International Journal of Fracture
110:
351–369, 2001.© 2001
Kluwer Academic Publishers. Printed in the Netherlands.
A cohesive model of fatigue crack growth
O. NGUYEN, E.A. REPETTO, M. ORTIZ and R.A. RADOVITZKY
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena CA 91125, USA
Received 15 December 1999; accepted in revised form 15 January 2001
Abstract.
We investigate the use of cohesive theories of fracture, in conjunction with the explicit resolutionof the neartip plastic ﬁelds and the enforcement of closure as a contact constraint, for the purpose of fatiguelife prediction. An important characteristic of the cohesive laws considered here is that they exhibit unloadingreloading hysteresis. This feature has the important consequence of preventing shakedown and allowing for steadycrack growth. Our calculations demonstrate that the theory is capable of a uniﬁed treatment of long cracks underconstantamplitude loading, short cracks and the effect of overloads, without
ad hoc
corrections or tuning.
Keywords:
Cohesive law, fatigue, ﬁnite elements, overload, short cracks.
1. Introduction
Fatigue life prediction remains very much an empirical art at present. Following the pioneering work of Paris et al. (1961) phenomenological laws relating the amplitude of theapplied stress intensity factor,
K
and the crack growth rate, d
a
/d
N
, have provided a valuableengineering analysis tool. Indeed, Paris’s law successfully describes the experimental dataunder ‘ideal’ conditions of smallscale yielding, constant amplitude loading and long cracks(Klesnil and Lukas, 1972; Anderson, 1995). However, when these stringent requirements arenot adhered to, Paris’s law loses much of its predictive ability. This has prompted a multitudeof modiﬁcations of Paris’s law intended to suit every conceivable departure from the idealconditions: socalled
R
effects (Gilbert et al., 1997; Wheeler, 1972); threshold limits (Laird,1979; Drucker and Palgen, 1981; Needleman, 1987); closure (Foreman et al., 1967); variableamplitude loads and overloads (Willenborg et al., 1971; Xu et al., 1995); small cracks (Elber,1970; El Haddad et al., 1979); and others. The case of short cracks is particularly troublesomeas Paris’s lawbased designs can signiﬁcantly underestimate their rate of growth (Tvergaardand Hutchinson, 1996).The proliferation of
ad hoc
fatigue laws would appear to suggest that the essential physicsof fatiguecrack growth is not completely captured by theories which are based on the stressintensity factors as the sole cracktip loading parameters. A possible alternative approach,which is explored in this paper, is the use of cohesive theories of fracture, in conjunctionwith the explicit resolution of the neartip plastic ﬁelds and the enforcement of closure as acontact constraint. Cohesive theories regard fracture as a gradual process in which separationbetween incipient material surfaces is resisted by cohesive tractions. Under monotonic loading, the cohesive tractions eventually reduce to zero upon the attainment of a critical openingdisplacement. The formation of new surface entails the expenditure of a welldeﬁned energyper unit area, known variously as speciﬁc fracture energy or critical energy release rate.A number of cohesive models have been proposed – and successfully applied–to date forpurposes of describing monotonic fracture processes (Needleman, 1990a,b; Neumann, 1974;
352
O. Nguyen et al.
Figure 1.
Cohesive law with irreversible behavior.
Pandolﬁ and Ortiz, 1998; Ortiz and Pandolﬁ, 1999; Camacho and Ortiz, 1996; von Euw et al.,1972; Donahue et al., 1972; Ortiz and Popov, 1985; Paris et al., 1972; Rice, 1967). In somemodels, unloading from – and subsequent reloading towards–the monotonic envelop is takento be linear, e.g., towards the srcin, and elastic or nondissipative (Camacho and Ortiz, 1996;Cuitiño and Ortiz, 1992), Figure 1. As it turns out, however, such models cannot be appliedto the direct cyclebycycle simulation of fatigue crack growth. Thus, our simulations revealthat a crack subjected to constantamplitude cyclic loading, and obeying a cohesive law withelastic unloading, tends to
shake down
, i.e., after the passage of a small number of cyclesall material points, including those points on the cohesive zone, undergo an elastic cycle of deformation, and the crack arrests.The centerpiece of the present approach is an irreversible cohesive law with unloadingreloading hysteresis. The inclusion of unloadingreloading hysteresis into the cohesive lawis intended to simulate simply dissipative mechanisms such as crystallographic slip (Atkinson and Kanninen, 1977; Kanninen and Popelar, 1985) and frictional interactions betweenasperities (Gilbert et al., 1995). Consideration of unloadingreloading hysteresis proves critical in one additional respect. Thus, the attainment of an elastic cycle is not possible if thecohesive law exhibits unloadingreloading hysteresis, and the possibility of shakedown –and the attendant spurious crack arrest – is eliminated altogether. Frictional laws exhibitingunloadingreloading hysteresis have been applied to the simulation of fatigue in brittle materials (Gylltoft, 1984; Hordijk and Reinhardt, 1991; Kanninen and Atkinson, 1980; Ziegler,1959).The plastic neartip ﬁelds, including the reverse loading that occurs upon unloading, arealso known to play an important role in fatigue crack growth (Rice and Beltz, 1994; Ortiz,1996; Leis et al., 1983; Suresh, 1991; Tvergaard and Hutchinson, 1996). Models based ondislocation pile–ups (Bilby and Swinden, 1965) or ‘superdislocations’ (Atkinson and Kanninen, 1977; Kanninen and Popelar, 1985) have been proposed to describe the plastic activityattendant tocrack growth. TheDugdale–Barenblatt (Barrenblatt, 1962; ElHaddad etal., 1980)strip yield model was used by Budiansky and Hutchinson (Budiansky and Hutchinson, 1978)to exhibit qualitatively the effects of closure, thus demonstrating the importance of the plasticwake in fatigue crack growth.
A cohesive model of fatigue crack growth
353Here, we propose to resolve the neartip plastic ﬁelds and the cohesive zone explicitly byrecourse to adaptive meshing. In particular, the plastic dissipation attendant to crack growthis computed explicitly and independently of the cohesive separation processes, and thereforeneed not be lumped into the crackgrowth initiation and propagation criteria. The materialdescription accounts for cyclic plasticity through a combination of isotropic and kinematichardening; and for ﬁnite deformations such as accompany the blunting of the crack tip. Crack closure is likewise accounted for explicitly as a contact constraint.In the present approach, crack growth results from the delicate interplay between bulk cyclic plasticity, closure, and gradual decohesion at the crack tip. Since the calculations explicitly resolve all plastic ﬁelds and cohesive lengths, the approach is free from the restrictionof smallscale yielding. This opens the way for a uniﬁed treatment of long cracks, shortcracks, and fullyyielded conﬁgurations. In addition, loadhistory effects are automaticallyand naturally accounted for by the pathdependency of plasticity and of the cohesive law.This effectively eliminates the need for
ad hoc
cyclecounting rules under variableamplitudeloading conditions, or for
ad hoc
rules to account for the effect of overloads.The paper is structured as follows. We begin by setting the basis of the ﬁnite elementmodel for fatigue simulation in Section 2. The cohesive law is deﬁned in Subsection 2.1.Then follows Subsection 2.2 on cyclic plasticity. Finally, Subsection 2.3 describes the ﬁniteelement implementation of the model. In Section 3, we present the results of validation testswhich establish the predictive ability of the model under a variety of conditions of interest.We begin by establishing that the model exhibits Parislike behavior under ideal conditions of long cracks, smallscale yielding and constant amplitude loading. Finally, we show that themodel captures the smallcrack effect and the effect of overloads without
ad hoc
correctionsor tuning.
2. Description of the model
We have developed a ﬁnite element model to simulate the fatigue behavior of a plane strainspecimen. The simulation was performed by an implicit integration of the equilibrium equations using a NewtonRaphson algorithm to resolve the nonlinear system of equations(Dafalias, 1984). The main constituents of the model are described in the next 3 Subsections,2.2, 2.1, and 2.3.2.1. A
COHESIVE LAW WITH UNLOADING

RELOADING HYSTERESIS
The centerpiece of the present approach is the description of the fracture processes by meansofanirreversible cohesive lawwithunloadingreloading hysteresis. Theinclusion ofunloadingreloading hysteresis within the cohesive law is intended to account, in some effective andphenomenological sense, for dissipative mechanisms such as frictional interactions betweenasperities (Gilbert et al., 1995) and crystallographic slip (Atkinson and Kanninen, 1977; Kanninen and Popelar, 1985). As noted earlier, consideration of loadingunloading hysteresisadditionally has the farreaching effect of preventing shakedown after a few loading cyclesand the attendant spurious crack arrest.We start by considering monotonic loading processes resulting in pure mode I opening of the crack. As the incipient fracture surface opens under the action of the loads, the openingis resisted by a number of materialdependent mechanisms, such as cohesion at the atomisticscale, bridging ligaments, interlocking of grains, and others (Anderson, 1995). For simplicity,
354
O. Nguyen et al.
we assume that the resulting cohesive traction
T
decreases linearly with the opening displacement
δ
, and eventually reduces to zero upon the attainment of a critical opening displacement
δ
c
(e.g., Paris et al., 1972; Camacho and Ortiz, 1996; Rice, 1967) Figure 1. In addition, separation across a material surface is assumed to commence when a critical stress
T
c
is reachedon the material surface. We note that, prior to the attainment of the critical stress, the openingdisplacement is zero, i.e., the potential cohesive surface is fully coherent. We shall refer to therelation between
T
and
δ
under monotonic opening as the monotonic cohesive envelop. Moreelaborate monotonic cohesive envelopes than the one just described have been proposed by anumber of authors (Needleman, 1992; Simo and Laursen, 1992; Xu and Needleman, 1994),but these extensions will not be pursued here in the interest of simplicity.The critical stress
T
c
may variously be identiﬁed with the macroscopic cohesive strength orthe spall strength of the material. In addition, the area under the monotonic cohesive envelop,
G
c
=
δ
c
0
T(δ)
d
δ
=
12
σ
c
δ
c
(1)equals twice the intrinsic fracture energy or critical energy release rate of the material. Ingeneral, the macroscopic or measured critical energyrelease rate may be greatly in excess of
G
c
by virtue of the plastic dissipation attendant to crack initiation and growth. In addition,
G
c
/
2 may itself be greatly in access of the surface energy owing to dissipative mechanismsoccurring on the scale of the cohesive process zone.For fatigue applications, speciﬁcation of the monotonic cohesive envelop is not enoughand the cohesive behavior of the material under cyclic loading is of primary concern. We shallassume that the process of unloading from–and reloading towards–the monotonic cohesiveenvelop is hysteretic. For instance, in some materials the cohesive surfaces are rough andcontain interlocking asperities or bridging grains (Gilbert et al., 1995). Upon unloading andsubsequent reloading, the asperities may rub against each other, and this frictional interaction dissipates energy. In other materials, the crack surface is bridged by plastic ligamentswhich may undergo reverse yielding upon unloading. Reverse yielding upon unloading mayalso occur when the crack growth is the result of alternating crystallographic slip (Atkinsonand Kanninen, 1977; Kanninen and Popelar, 1985). In all of these cases, the unloading andreloading of the cohesive surface may be expected to entail a certain amount of dissipationand, therefore, be hysteretic.Imagine, furthermore, that a cohesive surface is cycled at low amplitude after unloadingfrom the monotonic cohesive envelop. Suppose that the amplitude of the loading cycle is lessthan the height ofthe monotonic envelop at theunloading point, Figure 2. Weshall assume thatthe unloadingreloading response degrades with the number of cycles. For instance, repeatedrubbing of asperities may result in wear or polishing of the contact surfaces, resulting ina steady weakening of the cohesive response. A class of simple phenomenological modelswhich embody these assumptions is obtained by assuming different incremental stiffnessesdepending on whether the cohesive surface opens or closes, i. e.,
˙
T
=
K
−
˙
δ,
if
˙
δ <
0,
K
+
˙
δ,
if
˙
δ >
0,(2)where
K
+
and
K
−
are the loading and unloading incremental stiffnesses respectively. In addition, we take the stiffnesses
K
±
to be internal variables in the spirit of damage theories, and
A cohesive model of fatigue crack growth
355
Figure 2.
Cyclic cohesive law with unloadingreloading hysteresis.
their evolution to be governed by suitable kinetic equations. For simplicity, we shall assumethat unloading always takes place towards the srcin of the
T
−
δ
axes, i.e.,
K
−
=
T
max
δ
max
,
(3)where
T
max
and
δ
max
are the traction and opening displacement at the point of load reversal, respectively. In particular,
K
−
remains constant for as long as crack closure continues, Figure 2.By contrast, the reloading stiffness
K
+
is assumed to evolve in accordance with the kineticrelation:
˙
K
+
=
−
K
+
˙
δ/δ
f
,
if
˙
δ >
0
,(K
+
−
K
−
)
˙
δ/δ
f
,
if
˙
δ <
0
,
(4)where
δ
f
is a characteristic opening displacement. Evidently, upon unloading,
˙
δ <
0,
K
+
tends to the unloading slope
K
−
, whereas upon reloading,
˙
δ >
0,
K
+
degrades steadily,Figure 2. Finally, we assume that the cohesive traction cannot exceed the monotonic cohesiveenvelop. Consequently, when the stressstrain curve intersects the envelop during reloading, itis subsequently bound to remain on the envelop for as long as the loading process ensues.Evidently, the details of the kinetic equations for the unloading and reloading stiffnesses just described are largely arbitrary, and the resulting model is very much phenomenological innature. However, some aspects of the model may be regarded as essential and are amenable toexperimental validation. Consider, for instance, the following thought experiment. A cohesivesurface is imparted a uniform opening displacement
δ
0
< δ
c
and subsequently unloaded. Let
K
+
0
be the initial reloading stiffness after the ﬁrst unloading. The cohesive surface is thencycled between the opening displacements 0 and
δ
0
. Let
K
+
N
be the initial reloading stiffnessafter
N
cycles. A straightforward calculation using Equations (3) and (4) then gives:
K
+
N
+
1
=
λK
+
N
,
(5)where
λ
=
δ
f
δ
0
(
1
−
e
−
δ
0
/δ
f
)
2
+
e
−
2
δ
0
/δ
f
(6)