effect of triaxial state of stress

8/14/2019 effect of triaxial state of stress

1/13

International Journal of Fracture 88: 359371, 1997.

1997Kluwer Academic Publishers. Printed in the Netherlands.

On the effect of triaxial state of stress on ductility using nonlinear

CDM model

NICOLA BONORAIndustrial Engineering Dept., University of Cassino, Via Di Biasio 43, 03043, Cassino (FR), Italy;

e-mail: [email protected]

Received 28 July 1997; accepted in revised form 6 February 1998

Abstract. Ductility takes into account the material capability to plastically deform. This parameter is not only

modified by temperature but it is strongly affected by the stress triaxiality that, in the case of positive hydrostatic

stress, reduces the material strain to failure. Due to the importance of this parameter in engineering design many

attempts to predict the evolution of ductility with stress triaxiality have been done. Here, a nonlinear continuum

damage model, as proposed by the author, is used to obtain the evolution of material ductility with stress triaxiality.

The expression found relates the strain to failure in multi-axial state of stress regime only to the uniaxial strainto failure, to the damage strain threshold, to the material Poissons ratio, and, of course, to stress triaxiality. The

proposed model was successfully verified comparing the predicted evolution of material ductility with the experi-

mental data relative to several metals. The procedure for the damage parameters identification is also discussed in

details.

Keywords:Continuum damage mechanics, plasticity, triaxiality, ductile failure.

Introduction

The mechanics of ductile failure in metals is of significance in designing structure and compo-

nents against plastic collapse and fracture. In engineering design is not rare to find application,

such as metal-forming processes, where large plastic deformations need to be taken into ac-

count. Ductility characterizes the material capability to plastically deform and the knowledge

of this parameter is required in engineering design. What makes the choice of this parameter

difficult is that material ductility changes not only with temperature, that can be considered

constant in a large spectrum of application, but also it changes dramatically together with the

stress state. Then, the same material will fail at different strain levels if tested under uniaxial

or multi-axial state of stress as showed since the pioneering experimental work of von Karman

(1911), that underlined how tensile hydrostatic stress strongly reduces the material ductility

leading to premature failure.

Thus, material ductility and ductile failure process are intimately related. Under plastic

deformation, material failure occurs as a result of the microcavities nucleation and growth

process. The analytical work performed by McClintock (1968), Rice and Tracy (1969) andBudiansky et al. (1981), pointed out the exponential amplification of the growth rate of mi-

crovoids with stress triaxiality in elastic-perfectly plastic materials.

Hancock and Meckenzie (1976) and Thomson and Hancock (1984) extensively investi-

gated the dependence of the material ductility on the triaxiality state of stress. Their work

showed that an exponential decay of the ductility as a function of triaxiality as given by Rice

and Tracy (1969) is approximate and can give only a rough idea of the effective link between


2/13

360 N. Bonora

the ductility and triaxiality for an hardening material. Manjoine (1982) proposed a different

expression for the evolution of ductility, that is normalized by a not well defined tensile

elongation, with stress triaxiality as result of best fit procedure of experimental data.

In the recent years, continuum damage mechanics (CDM) come out to be a powerful tool

to approach ductile fracture related problems such as crack geometry and size effect on crack

resistance curve under large scale yielding, (Xia and Shih, 1995). Lemaitre and Chaboche

(1985) proposed a CDM model for plasticity damage where the damage variable is a linear

function of the effective accumulated plastic strain. The experimental results presented by

LeRoy et al. (1981) gave compelling evidence of the nonlinearity of damage evolution with the

effective accumulated plastic strain in metals. Recently, Bonora (1997) proposed a nonlinear

damage model that predicts very well ductile damage evolution with strain for a large class of

metals.

In the present paper, the nonlinear damage model proposed by the author is summarized

and its extension to the multi-axial state of stress is used to derive the evolution of the material

ductility as a function of stress triaxiality. The presented model is successfully verified com-

paring ductility evolution with triaxiality for several metals. It is worth to anticipate here that

the model presented needs to know only material parameters, such as uniaxial failure strain

and damage threshold strain, that can be easily measured with simple tensile test according tothe procedure described in the following sections.

The effect of the material Poissons ratio, together with the material damage parameters,

on the predicted material ductility in multi-axial state of stress is also presented and discussed.

1. Nonlinear damage model

In the framework of CDM, damage is addressed as one of the thermodynamics variables and

its evolution law is given as a general function of other state variables such as stress, plastic

strain, temperature and so on. From a general point of view, damage variable should be de-

scribed using a tensor formulation (Murakami 1987; Chaboche 1984). From the physical pointof view, damage variable indicates the progressive material deterioration due to nonreversible

deformation processes and can be expressed by the reduction of the nominal section area of a

given reference volume element (RVE) as a result of microvoids formation and growth

D(n)=1A

(n)eff

A(n)0

, (1)

where, for a given normaln,A(n)0 is the nominal section area of the RVE;A

(n)effis the effective

resisting section area that takes into account of the presence of microdefects, voids and their

mutual interactions.

Making the assumptions of isotropic damage, the variable does not depend on the directionnand the damage state can be completely characterized by a scalar quantity indicated with D.

Let us assume as valid thestrain equivalence hypothesis (Lemaitre, 1985), the strain asso-

ciated with a damage state under the applied stress is equivalent to the strain associated with

its undamaged state under the effective stress, (Simo and Ju, 1987). The deformation behavior

of the material is only affected by damage in the form of effective stress, i.e. the constitutive

equations of a damaged material are the same of the virgin material with no damage where


3/13

On the effect of triaxial state of stress 361

the stress is simply replaced by the effective stress. Thus, the following definition of damage

can be given

D=1 EeffE0

, (2)

where E0 and Eeffare the Youngs modulus of the undamaged and damaged material, re-spectively. In the framework of CDM, the existence of a damage dissipation potential FD is

assumed and, in the case of plasticity damage, the total dissipation potential is given as,

F= Fp(,R,X)+FD(Y,p,D), (3)whereYis the internal variable associated to damage andp the effective accumulated plastic

strain. Fp is the dissipation potential associated to plastic deformation that is function of the

actual stress tensor kinematic and hardening back stressX andR.If the expressions of the two potentials are known, the evolution law of the internal vari-

ables can be obtained through the normality rule.

For a isotropic hardening material we can derive the full set of constitutive equation asfollows.

Let assume strain decomposition as,

Tij= eij+pij, (4)

where elastic strain components are given by,

eij=1+

E

ij

1D

E

kk

1D ij. (5)

Standard isotropic plasticity associated with a Von Mises yield criterion leads to,

Fp(,R;D)=eq

1D R(r)y, (6)

where plastic strain rate components can be obtained as,

p

ij=Fp

ij=3

2

sij

1D1

eq, (7)

and

p

=

Fp

R =

1D, (8)

wheresijis the deviatoric part of the stress tensor, is the plastic multiplier and the dot indi-

cates time derivative. Damage equation can be obtained in the similar way from the damage

potential as,

D= FDY

. (9)


4/13

362 N. Bonora

The author (Bonora, 1997) proposed the following expression for the damage dissipation

potential,

FD=1

2

Y

S0

2 S0

1D (DcrD)(1)/

p(2+n)/n , (10)

whereDcris the critical value of the damage variable for which ductile failure occurs; is thedamage exponent that characterizes the shape of the damage evolution curve.S0 is a material

constant andn is the material hardening exponent. This choice of the damage potential leads

to the following damage kinetic evolution with the effective accumulated plastic strain,

dD= (DcrD0)1/

ln(f)ln(th)f

m

eq

(DcrD)(1)/

dp

p, (11)

where D0 is the initial damage in the material microstructure as a result of the presence of

inclusions or second phase precipitates, th andf are the threshold strain at which damage

process starts and the strain to failure in the uniaxial state of stress, respectively.

Damage is both sensitive to the shear and to the volumetric deformation energy since

microvoids are very sensitive to the hydrostatic state of stress. On these premises, Lemaitre

(1985) postulated that damage mechanism is governed by the total elastic strain energy We

decomposed in the distortion and volumetric change contributions. Assuming that damage

does not vary within the elastic range, the expression of the elastic strain energy, with the use

of (5), leads to the definition of an equivalent damage stress similarly to the equivalentstress in plasticity, by stating that deformation energy in multi-axial state of stress is equal to

that in an equivalent uniaxial state defined by, i.e.,

We = WD +Wm =

sijdDij+

mdm

=1

2 1+E sijsij1D +3 12E 2m1D , (12)that with the definition of the equivalent Mises stress can be rewritten as,

We = eq2(1D)

2

3(1+)+3(12)

m

eq

2=

2(1D) , (13)

where,

=eqf

m

eq

(14)

and

f

m

eq

= 2

3(1+)+3(12)

m

eq

2. (15)

Thus, the function f (m/eq)takes into account of the extension to triaxial state of stress

of the damage variable.


5/13


.Figure 1. Triaxiality along the minimum section of round notch tensile bar

Figure 2. Plastic strain along the minimum section of round notch tensile bar.

In the present model five material parameters needs to be identified: the threshold strain

th at which damage process starts, the failure strain fat which damage variable reaches its

critical value; the critical damage Dcr, the initial damage D0 and the damage exponent .

The identification of these parameters can be done in a uniaxial tensile test using an

appropriate hourglass shaped tensile specimen. The collocation of a small strain gauge in

the minimum specimen section, where plastic deformation is localized, monitors the varia-

tion of the material stiffness as a function of the imposed strain level performing a series

of loading-unloading ramps. The initial damage in the material D0 can be assumed equal

to zero for the virgin material. The determination of damage threshold strain needs some

attention as a result of the fact that its measure can be very scattered. The same attentionis required for the critical damage Dcr because close to failure damage variable quickly

accumulates with strain. Once the damage data as a function of strain are collected, the

damage exponent can be determined as the slope of the best fit line in the logarithmic

plane ln[(DcrD)/Dcr] ln[ln(/)/ln(/th)]. More experimental details on the procedurehow to measure damage parameters can be found in Lemaitre and Dufally (1987) and Bonora

et al. (1994).


6/13

364 N. Bonora

Figure 3. Comparison between the present model, Rice & Tracy and Lemaitres estimation for a Q1 steel tested

along LT direction, (exp. data from Thomson & Hancock, 1984).

Figure 4. Comparison between the present model, Rice & Tracy and Lemaitres estimation for a Q1 steel tested

along ST direction, (exp. data from Thomson & Hancock, 1984).

2. Triaxial state of stress and ductility

Hanckock and Meckenzie (1976) have largely investigated the effect of stress triaxiality on

failure strain in high strength steels. Their investigation was based on the Earl and Browns

(1976) analysis of round notch tensile bar specimen. This specimen geometry is peculiar

because allows to change stress triaxiality simply changing the radius of the notch and it

is characterized to have uniformly constant plastic strain on the minimum section once the

full section is yielded. In addition, stress triaxiality, that varies along the minimum section, is

load independent. In Figure 1, an example of the triaxiality distribution along the minimum

section for a round notch specimen, is shown (Bonora et al., 1992). In Figure 2 the distribution

of the equivalent plastic strain along the minimum section for different notch radii and fordifferent values of the Ludwiks power law exponent is given. Earl and Brown found that the

value of stress triaxiality on the specimen center can be given with good approximation by the

following Bridgmans solution,m

eq

max

= 13

+ln

1+ d2R

, (16)


7/13


wheredis the actual radius of the minimum cross-section and Ris the notch radius as depicted

in Figure 1. While the equivalent plastic strain along the minimum section is given by,

peq=2 ln

d0

d

, (17)

where d0 is the initial diameter of the reduced section. Hancock and Meckenzie found thatductility at failure and triaxiality are nonlinearly inversely proportional: the more stress state is

triaxial the less the strain at failure is. This result was in part anticipated by Rice and Tracy that

analytically found an exponential decrease of strain at failure with the increase of triaxiality,

f= exp

32

m

eq

. (18)

McClintock (1968), considering the growth of elliptical holes, found an analogous relation

as

f= (1n)ln(l0/b0)

sinh 32 (1n) a+beq + 34 b+aeq , (19)

where l0 andb0 identify the initial void and cell size; a,b are the principal stresses in the

remote matrix along the void axes directions and n is the material hardening exponent as

defined in the Ludwiks power-law,

eq=0(peq)n, (20)where 0 is the uniaxial yield stress. However, these expressions match the experimental

observations in a very approximate way as it is shown in Figures 3 and 4 for a Q1 steel

tested along the LT and ST directions.

In the framework of the continuum damage model proposed material ductility, neglecting

the elastic strain, is expressed by the effective accumulated plastic strain p that is defined asthe equivalent plastic strain effectively accumulated in the generic multi-axial state of stress

together with damage presence. In the uniaxial case pfdefinition coincides with f.

Using continuum damage mechanics, a damage model should be able to determine the

evolution law for the failure strain as a function of stress triaxiality only on the base of uniaxial

experimental data. The damage kinetic evolution law given in (11) can be integrated in the

uniaxial case(f(m/eq)=1)over the strain range[th, f]leading to

D=D0+(DcrD0)

1

1 ln(/th)ln(f/th)

(21)

and in the case of proportional loading(f(m/eq)=const)where we get,

D=D0+(DcrD0)

1 1 ln(p/pth)ln(f/th)

f meq

. (22)

Dividing Equation (21) by (22) we get the following relation between strainspf

p0

=

f

th

1/f (m/eq). (23)


8/13

366 N. Bonora

Figure 5. Comparison between the model proposed and experimental data relative to Marrel steel, (exp. data from

Thomson & Hancock, 1984).

Figure 6. Comparison between the model proposed and experimental data relative to a C-Mn steel, (exp, data

from Tai, 1990).

This relation states that effective strain to failure in multi-axial state of stress is function of

the uniaxial failure strain, stress triaxiality and of the Poissons ratio of the material. Equation

(23) needs the knowledge of the evolution of the plastic threshold strain p0, as a function

of stress triaxiality in order to be used. According to the Thomson and Hancocks (1984)

observations, the equivalent threshold strain p0 can be taken constant and equal to th. Thus,

(23) leads to the following evolution law for strain to failure in multi-axial state of stress,

pf= th

f

th

1/f (m/eq). (24)

The validity of this relation has been verified with experimental data of six different ma-terials. In Figures 3 and 4, the comparison between the experimental data relative to Q1 steel

tested along the LT and ST directions and the present model is given together with the Rice

and Tracy and Lemaitres linear damage model. These pictures show a very good agreement

between the model proposed and the experimental data and underlines differences with the

other models. Even though differences between the Rice and Tracy and CDM model do not

surprise, because the two models have been obtained in different frameworks, the differences


9/13


Figure 7. Comparison between the model proposed and experimental data relative to a welded steel 6013, (exp.

data from Tai, 1990).

Figure 8. Comparison between the model proposed and experimental data relative to a welded steel 6015, (exp.data from Tai, 1990).

Figure 9. Comparison between the present model and experimental data relative to a HY130 steel along the two

directions LT and ST, (exp. data from Hancock & Meckenzie, 1976).


10/13

368 N. Bonora

Figure 10. Damage threshold strain measurement for a Swedish pure steel in different triaxiality states of stress,

(data from Thomson & Hancock, 1984).

of the present model with the Lemaitres one put in evidence the importance of the nonlinear

description for damage process.

In Figures 59 the comparison of the present model with the experimental data relative

to Marrel steel, to a C-Mn steel, two welded steels, 6013 and 6015, and to HY130 steel are

given, respectively. In all the cases examined the model presented is always in a very good

agreement with experimental measurements. In Table 1, the damage parameters f, th and

the Poisson ratios for each material are also summarized.

3. Triaxiality and damage threshold strain

Damage threshold strain indicates the strain level at which damage process starts to take place.

At this stage voids nucleation and growth are large enough to affect the material stiffness. In

the framework of the proposed damage model, the knowledge of th in very important because

this parameter affects significantly damage evolution law. As already seen for failure strain,

th should be sensible to stress triaxiality in the same way. In the Lemaitres linear model

the assumption that triaxiality acts in the same way on pf andp0, is made, Lemaitre, 1985).

Thomson and Hancock (1984) have experimentally investigated the variation ofp0 with the

increase of stress triaxiality as depicted in Figure 10 for a Swedish pure iron. They underlined

two main results

(a) threshold strain is scarcely sensitive to stress triaxiality,

(b) experimental data are largely scattered for low triaxiality. Experimental data scatter at

low triaxiality can be explained with the observation that the less stress triaxiality is the

higher is the effect in damage localization due to inclusion shape, size and distribution.


11/13


Table 1. Material damage parameters

Material Uniaxial failure Threshold Poissons ratio

strainf strainth

Marrel 1.04 0.05 0.30

C-Mn steel 1.13 0.21 0.33

HY130 (LT) 0.97 0.05 0.30

HY130 (TS) 0.54 0.002 0.30

weld 6013 1.04 0.05 0.33

weld 6015 1.41 0.05 0.37

Q1 steel (LT) 1.05 0.05 0.30

Q1 steel (TS) 0.67 0.02 0.30

Figure 11(a). f effect on the triaxiality-ductility

curve Equation (22).

Figure 11(b). f effect on the triaxiality-ductility

curve Equation (22).

Figure 11(c).Poissons ratio effect on the triaxiality-

ductility curve Equation (22).

The verification of this hypothesis can be done using the results given in Figures 59. In

fact, the effective strain at failure evolution curve given in (24) leads to a very good comparison

with experimental data if the appropriate values of uniaxial thand f are provided.

Variation offleads to curves that shift on the left side of the diagram with consequent

shifting of the point that represent the uniaxial case. A variation of the p0 (=th)produces ashift to the right of the upper part of the curve that correspond to high triaxiality: in this case


12/13

370 N. Bonora

failure strain in multi-axial stress state is overestimated. In Figures 11(ac), the modification

of (24) due to the variation ofth,fand the Poissons ratio, is summarized.

According to this, threshold strain can be simply evaluated performing few tests only at

high triaxial stress state gaining two conditions: firstly, a pivoting point for (24) is found, sec-

ondly, experimental scatter in the measurements is drastically reduced avoiding the tests rep-

etition or the examination of the whole specimen minimum section as described in (Thomson

and Hancock, 1984).

Figure 12. Specimen redution for damage measurement in triaxial state of stress.

Threshold strain test measurements in multi-axial state of stress can be performed as sug-

gested by Bonora et al., (1996) using round notch tensile bar (RNB(T)) multiple specimen

technique. This method consists in pre-straining a number of RNB(T) specimens. The strain

level along the minimum section can be monitored according to (17). The stress triaxiality

along the minimum section can be calculated with finite element method once the mesh

has been calibrated comparing the load vs diameter reduction (P ) response. Once aspecimen has been prestrained, it has to be reduced to standard rectangular flat tensile spec-

imen as depicted in Figure 12 using electro-discharge cutting technique. Thus, damage canbe measured positioning a strain gauge at the center of the old minimum section, where the

triaxiality was constant during straining, and loading the specimen in the elastic range. Even

if this technique can be very expensive, it allows to monitor damage evolution in the triaxial

state of stress and, if a replication technique is used, it can be used to control the effective

microscopical modification at each damage stage.

4. Conclusions

In the present paper, ductile fracture has been approached using continuum damage concepts.

The nonlinear damage model, as proposed by the author, was reviewed and it has been used to

discuss, on the basis of experimental data, the effect of stress triaxiality on damage parameters.The predicted values of the effective strain to failure have been successfully compared with

experimental data relative to six different steels. The comparison with other models, such as

the Rice and Tracys approximated expression for cavities growth and the Lemaitres linear

damage model have been also presented and the differences have been highlighted.

The possibility to successfully derive without any additional hypothesis from a CDM

model the evolution of material ductility with stress triaxiality is an additional validation of


13/13


the damage model proposed. The experimental procedures to identify all the five damage

parameters necessary has been discussed in details.

References

Bonora, N. (1997). A nonlinear CDM model for ductile failure. Engineering Fracture Mechanics58(1-2), 1128.

Bonora, N., Marchetti, M., Milella, P.P., Maricchiolo, C. and Pini A. (1992). Ductile fracture criteria: An ap-

plication of local approach based on cavity growth theory in Pressure Vessel Fracture, Fatigue, and Life

Management(edited by S. Bhandari, P.P. Milella and W.E. Pennell) Book no. G00668, PVP Vol. 233, ASME

New York, 111115.

Bonora, N., Cavallini, M., Iacovello, F. and Marchetti, M. (1994). Crack initiation in Al-Li alloy using contin-

uum damage mechanics, inLocalized Damage III Computer-Aided Assessment and Control . (Edited by M.H.

Aliabadi, A. Carpinteri, S. Kalisky and D.J. Cartwright), Computational Mechanics Publication, Southampton

Boston, 657665.

Bonora, N., Gentile, D. and Iacoviello, F. (1996). Triaxiality and ductile rupture in round notch tensile bar (in

Italian),Proceedings of XII Italian Group of Fracture National Meeting , Parma, June 1996.

Budiansky, B., Hutchinson, J.W. and Slutsky, S. (1981). InMechanics of Solids(The Rodney Hill 60th Anniversary

Volume). (Edited by H.G. Hopkins and M.J. Sewell). Pergamon Press, Oxford, 1315.

Chaboche, J.L. (1984). Anisotropic creep damage in the framework of the continuum damage mechanics. Nuclear

Engineering and Design79, 309319.Earl, J.C. and Brown, D.K. (1976). Distribution of stress and plastic strain in circumferentially notched tension

specimens.Engineering Fracture Mechanics 8, 599611.

Hancock, J.W. and Meckenzie, A.C. (1976). On the mechanisms of ductile failure in high strength steels subjected

to multi-axial stress-states.Journal of the Mechanics and Physics of Solids 24, 147169.

Lemaitre, J. (1985). A continuous damage mechanics model for ductile fracture. Journal of Engineering Material

and Technology107, 8389.

Lemaitre, J. and Chaboche, J.M. (1985).Mechanics of Solids Materials, Cambridge Academic Press, (1985).

Lemaitre, J. and Dufally, J. (1987). Damage measurements, Engineering Fracture Mechanics 28(5/6), 643881.

Le Roy, G., Embury, J.D., Edward, G. and Ashby, M.F. (1981). A model of ductile fracture based on the nucleation

and growth of voids. Acta Metallurgica29, 15091522.

Manjoine, M.J. (1982). Creep-rupture behavior of weldments. Welding Research Supplement, 50s57s.

McClintock, F.A. (1968). A criterion for ductile fracture by the growth of holes.Journal of Applied Mechanics35,

363371.

Murakami, S. (1987). Anisotropic aspects of material damage and application of continuum damage mechanics.

CISM Courses and Lectures no. 295, (Edited by D. Krajcinovic and J. Lemaitre) Springer-Verlag, Wien-New

York, 91133.

Rice, J.R. and Tracy, D.M. (1969). On ductile-enlargement of voids in triaxial stress fields. Journal of Mechanics

and Physics of Solids 17, 210217.

Simo, J.C. and Ju, J.W. (1987). Stress and strain based continuum damage models I formulations. International

Journal of Solids and Structures23, 841869.

Tai, H.W. (1990). Plastic damage and ductile fracture in mild steels.Engineering Fracture Mechanics36, 853880.

Thomson, R.D. and Hancock, J.W. (1984). Ductile failure by void nucleation, growth and coalescence. Interna-

tional Journal of Fracture26, 99112.

Von Karman, T. (1991). Z. Vereins Deutscher Ing.,55, 17491757.

Xia, L.C. and Shih, C.F. (1995). Ductile crack growth-II. Void nucleation and geometry effects in macroscopic

fracture behavior.Journal of the Mechanics and Physics of Solids 43, 233259.

effect of triaxial state of stress

Documents