A modified micromechanics framework to predict shear involved ductile fracture in structural steels at intermediate and low-stress triaxialities

Journal Pre-proofs A Modified Micromechanics Framework to Predict Shear Involved Ductile Fracture in Structural Steels at Intermediate and Low-Stress Triaxialities Yazhi Zhu, Ravi Kiran, Jihui Xing, Zuanfeng Pan, Lei Li PII: DOI: Reference:

S0013-7944(19)30920-8 https://doi.org/10.1016/j.engfracmech.2019.106860 EFM 106860

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

26 July 2019 22 December 2019 23 December 2019

Please cite this article as: Zhu, Y., Kiran, R., Xing, J., Pan, Z., Li, L., A Modified Micromechanics Framework to Predict Shear Involved Ductile Fracture in Structural Steels at Intermediate and Low-Stress Triaxialities, Engineering Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.engfracmech.2019.106860

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A Modified Micromechanics Framework to Predict Shear Involved Ductile Fracture in Structural Steels at Intermediate and Low-Stress Triaxialities Yazhi Zhu1, Ravi Kiran2, Jihui Xing3, Zuanfeng Pan1, and Lei Li4 1

Dept. of Structural Engineering, Tongji University, Shanghai 200092, China. Dept. of Civil and Environmental Engineering, North Dakota State University, ND 58105, United States. 3 School of Civil engineering, Beijing Jiaotong University, Beijing 100044, China. 4 Dept. of Structural Engineering, University of California, San Diego, CA 92093, United States.

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Abstract: This paper employs a micromechanics framework to investigate the mechanisms and to predict the ductile fracture of structural steels at intermediate and low-stress triaxialities and under shear involved stress states. Unit cell-based micromechanical analyses are carried out to provide insights into the combined effects of triaxiality, Lode parameter, and shear stress component on the micro-mechanisms of ductile fracture. Based on the micromechanical analyses, an existing fracture model is modified to incorporate the influence of the shear stress component. Experimental investigations are carried out on three axisymmetric tension specimens, and a series of shear specimens made of Chinese Q460 steels to achieve uniaxial tension and shear dominated loading conditions, respectively. Finite element analyses are performed to evaluate the stress and strain fields in the tension and shear specimens. The modified model is calibrated by combining the experimental results with micromechanical analyses. Validation studies show that the predicted fracture initiation by the model agrees well with experimental results, implying a good performance of the proposed micromechanics framework for the prediction of the shear-dominated ductile fracture at intermediate and lowstress triaxialities. Keywords: Ductile fracture, Structural steels, Intermediate and low triaxialities, Shear stress, and Micromechanics.

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Introduction Ductile fracture is the main failure mode that governs the strength and ductility of steel

structural components, connections, as well as collapse behavior of steel structural systems under severe earthquakes or other extreme loading conditions [1]. An accurate prediction of ductile fracture in steel structures is thus a topic of practical importance. In metals, the ductile fracture is a multistep process associated with microscale void nucleation, growth, and coalescence, which ultimately leads to fracture at the macroscale [2]. Among these stages, void growth was considered as the most critical step during the entire process of ductile failure in some earlier studies [3-5], and the most important void dilation-controlling parameters are found to be the stress triaxiality (the ratio of the hydrostatic stress m and the effective stress e, i.e., T = m/e) and the equivalent plastic strain. As a result, a majority of past studies were focused on the void volumetric change at high and intermediate triaxialities. For instance, Hayden and Floreen [6], and Cox and Low [7] experimentally investigated the void surface in a two-dimensional plane and qualitatively analyzed the relationship between the void growth rate and the plastic strain and the stress triaxiality. Marini et al. [8] extended this body of work to the three-dimensional cases and arrived at similar conclusions about the roles of triaxiality and equivalent plastic strain on void dilation. Experimental studies conducted by Benzerga et al. [9] and Weck et al. [10] showed that the voids grow predominantly in one direction in the early stages of deformation and later transition into a mode of growth in a preferred direction, which ultimately leads to coalescence between neighboring voids. Recent micromechanical studies on void dilation again showed a strong sensitivity of void volumetric growth to triaxiality [11,12]. Unlike at high and intermediate triaxialities, void growth and coalescence are associated with mechanisms other than void dilation at low triaxialities. Previous investigations showed that the volume of a void in the regime of low triaxiality ceases to increase and even decreases in some cases [13-15]. Compared to the internal necking type of strain localization dominant at high and intermediate triaxialities as depicted in Fig. 1(a), void growth in this regime is mainly due to void distortion (shape changes) caused by void elongation, and rotation [14], as 2

shown in Fig. 1(b). The major factors that influence the extent of void distortion are the shear stress [16], and the interaction between the matrix and the particle inside the void. Herein, the shear stress can be either referred to as the applied shear stress components or the maximum shear stress (Tresca). The shear stress contributes to the rotation and flattening of voids, which may eventually lead to the closure of voids into micro-cracks [13]. Subsequently, either the severe void distortion or the shearing effect precedes the formation of a shear band. The strain localization across the shear band and the void locking caused by the void-particle interaction accelerate the softening of the matrix, increasing the interaction between the neighboring voids, and ultimately lead to the initiation of void coalescence [17]. During this process, the rate of void shape change is relatively insensitive to stress triaxiality, and the void dilation is minimal. The mode of void growth and coalescence in the low triaxiality regime referred to as internal shearing, is primarily controlled by shear-dominated stress states [16,18]. To model void growth under low triaxialities introduces another stress state parameter referred to as Lode parameter (L). The Lode parameter quantifies the shear stress state, and is used to describe the relative alignment of the three principal stresses (L = (22 - 1 - 3) / (1 - 2), where 1 ≥ 2 ≥ 3 are the three principal stresses). The sensitivity of void growth and coalescence on the Lode parameter has been extensively studied based on micromechanical modeling [19] and coupon-level experiments [20]. The currently available models to predict the ductile fracture over a wide range of triaxialities can be broadly classified into the extensions of Gurson-Tvergaard-Needleman (GTN) model and the Void Growth Model (VGM) [21-25]. These models typically incorporate the Lode parameter into the fracture criterion or damage equations to quantify its influence on ductile fracture. For structural steels, however, most of the existing models are only capable of predicting fracture at high triaxialities, and there are a limited number of models available that account for the dependency of ductile fracture on combined stress triaxiality and Lode parameter. In practical applications, structural steel components are often subjected to pure shear or combined shear, and tension loading conditions, which make them susceptible to shear dominated fracture that occurs at low and intermediate triaxiality ranges and with Lode 3

parameter varying between -1 (axisymmetric tension) and 1 (axisymmetric compression). Representative examples include bolt/ block shear fracture and bearing tear-out fracture in bolted connections [26,27], and global and local buckling induced fracture in braces [28,29]. Considering the above critical scenarios, models that can predict ductile fracture at low and intermediate triaxialities for structural steels are need of the hour. To investigate the ductile fracture in structural steels, numerous tests have been conducted in the past using axisymmetric smooth/ notched specimens [26,30,31]. In these tests, the ductile fracture was investigated under a limited range of stress states, i.e., at high and intermediate triaxialities (T > 1) and with a constant Lode parameter (L = -1). Kanvinde and Deierlein [32] carried out tests on ductile fracture initiation in smooth-notched, compact tension, and single end notched bending specimens made of ASTM A572 steels. Kiran and Khandelwal [33] calibrated the Void Growth Model (VGM) for ASTM A992 steels through the experimental results of smooth/ sharp notched tensile tests. Kiran and Khandelwal [34] calibrated the Gurson-Tvergaard-Needleman Model (GTN) for A992 steels to predict ductile fracture at high triaxialities. Other experimental studies were reported on ductile fracture at intermediate and low triaxialities and under various Lode parameters. For instance, Smith et al. [35] performed a series of tests on ductile fracture in ASTM A572 Gr. 50 steels covering a wide range of triaxialities (0.1 < T < 1.6). Jia et al. [36] conducted ductile fracture tests for SS400, SM490, and SM570 steels using pure shear and combined shear and tension plates, and numerically investigated shear-dominated fracture using the VGM model. Li et al. [37] experimentally investigated ductile fracture in Chinese Q460 high-strength structural steels at intermediate and low triaxialities. Huang et al. [31] conducted combined tension and shear tests on Chinese Q235 structural steels to study ductile fracture covering a wide range of triaxialities. To construct the fracture loci of structural steels, the crucial step in some of the existing models is to determine the relationship between the macroscopic strains at fracture of the specimens and the stress states. To this end, experimental tests are combined with finite element models to acquire stress and strain states during the loading history and at fracture. However, these conventional approaches have three shortcomings. First of all, although these 4

approaches can evaluate ductile fracture quantitatively, they provide limited insights into the underlying fracture mechanisms at microscales, particularly under complex stress states. Secondly, even when a test specimen has a relatively simple geometry, the stress states in the vicinity of material failure (crack) can be highly non-uniform and can evolve considerably during the loading history which should be considered. Thirdly, a significant number of test specimens and extensive computations are required to study the relationship between the fracture strains and stress triaxiality, and Lode parameter when the current approaches are employed. These inherent challenges should be inevitably addressed even for the prediction of ductile fracture in structural steels at intermediate and low triaxialities (e.g., [23,38,39]). Alternatively, computational analyses on microvoid evolution using finite element micromechanical modeling are preferable to investigate the stress state-dependent strains to fracture and for the development of the fracture locus. Previous attempts to use this approach were made by Kiran and Khandelwal [40] and Bomarito and Warner [15], in which satisfying results have been obtained with regard to the straightforward examination of the roles of stress states under complex loading conditions. However, such a micromechanics framework has rarely been used in the field of structural engineering. The available applications of this approach were limited to the failure of structural steels at high triaxialities [1,41]. Therefore, the main objective of this study is to employ a micromechanics framework to predict ductile fracture of structural steels at low and intermediate stress triaxialities. This work experimentally and numerically investigates the ductile fracture in structural steels subjected to uniaxial tensile, pure shear, as well as combined tensile and shear loadings to provide a more accurate prediction of tension and shear dominated fractures in structural steels. An existing fracture model is modified based on micromechanical analyses and is validated on coupon-level tests. The proposed approach has the potential to provide insights into the micro-mechanisms of void evolution and can be used to predict ductile fracture in structural steels in low and medium triaxiality ranges. First, an existing triaxiality and Lode parameter-dependent ductile fracture initiation criterion is presented associated with a damage evolution rule. Micromechanical modeling on unit cells is conducted to qualitatively and 5

quantitatively investigate the combined effects of triaxiality, Lode parameter, and a shear stress component on microvoid evolution at intermediate, and low triaxialities. Based on these micromechanical analyses, the previously presented fracture model is modified to incorporate the influence of the shear component which is found to be crucial at low and intermediate triaxialities. Experimental tests are carried out on cylindrical specimens, and shear plate specimens made of Chinese Q460 steels that cover a range of stress triaxialities between 0.33 and 1.00. Finite element simulations of the fracture specimens are performed to evaluate the evolution of strain and stress fields, particularly at the critical regions of each specimen. The parameters in the modified model are then calibrated by combining coupon-level test results with micromechanical modeling. Prediction of ductile fracture initiation in the specimens by employing the proposed modified model is validated by experimental results. Implementation procedure and drawbacks of the proposed framework are finally discussed. 2

Ductile fracture model As discussed previously, stress triaxiality and Lode parameter are the two main

dimensionless parameters that influence the local material failure in metals. The Lode parameter-dependent void shape changes are prominent at low triaxialities. Micromechanical modeling can provide quantitative insights into void growth and coalescence mechanisms at various triaxiality, and Lode parameter values [11,15,16,18]. In the micromechanical analysis, a unit cell with a single void or a representative volume element of the metal under consideration is the control volume. The unit cell is deformed by maintaining a constant triaxiality, and Lode parameter in the micromechanical analyses to evaluate the behavior of matrix and void, as well as the quantitative relationship between stress states and material response. Through a series of micromechanical analyses, Kiran and Khandelwal [40] investigated microscopic damage mechanisms at different triaxialities and developed a computational fracture criterion covering a wide range of triaxialities. The model consisted of a ductile fracture initiation criterion, and a damage evolution rule. This micromechanics based model was validated on axisymmetric tension specimens. This model is modified in the present

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study to incorporate the influence of the shear component and will be discussed further in the context of current work. 2.1 Micromechanics-based ductile fracture initiation criterion A piecewise function was introduced by Kiran and Khandelwal [40] to account for both triaxiality dependent void dilation and Lode parameter sensitive elongation. The general form of the proposed fracture locus is given as:

T    T0   T0      T p f

0  T  T0 T  T0

(1)

where  fp is the effective plastic strain at fracture, , ,  are three model parameters. T0 denotes the transition triaxiality, at which the dominant micro-mechanism leading to fracture is considered to switch from void dilation to void elongation or vice versa. The value of T0 is thus critical for determining the dominant damage mechanism responsible for fracture initiation. In the original model, the value of T0 was evaluated to be 0.75 for ASTM A992 steel through micromechanical analyses. Unlike the above model which was developed through the micromechanical analyses of unit cells, the VGM was analytically derived by deforming a single void in an infinite elasticperfectly plastic solid subjected to relatively high triaxialities, and presented a theoretical relationship between the rate of the void dilation and the rates of plastic strain and stress triaxiality. The fracture locus predicted by the VGM is found to be accurate for high triaxiality values, at which void dilation is dominant [32]. However, the performance of the VGM is questionable for relatively low triaxiality values as the theoretical derivation of the VGM involves studying void dilation at high triaxialities as previously mentioned. Fig. 2 illustrates the fracture loci under axisymmetric uniaxial tension cases constructed using the VGM, the Bao-Wierzbicki (B-W) model, and the Kiran-Khandelwal (K-K) model. It can be seen that at high triaxialities, B-W, VGM, and K-K models present a similar dependency of fracture locus on triaxiality. In contrast, macroscopic experimental evidence provided by Bao and Wierzbicki

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[42] supports the existence of two different microscopic damage mechanisms leading to ductile fracture (i.e., void dilation and elongation). Considering both micromechanics, and experimental evidence, Bao and Wierzbicki [42] and Kiran and Khandelwal [40] modified the dependency of fracture locus on the triaxiality in the regime of relatively low triaxialities and extended the applicability of the two models to shear dominated ductile fracture. Note that the transition triaxiality for 2024-T351 aluminum alloy (of the B-W model) is 0.40 and 0.75 for ASTM A992 steels, implying that the transition triaxiality is dependent on the metal under consideration. Both the B-W model and the K-K model significantly lower the predicted strains to fracture at intermediate and low triaxialities compared to the VGM model, while the latter often overestimates the fracture strains in the low triaxiality regime [43]. A brief note on the approach adopted in the K-K model to incorporate the influence of the Lode parameter is provided next. The Lode parameter (L = (22 - 1 - 3) / (1 - 2)) ranges between -1 and 1. Three values of Lode parameter, i.e., -1, 0, 1, represent axisymmetric tension, plane strain, and axisymmetric compression cases, respectively. Numerous experiments on ductile fracture in structural steels have been conducted using the axisymmetric tensile tests on round bars, for which the Lode parameter is equal to -1. To account for the dependence of the model parameters given in Eq. (1) on Lode parameter, Kiran and Khandelwal [40] introduced three correction factors k, k, k to modify the material parameters , ,  (see Eq. (1)) which are given as:

  k  L 1   k   L 1

(2)

  k  L 1 where L=-1, L=-1, L=-1 are the magnitudes of the material parameters , ,  at L = -1, respectively, which are calibrated from tests conducted on axisymmetric tension specimens. The correction factors at L = 0, and 1 are calculated using micromechanical analyses. Linear interpolation is used in K-K model to evaluate the correction factors for intermediate values of the Lode parameter.

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Macroscopic fracture is assumed to initiate when the indicator of ductile fracture initiation ID satisfies the following condition: ID  

p

0

d p 1  fp T , L 

(3)

This condition considers the effect of the loading path (i.e., the evolution of stress states) during deformation history. Under a constant triaxiality and Lode parameter, the plastic strain to fracture can be directly obtained by Eq. (1). 2.2 Damage evolution rule A damage evolution rule is employed after the material meets the fracture initiation criterion provided in Eq. (3) to describe the loss of material strength. To this end, a scalar damage variable d  [0, 1] is introduced along with the damage evolution rule. The effective stress in a damaged state  e is then related to the stress e in an undamaged state using the damage variable as  e  1  d   e , where d = 0 and 1 correspond to no damage and full damage conditions, respectively. Here full damage refers to the condition in which the material point (integration point) completely loses its loading carrying capacity. The evolution of the damage variable employed in the present paper is based on the concept of effective plastic displacement available in ABAQUS® [44], which is assumed to have an exponential form given as: u p u fp

1 e d 1  e 

(4)

where  and u fp are two parameters that need to be calibrated, u fp represents the effective plastic displacement at the instant of failure. This damage evolution rule is also used in many previous studies owing to its theoretical simplicity, and computational efficiency [45-47]. It is well-known that when a softening material is modeled as a classical elastoplastic continuum, numerical results will exhibit a strong dependence on the mesh [48] and the strain tends to localize into a region with zero size upon mesh refinement. To alleviate this spurious mesh dependency in finite element modeling, the effective plastic displacement is related to 9

the plastic strain as follows u p  l p

(5)

where, l is the characteristic length of the element and  p is the rate of change of plastic strain. By introducing a characteristic length to the damage evolution rule, the pathological mesh dependent solutions can be avoided. Alternative strategies to overcome such numerical instability can be found elsewhere (see refs. [46,49,50]). 3

Computational micromechanical analyses

3.1 3D cell model Micromechanical analyses are carried out in this section to quantify the effect of stress states on the microvoid evolution and to calibrate the model presented previously. Two assumptions are made (1) among the stages of ductile fracture, microvoid growth (dilation and elongation) is considered as the critical phase as it leads to microvoid coalescence and the subsequent material failure; and (2) the material matrix is idealized as a periodic array of unit cells. These two assumptions allow for the use of a unit cell as the representation of the metal microstructures. By enforcing periodic boundary conditions, the single cell can be studied instead of a 3D array of voids. Similar to the unit cells chosen in the previous studies [15,16,40,41], a unit cube with a pre-existing spherical void at its center is selected as the unit cell in this study. As such, the stage of void nucleation is excluded in the current study for the reasons mentioned in the literature [51,52]. Among various unit cell geometric parameters, the initial void size is the critical one as it considerably influences the fracture behavior as indicated by previous studies [11,18,41]. The size of the void is quantified by the void volume fraction (f), which is referred to the ratio of the volume of the void to the total volume of the unit cell. As stated by Yan et al. [41], the void volume fraction is very small for most structural steels. Kiran and Khandelwal [53] suggested that the initial void volume fraction (f0) can be taken between 1% to 2% for low-carbon structural steels. In the present study, the initial void volume fraction f0 is assumed as 1% for 10

the lack of experimental evidence. The volume average of the effective plastic strain  effp measured from the unit cell is considered as the macroscopic plastic strain Ep and is given as: Ep 

where  effp is calculated as  effp 

1 p  eff dV V

(6)

2 t p ε dt using the plastic strain rate ε p and V is the 3 0

current volume of the unit cell. Similarly, the macroscopic Cauchy stresses ij calculated by evaluating the volume averages of the Cauchy stresses (ij) over the current volume V of the unit cell are used as the measures of macroscopic stresses. The overall macroscopic stress state applied to the unit cell is the combination of three normal stresses (11, 22, and 33) and a shear stress component (12). In this case, the macroscopic hydrostatic stress and the macroscopic effective stress are given as: m 

11   22  33 3

 11   22    11  33     22  33  2

e 

2

2

2  612

(7)

2

Due to the introduction of shear stress Σ12, there are infinite combinations of macroscopic stresses to achieve a set of triaxiality and Lode parameter values. Following the work by Bomarito and Warner [15], only the critical combination that minimizes the macroscopic plastic strain to failure ( E fp ) is considered in the present study. To this end, the ratio of Σ12 to Σ22 (  2  12  22 ) is taken as an independent parameter. For a given triaxiality and Lode parameter, the range of ρ2 is discretized into increments of 0.05, and the one with the minimum E fp is chosen as ρ2 of interest. With a given triaxiality, Lode parameter, and ρ2, it is now

straightforward to determine the proportional relations among all the four macroscopic stress components. The geometry of the unit cell and the computational process of loading are explained next. The unit cell model is in a Cartesian coordinate system (X1, X2, X3) such that -D1/2 ≤ X1

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≤ D1/2, -D2/2 ≤ X2 ≤ D2/2, -D3/2 ≤ X3 ≤ D3/2, and the initial dimensions of the unit cell are D1 = D2/2 = D3 = D0 (see Fig. 3). Due to symmetry, only half of the unit cell (-D3/2 ≤ X3 ≤ 0) is considered in the micromechanical analyses. The unit cell has fully periodic boundary conditions. Each pair of boundary conditions is enforced by prescribing the translational displacements of the nodes at the surfaces, edges, and corners of the unit cell. According to the stress state described above, the displacements U1, U2, U3, and UT are used to describe the boundary conditions applied to the model as follows: D0 D D D and  0 : u1 ( 0 , X 2 , X 3 )  u1 ( 0 , X 2 , X 3 )  U1 2 2 2 2 D D u2 ( 0 , X 2 , X 3 )  u2 (  0 , X 2 , X 3 ) 2 2 D0 D0 u3 ( , X 2 , X 3 )  u3 (  , X 2 , X 3 ) 2 2 on X 2  D0 and  D0 : u1 ( X 1 , D0 , X 3 )  u1 ( X 1 ,  D0 , X 3 )  U T on X 1 

(8)

u2 ( X 1 , D0 , X 3 )  u2 ( X 1 ,  D0 , X 3 )  U 2 u3 ( X 1 , D0 , X 3 )  u3 ( X 1 ,  D0 , X 3 ) on X 3 0 and 

D0 : 2

u3 ( X 1 , X 2 ,0)  0 u3 ( X 1 , X 2 , 

D0 )  U3 2

where U1, U2, and U3 are three uniform normal displacements, and UT is a uniform tangent displacement. 3.2 Numerical implementation The above micromechanical modeling is conducted using ABAQUS® with the consideration of large deformation using an updated Lagrangian formulation. The unit cell is meshed using the 8-node linear isoparametric elements with reduced Gauss integration point (C3D8R). Three different mesh refinements (coarse, medium, and fine meshes) are considered to examine the mesh sensitivity of the unit cell mechanical behavior (for the case with T = 1, L= -1, and ρ2 = 0). The FE discretizations shown in Fig. 4(a) are coarse mesh - 7232, medium mesh – 14400, and fine mesh - 41220 elements. Mesh sensitivity studies are conducted to ensure convergence with respect to the macroscopic stress-plastic strain response of the unit

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cell. As observed in Fig. 4(b), the macroscopic stress-plastic strain responses in the elastic and strain hardening ranges are identical for the three different mesh discretizations. The softening behavior of the unit cell exhibits only a marginal difference between the medium and fine meshes. To save the computational time, the medium mesh is adopted for further micromechanical analyses. The matrix material is modeled using von Mises plasticity model with isotropic hardening. As an illustrative example in this section, a simplified power-law rule is employed:  y   0 1  E p  0  , where 0 = 400 MPa is the initial flow stress, E = 206,000 n

MPa is Young’s modulus, n = 0.1 is strain hardening exponent, y is flow stress, and εp denotes plastic strain. Note that the actual material stress-strain relationship obtained from standard material tests is required to capture the accurate effects of the stress states on the microvoid evolution and macroscopic material response for a given material. Execution of the unit cell simulations involves two main steps: (1) applying the periodic and homogeneous boundary conditions, and (2) applying the prescribed macroscopic stress states. To impose the corresponding displacements U1, U2, U3, and UT, a set of generalized forces P1, P2, P3, and PT are applied to the two corner nodes A1 and A2 as shown in Fig. 4(a). Nodes A1 and A2 are taken as the control nodes, and the displacements of the remaining nodes at the surfaces can be inferred based on the boundary conditions given in Eq. (8). Regarding the nodes on the faces X 1  D0 2 and X 2  D0 , the prescribed displacements of each pair of nodes are applied by using multiple linear constraint equations available in ABAQUS®. Zhu et al. [16] derived a relationship between the generalized forces, the macroscopic stresses and the prescribed displacements, which is given as:

 U P1 : P2 : P3 : PT   11  12 T D2 

 D2 D : 22 : 33 2 : 12  D3  D1

(9)

where D1, D2, and D3 are the current dimensions of the deformed unit cell. With this, the relationship between the macroscopic stresses can be determined for a given triaxiality and Lode parameter. In other words, prescribing a constant triaxiality, and Lode parameter requires enforcing proportional loads (P1, P2, P3, and PT). This is once again implemented through linear multi-points constraints (i.e., MPC subroutine) available in ABAQUS®, where the coefficients 13

of the constraint equation are related to the generalized loads. Since the proportional relations of the generalized loads are dependent on the current displacements and dimensions, MPC imposed on the model are not linear constraints for which the coefficients must be constant. To this end, a user-defined subroutine *MPC is used to update the coefficients at each analysis increment during the deformation. More details about the implementations of the user-defined MPC subroutine can be found in the literature [16]. 3.3 Numerical results Most previous micromechanical models for ductile fracture only considered the unit cell models subjected to triaxial stress states [40,54,55]. The significance of shear stress on void growth and coalescence at high triaxialities has been identified in the refs. [15,16,18]. In addition to triaxiality, and Lode parameter, shear stress can also be an independent parameter that influences the evolution of microvoid, particularly at intermediate and low triaxialities. As the mode of void growth and coalescence at these triaxiality regimes is relevant to shear-related stress states, it is necessary to incorporate a macroscopic shear stress component to the micromechanical modeling of the unit cell. Fig. 5 illustrates the effective plastic strain distribution at macroscopic strain Ep = 0.15 for the cases with a given T = 1, L = 0 and different shear ratios ρ2 = 0, 0.2, 0.4, and 0.6. Comparison among the models with and without shear stress component shows that there is a change in the deformation mode of the unit cell. In the case of ρ2 = 0, the deformation mode is symmetric, and the boundaries remain plane. While for the cases of ρ2 ≠ 0, deformation exhibits a combination of anti-symmetric and symmetric mode. The non-uniform part of deformation increases with ρ2, which subsequently influences the localization mode of the cell. As indicated by the contours of the effective plastic strain in Fig. 5, localization of the cell strongly depends on the value of ρ2. With the same Ep = 0.15, strain concentration, which occurs at a band across the void, becomes more prominent as ρ2 increases. As suggested by many past studies [15,16], while the localization in the unit cell subjected to a triaxial stress state with ρ2 = 0 is generally uniaxial, localization in the case of ρ2 ≠ 0 is biaxial. Because localization in the unit cell is widely recognized as the indicator of void coalescence and failure of ductile material, the onset of localization is assumed as a failure of the unit cell 14

in this study. Another important effect of shear stress is on the evolution of microvoid. At a given triaxiality and Lode parameter, the rotation and elongation of the void becomes more prominent with an increase in the value of ρ2. As investigated by Tvergaard [13,14] and Zhu et al. [16], the void subjected to intensive shearing ultimately collapses into a micro-crack inclined in the X1-X3 plane, ultimately leading the macroscopic cracks. This results in a shearing type of void coalescence. In conclusion, the shear stress influences the deformation mode of the cell and the void, resulting in distinctive modes of localization, void coalescence, and material failure. 3.4 Effect of shear stress Fig. 6 illustrates the macroscopic stress-plastic strain response of the unit cell with different shear stress ratios ρ2 = 0, 0.2, 0.4, and 0.6. Shear stress significantly influences the unit cell response at the stages before and after the peak macroscopic stress. The reasons for the differences of response among four cases shown in Fig. 6 include: (1) the change in the shear stress ratio ρ2 leads to different triaxial stress components, even for a given T = 1, and L = 0; (2) as discussed previously, different magnitudes of the shear stress component results in different changes in the volumes of the void and the cell; (3) the localization and void coalescence modes, strongly depend on the shear stress ratio in the cases of T = 1, and L = 0, significantly affecting the softening behavior of the unit cell. The box symbol on each curve in Fig. 6 indicates the onset of localization, i.e., the failure of the unit cell. For the case without the shear stress, the failure point coincides with the point where the macroscopic stress drops abruptly. For the remaining three cases where no abrupt change in the macroscopic stress is observed, the failure points are determined as the instant when deformation in the X1 direction stops (localization initiates). This is because the initiation of void coalescence corresponds to the shift of the uniform deformation mode to flow localization in the ligament between adjacent voids. This shift means a triaxial straining mode changing to a uniaxial one when ρ2 = 0, while to a biaxial one in the presence of shear stress. This shift of the deformation mode at the instance of failure matches those observed in earlier studies [15,16]. The deformed unit cell at failure for each case is also provided in Fig. 6. By observing the deformed unit cell, a transition 15

of void growth, and coalescence mode appears from internal necking to shearing with increasing ρ2. The macroscopic plastic strains to failure vary among different cases due to different macroscopic responses and void evolution modes, implying a significant influence of the shear stress component on the fracture locus. The influence of shear stress on the failure of the unit cell can be quantitatively evaluated by studying its influence on macroscopic plastic strains to failure E fp . As shown in Fig. 7, the strain to failure decreases with an increase in the shear stress ratio for ρ2 ≤ 0.35 and increases with an increase in the shear stress ratio for 0.35 < ρ2 ≤ 0.7. Note that the maximum shear stress ratio (ρ2max) is dependent on the parameters T and L, and ρ2max is equal to 0.7 for the given T = 1 and L = 0. The variation of the strain to failure with respect to ρ2 shown in Fig. 7 is similar to the finding reported by Bomarito and Warner [15]. The lowest strain to failure appears at ρ2 = 0.35 for the given T = 1 and L = 0, which is even lower than that at ρ2 = 0. In the case of a given triaxiality and Lode parameter, the unit cell subjected to a triaxial stress state without the shear component may not present the critical (lowest) strain to fracture. However, this section only studies limited cases with a given set of material properties, triaxiality, and Lode parameter. More in-depth studies are required to fully investigate the effects of multi-shear stress components on the evolution of microvoid at different triaxialities, and Lode parameters and for various matrix materials. Nevertheless, the results of the present study emphasize the necessity for the development of fracture locus while considering a triaxial stress state associated with shear stress components. More importantly, the shear stress ratio with the minimum E fp , defined as the critical magnitude 2c , needs to be quantitatively determined for every given T and L before constructing the fracture locus. The critical shear stress ratio 2c , for instance, is about ρ2max/2 for the present case with T = 1 and L = 0. 3.5 Construction of fracture locus Through micromechanical modeling, the strains to fracture are obtained for the cases with various T and L values, and the corresponding critical shear stress ratio 2c . By curve fitting, the model parameters are evaluated as L=-1 = 2.15, L=-1 = 0.18, L=-1 = -1.90, and the 16

correction factors are evaluated as k(L = 0) = 0.76, k(L = 0) = k(L = 0) = 1.00, k(L = 1) = 0.46, k(L = 1) = 1.19, k(L = 1) = 0.85. Besides this, the transition triaxiality in the present case is evaluated to be 0.70. The constructed fracture locus is illustrated in Fig. 8. These results are consistent with the findings by Kiran and Khandelwal [40] with respect to two main aspects: (1) the plastic strain to fracture monotonically decreases with increasing triaxiality only when T > T0 (T0 = 0.7 for this case), while the trend is opposite for T < T0. As previously discussed, the main reason for these differences is the switch in the void growth and coalescence modes at different triaxiality regimes, and (2) in the high triaxiality regime i.e., T > T0, the effects of Lode parameter and shear stress ratio on fracture locus are negligible. Stress triaxiality dominates the evolution of microvoid and the response of the unit cell over the other two parameters in this range of triaxiality. In this study, a combination of micromechanical studies, and experimental tests are used to calibrate the fracture loci. Results of the experimental studies are provided next. 4

Experimental studies

4.1 Test specimens Axisymmetric smooth bars and shear-tension plates were tested in this study. All the specimens were made from Chinese Q460 high-strength steels, which are increasingly used in steel structures in China. The geometry of all the specimens is provided in Fig. 9, and the dimensions of pure shear, and shear-tension specimens are given in Table 1. The designations of the specimens are as follows: TR denotes the axisymmetric round specimen under uniaxial tension; TGR denotes the axisymmetric notched specimen under uniaxial tension; PS represents the plate used to achieve macroscopic pure shear loading conditions; ST refers to the plate used to achieve combined tension and shear loading conditions. Among these specimens, the axisymmetric round specimen, designed according to ASTM E8/E8M-13a [56], was used to measure the mechanical properties of Q460 high-strength steels. Different magnitudes of stress triaxialities can be achieved in two axisymmetric specimens owing to the different sizes of notches. The different PS and ST specimens were designed to achieve 17

different stress triaxialities and Lode parameters at fracture initiation. 4.2 Test setup The tests were performed using an MTS 810 system, and the strain was measured using contact extensometers (see Fig. 10). The loading protocols were selected in accordance with ASTM E8/E8M-13a (ASTM 2013). TR was loaded at a strain rate of 0.015/min before yielding and 0.05/min after yielding. The notched round bars were loaded at a rate of 0.004/min. The pure shear and combined tension and shear specimens were loaded at a rate of 0.01/min. The elongations of axisymmetric bars and shear plates were measured over a gauge length of 25 mm, and 50 mm, respectively. 4.3 Test results The load at yield, the peak load, the load at fracture and the corresponding displacements measured from tests are given in Table 2. Note that ductile fracture is considered to initiate at the point where load carrying capacity drops abruptly caused by a fracture. The ductility is calculated as the ratio between the elongation to fracture and the gauge length. The results from the tests on TR are used to calculate the yield strength and nominal ultimate strength for Chinese Q460 steel. Despite TGR1 and TGR2 having identical critical cross sectional areas, they exhibitted different load-displacement responses and ductility. The two notched bars have different geometries and hence will result in different stress states at the critical cross-section (e.g., stress triaxiality), which in turn significantly influences the plastic flow and fracture behavior of each TGR specimen. 5

Finite element simulations of test specimens

5.1 FE models Finite element simulations of test specimens are performed using ABAQUS® to obtain the stress and strain fields. Taking advantage of geometrical symmetry, round smooth and notched specimens are modeled as axisymmetric models using the bilinear solid element 18

CAX4. Specimens PS and STs are modeled using the three-dimensional C3D8 solid elements. Mesh sensitivity studies are carried out by comparing the load-displacement responses and the local plastic strain quantities of the specimens with different FE discretizations, and are presented in Fig. 11. Two representative specimens, i.e., TR2 and PS, are used in the sensitivity studies. Six different mesh sizes (i.e., 0.8 mm, 0.6 mm, 0.4 mm, 0.2 mm, 0.1 mm, 0.05 mm) are considered for the FE discretization in the critical region of each specimen. The results presented in Fig. 11(a) indicate that the load-displacement curves for the axisymmetric specimen converge when the element size in the critical region is smaller than 0.2 mm. From Fig. 11 (b), the global load-displacement response of the pure shear specimen is insensitive to the mesh refinement. To examine the sensitivity of the local mechanical responses to the numerical discretization, the equivalent plastic strains at two nodes (node A at the center of TR, and node B at the center of notch surface of PS shown in Fig. 11(c)) are extracted. It is clear that the local equivalent plastic strains at both nodes converge again as the element size is reduced to less than 0.2 mm. In order to limit the computational costs without compromising the numerical accuracy, the element size in the critical regions is chosen as 0.2 mm while 1 mm elements are used for the remaining regions for all the FE models. The representative FE models are provided in Fig. 12. 5.2 Calibration of true stress and strain curves J2 plasticity associated with isotropic hardening was considered for the analysis. The true stress-plastic strain curve for Q460 steel was evaluated from the results of uniaxial tensile test engineering stress-strain curve for TR. The true stress-strain relationship before necking can be obtained using the engineering stress and strain quantities, which is given as:

  s 1  e    ln 1  e 

(10)

where  = true stress,  = true strain, s = engineering stress, e = engineering strain. To calibrate material properties in the post-necking regime, the modified weighted average (MWA) method developed by Jia and Kuwamura [30] along with an iterative procedure are employed to match 19

the predicted loading-displacement curve with the experimentally measured engineering stressstrain curve. For the true stress-strain relationship after necking, Jia and Kuwamura [30] have proposed two bounds (lower and upper bounds) to estimate the post-necking strain hardening properties. In this method, the lower bound is obtained by assuming that the true stress is constant at necking (necking = true stress at the onset of necking). Referring to the upper bound, the rate of strain hardening at a given stress and strain is considered equal to that at necking initiation, i.e., d / d = necking. In addition to this, WMA also assumes that the rate of strain hardening remains constant after necking. A linear interpolation of the rate of strain hardening between the upper and lower bounds is used to estimate the actual hardening modulus after necking. The post-necking true stress-true strain relationship employed in the MWA method can be expressed as:

   necking  w necking     necking 

(11)

where necking = true strain at the onset of necking, w is the weight factor of the MWA method. Numerical simulations as well as the iterative procedure are conducted to determine the optimal value of the parameter w, which provides the best fitting of the post-necking load-displacement curves of TR. Comparison of the load-displacement responses among test and numerical modeling using different values of the parameter w is shown in Fig. 13(a). The weight factor w is calibrated as 0.55 using the aforementioned approach. The entire calibrated true stressplastic strain curve along with the engineering stress-plastic strain curve are shown in Fig. 13(b). 5.3 Numerical results Fig. 14 illustrates the evolution of triaxiality, and Lode parameter at the estimated location of fracture initiation in each specimen calculated using FE simulations. The location of fracture initiation is assessed based on the distribution and evolution of the stress and strain over the critical cross-section. Referring to the axisymmetric smooth specimens (i.e., TR, TGR1, and TGR2), it is found that the stress triaxiality is much larger at the center of the bar when compared to the surface of the bar, and the gradient of triaxiality over the entire cross section 20

is relatively sharp. According to Kanvinde and Deierlein [32] and Zhu et al. [39], the location with maximum stress triaxiality is likely the site of fracture initiation in a circumferentially smooth specimen. In a shear-type specimen, however, the equivalent plastic strain is dominated over the stress parameters (e.g., stress triaxiality, Lode parameter, and shear stress ratio) in evaluating fracture initiation. Through numerical simulations of the PS/ ST specimens, concentration of plastic strain in the central region of the critical cross-section of each specimen is observed, where fracture initiation is likely to occur. More detailed approaches for estimation of fracture initiation and the subsequent verification can be found in the literature [39]. As seen from Fig. 14(a), stress triaxialities in Specimens PS, and STs vary within a limited range of 0.4 and 0.6. For the TR specimen, the triaxiality is initially constant at 0.33 and gradually increases to 0.68 at fracture initiation. The triaxiality varies within 0.7 and 1 for TGR1 and within 0.5 and 0.8 for TGR2. Fig. 14(b) shows the evolution of the Lode parameter for each specimen. For Specimens PS, and STs, Lode parameter ranges from -0.75 to 0.12, whereas for the threeround bars, Lode parameter is constant at -1 throughout the entire loading history. 6

Model calibration and validation

6.1 Calibration procedure The previously presented ductile fracture model has three groups of parameters to be calibrated, including (1) the parameters of the ductile fracture initiation criterion (i.e., L=-1,

L=-1, and L=-1), (2) the parameters for damage evolution rule (i.e., , and

p

uf

), and (3) the

Lode parameter correction parameters (i.e., k, k, and k) and the transition triaxiality T0. The first two groups of parameters are evaluated based on the load-displacement curves obtained from tests and the continuum stress and strain fields calculated using FE simulations. The group of the correction parameters is determined from micromechanical analyses. Once ductile fracture initiates at a given integration point in an FE model, damage evolution rule is activated. The procedure for model calibration and validation is summarized as follows (1) firstly, the location and instance of fracture initiation in each axisymmetric tension specimen is estimated. Determination of the instant of fracture initiation in the axisymmetric 21

specimen is relatively straightforward, which is commonly based on the experimental loaddisplacement curve. As supported by many studies [32,39,53], the instant at which the loading carrying capacity drops abruptly is considered the instance of ductile fracture initiation. As discussed previously, the location of fracture initiation is estimated by analyzing the stress and strain distributions over the cross-section. Note that the the notch radius of each axisymmetric notched specimen in the present study is relatively large as compared to the cross-section radius. This results in a high gradient of stress triaxiality and a low gradient of strain over the cross section. In such a situation of the stress/ strain gradients, the location of fracture initiation is likely the site with peak stress triaxiality (at the center of each round bar). With the available displacement (from experiments) and evolution of stress and strain fields until fracture (from numerical simulations of specimens), the plastic strain to fracture initiation for each specimen can be calculated by satisfying the criterion provided in Eq. (3), which can also be expressed as a function of the parameters L=-1, L=-1, and L=-1. Repeating the procedure for all the three axisymmetric specimens, i.e., TR, TGR1, and TGR2, establishes a system of nonlinear equations, the solutions to which are the values of parameters L=-1, L=-1, and L=-1. (2) the two damage evolution rule parameters (, u fp ) are calibrated using an iterative approach. The three axisymmetric specimens used in step (1) are used again in this exercise. In this step, the parameters are evaluated to accurately match the sudden loss of load-carrying capacity of specimens after fracture initiation. According to the studies by Kiran and Khandelwal [40], the parameter  is in the order of 1 ~ 10. The parameter u fp , which should be greater than l fp , is about l fp ~ 2l fp . An example regarding Specimen TR is provided in Fig. 15 to illustrate the calibration results of the damage evolution parameters. The considered values for  are in the estimated range from 1 to 10, and values for u fp are ranging from l fp to 2l fp . As shown in Fig. 15, three different values for each of the parameters (,

u fp ) are employed to calculate the load-displacement curves for TR. The parameters that result in a good fit of the load-displacement curve after several iterations are used as damage evolution parameters. (3) by employing the calibrated plastic-hardening curve for the matrix material of the unit 22

cell, the Lode parameter correction parameters for L = 0, 1, and the transition triaxiality T0 are evaluated through micromechanical analyses. Through micromechanical analysis of the unit cell, macroscopic strain to failure is obtained for a given set of stress triaxiality, Lode parameter and shear stress ratio (0 ≤ T ≤ 1 in this paper, L = -1, 0 ,1, and ρ2 = ρ2max/2). Repeating micromechanical analyses for vaious sets of stress parameters allows for the construction of the computational fracture loci. The computational fracture locus for each given L (L = -1, 0, and 1) is established by fitting the discrete data points (T, E fp ) in the plane of macroscopic strain to failure and stress triaxiality. Comparison of the computational fracture loci for L = -1, 0 with that for L = -1 is used to determine the correction parameters and the transition triaxiality. (4) as all the ductile fracture parameters are calibrated based on the estimations of fracture initiation in three axisymmetric specimens, further studies are required to verify the validity of the estimated location and instance of fracture initiation. In other words, fracture initiation is evaluated using the calibrated model, ensuring that predicted fracture locus agrees with the location and instant of fracture initiation assumed previously in step (1). Moreover, comparison between the predicted and the experimentally global load-displacement responses of each axisymmetric specimen is needed to further examine the calibrated damage evolution rule. (5) using all the three groups of the calibrated parameters, the model is then validated using the shear specimens (PS and STs). The focus of validation studies is not only on the fracture locus but also on the post-fracture initiation response with respect to the global loaddisplacement curves. Assuming that fracture initiates at the center of each axisymmetric specimen, the calibration procedure results in the model parameters as L=-1 = -0.45, L=-1 = 0.38 and L=-1 = -1.55, the damage evolution parameters as u fp = 0.1 mm and  = 6, the correction parameters as k(L = 0) = 0.7, k(L = 0) = k(L = 0) = 1, k(L = 1) = 0.61, k(L = 1) = 1.32, k(L = 1) = 0.86, and the transition triaxiality T0 = 0.72. Fig. 16(a) compares the load-displacement curves for the two axisymmetric notched specimens between numerical modeling and experiments. It can be observed that the predicted load-displacement response before fracture initiation, instant of 23

fracture initiation, and response after fracture initiation of each specimen agree well with experiments. The results indicate accurate FE modeling of the axisymmetric specimens and good estimations of the model parameters. 6.2 Validation of the calibrated model by shear specimens Fig. 16(b) compares the load-displacement curves of Specimens PS and STs between experiments and FE simulations. The predicted load-displacement response agrees well with tests until the maximum loading point. In general, the calibrated model overestimates the loadcarrying capacity and ductility of each specimen at the later stage of loading history. The possible reason is that the internal damage starts to accumulate even before ductile fracture initiation, which contributes to early stress degradation and cannot be simulated by the proposed model. As indicated by the open rectangles in Fig. 16(b), the predicted fracture initiation in PS and ST1 matches well with experiments. The calibrated model overestimates the displacements to fracture of ST2, ST3, and ST4 by 17%, 11%, and 18.5%, respectively. For the response after fracture initiation, the predicted results by the proposed model show the sudden drops of loading carrying capacity similar to the experimental results. 7

Conclusions This study employs a micromechanics framework in order to account for the influence of

shear component along with triaxiality, and Lode parameter for predicting the ductile fracture in structural steels at intermediate, and low-stress triaxialities. The important conclusions of this study are as follows 1. In this study, micromechanical analysis of the unit cell considers three normal stresses and a shear stress component, to investigate the combined effects of stress triaxiality, Lode parameter and macroscopic shear stress on the evolution of microvoid. The results indicate that in addition to triaxiality and Lode parameter, the macroscopic shear stress can be an independent parameter that influences the void growth process, particularly for sheardominated ductile fracture. The strains to failure obtained from shear stress component24

incorporated micromechanical analyses are considerably lower than the analyses conducted neglecting the effect of shear stress component. 2. As a part of the model calibration procedure, micromechanical analyses serve to determine the transition triaxiality and the Lode parameter correction factors. As indicated by the results of the micromechanical analyses, the shear component significantly affects the Lode parameter correction factors and the dependence of fracture locus on triaxiality. Consequently, the additional shear component lowers the predicted fracture strains when compared to the KK model, particularly in the range of intermediate and low triaxialities. This is supported by the fact that the effects of the Lode parameter and the shear component on microvoid evolution are more significant at intermediate and low triaxialities than at high triaxiality. In this case, incorporating the shear stress component may be of great importance for an accurate prediction of ductile fracture initiation at intermediate and low triaxialities. 3. The critical shear stress ratio 2c that governs the minimum strain to fracture should be quantitatively determined for every given combination of T and L, which is computationally expensive. It is impractical to obtain the value of 2c for every combination of T and L for the implementation of the model. Based on several micromechanical analyses, and past studies, a magnitude of ρ2max/2 is taken as a good estimator of 2c . By making this assumption for the value of 2c , a balance between the computational cost, and the predictive accuracy of the proposed model is achieved. 4. Validation studies provided in this study show that the proposed framework can predict the ductile fracture accurately for the shear dominated fractures that occur at intermediate and low triaxialities. Practical applications of the proposed model in the field of structural engineering include predicting fracture in tearing failure of plates, and bearing failure of bolts in bolted connections, and web shearing fracture in shear links in eccentrically braced frames. Acknowledgments The authors acknowledge the funding supports of National Key Research and Development Plan (Grant 2017YFC1500700), and National Natural Science Foundation of 25

China (Grant Nos. 51908416, 51778462 and 51878029). This work is also sponsored by Shanghai Pujiang Program (Grant 19PJ1409500). Any opinions, findings, and conclusions or recommendations provided in this paper are those of the authors and do not necessarily reflect the views of the sponsors. References [1] [2] [3] [4] [5]

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Captions to Tables and Figures Table 1. Dimensions of the PS and ST specimens (units: mm) Table 2. Summary of experimental results Fig. 1. Illustration of two different microvoid evolution modes Fig. 2. Comparison of fracture loci predicted by VGM, B-W, and K-K models Fig. 3. Illustration of the (a) geometry, and (b) stress states in the unit cell model Fig. 4. Finite element model of the unit cell, (a) mesh refinements of the unit cell, and (b) macroscopic stress versus macroscopic plastic strain curves three for mesh refinements Fig. 5. Contours of the effective plastic strain for the deformed unit cells under different shear ratios (T = 1, L = 0) Fig. 6. Macroscopic stress versus macroscopic plastic strain curves for cases with a given T = 1, L = 0 and different shear ratios (symbols denote the failure of the unit cell) Fig. 7. Macroscopic plastic strains to failure for cases under a given T = 1, L = 0 and different shear ratios Fig. 8. Fracture locus surface developed through micromechanical analyses Fig. 9. Geometries of (a) TR, (b) TGR (TGR1: r0 = 2 mm; TGR2: r0 = 4 mm), (c) PS, (d) ST, and (e) pictures of fracture specimens Fig. 10. Typical test setup for PS/ ST specimens Fig. 11. Mesh sensitivity analysis in terms of mechanical responses for TR and PS specimens, (a) load-displacement curves for TR, (b) load-displacement curves for PS, and (c) evolution of the equivalent plastic strain with mesh refinement Fig. 12. Typical FE mesh in the critical regions for (a) axisymmetric bar, and (b) shear plate Fig. 13. Result of calibration of the true stress-plastic strain curve for Chinese Q460 steel, (a) calibration of the weight factor, (b) true and engineering stress vs. plastic strain curves Fig. 14. Evolution of (a) stress triaxiality, and (b) Lode parameter at the location of fracture initiation Fig. 15. Comparison of the predicted post-fracture initiation response for TR using different damage evolution parameters, (a) effect of the parameter ω, and (b) effect of the parameter u fp Fig. 16. Comparison of load-displacement curves obtained from experiments and FE analyses (circle and box symbols denote ductile fracture initiation), (a) axisymmetric smoothnotched specimens, and (b) shear-type specimens

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Table 1. Dimensions of the PS and ST specimens (units: mm) Specimen PS ST1 ST2 ST3 ST4

b1 30 27 32.6 35 39.1

b2

70

b3

10

b4 20 21.6 22.3 21.2 14.6

b5 15 11.8 12.9 12.5 12.5

h1 70 72.5 74.5 76.2 77.7

h2 25 22.4 19.9 17.4 14.3

h3 10 12.6 15.1 17.5 20.6

h4

15

h5

70

s0

2

s1

10

s2 s3 s2+s3=20 6.7 4.6 10 4.5 4.8

r1

5

r2

2.5

r3



t

1.3

0 10 20 30 45

6.4

Table 2. Summary of experimental results Specimen Py (kN) Pu (kN) Pf (kN) Ductility (%) y (mm) u (mm) f (mm) TR 12.1 0.05 18.1 3.4 10.6 6.2 24.9% TGR1 5.3 0.03 7.3 0.3 5.2 0.7 3% TGR2 4.9 0.02 7.1 0.4 4.1 1.1 4.3% PS 8.1 0.13 13.1 2.3 11.8 2.8 5.6% ST1 7.4 0.13 13.7 3.2 11.7 3.4 6.8% ST2 8.3 0.12 13.4 1.7 12.2 2 4% ST3 9.8 0.14 14.7 1.2 13.2 1.8 3.5% ST4 11.6 0.11 16.3 0.7 9.6 1.2 2.4% Note that the measured quantities denote: Py = load at yield; y = elongation at yield; Pu = peak load; u = elongation at peak load; Pf = load at rupture; f = elongation to fracture. The mechanical properties of Q460 steel are obtained from the tests on TR: Yield strength = 428 MPa; nominal ultimate strength = 640 MPa.

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Fig. 1. Illustration of two different microvoid evolution modes 0.8 VGM B-W K-K pf=0.7exp(-1.5T)

0.6

pf 0.4

0.2

pf=0.2T-1.34

f =0.15/T p

f =1.9T -0.18T+0.21 p

2

f =T +0.12 p

0.0 0.0

0.2

6

0.4 0.6 Stress triaxiality, T

0.8

1.0

Fig. 2. Comparison of fracture loci predicted by VGM, B-W, and K-K models

Fig. 3. Illustration of the (a) geometry, and (b) stress states in the unit cell model

31

(b)1.6

e0

1.2

0.8

0.4 Coarse mesh Medium mesh Fine mesh

0.0 0.00

0.05

0.10

0.15 Ep

0.20

0.25

0.30

Fig. 4. Finite element model of the unit cell, (a) mesh refinements of the unit cell, and (b) macroscopic stress versus macroscopic plastic strain curves three for mesh refinements 1.4 1.3 1.2 1.0 0.9 0.8 0.7 0.6 0.5 0.3 0.2 0.1 0.0

2=0

2=0.2

2=0.4

2=0.6

Fig. 5. Contours of the effective plastic strain for the deformed unit cells under different shear ratios (T = 1, L = 0) 1.6 (1) (2)

1.2

(4)

e0

(3)

0.8

0.4

0.0 0.0

0.1

0.2 Ep

2=0

2=0.2

2=0.4

2=0.6

0.3

0.4

Fig. 6. Macroscopic stress versus macroscopic plastic strain curves for cases with a given T = 1, L = 0 and different shear ratios (symbols denote the failure of the unit cell)

32

0.5 0.4 0.3 Epf

0.2 0.1 0.0 0.0

0.2 0.4 0.6 Shear stress ratio, 2

0.8

Fig. 7. Macroscopic plastic strains to failure for cases under a given T = 1, L = 0 and different shear ratios

Fig. 8. Fracture locus surface developed through micromechanical analyses

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Fig. 9. Geometries of (a) TR, (b) TGR (TGR1: r0 = 2 mm; TGR2: r0 = 4 mm), (c) PS, (d) ST, and (e) pictures of fracture specimens

Fig. 10. Typical test setup for PS/ ST specimens

34

(a) 8

(b)20 16 Load (kN)

Load (kN)

6

12

Coincide

4 0.05 mm 0.1 mm 0.2 mm 0.4 mm 0.6 mm 0.8 mm

2

0 0.0

All six curves coincident

8 0.05 mm 0.1 mm 0.2 mm 0.4 mm 0.6 mm 0.8 mm

4

0 0.3

1.2

0.9 0.6 Elongation (mm)

0

1

2 Elongation (mm)

3

4

Equivalent plastic strain

(c)0.6

Center node A

0.4

0.2

Sufrace node B

PS specimen TR specimen

0.0 0.0

0.2

0.4 0.6 Element size (mm)

0.8

1.0

Fig. 11. Mesh sensitivity analysis in terms of mechanical responses for TR and PS specimens, (a) load-displacement curves for TR, (b) load-displacement curves for PS, and (c) evolution of the equivalent plastic strain with mesh refinement

Fig. 12. Typical FE mesh in the critical regions for (a) axisymmetric bar, and (b) shear plate

35

(a)20

(b)1200

16

12

True stress vs. plastic strain

stress /MPa

Load (kN)

900

8

600 Engineering stress vs. plastic strain

Test result w = 0.1 w = 0.3 w = 0.55 w = 0.7

4

300

0 0

2

4 Displacement (mm)

6

0 0.0

8

0.2

0.4 0.6 Plastic strain

0.8

1.0

Fig. 13. Result of calibration of the true stress-plastic strain curve for Chinese Q460 steel, (a) calibration of the weight factor, (b) true and engineering stress vs. plastic strain curves (b) 0.8

(a)0.8

PS ST1 ST2 ST3 ST4 TR TGR1 TGR2

0.6

Plastic strain

Plastic strain

0.6

PS ST1 ST2 ST3 ST4 TR TGR1 TGR2

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4 0.6 Stress triaxiality, T

0.8

0.0 -1.5

1.0

-1.0

-0.5 0.0 Lode parameter, L

0.5

1.0

Fig. 14. Evolution of (a) stress triaxiality, and (b) Lode parameter at the location of fracture initiation (b)20

16

16 16

12

Load (kN)

Load (kN)

(a)20

14

8

12

16

12

14

8

12

10

10 5.5

4

6.0

6.5

Test =1 =6  = 10 Modeling W/O damage

0 0

1

2

3 4 5 Elongation (mm)

6

5.5

4

6.5

upf = 0.1 mm

upf = 0.15 mm

upf = 0.2 mm

Modeling W/O damage

0 7

6.0

Test

0

1

2

3 4 5 Elongation (mm)

6

7

Fig. 15. Comparison of the predicted post-fracture initiation response for TR using different damage evolution parameters, (a) effect of the parameter ω, and (b) effect of the parameter u fp

36

(b)18

(a) 8

15 6

Load (kN)

Load (kN)

12 4

2

0 0.0

Test- TGR1 FE- TGR1 Test- TGR2 FE- TGR2

Test- PS FE- PS Test- ST1 FE- ST1 Test- ST2 FE- ST2 Test- ST3 FE- ST3 Test- ST4 FE- ST4

9 6 3 0

0.3

0.6 0.9 Elongation (mm)

1.2

0

1

2 Elongation (mm)

3

4

Fig. 16. Comparison of load-displacement curves obtained from experiments and FE analyses (circle and box symbols denote ductile fracture initiation), (a) axisymmetric smooth-notched specimens, and (b) shear-type specimens

37

he influence of shear stress on the micro-mechanisms of ductile fracture is identified.

ls.

38

Conflict of interest statement I would like to declare on behalf of my co-authors that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “A Modified Micromechanics Framework to Predict Shear Involved Ductile Fracture in Structural Steels at Intermediate and Low-Stress Triaxialities”.

Corresponding author: Zuanfeng Pan E-mail address: [email protected]

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