A Non-linear Continuous Fatigue Damage

Fatigue From Engng Muter. Strucf. Vol. I I , No. 1, pp. 1-17, 1988 8756-758X/88 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright 0 1988 Fatigue of Engineering Materials Ltd

A NON-LINEAR CONTINUOUS FATIGUE DAMAGE MODEL

J. L. CHABOCHE and P. M. LESNE Office National d’Etudes et de Recherches Airospatiales, BP 72. F-92322 Chiitillon Cedex, France

(Received in final form 10 September 1987)

Abstract-A non-linear cumulative fatigue damage model proposed previously is applied to different steels and various loading situations, including two-level tests and block-programs. Its ability to describe all the main features of fatigue damage and the ease of its practical use for engineers are discussed and several examples cited. The relationship with other formulations are pointed out together with the main advantages of the proposed model. A generalization is proposed for both cyclic temperature and multiaxial loading conditions.

NOMENCLATURE

D = damage variable

D, = damage value at micro-crack initiation D = damage variable consistant with Continuum Damage Mechanics

uM, 5 = maximum and mean tensile stresses Acp = plastic strain range

q = plastic strain memory variable (model with interactions) L =crack length N = current number of cycles

n, = number of cycles in block “i” of a block program

ulo = fatigue limit under reversed stress conditions o, = ultimate tensile stress

b. M,, a, b = coefficients of the NLCD model

NF. Ni = numbers of cycles to failure and to micro-initiation

a( ), M( ), u,( )=functions in the NLCD model

y , C = coefficients of the power fitting of S-N curves

( p = exponent of non-linearity in the two-level test predictions. ) = MacCauley bracket symbol. ( u ) = 0 if u -= 0, ( u ) = u if u > 0.

INTRODUCTION

The life of metallic structures is very often governed by the fatigue process caused by vibratory loading conditions. A large part of the life is related to a sequence of processes during which slip bands, localized strains, damage and microcracks initiate and develop until some macroscopic crack initiates. After that stage crack propagation is much more rapid and is usually taken into account through Fracture Mechanics analyses.

Non-linear cumulative fatigue damage theories were developed a long time ago and include the works of Henry [I], Gatts [2], Marco and Starkey [3], Manson [4] and many others. They are based either on the separation of fatigue life into two periods (initiation and propagation) [5] on the progressive decrease of fatigue limit [6], or on remaining life and continuous damage concepts [7].

The present paper reviews the main features of a model proposed in 1974 [7] to describe non-linear cumulative fatigue damage. It is called here the Non-Linear-Continuous-Damage (NLCD) model. The formulation is supported by Continuum Damage Mechanics [7,8] but its use

1

2 J. L. CHABOCHE and P. M. LESNE

under pure fatigue is mainly related to remaining life aspects. The model has also been used in high temperature situations. Its form allows for the cumulative effect of creep and fatigue damaging processes [9, 101.

The considered approach generalizes the model of Marco and Starkey [3] and the Damage curve approach of Manson [l I]. The connections with other approaches will be pointed out together with some detailed examples on different materials and for various loading conditions, including two-level tests and block-programs. Also the cases of varying temperature conditions will be considered.

Let us note that the model is developed as an engineering tool, no more difficult to use than the simple linear rule. As a consequency, the physical microprocesses, involving nucleation, initiation and short crack growth, are not considered in detail.

THE NON-LINEAR CONTINUOUS FATIGUE DAMAGE MODEL

Uniaxial formulation Fatigue life prediction in structures needs to take into account several effects, including

mean-stress influence and cumulative damage under non-periodic loading. A fatigue damage accumulation model called NLCD, was proposed in Ref. [7l to describe the progressive deterioration processes before the macroscopic crack initiation. Developed in the case of smooth fatigue specimens (tensionsompression), its main features are:

(a) Crack initiation is defined at an “engineering level” by the crack sufficiently long to become independent of the microstructure and defects existing near the initiation area [8]. The classical definition, corresponding to the stage I-stage I1 transition (one or two grains depth, 10-200 pm for metallic materials) is referred here as the “micro-initiation”, the main maqroscopic crack must have a well defined average geometry in order to be treated in the framework of fracture mechanics. Typically its depth is 1 mm (1 mm2 area) for metallic materials. Measurements of striation on various materials [12] have shown abrupt changes in the behaviour of microcracks for about 300-1000 pm, which could correspond to the present macro-initiation definition. Moreover, the problems posed by the treatment of short cracks when using fracture mechanics will be avoided: in fact this short crack regime, which can be the dominant period in fatigue, is included in the NLCD model.

(b) Microinitiation and micropropagation are included in the model through a continuous damage evolution, written in a differential form. Damage variable D is normalized to 0 for the initial undamaged state and to 1 for the macro-crack initiation ( N = NF). To facilitate the practical experimental treatment, this state ( 1 mm depth) is approximated by the rupture of the fatigue specimen, neglecting the phase of rapid macrocrack propagation between 1 and 2 or 3mm.

(c) The loading parameters considered are the maximum stress uM, and mean stress Ti in each cycle. The damage rate, expressed in terms of cycles N is:

dD = f(uM, 5, D)dN (1) (d) The damage rate in (1) depends on the present damage state, permitting a non-linear

evolution under periodic loading. This is similar to the dependence between fatigue crack growth rate and crack length. The non-linear evolution is not sufficient to describe non-linear accumulation.

(e) Moreover, the functionfhas unseparable variables uM and D, in order to describe non-linear damage accumulation and sequence effects. Separable variables always induce linear accumulation 1131.

A non-linear continuous fatigue damage model 3

( f ) The specific form chosen for equation (1) is [9, 101:

(g) Integrating equation (2) for constant dM and 5, between D = O and D = 1 , leads to ( D = 1 for N = NF):

(h) Due to the dependence between a and the loading parameters the damage evolution curves as a function of the life ratio N/NF, depend on oM and 5. As it is classical, this dependence leads to the non-linear accumulation and allows the description of sequence effects. For instance, considering two-level tests, one finds -by integrating (2) in two steps:

1 - a2

where NF, and NF, are the failure lives (on the S-N curve) for the two loading conditions, N , is the number of cycles at the first level, N2 the remaining life at the second level.

(i) The function M(E) is chosen so as to describe the well-known linear dependency [14], between mean stress and the fatigue limit (see below). Here, in terms of an equivalent amplitude, one writes:

M ( 5 ) = Mo( 1 - b5) (6) where Mo and b, as well as exponent p in (2), are coefficients depending on the material.

DIfSei-ent choices for function u

In Ref. [7], the function a was determined from damage measurements through the effective stress concept and Continuum Damage Mechanics [8], with a different-but equivalent-form for the damage rate equation (2). The measured values were fitted by:

a = l - u t ) 7-0 (7)

where 0, is the ultimate tensile strength, and y is the slope of the S-N curve (reversed stress conditions) in the intermediate region:

NF = COG’ (8)

a, B, and C = a’,/a are coefficients. The correlation p = 0.55 y was satisfactory for many materials [7]. Figure 1 gives the example of superalloy IN 100 used in turbine blades.

Identification of the NLCD model is easier if the form of function a is specified in the following way [9], which corresponds to the presently used formulation:


0 250 500

Maw stress aM MPa

Fig. 1. Values of a measured by the effective stress concept and by fitting various relationships for IN100 at 900 and 1000°C.

where oi0 is the fatigue limit for fully reversed conditions. o,(Z) is the fatigue limit for a non-zero mean-stress. It is expressed in terms of the maximum stress. Symbol ( ), defined as ( u ) = 0 if u c 0, ( u ) = u if u > 0, gives rise to o! = 1 when the maximum stress is lower than the fatigue limit a,(Z). With function (9), the number of cycles to failure is calculated as:

o , , - n M [.M-Z]-’ NF =

a ( b M - M ( Z ) In the two-level test condition, the remaining life is predicted by ( 5 ) with the exponent:

- I - xi

b M 2 - oi(Z2) Ou - OM,

CMi - ollfZl) Ou - oM2 p =-- -

The ultimate tensile stress o,, plays no role in the fatigue process under reversed and high cycle conditions. It is used here as a normalizing parameter, which is able to reproduce the asymptotic shape of the S-N curve in the regime of very short lives, especially under repeated conditions. The relation (1 1) shows the two limiting cases of NF = 0 for oM = o,, and NF = 00 for 0, = q(Z). It describes correctly S-N curves for different mean-stresses within a large range of cycles 10 c NF < lo’. Figure 2 gives a typical example. Cumulative effects are predicted by only specifying the fatigue limit and the ultimate tensile strength q,. The symmetry between High-Low and Low-High loading conditions, as shown by examples described later, should be especially noted. Such symmetry is evident from the damage rate equation (1).

Comparison to other models Equation (3) shows that the present damage approach gives a differential form to the damage

evolution curves considered earlier by Marco and Starkey [3]. Moreover the Damage Curve (DC) approach, developed recently by Manson [I 13 corresponds to a particular case of the present formulation. Combining (7) and (8), with f l = 0.55 y, leads to:


A 517

m 5 I A201 I

100 10 10’ 103 104 1 0s

Number of cycles to failure NF

Fig. 2. S-N curves for A201 and A517 steels together with NLCD model predictions.

and

In Ref. [ll], Manson uses 0.4 instead of exponent 0.45. The equivalence between the present approach and the DC model is clear for loads higher than the fatigue limit, even though the DC model introduces differently the damage growth equation [ll]. As pointed out in reference [14], equation (14) presents some limitations. One advantage of the present approach is to consider the damage produced by loading cycles below the fatigue limit (see later).

Let us mention also the method proposed by Subramanyan [15], defining a knee-point on the S-N curve in the region of the fatigue limit. In the two-levels situation, this method predicts (5 = 0):

which follows from the present approach if a is chosen as:

a = 1 -u(OM-O,) (16)

Also slightly different expressions have been used, for instance in Refs [16], [17] and [18], corresponding to:

where NY corresponds to the “knee point” on the S-N curve, with various definitions [15]. These expressions are equivalent for a S-N curve described by a power relationship. They give rise, for the two-level condition, respectively to:


Steel (2.35

1

0

Fig. 3. Prediction ofthe two-level test results for the C-35 steel with different a functions. Data from Ref. [HI.

Figure 3 shows the comparison of the different predictions for two-level tests on a C-35 steel [15] and demonstrates that the choice of equation (1 1) slightly overpredicts the non-linearity of the accumulation.

Other non-linear cumulative rules are based on the decomposition of fatigue process into crack initiation (Ni) and crack propagation stages. Reference [5] uses an exponential accumulation rule in the two regimes but, due to the low values taken for Ni/NF(Ni/NF = 1.421. lop3 NOF.42'), the model behaves mainly in the propagation regime. In that case the remaining life can be calculated at the second level by an equation similar to (1 5), with

(19) log(&, 10g(q,/NF,) lOg(Li,l&) W K , /NF, 1 P =

where Ni is the number of cycles to micro-crack initiation, Li the corresponding crack length (which depends on the life). Lf is the crack length at failure.

In Refs [4] and [19] the accumulation of damage is assumed linear during each of the two phases. They lead to different versions of a Double Linear Damage type of rule. As pointed out in Refs

1

275 MPa 334 MPa


Fig. 4. Schematic

1

0.5

Di 0

&Ni,/NF,-

comparison of the NLCD and the DLDR approaches to the evolution of damage.

[ 1 11 and [20], there exists qualitative and quantitative correspondences between the two approaches. As shown schematically in Fig. 4, in the DLDR rule the life ratio to crack initiation depends, for example, on the total life, leading to sequence effects. In the continuous damage models, it is the shape and the non-uniqueness of the damage evolution that dictates sequence effects. The correspondence between the two types of approach is possible by considering a critical damage Di (with a low constant value). On each continuous damage curve, Di corresponds to the micro-crack initiation (see Fig. 4). From equation (3) one finds the following relation, where the function a(&) is defined either by equations (7), (9), (13), (16), or (17):

Ni = NFD!-a(N~) (20) The damage accumulation models based on the progressive decrease of fatigue limit [l, 2,6]

cannot be deduced explicitly from the present approach. Here, the decrease of fatigue limit (or its complete annihilation) is only a consequence of the damage rate equation. Also the model proposed by Kramer [21] cannot be written in a differential form similar to (1). In that model, there is no symmetry between Low-High and High-Low two-level test conditions.

Determination of the NLCD model The constants are easily determined from conventional data, including the S-N curve: a, is

usually known, q,and b fit the results on the fatigue limits with the linear relation (10). Exponent p is obtained from the S-N curve for reversed conditions, by plotting aM as function of NF(cM - C T , ~ ) / ( C T ~ - oM), as deduced from relation (1 1). Coefficient M,a is obtained from one point of the S-N curve. The independent values of a and Mo have no importance as long as only fatigue damage is considered.

A check of the model is possible by the two-level tests. Figure 5 shows a typical example for steel AISI 4340 [22]. The symmetry between High-Low and Low-High conditions should be noted. Many other materials obey correctly to the model, as shown in Refs [7] and [9].

Comparison with damage measurements The above determination of the model and the remain life measurements are insufficient to

completely characterize the damage equation, that is the value of a. This indetermination has no


A :88MPa

1

Fig. 5. Prediction of two-level test results on 4340 steel using the NLCD rule and equation (16). Data from Ref. [22].

importance for fatigue damage accumulation, even under complex block programs, as shown later. If necessary the determination can be completed by using damage measurements such as:

The number of cycles to micro-crack initiation N,, with an arbitrary associated damage value Di, can yield an average value for coefficient Q, using together equations (19) and (9). The value of a can be obtained from microstructural damage measurements, using for instance the surface crack length L, normalized by a reference value & as in Refs [23] and

Measurements of damage can be obtained from the change in the stress-strain response during the fatigue process, using the effective stress concept and Continuum Damage Mechanics [8].

The measurements obtained from changes in the mechanical response suggested a different form for the damage rate equation [7]. The damage variable associated to such measurements is called D* and the rate equation writes:

(i)

(ii)

[24], D = L/Lf . (iii)

Clearly, due to the limit values 0 and 1 for D* as well as for D, there is a one-to-one

(22)

correspondence between equations (2) and (21), with:

D = 1 - (1 - D*)B+I


0

do

Fig. 6. Damage evolution curves as measured for IN 100 from the effective stress concept; prediction by the NLCD model and equation (23).

Therefore the two theories give rise to exactly the same response under any fatigue loading. The damage evolution under periodic loading becomes:

where NF obeys still to equation (4). The measurements of D* from the changes in the mechanical response allow values of a and

then an average value of coefficient a to be determined. Figure 6 shows the correctness of relation (23), especially for the order of damage evolution curves.

It is interesting to note that measurements from microcracks (ii) have been shown to fit approximately the relation (23). In fact, as shown in Ref. [20], there are implicit connections between the continuous damage parametrization and the crack length used in Fracture Mechanics. Reference [25] showed that the effective stress concept can be applied to quantify damage for materials showing various cyclic behaviours (stabilized, softening, hardening).

Remarks on the shori crack problem As mentioned previously, the present cumulative damage model incorporates in the same state

variable the micro-initiation and the micro-propagation periods. The growth of short cracks is then described in a global way, using D instead of the crack length measure. The fact that short crack propagation does not follow the long crack behaviour (see Fig. 8 of Ref. [26] for instance) is a good justification to use a more global parametrization, in the framework of Continuum Damage Mechanics instead of Fracture Mechanics.

The fact that the damaging processes are different during the micro-initiation, micro-propagation and macro-propagation phases is obviously recognized. For the sake of simplicity, in the present model, only one damage growth equation is used to describe the two first periods. A recent version of the NLCD model incorporates explicitly an initiation period [27] with different growth equations.

The present model describes to some extent the growth of small cracks below the fatigue limit, as it will be demonstrated in the next section. The evolution below the initial fatigue limit is possibly due to an initial cycling above the fatigue limit. However the detailed physical mechanisms are obviously not correctly reproduced with a single variable and the simple equation (2). For example, the decreasing rate for non-propagating short cracks [26,28] is not described.


In several works [19,29,30], the short crack regime is described in terms of a crack growth rate equation like

dL dN - = BL"At,B

where L is the crack length, Acp the plastic strain range. It is interesting to note the use of a global plastic strain measure (not the strain at the crack tip) which can be justified only in the frame of the Continuum Damage Mechanics (with multiple and homogeneously distributed micro cracks). Moreover, the similarities between equations (24) and (2) are evident. Let us remark that, below the fatigue limit the exponent a of equation (2) is equal to 1, which is the value for a often used in equation (24).

APPLICATIONS

For two -level loading conditions This is a direct application of equation (5). The non-linearity of damage accumulation is correctly

predicted, as shown in several materials [7, 1 1 , 16, 171. One advantage of the NLCD model is to allow the growth of damage below the initial fatigue

limit when the material is submitted to prior cycling above the fatigue limit. In the case of a two-level loading, integration of the model (with a = 1 under the initial fatigue limit) leads to the remaining life at the second level (below the fatigue limit):

Clearly the fatigue limit disappears, even for low initial damage. This effect is observed in the experiments, as shown in Refs [7], [26] and [27].

Block programs Fatigue under block programs very often shows life-ratio summations markedly lower than

unity. It is a common feature of the NLCD model and other models discussed previously to predict correctly this situation. Figure 7 shows the example of Maraging steel with two blocks per sequence (data from Ref. [4]).

A Block-program o Two-levels - Calculation

1 Z(n/NF)

Miner's rule t 0.5-

0 2 blocks

4 4 4 1 block '3 ' I

base block :

1300 88OOO

I n, /1300 = n2 /88M)O I I I-

0 1 2 3 4 5 Fraction of the "base block"

Fig. 7. Fatigue under block loading programmes and the prediction of life summation using the NLCD model; Maraging steel, data from Ref. [4].


Let us note that predicting fatigue under block-program is very easy through sequential calculations [7, 111. Such calculations are manageable with a pocket computer, using some variable change:

where Di is the value of damage at the end of the ith block in the program, with the associated value ai for a. In that block ni cycles are applied. In the case of loadings above the fatigue limit, integration of (2) easily leads to:

y = D!-% (26)

For a block 'T' under the fatigue limit integration is a little more difficult. With tl = 1 , one finds:

Di = Di_l exp(ni/N*) (28)

where NT denotes a fictitious reference number of cycles to failure:

After some manipulations, damage accumulation for the next block above the fatigue limit gives:

In equation (30) (1 - C X ~ + ~ ) / N I + may be evaluated through the product M , u - ' / ~ , without knowing precisely the values of a and Mo. This proves that complete measures of damage are not needed before using the NLCD model for any kind of loading programs, including changes in the mean-stress and load cycles which are below the initial fatigue limit. These last conditions are important when considering the superposition of a high frequency fatigue (high cycle) to a low-cycle fatigue loading.

Application to materials with cyclic hardening For some materials an interaction between hardening and fatigue damage plays an important

role, not taken into account in the above Non-Linear Continuous Damage model. Evidence of such interaction effects follows from one of the following experimental observations, obtained for instance in the room temperature fatigue of 316 stainless steel [33,34]:

A prior monotonic hardening increases markedly the fatigue life under stress control (factors above than loo), due to the change of microstructure, especially near the specimen surface [21]. The repeated stress control shows a better fatigue resistance for the same stress range, especially in the Low-Cycle fatigue domain, which contradicts the classically observed mean-stress influence. The lower damage rate is due to the prior monotonic hardening during the few first cycles (with large plastic strains, due to the lower yield stress). A number of high level cycles (load control) increases the life at a lower level, showing the opposite effect as compare the classical observations.

(i)

(ii)

(iii)


(I- -- 1 RU= - 0.485 1 Tests [‘::,lo 1

x R e = - 1 + R , = O

Calculations

l0Ol ’ ’ ” : ” ‘ : I ” : ’ , ‘ , : 10’ 103 10‘ 105 106

Number of cycles to failure NF Fig. 8. S-N curves for 316L stainless steel subjected to various mean stress levels. Calculations are by the

NLCD model including the hardening effect.

In such cases, the damage rate equation (1) is insufficient. An additional parameter plays role and one can write:

dD = f (oM, 5, Dq) d N (31)

where q is for instance a measure of prior hardening. An explicit model was proposed in Ref. [34] and applied to the 316 stainless steel, where q is a memory of maximum plastic strain range:

Acp M q = - 2

Figure 8 shows the possibilities of the model for experimental repeated conditions under stress control. Let us note the marked difference by comparing to other steels in Fig. 2. The two level test conditions, with an improved life at the second level was also correctly predicted due to the higher hardening during the first level [34].

In the common Low-Cycle Fatigue domain, equation (31) corresponds to a certain extent to a model with a mixed parametrization, stress oM and strain range A$, as in other approaches [35,36].

The model taking into account the interaction with hardening is more complex to determine and to use, due to the need of some constitutive equations (taking into account complex hardening memorization). In fact, it predicts situations similar to the ones described by damage models “with interaction” [21,37], where the damage evolution at the second level is modified by the initial loading. Figure 9 is a schematic of the behaviour of equations of type (3 1) for a two-level loading: the internal state at the beginning of the second level depends not only on the present damage but also on the hardening state, which is different after a high level. Let us note that models with interaction such as in Refs [21] or 1371 need also many additional experimental data.

Use under strain control In that case, the alternative is to use either the correspondence between stress and strain

parameters through the cyclic curve, or to define a similar model in terms of strain. The example


t"

1 - Evolution depending on stresses but without interaction (q, = q2 1 2 - Evolution with interaction (q, # q 2 )

Fig. 9. Schematic of the influence of the hardening variable on the evolution of damage in two-level tests.

of 316 L stainless steel, taken from reference [38] was treated with:

with using t, and t , as empirical parameters (here t, = 0.15%, cu = 3%). Figure 10(a) shows the two-level test conditions and Fig. 10(b) reports both predictions with the linear summation and with the non-linear one for block-programs (in that case the level change was controlled in order to maintain approximately a zero mean-stress for each level). The Marco-starkey model, used in Ref [38], and the present model [Fig. 10(b)] improve the prediction by a factor 5 as compared with the linear rule.

The situation under stress concentration is of a large practical interest. In that case the local stresses (maximum and mean stresses) can be obtained through the use of cyclic constitutive equations and a cycle by cycle analysis following the simplified Neuber's rule [39,40]. Due to stress redistributions, the non-linearity of damage accumulation is less pronounced 1381, but the predictions made with the NLCD model agree fairly well with experiments.

Use under varying temperature If temperature varies during each cycle (that is the case in thermal fatigue), damage growth

equations written in terms of cycle and of cyclic parameters (AE or oM3 5) need the use of an effective or equivalent temperature. The well-known method of Taira [41] implicitly supposes a linear damage accumulation, defining the effective temperature T* by integrating over one cycle:

I 1 dt - 1 -- N : NF(At , T*) = t NF(Ac, T( t ) ) (33)

T* is defined as the temperature (constant) for which the present strain range At produces the same number of cycles of failure N $ as the one calculated by relation (33). In the second member, NF is defined, for each temperature at each instant t , from conventional isothermal data.

F F . E . M S 1111 I

14

1 % '0.3% 0 1 % + 0.2%

'-\

J. L. CHABOCHE and P. M. LFXNE

103 104 1 0 3 104 105

Calculated life NF

Fig. 10. Predictions of fatigue life under two-level tests on 316 stainless steel using the modified NLCD model. (a) Comparison with a linear rule. (b) Fractional life plots; data from Ref. [34].

For the present non-linear damage model the above equation (33) is still applicable, considering linear accumulation only for the step of defining the effective temperature T*:

NF is defined for isothermal conditions by equation (1 1). After determining T* from the reciprocal of equation (34), the damage rate equation can be applied simply as:

dD = f(aM, 8, D,T*) dN (35) This method has been used systematically for the turbine blade superalloy IN 100, taking into

account both fatigue damage under varying temperature and creep damage [9,10,42].

Multiaxial conditions The Non-Linear-Continuous-Damage model was written in its uniaxial form, based on


maximum and mean stresses. Under multiaxial loading conditions, three successive problems are encountered:

The definition of the multiaxial stress parameters associated to the maximum and mean values, for instance under proportional loading. The checking of these parameters and of the damage rate equation for non proportional loading conditions, considering a scalar damage variable (isotropic damage). The anisotropic fatigue damage, that is its directionality, which can influence the damage growth under multilevel conditions with changing the loading direction. The out-of-phase loading conditions is one of the problems.

The generalization of the NLCD model to multiaxial conditions has been proposed in references [43] and [lo], and some partial experimental checking has been done in [36]. The theory is based on equation (2) and a judicious choice for stress invariants corresponding to the two limiting cases (fatigue limit criterion N + co, and monotonic rupture N + 0). This generalization is not discussed in the present paper.

(i)

(ii)

(iii)

CONCLUSIONS

The considered non-linear continuous fatigue damage model (NLCD) describes adequately the various phases of the deterioration processes, including, in a continuous form, the microinitiation and micropropagation stages. It is developed for design purposes and predicts correctly:

(i) The one level or two-level stress controlled fatigue tests on smooth specimens. (ii) The influence of mean stress for the whole domain of number of cycles. (iii) The remaining life, even under the initial fatigue limit. (iv) The block program loading conditions, where Miner’s rule always overpredicts the life. (v) Strain-controlled fatigue tests (even in the Low-Cycle-Fatigue range), including two-levels

and block program tests.

Several similarities with other rules have been pointed-out but the developed model presents some specific advantages; namely:

It describes the non-linear accumulation, even when changing the mean stress. There are no rate-discontinuities as in the models based on the separation between initiation and propagation. It describes the whole S-N curve (10 to lo’ cycles) with a unique set of coefficients, including the effect of mean-stress. The damage growth below the initial fatigue limit after prior damage, which is not the case with some other models. Material parameters are determined easily from the S-N curve and the model can be used for complex programs without difficulty. Interaction effects can be taken into account, using an additional hardening variable. Fatigue damage can be combined in a natural way with creep damage, under constant or varying temperature. A multiaxial generalization can be formulated, combining multiaxial fatigue-limit criteria to the Von-Mises equivalent stress.

Concerning the physical basis of the model, some deficiencies have to be pointed out:

(i) The definition of macro-crack initiation is conventional and some correction to the total rupture life has to be done in order to substract the final propagation, phase.


(ii)

(iii)

(iv)

There is no distinction between the rate equation for the initiation regime and the rate equation in the case of micropropagation. The fatigue limit disappears after an amount of initial damage produced by higher stress levels. The behaviour of short cracks below the initial fatigue limit is described only in an approximate global way, but the classical equations used in a Fracture-Mechanics-based approach have the same form. The Continuum Damage Mechanics approach which supports the above mentioned global measure of damage, presents two kinds of deficiencies: the surface character of fatigue damage and the fact that during the propagation period the number of large defects is small which is inconsistent with a continuum approach.

However, due to the above mentioned correct description of many experimental results in terms of cumulative fatigue damage, the NLCD model constitutes a good and simple approach. For practical engineering applications, it is in fact as simple to use as the linear Palgreem-Miner’s rule is.

Applicability of the Non-Linear Continuous Damage model to fatigue life prediction for stress concentration problems is discussed in Ref. [40]. Further researches will consider both the checking of the multiaxial form and its generalization for out-of-phase loading conditions.

(v)

REFERENCES

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83. 529-540. 3. Marco S. M. and Starkey W. L. (1954) A concept of fatigue damage. Trans. ASME 76(4), 627-632. 4. Manson S. S., Freche J . C. and Ensign C. R. (1967) Application of a double linear damage rule to

cumulative fatigue. NASA TN D 3839. 5. Miller K. J. and Zachariah K. P. (1977) Cumulative damage laws for fatigue crack initiation and stage

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Rev. Fr. M&. No. 5&51, pp. 71-82. English trans]. in Muter. Building Res., Ann de I’ITBTP 39 (1977),

8. Chaboche J. L. (1981) Continuous damage mechanics. A tool to describe phenomena before crack

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