An improved shear modified GTN model for ductile fracture of aluminium alloys under different stress states and its parameters identification
Graphical abstract
Introduction
In the past century, the prediction of ductile damage and fracture of metal materials under complex loadings has always been an important topic in many industries like the metal forming industry [1]. A lot of experimental evidences show that the triaxiality, which is defined as the ratio of the hydrostatic stress and the equivalent stress to depict the stress states of materials, plays an important role in material ductility [2]. Chen [3] showed that material ductility decreases with increasing stress triaxiality. Wierzbicki [4] found nonmonotonic fracture locus in space of fracture strain and stress triaxiality. Different micromorphologies of fracture surface indicate different fracture mechanism. At low triaxiality, elongated dimples indicate shear localization mechanism. At high triaxiality, deep and flat dimples represent void growth and coalescence mechanism [5], [6], [7]. It means that nucleation, growth and coalescence of microvoids in metallic materials are common ductile damage mechanisms [8]. Therefore, it is necessary to develop a constitutive model containing two damage mechanisms at the same time to capture mechanical and fracture properties of ductile materials under different triaxiality.
McClintock [9,10] and Rice and Tracey [11] first studied the mechanism of ductile damage based on microvoid theory. McClintock [9,10] thought that the fracture process of plastic materials can be described as the process of void growth and coalescence based on strain hardening of plastic materials, and then the expansion law with cylindrical and elliptical voids were obtained forming the McClintock model. Rice and Tracey [11] presented an approximate expansion law of isolated spherical voids in infinite ideal materials. It was pointed out that the volume expansion rate of voids would increase rapidly with the increasing stress triaxiality. Therefore, the growth rate formula of average radius R of voids was derived. Based on the research results of McClintock [9,10], and Rice and Tracey [11] and abandoning the assumption of infinite matrix, Gurson [12] in 1977 proposed a cellular model with microvoids in the finite matrix, established the relationship between the evolution of voids and plastic potential, and proposed a porous plastic damage model for the first time. The Gurson model received great attentions after it was put forward, and later scholars have made a lot of improvements to it for different applications. Tvergaard [13,14] considered the interaction between voids by introducing two adjustment parameters. Chu and Needleman [15] supplemented the strain controlled void nucleation criteria and stress controlled void nucleation criteria. Tvergaard and Needleman [16] put forward the concept of equivalent void volume fraction on the basis of considering the coalescence of voids, which is used to describe the rapid decrease of material strength in the final hardening stage, thus forming the classic Gurson-Tvergaard-Needleman meso-damage constitutive model, which is known as GTN model. Since the GTN model was put forward, it has been widely used [17], [18], [19], [20], [21], [22], [23].
Although the GTN model has been widely used, it still has some defects [24]. One of the obvious defects is that it cannot simulate the fracture mode dominated by shear mechanism under low stress triaxiality. To solve this problem, subsequent scholars have made shear improvements on the GTN model. Nahshon and Hutchinson [25] introduced a phenomenological damage law into the GTN model to account for the effective damage accumulation caused by shear mechanisms such as rotation and torsion of microvoids under shear loads. Xue [26] proposed an extended geometrically significant shear damage mechanism based on the through failure condition of voids in the internal shear band proposed by McClintock et al. [10], and introduced an extended stress state function into the shear damage mechanism to distinguish tensile and shear stress states. The volume fraction of voids and shear damage were unified and defined as a total damage variable used in the improved GTN model. Malcher et al. [27] proposed a shear modified GTN model with double damage variables based on the research results of Nahshon and Hutchinson [25] and Xue [26]. This model considered the hardening effect in the mixed stress state proposed in Refs. [27,28]. Zhou [29] thought that the shear damage was caused by the accumulation of plastic deformation and incorporated the Lemaitre principle [30,31] into the GTN model, forming a shear modified GTN model with double damage variables. Although some different shear improvements have been made on the GTN model up to now, each shear modified GTN model has its own advantages and disadvantages. The advantages and disadvantages of the above mentioned shear modified GTN models are summarized in Table 1. Another major disadvantage of these shear modified GTN models is that there are too many parameters, which are difficult to determine [32].
For overcoming the shortcomings and taking advantages of the merits of several existing shear modified GTN models in Table 1, an ISMGTN model and an identification method for identifying the damage parameters with high efficiency and accuracy. The ISMGTN model includes two independent damage mechanisms, i.e., a tensile damage mechanism and shear damage mechanism. Both of these two damage mechanisms include the void nucleation criterion and void coalescence principle, and the hardening effect under a mixed stress state is also considered. These improvements make up for the defects of the currently existing shear modified GTN models: only a single damage parameter used in the Nahshon and Hutchinson's model [25] and Xue's model [26] is difficult to express both tensile and shear damage mechanisms; the shear damage in Zhou's shear improvement model [29] does not have geometric significance; Malcher et al.’s shear improvement model [27] does not contain void coalescence principle. A method to identify these damage parameters of the improved shear damage GTN model is also proposed here. This identification method uses the finite element (FE) inverse identification method incorporating the Latin hypercube design, Kriging approximate model and NLPQL optimization method to determine the damage parameters according to their categories. This method is performed in the optimization software ISIGHT and thus reduces the difficulty for the damage parameter identification. To validate the accuracy of the ISMGTN model proposed in this work, it is applied to predict ductile fracture of aluminum alloy 6061 under different stress states.
The remainder of this paper is arranged as follows: Section 2 introduces the original GTN model and the proposed ISMGTN model with two independent damage mechanisms. A numerical iterative algorithm of the ISMGTN model, which is embedded into the commercial FE software LS-DYNA through the user-defined material subroutine UMAT, is presented in Section 3. Section 4 analyzes the influence of various damage parameters on damage evolution under different stress states and the strategy of damage parameters identification is introduced based on this analysis. Influence of different stress states on damage evolution through unit cell model is analyzed as well. In Section 5, the ISMGTN model and its parameters identification method is performed on the aluminum alloy 6061 as an example. Several shear tests and tensile tests are carried out, and the damage parameters are determined by the FE inverse identification method incorporating the Latin hypercube design, Kriging approximate model and NLPQL optimization method. The rationality of the ISMGTN model and the damage parameters are verified by notch tensile tests and fracture morphology analysis. Section 6 summarizes the whole research work.
Section snippets
GTN model
The GTN model is one of the most classical damage models regarding damage mechanics, which has a wide application and perfect development in studies of metal ductile fracture. It was originated from Gurson [12] and was later improved by Tvergaard and Needleman [13,14,16] by introducing an equivalent void volume fraction f* and two more parameters q1 and q2 into the yield function of Gurson's model to model the complete loss of load-carrying capacity at a realistic void volume fraction. The
Numerical iterative algorithm
In this work, the ISMGTN model proposed in this paper were implemented into the explicit FE solver LS-DYNA using the user material subroutine UMAT. At the beginning of each time step, the time step Δt and strain increment Δε will be provided by the explicit LS-DYNA code. The implicit stress integration algorithm proposed by Aravas [39] and Zhang [40] is then used to correct the stress and effective damage. The updated plastic stress/strain and damage will then be used as initial value of the
Sensitivity analysis of damage evolution with respect to damage parameters under different stress states
A unit cell model [26,41] is used in this section to analyze the influence of tensile damage parameters, shear damage parameters and hardening factors of the ISMGTN model on damage evolution under different stress states. The side length of the unit cell model is 1mm. By respectively applying two constant speeds in the X direction and Y direction on the YZ plane at the same time as shown in Fig. 3, the unit cell is under a certain stress state. Different stress states can be realized by
Mechanical properties test of aluminum alloy 6061
To verify the accuracy of the ISMGTN model proposed, it is applied to simulate mechanical properties of aluminum alloy 6061 under different stress states here. The corresponding material parameters for the aluminum alloy 6061 should be identified through several mechanical tests according to the identification strategy in Section 4.3.
A series of tensile tests were first carried out on the electronic universal tensile testing machine (see Fig. 14), including 0° shear, 30° shear, 60° shear, plate
Conclusion
In this paper, an ISMGTN model with two independent damage mechanisms is proposed by overcoming the shortcomings and combining the merits of several previous research works [[25], [26], [27],29]. A parameters identification strategy is proposed based on the damage evolution analysis performed on a unit cell model to determine the material parameters, in which a damage parameters identification method is proposed to determine the damage parameters including the tensile damage parameters, shear
Declaration of Competing Interest
No conflict of interest exits in the submission of this manuscript. All funding sources supporting the work and the institutional or corporate affiliations of the authors have been acknowledged in this work. I would like to declare on behalf of my co-authors that the work has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.
Acknowledgments
The authors would like to acknowledge the foundation from the National Natural Science Foundation of China (No. 51805339) and Fundamental Research Funds for Central Universities in China (Nos. 2020CDCGQC049 and YJ201825).
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