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Low-cycle fatigue properties of steel 42CrMo4

R. Kunc , I. Prebil

Faculty of Mechanical Engineering, Askerceva c. 6, SI-1000 Ljubljana, Slovenia

Received 22 November 2001; received in revised form 25 June 2002

Abstract

In structures subjected to heavy cyclic loading, fatigue damage of material during cyclic plasticity is one of the most frequentfailure mechanisms. For determining the low-cycle fatigue lifetime, it is essential to know the elasto-plastical response of the material

to a cyclic load, since the accumulated plastic strain has a direct influence on the lifetime of the material. The elasto-plastic response

of the material under a cyclic load is described here by using the constituti ve macroscopic model of small deformations. This model

considers isotropic and kinematic hardening or softening in connection with material damage. The procedure of determining the

material parameters by the described model is illustrated for normalised and tempered states of a low-alloy steel 42CrMo4 (ISO 683/

1).

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Low cycle fatigue; Cyclic plasticity; Material parameters; Damage mechanics; Kinematic hardening; Isotropic hardening

1. Introduction

The fatigue life of highly loaded engineering compo-

nents is influenced by repeated plastic deformation,

which causes the accumulation of plastic strain in

material. Shortening of fatigue life of such components

can be estimated by means of non-linear response of

material to a cyclic load. A plethora of numerical

models for describing material response have been

developed e.g. [1

/9]. The microscopic fatigue processeswithin the material (dislocations, sliding systems of the

principal sliding planes etc.) are described by means of

models of Bauschinger effect, cyclic hardening or soft-

ening, ratchetting and mean stress relaxation. There are

also several theories that have been developed for the

purpose of determining the failure moment of a material

under plastic load. The theory of continuum damage

mechanics attempts to compose several micromechani-

cal damage mechanisms into a single mechanisms e.g.

[1,3,7,9]. The damage parameter of a material and,

respectively, the continuum damage theory is in this case

included into the constitutive equation and thereby

influences the elasto-plastical response of the material.

An alternative to the damage mechanics of continuum is

a series of damage criteria that have no influence on the

constitutive equation e.g. [10].

To simulate the non-linear cyclic response of thematerial, we have decided to use the material model,

which unites non-linear kinematic hardening, isotropic

hardening or softening and simplified damage me-

chanics of continuum, where we have used the same

damage evolution for negative and positive stresses[1,4].

The decision to use such a model was based upon the

results of own experimental research of a low-alloy steel

42CrMo4 (ISO 683/1), which show the influence of

material damage on cyclic strain/stress response of the

material, as well as the results of experiments and

numerical simulation of material response [4] and the

quickest possible*/and thus cheap*/procedure of de-

Corresponding author. Tel.:/386-1-4771-128; fax: /386-1-2518-

567

E-mail addresses: [email protected] (R. Kunc ),

[email protected] (I. Prebil).

Materials Science and Engineering A345 (2003) 278/285

www.elsevier.com/locate/msea

0921-5093/02/$ - see front matter# 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 4 6 4 - 1
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]

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termining the material properties. The article demon-

strates the procedure of determining all the material

parameters of the steel 42CrMo4 (ISO 683/1) in normal-

ised state with hardness of 195 HV and tempered state

with hardness of 462 HV, by using the described

material model.

2. Modelling of cyclic plastification and damage

2.1. The mechanical principle

The existence of microscopic cracks and defects the

size of crystal grains is in macroscopic mechanics

referred to as material damage. The nucleation process,

growth and merging ofvoids that represent initiation of

macrocrack and induce progressive weakening of mate-

rial toughness and stiffness is referred to as damage

evolution. The state of damage in the material, D , istheoretically determined by the common influence of

size and configuration of microcracks and microvoids.

The actual macroscopic stress within the damaged

material s is determined by assuming that the nominal

cross-section A is reduced by the size of the damaged

area AD [1,3,7,10,11].

D(M; n; x)dADx

dA[ s

F

AAD

F=A

1AD=A

s

1D: (1)

The material damage concept is built into the

constitutive model of small deformation oij, which are

split into their elastic and their plastic part:

oijoeijo

pij; (2)

where the relationship between elastic stresssij strainoij,

damage D and elasticity modulus tensor Lijkl is deter-

mined by:

sij (1D)Lijkloekl: (3)

The elasticity modulus tensor is determined by

Youngs modulus Eand Poissons ratio n .

Lijkl E

1 n

1

2(dikdjldildjk)

n

1 2ndijdkl

: (4)

The rate of plastic strain is derived from the normality

rule:

opij l

@f

@sij; (5)

where l is the plastic multiplier, derived from the

consistency condition f0: The stress potential f [1]is a function of stress tensor sij, the components of

kinematics (Xij) and isotropic (R ) hardening and the

material damage D . For an isothermal state dT/dt/0,

the rheological model and the evolution equation of the

stress potential is determined by[1,4]:

f seq(Rk)0; (6)

seq ffiffiffiffiffiffiffiffiffiffiffiffiffi3

2 sijsijs

; sij

sij

1

3 dijsk;

sij sij

1DXij:

(7)

Taking the derivative ofEq. (5)yields:

f@f

@sijsij

@f

@XijXij

@f

@RR

@f

@DD; (8)

where:

@f

@sij

3

2

sij

(1D)seq;

@f

@R1; (9)

@f

@XDij

3

2

sij

seq;

@f

@D

3

2

sijsij

(1D)seq:

2.1.1. The isotropical model of hardening/softening

If the yield surface is enlarged or reduced during a

cycle loading process, the material is subject to cyclic

hardening or softening. The evolution of a yield surface

is described by scalars R and k. R represents the

variable that describes isotropic hardening or softening

stress of the material, while k represents the size of the

elastic area. The initial values for the size of the yield

surface are determining by using the criterionk/syand

R/0, wheresyrepresents the yield stress. The evolution

equation for isotropic hardening or softeningR has the

form:[1]

Rb(RR)l; (10)

where b represents the material parameter, which

determines the level of isotropic hardening or softening,

while the parameter R determines the boundary of

isotropic cyclic hardening or softening.

2.1.2. The kinematic hardening model

Kinematic hardening is essential for the description of

cyclic effect as ratchetting and mean stress relaxation as

well as the Bauschinger effect. It is described by the back

stress tensor Xij that determines the centre of the yield

surface in stress space and is taken to be the sum of three

non-dependent contribution:

XijX3n1

X(n)ij : (11)

To determine the values of each of the three-tensor

components Xij(n ), the evolution equations proposed by

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Armstrong, Frederick and Chaboche [12,14/18] have

been used. The boundary value of hardening is deter-

mined by the kinematic hardening coefficient X(n ),

whilst the value of plastic extension at which the

respective components Xij(n ) reach their boundary value

is determined by the kinematic hardening level g(n ). The

influence of the rate of reduction of the mean stressvalue is controlled by the Ohno/Wang material para-

meter mn .

X(n)ij

2

3g(n)X(n)

(1D)opl;cij

X(n)eq

X(n)

mnX

(n)ij g

(n)l; (12)

where the effectivevalue of the stress space centre tensor

is:

X(n)eq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3X

(n)ij X

(n)ij

s : (13)

2.1.3. The continuum damage mechanics

The continuum damage mechanics deals with the

material damage during cyclic loading. A damaged

material contains microcracks and defects that reduce

the load carrying capacity of material. The problem

arises when trying to determine the initiation and

growth of the material damage during loading process.

The material damage can only be determined with

sufficient accuracy by using the failure method, where

the size of the microcrack area is measured on arepresentative volume element. Even this method fails,

though, when the distribution of cracks and defects is

non-homogenous.

Experiments on low-cycle lifetime of sub-eutectiod

steels have verified the statements[1,3/6]that measure-

ment of the increase in material damage can be done by

using the indirect experimental method of observing the

change of the modulus of elasticity:

E(1D)E: (14)

The increase of the damage is described by an

evolution equation[1,12,13], which considers a propor-

tional influence of the effective plastic deformation onthe change of damage, of the form:

Ds2eq(2=3(1 n) 3(1 2n)(skk=3seq)

2)

2SE(1D)2 pa(p);

a(p) 1; for p]pD

0; for pBpD:

(15)

The initial damage threshold pD is determined by the

accumulated plastic deformation p :

pl

1D; pDmax(p(D0)): (16)

3. The experimental work

The test pieces were manufactured of a rolled plate

made of low-alloy tempering steel 42CrMo4 (ISO 683/1;

W.Nr. 1.7225). Chemical composition of the steel is

presented inTable 1.The principal axis of the test pieces

was perpendicular to the axis of rolling. The test pieceswere shaped as to ensure minimal stress concentration in

the areas of change in diameter [19]. The steel was

normalised at 840 8C and cooled down in a stove. The

tempered pieces were heated up to a temperature of

830 8C and quenched in oil and then tempered at

400 8C for 1 h.

The first series of tests were monotonic tensile/

compressive tests to failure, the second series of tests

were 20 cyclic uniaxial tests with control of extension

amplitude for determining the cyclic parameters of the

material. The extension was measured directly on the

test piece by means of a dynamical measuring device[19]. In practice, it is sufficient to conduct eight cyclic

tests for a good simulation of material response.

To avoid the local cross-section contraction within the

probe length of the test piece, the tensile/compressive

test was conducted at constant extension amplitude. The

mechanical properties were measured at 20 8C. Heating

of test pieces was prevented by the low loading

frequency of 2 Hz.

4. Determining the material parameters

4.1. The monotonic parameters

The described damaged model contains 17 material

parameters (Table 2). The basic material parameters like

the modulus of elasticity E, the Poisson ratio n and the

yield stress sy were determined by a series of standard

monotonic uniaxial tensile/compressive tests[19].

4.2. The parameters of isotropic and kinematic hardening

The parameters that describe kinematic hardening

were determined from thes /ocyclic curve at a constantamplitude of extension Do. Different extension ampli-

tudes after stabilisation give hysteresis loops (Ns) of

respective amplitudes of plastic extension Dop and

amplitudes of stress Ds (Fig. 1). The levels g1, g2 and

g3 and the boundary values X1 , X

2 and X3 of

kinematic hardening are derived from the course of

the experimentally determined stable hysteresis loop by

means of theEq. (16)[1].

The levels of kinematic hardening g(n ) determine the

values of plastic extension at which the respective

components of the kinematic hardening tensor Xij(n )

reach their boundary value X(n ) (Fig. 1).

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Ds

2 (R

k)X1

tanh

g1

Dop

2

X2

tanh

g3

Dop

2

X3

tanh

g3

Dop

2

: (17)

Simultaneously, the boundary value of isotropic soft-

ening R and the size of the elastic area k are

determined from the experimental stable hysteresis

loop and the mutual influence of the kinematic hard-

ening. The experiment shows the trend of reduction of

the maximal and minimal stress dependant on the

number of cycles which means that the material issubject to softening (Figs. 4 and 6). The values of the

Ohno/Wang factorsm1,m2and m3are determined from

the course of experimental curves of amplitudes and

from the mean value of the stress, dependant on the

number of cycles (Fig. 4). They depend on the rate of

reduction of the mean value of stress until stabilisation

of the hysteresis loops. To determine these factors, a

non-symetrical uniaxial cyclic test with a constant non-

zero extension amplitude is used.

The isotropic hardening or softening level is deter-

mined by the parameter b . It depends on the ratio of

experimental course of stress amplitudes smaxN

at aconstant strain amplitude, from the first cycle smaxl to

stabilisation of hysteresis loop smaxS (Fig. 2). It is

described byEq. (17) [1]:

bln[1 (sNmax s

lmax)=(s

Smax s

lmax)]

2DopN: (18)

4.3. The damage parameters

The experiments have shown, that in low-cycle

loading of a sub-eutectiod steel 42CrMo4 in the begin-ning of growth of the accumulated plastic deformation

p , the modulus of elasticity can decrease as much as 5%.

The cause of this decrease is in connection of micro-

plastifications with reversible travel of dislocations and

in forming of Luders bands, and not in the growth of

material damage [1,21]. This phenomenon is soon

saturated, therefore, the further change of the modulus

of elasticity is considered as the criterion of damage

growth.

In numerical observation of damage growth, we need

to know the initial damage threshold pD, the resistive

energy of the damageSand the damage critical valueDc

[1]. The value of the pD parameter equals 0 for the

chosen steel. This is proven by the experimentally

determined curve of damage growth (Fig. 3), which

continuously rises from thevery beginning of the plastic

deformation in the material. The average value of the

damage resistive energy S for the subeutectoid steel

42CrMo4 at different heat treatments is determined

experimentally (Fig. 3), by using the Eqs. (18) and (13)

[1]:

S s

2E(1D)2 dD=

dp; p2X

N

i1

Dop

i: (19)

Theoretically, the critical damage value Dc reaches 1

in the moment of material breakdown (Nf). In the

majority of cases, though, the material breaks down

instantly, which causes a fall in the value of Dc.

Determining the actual value of the Dc parameter is

based on the experiments and the Eq. (19). The

progressive damage growth immediately before the

failure limit represents the initiation of a macro crack

and is the critical limit of Dc (Fig. 3). The average

critical values for the 42CrMo4 steel are around 0.16 in

normalised state and around 0.06 in tempered state.

Dcmax(D(p)): (20)

5. Model verification

The numerical damage model has been incorporated

into a computer programme based on the finite element

method [20]. In programme development, a common

symbolic approach was used [20], which enables the

creation of an effective, quadratically convergent im-

plicit numerical scheme, even in cases with largenumbers of evolution equations and interconnected

non-linear systems.

From the comparison of numerical and experimental

results, avery good accordance of maximal and minimal

stress in relation to the number of load cycles is obvious

(Figs. 4 and 6), as well as matching in the size and the

shape of the hysteresis loops (Figs. 5 and 7). The

verification of the isotropic softening and kinematic

hardening runs from the start of loading to stabilisation

of the hysteresis loops (Figs. 4 and 6). The isotropic

softening is compared with the course of change in

maximal and minimal stress amplitudes, while the

Table 1

Chemical composition of 42CrMo4 steel (values in %)

C Si Mn P S Cr Ni Mo Cu Al Sn

42 CrMo4 0.43 0.26 0.65 0.015 0.021 1.07 0.19 0.16 0.16 0.021 0.006


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Table 2

Material parameters of the damage model for the 42CrMo4 steel

42CrMo4 (HV) E (Mpa) sy (MPa) n b R g1 g2 g3 X

1 (Mpa) X2 (Mpa) X

3 (Mpa)

195 2.05105 180 0.3 4 80 3700 860 30 100 100 400

462 2.11105 650 0.3 5 380 15 000 2000 190 200 250 900

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kinematic hardening is compared with the course of the

mean stress value (Fig. 6). After stabilisation of the

hysteresis loops or, respectively, after the decrease in

influence of kinematic and isotropical hardening, the

main influences on the stress/strain state is by the

material damage growth. The growth of damage causes

a slight decrease of the stress amplitude all the way to

the terminal failure.

The good match between the calculated strain/stress

response and the measured values is also obvious in the

course of the plastic elongation as it depends of the

number of oscillations (Figs. 8 and 9). The matching is

good in the surge at the beginning, as well as in the

further course of the plastic elongation, which is the

base for verifying the suitability of the model in

Fig. 1. Determination of the components of hardening parameters

X(n ) and g(n ) and the limit of isotropic hardening R from the

experimentally determined stable hysteresis loop for the normalised

steel 42CrMo4 with hardness of 195 HV. A symmetrical load case with

constant extension ofDo/1.2%.

Fig. 2. Determination of isotropic hardening level parameterb for the

normalised steel 42CrMo4 with hardness of 195 HV. Various

symmetrical load cases at a constant extension.

Fig. 3. Determination of material damage parameters (pD,Dcand dD /

dp ) for the normalised steel 42CrMo4 with hardness of 195 HV. A

symmetrical load case with a constant extension ofDo/1.2%.

Fig. 4. Comparison of the course of amplitudes in relation to the

number of cycles between the measured values and the numerical

calculations for the normalised steel 42CrMo4 with hardness of 195

HV. A symetrical load case at a constant extension ofDo/1.2%.

Fig. 5. A comparison of hystreresis loops in relation to the number of

cycles between measurement and numerical calculation for the

tempered steel 42CrMo4 normalised steel 42CrMo4 with hardness of

195 HV. A symmetrical load case at a constant extension ofDo/1.2%.


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following the damage growth and the accumulated

plastic strain.

Fig. 5 shows the phenomenon of natural yield stress

for the normalised 42CrMo4 steel with a sharp yield

stress boundary and a steep fall down to the so called

lower yield stress. This phenomenon is not that obvious

for the tempered state of the same steel (Fig. 7). The first

decrease in the modulus of elasticity for the values of up

to 5% actually occurs in the area of crossing the Lu ders

elongation. This initial decrease in the modulus of

elasticity, though, is not taken into account in determin-

ing the material damage growth. The chosen model is

not able to accurately describe the natural yield stress

phenomenon and thus the numerical non-linear re-

sponse in the first load cycles deviates from the real

material behaviour (Figs. 5 and 7).

6. Conclusions

Comparing the results obtained from the measure-

ments and from the chosen numerical model shows that

the simulation of the material response to a low-cycle

loading to destruction is very reliable. The numerical

model incorporates non-linear kinematic hardening,

isotropical hardening or softening and the continuum

damage mechanics.

Simulations of low-cycle material response to destruc-

tion have shown that all the required parameters of

normalised (195 HV) and tempered (462 HV) low alloy

steel 42CrMo4 can be accurately determined from a

number of experiments as low as ten.

Fig. 6. Comparison of the course of amplitudes in relation to the

number of cycles between the measured values and the numerical

calculations for the tempered steel 42CrMo4 with hardness of 462 HV.

A non-symetrical load case at a constant extension ofDo/1.4% and a

mean value ofomean/0.5%.

Fig. 7. A comparison of hystreresis loops in relation to the number of

cycles between measurement and numerical calculation the tempered

steel 42CrMo4 with hardness of 462 HV. A non-symmetrical load case

at a constant extension ofDo/1.4% and a meanvalue ofomean/0.5%.

Fig. 8. A comparison of plastic strain in relation to the number of


normalised steel 42CrMo4 with hardness of 195 HV.

Fig. 9. A comparison of plastic strain in relation to the number of


tempered steel 42CrMo4 with hardness of 462 HV.

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The experiments conducted to determine the low-

cycle lifetime of sub-eutectoid steels have shown that the

damage increment can also be determined with sufficient

accuracy by indirectly observing the change in the

modulus of elasticity.

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[3] J. Skrzypek, A. Ganczarsi, Modeling of Material Damage and

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[14] S. Bari, T. Hassan, Int. J. Plast. 16 (2000) 381/409.

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[16] N. Ohno, J.D. Wang, Int. J. Plast. 9 (1993) 391/403.

[17] P.J. Armstrong, C.O. Frederick, A mathematical representation

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[19] R. Kunc, J. Korelc, I. Prebil, M. Torkar, Mater. Technol. 35

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[20] J. Korelc, Theo. Comp. Sci. 187 (1997) 231/248.

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