A MODIFIED GTN MODEL FOR THE PREDICTION OF DUCTILE FRACTURE A T LOW STRESS TRIAXIALITIES

F. Reis1, L. Malcher1, F. M. Andrade Pires1, J. M. A. Cesar de Sa1

1IDMEC – Institute of Mechanical Engineering

Faculty of Engineering, University of Porto

Rua Dr. Roberto Frias, Porto 4200-465, Portugal

e-mail: fpires@fe.up.pt

ABSTRACT

In this contribution, the so-called GTN constitutive model[1] is extended in order to include a shear mechanism on thedamage evolution equation, which depends on the Lode angle.This mechanism is introduced in order to improve the dam-age prediction under shear dominated loads. From the numerical viewpoint, a fully implicit elastic predictor/return mappingalgorithm is developed and the associated consistent elasto-plastic tangent operator is presented. The numerical model isassessed through the simulation of pure shear and combined shear/tensile stress states. Some representative results showthe influence of the modifications on the constitutive model of several parameters such as porosity and accumulated plas-tic strain. A comparative study demonstrates that the proposed model provides better agreement with experimental evidence.

KEY WORDS: GTN model, Low stress triaxiality, Ductile materials, Shear tests, Lode angle dependence.

1. INTRODUCTION

Many classical ductile damage models have the abilityto predict the correct fracture location. However, mostof them is only effective within a specific range ofstress triaxialities. For instance, Lemaitre’s model [2]provides good predictions under pure shear or combinedshear/tensile stress states, where typically low stresstriaxialities are present.

However, at high stress triaxialities, Gurson’s originalmodel [3] is more accurate than Lemaitre’s. In thiscase, Gurson’s model predicts failure onset for a lowerdisplacement value than Lemaitre’s. As reported in aprevious contribution [4], this result is in agreement withexperimental evidence. This stems from the fact thatGurson’s original model is particularly able to capturespherical void growth which is the most relevant mecha-nism present at high triaxialities. However, under sheardominated stress states, failure mechanisms are drivenby shear localization of plastic strain of the inter-voidsligaments due to void rotation and distortion [5]. SinceGurson’s original model does not include such importantmechanisms, it is not able to capture the behavior ofthe material under these conditions, yielding on poorerpredictions as it will be clear in the following.

In a pure shear test, failure is predicted with reasonableaccuracy with Lemaitre’s model (see Figure 1). However,this is not the case when Gurson’s original model isadopted, since the value of the porosity remains constantonce we reach a prescribed displacement and neverreaches its critical value. This is a direct consequence of

Gurson’s constitutive theory that only takes into accountthe nucleation of micro-voids. Thus, we can concludethat, in order to capture the behavior of a ductile materialclose to rupture under shear dominated loads, a shearmechanism must also be included.The evolution laws for internal degradation in Lemaitre

Figure 1. Evolution of the damage and porosity param-eters in a pure shear test with Lemaitre’s and Gurson’smodels.

and Gurson material models are dependent on thepressure through the stress triaxiality parameter. Nev-ertheless, many researchers advocate that damageevolution depends on more than one parameter and thatits definition depending solely on the stress triaxiality isnot sufficiently accurate (e.g. Brunig et al. [6], Malcheret al. [4], Xue [5], Nahshon and Hutchinson [7]). Inthis context, several authors (e.g. Brunig et al. [6], Baoand Wierzbicki [8], Xue [5], Nahshon and Hutchinson[7]) have proposed the introduction of the effect of thethird invariant of the deviatoric stress tensor through the

1

so-calledLode angleon the set of damage evolutionequations.

The goal of this work is to improve the capability ofpredicting the correct fracture location for a wider rangeof stress triaxialities. The Lode angle parameter isdefined on the deviatoric plane as being the smallestangle between the line of pure shear and the projectionof the stress vector on theπ-plane. This parameter isresponsible for the shape of the yield surface and itseffect can be introduced into the constitutive model orinto damage evolution law through the so-called Lodeangle function.

More recently, Xue [5] has proposed an improvement onthe porosity evolution law aiming to capture the behaviorof the material at low stress triaxialities through a shearmechanism which is Lode angle dependent. However,although this model has shown significant improvements,its numerical implementation is extremely difficult.

Therefore, this paper has two main goals: (a) to describethe enhancements achieved by the introduction of a shearmechanism into the GTN model which was based in-spired in the work of Xue [5]; (b) to suggest a suitablemodification of this mechanism aiming to facilitate thenumerical implementation and increase efficiency of theconstitutive integration algorithm.

2. CONSTITUTIVE MODELING

Gurson’s [3] approach has been proposed to describematerial internal degradation in the presence of plasticstrains. This model, which was developed to provide the”bridge” between material plasticity and damage accu-mulation, allows the prediction of the loss of resistanceof porous materials due to the growth of spherical micro-voids. The main feature of Gurson’s model is the intro-duction of a yield function that is governed by first andsecond invariant of stress tensor and also by the dam-age variablef , which represents the volume fraction ofmicro-voids embedded in the material matrix.

2.1. GTN original model

One of the most commonly employed versions of Gur-son’s model is the Tvergaard-Needleman modification[1], commonly referred to as GTN model. The modelassumes isotropic hardening and isotropic damage (rep-resented by the effective porosityf∗). In the GTN model,the flow potential is generalized into the form:

Φ(σ, k, f) = J2 (S)

−

1

3

1 + q3.f∗2

− 2.q1.f∗

. cosh

„

3.q2.p

2.σ0

«ff

.σ2

0

(1)

whereJ2 represents the second invariant of the devia-toric stress tensor,p is the pressure,σ0 is the isotropic

hardening rule (which can be defined asσ0 = R − σy0)andR represents the isotropic hardening state variable.The parametersq1, q2 and q3 are introduced to bringthe model predictions into closer agreement with fullnumerical analyses of a periodic array of voids.

The effective porosity reproduces the mechanisms of nu-cleation, growth and coalescence of voids (which may oc-cur either simultaneously or successively):

f∗ =

{

f , f < fc

fc +(

1q1

− fc

)

(f−fc)(ff−fc)

, f ≥ fc

(2)

where fc is the value of void volume fraction thatdefines the beginning of coalescence phenomenon andff represents the porosity at fracture. The evolution offis a sum of the nucleation and growth mechanisms:

f = fN + fG (3)

The nucleation mechanism in this case is driven by theplastic strain:

fN =fN

sN .√

2π. exp

[

−1

2

(

εp − εN

sN

)2]

εp

(4)

wherefN represents the volume fraction of all second-phase particles with potential for micro-void nucleation,εN and sN are the mean strain for void nucleationand its standard deviation. The variableεp representsthe equivalent plastic strain andε

pis the rate of the

accumulated plastic strain.

The most significant contribution is the growth of exist-ing voids, denoted byfG, obtained from the condition ofplastic incompressibility of the matrix material:

fG = (1 − f) .tr (εp) = (1 − f) .εpv (5)

where εpv represents the rate of the volumetric plastic

strain.

2.2. Shear mechanism

In order to improve the GTN model ability to capture thebehavior of ductile materials close to rupture at low stresstriaxialities, Xue [5] has proposed the inclusion of a shearmechanism which is a function of the porosity, equivalentstrain and the Lode angle. Originally, the shear mecha-nism was developed considering geometrical considera-tions in a cell structure with a circular void at the centersubjected to a simple shear strain (for more details, seeXue [5]). After some straightforward algebra manipula-tion, the rate of this mechanism is expressed as:

fShear = q4.fq5 .g0.εeq.εeq (6)

whereq4 andq5 are parameters related to two- or three-dimensional problems, respectively set toq4 = 1.69 and

2

q5 = 1/2, q4 = 1.86 andq5 = 1/3. The parameterf rep-resents the porosity,εeq is the equivalent strain andg0 isa parameter responsible to incorporate the Lode angle de-pendence in the shear mechanism and that can be definedas:

g0 = 1 − 6. ‖θ‖π

(7)

whereθ is the Lode or azimuth angle, which can be de-termined as:

θ = tan−1

{

1√3

[

2.

(

S2 − S3

S1 − S3

)

− 1

]}

(8)

in whichS1, S2 andS3 are components of the deviatoricstress tensor in the principal plane.

The shear mechanism proposed by Xue can be added tothe GTN model which already features the mechanismsof nucleation and growth of micro-voids. Thus, the evo-lution of the porosity now reads:

f = fN + fG + fShear (9)

2.3. The proposed modification

The shear mechanism was suggested in order to intro-duce void elongation effect into the original GTN model.Therefore, the contribution of such mechanism is higherat low stress triaxialities and, through the definition ofthe parameterg0, for low values of the Lode angle. Inthis contribution, the evolution of the porosity due toshear effects will be a function of both the accumulatedplastic strain and the rate of the accumulated plasticstrains instead of the total strain and total strain rate. Thissimplification is resoanable in the majority of problemssince the elastic strains can be considered negligible.

fShear = q4.fq5 .g0.ε

p.εp (10)

The Lode angle function can also be rewritten as a func-tion of the normalized third invariant:

g0 = 1 −∥

∥θ∥

∥ (11)

whereθ represents the normalized Lode angle which is afunction of the normalized third invariant [8]:

θ = 1 − 6.θ

π= 1 − 2

πarccos ξ (12)

In the equation above,ξ represents the normalized thirdinvariant and can be calculated as:

ξ =27

2

det (εed)

(

32εe

d : εed

)32

(13)

whereεed represents the deviatoric elastic strain tensor

[9]. The initial value of the normalized Lode angle fordifferent stress conditions is represented in Figure 2 [9].As we can see in Figure 2, in pure shear conditions,θis around zero which correspond a value ofg0 around 1.

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Normalized Lode Angle

Str

ess

Tri

axia

lity

Equibiaxialtension

Notchedround bars,comp.

Cylinders,upsetting

Axial symmetry,compression

Flat groovedplates, tension

Torsionor shear

Plane Stress

Plastic Plane strainor Generalized shear

Plastic plane straincomp.

Axial symmetry,tension

Equibiaxialcomp.

Smooth roundbars, tension

Notched roundbars, tension

Figure 2. Triaxiality as a function of the Normalized Lodeangle for initial stress states [9].

In this case, the effect of the shear mechanism is addedto the damage variable of the original GTN model. Onthe other hand, in traction problems with round bars, theLode angle function is approximately zero and in thesecases, the shear mechanism proposed does not play anyrole.

2.4. Integration algorithm

As already pointed out, the proposed modification leadsus to a constitutive model which is easier to implementthan the model proposed by Xue. The constitutive equa-tions of the model are integrated implicitly by using a typ-ical elastic predictor/return mapping algorithm (a detaileddiscussion on return mapping algorithms can be foundelsewhere, e.g., Simo and Hughes [10], de Sousa Netoet al. [11], etc.). Straightforward (pseudo-)time dis-cretization of the constitutive equations leads to the fol-lowing system of equations:

r∆γ =1

2

ST rial : ST rial

1 + 2.G.∆γ2

−1

3

(

1 + q3.f2n+1 − 2.q1.fn+1. cosh

3.q2.pn+1

2.σ0

!)

.σ20

rp = pn+1 − pTrialn+1 + ∆γ.K.q1.q2.fn+1.σ0. sinh

„

3.q2.pn+12.σ0

«

rf = fn + 1 − fT rialn+1 − ∆fG − ∆fT rial

rR = Rn+1 − RT rialn+1 − ∆Rn+1

(14)which needs to be solved for∆γ, pn+1, fn+1 andRn+1.Here, we choose to apply the standard Newton-Raphsonmethod for the solution of the non-linear system. Theresidual system of equations, on the linearized form canbe given by:

∂r∆γ∂∆γ

∂r∆γ∂pn+1

∂r∆γ∂fn+1

∂r∆γ∂Rn+1

∂rp∂∆γ

∂rp∂pn+1

∂rp∂fn+1

∂rp∂Rn+1

∂rf∂∆γ

∂rf∂pn+1

∂rf∂fn+1

∂rf∂Rn+1

∂rR∂∆γ

∂rR∂pn+1

∂rR∂fn+1

∂rR∂Rn+1

k

.

[

δ∆γδpn+1δfn+1δRn+1

]k+1

= −[

r∆γ (∆γ, p, f, R)

rp (∆γ, p, f, R)

rf (∆γ, p, f, R)

rR (∆γ, p, f, R)

]k

(15)

3

where we need to determine the derivatives ofr∆γ , rp

andrR in respect to every unknown of the problem.

The associated consistent elastoplastic tangent operator,necessary to ensure quadratic convergence rates, can beexpressed in the following closed-form:

Dep

=2.G

(1 + 2.G.∆γ)

„

I −1

3

I ⊗ I

«

+

"

2.G

(1 + 2.G.∆γ)

#2

.εe triald n+1

⊗

2

4

0

@C11

∂r∆γ

∂εe T riald n+1

+ C13

∂rf

∂εe T riald n+1

1

AC12∂rp

∂εe T rialv n+1

I

3

5

− I ⊗

2

4

0

@C21∂r∆γ

∂εe T riald n+1

+ C23

∂rf

∂εe T riald n+1

1

A + C22∂rp

∂εe T rialv n+1

I

3

5

(16)

whereC11, C12, C21, C23 andC22 are constants associ-ated with the non-linear system. Furthermore,I andI arerespectively the fourth and second order identity tensors.

3. NUMERICAL RESULTS

In this section, the performance of the extended constitu-tive model is assessed by means of two examples. Bothwere simulated considering a finite strain formulation inan in-house finite element code.

3.1. Shear test

To illustrate the effectiveness of the proposed constitutivemodel, a shear specimen subject to a shear stress statehas been analysed [6].

A three-dimensional finite element mesh with 8-node lin-ear elements withFBARtechnology (for more details see[11]) has been employed in all analyses (see Figure 3).The material properties adopted are listed in Table 1.

Table 1. Material properties for the aluminum alloy (Al2024-T351).

Description Symbol Value

Density ρ 2.7 × 103 kg/m3

Elastic modulus E 7.115 × 104 [MPa]Poisson’s ratio v 0.3

Initial yield stress σy0 370 [MPa]

Hardening curve σy`

εp´ 908`

0.0058 + εp´0.1742

[MPa]q1 1.5q2 1q3 2.25

Volumetric fraction fN 0.04of micro-void for nucleation

for nucleationMean strain of εN 0.2void nucleation

Standard deviation strain SN 0.1for nucleation

Figure 4.a shows the evolution of porosity for thetwo Gurson based models. In this test, where thetriaxiality ratio attains a value around zero, we canobserve that the modified GTN model proposed hereinhas the ability of predicting the degradation associated

Z

YX

Figure 3. Finite element mesh for the shear specimenwith 3432 elements and 4785 nodes.

with shear effects. The original GTN model, however,provides an inaccurate response when the shear effectsare present. This can be concluded since porosityremains constant when the prescribed displacement isincreased. Thus, the proposed model has the ability topredict shear effects as well as nucleation and growthof micro-voids. Even though the final values of thedamage variable are considerable distinct, when we com-pare the distributions of damage in the shear specimenwe conclude that the distribution is similar (see Figure 5).

In Figure 4.b, the evolution of accumulated plastic strainis plotted. As expected, this internal variable is largerwhen the new model is considered since degradationis significantly higher if compared with GTN originalmodel that only contemplates nucleation and growthmechanisms. In contrast with the accumulated plasticstrain, we can see in Figure 4.c that the reaction forceassociated with the new model is smaller than the similarcurve associated with GTN model. Again, this differenceis a consequence of an higher internal damage. Oneinteresting aspect to note in Figure 4.c is the pronouncedeffect of softening in the reaction curve of the new modelfor the same prescribed displacement.

Moreover, the efficiency of the proposed algorithm isalso assessed. Table 2 shows a typical residual conver-gence observed during the analysis of the present exam-ple. Clearly, convergence exhibits a quadratic conver-gence rate due to the consistent tangent operator.

Table 2. Typical global convergence of proposed model.

Iteration Relative residual norm (%)1 4.575722 0.559380E-013 0.978862E-054 0.495718E-09

3.2. Flat-grooved test

In this section, a flat-grooved plate [12] in plane strain isanalysed under a tensile dominant load. The main goalof the numerical simulation is to verify the ability of the

4

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.05

0.1

0.15

0.2

0.25

Displacement (mm)

Dam

age

GTN ModelGTN Modified ModelCritical Void Volume Fraction

a)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Displacement (mm)

Acc

.P

l.Str

ain

GTN ModelGTN Modified Model

b)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

500

1000

1500

2000

2500

Displacement (mm)

Rea

ctio

n(N

)

GTN ModelGTN Modified Model

c)

Figure 4. (a) Evolution of damage; (b) evolution of accu-mulated plastic strain; (c) reaction curve

0 0.0163 0.0325

Damage

a)

0 0.0303 0.0607

Damage

b)

Figure 5. Distribution of damage a) GTN model; b) Newmodel.

proposed model to predict the45 degrees inclined shearbands observed in experimental testing [5]. The materialproperties are the same as the material in the precedingsection (see Table 1). Spatial discretization has beendone by using 8-node quadratic elements with a reducedintegration scheme (see Figure 6).

Figure 6. Finite element mesh for the plane strain speci-men with 1600 elements and 5033 nodes.

In Figure 6, we can clearly observe the effects of theincorporated shear mechanism. For the same applieddisplacement, the contours of the new damage variablehave correctly predicted the shear bands inclined 45degrees in respect to the loading direction. In the simula-tion with the original GTN model, however, damage hasconcentrated at the center of the specimen which is notin agreement with experimental results [5].

4. CONCLUSIONS

In this work, we have briefly revised the original GTNmodel and depicted its limitations when subject to low tri-axialities. For such model, the damage variable (volumefraction of voids dependent on pressure) does not reflectthe degradation associated with shear effects. The incor-poration of a modified shear mechanism into the constitu-

5

0 0.0141 0.0282

Damage

(a)

0 0.0304 0.0608

Damage

(b)

Figure 7. Damage contours at critical zone for the planestrain test: (a) original GTN model; (b) proposed model.

tive model has allowed a much more accurate predictionof failure for a wider range of triaxialities. Finally, themodified evolution of the porosity due to shear effectsproposed in this paper has provided an extremely effi-cient numerical implementation exhibiting high asymp-totic convergence rates.

ACKNOWLEDGEMENTS

Fabio Reis is supported by Portuguese Science and Tech-nology Foundation (FCT), under scholarship numberSFRH/BD/60887/2009. Lucival Malcher is supported byPortuguese Science and Technology Foundation (FCT),under scholarship number SFRH/BD/45456/2008. Theauthors acknowledge the support of Portuguese Scienceand Technology Foundation (FCT) under grant with ref-erence PTDC/EME-TME/71325/2006.

REFERENCES

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[3] A.L. Gurson. Continuum theory of ductile ruptureby void nucleation and growth - part I: Yield crite-ria and flow rule for porous media.Journal of Engi-neering Materials and Technology, 99:2–15, 1977.

[4] L. Malcher, F. M. Andrade Pires, J. M. A. Cesar deSa, and F. X. C. Andrade. Comparative study be-tween ductile damage constitutive model. InCOM-PLAS, Spain, Barcelona, 2009.

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[6] M. Brunig, O. Chyra, D. Albrecht, L. Driemeier,and M. Alves. A ductile damage criterion at variousstress triaxialities.International Journal of Plastic-ity, 24:1731–1755, 2008.

[7] L. Nahshon and J. W. Hutchinson. Modification ofthe gurson model for shear failure.European Jour-nal of Mechanics A/Solids, 27:1–17, 2008.

[8] Y. Bao and T. Wierzbicki. On fracture locus in theequivalent strain and stress triaxiality space.Inter-national Journal of Mechanical Sciences, 46(81):81–98, 2004.

[9] Y. Bai. Effect of Loading History on Necking andFracture. PhD thesis, Massachusetts Institute Tech-nology, 2008.

[10] J. C. Simo and T. J. R Hughes.Computational In-elasticity. Springer, 1998.

[11] E. A. de Sousa Neto, D. Peric, and D. R. Owen.Computational Methods for Plasticity: Theory andApplication. Wiley, 2008.

[12] X. Teng. Numerical prediction of slatn fracture withcontinuum damage mechanics.Engineering Frac-ture Mechanics, 75:2020–2041, 2008.

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