Applying the VCCT Technique to the XFEM-Based LEFM Approach

Modeling discontinuities, such as cracks, as an enriched feature:

can be based on the principles of linear elastic fracture mechanics (LEFM);
is more appropriate for brittle crack propagation;
can also be used for simulating fatigue crack growth; and
can be simultaneously used with the surface-based cohesive behavior approach (see Contact Cohesive Behavior) or the Virtual Crack Closure Technique (see Crack Propagation Analysis), which are best suited for modeling interfacial delamination.

This page discusses:

Crack Nucleation
Crack Propagation/Extension
Nonlocal Smoothing of the Crack Normals
Mixed-Mode Behavior
Enhanced VCCT Criterion
Fatigue Crack Growth Criterion Based on the Principles of LEFM
Viscous Regularization for the XFEM-Based LEFM Approach
Output
References

Crack Nucleation

If there is no pre-existing crack in the model, a crack can be nucleated as part of the analysis. The crack nucleation is governed by any of the available crack initiation criteria discussed in Crack Initiation and Direction of Crack Extension. After a crack nucleates in an enriched region, the XFEM-based LEFM criterion governs the subsequent propagation of the crack.

Input File Usage

Use the following option to specify the crack nucleation criterion as part of the material definition when there is no preexisting crack in an enriched region:

DAMAGE INITIATION, TOLERANCE= $f_{t o l}$

Abaqus/CAE Usage

Property module: material editor: Mechanical: Damage for Traction Separation Laws: Quade Damage, Maxe Damage, Quads Damage, Maxs Damage, Maxpe Damage, or Maxps Damage:  $f_{t o l}$

Crack Propagation/Extension

The formulae and laws that govern the behavior of the XFEM-based linear elastic fracture mechanics approach for crack propagation analysis are very similar to those used for modeling delamination along a known and partially bonded surface (see VCCT Criterion), where the strain energy release rate at the crack tip is calculated based on the modified Virtual Crack Closure Technique (VCCT). An existing crack is assumed to propagate when the energy released due to crack extension is equal to or greater than the fracture toughness of the material (energy required to create new crack surfaces). However, unlike the method described in VCCT Criterion, the XFEM-based LEFM approach can be used to simulate crack propagation along an arbitrary, solution-dependent path in the bulk material with or without an initial crack. You complete the definition of the crack propagation capability by defining a fracture-based surface behavior and specifying the fracture criterion in enriched elements.

Specifying When a Preexisting Crack Will Extend

For the discussion to follow, it is useful to define a ratio, f, of the energy release rate at the tip of a pre-existing crack to the fracture energy of the material. In the simple case of crack propagation in one of the pure modes (I, II, or III), this ratio may be defined as:

f = \frac{G_{j}}{G_{j C}},

where

j \in {I, I I, I I I}

is an index representing one of the pure modes. In the more general case of mixed-mode crack propagation,

G_{j}

and

G_{j C}

are replaced with

G_{e q u i v}

and

G_{e q u i v C}

, respectively, which are measures of the energy release rate and fracture energy that are more appropriate under mixed-mode conditions and are defined more precisely later in the section. The ratio, f, provides a quantitative measure of the fracture criterion.

If there is a preexisting crack in an enriched region, the crack extends after an equilibrium increment when the fracture criterion, f, reaches the value 1.0 within a given tolerance:

1.0 \leq f \leq 1.0 + f_{t o l} .

You can specify the tolerance $f_{t o l}$ . If $f > 1 + f_{t o l}$ , the time increment is cut back such that the crack extension criterion is satisfied. The default value of $f_{t o l}$ is 0.2.

Input File Usage

Use both of the following options:

SURFACE BEHAVIOR
FRACTURE CRITERION, TOLERANCE= $f_{t o l}$ , TYPE=VCCT

Abaqus/CAE Usage

Interaction module: Interaction Property Create , Contact, Mechanical Fracture Criterion , Tolerance: $f_{t o l}$

Fracture of Multiple Elements in an Unstable Crack Growth Analysis

For an unstable crack growth problem, sometimes it is more efficient to allow multiple elements at and ahead of a crack tip to fracture without excessively cutting back the increment size when the fracture criterion is satisfied. Abaqus/Standard activates this capability automatically if you specify an unstable growth tolerance, $f_{t o l}^{u}$ . In this case if the fracture criterion, f, is within the given unstable growth tolerance:

1.0 + f_{t o l} \leq f \leq 1.0 + f_{t o l}^{u},

where

f_{t o l}

is the tolerance described earlier in this section, Abaqus/Standard immediately reduces the time increment size by default to a very small value,

β ▵ t_{\min} .

Reducing the time increment size allows more elements to fracture until

f < 1

for all the elements ahead of the crack tip. After this limit is reached, the time increment size is recovered automatically to a larger value,

α ▵ t_{p r e}

, where

▵ t_{\min}

is the minimum time increment allowed;

▵ t_{p r e}

is the time increment size prior to the unstable crack growth; and

α

, and

β

are scaling parameters. The default values of

α

and

β

are 0.5 and 2.0, respectively.

Use of a large unstable growth tolerance may sometimes lead to an overshoot of the peak load in the load displacement response of a structure. To prevent this, you may optionally specify the maximum number of cutbacks allowed, $n$ , to be controlled by the regular tolerance, $f_{t o l}$ , prior to the activation of the unstable growth tolerance in an increment. The regular cutbacks ensure that the numerical solution does not over-predict the peak load that a structure containing a crack can sustain before the crack starts propagating. The default value of $n$ is 0.

If you do not specify a value for the unstable growth tolerance, the default value is infinity. In this case the fracture criterion, f, for unstable crack growth is not limited by any upper bound value in the above equation.

Input File Usage

Use both of the following options:

FRACTURE CRITERION, UNSTABLE GROWTH TOLERANCE= $f_{t o l}^{u}$ 
CONTROLS, TYPE=NO CUTBACK SCALING

Abaqus/CAE Usage

Use the following option to specify an unstable crack growth tolerance:

Interaction module: Interaction Property Create : Contact: Mechanical Fracture Criterion , Toggle on Specify tolerance for unstable crack propagation, select Specify value

Specifying the Crack Propagation Direction

You must specify the crack propagation direction when the fracture criterion is satisfied. The crack can extend at a direction normal to the direction of the maximum tangential stress, orthogonal to the element local 1-direction (see Conventions), or orthogonal to the element local 2-direction. By default, the crack propagates normal to the direction of the maximum tangential stress.

Input File Usage

Use one of the following options to specify the crack direction when the fracture criterion is satisfied:

FRACTURE CRITERION, NORMAL DIRECTION=MTS (default)

FRACTURE CRITERION, NORMAL DIRECTION=1

FRACTURE CRITERION, NORMAL DIRECTION=2

Abaqus/CAE Usage

Interaction module: contact property editor: Mechanical Fracture Criterion : Direction of crack growth relative to local 1-direction: Maximum tangential stress, Normal, or Parallel

Limiting the Crack Propagation Direction

If the crack direction normal to the maximum tangential stress is specified, you can limit the new crack propagation direction to within a certain angle (in degrees) of the previous crack propagation direction. The default is 85°.

Input File Usage

Use the following option to set the maximum allowed change in the crack propagation angle (in degrees):

FRACTURE CRITERION, ANGLEMAX=max_angle

Abaqus/CAE Usage

Limiting the crack propagation direction is not supported in Abaqus/CAE.

Nonlocal Smoothing of the Crack Normals

After the normals of the individual crack facets are obtained based on the fracture criterion defined above, you can use a moving least-squares approximation by polynomials to further smooth out the crack normals. The least-squares approximation is applied to the normals of the individual facets in elements along the crack front that satisfy the fracture criterion to obtain a more accurate crack propagation direction.

Input File Usage

FRACTURE CRITERION, POSITION=NONLOCAL

Abaqus/CAE Usage

Nonlocal smoothing of the crack normals is not supported in Abaqus/CAE.

Specifying the Approximation Used in the Least-Squares Approximation

You can use linear, quadratic, or cubic polynomial approximation for the moving least-squares approximation to smooth out the crack normals. You specify the number of terms in the polynomial.

Input File Usage

Use one of the following options:

FRACTURE CRITERION, NPOLY=4
FRACTURE CRITERION, NPOLY=7 (default)
FRACTURE CRITERION, NPOLY=10

Abaqus/CAE Usage

Nonlocal smoothing of the crack normals is not supported in Abaqus/CAE.

Specifying the Region of the Model Used for Nonlocal Smoothing of the Crack Normals

To control the range of elements used for nonlocal smoothing of the crack normals in the crack direction calculations, you can specify a radius, $r_{c}$ , within which the elements around the crack tip along the crack front are included. The default radius is three times the typical element characteristic length along the crack front in the enriched region.

Input File Usage

FRACTURE CRITERION, R CRACK DIRECTION= $r_{c}$

Abaqus/CAE Usage

Nonlocal smoothing of the crack normals is not supported in Abaqus/CAE.

Mixed-Mode Behavior

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

BK Law

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

G_{e q u i v 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_{e q u i v} = G_{I} + G_{I I} + G_{I I I} .

If there is a pre-existing crack in an enriched region, the crack extends after an equilibrium increment when the fracture criterion, $f = \frac{G_{e q u i v}}{G_{e q u i v C}}$ , reaches the value of 1.0 within a specified tolerance (as discussed earlier).

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

Input File Usage

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

Abaqus/CAE Usage

Interaction module: contact property editor: Mechanical Fracture Criterion : Mixed mode behavior: BK, and enter the critical energy release rates in the data table

Power Law

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

\frac{G_{e q u i v}}{G_{e q u i v 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}} .

G_{e q u i v} = G_{I} + G_{I I} + G_{I I I} .

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=VCCT, MIXED MODE BEHAVIOR=POWER

Abaqus/CAE Usage

Interaction module: contact property editor: Mechanical Fracture Criterion : Mixed mode behavior: Power, and enter the critical energy release rates in the data table

Reeder Law

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

G_{e q u i v 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},

G_{e q u i v} = G_{I} + G_{I I} + G_{I I I} .

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=VCCT, MIXED MODE BEHAVIOR=REEDER

Abaqus/CAE Usage

Interaction module: contact property editor: Mechanical Fracture Criterion : Mixed mode behavior: Reeder, and enter the critical energy release rates in the data table

Defining Variable Critical Energy Release Rates

You can define a VCCT criterion with varying energy release rates by specifying the critical energy release rates at the nodes.

If you indicate that the nodal critical energy rates will be specified, any constant critical energy release rates you specify are ignored and the critical energy release rates are interpolated from the nodes. The critical energy release rates must be defined at all nodes in the enriched region.

Input File Usage

Use both of the following options:

FRACTURE CRITERION, TYPE=VCCT, NODAL ENERGY RATE
NODAL ENERGY RATE

Abaqus/CAE Usage

Defining a VCCT criterion with varying energy release rates is not supported in Abaqus/CAE.

Enhanced VCCT Criterion

The formulae and laws governing the behavior of the enhanced VCCT criterion are very similar to those used for the VCCT criterion. However, unlike the VCCT criterion, the onset and growth of a crack can be controlled by two different critical fracture energy release rates: $G_{C}$ and $G_{C}^{P}$ . In a general case involving Mode I, II, and III fracture, when the fracture criterion is satisfied; i.e,

f = \frac{G_{e q u i v}}{G_{e q u i v C}} \geq 1.0,

the traction on the two surfaces of the cracked element is ramped down over the separation with the dissipated strain energy equal to the critical equivalent strain energy required to propagate the crack, $G_{e q u i v C}^{P}$ , rather than the critical equivalent strain energy required to initiate the separation, $G_{e q u i v C}$ . The formulae for calculating $G_{e q u i v C}^{P}$ are identical to those used for $G_{e q u i v C}$ for different mixed-mode fracture criteria.

Input File Usage

Use both of the following options:

SURFACE BEHAVIOR
FRACTURE CRITERION, TYPE=ENHANCED VCCT

Abaqus/CAE Usage

Specifying the enhanced VCCT criterion is not supported in Abaqus/CAE.

Fatigue Crack Growth Criterion Based on the Principles of LEFM

If you specify the fatigue crack growth criterion, progressive crack growth at the enriched elements subjected to sub-critical cyclic loading can be simulated. This criterion can be used only in the general fatigue crack growth approach (Linear Elastic Fatigue Crack Growth Analysis). A fatigue crack growth analysis step can be the only step, can follow a general static step, or can be followed by a general static step. You can include multiple fatigue crack growth analysis steps in a single analysis. If you perform a fatigue analysis in a model without a preexisting crack, you must precede the fatigue step with a static step that nucleates a crack, as discussed in Crack Propagation/Extension.

The onset and fatigue crack growth are characterized by using the Paris law, which relates the fracture energy release rate or the stress intensity factor to crack growth rates. Different forms of Paris law are available in Abaqus. For details, see Fatigue Crack Growth Using the Paris Law

Input File Usage

Use both of the following options:

SURFACE BEHAVIOR
FRACTURE CRITERION, TYPE=FATIGUE

Abaqus/CAE Usage

Specifying a fatigue crack growth criterion is not supported in Abaqus/CAE.

Linear Elastic Fatigue Crack Growth Analysis

The fracture energy release rates or the stress intensity factors at the crack tips in the enriched elements are calculated based on the above mentioned VCCT technique. The process of fatigue crack growth is based on a damage extrapolation technique that relies on fracturing at least one enriched element ahead of the current crack tip (once the onset criterion is satisfied) with a corresponding increase in the number of cycles. Specifically, if the criterion for the onset of the crack growth is satisfied at any crack tip in the enriched element at the end of a completed cycle, $N$ , Abaqus/Standard extends the crack length, $a_{N}$ , by fracturing at least one enriched element ahead of the crack tip/front. This increase in the crack length corresponds to an increase in the cycle count from $N$ to $N + Δ N$ , where $Δ N$ is computed based on the Paris law. For additional information on how a crack extends over an incremental number of cycles, see Damage Extrapolation Technique.

If $G_{m a x} > G_{p l}$ , the enriched elements ahead of the crack tips will be fractured by increasing the cycle number count, $Δ N$ , by one only.

For information on how to accelerate the fatigue crack growth analysis and to provide a smooth solution for the crack front, see Controlling Element Fracture.

Viscous Regularization for the XFEM-Based LEFM Approach

The simulation of structures with unstable propagating cracks is challenging and difficult. Nonconvergent behavior may occur from time to time. Localized damping is included for the XFEM-based LEFM approach by using the viscous regularization technique. Viscous regularization damping causes the tangent stiffness matrix of the softening material to be positive for sufficiently small time increments.

Input File Usage

Use one of the following options

FRACTURE CRITERION, TYPE=VCCT, VISCOSITY= $μ$

FRACTURE CRITERION, TYPE=ENHANCED VCCT, VISCOSITY= $μ$

Abaqus/CAE Usage

Interaction module: contact property editor: Mechanical Fracture Criterion : Viscosity: $μ$

Output

Whole element variables:

ENRRTXFEM: All components of strain energy release rate when linear elastic fracture mechanics with the extended finite element method is used.
CYCLEXFEM: Number of cycles to fracture at the enriched element.
CYCLEINIXFEM: Minimum number of cycles needed to satisfy the condition for the onset of fatigue crack growth at an enriched element.

Surface variables (available only for propagating cracks modeled with first-order solid continuum elements):

CRKDISP: Crack opening and relative tangential motions on cracked surfaces in enriched elements.
CSDMG: Damage variable on cracked surfaces in enriched elements.
CRKSTRESS: Remaining residual pressure and tangential shear stresses on cracked surfaces in enriched elements.

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.
Irwin, G. R., “Linear Fracture Mechanics Fracture Transition, and Fracture Control,” Engineering Fracture Mechanics, vol. 1, pp. 241–257, 1968.
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.