Damage Evolution and Ductile Fracture

CHAPTER 3

Damage Evolution and Ductile Fracture 3.1 INTRODUCTION Due to increasing demand for lightweight, high-strength, and superiorperformance products, manufacturing of precision products by using hard-to-deform materials has attracted increasing attention recently. Deformation-based materials processing is key to forming shapes and geometries and to tailoring the properties of the deformed parts. However, in deformation-based manufacturing, inappropriate thermalmechanical loadings are often applied and nonuniform deformation is frequently induced, resulting in the deterioration of material properties due to damage and fracture. Damage and fracture directly affect the performance of the deformation process and the quality of the deformed parts, which thus greatly cripples this otherwise advantageous technique. Understanding the mechanisms of damage evolution and ductile fracture, and accurately modeling the processes to avoid failure, are key to exploiting the potential of deformation-based materials processing to the fullest. In deformation-based materials processing, increase in void-volume fraction and shear-induced damage are the main failure mechanisms of materials. In the deformation process, the damage of materials undergoes a complex evolution with the change of load condition, which may result in undesirable deformation and further affect the formability of materials and the load-carrying capacity of products. Accurate prediction of damage evolution and ductile fracture is thus of practical importance in the design and optimization of deformation-based materials processing. Based on physical observation and micromechanical analysis, a large number of ductile fracture criteria (DFC) and failure diagrams have been developed to predict fracture, by which practitioners are able to capture and restrain further damage and ductile fracture, thereby improving forming limit and forming quality.

Deformation-Based Processing of Materials DOI: https://doi.org/10.1016/B978-0-12-814381-0.00003-0

© 2019 Elsevier Inc. All rights reserved.

85

86

Deformation-Based Processing of Materials

Accurate modeling of nonuniform deformation is a prerequisite for ductile fracture prediction, which was discussed extensively in Chapter 2, Deformation Inhomogeneity. Thus, this chapter presents the mechanisms that induce ductile fracture and examines the effects of void evolution and shear-induced damage on the deterioration of material properties. On this foundation, various modeling methods are presented to represent and reveal the mechanisms of damage evolution. Both coupled and uncoupled DFC and failure diagrams are exhibited and numerically implemented to predict and analyze ductile fracture. Applications in tube bending and cup drawing validate the models and confirm the feasibility of predicting damage evolution and ductile fracture occurrence, and provide guidance for design decision-making in deformation-based materials processing.

3.2 MECHANISM OF DAMAGE EVOLUTION AND DUCTILE FRACTURE Ductile fracture of metallic materials is a very complex phenomenon significantly influenced by many factors, such as material state, workpiece geometry, strain path, working temperature, and strain rate. The mechanism of damage evolution and ductile fracture is sophisticated and has garnered much attention. As is known, under tension-dominated deformation conditions, ductile fracture results from the nucleation, growth, and coalescence of microvoids. It is noteworthy that the ductile fracture mechanism under shearing-dominated deformation still has a dispute. One accepted interpretation is that damage and fracture mainly result from shear linkup of the voids. For deformation-based materials processing, ductile fracture mainly originates from a mixture of tensionand shear-induced damage. In this section, the common phenomena and mechanisms of damage evolution and ductile fracture during deformation-based materials processing are introduced and discussed.

3.2.1 Tension-Induced Void Initiation, Growth, and Coalescence Ductile fracture in metals and metallic alloys often originates from the initiation, growth, and coalescence of microscopic voids during plastic deformation [1
6]. The nucleation of voids usually takes place at the interfaces of inclusions and second-phase particles. The dissociation of these interfaces is regarded as the dominant mechanism of void nucleation. Once the voids nucleate, further plastic deformation enlarges the size of

Damage Evolution and Ductile Fracture

87

voids and distorts the shape, which is often called void growth [7]. As voids enlarge and distort substantially with plastic deformation, adjacent voids ultimately link up or coalesce with each other due to the localization of plastic strain in the intervoid matrix, forming the final fracture surface. Void coalescence can be observed directly by SEM analysis from fracture surfaces, since void coalescence is the final stage in ductile fracture [8]. Void initiation, growth, and coalescence mainly occur along the direction of maximum tension stress, as shown in Fig. 3.1. Under the uniaxial tensile load, damage evolution and ductile fracture of steel at room temperature (shown in Fig. 3.2 [9]) confirm this, as the fracture surface is perpendicular to the maximum tensile stress. From Fig. 3.2 necking, which is a mode of tensile deformation where relatively large amounts of strain localization appear in a small region of the material, occurs before ductile fracture. In general, necking can be

Figure 3.1 Void evolution under maximum tension stress.

Figure 3.2 Diagram of void initiation, growth, and coalescence [9].

88

Deformation-Based Processing of Materials

classified as diffuse necking and localized necking. The first one is the case where the uniform reduction of thickness in a relatively large range occurs, while the second is the case where the thinning of material concentrates in a localized region. During deformation-based materials processing, fracture caused by necking is an irrecoverable failure. Localized necking, rather than diffuse necking, is an important factor that determines the amount of useful deformation, so the point in which localized necking first occurs is regarded as an ideal critical point. It should be noted that fracture may occur without apparent necking for materials with low plasticity. For this phenomenon, the mechanism of damage evolution and ductile fracture is determined based on fractography analysis. For different materials, the mechanisms are different. Al-alloy contains a large-volume fraction of various intermetallic particles. In its microstructure, two types of inclusions and particles in brittle phases are dispersed in the Al-matrix: Fe-based intermetallics and Mg2Si intermetallics [4]. Due to the incompatibility of the Al-matrix and intermetallic particles with different properties, the voids are initiated or nucleated primarily at the grain boundaries once plastic deformation occurs. These characteristics are convenient for exploring the mechanisms of damage evolution and ductile fracture. As can be seen from situ tensile tests of the Al-alloy, dimple-dominant fracture usually occurs under the tensionstress-dominant condition for ductile materials, as shown in Fig. 3.3. Under tension-dominant deformation, a higher triaxiality accelerates void

Figure 3.3 Dimple-dominant fracture caused by the maximum tension stress [10].

Damage Evolution and Ductile Fracture

89

Figure 3.4 Schematic view of laminography setup [16]. (A)

(B) Void 1

Void 2

Void 3

0.5 0.45

Experimental Model

20 µm ε = 0.8

0.35 0.3 0.25

6056 6005A

0.2 0.15 0.1 0.05 0 0

6061

0.2 0.4 0.6 0.8

∋

ε = 0.4

Void volume fraction

0.4

ε=0

1

1.2 1.4 1.6 1.8

2

Loc

Figure 3.5 Evolution of (A) three chosen voids and (B) the void-volume fraction during deformation [13,17].

initiation and propagation, which results in a large number of dimples and cracks perpendicular to the maximum tensile stress. To describe the damage evolution during plastic deformation and predict ductile fracture, several indicators related to voids have been studied and used to model fracture. Void-volume fraction is widely used [1,3,11
13], especially the Gurson-type fracture model. As shown in the X-ray microtomography in Fig. 3.4, the evolution of voids and the voidvolume fraction during deformation can be obtained, as shown in Fig. 3.5. The volume fraction usually slowly increases first, before rapidly increasing, which can be attributed to the coalescence of voids under large plastic strain [12,13]. Rapid increase of the void-volume fraction signifies that necking has already appeared. Recently, the evolution of void shape and orientation during deformation has also been considered to improve the prediction accuracy of ductile failure under shear-dominated loadings [14,15]. Those have greatly promoted understanding of the effects of void evolution on ductile fracture.

90

Deformation-Based Processing of Materials

3.2.2 Shear-Stress-Induced Damage and Ductile Fracture Shear deformation often occurs during deformation-based materials processing. Nevertheless, the mechanism of shear-stress-induced damage and ductile fracture is still controversial. One of the main points is that void linking leads to damage and fracture under shearing-dominated loadings [8,18,19]. SEM analysis of fractured surfaces for DP980 steel and AA7075 alloy indicates that fractured voids tend to elongate along the direction of the maximum shear stress in various stress states [8], as shown in Fig. 3.6. In situ tensile tests of Al-6061 alloy also reveal that shear linkup of voids is the main cause of void coalescence [10]. As the voids nucleate and grow with a lower growth rate, shear linkup of voids is the main mechanism of fracture. A shear band with minor voids was observed, as depicted in Fig. 3.7. The tests confirmed that void linkup had significant effects on damage evolution. The voids’ deformation mode is very

Figure 3.6 Schematic illustration of shear linkup of voids.

Figure 3.7 Shear-deformation-induced ductile fracture [10].

Damage Evolution and Ductile Fracture

91

sensitive to stress triaxiality [20]. Void linkup is strongly affected by numerous factors: void shape, relative void spacing, the nucleation of secondary voids, and the surface contact of flattened voids at low or negative stress triaxiality [8].

3.2.3 Damage Caused by a Mixture of Tension and Shear Stress Under deformation-based processing, the materials are usually subjected to a mixture of tensile and shear stresses. The effects of the maximum tensile stress and shear stress on void evolution are different, which may lead to difference in macrofracture mode [10], as shown in Figs. 3.3 and 3.7. Under this mixed stress state, the voids elongate along the direction of the maximum tensile stress and shear linkup occurs simultaneously, as shown in Fig. 3.8. In situ tensile tests of Al-6061 alloy, as shown in Fig. 3.9, illustrate the presence of

Figure 3.8 Void evolution caused by a mixture of tension and shear stress.

Figure 3.9 Ductile fracture caused by a mixture of tension and shear stress [10].

92

Deformation-Based Processing of Materials

dimples with shear bands in the center region of the fractured surface. It should be noted that the mixture of tension- and shear-induced damage is the main mechanism of ductile fracture during deformation-based materials processing.

3.3 MODELING AND PREDICTION All the while, the accurate prediction of damage evolution and ductile fracture is one of the most challenging and contentious problems in deformation-based materials processing. Based on experimental observation and micromechanics, plenty of DFC based on continuum mechanics have been developed to capture and predict damage and fracture. Another typical method for predicting fracture is to establish failure diagrams, such as the forming limit diagram (FLD), which is widely used in the manufacturing industry. Recently, phase field modeling and crystal plasticity have come into use for analysis of ductile fracture [21
24]. Advances in understanding of the underlying physical mechanisms of these criteria have led to improvements or modifications to more accurately capture the physical essence of microscopic ductile fracture behavior. In this section, both DFC and FLDs are discussed according to the development of those models.

3.3.1 Ductile Fracture Criteria In recent decades, many DFC and their modified versions have become available for predicting and analyzing fracture in deformation-based materials processing. In view of how the models handle the yield behaviors of materials, DFC are classified into two categories: uncoupled, which neglects the effects of damage on materials’ yield surface, and coupled, which incorporates the damage accumulation into the constitutive equations of materials and shows how it affects the yield and damage. 3.3.1.1 Coupled Ductile Fracture Criteria Coupled DFC incorporate damage accumulation within their constitutive equations [1,11] and allow the materials’ yield surface to be modified with the change of the damage-induced void density. The most widely used coupled DFC in the forming process is the Gurson
TvergaardNeedleman (GTN) model, which is based on the fact that void growth has an effect on plastic deformation and ductile fracture. Other coupled DFC are continuum damage mechanics (CDMs) models and are based on

Damage Evolution and Ductile Fracture

93

a consistent thermodynamic framework [12,25,26], in which the evolution of the phenomenological damage parameter is obtained through thermodynamic dissipation potential. In this section, the GTN model and CDM-based Lemaitre modeling and their modification versions are introduced. 3.3.1.1.1 GTN-Type Ductile Fracture Criteria The GTN model of porous plasticity was first developed by Gurson [1] and further modified by Tvergaard and Needleman [2,3]. To improve the prediction accuracy of shear deformation, many modifications considering shear stress and change of stress triaxiality have been carried out. Recently, some GTN-based anisotropic fracture models have been developed to predict anisotropic ductile fracture during deformation-based materials processing. Those studied in detail are presented in the following section. • GTN model In the GTN model, the behavior of a void-containing solid is described by pressure-sensitive plastic flow. The void-volume fraction is introduced to explain void evolution in the deformation process, while some additional parameters (q1 ; q2 ; q3 ) are used to represent void nucleation, growth, and coalescence. The GTN model is designated as:
2
   σe 3 q2 σh φ5 (3.1) 1 2q1 f  cosh 2 2 11q3 f 2 5 0 2 σm σm where q1 , q2 , and q3 are material parameters to be determined by experiment; σe is the equivalent stress; σm is the flow stress of the undamaged matrix material; σh is the hydrostatic stress; and f  is the void-volume fraction function and is also considered as the damage variable.   f ; f # fc f 5 (3.2) fc 1 kðf 2 fc Þ; f . fc where fc is the voids’ critical volume fraction for coalescence, k 5 ðfu 2 fc Þ=ðff 2 fc Þ is the acceleration factor of void growth, ff is the void-volume fraction at fracture, and fu is the void-volume fraction with zero stress capacity. When f -ff and f  -fu , the material gradually loses its load-bearing capacity.

94

Deformation-Based Processing of Materials

In the original GTN model, the void-volume fraction f is dependent on the nucleation of the new voids and the growth of the existing voids, as shown in Eq. (3.3). f 5 dfnucleation 1 dfgrowth

(3.3)

Nucleation of voids can occur as a result of microcracking and/or a decohesion of the interface of inclusions and second-phase particles. The plastic-strain-controlled nucleation is given by Eq. (3.4). dfnucleation 5 Adεpm

(3.4)

where A is a material parameter related to the matrix’s total equivalent plastic strain (EPS), and dεpm is the increment of EPS. Assuming that the matrix material is plastically incompressible, the void growth rate is a function of the plastic volume change, as shown in Eq. (3.5). p

dfgrowth 5 ð1 2 f Þdεii

(3.5)

p

•

where dεii is the trace of the plastic strain tensor. Shear-modified GTN model One important limitation of the original GTN model is its inapplicability to localization and fracture with low-stress triaxiality [11] and shear-dominated deformations, such as shear bending. The damage increment is tied to the incremental growth of voids and thus requires a positive mean stress. Consequently, there is no damage in shearing deformations under zero mean stress, and neither localization nor material failure occurs. However, this scenario is unrealistic for pure shear deformation, in which the mean stress is zero but a shear crack is observed aligned with the plane of maximum shear stress. Motivated by these issues, there have been many modifications to the original GTN model [11,27]. Nahshon and Hutchinson introduced an additional term to quantify shearing effects at low triaxiality [11]. The shear damage term that assumes linear dependence on the current porosity and the deviatoric part of plastic strain rate can be expressed as dfshear 5 kω f ω

S:_εp σe

(3.6)

where kω is a parameter producing the damage rate in the shear loading and ω is a function of the stress state.

Damage Evolution and Ductile Fracture

95

•

GTN-based anisotropic damage model In recent years, anisotropic fracture has attracted a substantial amount of attention for damage and fracture analysis [28
30]. The GTN model is also used to capture this phenomenon, using anisotropic equivalent stress instead of Mises stress [31,32]. In the GTN-based anisotropy model, materials’ anisotropy is represented by modifying the contribution of each stress deviator. The voids are still assumed to be spherical and uniformly distributed in the matrix material with isotropic evolution. The plastic flow depends on both the void-volume fraction f and the accumulated plastic strain εplm of the matrix material. The total damage evolution, that is, variations in the void-volume fraction, is determined by Eq. (3.3). The evolution function of εplm is obtained by the effective plastic work theorem and designated as: ð1 2 f Þσm dεplm 5 σ:dεp

(3.7)

where dεplm is the accumulated plastic strain increment of matrix material and dεp is the macroplastic strain increment. According to the association flow rule, the plastic strain increment can be obtained:
 @φ 1 @φ @φ 1 p dε 5 dλ I1 n 5 Δεσh I 1 Δεσe n (3.8) 5 dλ 2 @σ 3 @σh @σe 3 where n 5 ð3=2σe ÞS is the flow direction and Δεσh 5 2 dλð@φ=@σh Þ, Δεσe 5 2 dλð@φ=@σe Þ. Removing dλ then yields the following: Δεσh

@φ @φ 1 Δεσe 50 @σe @σh

(3.9)

Eqs. (3.1) and (3.9) constitute a set of nonlinear functions, which can be solved by the Newton iteration method in discrete incremental form. The evolution functions of the two internal variables, εplm and f, can be derived by considering the directions of the static pressure and plastic flow. In addition, the stress tensor can be represented by 2 σ 5 2 σh I 1 σe n 3

(3.10)

96

Deformation-Based Processing of Materials

According to Eqs. (3.7), (3.8), and (3.10), the following equation is obtained: dεplm 5

2 σh Δεσh 1 σe Δεσe ð1 2 f Þσm

df 5 ð1 2 f ÞΔεσh 1 Adεplm

(3.11)

(3.12)

3.3.1.1.2 Continuum Damage Mechanics
Based Lemaitre Criteria The CDM-based damage model defines damage as the progressive deterioration of the materials, which is an irreversible deformation process in the microcosmic structure associated with void nucleation and growth, and the microcracking of brittle inclusions. Lemaitre proposed a damage model that incorporates the effective stress for the elastoplastic case. However, the model fails to explain void nucleation in damage evolution. To overcome this problem, a damage growth law for void nucleation and growth was proposed [33]. The Lode parameter is also introduced to improve the applicability of the Lemaitre model to various loading conditions [34,35]. In this section, the Lemaitre model and its modification are introduced and discussed. • Lemaitre damage model The Lemaitre criterion is one of the most important CDM-based models [25]. For the Lemaitre-based criterion, by introducing the scalar damage variable D, written in Eq. (3.13), the isotropic damage evolution is incorporated into the constitutive modeling equation to deal with the progressive weakening over the course of the failure development. Thus, the damage behavior is represented by the constitutive equation of the undamaged material, in which only the stress is replaced by the modified effective stress, as shown in Eq. (3.14). In addition, the model’s identification is straightforward and uses conventional mechanical tests such as tensile tests, which makes it useful for industrial applications with high-stress triaxiality. Dn 5 1 2

S~ SD 5 S S

(3.13)

where S~ is the effective resisting area, SD is the damaged area, and S is the overall section area of that element defined by its normal n.

Damage Evolution and Ductile Fracture

σ~ 5

σ σ σ~ ;ε5 5 12D E ð1 2 D Þ E

97

(3.14)

The damage indicator D is linearly related to the accumulated plastic strain, which is derived from the damage dissipation potential and the release rate of the damage strain energy, with an assumption of _ is as follows: strain equivalence. The increment of the damage value D ( "
2 # )s0 2 K 2 σh 2 _5 ð11ν Þ13ð122ν Þ D εpM ε_ p (3.15) 2ES0 3 σe •

•

where K and S0 are material parameters. Improved-modified Lemaitre damage model Under shear-dominated and complex loadings at low-stress triaxiality, the Lemaitre model often fails to provide good prediction of damage localization. In this case, the damage accumulation depends only on the von Mises equivalent stress. Accordingly, by considering the changes in stress triaxiality and Lode parameter, some modified Lemaitre models have been developed and applied to predict fracture with various mechanical tests, such as shear tests on butterfly specimen and torsion tests [34
36]. As changes in stress triaxiality and Lode parameter do not influence the damage accumulation under the uniaxial tension stress state, the improved Lemaitre damage model can be reduced to the original Lemaitre damage model under uniaxial tension. Lemaitre-based anisotropic damage model To capture the anisotropic ductile fracture, some researchers have coupled the Lemaitre model and the anisotropy yield criteria instead of the Mises criterion [31]. For the Lemaitre-based anisotropic damage model, the yield condition can be denoted as:  σ 2 e ~ 5 φðσÞ 2 σ2m # 0 (3.16) 12D where σe is the anisotropic equivalent stress.

3.3.1.2 Uncoupled Ductile Fracture Criteria In the uncoupled DFC category, damage accumulation is formulated empirically or semiempirically with the general function in Eq. (3.17) in terms of certain macroscopic variables, such as the EPS, principal stress, shear stress, and hydrostatic stress, all of which capture fracture initiation

98

Deformation-Based Processing of Materials

and propagation [37]. Those DFC are formulated according to experimental observation, micromechanical analysis, numerical results, or some combinations. Despite its limitations in representing the deterioration of damaged materials, the uncoupled approach has been used widely due to its simple formulation and ease of calibration [38]. ð εp f f ðσ; εp Þdεp $ C (3.17) 0

where εf is the EPS to fracture, εp is the EPS, σ is stress tensor, and C is the fracture threshold of materials. Considering the significant effect of stress state on ductility, most of the uncoupled criteria incorporate different sources of stress, such as hydrostatic stress, principal stress, and shear stress, into the modeling [39
42]. Furthermore, the Lode parameter is also taken into consideration to improve the prediction accuracy for ductile fracture [43
46]. Regardless of whether the Lode parameter is considered, the uncoupled DFC are further classified into two subcategories, as follows: • The uncoupled DFC without considering the Lode parameter Most uncoupled DFC without considering Lode parameter were proposed to predict ductile fracture in different deformation processes, such as extrusion, drawing, upsetting, and rolling. Those uncoupled DFC have a very simple form, as shown in Table 3.1. Determining the critical damage value Ci in these models is the foremost concern. In most situations, however, the determined critical damage value Ci varies across experiments due to different deformation histories and different stress states at the fracture locations. Moreover, almost all of these criteria were developed for specific processes with specific deformation paths. For complex loading paths and stress states, the above criteria must be improved to obtain the higher prediction accuracy. • The uncoupled DFC considering the Lode parameter By incorporating the Lode parameter, scholars have recently proposed a number of new DFC to improve the prediction accuracy of ductile fracture [43
49]. Compared with the uncoupled DFC listed in Table 3.2, the DFC fracture locus considering the Lode parameter is no longer simply a curve on a 2D plane, but a 3D surface. At the same time, more tests need to be done to calibrate the parameters of these new criteria, as the prediction accuracy is dependent on calibration tests.

Table 3.1 Uncoupled DFC Without Considering the Lode Parameter Criterion Formula

Freudenthal

Ð εf

Cockcroft & Latham (C&L)

Ð εf

Brozzo

Ð εf

0

σdεp 5 C1

Plastic work

0

hσ1 i=σdεp 5 C2

Based on the effects of the maximum tensile stress on void evolution

0

2hσ1 i dεp 5 C3 3hσ1 2 σm i σm σ dεP 5 C4   exp 32 σH σ dεp 5 C5

0

ð1 1 AσH σÞdεP 5 C6

0

Ayada

Background

Ð εf

Rice & Tracey (R&T)

Ð0ε f

Oyane

Ð εf

Tresca

τ max 5

Johnson
Cook

εf 5 ðD1 1 D2 expðD3 σ ÞÞð1 1 D4 ln ε_  Þð1 1 D5 T Þ

σ1 2 σ3 2

5 C7

Based on the effects of stress triaxiality on void evolution, neglecting void nucleation, and coalescence Shear linking up of voids Considering the effects of strain rate and temperature

Table 3.2 Uncoupled DFC Considering the Lode Parameter Criterion Formula

Modified Mohr
Coulomb (MMC) [43] Lou 2012 [18] Hosford
Coulomb [48] Hu 2017 [49]

8 2 1 0 0 113 " pffiffiffi >

# 2 n rffiffiffiffiffiffiffi2ffi
 h 6 2 2 3 : 2  b Ð εf a h113ηi ð 2τ σ Þ dεP 5 C8 max 0 2 n

  1 o21n εf 5 Kb 12 ðf1 2f2 Þa 1 ðf2 2f3 Þa 1 ðf1 2f3 Þa 1c ð2η1f1 2f3 Þ a   P Ð εf  a 1 b ð 2τ σ Þ 1 η2 dε 5 C9 max 0 3

Damage Evolution and Ductile Fracture

•

•

101

Anisotropic ductile fracture criteria The DFC reviewed above did not consider the anisotropic behavior in ductile fracture even though the anisotropy in ductile fracture ubiquitously and naturally exists in textured metals. To consider the impact of anisotropic plasticity, the plastic anisotropy was incorporated into the isotropic modified Mohr
Coulomb (MMC) fracture criterion [50]. Using a stress state-dependent weighting function with an anisotropic plastic-strain measure, an anisotropic MMC criterion was proposed and employed to predict the fracture of AA 6260-T6 [51]. Based on an isotropic strain-rate potential computed from an isotropic damageequivalent strain-rate vector, which is mapped from the plastic strain-rate vector by a fourth-order linear transformation tensor, an anisotropic ductile fracture criterion was proposed and applied to depict anisotropic ductile fracture of AA 6K21 in shear, uniaxial tension, and plane-strain tension along different loading directions and balanced biaxial tension [28]. These studies give an access to model and predict anisotropic ductile fracture in deformation-based materials processing. Strain-rate and temperature-dependent uncoupled DFC The ductile fracture of metal is not only dependent on the stress state, but also on strain rate and temperature. Many DFC, such as maximum shear stress, the Johnson
Cook model, and the constant fracture strain model, can be employed to predict ductile fracture under high strain rate. The Johnson
Cook model is the most widely used DFC. The influence of strain rate and temperature on isotropic ductile fracture was considered using power-law forms [40]. Based on the same principle, strain-rate and temperature terms were incorporated into the anisotropic fracture criterion to correlate the anisotropic fracture strength over a wide range of strain rate and temperature [52]. The Zener
Hollomon parameter is also known as the temperaturecompensated strain rate, which has been widely used for characterizing behaviors of materials under hot working conditions. Accordingly, the Zener
Hollomon parameter was also employed to represent the effect of temperature and strain rate on ductile fracture [53]. Those studies have greatly promoted the development of strain-rate and temperature-dependent ductile fracture models.

3.3.2 Failure Diagrams For sheet metal forming, failure diagrams are a commonly used method to evaluate materials invalidation. Failure diagrams are usually described

102

Deformation-Based Processing of Materials

with stress and strain state to fracture, and can directly reveal the formability of materials in the deformation process. Considering the difference of failure mechanism in deformation-based materials processing, the neckingand fracture-based failure diagrams are discussed in this section. 3.3.2.1 Necking-Based Failure Diagram In deformation-based materials processing, fracture caused by necking is an irrecoverable failure, and the necking prediction has always been an important topic. The FLD and its extension provide a way to predict sheet forming limit. 3.3.2.1.1 Forming Limit Diagram Necking can be divided into two categories, that is, diffuse necking and localized necking. With diffuse necking, the uniform reduction of thickness in a relatively large range occurs, while with localized necking the thinning of materials concentrates in a localized region. The FLD in the space of in-plane major strain and minor strain is an instrument widely used for the quantitative description of the sheet metal formability. The FLD, as shown in Fig. 3.10, is usually plotted from the condition of uniaxial tension to plane strain, and further to balanced biaxial tension, which can predict the failure behavior of materials during sheet metal forming.

Figure 3.10 Schematic of FLD for sheet metals [54].

Damage Evolution and Ductile Fracture

103

Various methodologies have been proposed for the experimental determination of the FLD. For instance, the uniaxial tension of sheet specimens with circular notches allows the exploration of the tension
compression range (the positive
negative domain of the FLD). By using relatively wide specimens, it is also possible to reach the planestrain point. In-plane biaxial tensile tests can be used to determine forming limit point of balanced biaxial tension. The positive
positive region of the FLD can be reproduced in a hydraulic bulging device equipped with die with circular or elliptic apertures. Different load paths belonging to the tension
tension domain are generated by varying the eccentricity of the elliptic aperture [55]. Other procedures used for the experimental determination of the FLC are those based on the punch stretching principle. The Nakazima test in which a hemispherical punch with a constant radius in combination with rectangular specimens with different widths is adopted is one of the most effective ways [56,57]. The Nakazima test shown in Fig. 3.11 can explore both the tension
compression and the tension
tension domains of FLD. By using circular specimens with lateral notches, the main disadvantage of the Nakazima test, namely the wrinkling of wide specimens, can be removed [58]. The Marciniak test is another method that can obtain different load paths by modifying the cross-section of the punch (circular, elliptic, or rectangular) [59]. In addition, due to the fact that the specimen is placed on the top of a carrier blank, it can reduce the frictional effects in the case of the flat punch drawing test. Different researchers use different methods (i.e., Nakazima test or Marciniak test) to define the onset of necking, resulting in different determined FLDs for the same material [59
61]. Hence, the above factors should be taken into account when determining the forming limit strains.

Figure 3.11 Schematic of (A) Marciniak test and (B) Nakazima test [60,61].

104

Deformation-Based Processing of Materials

Although standardized experimental FLD tests exist, the experimental determination of FLD suffers from a lack of reproducibility, which can be partly attributed to the high sensitivity of FLD to experimental factors [56]. Moreover, these specific tests are restricted to simple geometries and remain expensive and time consuming. To overcome these drawbacks, theoretical methods to determine FLD, based on the use of localization criteria, have been investigated for several decades. These contributions are based on various approaches, ranging from empirical observations to theoretically sound criteria, making them more or less general, applicable to various types of materials, and able to predict necking. One of the earliest analytical approaches used to predict formability of sheet metals is the maximum loading criterion [62], which is based on Considère’s empirical observation that diffuse necking begins in a bar when the maximum force F1 is reached during a tensile test and thus: dF1 5 0

(3.18)

Assuming that the sample material is rigid plastic, Eq. (3.18) can be written as follows: dσ1 5 σ1 dε1

(3.19)

An extension of Considère’s criterion is the Swift diffuse necking model [63], which assumes that if the proportional loads in these two directions (F1, F2) reach the maximum values, plastic instability will occur. dF1 5 0; dF2 5 0

(3.20)

An equivalent expression is as follows: dσ1 dσ2 5 σ1 ; 5 σ2 dε1 dε2

(3.21)

Although the diffuse necking theory can predict the formability of sheet metal from the uniaxial tension region to the balanced biaxial tension region, it is hard to clearly observe the occurrence of diffuse necking in experiments. Moreover, it is reported that the maximum allowable strains in industrial stampings are determined by localized necking instead of diffuse necking and thus the study of diffuse necking is of limited practical interest [64]. With the assumption that the

Damage Evolution and Ductile Fracture

105

Figure 3.12 Schematic diagram of the M
K model [69].

discontinuity of stress or velocity causes plastic instability, Hill proposed a criterion to predict materials’ localized necking in the form [65]: pffiffiffiffiffiffiffiffiffi θ 5 arctan 2 β (3.22) where θ is an angle between the direction of major strain and the normal direction of the neck and β is the strain ratio in the plane. Another important class of necking prediction is based on the existence of initial heterogeneities in sheet metal. A widely used prediction model is the Marciniak
Kuczynski theory [66], that is, the M
K theory, which assumes that there is a shallow groove in the specimen, as shown in Fig. 3.12. During the proportional loading process, the area outside the groove undergoes uniform proportional straining, whereas the area inside the groove is under larger plastic strain and eventually forms a neck. In the positive
positive domain of FLD, the direction of the groove is usually perpendicular to the orientation of the maximum principal stress, while in the positive
negative domain of the FLD, the direction of the groove is at a specific angle with the orientation of the maximum principal stress. It should be mentioned that the robustness of the M
K model is influenced significantly by factors [67,68] such as yield criterion, flow rule, hardening model, strain, and temperature. Therefore, in the analysis of material formability based on the M
K theory, all the key parameters should be taken into account. 3.3.2.1.2 Extension of the Forming Limit Diagram The FLD is significantly dependent on the strain path, as observed in both simulation and experiment. To take the influences of strain path into consideration, many researchers have sought out other methods to predict failure or the formability of sheet metals. Considering that steel, copper,

106

Deformation-Based Processing of Materials

and brass have an identical forming limit stress state, regardless of the type of prestrain imposed, a forming limit stress diagram (FLSD), depicting the in-plane minor and major stresses, has been used to predict failure under nonproportional loading [70
72]. The FLD can be converted into the FLSD based on the following derivation. Under plane stress assumption σ3 5 0, σ1 and σ2 can be calculated from experimental major strain (ε1) and minor strain (ε2). The ratio of the minor true strain, ε2, to the major true strain, ε1, is defined by the following parameter: ε2 ρ5 (3.23) ε1 The ratio of the minor true stress, σ2, to the major true stress, σ1, is defined by the parameter σ2 α5 (3.24) σ1 By using Eqs. (3.23) and (3.24), Mises yield criterion, and the associated flow rule, the relation between α and ρ is obtained as follows: α5

2ρ 1 1 21ρ

(3.25)

Based on the Mises yield criterion, the effective stress σ is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 5 σ21 1 σ22 2 σ1 σ2

(3.26)

This relation can also be expressed in terms of σ1 and α. σ 5 σ1 ξðαÞ3σ1 5

σ ξðαÞ

(3.27)

where ξ(α) is a function of material parameters and can be calculated as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ 5 1 1 α2 2 α (3.28) The relation between the effective stress and effective strain can then be formulated formally as: σ 5 σðεÞ

(3.29)

Based on Eqs. (3.23), (3.25), (3.27), (3.28), and (3.29), σ1 is obtained. Then σ2 is calculated by Eq. (3.24) and the FLSD is obtained.

Damage Evolution and Ductile Fracture

107

Although many researchers have proven that the FLSD is almost path-independent, a detailed study holds that the FLSD is more path-independent than the FLD when the prestrain is small, and if the prestrain is close to the strain at necking, the final forming limit stresses are then obviously different from those obtained from the proportional loading path [72]. A study of FLSDs and FLDs in two kinds of combined loadings shows that the FLSD of the combined loading, which has unloading between two paths, is identical to that of the proportional loading, while the FLSD of the combined loading, which has no unloading between two paths, is different from that of the proportional loading [70]. Some studies have revealed that work-hardening behaviors with different prestrains and loading conditions have important effects on FLSDs [71
73]. Although FLSDs have more advantages than strain-based FLDs, most researchers and industrial practitioners prefer to characterize formability in terms of strain, as stress cannot be measured directly in experiments. Moreover, a new strain-based FLD, in which the forming limit is the function of the EPS and strain-rate ratio, has been proposed to characterize the strain distribution in strain space [74]. The EPS-based FLD was further improved and expressed in polar coordinates [75]. These new approaches to predicting the formability of materials have been validated as being much less dependent on strain path than the traditional FLD. 3.3.2.2 Fracture-Based Failure Diagrams Both FLDs and FLSDs are based on the necking mechanism, so their stress triaxiality is confined to a scope of 1/3
2/3. If the fracture strains subsequent to necking are superimposed on the strain’s FLD axes, a fracture-forming limit diagram (FFLD) can be obtained, as shown in Fig. 3.13. Under certain conditions, tensile fracture can precede necking in traditional sheet metal forming processes, particularly when loading under biaxial tension, in which the FFLD rather than the FLD determines the achievable deformation [76]. The strain-based FFLD can be determined using Nakajima stretch-forming tests and the tests of samples under shear deformation [77], or obtained via the DFC covering from uniaxial compression to balanced biaxial tension [19,78]. The failure diagram in terms of effective plastic strain to fracture (εf ) and stress triaxiality (η), as shown in Fig. 3.14, can be plotted based on the FFLD. According to the DFC and experimental data, the fracture

108

Deformation-Based Processing of Materials

Figure 3.13 Schematic of the fracture-forming limit diagram [54].

Equivalent plastic strain to fracture

0.8

Exp. data points plane stress Axi-symmetric

η=0

0.7 η = 1/3

0.6

η = 1/sqrt(3)

0.5 σ2 = 0

0.4

σ3 = 0 η = 2/3

0.3 0.2

η = –1/3 (Cutoff value)

0.1 0.0 –0.50

–0.25

0.00

0.25

0.50

0.75

1.00

Stress triaxiality η

Figure 3.14 Schematic of failure diagram in terms of εf and η [18].

locus in the space of stress triaxiality, the Lode parameter, and the EPS, can be constructed and adopted to predict the formability of ductile materials [19,28,43]. In Fig. 3.15, it is seen that there is a significant difference in fracture strain under different stress triaxiality and Lode parameter. Recently, the stress-induced fracture map shown in Fig. 3.16, which articulates the relationship of size effect, fracture energy, and the expected

109

Damage Evolution and Ductile Fracture

Figure 3.15 Schematic of fracture surface in the space of ðη; L; εf Þ [43].

2.2

2.1 2

2 Fracture strain

1.9 1.8

1.8

1.7 1.6

1.6 1.5

1.4 1.4

80

1.1 1 Str es

0.9 ria xia lity

st

40 20

0.8 0.7 0

n rai

siz

µ e(

60 m)

G

Figure 3.16 Schematic of fracture surface in the space of ðη; L; εf Þ [79].

fracture strain in microforming, was proposed and constructed for fracture prediction in microforming processes [79,80]. The research thus helps understand the ductile fracture and facilitates deformation-based materials processing determination and application.

110

Deformation-Based Processing of Materials

3.3.3 Calibration Method for the Model Parameters For the above ductile fracture models, there are many model parameters that need to be calibrated. The calibration experiments can be carried out to determine those parameters directly. Moreover, the inverse method in which the parameters are determined by evaluating the goodness of fit of the force-displacement response between experiment and simulation can also be employed. It should be noted that the experiment-based methods can determine the model parameters and do not need extra simulation. 3.3.3.1 Experiment-Based Method For some DFC, the model parameters can be determined directly based on the experimental fracture strain, which is called the direct experiment method. However, for other DFC such as the Lemaitre model, the parameters are usually determined based on the change of other material parameters, which is called the indirect experiment method. These methods will be discussed in the following. 3.3.3.1.1 Direct Experiment Method Experiments reveals the true stress and strain states of materials under different loading conditions. The experimental fracture strain can be used for calibration of parameters of some uncoupled DFC, such as the Lou 2012 model, which can be reduced to a simple equation under certain conditions such as plane strain, pure shear, balanced biaxial tension, etc. On the other hand, the uniaxial tensile test, plane-strain tension test, pure shear test, and the balanced biaxial tensile test can produce those stress states and are widely employed to determine the parameters of uncoupled DFC [11,43]. The widely used samples for parameters calibration are shown in Fig. 3.17. The cruciform specimen shown in Fig. 3.18 is another widely

Figure 3.17 The widely used samples: (A) specimen with a central hole; (B) notched specimen; (C) in-plane shear specimens; (D) the Nakajima test specimens.

Damage Evolution and Ductile Fracture

111

Figure 3.18 Geometry of the cruciform specimen [69].

used specimen for biaxial tensile tests. These conditions, however, are ideal cases, which cannot be achieved in practical experiments due to necking and other factors. Moreover, more experimental data points are needed in construction of a fracture locus with high accuracy. Since an extensometer rather than a strain gauge is used for strain measurement, the measured true strain is the average value within the gauge length. The average value is less than the actual strain when localization occurs. Recently, the digital image correlation (DIC) method, which is a typical noncontact optical strain measurement method capable of capturing the entire strain field during deformation, has come into common use to determine fracture strain [28]. Its gauge can be conveniently determined after tension and can be very short. It thus overcomes the disadvantages of the fixed gauge of the traditional contact mechanical extensometer. As shown in Fig. 3.19, a stochastic pattern is first embedded onto the surfaces of the testing sample, and the speckle images before and after deformation are captured by two cameras, then analyzed by a specific pattern matching algorithm to determine the deformation of the testing sample. For the GTN-type model based on void effects, the metallographic test of deformed materials is typically used to determine the initial value of the void-volume fraction. Recently, X-ray microtomography has come into use for more accurately obtaining the evolution of the void-volume fraction during deformation [12,13], as shown in Fig. 3.20. It should be

112

Deformation-Based Processing of Materials

Figure 3.19 Digital image correlation method: (A) DIC principle [31]; (B) measured strain [19].

Figure 3.20 3D representation of the population of cavities inside the deforming sample in (A) its initial state and (B) just before fracture [6].

pointed out that the void-volume fraction parameter does not correspond well to the actual void size on equivalent materials. The void-volume fraction parameters were originally introduced as a phenomenological softening factor to represent the whole effect of voids on the constitutive

Damage Evolution and Ductile Fracture

113

relationship. Therefore, the measured value has a relatively large error and is only indirectly used as an initial parameter. In addition, the FLD, as well as failure diagram in terms of (εf , η) or (εf , η, θ), can also be expediently constructed using the direct experiment method. However, as fracture often occurs rapidly and is difficult to capture accurately, the model parameters usually have some error, which makes completely reproducing damage evolution and ductile fracture impossible. Therefore, an optimization method, namely the inverse method, is suggested to improve the accurate determination of the model parameters and is presented in Section 3.3.3.2. 3.3.3.1.2 Indirect Experiment Method ~ For the Lemaitre model, the damage variable is defined as Dn 5 1 2 S=S, and can be determined by the direct measurement consisting of the evaluation of the effective resisting area and the overall section area at mesoscale. However, the indirect experiment method is widely employed in practical applications. Ductile damage of materials is usually characterized by normalization indices, such as elastic modulus, microhardness, density, toughness indicator, resistivity, and so on [10,31]. Void-volume fraction can also be used to calibrate model parameters [12]. Those methods are based on the influence of damage on some physical or mechanically measurable properties. The elastic modulus, microhardness, and density methods are detailed in the following. • Elasticity modulus method The elastic modulus method requires first the preparation of specimens in order to conduct mechanical tests. It is assumed that the uniform and homogeneous damage exists in the specimen gauge section. Based on the influence of damage on elasticity and Eqs. (3.14) and (3.30): E~ 5 Eð1 2 DÞ

(3.30)

where E is Young’s modulus and E~ is the effective elasticity modulus of the damaged material that can be measured. The value of the damage may be derived from the measurement of E~ and is denoted as: D512

E~ E

(3.31)

114

•

Deformation-Based Processing of Materials

This technique may be used for analysis of any kind of damage as long as the damage is uniformly distributed in the deformation body on which the strain is measured. If the damage is too greatly localized, such as the high cycle fatigue of metals, another method must be used. This is the main limitation of this method. Microhardness method This is another indirect measurement based on the influence of damage on the plasticity of materials. Since microhardness can be measured by making a very small indentation on the surface of materials, the microhardness method based on the measurement of microhardness, may be considered as practically nondestructive. The test consists of inserting a diamond indenter into the material to obtain the hardness H. Theoretical analyses and many experimental results prove a linear relationship between H and flow stress σm in the following [81]: HV 5 aσm ðσm 5 KR ðεpl 1bÞnR Þ

(3.32)

where a is the material constant. For the damaged material, Eq. (3.32) should be designated as: HV aσm 5 12D 12D

(3.33)

Eq. (3.32) can be further formulated in the following format: HV 5 aσ~ m

(3.34)

Since microhardness is linearly proportional to effective stress, the following equation can be obtained: HV 5 KH ðεpl 1bÞnH

(3.35)

where KR and KH are the hardening coefficient and microhardness coefficient and nR and nH are the hardening exponent and microhardness exponents, respectively. To satisfy the linearity between HV and σeq , the value of nR should be equal to nH . By tensile test, both n and KR can be obtained. Through the microhardness test, the HV for the damaged material is measured. It is assumed that, for a small plastic pl strain ε0 , damage can be negligible and the following relation is thus maintained: HV 0 5 KH ðε0 pl 1bÞnH 5 HV 0

(3.36)

Damage Evolution and Ductile Fracture

115

By measurement of HV0 and ε0 pl , KH can then be calculated. For the undamaged material, HV can then be obtained. Therefore, the damage parameter D can be calculated based on Eq. (3.37): D512

HV HV

(3.37)

•

Density method This method is based on the influence of damage on the physical properties of materials. In the case of pure ductile damage, the defects are cavities, which are assumed to be roughly spherical [82]. This means that the void-volume fraction increases with damage. The corresponding decrease of density is measurable with apparatuses based on the Archimedean principle. The relative variation of density between the damaged state ρ~ and the initial nondamaged state ρ is defined as ðρ~ 2 ρÞ=ρ. Using micromechanics and considering a spherical cavity of radius r in a spherical representative volume element of initial radius R and mass m, the following relationships between the damage value D and the variation of density or porosity can be derived by assuming no residual microstress exists in the materials. m ρ5 (3.38) ð4=3ÞπR3 ρ~ 5

m ð4=3ÞπðR3 1 r 3 Þ

(3.39)

ρ~ 2 ρ R3 2 r3 5 3 2 1 5 ρ R 1 r3 R3 1 r 3

(3.40)


3 2=3 SD πr 2 r D5 5 5 3 2=3 3 3 S R 1r 3 πðR 1r Þ

(3.41)

Finally, the damage parameter D can be calculated by
 ρ~ 2=3 D 5 12 ρ

(3.42)

116

Deformation-Based Processing of Materials

3.3.3.2 Inverse Method For inverse method, the model parameters are determined via goodness evaluation of the fitting of the load
displacement response between experiment and simulation [31,36]. For the uncoupled criteria such as Freudenthal, C&L, Brozzo, Ayada et al., most of the critical damage values and material parameters have no clear physical meanings, but only the integration values of the damage functions. The critical damage values of those uncoupled criteria can be determined by directly comparing the simulation with the physical experiments in terms of the critical deformation level (load
displacement curve). The number of experiments needed for calibration of the uncoupled DFC is equal to the number of material parameters. For GTN, the Lemaitre model, and their modified versions, the initial values of the parameters are usually determined by the experiment-based method. The final damage parameters are determined by comparing the numerical predictions and experimental results in terms of the load
displacement curves [31,34,36]. An iterative fitting approach, as shown in Fig. 3.21, is often employed. Due to the relatively large errors from experimental measurement, there is a significant difference in the load
displacement curve between experiment and simulation. By modifying 35,000

30,000

Load (N)

25,000

20,000

Experimental data fc = 0.045, ff = 0.0475

15,000

fc = 0.09, ff = 0.095

10,000

5000

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Displacement (mm) Figure 3.21 The iterative fitting procedure for the GTN model [10].

4.5

5.0

Damage Evolution and Ductile Fracture

117

the model parameters, the simulated load
displacement curve is consistent with the experimental ones. The final model parameters are then confirmed. In addition, the plastic strain and stress to fracture can be obtained based on the inverse method, which provides a correction for the measured true strain that is the average value within the gauge length. The plastic strain and stress from simulation can be employed to construct the FLD, FLSD, and other failure diagrams. It should be noted that inverse method caused the physical meaning loss of model parameters. In general, the inverse method is an important supplement for experiment-based methods, and provides a convenient way to determine the model parameters.

3.4 CONTROL OF DAMAGE EVOLUTION AND DUCTILE FRACTURE In deformation-based materials processing, fracture is one of the most critical forming defects in the manufacture of high-strength and highperformance products. To delay materials failure and improve forming limit, damage and fracture need to be efficiently identified, restrained and prevented. In addition, materials selection, tools design, and process route determination and process parameters configuration must be carefully considered.

3.4.1 Materials Selection-Based Damage Control Due to the diversity in microstructure of different materials, damage evolution can differ even under the same processing conditions [12]. Accordingly, an important way to delay failure during processing is to choose materials with good ductility. Strength and ductility are key mechanical properties of metallic materials for developing energy-efficient and lightweight structural components across a variety of industries, including automotive and aerospace. Unfortunately, improvements to strength tend to result in a degradation in ductility and vice versa, an issue known as the strength-ductility tradeoff. Many scholars have devoted themselves to circumventing this tradeoff through various methods, such as dislocation hardening [83], minimal lattice misfit and high-density nanoprecipitation [84,85], and gradient hierarchical nanotwins [86]. The latest studies also show how the new and natural materials can overcome the tradeoff to achieve the unprecedented levels of damage tolerance within the respective material classes [87], as shown in Fig. 3.22. Progress in material composition makes it easier to delay failure in deformationbased materials processing.

118

Deformation-Based Processing of Materials

Figure 3.22 Strength
toughness relationships for engineering materials [87].

3.4.2 Die Design
Based Damage Control Die design also has a significant effect on materials’ damage evolution. The working conditions for dies of hot stamping become severe because sheets with an initial forming temperature up to 850°C are formed [88]. Friction at the sheet-die interface at such temperature becomes high. The tool wear is also comparatively large. At the same time, the mechanical properties of the forming parts deteriorate due to surface damage caused by high friction. Thus, new tool materials, optimal die design, and efficient lubrication condition also need to be developed to reduce friction and lower tool wear in deformation-based materials processing. The forming parts’ surface damage is expected to decrease and the forming limit and forming quality to improve. Another example of a tool designed to control damage is the rotary draw bending (RDB) of a tube. Research has shown that additional forces can reconstruct the nonuniform deformation distribution during bending [89]. These additional forces are achieved by designing additional tools and coordinating the process parameters. For the RDB, an additional tensile force is inevitably introduced, which increases damage to the outer wall of the tube and creates a greater risk of fracture. To avoid or weaken the tension-induced defects, an end-push tool and suitable push velocity can be designed to impose an additional compressive force, thus balancing the draw-tension force [90].

Damage Evolution and Ductile Fracture

119

3.4.3 Process-Optimization-Based Damage Control With suitable materials and optimized die design, the forming limit and forming quality are thus determined by the processing technology. The materials will have various stress and strain states under different processing technologies. FE simulation can identify the trends in damage evolution and predict material failure under different conditions. For example, multistage forming can decrease material damage and improve the forming limit when compared with the reduced step forming [91]. Fig. 3.23 shows a three-step deep drawing of A-G90 steel. A reduced two-step drawing process can be obtained by merging the second and third steps of this three-step process. The computed damage index profiles at the end of each drawing step along the AB line depicted in Fig. 3.24. Although, as expected, the damage values increase at the subsequent deformation stages in the three-step deep drawing process, as shown in Fig. 3.24B and C, the

Figure 3.23 Multistep deep drawing process of A-G90 steel [91]. (A) Blank; (B) first step; (C) second step; (D) third step.

120

Deformation-Based Processing of Materials

Figure 3.24 Numerical damage profiles of a multistep deep drawing along the AB line for (A) step 1 for the two- and three-step processes, (B) step 2 for the three-step process, (C) step 3 for the three-step process, and (D) step 2 for the two-step process [91].

maximum damage value is lower than the threshold of failure, that is, a sound part is finally attained. However, the damage criterion predicts excessively large values in the specific zones of the sheet (at the zone of contact with the punch perimeter), as shown in Fig. 3.24D. The proposed step reduction is thus precluded, illustrating that processing technologies can significantly delay materials damage and prevent fracture during deformation-based manufacture.

3.5 CASE STUDIES The structural integrity, complexity, and weight reduction of products have all become significant in manufacturing industries. For example, tubular components are attracting increasing attention as a lightweight part for structure load bearing or fluid transmission [92]. Sheet metals, including tubes, which undergo extensive plastic deformation during the rolling process, usually exhibit significant anisotropy in their mechanical

Damage Evolution and Ductile Fracture

121

properties. Due to the significant anisotropy deformation behaviors under complex loading conditions, overthinning, necking, and further cracking can easily occur in the deformation process, thereby significantly reducing the forming limit and forming quality of materials [93]. In this section, taking tube flaring and bending of high-strength stainless-steel tubes (HSSTs) as case studies, modeling and prediction of ductile fracture are discussed.

3.5.1 Calibration of the Ductile Fracture Models HSSTs have promising applications in many clusters as key lightweight materials. However, overthinning and further crack in plastic deformation are prone to occur due to limited strain hardening and high yield-strength ratio. To avoid this phenomenon, accurate prediction of the forming limit of HSST needs to be achieved by considering uneven deformationinduced fracture. The diffuse necking, limited hardening, and anisotropy evolution of the plastic strain are obtained by combining uniaxial tension with the DIC method. A piecewise hardening model has been established and further combined with the Hill-based model by considering r variation. The Hill-yield framework is also used to describe anisotropy plasticity and is incorporated with the GTN and Lemaitre models to examine anisotropy ductile fracture. The parameters of the GTN-based anisotropic ductile fracture model are determined by comparing the load
displacement curve of the tension test between experiment and simulation. By using FE simulation and microhardness tests, the inverse method can also be used to calibrate the three parameters in the Lemaitre-based anisotropy model. 3.5.1.1 Calibration of the GTN-Based Anisotropic Damage Model For the GTN-based anisotropic model, there are in total nine parameters, namely, the initial volume fraction of void f0 ; modification parameters q1 , q2 , and q3 (q3 5 q21 ); void nucleation parameters εN , sN , and fN ; the critical volume fraction of voids for coalescence fc ; and the void-volume fraction at fracture ff . The inverse method is used to calibrate the nine parameters. First, the metallurgical test is done to obtain the experimental data as the initial values. According to the effects of the damage parameters on the deformation behaviors, the properties of materials are then adjusted by comparing the experimental displacement
deformation force with the FE simulation results.

122

Deformation-Based Processing of Materials

Table 3.3 Damage Parameters for the GTN-Based Anisotropic Model of HSSTs [31] Parameters

q1

q2

q3

f0

fc

fF

fN

εN

SN

Initial values Final values

1.5

1.0

2.25

0.001

0.2

0.4

0.01

0.1

0.1

1.5

1.0

2.25

0.001

0.022

0.025

0.014

0.19

0.1

• • • •

•

Let q1 , q2 , and q3 (q3 5 q21 ) be specified as 1.5, 1, and 2.25, as shown in Table 3.3, for reflecting the interaction among the microvoids. The initial volume fraction of the voids f0 is a minor positive number and given with the value of 0.001 based on the image analysis. The initial parameters fc ; ff ; fN ; SN are obtained via fracture analysis as shown in Table 3.3. It is found that SN has little effect on the simulation results, and is assigned as 0.1. Thus, keeping the foregoing parameters unchanged, the vital parameters that need to be well determined are limited to p 5 pi 5 ðfc ; ff ; fN ; εN ÞT . As shown in Fig. 3.25, fc and ff are the two key parameters used to determine the variation trend of the curves. It is noted that the value of ff determines the coalescence process and has a minor effect on the plastic response of the material before coalescence actually occurs. Thus, the fc is firstly tuned, and then the ff is used to adjust the final results. All of these can be iteratively calculated as shown in Table 3.3.

3.5.1.2 Calibration of the Lemaitre-Based Anisotropic Damage Model For the Lemaitre-based anisotropic model, there are three parameters: Dc , εD , and εR . The microhardness test is adopted to obtain the initial values of the three parameters. The final damage parameters are determined by using the inverse method. • The Vickers indenter is first used to conduct the microhardness test on the surface of the fractured tensile specimen, and the plastic strain is marked along the tensile direction, as shown in Fig. 3.26. The fractured tensile specimen is sectioned by wire cutting, and further ground and polished. Then, several parallel lines are drawn as a reference and the measurements are done at the location with the minimum plastic strain region and the failure zone with the maximal plastic strain Sm. • The pressure for microhardness measurement is 1.96N. To improve the measurement accuracy, three repeating measurements are conducted at each mark, and the average value of the three measurements

Damage Evolution and Ductile Fracture

123

6

Tension loads (N)

5

4

3

Exp. fc = 0.2, ff = 0.4

2

fc = 0.01, ff = 0.03

1

fc = 0.022, ff = 0.03 fc = 0.022, ff = 0.025

0 0

2

4

6

8

10

12

Displacement (mm) Figure 3.25 The fitting procedure for the GTN-based anisotropic model [31].

Figure 3.26 Sampling method and specimen of microhardness test [31].

• •

is taken as the microhardness at the mark. Fig. 3.27A shows the curves of the measured microhardness versus plastic strain. By using Eq. (3.36), the damage parameter D can thus be obtained. From Fig. 3.27B, it is shown that the damage parameters Dc , εD , and εR are 0.21, 0.18854, and 0.51268, respectively. Taking the above values as the initial ones, the inverse method is used to obtain the final damage parameters, that is, Dc is 0.27, εD equals zero, and εR is 0.7.

124

Deformation-Based Processing of Materials

(A)600

(B) 0.25 HV

0.15

Damage value D

Micro-hardness (Vic)

D Fit line

0.20

H*V

550

500

450

0.10 0.05 0.00

400

–0.05 350

–0.10 0.0

0.1

0.2

0.3

Plastic strain

0.4

0.5

0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Plastic strain

Figure 3.27 Experimental data of (A) the microhardness test and (B) the damage parameter curves [31].

3.5.2 Modeling and Prediction of Ductile Fracture The four kinds of models, including the Hill-based model with piecewise hardening (Hill’48 model), Hill-based model considering R variation and piecewise hardening (AnisoPlas model), GTN-based damage model coupled with anisotropic plasticity (AnisoPlas-GTN model), and the Lemaitre-based damage model coupled with anisotropic plasticity (AnisoPlasLemaitre model), were employed to predict ductile fracture [31]. The flaring tests and the mandrel bending of the HSSTs were conducted and used to evaluate the above models. For the GTN-based anisotropic damage model and the Lemaitre-based anisotropic damage model, the corresponding nine parameters and three parameters are given in Section 3.5.1. As shown in Fig. 3.28, within the variations of the damage presented, the damage evolution has an obvious effect on macro-yield behavior. 3.5.2.1 Tube-Flaring Behaviors The flaring behavior of HSSTs is predicted with the aforementioned models. Fig. 3.29 shows the die geometry for flaring tests of HSSTs. The compression velocity is 2 mm/min. Under the assigned conditions, Fig. 3.30 shows that crack occurs in the part of the tube made of HSST. For FE modeling, the friction coefficient of 0.05 is given. Fig. 3.30 shows that after comparing the predictions using the piecewise hardening model and considering the variation in R, it is found that both the coupled models provide more accurate results. Similar to the results of tensile tests, the GTN-based coupling model gives the most accurate prediction of the displacement
load curve. As shown in Figs. 3.31 and 3.32, the uniform stress distribution and flaring deformation

125

Damage Evolution and Ductile Fracture

(A)

(B) Plastic strain: 0.2 1500

σy (MPa)

σy (MPa)

1500

1000

1000

Plastic strain: 0.2 f * = 0 (Hill' 48) f * = 0.009 f * = 0.018 f * = 0.027

500

500

D=0 D = 0.01 D = 0.2 D = 0.4

0

0 0

500

1000

0

1500

500

1000

1500

σx (MPa)

σx (MPa)

Figure 3.28 Yield locus of (A) GTN-based and (B) Lemaitre-based damage models [31].

Figure 3.29 Die geometry for flaring tests of tubular materials [31].

32,000

Experiments Vumat (Hill'48 model) Vumat (AnisoPlas model) Vumat (AnisoPlas-GTN) Vumat (AnisoPlas-Lemaitre)

28,000 24,000

Load (N)

20,000 16,000 12,000 8000 4000 0 0

4

8

12

16

20

24

28

Displacement (mm)

Figure 3.30 Comparison of the load
displacement curves in flaring tests [31].

Figure 3.31 Distribution of the equivalent stress in flaring before fracture [31]. (A) Hill’48 model; (B) AnisoPlas model; (C) AnisoPlas-GTN model; (D) AnisoPlas-Lemaitre model.

Damage Evolution and Ductile Fracture

127

Figure 3.32 Distribution of equivalent strain in flaring before fracture [31]. (A) Hill’48 model; (B) AnisoPlas model; (C) AnisoPlas-GTN model; (D) AnisoPlas-Lemaitre model.

are reproduced by the plasticity model without considering the damage effect. However, in actual flaring deformation, the uniform deformation is seldom encountered due to the nonuniform stress and strain distributions. Fortunately, such uneven stress and strain distributions are captured by the coupled models, especially for the GTN-based coupling model. Note that the predicted stress and strain by the GTN-based coupling model are larger than those by other models. By using the Lemaitre-based model, multiple cracks were predicted to occur during the flaring process, which actually did not occur in the experiments. In general, the coupling model used here can provide better predictions for anisotropic fracture. Based on the prediction results, the forming parameters can be optimized to improve the forming limit and forming quality during deformation-based processing.

128

Deformation-Based Processing of Materials

3.5.2.2 Tube-Bending Behaviors The above ductile fracture models are also applied to mandrel bending to predict failure and determine the bending limit of HSSTs. The principle of mandrel bending and the 3D FE model are shown in Fig. 3.33. Throughout, bending and ball-retracting processes are developed based on the ABAQUS/Explicit platform. Detailed modeling techniques can be found in the literature [94]. By using the GTN and Lemaitre-based coupling models, the damage evolution history during the mandrel bending of HSSTs is identified. Fig. 3.34 shows that the same evolution tendency is captured by both models—namely, the maximum void-volume fraction and the damage factor increase gradually with the bending angels, but become stable when the bending angle reaches about 30 degrees. For the GTN-based coupling model, the maximum void-volume fraction is less

Figure 3.33 Tube-bending procedure and the 3D FE model of HSST [31]. Vumat (AnisoPlas-GTN) Vumat (AnisoPlas-Lemaitre)

0.025

Maximum f

0.020

0.020

0.015

0.015

0.010 0.010

Maximum D

0.025

0.005 0.005 0.000

A –10

0

B 10

20

30

40

F

E

D

C

50

60

70

80

90

0.000 100

Bending angles (degrees)

Figure 3.34 Evolution of the damage variable at different bending moments [31].

Damage Evolution and Ductile Fracture

129

than the critical value fF of 0.025; as such, the fracture does not occur. For the Lemaitre-based coupling model, the fracture also does not occur because the maximum damage factor is far less than the critical value of 0.27. The damage evolution characteristics confirm the incremental and local deformation features in mandrel bending; in the early stage, the bending deformation is unequal, and when the bending angle reaches the critical value, the deformation becomes stable until the desired bending angle is complete. The push-assistant matching speed of the pressure die is the key parameter that influences the forming limit of tube mandrel bending [94]. By decreasing the push-assistant matching speed of the pressure die from 102% to 95%, fracture occurs at the bending angle of 16.7 degrees in bending of the HSST, as shown in Fig. 3.35. Considering the variation of

Figure 3.35 Distribution of the equivalent strain at the critical moment [31]. (A) Hill’48 model; (B) AnisoPlas model; (C) AnisoPlas-GTN model; (D) AnisoPlas-Lemaitre model.

130

Deformation-Based Processing of Materials

R and piecewise hardening, the GTN-based coupling model can accurately predict cracking, while in the simulation, the fracture occurs at 21.2 degrees because the voids’ volume fraction reaches the maximum of 0.025 at this bending moment. The predicted fracture location is near the clamp die, which further verifies the reliability of the GTN-based coupling model. By using the GTN-based coupling model, the ductile fracture-based formability of HSST is identified. Rd =D is used as an index to evaluate the bending limit of the tubular materials. By reducing the Rd =D by 0.25 from 2.0, the fracture-related bending limit is obtained. Under a bending radius of 1.75, the voids’ maximum volume fraction is 0.023, which is less than the voids’ critical volume fraction ff of 0.025. The maximum wall-thinning degree is 26.47%, which is far less than the tolerance requirement of 35% specified for this tubular part. Thus, stable bending can be achieved under a bending radius of 1.75D. Furthermore, with a bending radius of 1.5D, the voids’ maximum volume fraction exceeds 0.025 and the fracture occurs at a bending angle of 26.5 degrees. Therefore, the fracture-related bending limit of HSST is determined to be 1.75.

3.6 SUMMARY This chapter gave an introduction to the damage evolution and ductile fracture of materials in deformation-based processing. The initiation, growth, and coalescence of microscopic voids during the plastic deformation are the main causes of damage and fracture. Moreover, shear stress also has a significant effect on damage and fracture. Under different stress states, the voids’ evolution and shear-induced damage exhibit various characteristics for different materials. To predict damage and fracture in deformation-based materials processing, coupled and uncoupled DFC and failure diagrams were discussed. The GTN-type and Lemaitre models and their modified versions to accommodate various stress states were given for better prediction of anisotropic damage and fracture. The uncoupled DFC, which is formulated according to experimental observation, micromechanical analysis, numerical results, or some combination of the above, were divided into two subcategories based on whether the Lode parameter is considered or not. Furthermore, failure diagrams, which are widely applicable to manufacturing, were also presented. Using appropriate material selection and optimization design of tooling and processing

Damage Evolution and Ductile Fracture

131

technologies, the damage and ductile fracture of materials can be delayed and restrained, and thus the forming limit and forming quality be improved. Application of the GTN and Lemaitre models to tube bending and flaring confirmed the importance of modeling and predicting damage and fracture in the deformation process. In addition, choosing a reasonable DFC to predict ductile fracture in deformation-based processes with different fracture mechanisms can help industries optimize the design of products and forming processes. With the development of new materials—such as advanced highstrength steels, titanium
aluminum alloys, high-entropy alloys, etc., and the advent of new forming processes including heat-assistant, micro-, incremental-, and electromagnetic forming processes, different fracture mechanisms may exist in different forming scenarios and thus need to be explored and clarified. Moreover, the applicability of the developed fracture criteria need to be studied and determined. Furthermore, the nonuniform deformation-related anisotropic damage and fracture are not well known and exploration and study of them is necessary. Under the coupling effects of the temperature field and the complex loading condition, the effects of anisotropy and strain path on ductile fracture also need to be determined and more in-depth research on ductile fracture mechanism and modeling will promote and facilitate the wide applications of new forming processes and new and advanced materials.

REFERENCES [1] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I—yield criteria and flow rules for porous ductile media, J. Eng. Mater. Technol. 99 (1977) 2. [2] A. Needleman, V. Tvergaard, An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids 32 (1984) 461
490. [3] V. Tvergaard, Material failure by void growth to coalescence, Adv. Appl. Mech. 27 (1989) 83
151. [4] D. Lassance, D. Fabrègue, F. Delannay, T. Pardoen, Micromechanics of room and high temperature fracture in 6xxx Al alloys, Prog. Mater. Sci. 52 (2007) 62
129. [5] T.F. Morgeneyer, T. Taillandier-Thomas, A. Buljac, L. Helfen, F. Hild, On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3), J. Mech. Phys. Solids 96 (2016) 550
571. [6] E. Maire, O. Bouaziz, M. Di Michiel, C. Verdu, Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography, Acta Mater. 56 (2008) 4954
4964. [7] A.A. Benzerga, J. Besson, A. Pineau, Anisotropic ductile fracture: Part I: experiments, Acta Mater. 52 (2004) 4623
4638.

132

Deformation-Based Processing of Materials

[8] Y. Lou, J.W. Yoon, H. Huh, Q. Chao, J.-H. Song, Correlation of the maximum shear stress with micro-mechanisms of ductile fracture for metals with high strengthto-weight ratio, Int. J. Mech. Sci. 146
147 (2018) 583
601. [9] A. Pineau, A.A. Benzerga, T. Pardoen, Failure of metals I: brittle and ductile fracture, Acta Mater. 107 (2016) 424
483. [10] H. Li, M.W. Fu, J. Lu, H. Yang, Ductile fracture: experiments and computations, Int. J. Plast. 27 (2011) 147
180. [11] K. Nahshon, J.W. Hutchinson, Modification of the Gurson Model for shear failure, Eur. J. Mech. A/Solids 27 (2008) 1
17. [12] N. Bonora, A nonlinear CDM model for ductile failure, Eng. Fract. Mech. 58 (1997) 11
28. [13] F. Hannard, T. Pardoen, E. Maire, C. Le Bourlot, R. Mokso, A. Simar, Characterization and micromechanical modelling of microstructural heterogeneity effects on ductile fracture of 6xxx aluminium alloys, Acta Mater. 103 (2016) 558
572. [14] L. Morin, J.B. Leblond, V. Tvergaard, Application of a model of plastic porous materials including void shape effects to the prediction of ductile failure under sheardominated loadings, J. Mech. Phys. Solids 94 (2016) 148
166. [15] T.F. Guo, W.H. Wong, Void-sheet analysis on macroscopic strain localization and void coalescence, J. Mech. Phys. Solids 118 (2018) 172
203. [16] A. Buljac, L. Helfen, F. Hild, T.F. Morgeneyer, Effect of void arrangement on ductile damage mechanisms in nodular graphite cast iron: in situ 3D measurements, Eng. Fract. Mech. 192 (2018) 242
261. [17] C. Landron, E. Maire, O. Bouaziz, J. Adrien, L. Lecarme, A. Bareggi, Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels, Acta Mater. 59 (2011) 7564
7573. [18] Y. Lou, H. Huh, S. Lim, K. Pack, New ductile fracture criterion for prediction of fracture forming limit diagrams of sheet metals, Int. J. Solids Struct. 49 (2012) 3605
3615. [19] Y. Lou, L. Chen, T. Clausmeyer, A.E. Tekkaya, J.W. Yoon, Modeling of ductile fracture from shear to balanced biaxial tension for sheet metals, Int. J. Solids Struct. 112 (2017) 169
184. [20] H. Agarwal, A.M. Gokhale, S. Graham, M.F. Horstemeyer, Void growth in 6061aluminum alloy under triaxial stress state, Mater. Sci. Eng. A 341 (2003) 35
42. [21] C. Miehe, D. Kienle, F. Aldakheel, S. Teichtmeister, Phase field modeling of fracture in porous plasticity: a variational gradient-extended Eulerian framework for the macroscopic analysis of ductile failure, Comput. Methods Appl. Mech. Eng. 312 (2016) 3
50. [22] C. Miehe, F. Aldakheel, A. Raina, Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory, Int. J. Plast. 84 (2016) 1
32. [23] C.C. Tasan, J.P.M. Hoefnagels, M. Diehl, D. Yan, F. Roters, D. Raabe, Strain localization and damage in dual phase steels investigated by coupled in-situ deformation experiments and crystal plasticity simulations, Int. J. Plast. 63 (2014) 198
210. [24] H. Li, D. Huang, M. Zhan, Y. Li, X. Wang, S. Chen, High-temperature behaviors of grain boundary in titanium alloy: modeling and application to microcrack prediction, Comput. Mater. Sci. 140 (2017) 159
170. [25] J. Lemaitre, A continuous damage mechanics model for ductile fracture, J. Eng. Mater. Technol. 107 (1985) 83
89. [26] M. Brünig, S. Gerke, M. Schmidt, Damage and failure at negative stress triaxialities: experiments, modeling and numerical simulations, Int. J. Plast. 102 (2017) 70
82.

Damage Evolution and Ductile Fracture

133

[27] W. Jiang, Y. Li, Y. Shu, Z. Fan, Analysis of metallic ductile fracture by extended Gurson models, in: 13th Int. Conf. Fract., 2013, pp. 16
21. [28] Y. Lou, J.W. Yoon, Anisotropic ductile fracture criterion based on linear transformation, Int. J. Plast. 93 (2017) 3
25. [29] S.H. Li, J. He, B. Gu, D. Zeng, Z.C. Xia, Y. Zhao, et al., Anisotropic fracture of advanced high strength steel sheets: experiment and theory, Int. J. Plast. 103 (2018) 95
118. [30] Y. Lou, J. Whan, Anisotropic yield function based on stress invariants for BCC and FCC metals and its extension to ductile fracture criterion, Int. J. Plast. 101 (2017) 125
155. [31] H. Li, H. Yang, R.D. Lu, M.W. Fu, Coupled modeling of anisotropy variation and damage evolution for high strength steel tubular materials, Int. J. Mech. Sci. 105 (2016) 41
57. [32] A. Kami, B.M. Dariani, A. Sadough Vanini, D.S. Comsa, D. Banabic, Numerical determination of the forming limit curves of anisotropic sheet metals using GTN damage model, J. Mater. Process. Technol. 216 (2015) 472
483. [33] S. Dhar, R. Sethuraman, P.M. Dixit, A continuum damage mechanics model for void growth and micro crack initiation, Eng. Fract. Mech. 53 (1996) 917
928. [34] J. Lian, Y. Feng, S. Münstermann, A modified Lemaitre damage model phenomenologically accounting for the Lode angle effect on ductile fracture, Proc. Mater. Sci. 3 (2014) 1841
1847. [35] T.S. Cao, J.M. Gachet, P. Montmitonnet, P.O. Bouchard, A Lode-dependent enhanced Lemaitre model for ductile fracture prediction at low stress triaxiality, Eng. Fract. Mech. 124
125 (2014) 80
96. [36] L. Malcher, E.N. Mamiya, An improved damage evolution law based on continuum damage mechanics and its dependence on both stress triaxiality and the third invariant, Int. J. Plast. 56 (2014) 232
261. [37] T. Wierzbicki, Y. Bao, Y.W. Lee, Y. Bai, Calibration and evaluation of seven fracture models, Int. J. Mech. Sci. 47 (2005) 719
743. [38] Z. Xue, M.G. Pontin, F.W. Zok, J.W. Hutchinson, Calibration procedures for a computational model of ductile fracture, Eng. Fract. Mech. 77 (2010) 492
509. [39] M.G. Cockcroft, D.J. Latham, Ductility and the workability of metals, J. Inst. Met. 96 (1968) 33
39. [40] G.R. Johnson, W.H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Eng. Fract. Mech. 21 (1985) 31
48. [41] P. Wang, S. Qu, Analysis of ductile fracture by extended unified strength theory, Int. J. Plast. 104 (2018) 196
213. [42] S.A. Dizaji, H. Darendeliler, B. Kaftano˘glu, Prediction of forming limit curve at fracture for sheet metal using new ductile fracture criterion, Eur. J. Mech. A/Solids 69 (2018) 255
265. [43] Y. Bai, T. Wierzbicki, A new model of metal plasticity and fracture with pressure and Lode dependence, Int. J. Plast. 24 (2008) 1071
1096. [44] L. Mu, Y. Zang, Y. Wang, X.L. Li, P.M. Araujo Stemler, Phenomenological uncoupled ductile fracture model considering different void deformation modes for sheet metal forming, Int. J. Mech. Sci. 141 (2018) 408
423. [45] C. Defaisse, M. Mazière, L. Marcin, J. Besson, Ductile fracture of an ultra-high strength steel under low to moderate stress triaxiality, Eng. Fract. Mech. 194 (2018) 301
318. [46] S. Yan, X. Zhao, A fracture criterion for fracture simulation of ductile metals based on micro-mechanisms, Theor. Appl. Fract. Mech. 95 (2018) 127
142.

134

Deformation-Based Processing of Materials

[47] L. Xue, T. Wierzbicki, Ductile fracture initiation and propagation modeling using damage plasticity theory, Eng. Fract. Mech. 75 (2008) 3276
3293. [48] D. Mohr, S.J. Marcadet, Micromechanically-motivated phenomenological HosfordCoulomb model for predicting ductile fracture initiation at low stress triaxialities, Int. J. Solids Struct. 67
68 (2015) 40
55. [49] Q. Hu, X. Li, X. Han, J. Chen, A new shear and tension based ductile fracture criterion: modeling and validation, Eur. J. Mech. A/Solids 66 (2017) 370
386. [50] A.M. Beese, M. Luo, Y. Li, Y. Bai, T. Wierzbicki, Partially coupled anisotropic fracture model for aluminum sheets, Eng. Fract. Mech. 77 (2010) 1128
1152. [51] M. Luo, M. Dunand, D. Mohr, Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading—Part II: ductile fracture, Int. J. Plast. 32
33 (2012) 36
58. [52] A.S. Khan, H. Liu, Strain rate and temperature dependent fracture criteria for isotropic and anisotropic metals, Int. J. Plast. 37 (2012) 1
15. [53] X. Shang, Z. Cui, M.W. Fu, A ductile fracture model considering stress state and Zener-Hollomon parameter for hot deformation of metallic materials, Int. J. Mech. Sci. 144 (2018) 800
812. [54] Q. Hu, F. Zhang, X. Li, J. Chen, Overview on the prediction models for sheet metal forming failure: necking and ductile fracture, Acta Mech. Solida Sin. 31 (2018) 259
289. [55] A.J. Ranta-Eskola, Use of the hydraulic bulge test in biaxial tensile testing, Int. J. Mech. Sci. 21 (1979) 457
465. [56] F. Abed-Meraim, T. Balan, G. Altmeyer, Investigation and comparative analysis of plastic instability criteria: application to forming limit diagrams, Int. J. Adv. Manuf. Technol. 71 (2014) 1247
1262. [57] F. Ozturk, D. Lee, Experimental and numerical analysis of out-of-plane formability test, J. Mater. Process. Technol. 170 (2005) 247
253. [58] D. Banabic, L. Lazarescu, L. Paraianu, I. Ciobanu, I. Nicodim, D.S. Comsa, Development of a new procedure for the experimental determination of the Forming Limit Curves, CIRP Ann. 62 (2013) 255
258. [59] Z. Marciniak, K. Kuczyn´ ski, T. Pokora, Influence of the plastic properties of a material on the forming limit diagram for sheet metal in tension, Int. J. Mech. Sci. 15 (1973) 789
800. [60] J. Min, T.B. Stoughton, J.E. Carsley, J. Lin, An improved curvature method of detecting the onset of localized necking in Marciniak tests and its extension to Nakazima tests, Int. J. Mech. Sci. 123 (2017) 238
252. [61] J. Min, T.B. Stoughton, J.E. Carsley, J. Lin, Compensation for process-dependent effects in the determination of localized necking limits, Int. J. Mech. Sci. 117 (2016) 115
134. [62] A. Considère, Mémoire sur l’emploi du fer et de l’acier dans les constructions, Annales des Ponts et Chaussées 9 (1885) 574. [63] H.W. Swift, Plastic instability under plane stress, J. Mech. Phys. Solids 1 (1952) 1
18. [64] A.B.D. Rocha, F. Barlat, J.M. Jalinier, Prediction of the forming limit diagrams of anisotropic sheets in linear and nonlinear loading, Mater. Sci. Eng. A 68 (1985) 151
164. [65] R. Hill, On discontinuous plastic states, with special reference to localized necking in thin sheets, J. Mech. Phys. Solids 1 (1952) 19
30. [66] Z. Marciniak, K. Kuczynski, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 609. IN1, 613
612, IN2, 620. [67] H. Son, Y. Kim, Calculations of forming limit diagrams for changing strain paths on the formability of sheet metal, J. Mater. Sci. Technol. 17 (2001) 141
142.

Damage Evolution and Ductile Fracture

135

[68] P. Eyckens, A. Van Bael, P. Van Houtte, Marciniak-Kuczynski type modelling of the effect of through-thickness shear on the forming limits of sheet metal, Int. J. Plast. 25 (2009) 2249
2268. [69] W.N. Yuan, M. Wan, X.D. Wu, C. Cheng, Z.Y. Cai, B.L. Ma, A numerical M-K approach for predicting the forming limits of material AA5754-O, Int. J. Adv. Manuf. Technol. 98 (2018) 811
825. [70] K. Yoshida, T. Kuwabara, M. Kuroda, Forming limit stresses of sheet metal under proportional and combined loadings, AIP Conf. Proc. 778 A (2005) 478
483. [71] K. Yoshida, N. Suzuki, Forming limit stresses predicted by phenomenological plasticity theories with anisotropic work-hardening behavior, Int. J. Plast. 24 (2008) 118
139. [72] P.D. Wu, A. Graf, S.R. MacEwen, D.J. Lloyd, M. Jain, K.W. Neale, On forming limit stress diagram analysis, Int. J. Solids Struct. 42 (2005) 2225
2241. [73] J. He, D. Zeng, X. Zhu, Z. Cedric Xia, S. Li, Effect of nonlinear strain paths on forming limits under isotropic and anisotropic hardening, Int. J. Solids Struct. 51 (2014) 402
415. [74] R. Mesrar, S. Fromentin, R. Makkouk, M. Martiny, G. Ferron, Limits to the ductility of metal sheets subjected to complex strain-paths, Int. J. Plast. 14 (1998) 391
411. [75] T.B. Stoughton, J.W. Yoon, Path independent forming limits in strain and stress spaces, Int. J. Solids Struct. 49 (2012) 3616
3625. [76] P.A.F. Martins, N. Bay, A.E. Tekkaya, A.G. Atkins, Characterization of fracture loci in metal forming, Int. J. Mech. Sci. 83 (2014) 112
123. [77] S. Panich, M. Liewald, V. Uthaisangsuk, Stress and strain based fracture forming limit curves for advanced high strength steel sheet, Int. J. Mater. Form. 11 (2017) 643
661. [78] N. Park, H. Huh, S.J. Lim, Y. Lou, Y.S. Kang, M.H. Seo, Facture-based forming limit criteria for anisotropic materials in sheet metal forming, Int. J. Plast. 96 (2014) 1
35. [79] W.T. Li, M.W. Fu, S.Q. Shi, Study of deformation and ductile fracture behaviors in micro-scale deformation using a combined surface layer and grain boundary strengthening model, Int. J. Mech. Sci. 131
132 (2017) 924
937. [80] J.Q. Ran, M.W. Fu, A hybrid model for analysis of ductile fracture in micro-scaled plastic deformation of multiphase alloys, Int. J. Plast. 61 (2014) 1
16. [81] A. Mkaddem, F. Gassara, R. Hambli, A new procedure using the micro-hardness technique for sheet material damage characterisation, J. Mater. Process. Technol. 178 (2006) 111
118. [82] J. Lemaitre, A Course on Damage Mechanics, Springer, Berlin, Heidelberg, 1992. [83] B.B. He, B. Hu, H.W. Yen, G.J. Cheng, Z.K. Wang, H.W. Luo, et al., High dislocation density-induced large ductility in deformed and partitioned steels, Science 357 (2017) 1029
1032. [84] S. Jiang, H. Wang, Y. Wu, X. Liu, H. Chen, M. Yao, et al., Ultrastrong steel via minimal lattice misfit and high-density nanoprecipitation, Nature 544 (2017) 460
464. [85] D. Raabe, D. Ponge, O. Dmitrieva, B. Sander, Nanoprecipitate-hardened 1.5 GPa steels with unexpected high ductility, Scr. Mater. 60 (2009) 1141
1144. [86] Y. Wei, Y. Li, L. Zhu, Y. Liu, X. Lei, G. Wang, et al., Evading the strength
ductility trade-off dilemma in steel through gradient hierarchical nanotwins, Nat. Commun. 5 (2014) 3580. [87] R.O. Ritchie, The conflicts between strength and toughness, Nat. Mater. 10 (2011) 817
822.

136

Deformation-Based Processing of Materials

[88] K. Mori, P.F. Bariani, B.-A. Behrens, A. Brosius, S. Bruschi, T. Maeno, et al., Hot stamping of ultra-high strength steel parts, CIRP Ann.-Manuf. Technol. 66 (2017) 755
777. [89] H. Li, J. Ma, B.Y. Liu, R.J. Gu, G.J. Li, An insight into neutral layer shifting in tube bending, Int. J. Mach. Tools Manuf. 126 (2018) 51
70. [90] Y. Jing, Y. He, Z. Mei, L. Heng, Forming characteristics of Al-alloy large-diameter thin-walled tubes in NC-bending under axial compressive loads, Chin. J. Aeronaut. 23 (2010) 461
469. [91] M. Pacheco, D. Celentano, C. García-Herrera, J. Méndez, F. Flores, Numerical simulation and experimental validation of a multi-step deep drawing process, Int. J. Mater. Form. 10 (2015) 15
27. [92] H. Yang, H. Li, Z. Zhang, M. Zhan, J. Liu, G. Li, Advances and trends on tube bending forming technologies, Chin. J. Aeronaut. 25 (2012) 1
12. [93] H. Li, H.Q. Zhang, H. Yang, M.W. Fu, H. Yang, Anisotropic and asymmetrical yielding and its evolution in plastic deformation: titanium tubular materials, Int. J. Plast. 90 (2017) 177
211. [94] H. Li, H. Yang, M. Zhan, Z.C. Sun, R.J. Gu, Role of mandrel in NC precision bending process of thin-walled tube, Int. J. Mach. Tools Manuf. 47 (2007) 1164
1175.