computational mechanics of fatigue and life predictions for composite materials and structures

Computational mechanics of fatigue and life predictionsfor composite materials and structures

Jacob Fish *, Qing Yu

Department of Civil Engineering, Mechanical and Aerospace Engineering, 4042 Jonsson Eng. Center, Rensselaer Polytechnic Institute,

Troy, NY 12180-3590, USA

Received 24 January 2000; received in revised form 8 December 2001

Abstract

A multiscale fatigue analysis model is developed for brittle composite materials. The mathematical homogenization

theory is generalized to account for multiscale damage effects in heterogeneous media and a closed form expression

relating nonlocal microphase fields to the overall strains and damage is derived. The evolution of fatigue damage is

approximated by the first order initial value problem with respect to the number of load cycles. An efficient integrator is

developed for the numerical integration of the continuum damage based fatigue cumulative law. The accuracy and

computational efficiency of the proposed model for both low-cycle and high-cycle fatigue are investigated by numerical

experimentation.

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

1. Introduction

In recent years several fatigue models have been developed within the framework of continuum damagemechanics (CDM) [10,12,29,35,36]. Within this framework, internal state variables are introduced to modelthe fatigue damage. The degradation of material response under cyclic loading is simulated using con-stitutive equations which couple damage cumulation and mechanical responses. The microcrack initiationand growth are lumped together in the form of the evolution of damage variables from zero to some criticalvalue. Most of the existing CDM based fatigue damage models are based on the classical (local) continuumdamage theory even though it is well known that the accumulation of damage leads to strain softening andloss of ellipticity in quasi-static problems and hyperbolicity in dynamic problems (see, for example,[1,2,5,6,19,22]). To alleviate the deficiencies inherent in the local CDM theory, a number of regularizationtechniques have been devised to limit the size of the strain localization zone, including the nonlocal damagetheory [1,2] and gradient-dependent models [23,37]. Recent advances in CDM based theories [22,23,30,45]revealed the intrinsic links between the nonlocal CDM theory and fracture mechanics providing a

Comput. Methods Appl. Mech. Engrg. 191 (2002) 4827–4849

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* Corresponding author. Tel.: +1-518-276-6191; fax: +1-518-276-4833.

E-mail address: [email protected] (J. Fish).

0045-7825/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

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possibility for building a unified framework to simulate crack initiation, propagation and overall structuralfailure under cyclic loading.

When applying the CDM based fatigue model to life prediction of engineering systems, the couplingbetween mechanical response and damage cumulation poses a major computational challenge. This isbecause the number of cycles to failure, especially for high-cycle fatigue, is usually as high as tens ofmillions or more, and therefore, it is practically not feasible to carry out a direct cycle-by-cycle simulationfor the fully coupled models even with today’s powerful computers. An efficient integrator, or the so-called‘‘cycle jump’’ technique [9], for the integration of fatigue damage cumulative law must be developed for thefully coupled fatigue analysis. Several efforts have been made in this area as reported in [12] and [36].

For composite materials, the fatigue damage mechanisms are very complex primarily due to the inter-action between microconstituents [14,41–43]. Even though numerous CDM based macro- and microme-chanical models have been developed for static problems (see, for example, [1,11,20,21,25,44]), the researchon the CDM based multiscale fatigue life prediction model for heterogeneous media with consistent cou-pling of mechanical response and damage accumulation is very limited [11].

In this paper, we develop a fatigue analysis model based on the multiscale nonlocal damage theory forcomposites [20] with the fatigue damage cumulative law stated on the smallest scale of interest. In Section 2,the multiscale nonlocal damage theory based on the mathematical homogenization is summarized withemphasis on its application to fatigue of composites. Double-scale asymptotic expansions of damage anddisplacements lead to the closed form expressions relating local (microscopic) fields to overall (macro-scopic) strains and damage. In Section 3, a novel fatigue damage cumulative law is derived by extending theCDM based static damage evolution law proposed in [20]. The integration of fatigue law is approximatedby the first order initial value problem with respect to the number of load cycles. Adaptive modified Euler’smethod in conjunction with the step size control is used to integrate the initial value problem. Consistencyadjustment procedures are introduced to ensure that the integration of the initial value problem preservesthermo-mechanical equilibrium, compatibility and constitutive equations. Computational framework, in-cluding implicit stress update procedures, consistent linearization schemes, and adaptive solution of initialvalue fatigue problem are presented in Section 4. In Section 5, we study the computational efficiency of theproposed multiscale fatigue model and compare its performance with respect to available test data. Dis-cussion and future research directions conclude the manuscript.

2. Multiscale nonlocal damage theory for composites

The microstructure of a composite material is assumed to be statistically homogeneous with localperiodicity. The representative volume element (RVE) can be identified as shown in Fig. 1, where RVE isdenoted by H. The size of RVE is assumed to be small compared to the characteristic length of macro-

Fig. 1. Macroscopic and microscopic structures.

4828 J. Fish, Q. Yu / Comput. Methods Appl. Mech. Engrg. 191 (2002) 4827–4849

domain so that the asymptotic homogenization applies [7,38]. Let x be the macroscopic coordinate vectorin the macrodomain X and y � x=1 be the microscopic position vector in H. 1 denotes a very small positivenumber compared with the characteristic length of X and y � x=1 is a stretched coordinate vector in themicroscopic domain. Due to the high heterogeneity in the microstructure, the local oscillations exist for allthe mechanical quantities. In this respect, all quantities have two explicit dependencies: one on the mac-roscopic level x and the second one, on the level of microconstituents y � x=1. Using the classical no-menclature, any Y-periodic function can be represented as

f 1ðxÞ � f ðx; yðxÞÞ; ð1Þ

where superscript 1 denotes the Y-periodic function f.To model fatigue damage, we define a scalar damage variable x1 as a function of microscopic and

macroscopic position vectors, i.e., x1 ¼ xðx; yÞ. The constitutive equation on the microscale is derived fromthe strain-based continuum damage theory. Following [39], the free energy density for has the form of

Wðx1; e1ijÞ ¼ ð1� x1ÞWeðe1

ijÞ; ð2Þ

where x1 2 ½0; 1Þ is a scalar damage variable on the microscale; Weðe1ijÞ is elastic free energy density function

given as Weðe1ijÞ ¼ ð1=2ÞLijkle

1ije

1kl. Throughout this paper, the summation convention for repeated subscripts

is employed, except for the subscripts x and y. We assume that the microconstituents possess homogeneousproperties and satisfy the following boundary value problem:

r1ij;xjþ bi ¼ 0 in X; ð3Þ

r1ij ¼ ð1� x1ÞLijkle

1kl in X; ð4Þ

e1ij ¼ u1

ði;xjÞ in X; ð5Þ

u1i ¼ ui on Cu; ð6Þ

r1ijnj ¼ _tti on Ct; ð7Þ

where r1ij and e1

ij are components of stress and strain tensors; Lijkl denotes the elastic constitutive tensorcomponents; bi is a body force assumed to be independent of y; u1

i denotes components of the displacementvector; X is the macroscopic domain of interest with boundary C; Cu and Ct are the boundary portionswhere displacements ui and tractions _tti are prescribed, respectively, such that Cu \ Ct ¼ ; and C ¼ Cu [ Ct;ni denotes the normal vector component on C. We assume that the interface between the phases is perfectlybonded, i.e., ½r1

ijnnj� ¼ 0 and ½u1i � ¼ 0 at the interface, Cint, where nni denotes the normal vector to Cint and ½ �

is a jump operator; the subscript pairs with parenthesizes denote the symmetric gradients defined

as u1ði;xjÞ � ðu

1i;xj þ u1

j;xiÞ=2.

Since the discretization with the mesh size comparable to the scale of microscopic constituents is com-putationally prohibitive, mathematical homogenization theory is employed to account for microstructuraleffects without explicitly representing the details of microstructure in the global analysis. This is accom-plished by approximating the displacement field, u1

i ðxÞ ¼ uiðx; yÞ and the damage variable, x1ðxÞ ¼ xðx; yÞ,in terms of the double-scale asymptotic expansions on X�H:

uiðx; yÞ � u0i ðx; yÞ þ 1u1

i ðx; yÞ þ � � � ; ð8Þ

xðx; yÞ � x0ðx; yÞ þ 1x1ðx; yÞ þ � � � ; ð9Þ

J. Fish, Q. Yu / Comput. Methods Appl. Mech. Engrg. 191 (2002) 4827–4849 4829

where the superscripts denote the order of terms in the asymptotic expansion. With these expansions, wehave developed a nonlocal damage theory for brittle composites in [20]. In the rest of this section, we merelypresent the major results which are closely related to the fatigue problem of interest in this paper. Thecomplete derivation is referred to [20].

By inserting the expansion (8) and (9) into the boundary value problem (3)–(7), we can obtain a set ofequilibrium equations for the various orders of 1 starting from 1�2. The solution to these equations gives theasymptotic expansion of the strain field in RVE as

eijðx; yÞ ¼ AijmnðyÞ�eemnðxÞ þ GijmnðyÞdxmnðxÞ þOð1Þ; ð10Þ

where �eemnðxÞ is the elastic strain in macrodomain and dxmnðxÞ is a damage-induced macroscopic strain;

AijmnðyÞ is the so-called elastic strain concentration function [25] given by

Aijmn ¼ Iijmn þ Gijmn; Iijmn ¼ 12ðdimdjn þ dindjmÞ; ð11Þ

where dmk is Kronecker delta; GijmnðyÞ is termed as the local distribution function of damage-induced strain,which can be obtained by solving a linear boundary value problem in H with Y-periodic boundary con-ditions, i.e.,

fLklijðIijmn þ Hði;yjÞmnðyÞÞg;yl ¼ 0; ð12Þ

where HimnðyÞ is a Y-periodic third rank tensor with symmetry Himn ¼ Hinm, and GijmnðyÞ ¼ Hði;yjÞmnðyÞ. Thedamage-induced strain dx

mnðxÞ can be related to the elastic strain in macrodomain through a fourth ranktensor

dxmnðxÞ ¼ DklmnðxÞ�eemnðxÞ; ð13Þ

where DklmnðxÞ, which is determined by the damage state in each microconstituents, represents the influenceof the fatigue damage cumulation on the macroscopic response. Following [20], we have

DklmnðxÞ ¼ Iklst

�Xng¼1

BðgÞklstxðgÞðxÞ

!�1 Xng¼1

CðgÞstmnxðgÞðxÞ

!; ð14Þ

where g denotes different phases in RVE such thatSng¼1

HðgÞ ¼ H; xðgÞðxÞ represents the phase averagedamage; BðgÞijkl and CðgÞijkl are given by

BðgÞijkl ¼1

jHj ð~LLijmn � LijmnÞ�1

ZHðgÞ

GstmnLstpqGpqkl dH; ð15Þ

CðgÞijkl ¼1

jHj ð~LLijmn � LijmnÞ�1

ZHðgÞ

GstmnLstpqApqkl dH; ð16Þ

~LLijmn ¼1

jHj

ZHLijmndH and Lijkl ¼

1

jHj

ZHLijmnAmnkl dH; ð17Þ

where jHj is the volume of a RVE notice that Lijkl is the homogenized elastic stiffness tensor [33]. Accordingto (13) and (14), it is clear that the damage-induced strain dx

mnðxÞ vanishes when the microstructure is free ofdamage.

After solving for the local strain field in the RVE, the homogenized field can be obtained by the phaseaverage process. By integrating (10) over HðgÞ and making use of (13), we have

eðgÞij ¼1

jHðgÞj

ZHðgÞ

eij dH ¼ AðgÞijkl�eekl þ GðgÞijklDklmn�eemn þOð1Þ; ð18Þ


where

AðgÞijkl ¼1

jHðgÞj

ZHðgÞ

Aijkl dH and GðgÞijkl ¼1

jHðgÞj

ZHðgÞ

Gijkl dH: ð19Þ

The constitutive equating for the phase average field can be expressed as

rðgÞij ¼ ð1� xðgÞÞLðgÞijmneðgÞmn ; ð20Þ

where rðgÞij is the phase average stress, and the overall homogenized stress field turns into

�rrij ¼Xng¼1

vðgÞrðgÞij ; ð21Þ

where vðgÞ is the volume fractions for phase HðgÞ in RVE satisfyingPn

g¼1 vðgÞ ¼ 1. The phase free energy

density corresponding to the nonlocal constitutive Eq. (20) is given as

WðgÞðxðgÞ; eðgÞij Þ ¼ 12ð1� xðgÞÞLðgÞijmne

ðgÞmneðgÞij ð22Þ

and the corresponding phase damage energy release rate and the energy dissipation inequality [9,39] appliedto the phase average field can be expressed as

Y ðgÞ ¼ � oWðgÞ

oxðgÞ¼ 1

2LðgÞijmne

ðgÞmneðgÞij ; ð23Þ

Y ðgÞ _xxðgÞP 0: ð24ÞIt should be noted that the nonlocal character of the phase average damage xðgÞ and the constitutive Eq.(20) has been proved in [20]. This important feature validates our homogenization theory for simulating thedamage evolution in composite materials [1,2,5,23,30,37]. In the next two sections, a fatigue damage cu-mulative law for the two-phase composites as well as computational framework, including implicit stressupdate procedures, consistent linearization schemes, and adaptive integration of the initial value fatigueproblem are described.

3. Fatigue damage cumulative law

In cyclic fatigue process, the damage accumulation is usually dependent of previous damage history,loading sequence and frequency, material properties, and environmental effects. A large number of purelyphenomenological fatigue damage cumulative laws have been proposed since the middle of this century[15,16]. Within the framework of CDM, several fatigue damage laws for homogeneous materials weredeveloped recently (see, for example, [3,4,9,12,13,16,29,36]). In [35,42], the application of CDM to thefatigue of heterogeneous media is also explored. An obvious advantage of these CDM based damage cu-mulative laws is their consistency with continuum mechanics, with which a unified fatigue analysis modelcould be developed for numerical simulation. In this section, a new CDM based fatigue damage cumulativelaw for composites is developed following the scheme proposed in [29].

In our previous work [20], we defined the nonlocal ‘‘static’’ damage variable xðgÞ as an increasing functionof nonlocal phase deformation history parameter, jðgÞ, which characterizes the maximum deformation stateexperienced in the local neighborhood throughout the loading history. jðgÞ represents the evolving boundaryof a reversible domain, i.e., all strain states are either within this domain or on the boundary of this domain.The evolution of the nonlocal phase static damage at a given time can be expressed as

xðgÞðx; tÞ ¼ UðhjðgÞðx; tÞ � �##ðgÞini iþÞ and

oUðh �##ðgÞðx; tÞ � �##ðgÞini iþÞ

ojðgÞP 0; ð25Þ


where xðgÞ 2 ½0; 1Þ; the operator h iþ denotes the positive part, i.e., h iþ ¼ supf0; g; the phase deformationhistory parameter jðgÞ is determined from the evolution of the phase damage equivalent strain, denoted by�##ðgÞ

jðgÞðx; tÞ ¼ maxf �##ðgÞðx; sÞjðs6 tÞ; �##ðgÞini g; ð26Þwhere �##

ðgÞini represents the threshold value of damage equivalent strain prior to the initiation of phase

damage; �##ðgÞ is defined as the square root of the phase damage energy release rate [39]

�##ðgÞ ¼ffiffiffiffiffiffiffiffiY ðgÞp

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12LðgÞijkle

ðgÞij eðgÞkl

q: ð27Þ

Since LðgÞijkl is assumed to be a positive definite fourth order tensor it follows that Y ðgÞP 0 and consequently,_xxðgÞP 0 must hold due to the energy dissipation inequality in (24).

To generalize the above static damage evolution law to cyclic fatigue, we first reformulate the static law.The key is to preserve the irreversible character of the internal state variable xðgÞ and to relate it to thenonlocal phase deformation history. Without loss of generality this can be accomplished by reformulating(25) and (26) as

xxðgÞðx; tÞ ¼ Uðh �##ðgÞðx; tÞ � �##ðgÞini iþÞ and

oUðh �##ðgÞðx; tÞ � �##ðgÞini iþÞ

o �##ðgÞP 0; ð28Þ

xðgÞðx; tÞ ¼ maxfxxðgÞðx; sÞjðs6 tÞg; ð29Þwhere is xxðgÞ 2 ½0; 1Þ termed here as a nonlocal ‘‘pseudo-damage’’ parameter. By setting the so-called‘‘gauge function’’ [29] as

f ¼ xxðgÞ=xðgÞ ¼ UðgÞ=xðgÞ ð30Þthe rate form of the static phase damage evolution law (29) turns into

_xxðgÞðx; tÞ ¼0 f < 1;oUðgÞ

o �##ðgÞ_�##�##ðgÞ

D Eþ

f ¼ 1;

(ð31Þ

where UðgÞ � Uðh �##ðgÞðx; tÞ � �##ðgÞini iþÞ can be expressed as an arctangent form damage evolution proposed in

[20]:

UðgÞ ¼atan aðgÞ

h �##ðgÞ � �##ðgÞini iþ

�##ðgÞ0

!� bðgÞ

" #þ atanðbðgÞÞ

p2þ atanðbðgÞÞ

; ð32Þ

where aðgÞ, bðgÞ, �##ðgÞ0 are material constants; �##ðgÞ0 denotes the threshold of the strain history parameter beyond

which the damage will develop very quickly. The calibration of these parameters can be performed by usingthe quasi-static uniaxial loading test with the specimen made of phase materials [27]. The phase endurancelimit �##

ðgÞini can also be calibrated by uniaxial tests while the triaxial stress test may give better results [4].

According to [29,39], it can be proved that gauge function f has the classic yield surface properties.Indeed, with the definition (31), there exists a closed reversible domain in the strain space bounded by thegauge function such that the damage does not increase from any interior point but may develop from thestate on the boundary. The damage loading/unloading (inelastic/elastic) condition for any state onthe boundary is determined by the sign of _�##�##

ðgÞ. From (28) and (31) it can be seen that sgnð _�##�##ðgÞÞ ¼ sgnð _ff Þ. To

extend this static damage evolution law to account for the fatigue damage cumulation, Marigo [29] pro-posed to drop the yield surface concept and to replace it by an irreversible loading/unloading concept. Inthis respect, the boundary of the strain space domain is assumed to be fixed and the strain state is permitted


to penetrate it in the loading history, and accordingly, an extension of (31) along the lines of the power lawin viscoplasticity [34] is in the form of

_xxðgÞðx; tÞ ¼0 �##ðgÞ < �##

ðgÞini ;

UðgÞ

xðgÞ

� �cðgÞ

oUðgÞ

o �##ðgÞh _�##�##ðgÞiþ �##ðgÞP �##

ðgÞini ;

8><>: ð33Þ

where �##ðgÞini can be interpreted as the endurance domain within which the change of strain state along any

loading path does not lead to the growth of damage; cðgÞ is a stress dependent parameter. When cðgÞ ! 1,the fatigue damage power law reduces to the static damage law (31) in the sense that the damage evolutionis controlled by the value of f ¼ UðgÞ=xðgÞ. The difference between the two models is schematically illus-trated in Fig. 2.

Based on the same concept, two similar fatigue damage cumulative laws have been developed in [35] forhomogeneous material and [36] for concrete on macroscopic scale. It has been proved in [36], by assumingconstant exponent in the power law, that fatigue damage cumulation law in the form of (33) can be reducedto the modified Palmgreen–Miner’s model [16]. More sophisticated fatigue damage model reveals that theexponent is better off defined as a stress dependent parameter, i.e., the function of the maximum stress andmean stress value in uniaxial cyclic loading [10,35]. In this sense, we assume that parameter cðgÞ depends onthe mean and the maximum values of principal phase stresses, i.e.,

cðgÞ ¼ gðcðgÞi ; rðgÞmax; rðgÞmeanÞ; ð34Þ

where cðgÞi is a set of material constants; rðgÞmax and rðgÞmean are dimensionless quantities defined as

rðgÞmax ¼rrðgÞ1 max

2rðgÞu

and rðgÞmean ¼rrðgÞ1 max þ rrðgÞ1 min

2rðgÞu

; ð35Þ

where rrðgÞ1 max and rrðgÞ1 min are the maximal and minimal principal phase average stresses at a given global

position; rðgÞu is the ultimate strength of the phase material. The calibration of the material constants in (34)is not trivial, since cðgÞ is a function of the phase average stresses, whereas the only experimental dataavailable is the number of cycles to failure, or fatigue life Ncr. Thus to calibrate material constants we set the

following inverse problem: Find the material constants, cðgÞi , so that jjN�crðcðgÞi Þ �N crjj2 is minimal, where

N cr ¼ ½N 1cr; . . . ;N

kcr�

Tis a set of experimental fatigue life predictions obtained with various cyclic initial

loading conditions; N�crðcðgÞi Þ is a set of predicted fatigue life. The Jacobian matrix for the least square

analysis could be evaluated using the finite difference method.

Fig. 2. Comparison between static and fatigue damage cumulation laws.


Another important issue in modeling the fatigue damage cumulation is the different effect of tension andcompression. In fatigue process, it is well recognized that the damage growth and stop are not strictly inaccordance with the tension and compression [15,31,41]. We notice that the definition of damage ‘‘drivingforce’’ in (27) cannot account for such kind of phenomena. To remedy this incapability, we redefine thedamage equivalent strain as �##ðgÞ as follows:

�##ðgÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12LLðgÞijklee

ðgÞmnF

ðgÞijmnee

ðgÞst F

gklst

q; ð36Þ

where LLðgÞijkl and eeðgÞij represent the elastic phase constitutive tensor and nonlocal phase strain tensor, re-spectively, both expressed in principal directions; F ðgÞijkl denotes the phase strain weighting tensor. In matrixnotation Eq. (36) is given as

�##ðgÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ðFðgÞeeðgÞÞTLLðgÞðFðgÞeeðgÞÞ

q; ð37Þ

where the superscript T represents the transpose; the principal nonlocal phase strain can be written aseeðgÞ ¼ ½eðgÞ1 ; eðgÞ2 ; eðgÞ3 �

Tand the phase strain weighting matrix is defined as

FðgÞ ¼hðgÞ1 0 0

0 hðgÞ2 0

0 0 hðgÞ3

264

375; ð38Þ

where hðgÞn � hðeðgÞn Þ, n ¼ 1, 2, 3 are the weighting functions for each component of principal strain. Thedefinition of weighting function hðeðgÞn Þ is dependent of material properties and environmental conditions.The detail exploration on the effect of tension and compression on fatigue damage cumulation prior to themacrocrack initiation is out of the content of this paper. We refer the interested readers to [41] for thecomprehensive discussion. Here, we merely introduce a heuristic expression in the form of

hðgÞn � hðeðgÞn Þ ¼1

2þ 1

patan½a1ðeðgÞn � a2Þ�; n ¼ 1; 2; 3; ð39Þ

where the constants a1 and a2 are schematically illustrated in Fig. 3. In the limit, as a1 !1 and a2 ¼ 0, theweight function reduces to hðeðgÞn Þ ¼ he

ðgÞn iþ=e

ðgÞn , which corresponds to the case where phase strain in

compression does not promote damage.

Fig. 3. Phase strain weighting function.


4. Computational issues

In this section, we focus on computational aspects of the CDM based multiscale fatigue model developedin the previous two sections. We assume that the composite material consists of two phases, matrix ðg ¼ mÞand reinforcement ðg ¼ f Þ, denoted by HðmÞ and Hðf Þ such that H ¼ HðmÞ [Hðf Þ. For simplicity, we assumethat fatigue damage occurs in the matrix phase only, i.e., xðf Þ � 0. The volume fractions for matrix andreinforcement are denoted as vðmÞ and vðf Þ, respectively, such that vðmÞ þ vðf Þ ¼ 1. The overall elastic prop-erties (17) are given as

Lijkl ¼ vðmÞLðmÞijmnAðmÞmnkl þ vðf ÞLðf ÞijmnA

ðf Þmnkl ð40Þ

and the overall stress defined in (21) reduces to

rij ¼ vðmÞrðmÞij þ vðf Þrðf Þij ; ð41Þ

where the nonlocal phase average stresses rðmÞij and rðf Þij are defined by (20). Due to nonlinear character ofthe problem an incremental finite element analysis is employed. Prior to nonlinear analysis elastic strainconcentration factors, AijklðyÞ, are precomputed using (12) and (11) in the microscopic domain (RVE) byeither finite element method or, if possible, by solving an inclusion problem analytically. Subsequently, thephase elastic strain concentration factors AðgÞijkl ðg ¼ m; f Þ and damage strain concentration factors GðgÞijkl areprecomputed using (19).

In the remaining of this section we focus on three computational issues: (i) implicit micro- and macro-stress update (integration) procedures, (ii) consistent linearization, and (iii) integration of the phase fatiguedamage cumulative law.

4.1. Stress update procedures

Given: The displacement vector tum; the overall strain t�eemn; the damage variable txðmÞ; the displacementincrement Dum calculated from the incremental finite element analysis of the macroproblem. The leftsubscript denotes the increment step, i.e., tþDt� is the variables at the current increment, whereas t� is aconverged variable from the previous increment. For simplicity, we will often omit the left subscript for thecurrent increment, i.e., � � tþDt�.

Find: The displacement vector um � tþDtum ¼ tum þ Dum; the overall strain �eemn; the nonlocal phase strainseðmÞmn and eðf Þmn ; the nonlocal phase damage variable xðmÞ; the overall stress rmn and the nonlocal phase stressesrðmÞmn and rðf Þmn .

The stress update procedure consists of the following steps:

(i) Calculate the macroscopic strain increment, D�eemn ¼ Duðm;xnÞ and update the macroscopic strains by�eemn ¼ t�eemn þ D�eemn.

(ii) Compute the principal components of eðmÞmn by (18) and the damage equivalent strain �##ðmÞ by (37) interms of txðmÞ and �eemn.

(iii) Check the inelastic/elastic process conditions by (33) where _�##�##ðmÞ

is integrated as tþDtD �##ðmÞ ¼ �##ðmÞ � t�##ðmÞ.

If inelastic process, i.e., �##ðmÞ > t�##ðmÞ and �##ðmÞP �##

ðmÞini then update xðmÞ through integration of (33).

The integration of the fatigue damage cumulative law (33) is carried out using the backward Eulerscheme such that


@ðmÞ � xðmÞ � txðmÞ �

tþDt

UðmÞ

xðmÞ

� �cðmÞ

tþDt

oUðmÞ

o �##ðmÞ

" #ð �##ðmÞ � t

�##ðmÞÞ ¼ 0: ð42Þ

Since �##ðmÞ is governed by the current average strains in the matrix phase which in turn depend on thecurrent damage variable, it follows that (42) is a nonlinear function of xðmÞ. Newton method is used to solvefor xðmÞ:

iþ1xðmÞ ¼ ixðmÞ � o@ðmÞoxðmÞ

� ��1

vðmÞ" #��

ixðmÞ

; ð43Þ

where the derivation of the derivative term in (43) is given in Appendix A.Otherwise, for elastic process: xðmÞ ¼ txðmÞ.

(vi) Update the nonlocal phase strains eðmÞmn and eðf Þmn by (18).(v) Calculate the nonlocal phase stresses rðmÞij and rðf Þij using (20) and update the macroscopic stresses rij

defined by (41).

4.2. Consistent tangent stiffness

In this subsection, we focus on the derivation of the consistent tangent stiffness matrix for the macro-problem. We start by substituting (18) into (20) and then take material derivative of the incremental form of(20) in the matrix domain ðg ¼ mÞ, which yields

_rrðmÞij ¼ P ðmÞijmn_�ee�eemn þ QðmÞijmn�eemn _xx

ðmÞ; ð44Þ

where

P ðmÞijmn ¼ ð1� xðmÞÞLðmÞijklðAðmÞklmn þ GðmÞklstD

ðmÞstmnÞ; ð45Þ

QðmÞijmn ¼ ð1� xðmÞÞLðmÞijklGðmÞklstR

ðmÞstmn � LðmÞijklðA

ðmÞklmn þ GðmÞklstD

ðmÞstmnÞ: ð46Þ

The fourth order tensor RðmÞstmn is obtained by taking derivative of the nonlocal matrix strain eðmÞkl defined in(18) with respect to xðmÞ such that

oeðmÞkl

oxðmÞ¼ GðmÞklstR

ðmÞstmn�eemn; ð47Þ

RðmÞstmn ¼ ðIstpq � BðmÞstpqxðmÞÞ�2CðmÞpqmn: ð48Þ

To obtain _xxðmÞ, we make use of damage cumulative law (33) with the inelastic/elastic process conditionsdefined in Section 3. In the case of elastic process, we have _xxðmÞ ¼ 0. For inelastic process, the derivation of_xxðmÞ is detailed in Appendix A and the final result can be expressed as

_xxðmÞ ¼ W ðmÞij

_�ee�eeij: ð49Þ

Substituting (49) into (44) and manipulating the indices, we get the following relation between the rate ofthe overall strain and nonlocal phase stresses in the matrix domain

_rrðmÞij ¼ }ðmÞijmn

_�ee�eemn; ð50Þ


where

}ðmÞijmn ¼ P ðmÞijmn þ QðmÞijst �eestW

ðmÞmn : ð51Þ

A similar result relating the rate of the nonlocal reinforcement stress and the overall strain rate can beobtained by substituting (18) into (20) and then taking material derivative of (20) in the reinforcementdomain ðg ¼ f Þ

_rrðf Þij ¼ }ðf Þijmn

_�ee�eemn; ð52Þ

where

}ðf Þijmn ¼ P ðf Þijmn þ Qðf Þijst�eestW

ðmÞmn ; ð53Þ

P ðf Þijmn ¼ Lðf ÞijklðAðf Þklmn þ Gðf ÞklstD

ðmÞstmnÞ; ð54Þ

Qðf Þijst ¼ Lðf ÞijklGðf ÞklmnR

ðmÞmnst: ð55Þ

Finally, the overall consistent tangent stiffness is constructed by substituting (50) and (52) into the rate formof the overall stress–strain relation (41)

_rrij ¼ }ijmn _�ee�eemn; ð56Þ

}ijmn ¼ vðmÞ}ðmÞijmn þ vðf Þ}ðf Þijmn: ð57Þ

4.3. Integration of fatigue damage cumulative law

To develop an efficient accelerating technique for the integration of fatigue damage cumulative law, aconstant amplitude cyclic loading history is typically subdivided into a series of load cycle blocks, and eachblock consists of several load cycles. One of the integration schemes developed in [36] assumes that in eachblock of cycles, the mechanical response is independent of fatigue damage cumulation until the localrupture occurs. In another model developed for homogeneous materials [12], the first cycle in the block inwhich the damage increment is caused by inelastic deformation, is used to compute a constant rate offatigue damage growth in that block. The major shortcomings of this model are threefold: (i) the deviationfrom the equilibrium path caused by the integration of fatigue damage cumulation law, (ii) the difficulty inestimating an adequate block size, especially in the initial and near-rupture loading stages where the growthof fatigue damage is very rapid, and (iii) applicability to heterogeneous materials.

In what follows the damage cumulative law will be approximated by the first order initial value problemwith respect to the number of load cycles, and subsequently solved using the adaptive modified Euler’smethod with the maximum damage increment control and consistency adjustment.

Let us return to the fatigue damage cumulative law defined in (33). Since this fatigue law is stated in therate form, it is necessary to integrate it along the loading path to obtain the current damage state. Thenonlocal matrix phase damage increment in one load cycle can be expressed asZ ðtþs0Þ

t_xxðmÞ dt ¼

Z ðtþs0Þ

t

UðmÞ

xðmÞ

� �cðmÞoUðmÞ

o �##ðmÞh _�##�##ðmÞiþ dt; ð58Þ

where t is the time at the beginning of a load cycle and s0 is the period of the cyclic loading. The aboveintegration has to be carried at each Gauss point in the macrodomain. Assuming that the increment ofphase damage in one load cycle is very small, we can approximate the derivative of the nonlocal damageparameter with respect to the number of load cycles as


dxðmÞ

dN

��K

�Z ðtþs0Þ

t_xxðmÞ dt � DxðmÞjK ¼ xðmÞjK � xðmÞjK�1; ð59Þ

where N denotes the number of load cycles; xðmÞjK is the phase damage at the end of load cycle K which canbe obtained by the incremental finite element analysis for this cycle with initial damage xðmÞjK�1 and thecorresponding initial strain/stress conditions.

Using the forward Euler’s method the nonlocal phase damage after DNK cycles from the current loadcycle K can be approximated by

xðmÞjðKþDNk ;DNK Þ ¼ xðmÞjK þ DNKDxðmÞjK ; ð60Þ

where xðmÞjðKþDNk ;DNK Þ represents the approximate solution of the nonlocal phase damage at the end of loadcycle K þ DNK with the block size DNK and the initial nonlocal damage xðmÞjK .

It is important to note that updating the damage variable while keeping the rest of the fields fixed vi-olates constitutive equations. This inconsistency is subsequently alleviated in two steps: (i) update thenonlocal phase stresses using the overall strain from the end of cycle K, �eeijjK , (ii) carry out nonlinear finiteelement analysis to equilibrate discrete equilibrium equations. We will refer to this two-step process as theconsistency adjustment.

For forward Euler’s one-step method the block size DNK should be selected to ensure accuracy. Theblock size can be adaptively selected by keeping the nonlocal phase damage increment sufficiently smallwhen the damage increases rapidly and vice versa. This can be expressed as follows:

DNK ¼ int DxðmÞa =maxgaussðDxðmÞÞjK

� �; ð61Þ

where operator intf g denotes the truncation to the decimal part; DxðmÞa is a user-defined allowable tol-erance of phase damage increment per cycle; maxð Þ is computed with respect to all integration points inthe macroproblem. There are two major reasons to monitor the value of maximum damage increment.First, is to ensure the existence of the initial value problem, i.e., if the damage growth rate in cycle K in atleast one of the Gauss points is very high, the approximation of the initial value problem might be inac-curate, and thus the block size DNK evaluated by (61) should be set to zero. In this case, the method reducesto the direct cycle-by-cycle approach. The second reason is to ensure accuracy of the aforementionedconsistency adjustment process.

The fatigue life, denoted as Nmax, can be expressed as

Nmax ¼ nþXnK¼1

DNK ; DNK P 0; ð62Þ

where n is the number of the cycle blocks in the loading history which is also the actual number of the cyclescarried out in the case of the direct simulation. The maximal value of n is intf1=ðDxðmÞa Þg provided that thefailure occurs when xðmÞ reaches one at the critical Gauss point.

To control solution accuracy of the initial value problem we adopt the modified Euler’s integrator [40]with the initial block size determined by (61). The nonlocal phase damage at load cycle K þ DNK (60) is thendefined as

xðmÞjðKþDNK ;DNK Þ ¼ xðmÞjK þDNK

2ðDxðmÞjK þ DxðmÞjKþDNK

Þ; ð63Þ

where DxðmÞjK is evaluated by (59) while DxðmÞjKþDNKis also obtained by (59) after substituting K þ DNK for

K; xðmÞjKþDNKis the first order approximation defined in (60).


The problem of integration of phase fatigue damage cumulative law can be stated as follows:Given: The tolerance err; allowable damage increment per load cycle DxðmÞa ; initial nonlocal phase

damage in the matrix phase xðmÞjK�1, and the overall strain �eemnjK�1 at the beginning of load cycle K.Find: The size of the block DNK ; fatigue life Nmax; nonlocal phase damage and overall strain at the end of

cycle K þ DNK .The adaptive scheme is summarized as follows:

(i) Carry out the incremental finite element analysis, as described in Sections 4.1 and 4.2, for one loadcycle with initial nonlocal phase damage xðmÞjK�1 and the overall strain �eemnjK�1. Denote the nonlocalphase damage at the end of this cycle as xðmÞjK , and the overall strain as �eemnjK . At each integration pointin the macrodomain estimate the rate of nonlocal phase damage in the current load cycle, DxðmÞjK , asdefined by (59).

(ii) Calculate the initial block size using (61).(iii) At each integration point in the macrodomain, compute the approximate solution xðmÞjðKþDNK ;DNK Þ with

the block size DNK using the modified Euler’s method (63); then using the block size DNK=2, computexðmÞjðKþDNK ;DNK=2Þ by two successive uses of (63).

(iv) Find the maximum error among all the integration points and check the convergence by

maxgaussfjxðmÞjðKþDNK ;DNK Þ � xðmÞjðKþDNK ;DNK=2jg6 err: ð64Þ

If (64) is false, DNK DNK=2 set and go back to (iii). Otherwise, update the fatigue life by (62), i.e.,NmaxjK ¼ NmaxjK�1 þ DNK þ 1; update the approximation of the nonlocal phase damage at the end ofload cycle K þ DNK , xðmÞjðKþDNK ;DNK=2Þ and the overall strain �eemnjK . Then, compute the nonlocal phasestrains eðmÞmn jK and eðf Þmn jK using (18).

(v) Perform consistency adjustment: (i) calculate nonlocal phase stresses rðmÞij jK and rðf Þij jK by (20), and mac-roscopic stresses rijjK by (41); (ii) equilibrate discrete solution using nonlinear finite analysis. Finally,set K K þ 1 and go to (i) for the next block of cycles.

5. Numerical examples

5.1. Qualitative examples for two-phase fibrous composites

The first set of numerical examples investigates the computational efficiency and accuracy of the pro-posed fatigue model. We consider the classical stress concentration problem––a thin plate with a centeredsmall circular hole, as shown in Fig. 4. The plate is assumed to be composed of 0=0 ply of fibrous com-posite. The plate is subjected to uniaxial tension perpendicular to the fiber direction. The fiber direction isaligned along the Z axis whereas the two transverse directions coincide with the X and Y axes. Theproperties of the two microphases are as follows:

Matrix: vðmÞ ¼ 0:733; EðmÞ ¼ 69 Gpa; lðmÞ ¼ 0:33;

Fiber: vðf Þ ¼ 0:267; Eðf Þ ¼ 379 Gpa; lðf Þ ¼ 0:21;

where E, l and G denote Young’s modulus, Poisson ratio, and shear modulus, respectively. The parametersof the damage evolution law are chosen as aðmÞ ¼ 8:2, bðmÞ ¼ 10:2 and �##

ðmÞ0 ¼ 0:05 (MPa)1=2. For simplicity,

cðmÞ is assumed to be constant, and set cðmÞ ¼ 4:5 for low-cycle fatigue, and cðmÞ ¼ 15 for high-cycle fatigue.


The static loading capacity of the plate is 103.6 N as shown in Fig. 5.The cyclic loading is designed as a tension-to-zero loading with amplitude of 90 N. The evolution of

fatigue damage cumulation and the nonlocal matrix strain at the critical point are illustrated in Figs. 6–8.For low-cycle fatigue, the direct cycle-by-cycle simulation serves as a reference solution. Several al-

lowable damage increments per cycle were selected to study the convergence of the method. Indeed, theresults summarized in Figs. 6–8 demonstrate excellent convergence characteristics of the proposed fatiguemodel. For high-cycle fatigue problems the solution obtained by the forward Euler method with very smallDxðmÞa is used as a reference solution instead of the direct simulation which is computationally prohibitive.Similar observations can be made for high-cycle problem.

5.2. Large scale fatigue analysis for woven composites

In this section, we consider the tailcone exhaust structure made of Techniweave T-Form Nextel 312/Blackglas Composite System as shown in Fig. 9. The fabric designs used 600 denier bundles of Nextel 312

Fig. 4. FE model of RVE and macrodomain.

Fig. 5. Static loading capacity.


fibers surrounded by Blackglas 493C matrix material [8]. The bundles are assumed to be linear elastic(damage-free) throughout the analysis.

Here, we assume the woven composite to be a periodic two-phase material composed of bundles andmatrix. The phase properties of RVE are summarized below:

Blackglas matrix: vðmÞ ¼ 0:565; EðmÞ ¼ 38:61 GPa; lðmÞ ¼ 0:26;

Bundle: vðf Þ ¼ 0:435; Eðf ÞA ¼ 114:28 Gpa; Gðf ÞA ¼ 45:19 Gpa; lðf ÞA ¼ 0:244;

Eðf ÞT ¼ 112:10 GPa; Gðf ÞT ¼ 44:95 GPa;

where the subscripts A and T represent the axial and transverse directions for transversely isotropic ma-terial.

Fig. 7. Strain softening for low-cycle and high-cycle fatigue.

Fig. 6. Fatigue damage cumulation for low-cycle and high-cycle fatigue.


The microstructure of RVE is discretized with 20,558 nodes and 98,282 elements totaling 61,650 degreesof freedom as shown in Fig. 9, where the matrix phase has been removed in order to give a clear view ofboundles. The compressive principal strains have been observed to have little effect on the damage cu-mulation, so the constants in (38) are selected as a1 ¼ 107 and a2 ¼ 0. The material constants aðmÞ, bðmÞ and�##ðmÞ0 in (32) have been calibrated based on the tensile test under quasi-static uniaxial loading in the weave

plane [8], which gives aðmÞ ¼ 7:6, bðmÞ ¼ 10:9 and �##ðmÞ0 ¼ 0:24 (MPa)1=2. The endurance limit is taken as

�##ðmÞini ¼ 0. The predicted ultimate strength in weave plane was 105.8 MPa with 0.178% ultimate strain. In the

direction normal to the weave plane, the ultimate strength is 69.1 MPa and the ultimate strain 0.21%.Material constants were selected so that numerical results at ultimate points were in good agreement withthe test data.

Following the procedure described in Section 3, we assumed that the fatigue parameter cðmÞ (34) is in theform of

cðmÞ ¼ ðrðmÞmaxÞc1 c2

nþ c3ðrðmÞmax � rðmÞmeanÞ þ c4ðrðmÞmax � rðmÞmeanÞ

2o; ð65Þ

Fig. 8. Local stress relaxation for low-cycle and high-cycle fatigue.

Fig. 9. Geometric model of the Techniweave T-Form Woven microstructure.


where c1–c4 are material constants. The calibration of the material constants in (65) is performed for theuniform cyclic tension–tension loading in the weave plane. The minimal tensile loading is ten percent of themaximum value.

Fig. 10, compares the computed fatigue life with material constants c1 ¼ 0:5, c2 ¼ 0:554, c3 ¼ 1:35,c2 ¼ �2:68 and test data. The ultimate strength of the matrix phase is rðmÞu ¼ 69:1 MPa. Fig. 11 illustratesthe evolution of nonlocal matrix damage parameter and the nonlocal equivalent matrix stress.

Once the fatigue damage model has been calibrated, we turn to the evaluation of fatigue life of theexhaust tailcone structure of a aircraft engine. The finite element mesh of one-eighth of the tailcone modelin the neighborhood of the attachment hole is shown in Fig. 12. It consists of 3154 nodes, 3242 thin shellelements and 385 spring elements totaling 17,766 degrees of freedom. The cyclic internal pressure is appliedto the shell structure with the minimum pressure being of the 10% of the maximum pressure. Since themodel is the thin shell structure, membrane force is the dominant internal force which is approximately in

Fig. 10. Predicted fatigue life and test data.

Fig. 11. Damage evolution and stress relaxation in uniform tension–tension cyclic loading.


the plane of weave. It can be seen that the damage initiates at the location of attachment and then quicklyspreads around the supporting ring causing the overall structural failure. The critical state is reached after1.12 million cycles. No experiments have been conducted up to failure to verify this result.

6. Conclusions and future research directions

Traditionally, life predictions for macrocrack initiation are carried out using S–N or e–N curves inconjunction with some parameters designated to take into account the differences between actual com-ponents and test specimens such as geometry, fabrication, environmental conditions, etc. Due to thecoupled nature of the present multiscale CDM based fatigue model, mechanical and fatigue damage cu-mulation analyses are carried out simultaneously without relying on S–N or e–N curves. The novel ac-celerating technique for the integration of CDM based fatigue damage cumulative law make it possible tosimulate the damage growth by fully coupled finite element analysis for the real structure. Thus largeamount of specimen tests can be avoided, while complex geometrical features, material imperfections,multiaxial loading conditions, and material data can be readily incorporated into the computational model.

For macrocrack propagation problem, the present model has certain limitations. As observed in [22,23],the nonlocal damage theory may lead to the spurious damage zone widening phenomenon, especially whenthe crack opening is accompanied with large strains. Physically, with the evolution of macroscopic cracksðx1 ¼ 1Þ, the nonlocal domain in the vicinity of the macrocrack should be finally collapsed into the lo-calized discontinuity, i.e., crack line. As noted in Section 3, however, the size of characteristic volume isassumed to be constant and the value of the damage variable in our model is not allowed to reach one toensure the regularity of the solution. As a result, the localized discontinuity can never be formed and thewidening of the damaged zone is unavoidable. In the future work we will explore a transient–gradientdamage model [22,23], in which the characteristic length is assumed to be history dependent. As an al-ternative, we will also consider a possibility of switching to the fracture mechanics approach [32,41], and a

Fig. 12. Damage distribution in critical region of the tailcone.


unified methodology linking the nonlocal damage theory and fracture mechanics will be also explored [30].Finally, we notice that our model is proposed for the brittle composite material and interfacial damage isnot taken into account.

7. Uncited reference

[17,18,24,26,28]

Acknowledgement

This work was supported in part by the Allison Engines and the Office of Naval Research through grantnumber N00014-97-1-0687.

Appendix A

In this section we present the details of the derivations of two derivatives, o@ðmÞ=oxðmÞ in (43) and _xxðmÞ in(49). We start with the first. Taking derivative of (42) gives

o@ðmÞoxðmÞ

¼ 1þ cðmÞ

xðmÞ

� �UðmÞ

xðmÞ


o �##ðmÞð �##ðmÞ � t

�##ðmÞÞ � 1

xðmÞ

� �cðmÞ

cðmÞðUðmÞÞcðmÞ�1 oUðmÞ

o �##ðmÞ

!224

þ ðUðmÞÞcðmÞ o2UðmÞ

o �##ðmÞ2

!35 o �##ðmÞ

oxðmÞð �##ðmÞ � t

�##ðmÞÞ �

0@� UðmÞ

xðmÞ


o �##ðmÞo �##ðmÞ

oxðmÞ

1A; ðA:1Þ

where the terms, oUðmÞ=o �##ðmÞ, o2UðmÞ=o �##ðmÞ2

and o �##ðmÞ=oxðmÞ are subsequently computed. From (32), we canget

oUðmÞ

o �##ðmÞ¼ aðmÞ �##ðmÞ0

½p=2þ atanðbðmÞÞ�½ð �##ðmÞ0 Þ2 þ ðaðmÞ �##ðmÞ � bðmÞ �##ðmÞ0 Þ

2�ðA:2Þ

and

o2UðmÞ

o �##ðmÞ2 ¼ �

�2aðmÞðaðmÞ �##ðmÞ � bðmÞ �##ðmÞ0 Þð �##ðmÞ0 Þ

2 þ ðaðmÞ �##ðmÞ � bðmÞ �##ðmÞ0 Þ2

oUðmÞ

o �##ðmÞ: ðA:3Þ

The derivation of o �##ðmÞ=oxðmÞ is not trivial since the principal components of the nonlocal matrix strains areused to define the matrix damage equivalent strain �##ðmÞ in (36). Differentiating (37) with respect to xðmÞ

yields

o �##ðmÞ

oxðmÞ¼ ðbðmÞÞT oðF

ðmÞeeðmÞÞoxðmÞ

; ðA:4Þ

where the vector bðmÞ takes following form

ðbðmÞÞT ¼ 1

2 �##ðmÞðFðmÞeeðmÞÞTLLðmÞ ðA:5Þ


and by using the definition of FðmÞ in (38), the derivative in (A.4) can be expressed as

oðFðmÞeeðmÞÞoxðmÞ

¼ oðhðmÞ1 eeðmÞ1 ÞoxðmÞ

oðhðmÞ2 eeðmÞ2 ÞoxðmÞ

oðhðmÞ3 eeðmÞ3 ÞoxðmÞ

" #T

: ðA:6Þ

Since the three vector components in (A.6) have the same structure, we denote them as oðhðmÞn eeðmÞn Þ=oxðmÞwith n ¼ 1; 2; 3 and then by using (38) we have

oðhðmÞn eeðmÞn ÞoxðmÞ

¼ohðmÞn

oeeðmÞn

eeðmÞn

24 þ hðmÞn

35 oeeðmÞn

oxðmÞ; n ¼ 1; 2; 3; ðA:7Þ

ohðmÞn

oeeðmÞn

¼ a1=p

1þ a21ðeeðmÞn � a2Þ2

: ðA:8Þ

To this end we need to compute the derivative of each component of the principal strain eeðmÞ with respect tothe nonlocal damage parameter xðmÞ. The principal components of a second order tensor satisfy Hamilton’sTheorem, i.e.,

ðeeðmÞn Þ3 � I1ðeeðmÞn Þ

2 þ I2eeðmÞn � I3 ¼ 0; ðA:9Þ

where I1, I2, I3 are the three invariants of eðmÞij or eeðmÞ which can be expressed as

I1 ¼ eðmÞii ¼ eeðmÞ1 þ eeðmÞ2 þ eeðmÞ3 ; ðA:10Þ

I2 ¼ 12ðeðmÞii eðmÞjj � eðmÞij eðmÞji Þ ¼ eeðmÞ1 eeðmÞ2 þ eeðmÞ2 eeðmÞ3 þ eeðmÞ3 eeðmÞ1 ; ðA:11Þ

I3 ¼ 16ð2eðmÞij eðmÞjk eðmÞki Þ � ð3eðmÞij eðmÞji eðmÞkk þ eðmÞii eðmÞjj eðmÞkk Þ ¼ eeðmÞ1 eeðmÞ2 eeðmÞ3 : ðA:12Þ

Differentiating (A.9) with respect to xðmÞ gives

oeeðmÞn

oxðmÞ¼ ½3ðeeðmÞn Þ

2 � 2I1eeðmÞn þ I2��1 oI1

oxðmÞðeeðmÞn Þ

2

�� oI2oxðmÞ

eeðmÞn þoI3

oxðmÞ

�; ðA:13Þ

where the derivative of the invariants with respect to xðmÞ can be obtained by using (A.10)–(A.12)

oI1oxðmÞ

� E½1�ijoeðmÞij

oxðmÞ¼ dikdjk

oeðmÞij

oxðmÞ; ðA:14Þ

oI2oxðmÞ


oxðmÞ¼ ðeðmÞmmdikdjk � eðmÞij Þ

oeðmÞij

oxðmÞ; ðA:15Þ

oI3oxðmÞ


oxðmÞ¼�

eðmÞik eðmÞkj � eðmÞmmeðmÞij �1

2eðmÞmn eðmÞnm dikdjk þ

1

2eðmÞmmeðmÞnn dikdjk

�oeðmÞij

oxðmÞ: ðA:16Þ

Combining (A.7), (A.13)–(A.16), we get

oðhðmÞn eeðmÞn ÞoxðmÞ

¼ ZðmÞnij

oeðmÞij

oxðmÞ; n ¼ 1; 2; 3; ðA:17Þ

ZðmÞnij �ohðmÞn

oeeðmÞn

eeðmÞn

24 þ hðmÞn

35½3ðeeðmÞn Þ

2 � 2I1eeðmÞn þ I2��1½E½1�ij ðee

ðmÞn Þ

2 � E½2�ij eeðmÞn þ E½3�ij �: ðA:18Þ


Finally the derivative o �##ðmÞ=oxðmÞ in (A.4) can be written in a concise form by using (A.5) and (A.17)

o �##ðmÞ

oxðmÞ¼

X3

n¼1

bðmÞn ZðmÞnij

!oeðmÞij

oxðmÞ; ðA:19Þ

where the derivative on right hand side can be evaluated using (48), i.e.,

oeðmÞij

oxðmÞ¼ GðmÞijklR

ðmÞklmn�eemn: ðA:20Þ

To complete the derivations of o@ðmÞ=oxðmÞ we combine the results of equations (A.1)–(A.3), (A.19) and(A.20). The time derivative of the nonlocal matrix damage variable, _xxðmÞ, can be obtained by making use ofo@ðmÞ=oxðmÞ. From the fatigue damage cumulative law (33), the material derivative of the nonlocal matrixdamage parameter (in the case of damage process) can be written as

_xxðmÞ ¼ UðmÞ

xðmÞ


o �##ðmÞ_�##�##ðmÞ

; ðA:21Þ

where _�##�##ðmÞ

is derived in the similar way to o �##ðmÞ=oxðmÞ, which yields

_�##�##ðmÞ ¼

X3

n¼1

bðmÞn ZðmÞnij

!_eeðmÞij ; ðA:22Þ

where the rate of matrix nonlocal strain can be obtained by taking time derivative of both sides of (18),which gives

_eeðmÞij ¼ ðAðmÞijmn þ GðmÞijklDklmnÞ_�ee�eemn þ GðmÞijklR

ðmÞklmn�eemn _xx

ðmÞ: ðA:23Þ

Substituting equation (A.21) and (A.22) into (A.23) yields

_xxðmÞ � W ðmÞmn

_�ee�eemn ¼SðmÞij ðA

ðmÞijmn þ GðmÞijklDklmnÞ

1� SðmÞij GðmÞijklRðmÞklmn�eemn

_�ee�eemn; ðA:24Þ

where

SðmÞij ¼UðmÞ

xðmÞ


o �##ðmÞ

X3

n¼1

bðmÞn ZðmÞnij

!: ðA:25Þ

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computational mechanics of fatigue and life predictions for composite materials and structures

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