A method for determining the ductile damage parameters of high strength steels and weld metal

The microvoid based Gurson-Tvergaard-Needleman (GTN) model is a powerful tool for predicting ductile fracture behavior, and application of such model to steels and welds needs the identification of microvoid related damage parameters. Currently, there is no standard damage parameter identification method available. In this study the previously proposed complete Gurson model (CGM) where a physical void coalescence mechanism is incorporated into the GTN model is revisited. According to the CGM, the void nucleation process dictates ductile fracture. By adopting the cluster nucleation model with an effective initial void volume fraction as the only controlling parameter, a method is proposed to explicitly determine the effective initial void volume fraction from the strain at maximum load and strain at fracture of a specially designed notched tensile specimen. The proposed equation has been experimentally verified by applying to three high strength materials, including a X80 pipeline steel and associated weld metal, and a 15CrMo steel. A general procedure for damage parameter identification is also suggested. It is argued that the obtained effective initial void volume fraction can be treated as a type of material ductility indicator.


Introduction
Structural components made of high strength steels may fail by ductile fracture. It is well recognized that ductile fracture of steels results from the nucleation, growth and coalescence of microvoids. The most widely used model for ductile fracture was originally developed by Gurson (1977), and further modified and improved by Needleman (Tvergaard, 1981, 1982;Tvergaard and Needleman, 1984). The application of this ductile damage model, also well-known as the GTN model, needs the identification of as many as 17 parameters (Zhang et al., 2018), including the material-specific void (nucleation and coalescence) parameters, constitutive parameters, characteristic length parameter(s) as well as other materialindependent model parameters. It is a common practice to calibrate the parameters of the GTN model through a combination of experiments and numerical simulations (Sun et al., 1989;Springmann and Kuna, 2005;Abendroth and Kuna, 2006;Seupel et al., 2020;Miloud et al., 2019).
Smooth and notched tensile specimens where the stress/ strain gradients were not strong and finite element mesh size does not play a significant role, are often used to determine the void parameters (Zhang et al., 2000). When the GTN model is applied to cracked components where severe stress gradient exists in the vicinity of crack tip, the predicted failure becomes mesh size dependent. In such cases, a so-called non-local characteristic length parameter(s) should be considered (Zhang et al., 2018). Fracture mechanics specimens such as compact tension (Acharyya and Dhar, 2008;Dotta and Ruggieri, 2004) single edge notched tension (Chen and Lambert, 2003), single edge notched bending (Qiang and Wang, 2019;Sarzosa et al., 2016), drop weight tear testing (Scheider et al., 2014;Nonn and Kalwa, 2012) have been used for the calibration of both the void and characteristic length parameters either separately or simultaneously. It is interesting to note that machine learning methods have also been recently applied to the identification of GTN parameters (Chen et al., 2021;Ouladbrahim et al., 2021). In order to limit the scope, this study will focus only on the void parameters.
As far as void parameters concerned, it has been noted by Zhang (1996) more than two decades ago that when comparing with a set of experimental results, the fitted GTN damage parameters are not unique. This nonuniqueness problem implies that different combinations of void nucleation and coalescence parameters may lead to indistinguishable macroscopic responses. This nonuniqueness problem may not have severe consequence if the purpose is to simulate the macroscopic behavior of a specific structural component. However, it does have serious implications in terms of the physical interpretation, transferability of the void parameters and classification of materials deformation and fracture behavior based on damage parameters.
The non-uniqueness problem partly attributes to the lack of a physical void coalescence mechanism in the GTN model which treats the void nucleation and coalescence as two independent events. As a viable solution to the problem, by incorporating a physical void coalescence mechanism based on Thomason's plastic limit load model (Thomason, 1990) into the GTN model, a socalled complete Gurson model (CGM) has been proposed by Zhang (Zhang et al., 2000). The key feature of the CGM is that the void coalescence is a natural result of the plastic deformation of a material with nucleated voids, and the so-called critical void volume fraction at the beginning of coalescence f c , is thus not a material constant and does not need to be fitted or settled beforehand. The great advantage of the CGM is that in specimens where the stress/strain gradients are not significant, ductile fracture is solely controlled by the void nucleation process. If the ductile fracture behavior of these specimens is known experimentally, the void nucleation parameters can be inversely determined from the measured loaddisplacement curves.
In the literature, it has been a common practice to estimate the initial volume fraction f 0 , from the volume fraction of primary particles such as MnS in steels (Tanguy and Besson, 2002;Tanguy et al., 2005). However, the problem is that normally not every particle, and only an unknown fraction of these particles will nucleate voids during the plastic deformation. In this study, we hypothesize that the initial void volume fraction can be obtained inversely through tests and simulations. By adopting the cluster nucleation model and assuming that the void nucleation is represented by an effective initial void volume fraction, in this paper, the CGM is applied to study the relationship between the inputted f 0 and the tensile properties of a specially designed notched tensile specimen for the purpose to develop a method for inversely determining the f 0 .
In the following, Section 2 briefly revisits the CGM. Section 3 describes the specially designed notched tensile specimen used. In Section 4, the numerical simulation procedure, the correlation data and the obtained equation to explicitly calculate the f 0 are presented. The proposed equation is experimentally verified by applying to three engineering materials with different plastic strain hardening ability and ductility, including a high strength pipeline steel X80, a X80 associated weld metal and a 15CrMo steel. A recommended procedure for determining the f 0 is suggested. Potential to treat the obtained the f 0 as a ductility indicator and use the f 0 to classify the deformation and fracture behavior of steels and welds together with suggested future work and main conclusions are presented in Section 5.

Revisit of the complete Gurson model
The special feature of the complete Gurson model is the introduction of a physically based coalescence criterion. In the following the GTN (Gurson, 1977;Tvergaard, 1981Tvergaard, , 1982Tvergaard and Needleman, 1984) model will be introduced first. The GTN model describes the homogeneous yielding and void growth behavior of a unit cell with a void located in the centre. The yielding function is expressed as: where q is the von Mises stress, σ h is the mean stress, σ is the flow stress, f is the current void volume fraction. q 1 and q 2 are constants introduced by Tvergaard, 1981Tvergaard, , 1982 to modify the Gurson model. Fixed values of q 1 ¼ 1:5 and q 2 ¼ 1:0 have been used in this study. The application of the GTN model requires the information about the void nucleation process. In general, void nucleation can be cluster-based or of statistical nature (Zhang et al., 2000). The cluster model which usually assumes that the micro-voids are nucleated upon the onset of plastic deformation, is one of the most commonly used models to describe void nucleation for ductile metals. The cluster nucleation model is adopted in this work. The advantage of the cluster void nucleation model is that there is only one controlling parameter, namely the initial void volume fraction f 0 . The growth of existing voids can be expressed by assuming that the matrix material is incompressible, where ε P is the plastic strain tensor, I is the second-order unit tensor. When the void volume fraction f grows to a critical value, f c , void coalescence occurs. Once the void coalescence has started, Tvergaard and Needleman (1984) introduced a function to simulate the process of void coalescence by accelerating the void growth: (1). When f reaches the failure volume fraction f F , the void coalescence ends. It has been shown that the minimum value of f F is close to 0.15 and an approximation f F ¼ 0:15 þ 2f 0 has been proposed by (Zhang et al., 2000). In most of the literature, an arbitrary or fitted constant f c has been used and no physical coalescence criterion is considered. (Thomason, 1990) proposed a physically-based plastic limit load model to describe void coalescence. Once the void coalescence mechanism is implemented, the void coalescence is thus a result of the competition between the homogeneous deformation mode described by the GTN model and localized deformation mode governed by the plastic limit load model (Zhang and Niemi, 1994a).
By incorporating the GTN model for void growth and Thomason's plastic limit load model for void coalescence, the CGM is obtained. It is called "complete" in a sense that once the void nucleation process is known, the CGM now can physically capture the complete void growth and coalescence process without a need of using any artificial void coalescence criterion or constant Niemi, 1994a, 1994b). It is worth emphasizing that in the CGM void coalescence is not determined by the so-called critical void volume fraction f c . With the CGM and in the absence of stress/strain gradient, ductile failure is intrinsically controlled by the void nucleation. The CGM has been implemented into the ABAQUS using a user subroutine UMAT developed by Zhang with dedicated efficient numerical integration algorithms (Zhang, 1995a(Zhang, , 1995bZhang and Niemi, 1995), a free copy of the UMAT source code for the implementation can be obtained from the corresponding author.
It must be noted that the CGM treats the voids from the nucleation to the end of coalescence as spherical and the void shape change during the plastic deformation is not considered. The interaction between the voids is also neglected. Thomason's plastic limit load model is not the only one which can be used for describing void coalescence. Other sophisticated coalescence models which consider both plasticity and void anisotropy can be found in (Benzerga and Leblond, 2010).
The "magic" notched tensile specimen for determining both the stress-strain curve and damage parameters Standard tensile bar specimen used for determining materials' stress-strain curve has several limitations in terms of the size of specimen required, unpredictable necking position and the need to perform Bridgman correction (Zhang et al., 2002). Notched round bar tensile specimens are good alternatives to the conventional smooth tensile specimen for characterizing material tensile properties. Notched tensile specimens possess several advantages compared with the smooth one. Firstly, unlike the smooth specimen where necking often occurs in a random manner, the necking in the notched specimens always appear at the intended location. This is especially appealing for the case when the volume of the material to be tested is too small for machining a standard tensile specimen, for example the heat affected zone or weld metal in a steel weldment. Secondly, recent studies show that material's plastic flow stress-strain curve can be conveniently obtained from notched tensile specimens. In particular, there exists a "magic" notched tensile specimen (Tu et al., 2017), from which the load-minimum cross-sectional area reduction curve recorded can be directly converted to material's equivalent stress-strain or plastic flow stress-strain curve without the need of performing the nontrivial Bridgman correction (Tu et al., 2017(Tu et al., , 2018(Tu et al., , 2019(Tu et al., , 2020.
For a general notched round bar tensile specimen sketched in Figure 1(a), L and D 0 are the initial length and diameter, a 0 is initial radius of minimum cross section, R 0 is initial radius of the notch. Zone W and Zone B are used to distinguish the weld and base metal of a weldment, and H is the width of the weld metal. Early studies have shown that when H is larger than 0.5 D 0 , the properties of the Zone B will not influence the results of the measured weld metal properties (Tu et al., 2017(Tu et al., , 2018(Tu et al., , 2020. The average true strain of a notched tensile bar specimen is calculated from the reduction in the minimum cross-sectional area: ε t ¼ 2lnða 0 =aÞ, where ε t is the average true strain, a is the current radius of minimum cross section. The average engineering stress and true stress of a notched tensile specimen are calculated by the applied load divided by the respective initial and current minimum cross-sectional area, σ e ¼ P=ðπa 0 2 Þ and σ t ¼ P=ðπa 2 Þ where σ e is the engineering stress, σ t the true stress, P is applied load. The engineering stress at the maximum load is marked as σ m , and the true stress at fracture is denoted as σ f . The true strains corresponding to σ m and σ f are marked as ε m and ε f , respectively.
The ratio of minimum cross section radius to notch radius, a 0 =R 0 is a key characteristic of the notched tensile specimen. Our previous studies have shown that among all the notched specimens, there is one "magic" notch geometry, a 0 =R 0 ¼ 2, which possesses a special feature. It was found that by normalizing the true stress of the "magic" notched tensile specimen by a geometry factor G m ¼ À0:824n þ 1:6189, where n is the plastic strain hardening exponent and can be approximated by the strain at the maximum load ε m , the normalized true stresstrue strain curve outputted from the notched specimen, σ t G m À ε t , represents the equivalent stressstrain relation of the material. Details about the "magic" notched tensile method can be found in (Tu et al., 2017(Tu et al., , 2018(Tu et al., , 2019(Tu et al., , 2020. In this study, L = 180 mm, D 0 = 16 mm, a 0 = 4 mm and R 0 = 2 mm have been used in both the numerical simulation and experiments.

Inversely determining the initial void volume fraction
In the following we apply the CGM to analyze the "magic" notched tensile specimen made of materials with different plastic hardening exponents to generate the correlation data between the fracture strain and inputted initial void volume fraction. An explicit equation will be derived to inversely calculate the initial volume fraction from the data points. The obtained initial void volume fraction will then be applied to compare the predicted fracture strains with the experimental values. A recommended procedure to utilize the method for parameter identification is also suggested.

The numerical procedure
In order to establish the correlation between the inputted damage parameters and the predicted fracture strain, a 1/4 of the axisymmetric notched specimen was modeled with CAX4 element in ABAQUS, as shown in Figure 1(b). The fracture strain was calculated from the predicted crosssectional area reduction at failure using the CGM. A remote homogenous displacement (clamped boundary condition) was applied at the end of the specimens. The NLGEOM option was activated to take the large-strain effect into account. A general power-law plastic hardening model was used in the parametric study: where σ is the flow stress, σ 0 is the yield stress, ε p is the equivalent plastic strain, ε 0 is the yield strain, n is the strain hardening exponent. In all the analyses, Young's modulus E ¼ 210 GPa and Poisson ratio υ ¼ 0:3 have been applied. Strain hardening exponent n from 0.02 to 0.2 and the initial void volume fraction f 0 in the range from 0.000,001 to 0.01 have been analyzed. The yield stress σ 0 varied from 400 to 1000 MPa, however, it can be expected that the yield stress does not have a remarkable influence on the predicted fracture strain. Because the notch induces a stress concentration, the influence of element size on the ductile fracture needs to be assessed. The local element size along the middle section was set as l 0 , and a transition mesh was used to bridge the fine and coarse mesh. A range of finite element size from 0.05 mm to 0.5 mm was analyzed. Some preliminary analyses show that the range of element size utilized does not have any visible effect on the load-displacement curves of the notched tensile specimen until the fracture point. It can also be noted that the element size used does not noticeably influence the true strain at the maximum load, ε m . The general observation is that the larger the element size, the larger the resulted fracture strain. In order to better illustrate the effect of mesh size, the fracture strain versus normalized element size is plotted in 2a for constant f 0 ¼ 0:0001 and Figure 2  hardening exponent n. Figure 2 shows that when the element size is large, the fracture strains in both cases with constant n and constant f 0 are an increasing function of the mesh size. When the element size is small enough, for example, when a 0 =l 0 is larger than 20, the fracture strain is practically independent of the element size. Based on these observations, a default value of a 0 =l 0 = 40 (l 0 ¼ 0:1 mm) has been used in the following analyses unless indicated otherwise.

The correlation data and explicit equation
When the CGM is used, the fracture strain of the notched tensile specimen depends on two key factors, the plastic strain hardening exponent n and initial void volume fraction f 0 . For the given notch geometry, the fracture strain ε f can be expressed as follows: where the fracture strain ε f and the plastic hardening exponent (strain at the maximum load) n = ε m can be obtained directly from the notched tensile test. Figure 3(a) shows the predicted fracture strain vs the strain hardening exponent n for the cases with f 0 in the range 10 À6 -10 À2 , which should cover most of the engineering materials. It is interesting to note that for a given value of f 0 , fracture strain ε f increases approximately in a linear manner with the hardening exponent n. The relationship between the initial void volume fraction f 0 and the fracture strain ε f is shown in Figure 3(b) with n in the range from 0.02 to 0.2. As f 0 increases from 10 À6 to 10 À3 , the ε f decreases drastically, and the decrease slows down to a nearly linear manner when f 0 is larger than 10 À3 .  For a given material the plastic hardening exponent n, the initial void volume fracture f 0 and fracture strain ε f are uniquely correlated, the equation (5) can be rewritten as: Various forms of equations have been used to fit the correlation data shown in Figure 3. It is found that the data shown in Figure 3 can be best fitted by the following equation: where α, β are functions of ε m : γ ¼ 17:7 Á expðÀ0:165 Á ε m Þ À 14:73 (10) Figure 4 compares the initial void volume fraction calculated from equation (7) with the original data obtained from the finite element analyses. As can be observed from Figure 4, equation (7) fits the data points quite well. Even for f 0 in the range from 10 À6 to 10 À3 , a reasonable agreement is obtained. It should be noted that the applicability of equation (7) is limited to ε m ¼ 0:02 ∼ 0:2, ε f ¼ 0:07 ∼ 1:3.

Application to high strength steels and weld metal
The proposed equation (7) for inversely calculating the initial void volume fraction has been verified against three engineering materials with different plastic hardening ability and ductility. The three materials considered are: X80 pipe steel (marked as Specimen 1#), X80 girth weld (Specimen 2#) and 15CrMo steel (Specimen 3#). Specimen 1# and Specimen 2# were derived from a gas pipeline with a diameter of 1016 mm and a wall thickness of 18.4 mm. A total of 9 uniaxial tensile tests using the "magic" notch geometry have been conducted.
The experimental setup is displayed in Figure 5. The experiments were performed using a MTS 810-25 testing machine. The displacement-controlled mode was adopted, and loading speed was set to be 0.3 mm/min. The loadcross head displacement curves of the 9 specimens are presented in Figure 6. It can be seen that there are relatively small deviations in the load -displacement curves of the X80 steel and 15CrMo steel. As it can be expected, the results of the X80 girth weld (Specimen 2#) shows larger variation in both the load carrying capacity and fracture strain.
During the tensile testing, the deformations of notched specimens were captured every second by using an industrial digital camera. A python code was developed to retrieve the evolution of the minimum cross section diameter, from which the engineering stress and true strain of the notched tensile specimen can be evaluated. The engineering stress versus true strain curves of the 9 specimens are grouped into 3 figures shown in Figure 7. From these results, the corresponding fracture strain and strain Figure 4. Comparison of the initial void volume fraction calculated from equation (7) and original data from Figure 3. Curves are from equation (7) and dots are the data from Figure 3. hardening exponent can be identified, and the effective initial void volume fraction is then calculated by applying equation (7). The measured yield stress, hardening exponent and fracture strain together with the calculated initial void volume fraction are summarized in Table 1. It is interesting to compare the X80 weld metal with the X80 base metal. The hardening ability of the weld metal is slightly higher than its base metal, however, the yield stress of the weld metal is significantly lower than that of the base metal and the weldment is a so-called undermatched weldment. Table 1 shows the inversely calculated average initial void volume fractions for the three materials tested. The X80 steel has the smallest initial void volume fraction, with a mean value between 0.01% to 0.02%. It is interesting to observe from Table 1 and Figure 7 that although the fracture strain of the X80 weld metal is in the similar level of the X80 base metal, the average initial void volume fraction of the weld metal is much higher than, almost 3 times, those of the base metal. Figure 7 also shows that the fracture strain of the 15CrMo steel is the lowest among the three materials tested, and the average initial void volume fraction is approximately 15 times larger than the X80 steel. This finding indicates that the initial void volume fraction of a material is not solely determined by the ductility but is strongly dependent on the plastic strain hardening ability. For materials with identical fracture strain, the material with higher hardening ability will yield larger initial void volume fraction.
We note that the equation (7) was obtained based on the analysis of the ideal power-law hardening materials. It is interesting to verify that the application of the initial void volume fraction calculated from equation (7) will result in the same ductile fracture behaviour of the engineering materials. The measured yield stress and plastic strain hardening exponent and the initial void volume fraction shown in Table 1 have been taken as inputs to the finite element analyses, and the outputs from the finite element analyses have been compared with the experimental results in Figure 7. The comparison shows that the finite element results are in well accordance with the test results, especially the prediction of fracture strain is captured quite  accurately. The good agreement confirms that the validity of equation (7).

Recommended procedures for identifying initial void volume fraction
In this work an explicit equation, equation (7), for inversely determining the initial void volume fraction for a given material is obtained. It has been shown that the equation possesses a good accuracy for the materials tested. Following recommended procedure is suggested.
(1) Prepare the "magic" notched tensile specimen as shown in Figure 1(a). It is particularly important that the notch geometry fulfils the requirement a 0 =R 0 ¼ 2. At least three parallel specimens should be prepared for each material.
(2) Perform uniaxial notched tensile tests. Both the applied load and the reduction of the minimum cross section diameter should be measured. Unlike the smooth tensile specimen, the elongation within a gauge length is not useful for the notched tensile specimen.
(3) Sketch the engineering stress-true strain curves of the notched tensile specimen (σ e À ε t ) to identify the strain at the maximum load and the strain at fracture. (4) Compute the initial void volume fraction f 0 with Equation (7) as a function of the hardening and fracture strain obtained in step (3).
When equation (7) is applied to the case where the volume of material of interest (weld metal or heat affected zone in a weldment) is limited, the zone size of the homogenous material to be tested should be as large as possible compared with the minimum cross section diameter H=D 0 ≥ 0:5. When the material zone size is very small, the size of the notched tensile specimen can be scaled down accordingly to satisfy this geometrical requirement.

Discussion and conclusions
The "magic' notched tensile specimen has been shown previously to possess a special function, namely, the true stress-strain curve obtained from the tensile test using this special specimen can be directly converted into materials equivalent stress-strain curve without the need of performing a Bridgman correction. This study extends the function of the "magic" notched tensile specimen to determine the ductile damage parameterthe initial void volume fraction f 0 . The CGM model has been applied to analyze the correlation between the predicted fracture behavior of the notched tensile specimen and its input parameter f 0 . It is shown that the ductile fracture of the notched tensile specimen mainly depends on two parameters, the plastic strain hardening exponent and the initial void volume fraction. Because the fracture strain and plastic strain hardening exponent can be conveniently obtained from the strain-strain curve, an explicit equation to calculate the initial void volume fraction is proposed. It is important to emphasize that the obtained the initial void volume fraction physically represents the effective initial void volume fraction of the tested material. The obtained effective initial void volume fraction ideally can be treated as a material ductility indicator, and transferred to the other cases with different stress triaxiality. However, the transferability may be material specific and remains to be experimentally verified in the near future. Only one notch geometry was considered in this study. Similar equations for other notched tensile specimens can be derived using the same procedure reported in this study. When the effective initial void volume fractions obtained from two notched tensile specimens differ significantly, question regarding the choice of the cluster void nucleation model should be raised. If that is the case, other void nucleation models, such as continuous void nucleation or statistical nucleation model may be considered [27]. There is only one parameter associated with the continuous void nucleation model. However, more than one parameter must be involved in the statistical void nucleation model. A good Table 1. Mechanical properties and the inversely calculated initial void volume fraction. σ 0 (MPa) n (ε m ) ε f f 0 from equation (7) Average f 0 Notes advice is to fix as many parameters as possible and leave only one parameter to be identified. In summary, based on the CGM model and by adopting a cluster void nucleation model and assuming that the void nucleation can be described by an effective initial void volume fraction, an explicit equation to calculate the effective initial void volume fraction using a special notched tensile specimen is presented. A recommended procedure for using the method is proposed. The method is particularly applicable to the cases where the size of the material zone of interest is limited, and it is impractical to machine the material to fabricate standard specimens for mechanical characterization. The method has been verified against three engineering materials including two types of high strength steels and one weld metal. It is expected that the method will facilitate the identification of damage parameters.
Highlights · A method is proposed to explicitly calculate the effective initial void volume fraction from the strain at the maximum load and the strain at fracture of a specially designed notched tensile specimen. · The method is well-suited for weldments where material volume of homogenous microstructure is limited. · The proposed method has been experimentally verified by applying to three engineering materials including both high strength steels and an associated weld metal. · This effective initial void volume fraction can facilitate the classification of deformation behavior of metallic materials.