Fatigue Crack Growth Laws for Linear Elastic Materials

A linear elastic fatigue crack growth criterion can be used to:

model propagation of a discrete crack in a brittle material along an arbitrary, solution-dependent path, without any remeshing, and using the extended finite element method (XFEM); and
model progressive delamination growth along a predefined path at brittle material interfaces in laminated composites.

This page discusses:

Linear Elastic Fatigue Crack Growth Behaviors
Mixed-Mode Behavior
Onset of Fatigue Crack Growth
Fatigue Crack Growth Using the Paris Law
Elements
References

Linear Elastic Fatigue Crack Growth Behaviors

The onset and fatigue crack growth are both characterized by means of the Paris law, which relates either the onset or the rate of growth of a fatigue crack to the relative fracture energy release rate, as illustrated in Figure 1. The Paris law can also be expressed in terms of the equivalent relative stress intensity factor in some cases. An alternative form to the original Paris law, which better accounts for the mixed mode fatigue crack growth, and that depends on the maximum energy release rate (instead of the relative value) is also available. The alternative form is illustrated in Figure 2. The detailed equations for the different forms of the Paris law that are available in Abaqus are described later in this section.

The Paris regime is bounded by lower and upper limits, $G_{t h r e s h}$ and $G_{p l}$ , respectively. The lower limit represents the energy release rate threshold, $G_{t h r e s h}$ (or the equivalent stress intensity factor threshold, $K_{t h r e s h}$ ), below which there is no consideration of fatigue crack initiation or growth. The upper limit, $G_{p l}$ (or the equivalent stress intensity factor limit, $K_{p l}$ ) , represents the energy release rate above which the fatigue crack grows at an accelerated rate. You can specify the ratio of $\frac{G_{t h r e s h}}{G_{C}}$ and the ratio of $\frac{G_{p l}}{G_{C}}$ to define the lower and upper limits of the Paris regime. The quantity $G_{C}$ in these ratios represents the critical equivalent strain energy release rate, calculated based on the pure mode fracture strengths of the bulk material and the user-specified mode-mix criterion. The default values of these bounds are $\frac{G_{t h r e s h}}{G_{C}} = 0.01$ and $\frac{G_{p l}}{G_{C}} = 0.85$ , respectively.

Input File Usage

FRACTURE CRITERION, TYPE=FATIGUE

Abaqus/CAE Usage

The onset and growth criterion for a fatigue crack are not supported in Abaqus/CAE.

Mixed-Mode Behavior

Abaqus provides three common mode-mix formulas for computing the equivalent fracture energy release rate, $G_{C}$ : the BK law, the power law, and the Reeder law. The choice of model is not always clear in any given analysis; an appropriate model is best selected empirically.

BK Law

The BK law in Abaqus follows Benzeggagh and Kenane (1996) and is described by the following formula:

G_{C} = G_{I C} + (G_{I I C} - G_{I C}) {(\frac{G_{I I} + G_{I I I}}{G_{I} + G_{I I} + G_{I I I}})}^{η} .

To define this model, you must provide $G_{I C}, G_{I I C},$ and $η$ . This model provides a power law relationship combining the energy release rates in Mode I, Mode II, and Mode III into a single scalar fracture criterion.

Input File Usage

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=BK

Abaqus/CAE Usage

Interaction module: Create Interaction Property: Contact, MechanicalFracture Criterion, Type: FATIGUE, Mixed mode behavior: BK

Power Law

The power law model in Abaqus follows Wu and Reuter (1965) and is described by the following formula:

\frac{G_{I} + G_{I I} + G_{I I I}}{G_{C}} = {(\frac{G_{I}}{G_{I C}})}^{a_{m}} + {(\frac{G_{I I}}{G_{I I C}})}^{a_{n}} + {(\frac{G_{I I I}}{G_{I I I C}})}^{a_{o}} .

To define this model, you must provide $G_{I C}, G_{I I C}, G_{I I I C}, a_{m}, a_{n},$ and $a_{o}$ .

Input File Usage

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=POWER

Abaqus/CAE Usage

Interaction module: Create Interaction Property: Contact, MechanicalFracture Criterion, Type: FATIGUE, Mixed mode behavior: Power

Reeder Law

The Reeder law model in Abaqus follows Reeder et al. (2002) and is described by the following formula:

G_{C} = G_{I C} + (G_{I I C} - G_{I C}) {(\frac{G_{I I} + G_{I I I}}{G_{I} + G_{I I} + G_{I I I}})}^{η} +

(G_{I I I C} - G_{I I C}) (\frac{G_{I I I}}{G_{I I} + G_{I I I}}) {(\frac{G_{I I} + G_{I I I}}{G_{I} + G_{I I} + G_{I I I}})}^{η} .

To define this model, you must provide $G_{I C}, G_{I I C}, G_{I I I C},$ and $η$ . The Reeder law is best applied when $G_{I I C} \neq G_{I I I C}$ . When $G_{I I C} = G_{I I I C}$ , the Reeder law reduces to the BK law. The Reeder law applies only to three-dimensional problems.

Input File Usage

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=REEDER

Abaqus/CAE Usage

Interaction module: Create Interaction Property: Contact, MechanicalFracture Criterion, Type: FATIGUE, Mixed mode behavior: Reeder

Onset of Fatigue Crack Growth

The onset of fatigue crack growth is characterized by $Δ G$ , which is the relative fracture energy release rate when the structure is loaded between its maximum and minimum values. The fatigue crack growth initiation criterion is defined as

f = \frac{N}{c_{1} Δ G^{c_{2}}} \geq 1.0,

where

c_{1}

and

c_{2}

are assumed to be material constants and

N

is the cycle number for onset. The crack front is not propagated unless the above equation is satisfied, and the maximum fracture energy release rate,

G_{\max}

(which corresponds to the cyclic energy release rate when the structure is loaded up to its maximum value) is greater than

G_{t h r e s h}

. If you do not specify the onset criterion, Abaqus/Standard assumes that the onset of fatigue crack growth is satisfied automatically.

Fatigue Crack Growth Using the Paris Law

Once the onset criterion is satisfied at nodes along the crack front, the crack growth rate, $d a / d N$ , can be calculated based on the relative fracture energy release rate, $Δ G$ . If $G_{t h r e s h} < G_{m a x} < G_{p l},$ , the rate of crack growth per cycle is given by the Paris law:

\frac{d a}{d N} = c_{3} Δ G^{c_{4}},

where $c_{3}$ and $c_{4}$ are assumed to be material constants.

Ratcliffe and Johnston (2014) and Deobald et al. (2017) proposed the following alternate form of the fatigue law, which better accounts for mixed-mode fatigue crack growth:

\frac{d a}{d N} = c_{3} G_{T M a x}^{c_{4}} .

In the above expression, $G_{T M a x}$ is the total maximum strain energy release rate as opposed to the strain energy release rate range over a cycle, and the Paris law parameters, $c_{3}$ and $c_{4},$ are assumed to be material parameters that depend on mode-mix and stress ratios. Abaqus does not support the above form of the crack growth rate equation directly; instead, it allows you to specify $d a / d N$ as a tabular function of $G_{T M a x}$ , the mode-mix ratio, and the stress ratio.

In addition, you can utilize user subroutine UMIXMODEFATIGUE to implement a user-defined fatigue crack growth law.

For linear elastic materials, the fracture energy release rate is directly related to the stress intensity factors by the following relationship:

G = \frac{K_{I}^{2}}{E^{'}} + \frac{K_{I I}^{2}}{E^{'}} + \frac{K_{I I I}^{2}}{2 μ},

where

E^{'}

is the Young's modulus (which is equal to

\frac{E}{1 - v^{2}}

under plane strain and full three-dimensional conditions),

μ

is the shear modulus, and

ν

is the Poisson's ratio. A form based on the stress intensity factor,

K

that is equivalent to the fracture energy release rate–based form above, is also available:

\frac{d a}{d N} = c_{3} Δ K_{e f f}^{c_{4}},

where

c_{3}

and

c_{4}

are assumed to be material constants, and

Δ K_{e f f}

is the effective stress intensity factor range of a load cycle.

Abaqus uses the Irwin (1968) definition of an effective stress intensity factor that is applicable under mixed-mode conditions and is given by:

Δ K_{e f f} = \sqrt{A * Δ K_{I}^{2} + B * Δ K_{I I}^{2} + C * \frac{1}{1 - ν} Δ K_{I I I}^{2}},

where A, B, and C are user-defined material constants.

For the fatigue crack growth criterion, the following forms based on the stress intensity factor are also available:

A tabular form to support multiple piecewise linear log-log ( $\frac{d a}{d N}$ versus $Δ K_{e f f}$ ) segments.
A user-defined crack growth criterion using user subroutine UMIXMODEFATIGUE.

Input File Usage

Use one of the following options to define the fatigue onset and crack growth criterion based on the original form of the Paris law:

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=BK (default)

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=POWER

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=REEDER

Use the following option to define the fatigue onset and crack growth criterion based on the alternative form of the Paris law:

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=TABULAR

Use the following option to define a fatigue crack growth criterion using user subroutine UMIXMODEFATIGUE:

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=USER

Use one of the following options to define a stress intensity factor–based fatigue crack growth criterion.

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=IRWIN

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=TABULAR, K-BASED

FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=USER, K-BASED

Abaqus/CAE Usage

The onset and growth criterion for a fatigue crack are not supported in Abaqus/CAE.

Elements

In general, any of the stress/displacement elements in Abaqus/Standard can be used in a fatigue crack growth analysis (see Choosing the Appropriate Element for an Analysis Type). However, when modeling fatigue crack growth with the extended finite element method based on the principles of linear elastic fracture mechanics, only first-order continuum stress/displacement elements and second-order stress/displacement tetrahedral elements can be associated with an enriched feature (see Modeling Discontinuities as an Enriched Feature Using the Extended Finite Element Method).

Output Variables

In addition to the standard output identifiers available in Abaqus/Standard (see Abaqus/Standard Output Variable Identifiers), whole element and surface variables are available.

The following whole element variables are available with the extended finite element method:

CYCLEINIXFEM: Number of cycles to initialize the fatigue crack growth at the enriched element.
CYCLEXFEM: Number of cycles to fracture at the enriched element.
ENRRTXFEM: All components of strain energy release rate at the enriched element.

The following additional surface output variables can be also requested along a predefined path at the interfaces:

CSDMG: Overall value of the scalar damage variable.
BDSTAT: Bond state. The bond state varies between 1.0 (fully bonded) and 0.0 (fully unbonded).
CYCLE: Number of cycles to debond.
CRKLENGTH: Accumulated crack length. It is measured starting from the unbonded nodes immediately behind the initial crack front. As the crack tip nodes debond, debonded element length will be added to the accumulated crack length.
ENRRT: All components of strain energy release rate.

References

Benzeggagh, M., and M. Kenane, “Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus,” Composite Science and Technology, vol. 56 439, 1996.
Deobald, L., G. Mabson, S. Engelstad, M. Rao, M. Gurvich, W. Seneviratne, S. Perera, T. O'Brien, G. Murri, J. Ratcliffe, C. Davila, N. Carvalho , and R. Krueger, “Guidelines for VCCT-Based Interlaminar Fatigue and Progressive Failure Finite Element Analysis,” NASA/TM-2017-219663, 2017.
Irwin, G. R., “Linear Fracture Mechanics Fracture Transition, and Fracture Control,” Engineering Fracture Mechanics, vol. 1, pp. 241–257, 1968.
Ratcliffe, J., and W. Johnston, “Influence of Mixed Mode I-Mode II Loading on Fatigue Delamination Growth Characteristics of a Graphite Epoxy Tape Laminate,” Proceedings of American Society for Composites 29th Technical Conference, 2014.
Reeder, J., S. Kyongchan, P. B. Chunchu , and D. R.. Ambur, “Postbuckling and Growth of Delaminations in Composite Plates Subjected to Axial Compression” 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, vol. 1746, p. 10, 2002.
Wu, E. M., , and R. C. Reuter Jr., “Crack Extension in Fiberglass Reinforced Plastics,” T and M Report, University of Illinois, vol. 275, 1965.