal stress and maximum shear stress can be easily introduced to stress triaxiality and Lode parameter, and the proposed fracture criterion can be transformed into the space of (𝜂, 𝜇,𝜀̅ ) as: 𝐶

𝜀̅ = (η +

3−µ 1 3 + √3𝐶 + ) ( −𝐶 ) 3 ∗ √µ + 3 √µ + 3 √µ + 3

Where (η, μ) are the stress triaxiality and Lode parameter respectively.

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(20)

On the microscopic view, the proposed fracture criterion is introduced in form of function of stress triaxiality and Lode parameter to evaluate micro fracture mechanism of material as shown in Eq. (20). Additional, according to discussion of micro fracture mechanism void shape and size in void growth stage change under the influence of both the maximum principal stress and the normalized maximum shear stress, and these can be expressed as term of both stress triaxiality and Lode parameter (𝜂, 𝜇) and associated with (η +

∗√

+

√

) . The effect of secondary

voids and rotation of voids in void coalescence are governed by maximum shear stress is introduced in the proposed fracture model as a function of Lode parameter that is linked with (

√ √

− 𝐶 ).

2.3.2 Damage evolution rule A model extension for non-proportional loading should be included in this paper because of inevitable stress state variations in many fracture experiments. Bai and Werzibicky [13] show important effects of non-linear loading paths on ductile fracture during the crushing of prismatic columns and proposed the phenomenological modification of the accumulation rule of damage indicator models to account for the effect of non-proportional loading histories on the onset of ductile fracture. This damage evolution rule has been adopted in various studies. In this work, a linear incremental relationship is utilized between the damage indicator D and the equivalent plastic strain to fracture. ̅

D(ε̅ ) = ∫

dε̅ 𝜀̅ [𝜂, 𝜇]

(21)

Where 𝜀̅ denoting the equivalent plastic strain to fracture, the stress state parameter 𝜂 (ε̅ ), 𝜇(ε̅ ) are unique functions of the equivalent plastic strain. Then, Eq. (20) is presented under integral form like other criteria as below: ̅

D(ε̅ ) = ∫

1 3−µ 1 3 + √3𝐶 (η + + ) ( − 𝐶 ) dε̅ 𝐶 3 ∗ √µ + 3 √µ + 3 √µ + 3

(22)

When the damage indicator is reached (𝐷(𝜀̅ ) = 1) the material element is considered to fail and delete. The integral form of the proposed fracture criterion can be easily implemented into numerical analysis to describe ductile fracture in complex strain paths. Besides, to construct the 14

fracture locus map by the average value of stress state parameters during the strain history, the average value of stress state parameters (η, μ) are calculated by: 𝜂

=

1 ̅ ∫ 𝜂 (𝜀̅)𝑑𝜀̅ 𝜀̅

(23) 1 ̅ µ = ∫ µ(𝜀̅)𝑑𝜀̅ 𝜀̅ The linear damage rule in Eq. (21) has been widely used in engineering practice. However, in the case of complicated loading paths such as cyclic loading paths, a nonlinear damage rule should be considered as shown in reports of Xue [5], Bai and Werzibicky [13] and Wen and Mahmoud [27]. 2.4 Parametric study The fracture behaviors associated with sheet metal forming are mainly described in terms of the strain-loading path. Therefore, the fracture locus in the (η, 𝜀̅ ) space can be transformed into the fracture forming limit curve (FFLC) in the (ε1, ε2) space by using the theory of sheet metal forming under the plane stress condition. Material constants in the proposed ductile fracture criterion modulate the effects of void nucleation, void growth, and void shear coalescence on the plastic strain to fracture. Their effects on the FFLC and fracture surface will be investigated to provide a comprehensive understanding of the proposed ductile fracture criterion. 2.4.1. Effect of material constant C1 The material parameter C1 is the simplest of all the three material parameters. To study effect of parameter C1, the FFLC and the fracture surface are plotted with C2 and C3 unchanged; as a result, the value of C1 increases remarkably (C1=1.0, 1.2, 1.4, and 1.6). As shown in Fig. 4, the parameter C1 controls only the magnitude of the FFLC without influencing changes in the shape of the FFLC. Figure 5 shows four different fracture surfaces constructed using the above parameters; the fracture surfaces move upward as the value of parameter C1 increases.

15

Fig. 4. Influence of parameter C1 on fracture strain in (ε1, ε2) space.

Fig. 5. Influence of parameter C1 on fracture surface strain in (𝜂, 𝜇,𝜀̅ ) space: C2 = 0.55876 and C3 = 3.68514 2.4.2. Effect of material constant C2 The material parameter C2 governs effect of the maximum principal stress and the maximum shear stress in the important micro mechanisms of void growth and void coalescence. With C1 and C3 unchanged, C2 (C2 = 0.1, 0.3, 0.5, and 0.7) is increased. In the case of plane stress condition, the FFLC at point F does not depend on C2, where (η + (3 − µ)⁄(3 ∗ √µ + 3) + 1⁄√µ + 3 = 1), as shown in Fig. 6. When stress triaxiality η is larger than ηF (stress triaxiality 16

at point F), void growth under the tension mode dominates, and the influence of void growth in the tension mode is distinct under high-stress triaxiality as C2 increases, which results in a smaller FFLC. In the compression mode with stress triaxiality η less than ηF, void growth under tension is suppressed, and the progress of void growth under shear stress alone is slower, which results in a larger FFLC in this zone. Figure 7 shows the effect of the parameter C2 on the fracture surface of the material. As C2 is changed, the fracture surface rotates around the line where the fracture surface is not affected by C2.

Fig. 6. Influence of parameter C2 on fracture strain in (ε1, ε2) space: C1 = 0.45818 and C3 = 3.68514.

Fig. 7. Influence of parameter C2 on fracture strain in (𝜂, 𝜇,𝜀̅ ) space: C1 = 0.45818 and C3 = 3.68514.

17

2.4.3. Effect of material constant C3 As discussed in the above sections, when void coalescence occurs, the secondary void band significantly influences the final fracture stage. Parameter C3 is included in the fracture equation to represent the secondary void band of void coalescence. With C1 and C2 unchanged, C3 is varied, and the FFLC and the fracture surface constructed with the proposed ductile fracture model are shown in Figs. 8 and 9 for four cases of C3 (C3 = 3.0, 4.0, 5.0, and 6.0). Parameter C3 significantly influences changes in the shape of the FFLC and the fracture surface. Under the uniaxial (η = 1/3, µ = -1) and bi-axial (η =2/3, µ = 1) tension modes, void growth occurs under both the principal tension stress in the longitudinal direction and enhanced shear stress along the thickness direction, which results in the FFLC becoming larger with increasing C3. Meanwhile, in the pure shear tension mode and the plane strain mode, the value of the Lode parameter is the minimum (µ=0), and the FFLC remains unchanged even when C3 increases. Figure 9 shows the effect of C3 on the fracture surface; the fracture strain near the normalized plane strain mode remains constant, while its value in the others areas increases as C3 increases.

Fig. 8. Influence of C3 on fracture strain in (ε1, ε2) space: (C1 = 0.45817 and C2 = 0.55867).

18

Fig. 9. Influence of C3 on fracture strain in (𝜂, 𝜇,𝜀̅ ) space: (C1 = 0.45817 and C2 = 0.55867). 2.5. Calculation of material constants At least three simple experiments were performed with sheet metals to determine the three parameters employed in the proposed ductile fracture criterion; the tests considered for this purpose were the uniaxial tension, pure shear tension, and plane strain tension, and the equibiaxial tension tests. For isotropic materials, the proposed fracture criterion has the following simple forms under the pure shear condition, uniaxial tension, plane strain condition, and equibiaxial tension: Pure shear condition (η = 0, µ = 0):

2 ( ) √3

=ε

(24)

√3

Uniaxial tension (η = 1/3, µ = -1):

3 ( 2)

3 + √3 ( 2

=ε −

(25)

)

19

Plane strain condition (η = 1⁄√3, µ = 0):

(√3) √3

=ε

(26)

Equi-biaxial tension (η = 2/3, µ = 1):

3 ( 2)

3 + √3 ( 2

Where ε

,ε

=ε −

(27)

)

, ε , and ε

are the equivalent plastic strain in the pure shear condition,

uniaxial tension, plane strain condition, and equi-biaxial tension, respectively. Any three equations from among Eqs. (24)–(27) can be used to explicitly identify the three parameters of the proposed model. From Eq. (24) and Eq. (26) C2 can be calculated as follows: 𝐶 =

nε

(28)

− nε n3 − n2

Then, from Eq. (26), C1 can be defined as follows: (29)

𝐶 = ε (√3) √3 Finally, C3 can be calculated by substituting C1 and C2 into Eq. (25). However, in almost experiments testing on various types of specimens, the stress state

parameters stress triaxiality and Lode parameter are not stable as in perfect condition. It is an intrinsic difficult issue in calibrating material parameters to construct fracture locus. So, the average stress triaxiality and Lode parameter value from Eq. (23) are used in the calibration process. The calibration consists of finding a set of parameters (C1, C2, C3) such that the model describes correctly the onset of fracture in all calibration experiments. The fracture parameter can be determined by minimizing the relative error function, see the following equation:

n(

,

,

) (𝐸𝑟𝑟𝑜𝑟)

=

n(

,

,

)*

∑

(̅

|

,

̅

Where N = number of calibration fracture tests analyzed, 𝜀̅

,

̅

, ,

3, with 𝜀̅

)

(30)

|+ ,

and

denoting the respective predicted and measured fracture strain corresponding to

the same test.

20

The above calibrate procedure has been first applied to identify fracture of Al 2024-T351 by Bao et al [44]. Then, various good results achieved in applications by Bai and Wierzbicki [13], Mohr and Marcadet [18], Wen and Mahmoud [25] demonstrate the reliability of this estimation approach. 3. Revisiting the results obtained with Aluminum 2024-T351 and TRIP RA-K40/70 steel The aluminum alloy AL2024-T351 is widely used in many practical applications. Hence, in this study, Al2024-T351 is employed to evaluate the proposed fracture criterion. Many experiments involving this material have been performed to determine the equivalent plastic strain at fracture, and the resulting experimental datasets have been used in many studies for verifying various proposed ductile fracture criteria. As the Al2024-T351 test data, the experimental data obtained by Bao and Wierzbicki [12] in a stress triaxiality range of -0.2780–0.6030 and the test data of Khan and Liu [20] in a stress triaxiality range of less than -1/3 are listed in Table 1. Table 1. Testing data of AL2024-T351 (σ = 74 ε Test No.

Specimens

) in a wide range of stress triaxiality η

̅ θ

µ

ε̅

1

Cylinder compression (d0/h0 = 0.5)

-0.2780

-0.8215

0.7946

0.4505

2

Cylinder compression (d0/h0 =0.8)

-0.2339

-0.6809

0.6451

0.3800

3

Cylinder compression (d0/h0 =1)

-0.2326

-0.6794

0.6435

0.3563

4

Cylinder compression (d0/h0 =1.5)

-0.2235

-0.6521

0.6155

0.3410

5

Round notched, compression

-0.2476

-0.7141

0.6796

0.6217

6

Pure shear

0.0124

0.0355

-0.0322 0.2107

7

Shear and tension

0.1173

0.3381

-0.3099 0.2613

8

Tension, plate with hole

0.3431

0.9661

-0.9594 0.3099

9

Tension, dog-bone

0.357

0.9182

-0.9034 0.4798

10

Tension, pipe

0.3557

0.9286

-0.9155 0.3255

11

Pure torsion

0

0

12

Torsion with constant axial tension

0.1910

0.5560

-0.5190 0.2530

13

Tension, thin notched (r = 0.396 mm)

0.5648

0.0942

-0.0855 0.1951

14

Tension, Thin notched (r = 1.984 mm)

0.4974

0.4486

-0.4145 0.2204

21

0

0.2880

15

Tension, Thin notched (r = 4.763 mm)

0.4319

0.6960

-0.6607 0.2441

16

Combined tension and torsion

0.1585

0.4780

-0.4428 0.2656

17

Tension, flat-grooved

0.6030

0.0754

-0.0684 0.2100

18

Bi-axial compression

-0.496

-0.398

0.367

0.349

Bai and Wierzbicki [14] used TRIP RA-K40/70 (or TRIP690) sheets in their study. Author performed fracture tests over a wider range of stress triaxiality with five upsetting specimens, namely, dog-bone specimen and flat specimen with cutouts subjected to tension, disk specimen subjected to equi-biaxial tension, and butterfly specimen subjected to plane strain tension and simple shear tension, to determine the fracture strain, as shown in Table 2. Table 2. Summary of fracture testing results on TRIP 690 steel specimens. No.

Specimen description

1 2 3 4 5

Dog-bone, tension Flat specimen with cutouts, tension Disk specimen, equi-biaxial tension Butterfly specimen 1, tension Butterfly specimen 2, simple shear

η

̅ θ

ε̅

0.379 0.472 0.667 0.577 0

1.0 0.496 -0.921 0 0

0.751 0.394 0.950 0.46 0.645

Additionally, the extended L–H criterion and the MMC criterion were selected for comparison with the proposed ductile fracture criterion. All the three abovementioned criteria were expressed in the (η, µ, 𝜀̅ ) space. The detailed formulations of these fracture criteria are given in Appendix A. For Al2024-T351, the constant material parameters of the proposed fracture criterion (C1 = 0.4285, C2 = 0.4377, and C3 = 5.3161) and the MMC model (C1 = 0.041, C2 = 339 MPa, C3 = 0.9739, A = 740 MPa, and n = 0.15) were calculated with an optimization procedure by using all the experimental data given in Table 1, while the material parameters of the extended L–H criterion (C1 = 4.0983, C2 = 0.4316, C3 = 0.3914, and C = 1/3) were calculated by Lou et al. [21]. For TRIP 690 steel, the constant material parameters of all three criteria, namely, MMC (C1 = 0.12, C2 = 720 MPa, C3 = 1.095, A = 1275.9 MPa, and n = 0.2655), extended L–H model (C1 = 4.3129, C2 = 0.6887, and C3 = 0.9225), and the proposed model (C1 = 1.2595, C2 = 0.83365, C3 = 4.2652) were optimized using all the experimental data given in Table 2.

22

To evaluate the accuracy of these ductile fracture criteria, we defined an error fuction as the absolute relative error between the predicted fracture strain value and the experimental fracture strain value for each test as follows: (𝜀̅ ) = |

̅ ̅

− 1|

1

(31)

A more useful error value, the average error across different tests, was defined and used, as follows: 𝐸𝑟𝑟 =

∑ (̅ )

(32)

where N is the number of tests conducted for model validation. Figure 10(a) shows a comparison of the predictions made using the three fracture models with the experimental results obtained using Al2024-T351 specimens. The results of the MMC, extended L–H, and proposed fracture criteria exhibit good fits with the experimental data, not only under intermediate stress triaxiality but also under low-stress triaxiality. All fracture loci are close to the experimental data under stress triaxiality from zero to 2/3. However, the loci obtained using the three criteria differ under low-stress triaxiality. The extended L–H and the MMC criteria overestimate the fracture strain at most experimental points under low-stress triaxiality (see Zone A), while the fracture locus obtained with the proposed ductile criterion showed a good fit to these experimental points. For TRIP 690 steel, the fracture loci predicted by three fracture criteria showed good agreement with the experimental data. Especially, by using three experimental data points under pure shear tension, uniaxial tension, and plane strain tension, these fracture criteria predicted the onset of fracture at the equi-biaxial tension position. Nevertheless, the facture strain values predicted with the MMC model and the proposed model at the equi-biaxial tension position were higher than the corresponding experimental data. By contrast, the fracture strain predicted with the extended L–H criterion was lower than the corresponding experimental result, as shown in Table 4. The extended L–H and the proposed fracture criteria could accurately predict fracture strain in the plane strain tension mode relative to the experimental data, whereas the value predicted with the MMC criterion was lower than the corresponding experimental value, as shown in Fig. 10(b). In addition, Fig. 11 illustrates the fracture surface obtained with the proposed criterion in the (𝜂, 𝜇,𝜀̅ ) space for Al2024-T351 and TRIP 690 steel. The proposed fracture surface shows a good

23

fit with several experimental points for stress triaxiality ranging from low to intermediate and high in the 3D stress state. Furthermore, the values of the error functions of the extended L–H, MMC, and proposed criteria under the plane stress condition are listed in Tables 3 and Table 4 for two materials. The average error values are as follows: ErrExtended L–H = 13.189%, ErrMMC = 13.333%, and Errproposed = 10.881% for Al2024-T315; ErrExtended L–H = 8.158%, ErrMMC = 9.696%, and Errproposed = 7.586% for TRIP 690 steel. The average error values of the proposed ductile fracture criterion are the lowest among the three fracture criteria compared herein. This proves that the fracture locus and the fracture surface predicted with the proposed fracture criterion for different materials are fairly accurate relative to the experimental data over a broad range of stress triaxiality.

(a)

24

(b) Fig. 10. Comparison of fracture locus constructed with MMC, extended Lou–Huh, and proposed fracture criteria for (a) aluminum 2024-T351; (b) TRIP 690 steel.

(a)

25

(b) Fig. 11. Fracture surface constructed with proposed fracture criterion in (𝜂, 𝜇,𝜀̅ ) space for (a) Al2024-T351 (C1 = 0.42848, C2 = 0.43769, and C3 = 5.3161) and (b) TRIP690 steel (C1 = 1.2595, C2 = 0.83365, and C3 = 4.2652). Table 3. Error value of fracture strain predicted with different ductile fracture criteria under plane stress condition for Al 2024-T351. Test 1 2 3 4 5 6 7 8 9 10 11 12 13

Experimental data η 𝜀 -0.278 -0.2476 -0.2339 -0.2326 -0.2235 0.000 0.0124 0.1173 0.1585 0.191 0.3431 0.3557 0.357

0.4505 0.6217 0.38 0.3563 0.341 0.288 0.2107 0.2613 0.2656 0.253 0.3099 0.3255 0.4798

Extended Lou–Huh Error 𝜀 0.5175 0.4501 0.4255 0.4233 0.4087 0.2559 0.2539 0.2538 0.2625 0.2734 0.3751 0.3557 0.3538

14.8724 27.6017 11.9737 18.8044 19.8534 11.1458 20.5031 2.8703 1.1672 8.0632 21.039 9.278 26.2609

26

𝜀

MMC Error

0.5326 0.4611 0.4348 0.4324 0.4168 0.2537 0.2517 0.2545 0.2656 0.2793 0.4039 0.3786 0.3761

18.2242 25.8324 14.4211 21.3584 22.2287 11.9097 19.4589 2.6024 0.0002 10.3953 30.3324 16.3134 21.6132

𝜀 0.4804 0.4077 0.3829 0.3807 0.3664 0.2323 0.2308 0.2345 0.2448 0.258 0.4219 0.3858 0.3825

New Error 6.6371 34.4217 0.7632 6.8482 7.4487 19.3403 9.5396 10.2564 7.8313 1.9763 36.1407 18.5253 20.2793

14 15 16

17 18

0.4319 0.4974 0.5648 0.603 -0.496

0.2441 0.2204 0.1951 0.21 0.349

0.2699 0.2274 0.2076

10.5694 3.176 6.407

0.2699 0.2184 0.1951

10.5694 0.9074 0.0005

0.2614 0.2146 0.1951

7.0873 2.6316 0.0002

0.2102 0.4318

0.0952 23.7249

0.1981 0.3775

5.6667 8.1662

0.1976 0.3482

5.9048 0.2292

Table 4. Error value of fracture strain predicted with different ductile fracture criteria under plane stress condition for TRIP 690. Test 1 2 3 4 5

Experimental data η 𝜀

Extended Lou–Huh Error 𝜀

0.379 0.472 0.6667 0.577 0

0.7508 0.5432 0.9225 0.4599 0.6451

0.751 0.394 0.95 0.46 0.645

0.0005 37.868 2.8947 0.0217 0.0155

27

MMC Error

𝜀 0.7671 0.5092 1.0048 0.4153 0.6346

2.1438 29.2386 5.7684 9.7174 1.6124

New Error

𝜀 0.75 0.5353 0.966 0.4594 0.6442

0.1332 35.8629 1.6842 0.1304 0.124

4. Application of proposed ductile fracture to predict failure of Aluminum 6014-T4 Aluminum shows various advantages such as lightweight material, highly corrosion-resistant, readily formable and this material is a very attractive material for use in sheet metal forming field of the automotive industry. So, an automotive material Al6014-T4 (For example ABS body part, longitudinal member front reinforcement part) is selected to introduce a complete procedure from calibrating material parameters with several standard specimens to examine the accuracy of model in a typical sheet forming experiment with rectangular cup drawing test. Three constant material parameters can be identified after the calibration process. Then the new fracture criterion is introduced into Abaqus/Explicit via a VUMAT subroutine to simulate a rectangular cup drawing test. The comparison in force-displacement curve indicates that the new ductile fracture criterion is acceptable in predicting onset of ductile fracture in sheet metal forming.

4.1 Material and geometry of fracture testing specimens In this study, we analyzed the aluminum alloy 6014-T4. All specimens were extracted from 1.43mm-thick metal sheets with the longitudinal axis along the rolling direction. For sheet metal forming, six types of specimens were utilized to construct fracture loci and identify fracture parameters, as shown in Fig. 12. The uniaxial tension-specimen (UT) featured a 6-mm-wide and 30-mm-long gage section. Throughout the experiments, the displacement and the strain field of the surface specimens were monitored using the GOM-Correlate Professional digital image correlation (DIC) system. Other tension experiments were performed on central hole (CH) specimens and circular cut-out notched tension (NT) specimens. The CH specimens featured a 20-mm-wide gage section with an 8-mm-diameter hole in the center. The NT specimens labeled “NT5,”, “NT10,” and “NT20” refer to notch radii of R = 5 mm, 10 mm, and 20 mm, respectively. The in-plane shear tension specimen geometry, which is generally employed in fracture tests at low-stress triaxiality, was used herein for quasi-static shear testing. Six different geometry specimens are typically utilized in order to identify two main essential information for fracture testing. Firstly, the basic mechanical properties required in FEM simulation are stress-strain relations are obtained from UT specimen. And, five types of specimens have covered a large range of stress triaxiality from zero to intermediate and highstress triaxiality are used to calibrate fracture model and investigate the effect of stress triaxiality and Lode parameter on ductile fracture in proposed fracture criterion.

28

Fig. 12. Different types of specimens: Uniaxial tension (UT), shear tension (In-plane shear), central-hole tension (CH), and notched tension (NT5, NT10, and NT20). 4.2 Plastic model In a uniaxial tension test with a dog-bone specimen, localized necking occurs at a relatively small strain of approximately 20%. However, the current hardening laws have been verified only up to the necking strain. Therefore, the strain-hardening behavior of tAl6014-T4 under large deformation is determined by using an appropriate extrapolation strategy. An inverse approach reported by Luo et al. [45], Wang and Wierzbicki [46], and Roth and Mohr [47] was employed to extend the hardening curve to larger strains based on the uniaxial tensile test. The strain-hardening behavior was modeled through a linear combination of the exponential Voce law [48] and the power Swift law [49], as follows: 𝜎

= ( (𝜀 + 𝜀̅ ) ) + (1 − ) (

Where α is a weighting factor,

+

(1 − 𝑒

(− 𝜀̅ )))

(33)

[ ,1].

Exponential law: 𝜎

=

+

(1 − 𝑒

(34)

(− 𝜀̅ ))

Power law: 𝜎

(35)

= (𝜀 + 𝜀̅ )

The extrapolation stress-strain curve of the material was fitted with the Swift law, Voce law, and optimized linear Voce–Swift law, as shown in Fig. 13a. Then, the force-displacement curve obtained via the in plane-shear test was used to verify the accuracy of the hardening law for 29

large-strain conditions. The simulation results of force-displacement obtained via Swift law extrapolation overestimated the experimental data of the shear tension test, while the curve plotted using the Voce exponential law underestimated the experimental data. The linear Voce– Swift hardening law provided an accurate prediction of the measured force-displacement curve up to fracture initiation, as can be seen in Fig. 13b. In the following sections, the optimized linear hardening curve is used in all simulations. A summary of the hardening parameter of Al6014-T4 is given in Table 5.

(a) (b) Fig. 13 (a) Stress-strain curves extrapolated with different hardening laws, and (b) comparison of force-displacement curves of in-plane shear specimens predicted with three stress-strain curves. Table 5: Mechanical properties of AA6014-T4 at room temperature Swift parameter E [GPa] υ [-] A [MPa] Al6014-T4

69

0.3

384.124

Voce parameter

ε0 [-]

n [-]

k0 [MPa]

Q [MPa]

β [-]

α [-]

0.0037

0.234

122.369

149.570 12.342 0.231

4.3. Fracture model calibration and simulation 4.3.1 Fracture testing Five types of different geometry specimens that cover a large scope of stress triaxiality from zero to intermediate and high value of stress triaxiality has been tested in this study. The proposed ductile fracture criterion should be studied in this range to confirm the effect of stress triaxiality and Lode parameter on ductile fracture and accuracy of the proposed fracture criterion. Besides, the calibrate test used to determine the parameters in ductile fracture model is similar to the deformation process which needs to predict, the prediction would be accurate. So, the in-plane shear specimen presents low-stress triaxiality, CH specimen presents intermediate stress 30

triaxiality of 0.333, and the NT5 specimens indicate the stress state near the plane strain condition with high-stress triaxiality of 0.577 are frequently performed for calibrating fracture model. According to Dunand and Mohr [50], Lou et al [45] all experiments are carried out under quasistatic conditions with displacement control at crosshead velocity and 0.001 of strain rate. The Digital Image Correlation (DIC) system is used to measure the relative displacement of specimens. A virtual extensometer with 40mm of the gage length has been used in the DIC during the experiment process with all fracture specimens. 4.3.2 Fracture model calibration 4.3.2.1 Finite-element model Explicit finite-element (FE) simulations were performed for each fracture test detailed in Sect. 4.3.1 with full-size specimens. A 3D-FE model (FEM) comprising C3D8R elements with reduced integration was taken from the ABAQUS element library. To reduce the effect of mesh size, element size in the fracture area was set to 0.1 mm. The gage length of 40 mm used in the FEM model was consistent with the positions of the virtual extensometers used in the DIC to obtain the displacement of all fracture testing specimens. Figure 14 shows very fine meshes of in front of view of the upper right quarter of specimens with 15 solid elements through the thickness. The basic mechanical properties of Al6014-T4 material for FEM simulation, the linear VoceSwift hardening law as shown in table 5, is used in all fracture test simulation and square cup drawing simulation in next subsection. Moreover, the constitutive equations in section 2 were implemented into ABAQUS/Explicit using a VUMAT subroutine, which did not cover any fracture criteria in the code, to identify the average stress triaxiality and the average Lode parameter. This subroutine has been used to perform all calibrate simulations of tests. In calibrate simulation, the equivalent plastic strain to fracture, stress triaxiality, and Lode parameter are extracted from the corresponding simulation at fracture stroke.

31

Fig. 14. Detail of meshes of fracture test specimens. (a) in-plane shear, (b) Central hole, (c) NT 20, (d), NT10, (e) NT5 4.3.2.2 Calibration of fracture model A hybrid experimental and numerical method was applied to calibrate the fracture model. The force-displacement curves were extracted from each simulation and compared with the experimental measurements. To identify the material parameter of the proposed ductile fracture criterion for Al6014-T4, the equivalent plastic strain until fracture was measured by conducting experiments with at least three different specimens, namely, in plane-shear, CH, and plane strain (NT5) specimens. Besides, the ductile fracture criterion represents a general form of the equivalent plastic strain. The equivalent plastic strain is known as a critical factor in most ductile fracture criteria, even though various weight functions have been proposed based on different assumptions, as discussed in Section 1. This is because the fracture strain is simply approximated using the maximum equivalent plastic strain predicted via numerical analysis at a fracture stroke for these 32

specimens. The fracture strain, in the present study, was identified as the critical element subjected to the highest equivalent plastic strain for each test case. For the in-plane shear specimens, the selected elements were located at center of the shear band on the surface of each specimen. Because of without thickness reduction behavior, thickness did not influence the calculation of equivalent plastic strain, and fracture occurred on the surface of the specimens. For the NT5, NT10, and NT20 specimens, the selected elements were located at the center of each specimen in the simulation. The selected element in the CH specimen was located 0.4 mm away from the edge of the hole at the center of the specimen. Notably, average stress triaxiality and average Lode parameter as shown in Eq. (23) are extracted from the corresponding selected element in each calibrate testing simulation. Figures 15(a)–(c) show the hybrid simulation–experimental results obtained in tests with the inplane shear, CH, and plane strain (NT5) specimens. In this study, fracture strokes were determined at the point when the experimental loading curve dipped abruptly. In addition, the evolution of surface strain was compared between the DIC-measured results and the simulation results at selected points on the surface of the specimens. The comparison results indicated that the FEM model was adequately accurate for determining the plastic strain to fracture of the specimens. The maximum equivalent plastic strain until fracture, average stress triaxiality and average Lode parameter are introduced via the solution-dependent state variable (SDV) in the VUMAT subroutine such as SDV1, SDV2 and SDV3 respectively. At the fracture stroke, these fracture information were extracted from the numerical simulation for in-plane shear, central hole and plane strain specimens as shown in Table 6. Table 6. Plastic fracture strains of Al6014-T4 and the corresponding average stress triaxiality and Lode parameter. Specimen

In plane-shear

Central hole

Plane strain

Fracture strain

1.150

0.849

0.635

Triaxiality

0.098

0.432

0.610

Lode parameter

-0.257

-0.617

0.188

The experimental results given in Table 6 were utilized to calibrate three material parameters (C1, C2, and C3) of the proposed ductile fracture criterion. Notably, the average values of stress triaxiality and the Lode parameter are imperfect at the pure shear, uniaxial tension, and plane 33

strain positions. To identify the three fracture parameters, a Matlab code is written to perform the optimization procedure shown in Eq. (30). After several runs, the three fracture parameter are obtained with (C1 = 3.2283, C2 = 2.0271, and C3 = 3.2844). The new ductile fracture locus and the fracture surface of Al6014-T4 are plotted in the full-stress space of (η, µ,𝜀̅ ) by using the three calibrated material parameters as shown in Fig.16.

(a)

(b)

34

(c) Fig. 15. Determination of equivalent plastic strain to fracture by using hybrid experimental– numerical method: (a) in-plane shear, (b) central hole, and (c) plane strain.

Fig. 16. Fracture locus of Al6014-T4 predicted using the proposed criterion in (η, µ,𝜀̅ ) space (C1 = 3.2283, C2 = 2.0271, and C3 = 3.2844). 4.3.3 Prediction of proposed ductile fracture criterion To verify the accuracy of proposed ductile fracture criterion, all the fracture experiment tests described above were conducted to predict the onset of fracture behavior. The proposed fracture 35

criterion was implemented in ABAQUS/Explicit by using the VUMAT subroutine. Figures 17(a)–(e) show a comparison of the force-displacement results obtained in FEM simulations and the experimental results obtained with the in-plane shear, CH, plane strain NT5, notched NT10, and notched NT20 specimens. The initial fracture points predicted using the proposed ductile criterion in each case are marked with red squares. The predicted fracture stroke is close to the experimental fracture stroke. The average error function in Eq. (32) for comparing the simulation fracture stroke with the experimental fracture stroke was utilized to evaluate the accuracy of the proposed criterion in five types of tests. The resulting error values are listed in Table 7, with all values being lower than 2%. The comparison results indicate that the proposed fracture criterion can accurately predict the initial fracture of all five types of specimens over a broad range of stress triaxiality from zero to 2/3 under plane stress conditions. Furthermore, the fracture strain on the surface of each specimen, as measured with DIC, and the equivalent plastic strain to fracture (PEEQ) determined using the FEM model are shown in Fig. 17. The strain distributions in the fracture zone obtained via simulation and experiments are similar. However, the fracture strain measured with DIC is slightly lower than the PEEQ value obtained in the simulation. This discrepancy can be ascribed to the reduction in specimen thickness during the experiments; the DIC system can only measure the strain field on the surface of specimens, while failure occurs at the center of specimens. Notably, the experimental results of in-plane shear are close to the simulation results because the initial fracture occurs on the specimen surface. Table 7. Summary of predicted fracture strokes (Pre. Stroke) and experimental fracture strokes (Exp. Stroke) Al6014-T4

Shear

Central hole

NT20

NT10

NT5-Plane strain

Exp. Stroke [mm]

2.504

2.104

2.875

2.374

2.097

Pre. Stroke [mm]

2.469

2.082

2.851

2.345

2.075

Error [%]

1.398

1.045

0.835

1.221

1.049

36

(a) In plane-shear

(b) Central hole

(c) NT20

(d) NT10

(e) NT5 Fig. 17. Comparison of load-displacement curves: (a) in plane-shear; (b) CH; (c) NT20; (d) NT10, and (e) NT5-plane strain.

37

4.4 Cup drawing test of Al6014-T4 4.4.1 Square cup drawing test Sheet metal forming with deep drawing is a widely used manufacturing process, especially in the automotive industry. In this study, a cup drawing test was performed to characterize the complex forming state of the proposed ductile fracture criterion. Square plates (100 mm × 100 mm) were cut from the same metal sheet of Al6014-T4 for punch testing. Figure 18 depicts a schematic of the experimental setup, which consisted of a cup square punch with a width of 50 mm and an edge radius of 5 mm, and a die with a width of 54 mm and an edge radius of 5 mm. The sheet was arranged along its rolling direction in the coordinate system of the punch. During the entire experimental operation, a constant blank holding force (BHF) of 5 ton was applied on the aluminum sheet metal, while the punch moved upward and drew the blank sheet. A certain lubricant was applied to all contact surfaces. We evaluated the accuracy of the proposed ductile fracture criterion in predicting failure behavior in a real forming process.

Fig. 18. Setup of cup drawing experiment, along with applied blank holding force (BHF). 4.4.2 Finite element results The FE simulation of the cup-drawing test was performed in the ABAQUS/Explicit environment. Because of the symmetric load and geometry, a quarter of the square cup geometry of the specimens was simulated to reduce simulation time. Solid C3D8R elements with reduced integration, available from the ABAQUS element library, were used in the simulation. Mesh size influences the prediction of the initial fracture moment during drawing. Park et al. [51] presented 38

simulation results for square cup drawing with various mesh sizes for two groups of FE models: quarter model and half model. The relationship between the ratio of mesh size per punch radius and the punch stroke to fracture was used to evaluate the dependence of mesh size on initial fracture in the cup drawing process. A comparison of the simulation results with experimental results indicated that in case of the quarter model, a mesh size per punch radius ratio of 0.04 provides good accuracy. This is because the element size in the fracture zone was set to 0.2 mm with a punch radius of 5 mm in this study. The plasticity model and the proposed ductile fracture criterion were implemented in the VUMAT subroutine to simulate the square cup drawing model. Crack initiation and propagation were simulated using the element-deletion technique. The friction coefficient used in the simulation was determined using a combination of physical and numerical factors. For cold rolling with the lubricant employed herein, a friction coefficient of 0.11 is generally used in simulations. The material properties for FEM model for Al6014-T4 have been introduced in the previous section. The stress train relation is described by the linear Voce-Swift hardening law. And the proposed ductile fracture criterion is implemented into VUMAT subroutine with three fracture parameter which has been calibrated. Figure 19 presents the flowchart of the stress integration algorithm of proposed ductile fracture criterion.

39

Fig. 19. VUMAT subroutine flowchart of the proposed ductile fracture criterion. Figure 20 illustrates a comparison between the experimentally obtained and simulated forcestroke curves. The experimental force-stroke curve, simulated force-stroke curve without implementing the proposed fracture criterion, and simulated force-stroke curve with implementation of the proposed fracture criterion are represented by black-colored dots, greencolored line-dots, and red-colored line-dots, respectively. In the experiment, the force-stroke curve was stopped at 14.825 mm when the reduction in force was greater than setting value of the machine. To evaluate the effect of predicting the initial fracture state on the square cup drawing test, simulation results obtained without implementation of the proposed fracture criterion and with implementation of the proposed fracture criterion were compared for punch travel of 17 mm. Without implementation of the fracture criterion in the cup drawing test, the force-stroke curve after the experimental fracture stroke extended the curve until a stroke of 17 mm without any decrease in the force value. By contrast, with implementation of the proposed

40

fracture criterion, the simulation results are consistent with the experimental data. These results indicate that the proposed fracture criterion performs well in terms of predicting fracture behavior in the square cup drawing test. Furthermore, the crack occurred at the corner-drawn area near the contact between the punch and the sheet metal in the experimental and the simulation results. Based on the simulation results, the fracture is initiated on the outer layer of material, which undergoes bending and stretching during the experimental process, as shown in Figure 21. Notably, loading direction and geometry of the deformation area of application has strong effect on fracture behaviors. In other words, depending on the tendency of the plastic deformation direction of the forming process, the failure of material occurs under negative stress triaxiality area, low, intermediate or high-stress triaxiality scopes. In negative range of stress triaxiality ductile fracture can be found in area where shear stress dominate. For fracture in structural components, Wen and Mahmoud [26] report the shear fracture phenomenon in bolted connections. On the other hand, it is clear that in rectangular cup drawing test the corner element deformed under both stretching and bending. The initial fracture happens near high scope of stress triaxiality from plane strain mode to biaxial tension mode.

Fig. 20. The comparison result of force-punch stroke for square cup drawing test

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Fig. 21. Localized equivalent plastic strain distribution at point of initial fracture in square cup drawing test. 5. Conclusions In this study, a new uncoupled ductile fracture criterion based on the micro mechanisms of void nucleation, void growth, and evolution of void coalescence was developed. The proposed ductile fracture criterion can be described under two forms of ductile fracture. In macro fracture mechanism form, the proposed fracture model carried out fracture phenomenon under effect of maximum shear stress and maximum tension stress. And in micro fracture mechanism form, the proposed fracture model can be transformed into function of both stress triaxiality and Lode parameter to describe the effect of micro fracture mechanism on ductile fracture. In the proposed fracture criterion, the void nucleation is controlled by equivalent plastic strain, void growth is further extended and presented as function of both stress triaxiality and Lode parameter, and the effects of the secondary voids and rotation of voids in coalescence stage is incorporated into function of Lode parameter. Several sets of revised experimental data of aluminum Al2024-T351 and TRIP 690 steel were used to determine the accuracy of the proposed ductile fracture criterion, MMC, and extended L–H fracture criterion. A comparison of the average error values of the three criteria indicated that the proposed ductile fracture criterion yields an accurate fracture locus and a fracture surface with acceptable errors. The hybrid experimental–numerical method was used to identify fracture strain in five fracture tests conducting using aluminum 6014-T4 specimens. Under the assumption of material isotropy, the fracture locus and the fracture surface were successfully constructed with the proposed ductile fracture criterion. Furthermore, a square cup drawing test was performed to prove the efficiency of the proposed ductile fracture criterion for aluminum 6014-T4. A comparison of the 42

experimental and simulated force-strokes in the cup drawing test showed that the proposed ductile fracture criterion performs well in predicting initial fracture. Therefore, we concluded that the proposed ductile fracture criterion can adequately predict fracture initiation in the sheet metal forming process over a broad range of stress states. Declaration of Conflict of Interest I declare that this manuscript is original, has not been submitted to, nor is under review at, another journal or other publishing venue. The manuscript has been read and approved by all authors.

ACKNOWLEDGEMENT This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019R1A2C1011224).

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Appendix A. Summary of listed typical ductile fracture criteria. Cockcroft–Latham criterion [8] ̅

∫ 𝜎 𝑑𝜀̅ = 𝐶

(A.1)

Rice-Tracey criterion [10]: ̅

3 283exp ( 𝜂) 𝑑𝜀̅ = 𝐶 2

∫

(A.2)

Oh criterion [40]: ̅

〈𝜎 〉 𝑑𝜀̅ = 𝐶 𝜎̅

∫

(A.3)

Maximum shear stress criterion [42]: 𝜎 −𝜎 𝜏 = =𝜏 2 Ko-Huh criterion [41]:

(A.4)

̅

∫

𝜎 (〈1 + 3𝜂〉)dε̅ = 𝐶 𝜎̅

(A.5)

Extended Lou-Huh criterion [21]: ̅

∫ (

,

2 õ + 3

1 (⟨ 1+𝐶 ,

)

(𝜂 +

3−µ 3√µ + 3

,

+𝐶

,

)⟩)

𝑑𝜀̅ = 𝐶

(A.6)

,

Bai and Wierzbicki [13]: 1 ̅ 2 + 1 (𝐷 e−𝐷2 𝜂 − 𝐷 e−𝐷6 𝜂 )𝜃 ̅ + 𝐷 e−𝐷4 𝜂 𝜀̅ = * (𝐷1 e−𝐷2 𝜂 + 𝐷5 e−𝐷6 𝜂 ) − 𝐷3 e−𝐷4 𝜂 + 𝜃 5 3 2 2 1

(A.7)

Wen-Mahmoud criterion [25]: , 𝜋 𝜂) *𝑐𝑜𝑠 ( 𝜃̅)+ 6 Modified Mohr–Coulomb criterion in the (η, µ, 𝜃̅) space [14]:

𝜀̅ = 𝐶

exp(𝐶

,

,

(A.8) (A.9)

𝜋𝜃̅ 1+𝐶 𝜋𝜃̅ 1 𝜋𝜃̅ (1 − 𝐶 ) *𝑠𝑒𝑐 ( ) − 1+- {√ ,𝐶 + 𝜀̅ = 𝑐𝑜𝑠 ( ) + 𝐶 *𝜂 + 𝑠𝑖 ( )+} 𝐶 6 3 6 3 6 2 − √3 { } √3

Where A and n are the strength coefficient and the hardening exponent, µ and 𝜃̅ are the Lode parameter; η is stress triaxiality; 𝜎 , 𝜎 , 𝜎 are principal stresses; and 𝐶 𝐶

,

,𝐶

,

, D1-D6 are material parameters. 44

,𝐶 ,𝜏 ,𝐶

,

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Graphical abstract. 48