a cohesive model for fatigue failure of polymers

Engineering Fracture Mechanics 72 (2005) 691–708

www.elsevier.com/locate/engfracmech

A cohesive model for fatigue failure of polymers

Spandan Maiti, Philippe H. Geubelle *

Department of Aerospace Engineering and Beckman Institute, University of Illinois at Urbana-Champaign,

306 Talbot Laboratory, 104 S. Wright Street, Urbana, IL 61801, USA

Received 19 April 2004; received in revised form 25 June 2004; accepted 27 June 2004

Available online 17 September 2004

Abstract

A cohesive failure model is proposed to simulate fatigue crack propagation in polymeric materials. The model relies

on the combination of a bi-linear cohesive failure law used for fracture simulations under monotonic loading and an

evolution law relating the cohesive stiffness, the rate of crack opening displacement and the number of cycles since the

onset of failure. The fatigue component of the cohesive model involves two parameters that can be readily calibrated

based on the classical log–log Paris failure curve between the crack advance per cycle and the range of applied stress

intensity factor. The paper also summarizes a semi-implicit implementation of the cohesive model into a cohesive-vol-

umetric finite element framework, allowing for the simulation of a wide range of fatigue fracture problems.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Cohesive model; Fatigue failure; Cohesive finite element; Polymeric materials

1. Introduction

Polymeric materials subjected to cyclic loading are highly susceptible to fatigue failure, which often pre-

cedes monotonic or creep failures at similar applied load levels [19]. Substantial effort has thus been devotedto understand the mechanisms underlying the fatigue failure of this class of materials and to obtain predic-

tive capabilities for their service life. Early attempts to characterize the fatigue failure in polymers were

inspired by similar works for metals and were generally empirical in nature. Fatigue life test results were

generally presented in the form of S�N curves. But these curves are of limited use in practical situations

as, unlike metals, the crack initiation life for these materials are often short compared to the fatigue crack

propagation life [15]. Subsequently, different relationships were proposed to relate the per cycle crack

0013-7944/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2004.06.005

* Corresponding author. Tel.: +1 217 244 7648; fax: +1 217 244 0720.

E-mail address: [email protected] (P.H. Geubelle).

mailto:[email protected]

692 S. Maiti, P.H. Geubelle / Engineering Fracture Mechanics 72 (2005) 691–708

advance with various loading and fracture parameters, such as the range of energy release rate, the range of

the stress intensity factor (DK), or the mean stress intensity factor [21]. It has since been realized that the

Paris power relation between the crack advance rate per cycle (da/dN) and DK, which is extensively used to

represent the fatigue failure of metals, can also correlate the experimental data well for different polymers

for various loading and material parameters. But the slope of the Paris log–log curve, i.e., the exponent mof the power relationship between da/dN and DK in da/dN = CDKm, is generally much higher than that for

metals, typically exceeding a value of 6, indicating a much higher sensitivity to the applied loading.

Extensive experiments performed on different polymers established further differences between the mech-

anisms responsible for the crack growth in metals and glassy polymers. While it is mostly plasticity and

distributed damage ahead of the crack tip in the case of metals, the primary mechanism for crack growth

in polymers is the crazing of the material in a thin strip ahead of the crack tip. Though brittle to ductile

transition is widely reported for this class of materials [22], most polymers tend to fail in a brittle manner

under tension, and almost all of them fail in a macroscopically brittle way under cyclic loading provided thestress level is above some minimum value [18]. It has also been noted that, in the case of cyclic loading, in

addition to continuous per cycle crack growth as in metals, an additional crack propagation mechanism is

at work. In many polymers, the craze zone may grow over a number of cycles before the fibrils fail and the

crack jumps through two thirds of the craze length, giving rise to discontinuous crack growth [20,8].

One of the most successful techniques used to model crack propagation in polymers relies on the cohe-

sive modeling of fracture, which assumes the fracture process to take place in a vanishingly thin region

ahead of the crack tip. This approach seems indeed particularly appropriate to capture the failure of pol-

ymeric systems where, as mentioned earlier, the appearance of a thin crazing zone is a key component [7].Models based on cohesive technique have appeared in the literature for quasi-static [14] and dynamic crack

propagation [23] in polymers. Though there is a wealth of experimental results on fatigue crack propagation

in polymers, very few attempts have been undertaken to model fatigue crack propagation by cohesive tech-

nique in this class of materials.

Cohesive modeling technique has been attempted for fatigue crack growth in metals [3,13,5,4], along

interfaces [17] and in quasi-brittle materials [24]. In one of the early models proposed in [3], no distinction

was made in the model between the loading and unloading paths, but a damage parameter was assumed.

The evolution of this parameter with the number of load cycles was prescribed explicitly in the model. Thepresence of plasticity in the bulk material around the crack tip influenced the crack closure and hence the

failure of the material. But it was found that the crack ceases to grow after a few cycles due to plastic shake-

down [13]. It has since been identified that a distinction needs to be made between the loading and unload-

ing paths allowing for hysteresis so that subcritical crack growth becomes possible. Nguyen and co-workers

has worked out a one-parameter cohesive model for metals [13], which is able to capture experimental Paris

curves quite well and, in particular, the slope m of the curves (equal to approximately 3 for most metals).

However, this one parameter model cannot yield steeper Paris curves, thus precluding the glassy polymers

where the slope is at least 6. Another study deals with near threshold crack growth along a metal-rigid sub-strate interface and in a single crystal [5]. In this work, the unloading path is assumed to be parallel to the

previous loading path, thus leaving certain amount of residual separation in the cohesive zone after each

cycle. It is argued that the environment-assisted oxidation of the crack faces can give rise to this kind of

behavior. But this type of crack closure effect is not very common in polymers. The fatigue model for inter-

face cracks presented by Roe and Siegmund [17] is based on damage mechanics, where a history-dependent

damage parameter gives rise to irreversibility. In this model also, the unloading path is not toward the ori-

gin. Values of the slope m of the Paris curves up to 3.1 have been reported in that study. Finally, fatigue

crack growth in quasi-brittle materials has also been studied by cohesive techniques in [24], where the irre-versibility of the loading and unloading paths is taken into account. A polynomial expression for the cyclic

behavior is postulated in that study. A special case of Paris law, where the multiplicative constant is func-

tionally dependent on the maximum loading, has been reported by these authors.

S. Maiti, P.H. Geubelle / Engineering Fracture Mechanics 72 (2005) 691–708 693

In the present study, we formulate a new cohesive model specialized for polymers with higher fatigue

crack growth sensitivity on the range of cyclic loading. Special emphasis is placed on the mode I fatigue

failure of quasi-brittle glassy polymers such as PMMA, PC, epoxy etc. Our approach is inspired by the

work presented in [13,24,17], where all the nonlinear effects associated with the failure process are

accounted for in the cohesive zone model. We also describe the implementation of the cohesive model ina finite element framework (referred to as the Cohesive Volumetric Finite Element or CVFE scheme) allow-

ing for the solution of a large set of structural problems. For simplicity, the examples presented in the pre-

sent study involve its incorporation in a 1-D Euler–Bernoulli beam bending model of a double cantilever

beam (DCB) specimen.

The paper is organized as follows: To describe the cohesive model for fatigue failure, we start in Section 2

with a brief review of its foundation, i.e., a bi-linear cohesive model used to simulate quasi-static crack

propagation under monotonic loading, and its implementation in the cohesive finite element scheme. We

then discuss in Section 3 the formulation and implementation of the proposed cohesive model for fatiguecrack propagation in polymeric materials. Finally, we present in Section 4 a detailed discussion of the re-

sults associated with the proposed model, including a comparison with some experimental results obtained

on epoxy.

2. Cohesive model for monotonic loading

2.1. Cohesive failure law

The cohesive model that serves as the foundation for the proposed cohesive modeling of fatigue crack

propagation is characterized by a bi-linear, rate-independent, damage-dependent failure law between the

cohesive traction vector T and the displacement jump vector D acting across the cohesive surfaces Cc. This

particular cohesive model has been successfully used to simulate various fracture events in brittle media,

namely, the impact-induced delamination of composite materials [6], quasi-static fiber pushout in a model

composite [10], the instability of dynamic crack path in ceramic materials [11], and the dynamic fragmen-

tation of ceramics [12]. In the tensile (mode I) case, which is the focus of the work presented hereafter, thiscohesive model takes the simple form

T n ¼S

1�S

Dn

Dnc

rmax

Sinit

; ð1Þ

where Tn and Dn respectively denote the normal component of the cohesive traction and crack opening dis-

placement vectors, rmax is the tensile cohesive failure strength, and Dnc the critical opening displacement

jump (Fig. 1). The evolution of the damage process is quantified by the monotonically decreasing damageparameter S defined as

S ¼ min Sp; h1� Dn=Dnci� �

; ð2Þ

where hai = a if a > 0 and =0 otherwise, and Sp denotes the previously achieved S value.

As the material starts failing along its cohesive interface, the value of S gradually decreases from an

initial value Sinit (chosen close to unity) to zero, point at which complete failure is achieved. The monoto-nicity of S is ensured by (2): the accumulated damage is preserved and no healing of the cohesive zone

occurs due to unloading. As schematically indicated in Fig. 1, upon reloading, the cohesive stiffness main-

tains its most recent value until further failure is achieved (i.e., when the cohesive failure envelope is once

again reached).

A contact algorithm based on the penalty method has been incorporated in the numerical model to pre-

vent overlapping of the crack faces and thus simulate the crack closure effects. To that effect, the damage

Ic

loading

unloading

∆

T n

σmax

nc

∆ n

G

Fig. 1. Cohesive traction–separation law for tensile failure described by (1). After an unloading phase, reloading occurs along the

unloading path rather than the initial loading path, thereby preserving the previously achieved damage level.


parameter S is kept to its initial value Sinit close to unity when the normal displacement jump Dn becomes

negative. As illustrated in Fig. 1, this approach results in very high repulsive normal traction across the con-

tacting fracture surfaces.

2.2. Implementation

The cohesive model described by (1) and (2) can readily be implemented in a finite element framework,

referred to as the cohesive volumetric finite element (CVFE) scheme, using the following form of the prin-

ciple of virtual work
ZXS : dE dX|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}internal
�Z

Cex

Tex � dudCex|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}external

�Z

Cc

T ndDn dCc|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}cohesive

¼ 0; ð3Þ

where u is the displacement vector, S and E denote the internal stress and strain tensors, respectively, Tex is

the externally applied traction, and X, Cc and Cex respectively denote the volume, cohesive boundary andexterior boundary of the deformable body. The last term in (3) corresponds to the virtual work done by

cohesive traction Tn for a virtual separation dDn and is associated with the contribution of the cohesive ele-

ments. The expression of the internal component of the virtual work (first term in (3)) depends on the type

of volumetric element used in the analysis, which, in turn, affects the expression of the stress S and virtual

strain dE. As mentioned in the introductory section, we use in the present study classical 2-node Euler–

Bernoulli beam elements as volumetric elements since the primary focus of this work is the simulation of

fatigue delamination failure in a double cantilever beam (DCB) specimen. The corresponding finite element

formulation can be found in Appendix A.To ensure the spatial convergence of the CVFE scheme, it is critical to capture the fracture process accu-

rately, i.e., to have a sufficient number of cohesive elements in the active cohesive zone. A rough estimate of

the cohesive zone size Lcoh is given by [16]

Lcoh ¼p8

E1� m2

GIc

r2ave

;


where E is the Young�s modulus, GIc the mode I fracture toughness (=rmaxDnc/2 for the bi-linear cohesive

model described earlier), and rave the average stress in the cohesive zone (taken as rmax/2 for the bi-linear

cohesive law). Our experience with cohesive element modeling suggests that at least 4 cohesive elements

must be present in the active cohesive zone.

2.3. Verification and validation

To verify the CVFE scheme for crack propagation in a DCB specimen subjected to monotonic loading,

we compare our numerical results to the theoretical predictions obtained in the limiting case of a vanish-

ingly small cohesive zone size. For reference, the analytical expression of the evolution of the crack length

a and load P for a displacement-controlled DCB specimen (of initial uncracked length b0 and initial crack

length a0) can be found in Appendix B. In the example shown hereafter, b0 = 100mm, the Young�s modulus

E = 3.4GPa, the fracture toughness GIc = 88.97J/m2, and the cohesive strength rmax = 35MPa. In Fig. 2,the reaction force developed in the beam at the loading point is plotted against the applied displacement

D for three values of the initial crack length a0, namely 50, 100 and 200mm. Excellent agreement is achieved

between numerical and theoretical curves, especially for the larger value of a0 for which the assumptions

underlying the theoretical solution are increasingly applicable. This agreement between theoretical and

numerical predictions is also observed in the evolution of the crack length presented in Fig. 3. As predicted

by the theory, the crack remains stationary until sufficient energy is stored in the deformed beam elements

to allow for crack propagation. Once again, the longer the initial crack, the better this agreement.

Despite its simplicity, the CVFE model of DCB delamination can be used very efficiently to model theprogressive failure of actual DCB specimens. To validate the CVFE scheme, we choose to model the mode I

delamination experiments performed by Borg and co-workers [1] on HTA/6376C composite DCB speci-

mens. The corresponding constitutive and failure properties are E = 146GPa and GIc = 259J/m2 as reported

in their paper. For the simulation of this problem, the number of cohesive elements ahead of the pre-crack

is chosen as 220. The cohesive strength rmax is taken to be 35MPa, and, as shown in Fig. 4, the cohesive

finite element model is able to capture quite well the evolution of both the reaction force and the crack

length (a0 = 35mm).

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

100

200

300

400

500

600

700

800

900

1000

theoretical

a0 = 200

a0 = 100

a0 = 50

∆ (mm)

P(N

)

mm

mm

mm

Fig. 2. Comparison between theoretical (dashed curve) and numerical (solid curves) evolution of the reaction force P at the tip of a

DCB specimen under prescribed displacement D, for 3 values of the initial crack length a0.

0 5 10 15 20 25 30 350

50

100

150

200

250

∆ (mm)

a (m

m)

theoretical

a0 = 200 mm

a0 = 50 mm

a0 = 100 mm

Fig. 3. Crack length a versus prescribed displacement D: comparison between numerical (solid curves) and theoretical (dashed curve)

results obtained for 3 values of the initial crack length a0.

0 1 2 3 4 5 60

10

20

30

40

50

60

∆ (mm)

Cra

ck E

xten

sion

(m

m)

P (N

)

60

50

40

30

20

10

0

experimental reaction

computed reaction

computed crack extension

experimental crack extension

Fig. 4. Delamination of a composite DCB specimen: experimental measurements [1] and numerical predictions of the reaction and

crack extension versus D curves.


3. Cohesive model for fatigue

3.1. Model description

After describing, verifying and validating the cohesive model for mode I crack propagation under mono-

tonic loading, we now turn our attention to the main topic of the present paper: the formulation and imple-mentation of a cohesive model for fatigue crack propagation in polymers. As mentioned earlier, although it

prevents healing of the fracture surfaces through the enforcement of monotonic decay of the damage

parameter S, the cohesive model discussed in the previous section leads to similar unloading and reloading

paths in the traction–separation curve. This characteristic prevents crack growth under subcritical cyclic

loading due to the progressive degradation of the cohesive properties in the cohesive failure zone. This lim-

∆nc ∆n

σ

Tn

max

Fig. 5. Schematic evolution of the cohesive stiffness during cyclic loading, assuming unloading towards the origin and degradation of

the cohesive stiffness during reloading.


itation suggests the need for an evolution law to describe the changes incurred by the cohesive strength

under fatigue [24,17,13,5,4].

As mentioned earlier, complex physical phenomena take place in the craze (or cohesive) zone ahead of

the advancing crack with the appearance of irreversible processes associated with the creation, extension

and failure of fibrils. A phenomenological model of such processes involves the progressive degradationof the cohesive zone strength during reloading events as illustrated schematically in Fig. 5.

The evolution law of the instantaneous cohesive stiffness kc, i.e., the ratio of the cohesive traction Tn to

the displacement jump Dn, during reloading can be expressed in the general form

kc ¼dT n

dDn

¼ FðN f ; T nÞ; ð4Þ

where Nf denotes the number of loading cycle experienced by the material point since the onset of failure,

i.e., at the time the cohesive traction Tn first exceeded the failure strength rmax. Simplifying (4), we adopt

the separable form

dT n

dDn

¼ �cðN fÞT n; ð5Þ

leading to an exponential decay of the cohesive strength, with the rate of decay controlled by the parameter

c. In the present study, we use the following two-parameter power-law relation

c ¼ 1

aN�b

f ; ð6Þ

where a and b are material parameters describing the degradation of the cohesive failure properties. The

physical significance of these two parameters is discussed in Section 4 in terms of their effect on the resulting

Paris� fatigue curve. Note that the parameter a has the dimension of length and b denotes the history

dependence of the failure process. When it vanishes, we retrieve the one-parameter model of Nguyen

et al. [13].


The proposed evolution law for the cohesive model can also be expressed in terms of the rate of change

of the cohesive stiffness _kc as follows:

_kc ¼ � 1

aN�b

f kc _Dn if _Dn P 0;

¼ 0 if _Dn 6 0;

ð7Þ

where _Dn is the rate of change of the normal separation. The second equation simply states that the cohesivestiffness is assumed to remain constant during unloading.

3.2. Implementation issues

During the reloading phase, (7) can be described in its discretized form as

kjþ1c � kjc ¼ � 1

aðNj

fÞ�bkjcðDjþ1

n � DjnÞ;

where the superscripts j and j + 1 stand for loading steps j and j + 1, respectively. Recalling the expression

for kc obtained from (1), the updated value of the cohesive stiffness kjþ1c can be written as

kjþ1c ¼ Sj

1�Sj

1

Dnc

rmax

Sinit

1� 1

aðNj

fÞ�bðDjþ1

n � DjnÞ

� �; Djþ1

n P Djn: ð8Þ

A quasi-implicit load stepping scheme is used in this investigation to form the equations of equilibrium. In

this simple DCB implementation, the stiffness matrix arising from the volumetric (beam) elements is con-

stant throughout the simulation. The only nonlinear component of this problem is associated with the cohe-

sive component of the stiffness matrix, which is updated at each load step based on the displacement jump

values achieved at the previous loading step.

As shown in relation (8), during the reloading phase, the cohesive stiffness at each material point along

the cohesive zone gradually reduces proportionally to the increment in crack opening displacement. This

proportionality factor c evolves with the cycles to failure Nf in accordance with (6) and thus gives a measureof the total accumulated damage in the failure process. Note that the second term in the square-bracketed

quantity in (8) can be viewed as a damage parameter which depends on the increment of deformation and

starts accumulating only after the parameter S goes below its initial valueSinit. Therefore, the material can

cycle infinitely without failure when the cohesive traction is below its critical value rmax. The basic tenets of

damage evolution laws are thus also satisfied [9].

To further elaborate on the difference between subcritical and critical failure, we consider in Fig. 6 the

evolution of the failure process taking place in the cohesive zone ahead of a delamination crack propagating

in a DCB specimen subjected to a prescribed sinusoidal displacement loading. The DCB specimen is150mm long and has an initial crack length a0 equal to 50.1mm. The cyclic displacement D applied at

the end of the beam has a maximum amplitude of 1.5mm and the ratio of minimum to maximum amplitude

of loading R = 0. The specimen is made of epoxy with a Young�s modulus E = 3.9GPa, and a mode I frac-

ture toughness GIc = 88.97J/m2, or, equivalently, a critical stress intensity factor KIc = 0.55MPaffiffiffiffim

p. The

cohesive strength rmax is taken as 35MPa and the values of the two fatigue parameters a and b are chosen

as 4lm and 0.3 respectively. Since the estimated cohesive zone size is approximately 0.45mm, the cohesive

element size is chosen as 0.1mm. Fig. 6 represents the evolution of the failure process at a point located

ahead of the pre-existing crack, more precisely, at the location of the first Gauss point of the tenth cohesiveelement. During the initial phase of the failure process, the stress concentration present at the crack tip

leads to a critical fatigue failure, as the material degradation curve follows the envelope of the bi-linear fail-

ure law. The failure process then becomes subcritical as the degradation leads to peak cohesive traction val-

ues well below the critical value (denoted by the dashed curve). In the event of an overload, the fatigue

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

∆ n /∆ nc

Tn/σ

max

15 20 25–1

0

1

2

x 105

Cycles

∆(m

)

Fig. 6. Subcritical versus critical failure: evolution of the traction–separation curve for a point located in the path of a fatigue crack.

This plot illustrates the initial critical failure followed by subcritical degradation of the cohesive properties, with a transient critical

cycle associated with an overload (inset) in applied displacement.


failure may become critical again and follow the failure envelope until the next unloading phase, as illus-

trated in Fig. 6. In all cases, the unloading portion of the cohesive failure curve points to the origin.

4. Results and discussion

4.1. Characteristics of fatigue crack propagation

In this section, we comment on some basic characteristics of the fatigue crack propagation results ob-

tained from the numerical model developed in Section 3. As a basis for this discussion, we consider the

fatigue failure of the DCB specimen whose geometry and constitutive and failure properties have been de-scribed at the end of Section 3.2. The cyclic displacement applied at the tip of the beam has the same ampli-

tude (1.5mm) but its minimum value is chosen as 0.15mm, resulting in an amplitude ratio R = 0.1. In this

initial simulation, the parameters a and b are taken as 4lm and 0.3 respectively.

Fig. 7 presents the evolution of the crack tip and cohesive zone tip locations as a function of the number

of loading cycles. As expected, in this prescribed displacement problem, the crack driving force decreases

with crack advance, and the crack extension rate decreases with the number of loading cycles giving rise to

a stable crack growth. The cohesive zone tip follows a similar evolution, but, as the crack extension rate

decreases, the cohesive zone length increases significantly, from an initial size of 1.1mm to a final size of2.4mm.

Using the analytical solution of the DCB fracture problem summarized in Appendix B, the crack length

can be related to the applied stress intensity factor K, in terms of its maximum applied value (Kmax) or its

range (DK). As mentioned earlier, the proposed model leads to subcritical material degradation character-

ized with energy levels well inferior to the (monotonic) fracture toughness GIc. This fact is illustrated in Fig.

8, which presents the variation of the expended energy G (i.e., the area under the fatigue cohesive failure

curve) with the applied Kmax value (normalized by KIc). As apparent in that figure, the fracture toughness

0 1000 2000 3000 4000 500050

55

60

65

70

75

80

Cycles

Cra

ck a

nd c

ohes

ive

zone

tip

loca

tion

(

mm

)

crack tip

cohesive zone tip

Fig. 7. Evolution of crack and cohesive zone tip locations with number of loading cycles, a = 4lm and b = 0.3.

0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

G/G

IC

Kmax /KIc

0.4 0.5 0.6 0.7 0.8 0.9 10

1000

2000

3000

4000

5000

6000

7000

Cyc

les

to fa

ilure

Fig. 8. Variation of the fatigue fracture toughness G and the number of cycles to failure with the applied Kmax, for a = 4lm and

b = 0.3. G and Kmax both are normalized by their monotonic counterparts, GIc and KIc, respectively.


under fatigue loading can represent a very small fraction of its monotonic counterpart, with values inferior

to 5% of GIc for Kmax < 0.4KIc. For higher values of the applied cyclic load, the required energy for failure

increases rapidly, especially as the applied Kmax approaches 0.9KIc. The figure also illustrates the number of

cycles to failure as a function of Kmax/KIc for the same fatigue crack propagation problem, showing a rap-

idly decaying trend down to zero for Kmax = KIc.

4.2. Effect of the parameters on fatigue crack propagation

We now turn our attention on the effect of the parameters a and b on the predicted fatigue failure. Fig. 9

presents the evolution of the total crack extension Da with the number of loading cycles for three values of

a, with the parameter b kept constant at 0.3. As apparent there, the crack growth rate decreases with

increasing a: for a = 4lm, the crack extends by 26mm after 4000 loading cycles, whereas it has extended

by only 20mm and 17mm for a = 6lm and 8lm, respectively. Differentiating these curves, we can charac-terize the effect of a on the fatigue response in terms of the evolution of the crack advance per cycle (da/dN)

0 1000 2000 3000 4000 5000 6000 7000 800050

55

60

65

70

75

80

Cycles

Cra

ck e

xten

sion

(m

m)

∆f = 4 µ m

∆f = 6 µ m

∆f = 8 µ m

Fig. 9. Evolution of the crack extension with the number of cycles for three values of a, with b = 0.3.


versus the applied range of stress intensity factor, DK (normalized with KIc), better known as the Paris fa-

tigue failure law (Fig. 10). As apparent in that figure, all the curves have the same slope but different inter-cepts, suggesting that a unit change in DK produces an equal amount of change in the crack growth rate,

though the actual value of da/dN can be different.

10–0.4

10–0.3

10–0.2

10–0.1

10–3

10–2

∆ K/KIC

da/d

N

(m

m/c

ycle

)

α = 4 µm

α = 6 µm

α = 8 µm

Fig. 10. Effect of the parameter a on the fatigue curve, showing that a affects the intercept but not the slope of the curves (b = 0.3).

0 2000 4000 6000 8000 10000 12000 14000 1600050

55

60

65

70

75

80

85

Cycles

Cra

ck e

xten

sion

(m

m)

β = 0.0

β = 0.3

β = 0.5

Fig. 11. Effect of b on the evolution of the crack extension (a = 8lm).


The effect of the parameter b on the fatigue crack propagation is more pronounced, as illustrated byFig. 11, which presents the crack extension versus loading cycles for b = 0.0, 0.3 and 0.5, with a kept con-

stant at 8lm. As apparent in that figure, b affects the crack growth rate, leading to Paris failure curves

showing very different slopes (Fig. 12). This parameter plays therefore a key role in capturing the fatigue

response of polymeric materials, where the rate of crack growth per cycles may vary substantially with

the crack driving force from material to material. The effect of the parameter b on the slope m of the Paris

fatigue failure curve is summarized in Fig. 13: for lower values (b < 0.4), the predicted value for the expo-

nent m is about 3, which is characteristic of the fatigue failure of metals. As b increases, increasing values of

the exponent m of the Paris fatigue law are obtained.

4.3. Fatigue failure of epoxy—comparison with experiments

To conclude this result discussion, we simulate in this last section the fatigue crack growth in epoxy and

compare it with experimental results [2]. As many polymeric materials, epoxy has a very high sensitivity of

the crack growth rate with the change in stress intensity factor, hence a large slope of the Paris curve. The

slope m of the Paris fatigue curve for epoxy is about 10 compared to about 3 for most metals. So the change

in fatigue crack growth rate is strongly correlated with the crack driving force. The experimental values ofthe crack growth rate da/dN were obtained by Brown [2] with a tapered cantilever beam specimen made of

neat EPON 828 epoxy resin with 12 pph Ancamine DETA (diethylenetriamine) curing agent subjected to a

range of cyclic loading conditions. Owing to the tapered configuration, the stress intensity factor remained

constant during the experiments. Therefore, every test led to a single data point on the da/dN versus DKplot. In our numerical simulations, we use a cantilever beam with constant cross section so that the stress

intensity factor varies continuously with the propagating crack, allowing us to obtain in one simulation the

entire fatigue failure curve. The maximum amplitude of the applied displacement Dapplied is taken to be

1.8mm resulting in an initial value of the maximum stress intensity factor 0.6MPaffiffiffiffim

p, so that the crack

quickly establishes itself. Once the crack starts growing, the stress intensity factor goes down and fatigue

crack propagation is established. The load ratio R is taken to be 0.1.

Fig. 14 presents the comparison between the experimental points and numerical simulations. The oscil-

lations apparent in the da/dN versus DK curve for low values of DK are associated with the difficulty of

differentiating the crack evolution curve for very small crack advances. The two parameters of the cohesive

fatigue model are taken to be a = 8lm and b = 0.65. The value of b has been chosen based on the results

10–0.4

10–0.3

10–0.2

10–4

10–3

10–2

10–1

∆ K/KIC

da/d

N

(m

m/c

ycle

)β = 0.0

β = 0.3

β = 0.5

Fig. 12. Effect of b on the Paris fatigue curve, with a = 8lm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72

4

6

8

10

12

14

16

18

20

22

Parameter β

Slo

pem

Fig. 13. Variation of slope of the Paris fatigue curve m with the parameter b.


presented in Fig. 13. The proposed two-parameter model captures the entire fatigue failure response curve

quite well, including in the higher crack growth rate regime, where the numerical results show the rapidacceleration of the fatigue crack for DK/KIc > 0.8.

1.00.4

10–5

10–4

10–3

∆ K/KIC

da/d

N

(m

m/c

ycle

)

Fig. 14. Comparison between experimental (symbols) and computed (solid curve) fatigue response curves for epoxy. The experimental

results are taken from [2].


For low values of the applied displacement, the proposed model naturally introduces a threshold value

of DK below which no fatigue crack propagation is possible. For the simple DCB problem described earlier,

this threshold value can be obtained analytically by solving the linear problem of a beam on an elastic foun-

dation over a portion (L � a0) of its length L, with a0 denoting the initial crack length. In the absence of any

failure, the stiffness of the elastic foundation is constant and is equal to

kf ¼1

1�Sinit

rmax

Dnc

: ð9Þ

The displacement d at the initial crack tip is related to the half applied displacement D at the tip of the beamby

d ¼ Kða0; kf ; L;E; IÞD; ð10Þ
where the coefficient of proportionality K is given in Appendix C.
The onset of fatigue failure will be reached when

d ¼ ð1�SinitÞDnc=2; ð11Þ
which, combined with (9) and (B.2), leads to the following expression of the threshold value of the applied
DK at which the first cohesive element enters the ‘‘softening’’ portion of the cohesive model

DK th ¼3Eð1� RÞ

a2

ffiffiffiIb

rð1�SinitÞDnc=2K: ð12Þ

10–1

100

10–3

10–2

10–1

100

∆ K/KIC

da/d

N

(

mm

/cyc

le)

Fig. 15. Paris plot showing the upper and lower thresholds of the fatigue crack propagation with a = 4lm and b = 0.3. The right

vertical arrow denotes DK normalized with KIc corresponding to Kmax = KIc and the left vertical arrow is the theoretical lower

threshold.


For the geometry and material parameters used in the DCB problem described in Section 3, with R = 0.1

and Sinit ¼ 0:98, the computed value of DK th=KIc ¼ 0:09MPaffiffiffiffim

p. Due to the high fatigue sensitivity of

epoxy (m � 10), this threshold value corresponds to an extremely small value of the crack extension per

cycle (da/dN < 10�6mm/cycle), making its numerical capture excessively expensive. However, if we consider

a material system with a = 4lm and b = 0.3 (i.e., m � 3), the cohesive modeling of the fatigue crack prop-

agation yields the results shown in Fig. 15, which clearly illustrates the near-threshold behavior. As DK/KIc

decreases and approaches the threshold value (DKth = 0.09KIc) indicated by a vertical arrow, the crack ad-

vance slows down to a halt. The difference between theoretical and numerical values of DKth is due to

numerical error associated with the finite element approximation of the beam on (nonlinear) elastic

foundation.

5. Conclusion

We have presented a cohesive failure model specially developed for the simulation of fatigue crack prop-

agation in polymeric structures subjected to mode I cyclic loading. The phenomenological model is based

on a two-parameter evolution relation between the cohesive stiffness, the crack opening displacement rate

and the number of cycles since the onset of failure. The fatigue evolution law has been combined with a

bi-linear cohesive model for crack propagation under monotonic loading. The two parameters (a and bin (7)) entering the evolution law define the irreversible degradation process during each reloading cycle


and can be readily calibrated based on the slope and intercept of the Paris fatigue failure curve character-

izing the material.

The semi-implicit implementation of the cohesive model into a cohesive-volumetric finite element

(CVFE) framework has also been described. Although the simulations presented in the present paper have

used a simple 1-D CVFE scheme combining cohesive elements with conventional Euler–Bernouilli beamelements, the formulation presented here applied to other volumetric finite elements and therefore allows

for the simulation of a wide range of fatigue failure problems.

The cohesive model has been shown to be quite successful in capturing the fatigue response of an epoxy,

not only within the range of loading amplitudes where the Paris fatigue law is applicable, but also with re-

gards to the low and high amplitude regimes.

Acknowledgments

The authors gratefully acknowledge funding for this project provided by AFOSR Aerospace and Mate-

rials Science Directorate (Grant # F49620-02-1-0080). They also wish to thank Dr. Eric Brown and Prof.

Nancy Sottos for helpful discussions.

Appendix A. Cohesive element formulation

The expression of the local stiffness matrix kcoh of the cohesive element for this DCB problem is derived

from the cohesive component of the principle of virtual work described by (3). Let d denote the vector

containing the nodal deflection (w1 and w2) and rotation (h1 and h2) degrees of freedom associated with

the 2-node Euler–Bernouilli (C1 continuous) element. Only the upper half of the DCB specimen has been

considered for computations due to symmetry. The normal displacement jump or crack opening displace-

ment along that cohesive element (of length l) is approximated by

DnðsÞ ¼ 2d �N ; ðA:1Þ

where the vector N contains the four cubic shape functions of the beam element. The introduction of the

cohesive failure law described by (1) leads to

kcoh ¼ 2

Z l

0

NTkcN dCc; ðA:2Þ

where kc denotes the cohesive stiffness and, for the bi-linear cohesive relation (1), is given by

kc ¼S

1�S

1

Dnc

rmax

Sinit

: ðA:3Þ

A 3-point Gauss quadrature scheme is used to evaluate the integral appearing in (A.2).

Appendix B. LEFM solution for a DCB

We summarize here the classical linearly elastic fracture mechanics (LEFM) solution for the delamina-

tion of a double cantilever beam (DCB) specimen with an initial crack length a0 and subjected to an end

displacement 2D. This solution is used in Section 2 to verify the CVFE scheme in the limit of a very small


cohesive zone length (Lc � a0), and in Sections 3 and 4 to extract the range (DK) and maximum value

(Kmax) of the applied stress intensity factor K during fatigue crack propagation.

Based on the expression of the compliance C of the DCB specimen,

Fig. 16

tip.

C ¼ 2D=P ¼ 2a3

3EI; ðB:1Þ

the energy release rate G associated with a crack of length a is given by

G ¼ K2I

E¼ P 2

2bdCda

¼ 9EI

ba4D2: ðB:2Þ

In (B.1) and (B.2), E, I and b respectively denote the Young�s modulus, moment of inertia and width of the

two beams.

Under steady-state crack propagation, G = Gc, which leads to the classical relations between the crack

length a, the reaction force P and the applied displacement D:

a ¼ 9EIbGc

� �0:25

D0:5; P ¼ ðEIÞ0:25ðbGcÞ0:75

ð3DÞ0:5: ðB:3Þ

Appendix C. Threshold limit for the fatigue crack propagation

For the problem of an elastic beam on a partial elastic foundation shown in Fig. 16, Relation (10) in

Section 4 can be restated as

d ¼ KD ¼ A

BD; ðC:1Þ

A ¼ 3½e4�bð1��aÞð1þ �b�aÞ � 2e2�bð1��aÞðsinð2�bð1� �aÞÞ � �b�aþ 2�b�acos2ð�bð1� �aÞÞ � ð1� �b�aÞ�; ðC:2Þ

B ¼ e4�bð1��aÞð3þ 6�b�aþ 6�b

2�a2 þ 2�b

3�a3Þ þ e2

�bð1��aÞ½�6 sinð2�bð1� �aÞÞ þ 12�b�að1� 2�b�acos2ð�bð1� �aÞÞÞ

þ 12�b2�a2 sinð2�bð1� �aÞÞ þ 4�b

3�a3ð1þ 2cos2ð�bð1� �aÞÞ� þ ð�3þ 6�b�a� 6�b

2�a2 þ 2�b

3�a3Þ; ðC:3Þ

with

b ¼ ðkf=4EIÞ0:25; �b ¼ bL and �a ¼ a0=L:

a L a0

δk f

∆

0–

. Schematic of a cantilever beam of length L partially on an elastic foundation of stiffness kf subjected to a displacement D at the


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a cohesive model for fatigue failure of polymers

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