fatigue life assessment of structural component with a non-uniform stress distribution based on a...

Fatigue life assessment of structural component with a non-uniform stress distribution

based on a probabilistic approach

Aleksander KAROLCZUK.Thierry Palin-Luc

Department of Mechanics and Machine Design, Opole University of Technology, Poland

17th European Conference on Fracture (ECF17) Brno. 2 – 5 September 2009

Arts et Métiers ParisTech, Université Bordeaux 1, France

2

Plan of the presentation

• Introduction

• A short review of the fatigue life calculation approaches applied for structural elements with non-uniform stress distributions - summary

• A Weibull based mathematical model

• Implementation of the model for fatigue life calculations

• Experimental tests and results

• Numerical simulations and results

• Conclusions

3

Introduction

Element with a hole

Elements underbending

Elements with a non-uniform stress distribution

Spherical defect

Alternating heterogeneous stress distribution ….complex mechanisms of fatigue failure …

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-400

-200

0

200

400

Time, s

xx

, MP

a

Yield strength

4

A short review of approaches applied for elements with non-uniform stress distributions

(i) Empirical approachesMurakami i Endo (defects):

]2/)1[()/()120(43.1 6/1 RareaHVaf

(ii) Approaches based on fatigue notch factorMitchel (defects): )]/'1/()1(1/[0 CKtafaf

Deterministic concepts

(iii) Non-local approaches

The so-called non-local approaches consider the interaction between locally damaged points located in the vicinity of the hot spot by normal or weighted integration process of local damage parameters.

Line: = LArea: = AVolume: = V

dzyxzyxyyeff

eff ),,(),,(1

5

Probabilistic concepts

The fatigue failure could start at any elementary domain in element but the probability of such event depends on the local stress history. Thus. the fatigue failure of the whole element is a function of the cyclic stress volumetric distribution and dimensions of the considered structural element. This mechanism is usually taken into account by probabilistic methods based on the weakest link concept.

V

m

af

eqa dVzyx

V

f VP

),,(1

021)(

(i) Bomas, Linkewitz, Mayr (for fatigue limit regime)

m

af

eqa

V

V

f VP

021)(

V

m

u

gg dVW

WW

V

f eVP

*1

01)(

(ii) Thomas Delahay, Thierry Palin-Luc

A short review of approaches applied for elements with non-uniform stress distributions

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

eqa

, MPa

PzPf

6

• Deterministic concepts do not consider the statistical nature of fatigue

• Most of the probabilistic concepts based on weakest link concept are limited to fatigue strength regime

• There is necessity (and possibility) to build a probabilistic model for fatigue life calculation in the fatigue crack initiation regime

Summary of review

105

106

0

0.2

0.4

0.6

0.8

1

Pz

N, cykli Ncal

Pz=0,632

105

106

0

0.2

0.4

0.6

0.8

1

Pz

N, cykli Ncal

Pz=0,632

Pf

Pf = 0.632

N, cycles

7

A Weibull based mathematical model

Aim:

Fatigue life calculation Ncal using probability approach under arbitrary fatigue regime (LCF-HCF) for a given (or requested) failure probability level Pf.

Main hypothesis:

The weakest link concept might be applied in any arbitrary fatigue regime if the appropriate crack length defining the failure as a crack initiation is specified.

8


Weibull model

The usual (Weibull - 1939) form of failure probability Pf =1-Ps is as follows

dg

f eP)(

1

01Where: 0 reference domain (surface or

volume).

0, u, m – parameters of stress shift, stress

scale and shape

m

u

g

0)(

New concept

The general form of probability distribution is analogous to the Weibull expression. However. the former stress function of ”risk of rupture” g() becomes also function of lifetime N and the failure probability takes the general form as follows

General form:

dNh

f eP),(

1

01

The failure probability Pf increases with increasing the stress level a but Pf (for a given a) also increases with the number of cycles N. The longer structural element is in service the failure probability is higher.

9

Some researchers lean towards a view that the Weibull distribution describes the scatter of fatigue life (under a given stress amplitude a) in logarithmic scale fairly well. It could be expresses by the following equation

mN

f eP

)log(

1

where is the lifetime scale parameter and m is the shape parameter.


105

106

100

150

200

250

300

350

400

Nf, cykli

a,

MP

a

Nf

Charakterystyka zmęczeniowa

a-N

f

Fatigue characteristic

= log(Nf)m

fN

N

f eP

)log(

)log(

1

For another stress level a the mean value of fatigue lives N(a ) would be moved. Thus.the scale parameter would also change its value. Relation between expected fatigue life Nf and stress level is captured by fatigue characteristic, e.g. Wöhler curve. As a result the scale parameter becomes function of expected fatigue life Nf taken from fatigue reference curve:

10

++++ + +

+ + ++ + +

a

Nf

)log()()(

ff N

pNmm

where p is a constant parameter.

The coefficient p can be seen as a quality factor of both element manufacturing process and material (internal defect for instance). Finally, the failure probability distribution of the component, before N under the stress amplitude a, takes the form

dN

N

z

fN

p

feP

)log(

0 )log(

)log(1

1

where =V (volume) or =A (surface).

Some experimental results shows that the magnitude of fatigue life scatter depends on the stress level a. Thus, the scale parameter m should be function of the stress level a or expected fatigue life Nf taken from reference fatigue curve.This relation is modeled by a proposed simple function

(1)


11

Fig. 1. (a) Simulated two-dimensional distribution of the failure probability Pf for the element made of 18G2A steel (p=580); (b) Fatigue reference curve a-Nf with experimental points and fatigue scatter bands for Pf = 0.05 and Pf = 0.95 (p=580).

104

105

106

107

108

150

200

250

300

Nf , cycles

a, MP

a

+ experimental points reference curve

Pf=0,63

Pf=0,05

Unbroken specimens

a =

a*

a =

af

Pf=0,95

Reference curvea-Nf

104 10

5 106 10

7 108

100

200

300

400

0

0.2

0.4

0.6

0.8

1

Nf

a , MPa

Pf

a=

a*Pf=0,05

Pf=0,95

In the case of a uniform stress distribution, the previous equation reduces to

)log(

)log(

)log(

1

fN

p

fN

N

f eP

Figure 1a shows an example of two-dimensional failure probability distribution according to Eq. (2). using the fatigue reference curve (a-Nf) of 18G2A steel with p = 580. Fig. 1b illustrates experimental fatigue test data used to identify the reference Wöhler curve a-Nf along with scatter band obtained for Pf = 0.05 and Pf = 0.95.

(2)


12

Implementation of the model for fatigue life calculations

104

105

106

107

108

150

200

250

300

350

Nf , cycles

a, MP

a

105

0

0.5

1

105

106

0

0.5

1

105

106

0

0.5

1

Ps

105

106

0

0.5

1

Ps

Ps(1)

Ps(2)

Ps(1)*Ps

(2)

105

106

0

0.5

1

Pf

Pf=1-Ps(1)*Ps

(2)

A(1)

eqa(1)

N

Nf(1) N

f(2)

Reference curve

Survival probabilitydistributions ofsubdomains: (1) and (2)

(a) (b) (c)

N

Ps(1)

NN

cal(P

f )

A(2)

Multiaxial fatiguecriterion


Ps(2)

eqa(2)

kl(1)(t)

kl(2)(t)

• The free surface of the considered element is divided into sub-domains A(i) of sizes which allows appropriate integration process

• In each sub-domain A(i) multiaxial stress state is reduced to an equivalent stress amplitude state by using a multiaxial fatigue crack initiation criterion

• The equivalent stress amplitude and the fatigue reference curve a-Nf are used for calculations of a number of cycles to failure for each sub-domain A(i) Then. the survival probability distribution is determined as follows

13


• For each fatigue life N. exponents of e natural logarithm are summed along all the sub-domains A(i) and the survival probability distribution Ps(N) for the whole structural element is obtained

104

105

106

107

108

150

200

250

300

350

Nf , cycles

a, MP

a

105

0

0.5

1

105

106

0

0.5

1

105

106

0

0.5

1

Ps

105

106

0

0.5

1

Ps

Ps(1)

Ps(2)

Ps(1)*Ps

(2)

105

106

0

0.5

1

Pf

Pf=1-Ps(1)*Ps

(2)

A(1)

eqa(1)

N

Nf(1) N

f(2)

Reference curve

Survival probabilitydistributions ofsubdomains: (1) and (2)

(a) (b) (c)

N

Ps(1)

NN

cal(P

f )

A(2)



Ps(2)

eqa(2)

kl(1)(t)

kl(2)(t)

• The fatigue life calculation Ncal is performed for Pf(Ncal) = 0.63. Fatigue life for any other probability level, i.e. the scatter of results, can be calculated in a similar way.

14

Parameters identification


Taking advantage of the empirical analytic equation of the reference Wöhler curve a-Nf. the two-dimensional distribution parameters has only two parameters to be identified. i.e. A0 and p. The reference surface area A0 is the free surface area of the specimen applied for determining the reference Wöhler curve.

The parameter p responsible for the distribution of the fatigue life scatter can be identified from the tests of specimens having the same distribution of defects (kind and morphology) as the considered element. However, manufacturing qualities of elements and specimens are usually different. In such a case, distribution parameters should be fitted based on one series of tests of a real element subjected to simple fatigue loadings.

In the present paper, the authors applied different values of the parameter p to find the best correlation between experimental fatigue lives Nexp and the calculated fatigue lives Ncal.

15

(i) In the first set of experiments. cruciform specimens made of 18G2A steel with a central hole as stress concentrator were subjected to biaxial fatigue loading.

Experimental tests and results

Experimental results obtained from testing two steels with different specimen geometries were used for analyzing and verification of the proposed probabilistic method.

16

Test conditions and results

Specimen

d,mm

h,mm

Fxa

kN

Fya

kN

Ni

cycles

Ai,

mm

P02 3.0 1.40 13.30 13.10 39700 0.22

P03 3.0 1.54 13.50 13.30 31100 0.37

P04 3.0 1.86 13.55 13.30 60048 0.07

P05 2.5 1.50 10.21 9.90 246695 0.25

P08 2.4 1.20 9.30 9.10 167050 0.10

P07 3.0 1.75 11.20 10.80 140700 0.20

Ni is the numbers of cycles to crack initiation which corresponds to the crack length ai


6969 6969.1 6969.2 6969.3 6969.4 6969.5 6969.6 6969.7 6969.8 6969.9 6970-15

-10

-5

0

5

10

15

Czas, s

Siła

, kN

F

x

Fy

6969 6969.1 6969.2 6969.3 6969.4 6969.5 6969.6 6969.7 6969.8 6969.9 6970-15

-10

-5

0

5

10

15

Czas, s

Siła

, kN

F

x

Fy

Time, s

For

ce, k

N

17

The second set of experimental results comes from the work of Fatemi et al [1]. Circumferentially notched round bar made of AISI 1141 steel, in both the as-forged (AF) and quenched and tempered (QT) conditions were subjected to tension-compression loading.

[1] Fatemi A., Zeng Z., Plaseied A. (2004), Fatigue behavior and life predictions of notched specimens made of QT and forged microalloyed steels, Int. J. Fatigue 26, pp. 663–672.

Geometry with two notch radius were tested R=0.529 mm and R=1.588 mm.


18

Numerical simulations and resultsThe strain and stress distribution in the specimens were calculated using the 3D finite element analysis on COMSOL software. In all the computations a cyclic constitutive model with linear kinematic hardening was applied.

The Lagrange elements (tetrahedrons) of order 2 with higher mesh density in the vicinity of the notch were used in the computations. Because of the symmetry of loading and geometry of the specimens, only 1/8 part of the cruciform specimens was modeled (Fig) and 1/32 part of round notched specimen was modeled. Careful analysis of mesh size influence on stress and fatigue life calculation has been performed.

19

where ni is the unit normal vector to the plane experiencing the maximum normal stress or strain

1/8 of geometry of the cruciform specimen with distribution of the equivalent stress amplitude eqa on the notch surface (specimen P05, Fxa=10.21 kN, Fya=9.90 kN)

jiijneqjiijneq nntttnnttt )()()(,)()()(

1/32 of geometry of the notched round bar with the distribution of the equivalent strain amplitude eqa on the notch surface (state AF, R=0.529 mm, under nominal stress amplitude in net section anom = 225 MPa)

The criterion of maximum normal stresses (cruciform specimens) or strains (round specimen) on the critical plane was assumed as the criterion of multiaxial fatigue crack initiation. The equivalent stresses or strains are calculated according to the following equations

Numerical simulations and results

20


103

104

105

106

103

104

105

106

Nexp

, cycles

Nca

l, cyc

les

2.0

2.03

3

0.05 0.63 0.9510

4

105

106

Pf

18G2A steelp = 400

Pf=0,95

Pf=0,63

Pf=0,05

103

104

105

106

103

104

105

106

Nexp

, cycles

Nca

l, cyc

les

2.0

2.0

18G2A steelp = 580

Comparison of the experimental fatigue lives Nexp with the calculated lives Ncal for (a) p = 400 and (b) p = 580 for the cruciform specimens in 18G2A steel under biaxial tension

(a)

(b)

Fatigue lives to crack initiation Ncal were calculated for three failure probability levels: Pf = {0.05; 0.63; 0.95} and for different values of the p parameter

Additional two scatter bands (2 and 3) around the solid line Ncal=Nexp of perfect results consistency are also shown in figures

21


103

104

105

106

107

103

104

105

106

107

Nexp

, cycles

Nca

l, cyc

les

2.0

2.0

AISI 1141 steel p = 200

AF QT

103

104

105

106

107

103

104

105

106

107

Nexp

, cyclesN

cal, c

ycle

s

2.0

2.0

Pf=0,95

Pf=0,63

Pf=0,05

AF QT

AISI 1141 steel p = 340

Comparison of the experimental lives Nexp with the calculated lives Ncal for (a) p = 200 and (b) p = 580 for the notched round specimens in AISI 1141 steel under fully reversed tension

(a) (b)

22

Conclusions

A procedure for determining the two-dimensional failure probability distribution Pf-N-/ of structural elements has been proposed. It allows the calculation of fatigue life of structural elements under any stress amplitude and probability levels.

The presented approach allows calculating global fatigue life at any requested probability levels for elements with heterogeneous stress fields.

The calculated fatigue lives Ncal for failure probability equal to 63% are well correlated with the experimental fatigue lives Nexp for notched round specimens made of AISI 1141 steel (with p=340) independently on the notch radius and material state (QT or AF); and for the notched cruciform specimens made of 18G2A steel (with p=580).

The proposed probability distribution function to the fatigue macro crack initiation needs to identify only one additional parameter (p). This parameter can be seen as a quality factor of both element manufacturing process and material (internal defects for instance).