application of mohr coulomb criterion to ductile failure

Int J Fract (2010) 161:1–20DOI 10.1007/s10704-009-9422-8

ORIGINAL PAPER

Application of extended Mohr–Coulomb criterion to ductilefracture

Yuanli Bai · Tomasz Wierzbicki

Received: 7 May 2009 / Accepted: 15 October 2009 / Published online: 12 November 2009© Springer Science+Business Media B.V. 2009

Abstract The Mohr–Coulomb (M–C) fracture cri-terion is revisited with an objective of describingductile fracture of isotropic crack-free solids. Thiscriterion has been extensively used in rock and soilmechanics as it correctly accounts for the effects ofhydrostatic pressure as well as the Lode angle param-eter. It turns out that these two parameters, whichare critical for characterizing fracture of geo-materi-als, also control fracture of ductile metals (Bai andWierzbicki 2008; Xue 2007; Barsoum 2006; Wilkinset al. 1980). The local form of the M–C criterionis transformed/extended to the spherical coordinatesystem, where the axes are the equivalent strain tofracture ε f , the stress triaxiality η, and the nor-malized Lode angle parameter θ . For a proportionalloading, the fracture surface is shown to be an asym-metric function of θ . A detailed parametric study isperformed to demonstrate the effect of model param-eters on the fracture locus. It was found that theM–C fracture locus predicts almost exactly the expo-nential decay of the material ductility with stress tri-axiality, which is in accord with theoretical analysis ofRice and Tracey (1969) and the empirical equation ofHancock and Mackenzie (1976), Johnson and Cook

Y. Bai (B) · T. WierzbickiImpact and Crashworthiness Laboratory, MassachusettsInstitute of Technology, Cambridge, MA 02139, USAe-mail: [email protected]

Present Address:Y. BaiGeneral Electric Global Research Center, Niskayuna, NY, USA

(1985). The M–C criterion also predicts a form ofLode angle dependence which is close to parabolic.Test results of two materials, 2024-T351 aluminumalloy and TRIP RA-K40/70 (TRIP690) high strengthsteel sheets, are used to calibrate and validate the pro-posed M–C fracture model. Another advantage of theM–C fracture model is that it predicts uniquely theorientation of the fracture surface. It is shown thatthe direction cosines of the unit normal vector to thefracture surface are functions of the “friction” coeffi-cient in the M–C criterion. The phenomenological andphysical sound M–C criterion has a great potential tobe used as an engineering tool for predicting ductilefracture.

Keywords Mohr–Coulomb criterion ·Ductile fracture · 3D Fracture locus · Crack direction

1 Introduction

The Mohr–Coulomb (M–C) fracture criterion (Coulomb1776; Mohr 1914) has been widely used in rock andsoil mechanics (e.g. Zhao (2000); Palchik (2006)) andother relatively brittle materials (e.g. Lund and Schuh(2004)). This is a physically sound and simple fracturemodel. Fundamentals and applications of this modelcan be found in many textbooks, monographs andresearch papers. As a stress-based criterion, the Mohr–Coulomb model has good resolution for materials

123

2 Y. Bai, T. Wierzbicki

that fail in the elastic range and/or under small strainplasticity, such as rock, soil, concrete and so on.

There have recently been several successful applica-tions of this model for predicting fracture of ceramicsunder static and dynamic loading (Fossum and Brannon2006). The M–C criterion is a special case of the San-dia GeoModel (Fossum and Brannon 2005). A uniquefeature of the M–C model is an explicit dependence onthe Lode angle parameter, which is missing in almostall existing models of ductile fracture. The purpose ofthe present paper is to demonstrate the applicabilityof the M–C criterion to ductile fracture of uncrackedbodies.

Meanwhile, the ductile fracture community took adifferent path. In search for a physically based fracturemodel, the mechanism of nucleation, growth and coa-lescence of void was identified and extensively stud-ied. Due to fundamental work by McClintock (1968),Rice and Tracey (1969), Gurson (1975), Tvergaard andNeedleman (1984), it was determined that ductile frac-ture is mostly affected by the hydrostatic pressure.Accordingly, the equivalent strain to fracture, whichis a measure of material ductility, was made dependenton the first invariant of the stress tensor. The mixedstress–strain formulation of a fracture criterion is jus-tifiable because, well into the plastic range, the resolu-tion of strains is much larger than stresses, as explainedin Fig. 1. Another feature of the Gurson-Tvergaard-Needleman (GTN) model (Tvergaard and Needleman1984) is that it describes well the predominate tensilefracture, characterized by relatively high stress triax-iality, but fails to predict shear fracture. Attempts arecurrently underway to extend the void growth and coa-lescence model to describe shear fracture (Xue 2007;Nahshon and Hutchinson 2008). The M–C criterion isan extension of the maximum shear stress fracture cri-terion and therefore it is well poised to predict shearfracture.

In parallel with the “physically based” models ofductile fracture, a number of empirical fracture mod-els have earned a permanent place in the literature(Cockcroft and Latham 1968; Hancock and Macken-zie 1976; Wilkins et al. 1980; Johnson and Cook 1985;Bao and Wierzbicki 2004; Wierzbicki and Xue 2005;Bai and Wierzbicki 2008). These models were based onextensive test programs on bulk material and/or sheets.One of the most comprehensive series of experimentsinvolving tensile tests on unnotched and notched roundbars, upsetting tests and shear tests was reported by

σ

ε

σ

σ

εε

Brittlefracture

Ductilefracture

Fig. 1 Different resolution quality of strain and stress param-eters: the stress parameters have good resolutions in the elasticregion, and the strain parameters have good resolutions far in theplastic region

Bao (2003), Bao and Wierzbicki (2004). Even morerecently Wierzbicki et al. (2005a), performed a series offracture tests on specially designed butterfly specimensunder combined tension/shear/compression loading.Results of bi-axial fracture tests on tubular specimensin the tension/torsion loading frame was published byBarsoum and Faleskog (2007), while Korkolis and Kyr-iakides (2008) studied fracture of 6260-T4 aluminumtubes subjected to internal pressure and axial tension orcompression. All of these recent tests have proven thatthe material ductility depends on both stress triaxialityand the Lode angle parameter. These two effects areactually captured by the M–C model, as will be shownin the paper.

A particular case of Mohr–Coulomb criterion is themaximum shear stress criterion. It has been shown byLee (2005) that the maximum shear stress criterionpredicts well plane stress fracture for 2024-T351 alu-minum alloy, see Fig. 2. A comparison of maximumshear stress criterion with other models was reportedby Wierzbicki et al. (2005b). One main shortcoming ofthe maximal shear stress fracture criterion is the miss-ing pressure dependence. The M–C criterion removesthis shortcoming. To increase resolution of the duc-tile fracture prediction, the M–C criterion is trans-formed/extended to a strain-based representation underthe assumption of monotonic loading. A parametricstudy is performed to get a better insight into this frac-ture model. The experimental results of two materi-als, 2024-T351 aluminum alloy (Bao 2003; Bao andWierzbicki 2004; Wierzbicki et al. 2005b) and TRIP

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Application of extended Mohr–Coulomb criterion to ductile fracture 3

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

full point used for calibration

_

_

0.8

0.4

Maximum shear stress τmax

= const.

experiment (plane stress)experiment (axisymmetric)

Al2024-T351

σm/σ

εf

Fig. 2 Application of maximum shear stress fracture criterion toductile fracture for plane stress condition (Lee 2005; Wierzbickiet al. 2005b)

RA-K40/70 steel sheets, are used to validate the newform of the Mohr–Coulomb criterion.

2 Characterization of the stress state

The three invariants of a stress tensor [σ ] are definedrespectively by

p = −σm = −1

3tr([σ ]) = −1

3(σ1 + σ2 + σ3) (1)

q = σ =√

3

2[S] : [S]

=√

1

2

[(σ1 − σ2)2+(σ2 − σ3)2+(σ3 − σ1)2

](2)

r =(

9

2[S] · [S] : [S]

)1/3

=[

27

2det([S])

]1/3

=[

27

2(σ1 − σm)(σ2 − σm)(σ3 − σm)

]1/3

(3)

where [S] is the deviatoric stress tensor defined by,

[S] = [σ ] + p[I ], (4)

[I ] is the identity tensor and σ1, σ2 and σ3 denote prin-cipal stresses. It is assumed that σ1 ≥ σ2 ≥ σ3. Notethat the parameter p is positive in compression, but σm

is positive in tension. It is convenient to work with thedimensionless hydrostatic pressure η, defined by

η = −p

q= σm

σ. (5)

The parameter η, often referred to as the triaxialityparameter, has been extensively used in the literature

σ

2σ

Deviatoric plane ( plane)

z

2'

3O P σ=

'oθ

ρ

P(Lode angle)

ϕ Aπ

1σ

3σ

o' 3 mOO σ=

( )1 2 3, ,σ σ σCartesian coordinate system

( ), ,mσ σ θCylindrical coordinate system

( ), ,σ η θ“Spherical” coordinate system

( ), , zρ θ

( ), ,ϕ θ ρ

Fig. 3 Three types of coordinate system in the space of principalstresses

on ductile fracture (McClintock 1968; Rice and Tracey1969; Hancock and Mackenzie 1976; Mackenzie et al.1977; Johnson and Cook 1985; Bao 2003). The sec-ond important parameter is the Lode angle θ , which isrelated to the normalized third invariant ξ through

ξ =(

r

q

)3

= cos(3θ). (6)

One can see that that the normalized third devia-toric stress invariant can be expressed in terms of theLode angle θ (see Malvern (1969), Xu and Liu (1995),ABAQUS 2005; the derivation is also summarized inSect. 3.1). Since the range of the Lode angle is 0 ≤ θ ≤π/3, the range of ξ is −1 ≤ ξ ≤ 1. The geometricalrepresentation of Lode angle is shown in Fig. 3.

As shown in Fig. 3, one can think of three types ofcoordinate systems to describe the stress state. The firstis the Cartesian coordinate system (σ1, σ2, σ3), the sec-ond is the cylindrical coordinate system (σm , σ , θ), andthe third is the spherical coordinate system (σ , η, θ).The equivalent stress σ is related to the equivalent strainε through the strain hardening function of a material.The coordinate ϕ is related to the stress triaxiality η bythe following equation,

η = σm

σ=

√2

3cotanϕ. (7)

Furthermore, the Lode angle can be normalized,

θ = 1 − 6θ

π= 1 − 2

πarccos ξ. (8)

So that the range of θ is −1 ≤ θ ≤ 1. The parameterθ will be called the Lode angle parameter hereinaf-ter. Now, the direction of every stress vector (or load-ing condition) in the space of principal stresses can becharacterized by the above defined set of parameters

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(η, θ). It is easy to show (Wierzbicki and Xue 2005;Bai and Wierzbicki 2008) that θ = 1 corresponds to theaxisymmetric tension, θ = 0 corresponds to the gener-alized shear (or plastic plane strain) loading condition,and θ = −1 corresponds to the axisymmetric compres-sion or equi-biaxial tension. Special attention is givento the plane stress state. It was shown by Wierzbickiand Xue (2005), Bai and Wierzbicki (2008) that theplane stress condition, σ3 = 0, uniquely relates theparameters η and ξ or θ through

ξ = cos(3θ) = cos[π

2(1 − θ )

]= −27

2η

(η2−1

3

).

(9)

In this paper, the Mohr–Coulomb model will be refor-mulated in the spherical coordinate system.

3 Transformation of Mohr–Coulomb to the spaceof (ε f , η, θ)

Consider a material element subjected to three princi-pal stresses σ1, σ2, and σ3. At an arbitrary cutting planedefined by the unit normal vector (ν1, ν2, ν3), the shearstress and the corresponding normal stress are given by

τ =√

ν21ν2

2 (σ1−σ2)2 +ν22ν2

3 (σ2−σ3)2 +ν23ν2

1 (σ3−σ1)2,

(10)

σn = ν21σ1 + ν2

2σ2 + ν23σ3, (11)

where the three components ν1, ν2 and ν3 are con-strained by ν2

1 + ν22 + ν2

3 = 1.The Mohr–Coulomb fracture criterion says that frac-

ture occurs when the combination of normal stress andshear stress reach a critical value, according to

(τ + c1σn) f = c2, (12)

where c1, c2 are material constants. The constant c1 isoften referred to as a “friction” coefficient, and c2 isshear resistance. The ranges of c1 and c2 are c1 ≥ 0and c2 > 0. In the limiting case of c1 = 0, the M–Ccriterion reduces to the maximal shear stress criterion.

The applicability of the M–C criterion to captureductile fracture will be shown in Sect. 8. In order tofind on which cutting plane the M–C criterion will bemet first, one must solve the following maximum valueproblem:⎧⎪⎪⎨⎪⎪⎩

Max

{√ν2

1ν22 (σ1−σ2)

2 +ν22ν2

3 (σ2−σ3)2 +ν2

3ν21 (σ3−σ1)

2

+c1(ν2

1σ1+ν22σ2+ν2

3σ3)}

Subject to ν21 + ν2

2 + ν23 = 1

.

(13)

Recalling that σ1 ≥ σ2 ≥ σ3, the solution of themaximum value problem using the Lagrangian mul-tiplier technique is⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ν21 = 1

1+(√

1+c21+c1

)2

ν22 = 0

ν23 = 1

1+(√

1+c21−c1

)2

, (14)

where ν1, ν2, and ν3 are direction cosines of maxi-mum, intermediate, and minimum principal stresses,respectively. Substituting Eq. 14 into Eq. 12, the M–Ccriterion can be expressed in terms of principalstresses,(√

1 + c21 + c1

)σ1 −

(√1 + c2

1 − c1

)σ3 = 2c2,

(15)

provided that σ1 ≥ σ2 ≥ σ3. The plane stress represen-tation of the above fracture criterion is shown in Fig.20. In view of enormous literature on the M–C crite-rion, the solution to the above maximum value problemmust have been published earlier. However, the pres-ent authors were unable to find any reference on thistopic. Here, only the final results are given. It should benoted that the orientation of the fracture plane dependsonly on the friction coefficient c1, while the onset offracture is controlled by both c1 and c2. In order totransform the Mohr–Coulomb criterion to the spaceof (ε f , η, θ ), one needs to express principal stressesin terms of σm , η and θ . Similar transformation equa-tions can be found, for example, in Malvern (1969).However, for the consistency of notation, the requiredtransformation is derived here from the geometricalconstruction in Sect. 3.1.

3.1 Representation of the M–C criterion in terms ofσ , η and θ

The principal stresses and principal deviatoric stessescan be geometrically represented on the deviatoricplane (π plane), as shown in Fig. 4. Note that all thecomponents are scaled because of the inclined anglebetween the deviatoric plane and the principal axis (seeFig. 3). From the geometrical construction, one can eas-ily obtain the expressions of the deviatoric principalstresses in terms of σ and θ ,

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1

2

3

1

2

3σ

1

3

2s

θ 60

2

3σ

3

3

2s

2

3

2s

2

2

3σ

3

2

3σ

60P'O

Fig. 4 Geometrical representation of principal stresses(σ1, σ2, σ3), deviatoric stresses (s1, s2, s3), equivalent stress (σ )and Lode angle (θ ) on the deviatoric plane (π plane)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

√3

2s1 =

√2

3σ cos θ√

3

2s2 =

√2

3σ cos

(2

3π − θ

)√

3

2s3 =

√2

3σ cos

(4

3π − θ

)

⇒

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

s1 = 2

3σ cos θ

s2 = 2

3σ cos

(2

3π − θ

)

s3 = 2

3σ cos

(4

3π − θ

) . (16)

It should be noted that the constraint of deviatoricprincipal stresses,

s1 + s2 + s3 = 2

3σ

[cos θ + cos

(2

3π − θ

)

+ cos

(4

3π − θ

)]= 0,

(17)

is satisfied automatically. Using Eq. 16, one can expressthe three principal stresses in terms of σm , η and θ .⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σ1 = σm + s1 = σm + 23 σ cos θ =

[1 + 2 cos θ

3η

]σm

σ2 = σm + s2 = σm + 23 σ cos

( 23π − θ

)

=[

1 + 2 cos(

23 π−θ

)3η

]σm

σ3 = σm + s3 = σm + 23 σ cos

( 43π − θ

)

=[

1 + 2 cos(

43 π−θ

)3η

]σm

(18)

Since the range of Lode angle is 0 ≤ θ ≤ π/3,it can be proved that σ1 ≥ σ2 ≥ σ3 is satisfied forthe three principal stresses in Eq. 18. Using Eq. 18, onecan also prove the relationship of the normalized thirddeviatoric invariant ξ and the Lode angle θ expressedearlier in Eq. 6. Also, the condition of plane stress(Eq. 9) can be easily proved by using the above trans-formation formulas.

Substituting Eq. 18 into Eq. 15, one can express theM–C criterion in terms of σ , η and θ ,

σ = c2

⎡⎣√

1 + c21

3cos(π

6− θ)

+ c1

(η + 1

3sin(π

6− θ))]−1

. (19)

This fracture envelope forms a surface in the space ofnormalized stress invariants, which is also the “spher-ical” stress coordinate system.

3.2 Extended plasticity model

As explained in Fig. 1, the resolution of detecting theonset of ductile fracture is much better if a strain repre-sentation is used. Furthermore, fracture is a local phe-nomenon with large stress gradients. There is no easyway of measuring components of the stress tensor andevaluate σ directly from tests. This difficulty is over-come if the equivalent stress is expressed in terms ofother measurable parameters.

The quadratic yield condition, on which the powertype relation between the equivalent stress and theequivalent strain is based, σ = Aεn , is not sufficient topredict correctly material yield strength and directionsof plastic flow. Hosford (1972) proposed a non-qua-dratic yield function with one more parameter k,

2σ 2ky = (σ1 − σ2)

2k + (σ2 − σ3)2k + (σ3 − σ1)

2k .

(20)

Equation 20 reduces respectively to the von Mises yieldcondition for k = 1 and to the Tresca yield locus fork → ∞. The actual value of the exponent 2k can beadjusted to test fit experimental data on biaxial testing.For example 2k = 8 was quoted to fit the initial yieldsurface of a class of aluminum alloys. Further extensionof the Hosford yield condition for isotropic and aniso-tropic solids and sheets can be found in the papers byKarafillis and Boyce (1993), Barlat et al. (2007).

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Recent experimental results (Bai and Wierzbicki2008; Yang et al. 2009; Gao et al. 2009) have shown thatboth the hydrostatic pressure and Lode angle param-eter should be taken into account in metal plasticity.Bai and Wierzbicki (2008) proposed a generalizedhardening rule with pressure and Lode angle depen-dence in the form,

σ = Aεn [1 − cη(η − η◦)] [

csθ + (cax

θ − csθ )γ], (21)

caxθ =

{1 for θ ≥ 0ccθ for θ < 0

. (22)

where A is a material constant, n is the strain hard-ening exponent, and cη, η◦, cs

θ and ccθ are parameters

to describe both the pressure dependence and Lodeangle dependence of the material plasticity. Altogether,there are six parameters defining the plasticity model.The parameter γ in Eq. 21 is related to the Lode angleparameter by

γ =√

3

2 − √3

[sec(θ − π

6

)− 1]

=√

3

2 − √3

[sec

(θπ

6

)− 1

]. (23)

Some limiting cases of the general yield function areobtained by suitably choosing model parameters. It isnoted that by fixing the parameters cc

θ = 1 and cη =0, there is a close analogy between the non-quadraticHosford yield function and the new representation ofyield function in terms of σ and θ , Eq. 21. For example,csθ = 1 corresponds to the von-Mises yield condition;

while csθ = √

3/2 gives the Tresca yield condition.Thus, there is a one-to-one correspondence betweenthe exponent k in Eq. 20 and parameter cs

θ in Eq. 21.The presence of the parameter cc

θ gives additional flex-ibility to the size and shape of the yield surface, asillustrated in Fig. 5.

3.3 Representation of the M–C criterion in terms ofε f , η and θ

Consider the case of a monotonic loading, and denotethe equivalent stress and strain at the point of frac-ture by σ f and ε f . Substituting Eq. 21 and Eq. 23 intoEq. 19, the Mohr–Coulomb fracture criterion is trans-formed from the stress-based form into the mixed space

Fig. 5 Examples of yield loci for plane stress condition. (Here,the effect of hydrostatic pressure is deactivated, cη = 0)

of (ε f , η, θ ).

ε f ={

A

c2

[1−cη(η−η◦)

]

×[

csθ+

√3

2−√3(cax

θ −csθ )

(sec

(θπ

6

)−1

)]

⎡⎣√

1+c21

3cos

(θπ

6

)+c1

(η+1

3sin

(θπ

6

))⎤⎦⎫⎬⎭

− 1n

.

(24)

A total of eight parameters (A, n, cη, η◦, csθ , cc

θ , c1, c2)need to be found, but only two have to be calibratedfrom fracture tests. If a von Mises yielding function isused (cη = 0, cs

θ = ccθ = 1), then Eq. 24 reduces to

ε f =⎧⎨⎩

A

c2

⎡⎣√

1 + c21

3cos

(θπ

6

)+ c1

(η + 1

3sin

(θπ

6

))⎤⎦⎫⎬⎭

− 1n

.

(25)

In the case of Tresca yield function (cη = 0, ccθ = 1,

csθ = √

3/2 ), and Eq. 24 reduces to

ε f =⎧⎨⎩

A

c2

⎡⎣√

1+c21

2+c1

√3

2sec

(θπ

6

)(η+ 1

3sin

(θπ

6

))⎤⎦⎫⎬⎭

− 1n

.

(26)

In the above two limiting cases, the simplified plastic-ity model depends on only two parameters, A and n,which can be found from the same tests as the fracturetests. From the application point of view, the ease andpracticality of the M–C criterion in terms of findingmodel parameters from experiments are obvious.

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Fig. 6 3D geometry representation of maximum shear fracturecriterion (A = 740 MPa, n = 1/6, cη = 0, cs

θ = ccθ = 1,

c1 = 0.0, and c2 = 330 MPa). The curve on the surface repre-sents the plane stress curve

4 A limiting case: maximum shear stress criterion

In the limiting case of c1 = 0, the Mohr–Coulombfracture criterion reduces to the maximum shear stressfracture criterion. For example, if the von Mises yieldfunction is used, then Eq. 25 reduces to

ε f =[√

3

3

A

c2cos

(θπ

6

)]− 1n

, (27)

which is pressure independent. At the same time,the direction cosines of the fracture plane, defined

by Eq. 14, are constant and equal to(

1√2, 0, 1√

2

). A

geometric representation of this criterion is shown inFig. 6. The fracture locus depends only on the Lodeangle parameter and forms a half tube in the space of(ε f , η, θ ). The corresponding fracture locus for planestress condition (Eq. 9) is a three-branch curve lying onthe half tube. The projection of the plane stress fracturelocus onto the plane of equivalent strain to fracture andthe stress triaxiality is shown in Fig. 2, on which onlytwo branches are plotted.

If the Tresca yielding condition is used, the maxi-mum shear stress criterion, Eq. 26, reduces to the con-stant fracture strain criterion.

ε f =(

A

2c2

)− 1n

. (28)

Fig. 7 3D geometry representation of Mohr–Coulomb fracturemodel (A = 740 MPa, n = 1/6, cη = 0, cs

θ = ccθ = 1, c1 = 0.1,

and c2 = 330 MPa)

5 Representation of Mohr–Coulomb fracturecriterion in 3D space of invariants

The new form of Mohr–Coulomb fracture model, Eq.24, can be geometrically represented in the 3D spaceof (ε f , η, θ ), see Fig. 7. Here, an example group ofparameters is used: A = 740 MPa, n = 1/6, cη = 0,csθ = cc

θ = 1, c1 = 0.1, and c2 = 330 MPa. The 3Dfracture locus is seen to be a monotonic function ofthe stress triaxiality and an asymmetric function withrespect to the Lode angle parameter θ .

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

η

εf

Mohr-Coulomb: A=740, n=1/6, c1=0.1, c

2=330

Rice and Tracey: εf=0.2143e-0.9796η

Fig. 8 Mohr–Coulomb criterion in the space of equivalent strainto fracture and stress triaxiality (assuming θ = 0)

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5.1 Pressure dependence

If a von-Mises yielding function is used (cη = 0, csθ =

ccθ = 1), and the Lode angle parameter θ in Eq. 24 is

fixed at a certain value, for example θ = 0, then Eq. 24reduces to

ε f =⎡⎣ A

c2

⎛⎝√

1 + c21

3+ c1η

⎞⎠⎤⎦

− 1n

. (29)

which is a nonlinear hyperbolic function of stress triax-iality η. An example plot of Eq. 29 is shown in Fig. 8.It is found that an exponential function D3e−D4η fitsEq. 29 for a wide range of stress triaxiality. So, theMohr–Coulomb criterion provides an interesting phys-ical interpretation of the effect of stress triaxiality onductile fracture, which is originally based on the the-ory of void growth (McClintock 1968; Rice and Tracey

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

θ

εf

Mohr-CoulombParabolic fitting

Fig. 9 Mohr–Coulomb criterion on the space of equivalent strainto fracture and the Lode angle parameter (assuming η = 0)

1969). This is also in accord with the empirical resultsof Hancock and Mackenzie (1976), Johnson and Cook(1985).

5.2 Lode angle dependence

If the von-Mises yielding function is used again (cη =0, cs

θ = ccθ = 1), and the stress triaxiality η in Eq. 24 is

fixed at a certain value, for example η = 0, then Eq. 24becomes

ε f =⎧⎨⎩

A

c2

⎡⎣√

1 + c21

3cos

(θπ

6

)+ c1

3sin

(θπ

6

)⎤⎦⎫⎬⎭

− 1n

.

(30)

An example plot of Eq. 30 is shown in Fig. 9. A par-abolic function controlled by three points, ε f (θ = 1),ε f (θ = 0) and ε f (θ = −1), is used to fit the curve.It follows from Fig. 9 that a parabolic function pro-vides a good functional dependence of the Lode angleparameter on ductile fracture. It should be noted thatthe minimum value of the function does not occur atθ = 0 but it is slightly shifted, which makes the 3Dfracture locus an asymmetric function.

6 Parametric study

There are eight parameters (A, n, c1, c2, cη, η◦, csθ

and ccθ ) in the Mohr–Coulomb fracture criterion (see

Eq. 24). In this section, a parameter study is performedto get a better understanding of new forms of Mohr–Coulomb criterion. In particular, some qualitative fea-tures of model parameters on the fracture locus aredemonstrated. As a starting point, a combination ofmodel parameters, A = 740 MPa, n = 1/6, cη = 0,

Fig. 10 Effect of c1 on thenew form of theMohr–Coulomb criteria

123


η◦ = 0, csθ = cc

θ = 1, c1 = 0.1 and c2 = 330 MPa, isconsidered.

6.1 Effect of c1

Keeping other parameters unchanged, the new form ofthe Mohr–Coulomb criterion is plotted for three val-ues of c1 (c1 = 0, 0.1, and 0.2) on the same planes asthose in Sects. 5.1 and 5.2, see Fig. 10. It is found thatas c1 increases, the fracture strain becomes more pres-sure dependent (see Fig. 10a), and the fracture locusbecomes more asymmetric (see Fig. 10b). The limitingcase c1 = 0, which corresponds to the maximum shearstress criterion, implies a symmetric fracture locus withrespect to Lode angle parameter θ .

6.2 Effect of c2

Keeping other parameters unchanged, two values of c2

are used to plot the new form of the Mohr–Coulombcriteria, see Fig. 11. It is found that the value of c2

affect only the “height” of the fracture locus while theshape of the fracture locus is unchanged. A larger valueof c2 scales up the fracture locus.

6.3 Effect of A

Similarly, two values of A are used to plot the newform of the Mohr–Coulomb criteria, see Fig. 12. It isfound that the parameter A has a similar effect as thatof parameter c2, but the effect of A on fracture locusis in the opposite direction of that of c2. A larger valueof A scales down the fracture locus. In other words,if a material gains more strength in the plastic range,then it loses some of its ductility. This property hasbeen observed in many advanced high strength steels(Pfestorf 2005).

6.4 Effect of strain hardening exponent n

Again, two values of the strain hardening exponent, n,are used to plot the new form of the Mohr–Coulombcriteria, see Fig. 13. One obvious effect of the strainhardening exponent n is that it raises the fracture locus.

Three values of n ( 16 , 1

4 and 12 ) are chosen in Fig. 14

to further demonstrate the effect of n. In Fig. 14a, allthree curves are normalized with respect to ε f (η = 0),

and in Fig. 14b, all curves are normalized with respectto ε f (θ = 1). From Fig. 14a, one can see that a highervalue of n decreases the pressure dependence of frac-ture locus. From Fig. 14b, one can see that a highervalue of n decreases the dependence of fracture locuson the Lode angle parameter, which was first noted byXue (2007) using the Stören and Rice necking criterion(Storen and Rice 1975).

6.5 Effect of cη

Taking into account the pressure effect on material plas-ticity, for example, cη = 0.09 and η◦ = 0 with otherparameters kept unchanged, a comparison of fracturelocus is shown in Fig. 15. It is found that changing theparameter cη has no effect on the Lode angle depen-dence of the fracture locus (see Fig. 15b), but increasingcη will decrease the pressure dependence of the frac-ture locus (see Fig. 15a). In other words, if a materialhas more pressure dependence on plasticity, then it willhave less pressure dependence on fracture. This resultshould also be further investigated experimentally ondifferent materials.

6.6 Effect of csθ

Assuming ccθ = 1, three values of cs

θ (1.0, 0.93 and√3

2 = 0.866) are used to plot the fracture locus, seeFig. 16. One can see that decreasing the parameter cs

θ

will raise the fracture locus (see Fig. 16a). On the otherhand, decreasing cs

θ will decrease the Lode dependenceof the fracture locus (see Fig. 16b). In other words, ifa material has stronger Lode dependence on plasticity,then it will have weaker Lode dependence on fracture.This point has been confirmed by comparing fracturedata of two steels (1045 steel and DH36 steel), see (Baiet al. 2009). The 1045 steel has no Lode angle depen-dence on plasticity but exhibits a Lode angle depen-dence on fracture locus. On the other hand the DH36exhibits a Lode angle dependence on plasticity but noton fracture.

6.7 Effect of ccθ

In Eq. 21, the parameter, ccθ , controls the asymme-

try of a yield surface, so this parameter will affect

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Fig. 11 Effect of c2 on thenew form of theMohr–Coulomb criteria

Fig. 12 Effect of A on thenew form of theMohr–Coulomb criteria

Fig. 13 Effect of n on thenew form of theMohr–Coulomb criteria

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Fig. 14 A normalized plotto show the effect of n onthe new form of theMohr–Coulomb criteria

Fig. 15 Effect of cη on thenew form of theMohr–Coulomb criteria

Fig. 16 Effect of csθ on the

new form of theMohr–Coulomb criteria.This parameter controls theamount of Lode angledependence of the fracturelocus

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Fig. 17 Effect of ccθ on the

new form of theMohr–Coulomb criteria.This parameter controls theasymmetry of fracture locuswith respect to the Lodeangle parameter

Fig. 18 A proposed shapeof cutoff region of fractureone the plane of (η, θ) withc1 = 0.57

the asymmetry of material’s fracture locus. Assumingcsθ = 1, fracture loci with three values of cc

θ (1.0, 0.95and 1.05) are plot in Fig. 17. One can see that decreas-ing the parameter cc

θ will raise the fracture strains atθ = −1 (see Fig. 17b), which changes the symmetry ofthe fracture locus. For some materials, fracture strainunder equi-biaxial tension (θ = −1) is much higherthan that of uni-axial tension (θ = 1), which would becontrolled by the parameter cc

θ . On the other hand, theparameter cc

θ does not change the pressure dependenceof fracture locus (see Fig. 16a). Referring to Sects. 6.4and 6.6, it is found that the parameters, cs

θ , ccθ and n,

are the key parameters controlling the Lode angle effecton the fracture locus. This conclusion emphasizes the

importance of developing an accurate plasticity modelto predict fracture.

6.8 Existence of a cutoff region

The existence of a cutoff value in the low stress triaxi-ality region has been revealed by Bao (2003), Bao andWierzbicki (2005) and Teng (2004). Based on anal-ysis of upsetting tests and Bridgman’s tests (Bridg-man 1952), Bao and Wierzbicki (2005) discovered thatthe cutoff value for fracture occurs at ηcutoff = − 1

3 .No fracture can occur below this critical value. Teng(2004) confirmed the importance of a cutoff value inhigh velocity impact simulation.

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Shear Force

Normal Force

Die-cone

nσ

τ

Total ForceF

Fig. 19 A physical interpretation of existence cutoff regionusing the concept of die-cone in friction

Introducing the Lode angle parameter θ into the duc-tile model, the cutoff region in the plane of the stressinvariants (η, θ) of the 3D fracture locus appears auto-matically as a consequence of the M–C model. FromEq. 24, it is found that the fracture strain will go toinfinity when the following condition is satisfied,√

1 + c21

3cos

(θπ

6

)+ c1

(η + 1

3sin

(θπ

6

))≤ 0.

(31)

This cutoff region in the stress state plane (η, θ) isshown in Fig. 18 with a shaded area. In Fig. 18, anarbitrary value c1 = 0.57 is assumed.

The above property of the cutoff region can be justi-fied by revoking the physical concept of die-cone in thefriction force, see Fig. 19. When the stress triaxiality η

is less than a certain value, the vector of total force willbe contained within the die-cone, under which con-dition slips will not occur during the material defor-mation. This gives an interesting physical interpreta-tion of cutoff region using the Mohr–Coulomb criteriontogether with the concept of die-cone. However, “unre-alistically” large values of the parameter c1 would beneeded in Eq. 31. Challenging tests should be designedand carried out in the future to explore this idea.

7 Crack directions in the plane stress and uni-axialstress

An important feature of M–C criterion is that it not onlytells when a material point fails but also in which direc-tion the material element cracks. In the general case,the crack plane is defined by Eq. 14. For the plane stresscondition, it is convenient to consider a 2D represen-tation of fracture planes, see Fig. 20. This picture wasconstructed taking c1 = 0.1 as an example. The asym-

metric hexagon represents the locus of crack directions.For example, in the first quadrant, there will be slantfracture through thickness, but in the planar view, thecrack forms perpendicular to the maximum principalstress. In the second and fourth quadrant, the throughthickness crack is normal to the middle surface of asheet, but follows an inclined planar direction. It isinteresting to note that the planar vector of the crackdirection is perpendicular to unit normal vector of theM–C fracture surface.

To simulate the crack propagation, the currently usedtechnique in finite element simulation is based either onelement deletion or element split along element edges.Neither of these two methods correctly simulates theboundary condition after a crack initiates, which is criti-cal to predict the crack propagation, especially the slantfractures. It is suggested touseanextendedelement splittechnique in the spirit of an enriched FE shell elementmodel suggested by Belytschko (for example, Areiasand Belytschko (2005)). The direction of local elementsplit surface will then follow from the M–C criterion.This is a technique requiring the joint efforts of bothphysical fracture modeling and finite element coding.

8 Experiment calibration and verification

8.1 A form of Mohr–Coulomb fracture locusproposed for application

Parameter study in Sect. 6 provides an insight under-standing of the parameters in the Mohr–Coulomb frac-ture locus (Eq. 24). Since the effects of cη and c1 aresimilar in term of stress triaxiality, the term of pressuredependence on yield surface is neglected for simplicity.The following form of fracture locus is suggested forapplication.

ε f ={

A

c2

[csθ +

√3

2 − √3

(caxθ − cs

θ

) (sec

(θπ

6

)− 1

)]

⎡⎣√

1 + c21

3cos

(θπ

6

)+ c1

(η + 1

3sin

(θπ

6

))⎤⎦⎫⎬⎭

− 1n

,

(32)

where caxθ is defined in Eq. 22. There are a total of six

parameters (A, n, c1, c2, csθ , cc

θ ) that need to be found.The first two parameters, A and n, are parameters ofmaterial strain hardening, which can be calibrated fromcurve fitting of the stress–strain curve using power

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Fig. 20 According toMohr–Coulomb fracturecriterion, a crack directionlocus is proposed for planestress conditions

Fig. 21 Specimens used byBao to calibrate the fracturelocus for the wide range ofstress triaxialtiy, see Table 1for details

function. The two basic Mohr–Coulomb parameters,c1 and c2, need to be determined from material testscarried up to fracture. There are two additional param-eters, cs

θ and ccθ . The parameter cs

θ controls the amountof Lode angle dependence, and the parameter cc

θ affectsthe asymmetry of the fracture locus. Their default val-ues are 1.0 if no additional test data are available. Itshould be noted that these two parameters, cs

θ and ccθ ,

could be determined from careful plasticity tests anddetermination of the shape of the yield surface. Alterna-tively, one can leave those parameters undetermined in

the plasticity model and determine all four parameters(c1, c2, cs

θ , ccθ ) from best fit of the fracture data points.

In the following two subsections, test data of twoexample materials are used to calibrate and verifythe proposed Mohr–Coulomb fracture locus. One setof material data is taken from Bao and Wierzbick-i’s tests on aluminum alloy 2024-T351 (Bao 2003;Bao and Wierzbicki 2004; Wierzbicki et al. 2005b).The second set of material data on a TRIP steel sheet(RA-K40/70) from ThyssenKrupp was obtained by thepresent authors.

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8.2 Example 1, Aluminum alloy 2024-T351

Bao and Wierzbicki (Bao 2003; Bao and Wierzbicki2004; Wierzbicki et al. 2005b) performed a series oftests to calibrate the fracture locus of 2024-T351 alu-minum alloy in a wide range of stress triaxiality, seeFig. 21. All the data from the 15 types of tests doneby Bao and Wierzbicki are re-processed to calculatethe Lode angle parameter θ , which was not introducedby Bao and Wierzbicki. Since in general the two stressstate parameters are variable during the loading pro-cess, average values are used, according to the defini-tions given in Sect. 9. A list of the stress state param-eters, η and θ , and the equivalent strain to fracture ε f

is shown in Table 1. The corresponding specimens ofall the data points are labeled in Fig. 21. These tests,except test No.9 (where there were no obvious observedcracks), will be revisited using the new form of theMohr–Coulomb criterion.

8.2.1 Plasticity

The parameters of the basic hardening curve, A = 740MPa, n = 0.15, were found from the best fit of thestress–strain curve corresponding to the upsetting test(No. 6). In order to quantify the effect of Lode angleon plasticity (parameter cs

θ and ccθ ), axial-symmetric

tension (test No. 1), axial-symmetric compression (testNo. 6) and plane strain (test No. 4) experiments shouldbe compared with each other, as described by Bai andWierzbicki (2008). Such a comparison was performedby Bao (2003) showing that there is a very little Lodeangle effect on plasticity. Therefore, for the purposeof the present analysis, the two parameters are set tounity (cs

θ = ccθ = 1). This completes determination of

plasticity constants.

8.2.2 Fracture

The M–C fracture criterion involves two basic mate-rial constants, so two tests are needed to calibrate thiscriterion. For that purpose, fracture test No. 6 and 10of Table 1 are used. Substituting the experimental data(ηav, θav and ε f ) into Eq. 32, a set of two nonlinear alge-braic equations for c1 and c2 are obtained. The solu-tion of this system yields c1 = 0.0345 and c2 = 338.6MPa. Now all six parameters in Eq. 32 have been deter-mined. The plot of the resulting fracture locus is shown

Fig. 22 3D geometric representation of Mohr–Coulomb fracturelocus for 2024-T351 aluminum alloy. (A = 740 MPa, n = 0.15,c1 = 0.0345, c2 = 338.6 MPa, cs

θ = ccθ = 1.0)

in Fig. 22. Tests No. 6 and 10, used for fracture calibra-tion, are displayed as diamonds. The “half tube” surfacepasses exactly through those two points. Points corre-sponding to the remaining twelve tests are denoted bycircles. They appear to be very close to the 3D surface,except for three points corresponding to unnotched andnotched round bars. It has been shown by many investi-gators that the fracture in round bars in tension involvesa mechanism of void growth and linkage. Clearly thisis not well captured by the M–C fracture criterion. Inother words, the M–C model describes shear type offracture well but not fracture produced by void growthand linkage. It can be concluded that the M–C modelis able to capture, with an engineering accuracy, shearfracture of 2024-T351 aluminum in a wide range ofstress triaxiality and Lode angle parameter.

Most of the tests done by Bao and Wierzbicki werein plane stress condition, except for three round barstensile tests (No. 1, 2 and 3). It follows from Eq. 9 thatthe plane stress data points lie on the “s”-shaped curveof plane stress condition. By substituting Eq. 9 intoMohr–Coulomb criterion, Eq. 32, the fracture locus ofplane stress condition can be plotted on the plane ofequivalent strain to fracture and the stress triaxiality(refer to Fig. 23). One can see that the fracture locusof plane stress consists of three half-cycles. This phe-nomenon was first revealed by Wierzbicki and Xue(Wierzbicki and Xue 2005) from a symmetric 3D frac-ture locus. The experimental data points used for cali-bration are marked in Fig. 23 by full circles. It is foundthat the calibrated Mohr–Coulomb criterion predictsthe trends of experimental results very well. At the same

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Table 1 A summary of Baoand Wierzbicki’s test resultson 2024-T351 aluminum,[experimental data after Bao(2003), Wierzbicki et al.(2005b)]

No. Specimen description ηav θav ε f

1 Smooth round bar, tension 0.4014 0.9992 0.4687

2 Round large notched bar, tension 0.6264 0.9992 0.2830

3 Round small notched bar, tension 0.9274 0.9984 0.1665

4 Flat grooved, tension 0.6030 0.0754 0.2100

5 Cylinder(d◦/h◦ = 0.5), compression −0.2780 −0.8215 0.4505




9 Round notched, compression −0.2476 −0.7141 0.6217

10 Simple shear 0.0124 0.0355 0.2107

11 Combination of shear and tension 0.1173 0.3381 0.2613

12 Plate with a circular hole 0.3431 0.9661 0.3099

13 Dog-bone specimen, tension 0.3570 0.9182 0.4798

14 Pipe, tension 0.3557 0.9286 0.3255

15 Solid square bar, tension 0.3687 0.9992 0.3551

-2/3 -1/3 0 1/3 2/3 10

0.2

0.4

0.6

0.8

1

η

εf

Plastic plane strain or shear

Best fit of Rice-Tracey

Plane stress

Fig. 23 Mohr–Coulomb fracture locus for 2024-T351 on theplane of equivalent strain to fracture and stress triaxiality. Onlythe test data under plane stress condition are plotted

time the Rice–Tracey fracture model (Rice and Tracey1969) can not predict the trend of plane stress fracture.

8.3 Example 2: TRIP RA-K40/70 steel sheet

The second example material is a TRIP steel sheet(RA-K40/70, or called TRIP690) provided by Thys-senKrupp Steel. The material comes in sheets, so alltests correspond to the plane stress condition. Thematerial shows small amount of anisotropy in bothplasticity and fracture properties. Dog-bone speci-mens were cut from three directions (0, 45 and 90degrees) and tested under simple tension. The initialyield stresses are σ0 = 446.4 MPa, σ45 = 457.7 MPa,and σ90 = 455.5 Mpa. Like the small difference ininitial yielding, the subsequence material strain hard-ening in three directions are also very similar. Themeasured r-ratios in three directions are r0 = 0.79,r45 = 0.97, and r90 = 1.02, which shows some mate-rial anisotropy in the plastic flow. These tension testswere carried all way to fracture, the measurement ofarea reductions at fracture cross-sections gave esti-mations of the equivalent fracture strains, which are

Table 2 A summary of testresults on a TRIP steel

No. Specimen description ηav θav ε f

1 Dog-bone, tension 0.379 1.0 0.751

2 Flat specimen with cutouts, tension 0.472 0.4960 0.394

3 Disk specimen, equi-biaxial tension 0.667 −0.921 0.950

4 Butterfly specimen1, tension 0.577 0.0 0.460

5 Butterfly specimen2, simple shear 0.0 0.0 0.645

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Fig. 24 Mohr–Coulomb fracture locus for a TRIP steel sheeton the plane of equivalent strain to fracture and stress triaxial-ity. (A = 1275.9 MPa, n = 0.2655, c1 = 0.12, c2 = 720 MPa,csθ = 1.095, cc

θ = 1.0)

ε f,0 = 0.703, ε f,45 = 0.754, and ε f,90 = 0.744.One can see that the difference of fracture propertiesin three directions is also small. In this paper, mate-rials are assumed to be isotropic. For simplicity, theplasticity of the TRIP steel sheet are described by anpower hardening curve, σ = Aεn = 1275.9ε0.2655

MPa.In total, five types of specimens are used for fracture

calibration: dog-bone specimen, flat specimen with cut-outs, punch test, butterfly specimen in tension and but-terfly specimen in simple shear. Similarly, the averagevalue of two stress state parameters (η, θ ) are calculatedfrom the numerical simulation using shell elements. Adetail experimental and simulation results of those testsare described in the reference (Bai et al. 2008). A sum-mary of the stress state parameters, η and θ , and theequivalent strain to fracture ε f is shown in Table 2.The corresponding specimens of the data points arelabeled in Fig. 24.

The first two parameters in the fracture locus (Eq. 32)are determined from the material plasticity, A =1275.9 MPa and n = 0.2655. The remaining fourparameters (c1, c2, cs

θ , ccθ ) are found from the best fit of

equivalent strain to fracture for all five fracture tests.An approach of parameter optimization using Matlabcode was described in reference (Bai and Wierzbicki2008). Utilizing the same approach, the remaining fourparameters are found as: c1 = 0.12, c2 = 720 MPa,csθ = 1.095, cc

θ = 1.0. A comparison of test results withthe calibrated M–C criterion is shown in Fig. 24. Again,the M–C criterion was able to capture all features of thefracture locus of metal sheets.

9 Damage evolution rule

Besides the fracture locus, discussed extensively in thepreceding section, the rule of damage evolution is anintegral part of the fracture predictive technique. Thefracture locus is defined under monotonic loading con-ditions, so a linear incremental relationship is assumedhere between the damage indicator, D, and the equiv-alent plastic strain εp,

D(εp) =

εp∫

0

d εp

f(η, θ) , (33)

where the stress direction parameters,η(εp), and θ (εp),are unique functions of the equivalent plastic strain. Amaterial element is considered to fail when the limit ofductility is reached, εp = ε f , so that D(ε f ) = Dc = 1.In the limiting case (for example proportional loading),when the parameters (η, θ) are constant over the load-ing cycle, Eq. 33 can be integrated to give

ε f = f(η, θ) = ε f

(η, θ), (34)

which reduces to the 3D fracture locus ε f(η, θ). The

best tests for calibration are the ones in which the stressparameters η and θ are held constant. This is the casewith the equi-biaxial punch test (No. 3 in Fig. 24), theshear test (No. 5), and to a certain extent with the flatnotched specimens (No. 2) (Beese et al. 2009). In thegeneral case of variable stress parameters, the integralrepresentation of the fracture criterion must be used.

The functions, η(εp) and θ (εp), are known fromthe numerical simulation of the tests, and the equiv-alent strain to fracture ε f is determined from inversemethod of mapping the measured displacement to frac-ture into the calculated strain to fracture. Alternatively,fracture strains can be measured directly by an opti-cal measurement system and the stress invariants canbe calculated from a suitable plasticity model (Dunandand Mohr 2009). Then the unknown fracture parame-ters c1, c2, cs

θ , and ccθ can be determined by minimizing

the residue R defined by Eq. 35,

R =ε f∫

0

d εp

f(η, θ) − 1. (35)

Yet, the first method in which the equivalent strain tofracture is found directly from measurements and thevalues of stress triaxiality and Lode angle parameterare constant and known beforehand is by far superior.

123


In these cases, fracture calibration bypasses the needfor a plasticity model which may introduce additionalerrors.

A linear incremental dependence of the damagefunction D

(εp)

on the equivalent plastic strain (Eq. 33)was shown to work well for monotonic loading. In thecase of reverse straining or more complicated loadingpaths, a nonlinear incremental rule must be considered,as proposed by Bai (2008).

10 Discussion and conclusion

In this paper, the transformation formulas between theprincipal stresses and the stress state parameters (thestress triaxiality η and the Lode angle parameter θ ) arederived. For monotonic loading conditions, the Mohr–Coulomb criterion is transformed from a local repre-sentation in terms of a shear stress and a normal stress,to the mixed strain-stress representation of (ε f , η, θ ).The corresponding fracture locus of the Mohr–Cou-lomb criterion is shown to be described by a mono-tonically decreasing function of the stress triaxialitycoupled with an asymmetric function of Lode angleparameter. A parametric study on the new form ofthe Mohr–Coulomb criterion was performed and theresults are summarized as follows.

• Increasing the friction parameter c1 increases boththe dependence of fracture locus on pressure andthe asymmetry of the fracture locus.

• Increasing the shear resistance parameter c2 shiftsthe fracture locus upward without changing itsshape.

• Increasing the amplitude of the power hardeninglaw, parameter A, shifts the fracture locus down, butits shape remains the same. In other words, increas-ing the material’s strength will decrease its ductility.

• Increasing the power exponent n shifts the frac-ture locus upward, but decreases both the pressuredependence and the Lode angle dependence of thefracture locus. In other words, less strain hardeningmaterials will have more pressure and Lode angledependence on fracture.

• Increasing the parameter cη will decrease the pres-sure dependence of the fracture locus, but leave noeffect on the shape of the fracture locus.

• Decreasing the parameter csθ will increase the whole

height of the fracture locus, and decrease the effectof Lode angle on the fracture locus. In other words,

if a material exhibits more Lode dependence onplasticity, then it has less Lode dependence on frac-ture, and vice versa.

• Decreasing the parameter ccθ will increase the frac-

ture strains under axial symmetric compressionconditions (θ = −1), which controls the asymme-try of the fracture locus.

• There exists a cutoff region on the stress state plane(η, θ), see Eq. 31, where fracture will not occur.

Bao and Wierzbicki’s experimental results on 2024-T351 aluminum alloy are revisited using the new formof the Mohr–Coulomb criterion. Using two types oftest to calibrate the basic Mohr–Coulomb parameters,the model predicts the remaining nine tests with goodaccuracy, especially for plane stress fracture. While theMohr–Coulomb criterion predicts most of the shearingdominated fracture well, there are still some limitationsof this fracture criterion. For example, it does not pro-vide a good prediction of the fracture of round bars intension, which satisfy the axial symmetry condition. Atthe same time, fracture in upsetting tests occurs at theequatorial region of the outer surface. This is a planestress fracture, and it is predicted well by the M–Ccriterion.

As a second example, the test results on one type ofadvanced high strength steel, TRIP RA-K40/70 steelsheet, are provided to demonstrate the applicabilityof the proposed M–C criterion to metal sheets. TheTRIP steel sheet shows small amount of anisotropy. Itshould be noted that the current M–C criterion assumesmaterial isotropy, and it should be further extended todescribe the fracture anisotropy, which is the on-goingresearch.

The M–C criterion predicts not only crack initia-tion sites but also crack directions. Providing a givenmaterial element crack direction in conjunction withthe arbitrary element splitting technique can predictcrack path with greater accuracy. With the M–C cri-terion’s clear and simple physical meaning, this ductilefracture model has great potential for many engineeringapplications.

Acknowledgments Thanks are due to Dr. Yaning Li, Dr. CareyWalters, and Mr. Meng Luo for their assistance in tests on theTRIP steel, and to Professor Tony Atkins of University of Read-ing for his valuable comments. The authors gratefully appreci-ate the partial financial support from the AHSS MIT industryconsortium and the NSF/Sandia alliance program. We greatlyappreciate the Altair Company for the continuous support withthe program HyperMesh.

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