A continuous low cycle fatigue damage model and its application in engineering materials

Int. J. Fatigue Vol. 19. No. IlL pp. 6,87-692, 1997 t:) 1998 Elsevier Science l.td. All rights reserved Prinled in Great Britain 0142--1123/97/$17.00+.00

ELSEVIER

PII: S0142-1123(97)00102-3

A continuous low cycle fatigue damage model and its application in engineering materials Xiaohua Yang*t, Nian Lit, Zhihao Jint and Tiejun Wang¢ tResearch Institute for Strength of Metals, Xi'an Jiaotong University, Xi'an, 710049, People's Republic of China tDepartment of Engineering Mechanics, Xi'an Jiaotong University, Xi'an, 710049, People's Republic of China (Received 7 April 1997; revised 15 July 1997; accepted 29 July 1997) In this study, the low cycle fatigue (LCF) damage evolution of the engineering materials is studied by use of continuum damage mechanics (CDM) theory. Based on thermodynamics, on a continuum damage variable, D, and on the effective stress concept, a continuum damage model of isotropic LCF is derived and is used to analyze the strain-controlled LCF damage evolution of steam turbine blade material 2Cr13 steel. The damage variable D = 1 - Ao'/A(ro is selected and measured during strain-controlled LCF tests to verily the model, which is in good agreement with the results. The parameters in the model have clear physical meaning and can easily be determined. The evolution of microstructure during fatigue is observed by transmission electron microscopy, which gives the microscopic explanation for LCF damage evolution law of 2Cr13 steel. © 1998 Elsevier Science Ltd. All rights reserved (Keywords: LCF; CDM;

fatigue damage)

INTRODUCTION

theoretical basis tk~r describing damage quantitatively. Phenomenological methods are adopted in the framework of thermodynamics to deal wilh the collective effect of distributed crack-like defects on macro-mechanical properties by defining one or several, scalar or tensorial, continuum damage variables as the measures of the degradation of materials before the initiation of the macrocrack. It has been used in many areas, that is, ductile plastic damage '~, high cycle fatigue damage m, creep damage ~J, creep-fatigue damage ~-' and composite materials damage ~3, etc. There are still some barriers on the application of CDM to study fatigue. It is a great subject to set up appropriate damage evolution equation which can reflect the damage character of engineering materials. Also, more attention will be paid to the study of the relationship between the meso-scale and micro-scale structure and macro-mechanical behavior. Some low cycle fatigue (LCF) damage models have been proposed j<~5, but their validity is limited in some particular cases, the physical meaning of the parameters in the model is not clear, and the microstructure analysis is not given. The purpose of the work is to develop a CDM model for LCF damage from a new dissipation potential chosen by the present author. The parameters in the model has clear physical meaning and can easily be determined. The model is identified by straincontrolled LCF tests of steam turbine blade material 2Cr13 steel, whose failure is often caused by fatigue

The fatigue damage evolution has been studied in aspects of theory, numerical methods and experiments ~ ~. Generally speaking, the fatigue fracture process consists of three stages, that is, nucleation of cracks, growth of cracks and rupture. Fracture mechanics deals with the mechanical behavior and failure criterion of materials containing a crack during load supporting 4. The presentation given by fracture mechanics is the governing variables representative of the stress-strain field intensity of the material medium at the crack tip. It solves the problem of failure under low stress, which cannot be explained by conventional strength theory. In addition, the valuable Paris equation can be used to describe the growth rate of fatigue crack and has been verified by many experiments. However, the time for nucleation of fatigue crack is often very long, c a 80% or more of fatigue life for many components in service. The deterioration of microstructures will lead to the lowering of load supporting capacity of materials. Therefore, it is very important to describe damage quantitatively for predicting the performance and life of engineering structures. Continuum damage mechanics (CDM), originally introduced by Kachanov 5 and Robotnov 6, and later developed by Lemaitre and co-workers 7'8, gives the *Author for correspondence.

687

688

X i a o h u a Y a n g et al.

during service. In addition, the microstructure evolution during fatigue is observed by transmission electron microscopy (TEM) in order to understand the damage evolution mechanism of the steel. THE THEORETICAL ELEMENTS OF CDM AND THERMODYNAMICS Krajcinovic and Lemaitre 8 have proposed the description of damage evolution in the thermodynamical terms. Only the brief information about the damage coupled constitutive equations is introduced here in order to understand the model easily. The elastic constitutive equations of damage materials can be derived from the damage coupled elastic potential = ~ ( ~ . , T, D, ~r)

(r~i = p Oe~i ]+p

E(J

(4)

where o-H is the hydrostatic stress defined by

1 3 o7~

where O-~qis the Von Mises equivalent stress defined by 1

t

where S~j is the stress deviator defined by S~j ~- O'ij -- O'H(~0

Assuming plastic deformation and microplastic deformation to cause damage and internal energy dissipation, the dissipate potential ¢h is

dp=c~p(o',R;D)+C~D(Y,h,#;T,e~,D)+ff) ~

(5)

where R is the isotropic hardening scalar variable associated with accumulated plastic strain, p, and ~bp is the Von Mises plasticity function coupled with damage O-~q - R IJ¥ --

i 7

O

O'y

(7b)

i:v;~vi -"=¢']

(7c)

b = - A 84'1,

(7d)

8Y

where ?t is non-negative proportion factor, which can be obtained from consistency condition. 4~p = 0.

r~ /, 4, = 2s,, (1 - D)',,

(8)

the damage character of LCF is consided and a dissipation potential #5 is chosen as follows: y2 ~/., &=2So

(

N]"

X,"'

(9)

1 - N;/

+ u)+3(1 - 2u) ( O-H1-~ \O'~q/

S u Sii

04¥ A A OR - ( 1 - D)

S~i o-~,,

(3)

where Rv is the stress triaxiality factor, which is expressed as

O-~q=

p=-

A (1 2 D)

Fatigue damage is mainly caused by accumulated plastic strain. According to CDM theory, LCF damage evolution law can be described by a suitable dissipation potential. At the basis of the damage potential function Equation (8), which is sufficient to model all the main properties within the hypothesis of isotropy damage s

p

O~ o-~qRv Y= - P ~D = - 2E(1 ~ / ) ) 5

O'H =

3

i~}} =A Oo.i = 2

LCF DAMAGE MODEL

where P is the density, o- is the stress tensor, e is strain tensor, E and u are Young's modules and Poisson's ratio, respectively. The damage strain energy release rate variable Y associated with D is defined by

R v = ~2( l

Oqb~,

(7a)

(2)

or

-

C/ = ~i + El)

(1)

where e~ is the elastic strain tensor, T is the temperature, D is an isotropic damage scalar variable and ~" is the microplasticity accumulated strain. For linear elasticity and isotropic damage, coupled damage constitutive equations are 8

e,, = E ( 1 " Di

with Ov being the initial yield stress of the material. There is little information about microplastical dissipation potential ~b~, which is not considered here. Then the coupled damage constitutive equations and the dynamic damage evolution law can be derived from the plastic dissipated potential 4)p and the damage dissipated potential &D as follows:

(6)

where So is temperature dependent material constants which can be evaluated from the law of Mason-Coffin of the material s, Ap is the range of accumulated plastic strain per cycle, N is the number of cycles to produce an amount of damage D, Nr is the number of cycles to produce fatigue crack resulting in failure, which is defined as the fatigue life. The term (1 - N/NO, other than (1 - D) in Ref. s, reflects the influence of accumulated plastic strain, a is a parameter which describes the extent of accumulated damage, here it is the plastic strain increment per cycle, which can be determined from monotonic tensile and cyclic tensile stress-strain curve. From Equation (7)d we call easily get

D-

a Y=

-so

~ - N,I

Substituting Equation (4) into Equation (10), gives = -

2ESo(I - D ) e

(

N) ~' ,%'" 1 -

Nt,

(11)

A continuous low cycle fatigue damage model and its application in engineering materials For fatigue load s

VALIDATION OF THE MODEL

2 ,, ,{ AO'H ~2 Rv = ~ (1 + v) + 3(1 - z v ) ~ A o , l

(12)

According to Lemaitre's hypothesis of strain equivalence s , the cyclic stress-strain relationship coupled with damage should be written as follows: AO'eq

i 219 = K(&P)M

(13)

where K and M are material constants. Equations (11) and (13) give the general constitutive equation for LCF damage

D=

K2Rv

A p TM

2ESo .(I - Nt'. - Nr~ \ N,./

±P

(14)

In the case of the proportional loading per cycle, Rv can be considered as constant with respect to time, and the damage during one cycle may be obtained through the integration of Equation (14) AP TM + I ~ N~ i iNr,~ (2M + 1)fl - N,.)

6D K2Rv 6N = 2ES,~

(15)

Integrating Equation (15) with the initial conditions

DIN = N,, = Do. DIN =N, = 1 where No is the number of cycles to produce initial damage Do, gives 1 - Do=

K2Rv

Ap2M+ i

1

2E&(

(2M+ l)

OL

Experiment In this study, the material chosen for the experiments is 2Cr13 matensitic stainless steel from a steam turbine blade. The chemical composition of the steel is shown in Table 1. The heat treatment schedule is austenitizing at 980-1000°C for 2h, cooling in oil, tempering at 680-700°C for 6 h followed by air cooling. An MTS880 material testing machine is used for strain-controlled fatigue tests. Smooth cylinder specimens are used with diameter of 6 mm. The range of strain is + 0.35, + 0.5, + 0.6 and _+ 0.7%, respectively. The stress amplitudes A~ at different cycles are recorded by an X - Y recordmeter. Foils of different cycles at strain +_ 0.5% for TEM observation are cross cut at the work part by a spark cutting machine. The TEM specimens were prepared by a preliminary mechanical polish and then by electropolish in a solution of 5% chloric acid in ethyl alcohol at - 50°C. The evolution of microstructure during fatigue is observed under TEM. Damage measurement and model identification Measurement of LCF damage may be performed by means of the variation of the elastic modules, but a most convenient method here is to evaluate the value of damage from its coupling effect with cyclic plasticity. One may use the one-dimensional relation between the stress range A~ and the strain range Ae at cycles. This relation may be derived from the Ramber-Osgood strain hardening law and from Masing's rule (symmetry with a ratio of two between the tension and compression curves) / J o q M, ALiKe)

(16)

and

D-

689

(21)

where K~ and M~ are material parameters identified from the cyclic hardening curve. The couple with damage is introduced through the principle of strain equivalence with the effective stress:

K2Rv Ap2a4 + I Do= 2ESo ( 2 M + 1) (17)

Comparison of Equation (16) with Equation (17) gives the general LCF damage accumulation law D= 1 -(1

-Do)

)

N %" 1 -,,/vr/

(l g)

In the model, the parameter a can be determined as follows: first, the specimen is in monotonic tension at the first quarter cycle during strain-controlled fatigue. The plastic strain at that moment is expressed by Eg, after that the cycle plastic strain increases with cycles. Assuming that the plastic strain is E~ when cycling to Nf, then the increment of plastic strain from second cycle to fatigue failure is A O' = ¢ - el;

(20)

From this equation the parameter a can easily be determined.

K,(I " D)

(22)

If a cyclic test at constant amplitude of strain A~, is considered at first cycle being Ao-o tot a material of cyclic softening, the damage may be assumed to be zero, hence 1

Ao'o = K,.AeM

(23)

from Equation (22) I

(24)

Ao- = ( 1 - D)K~.AE M

Then, from the two relations Equations (23) and (24) D = 1-

(19)

The increment of plastic strain in one cycle can be derived from Equation (19) 6D Ao' = c~ aN = N r 1

A~=

Table 1 C

0.18

A~r

(25)

Ao'o

Composition of 2Cr13 stccl Cr

Si

Mn

P

13.18

0.22

0.23

0.015

S

0.0071

Xiaohua Yang et

690

W h e n the cycle stress amplitude is up to stabilization Ao-*, the damage is assumed to be Do Do = 1 -

Ao-

(26)

Ao-*

al.

For E =_+ 0.6% Do = 0.08, NI = 1289, the model is D=

Reference 16 shows that the cyclic softening occurs during strain-controlled L C F for the 2Cr13 steel. The relationship between the cycle life fraction N/Nr and the damage variable D can be measured easily as shown in Figure 1. For 2Cr13 steel under total strain controlled with • = _+ 0.35% per cycle, from cyclic test results at constant amplitude of strain and Equations (16), (17), (20) and (26) give Do = 0.094, Nr = 6230, Nroe = 0.058 the model is

N],,o.s ~ D=I

-0.906

1

(27)

6230/

For E = _ + 0 . 5 % Do = 0.097,

Nr = 1950, Nro~ = 0.064

the model is D = 1 - 0.903 1 -

N ]o.o64 1950]

(28)

the model is N O.Ll3 D = 1 -- 0.923 1 - 844/

(30)

The values of D calculated from Equations (26)(29) and measured by experiment at cyclic strain E = _+ 0.35%, E = + 0.5%, e = +_ 0.6% and e = _+ 0.7% are plotted against the cycle fraction N/Nr in Figure 1, which are in good agreement with each other. The curves show that D increases quickly at the early and the last stage of the whole cycling and slowly at the middle stage from 10 to 80% of the total cycles, which is characteristic of LCF damage.

Explanation of parameters Figure 2 shows the microstructures cycling to different cycles. The dislocation cell structure has formed 1.0 -

0.8

0.8

1 I

0.6 D 0.4

0.2

J

0.2 -

0.0-

__... •

I I II

•

71

0.0

,

, ,

0.2

I

,

, ,

0.4

I

, ,

0.6

,

I ,

,

, I

0.8

0.0 0.0

1.0

~

J

I

0.2

~

_...It.---. - - - - ' - ' ' ~ ~

!~

~

~1

0.4

N/Nf

0.6

~

~

tl

~

0.8

~

I

1.0

N/Nf

(a) e= ±0.35% lo

(29)

1289/

N,-= 844, N, oe = 0.113

Do = 0.077,

1.0

04:

(

l - 0.92 I -

For • = + 0 . 7 %

0.6

c'-,

Nr~ = 0.076

(b) ~'= ±0.5%

•

:.

1.0--

0.8

•

0.8

0.6

0.6 ¢-,

D 0.4

0.4 0.2

0.0

J

0.2 .... 0.0

I, 0.2

,

,I,

,,

0.4

I, 0.6

,,

I 0.8

, ,

,

I 1.0

0.0 0.0

0.2

0.4

N/Nf

(c) , = _+o.oeojo

Figure 1

Comparison (d) ~ = +_ 0.7eYe

of the curves derived from the model with experimental

0.6

0.8

1.0

N/Nf

(d) ~ : +o.o7Ojo r e s u l t s (a) E = _+ 0 . 3 5 % ,

(b) E = + ().5~;, (c) E = _+ ( ) . 6 ~

and

A continuous low cycle fatigue damage model and its application in engineering materials

691

!

i

Figure2 TEM photographs of 2Cr13 steel during cycling to different cycles at E = _+_0.5c~. (a) 0.25, (d) N/N, = 0.5, (e) N/N, = 0.75 and (f) N/N, = I

at 1/10N~ cycles when the stress reaches saturation. The damage evolution before that cycles is mainly caused by changes of dislocation substructures. For a given material, dislocation substructure gets stable in early stage of fatigue w. Even if their initial microstructure is different, the stable substructure is almost the same for a given cycle strain. The evolution of dislocation substructures causes the increase of D in 2Cr13 steel but the stable substructure has strong resistance to fatigue failure at cycles later. It is thought that the evolution of substructure here brings no damage to the material but improves the resistance to fatigue failure. The increase of damage value due to the evolution

N/Nj -

0.05, (bt

N/N~

().1, (c)

N/Nt =

of dislocation substructure is defined as the initial damage D~.

Summary The analysis above indicates that the initial damage is caused by evolution of dislocation substructure during strain-controlled fatigue. After f~wmation of the stable substructure, the damage value increases very slowly until 80% Nr cycles. It indicates that the substructure has high fatigue resistance. D begins to increase quickly at cycles over 80% Nr, which indicates the beginning of damage localization and the formation of fatigue microcracks.

Xiaohua Yang et al.

692

The CDM model expressed in Equation (17) can be used to described the damage evolution of the engineering material 2Cr13 steel satisfactorily. The model is not complex and the parameters in which has clear physical meaning and can easily be determined. The result can be available for reference to predict the fatigue life of steam turbine blade made of 2Cr13 steel.

3 4 5 6 7

CoNcLusION A continuum damage model of isotropic LCF is derived on the basis of CDM theory, which is a powerdependence upon cyclic fraction. Its range of validity is limited by the hypothesis of isotropy of damage and of constant triaxiality ratio during loading, that is proportional loading per cycle. The present model can describe the LCF damage evolution of engineering material 2Cr13 steel perfectly. The microscopy analysis shows that the initial increment of damage value during fatigue is caused by the evolution of dislocation substructure. After the formation of stable dislocation cell structure, the damage increases very slowly until 80% N~, then it increases quickly because of the nucleation of fatigue microcracks. At the end of cycling the damage value is 1. REFERENCES Lemaitre, J., Local approach of fracture. Engineering Fracture Mechanics, 1986, 25(5/6), 523-537. Chaboche, J. L., Continuous damage mechanics--a tool to

8

9

10 I1 12 13

14

15

16

17

describe phenomena before crack initiation. Nuclear EngineeriJ~g and Design, 1981, 64, 233-247. Lemaitre, J. and Dufailly, J., Damage measurements. Engim,~'ring Fracture Meehanics, 1987, 28(5/6), 643-661. Jansom J. and Hult, J., Fracture mechanics and damage mcch anics. Journal of Applied Mechanics, 1977. 44(I), 69 84. Kachanov, L. M., Time of the rupture process under creep conditions, lsv. Akad. Nauk. SSA Otd. Tekh., 1958, 8, 26 31. Robotnov, Y. N., Creep Problems in Stnu'ture Memher~. North Holland, Amsterdam, 1969. Lemaitrc, J. and Chaboche, J. L., Mechanics, o[' Solid Materials. Cambridge University Press, Cambridge, 1990. Krajcinovic, D. and Lemaitre, J. Continuum Damage Me~qumi¢,s: Theory and Application. Springer Verlag, Berlin, 1987. pp. 37 89. Lemaitre, J., A continuum damage mechanics model for ductile fracture. Journal q[" Engineering Materials Technology. 1985, 107, 83-89. Lemaitrc, J.. How to use damage mechanics. Nucl~,ar En~iJle~ring and l)eMgn, 1984, 80, 233 247. Lemaitre, J., Computer Methods in Applied Mechanics Engineer ing, 1985, 51, 31~-9. Lemaitre, J. and Krajciniovic, D., Jonrnol o/ l~/u,,ineeri11~ Materials Technology, 1979, 101, 284-292. Fong, J. T. What is fatigue damage? Damage m composite materials. In ASTM STP775, ed. K. L. Reifsnider. ASTM. Philadelphia, 1982, pp. 243-266. Chaboche, J. L. and Lesne, P. M., A non-linear continuous fatigue damage model. F~ltigue Frael. Engng. Mater. ,~'IFH~I,, 1988, 11(1), 1--17. Wang, T. J. and Lou, Z. W., A continuum damage model I).w weld heat affected zone under low cycle fatigue loading, b,'n~in eering Fracture Mechanics, 1990, 37(4), 825-829. Yang, X. H.. Li. N. and Bu, B. P., Research on microsU'uctures and mechanical properties for a failed blade. Journal O~ X.HI,'. 1996. 30(10), 94 99. Suresh. S. Fatigue o[ Material,~. Cambridge Univcrsitv Press, Cambridge, 1991.