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Investigation and prediction of tearing failure during extrusion based on a modified shear damage model
Accepted Manuscript
Investigation and prediction of tearing failure during extrusion based on a modified shear damage model P.J. Zhao , Z.H. Chen , C.F. Dong PII: DOI: Reference:
S0167-6636(16)30533-6 10.1016/j.mechmat.2017.05.008 MECMAT 2741
To appear in:
Mechanics of Materials
Received date: Revised date: Accepted date:
30 November 2016 26 February 2017 24 May 2017
Please cite this article as: P.J. Zhao , Z.H. Chen , C.F. Dong , Investigation and prediction of tearing failure during extrusion based on a modified shear damage model, Mechanics of Materials (2017), doi: 10.1016/j.mechmat.2017.05.008
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Metallurgical microstructure observed by OM and SEM were analyzed to investigate the mechanism of tearing failure. The new shear GTN damage model was used to predict material damage and tearing fracture during extrusion. Cohesive zone model was applied to the RVE to simulate the de-bonding of the grain interfaces Material evolution in severe shear zone of extrusion process was studied. The deformation and propagation mechanism of shear band during extrusion were discussed.
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Investigation and prediction of tearing failure during extrusion based on a modified shear damage model P.J. Zhao a,*,1Z.H. Chen a, and C.F. Dong b
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a School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China b Institute for Advanced Materials and Technology, University of Science and Technology Beijing, Beijing 100083, China
Abstract: In this study, a recent modified shear GTN model suitable for various triaxialities was implemented and validated, which combines the porous plasticity model with a damage mechanics concept. The original Gurson model was modified by coupling the shear damage
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and void volume fraction parameters into the yield surface. The stress update algorithm was developed via a user-material subroutine. With the help of representative volume elements (RVEs), a connection between the formation of macroscopic cracks and the microstructure was established. The modified GTN model was applied to predict the damage and tearing fracture. Additionally, the cohesive zone model was applied to the RVE to simulate the de-
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bonding of the grain interfaces from a microscopic perspective. Metallurgical inspection by means of optical and scanning electron microscopy was performed to explore the failure
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mechanism and validate the numerical computation results. Furthermore, the deformation of the shear band and the damage propagation mechanism during the extrusion process were
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discussed.
Keywords: GTN model; shear damage; representative volume element; cohesive model;
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extrusion
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1. Introduction
Extrusion processes are commonly used to form metallic materials because they
typically result in a good dimensional accuracy, a low material consumption and a high productivity. Due to the improved quality of parts and the economic advantages offered by this advanced processing technique, it has been widely used to produce a variety of different parts, e.g., automobile components, camera components, and other small items used in 1*
Corresponding author. E-mail address: [email protected] (P.J. Zhao). 2
ACCEPTED MANUSCRIPT measuring equipment. A number of interesting extrusion experiments and numerical simulations of extrusion processes have been performed. For instance, Che et al. [1] investigated the plastic deformation and damage behavior of Al alloys during the extrusion process under crash loading conditions. Liang et al. [2] analyzed the extrusion of metallic materials at evaluated temperatures by developing a constitutive equation with strain compensation. Jung et al. [3] applied the finite element (FE) method to study the plastic deformation behavior during the
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tandem process of simple shear extrusion and twist extrusion. Hassan et al. [4] investigated the texture evolution during a simple shear extrusion process using a crystal plasticity finite element model, and Jie et al. [5] studied the effects of hot extrusion on the microstructure evolution and mechanical properties. Since the typical failure modes at an extrusion surface (as shown in Fig. 1) are ductile fracture and shear tearing, it is vital to explore the failure
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mechanism in detail and develop a methodology to predict the optimum manufacturing
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technique and parameters of the extrusion process.
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Fig. 1. Example of a fracture surface of an extruded workpiece. The ductile failure mechanism of metallic materials generally includes three phases, i.e., micro-void nucleation, micro-void growth and coalescence. In order to describe the damage
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and failure behavior of ductile materials, Gurson [6] first proposed the classical spherical void growth model for elastic-plastic metallic materials. Tvergaard and Needleman [7] introduced three corrective parameters into the damage model to explain the effects of material strain hardening and void interaction, resulting in the so-called GTN model. The GTN model allows for a good prediction of ductile fracture and thus has attracted the attention of numerous research groups, especially in recent years, such as [8-10]. Although the GTN model has achieved tremendous success, there still exist many limitations. For instance, it fails to accurately predict the damage behavior under sheardominated loading conditions, because it is unsuitable for describing void growth and 3
ACCEPTED MANUSCRIPT damage in case of low triaxiality. A modified GTN model was therefore proposed by Nahshon and Hutchinson [11] by introducing a Lode angle and the stress triaxiality parameter. Xue [12] also proposed a damage model based on the solution for void coalescence in the shear band. Mohamed et al. [13] predicted the ductile damage under shear dominated loads based on an advanced Gurson model. Quan et al. [14] applied a GTN-type model to analyze crack initiation and crack propagation under cold rolling conditions. Mohamed et al. [15] studied the damage evolution and the effect of punch-die clearance to
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validate the predictive capacity of the modified model. Although these modified models can be considered great improvements with respect to the prediction of the failure behavior of metallic materials, they tend to overestimate the void growth, which consequently leads to inaccurate numerical results under shear-dominated conditions. To overcome the limitations of existing models, Zhou et al. [16] proposed a new modified GTN model by combining the
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damage mechanics concept of Lemaitre [17] with the void damage model.
In recent years, the damage models combined with representative volume element (RVE) for ductile metallic material were proposed. For instance, Vitoon et al. [18] described the influence of the multiphase microstructure on the complex failure mechanism as well as
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mechanical properties by using RVE model within the framework of continuum damage mechanics. Vitoon et al. [19] studied the micro-crack formation in multiphase steel using
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RVE model and damage mechanics. Napat et al. [20] employed XFEM and damage curve derived to study the interaction between failure modes in DP steels by means of RVE. Ling et al. [21] used the RVE model to discuss the crystallographic texture and morphological
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features in the FE simulation of nanoindentation. Hosseini et al. [22] predicted stress–strain response of dual phase steel using the FE analyses of selected RVEs from SEM images.
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Ayatollahi et al. [23] investigated failure and damage evolution of three RVE models in DP steels by using experimental and numerical methods.
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In this study, the recent modified shear GTN model proposed by Zhou suitable for various types of stress triaxiality was deduced and programmed. The correlation between the material microstructure and the resulting macroscopic failure behavior was established by means of RVE. In addition, the de-bonding of the interface between metallic grains was simulated by employing a phenomenological cohesive zone model (CZM) based on a separation law. The modified GTN model and the CZM model were then applied to the extrusion process to describe the ductile failure behavior of DP600 steel, as well as crack propagation. The implementation of the constitutive equations into a numerical algorithm was achieved by developing a VUMAT subroutine. The surfaces of extruded specimens were observed via optical microscopy (OM) and scanning electron microscopy (SEM) to 4
ACCEPTED MANUSCRIPT investigate the tearing failure mechanism occurring in the shear zone. The validity of the numerical model was then verified by comparison with the experimental results.
2. Constitutive model 2.1. The modified shear GTN model
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As a micro-mechanical ductile damage model, the GTN model was used to determine the damage onset or crack initiation mechanism in metallic materials. A spherical void in the plastic matrix material was proposed by Gurson [6] to describe the yield potential for a porous plastic material. Tvergaard and Needleman [7] modified the original model by introducing two coefficients, resulting in the so-called GTN model. Nevertheless, the GTN
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model is still not able to accurately predict the damage behavior under shear-dominated loading conditions. To solve the problems, Zhou et al. [16] proposed a new modified GTN model which combined the continuum damage mechanics proposed by Lemaitre [24, 25] with micro-void damage mechanics. The modified model is typically expressed as follows: 2
(1)
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3q2 m * ) 1 D 2 2 Ds 0 2q1 f cosh ( 2 Y Y
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where Y denotes the yield stress,. m kk / 3 is the mean stress, 3 / 2 S : S is the von Mises equivalent stress of the matrix material, and S is the deviatoric stress tensor. The total damage can then be defined as
D q1 f * Ds
(2)
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where the total damage term contains an additional shear damage state variable Ds . The load carrying-capacity of the material is lost completely once the total damage D is equal to unity. The modified model will degenerate to the original GTN model in the absence of a shear load; the model will have the same form as the CDM of Lemaitre if there is only shear stress. The shear damage term can be written as n
p Ds s (3) f where n is a weakening exponential. sf denotes the failure strain under pure shear loading conditions. Ds is equal to one when p reaches sf . The softening effect is small at the early stage of plastic deformation where n is greater than one and becomes larger as the material approaches failure. The incremental expression is given by
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Ds
nDs n
sf
p
(4)
To extend the Eq. (4) to all stress states, the stress triaxiality T * was introduced and the Lode angle has been added as weight factor [26, 27] n 1
Ds , T *
nDs n
sf
p
(5)
with g 1
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and
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1 s2 s3 1 2 3 s1 s3
tan 1
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where the weight function ,T * is usually expressed as follows: T 0 g , T * g 1 k k T 0
(6)
(7)
(8)
where S k ( k 1, 2, 3) denotes the principal deviatoric stress.
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2.2. Representative volume element
(a)
(b)
Fig. 2. Illustration of (a) the 2D Voronoi tessellation and (b) the cohesive elements or interface elements which were applied to simulate the structure at the grain level In this work, 2D Voronoi tessellations were applied to simulate the grain-level structures required for describing the properties of a polycrystalline material. In order to reduce the computation time, the representative volume element (RVE) was introduced into the mechanical models as a representative part of the entire model to describe the failure 6
ACCEPTED MANUSCRIPT behavior at the micro-scale. Fig. 2(a) illustrates the 2D Voronoi tessellation with the embedded cohesive elements, which were generated using the Neper software [28]. The different colors in the RVE 2 represent the different grains. The square RVE with a size of 0.5 × 0.5 mm contains 100
grains and the mean grain size is 0.05 mm. Cohesive elements (COHAX4) were then introduced at the grain interfaces. Fig. 2(b) shows the cohesive elements in the RVE, which
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represent the potential crack zone. As a pre-process, the COPYNODE algorithm was programmed in Python to automatically embed the cohesive elements, as shown in Fig. 3. The program consisted of the following steps. Firstly, the information regarding the nodes at the grain boundaries was collected and then defined as a set. Secondly, the surface element nodes were updated to
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assure that each grain had a unique node number. Finally, the new nodes were combined with the surface element nodes and the cohesive elements were generated. Original RVE Voronoi tesselation mesh data
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Obtain the information of the surface nodes
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Copy and update the surface nodes
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Generate 2D cohesive elements with zero thickness
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Fig. 3. The flowchart of the COPYNODE program used to embed the cohesive elements into the 2D RVE.
In order to simulate the interface de-bonding between the grains, the cohesive zone
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model (CZM) was employed. The stiffness of the interface elements degenerates with increasing accumulated damage until the elements between the continuum bodies separate and fracture occurs. The traction-separation law was applied to the finite element model to simulate the decohesion of the interface elements and the crack initiation process. A localized crack nucleation criterion was used to determine the time of separation of the grain boundary. In this study, the failure criterion with the critical energy 0 was applied to the RVE, with the energy being dissipated from the cohesive elements. According to Vitoon et al. [29], the crack nucleation criterion for a cohesive interface can be expressed as follows: 7
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0 T d
(9)
0
The failure law depends on three material parameters: the cohesive energy 0 , the cohesive stress0 and the critical separation at material de-cohesion 0 . 2.3. Numerical algorithms The stress update algorithm of the constitutive equations is briefly described in this
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section. The modified model, which is suitable for various stress states, was implemented into the commercial FE software ABAQUS through developing the required user-material subroutine. The integration of the constitutive equations at the Gauss points was carried out using the backward Euler return mapping algorithm. The implementation procedure of the numerical integration algorithm was as follows:
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Step (1): Determination of the initial values of t , t , H , t t ( t = 0,, ti ) for each
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incremental time step. where t is the stress tensor, t is the strain tensor, t t is the incremental strain tensor, p and H ( =1,2,3) denotes the three different state variables , f and Ds . The equivalent p plastic strain of the matrix , the void volume fraction f and the shear damage Ds were treated as scalar internal variables. Step (2): Calculation of the total trial stress.
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It is assumed that the strain increment is completely elastic. Then, the trail elastic stress is given by
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e tet t Cijkl : t t
2 e Cijkl 2G ik jl K G ij kl 3
(10) (11)
e where Cijkl denotes the fourth-rank elastic modulus. The superscript e indicates the trail
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elastic state. Here, G E / 2(1 v) is the shear modulus, K E / 3(1 2v) is the bulk
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modulus, E is Young’s modulus and v is Poisson’s ratio. Next, the equivalent stress was calculated through
and the mean stress through
e
3 e e S :S 2
1 3 Step (3): Solving the yield function to ascertain the yield state.
(12)
me tet : I
(13)
tet e , me , H1, H 2 , H 3
(14)
e e If t t 0 , the trial stress results in elastic deformation, t t t t , and the algorithm
proceeds to Step (5) 8
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Step (4). Step (4): Calculation of the plastic deformation and correction.
1 p mp I eqp n 3
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Here, the subscript t t is ignored to simplify the notation. The flow direction can be expressed as 3S e (15) n 2 and the flow rule as
(17)
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Furthermore, the volumetric strain increment is given by 3q2 m 1 * Dm mp 3q1q2 f sinh 2 Y Y m
(16)
where denotes the plastic strain-rate multiplier, and the equivalent plastic strain increment can be written as 2 Deq eqp (18) 2 Y Then, the two following functions had to be satisfied:
Deq 0 m
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f1 Dm
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f 2 , m , H1, H 2 , H 3 0
(19) (20)
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The nonlinear equations were solved employing the Newton-Raphson iterative method according to the above expressions. When f1 and f 2 < Tolerance (equal to 1.0-8), the convergence conditions were satisfied. In this case, the algorithm proceeded to Step (5). Otherwise, the iteration process was terminated. Step (5): Updating the variables Dm , Deq , the hydrostatic stress m and the equivalent stress Dmk 1 Dmk Dm k 1 k Deq Deq Deq k 1 e k 1 m m K tDm k 1 e k 1 3GtDeq
(21)
(22)
Next, the stress tensor and the three state variables h ( = 1, 2, 3) were updated as follows: 2 tn 1 m I n (23) 3
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(24)
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mk 1Dmk 1 Deqk 1 k 1 k 1 p h1 H1 1 f Y m Dm Deq k 1 k 1 h2 H 2 f 1 f Dm I Deq n AN 1 f Y n 1 n k 1 k 1 * nDs p h3 H 3 Ds , T s M f Step (6): Beginning of the next circulation.
The detailed numeralization procedure for the stress update algorithm of the modified model is illustrated in Fig. 4. The partial derivatives of each term in the above equations are provided in the appendix. Start of VUMAT
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Known t , t , H , t t e e e Calculate t t , m ,
( me , e , H ) 0
Yes
No Initial value k 0, h 0 Calculate Dm , Deq , f1 , f 2
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Newton iterative
f1 toler and f 2 toler
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No kN No Iteration not converge stop ABAQUS
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Yes
Yes Update stress t t Calculate Dm , Deq , m , Update state variables h , H , D End of VUMAT
algorithm of the modified GTN model.
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Fig. 4. Schematic diagram illustrating the numeralization procedure for the stress update
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3.1. Extrusion experiment
Fig. 5. Photographs showing the equipment used for the extrusion experiment.
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Table 1
Chemical composition of DP600 steel (in wt%). Si
0.072
1.577
0.246
Al
P
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Mn
0.032
0.015
S
Ni
V
Cr
Cu
0.001
0.024
0.011
0.552
0.01
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The extrusion experiment was carried out on a 315 t four column hydraulic press. The
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experimental setup is shown in Fig. 5. DP600 steel with a thickness of 5 mm was used as sample material, and its chemical composition is given in Table 1. The schematic diagram
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used in this study to describe the extrusion process is shown in Fig. 6. In the figure, FG is the counter force and FR is the holder force. The cutting clearance is c 0.025 mm, i.e., 0.5% of the material thickness. The punch velocity was kept constant at 10 mm/s. The extrusion process was regarded as quasi-static so that the effect of the strain rate could be ignored. Different punching depths were used during the extrusion experiments to verify the numerical simulation results.
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Fig. 6. Schematic illustration of the extrusion process (length units in mm).
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3.2. Numerical simulation
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Fig. 7. Illustration of the finite element mesh in which the cohesive element was embedded around the local area.
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A 2D axisymmetric model was used to simulate the extrusion process. The RVE with a
2 size of 0.5×0.5 mm was embedded in the specimen to investigate the change of the
microstructure of the material during the extrusion process, as shown in Fig. 7. A dense mesh was applied to the large deformation area to improve the precision of the numerical calculation, and a coarser mesh was applied to the other areas to minimize the computational cost. The arbitrary Lagrangian Euleurian (ALE) technique was adopted to avoid the distortion of elements in those regions where severe distortion may occur. The Coulomb friction model was used to simulate the contact between the workpiece and the dies. The friction coefficient was fixed at 0.1. The dies and the punch were regarded as rigid bodies 12
ACCEPTED MANUSCRIPT and the metal sheet was defined as an elastic-plastic body. For the deformed body, 3-node linear axisymmetric triangle elements (CAX3) were used and 4-node axisymmetric cohesive elements (COHAX4) were used to model the grain interfaces. The corner radius of the die and the punch were set to be 0.1 mm. Table 2 lists all parameters used to simulate the extrusion of a DP600 steel workpiece, including the cohesive zone model parameters employed by Vitoon et al [30] and the modified GTN model parameters used by Zhao et al [31]. Just to visualize the micro-cracks
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occurred in extrusion process from a microscopic perspective, cohesive zone model parameters were directly quoted into simulation due to the same material DP600 steel and the identical stress-strain relationship. In this work, the modified GTN model and the cohesive zone model were investigated to determine their capability to accurately predict the
Table 2
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extrusion process.
Parameters of the cohesive zone model and the modified GTN model to simulate the extrusion of a DP600 steel workpiece.
[N/mm] 100
1.5
q2 1
fo
fN
0.0008
0.02
SN
N
0.1
0.2
fc
ff
sf
0.028
0.09
0.5
n 5
k 0.5
K 863
0 0.011
N 0.165
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1500
q1
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T0 [MPa]
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4. Results and discussion
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4.1. Metallographic inspection
(a)
(b)
Fig. 8. Photographs showing the specimens after the extrusion experiments at different pre13
ACCEPTED MANUSCRIPT set punching depths: (a) sectioned specimens, (b) polished and etched specimens. In order to study the microscopic surface topography of the DP600 steel specimens, interrupted extrusion experiments were performed at different pre-set punching depths. The blank specimens were cut manually to avoid the damage and thermal effects typically generated during an electrical wire cutting process. Then, all parts were polished and etched
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with Nital. Photographs of the fine-blanked specimens are shown in Fig. 8.
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Subsequently, the microstructure of the metal samples was observed by OM and SEM, and the results are shown in Fig. 9. When the sample was penetrated to 33.3% of its thickness (Fig. 9a), the shear band gradually formed. The deformation was localized in the narrow clearance zone between the die and the punch. The serious shear deformation resulted in an intensive slip and dislocation, and the particles adjacent to the die edge were found to be elongated and compressed.
When the sample was penetrated to 58.3% of its thickness (Fig. 9b), the localized shear deformation was found to have further propagated parallel to the punching direction and
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highly rotated and elongated metallic grains appeared in the shear band. The growth and coalescence of voids were observed to be limited to the clearance zone. A few micro-cracks and micro-voids were found surrounding inclusions extended along the shearing direction, which indicates that the local damage triggered the initiation of fracture. At the final stage (Fig. 9c), a macroscopic crack occurred near the die edge and the
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crack propagated at an angle of 135°. In the vicinity of the die, the tearing failure of the specimen occurred along the direction perpendicular to the principal stress due to the
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combined action of interfacial friction and axial tensile stress, as illustrated in Fig. 9d. In conclusion, the observed material damage suggests that the ductile fracture mechanism can
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be applied to describe the extrusion process.
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4.2. Numerical results
(a)
(b)
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(c)
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Fig. 10 shows the distribution of the hydrostatic stress at three different penetration depths. In order to more clearly reveal the stress distribution, the 2D contours were rotated at a small angle of 15°. In the early stage of the extrusion process, the material close to dies is compressed and the hydrostatic stress propagates along the entire large deformation region, which suppresses the formation of micro-cracks. At a penetration depth of 30–70% of the sheet thickness, severe plastic deformation was observed localized around the shearing band. The intensive shear localization contributes to the elongation of grains and the accumulation
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of micro-defects.
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Fig. 11. Evolution of the shear stress with the penetration depth.
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Fig. 11 shows the shear stress distribution in three specified elements of the workpiece. At the beginning, the shear stress monotonically increases due to the effects of strain hardening. The largest value in the entire deformation area is close to the punch edge. As the deformation proceeds, the shear stress close to the corner of the punch increases to 557 MPa, which is the peak value in the whole specimen.
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Fig. 12. Comparison between the numerical simulation and the experimental results obtained for the variation of the punching load with the penetration depth.
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A comparison between the numerical simulation and the experimental results obtained for the variation of the punching load with the penetration depth is shown in Fig. 12. The data acquired through the numerical simulation shows the same trend as the experimental data. The punching process can be divided into three stages. First, the punching load
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monotonically increases over the interval marked by A and B as a result of strain hardening. Subsequently, the punching load monotonically decreases after reaching its peak value in point B. The decline may be connected with the decrease of the shearing area and the plastic instability. The load tends to be stable over the interval marked by C to D, with the
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deformation being mainly caused by the growth of micro-cracks and microstructural effects.
(a) at 30% penetration
at 50% penetration
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at 70% penetration
(b) at 30% penetration
at 50% penetration
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at 70% penetration
Fig. 13. Distribution of the state variables for different penetration depths: (a) equivalent
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plastic strain, (b) total damage value.
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Fig. 13 illustrates the evolution of the equivalent plastic strain and the total damage as predicted by the modified GTN model for different penetration depths. As shown in Fig. 13(a), the plastic deformation mainly occurs in the narrow clearance area, and the strain hardly changes outside the shear band. The distribution of the equivalent plastic strain is
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more localized than that of the shear stress. The maximal plastic deformation occurs in the clearance area close to the die and the punch edge, where the crack formation occurs first. As shown in Fig. 13(b), at a penetration depth equal to 30% of the sheet thickness, the damage begins to accumulate and propagate along the shear deformation direction. At a
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penetration depth equal to 50% of the sheet thickness, the damage reaches its critical value, which suggests that the material in the shear band has lost its load-carrying capacity and thus the formation of cracks is triggered. Then, a crack appears at the cutting edge of the die and propagates towards the center of the thickness, which leads to fracture of the material when the penetration depth reaches 70% of the sheet thickness.
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Fig. 14. Variation of the equivalent plastic strain with the penetration depth for two
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specified elements near the die corner.
Fig. 15. Evolution of the total damage in the shear zone.
Fig. 14 illustrates the variation of the plastic strain for two specified elements close to the die corner, which indicates that the deformation near the punch corner is slightly larger than that near the die corner. Fig. 15 illustrates the evolution of the total damage in the shear band with the penetration depth. The increase of the punching depth leads to a localization of the damage because the plastic strain mainly occurs in the shear zone. The amount of damage at a local area increases to a sufficiently large value which eventually results in material failure and fracture along the punching direction. 19
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Fig. 16. Distribution of (a) the von Mises stress and (b) the damage variable in the RVE at
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the site and moment of crack initiation.
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Fig. 16 shows the distribution of the von Mises stress and the damage variable at the site and moment of crack initiation. The maximum stress occurs near the die corner and extends along the direction with a horizontal angle of 135°. As shown in Fig. 16b, because the cohesive elements are defined to be located between the continuum elements, when the damage exceeds the critical value and the elements lose their stiffness at the moment of failure, a number of micro-cracks occur, which propagate at an angle of 45°, which is consistent with the experimental results shown in Fig. 9.
Fig. 17. Schematic illustration of the ductile damage evolution mechanism.
Fig. 17 schematically illustrates the ductile damage evolution mechanism for metallic materials. It generally consists of the following four stages: When the damage exceeds a critical value, micro-voids will nucleate in the region that contains the second phase particles or inclusions. The growth of the micro-voids is caused by the further plastic deformation and high hydrostatic stress, and some new micro-voids will nucleate. Once the 20
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number of micro-voids exceeds a certain threshold value, void coalescence occurs and the
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voids will form connections, which finally leads to the fracture of material.
Fig. 18. Comparison of the results of the numerical simulation of the extrusion process with
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the experimental results.
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Fig. 18 compares the FE simulation and the experimental results calculated using the modified damage model. It can be seen that the shear plane of the fine-blanked specimen can be sectioned into four zones: the burr zone, the fracture zone, the shear zone and the rollover zone. The predicted fracture position and shear surface are a good agreement with the experimental observations.
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5. Conclusions
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The objective of this study was to implement and validate a recent modified GTN model established to simulate an extrusion process, which explains the shear-induced damage by combining the void damage model with the CDM concept. The yield function includes two damage variables, i.e., the void volume damage and the shear damage. The implementation of the algorithm was achieved by developing a VUMAT subroutine. The RVE was used to account for the influence of the microstructure and fracture mechanisms. The grain morphology is created by the Voronoi tessellation method. A 2D RVE was modeled to simulate the propagation of phenomenological damage at the grain boundaries of the DP600 steel material. The CZM model and the traction-separation law were also used to describe the de-bonding at the grain interface. The degradation of the interface was achieved using the cohesive elements. The fracture surfaces of the DP600 steel as observed by OM and SEM indicate that the elongated grains in the large deformation area propagate along the shear direction and that the crack initiates from the area near the die corner. The profile of the shear plane simulated using the FEM was found to match the experimental observations. 21
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Acknowledgments
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The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Grant No. 51671029). The related experiments were performed at the Beijing Research Institute of Mechanical and Electrical Technology.
Appendix
The stress integration algorithm of the constitutive equation is described below. First, the yield surface of the modified GTN model can be expressed as 2
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3q2 m * ) 1 D 2 2 Ds 0 2q1 f cosh ( 2 Y Y
(25)
The three internal variables are given by
H1 p , H 2 f , and H 3 Ds .
(26)
(27)
The partial derivatives can then be expressed as follows: 2 Y2
(28)
3q 3q1q2 f * sinh 2 m m Y 2 Y
(29)
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The evolution functions of the three internal variables can be written as m Dm Deq h1 H1 1 f Y p h2 H 2 1 f Dm AN 1 f Dm AN h1 n 1 n 1 h H g n D n p ng D n h 3 3 s 1 sf s sf
2 0 m
(30)
2 2 2 2 Y
(31)
3q2 m 2 9q1q22 f * cosh m2 2 Y2 2 Y 2 2 H1 H1 Y2
4 Y 4 3 H 3 Y H1 Y
2 0 H 2
22
(32) (33) (34)
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(35)
3q2 m 3q2 m 3q 3q1q2 f * H 2 cosh 2 m sinh 2 m H1 2 Y 2 Y 2 Y 2 Y
3q 3q q f * 2 1 2 m sinh 2 m m H 2 Y f 2 Y
(37)
(38)
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, f fC 1 f * 1 / q1 f C f f f , f fC F C 2 0 m H 3
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3q 3q2 m * 2 sinh 2 m H 2 H 2q1 f 2 H1 Y Y 2 Y 2 Y 3q 2 H 3q q f * H 3 1 2 2 m sinh 2 m Y 2 Y 2 Y
3q2 m 2 * f * f * f * 2q1 cosh 2q1Ds 2q1 f H 2 f f f 2 Y 2q1 f * 2 Ds 2 H 3
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(36)
(39)
(40)
(41) (42)
The partial derivatives of the internal variables can be obtained by solving Eq. (43) and Eq. (44):
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h1 h1 h h H t 1 t 1 1 1 t H1 H 2 H 3 Dm Dm h h2 h h H 1 t 2 t 2 2 t 2 t H1 H 2 H 3 Dm Dm h3 h h h H t 3 t 3 1 t 3 3 H1 H 2 H 3 Dm Dm h1 h1 h1 h1 H1 1 t t t D H H H eq Deq 1 2 3 h h2 h2 h2 H 2 2 t 1 t t t H1 H 2 H 3 Deq Deq h3 h3 h3 H 3 h3 t t 1 t H1 H 2 H 3 Deq Deq
where
h1 m h tK 2 m h tK 3 m h 3Gt 1 h 3Gt 2 h3 3Gt tK
(43)
(44)
m h1 Dm 1 f Y
(45)
h2 h1 1 f AN Dm Dm
(46)
23
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(47)
Dm h1 m 1 f Y
(48)
h2 h1 AN m m
(49)
h3 ng nn1 h1 Ds m sf m
(50)
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m Dm Deq h1 h1 Y H H1 Y H1 1 f Y2
(51)
A h2 AN h1 h AN 1 h1 N H1 H1 H1 H1
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h p N p N p 1 AN AN c20 h1 sN2 sN2 H1 n 1 h3 ng n h1 Ds H1 sf H1
D Deq h1 1 m m H 2 1 f Y 1 f
2
h1 1 f
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(55)
h3 ng nn1 h1 Ds H 2 sf H 2 h1 0 H 3 h2 0 H 3
(56)
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p
h1 Deq 1 f Y h2 h AN 1 Deq Deq
(53) (54)
h2 h1 Dm AN H 2 H 2
ng n 1 nn11 h3 Ds H 3 sf n
(52)
(57) (58)
n 1 g D 1n h sf
s
1
(59) (60) (61)
ng nn1 h1 h3 Ds Deq sf Deq
(62)
Deq h1 1 f Y
(63)
h2 h AN 1 h3 ng nn1 h1 Ds sf 24
(64) (65)
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The iterative variables can be calculated by solving 1 Dm K11 K12 R1 Deq K21 K22 R2 where R1 f1 Dm
(66)
Deq m
(67)
R2 f 2 m , , H1, H 2 , H 3 3 f1 2 2 H i Dm tK Dm m i 1 H i Dm
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K11
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3 2 2 H i Deq tK 2 m i 1 mH i Dm 3 f 2 2 H i K12 1 Dm 3Gt Deq m 2 i 1 H i Deq 3 2 2 H i Deq 3Gt m i 1 mH i Deq 3 f H i K21 2 tK Dm m i 1 H i Dm 3 f H i K22 2 3Gt Deq i 1 H i Deq
References
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Investigation and prediction of tearing failure during extrusion based on a modified shear damage model P.J. Zhao , Z.H. Chen , C.F. Dong PII: DOI: Reference:
S0167-6636(16)30533-6 10.1016/j.mechmat.2017.05.008 MECMAT 2741
To appear in:
Mechanics of Materials
Received date: Revised date: Accepted date:
30 November 2016 26 February 2017 24 May 2017
Please cite this article as: P.J. Zhao , Z.H. Chen , C.F. Dong , Investigation and prediction of tearing failure during extrusion based on a modified shear damage model, Mechanics of Materials (2017), doi: 10.1016/j.mechmat.2017.05.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Highlights
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Metallurgical microstructure observed by OM and SEM were analyzed to investigate the mechanism of tearing failure. The new shear GTN damage model was used to predict material damage and tearing fracture during extrusion. Cohesive zone model was applied to the RVE to simulate the de-bonding of the grain interfaces Material evolution in severe shear zone of extrusion process was studied. The deformation and propagation mechanism of shear band during extrusion were discussed.
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Investigation and prediction of tearing failure during extrusion based on a modified shear damage model P.J. Zhao a,*,1Z.H. Chen a, and C.F. Dong b
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a School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China b Institute for Advanced Materials and Technology, University of Science and Technology Beijing, Beijing 100083, China
Abstract: In this study, a recent modified shear GTN model suitable for various triaxialities was implemented and validated, which combines the porous plasticity model with a damage mechanics concept. The original Gurson model was modified by coupling the shear damage
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and void volume fraction parameters into the yield surface. The stress update algorithm was developed via a user-material subroutine. With the help of representative volume elements (RVEs), a connection between the formation of macroscopic cracks and the microstructure was established. The modified GTN model was applied to predict the damage and tearing fracture. Additionally, the cohesive zone model was applied to the RVE to simulate the de-
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bonding of the grain interfaces from a microscopic perspective. Metallurgical inspection by means of optical and scanning electron microscopy was performed to explore the failure
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mechanism and validate the numerical computation results. Furthermore, the deformation of the shear band and the damage propagation mechanism during the extrusion process were
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discussed.
Keywords: GTN model; shear damage; representative volume element; cohesive model;
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extrusion
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1. Introduction
Extrusion processes are commonly used to form metallic materials because they
typically result in a good dimensional accuracy, a low material consumption and a high productivity. Due to the improved quality of parts and the economic advantages offered by this advanced processing technique, it has been widely used to produce a variety of different parts, e.g., automobile components, camera components, and other small items used in 1*
Corresponding author. E-mail address: [email protected] (P.J. Zhao). 2
ACCEPTED MANUSCRIPT measuring equipment. A number of interesting extrusion experiments and numerical simulations of extrusion processes have been performed. For instance, Che et al. [1] investigated the plastic deformation and damage behavior of Al alloys during the extrusion process under crash loading conditions. Liang et al. [2] analyzed the extrusion of metallic materials at evaluated temperatures by developing a constitutive equation with strain compensation. Jung et al. [3] applied the finite element (FE) method to study the plastic deformation behavior during the
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tandem process of simple shear extrusion and twist extrusion. Hassan et al. [4] investigated the texture evolution during a simple shear extrusion process using a crystal plasticity finite element model, and Jie et al. [5] studied the effects of hot extrusion on the microstructure evolution and mechanical properties. Since the typical failure modes at an extrusion surface (as shown in Fig. 1) are ductile fracture and shear tearing, it is vital to explore the failure
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mechanism in detail and develop a methodology to predict the optimum manufacturing
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technique and parameters of the extrusion process.
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Fig. 1. Example of a fracture surface of an extruded workpiece. The ductile failure mechanism of metallic materials generally includes three phases, i.e., micro-void nucleation, micro-void growth and coalescence. In order to describe the damage
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and failure behavior of ductile materials, Gurson [6] first proposed the classical spherical void growth model for elastic-plastic metallic materials. Tvergaard and Needleman [7] introduced three corrective parameters into the damage model to explain the effects of material strain hardening and void interaction, resulting in the so-called GTN model. The GTN model allows for a good prediction of ductile fracture and thus has attracted the attention of numerous research groups, especially in recent years, such as [8-10]. Although the GTN model has achieved tremendous success, there still exist many limitations. For instance, it fails to accurately predict the damage behavior under sheardominated loading conditions, because it is unsuitable for describing void growth and 3
ACCEPTED MANUSCRIPT damage in case of low triaxiality. A modified GTN model was therefore proposed by Nahshon and Hutchinson [11] by introducing a Lode angle and the stress triaxiality parameter. Xue [12] also proposed a damage model based on the solution for void coalescence in the shear band. Mohamed et al. [13] predicted the ductile damage under shear dominated loads based on an advanced Gurson model. Quan et al. [14] applied a GTN-type model to analyze crack initiation and crack propagation under cold rolling conditions. Mohamed et al. [15] studied the damage evolution and the effect of punch-die clearance to
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validate the predictive capacity of the modified model. Although these modified models can be considered great improvements with respect to the prediction of the failure behavior of metallic materials, they tend to overestimate the void growth, which consequently leads to inaccurate numerical results under shear-dominated conditions. To overcome the limitations of existing models, Zhou et al. [16] proposed a new modified GTN model by combining the
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damage mechanics concept of Lemaitre [17] with the void damage model.
In recent years, the damage models combined with representative volume element (RVE) for ductile metallic material were proposed. For instance, Vitoon et al. [18] described the influence of the multiphase microstructure on the complex failure mechanism as well as
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mechanical properties by using RVE model within the framework of continuum damage mechanics. Vitoon et al. [19] studied the micro-crack formation in multiphase steel using
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RVE model and damage mechanics. Napat et al. [20] employed XFEM and damage curve derived to study the interaction between failure modes in DP steels by means of RVE. Ling et al. [21] used the RVE model to discuss the crystallographic texture and morphological
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features in the FE simulation of nanoindentation. Hosseini et al. [22] predicted stress–strain response of dual phase steel using the FE analyses of selected RVEs from SEM images.
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Ayatollahi et al. [23] investigated failure and damage evolution of three RVE models in DP steels by using experimental and numerical methods.
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In this study, the recent modified shear GTN model proposed by Zhou suitable for various types of stress triaxiality was deduced and programmed. The correlation between the material microstructure and the resulting macroscopic failure behavior was established by means of RVE. In addition, the de-bonding of the interface between metallic grains was simulated by employing a phenomenological cohesive zone model (CZM) based on a separation law. The modified GTN model and the CZM model were then applied to the extrusion process to describe the ductile failure behavior of DP600 steel, as well as crack propagation. The implementation of the constitutive equations into a numerical algorithm was achieved by developing a VUMAT subroutine. The surfaces of extruded specimens were observed via optical microscopy (OM) and scanning electron microscopy (SEM) to 4
ACCEPTED MANUSCRIPT investigate the tearing failure mechanism occurring in the shear zone. The validity of the numerical model was then verified by comparison with the experimental results.
2. Constitutive model 2.1. The modified shear GTN model
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As a micro-mechanical ductile damage model, the GTN model was used to determine the damage onset or crack initiation mechanism in metallic materials. A spherical void in the plastic matrix material was proposed by Gurson [6] to describe the yield potential for a porous plastic material. Tvergaard and Needleman [7] modified the original model by introducing two coefficients, resulting in the so-called GTN model. Nevertheless, the GTN
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model is still not able to accurately predict the damage behavior under shear-dominated loading conditions. To solve the problems, Zhou et al. [16] proposed a new modified GTN model which combined the continuum damage mechanics proposed by Lemaitre [24, 25] with micro-void damage mechanics. The modified model is typically expressed as follows: 2
(1)
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3q2 m * ) 1 D 2 2 Ds 0 2q1 f cosh ( 2 Y Y
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where Y denotes the yield stress,. m kk / 3 is the mean stress, 3 / 2 S : S is the von Mises equivalent stress of the matrix material, and S is the deviatoric stress tensor. The total damage can then be defined as
D q1 f * Ds
(2)
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where the total damage term contains an additional shear damage state variable Ds . The load carrying-capacity of the material is lost completely once the total damage D is equal to unity. The modified model will degenerate to the original GTN model in the absence of a shear load; the model will have the same form as the CDM of Lemaitre if there is only shear stress. The shear damage term can be written as n
p Ds s (3) f where n is a weakening exponential. sf denotes the failure strain under pure shear loading conditions. Ds is equal to one when p reaches sf . The softening effect is small at the early stage of plastic deformation where n is greater than one and becomes larger as the material approaches failure. The incremental expression is given by
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Ds
nDs n
sf
p
(4)
To extend the Eq. (4) to all stress states, the stress triaxiality T * was introduced and the Lode angle has been added as weight factor [26, 27] n 1
Ds , T *
nDs n
sf
p
(5)
with g 1
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and
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1 s2 s3 1 2 3 s1 s3
tan 1
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where the weight function ,T * is usually expressed as follows: T 0 g , T * g 1 k k T 0
(6)
(7)
(8)
where S k ( k 1, 2, 3) denotes the principal deviatoric stress.
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2.2. Representative volume element
(a)
(b)
Fig. 2. Illustration of (a) the 2D Voronoi tessellation and (b) the cohesive elements or interface elements which were applied to simulate the structure at the grain level In this work, 2D Voronoi tessellations were applied to simulate the grain-level structures required for describing the properties of a polycrystalline material. In order to reduce the computation time, the representative volume element (RVE) was introduced into the mechanical models as a representative part of the entire model to describe the failure 6
ACCEPTED MANUSCRIPT behavior at the micro-scale. Fig. 2(a) illustrates the 2D Voronoi tessellation with the embedded cohesive elements, which were generated using the Neper software [28]. The different colors in the RVE 2 represent the different grains. The square RVE with a size of 0.5 × 0.5 mm contains 100
grains and the mean grain size is 0.05 mm. Cohesive elements (COHAX4) were then introduced at the grain interfaces. Fig. 2(b) shows the cohesive elements in the RVE, which
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represent the potential crack zone. As a pre-process, the COPYNODE algorithm was programmed in Python to automatically embed the cohesive elements, as shown in Fig. 3. The program consisted of the following steps. Firstly, the information regarding the nodes at the grain boundaries was collected and then defined as a set. Secondly, the surface element nodes were updated to
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assure that each grain had a unique node number. Finally, the new nodes were combined with the surface element nodes and the cohesive elements were generated. Original RVE Voronoi tesselation mesh data
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Obtain the information of the surface nodes
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Copy and update the surface nodes
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Generate 2D cohesive elements with zero thickness
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Fig. 3. The flowchart of the COPYNODE program used to embed the cohesive elements into the 2D RVE.
In order to simulate the interface de-bonding between the grains, the cohesive zone
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model (CZM) was employed. The stiffness of the interface elements degenerates with increasing accumulated damage until the elements between the continuum bodies separate and fracture occurs. The traction-separation law was applied to the finite element model to simulate the decohesion of the interface elements and the crack initiation process. A localized crack nucleation criterion was used to determine the time of separation of the grain boundary. In this study, the failure criterion with the critical energy 0 was applied to the RVE, with the energy being dissipated from the cohesive elements. According to Vitoon et al. [29], the crack nucleation criterion for a cohesive interface can be expressed as follows: 7
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0 T d
(9)
0
The failure law depends on three material parameters: the cohesive energy 0 , the cohesive stress0 and the critical separation at material de-cohesion 0 . 2.3. Numerical algorithms The stress update algorithm of the constitutive equations is briefly described in this
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section. The modified model, which is suitable for various stress states, was implemented into the commercial FE software ABAQUS through developing the required user-material subroutine. The integration of the constitutive equations at the Gauss points was carried out using the backward Euler return mapping algorithm. The implementation procedure of the numerical integration algorithm was as follows:
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Step (1): Determination of the initial values of t , t , H , t t ( t = 0,, ti ) for each
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incremental time step. where t is the stress tensor, t is the strain tensor, t t is the incremental strain tensor, p and H ( =1,2,3) denotes the three different state variables , f and Ds . The equivalent p plastic strain of the matrix , the void volume fraction f and the shear damage Ds were treated as scalar internal variables. Step (2): Calculation of the total trial stress.
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It is assumed that the strain increment is completely elastic. Then, the trail elastic stress is given by
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e tet t Cijkl : t t
2 e Cijkl 2G ik jl K G ij kl 3
(10) (11)
e where Cijkl denotes the fourth-rank elastic modulus. The superscript e indicates the trail
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elastic state. Here, G E / 2(1 v) is the shear modulus, K E / 3(1 2v) is the bulk
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modulus, E is Young’s modulus and v is Poisson’s ratio. Next, the equivalent stress was calculated through
and the mean stress through
e
3 e e S :S 2
1 3 Step (3): Solving the yield function to ascertain the yield state.
(12)
me tet : I
(13)
tet e , me , H1, H 2 , H 3
(14)
e e If t t 0 , the trial stress results in elastic deformation, t t t t , and the algorithm
proceeds to Step (5) 8
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Step (4). Step (4): Calculation of the plastic deformation and correction.
1 p mp I eqp n 3
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Here, the subscript t t is ignored to simplify the notation. The flow direction can be expressed as 3S e (15) n 2 and the flow rule as
(17)
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Furthermore, the volumetric strain increment is given by 3q2 m 1 * Dm mp 3q1q2 f sinh 2 Y Y m
(16)
where denotes the plastic strain-rate multiplier, and the equivalent plastic strain increment can be written as 2 Deq eqp (18) 2 Y Then, the two following functions had to be satisfied:
Deq 0 m
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f1 Dm
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f 2 , m , H1, H 2 , H 3 0
(19) (20)
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The nonlinear equations were solved employing the Newton-Raphson iterative method according to the above expressions. When f1 and f 2 < Tolerance (equal to 1.0-8), the convergence conditions were satisfied. In this case, the algorithm proceeded to Step (5). Otherwise, the iteration process was terminated. Step (5): Updating the variables Dm , Deq , the hydrostatic stress m and the equivalent stress Dmk 1 Dmk Dm k 1 k Deq Deq Deq k 1 e k 1 m m K tDm k 1 e k 1 3GtDeq
(21)
(22)
Next, the stress tensor and the three state variables h ( = 1, 2, 3) were updated as follows: 2 tn 1 m I n (23) 3
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(24)
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mk 1Dmk 1 Deqk 1 k 1 k 1 p h1 H1 1 f Y m Dm Deq k 1 k 1 h2 H 2 f 1 f Dm I Deq n AN 1 f Y n 1 n k 1 k 1 * nDs p h3 H 3 Ds , T s M f Step (6): Beginning of the next circulation.
The detailed numeralization procedure for the stress update algorithm of the modified model is illustrated in Fig. 4. The partial derivatives of each term in the above equations are provided in the appendix. Start of VUMAT
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Known t , t , H , t t e e e Calculate t t , m ,
( me , e , H ) 0
Yes
No Initial value k 0, h 0 Calculate Dm , Deq , f1 , f 2
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Newton iterative
f1 toler and f 2 toler
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No kN No Iteration not converge stop ABAQUS
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Yes
Yes Update stress t t Calculate Dm , Deq , m , Update state variables h , H , D End of VUMAT
algorithm of the modified GTN model.
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Fig. 4. Schematic diagram illustrating the numeralization procedure for the stress update
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3.1. Extrusion experiment
Fig. 5. Photographs showing the equipment used for the extrusion experiment.
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Table 1
Chemical composition of DP600 steel (in wt%). Si
0.072
1.577
0.246
Al
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Mn
0.032
0.015
S
Ni
V
Cr
Cu
0.001
0.024
0.011
0.552
0.01
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The extrusion experiment was carried out on a 315 t four column hydraulic press. The
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experimental setup is shown in Fig. 5. DP600 steel with a thickness of 5 mm was used as sample material, and its chemical composition is given in Table 1. The schematic diagram
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used in this study to describe the extrusion process is shown in Fig. 6. In the figure, FG is the counter force and FR is the holder force. The cutting clearance is c 0.025 mm, i.e., 0.5% of the material thickness. The punch velocity was kept constant at 10 mm/s. The extrusion process was regarded as quasi-static so that the effect of the strain rate could be ignored. Different punching depths were used during the extrusion experiments to verify the numerical simulation results.
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Fig. 6. Schematic illustration of the extrusion process (length units in mm).
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3.2. Numerical simulation
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Fig. 7. Illustration of the finite element mesh in which the cohesive element was embedded around the local area.
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A 2D axisymmetric model was used to simulate the extrusion process. The RVE with a
2 size of 0.5×0.5 mm was embedded in the specimen to investigate the change of the
microstructure of the material during the extrusion process, as shown in Fig. 7. A dense mesh was applied to the large deformation area to improve the precision of the numerical calculation, and a coarser mesh was applied to the other areas to minimize the computational cost. The arbitrary Lagrangian Euleurian (ALE) technique was adopted to avoid the distortion of elements in those regions where severe distortion may occur. The Coulomb friction model was used to simulate the contact between the workpiece and the dies. The friction coefficient was fixed at 0.1. The dies and the punch were regarded as rigid bodies 12
ACCEPTED MANUSCRIPT and the metal sheet was defined as an elastic-plastic body. For the deformed body, 3-node linear axisymmetric triangle elements (CAX3) were used and 4-node axisymmetric cohesive elements (COHAX4) were used to model the grain interfaces. The corner radius of the die and the punch were set to be 0.1 mm. Table 2 lists all parameters used to simulate the extrusion of a DP600 steel workpiece, including the cohesive zone model parameters employed by Vitoon et al [30] and the modified GTN model parameters used by Zhao et al [31]. Just to visualize the micro-cracks
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occurred in extrusion process from a microscopic perspective, cohesive zone model parameters were directly quoted into simulation due to the same material DP600 steel and the identical stress-strain relationship. In this work, the modified GTN model and the cohesive zone model were investigated to determine their capability to accurately predict the
Table 2
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extrusion process.
Parameters of the cohesive zone model and the modified GTN model to simulate the extrusion of a DP600 steel workpiece.
[N/mm] 100
1.5
q2 1
fo
fN
0.0008
0.02
SN
N
0.1
0.2
fc
ff
sf
0.028
0.09
0.5
n 5
k 0.5
K 863
0 0.011
N 0.165
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1500
q1
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T0 [MPa]
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4. Results and discussion
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4.1. Metallographic inspection
(a)
(b)
Fig. 8. Photographs showing the specimens after the extrusion experiments at different pre13
ACCEPTED MANUSCRIPT set punching depths: (a) sectioned specimens, (b) polished and etched specimens. In order to study the microscopic surface topography of the DP600 steel specimens, interrupted extrusion experiments were performed at different pre-set punching depths. The blank specimens were cut manually to avoid the damage and thermal effects typically generated during an electrical wire cutting process. Then, all parts were polished and etched
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with Nital. Photographs of the fine-blanked specimens are shown in Fig. 8.
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Subsequently, the microstructure of the metal samples was observed by OM and SEM, and the results are shown in Fig. 9. When the sample was penetrated to 33.3% of its thickness (Fig. 9a), the shear band gradually formed. The deformation was localized in the narrow clearance zone between the die and the punch. The serious shear deformation resulted in an intensive slip and dislocation, and the particles adjacent to the die edge were found to be elongated and compressed.
When the sample was penetrated to 58.3% of its thickness (Fig. 9b), the localized shear deformation was found to have further propagated parallel to the punching direction and
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highly rotated and elongated metallic grains appeared in the shear band. The growth and coalescence of voids were observed to be limited to the clearance zone. A few micro-cracks and micro-voids were found surrounding inclusions extended along the shearing direction, which indicates that the local damage triggered the initiation of fracture. At the final stage (Fig. 9c), a macroscopic crack occurred near the die edge and the
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crack propagated at an angle of 135°. In the vicinity of the die, the tearing failure of the specimen occurred along the direction perpendicular to the principal stress due to the
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combined action of interfacial friction and axial tensile stress, as illustrated in Fig. 9d. In conclusion, the observed material damage suggests that the ductile fracture mechanism can
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be applied to describe the extrusion process.
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4.2. Numerical results
(a)
(b)
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(c)
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Fig. 10 shows the distribution of the hydrostatic stress at three different penetration depths. In order to more clearly reveal the stress distribution, the 2D contours were rotated at a small angle of 15°. In the early stage of the extrusion process, the material close to dies is compressed and the hydrostatic stress propagates along the entire large deformation region, which suppresses the formation of micro-cracks. At a penetration depth of 30–70% of the sheet thickness, severe plastic deformation was observed localized around the shearing band. The intensive shear localization contributes to the elongation of grains and the accumulation
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of micro-defects.
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Fig. 11. Evolution of the shear stress with the penetration depth.
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Fig. 11 shows the shear stress distribution in three specified elements of the workpiece. At the beginning, the shear stress monotonically increases due to the effects of strain hardening. The largest value in the entire deformation area is close to the punch edge. As the deformation proceeds, the shear stress close to the corner of the punch increases to 557 MPa, which is the peak value in the whole specimen.
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Fig. 12. Comparison between the numerical simulation and the experimental results obtained for the variation of the punching load with the penetration depth.
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A comparison between the numerical simulation and the experimental results obtained for the variation of the punching load with the penetration depth is shown in Fig. 12. The data acquired through the numerical simulation shows the same trend as the experimental data. The punching process can be divided into three stages. First, the punching load
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monotonically increases over the interval marked by A and B as a result of strain hardening. Subsequently, the punching load monotonically decreases after reaching its peak value in point B. The decline may be connected with the decrease of the shearing area and the plastic instability. The load tends to be stable over the interval marked by C to D, with the
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deformation being mainly caused by the growth of micro-cracks and microstructural effects.
(a) at 30% penetration
at 50% penetration
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at 70% penetration
(b) at 30% penetration
at 50% penetration
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at 70% penetration
Fig. 13. Distribution of the state variables for different penetration depths: (a) equivalent
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plastic strain, (b) total damage value.
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Fig. 13 illustrates the evolution of the equivalent plastic strain and the total damage as predicted by the modified GTN model for different penetration depths. As shown in Fig. 13(a), the plastic deformation mainly occurs in the narrow clearance area, and the strain hardly changes outside the shear band. The distribution of the equivalent plastic strain is
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more localized than that of the shear stress. The maximal plastic deformation occurs in the clearance area close to the die and the punch edge, where the crack formation occurs first. As shown in Fig. 13(b), at a penetration depth equal to 30% of the sheet thickness, the damage begins to accumulate and propagate along the shear deformation direction. At a
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penetration depth equal to 50% of the sheet thickness, the damage reaches its critical value, which suggests that the material in the shear band has lost its load-carrying capacity and thus the formation of cracks is triggered. Then, a crack appears at the cutting edge of the die and propagates towards the center of the thickness, which leads to fracture of the material when the penetration depth reaches 70% of the sheet thickness.
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Fig. 14. Variation of the equivalent plastic strain with the penetration depth for two
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specified elements near the die corner.
Fig. 15. Evolution of the total damage in the shear zone.
Fig. 14 illustrates the variation of the plastic strain for two specified elements close to the die corner, which indicates that the deformation near the punch corner is slightly larger than that near the die corner. Fig. 15 illustrates the evolution of the total damage in the shear band with the penetration depth. The increase of the punching depth leads to a localization of the damage because the plastic strain mainly occurs in the shear zone. The amount of damage at a local area increases to a sufficiently large value which eventually results in material failure and fracture along the punching direction. 19
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Fig. 16. Distribution of (a) the von Mises stress and (b) the damage variable in the RVE at
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the site and moment of crack initiation.
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Fig. 16 shows the distribution of the von Mises stress and the damage variable at the site and moment of crack initiation. The maximum stress occurs near the die corner and extends along the direction with a horizontal angle of 135°. As shown in Fig. 16b, because the cohesive elements are defined to be located between the continuum elements, when the damage exceeds the critical value and the elements lose their stiffness at the moment of failure, a number of micro-cracks occur, which propagate at an angle of 45°, which is consistent with the experimental results shown in Fig. 9.
Fig. 17. Schematic illustration of the ductile damage evolution mechanism.
Fig. 17 schematically illustrates the ductile damage evolution mechanism for metallic materials. It generally consists of the following four stages: When the damage exceeds a critical value, micro-voids will nucleate in the region that contains the second phase particles or inclusions. The growth of the micro-voids is caused by the further plastic deformation and high hydrostatic stress, and some new micro-voids will nucleate. Once the 20
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number of micro-voids exceeds a certain threshold value, void coalescence occurs and the
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voids will form connections, which finally leads to the fracture of material.
Fig. 18. Comparison of the results of the numerical simulation of the extrusion process with
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the experimental results.
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Fig. 18 compares the FE simulation and the experimental results calculated using the modified damage model. It can be seen that the shear plane of the fine-blanked specimen can be sectioned into four zones: the burr zone, the fracture zone, the shear zone and the rollover zone. The predicted fracture position and shear surface are a good agreement with the experimental observations.
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5. Conclusions
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The objective of this study was to implement and validate a recent modified GTN model established to simulate an extrusion process, which explains the shear-induced damage by combining the void damage model with the CDM concept. The yield function includes two damage variables, i.e., the void volume damage and the shear damage. The implementation of the algorithm was achieved by developing a VUMAT subroutine. The RVE was used to account for the influence of the microstructure and fracture mechanisms. The grain morphology is created by the Voronoi tessellation method. A 2D RVE was modeled to simulate the propagation of phenomenological damage at the grain boundaries of the DP600 steel material. The CZM model and the traction-separation law were also used to describe the de-bonding at the grain interface. The degradation of the interface was achieved using the cohesive elements. The fracture surfaces of the DP600 steel as observed by OM and SEM indicate that the elongated grains in the large deformation area propagate along the shear direction and that the crack initiates from the area near the die corner. The profile of the shear plane simulated using the FEM was found to match the experimental observations. 21
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Acknowledgments
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The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Grant No. 51671029). The related experiments were performed at the Beijing Research Institute of Mechanical and Electrical Technology.
Appendix
The stress integration algorithm of the constitutive equation is described below. First, the yield surface of the modified GTN model can be expressed as 2
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3q2 m * ) 1 D 2 2 Ds 0 2q1 f cosh ( 2 Y Y
(25)
The three internal variables are given by
H1 p , H 2 f , and H 3 Ds .
(26)
(27)
The partial derivatives can then be expressed as follows: 2 Y2
(28)
3q 3q1q2 f * sinh 2 m m Y 2 Y
(29)
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The evolution functions of the three internal variables can be written as m Dm Deq h1 H1 1 f Y p h2 H 2 1 f Dm AN 1 f Dm AN h1 n 1 n 1 h H g n D n p ng D n h 3 3 s 1 sf s sf
2 0 m
(30)
2 2 2 2 Y
(31)
3q2 m 2 9q1q22 f * cosh m2 2 Y2 2 Y 2 2 H1 H1 Y2
4 Y 4 3 H 3 Y H1 Y
2 0 H 2
22
(32) (33) (34)
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(35)
3q2 m 3q2 m 3q 3q1q2 f * H 2 cosh 2 m sinh 2 m H1 2 Y 2 Y 2 Y 2 Y
3q 3q q f * 2 1 2 m sinh 2 m m H 2 Y f 2 Y
(37)
(38)
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, f fC 1 f * 1 / q1 f C f f f , f fC F C 2 0 m H 3
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3q 3q2 m * 2 sinh 2 m H 2 H 2q1 f 2 H1 Y Y 2 Y 2 Y 3q 2 H 3q q f * H 3 1 2 2 m sinh 2 m Y 2 Y 2 Y
3q2 m 2 * f * f * f * 2q1 cosh 2q1Ds 2q1 f H 2 f f f 2 Y 2q1 f * 2 Ds 2 H 3
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(36)
(39)
(40)
(41) (42)
The partial derivatives of the internal variables can be obtained by solving Eq. (43) and Eq. (44):
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h1 h1 h h H t 1 t 1 1 1 t H1 H 2 H 3 Dm Dm h h2 h h H 1 t 2 t 2 2 t 2 t H1 H 2 H 3 Dm Dm h3 h h h H t 3 t 3 1 t 3 3 H1 H 2 H 3 Dm Dm h1 h1 h1 h1 H1 1 t t t D H H H eq Deq 1 2 3 h h2 h2 h2 H 2 2 t 1 t t t H1 H 2 H 3 Deq Deq h3 h3 h3 H 3 h3 t t 1 t H1 H 2 H 3 Deq Deq
where
h1 m h tK 2 m h tK 3 m h 3Gt 1 h 3Gt 2 h3 3Gt tK
(43)
(44)
m h1 Dm 1 f Y
(45)
h2 h1 1 f AN Dm Dm
(46)
23
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(47)
Dm h1 m 1 f Y
(48)
h2 h1 AN m m
(49)
h3 ng nn1 h1 Ds m sf m
(50)
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m Dm Deq h1 h1 Y H H1 Y H1 1 f Y2
(51)
A h2 AN h1 h AN 1 h1 N H1 H1 H1 H1
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h p N p N p 1 AN AN c20 h1 sN2 sN2 H1 n 1 h3 ng n h1 Ds H1 sf H1
D Deq h1 1 m m H 2 1 f Y 1 f
2
h1 1 f
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(55)
h3 ng nn1 h1 Ds H 2 sf H 2 h1 0 H 3 h2 0 H 3
(56)
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p
h1 Deq 1 f Y h2 h AN 1 Deq Deq
(53) (54)
h2 h1 Dm AN H 2 H 2
ng n 1 nn11 h3 Ds H 3 sf n
(52)
(57) (58)
n 1 g D 1n h sf
s
1
(59) (60) (61)
ng nn1 h1 h3 Ds Deq sf Deq
(62)
Deq h1 1 f Y
(63)
h2 h AN 1 h3 ng nn1 h1 Ds sf 24
(64) (65)
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The iterative variables can be calculated by solving 1 Dm K11 K12 R1 Deq K21 K22 R2 where R1 f1 Dm
(66)
Deq m
(67)
R2 f 2 m , , H1, H 2 , H 3 3 f1 2 2 H i Dm tK Dm m i 1 H i Dm
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K11
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3 2 2 H i Deq tK 2 m i 1 mH i Dm 3 f 2 2 H i K12 1 Dm 3Gt Deq m 2 i 1 H i Deq 3 2 2 H i Deq 3Gt m i 1 mH i Deq 3 f H i K21 2 tK Dm m i 1 H i Dm 3 f H i K22 2 3Gt Deq i 1 H i Deq
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