Complete GURSON material model

Z. L. Zhang, A complete Gurson Model, in Nonlinear Fracture and Damage Mechanics, edited by M. H. Alibadi, WIT Press Southampton, UK, 2001, p 223-248.

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Nonlinear Fracture and Damage Mechanics, edited by M. H. Alibadi, WIT Press Southampton, UK, 2001. Chapter 8 A complete Gurson Model Z. L. Zhang SINTEF materials Technology, Trondheim, Norway Email: [email protected] Abstract For polycrystalline metals ductile fracture most often occurs by nucleation, growth and coalescence of microvoids. Void nucleation and growth decreases the load carrying capacity of a homogeneously deformed material, while void coalescence terminates the macroscopically homogenous dilatational deformation by shifting to a localized mode. In the past decades much research effort has been devoted to understanding the void mechanisms and to developing micro-mechanical models for better describing ductile fracture. Probably the best-known dilatational plasticity model is the one introduced by Gurson, later modified by Tvergaard and Needleman. The Gurson model was derived based on the assumption that the deformation mode of the matrix material surrounding a void is homogenous. It can therefore predict the material softening behaviour due to the nucleation and growth of voids, but has no intrinsic ability to predict the shift of a homogenous deformation mode to a localized mode by void coalescence. An empirical criterion, for example, the critical void volume fraction criterion, has to be used. In this study, the plastic limit load model introduced by Thomason has been modified and incorporated into the Gurson model. According to Thomason, the virtual plastic localization mode by coalescence exists once a void is nucleated, it can only be realized, however, when the plastic limit load has been reached. By integrating the Gurson model and the plastic limit load model, the Gurson model becomes complete. The complete Gurson model can predict both the homogenous and the localized dilatational deformation phases caused by the presence of voids. It is accurate for a wide range of stress triaxiality and strain hardening. Based on the complete Gurson model, a method for uniquely determined material’s void nucleation parameter is proposed. The advantages of the complete Gurson model are discussed in detail. 1 Introduction Unlike yielding where the yield strength of a material can be transferred from one geometry to another, ductile fracture is a geometry dependent event. The ductility or fracture toughness of a material varies with the change of geometry constraint level and can not be directly transferred from one geometry to another. It is now generally understood that conventional fracture mechanics, which uses global fracture parameters, such as J or CTOD, works only in some

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limited cases. For polycrystalline metals, it has been observed for a long time that ductile fracture is controlled by nucleation, growth and coalescence of microvoids. Fig.1 shows the dimples observed in the fracture surface of a 6082 aluminum alloy. It is therefore natural to link materials fracture behaviour to the parameters that describe the evolution of microvoids, rather than the conventional global fracture parameters. In the past decades, substantial efforts have been devoted to the understanding of void mechanisms. Micro-mechanical model-based approaches for describing ductile fracture as a consequence of void nucleation, growth and coalescence have received considerable attention. For review articles, refers to Tvergaard [1] and Needleman and Tvergaard [2]. For most engineering alloys, voids can be nucleated from large inclusions and second phase particles by particle fracture or interfacial decohesion [3]. Once a void has been nucleated, it will grow under plastic deformation and hydrostatic stress. Eventually the voids will connect and ductile fracture by void coalescence will appear. The ductile fracture process due to the presence of voids can be separated into two phases, the homogenous deformation with void nucleation and growth, and the localized deformation due to void coalescence. Much effort has been devoted to developing micro-mechanical model-based constitutive equations to simulate the ductile fracture process. One of the best known micro-mechanical models is that of Gurson [4]. The Gurson model was derived by assuming a homogenous deformation field in the matrix material surrounding a void.

Figure 1: Dimple type fracture surface of a 6082 aluminium alloy. Although Gurson’s original objective was to develop a mathematical model which could describe the whole ductile fracture process, Gurson model succeeded only in describing the first deformation phase of ductile fracture. Thomason criticized the Gurson model for actually only representing a weak response (homogenous deformation) of the material [5]. Because it is associated with a homogenous deformation field, it has no intrinsic ability to predict void coalescence – the shift from a homogenous deformation mode to a localized one. Instead, he derived a so-called strong dilatational model by assuming a localized deformation mode (Thomason [5-7]). The strong response denotes void coalescence. Thomason argued that the possibility of void coalescence exists in the material when a void has been nucleated; it can only be realized, however, when a plastic limit load criterion has been reached. A dual

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deformation model, which includes both the homogenous dilatational deformation due to void nucleation and growth and the localized dilatational deformation by void coalescence, should thus be adopted for modeling ductile fracture. It has been demonstrated by the author that, with a modification, the plastic limit load model by Thomason can be fully compatible with the Gurson model (Zhang [8-11]). In this study, by integrating the Gurson model and the plastic limit model by Thomason, a so-called complete Gurson model is proposed. The complete Gurson model includes both the homogenous and localized deformation modes, it has the ability, therefore, to predict the two phases involved in a ductile fracture process by void nucleation, growth and coalescence. In comparison with the FEM results of axisymmetric and plane strain cell models, it has been shown that the void coalescence predicted by the complete Gurson model is fairly accurate. One of the important features of the complete Gurson model is that void coalescence is controlled by a physical mechanism, rather than by a critical void volume fraction, which is not a material constant, but a function of stress triaxiality, strain hardening and the initial void volume fraction. According to the complete Gurson model, the ductile fracture is completely determined once the void nucleation is known.

Figure 2: Schematic plot of ductile fracture mechanisms: (a) void nucleation, (b) void growth, (c) the beginning of void coalescence and (d) the end of coalescence. Processes (a) and (b) are dilatational and macroscopically homogenous. Process (c) shifts the macroscopically homogenous deformation to a localized dilatational deformation mode.

(a) (b)

Homogenous deformation phase

(c) (d)

Localized deformation phase

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Figure 3: Numerical simulation of the void growth and coalescence. Coalescence starts at (e) after which all the deformation take place vertically. In the following, ductile fracture mechanisms will be briefly discussed first, in order to emphasize the two plastic deformation phases of ductile fracture of void containing materials. Modeling of the homogenous deformation phase of ductile fracture by the Gurson model will be addressed in detail in Section 3. Some new results about the void volume fraction at final fracture and the hardening effect on void growth are presented. In Section 4, Thomason’s plastic limit load model for the localized deformation phase of ductile fracture is introduced and modifications of the plastic limit load model for the Gurson model are discussed. The complete Gurson model and its verifications are presented in Section 5. The special features of the complete Gurson model and several other related issues are discussed in section 6. 2 Ductile fracture mechanisms for metals Engineering alloys usually contain inclusions, for example manganese sulfide and aluminum oxide in steels, and second-phase particles, for example, carbides in steels and intermetallic phases in aluminum alloys. It has been observed for a long time that the mechanisms of ductile fracture for engineering alloys include the following three stages. Once a critical condition has been reached, microvoids will nucleate from the inclusions or second phase particles by either

0.3σ

σ

a) b) c)

d) e) f)

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decohesion of the particle-matrix interface or by particle cracking (Fig. 2a). Further plastic deformation and hydrostatic stress will cause the nucleated voids to grow and some new voids will possibly nucleate during the plastic deformation increment (Fig. 2b). Finally, when the voids grow to a certain extent they will connect together and void coalescence occurs, which results in the separation of material. It should be noted that before void coalescence, the plastic deformation of a void containing material is dilatational and macroscopically homogenous. The homogenous deformation (Fig. 2a and 2b) terminates and shifts to a localized plastic deformation once the coalescence process begins (Fig. 2c). Fig. 3 shows the results of a numerical simulation of the growth and coalescence process of a 2D plane strain void with an initial volume fraction 1.2% under a bi-axial stress loading (1/4 model) – the horizontal stress is 30% of the axial stress. From Fig. 3 we can see that an originally cylindrical void in general will change shape, and void coalescence occurs by necking of the intervoid matrix. When necking begins (Fig. 3e) all the deformation goes in the vertical direction, with no additional deformation in the horizontal direction.

3 Gurson model for homogenous deformation by void nucleation and growth For a pure solid material, plastic deformation will not change material volume. However, when a material contains voids, the voids will grow and material volume will change under plastic straining and hydrostatic tension stress. The plastic flow thus becomes dilatational. The direct consequence of dilatational plastic flow is that material will soften and the load carrying capacity will be decreased. In order to model the dilatational plastic flow response, the highly non-linear relation between void growth and material softening should be considered. Many studies on void growth have been carried out in the literature; one of the well-known models is the one by Rice and Tracey [12]. Rice and Tracey studied the growth of a void in a remote straining field and derived the void growth rate as a function of plastic strain rate and an exponential function of hydrostatic tension stress. The Rice-Tracey equation, however, does not account for the softening resulting from void growth. Figure 4: Void-matrix aggregate and average measure of void volume fraction in the Gurson model.

fdVdV

Voids

Cell=

a) b)

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3.1 The Gurson model Based on the work by Rice and Tracey [12] and McClintok [13], Gurson further studied the plastic flow of a void containing material [4]. In the model by Gurson the softening effect due to the presence of voids was reflected in a yielding function. By idealizing the true void distribution into a unit cell containing one spherical void and carrying out the rigid-plastic upper bound analysis, Gurson [4] obtained the following yield function,

( ) 0)(1)2

3cosh(2,, 21

212

2

=−−+= fqqfqqfq m

σσ

σσφ (1)

where f is the void volume fraction, which is the average measure of a void-matrix aggregate (see Fig. 4). In Eq. (1), q1, q2 are constants introduced by Tvergaard [14,15], σm is the mean normal stress, q is the conventional von Mises equivalent stress, σ the flow stress of the matrix material. In the paper, the Gurson model means Eq. (1) with appropriate q1, q2 parameters. It is obvious from Eq. (1) that the material yielding is coupled with damage (f) and the hydrostatic stress. Unlike many soil material models where yielding is also dependent on hydrostatic stress but yield surface is fixed in stress space, the yield surface of the Gurson model decreases with the increase of damage until the complete loss of load-carrying capacity. For a material without voids, the Gurson model is exactly the same as the conventional von Mises model. 3.2 Critical void volume fraction In the original Gurson model ( 121 == qq ), material softening with the increase of void volume fraction is a continuous process, and complete loss of load carrying capacity would occur only when the void has grown to the ultimate value f=100%. This is an unrealistic situation – material should completely disappear. Tvergaard compared the bifurcation predictions (both 2D and axisymmetric) based on the Gurson model and his numerical studies for material containing periodic distribution of voids and suggested a modification of the Gurson model by taking 5.11 =q and 12 =q [14,15]. Even with this modification, the void volume fraction at which the Gurson model will lose load carrying capacity is still unrealistically large, 1/1 qf = . Both experimental observations [16] and results of cell model analysis by Koplik and Needleman [17] show that the volume fraction of voids at which void coalescence starts, cf , is much smaller than 1/1 qf = and is usually less than 15%. It therefore indicates that the Gurson model can not naturally predict void coalescence and an extra void coalescence criterion should be used. A so-called critical void volume fraction criterion for void coalescence has been commonly used in the literature [14-15,18-23]. The criterion assumes that void coalescence appears when a critical void volume fraction, cf is reached, irrespective of stress triaxiality. The value of

cf , was supposed to be a material constant and can be determined experimentally or

numerically [18], or by just taking cf =0.15 [23], or cf =0.1 [19,20]. A brief review on

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development of this criterion has been made in [8]. The critical void volume fraction was also assumed to be independent of the initial value of the void volume fraction, i.e. it does not matter whether it nucleates from large or small inclusions. In the beginning,

15.0=cf independent of the material has been adopted. Later on, with the advances of the study by Koplik and Needleman [17], Tvergaard commented that the critical void volume fraction should depend on the initial void volume fraction [1], i.e. it should depend on the material. Koplik and Needleman have also investigated the dependence of cf on stress triaxiality and observed that the dependence can be neglected when the initial void volume fraction is very small. On the other hand, if the initial void volume fraction is large (>1%), the

cf at high stress triaxiality is significantly lower than that at low stress triaxiality.

3.3 Simulation of void coalescence in the Gurson model Even though the Gurson model itself can not predict the void coalescence, once the void coalescence has been determined to occur according to a criterion, the void coalescence process can be simulated by the following function introduced by Tvergaard and Needleman [23]:

ff

ff ff f

f ffor f ffor f fc

u c

F cc

c

c

∗ ∗= +−−

−

⎧⎨⎪

⎩⎪

≤>( ) (2)

where Ff is the void volume fraction at final fracture and 1* /1 qfu = . The above equation

simply implies that before void coalescence, the void volume fraction and the decrease of load carrying capacity take the “normal” way by the Gurson model. After the void coalescence has started, the void volume fraction will be amplified to represent the sudden loss of load carrying capacity (see Fig. 5).

Figure 5: Schematic plot of Equation (2) – response of a material point in FE analyses with and without Equation (2).

f c f F

f c

f qu* /= 1 1

*f

f

A

B C

B

A

A

B

C

C

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It must be noted that the void coalescence is a process from beginning to end. Physically, the void coalescence (from the starting of void coalescence (Fig. 2c), to the complete separation of material (Fig. 2d) is a very rapid process with large void volume expansion, resulting in much larger void volume fraction at the end of coalescence than at the beginning. Both Koplik and Needleman [17] and the results in Section 5 have demonstrated that the effect of void coalescence can be well simulated by the function, though Eq. (2) becomes less accurate for high stress triaxiality cases. The void coalescence accompanied by accelerated decreasing of load-carrying capacity can also be numerically simulated using the element vanishing technique by Tvergaard [1], which reduces the traction of the element to zero in a 10-20 increment release step. A similar technique has recently been used by Kopenhoefer et al. [24]. However, this technique is not available in many of the commercial FEM programs and may involve difficulty for the case where the crack growth path is not known beforehand. Numerically, Eq. (2) provides a practical and convenient way to redistribute the load after the material has been damaged. 3.4 Void growth in the Gurson model The average void volume fraction of a material during a plastic strain increment will change, due partly to the growth of existing voids and partly to the nucleation of new voids. In the application of the Gurson model, a “homogenization” process is used. During a load increment, the increase of volume fraction from existing voids and newly nucleated voids will be added together and homogenized as “one” void for the next load increment (see Fig. 6). Fig. 6 shows that one void volume fraction parameter is used for describing the increase of volume fraction of more than one population of voids. Because of the incompressibility of the matrix material, the growth of the existing void under plastic straining can easily be written,

df f dgrowthp= −( ) :1 ε I . (3)

where ε p is the plastic strain tensor and I is the 2nd order unit tensor. For a given stress triaxiality void growth in the Gurson model with fixed q1, q2 is directly linked to the plastic strain and independent of material hardening exponent. This is a consequence of the assumption used in the Gurson model: the void is always assumed to be spherical. In the original Gurson model, the matrix material surrounding the voids is taken as rigid plastic obeying 2J flow rule. So, strictly speaking, the Gurson model “only” works for very low strain hardening materials. Fig. 7 shows the FE results of a plane strain cell model with fixed horizontal/vertical stress ratio 0.3 for different hardening materials. Low hardening results in higher void growth than high hardening. If the Gurson model is used to simulate the void growth for different materials shown in Fig. 7, we would fit a higher 1q (q2 =1), for high hardening material than for low hardening material. It seems the Gurson model is more “suitable” for high hardening material than low hardening material. Similar results can be seen in the work by Søvik [25]. Søvik tried to fit the Tvergaard parameter 1q as a function of

hardening exponent and found that 1q deviates from 1 when a material’s hardening ability

decreases, i.e. 1q =1 (original Gurson model) is “correct” for very strong hardening material. This finding contradicts with the common belief regarding the Gurson model. The reason for this behavior is that void shape change is much greater for a low hardening material than for a

231

high hardening material. In the Gurson model, the void is assumed to be always spherical (3D) or cylindrical (2D plane strain). The deviation to the assumption of non-hardening is compensated for by the smaller void shape change. In many practical applications of the Gurson model, the strain hardening effect on void growth can be neglected by choosing a fixed pair of (q1, q2 ). Results by Koplik and Needleman [17] show that q1=1.25, and q2 =1.0 are reasonable choices. Figure 6: Homogenization treatment of void nucleation and growth in the Gurson model.

Figure 7: Example of the effect of hardening on void growth, where n is the material hardening exponent. Plane strain model shown in Fig. 3a was used.

=Increment

t n + 1

t n

a) b) c¨)

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3.5 Modeling of void nucleation Void growth may be treated independently of material (hardening); however, void nucleation is a highly material- dependent process. This justifies the fact that void nucleation is the least understood part of ductile fracture [21]. Void nucleation belongs to material intrinsic properties and governs material failure behavior. In general it depends on particle strength, size and shape, and the hardening exponent of the matrix material. The nucleation mechanism can be strain controlled [3,26], or stress controlled [27] where the hydrostatic tension stress plays a role. Many alloys contain a bimodal or trimodal distribution of particles. Large inclusions usually tend to nucleate voids earlier than small particles, and inclusions with different length scales may have different nucleation criteria, for example dislocation-based or continuum mechanics criteria [5]. Figure 8: Three void nucleation models, (a) cluster nucleation, (b) continuous nucleation and (c) statistical nucleation model. In numerical modeling, the void nucleation law can be generally written as

)(. σσε ddBAddf mp

nucl ++= (4) where A and B are strain and stress controlled void nucleation intensity, respectively, Pε is the equivalent plastic strain, σ is the Mises equivalent stress and mσ the mean normal stress. Fig. 8 shows the three possible nucleation laws, where S stands for a stress or strain quantity. The first one is a cluster nucleation model. Cluster of voids will nucleate when some critical condition has been reached. For this model, it is usually assumed that the voids will be nucleated in the beginning of plastic deformation. The parameter for this model is the initial void volume fraction, 0f . The second model is called a continuous nucleation model. Gurland et al. [26] found that for some materials the number of voids increases with increasing plastic strain. There is also only one parameter in this nucleation model, pddfA ε=0 . The third model is a statistical treatment of the first model, which indicates that most of the voids will be nucleated around the critical quantity Sc . Chu and Needleman [28] have proposed the following equations for strain controlled nucleation intensity A:

S

a) 0f

S

b)0A

S Sc

c) A

233

⎥⎦

⎤⎢⎣

⎡ −−= 2)(

21

2 N

Np

N

N

Sexp

SfA εε

π. (5)

In the above equations, f N is the volume fraction of particles available for void nucleation, ε N the mean void nucleation burst strain, and SN the corresponding standard deviation. Similar expression can be written for stress controlled nucleation intensity B. The major difference between the strain and stress controlled nucleation mechanisms is that the hydrostatic tension stress has been taken into account in the latter. If the hydrostatic tension stress is taken out of Eq. (4), the stress-controlled mechanism returns to the strain controlled. In a material, the three mechanisms may exist spontaneously– the material’s nucleation law may be a combination of the models shown in Fig. 8. In modeling of ductile fracture, the cluster nucleation model assuming voids nucleated at the beginning of plastic deformation, and the continuous nucleation model are often preferred than the statistical model. The statistical model may have a physically sound basis, it involves, however, three unknowns which are difficult to determine and contribute to the so-called non-uniqueness problem (see the discussions in section 5). In this paper, the statistical model is not recommended. 3.6 Void volume fraction at final fracture, Ff Even though the void volume fraction at final fracture, Ff , has been considered as an unimportant parameter, it is interesting to know whether it is a constant. Fig. 9 shows the void volume fraction versus neck development from the results of plane strain cell model analysis with horizontal vertical stress ratio 0.3 for three different initial void volume fractions. We observe that the void volume fraction at final fracture (necking down to zero) is strongly dependent on the initial value of the void volume fraction, 0f . Compared with the difference in the initial values of the void volume fraction, the difference in the void volume fraction at final fracture is nearly doubled.

Figure 9: Results of void volume fraction versus neck development from 2D plane strain cell model FEM analyses.

0.3σ

σ

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The void volume fraction at final fracture is also influenced by stress triaxiality T, σσ mT = . Fig. 10 shows the results for two initial void volume fractions at different stress

triaxility. High stress triaxility will result in a higher void volume fraction at final fracture. Figs. 9 and 10 show that the void volume fraction observed at a fracture surface is much larger than the one at which void coalescence starts, it can not be used, therefore, for determining the “critical” void volume fraction at void coalescence. It can be extrapolated from Figs. 9 and 10 that the void volume fraction at final fracture, Ff , should be at least equal to or large than 0.15. Based on the observations, the following approximate equation can be obtained,

0215.0 ff F += . (6)

Figure 10: Results of void volume fraction versus neck development from 2D plane strain cell model FEM analyses. In the legend, T is the stress triaxiality. 4 Thomason plastic limit load model for localized deformation by void coalescence As discussed in the last two sections, deformation of void-containing materials displays two distinct phases, the homogenous phase and localized phase. Based on the observation that void coalescence is caused by localized necking of the intervoid matrix, Thomason developed a so-called dual dilatational constitutive equation theory for ductile fracture [5-7]. For a void-containing material, Thomason argues, the two deformation modes are in competition. Both deformation modes are dilatational, i.e. plastic deformation will result in change of material volume, and the material will always follow the deformation mode which needs less energy. After a detailed analysis of the mechanics of ductile fracture by void coalescence, Thomason found that the localized deformation mode by intervoid matrix necking can be described by a plastic limit load model. It is important to note that the plastic limit load of a void-containing cell is not fixed but is strongly dependent on the void/matrix geometry. For a material without

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void, the plastic limit load is infinite. At the starting of plastic deformation of a void-containing material, the void dimension is very small and the corresponding plastic limit load is very large. The homogenous deformation mode thus prevails. When a void starts to grow, the plastic limit load decreases, which indicates that the possibility of plastic localization increases. The homogenous deformation mode will be terminated once the localized mode of deformation becomes possible. The localized deformation mode is characterized by the maximum principal stress, σ1

Localized , which represents the micro-capacity of a voided material to resist the localized deformation, and the homogenous deformation mode is represented by the applied maximum principal stress, σ1

Homogenous , at the current yield surface. The condition for void coalescence by internal necking of the intervoid matrix can be written

LocalizedHomogenous11 σσ = . (7)

The above equation implies that the applied load has reached the capacity of the material to resist the void coalescence; then an unstable localized deformation mode will prevail and void coalescence occurs. For a real 3D problem with x, y and z representing the minor, medium and maximum principal stress/strain directions, it is very complicated to derive a plastic limit solution. In order to simplify the problem, an equivalent 3D unit cell with an ellipsoidal void ( zyx RRR ≠= , 2/yxeYX εε +== , where xε , yε are the minor and medium principal

strains) was used by Thomason (see Fig. 12a). By approximating the ellipsoidal void by an equivalent prismatic void and assuming two kinematically admissible velocity fields, Thomason obtained the following closed form empirical expression for the plastic limit load stress [5] of Fig. 12a,

0

2

4

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

σσ

1Homogenous

σσ

1Localized

Equivalent plastic strain

Figure 11: The competition of the two deformation modes in the Thomason theory.

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⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛

+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= 2

2

5.021 12.11.0

XR

XR

RXR

x

x

x

z

Localized πσ

σ (8)

where σ is the yield stress of the matrix material. For 2D plane strain problem with void matrix geometry shown in Fig. 12b, the closed form plastic limit load stress proposed by Thomason [5] is

6.0)/(

3.01 +−

=xx

Localized

RXRσσ . (9)

Eq. (7) has been examined by comparison with the axisymmetric finite element results by Koplik and Needleman for different void matrix geometry [8,9]. Koplik and Needleman have analyzed cell models with the same initial void volume fraction but different void spacing, and found that material load deformation response is primarily a function of the void volume fraction, while the void coalescence primarily depends on void spacing. It has been shown that even though Eq. (8) is empirical in nature, it is reasonably accurate for predicting the void coalescence [9]. Søvik has also verified Thomason’s model with his axisymmetric finite element results for different initial void shapes [25]. Similar conclusions have been drawn. 5. A complete Gurson model The main idea of Thomason’s ductile fracture theory is that void coalescence coincides with the plastic limit load condition for localized deformation of the intervoid matrix. In the original Thomason theory, the Mises model has been used for characterizing the homogenous deformation, and the change of void matrix geometry calculated by the Rice-Tracey equation is used for evaluating the plastic limit load condition. Although Thomason didn’t advocate the Gurson model, he commented that the Gurson model can be improved by treating the Gurson model as the one for describing homogenous deformation in his theory. 5.1 A complete Gurson model In this paper, Thomason’s plastic limit load model is modified by assuming that the void is always spherical (3D) or cylindrical (2D plane strain). This modification makes the plastic limit load model fully compatible with the Gurson model. Because of this assumption, the void/matrix geometry can be directly determined from the current void volume fraction and the current principal strains. For a 3D cell shown in Fig. 12a, zyx RRR == = 3

43 zyxef εεεπ

++ ,

X=Y= 2yxe εε + . By incorporating the modified plastic limit load model into the Gurson model as the void coalescence criterion (see Fig. 13), the Gurson model in a sense becomes complete. The complete Gurson model can be written as:

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• Homogenous yielding by the Gurson model:

( ) 0)(1)2

3cosh(2,, 2*1

2*12

2* =−−+= fqqfqqfq m

σσ

σσφ

from which the maximum principal stress at a material point, zσ can be calculated. The minor and medium stress/strain directions are represented by x and y. • Void growth and nucleation:

I:)1( pdfdf ε−= for the cluster nucleation model with parameter 0f , pp dAdfdf εε 0:)1( +−= I for the continuous nucleation model with parameter 0A .

• Void coalescence condition for 3D problem with and without hardening:

If ( )22

1 12.1111.0 rrr

πσσ

−⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ −< , no coalescence and ff =* ,

if ( )22

1 12.1111.0 rrr

πσσ

−⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ −= , coalescence starts and set ff c = ,

where XRr x /= = ⎟⎠⎞⎜

⎝⎛ +++ 2

43

3 yxzyx eef εεεεε

π, σ the current flow stress.

• Void coalescence condition for 2D plane strain

if 6.0113.01 +⎟⎠⎞

⎜⎝⎛ −<

rσσ

, no coalescence and ff =*

if 6.0113.01 +⎟⎠⎞

⎜⎝⎛ −=

rσσ

, coalescence starts and set ff c =

• Post-coalescence response:

)( ccF

cuc ff

ffff

ff −−−

+=∗

∗

where 0215.0 ff F += can be used.

238

It must be noted that Thomason plastic limit load model here has been extended to hardening material without strictly verification, exactly the same as what we have done to the Gurson model. Some recently modifications of Thomason plastic limit load model made in the literature will be discussed at the end of the paper.

Figure 12: An equivalent unit cell for 3D (a), and 2D plane strain unit cell (b). X, Z are the current dimensions. The x, y, z axes represent the minor, medium and maximum principal stress /strain directions. 5.2 Implementation of the complete Gurson model Because the yield surface of the Gurson model is changing with the increase of damage, the computer implementation of the Gurson model is complicated, especially for the finite element programs which use implicit algorithms. Aravas first presented an Euler backward algorithm for the Gurson model and derived an equation for calculating the so-called consistent-tangent-moduli [29]. However, this equation involves matrix inversion which is not efficient and not possible for non-hardening materials. A substantial study on the numerical treatment of the Gurson model has been performed by the author (Zhang and Niemi [30], Zhang [31-32]), and a family of the generalized-midpoint algorithms was proposed. An explicit equation with 7 constants for calculating the consistent tangent moduli has been presented [32]. This equation does not involve any matrix inversion and has thus avoided the singularity problem. The accuracy and efficiency of various algorithms have been assessed. It is concluded that the Euler backward algorithm may not be the most accurate but may be the best choice for many practical applications in terms of the balance between accuracy and efficiency. The complete Gurson model has been implemented into the ABAQUS using the algorithms developed by the author via the material user subroutine UMAT. A free copy of the implementation can be obtained from the author.

a) b)

y

z

x 2X

R z

Rx

2X

2Z R y

X

Z

RR

zx

239

x

z

2R

Homogenousσ1

Homogeneousdeformation bythe Gurson model

σ2

2R

Localizedσ1

Localizeddeformation

σ2z

rx

Figure 13: The complete Gurson model. Void is always assumed to be spherical (3D) or cylindrical (2D plane strain). 5.3 Verification of the complete Gurson model The prediction of void coalescence strain versus stress triaxility by the complete Gurson model has been compared with the axisymmetric finite element results by Koplik and Needleman [17]. The comparison is shown in Fig. 14 for two initial void volume fractions with non-hardening material. For the smaller void volume fraction, 0f =0.13%, excellent agreement is observed, and reasonably good agreement is also obtained for the large void volume fraction

0f =1.04%. The complete Gurson model has been verified for the 2D plane strain problem. Fig. 15 shows the comparison of the prediction by the Gurson model and the finite element results for the material with hardening exponent n=0.1 at stress triaxiality T=0.57. Two initial void volume fractions have been analyzed. Both Figs. 14 and 15 show that the complete Gurson model is fairly accurate. In general, if the void growth and load deformation response are reasonably predicted by appropriately choosing 1q and 2q parameters, the complete Gurson model will predict the void coalescence with a reasonable accuracy. Further verification of the complete Gurson model for 2D plane strain at higher stress triaxiality and a real 3D problem will be carried out in the near future. 5.4 cf predicted by the complete Gurson model In the complete Gurson model, void coalescence is not determined by the so-called critical void volume fraction, cf . On the contrary, the critical void volume fraction is just the material response at void coalescence. It is interesting to see how the critical void volume fraction depends on the stress triaxiality and the initial value of void volume fraction. Fig. 16 shows the critical void volume fraction predicted by the complete Gurson model for two initial void

240

volume fractions. It can be seen that for a small initial void volume fraction, cf is

approximately independent of the stress triaxiality; however, the cf is strongly dependent on the stress triaxiality when the initial void volume fraction is large. FEM results by Koplik and Needleman [17] show the same trend.

a) b)

Figure 14: Comparison of the prediction by the complete Gurson model and FEM results by Koplik and Needleman, (a) for f0 =0.0013 and (b) f0 =0.0104. The equivalent strain is the strain of the cell model at void coalescence

Figure 15: Comparison of the prediction by the complete Gurson model and the results of 2D plane strain cell model FEM analysis. The stress triaxiality is T=0.57, 5.11 =q and 2.12 =q , hardening exponent n=0.1.

241

Although for small initial void volume fraction, the dependence of cf on stress triaxility can be neglected, according to the complete Gurson model, the cf is a strong function of the initial void volume fraction. Fig. 17 shows the predicted cf for a plane strain model with hardening exponent n=0.1, 1q =1.5 and 2q =1.2 at two stress triaxility, T=0.57 and T=1.0. Some finite elements results from different authors compiled by Tvergaard [1] are also presented. The results in Figs. 16 and 17 clearly shows that in the application of the Gurson model there is no basis to assume a material independent critical void volume fraction.

0

0,02

0,04

0,06

0,5 1 1,5 2 2,5 3 3,5Stress triaxiality

f0=0.0013f0=0.0104

f c

Figure 16: Critical void volume fractions versus stress triaxility [9].

Figure 17: Critical void volume fractions versus initial void volume fraction, predicted by the complete Gurson model, together with FEM results from different authors compiled by Tvergaard [1].

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5.5 A method for determining the void nucleation parameters In the literature, void nucleation parameters are determined first, and the critical void volume fraction is fitted by comparing with the prediction of the Gurson model and the experimental results [18]. In the complete Gurson model, because void coalescence is automatically determined, material failure is directly linked to the void nucleation parameters. The nucleation parameters, rather than the critical void volume fraction, are to be fitted. For a single tensile specimen, by choosing a void nucleation law, the corresponding void nucleation parameter can be determined by selecting the value which best fits the load-displacement curve, usually the load-diameter reduction curve, see Fig. 18. In Fig. 18, three values have been tried, and the value “0.005” is the “right” nucleation parameter for the material. It must be noted that for single specimens, the nucleation law is usually not unique, for example, both cluster nucleation model, 0f , and continuous nucleation model 0A , can almost do the same job in Fig. 18.

0

4

8

12

16

20

0 1 2 3Diameter reduction [mm]

Tens

ile lo

ad [k

N]

0.0010.0050.01EXP

Figure 18: Single specimen approach: unique f 0 can be determined by comparing the prediction of the complete Gurson model and the experimental results.

Figure. 19: Multi-specimen approach: a schematic plot on the ductility diagram used in determining the nucleation parameter, fε is the fracture strain.

fε

0005.00 =f002.00 =f005.00 =f

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In order to solve the non-uniqueness problem and enhance the transferability of the nucleation parameter, a ductility diagram based on multi-specimens is proposed for fitting the nucleation parameter, Fig. 19. In this method, tensile specimens with different notch radius are used. Smooth specimens usually display high ductility, while specimens with notch reduce the ductility. Ductility reduction versus increasing stress triaxiality is different for different material. According to the complete Gurson model, different nucleation model will result in different curvature of the ductility curve [33]. Therefore, by using a ductility diagram, a unique nucleation model and its corresponding parameter may be determined. Fig. 21 shows the example for a pipeline steel and an aluminium alloy. In general, material tensile ductility can be written:

a) b)

Figure 20: a) Ductility for an Al-4.3%Si alloy, where cluster nucleation model with 0f =0.002 fits best with the experimental data; ductility diagram for a X65 pipeline steel where the continuous nucleation model with 0A =0.0008 fits best [33].

⎟⎠⎞

⎜⎝⎛= nf m

ff ,,0 σσεε (10)

where n is material’s strain hardening exponent, and 0f can be substituted by 0A . 5.6 Critical length parameter cl The tensile specimens proposed in the ductility diagram have relative homogenous stress state. Numerical analyses have shown that the effect of critical length parameter is not significant. For the sake of simplicity, here the length parameter means the mesh size at the notch/crack tip. When the complete Gurson model is applied to cracked specimens, crack resistance curve is controlled by both the void nucleation parameter and the critical length parameter. The general approach we propose is fitting the nucleation parameter from tensile specimens first. Then fit the critical length parameter by comparing numerical prediction with a J-R curve from a fracture mechanics test. Once the critical length parameter cl has been fitted, all the material parameters are determined. Similar idea has been shown in the recent paper by Grange et al. [34]. Fig. 21 shows the transferability diagram by the complete Gurson model.

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Figure 21: Transferability by the conventional ductile fracture mechanics (dashed arrow) and by the complete Gurson model (solid arrows). 6 Discussions In this paper, a complete Gurson model has been introduced. The word “complete” means that the modified Gurson model is able to simulate the complete process of ductile fracture – void nucleation, growth and coalescence. It does not mean, however, the model is perfect. The coalescence criterion by Thomason used in the paper have been verified for 3D with non-hardening material, and for 2D plane strain with hardening material but relatively low stress triaxiality. In general, it underestimates coalescence strain for 3D void with hardening matrix material and 2D plane strain with high stress triaxiality. Very recently, Pardoen and Hutchinson have studied the effect of void shape, void spacing and matrix strain hardening effect on coalescence behavior of axisymmetric voids [35]. It was concluded that Thomason’s plastic limit load model with a modification for strain hardening is quite accurate for predicting void coalescence. The effect of strain hardening on the plastic limit load model, Eq. (8) can be considered by modifying the two coefficients [35]:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠

⎞⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= 2

2

5.021 1

XR

XR

RXR

n x

x

x

z

Localized πβασ

σ (11)

where the two coefficients α and β were fitted from finite element cell model analyses,

24.1=β , ( ) 283.4217.01.0 nnn ++=α for 3.00 ≤≤ n . Eq. (11) will be implemented into the complete Gurson model and the effect of predicting void coalescence will be reported elsewhere.

Fracture mechanics Specimens

Components & structures

clnf ,,, 00 σ

Tensile or impact Specimens

J R−

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7 Summary A micro-mechanical model of ductile fracture should be able to describe the two deformation phases caused by voids – the homogenous deformation phase due to void nucleation and growth and the localized deformation phase by void coalescence. In this paper, a complete Gurson model that incorporates the Gurson model and the plastic limit load model for void coalescence is presented. The complete Gurson model can well predict the void growth and automatically and accurately predict void coalescence. Void growth and coalescence can be a material-independent process. It is mainly controlled by the plastic deformation and hydrostatic stress. However, void nucleation is a highly material-specific process. It is the void nucleation that controls material toughness. With the complete Gurson model, ductile fracture has been directly linked to the void nucleation parameters. Material ductility can be characterized according to the void nucleation parameter. The void nucleation parameter can be numerically fitted by using the complete Gurson model from the so-called ductility diagram of a series of smooth and notched tensile tests. The remaining parameter for ductile fracture, the critical length parameter, can be determined by comparing the FE analysis results with the experimental J-R curve of a fracture mechanics specimen. According to the complete Gurson model, the ductile fracture parameters are the initial void volume fraction and critical length parameter together with materials stress strain curve, which includes the yield stress and strain hardening. Acknowledgements The financial support from the Norwegian Research Council (NFR) via a Strategic Institute Program and also the PRESS project is appreciated. References: [1. ] Tvergaard, V. Advances in Applied Mechanics. J. W. Hutchinson & T. Y. Wu (eds.),

pp. 83-151, Academic Press, 1990. [2. ] Needleman, A. & Tvergaard, V. Analysis of plastic flow localization in metals. Applied

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Coalescence of Micro-Voids. Acta Metallurgica, 33, pp. 1087-1095, 1985.

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