Characterization of ductile damage for a high carbon steel using 3D X-ray micro-tomography and mechanical tests – Application to the identification of a shear modified GTN model

Computational Materials Science 84 (2014) 175–187

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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Characterization of ductile damage for a high carbon steel using 3D X-ray micro-tomography and mechanical tests – Application to the identification of a shear modified GTN model T.-S. Cao a,⇑, E. Maire b, C. Verdu b, C. Bobadilla c, P. Lasne d, P. Montmitonnet a, P.-O. Bouchard a a

Mines ParisTech, CEMEF – Center for Material Forming, CNRS UMR 7635, BP 207, 1 rue Claude Daunesse, 06904 Sophia Antipolis Cedex, France INSA-Lyon, MATEIS UMR 5510, 25 av. Capelle, 69621 Villeurbanne, France ArcelorMittal Long Carbon R&D, Gandrange, France d TRANSVALOR S.A., Parc de Haute Technologie, Sophia Antipolis, 694, av. du Dr. Maurice Donat, 06255 Mougins Cedex, France b c

a r t i c l e

i n f o

Article history: Received 13 November 2013 Accepted 2 December 2013 Available online 30 December 2013 Keywords: Ductile damage Micro-tomography Modeling GTN model Mechanical tests

a b s t r a c t The present paper deals with the identification of the parameters of a generalized GTN model and the characterization of ductile damage for a high carbon steel by both X-ray micro-tomography and ‘‘macroscopic’’ mechanical tests. First, in situ X-ray micro-tomography tensile tests are performed and the results are used for the modeling of ductile damage mechanisms (voids nucleation, growth and coalescence) using analytical formulations. Interrupted in situ SEM tensile test is also carried out to examine the microstructure evolution. The damage process during in situ X-ray micro-tomography tensile tests is the result of continuous nucleation of small voids and significant growth of large voids; whereas the coalescence takes place locally. In addition, tomography results combined with the results of macroscopic mechanical tests at different loading configurations are used to identify the Gurson–Tvergaard–Needleman model extended for shear loading by Xue (2008). It proved necessary to propose an improvement to account for the influence of the stress triaxiality level on the nucleation formulation of the GTN model. This new formulation is then identified via experimental tests. The results show that, with the parameters obtained from both microstructure measurements and macroscopic considerations, the modified GTN model can reproduce quite accurately the experimental results for different loading configurations. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Understanding and modeling of ductile damage mechanisms has been a subject for numerous studies in the literature [37,19,43,5]. Microscopically, damage is associated with voids nucleation, growth and coalescence in high and moderate stress triaxiality or shear band formation in low stress triaxiality. Macroscopically, damage is represented as the progressive degradation of material, which exhibits a decrease in material stiffness and strength. Rice and Tracey [37] studied the evolution of spherical voids in an elastic–perfectly plastic matrix and showed that the voids growth is governed by stress triaxiality, the ratio between the mean stress and the von Mises equivalent stress. In this study, the interaction between microvoids, the coalescence process and the hardening effects were neglected and failure was assumed to occur when the cavity radius would reach a critical value specific

⇑ Corresponding author. Current address: Mines ParisTech, CNRS UMR 7633, Centre des matériaux, BP 87, F-91003 Evry cedex, France . Tel.: +33 668264611. E-mail address: [email protected] (T.-S. Cao). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.12.006

for each material. Gurson [19], in an upper bound analysis of a finite sphere containing an isolated spherical void in a rigid perfectly plastic matrix, employed the void volume fraction f (or porosity) as an internal variable to represent damage and its softening effect on material strength. This model was then improved to account for different aspects: prediction accuracy [41], void nucleation [15], void coalescence [34,43], void shape effect (e.g. [18,35,28], void size effect (e.g. [46]), void/particle interaction (e.g. [38]), strain hardening (e.g. [27,32]), plastic anisotropy (e.g. [6,7]), rate dependency (e.g. [42]), shear effect (e.g. [33,47]). These extensions were developed to make the model able to describe the real mechanical behavior of the studied materials. However, except the Gurson–Tvergaard–Needleman (GTN) model, very few above-mentioned extensions were applied to real scale structural modeling. These authors often validated their models based on cell computations. Regarding the GTN model, although it is well known to give accurate results in moderate to high stress triaxiality, it still suffers some limitations, among which the inability to capture shear-driven ductile damage. In addition, since the GTN model is derived from microstructure consideration, special attention has to be paid to the identification

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of its parameters. Many authors in the literature carried out the identification of the GTN model parameters on different materials based principally on ‘‘macroscopic’’ mechanical tests. The identification of the original GTN model is often based on tensile tests on notched round bar or compact tension (CT) specimens, where the deformation is localized in the notch area and the stress triaxiality is relatively high. An inverse analysis methodology is often used to identify this set of parameters (e.g. [1]), based on stress–strain curves (or load–displacement curves). However, in these identifications, the 3 constitutive parameters q1 ; q2 ; q3 are often taken from the study of Needleman and Tvergaard [34], in which: q1 ¼ 1:5; q2 ¼ 1 and q3 ¼ ðq1 Þ2 . Note that a high value of q1 or q2 leads to a high retraction of the yield surface at a given value of f. The nucleation parameters (fN ; N ; SN ) are also often fixed: as they link directly to microstructural evolution, the measurement of these parameters is not straightforward. Recently, new and advanced experimental techniques allow identifying more accurately the values of the GTN model parameters. He and coworkers [20] among others, used in situ tensile tests with scanning electron microscope (SEM) to determine these parameters. This identification was based on the counting of void volume fraction at three damage states. This method gave them relatively exact values of initial void volume fraction as well as void volume fractions at 3 instants of measurement. However, this method could not give a continuous increase of void volume fraction with the plastic strain and could not distinguish the two different damage mechanisms: void nucleation and growth. Moreover, the 3 values of q1 ; q2 ; q3 were also fixed as in Needleman and Tvergaard [34]. The characterization of each stage of ductile damage requires the continuous monitoring of damage during deformation, which can be done thanks to X-ray tomography measurements [45,29,24,16]. Bettaieb, Fansi and their co-workers [9,17] used this technique to identify the parameters of an extended GTN model with kinematic hardening and anisotropic matrix for a DP steel. Accurate results were obtained by these authors by applying the extended model to microstructural simulations, but the authors did not present the applications to any macroscopic simulations. More recently, Thuillier et al. [40] identified the GTN model parameters for an aluminum alloy, based on a hybrid study of microtomography observations and numerical modeling. However, in their numerical simulation with Abaqus, the coalescence was not taken into account and no fracture criterion was defined. In addition, none of the above-mentioned studies analyze the consistency between microscopic and macroscopic experimental results. In the present paper, ductile damage of a high carbon steel is studied both from microscopic consideration of damage (voids evolution) and its macroscopic impact (load–displacement curves). The identification of the GTN and modified GTN models for this material is then detailed. The calibration process is based on both in situ X-ray micro-tomography tensile test (conducted at the European Synchrotron Radiation Facility – ESRF, in Grenoble, France) and the macroscopic mechanical tests for different loading configurations. In addition, an in situ SEM interrupted tensile test was also carried out to examine the microstructure evolution. First, a short presentation of the experimental and numerical techniques used is given. Then, the analyzed results are divided into two parts. In the first part, the experimental results from in situ X-ray tomography observation are exploited. Analytical formulations are used to model different mechanisms of ductile damage (e.g. void growth and nucleation). In the second part, the identification of the GTN model is carried out, using both in situ tensile test and macroscopic mechanical tests. A formulation for nucleation strain is proposed to account for the influence of the stress triaxiality on the nucleation process. Finally, the modified GTN model for shear-dominated loading by Xue is calibrated based on the torsion test.

2. Experimental techniques and numerical models 2.1. Material and experimental tests 2.1.1. Material The material used in the present study is a high carbon steel grade, which presents a fine pearlite structure after a patenting process. All the specimens used in the mechanical tests are extracted from the longitudinal direction of steel rods of maximum diameter of 17 mm. The mechanical property of this steel grade can be considered isotropic at patented state (see e.g. [31]). The fracture surface of a tensile specimen is shown in Fig. 1, revealing dimpled fractured surface. Moreover, from this figure, two populations of voids with different sizes can be detected. 2.1.2. Tomography observations used in in situ micro-tensile test X-ray micro-tomography was used to quantify damage during in situ tensile tests on notched round bars. In this experiment, the characterized object rotates around a single axis while a series of 2D X-ray absorption images is recorded. This series of images is used to reconstruct a 3D digital image where each voxel (volume element or 3D pixel) represents the X-ray absorption at that point. The experimental setup is shown in Fig. 2a, on ID15A beamline at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. This beamline along with ID19 beamline have been used in the present study. The ID15A beamline has a resolution of 1.1 lm voxel size and each scan lasts 5 s, while the ID19 one has a resolution of 0.7 lm voxel size but more time is needed to acquire each scan. The ID15A was used principally in the present study and the results with ID19 confirmed ID15A observations. An interrupted in situ test was carried out on ID19 beamline while continuous testing was used on ID15A. For the interrupted method, the deformation is stopped but maintained constant during imaging; while for the continuous characterization, imaging is carried out continuously during the deformation without interruption. The latter procedure is obviously better to characterize the evolution of the sample during the tensile test, but the drawback is that the scan time has to be small compared to the deformation speed to avoid blurring of the images [23]. For this reason, this method is more suitable for the ID15A beamline, in which the scanning time is shorter than the ID19. It was demonstrated that the results obtained with the two methods (interrupted or continuous) are similar (see Maire et al. [30] and Suéry et al. [39] for more details and see also Appendix B for the comparison obtained in this study between these two beamlines). The geometry of the tensile specimen is presented in Fig. 2b (as in Landron et al. [24]), which gives a value of the initial stress triaxiality of 0.425 according to Bridgman’s formula:

1 3



g ¼ þ ln 1 þ

a 2R

ð1Þ

where g is the stress triaxiality, a (mm) is the minimum radius of cross section, R (mm) is the notch radius. Due to the strong localization of deformation and high triaxiality in the central area of notch, this area is assumed to have the highest level of damage. Therefore, only a central cubic volume of ð385 lmÞ3 was chosen for the quantification of damage. During the in situ tensile test,1 the area of minimal cross section S was measured to calculate the average true strain as well as the true stress in the minimal cross section of the specimen:

loc ¼ ln

  S0 ; S

rtrue ¼

F S

ð2Þ

1 Throughout the present study, the term ‘‘in situ tensile test’’ refers to ‘‘in situ X-ray tomography tensile test’’, unless otherwise indicated.

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Fig. 1. Fractured surface of tensile specimen observed with SEM showing dimples.

1

Notched round bar Flat grooved specimen

Triaxiality

0.5

Plane strain

0

-0.5 Round bar Torsion Fig. 2. X-ray tomography experimental setup on ID15A: (a) tensile machine is mounted on the rotation stage of tomograph to achieve the in situ tensile test and (b) axisymmetric tensile sample: 2.5 mm radius notched specimen. All dimensions are in mm (figures adapted from Landron et al. [24]).

2

where S0 is the initial minimum cross section (mm ); F is the measured force (kN). By measuring the curvature radius  pffiffiffiffiffiffiffiffiffiof notch (R) and the radius of the minimal cross section a ¼ S=p , the stress triaxiality ratio can be calculated via Eq. (1), which was validated by numerical simulations (see [11]). Therefore, the evolution of the stress triaxiality with the equivalent plastic strain can be obtained from experiments with one of the two beamlines, ID15A or ID19. 0.55

Stress triaxiality

0.5 0.45 0.4 0.35 0.3 ID15A ID19

0.25 0.2 0

0.1

0.2

0.3

Equivalent plastic strain Fig. 3. Evolution of the stress triaxiality with the equivalent plastic strain during the in situ tensile test with two beamlines: ID15A and ID19.

-1 -1

0

1

Lode parameter Fig. 4. Representations of macroscopic mechanical tests performed in the space of initial stress triaxiality and Lode parameter.

From Fig. 3, the results obtained with two beamlines are in good agreement, although with the ID19, only three scans could be performed. In addition, during the tensile test, the stress triaxiality measured varies slightly (around the initial value of 0.425 – Eq. (1)); the loading can thus be considered as nearly proportional. 2.1.3. Mechanical tests The objective of the experimental program is to identify damage model parameters with a series of tests which covers a large positive range of stress triaxiality and Lode parameter. The latter parameter has been shown to have an important influence on material ductility (e.g. [4]). Detailed formulations for the stress triaxiality and the Lode parameter are given in Appendix A. The tests studied are tensile tests on axisymmetric specimens (axisymmetric stress state) and the torsion test. In addition, fracture tests (tensile tests on flat grooved specimens) are also conducted to construct the fracture locus (i.e. the strain to fracture as a function of the stress triaxiality and the Lode parameter). Fig. 4 represents all the tests in the space of theoretical stress triaxiality and Lode parameter. Note that during these tests, except the torsion, the stress triaxiality and the Lode parameter vary with plastic strain, especially after necking in tensile tests. These tests are referred

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Fig. 5. Geometry and dimensions of the specimens used in tensile tests: (a) notched round bar – NRB specimen; (b) round bar – RB specimen; and (c) flat grooved – FG specimen. For the two cases (a) and (c), specimens with three different notched radii are used. All dimensions are in mm.

to as ‘‘macroscopic’’ tests in comparison with the ‘‘microscopic’’ in situ tensile test presented above.2 The Young’s modulus of this material is E ¼ 210 GPa and the Poisson ratio is m ¼ 0:3. The dimensions of tensile specimens are shown in Fig. 5. For the torsion test, the specimen diameter and gauge length are 6 mm and 30 mm respectively. Tests velocities were defined to obtain strain rates in the range [0.7–1] s1 (obtained from the preliminary analytical analysis as well as FE analysis). Due to the heterogeneous deformations of these tests, the stress–strain curves are difficult to exploit correctly for the identification process. For this reason, the load–displacement curves obtained from tensile as well as the [torque-number of rotation] curve from torsion test were used for the identification procedure.

f_ nucleation ¼

"  2 #   fN 1 p  N pffiffiffiffiffiffiffi exp  _p ¼ A p _p 2 S N SN 2 p

with:  p : the equivalent plastic strain,  N : value of mean plastic strain at maximal nucleation,  SN : standard deviation of Gaussian distribution corresponding;  fN : the volume fraction of inclusions at which damage can be nucleated [8]. This parameter is determined so that the total void volume nucleated is consistent with the volume fraction of particles [15].3 The evolution of plastic strain is obtained through the equivalence of the plastic work, overall and in matrix material:

2.2. Numerical models

ð1  f Þr0 dp ¼ r : dp ! Dp ¼ 2.2.1. The GTN model The yield function of the GTN model [34] writes:

/ ¼ /ðp; q; H a Þ ¼



q

r0

2

ð3Þ

where q; p are respectively the von Mises equivalent stress and the hydrostatic pressure; r0 is the flow stress of matrix material; q1 ; q2 ; q3 ¼ ðq1 Þ2 are material constants; f  is the effective void volume fraction, which accounts for the voids’ linkage:

( 

f ¼

f

; if f < fc f  f

fc þ fu fcc ðf  fc Þ ; if f > fc

ð4Þ

f

where fc represents the critical value of f at which void coalescence pffiffiffiffiffiffiffiffiffiffi q  q2 q begins, ff its value at ductile failure, and fu ¼ 1 q3 1 3 the corresponding value of f  at failure. The evolution of void volume fraction is described as:

f_ ¼ f_ nucleation þ f_ growth

ð5Þ

where f_ growth is defined as f_ growth ¼ ð1  f Þtrð_ p Þ and f_ nucleation is often described by a Gaussian curve which was introduced by Chu and Needleman [15]: 2

r : Dp ð1  f Þr0

ð7Þ

where r and Dp are the stress tensor and the increment of the plastic strain tensor. The evolutions of the two internal state variables (p and f) are summarized as:

  3q p 2  1  q3 f  þ 2q1 f  cosh  2 2 r0

¼0

ð6Þ

In the present context, the terms ‘‘macroscopic’’ and ‘‘microscopic’’ refer to the size of specimens used.

1

Dp ¼ DH1 ¼ h ¼ 2

r : Dp ð1  f Þr0

Df ¼ DH2 ¼ h ¼ ð1  f ÞDp þ A

ð8Þ  

 p D p

ð9Þ

where p is the volumetric part of the plastic strain tensor. For a full description of the Gurson and GTN models, see Gurson [19] and Needleman and Tvergaard [34]. The implementation of the GTN model for a mixed velocity–pressure FE formulation using the P1 þ =P1 element was detailed in Cao et al. [14]. 2.2.2. Modified GTN model for shear proposed by Xue [47] Xue [47] carried out an analytical development for a shearing case, where a square cell having length L containing a cylindrical void of radius R at the center, is subjected to a simple shear straining. The author defined the damage associated with the void shearing as:

D_ shear ¼ q3 f q4 p _ p

ð10Þ

R1   3 The volume fraction that can be nucleated is equal to 0 A p dp , whereas R1   R1   fN ¼ 1 A p dp . If SN is small enough with respect to N ; f N ¼ 1 A p dp  R1   0 A p dp .

T.-S. Cao et al. / Computational Materials Science 84 (2014) 175–187

pffiffiffiffi q3 ¼ 6= p ¼ 1:69,

q4 ¼ 0:5 where for 2D case, and  1=3 q3 ¼ 3 p6 ¼ 3:722; q4 ¼ 1=3 for 3D problem. For an arbitrary loading case, Xue introduced a Lode angle dependence function g h into the above equation:

D_ shear ¼ q3 f q4 p _ p g h

ð11Þ

where g h is a linear function of the absolute value of Lode angle jhL j:

  1 2r2  r1  r3 hL ¼ tan1 pffiffiffi r1  r3 3 6jhL j gh ¼ 1 

p

ð12Þ ð13Þ

  In generalized tension hL ¼  p6 ; g h ¼ 0 there is no contribution of shear damage, while in generalized shear, hL ¼ 0; g h ¼ 1, Eq. (10) is resumed. Note that in Xue [47], the form of Lode angle dependence function was chosen phenomenologically. In the pres k ent authors point of view, any function g h ¼ 1  6jphL j could be employed (all these functions satisfy the condition g h ¼ 0 in generalized tension and g h ¼ 1 in generalized shear). Xue proposed a new damage variable (D) which accounts for shear damage:

  D_ ¼ dD q1 f_ þ D_ shear

ð14Þ

where dD is the damage rate coefficient. For the GTN model, which accounts for coalescence (i.e. using fc to define the start of coalescence stage), this coefficient is defined as:

( dD ¼

1; fu fc ff fc

if D 6 Dc ¼ q1 fc ; if Dc < D 6 1

ð15Þ

On the other hand, if the coalescence is not accounted for in the original GTN model, this coefficient can be defined as:

( dD ¼

1; fu fc ff fc

if D 6 Dc ¼ q1 fc ; if Dc < D 6 1

ð16Þ

where fc is not a physical parameter, but rather a phenomenological parameter to describe the acceleration of damage parameter. The fracture is assumed to occur when D ¼ 1. This modified damage variable can be considered as an indicator with no influence on yield function (damage counter approach) or as a damage yielding variable where the effective void volume fraction is replaced by this variable in the yielding condition. In the present study, the damage counter approach is employed. 2.2.3. Numerical implementation Throughout this study, the FE software Forge2009Ò, which is based on mixed formulations of velocity and pressure, is used. In this software, the updated Lagrangian formulation is adopted, which allows using the small strain approach. The local integration of constitutive equations is solved by backward Euler method (return mapping algorithm). The present simulations are carried out with the 3D solver (Forge3), in which the so-called MINI element (P1þ =P1) is used. This linear isoparametric tetrahedron element has a velocity node added at its center, which ensures the stability condition – the Brezzi/Babuska condition of existence and uniqueness of solution [3]. The GTN model was implemented in this software and details can be found in Cao et al. [14]. For Xue’s extension for shear loading, we adopted the ‘‘damage counter approach’’, which means the damage parameter does not influence elastoplastic properties (the yield surface is still the GTN surface). For such an approach, the damage parameter is calculated through a user-subroutine, which carries out the calculation at the end of time increment once all state variables are updated. To avoid partly

179

the mesh-dependency due to the use of a local coupled damage model, mesh size in the gauge section of the macroscopic mechanical tests is fixed (0.25 mm). The mesh size used for the in situ tensile test is chosen to have the same ‘‘mesh gradient’’ since the size of this specimen is significantly smaller than that of macroscopic specimens. It is worth emphasizing a difficulty in torsion simulation: with a few rounds of rotation, if remeshing is not used during the simulation, the mesh at the end of the simulation is strongly distorted, unrealistic results are obtained. However, if remeshing is used, the number of remeshing must be high (since about 1.6 rotations are simulated), errors of variables transfer are high. For this reason, a method was proposed by Transvalor – Forge Òsoftware company, which can be considered as a modification of the Arbitrary Lagrangian–Eulerian (ALE) method. The results are more consistent although some noise is still observed (see Section 4.4). 3. Tomography result analyses 3.1. Damage observation and quantification In order to quantify damage in the studied steel, the same procedure as in Maire et al. [29] and Landron et al. [24] was used in the present study. The image processing was performed with the ImageJ freeware [2]. Damage can be quantitatively observed in 3D as shown in Fig. 6 for different strain levels. ‘‘Local’’ void coalescence can be observed in Fig. 6b where voids start to link together. Fig. 6c corresponds to the last scan just before fracture, where coalescence can be observed: voids linked together to form series of voids about 45° with respect to loading direction. However, it will be shown in the following section that the coalescence is made of ‘‘local’’ events since no clear influence of coalescence on the nucleation and growth was observed (see Fig. 8a and b in the following section). This level of strain (0.32) is the strain of the last tomographic scan before fracture. The real fracture strain is therefore somewhat larger than 0.32. However, this value will be taken as the strain at fracture, which, considering the frequency of scans (see Fig. 3 for instance), should not introduce a large error. The results from the ID19 beamline, with higher resolution, help to complement the observations from the ID15A beamline. Fig. 7a shows the inclusion observed and cavities at a deformation of 0.25. Only one inclusion was detected for this resolution, at which the matrix-particle decohesion was not noticeable. Nevertheless, several long cavities were observed, which formed an angle with tensile direction, which also confirmed the void orientation in Fig. 6b and c. An in situ SEM interrupted tensile test was also carried out to examine the microstructure evolution. At the strain level of 0.2, cavities can also be observed inside the slip bands (without inclusion), which are about 45° with respect to loading direction (horizontal direction in Fig. 7b). 3.2. Voids nucleation and growth modeling The quantification of void nucleation process was carried out by counting the variation of the number of voids in the controlled volume (Fig. 8a) while the void growth process was characterized by the variation of the average diameter of the 20 largest voids of the population (Fig. 8b). Simple models can be used to simulate separately these two mechanisms. For void growth prediction, the Rice and Tracey (R&T) criterion (Eq. (17) – Rice and Tracey [37]) or Huang’s formulation (Eq. (18) – Huang [21]) can be used as in several publications in the literature, e.g. Maire et al. [29], Landron et al. [26], and Thuillier et al. [40]:

  dR 3 ¼ aRT exp g dp R 2

ð17Þ

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Fig. 6. 3D view of damage in the controlled volume at the specimen center at different strain levels: (a) loc ¼ 0:02, (b) loc ¼ 0:25, and (c) loc ¼ 0:32 (see Eq. (2)). Zoomed pictures show largest voids observed. The loading was applied in the vertical direction.

Fig. 7. Observation of voids orientation: (a) inclusion and elongated cavities obtained with the ID19 beamline observation at 0.25 strain level (load was applied in vertical direction) and (b) SEM observation at strain level of 0.2, showing cavity development inside slip bands (load was applied in horizontal direction).

(b) 25 Average diameter ( m)

Void density (mm-3)

(a) 14000 Experimental result

12000

Result with nucleation law

10000 8000 6000 4000 2000 0

All voids (Experiment) 20 largest voids (Experiment) R&T formulation Modified R&T formulation

20 15 10 5 0

0

0.1

0.2

0.3

Equivalent plastic strain

0

0.1

0.2

0.3

Equivalent plastic strain

Fig. 8. Tomography X results of voids evolution: (a) evolution of the void density, which characterizes the void nucleation process and (b) evolution of average diameter of the 20 largest voids and average diameter of all voids.

dR ¼ R

(





aHuang g0:25 exp 32 g dp ; if g 6 1   aHuang exp 32 g dp ; if g > 1

ð18Þ

where aRT ¼ 0:283 and aHuang ¼ 0:427 as proposed initially by Rice and Tracey [37]4 and Huang [21]; R is the radius of an isolated void in perfectly plastic matrix [37], which can be approximated by the average radius of the 20 largest voids [26]. Void nucleation can be phenomenologically modeled by different types of functions of equivalent plastic strain: polynomial function Bouaziz et al. [10], exponential function Maire et al. [29], or 3  The original formulation of R&T is: dR R ¼ CðmL Þ exp 2 g dp , where mL is the ‘‘Lode variable’’, which is defined by the three principal components of remote strain field: mL ¼  _13_2_3 , where _1 P _2 P _3 . CðmL Þ can be approximated as: CðmL Þ  0:279þ 0:004mL . In uniaxial tensile state, mL ¼ 1; CðmL Þ is thus reduced to aRT , which equals 0.283. 4

based on Argon criterion of decohesion Landron et al. [25]. These procedures were successfully applied for several recent studies, e.g. Maire et al. [29] and Landron et al. [24,26]. However, the 3 stages of ductile damage may occur simultaneously and affect each other. Consequently, care must be taken when applying the separate forms to simulate each process. In order to account for the influence of void nucleation on void growth, Bouaziz et al. [10,29] modified the R&T formulation (Eq. (17)) as:

  dR 3 1 dN ¼ aRTm exp g R ðR  R0 Þ dp 2 N dp

ð19Þ

where dN and N are the nucleation rate and the void density respectively; R0 is the radius of cavities just after their nucleation. This equation is referred to as ‘‘the modified R&T formulation’’ hereinafter.

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The evolution of the number of cavities per unit volume (or the void density) can be modeled by a product of linear and exponential functions of the equivalent plastic strain:

p N¼A N

exp



p N

 ð20Þ

where A is a material constant (mm3); N is the critical strain, at which the nucleation starts to occur, which is assumed as a function of the stress triaxiality (g):

N ¼ N

0

expðgÞ

ð21Þ

In the present study, in order to isolate the void growth process, the evolution of the average diameter of the 20 largest voids was modeled, using Eq. (17), while the void nucleation process was modeled separately using Eq. (20). The equivalent diameter was calculated from the total volume of cavity, which was measured by counting the number of voxels attributed to each cavity. The absolute error of this measurement of diameter is thus constant (of the order of several voxels) and the relative error depends on the diameter itself. It is rather important for small cavities [29]. The comparisons between the experimental and numerical results are presented in Fig. 8a and b for void nucleation and void growth processes respectively. As one can observe in Fig. 8b, the evolution of the average diameter of the 20 largest voids can be predicted by the R&T formulation but it is not the case for the average diameter of all voids, which varies slightly with the plastic strain. As mentioned above, this tendency can be explained by the combination between void growth and void nucleation mechanisms: the nucleation of new voids (small size) reduces the average diameter while the growth of voids increases the average diameter of all voids. Using the modified R&T formulation (Eq. (19)) allows reproducing numerically the evolution of the average diameter of all voids (black curve in Fig. 8b). The identified parameters are reported in Table 1. Note that the modified R&T formulation was used to simulate the evolution of the average diameter of all voids but this measurement was not the key parameter. In reality, large voids are often the origin of final fracture. The average diameter of the 20 largest voids is thus more important to be followed and modeled. In addition, continuous increase of the void density can be observed (also the rate of void nucleation), which suggests that void nucleation dominates void coalescence (when voids are linked, the total number of voids decreases). In fact, coalescence took place locally as shown in Fig. 6c but its influence on the void evolution was not noticeable (Fig. 8a and b). If the coalescence phase becomes important, it will slow down the increasing rate of void density and the average diameter of voids could increase rapidly (as shown in Landron et al. [24] and Landron[23]).

kmean ¼

1 N1=3

 Deq

ð22Þ

where N is the void density; Deq is the average diameter of all voids. Apparently, this average value cannot characterize the local event but it can be used as a fracture indicator: when the average intercavities distance is smaller than 40 lm, fracture initiation is liable to occur. In order to verify this observation, the evolution of kmean with the equivalent plastic strain is measured and presented in Fig. 9. As can be observed, this value is about 37 lm (<40 lm) at the strain level of 0.32 just before fracture, which is in good agreement with Landron [23]. 3.4. Discussion  Influence of resolution: apparently, with the X-ray tomography technique, the void nucleation result is strongly influenced by the resolution since very small voids can only be detected under high resolution. In Landron et al.[25], the authors observed no significant influence of the resolution on the void growth process for a dual-phase steel. In the present study, two resolutions (with ID15A and ID19 beamlines) were also compared in terms of the overall evolution of the void volume fraction with the plastic strain. No significant difference was noticed (see Appendix B). However, the results of mean inter-cavity distance and void density are sensitive to the resolution.  Non-spherical cavities: as revealed in Fig. 7a and b, the cavities observed are elongated and make an angle about 45° with respect to loading direction. This orientation of voids is most likely linked with the orientation of the slip planes created during the deformation (see Fig. 10). The Rice and Tracey formulation for void growth as well as the formulation proposed by Maire and co-workers for voids nucleation [29] were based on spherical void assumption. In our particular case, void is far from being spherical. The use of these formulations must therefore be taken with caution. More complex void growth and nucleation formulations accounting for particular void shape are necessary in a future study (e.g. [28]).  In the above section, the void growth and void nucleation processes have been successfully modeled (both separately and simultaneously). These formulations can be integrated into the GTN model as in Bettaieb et al. [9] and Fansi et al. [17], where the authors considered that the modeled material contained one isolated spherical void, which was equivalent in volume as all the voids in real material. The real voids of material were replaced by N identical voids with the same radius R equivalent in volume to all the voids, which were then approximated by one single void with the equivalent radius equal to: pffiffiffiffi Req ¼ 3 N R. The parameters identification needs the real

3.3. Empirical fracture prediction

Table 1 Identified parameters for voids nucleation and growth. R&T

aRT 2.657

Nucleation law A (mm3) 202.1

0

0.176

aRTm 2.078

120

80

40

0

Modified R&T formulation

N

160

Average inter-cavity distances ( m)

In Landron [23], the authors carried out the measurement of the mean distance between cavities (kmean ) inside the controlled subvolume (385 lm)3 for 5 types of steel and observed that the values of this variable just before fracture was in the interval ½20 lm;40 lm. By assuming that cavities are homogeneously distributed in the controlled volume, this parameter is calculated as:

R0 ðlmÞ 2.43

0

0.1

0.2

0.3

0.4

Equivalent plastic strain Fig. 9. Evolution of mean inter-cavities distance with the equivalent plastic strain.

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T.-S. Cao et al. / Computational Materials Science 84 (2014) 175–187

Fig. 10. The formation of slip plane across a grain during the in situ tensile test at three different strain levels: 0, 0.1 and 0.2. The arrows indicate the loading direction.

4. Identification of the GTN model and its extension for shear loading The parameters of the original GTN model can be classified into three groups. First, the flow stress of material matrix, which is supposed to be isotropic. The number of parameters depends on the hardening laws used. Second, the constitutive parameters q1 ; q2 (q3 ¼ q21 ) in the yield function. Third, the parameters related to the mechanisms of damage: initial void volume fraction (f0 ), nucleation ðfN ; N ; SN Þ, coalescence and failure ðfc ; ff Þ.

1400

Equivalent stress (MPa)

evolution of voids density as well as voids size during the calibration tests. In the present study, the parameters calibration for the modified GTN model is carried out using combined in situ X-ray tomography tensile tests on notched round bar and other macroscopic mechanical tests (tensile test on RB and NRB, torsion test). The objective is to validate the identified model for both microscopic and macroscopic tests. The identification of this model is based on the real evolution of porosity from in situ X-ray micro-tomography tensile test (microscopic observation) as well as the load–displacement curves from other classical tensile tests on RB and NRB (macroscopic observation). The identification of the parameter for the shear extension of GTN model is achieved through the calibration from torsion test on cylinder specimens.

1200 1000 800 600 400

Experimental result of in situ tomography tensile test

200

Identified modified Voce law - Mechanical tests

0 0

0.1

0.2

0.3

Equivalent strain Fig. 11. Comparison between the experimental result and numerical simulation with the modified Voce law.

comparison between the experimental stress–strain curve and the numerical curve with the modified Voce law. The result obtained with the identified modified Voce law is quite accurate although this law was identified from macroscopic tests. A difference can be observed at the beginning of plastic zone ðp < 0:03Þ. However, at this level of strain, damage (i.e. void volume fraction) is still negligible; this difference thus has minor influence on the identification of damage model parameters.

4.1. Hardening law parameters identification

4.2. GTN model for in situ tensile test

The matrix material is considered to be isotropic and the plastic deformation is isochoric, obeying the J 2 plasticity theory. The hardening identification of the studied steel grade was detailed in Cao et al. [12]. It was based on the cylinder compression, tensile test on round bars and torsion test. The hardening law used was a modified form of Voce law [44], in which the flow stress is defined as:

The evolution of the void volume fraction with the equivalent plastic strain is presented by the blue diamond symbol in Fig. 12a, in which each point represents the value at each scan. As mentioned above, in the present in situ tensile test, the nucleation mechanism seems dominant due to the fact that no clear influence of coalescence was found on the evolutions of the numbers of void and the average size of voids – see Fig. 8. However, ‘‘local’’ coalescence was observed and we did not have any information concerning the real void evolution in ‘‘macro’’ tensile tests. For this reason, two different identifications were carried out for the GTN model: one without coalescence criterion (which is, in our opinion, more suitable for this particular material) and the other with coalescence (using parameter fc ). Moreover, the last scan before fracture showed a void volume fraction of 0.0024. This value can be considered as the void volume fraction at fracture for the present in situ tensile test. In addition, the initial porosity f0 can be measured thanks to the X-ray tomography observation. For the constitutive parameters q1 and q2 , they have the same influence on the yield surface: the increase of these parameters leads to the retraction of the yield surface. Moreover, the q1 parameter affects the coalescence behavior (if coalescence is activated fu ¼ 1=q1 ) and the q2 parameter appears in the exponential term; it is often kept equal to 1 in the literature. With





r0 ¼ rp0 þ rps  rp0 þ K 2 p ð1  expðnp ÞÞ

ð23Þ

where r0 is the flow stress of material matrix, n and K 2 are material   parameters; rp0 ¼ r0 ðp ¼ 0Þ; rps ¼ r0 p ! 1 if K 2 ¼ 0. The unit of K 2 ; rp0 and rps is MPa. The identified hardening law was validated for different mechanical tests: compression, torsion and tensile tests. The identified parameters are reported in Table 2. From the in situ tensile test, the value of equivalent stress and equivalent strain can be extracted from each scan via Eq. (2). Fig. 11 shows the

Table 2 Identified hardening parameters of the modified Voce law for the high carbon steel [12].

rp0 ðMPaÞ

rps ðMPaÞ

n

K 2 ðMPaÞ

601.146

1113.194

37

100

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T.-S. Cao et al. / Computational Materials Science 84 (2014) 175–187

(b) Experimental result Numerical result - Without coalescence Numerical result - With coalescence

0.002 0.0015 0.001 0.0005

1.2

Minimum cross section diameter (mm)

Void volume fraction

(a) 0.0025

1 0.8 0.6 0.4

Experiment Numerical result - Without coalescence

0.2

0

Numerical result - With coalescence

0 0

0.1

0.2

0.3

0

Equivalent plastic strain

0.1

0.2

0.3

Equivalent plastic strain

Fig. 12. Comparison between experimental and numerical results obtained with identified GTN parameters using Forge2009Ò: (a) evolution of the void volume fraction with the local equivalent plastic strain and (b) variation of the radius of minimum cross section during the in situ micro-tomography tensile test.

 A better result is obtained for a model without coalescence (Fig. 12a). Moreover, the identified value fc ¼ 0:002 for the coalescence case is near the final value at fracture. It means, if coalescence takes place, post-coalescence leading to fracture is quite fast. Since the numerical simulation without coalescence gives a better result and is in good agreement with the identification for damage mechanisms in Section 3.1 (where no clear influence of coalescence was observed), this model will be used hereafter.  The identified parameter q1 ¼ 1:494 is consistent with the value of 1.5 proposed by Tvergaard and Needleman [43] and Needleman and Tvergaard [34].  As mentioned above, in the present in situ micro-tomography tensile test, the nucleation phase is dominant. The value of N is found equal to 0.29 ðSN ¼ 0:0054; fN ¼ 0:00237Þ, which is in relatively good agreement with Fig. 8a, where the void nucleation increases rapidly around 0.26–0.3 of equivalent plastic strain. Note that the increase of void volume fraction was due principally to the combination of void growth and nucleation.  The values of f0 ¼ 4:92  106 and ff ¼ 0:0024 identified from in situ observations are quite small, which is probably due to this particular material. Moreover, it must be recalled that the value ff ¼ 0:0024 corresponds to the last scan before fracture in the in situ test (but in the present authors’ opinion, the true value at fracture is of the same order of magnitude).

4.3. GTN model for macroscopic tensile tests

of these parameters from one test to another is questionable since some parameters might depend on the stress state (e.g. N or ff ). In Kao et al.[22], the authors carried out tensile tests on axisymmetric steel specimens under superimposed hydrostatic pressure and proved that for high triaxiality (small superimposed pressure), the nucleation process took place earlier. In the present study, we suggest that the parameter N (strain at which maximum nucleation occurs) of the GTN model depends on the stress triaxiality through an exponential function (similar to Eq. (21)):

N ¼ N

0

expðBgÞ

ð24Þ

where g is the stress triaxiality, which can be taken as the initial stress triaxiality if the loading is nearly proportional; B and N0 are two parameters to be identified. 4.3.1. Discussion on material ductility Let us discuss first the ductility of this material, represented by the local strain to fracture at different stress triaxiality levels. As shown in Fig. 13, the strains to fracture for different ‘‘monotonic’’ tests were plotted as a function of the initial stress triaxiality: the blue symbols represent the tensile test on axisymmetric specimens ðh ¼ 1Þ; the green symbols represent the plane strain tensile test (h ¼ 0) and torsion test ðh ¼ 0; g ¼ 0Þ. One can observe that the red point, from in situ X-ray microtomography tensile test on notched round bar, lies below the extrapolated curve of strain to fracture at this axisymmetric stress

Equivalent plastic strain at fracture

the latter choice, the cosh term function in Eq. (3) reduces to  expð3=2gÞ at high stress triaxiality, which is consistent with the equation proposed by Rice and Tracey (Eq. (17)). It is probably the reason why q2 is kept equal to 1 in the literature. Therefore, q2 is assumed equal to unity while q1 is a parameter to be identified. Finally, 4 parameters q1 ; f N ; N ; SN need to be calibrated from the results of in situ micro-tomography tensile test. The evolutions of the porosity and the minimum cross section radius as functions of the equivalent plastic strain are used for the identification process by inverse analysis.5 The comparisons between the experimental and numerical results are shown in Fig. 12. The results show a good agreement between the experiment and the numerical simulation.

0.7

Plane strain tensile tests + Torsion

y = 1.5096e-2.61x R2 = 0.9843

0.6

Axisymmetric tensile tests In-situ tensile test

0.5 0.4 0.3 0.2 y = 0.5817e-1.912x R2 = 0.9841

0.1 0 0

0.2

0.4

0.6

0.8

1

Stress triaxiality This identified set of parameters was then used for the tensile tests on RB and NRB as well as the torsion test. The transferability 5 The evolution of the minimum cross section diameter was used principally for verification purpose since at this small value of porosity, this diameter mostly depended on the hardening law used.

Fig. 13. The experimental local strain at fracture is plotted against the initial stress triaxiality for different tests: tensile test on axisymmetric specimens ðh ¼ 1Þ, plane strain tensile test ðh ¼ 0Þ, torsion test ðh ¼ 0; g ¼ 0Þ as well as the in situ tensile test. The approximate exponential functions are presented with their corresponding coefficients of determination R2 . Note that the x variable represents the stress triaxiality while the y variable stands for the local fracture strain.

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40 30 30

Load (kN)

Load (kN)

25 20 15 10

20 NRB-R9-EXP NRB-R9-GTN NRB-R6-EXP NRB-R6-GTN NRB-R4-EXP NRB-R4-GTN

10 RB-EXP

5

RB-GTN

0

0 0

2

4

6

8

0

0.5

Displacement (mm)

1

1.5

Displacement (mm)

Fig. 14. Experimental and numerical load–displacement curves of tensile tests with the identified GTN model.

Table 3 Comparison between experimental and numerical displacements to fracture for different tensile tests. Tensile test

RB

NRB-R4

NRB-R6

NRB-R9

Experimental displacement to fracture (mm) Numerical displacement to fracture (mm) Relative error (%)

7.576 7.6 0.317

0.856 0.84 1.869

1.124 1.12 0.356

1.526 1.42 6.946

sensitive to experimental setup (alignment) and it might be a source of the inconsistency between macroscopic and microscopic specimens. Extensive in situ tests in the future could help to confirm the reproducibility and to better understand the difference between micro and macro tests if it exists.

4.3.2. Applications of the GTN model to macroscopic tensile tests The parameters of the GTN model identified in Section 4.2 and the nucleation strain defined as N ðgÞ ¼ N0 expðBgÞ are applied to the FE simulations of macroscopic tensile tests. The function N ðgÞ was identified to minimize the discrepancy between the experimental and numerical load–displacement curves of tensile tests (see Fig. 14a and b). Note that for proportional or nearly proportional loadings, the value of the stress triaxiality in the function N ðgÞ can be approximated as its initial value. The identification gave N0 ¼ 1:412 and B ¼ 2:712. The comparison between experimental and numerical displacements to fracture is presented in Table 3. The numerical displacement to fracture is the applied displacement at which the

0.003 RB

Void volume fraction

state (i.e. the approximate curve6 for the blue diamond symbols in Fig. 13). The red arrow in this figure indicates the expected fracture strain at this stress triaxiality level. This result shows that the studied material is less ductile in the in situ tensile test than in macroscopic tensile tests. This would mean that at small scale, the influence of the increase of porosity on material ductility seems more important than at large scale. The early fracture in the in situ test is assumed to come from the early maximum nucleation. It is defined through the value of plastic strain at which the nucleation rate is maximum (i.e. N ). However, this observation is strongly qualitative, just based on two in situ tensile tests with two different resolutions. In addition, from Fig. 13, one can observe that the ductility of this material depends on the Lode parameter and a damage model based only on the stress triaxiality is not adapted to predict fracture for such a material. Regarding the size effect, since the size of cavities and second phase particles is significantly smaller than that of ‘‘micro’’ specimen (1 mm diameter), there is no clear reason for earlier fracture of small specimens. This fact also raises a question on the link between the measurement of the physical quantity representing damage (i.e. void) and its real effect on mechanical properties (e.g. strength and rigidity). Let us consider another microstructural size, for example, the grain size. This size is still significantly smaller than that of the specimen and the same remark as above can be drawn. However, several authors showed the influence of the ratio of the number of grains on surface to the grains in volume (e.g. [36]). For a small specimen, this ratio is higher than that of large specimen and thus can have different effects on fracture behavior. However, from the present authors’ point of view, this effect can be noticeable if ductile crack initiates on specimen surface. In the in situ tensile test as well as macroscopic tensile tests, cracks initiate at the specimen center and thus the explanation of the difference in terms of ductility between ‘‘micro’’ and ‘‘macro’’ specimens based on the above-mentioned ratio is not convincing. Moreover, small specimens are strongly

0.0025

NRB-R4 NRB-R6

0.002

NRB-R9

0.0015 0.001 0.0005 0 0

0.2

0.4

0.6

Equivalent plastic strain 6 For interpretation of color in Fig. 13, the reader is referred to the web version of this article.

Fig. 15. Evolution of the void volume fraction at tensile specimens’ centers obtained with the GTN model with the modified nucleation law.

T.-S. Cao et al. / Computational Materials Science 84 (2014) 175–187

185

40

Torque (KN.mm)

35 30 25 20 15 10 Numerical result with identified GTN model

5

Experimental data 0 0

0.4

0.8

1.2

1.6

Number of rotation (rounds)

Fig. 16. Damage variable of: (a) GTN model; (b) Xue’s modification for GTN model at the end of torsion test simulation with identified parameters; and (c) comparison between the numerical and experimental torque-number or rotations curves.

maximum of damage parameter (i.e. the void volume fraction) reaches its critical value ðff  0:0024Þ. The relative error of displacement to fracture is quite small, which shows the validity of the proposed formulation for the GTN model to predict fracture in these tensile tests. In addition, the comparison between numerical and experimental load– displacement curves for tensile tests are presented in Fig. 14a and b and show a good agreement.7 The evolution of the void volume fraction with the equivalent plastic strain at the centers of tensile specimens is also plotted (Fig. 15). The curves show that at high stress triaxiality (small notch radius), nucleation takes place earlier, which leads to earlier fracture (in terms of the equivalent plastic strain).

4.4. Identification of the extended GTN models using torsion tests For this extension, the parameter q4 is derived from analytical calculation: q4 ¼ 1=3 for 3D case. The phenomenological parameter fc must be identified since in the present study, we assumed that the increase of voids is due principally to nucleation and growth. By re-analyzing Xue’s analytical developments, the parameter q4 ¼ 1=3 seems correct and the parameter q3 was adjusted to fit our experimental results. 7 To validated the prediction ability of a damage model, we can base on either local measurements (e.g. local strain at fracture) or global measurements (e.g. displacement to fracture). For a detailed discussion about these two measurements, see Cao et al. [13].

The two parameters fc and q3 were identified by inverse analysis to have D ¼ Dc ¼ 0:98 8 at the end of simulation9 (i.e. fracture at the end of test was predicted), which gave fc ¼ 0:0019 and q3 ¼ 0:85. The result of damage variable at the end of simulation is shown in Fig. 16a (GTN model) and b (modified GTN by Xue). As expected, the GTN model failed to predict damage in torsion test (maximum value of porosity at the end of test is less than 5:9  106 , to be compared with the critical value ff ¼ 0:002410). For the extended GTN model by Xue, note that because of the difficulty in torsion simulation (mentioned in Section 2.2.3), damage field is not really homogeneous. The torque-number of rotations curve is also represented in Fig. 16a, where the numerical result with the GTN model is compared to the experimental result. In this figure, the curve obtained with the GTN model modified by Xue is not plotted because it is the same as the one obtained with the GTN model. The reason is that because the damage counter approach was used for the implementation of the modified GTN model by Xue, the damage variable does not impact the material strength, only the void volume fraction affects the yield surface. A negligible influence of this

8 The critical value is unity. However, to avoid numerical instability when D ¼ Dc (in case of coupling, the GTN yield surface reduces to the origin), we fixed the maximum value of damage equal to 0.98 instead of 1. 9 The torque-number of rotations curve was also used but since the damage counter approach was employed, only f influenced the yield surface, and in a negligible way. The torque-number of rotations curve was therefore less sensitive to D. 10 It should be noted that the critical value of the void volume fraction at fracture might also depend on the stress triaxiality ratio, which is out of scope of the present study.

T.-S. Cao et al. / Computational Materials Science 84 (2014) 175–187

Void volume fraction

(a)

(b)

0.0025 ID15A ID19

0.002 0.0015 0.001 0.0005 0 0

0.1

0.2

0.3

Average diameter of 20 largest voids ( m)

186

25 ID15A 20

ID19

15 10 5 0 0

Equivalent plastic strain

0.1

0.2

0.3

Equivalent plastic strain

Fig. B.17. Comparison between the two beamlines ID15A and ID19: (a) evolution of the void volume fraction and (b) evolution of the average diameter of 20 largest voids.

modification on the result of tensile tests was also verified (moderate to high stress triaxiality). 5. Closure remarks In this work, 3D X ray micro-tomography is used to characterize the ductile damage of a high carbon steel and is combined with macroscopic mechanical tests to identify the modified GTN model by Xue. Main results are summarized here:  Fracture in the in situ tensile tests is the result of a strong nucleation and considerable void growth, which can be modeled separately or jointly using analytical formulations. An empirical criterion based on the mean inter-cavities distance derived in Landron [23] is adopted to qualitatively predict fracture in the in situ tensile test. At the last scan just before fracture, the mean inter-cavities distance is lower than the ‘‘threshold’’ value of 40 lm, which is consistent with the results of Landron [23].  In the second part, the results of tomography observation are used to identify the GTN model. The model thus gives correct prediction of fracture for the in situ tensile test with the parameters identified. However, using these parameters for macroscopic tensile tests leads to inconsistent results. An improvement for the nucleation law of the GTN model is then proposed to account for the important influence of the stress triaxiality on the nucleation process, which was observed experimentally. With the formulation proposed for the nucleation strain and the parameters identified from the in situ tensile tests, the GTN model predicts quite accurately the displacement to fracture for all macroscopic tensile tests. Nevertheless, it cannot predict the fracture for the torsion test.  The extension of the GTN model for shear loading proposed by Xue with two additional parameters is then used and gives accurate results in terms of the fracture prediction and also the torque-number of rotation curve.  For the first time, the damage mechanisms of this high carbon steel are quantified using the combined analyses of both microscopic and macroscopic properties, as well as an enhanced micro-mechanical GTN damage model. The present study also reveals and discusses about the size effect on material ductility. For the moment, no clear explanation has been given. It is worth noting that voids observed in this material are far from being spherical. The study suggests that extensive tensile tests on small specimens should be carried out to clarify the inconsistency in ductility observed.

Acknowledgments The authors would like to acknowledge C. Le Moal and Y. Cassone for their contributions on experimental results. The financial support from ArcelorMittal, Cezus-Areva and Ugitech via the METAL project is appreciated. Appendix A. Characterization of stress states For an isotropic material, the stress state is characterized by the symmetric stress tensor (6 components) or its eigenvalues (3 principal stresses: r1 ; r2 ; r3 ). Material models can also be formulated in terms of the first stress invariant together with the second and the third deviatoric stress invariants, which are defined as: 1 1 I1 p ¼ rm ¼  trðrÞ ¼  ðr1 þ r2 þ r3 Þ ¼  3 3 3 ðA:1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i p ffiffiffiffiffiffiffi 3 1 q¼r¼ S:S¼ ðr1  r2 Þ2 þ ðr2  r3 Þ2 þ ðr3  r1 Þ2 ¼ 3J2 2 2 ðA:2Þ  1=3  1=3 27 27 27 1=3 r¼ detðSÞ ¼ ðr1  rm Þðr2  rm Þðr3  rm Þ ¼ ð J3 Þ 2 2 2 ðA:3Þ

The stress triaxiality is linked with the ratio between the first stress invariant and the second deviatoric stress invariants:

g¼

rm r

ðA:4Þ

The Lode angle h ð0 6 h 6 p=3Þ is defined through the normalized third stress invariant:

n¼

 3 r ¼ cosð3hÞ q

ðA:5Þ

The normalized Lode angle or the Lode parameter h is defined as:

h¼1

6h

p

 3 ! r ; ¼ 1  arccos q p 2

1 6 h 6 1

ðA:6Þ

Appendix B. Comparison between ID15A and ID19 beamlines This section presents a short comparison between the results of in situ X-ray micro-tomography tensile tests obtained with two different beamlines ID15A and ID 19 at the ESRF. Fig. B.17a shows the comparison of the void volume fraction evolution. For the ID19,

T.-S. Cao et al. / Computational Materials Science 84 (2014) 175–187

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