Fatigue life prediction using 2-scale temporal asymptotic ...jfish/fatigue-ijnme-2004.pdf · 330 C....

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2004; 61:329–359 (DOI: 10.1002/nme.1069)

Fatigue life prediction using 2-scale temporalasymptotic homogenization

Caglar Oskay and Jacob Fish∗,†

Civil and Environmental Engineering Department, Rensselaer Polytechnic Institute, Scientific ComputationalResearch Center, 110-8th Street, Troy, NY, 12180, U.S.A.

SUMMARY

In this manuscript, fatigue of structures is modelled as a multiscale phenomenon in time domain.Multiple temporal scales are introduced due to the fact that the load period is orders of magnitudesmaller than the useful life span of a structural component. The problem of fatigue life prediction isstudied within the framework of mathematical homogenization with two temporal co-ordinates. By thisapproach the original initial boundary value problem is decomposed into coupled micro-chronological(fast time-scale) and macro-chronological (slow time-scale) problems. The life prediction methodologyhas been implemented in ABAQUS and validated against direct cycle-by-cycle simulations. Copyright� 2004 John Wiley & Sons, Ltd.

KEY WORDS: fatigue life prediction; crack propagation; temporal homogenization; brittle damage;cohesive model

1. INTRODUCTION

Fatigue of structures is a multiscale phenomenon in space and time. It is multiscale in spacebecause a crack (or microcrack) size may be of several orders of magnitude smaller than astructural component of interest. It is multiscale in time because the load period could bein the order of seconds whereas the component life may span years. Given this tremendousdisparity of spatial and temporal scales, life predictions pose a great challenge to mechanicsand materials science communities.

Fatigue life prediction methods range from experimentation, to modelling and computationalresolution of spatial scales. The basic design tool today is primarily experimental, based onso called S-N curves, which provide the component life versus cyclic stress level. Since there

∗Correspondence to: Jacob Fish, Civil and Environmental Engineering Department, Rensselaer Polytechnic Insti-tute, Scientific Computational Research Center, 110-8th Street, Troy, NY, 12180, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: Office of Naval Research; contract/grant number: N00014-97-1-0687

Received 22 August 2003Published online 29 July 2004 Revised 12 January 2004Copyright � 2004 John Wiley & Sons, Ltd. Accepted 28 January 2004

330 C. OSKAY AND J. FISH

is a scatter of fatigue life data, a family of S-N curves with probability of failure known asS-N-P plots are often used. Fatigue experiments are generally limited to specimens or smallstructural components, and therefore, boundary and initial conditions between the component orinterconnect of interest and the remaining structure involves some sort of modelling. Typicallyfinite element analysis is carried out to predict the ‘far fields’ acting on the critical componentor interconnect. These types of calculations do not take into account force redistribution causedby accumulation of damage taking place in the critical components or interconnects.

Paris law [1] represents one of the first attempts to empirically model fatigue life. It statesthat under ideal conditions of high cycle fatigue (or small scale yielding) and constant ampli-tude loading, the growth rate of long cracks depends on the amplitude of the stress intensityfactors. Models departing from these ideal conditions have also been documented (e.g. Refer-ences [2–6]). Various crack growth ‘laws’ are being used in conjunction with multiple spatialscales methods which allow for propagation of arbitrary discontinuities in a fixed mesh (e.g.References [7, 8]).

An attempt to resolve the temporal scales, i.e. to carry out cycle-by-cycle simulation, inconjunction with a cohesive law crack model based on unloading–reloading hysteresis has beenrecently reported by Nguyen et al. [9]. Cycle-by-cycle simulation, however, may not be feasiblefor large-scale systems undergoing high cycle fatigue. Nevertheless, an attractive feature of thisapproach is that it allows for a unified treatment of long and short cracks and the effects ofoverloads.

The first multiscale computational technique in the time domain, known as the ‘cycle jump’technique [10, 11], has been proposed in the context of continuum damage mechanics. By thisapproach a single load cycle is used to compute the rate of fatigue damage growth at eachspatial integration point and then to construct an ordinary differential equation [12]:

d�(xa)

dN= �K(xa) − �K−1(xa) (1)

where �(xa) is the damage variable at the integration point a; N is the cycle variable; K isthe cycle count; and, �K and �K−1 are the states of the damage variable at the end of Kthand (K − 1)th cycles, respectively. Adaptive integrators have been used to control the accuracyof the integration [12]. Unfortunately, while advancing the state variables (damage variables)the constitutive equations are violated requiring adjustment of governing equations, which maynot be unique.

In this paper, we present a new paradigm for multiscale modelling of fatigue based onmultiple temporal scale asymptotic analysis. The proposed methodology hinges on the hypothesisof local periodicity in the time domain—the concept that serves the foundation of the spatialhomogenization theory [13]. In the temporal homogenization proposed here, both the appliedloads and the response fields are assumed to depend on a slow time co-ordinate, t (slowdegradation of material properties due to fatigue), as well as on the fast time co-ordinate, � (dueto locally periodic loading). The concept of local periodicity implies that at the neighbouringpoints in the time domain (same t), homologous by periodicity (same �), the value of thefunction is the same, but at points homologous by periodicity but separated in t , the value ofthe function can be different. In case of overloads, temporal scales cannot be separated, andthus, the solution has to be resolved in the portion of the time domain where the overloadsare present, while elsewhere in the time domain the temporal homogenization theory developedhere can be exercised.

Copyright � 2004 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:329–359

FATIGUE LIFE PREDICTION USING 2-SCALE TEMPORAL ASYMPTOTIC HOMOGENIZATION 331

The theory is developed for the case of accumulation of distributed damage (microvoids ormicrocracks) and propagation of macrocracks up to failure. For the latter, a cohesive crackmodel in the form of continuum damage mechanics exhibiting unloading–reloading hysteresisdeveloped by Fish and Yu [12] is adopted. Classical crack growth models cannot be applied todirect cycle-by-cycle simulation due to the non-dissipative nature of these models for subcriticalcyclic loading. Crack closure is enforced by selecting appropriate parameters within the damagemodel that effectively serve as a penalty function. In the case of distributed damage, continuumdamage mechanics exhibiting unloading–reloading hysteresis [12] is employed. Damage evolu-tion in the form of a power law along the lines of viscoplasticity is used as a regularizationof local softening behaviour.

2. PROBLEM STATEMENT FOR SOLIDS UNDERGOING FATIGUE DAMAGE

In the present study, multiple temporal scales are introduced to model slow degradation ofmaterial properties due to fatigue, as well as to resolve the deformation within a single loadcycle. The macro-chronological scale is denoted by the intrinsic time co-ordinate, t , whilethe micro-chronological scale is denoted by the fast time co-ordinate, �. A local periodic-ity (�-periodicity) assumption is made for field variables (unless otherwise stated), similar toY -periodicity in spatial homogenization theory. The two scales defined above are related by asmall positive scaling parameter, �:

� = t

�, �>1 (2)

The periodic response fields, �, are defined to accommodate the presence of multiple temporalscales:

��(x, t) = �(x, t, �(t)) (3)

where x represents the spatial co-ordinates. Time differentiation in the presence of multipletemporal scales is given by the chain rule:

��(x, t) = �,t (x, t, �) + 1

��,�(x, t, �) (4)

in which a comma followed by a subscript co-ordinate denotes partial derivative, and superposeddot denotes the total time derivative.

2.1. Failure by accumulation of distributed damage

The failure mechanism of some brittle materials, such as ceramics and concrete, is charac-terized by only minor plastic flow, and is dominated by degradation of material propertiesdue to nucleation, growth and coalescence of microcracks [14, 15]. Furthermore, the resultingpropagation of large cracks usually takes place at a significantly higher rate and constitutes asmall portion of the fatigue life. Continuous damage mechanics (CDM) provide a theoreticalframework to model this type of failure [16]. In CDM, a continuous tensorial quantity, M, isdefined to represent the state of damage, and introduced into the constitutive laws using thethermodynamics of irreversible processes. The order of M generally depends on the properties



of the material as well as the structure and geometry of the microdefects [14]. In the case ofisotropy, the state of damage may be represented by a scalar variable, �� = �(x, t, �) ∈ [0, 1),such that M = (1 − ��)I, in which I is the fourth-order identity tensor. Damage process maybe introduced by considering the free energy density, ��, of the form [17]:

��(��, ��) = (1 − ��)��e(�

�) (5)

where ��e is elastic free energy density. For small elastic deformations:

��e = 1

2�� : L : �� (6)

where �� is the strain tensor, and L is the fourth-order elastic isotropic constitutive tensor.For an isothermal process (i.e. constant temperature), the Clausius–Duhem inequality takes the

form; �� : ��. Differentiating (5) and using the Clausius–Duhem inequality, the following

constitutive equation and the condition of irreversible damage is obtained:

�� = ��(��, ��)

��= (1 − ��)L : ��, ��

e � 0 (7)

in which �� is the stress tensor.The initial boundary value problem (IBVP) for a continuous damage process may then be

expressed as:

Equilibrium equation: ∇ · �� + b(x, t, �) = 0 on � × (0, t0) × (0, �0) (8a)

Constitutive equation: �� = (1 − ��)L : �� on � × (0, t0) × (0, �0) (8b)

Kinematic equation: �� = 12 (∇u� + u�∇) on � × (0, t0) × (0, �0) (8c)

Initial condition: u� = u(x) on � (8d)

Boundary conditions: u� = u(x, t, �) on �u × (0, t0) × (0, �0) (8e)

��n = f(x, t, �) on �f × (0, t0) × (0, �0) (8f)

in which ∇ is the vector differential operator given by ∇ = (�/�x1, �/�x2, �/�x3) in Cartesianco-ordinate system; b is the body force; u� is the displacement vector; t0 is the observation timein macro-chronological (slow) time co-ordinate; �0 is the load period in micro-chronological(fast) time co-ordinate; � and � are the spatial problem domain and its boundary, respectively;u is the initial displacement field; u and f are prescribed displacements and tractions on theboundaries �u and �f , respectively, where � = �u ∪ �f and �u ∩ �f = 0.

When monotonic loading is considered, the evolution of the isotropic continuum damagevariable, ��, is chosen to be a function of a history variable, �, which is characterized by theKuhn–Tucker conditions:

� � 0, �� − � � 0, �(�� − �) = 0 (9)

in which damage equivalent strain, ��, is generally taken to be some measure of strains.Equation (9) represents the sufficient conditions of the irreversible damage accumulation. For



locally periodic loading, unloading–reloading cycles at subcritical load levels also contribute tothe accumulation of damage and must be reflected in the description of damage variable, ��.Such a model, previously developed by Fish and Yu [12] was adopted in this study.

The viscoplasticity like damage evolution model [12] represents the accumulation of fatiguedamage due to cyclic loading:

��(x, t) =

⎧⎪⎪⎨⎪⎪⎩

0 �� < �ini(��

��

)��

��〈��〉+ �� ini

(10)

where 〈·〉+ = [(·) + | · |]/2 are MacCauley brackets; | · | denote the absolute value operator; �iniis a threshold value of �� which represents a domain in strain space where no damage can beaccumulated regardless of the loading path; is a material parameter representing sensitivity tocyclic loading, and �� is a function of damage equivalent strain, which determines the evolutionof damage. In this study, a smooth damage evolution model [12] was adopted:

��(x, t) =arctan

[

(〈�� − �ini〉+

�0

)− �

]+ arctan �

(�/2) + arctan �(11)

where , � and �0 are material constants, and �� is the damage equivalent strain defined asthe square root of the energy release rate [17]:

�� =√

12�� : L : �� (12)

Tensile loading generally leads to a higher rate of damage accumulation when compared tocompression loading [18]. This effect may be modelled by introducing a weighting tensor, F�,and by altering (12) such that

�� =√

12 (F��

)L(F��) (13)

where �� is the principal strain vector; L ≡ L � = L �� is the elastic isotropic constitutivetensor projected onto the principal directions ( , � = 1, 2, 3 and no summation is implied forGreek indices). The strain weighting matrix, F�, is defined as

F� =

⎡⎢⎢⎢⎣

h�1 0 0

0 h�2 0

0 0 h�3

⎤⎥⎥⎥⎦ (14)

h�(��) = 1

2+ 1

�arctan[a1(�

� − a2)] (15)

in which h� is a vector composed of the diagonal components of F�, and a1 and a2 are materialparameters.



A bifurcation analysis was conducted to study the localization characteristics of the fatiguedamage cumulative law under subcritical periodic loading conditions. The analysis, describedin Appendix B, shows that the proposed model acts as a localization limiter when the materialparameter, , is taken to be a function of the strain tensor and the loading history.

2.2. Failure by crack propagation

The response of quasi-brittle and ductile materials is generally dominated by the propagationof a distinct macrocrack [19]. Cohesive theories of fracture have been applied to describesuch processes under monotonic (e.g. Reference [20]) as well as cyclic (e.g. Reference [9])loading. In this paper such a cohesive law was used to model the propagation of macrocracksunder fatigue loads. The direction of propagation is predefined and only small-scale yieldingis considered. Under these conditions, the IBVP may be posed by replacing the constitutiveequation (8b), by a linear elastic law (�� = L : ��) and considering a predefined cohesivezone; �c.

Following Ortiz and Pandolfi [21] a free energy density in the cohesive zone takes a generalform (for an isothermal process):

��c = ��

c(d�, ��) (16)

where d� is the crack opening displacement at the cohesive zone. We assume a linear depen-dence between �� and ��

c:

��c = 1

2 (1 − ��)k(d�)2 (17)

d� = ‖d�‖ (18)

in which k is the initial loading stiffness, and ‖ · ‖ is the Euclidean norm. Using the Clausius–Duhem inequality and arguments presented in the previous section, it can be shown that:

t� = ��c

�d�(19)

where t� is the traction along �c. Inserting (17) into (19) we obtain the constitutive equationat the cohesive zone:

t� = 1

d�

��c

�d�d� (20)

The evolution of damage is modelled using a fatigue cumulative damage law, similar to thatpresented in the previous section. In this case, the damage equivalent strain is expressed interms of the crack opening displacement:

��c =

√12 k(hd

�n + wd

�s )

2 (21)

where d�n = d�n; d

�s = d� −d

�nn; n is the unit normal to �c; h(d

�n) and w(d

�s ) are the weighting

functions (similar to (15)) of crack opening and sliding modes, respectively.



3. TWO-SCALE TEMPORAL ASYMPTOTIC ANALYSIS OF SOLIDS SUBJECTEDTO PERIODIC LOADING

We start by representing the displacement field, u�, using a two-scale asymptotic expansion:

u� = ∑m=0,1,...

�mum(x, t, �) (22)

where um are periodic functions. Using (22) and the chain rule, strain and strain rate fieldsmay be expressed as

�� = ∑m=0,1,...

�m�m(x, t, �), �m = 12 (∇um + um∇) (23)

�� = ∑m=0,1,...

�m−1�m−1(x, t, �), �−1 = �0,�, and �m = �m

,t + �m+1,� (24)

The fatigue damage variable, ��, and the stress field, ��, are also approximated using expansionsin the form of (22):

�� = ∑m=0,1,...

�m�m(x, t, �), �� = ∑m=0,1,...

�m�m(x, t, �) (25)

Considering failure by accumulation of distributed damage and inserting the asymptotic repre-sentations of the strain and stress fields, and damage variable into (8b), various orders of theconstitutive equation are obtained (m = 0, 1, . . .):

O(�0): �0 = (1 − �0)L : �0 (26)

O(�m+1): �m+1 = (1 − �0)L : �m+1 −m+1∑s=1

�sL : �m+1−s (27)

Alternatively, the constitutive equations may be represented in the rate form by using (24) andthe time derivatives of (26) and (27). The leading order constitutive equation becomes:

O(�−1) : �0,� = (1 − �0)L : �0

,� − �0,�L : �0 (28)

and higher order equations may be written in the following form:

O(�m): �m,t = L : �m

,t −m∑

s=0�sL : �m−s

,t −m∑

s=0�s

,tL : �m−s (29a)

�m+1,� = L : �m+1

,� −m+1∑s=0

�sL : �m+1−s,� −

m+1∑s=0

�s,�L : �m+1−s (29b)



Various orders of the equilibrium equation are obtained by substituting the expanded stress fieldinto (8a):

O(1): ∇ · �0 + b(x, t, �) = 0 on � × (0, t0) × (0, �0) (30)

O(�m+1): ∇ · �m+1 = 0 on � × (0, t0) × (0, �0) (31)

Similarly, the initial and boundary conditions may be described using the expanded represen-tations of displacement and stress fields along with (8d)–(8f):

O(1): u0(x, t = � = 0) = u(x) on � (32)

u0 = u(x, t, �) on �u × (0, t0) × (0, �0) (33)

�0n = f(x, t, �) on �f × (0, t0) × (0, �0) (34)

The initial and boundary conditions for higher order problems are trivial.The equilibrium and constitutive equations, along with the initial and boundary conditions

of equal orders are sought to be decomposed into micro- and macro-chronological IBVPs. Atemporal smoothing operator is, therefore, introduced [22]:

〈·〉 = 1

�0

∫ �0

0· d� (35)

The decomposition of the displacement, strain and stress fields are evaluated using the temporalsmoothing operator:

um(x, t, �) = 〈um〉(x, t) + �m(x, t, �) (36a)

�m(x, t, �) = 〈�m〉(x, t) + �m(x, t, �) (36b)

�m(x, t, �) = 〈�m〉(x, t) + �m(x, t, �) (36c)

in which 〈·〉 denote the macro-chronological fields; �m, �m, �m represent the micro-chrono-logical portion of the displacement, strain and stress fields, respectively. Damage variable, ��,is cumulative and irreversible in nature and, therefore, not �-periodic (〈�m

,�〉 = 0). Each unload-reload cycle (hence, each unit cell in time) increases the value of the damage variable. It isconvenient to introduce a somewhat different decomposition to the damage variable:

�m(x, t, �) = �m(x, t) + �m(x, t, �) (37)

in which �m and �m are macro- and micro-chronological strain induced damage variables,respectively, to be subsequently described.

The micro-chronological portion of the O(1) constitutive equation may then be obtained byapplying the strain, stress, and damage variable decompositions, (36b)–(37) to (28). At a giveninstant of the slow time co-ordinate, t :

�0,� = (1 − �0)L : �0

,� − �0,�L : �0 − �0L : �0

,� − �0,�L : 〈�0〉 on � × (0, �0) (38)



ωο )(x t τ, ,ωο )(x t τ, ,

x t τ, ,Λο( )1

x t,( )

x t τ, ,Λο ( )1

zoom

τοτ0οω x t,( )~

+οω

οω~

~

+οω~

t

Figure 1. Schematic interpretations of the micro- and macro-chronologicalstrain induced damage variables.

The macro-chronological portion of the O(1) constitutive equation is obtained by applyingthe averaging operator on (29a) and exploiting the definitions given in (36b)–(37):

〈�0〉,t = (1 − �0)L : 〈�0〉,t − �0,tL : 〈�0〉 − 〈�0L : �0〉,t on � × (0, t0) (39)

The last term in (39) may be evaluated using a Taylor expansion in fast time co-ordinate, �:

〈�0(x, t, �)L : �0〉 = �0(x, t, �1)L : 〈�0〉 + �0,�(x, t, �1)L : 〈(� − �1)�

0〉 + · · · (40)

where �1 ∈ (0, �0). Furthermore, for purely tensile or compressive loading; i.e. when:

�0(x, t, �+)�0(x, t, �−) � 0 ∀� ∈ (0, �o) (41)

where �± = � ± � and �>1 is a small positive constant. Using mean value theorem, it canbe shown that there exists a �1 for which 〈(� − �1)�0〉 = 0 (�1 = �0/2 in case of symmetry).Inserting (40) into (39) and using the above argument:

〈�0〉,t = (1 − �0 − �0(x, t, �1))L : 〈�0〉,t − (�0,t + �0

,t (x, t, �1))L : 〈�0〉 (42)

Figure 1 displays the graphical interpretations of the micro- and macro-chronological straininduced damage variables.

Taking a total time derivative of the asymptotic expansion of the damage variable given in(25) yields:

�� = ∑m=0,1,...

�m−1�m−1(x, t, �), �−1 = �0,� and �m = �m

,t + �m+1,� (43)



Similarly,

�� = ∑m=0,1,...

�m�m(x, t, �), �� = ∑m=0,1,...

�m�m(x, t, �) (44)

�� = ∑m=0,1,...

�m−1�m−1(x, t, �), �−1 = �0,� and �m = �m

,t + �m+1,� (45)

Using the above approximations, the asymptotic expansion of the damage evolution law pre-sented in (10) can be expressed as (for a damage process; i.e. �0 � �ini, �1 � 0 and �0

,� > 0):

� = 1

��0

,� + �0,t + �1

,� + O(�)

=[(

�0

�0

)

+ �

(�0

�0

) (�1

�0 − �1

�0

)+ O(�2)

]

×[

1

�

��0

��0�0,� + ��0

��0�0,t + ��1

��0�0,� + ��1

��1�1,� + O(�)

](46)

where the material parameter is assumed to be constant. The equation is trivial for unloadingand elastic loading (i.e. when �0 < �ini or �1 < 0 or �0

,� < 0). The fatigue damage model of

order O(�−1) may then be approximated by matching lowest order terms in (46):

�0,�(x, t, �) = �0

,�(x, t, �) =

⎧⎪⎪⎨⎪⎪⎩

0 �0 < �ini(�0

�0

)��0

��0〈�0

,�〉+ �0 � �ini

(47)

Furthermore, order O(1) fatigue model may be obtained similarly. For a damage process,matching the order O(�−1) terms in (46):

�0,t + �1

,� =(

�0

�0

)

(�0,t + �1

,�) +

(�0

�0

) (�1

�0 − �1

�0

)�0,� (48)

where

�0,t = ��0

��0�0,t and �1

,� = ��1

��0�0,� + ��1

��1�1,� (49)

To solve for (48), we assume the following decomposition for the higher order damage term,�1:

�1 =(

�0

�0

)

�1 (50)



Substituting (50) into (48) the higher order terms of the above equation are eliminated. Fur-thermore, in the absence of micro-chronological loading, (48) reduces to

�0,t =

(�0

�0

)��0

��0 �0,t (51)

where �0 ≡ �0(〈�0〉) has the same form as in (13).Various orders of the damage equivalent strain may be obtained by the asymptotic expansion

of (13). Exploiting the symmetry of L:

[�0 + ��1 + O(�2)]2 = 12 [F0�0 + �(F0�1 + F1�0

) + O(�2)]L

× [F0�0 + �(F0�1 + F1�0) + O(�2)] (52)

The first two orders of �� may then be given as

O(1): �0 =√

12 (F0�0

)L(F0�0) (53a)

O(�): �1 = 1

2�0 (F0�1 + F1�0)L(F0�1 + F1�0

) (53b)

The diagonal components of the strain weighting matrix, F�, may be expressed in terms ofvectors hm. For instance:

h0 = 1

2+ 1

�arctan[a1(�

0 − a2)], h1 = �1/�

1 + (a1�0 − a2)2

(54)

Expressions for �m can be obtained by the asymptotic expansion of (11) (when �0 � �ini,and �m � 0):

�0 + ��1 + O(�2) =arctan

[

(�0 + ��1 + O(�2) − �ini

�0

)− �

]+ arctan �

(�/2) + arctan �(55)

Matching the equal order terms in (55):

O(1): �0 =arctan

[

( 〈�0 − �ini〉+�0

)− �

]+ arctan �

(�/2) + arctan �(56a)

O(�): �1 =(

�0

(�/2) + arctan(�)

)�1

�20 + [(�0 − �ini) − ��0]2

(56b)

The higher order terms of ��, h� and �� may be evaluated using a similar algebra.



The corresponding equilibrium equation, initial and boundary conditions of order O(1) macro-chronological IBVP are obtained by averaging (30), (32)–(34) over one load cycle using (35).The macro-chronological problem may then be posed as follows:

Equilibrium equation: ∇ · 〈�0〉 + 〈b〉(x, t) = 0 on � × (0, t0) (57a)

Constitutive equation: 〈�0〉,t = (1 − �0 − �0(x, t, �1))L : 〈�0〉,t− (�0

,t + �0,t (x, t, �1))L : 〈�0〉 on � × (0, t0) (57b)

Initial condition: 〈u0〉(x, t = 0) = u on � (57c)

Boundary conditions: 〈u0〉 = 〈u〉(x, t) on �u × (0, t0) (57d)

〈�0〉n = 〈f〉(x, t) on �f × (0, t0) (57e)

The equilibrium equation, initial and boundary conditions of the micro-chronological problemmay be obtained by subtracting those of the macro-chronological problem from (30), (32)–(34), respectively. The micro-chronological IBVP of order O(1) at a given instant of slow timeco-ordinate, t , may then be posed as follows:

Equilibrium equation: ∇ · �0 + b − 〈b〉 = 0 on � × (0, �0) (58a)

Constitutive equation: �0,� = (1 − �0)L : �0

,� − �0,�L : �0

− �0L : �0,� − �0

,�L : 〈�0〉 on � × (0, �0) (58b)

Initial condition: �0 = 0 on � (58c)Boundary conditions: �0 = u(x, t, �) − 〈u〉 on �u × (0, �0) (58d)

�0n = f − 〈f(x, t, �)〉 on �f × (0, �0) (58e)

The constitutive equations of the micro- and macro-chronological problems clearly show atwo-way coupling. Hence, the micro-chronological initial boundary value problem defined by(38), (58c)–(58e) has to be solved at every time step of the macro-chronological problem.Computational aspects and solution procedures of the order O(1) IBVPs are discussed in thefollowing section.

The higher order initial boundary value problems may be obtained using a scheme similarto the formulation of order O(1) problems. The macro-chronological IBVP of order O(�m+1)

is given as:

Equilibrium equation: ∇ · 〈�m+1〉 = 0 on � × (0, t0) (59a)

Constitutive equation: 〈�m+1〉,t = L : 〈�m+1〉,t −m+1∑s=0

�s(x, t, �1)L : 〈�m+1−s〉,t

−m+1∑s=0

�s,t (x, t, �1)L : 〈�m+1−s〉 on � × (0, t0)

(59b)



where �m(x, t, �1) = �m+�m(x, t, �1). Initial and boundary conditions for higher order macro-chronological problems are trivial.

The micro-chronological IBVP of order O(�m+1) at a given instant of slow time co-ordinate,t , is expressed as:

Equilibrium equation: ∇ · �m+1 = 0 on � × (0, �0) (60a)

Constitutive equation: �m+1,� = L : �m+1

,� −m+1∑s=0

(�s + �s)L : �m+1−s,�

−m+1∑s=0

�s,�L : �m+1−s −

m+1∑s=0

�s,�L : 〈�m+1−s〉

on � × (0, �0) (60b)

Similar to the macro-chronological problem, the initial and boundary conditions are trivial. Thehigher order problems defined in (59) and (60) are fully coupled in addition to the contributionsfrom the lower order terms.

In this section, a two scale asymptotic analysis was conducted to resolve the temporal scalesfor the failure processes with accumulation of distributed damage. Asymptotic analysis of thefailure mechanisms with propagation of macrocracks is somewhat similar and will be skipped.

4. COMPUTATIONAL ISSUES

Finite element models of order O(1) micro- and macro-chronological problems were imple-mented and details of the implementation are discussed herein. The constitutive equations ofthe micro- and macro-chronological problems ((58b) and (57b), respectively) were shown to becoupled in the previous section. In this study, these two problems were evaluated in a staggeredmanner. Figure 2 displays a sketch of the general structure of the algorithm. An external batchfile controls the execution of the algorithm which in turn invokes a conventional finite elementprogram. The entire micro-chronological IBVP, which represents a single cycle of loading, isevaluated prior to the execution of each time step of the macro-chronological problem. Thestate variable information transfer between the two problems is conducted through external filesstored on a hard disk. This specific structure of the algorithm is selected to accommodate theuse of commercial finite element packages along with the appropriate user supplied subroutines(e.g. UMAT in ABAQUS).

4.1. Stress update procedure

4.1.1. Micro-chronological problemGiven: Strain and stress tensors, ��0 and ��0, respectively; micro-chronological strain in-

crement, ��0, micro-chronological damage variable, ��0, at the beginning of the time step;

macro-chronological strain tensor and damage variable, 〈�0〉 and �0, respectively. �(·) and�+��(·) denote the state of field variables at the beginning of the time step and their currentvalues, respectively. For simplicity we will often omit the subscript for the current values; i.e.(·) ≡�+�� (·).



External Batch

Commercial FEM software

External file processing

transfer smooth strains;

update time step;

ε0

update micro- damage;

update macro- damage;

Λ

/ FEM meshGeometric model

IBVP at step, tmacro-chronological

Solve

for each macro-chronologicaltime step, t

micro-chronologicalIBVP

Solve entire

0~ω

0

∆t

Figure 2. Schematic of the program architecture for multi-scale fatigue life prediction analysis.

Compute: Stress and strain tensors, �+��0 and �+��

0, respectively, and micro-chronologicalstrain induced damage variable, �+��

0.The stress update procedure for the micro-chronological problem is outlined below:

1. Update strain tensor: �0 = ��0 + ��0.2. Compute the current total strains, �0 = �+��

0 + 〈�0〉, and project the total strain tensorto the principle frame.

3. Compute the damage equivalent strain, �+��0, using (53a) along with L and �0.4. In case �+��0 > ��0 and �+��0 � �ini, update, �0 using (47). An implicit backward Euler

algorithm was used to integrate the fatigue damage law:

� ≡ �+��0 − ��

0 − �+��

(�0

�0 + �0

)

�+��

(��0

��0

)(�+��

0 − ��0) = 0 (61)

where,

��0

��0=

(�0

(�/2) + arctan �

)1

(�0)2 + [(�0 − �ini) − ��0]2 (62)



The above equation was solved for �+��0 using Newton method:

i+1�0 = i�0 −[(

��

��0

)−1

�

]∣∣∣∣∣i�0

(63)

where,

��

��0 = 1 +(

�0 + �0

) (�0

�0 + �0

) (��0

��0

) (�+��

0 − ��0) (64)

5. In case �+��0 � ��0 or �+��0 < �ini, no damage is accumulated at the current step:

�+��0 = ��

0 (65)

6. Compute the stress increment ��0 using (38) and update stress tensor:

�+��0 = ��

0 + ��0 (66)

4.1.2. Macro-chronological problemGiven: Strain and stress tensors, t 〈�0〉 and t 〈�0〉, respectively; macro-chronological strain

increment, �〈�0〉, micro- and macro-chronological strain induced damage variables, t�0(�1)

and t �0, respectively; macro-chronological time step, �t , which is computed adaptively. t (·)

and t+�t (·) denote the state of field variables at the beginning of the time step and their currentvalues, respectively. Similar to the previous section, quantities without left subscript denote thecurrent values; i.e. (·) ≡ t+�t (·).

Compute: Stress and strain tensors, t+�t 〈�0〉 and t+�t 〈�0〉, respectively, and current micro-and macro-chronological strain induced damage variables, t+�t�

0(�1) and t+�t �0, respectively.

The stress update procedure for the macro-chronological problem is summarized below:

1. Update strain tensor: 〈�0〉 = t 〈�0〉 + �〈�0〉.2. Update the micro-chronological strain induced damage variable. In this study, �0 is taken

to be a piecewise linear function of the slow time co-ordinate, t :

t+�t�0(�1) = t�

0(�1) + m0�t (67)

where m0 = [t�0(�0)− t�0(0)]/�o, is the rate of micro-chronological damage growth and

is computed during the evaluation of micro-chronological problem.3. Compute the principal components of the macro-chronological strains, 〈�0〉.4. Compute the damage equivalent strain t+�t �

0 using an Euler Backward algorithm and(51), along with L and 〈�0〉.

5. In case, t+�t �0 > t �

0 and t+�t �0 � �ini, update macro-chronological damage variable, �0

using a similar implicit backward Euler algorithm outlined in the stress update procedureof the micro-chronological problem, by replacing fast time co-ordinate � with t , and �0

with �0 (except for the term with power, , where �0 is omitted).



6. In case t+�t �0 � t �

0 or t+�t �0 < �ini, no damage is accumulated at the current step:

t+�t �0 = t �

0 (68)

7. Compute the stress increment �〈�0〉 using (42) and update stress tensor:

t+�t 〈�0〉 = t 〈�0〉 + �〈�0〉 (69)

4.2. Consistent tangent stiffness

The constitutive equations of the first-order micro- and macro-chronological IBVPs were pre-sented in the previous sections ((38) and (42), respectively). The consistent tangent stiffnessmatrices are developed herein.

4.2.1. Micro-chronological problem. Constitutive equation of the micro-chronological problemmay be rewritten in the following form:

�0,� = (1 − �0 − �0)L : �0

,� − �0,�L : (�0 + 〈�0〉) (70)

In the case of an elastic process, (�0 < �ini) or unloading (〈�0,�〉+ = 0), no damage is

accumulated (�,� = 0). Therefore, (70) reduces to

�0,� = (1 − �0 − �0)L : �0

,� (71)

For a damage process (i.e. �0 � �ini and 〈�0,�〉+ > 0), �0

,� may be expressed in terms ofmicro-chronological strains, �0

,�, using the cumulative damage law defined in (47):

�0,� = S : �0

,� (72)

where S is a second-order tensor. Details of the derivation of (72) is presented in AppendixA. Inserting (72) into (70) and rearranging the terms yields:

�0,� = P : �0

,� (73)

and

P = L : [(1 − �0 − �0)I − (�0 + 〈�0〉) ⊗ S] (74)

in which P is the tangent stiffness tensor for a damage process in the micro-chronologicalscale, and, ⊗ represents a tetradic product of two second-order tensors (e.g. Aijkl = BijCkl forcartesian co-ordinate system).



4.2.2. Macro-chronological problem. The constitutive equation of the first-order macro-chrono-logical problem may be written in the following form:

〈�0〉,t = (1 − �0(x, t, �1))L : 〈�0〉,t − �0,t (x, t, �1)L : 〈�0〉 (75)

In the case of an elastic process and unloading in the slow time scale, �0,t (x, t, �1) = 0 and

constitutive equation reduces to

〈�0〉,t = (1 − �0(x, t, �1))L : 〈�0〉,t (76)

When the macro-chronological loading is at the inelastic range (damage process):

�0,t (x, t, �1) = R : 〈�0〉,t (77)

where R is a second-order tensor. The derivation of R is much similar to the derivation of S,and details are presented in Appendix A. Substituting (77) into (75) and rearranging the termsyields:

〈�0〉,t = K : 〈�0〉,t (78)

and

K = L : [(1 − �0(x, t, �1))I − 〈�0〉 ⊗ R] (79)

where K is the tangent stiffness tensor for a damage process in the macro-chronological scale.

4.3. Evaluation of macro-chronological step size, �t

The accuracy of the proposed two-scale evaluation of fatigue damage evolution depends on thesize of the time incrementation of the macro-chronological problem. Selection of an appropriatetime step size is a trade-off between the resolution of the damage variable or the crack growthrate, and computational cost. We address this problem by employing an adaptive modified Euleralgorithm with maximum damage control. By this approach, the time step for the macro-scaleproblem is partly computed during the micro-chronological analysis.

Let �0|t be the micro-chronological strain induced damage variable, at slow time instant, t .The increment of �0 per one load-cycle may be expressed as

��0|Nt = �0|Nt+1 − �0|Nt (80)

where Nt denotes the load cycle number; �0|Nt = �0(t, 0) and �0|Nt+1 = �0(t, �o) are the stateof damage variable at the beginning and end of the micro-chronological problem, respectively.Using the modified Euler integrator [23], micro-chronological damage variable, �0|Nt+�Nt

, atload cycle, Nt + �Nt , may be defined as

�0|(Nt+�Nt ;�Nt ) = �0|Nt + �Nt

2

(��0|Nt + ��0|Nt+�Nt

)(81)



where � and � refers to the quantities related to the micro- and macro-chronological scales,respectively. ��0|Nt+�Nt

may be obtained by substituting Nt + �Nt for Nt in (80)

��0|Nt+�Nt= �0|Nt+�Nt+1 − �0|Nt+�Nt

(82)

where a first approximation of �0|Nt+�Nt−1 is obtained using a forward Euler scheme:

�0|Nt+�Nt= �0|Nt + �Nt��0|Nt (83)

The value of macro-chronological step size �t is a linear function of �Nt (�t = �0�Nt ),and �Nt is selected heuristically to control accuracy. For instance, the value of �Nt must besufficiently small when the rate of increase of �0 or the crack length is high, and vice versa.This condition may be achieved by constraining the maximum damage accumulation within themacro-chronological time step, �t at all integration points:

�Nt = int

[��0

all

‖��0|Nt ‖k

], k = 1 or ∞ (84)

in which ��0all is the maximum allowable accumulation of damage within a micro-chronological

problem.The adaptive scheme for the macro-chronological step size evaluation is summarized below:Prior to the t th step of the macro-chronological problem:

1. Carry out the micro-chronological IBVP, as described in Section 4.1.1 to obtain �0|Nt+1.Compute the change in damage variable per cycle, ��0|Nt using (80) at each integrationpoint of the mesh.

2. Compute the initial block size �Nt using (84).3. Set the block size to �Nt and compute �0|(Nt+�Nt ;�Nt ) using modified Euler algorithm

defined in (80)–(83).4. Set the block size to �Nt/2 and compute �0|(Nt+�Nt ;�Nt/2) by two successive invocation

of the modified Euler algorithm (note that steps 3 and 4 require the evaluation of micro-chronological model three additional times prior to every macro-chronological step).

5. Compute maximum error at each integration step and compare to the predefined errortolerance etol

k such that

‖�0|(Nt+�Nt ;�Nt ) − �0|(Nt+�Nt ;�Nt/2)‖k � etolk , k = 1 or ∞ (85)

using the values computed at all integration points.6. If (85) holds, set the macro-chronological problem step size at time t to �t = �0�Nt

and set the micro-scale contribution of damage, �0|(Nt+�Nt ) = �0|(Nt+�Nt ;�Nt/2). Printdamage information into external transfer files and exit adaptive algorithm. If (85) doesnot hold, set �Nt = �Nt/2 and go back to step 3.

RemarkAn analogous adaptive algorithm can be applied in the macro-chronological problem to controlthe growth of the macro-chronological strain induced damage, �0.



5. NUMERICAL EXAMPLES

5.1. Four-point bending beam

A four-point bending problem with a configuration defined in Figure 3 is considered for analysisof fatigue failure mechanisms by accumulation of distributed damage. The beam is made ofan isotropic brittle material. Microdefects are assumed to be homogeneously distributed alongthe beam and fatigue damage evolution law is applied to the entire geometry. The materialproperties used to conduct the numerical simulations are outlined in Table I. Nodal loads areapplied at the crossheads of the beam (Figure 3):

f = F� sin(�t) + Ft [1 − exp(−t/td)] (86)

Figure 3. Configuration and finite element mesh of the four-point bending beam model.

Table I. Material properties used in the four-point bendingbeam analysis.

Material property Value

Young’s modulus, E 50 GPaPoisson’s ratio, � 0.3�0 0.05�ini 0.0 8.2� 10.2 4.5a1 1e7a2 0.0



Table II. Loading parameters used in the four-point bendingbeam analysis.

Loading parameter Value

F� 4Ft 35� 20 � rad/htd 3.2 h

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

number of cycles

dam

age

para

met

er, ω

reference solutionmulti-scale solution

Figure 4. Fatigue damage accumulation at the midspan of the beam.

in which F� and Ft are the amplitudes of the periodic and smooth loads, respectively; �is circular frequency; and td is a constant in the order of the characteristic length of themacro-chronological scale. The values used in our simulations are presented in Table II.

Figure 4 illustrates the evolution of damage variable, �0 computed using the proposedadaptive multi-scale algorithm and cycle-by-cycle (reference solution) approach. In cycle-by-cycle approach, each cycle of the loading is resolved and evaluated by a direct numericalalgorithm based on a backward Euler scheme. The depicted damage history is at an integrationpoint at the mid-span of the mesh. Figures 5–7 show the damaged configurations of the beamfor multi-scale and reference algorithms after 1000 load cycles. Stress, strain and damagevariable histories revealed a close agreement between the proposed multi-scale and the referencesolutions.

5.2. Beam under periodic loading

Failure evolution by propagation of macrocracks is investigated on a beam problem as shown inFigure 8. Cohesive elements are placed at the midsection, and an initial flaw is introduced at thecenter bottom of the beam. The material properties used to conduct the numerical simulations



Figure 5. Damage distribution at the failure stage in the four-point bending beam problem.



Figure 6. Stress distribution at the failure stage in the four-point bending beam problem.



Figure 7. Strain distribution at the failure stage in the four-point bending beam problem.



Figure 8. Configuration and the finite element mesh of the beam under periodic loading.

Table III. Material properties used in the beam underperiodic loading example.

Material property (�c ) Value

Young’s modulus, E 5 GPaPoisson’s ratio, � 0.3�0 0.05�ini 0.0 8.2� 10.2 4.5a1 1e7a2 0.0w 0.0

Material property (� ∼ �c) ValueYoung’s modulus, E 70 GPaPoisson’s ratio, � 0.3

are summarized in Table III. Plane strain conditions are considered and the beam is subjectedto periodic displacements (Figure 8):

u = u�

2(1 + cos(�t − �)) (87)

where u� is amplitude of the periodic prescribed displacements, and � is circular frequency.The values used in our simulations are presented in Table IV.

Figure 9 illustrates a close agreement between the fatigue life curves evaluated using theproposed multiscale and the reference (cycle-by-cycle) approach. The stress distributions alongthe beam after 40 000 cycles (when the crack is 15 mm long) and after 150 000 cycles (when



Table IV. Loading parameters used in the beam underperiodic loading example.

Loading parameter Value

u� 2.5 × 10−4

� 20 � rad/h

0 5 10 15

x 104

0

5

10

15

20

25

30

35

number of cycles

crac

k le

ngth

, a (

mm

)

reference solutionmulti-scale solution

Figure 9. Fatigue life curve of the beam under periodic loading.

the crack is 33 mm long) are presented in Figure 10. As can be observed from this figure, acompressive region develops along the path of the crack due to force redistribution along thebeam. This compressive region causes the crack arrest shown in Figure 9.

6. CONCLUSION

This paper presented a new multiscale approach to the prediction of structural fatigue life basedon asymptotic analysis with multiple temporal scales. This approach attempts to resolve thescale disparity between the time span of the period of loading and life span of a structuralcomponent. The theory was developed to analyse the failure mechanisms with the accumulationof distributed damage, and propagation of macrocracks up to failure based on the assumption ofsmall-scale yielding. The effects of the plastic deformations near the crack tip will be addressedin a future publication. The proposed methodology was implemented using a commercial finiteelement package (ABAQUS) and validated against cycle-by-cycle approach.



Figure 10. Stress distribution at the crack tip of the beam under periodic loading.



APPENDIX A. CONSISTENT TANGENT STIFFNESS

In this appendix, the details of the derivation of (72) is presented. The macro-chronologicaldamage evolution equation is expressed using (47). In case of a damage process (i.e. �0 � �ini

and �0,� > 0), �0

,� may be defined as (exploiting the symmetry of L):

�0,� = 1

2�(F0�0

)L(F0�0),� (A1)

where

(F0�0),� = (h0

�0 ),� =

(�h0

��0

�0 + h

)��0

��, = 1, 2, 3 (A2)

in the cartesian co-ordinate system. Double indices do not imply summation convention inthe above equation (and whenever Greek letters are used as indices). Differentiating (15) withrespect to principal strain components:

�h0

��0

= a1/�

1 + a21(�0

− a2)2(A3)

The derivative of principal components of total strain tensor with respect to fast time co-ordinate, �, may be computed using Hamilton’s Theorem:

(�0 )

3 − I1(�0 )

2 + I2�0 − I3 = 0 (A4)

where I1, I2 and I3 are the invariants of the strain tensor:

I1 = tr(�0) = �0ii (A5a)

I2 = 12 (�0 : �0 − I 2

1 ) = 12 (�0

ij �0ij − �0

ii�0jj ) (A5b)

I3 = det(�0) = 16 eijkepqr�

0ip�0

jq�0kr (A5c)

in which tr(·) and det(·) denote trace and determinant of a second-order tensor, respectively,and eijk is permutation symbol. Differentiating (A4) with respect to � yields to the followingequation:

��0

��= (�I1/��)(�0

)2 − (�I2/��)�0

+ �I3/��

3(�0 )

2 − 2I1�0 + I2

(A6)

The fast time derivatives of the invariants may be obtained using (A5):

�I1

��= �ki�kj

��0ij

��= E

[1]ij

��0ij

��(A7a)

�I2

��= (�0

mm�0ki�

0kj − �0

ij )��0

ij

��= E

[2]ij

��0ij

��(A7b)



�I3

��=

(�0ik�

0kj − �0

kk�0ij − 1

2�0mn�

0nm�ik�jk + 1

2�0mm�0

nn�ik�jk

) ��0ij

��= E

[3]ij

��0ij

��(A7c)

Decomposing the total strain vector using (36b), exploiting the definition that macro-chrono-

logical strain tensor 〈�0〉 to be independent of �, ��0 /�� in (A7) may be replaced by ��

0 /��.

Combining (A2), (A3), (A6) and (A7), we obtain

�h0 �

0

��= Z0

ij

��0ij

��, = 1, 2, 3 (A8)

and

Z0 ij =

(a1/�

1 + a21(�0

− a2)2�0 + h0

)E

[1]ij (�0

)2 − E

[2]ij �0

+ E[3]ij

3(�0 )

2 − 2I1�0 + I2

(A9)

The fast time derivative of the damage equivalent strain, �0, may be expressed by combining(A1), (A8) and (A9). In the vectorial form:

�0,� = 1

2�0 (F0�0)LZ0 : �0

,� (A10)

where Z0 is a third-order tensor given by (A9). Finally, substituting the above equation into(47) yields

�0,� = S : �0

,� (A11)

where

S =(

�0

�0

)��0

��0

[1

2�0 (F0�0) : LZ0

](A12)

The derivation of R is similar to the derivation of S and will not be presented separately. Rmay be obtained in the same way as S by replacing fast time variable, �, by t , and evaluatingthe invariants of the strain tensor at � = �1.

APPENDIX B: LOCALIZATION

In this appendix, the localization effect on the fatigue damage cumulative law is investigated.A simple bifurcation analysis outlined by Pan et al. [24] is employed. Given a homogeneouslydeformed body subjected to an incrementally applied static loading, an additional solution(alternative to the homogeneous one) is sought in which the incremental field quantities suchas strain rate, � are discontinuous along a plane with a normal, n. The general condition ofbifurcation may be expressed as [25]:

(D : n ⊗ n)m = Am = 0 (B1)

where D is fourth-order tangent stiffness tensor; m is a unit vector in the direction of thevelocity field, and A(n) is a second-order tensor. Localization occurs when the conditions ofthe above equation is met.



The constitutive equation of a material with continuous fatigue damage was given previouslyas (8b). Rewriting this equation in the rate form (omitting superscript �):

� = (1 − �)L : � − �L : � (B2)

In case of damage process, � � 0 and � > �ini, the damage law was expressed in rate formusing:

� =(

�

�

) ��

�� (B3)

The damage equivalent strain � is expressed using (12) or (13) and it is a function of the straintensor only. Hence using the chain rule on (B3), combining (B2) and (B3), and exploiting thesymmetry of L:

� = D : � (B4)

where

D = L :[(1 − �)I −

(�

�

)

� ⊗ ��

��

](B5)

B.1. Analysis of a one-dimensional fatigue model

A simplified analysis is conducted to evaluate the localization characteristics of the fatiguedamage cumulative law. In a simple one-dimensional problem, the rate relation reduces to

� = E�

⎧⎪⎨⎪⎩

(1 − �) if |�| < �ini(1 − � −

(�

�

) ��

�|�| |�|H(�)

)if |�| � �ini

⎫⎪⎬⎪⎭ = Et � (B6)

where � and � are stress and strain rates, respectively; � is strain, E and Et are elastic andtangent stiffnesses, respectively; �ini = �ini/

√(0.5E) is a threshold strain value, and H(·) is

the Heaviside step function operator. When the spring is subjected to monotonically increasingstrain, the fatigue damage model reduces to a static damage law for a strain softening material.Hence, in this case, there exists a critical strain level, �scr � �ini such that the tangent modulusis vanished

Et =(

1 − �scr − ��

��

∣∣∣∣�scr

�scr

)E = 0 (B7)

in which �scr = �(�scr) is damage variable at the critical strain level. It is also worth mentioning

that the ultimate strength of the material for monotonic loading is reached when � = �scr.For subcritical periodic loading with a constant strain amplitude, �cm < �scr:

Et

E�� ≡ 1 − �(N, �) −

(�

�

) ��

��

∣∣∣∣�cm

�cm (B8)



where N is current number of load cycles. In this case, condition of localization reduces to� = 0 for positive elastic stiffness, E. Noting that � ∈ [0, 1) and �/� � 1, and considering ≡ (N, �) such that:

(N, �) >ln(1 − �) − ln(cm)

ln(�) − ln(�), cm = ��

��

∣∣∣∣�cm

�cm (B9)

the fatigue damage cumulative law acts as a localization limiter, if � < �. In case, � = �, inaddition to (B9), � in (B6) is replaced by � such that:

� = min(tol �, �) (B10)

in which tol ∈ (0, 1).

ACKNOWLEDGEMENT

This work was supported by the Office of Naval Research through grant number N00014-97-1-0687.This support is gratefully acknowledged.

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