A Nonlocal Multiscale Fatigue Model

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A Nonlocal Multiscale Fatigue ModelJacob Fish a & Caglar Oskay aa Civil and Environmental Engineering Department , Rensselaer Polytechnic Institute , Troy,NY, USAPublished online: 24 Feb 2007.

To cite this article: Jacob Fish & Caglar Oskay (2005) A Nonlocal Multiscale Fatigue Model, Mechanics of Advanced Materialsand Structures, 12:6, 485-500, DOI: 10.1080/15376490500259319

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Mechanics of Advanced Materials and Structures, 12:485–500, 2005Copyright c© Taylor & Francis Inc.ISSN: 1537-6494 print / 1537-6532 onlineDOI: 10.1080/15376490500259319

A Nonlocal Multiscale Fatigue Model

Jacob Fish and Caglar OskayCivil and Environmental Engineering Department, Rensselaer Polytechnic Institute, Troy, NY, USA

A nonlocal multiscale model in time domain is developed forfatigue life predictions. The method is based on the mathematicalhomogenization theory with almost periodic fields. The almost pe-riodicity reflects the effects of irreversible deformations in timedomain in the form of accumulation of damage. Multiple tem-poral scales are introduced to decompose the original boundaryvalue problem into micro-chronological (temporal unit cell) andmacro-chronological (homogenized) problems. A nonlocal Gursontype constitutive law is revisited for cyclic loading, calibrated andvalidated against fatigue crack propagation experiments on 316Laustenitic stainless steel specimens.

1. INTRODUCTIONFatigue of solids and structures is a multiscale phenomenon

in space and time. The existence of multiple spatial scales isevident due to the presence of cracks and/or voids which maybe orders of magnitude smaller than the structural component.Multiple temporal scales exist because of disparity between theperiod of loading and the overall structural fatigue life.

The primary phenomenological fatigue life prediction tooltoday is the so-called “total life” approach. By this approach,fatigue life is characterized by the stress-life (S-N) curves whichrelate the range of applied cyclic stresses to the number of loadcycles to failure. When significant yielding is expected arounda crack tip, the strain-life (ε-N) curves are commonly used in-stead. The ε-N curves relate the range of applied cyclic strains(total or plastic) to the number of load cycles to failure. Suchexperimental characterizations are generally limited to smallstructural components or specimens. For larger assemblies, soleexperimentation may not be feasible and far fields are typi-cally computed numerically. The drawback of such a mixedexperimental-computational approach is that it fails to accountfor force redistribution caused by damage accumulation.

One of the most widely used empirical fatigue life modelsis due to Paris and Erdogan [1]. The so-called Paris’ law re-lates the crack growth rate to the range of stress intensity factors

Received 28 April 2005; accepted 13 July 2005.Address correspondence to J. Fish, Engineering Department of Me-

chanical, Aerospace and Nuclear Engineering, Rensselaer PolytechnicInstitute, Troy, NY 12180, USA. E-mail: [email protected]

using a power representation. Paris’ law, which was originallydeveloped for the ideal condition of small scale yielding, was en-hanced and generalized to incorporate various mechanisms suchas R-effects, closure, threshold limits and others (e.g., see [2–5]). Unfortunately, the issues of modeling short cracks and thenecessity for embedding initial macrocracks remain unresolvedat large.

Direct Numerical Simulations (DNS) of fatigue crack growthby means of cycle-by-cycle simulations have been attemptedin the context of the cohesive theory (e.g., [6, 7]). Obviously,DNS is not a feasible approach for simulating large scale sys-tems subjected to high cycle fatigue. The so-called Cycle JumpSimulation (CJS) represents one of the first attempts to approxi-mate the response of the direct cycle-by-cycle simulation [8–10]using coarser time scales. The CJS has been found to performreasonably well for relatively simple constitutive models such asthose based on isotropic continuum damage theory. However, amathematical framework based on the CJS approach that wouldsatisfy governing equations in space and time for a general classof constitutive models remains an elusive task.

In this article we present an alternative approach based onthe mathematical homogenization theory in time domain withalmost periodic fields. Almost periodicity is a byproduct of ir-reversible processes, such as damage accumulation, which vi-olates the condition of temporal periodicity. We assume, how-ever, that the non-periodic contribution is a perturbation of theperiodic part. In addition, we assume that micro- and macro-chronological displacement fields are of the same order of mag-nitude. This is in contrast to the classical (spatial) mathemati-cal homogenization where the microscale displacement field istaken as a perturbation of the macroscale field. Spatial homog-enization in the presence of non-periodic conditions has beenpreviously investigated using stochastic [11–14] and determin-istic [15–21] methods. Mathematical analysis of the spatial ho-mogenization theory with almost- and non-periodic fields hasbeen conducted in Refs. [22–26].

The Gurson-Tvergaard-Needleman (GTN) [27] model servesas a basis for modeling of microvoid nucleation, growth, andcoalescence processes, and subsequent propagation of macro-cracks up to failure. Kinematic hardening and irreversible dam-age are employed to generate the hysteretic behavior, to preventshakedown and premature crack arrest. It is well known that the

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486 J. FISH AND C. OSKAY

degradation of material properties caused by progressive cavi-tation mechanism in the GTN model ultimately causes spuriousmesh sensitivity of the response fields, rendering its predictionsquestionable. For monotonic loading, such mesh sensitive localmodels are commonly regularized using localization limitersincluding nonlocal gradient (e.g., [28]) and integral (e.g., [29])type formulations, micro-polar continuum (e.g., [30]), and sev-eral others (e.g., [31–33]).

For cyclic loading, there are two scenarios that may giverise to mesh sensitivity: (i) loss of ellipticity at the progressivestages of damage growth, and (ii) unbounded damage growthrate caused by crack tip singularity. The latter has been inves-tigated by Peerlings et al. [34] in the context of high cycle fa-tigue for quasi-brittle materials. They showed that for isotropicdamage mechanics model, which preserves ellipticity of govern-ing equations up to failure (damage parameter equals to one),mesh sensitivity is caused by condition ii. They concluded thata gradient type nonlocal regularization is effective as a local-ization limiter for high cycle fatigue of quasi-brittle materials.The GTN model considered in the present article introduces ad-ditional challenges. In addition to the unbounded growth rateat the crack tip, the presence of plastic deformation may resultin loss of ellipticity of the governing equations. To the best ofthe authors’ knowledge, the present study is the first attempt toassess the performance of the nonlocal GTN model under theconditions of cyclic loading. In this article, we focus on the de-velopment, calibration and validation of the nonlocal multiscalemodel of fatigue based on the integral version of the nonlocalGTN model, which is an extension of the local fatigue modeldeveloped in [35, 36].

The remainder of this article is organized as follows. Section 2reviews the fundamental aspects of the temporal homogeniza-tion in the presence of almost periodic fields. The nonlocal GTNmodel and the associated boundary value problem are presentedin Section 3. Section 4 presents the formulation of the nonlo-cal multiscale model. Section 5 focuses on the computationalaspects and the implementation details. The local and nonlocalmultiscale models are compared in Section 6. In Section 7, thenonlocal multiscale fatigue model is first calibrated using fa-tigue experiments on 316L austenitic stainless steel specimens.Subsequently, the calibrated model is validated against a sep-arate and independent set of fatigue crack propagation experi-ments. Conclusions and future research directions are discussedin Section 8.

2. MULTIPLE TEMPORAL SCALESAND ALMOST PERIODICITYIn a typical fatigue process, accumulation of damage, and

the resulting macrocrack initiation and propagation is relativelyslow, compared to rapid fluctuations of displacements withineach load cycle. The disparity between the two characteristictime scales naturally introduces multiple time coordinates. Inthis study, we consider a two scale decomposition of time by

defining a macro-chronological scale denoted by the intrinsictime coordinate, t , and a micro-chronological scale denoted bythe fast time coordinate, τ. These two scales are related througha scaling parameter

τ = t

ζ; 0 < ζ 1 (1)

in which, ζ (= τo/tr), is defined by the ratio of the characteristiclengths in micro- and macro-chronological time, denoted byτo and tr, respectively. The response fields, φ, are assumed todepend on two temporal scales

φζ(x, t) = φ(x, t, τ(t)) (2)

where, φζ is a rapidly oscillating function in time; and x de-notes spatial coordinates. Time differentiation in the presenceof multiple temporal scales is obtained using Eq. (1)

φζ(x, t) = φ(x, t, τ) = φ,t (x, t, τ) + 1

ζφ,τ(x, t, τ) (3)

in which, the comma followed by a subscript coordinate denotespartial derivative; and superposed dot is a total time derivative.

In this study, response fields are assumed to be almost pe-riodic in time domain to account for the irreversible processesof damage accumulation. Almost periodicity implies that at theneighboring points in a spatial or temporal domain homologousby periodicity, the change in the response function is small butnonzero (in contrast to the case of local periodicity in which thechange in response functions is zero). A typical almost periodicfunction is depicted in Figure 1.

The space of almost periodic functions, T , is defined as:

T := φ|φ(x, t, τ + kκ) − φ(x, t, τ) = O(ζ) ∀k ∈ Z (4)

A function φ ∈ T is said to be κ-almost periodic if it belongsto the space of almost periodic functions. Attention is restrictedon a subspace of T denoted by T

T := φap|φap(x, t, τ) = φp(x, t, τ) + ζτφ(x) (5)

where, φ is an arbitrary function constant in time; and φp is aκ-periodic function

φp(x, t, τ + kκ) = φp(x, t, τ) (6)

Setting φζ

p(x, t) = φp(x, t, t/ζ), the well known weak con-vergence of the periodic fields is given as [37]:

limζ→0

∫T

φζ

p (x, t) dt →∫

T〈φp(x, t, τ)〉 dt (7)

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NONLOCAL MULTISCALE FATIGUE MODEL 487

FIG. 1. Almost periodic function.

for any subset T in R. The periodic temporal homogenization(PTH) operator, 〈·〉, is given by:

〈·〉 := 1

|κ|∫

κ

(·)dτ (8)

Defining φζap(x, t) = φap(x, t, t/ζ), it can be easily observed

that Eq. (7) does not hold for almost periodic fields

limζ→0

∫T

φζ

apdt = limζ→0

∫T

φζ

pdt + φt2

2

∣∣∣∣T

(9a)

∫T〈φap〉dt =

∫T〈φp〉dt + ζ

2

φ

|κ| τ2∣∣κt∣∣T

(9b)

The above serves as a motivation for an alternative definitionof the averaging operator that would satisfy the weak conver-gence relation (Eq. (7)) for almost periodic functions in T . Suchan almost periodic temporal homogenization (APTH) operatorcan be constructed in the rate form

M(φap),t (x, t) := 〈φap〉(x, t) (10)

so that the weak convergence relation (Eq. (7)) is satisfied foralmost periodic functions in T

limζ→0

∫T

φζ

ap(x, t)dt →∫

TM(φap(x, t, τ))dt (11)

Equation (11) may be verified by substituting the definition ofthe almost periodic functions given by Eq. (5) into the right hand

side, and using Eq. (7)

∫T

M(φap)dt =∫

T

( ∫ t

0

[1

ζ〈φap,τ〉 + 〈φap〉,s

]ds

)dt

=∫

T

( ∫ t

0

[φ + 〈φp〉,s

]ds

)dt

=∫

T(tφ + 〈φp〉)dt = φ

t2

2

∣∣∣∣T

+∫

T〈φp〉dt (12)

The APTH and PTH operations are equivalent if applied to pe-riodic fields.Remark 1: To further clarify the differences between the PTHand APTH operators, consider an illustrative example:

φ(t, τ) = a0 sin

(2π

ζτ0τ

)+ τφ + a1[1 − exp(−t/td )] (13)

where, the first term represents the rapidly oscillatory periodicfunction; the second reflects the non-periodic linear variation;and the last term depicts the slow macro-chronological variation.Figure 2 compares the functions M(φap) and 〈φap〉 with theoriginal function φζ as ζ → 0. Parameter values, a0 = 2.5,a1 = 10.0, τ0 = 1, and φ = 0.07 were used in the illustration.The figure clearly shows that 〈φap〉 does not adequately trackthe average response of the almost periodic function, whereasM(φap) does precisely that.

3. NONLOCAL GTN MODELProgressive failure of ductile materials is commonly ideal-

ized using phenomenological elastoplastic constitutive models

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FIG. 2. Decomposition of the response fields with respect to periodic andalmost periodic temporal homogenization operators.

with a cumulative damage mechanism. One of the most widelyknown models of this type is due to Gurson [38]. The originalGurson’s model is based on the limit analysis of a spherical unitcell with a spherical void inclusion at its center, and a Von-Misestype matrix material. A number of improvements to the origi-nal model have been proposed including the incorporation ofvoid nucleation and coalescence mechanisms [39, 40], effectsof non-spherical void inclusions [41], and others. The so-calledGurson-Tvergaard-Needleman (GTN) model, which accountsfor the nucleation and coalescence effects, has been validatedfor monotonic [42] and cyclic [43] loading conditions.

From the numerical point of view, the GTN model is knownto exhibit spurious mesh sensitivity when loading extends to thesoftening regime. Mesh sensitivity is characterized by localiza-tion of strains to the smallest possible volume admitted by thefinite element mesh. We defer the discussion on the localizationphenomenon and various localization limiters to the referencesdiscussed in the introduction of this article. In the present study,we revisit the integral type of nonlocal formulation of the GTNmodel by defining a nonlocal consistency parameter which gov-erns the evolution equations of hardening, void growth, and voidnucleation.

We start by defining the boundary value problem

∇ · σζ + bζ = 0 on Ω × (0, to) (14)

σζ = L : (εζ − µζ) on Ω × (0, to) (15)

εζ = 1

2(∇uζ + uζ∇) on Ω × (0, to) (16)

uζ = u′ on Ω (17)

uζ = uζ on Γu × (0, to) (18)

nσζ = tζ on Γt × (0, to) (19)

where, uζ is the displacement vector; σζ is the stress tensor; εζ,and, µζ are the total and plastic strain tensors, respectively; L isthe tensor of elastic moduli; ∇ is the vector differential operatorgiven by ∇ = (∂/∂x1, ∂/∂x2, ∂/∂x3) in Cartesian coordi-nates; (·∇) = (∇·)T, superscript T indicates the transpose; bζ

is the body force; and are the spatial problem domain andits boundary, respectively; to is the temporal problem domain;u′ is the initial displacement field; uζ and tζ are the prescribeddisplacements and tractions on the boundaries u and t , re-spectively, where = u ∪ t and u ∩ t = ∅. The analysisis restricted to small deformations.

The yield function of the GTN model is expressed as:

φζ : = φζ(σζ, Hζ) = (qζ)2(σ

ζ

F

)2 + 2 f ζ cosh

(− 3

2

pζ

σζ

F

)

−1 − ( f ζ)2 = 0 (20)

where, H denotes the set of internal state variables

qζ =√

3

2sζ : sζ, sζ = Bζ + pζδ, Bζ = σζ − αζ,

pζ = −1

3Bζ : δ (21)

αζ is the center of the yield surface; and δ is the second orderidentity tensor. The radius of the yield surface, σζ

F , is definedas [44]:

σζ

F = (1 − b)σy + bσζ

M (22)

in which, σy and σζ

M are the initial yield stress and matrix flowstress, respectively; and b ∈ [0, 1] is a constant. b = 1, b = 0,and b ∈ [0, 1] correspond to the pure isotropic, kinematic, andmixed hardening conditions, respectively.

The void coalescence function is modeled using the piecewiselinear function of the void volume fraction [40]

f ζ =

f ζ f ζ < fc

fc +(

1

q1− fc

)f ζ − fc

f f − fcf ζ ≥ fc

(23)

in which, fc and f f are material parameters; and q1 is the Tver-gaard constant. When f ζ approaches f f , f ζ → 1/q1, and thematerial looses its stress carrying capacity.

The flow of plastic strains is chosen to follow the normalityrule

µζ = λζ∂φζ

∂σζ; λζ ≥ 0 (24)

in which, λζ is the consistency parameter.The nonlocal GTN model is formulated by defining the non-

local consistency parameter, λζ, using the well known integral

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equation (e.g., [45])

λζ(x, t) = 1

W (x)

∫

w(x, s)λζ(s, t)ds (25)

where,

W (x) =∫

w(x, s)ds x ∈ (26)

and, w(x, s) is an attenuation function. In this study, we use

w(x, s) = 1[1 +

(‖x − s‖lc

)c1]c2

(27)

in which, lc is the characteristic material length. c1 and c2 definethe shape of the attenuation curve and they are set to 8 and2, respectively, as suggested in [46]. The nonlocal evolutionequations of the GTN model are defined based on the nonlocalconsistency parameter.

f ζ = f ζ

gr + f ζ

nuc (28)

The void volume fraction is allowed to grow only af-ter the onset of plastic yielding, and under tensile loadingconditions

f ζ

gr = q1λζ(1 − f ζ)

⟨∂φζ

∂σζ: δ

⟩+

(29)

where, 〈·〉+ = [(·) + | · |]/2 are MacCauley brackets. A straincontrolled void nucleation mechanism is adopted [39]

f ζ

nuc = Aζ(ρζ)ρζ (30)

where, Aζ is taken to have a normal distribution

Aζ(ρζ) = q1fN

sN

√2π

exp

− 1

2

(ρζ − εN

sN

)2(31)

if∂φζ

∂σζ: δ > 0; Aζ = 0 otherwise

in which, ρζ is the equivalent plastic strain; fN , is the vol-ume fraction of void nucleating particles; εN is the meanstrain for nucleation; and sN is the standard deviation ofnucleation.

The evolution of the effective plastic strain is given by:

ρζ = λζ

(1 − f ζ)σζ

F

Bζ :∂ζ

∂σζ(32)

The equivalent plastic strain rate is related to the matrix flowstrength

σζ

M =(

EEt

E − Et

)ρζ = Et ρζ (33)

in which E is the Young’s modulus; and Et is the tangent to theuniaxial stress-strain diagram at a given stress level.The evolution equation for the center of the yield surface is:

αζ = λζQζBζ; Qζ ≥ 0 (34)

The value of Qζ is determined using the consistency condition.

4. MULTIPLE TEMPORAL SCALE ANALYSISWe define the following decomposition of the almost periodic

fields

φ(x, t, τ) = M(φ)(x, t) + φ(x, t, τ) (35)

in which, φ denotes the displacement, strain, stress or inter-nal state variable fields; M(φ) and φ correspond to the macro-chronological (homogenized) and the micro-chronological (os-cillatory) parts of the response fields, respectively. Employingthe definition of the APTH operator, the following relation isobtained

〈φ〉,t = −1

ζ〈φ,τ〉 (36)

In view of Eq. (35), we seek to decompose the boundaryvalue problem defined in Section 3 to a set of coupled micro- andmacro-chronological problems. Applying the APTH operator toEq. (14), the macro-chronological equilibrium equations maybe obtained

∇ · M(σ) + M(b) = 0 on × (0, to) (37)

Decomposing the stresses and the body forces using Eq. (35)and subtracting Eq. (37) from the resulting equation yields theequilibrium equations for the micro-chronological problem. At amacro-chronological time instant, t , we have a unit cell problemin time domain

∇ · σ + b − M(b) = 0 on × (0, τo) (38)

The original constitutive relation may be decomposed by ap-plying Eq. (35) to the stress, strain, and plastic strain tensors,substituting the resulting terms into Eq. (15), and gathering theterms of equal order

O(ζ−1) : σ,τ = L : (ε,τ − µ,τ) on

× (0, τo) (39a)

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O(1) : M(σ),t + σ,t = L : (M(ε),t − M(µ),t+ε,t − µ,t ) (39b)

The micro-chronological constitutive relation is given byEq. (39a). Applying the PTH operator to Eq. (3) and usingEq. (36), the constitutive equation for the macro-chronologicalproblem is obtained

M(σ),t = L : (M(ε),t − M(µ),t ) on × (0, to) (40)

The kinematic equation may be decomposed into a micro-and macro-chronological parts by applying Eq. (35) to the dis-placement field, substituting the resulting terms into Eq. (16),and exploiting the linearity of the APTH operator

ε = 1

2(∇u + u∇) (41)

M(ε) = 1

2(∇M(u) + M(u)∇) (42)

The flow rules of the micro- and macro-chronological prob-lems are formulated by considering a two-scale decompositionof the local consistency parameter. At a material point, x, we have

λ(t, τ) = 1

ζλ1(t, τ) + λo(t) (43)

where, λ1 and λo may be interpreted as the consistency param-eters induced by the micro- and macro-chronological loadings,respectively. Applying Eq. (35) to the plastic strain tensor,substituting the resulting terms and the above decompositionof the consistency parameter to the flow rule (Eq. (24)), andgathering equal order terms yield:

O(ζ−1) : µ,τ = λ1 ∂Φ∂σ

(44a)

O(1) : M(µ),t + µ,t = λo ∂Φ∂σ

(44b)

The flow rule for the micro-chronological problem is given byEq. (44a). Applying the PTH operator to Eqs. (44), and usingEq. (36) gives:

M(µ),t = λo

⟨∂φ

∂σ

⟩+ 1

ζ〈µ,τ〉 (45)

where, the yield condition is expressed in terms of the micro-and macro-chronological fields using the relation

φ(σ, H) = φ(M(σ), σ, M(H), H) (46)

The evaluation of the micro-chronological plastic strains for thenonlocal GTN model is detailed in Appendix A. The macro-chronological plastic strains, stresses, total strains and inter-nal state variables are evaluated using a two-step algorithm de-scribed in Section 5.

The evolution equations of the internal state variables(Eqs. (22), (29), (30), (32)–(34)) may be expressed in a gen-eral functional form as:

H = h(σ, λ, H) (47)

or in terms of the macro- and micro-chronological fields

H = h(M(σ), σ, λo, λ

1, M(H), H) (48)

where,

λi(x, t) = 1

W (x)

∫

w(x, s)λi (s, t)ds, i = 0, 1 (49)

The evolution equations may be decomposed as:

h = 1

ζh1 + ho (50)

Decomposing the internal state variable fields based on Eq. (35)and using Eq. (3) yields:

H,τ = h1(M(σ), σ, λ1, M(H), H) (51a)

M(H),t + H,t = ho(M(σ), σ, λo, M(H), H) (51b)

The micro-chronological evolution equation for the internal statevariables is given by Eq. (51a). Applying the PTH operator toEqs. (51) and using Eq. (36) gives the evolution equations of themacro-chronological problem

M(H),t = 〈ho〉 + 1

ζ〈H,τ〉 (52)

The evolution equations for the nonlocal GTN model are pre-sented in Appendix B.

Substituting the displacement and stress decompositions intoEqs. (17)–(19), and applying the APTH operator yields the ini-tial and the boundary conditions for the macro-chronologicalproblem

M(u)(x, t = 0) = u′(x) on (53)

M(u) = M(u(x, t, τ)) on u × (0, to) (54)

nM(σ) = M(t(x, t, τ)) on t × (0, to) (55)

Similarly, the micro-chronological initial and boundary con-ditions are obtained by applying Eq. (35) to the displacement andstress fields, substituting the resulting equation into Eqs. (17)–(19) and subtracting Eqs. (53)–(55) from the resulting equations

u(x, t, τ = 0) = 0 on (56)

u(x, t, τ) = u − M(u) on u × (0, τo) (57)

nσ(x, t, τ) = t − M(t) on t × (0, τo) (58)

The coupled macro-chronological and the micro-chrono-logical problems are summarized in Table 1.

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TABLE 1Governing equations of the macro- and micro-chronological problems

Macro-chronological problem:

Equilibrium equation: ∇ · M(σ) + M(b) = 0 on × (0, to)

Constitutive equation: M(σ),t = L : (M(ε),t − M(µ),t ) on × (0, to)

Kinematic equation: M(ε) = 1

2(∇M(u) + M(u)∇) on × (0, to)

Initial condition: M(u)(x, t = 0) = u′(x) on

Boundary conditions: M(u) = M(u(x, t, τ)) on u × (0, to)

nM(σ) = M(t(x, t, τ)) on t × (0, to)

Flow rule: M(µ),t = λo

⟨∂φ

∂σ

⟩+ 1

ζ〈µ,τ〉

Evolution equations: M(H),t = 〈ho〉 + 1

ζ〈H,τ〉

Micro-chronological problem:

Equilibrium equation: ∇ · σ + b − M(b) = 0 on × (0, τo)

Constitutive equation: σ,τ = L : (ε,τ − µ,τ) on × (0, τo)

Kinematic equation: ε = 1

2(∇u + u∇) on × (0, τo)

Initial condition: u(x, t, τ = 0) = 0 on

Boundary conditions: u(x, t, τ) = u − M(u) on u × (0, τo)

nσ(x, t, τ) = t − M(t) on t × (0, τo)

Flow rule: µ,τ = λ1 ∂φ

∂σ

Evolution equations: H,τ = h1(M(σ), σ, λ1, M(H), H)

5. NUMERICAL ASPECTS

5.1. Nonlocal Consistency ParameterThe introduction of the nonlocal consistency parameter leads

to the integro-differential equation for the local consistencyparameter

λ(x, t) =∂φ

∂σ: L : ε

∂φ

∂σ: L :

∂φ

∂σ

− H∂φ

∂σ: L :

∂φ

∂σ

1

W (x)

×∫

w(x, s)λ(s, t)ds (59)

The expression for H and the details of the derivation of theabove equation are presented in Appendix A. Solution of theboundary value problem with such nonlocal characterization ofinternal state variables requires somewhat cumbersome integra-tion techniques [47]. To reduce the computational complexity

we use the following approximation of the nonlocal consistencyparameter

tn+1λ(xi ) = gi

Witn+1λ(xi ) + 1

Wi

ngp∑j=1j =i

g jw(xi , x j )tnλ(x j ) (60)

where, Wi = W (xi ); xi denotes the spatial coordinates of the i thintegration point; ngp is the total number of integration pointsin the mesh; gi are the weight factors for the numerical inte-gration; and the left subscript denote temporal discretization.Discretizing Eq. (59) and substituting into Eq. (60) yields

tn+1λ(xi ) =

∂φ

∂σ: L : ε

∣∣∣∣∣tn+1

gi

WiH + ∂φ

∂σ: L :

∂φ

∂σ

∣∣∣∣∣tn+1

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+ H/Wi

gi

WiH + ∂φ

∂σ: L :

∂φ

∂σ

∣∣∣∣∣tn+1

× (61)

ngp∑j=1j =i

g jw(xi , x j )tnλ(x j )

It can be observed that the local consistency parameter, tn+1λ,may be evaluated locally using Eq. (61). Similar methods havebeen previously employed for the nonlocal characterization ofother variants of the GTN model (e.g., [46]).

5.2. Adaptive Multi-Scale MethodologyThe micro- and macro-chronological boundary value prob-

lems formulated in the previous section and summarized inTable 1 are coupled through the constitutive relationships. Inthis study, we adopted a staggered solution algorithm with adap-tive macro-chronological time stepping control. The proposedalgorithm is convenient in the sense that it facilitates the use ofcommercially available finite element packages solely by invok-ing user supplied material subroutines.

The proposed adaptive algorithm for evaluating the micro-and macro-chronological problems is summarized in Table 2.The objective of the algorithm is to evaluate the micro- andmacro-chronological response fields denoted by φ(x, t, τ) andM(φ)(x, t), respectively, at time t = tn+1, given the responsefields at time t = tn (tn < tn+1). The solution at macro-chronological time tn+1 is computed adaptively. From a qual-itative viewpoint, the macro-chronological step size, t (=tn+1−tn) is chosen to be relatively large if the computed responseis smooth, otherwise, the step size is reduced to maintain accu-racy. To this extent, a vector of control variables, ω(σ, µ, H),is defined as a function of the response fields. Given a con-verged state at t = tn , the control variable vector at t = tn+1

is computed twice at each integration point; first using a singlemacro-chronological time step, t (to compute (tn+1;t )ω), thenusing two successive steps with macro-chronological step sizet/2 (to compute (tn+1;t/2)ω). The error at t = tn+1 is thengiven by

Eω = |(tn+1;t )ω − (tn+1;t/2)ω| ≤ Etol (62)

in which, Etol is the vector of predefined error tolerances for thecontrol variables; and the left subscript ( · ; · ) denotes the currenttime and the time step size used in the stress update. If the com-puted error exceeds the error tolerance vector, Etol , at any inte-gration point, the macro-chronological time step size is reducedbased on the value of the errors and the procedure is repeateduntil the error tolerances are met. The micro-chronological timestep size δτ = τk+1 − τk is controlled by the accuracy of thestress update procedure.

Remark 2: In this study, the vector of the control variables in-cludes the void volume fraction, f , and the Euclidian norm ofthe plastic strain deviator, sµ

ω = f, ‖sµ‖2T; sµ = µ − 1

3δ : µ (63)

For a given macro-chronological time step size, t , the inte-gration of the macro-chronological fields is conducted based ona two-step procedure. This procedure ensures the equilibriumequations and the consistency condition to be satisfied at everymacro-chronological step. Figure 3 illustrates the basic structureof the proposed two-step procedure. The micro-chronologicalloading induced macro-chronological fields are introduced tothe converged state of the macro-chronological fields at t = tn ,to obtain an intermediate configuration, tn+1M(φ)∗ (step 5 ofTable 2). The macro-chronological loading is subsequently ap-plied to the intermediate configuration to evaluate the currentstate of the macro-chronological fields, tn+1M(φ). The stressupdates of the micro- and macro-chronological problems are car-ried out using a return mapping algorithm based on the methodfirst proposed in [48] for isotropic hardening, and further ex-tended to account for kinematic hardening in [49].

The aforementioned adaptive algorithm requires data transferbetween the micro- and macro-chronological problems at eachtime step. In this study, the information between the micro- andmacro-chronological problems is communicated through an ex-ternal batch processing to facilitate the use of commercial finiteelement software. The ABAQUS analysis engine incorporatinguser supplied subroutines (UMAT) was employed to conduct the

FIG. 3. Schematic of the proposed two step update procedure.

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TABLE 2Adaptive algorithm

Input: tn M (φ) (x) and tn φ (x, τ)Output: t , tn+1M (φ) and tn+1φ (τ).1. Initialize: I Dtol ← false, EF ← 12. Compute tn ω ≡ ω(tn σ,tn H,tn µ) and tn δω at each integration point

tn δω (xk) =tn ω (xk, τo) −tn ω (xk, 0); ∀k ∈ 1, 2, . . . , nip

(* nip: number of integration points)3. Compute initial estimate of t :

t = minnωdj=1

(δωall

j /nip

maxk=1

[δω j (xk, t)

])τo

(* nωd : number of dimensions of the control variable vector).(* δωall : maximum allowable accumulation within a load cycle).4. while I Dtol = false5. Set t ← t/EF :

(tn+1;t)M (φ)∗ =tn M (φ) + tτo

[tn φ (x, τo) −tn φ (x, 0)

]6. Apply the macro-chronological external force increments and solve for (tn+1;t)M (φ), using (tn+1;t)M (φ)∗ as the initial

values of a standard update procedure.7. Solve micro-chronological IBVP for (tn+1;t)φ using (tn+1;t)M (φ)8. Set t ← t/2, evaluate steps (5)–(7) in two increments to compute (tn+1;t/2)M (φ) and (tn+1;t/2)φ

9. Compute (tn+1;t)ω using (tn+1;t)M (φ) and (tn+1;t)φ

10. Compute (tn+1;t/2)ω using (tn+1;t/2)M (φ) and (tn+1;t/2)φ.11. Evaluate (tn+1;t)δω and (tn+1;t/2)δω at each integration point.12.

Eωj = max

nip

k=1

(‖(tn+1;t)ω j (xk, τ) −(tn+1;t/2) ω j (xk, τ) ‖W)

; ∀ j ∈ 1, 2, . . . , nωd

‖ · ‖W = (∫ τo

0 (·)W dτ)1/W

; if 1 < W < ∞, ‖ · ‖W = max (·) (xk, τ) ; if W = ∞

13. if Eωj ≤ Etol

j ; ∀ j ∈ 1, 2, . . . , nωd then I Dtol ← true; return14. else

EF = min(

maxnωdj=1

[Eω

j /Etolj

], EL

)

return(* EL : maximum allowable time step size reduction factor).15. tn+1 ← tn + t

numerical simulations. The response information of the macro-and micro-chronological problems was stored on a hard disk.

6. MESH SENSITIVITY STUDIESThe localization characteristics of the nonlocal multiscale

model are assessed using fatigue crack propagation simulationsin a rectangular panel with a blunted notch. Figure 4 displays

the geometry of the panel and the applied vertical displacements.Plane strain conditions were assumed and a quarter of the panelwas modeled due to symmetry. Three finite element mesheswere considered with increased refinement along the path of thecrack. Far fields were taken to be identical for all meshes. Thepath of the crack propagation was discretized uniformly with anaverage element size of h = 0.1, 0.05, and 0.025 units, and a

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FIG. 4. Geometry and the far field mesh of the panel with a blunted notch.

small mesh transition zone is placed between the far fields andthe crack propagation path. Figure 5 shows the discretization atthe tip of the blunted notch for each mesh.

The propagation of a fatigue crack is modeled by effectivelyeliminating (a small fraction of the residual stiffness is retainedfor stability) the elements for which the void volume fractionreaches 90% of the critical void volume fraction. Since the voidvolume fraction growth in GTN model is controlled by the plas-tic process, the issue of spurious damage growth and increasein damage zone well beyond the size of the characteristic ma-terial length (see [50]) was not observed. Reduced integration

FIG. 5. Crack tip discretizations used in the fatigue simulations of the panelwith a blunted notch.

TABLE 3Material properties of the panel with a blunted notch

Poisson’s Hardening Initial voidratio parameter volume fraction

E/σY = 300 ν = 0.3 b = 0.0 f0 = 0.0053

Tvergaard Coalescence, Nucleation, Hardening exponentconstant, fc = 0.15 fN = 0.04 n = 10q1 = 1.5 f f = 0.25 sN = 0.1

εN = 0.3

with hourglass control is employed to avoid partially vanishingelements.

Mechanical properties of the panel and the GTN model pa-rameters are summarized in Table 3. The characteristic length,lc, is chosen to be 0.13 which is slightly larger than the elementsize of the coarsest mesh. The effects of far fields on the nonlo-cal weight function are negligible and were ignored by settingw(x, s) to zero for ‖x − s‖ ≥ 1.5lc.

The void coalescence mechanism is considered using a thirdorder void coalescence function

f ζ = f ζ + 6

f 3f

(1

q1− f f

)

×[(

f f

2− fc

)+

(fc

f f− 1

3

)f ζ

]( f ζ)2 (64)

The constants of the third order void coalescence functionwere chosen to provide the best curve fit of a piecewise linearfunction of Eq. (23), and to satisfy the boundary conditions.Similar to the piecewise linear function, material loses its stresscarrying capacity as void volume fraction approaches the criticalvalue ( f ζ → 1/q1 as f → f f ).

The uniaxial stress-strain curve of the matrix material wasmodeled using a piecewise power law

ε =

σ

Eσ ≤ σY

σY

E

(σ

σY

)n

σ > σY

(65)

where, n is the strain hardening exponent. The multiscalesimulations were performed using the algorithmic parameters,EL = 4, δωall = 0.01, 0.01, and with error tolerances ofEtol = 0.0015, 0.0015.

Figure 6 displays the path of the fatigue crack simulated us-ing the local GTN model as obtained after 500 load cycles. Thedamage zone clearly localizes to smaller regions and the rate ofcrack propagation increases as the mesh is refined. The fatiguecrack path obtained under identical loading and topological con-ditions but with the nonlocal GTN model is shown in Figure 7.It can be seen that the results obtained after 500 and 750 cycles

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FIG. 6. Local simulations.

are insensitive to the mesh size except for the coarsest mesh.The damage zone (i.e., the region with high void volume frac-tion) localizes to a finite region with a size comparable to thecharacteristic material length. The discrepancy between the re-sults of the coarsest mesh and the finer meshes is attributed tothe fact that the size of the elements in front of the crack tip iscomparable to the characteristic material length.

7. CALIBRATION AND VALIDATIONOF THE GTN MODELVerification studies of the local multiscale fatigue model

against cycle-by-cycle simulations were conducted in [35] forbrittle materials and in [36] for ductile materials. In the presentarticle attention is restricted to calibration and experimental val-idation of the nonlocal multiscale model for ductile metals. We

FIG. 7. Nonlocal simulations.

consider two independent sets of fatigue experiments conductedon 316L austenitic stainless steel specimens with similar chem-ical compositions. The first set is used for the calibration ofparameters of the GTN model. The calibrated parameters weresubsequently used to predict the propagation characteristics of asingle macrocrack, and the results were compared to the obser-vations of the second set of experiments. The model calibrationwas conducted using direct cycle-by-cycle simulations. Afterappropriate parameter values of the model were determined, thevalidation simulations were conducted using the nonlocal mul-tiscale approach.

7.1. Model CalibrationThe fatigue response of the 316L stainless steel was previ-

ously investigated experimentally by Shi and Pluvinage [51]under isothermal and thermomechanical loading conditions.

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FIG. 8. Geometry and discretization of the cylinderical fatigue test specimen made of 316L stainless steel.

Cylindrical test samples were taken from a plate which hasbeen solution treated and subsequently water quenched. Fig-ure 8 displays the geometry and dimensions of the cylindricalfatigue specimens. The specimens were subjected to displace-ment controlled mechanical cycles of various magnitudes usinga servo-hydraulic test machine. The initial and stabilized cyclictotal stress and plastic strain ranges were recorded.

The GTN model is calibrated against the results of twoisothermal fatigue experiments reported in [51]. The void vol-ume fraction of the virgin material, f0, was computed a pri-ori based on the chemical composition of the material usingFranklin’s empirical formula ( f0 = 0.054(%S − 0.001%Mn)).Using the S and Mn contents of the 316L stainless steel speci-mens given as 0.01 and 1.8, respectively, the initial void volumefraction is computed to be f0 = 0.022%. The 0.2% proof stressof the material was obtained from the values suggested by [52]as 173 MPa. The nonlocal characteristic length was chosen tobe of the order of the grain size (lc = 80 µm) as suggestedin [34]. The void nucleation and coalescence mechanisms wereexcluded by setting A = 0, and f = f .

Calibration of the remaining constitutive parameters whichare the mixed hardening parameter, b, the Tvergaard constant,q1, and the hardening curve of the GTN model is conductedusing the simulations of two isothermal fatigue experimentsunder prescribed cyclic displacements with maximum ampli-tudes of 0.012 mm (specimen CLB1) and 0.016 mm (speci-men CLB2) which corresponds to 1.2% and 1.6% engineeringstrains, respectively. The loading was applied symmetrically inthe compressive-tensile direction with a triangular waveform.Simulations are conducted under the assumption of axisymmet-ric conditions and half of the cross-section is discretized due tosymmetry as shown in Figure 8. A small imperfection was in-troduced around the circumference of the specimen. The imper-fection is a semi-circular region in the plane of the cross-section(0.3 mm radius) with void concentrations linearly increasing

toward the surface of the specimen ( f0 = 0.1% in elementsadjacent to the surface).

The hardening curve, the mixed hardening parameter andthe Tvergaard constant were evaluated by minimizing the dis-crepancy between the simulated and experimental values of theinitial and stabilized stresses and plastic strains, and the overallfatigue lives of the specimens in a least squares sense. The Tver-gaard constant, the mixed hardening parameter, and the strainhardening exponents were identified to be 0.22, 1.1, and 3.7,respectively.

Figure 9 illustrates the evolution of the stress ranges (σ =|σmin|+|σmax|) computed using the numerical simulations forspecimens CLB1 and CLB2. The corresponding fatigue liveswere predicted to be 500 and 330 cycles for specimens CLB1and CLB2, respectively. The observed experimental fatiguelives of the two specimens were given as 550 and 295 cycles,respectively. The observed values for the stabilized stress ranges

FIG. 9. Variation of the stress range throughout simulations, CLB1 and CLB2,compared to experimental observations.

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FIG. 10. Variation of the plastic strain range throughout simulations, CLB1and CLB2, compared to experimental observations.

are given as 686 and 720 for CLB1 and CLB2, respectively,which are in reasonable agreement with the stress ranges pre-dicted by the GTN model. The plastic strain ranges for thespecimens CLB1 and CLB2 as predicted by the numerical sim-ulations, and the observed experimental values are plotted inFigure 10. Similarly, the plastic strains were predicted to be inthe range of the experimental observations (within 8%). Thediscrepancies between the experimental observations and theresults of the numerical simulations are attributed to the limita-tion of the hardening laws used in the simulations (i.e., mixedkinematic-isotropic hardening). For low cycle fatigue, moreelaborate hardening laws and, possibly, bounding surface andmulti-yield surface approaches need to be investigated.

7.2. Model ValidationThe calibrated GTN model is then employed to predict the

growth of fatigue cracks on specimens made of 316L stainlesssteel. Wheatley et al. [53] previously conducted experimentsto investigate the propagation characteristics of cracks underfatigue loading and in the presence of overloads. They testedcompact-tension (CT) specimens made of 316L stainless steelwith chemical compositions similar to that of the cylindricalspecimens used in the calibration of the GTN model. The CTspecimens were 6 mm thick and 40 mm wide, and the initialnotches were 12 mm long parallel to the bar length. Prior to theexperiments, a 6 mm fatigue pre-crack was produced by highamplitude cyclic loading. Figure 11 illustrates the geometry ofthe CT specimens used in the experiments. A sinusoidal wave-form with maximum amplitude of 3 kN and an R-ratio of 0.1was applied in the tensile direction. In the overload experiment,a single cycle overload with maximum amplitude of 5 kN wasapplied to the pre-cracked specimen and the maximum ampli-tude of the loading was reduced to 3 kN in the subsequent cycle.Growth of the cracks was observed during the loading periodand crack growth curves are provided.

FIG. 11. Geometry and discretization of the CT specimen made of 316L stain-less steel.

We conducted numerical simulations using the GTN modelin the framework of the proposed multiscale life predictionmethodology to predict the crack growth curves of the fatigueand overload specimens. Plane stress conditions were assumedand half of the geometry was modeled due to symmetry. The fi-nite element mesh is illustrated in Figure 11. The multiscalesimulations were performed using the algorithm parameters,EL = 4, δωall = 0.05, 0.05, and with error tolerances ofEtol = 0.0075, 0.0075. The multiscale algorithm parametersare chosen to be relatively small to attain high levels of accu-racy. The effects of the choice of algorithm parameters on theaccuracy and performance of the model is presented in [36].The initial void volume fraction is obtained using Franklin’sempirical formula to be f0 = 0.012%. The calibrated modelparameters for fatigue life which are the Tvergaard constant, q1,the mixed hardening parameter, b, and the strain hardening ex-ponent, n, are set to the values identified in the calibration phase(q1 = 1.1, b = 0.22, and n = 3.7, respectively). The value of0.2% proof stress is provided by [53] as 334 MPa.

Figure 12 illustrates the crack growth curves of the fatigueand overload specimens as predicted by the simulations alongwith the experimental observations. A reasonable overall matchwas observed between the experimental observations and thenumerical predictions of both fatigue and overload specimens.The total life of the fatigue and overload specimens were 70000

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FIG. 12. Experimental and predicted crack growth curves of the fatigue andoverload specimens.

and 130000 cycles, respectively. The proposed multiscale tech-nique required the resolution of only 2530 and 2650 load cyclesto evaluate the crack growth on the fatigue and overload speci-mens, respectively.

8. DISCUSSION AND CONCLUSIONSThe article presents a computational fatigue life prediction

methodology based on the mathematical homogenization theorywith almost periodic fields. The almost-periodicity is a byprod-uct of irreversible processes, such as damage accumulation,which violates the condition of local periodicity. The nonlo-cal multiscale model has been shown to be insensitive to themesh size as long as the characteristic size is smaller than theelement size. Calibration studies revealed certain shortcomingsof the model, particularly in the modeling of the stress-strainloops under high amplitude loading (very low cycle fatigue).Some of these deficiencies could be circumvented by employ-ing multi-yield surfaces and/or bounding surface plasticity the-ories. Nevertheless, the calibrated nonlocal multiscale modelperformed reasonably well in validation studies more than justi-fying its usefulness due to significant computational cost savings(approximately 90–94% cost reduction) compared to the directcycle-by-cycle simulation.

ACKNOWLEDGEMENTThis work was supported by the Office of Naval Research

through grant number N00014-97-1-0687. This support is grate-fully acknowledged.

REFERENCES1. P. C. Paris and F. Erdogan, A Critical Analysis of Crack Propagation Laws,

J. Basic Engng., vol. 85, pp. 528–534, 1963.2. R. G. Foreman, V. E. Keary, and R. M. Engle, Numerical Analysis of Crack

Propagation in Cyclic-loaded Structures, J. Basic Engng., vol. 89, pp. 459–464, 1967.

3. C. Gilbert, R. Dauskardt, and R. Ritchie, Microstructural Mechanisms ofCyclic Fatigue-crack Propagation in Grain Bridging Ceramics, CeramicsInternational, vol. 23, pp. 413–418, 1997.

4. C. Laird, Mechanisms and Theories of Fatigue, pp. 149–203. AmericanSociety of Metals, 1972

5. O. E. Wheeler, Spectrum Loading and Crack Growth, J. Basic Engng.,vol. 94, pp. 181–186, 1972.

6. O. Nguyen, E. A. Repetto, M. Ortiz, and R. A. Radovitzky, A CohesiveModel of Fatigue Crack Growth, Int. J. Fracture, vol. 110 no. 4, pp. 351–369, 2001.

7. K. L. Roe and T. Siegmund, An Iirreversible Cohesive Zone Model forInterface Fatigue Crack Growth Simulation, Engng. Fracture Mech., vol. 70,pp. 209–232, 2003.

8. J. L. Chaboche, Continuum Damage Mechanics I: General Concepts & II:Damage Growth, Crack Initiation and Crack Growth, J. Appl. Mech., vol. 55,pp. 59–72, 1988.

9. C. L. Chow and Y. Wei, A Model of Continuum Damage Mechanics forFatigue Failure, Int. J. Fatigue, vol. 50, pp. 301–306, 1991.

10. J. Fish and Q. Yu, Computational Mechanics of Fatigue and Life Predictionsfor Composite Materials and Structures, Comput. Meth. Appl. Mech. Engng.,vol. 191, pp. 4827–4849, 2002.

11. M. Kaminski and M. Kleiber, Numerical Homogenization of n-componentComposites Including Stochastic Interface Defects, Int. J. Numer. Meth.Engng., vol. 47, pp. 1001–1027, 2000.

12. G. W. Milton, The Theory of Composites. Cambridge Univ. Press,Cambridge, UK, 2002.

13. S. Torquato, Random Heterogeneous Materials: Microstucture and Macro-scopic Properties, Springer-Verlag, New York, 2002.

14. K. Sab, On the Homogenization and the Simulation of Random Materials,Eur. J. Mech., A/Solids, vol. 11, pp. 585–607, 1992.

15. J. Fish and A. Wagiman, Multiscale Finite Element Method for a LocallyNonperiodic Heterogeneous Medium, Comp. Mech., vol. 12, pp. 164–180,1993.

16. S. Ghosh, K. Lee, and S. Moorthy, Multiple Scale Analysis of Heteroge-neous Elastic Structures using Homogenization Theory and Voronoi CellFinite Element Method, Int. J. Solids Structures, vol. 32, pp. 27–62, 1995.

17. S. Ghosh, K. Lee, and S. Moorthy, Two Scale Analysis of Hetero-geneous Elastic-Plastic Materials with Asymptotic Homogenization andVoronoi Cell Finite Element Method, Comput. Methods Appl. Mech. En-gng., vol. 132, pp. 63–116, 1996.

18. J. T. Oden and T. I. Zohdi, Analysis and Adaptive Modeling of HighlyHeterogeneous Elastic Structures, Comput. Methods. Appl. Mech. Engng.,vol. 148, pp. 367–391, 1997.

19. T. I. Zohdi, J. T. Oden, and G. J. Rodin, Hierachical Modeling of Heteroge-neous Bodies, Comput. Meth. Appl. Mech. Engng., vol. 138, pp. 273–298,1996.

20. J. Aboudi, The Generalized Method of Cells and High-fidelity General-ized Method of Cells Micromechanical Models—A Review, Mech. Adv.Materials and Structures, vol. 11, nos. 4–5, pp. 329–366, 2005.

21. F. G. Yuan and N. J. Pagano, Size Scales for Accurate Homogenization in thePresence of Severe Stress Gradients, Mech. Adv. Materials and Structures,vol. 10, no. 4, pp. 353–365, 2003.

22. N. Ansini and A. Braides, Separation of Scales and Almost-Periodic Effectsin the Asymptotic Behavior of Perforated Media, Acta Applicandae Math.,vol. 65, pp. 59–81, 2001.

23. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,Oxford Lecture Series in Mathematics and Its Applications. Oxford Univ.Press, Oxford, 1998.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

olor

ado

- H

ealth

Sci

ence

Lib

rary

] at

15:

04 2

5 Se

ptem

ber

2014


24. G. Nguetseng and H. Nnang, Homogenization of Nonlinear Monotone Op-erators Beyond the Periodic Setting, Electronic J. Diff. Eqn., vol. 36, pp. 1–24, 2003.

25. O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problemsin Elasticity and Homogenization, North-Holland Press, Amsterdam, 1992.

26. V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Dif-ferential Operators and Integral Functionals, Springer-Verlag, New York,1994.

27. V. Tvergaard, Material Failure by Void Growth to Coalescence, Adv. Appl.Mech., vol. 27, pp. 83–151, 1990.

28. R. de Borst and H. B. Muhlhaus, Gradient Dependent Plasticity: For-mulation and Algorithmic Aspects, Int. J. Numer. Meth. Engng., vol. 35,pp. 521–539, 1992.

29. Z. P. Bazant, T. B. Belytschko, and T. P. Chang, Continuum Theory forStrain Softening, J. Engng. Mech., vol. 110, pp. 1666–1691, 1984.

30. R. de Borst and L. J. Sluys, Localization in Cosserat Continuum under Staticand Dynamic Loading Conditions, Comput. Methods Appl. Mech. Engng.,vol. 90, pp. 805–827, 1991.

31. K. Garikipati and T. J. R. Hughes, A Variational Multiscale Approach toStrain Localization Formulation for Multidimensional Problems, Comput.Methods. Appl. Mech. Engng., vol. 188, pp. 39–60, 2000.

32. J. C. Simo and J. Oliver, A New Approach to the Analysis and Simulationof Strain Softening in Solids, in Z. P. Bazant, Z. Bittnar, M. Jirasek, andJ. Mazars, (eds), Fracture and Damage in Quasibrittle Structures. E&FNSpan, London–New York, 1994.

33. F. Zhou and J. F. Molinari, Dynamic Crack Propagation with CohesiveElements: A Methodology to Address Mesh Dependency, Int. J. Numer.Meth. Engng., vol. 59, pp. 1–29, 2004.

34. R. H. J. Peerlings, W. A. M. Brekelmans, R. de Borst, and M. G. D. Geers,Gradient-enhanced Damage Modelling of High-Cycle Fatigue, Int. J. Nu-mer. Meth. Engng., vol. 49, pp. 1547–1569, 2000.

35. C. Oskay and J. Fish, Fatigue Life Prediction using 2-scale TemporalAsymptotic Homogenization, Int. J. Numer. Meth. Engng., vol. 61, pp. 329–359, 2004.

36. C. Oskay and J. Fish, Multiscale Modeling of Fatigue for Ductile Materials,Int. J. Comp. Multiscale Engng., vol. 2, 2004.

37. D. Cioranescu and P. Donato, An Introduction to Homogenization, vol-ume 17 of Oxford lecture series in mathematics and its applications, OxfordUniv. Press, Oxford, UK, 1999.

38. A. L. Gurson, Continuum Theory of Ductile Rapture by Void Nucleationand Growth: Part I- Yield Criteria and Flow Rules for Porous Ductile Media,J. Engng. Mater. and Technol., vol. 99, pp. 2–15, 1977.

39. C. C. Chu and A. Needleman, Void Nucleation Effects in Biaxially StretchedSheets, J. Engng. Mater. and Technol., vol. 102, pp. 249–256, 1980.

40. V. Tvergaard, Material Failure by Void Coalescence in Localized ShearBands, Int. J. Solids Structures, vol. 18, pp. 659–672, 1982.

41. M. Gologanu, J.-B. Leblond, G. Perrin, and J. Devaux, Recent Extensionsof Gurson’s Model for Porous Ductile Metals, in Suquet (ed.), ContinuumMicromechanics, Springer-Verlag, New York, 1997.

42. V. Tvergaard and A. Needleman, Analysis of the Cup-cone Fracture in aRound Tensile Bar, Acta Metall., vol. 32(1), pp. 157–169, 1984.

43. J. Llorca, S. Suresh, and A. Needleman, An Experimental and NumericalStudy of Cyclic Deformation in Metal-Matrix Composites, Metall. Trans.A, vol. 23A, pp. 919–934, 1992.

44. M. E. Mear and J. W. Hutchinson, Influence of Yield Surface Curvature onFlow Localization in Dilatant Plasticity, Mech. Mater., vol. 4, pp. 395–407,1985.

45. G. Pijaudier-Cabot and Z. P. Bazant, Non-local Damage Theory, J. Eng.Mech., vol. 113, pp. 1512–1533, 1987.

46. V. Tvergaard and A. Needleman, Effects of Nonlocal Damage in PorousPlastic Solids, Int. J. Solids Structures, vol. 32, pp. 1063–1077, 1995.

47. J. C. Simo, Strain Softening and Dissipation: A Unification of Approaches,In J. Mazars and Z. P. Bazant, (eds), Cracking and Damage: Strain Local-ization and Size Effect, pages 440–461. Elsevier, New York, 1989.

48. N. Aravas, On the Numerical Integration of a Class of Pressure-dependentPlasticity Models, Int. J. Numer. Meth. Engng., vol. 24, pp. 1395–1416,1987.

49. J. H. Lee and Y. Zhang, On the Numerical Integration of a Class of Pressure-dependent Plasticity Models with Mixed Hardening, Int. J. Numer. Meth.Engng., vol. 32, pp. 419–438, 1991.

50. M. D. D. Geers, R. de Borst, W. A. M. Brekelmans, and R. H. J. Peerlings,Strain-based Transient-gradient Damage Model for Failure Analyses, Com-put. Methods. Appl. Mech. Engng., vol. 160, pp. 133–153, 1998.

51. H. J. Shi and G. Pluvinage, Cyclic Stress-strain Response During Isothermaland Thermomechanical Fatigue, Int. J. Fatigue, vol. 16, pp. 549–557, 1994.

52. H. J. Shi, Z. G. Wang, and H. H. Su, Thermomechanical Fatigue of a 316lAustenitic Steel at Two Different Temperature Intervals, Scripta Materialia,vol. 35, pp. 1107–1113, 1996.

53. G. Wheatley, R. Niefanger, Y. Estrin, and X. Z. Hu, Fatigue Crack Growthin 316l Stainless Steel, Key Engng. Matl., vol. 145–149, pp. 631–636, 1998.

54. V. Tvergaard, Effect of Yield Surface Curvature and Void Nucleation onPlastic Flow Localization, J. Mech. Phys. Solids, vol. 35, no. 1, pp. 43–60,1987.

APPENDIX

A Evaluation of the Consistency Parameterfor the Nonlocal GTN Model

The evaluation of the consistency parameter of the micro-chronological problem is similar to the computation of the con-sistency parameter of a single scale model (e.g., the boundaryvalue problem defined in Eqs. (14)–(19). To simplify the nota-tion, we therefore present the evaluation of the consistency pa-rameter using the response fields in total form. The consistencycondition for the micro-chronological problem is defined as:

Φ,τ = 0 (A66)

The local consistency parameter, λ, is evaluated based on theapproach proposed in Ref [54]. To this extent, we define afictitious yield surface, ΦG

ΦG = ΦG(σG, f, σM ) = (qG)2

σ2M

+2 f cosh

(−3

2

pG

σM

)−1−(

f)2

(A67)

The fictitious stress components, σG , and the fictitious local rateof plastic strains chosen to be:

σG

σM= B

σF(A68a)

µ,τ = µG,τ (A68b)

Equation (68a) ensures that the yielding occurs simultaneouslyin Φ and ΦG (i.e., Φ = 0 ⇒ ΦG = 0). Equation (68b) statesthat the local rate of plastic strains is taken to be identical to thatevaluated using the fictitious yield surface. Eq. (68b) must besatisfied at all material points in

λ∂φ

∂σ= λ

G ∂φG

∂σG(A69)

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The consistency condition (Eq. (66)) along with Eqs. (68)and (69) may be used to obtain an expression for the local con-sistency parameter in the form of an integral equation

λ (x, t) =∂φ

∂σ: L : ε,τ

∂φ

∂σ: L :

∂φ

∂σ

− H∂φ

∂σ: L :

∂φ

∂σ

× 1

W (x)

∫

w (x, s) λ (s, t) ds (A70)

The plastic modulus, H , is given by:

H = −σM

σF

∂Φ∂ f

(1 − f )

(δ :

∂Φ∂σ

)

+[σM

σFA∂φ

∂ f+ ∂φ

∂σF

]Et

1 − f

1

σM

(Θ :

∂Φ∂σ

)(A71)

B Evolution Equations for the Nonlocal GTN ModelThe evolution equations of the internal state variables of the

GTN model (i.e., void volume fraction, equivalent plastic strain,matrix flow strength, and the center of the yield surface) formicro- and macro-chronological problems is derived herein. Tosimplify the presentation, we define:

µ,τ = λ1 ∂Φ∂σ

(A72a)

M (µ),t = λo⟨∂Φ∂σ

⟩+ 1

ζ

⟨µ,τ

⟩(A72b)

Decomposing the void volume fraction using Eq. (35), andintroducing the resulting fields as well as Eqs. (72) into theoriginal evolution equation of the void volume fraction

O(ζ−1) : f,τ =×q1

(1 − M ( f ) − f

)δ

: µ,τ + Aρ,τ (A73)

O(1) : M ( f ),t + f,t =×q1

(1 − M ( f ) − f

)δ

:(M (µ),t + µ,t

)+A

(M(ρ),t + ρ,t

)(A74)

The micro-chronological evolution equation for the void volumefraction is given by Eq. (73). Applying the PTH operator to Eqs.(73) and (74) and using Eq. (36) yields:

M ( f ),t = q1(1 − M ( f ) − ⟨

f⟩)

δ : M (µ),t

−q1

⟨f δ :

(1

ζµ,τ + µ,t

)⟩

+ 〈A〉 M(ρ),t −⟨A

(1

ζρ,τ + ρ,t

)⟩(A75)

which is the evolution equation of the macro-chronological voidvolume fraction.

The evolution equations of the micro-chronological andmacro-chronological equivalent plastic strains may be derivedusing a similar algebra

ρ,τ = (M(B) + B) : µ,τ

[1 − M( f ) − f ][M(σF ) + σF ](A76)

M(ρ),t =⟨

(M(B) + B)

[1 − M( f ) − f ][M(σF ) + σF ]

⟩: M(µ),t

+⟨

(M(B) + B)

[1 − M( f ) − f ][M(σF ) + σF ]

:

(1

ζµ,τ + µ,t

)⟩(A77)

The evolution equations of the micro- and macro-chronological matrix flow strength and the radius of the yieldsurface may be expressed as:

σM,τ = Et ρ,τ (A78)

M (σM ),t = ⟨Et

⟩M (ρ),t +

⟨Et

(1

ζρ,τ + ρ,t

)⟩(A79)

and,

σF,τ = bσM,τ (A80)

M (σF ),t = bM (σM ),t (A81)

in which, Eqs. (78) and (79) are the micro- and macro-chronological evolution equations of the matrix flow strength,respectively. Equations (80) and (81) correspond to the evolu-tion equations of the yield surface radius for the micro- andmacro-chronological problems, respectively.

Decomposing the yield surface center, α, according toEq. (35), and introducing the micro- and macro-chronologicalconsistency parameters to Eq. (34)

O(ζ−1) : α,τ = λ1Q(M(B) + B) (A82)

O(1) : M (α),t + α,t = λoQ(M(B) + B) (A83)

in which,

Q = (1 − b)

(B :

∂Φ∂σ

)−1[H + σy

σF

∂Φ∂ f

q1(1 − f )

(δ :

∂Φ∂σ

)

+ Et

1 − f

1

σF

(B :

∂Φ∂σ

)](A84)

The micro-chronological evolution equation of the yield sur-face center is given as Eq. (82). Applying the PTH operatorto Eqs. (82) and (83) yields the macro-chronological evolutionequation of the yield surface center

M (α),t = λo (〈Q〉 M (B) + ⟨

QB⟩) + 1

ζ

⟨α,τ

⟩(A85)

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A Nonlocal Multiscale Fatigue Model