On the numerical implementation of a shear modified GTN damage ...

Int J Mater Form DOI 10.1007/s12289-017-1362-7

ORIGINAL RESEARCH

On the numerical implementation of a shear modified GTN damage model and its application to small punch test Liang Ying 1 & Dan-tong Wang 1

&

Wen-quan Liu 2 & Yi Wu 1 & Ping Hu 1

Received: 28 October 2016 / Accepted: 29 May 2017 # Springer-Verlag France 2017

Abstract Gurson-type models have been widely used to predict failure during sheet metal forming process. However, a significant limitation of the original GTN model is that it is unable to capture fracture under relatively low stress triaxiality. This paper focused on the fracture prediction under this circumstance, which means shear-dominated stress state. Recently, a phenomenological modification to the Gurson model that incorporates damage accumulation under shearing has been proposed by Nahshon and Hutchinson. We further calibrated new parameters based on this model in 22MnB5 tensile process and developed the corresponding numerical implementation method. Lower stress triaxiality were realized by new-designed specimens. Subsequently, the related shear parameters were calibrated by means of reverse finite element method and the influences of new introduced parameters were also discussed. Finally, this shear modified model was utilized to model the small punch test (SPT) on 22MnB5 high strength steel. It is shown that the shear modification of GTN model is able to predict failure of sheet metal forming under wide range of stress state.

Keywords GTN damage model . Shear modification . Stress triaxiality . Sheet metal forming . Numerical simulation

* Dan-tong Wang [email protected]

1

School of Automotive Engineering, Dalian University of Technology, Linggong Road 2#, Ganjingzi District, Dalian, Liaoning Province 116024, China

2

Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China

Introduction Recent experimental evidence points to importance in characterizing the critical strain in ductile fracture on the basis of stress state [1, 2]. This is brought into the sharpest relief by the fact that many sheet metal forming process have a fracture under shear-dominated stress state, at zero or low stress triaxiality, which is obviously lower than that under axisymmetric stressing at higher triaxiality. As currently formulated, GTN model excludes the possibility of capturing shear localization and fracture under conditions of low triaxiality while void nucleation is not invoked. To understand the fundamental of shear failure, experimental researches as well as the corresponding theoretical analyses have been done previously [3, 4]. However, these analytical solutions are only applicable to several specified situations where significant shearing deformation is the only reason causing the ductile failure of metals. This has highlighted the need for micromechanically-based models which can accurately capture material damage and failure under both low and high stress triaxiality (T = σh/ σeq, σh and σeq are hydrostatic stress and equivalent stress) in the context of actual sheet metal forming finite element simulations. Gurson [5] deduced a flow potential for void growth on an ideal plastic material, which employs spherical voids and based on the void growth mechanics, following the approach introduces by Rice and Tracey [6]. Then, the model was extended by Tvergaard and Needleman (the GTN model) [7–10] taking the interactions between the neighboring voids into account when the localized internal necking of the matrix occurs. However, these extensions are also mainly based on solutions for voids subject to axisymmetric stressing. As originally formulated, The Gurson Model only identifies a single damage parameter f, as the average, or effective, void volume fraction. An increase in f due to the growth of voids requires a positive mean stress. Thus, in shearing deformations under

Int J Mater Form

zero mean stress, the model predicts no increase in the damage parameter if continuous void nucleation is not invoked. Hence, an important limitation of the original GTN model is that it is unable to capture fracture under relative low or even negative stress triaxiality [11, 12]. A modified Gurson model was recently proposed by Nahshon and Hutchinson [4] (N-H model) to account for effective damage accumulation due to void distortion and inter-void linking under shearing. In parallel, L. Xue [13] has also developed a constitutive model where the accumulative damage of voided solids due to shear deformation was treated in a phenomenological way. Using the Gurson model including a shear modification, Soyarslan et al. [14] investigated the crack initiation at the convex surface and its propagation during bending. Dunand and Mohr [15] evaluated the predictive capabilities of the shear modified Gurson model in the tensile and the punch experiments over a wide range of stress triaxiality and Lode angles. Subsequently, the parameters in the Nahshon shear modification were calibrated

Fig. 1 Flow chart of numerical implementation for shear modification

by Z. Xue et al. [16]. K. Nahshon [17] included the void nucleation into the shear term for further modification to improve its accuracy. V. Tvergaard et al. [18] compared the fracture predicted by the shear modified model and micro-mechanical model. Although the above shear modification can predict the fracture when the stress triaxiality is low, some researchers found that the influence of this modification is rather large under some specific stress state when the stress triaxiality is high [19, 20]. Therefore, a further extension including an interpolation function was developed to decrease this influence by Nielsen and Tvergaard (N-T model) [20]. However, few publications have utilized this model in the numerical implementation of sheet metal forming. The method to calibrate the parameters in N-T model and experimental verification are not introduced, either. In the present paper, an extension of the GTN model incorporates nucleation of voids under low triaxiality by Nielsen and Tvergaard is employed. Based on this modified model, the yield surface is modified and the

Int J Mater Form Fig. 2 Sketches of experimental specimens for triaxiality

extension makes it possible to capture shear-dominated failures. By investigating the fracture behavior for a

high strength steel 22MnB5 under low stress triaxiality, several groups of experiments were conducted, relate material parameters were also identified. Moreover, a study of quasi-static SPT (small punch test) of 22MnB5 is performed in order to verify the modified model under a wider stress state. It is shown that the modified model employed in this paper can be applied to simulate complex sheet metal forming problems involving low stress triaxiality.

Modified GTN model Yield surface

Fig. 3 Force vs. displacement curves obtained by pure shear and sheartension combined tests and random texture detected after deformation

The micromechanical damage model developed by Gurson takes into account the degradation of the load carrying capacity due to the presence of porosity in isotropic materials. Tvergaard and Needleman modified the original Gurson model by concerning

Int J Mater Form Fig. 4 Flow chart for parameter calibration

the coalescence of voids. The yield surface is given by: !2 !  σeq 3q2 σh  * þ 2 f q1 cosh − 1 þ q3 f *2 ¼ 0 ð1Þ Φ¼ σm 2σm Where, σeq is macro Von Mises equivalent stress tensor; σh ¼ − 13 σkk is macro hydrostatic stress; σm is yield stress of matrix material; q1 , q2 , q3 is modified parameters introduced by Tvergaard and Needleman; f∗ is equivalent void volume fraction. 8 f ≤fc < T −T 2 ΩðT Þ ¼ > T 2 −T 1 : 0

T < T1 T1 ≤T ≤T2

ð10Þ

T > T2

Where, T is stress triaxiality: Shear modification

T¼

When tress triaxiality is rather small, the fracture cannot be predicted by GTN model. Therefore, to extend the applicability, K. Nahshon et al. introduced a modification term into the GTN model [4]:

σh σeq

ð11Þ

Include the interpolation function into kω, then: 0

k ω ¼ k ω Ω ðT Þ 0

sij dεp df s ¼ k ω ωðσÞ eq σ

df s ¼ k ω ΩðT ÞωðσÞ ð7Þ

Where, dfs is increment of equivalent void volume fraction caused by shear stress, kω is a shear equivalent coefficient that needs further calibration. It equivalents the void deformation, which can also cause damage in the material, to void growth. sij is deviatoric stress tensor; dεp is increment of plastic strain; ω(σ) depicts the stress state: !2 27 J 3 ωðσÞ ¼ 1− ð8Þ 2σ3eq J3 = det(sij) is the third invariant of deviatoric stress. Obviously, 0 ≤ ω(σ) ≤ 1. ω(σ) = 0 means there is no shear deformation while ω(σ) = 1 means there is only shear deformation. The increment of void volume fraction is expressed as follows after shear modification term included: df ¼ df g þ df n þ df s

ð9Þ

However, some researches show that the effect of this modification term is rather strong in some certain stress states, where the tress triaxiality is relatively high. In these cases, the predicted fracture occurs earlier than that in experiment. Therefore, a further modification by introducing an interpolation function developed by Nielsen and Tvergaard [20] was employed in this work, which is expressed as follows:

Fig. 6 Finite element model for tensile simulation

ð12Þ sij dε σeq

p

ð13Þ

Where, kω' is a constant shear equivalent coefficient. It can be seen that, when T < T1, the model is Nahshon modified model; when T1 ≤ T ≤ T2, kω decreases with the increase of stress triaxiality, so the effect of modification term weakens; when T > T2, the model degrades into the original GTN model. It is obvious that the shear equivalent coefficient kω should be different in different stress states and can be calibrated by comparing experimental and simulated results. During the deformation process under sheet metal forming, it should be ensured that the total influence of shear modification term is constant after the interpolation function included. The following equation should be satisfied:   0 Df ∫0 k ω Ω T f dD ¼ k iω D f

ð14Þ

Where, Tf is the stress triaxiality at the time point where fracture first occurs; D is displacement; Df is fracture displacement; k iω is the shear equivalent coefficients under different stress triaxiality.

Numerical implementation In this paper, the corresponding numerical implementation method of the modified model was developed as a usermaterial constitutive subroutine in ABAQUS/Explicit, as shown in Fig. 1. The detailed implementation procedures of the above mentioned algorithm are illuminated as follows:

Int J Mater Form Fig. 7 True stress vs. strain curves resulted from tensile test and simulation a response values on experimental curve b comparison of curves resulted from experiment, GTN model and SM (shear modified GTN model)

1. Initial values at t = 0, …, ti

Newton-Raphson iterative method is employed and iteration continues until both │f1│ and │f2│ < tolerance.

σt ; εt ; H αt ; dεtþdt Where, H αt are state variables, H 1t is the void volume fraction and H 2t is the matrix equivalent plastic strain. 2. Assuming the strain increment is purely elastic to obtain the trail elastic state (with a superscript e)

σetþdt ¼ σt þ C : dεtþdt

f 1 ¼ kþ1 dεh

∂Φ kþ1 eq ∂Φ þ dε ∂σeq ∂σh

ð17Þ

  h eq α f 2 ¼ Φ kþ1 σ ; kþ1 σ ; kþ1 H

ð18Þ

Update the hydrostatic stress, equivalent stress, stress and statement variables.

ð15Þ

Parameters calibration Where C is the fourth rank elastic modulus. Experimental set-up and parameters identification

3. Then the trail yield potential Φetþdt is calculated.   eqe α Φetþdt ¼ Φ σhe tþdt ; σtþdt ; H t

ð16Þ

If Φetþdt ≤ 0, the current time step is elastic, then update hydrostatic stress, equivalent stress, stress and statement variables. If Φetþdt > 0, the current time step is plastic, then begin the plastic calculation, go to Step 4. 4. Plastic calculation In this step, the subscript t + dt is omitted for simplification.

To calibrate the parameters in shear modified GTN model, tensile specimen was designed according to the national standard GB/T 228.1–2010. The experiments covering low triaxiality state was designed by employing pure shear specimen and shear-tension combined specimen. The ends of these shear-dominated specimens are similar to the tensile specimen to fit the fixture. Fig. 2 shows the sketches of above mentioned specimens and their stress triaxiality. All the tests mentioned in Fig. 2 were performed on a universal tensile machine. As for the tensile test, true stress vs. strain curve was obtained. A contact-type extensometer with the gauge length of 50 mm was employed to measure the extension and the strain rate was 0.1 s−1. As for the other shear-dominated tests, force vs.

Table 3

Average stress triaxiality of shear-dominated specimens

Specimen Table 2

GTN damage parameters for 22MnB5

Damage parameter

f0

fn

fc

ff

Value

0.002

0.0155

0.05

0.13

Pure shear Shear-tension Verification

Average stress triaxiality Central point

Fracture point

−0.0055 0.2109 0.1292

0.2871 0.3346 0.3920

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Fig. 8 Force vs. displacement curves resulted from experiment, GTN model and NM

Fig. 9 Force vs. displacement curves of pure shear and shear-tension combined specimens resulted from shear modified model, GTN model and experiment

displacement curves were achieved with a tensile speed of 30 mm/min. A Digital Image Correlation (DIC) method was employed to measure the strain distribution of the two sheardominated samples. The shutter speed was 4 s−1. The shear test was to investigate the fracture behavior of material under low triaxiality. For the improvement of accuracy and error minimization, the tests were at least repeated 3 times for each specimen. After fracture of specimen, the load vs. displacement curves were also obtained. Fig. 3 shows the average force vs. displacement curves resulted from pure shear and shear-tension combined tests and the image of random texture is also shown. The flow chart for parameter calibration employed in this paper is shown in Fig. 4. To determine the strain hardening behavior of material, the plastic strains are calculated by subtracting the elastic strains from the total strains, as shown in Eq. (19):

GTN parameters

εp ¼ εt −εe

There are 9 parameters in GTN damage model that needs to be determined firstly. Based on the research of Tvergaard [19], q1 = 1.5, q2 = 1.0, q3 ¼ q21 are fit for most metal materials. εN and SN can also be constant [20], εN = 0.3, SN = 0.1. As for f0 , fn , fc , ff, a finite element reverse method is employed in this paper for identification. First, the true stress vs. strain curve was obtained by tensile experiment. Then a corresponding finite element model was established as shown in Fig. 6. Model parts were discretized using C3D8R (continuum, 3D stress, 8-node linear brick, reduced integration) elements type and the central part was finer meshed for higher accuracy. Both ends were coarser meshed for higher calculating speed as well. The right side of this model was subject to a displacement constrain while the left

ð19Þ

Where, εt is the total strain, εe is the elastic strain. Hollomon hardening law, σ = KεN, was adopted to omit the effect of damage during tensile test. The fitting curve is shown in Fig. 5 together with the tensile curve and the fitting parameters are listed in Table. 1.

Table 4 Parameters of interpolation function

kω′

T1

T2

2.7

0.1998

0.5769

Fig. 10 Force vs. displacement curves of verification model resulted from experiment, GTN model and SM

Int J Mater Form Fig. 11 Strain distribution resulted from experiment and simulation (a) pure shear (b) shear-tension combined (c) verification

side was fixed. Numerical simulation was carried out by employing ABAQUS/Explicit. A four factors and three level CCD (central composite experimental design) was conducted, which contains 28 combinations of simulations. As shown in Fig. 7a, the coordinates of peak point and fracture point were obtained from each simulated true stress vs. strain curve and were treated as four response values (R1, R2, R3, and R4). Based on these data, a RSM (response surface model) was established. In terms of the corresponding coordinates obtained by experiment and by adopting the Genetic Algorithm (GA) method [21], the four damage parameters fit for the experimental material were identified and listed in Table. 2. For verification, the calibrated damage parameters were used for simulation as well. The simulated result was compared with experimental one. As shown in Fig. 7, it can be seen that the result obtained from simulation has a good agreement with that resulted from tensile test. Table 5

Parameters of shear modification According to Fig. 2, specimens with different shapes were designed to obtain different stress triaxiality. Corresponding finite element models were then established and the central parts were also finer meshed for better accuracy. By using the finite element software ABAQUS, the average stress triaxiality of the central points and fracture points were derived for each specimen, as listed in Table 3. It can be found from the central points that these shear-dominated specimens have a relatively lower stress triaxiality compared with tensile sample during deformation. The stress triaxiality of fracture points is employed for parameter calibration, because fracture should onset at the same position as modification is not included. The method to calculate average stress triaxiality is expressed as Eq. (20). T av ¼

1 Df ∫ TdD Df 0

ð20Þ

Where, T is stress triaxiality; D is displacement; Df is the displacement when fracture occurs.

Comparison of curves and maximum strain resulted from SM model, GTN model simulation and experiment

Specimen

Dis/mm

Pure shear Experiment GTN model SM Shear-tension combined Experiment GTN model SM Verification Experiment GTN model SM

Error/%

Force/kN

Error/%

Strain

Error/%

4.47 / 4.36

-2.5

4.540 / 4.652

2.5

0.260 0.258 0.260

0.8