Experimental and numerical investigation of ductile fracture using GTN damage model on in-situ tensile tests

Experimental and numerical investigation of ductile fracture using GTN damage model on in-situ tensile tests

Journal Pre-proof Experimental and numerical investigation of ductile fracture using GTN damage model on in-situ tensile tests H. Gholipour , F.R. Bi...

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Experimental and numerical investigation of ductile fracture using GTN damage model on in-situ tensile tests H. Gholipour , F.R. Biglari , K. Nikbin PII: DOI: Reference:

S0020-7403(19)32302-1 https://doi.org/10.1016/j.ijmecsci.2019.105170 MS 105170

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

26 June 2019 31 August 2019 16 September 2019

Please cite this article as: H. Gholipour , F.R. Biglari , K. Nikbin , Experimental and numerical investigation of ductile fracture using GTN damage model on in-situ tensile tests, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105170

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Highlights •







The objective is to investigate the mechanisms of ductile fracture under different stress states. A novel GTN based model, which is a micromechanical based damage model, was used for numerical simulations of SAE 1010 plain carbon steel for the first time. The input parameters of GTN model were obtained during this study by response surface method (RSM) through minimizing the difference between numerical and experimental results. The void related parameters of GTN model were determined 0.00107, 0.00716, 0.01 and 0.15 for (𝑓0,𝑓𝑁,𝑓𝑐,𝑓𝑓) respectively. The experimental tests and the calibrated model were employed to study the ductile failure in six in-situ tensile specimens with different notch shapes. The fractographic analysis was performed to identify the ductile fracture mechanism under different stress conditions. The effect of geometry variation and stress state in deformation zone on the shape of resulting fractured edges was studied by several experiments and FE simulations and showed a remarkable accordance. The extracted numerical results showed a good agreement with experimental observations comparing load-displacement plots with a margin of error within 5%. The location of fracture initiation, crack growth orientation and the displacement at final fracture for numerical studies has showed a close correspondence to the experimental results.

1

Experimental and numerical investigation of ductile fracture using GTN damage model on in-situ tensile tests H. Gholipoura, F. R. Biglaria, ⃰, K. Nikbinb a b

Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran. Mechanical Engineering Department, Imperial College London, London SW7 2AZ, UK

Abstract The present work is devoted to experimental and numerical investigation of in situ tensile tests to recognize the mechanisms of ductile fracture under different stress states. The GTN model, which is a micromechanical based damage model, was used for numerical simulations. The void related parameters of GTN model for SAE 1010 plain carbon steel were identified by response surface method (RSM) through minimizing the difference between numerical and experimental results of tensile test on a standard specimen. The void related parameters of GTN model were determined 0.00107, 0.00716, 0.01 and 0.15 for 𝑓 , 𝑓 , 𝑓 and 𝑓 respectively. After calibrating the damage model for the studied material, the tensile tests were carried out on the in-situ specimens with different geometries. The fractographic analysis was performed to identify the ductile fracture under wide range of stress states and two failure mechanisms were observed. The calibrated damage model was applied to FE simulations of in-situ tensile tests for numerical study of the experimentally observed fracture phenomenon. The extracted numerical results showed a good agreement with experimental observations comparing load-displacement plots with a margin of error within 5%. A better ductile fracture predictions were captured in 90o specimens. The location of fracture initiation, crack growth orientation and the displacement at fracture zone in numerical studies also showed close correspondence with experiments.

Keywords: Ductile fracture, GTN model, Void nucleation, Growth and coalescence, Stress state 1. Introduction In the context of fracture, there are two methods to formulate the constitutive equations of ductile damage. The first is known as continuum damage mechanics (CDM), which represents a phenomenological definition of damage. Lemaitre [1] suggested a representative model for this approach by taking into account the notion of effective stress. The phenomenological approach is based on a thermodynamic framework where the micro mechanical behavior and micro defects of material is not considered. The second method relies on the micromechanical description of material. From the micromechanical description of material, a common mechanism of ductile failure of metals and alloys during plastic deformation is generally identified as the micro-voids nucleation, growth and coalescence (VNGC). McClintock (1968) [2], Rice and Tracey (1969) [3] and Hancock and Mackenzie (1976) [4] carried out early studies of voids nucleation, growth and coalescence on cylindrical and spherical voids in infinitely large, plastic solids. Since then many studies have been carried out by different researchers to develop mechanism-based micromechanical models that describe the complex ductile failure process by considering 2

voids evolution phenomenon. Gurson (1977) [5] suggested a micromechanical based model which describes the influence of growth of spherical cavities on the behavior of voidcontaining materials. Rousselier (1987) [6] developed the constitutive relations for mechanical behavior of voided materials by coupling the damage parameter (porosity) with stress components using thermodynamic and plastic potentials. Many researches have been focused on extending Gurson model to cover the ductile fracture under various conditions. Chu and Needleman (1980) [7] proposed void nucleation models controlled by the local stress or plastic strain with a two parameter void nucleation criterion. Tvergaard (1981, 1982) [8, 9] developed the Gurson model application by introduction of two adjustment parameters accounting for the effect of void interaction and material strain hardening. Gologanu et al. (1993, 1994) [10, 11] extended the Gurson model and derived approximate models for ductile metals containing non-spherical voids. Tvergaard and Needleman (1984) [12] incorporated final material failure by void coalescence into the constitutive model via dependence of the yield function on the void volume fraction. The developed Gurson model by Tvergaard and Needleman, is known as the GTN model which is one of the most referenced void nucleation, growth and coalescence models. Since the introduction of GTN damage model by Tvergaard and Needleman (1984), many studies have been conducted on the failure and fracture analysis of different materials using the capability of this model. Nègre et al. [13] studied the crack extension in aluminum welds by using the GTN model. Nonn et al. [14] carried out a GTN based numerical modelling of damage behavior of laser-hybrid weld to study its effect on deformation and failure. In order to analyze ductile failure in a NiCr steel, Benseddiq et al. [15] performed numerical modelling with GTN model and experimental tests on notched round and CT specimens. Uthaisangsuk et al. [16] utilized the GTN model to predict formability in sheet metal forming. In an attempt to show the influence of the multiphase microstructure on the complex failure mechanism of multiphase steels, Uthaisangsuk et al. [17] simulated the dimple failure using GTN model. Butcher et al. [18] implemented the GTN model to numerical simulations of straight tube hydroforming of DP600. A consistent methodology to determine the GTN model parameters was developed by Cuesta et al. [19] and the model was used to simulate small punch tests. Liao et al. [20] employed the GTN model to investigate plastic deformation and damage behavior in the tensile bars and threaded bolts. The prediction of forming limit stress diagram of Aluminum Alloy 5052 based on GTN model was carried out by Min et al. [21]. Uthaisangsuk et al. [22] investigated damage and failure in multiphase high strength steels and applied GTN damage model to describe the ductile damage occurring mostly in the softer ferritic phase. Abbasi et al. [23] applied the response surface methodology to gain the GTN model parameters and utilized the calibrated model to determine the forming limit diagram of tailor welded blanks. In their work, a novel inverse procedure based on response surface method aimed to estimate 6 material parameters of the GTN model by optimizing 4 specified responses. They used a central composite design for determining the arrangements of experiments to study six factors at three levels. They utilized quadratic polynomial regression model for predicting the responses in terms of the six independent variables. Abbasi et al. implemented the analysis of variance (ANOVA) to evaluated the statistical significance of the full predicted quadratic models. Li et al. [24] performed a study on the parameter identification of GTN model using response surface methodology for high strength steel BR1500HS. They identified the four vital parameters in GTN model by performing several numerical simulations specified by central composite design. In order to determine the optimum parameters, they established the error evaluation functions (R) using four GTN parameters. Thereafter, the four parameters of GTN model were determined by minimizing four values of R by using response surface methodology and least square method. Fethi et al. [25] identified the GTN model parameters by using artificial 3

neural network and used the model to study the damage progress on the notched tensile specimen of sheet metals. Micromechanical modelling of damage behavior in order to understand the effect of microstructural morphology in mechanical properties and failure mechanism of Ti–6Al–4V was carried out by Katani et al. [26] implementing GTN model. The formation of edge cracks of silicon steel during tandem cold rolling process was investigated by Yan et al. [27] based on GTN model. An investigation of local deformation and damage of dual phase steel was conducted by Sirinakorn et al. [28] and GTN model was used to describe damage evolution in the microstructure. The tearing failure in blanking process of ultra-thin sheet-metal was analyzed by Wang et al. [29, 30] based on GTN model. Wu et al. [31] performed the analysis of damage evolution during spinning forming using GTN model. Safdarian [32] used GTN damage model in a numerical simulation of rotary draw bending process for investigation of tube fracture. Following the reviewed literature, in the present work, the ductile fracture phenomenon in SAE 1010 steel sheets has been investigated using GTN damage model. In order to find GTN parameters, the response surface method has been utilized. In-situ tensile tests with different specimen shapes have been conducted to calibrate the model parameters and to study the fracture.

2. Constitutive modeling of porous materials 2.1. Gurson model As one of the first micromechanical based models, Gurson (1997) proposed his model for description of ductile damage and fracture by using the upper bound plasticity theorem. The governing equations were derived by assuming a spherical cavity embedded in a cubic rigidplastic matrix without hardening. The yield function of Gurson model including isotropic hardening behavior is expressed by: (1) (

𝑓)

( )

{

𝑓

𝑓

(

)}

where , 𝑓 , and represent the second invariant of the deviatoric stress tensor , void volume fraction, hydrostatic pressure and isotropic hardening rule respectively. The isotropic hardening rule can be defined as , where represents the thermodynamical force associated to the isotropic hardening state variable and is initial yield stress. The void volume fraction evolution in Gurson model was achieved by considering the requirements for mass conservation of a rigid plastic material and plastic incompressibility. The density of a representative volume element with a void can be written as: (

(2)

𝑓)

where and represent the densities of the RVE and the matrix of the material respectively. After time differentiation, the density rate of the RVE can be expressed as follow: (3) ̇ ̇ ( 𝑓) 𝑓̇ By neglecting the elastic volumetric strains, the principals of mass conservation and plastic incompressibility require that ̇ . Thus the following expression can be obtained: 4

̇

̇

𝑓̇

(

(4)

𝑓)

The volumetric strain rate can be determined by considering the principal of mass conservation. ̇

̇

̇ ̇

(5)

where ̇ and ̇ are the elastic and plastic strain rate contributions respectively. Due to the assumption of rigid-plastic material in Gurson model, after neglecting the elastic strain rate, the void volume fraction rate can be expressed as: ̇

𝑓̇

(

̇

𝑓)

(

(6)

𝑓) ̇

According to the plastic flow rule and considering Gurson yield function: ̇ ̇

̇

̇𝑓

where ̇ , ̇ , ̇

and ̇

(

)

̇

̇

(7)

represent the plastic multiplier, the deviatoric plastic strain, the

volumetric plastic strain and the plastic strain rate tensor respectively;

is the vector in

deviatoric space normal to the yield surface. By replacing ̇ from the above equation, the evolution law for void volume fraction is obtained: 𝑓̇

(

𝑓) ̇

(𝑓

𝑓) ̇

(

(8)

)

2.2. The GTN damage model In the Gurson model, the evolution of void volume fraction is not dependent on the material ), no void strain history and in a case that the initial void volume fraction is zero (𝑓 volume fraction accumulation will be predicted. In order to eliminate this shortcoming of Gurson model, several void nucleation mechanisms have been introduced to trigger void nucleation depending on the strain history. Chu and Needleman [7] proposed a void nucleation law which was utilized in Gurson model by Tvergaard and Needleman [12]. The mechanism of void nucleation which is a result of fracture of inclusions and/or de-cohesion of inclusions (or second phase particles) from the surrounding matrix, can be controlled by plastic strain or hydrostatic pressure. The definition of void nucleation based on plastic strain is as follows: 𝑓̇

𝑓

[

(

̅

) ] ̅̇

(9)

√ where 𝑓 is the volume fraction of all particles with potential for void nucleation, is the mean plastic strain for void nucleation and is the standard deviation of the distribution. The variables ̅ and ̅̇ represent the equivalent plastic strain of matrix and the rate of the accumulated plastic strain of matrix respectively. Consequently, the change in the void volume fraction rate is due to the nucleation of new voids and growth of the existing voids.

5

𝑓̇

𝑓̇

(10)

𝑓̇

The complete loss of load carrying capacity in the Gurson yield function occurs when 𝑓 . This is untruly larger than the experimental observations. Tvergaard and Needleman [12] adopted the effective void volume fraction, 𝑓 , to model the rapid loss of load carrying capacity at a realistic level of the void volume fraction, which is determined by the following function: 𝑓 𝑓

{ 𝑓

(

𝑓 𝑓) 𝑓

𝑓 𝑓

𝑓

𝑓

𝑓

𝑓

𝑓

𝑓

(11)

where 𝑓 is the critical void volume fraction at which voids begin coalescence and 𝑓 represents the void volume fraction at fracture. The damage evolution comprises three mechanisms of nucleation, growth and coalescence of voids. When the void volume fraction is less than 𝑓 , the effective void volume fraction, 𝑓 , includes both nucleation and growth mechanisms while the coalescence mechanism is activated if the void volume fraction is higher than 𝑓 . In order to improve the model predictions and bring the results into a closer compatibility with the experiment and also considering void interaction effect due to multiple void arrays, the adjustment parameters , and were introduced by Tvergaard [8, 9]. With the all above mentioned modifications the yield function of Gurson model was changed to the equation (12) which is known as Gurson-Tvergaard-Needleman (GTN) yield function assuming isotropic hardening and isotropic damage. (

𝑓)

( )

{

𝑓

𝑓

(

)}

(12)

3. Identification of material parameters The application of GTN model to failure analysis needs determination of 14 material parameters including elastic parameters (E and ), hardening parameters (K, and n), coefficients of yield function ( , and ), void nucleation parameters (𝑓 𝑓 ) and void volume fractions of coalescence and fracture (𝑓 𝑓 ). In order to identify these parameters, uniaxial tensile tests have been carried out on the flat specimens taken from the SAE 1010 plain carbon steel sheet (chemical composition in Table 1) with the thickness of 1 mm in the rolling direction. The used material with the manufacturer designation St 12 (SAE 1010 plain carbon steel) has been undergone annealing heat treatment to increase its formability. The mean elongation and hardness according to the manufacturer are 42% and 105 HV respectively. The dimensions of specimens which were prepared according to ASTM E8, are illustrated in the Fig. 2. The tests were performed on an extensometer equipped hydraulic testing machine with the set speed of 1 mm/min on a gage length of 50 mm (Fig. 1). The tests were repeated 3 times in order to achieve accurate results. The procedures to identify 14 parameters are descried in the following sections.

6

Table 1. Chemical composition of SAE 1010 plain carbon steel sheet (% in weight)

C

Mn

Si

P

S

Al

0.1

0.5

0.16

0.025

0.025

0.02

Fig. 1. The flat tension specimen mounted on the extensometer equipped testing machine

Fig. 2. Flat tensile test specimen dimensions (mm) used for parameters determination

3.1. The hardening parameters In order to determine the mechanical and hardening behaviors of SAE 1010 plain carbon steel sheet, the uniaxial tensile test carried out as described in the previous section. The tension test undergone specimen, the consequent force-displacement plot and the extracted engineering and true stress-strain plots are presented in the Fig. 3 and Fig. 4. In the present study for describing the isotropic hardening behavior of material, the well-known Swift’s law [33] has been applied: ( ̅ )

(

(13)

̅ )

where , and are material hardening constants determined using experimental stressstrain curve extrapolation with the least squares method (Fig. 5). The quantities identified for the material elastic-plastic parameters are listed in the Table 2. In the present work, our study is focused on the rolling direction and all the uniaxial tensile specimens have been prepared from that direction, so the isotropic hardening law is applied. If the specimens are cut from different direction and the pulling direction vary, the use of isotropic hardening law is 7

problematic for accurate characterization of plasticity in blind assessments as pointed out by Boyce et al. [34-36] and the effect of anisotropy should be considered.

Fig. 3. Tensile test specimen 3500

400

3000

300

Stress (Mpa)

2000 1500

200

1000

True Stress-Strain Engineering Stress-Strain

100 500 0 0

5

10

15

20

25

0 0.0

Displacement (mm)

0.1

0.2

0.3

Strain

(a) (b) Fig. 4. (a) force-displacement plot and (b) engineering and true stress-strain plots 400 350

True Stress (Mpa)

Force (N)

2500

Experimental Swift's law fitting

300 250 200 150 100 50 0.00

0.05

0.10

Equation

K*(e+x)^n

K

480.53217

e

0.0109

n

0.24437

R-Square

0.99987

Adj. R-Square

0.99986

0.15

0.20

0.25

0.30

0.35

True Plastic Strain

Fig. 5. The true plastic stress-strain curve approximation to achieve hardening parameters

8

0.4

0.5

Table 2. Elastic-plastic material parameters of SAE 1010 plain carbon steel sheet (% in weight)

E ( MPa)

K( MPa)

200000

0.3

480.53

0.0109

0.24

3.2. The yield surface coefficients As mentioned earlier, Tvergaard [8, 9] introduced three constants as the void interaction parameters to improve the model functionality in giving more compatible results with the experiments. These parameters include , and where represents the loss of strength due to void interaction however and take into account the effect of stress triaxiality and void volume fraction, respectively. In order to better understand GTN model parameters in terms of quantity range and dependency on the material type, a literature review was performed and its results have been presented in Table 3. Since the most of references in Table 3 have been adopted the same quantities proposed by Tvergaard et al. [12] for the yield surface coefficients, it is reasonable to follow the instruction. So in the present study the yield function parameters are assumed to be , and . Table 3. The GTN model parameters in literature

Author

Material

𝑓

𝑓

𝑓

𝑓

Tvergaard et al. [12]

-

1.5

1

2.25

0.3

0.1

0

0.04

0.15

0.25

Hambli [37]

Carbon steel

1.5

1

2.25

0.3

0.1

-

0.04

-

-

20MnMoNi55

1.5

1

2.25

0.3

0.1

0

0.002

0.06

0.212

Steels DD13, X6Cr17

1.5

1

2.25

0.3

0.1

-

0.04

0.1

0.101

Steel

1.5

1

2.25

0.3

0.1

0.001

0.01

0.01

0.15

1.5

1

2.25

0.2

0.1

-

0.04

0.15

0.25

1.5

1

2.25

0.3

0.1

-

0.04

0.11

0.12

S235JR steel

1.91

0.79

3.65

0.3

0.05

0.0017

0.04

0.06

0.6

ASTM A992

1.5

1

2.25

0.45

0.05

0

0.02

0.03

0.5

HSLA

1.5

1

2.25

0.3

0.1

0

0.04

0.1

0.15

HSLA

1.2

0.8

1.44

0.2

0.1

0.0015

0.02

0.08

0.13

Zircaloy-4

1.5

1

2.25

0.1

0.02

0

0.012

0.03

0.08

AA6016-T4

1.5

1

2.25

0.3

0.1

0.00035

0.05

0.05

0.15

DP600

1.5

1

2.25

0.2

0.1

0.0008

0.02

0.028

0.09

Schmitt et al. [38] Rachik et al. [39] Springmann et al. [40] Lemiale et al. [41] Marouani et al. [42] Kossakowski et al. [43] Kiran et al. [44] Achouri et al. [45] Achouri et al. [46, 47] Zhou et al. [48] Kami et al. [49] Zhao et al. [50]

Mild steel XES FeSi (3wt.%) steel

9

Jiang et al.[51]

Al 2024 T3

1.5

1

2.25

0.2

0.1

0.004

0.02

0.025

0.15

3.3. The void related parameters In general, there are two approaches to determine void related parameters of GTN model; the direct and indirect/inverse methods. The direct approach involves the investigation of the microstructure of fractured specimens by the optical microscopy or SEM and the interpretation of microscopic images, direct current potential drop (DCPD) [16] and unit cell model simulation [52, 53]. The determination of parameters by direct or microstructural approach is completely dependent on the location of RVE chosen from the material. Furthermore some investigation [54, 55] have shown that the extracted inclusion volume fraction from microstructural investigation cannot be directly used in the GTN due to the fact that not all of the inclusions will participate in the void nucleation. For these reasons, the identification of GTN model parameters by direct approach is not much recommended. Consequently, it is suggested to determine the parameters by the indirect/inverse/macroscopic investigations. The invers approach [23, 49, 56] relies on the calibration of the GTN model parameters by fitting the finite element and experimental results of a process such as uniaxial tension test carried out on a standard specimen. In the present study the inverse calibration method was utilized to determine the void related parameters of GTN model. After deciding on the yield function coefficients, the remained GTN model parameters can be ) and the void divided to two categories: the nucleation rate function parameters ( volume fraction parameters (𝑓 𝑓 𝑓 𝑓 ). Since the simultaneous and exact determination of several parameters by inverse approach is a challenging process, in the present study it was decided to reduce the number of parameters to be determined so we could be able to put more attention on the precise identification of the remained parameters. Therefore, the amounts of ) was treated according to literature. By nucleation rate function parameters ( investigating the expressed values for these two parameters from the literature in Table 3, the amounts of 0.3 and 0.1 were specified for and respectively. The correct identification of the remained GTN model parameters (𝑓 𝑓 𝑓 𝑓 ) is a necessity for prosperous analysis of the ductile fracture by implementation of this model. In the present study in order to achieve this goal, the inverse calibration procedure using response surface method (RSM) has been applied. RSM uses a set of designed experiments to discover the relationships between several influencing variables and the response variable for achieving an optimal response. In this method, some techniques are used for developing the functional relationship between a response, y, and a number of associated control variables denoted by . In general, such a relationship is unknown but can be approximated by a low-degree polynomial model of [57] 𝑓( )

(14)

( ) 𝑓( ) is a vector function of elements that consists of powers where ) is a and cross products of powers of up to a certain degree denoted by ( vector of unknown constant coefficients referred to as parameters, and is a random experimental error assumed to have a zero mean. This is conditioned on the belief that model of equation (14) provides an adequate representation of the response. In this case, the quantity 𝑓 ( ) represents the mean response, that is, the expected value of , and is denoted by ( ) [57]. Two important models are commonly used in RSM named as first-degree and seconddegree models. The first-degree model ( ) is as follows 10

(15)



where is the main effect level, is the main effect and is the error. Also, the relationship between response and variables can be estimated by a second-degree polynomial model of the form ∑

∑∑

(16)



In this model, is the interaction effect and is the curvature effect. The seconddegree model has been applied for the present study. More details about the RSM and a case study of its application can be found in [57] and [58] respectively. In order to apply RSM optimization, four void volume fraction parameters (𝑓 𝑓 𝑓 𝑓 ) were treated as continuous variables and a central composite design was implemented to get experiments configuration. By looking into Table 3, the minimum and maximum ranges of these continuous parameters were determined and are presented in Table 4. A plan of 31 experiments including six replicates in central point was created by central composite design with face centered alpha design which are listed in Table 5. These experiments are numerical simulations of the uniaxial tensile test (described in previous section) with the specified combinations of parameters in Table 5. These numerical simulations were carried out in ABAQUS/Explicit with the geometry of Fig. 2 and the other conditions are detailed in section 4.3. The optimum minimum element size in the fracture zone for the flat specimen was determined to be 0.5*0.5 mm by performing mesh sensitivity studies. In the present study, the force-displacement curve was chosen as a response variable. Therefore, six components related to three critical points on force-displacement curve were regarded to obtain responses from simulation runs. The responses include , , , , and which are corresponded to extension at fracture, force at fracture, extension at maximum force, maximum force, extension of 50% and the force at that extension respectively. The quadratic polynomial regression model was selected in RSM for estimating the relations between the responses and four influencing variables. The dependency of responses to influencing variables are according to following equations: 𝑓 𝑓𝑓 𝑓𝑓

𝑓

𝑓

𝑓

𝑓𝑓

𝑓 𝑓𝑓

𝑓

𝑓

𝑓

𝑓

𝑓

𝑓 𝑓𝑓

𝑓 𝑓𝑓

𝑓 𝑓𝑓

𝑓𝑓

𝑓 𝑓𝑓

11

𝑓 𝑓𝑓

𝑓

𝑓𝑓

𝑓

𝑓𝑓

𝑓

𝑓 𝑓𝑓

(17)

(18) 𝑓𝑓 𝑓𝑓

(19) (20)

𝑓 𝑓𝑓

𝑓

𝑓 𝑓𝑓

𝑓 𝑓𝑓

𝑓

𝑓

𝑓

𝑓𝑓

𝑓𝑓

𝑓

𝑓

𝑓

𝑓 𝑓𝑓 𝑓𝑓

(21)

(22)

𝑓𝑓

The analysis of variance (ANOVA) was carried out to determine the statistical importance and effectiveness of full quadratic models. The results for (R-squared) and predicted are given in Table 6. R-squared is used in regression analysis to indicate how well the model fits your data while the predicted indicates how well the model predicts responses for new observations. The results show high capability of six quadratic response models for fitting the data and predicting the responses for new cases. Table 4. Void volume fraction parameters range

Parameter Range

0-0.004

0.002-0.05

0.01-0.15

0.15-0.6

Table 5. Central composite design test arrangement Run

Run

1

0.002

0.026

0.01

0.375

17

0.000

0.050

0.01

0.600

2

0.000

0.002

0.15

0.600

18

0.000

0.026

0.08

0.375

3

0.002

0.026

0.08

0.375

19

0.002

0.050

0.08

0.375

4

0.002

0.026

0.08

0.375

20

0.002

0.026

0.08

0.600

5

0.002

0.026

0.08

0.375

21

0.000

0.050

0.15

0.150

6

0.002

0.026

0.08

0.375

22

0.000

0.050

0.15

0.600

7

0.002

0.026

0.15

0.375

23

0.004

0.002

0.01

0.150

8

0.004

0.002

0.01

0.600

24

0.004

0.050

0.15

0.600

9

0.002

0.026

0.08

0.375

25

0.002

0.026

0.08

0.375

10

0.000

0.002

0.01

0.600

26

0.002

0.002

0.08

0.375

11

0.004

0.050

0.01

0.150

27

0.000

0.050

0.01

0.150

12

0.002

0.026

0.08

0.150

28

0.000

0.002

0.01

0.150

13

0.004

0.050

0.15

0.150

29

0.004

0.002

0.15

0.600

14

0.002

0.026

0.08

0.375

30

0.000

0.002

0.15

0.150

15

0.004

0.026

0.08

0.375

31

0.004

0.050

0.01

0.600

12

16

0.004

0.002

0.15

0.150

Table 6. Values of

(R-squared) % Predicted

%

and predicted

for six response variables

99.34

91.9

97.77

99.93

99.19

85.03

96.25

84.69

94.92

99.66

96.25

79.58

Finally, the optimum values of influencing variables (𝑓 𝑓 𝑓 𝑓 ) were identified by minimizing the difference between numerical and experimental force-displacement plots of tensile test. To achieve this goal, the amounts for six responses from the experimental forcedisplacement curve were entered as the targets into response optimizer of RSM, and the optimum values of (𝑓 𝑓 𝑓 𝑓 ) were determined by minimizing the differences of six RSM responses and target values simultaneously. The identified optimum values by implying inverse calibration through RSM were 0.00107, 0.00716, 0.01 and 0.15 for 𝑓 , 𝑓 , 𝑓 and 𝑓 respectively. The composite desirability of these optimum values is 0.924, which indicates the determined parameters well satisfy the defined targets for the responses. In order to validate the optimization process, the uniaxial tensile test was numerically simulated with the optimized parameters of GTN model and the force-displacement plot was compared with the experimental one. The results are presented in Fig. 6 which manifest an excellent agreement. Therefore, the optimization process was successful in providing reliable GTN model parameters. To conclude the GTN model input parameters identification procedure, the 9 determined parameters for SAE 1010 plain carbon steel sheet are given in Table 7.

4000 3500

Force (N)

3000 2500 2000

Experimental Numerical Simulation

1500 1000 500 0 0

5

10

15

20

25

Displacement (mm)

Fig. 6. Comparison of force-displacement plots from the experiment and numerical analysis with the optimum parameters.

13

Table 7. The GTN model parameters for SAE 1010 plain carbon steel sheet

1.5

1

2.25

0.1

0.3

0.00107

0.00716

0.01

0.15

4. Investigation of the fracture behavior of in-situ tensile tests In the present section, the GTN model, which was elaborated in the second section and the calibration procedure with the standard tension specimens, was explained in the third section, is applied to study damage and predict the fracture of in-situ tensile tests on specimens with various geometries. 4.1. In-situ tensile tests In order to study the ductile fracture behavior under different stress states, the in-situ tensile tests were conducted on six specimens of rectangular cross section with different shapes. The specimens No. 1 to 5 with different radius in deformation zone have fracture zone perpendicular to the applied test load direction and are called specimens. While the specimen No. 6, which has fracture zone inclined to the test load direction, is called specimen. The dimensions of the specimens which were cut from the SAE 1010 plain carbon steel sheet with the thickness of 1 mm in the rolling direction, are presented in Fig. 7. In order to ensure the repeatability of the tests and also achieve accurate results, the test on each shape was repeated three times. All tests were carried out on an extensometer equipped hydraulic testing machine with the same setup of speed and gage length as stated for the standard tensile specimen in section 3. The force-displacement plots for specimens 1 to 5 are presented in Fig. 8 and for specimen 6 is shown in Fig. 9.

No. 1

No. 2

14

No. 3

No. 4

No. 5 Fig. 7. Six tensile test specimens with different shapes

No. 6

3500 3000

Force (N)

2500 2000 1500 R 20 R 10 R5 R2 R 0.5

1000 500 0 0

1

2

3

4

Displacement (mm)

Fig. 8. Force-displacement plots for specimen number 1 to 5

15

5

2500

Force (N)

2000

1500

1000

No. 6 500

0 0

1

2

3

4

5

Displacement (mm)

Fig. 9. Force-displacement plot for number 6 specimen

4.2. Fractographic analysis The degradation of material properties during ductile fracture is controlled by ductile damage mechanism. From the micromechanical aspect, the ductile damage mechanism generally includes initiation, growth and coalescence of micro-defects, which results in failure and ductile rupture. Micro-cracks and micro-voids are two main micro-defects in engineering metals. In the present study, the in-situ tensile tests with different geometries were performed to evaluate the fracture evolution under various stress conditions. To understand the fracture mechanism of the material, the microstructure and deformation process of in-situ samples were observed. The microstructure image of complete sheet thickness and a part of material in rolling direction before deformation is shown in Fig. 10. Fig. 11 Presents the fractured edges of six in-situ tensile specimens. It shows the effect of geometry variation and consequently the triaxiality condition in deformation zone on the shape of resulting fractured edges. Fig. 12 a and b show the deformation zones prior to final fracture in a and the in – situ tensile specimens and Fig. 12 c and d reveal the fracture mechanism respectively. As it can be seen the fracture in specimens is caused by the occurrence of micro-voids coalescence while in the specimen the fracture is handled by both micro-voids and micro-cracks. In the specimens, the initiation of new voids and growth of the existing voids under tensile loads in the region of the minimum section results in a high concentration of voids. The coalescence of these voids leads to crack formation and fracture perpendicular to the tensile direction. In the specimen, the void nucleation and growth occur at lower rates at the inclined deformation zone to the loading direction. Instead, the shear fracture which was activated by the micro-cracks at deformation zone accompanies the void coalescence and the mixed micro-voids as well as a dominant micro-crack fracture would occur at the same time.

16

(a)

(b)

Fig. 10. Optical images of microstructure (a) complete sheet thickness and (b) a part of material in rolling direction

R20

R10

R5

R2

R0.5

45◦

Fig. 11. Six tensile test specimens with different shapes

17

(a)

(c)

(b)

(d)

Fig. 12. Optical microscope images of (a) a in–situ tensile specimen and (b) the in–situ tensile specimen prior the final fracture compared to SEM fractographs [59] of tensile test specimens with the fracture zones, (c) perpendicular and (d) inclined to tension direction.

4.3. Finite element model In the present work, the finite element simulations have been carried out for parameter calibration in RSM optimization (section 3.3) and for fracture studies on in-situ tensile specimens using GTN damage model in the ABAQUS/Explicit. In the numerical simulations of the uniaxial tensile test on the standard specimen (described in section 3) with different combinations of GTN parameters, the geometry was modeled and meshed with the C3D8R type elements. In order to better capture the localized deformation and damage and also reduce the computational times, the refined and coarse meshes have been used within and outside of the central region respectively. For the numerical simulation of fracture studies on six specimens, the complete geometry of each specimen was modeled. The same mesh type and mesh refining strategy for the standard specimen were also applied to these specimens. The generated mesh pattern for the numerical studies on six specimens has been illustrated in Fig. 13. The optimum minimum element size in the fracture zone for the specimens R20, R10, R5, R2, R0.5 and were determined to be 0.5*0.5 mm, 0.45*0.45 mm, 0.31*0.31 mm, 0.2*0.2 mm, 0.14*0.14 mm and 0.06*0.06 mm respectively by performing mesh sensitivity studies. In order to resemble tensile test 18

condition, the bottom of the specimens was fixed and the displacement constraint was applied in the top region.

19

Fig. 13. The finite element mesh implemented on different tension specimens

5. Results and discussion The results of numerical simulations of in-situ tensile specimens using GTN damage model with the parameters determined in section 3 are presented in the following section. Fig. 14 exhibits the experimental force-displacement plots in addition to the numerical predictions which were extracted from FE simulations. It can be seen from Fig. 14 a to e that there is a good agreement between numerical and experimental force-displacement plots for specimens. The extracted numerical results show a good agreement with experimental observations comparing load-displacement plots with a margin of error within 5%. A better ductile fracture predictions were captured in 90o specimens. These specimens due to their geometry and the direction of applied load undergo medium to high triaxiality and the tensile stress state is dominant. The failure mechanisms of GTN model including void nucleation, growth and coalescence act well under tensional stress conditions. The force-displacement plots for specimen (Fig. 14 f) exhibit some discrepancy between experimental and numerical results. This is attributed to the fact that there is a combination of shear and tensile stress state in this specimen (the mixed fracture mechanism as discussed in section 4.2) but the shear contribution to failure cannot be captured by the GTN model with void nucleation, growth and coalescence mechanism. This is one of the main drawbacks of GTN model addressed by some researchers [48, 51, 60-62]. Therefore, the fracture in experimental tests occurs earlier relative to the numerical predictions. The comparison of fracture edges of experimental in-situ tensile tests and numerical simulations for six specimens are depicted in Fig. 15. As it is obvious, the overlapped pictures show remarkable agreement between the experimental and numerical fracture edges for all samples. The stress triaxiality, equivalent plastic strain and void volume fraction at fracture initiation are presented in Fig. 19, Fig. 20 and Fig. 21 respectively. In these figures, fracture initiation is decided when void volume fraction of an element has reached to the value of 0.15 which is the void volume fraction at fracture (𝑓 ) and was determined by the RSM. In specimens, with the specimen notch radius varying from 20 mm to 0.5 mm the triaxiality is increased. Higher values of stress triaxiality accelerates the void initiation, growth and coalescence phenomenon and the material failure occur earlier at lower amounts of fracture strain and fracture displacement. In specimens the initiated crack grows perpendicular to the loading direction and the fracture occurs in a plane normal to the specimen axis. In the 20

1.0

1.0

0.8

0.8

Normalized Force

Normalized Force

specimens R20, R10, and R5 the triaxiality value has its maximum in the center of specimen thus void volume fraction reaches the critical value at this point earlier and the crack initiates in the center propagating toward the specimen surface. In contrast, in the cases related to R2 and R0.5 specimens, the cracks start near the surface of minimum cross section region of the specimen from both sides and grow toward the center of specimen. For specimen the mixed micro-voids and micro-cracks fracture occurs at an inclined fracture plane with the angle of to the loading direction. In this specimen, due to the higher stress triaxiality values at the two corners of the inclined fracture plane compared to other areas of the same specimen, two cracks initiate at these regions and develop through the fracture plane at the same time. The final failure happens when these cracks meet at the center of fracture plane. Fig. 22 represents the evolution of the void volume fraction with equivalent plastic strain in the central element of in-situ tensile test specimens. As understood, due to the variant dominant stress states in specimens with different geometries, the evolution of void volume fraction in specimens with smaller notch radius occurs faster so that the void damage catches up with the critical value earlier for R0.5 specimen. According to the hypothesis of GTN model, the nucleation and growth of voids only occur under tensile stress condition. As it is obvious from the figure, the evolution of void volume fraction in specimen due to weakened tensile stress state on the inclined fracture plane occurs much slower relative to specimens. Instead, the condition of low stress triaxiality and activated shear failure mechanism plays an important role in fracture as well as the void damage. Fig. 23 reveals the variation of fracture strain and displacement at final fracture in terms of stress triaxiality for the first failed element of in-situ tensile specimens. As it can be seen, the specimen with mixed mode fracture mechanism has the lowest fracture strain and displacement at final fracture in comparison to the other tensile samples. In specimens, the fracture strain and displacement at final fracture is decreased by the reduction of notch radius resulting in high stress triaxiality condition.

0.6

Numerical Experimental

0.4

0.6

0.4

Numerical Experimental

0.2

0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

0.4

0.6

0.8

Normalized Displacement

Normalized Displacement

(a)

(b)

21

1.0

1.0

0.8

0.8

Normalized Force

Normalized Force

1.0

0.6

0.4

Numerical Experimental

0.2

0.6

0.4

Numerical Experimental

0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

Normalized Displacement

0.4

0.8

1.0

(d)

1.0

1.0

0.8

0.8

Normalized Force

Normalized Force

(c)

0.6

0.4

0.6

Normalized Displacement

Numerical Experimental

0.2

0.6

0.4

Numerical Experimental

0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

Normalized Displacement

0.2

0.4

0.6

0.8

1.0

1.2

Normalized Displacement

(e)

(f)

Fig. 14. Comparison of experimental and numerical force-displacement plots: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.5 and (f) specimen

(a)

(b)

22

(c)

(d)

(e)

(f)

Fig. 15. Comparison of fracture edges of experimental in-situ tensile tests and numerical simulations for six specimens: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.5 and (f) specimen

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 16. Comparison of fracture edges of experimental in-situ tensile tests and numerical simulations for six specimens: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.5 and (f) specimen

(a)

(b)

23

(c)

(d)

(e)

(f)

Fig. 17. Comparison of fracture edges of experimental in-situ tensile tests and numerical simulations for six specimens: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.5 and (f) specimen

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 18. Comparison of fracture edge boundary shape of experimental in-situ tensile tests and numerical simulations for six specimens: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.5 and (f) specimen

(a)

(b)

24

(c)

(d)

(e)

(f)

Fig. 19. Contour plots of stress triaxiality for in-situ tensile specimens at fracture initiation: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.5 and (f) 45◦ specimen

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 20. Contour plots of equivalent plastic strain for in-situ tensile specimens at fracture initiation: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.5 and (f) 45◦ specimen

25

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 21. Contour plots of void volume fraction for in-situ tensile specimens at fracture initiation: (a) R20, (b) R10, (c) R5, (d) R2, and (e) R0.55 and (f) 45◦ specimen

26

Void volume fraction

0.20 R 20 R 10 R5 R2 R 0.5

0.15

45'

0.10

0.05

0.00 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Equivalent plastic strain

Fig. 22. The evolution of void volume fraction in central element of tensile specimens

1.00 1.0

R 20

Fracture Strain

0.9

Normalized Displacement

0.90

Fracture Strain

0.8

R 10

0.85

0.7

0.80

0.6

R2

0.75

0.5

0.70

0.4

R5

0.65

0.3

45' 0.60

Normalized Displacement

0.95

0.2

R 0.5

0.55

0.1

0.50

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Triaxiality

Fig. 23. The fracture strain and normalized displacement at fracture for six specimens

6. Conclusion In this study, the GTN model was implemented to investigate the ductile fracture in a low carbon steel. The study has been organized in two main parts including the identification of GTN model parameters for the under study material and then investigation of the fracture behavior by in-situ tensile tests. In the first part, the uniaxial tensile tests were carried out on the flat specimens taken from the SAE 1010 plain carbon steel sheet and resultant strain-stress plots were used to determine 27

the elastic-plastic mechanical parameters. Afterwards, the response surface method (RSM) was implemented to determine the GTN damage model parameters. The void related parameters of GTN model were determined 0.00107, 0.00716, 0.01 and 0.15 for 𝑓 , 𝑓 , 𝑓 and 𝑓 respectively. The uniaxial tensile test was numerically simulated by entering the identified parameters to the GTN model and the force-displacement predictions was compared to the experimental plot where the results showed an excellent agreement. In the second part, the experimental tests and the calibrated model were employed to study the ductile failure in six in-situ tensile specimens with different shapes. The effect of geometry variation and triaxiality condition in deformation zone on the shape of resulting fractured edges was taken from experiments and the corresponding shape extracted from numerical simulations exhibited a remarkable accordance. The micro-voids coalescence and the mix mode by both micro-voids and micro-cracks was analyzed to be the main fracture mechanisms in 90° and 45° specimens respectively. The location of fracture initiation, crack growth orientation and the displacement at final fracture for numerical studies has also showed close correspondence to the experimental results. The extracted numerical results showed a good agreement with experimental observations comparing load-displacement plots with a margin of error within 5%. A better ductile fracture predictions were captured in 90o specimens.

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