Crack Propagation Analysis

For fracture criteria based on the critical stress, critical crack opening displacement, or crack length versus time, it is possible to bond the nodes in the node set (or the contact pair if a node set is not defined) only in the normal direction. In this case the nodes are allowed to move freely tangential to the contact surfaces. Friction (Frictional Behavior) cannot be specified if the nodes are bonded only in the normal direction.

Bonding only in the normal direction is typically used to model bonded contact conditions in Mode I crack problems where the shear stress ahead of the crack along the crack plane is zero.

INITIAL CONDITIONS, TYPE=CONTACT, NORMAL

Cracks can propagate from either a single crack tip or multiple crack tips. The crack propagation capability in Abaqus/Standard requires that the surfaces be initially partially bonded so that the crack tips can be identified. A contact pair can have crack propagation from multiple crack tips. However, only one crack propagation criterion is allowed for a given contact pair. Crack propagation along several contact pairs can be modeled by specifying multiple crack propagation definitions.

This criterion is available only in Abaqus/Standard.

If you specify a critical stress criterion at a critical distance ahead of the crack tip, the crack-tip node debonds when the local stress across the interface at a specified distance ahead of the crack tip reaches a critical value.

This criterion is typically used for crack propagation in brittle materials. The critical stress criterion is defined as

where ${\sigma }_n$ is the normal component of stress carried across the interface at the distance specified; ${\tau }_1$ and ${\tau }_2$ are the shear stress components in the interface; and ${\sigma }^f$ and ${\tau }_1^f$ are the normal and shear failure stresses, which you must specify. The second component of the shear failure stress, ${\tau }_2$ , is not relevant in a two-dimensional analysis; therefore, the value of ${\tau }_2^f$ need not be specified. The crack-tip node debonds when the fracture criterion, f, reaches the value 1.0.

If the value of ${\tau }_1^f$ is not given or is specified as zero, it will be taken to be a very large number so that the shear stress has no effect on the fracture criterion.

The distance ahead of the crack tip is measured along the secondary surface, as shown in Figure 1. The stresses at the specified distance ahead of the crack tip are obtained by interpolating the values at the adjacent nodes. The interpolation depends on whether first-order or second-order elements are used to define the secondary surface.

Figure 1. Distance specification for the critical stress criterion.

FRACTURE CRITERION, TYPE=CRITICAL STRESS, DISTANCE=n

This criterion is available only in Abaqus/Standard.

If you base the crack propagation analysis on the crack opening displacement criterion, the crack-tip node debonds when the crack opening displacement at a specified distance behind the crack tip reaches a critical value. This criterion is typically used for crack propagation in ductile materials.

The crack opening displacement criterion is defined as

where $\delta$ is the measured value of crack opening displacement and ${\delta }_c$ is the critical value of the crack opening displacement (user-specified). The crack-tip node debonds when the fracture criterion reaches the value 1.0.

You must supply the crack opening displacement versus cumulative crack length data. In Abaqus/Standard the cumulative crack length is defined as the distance between the initial crack tip and the current crack tip measured along the secondary surface in the current configuration. The crack opening displacement is defined as the normal distance separating the two faces of the crack at the given distance.

You specify the position, n, behind the crack tip where the critical crack opening displacement is calculated. The value of this position must be specified as the length of the straight line joining the current crack tip and points on the secondary and main surfaces (Figure 2).

Figure 2. Distance specification for the critical crack opening displacement criterion.

Abaqus/Standard computes the crack opening displacement at that point by interpolating the values at the adjacent nodes. The interpolation depends on whether first-order or second-order elements are used to define the secondary surface. An error message will be issued if the value of n is not within the end points of the contact pair.

FRACTURE CRITERION, TYPE=COD, DISTANCE=n

FRACTURE CRITERION, TYPE=COD, DISTANCE=n, SYMMETRY

This criterion is available only in Abaqus/Standard.

To specify the crack propagation explicitly as a function of total time, you must provide a crack length versus time relationship and a reference point from which the crack length is measured. This reference point is defined by specifying a node set. Abaqus/Standard finds the average of the current positions of the nodes in the set to define the reference point. During crack propagation the crack length is measured from this user-specified reference point along the secondary surface in the deformed configuration. The time specified must be total time, not step time.

The fracture criterion, f, is stated in terms of the user-specified crack length and the length of the current crack tip. The length of the current crack tip from the reference point is measured as the sum of the straight line distance of the initial crack tip from the reference point and the distance between the initial crack tip and the current crack tip measured along the secondary surface.

Referring to Figure 3, let node 1 be the initial location of the crack tip and node 3 be the current location of the crack tip. The distance of the current crack tip located at node 3 is given by

where $l_1$ is the length of the straight line joining node 1 and the reference point, $\mathrm{\Delta }{l}_{12}$ is the distance between nodes 1 and 2, and $\mathrm{\Delta }{l}_{23}$ is the distance between nodes 2 and 3 measured along the secondary surface.

Figure 3. Crack propagation as a function of time.

The fracture criterion, f, is given by

where l is the length at the current time obtained from the user-specified crack length versus time curve. Crack-tip node 3 will debond when the failure function f reaches the value of 1.0 (within the user-defined tolerance).

If geometric nonlinearity is considered in the step (Defining an Analysis), the reference point may move as the body deforms; you must ensure that this movement does not invalidate the crack length versus time criterion.

Abaqus/Standard does not extrapolate beyond the end points of your crack data. Therefore, if the first crack length specified is greater than the distance from the crack reference point to the first bonded node, the first bonded node will never debond and the crack will not propagate. In this case Abaqus/Standard will print warning messages in the message (.msg) file.

FRACTURE CRITERION, TYPE=CRACK LENGTH, NSET=name

This criterion is available in both Abaqus/Standard and Abaqus/Explicit.

The Virtual Crack Closure Technique (VCCT) criterion uses the principles of linear elastic fracture mechanics (LEFM) and, therefore, is appropriate for problems in which brittle crack propagation occurs along predefined surfaces.

VCCT is based on the assumption that the strain energy released when a crack is extended by a certain amount is the same as the energy required to close the crack by the same amount. For example, Figure 4 illustrates the similarity between crack extension from $i$ to $j$ and crack closure at $j$ .

Figure 4. Mode I: The energy released when a crack is extended by a certain amount is the same as the energy required to close the crack.

In Figure 5 nodes 2 and 5 will start to release when

where $G_I$ is the Mode I energy release rate, $G_{IC}$ is the critical Mode I energy release rate, b is the width, d is the length of the elements at the crack front, $F_{v,2,5}$ is the vertical force between nodes 2 and 5, and $v_{1,6}$ is the vertical displacement between nodes 1 and 6. Assuming that the crack closure is governed by linear elastic behavior, the energy to close the crack (and, thus, the energy to open the crack) is calculated from the previous equation. Similar arguments and equations can be written in two dimensions for Mode II and for three-dimensional crack surfaces including Mode III.

Figure 5. Pure Mode I modified.

In the general case involving Mode I, II, and III the fracture criterion is defined as

where $G_{equiv}$ is the equivalent strain energy release rate calculated at a node, and $G_{equivC}$ is the critical equivalent strain energy release rate calculated based on the user-specified mode mix criterion and the bond strength of the interface. The crack-tip node will debond when the fracture criterion reaches the value of 1.0.

Abaqus provides three common mode mix formulas for computing $G_{equivC}$ : 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.

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

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

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

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

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

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

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

This criterion is available only in Abaqus/Standard.

The enhanced VCCT criterion is very similar to the original VCCT criterion described above. As in the original VCCT criterion, the fracture criterion in the general case involving Mode I, II, and III is defined as

The crack-tip node debonds when the fracture criterion reaches the value of 1.0. However, unlike the original VCCT criterion, you can specify two different critical fracture energy release rates: $G_C$ for the onset of a crack and $G_C^P$ for the growth of a crack. When the enhanced VCCT criterion is used in the general case involving Mode I, II, and III fracture, the amount of energy dissipated associated with the release of the debonding force is controlled by the critical equivalent strain energy release rate required to propagate the crack, $G_{equivC}^P$ , rather than by the critical equivalent strain energy release rate required to initiate the crack, $G_{equivC}.$ The formulas for calculating $G_{equivC}^P$ are identical to those used for $G_{equivC}$ for different mixed-mode fracture criteria.

FRACTURE CRITERION, TYPE=ENHANCED VCCT

Interaction module: Create Interaction Property: Contact, MechanicalFracture Criterion, Type: Enhanced VCCT

The Progressive Release eXplicit - Virtual Crack Closure Technique (PRX-VCCT) criterion was proposed by Carvalho et al. (2022) to simulate crack propagation and material delamination in a quasi-static analysis. It is available only in Abaqus/Standard.

The PRX-VCCT criterion is a modified VCCT criterion that incorporates (i) a progressive nodal release strategy within a conventional VCCT framework and (ii) an ‘explicit’ computation of the delamination increment based upon energy release rates of the previous increment. The PRX-VCCT method has certain advantages over the conventional VCCT method, such as:

• a smoother delamination front for non-self-similar crack propagation;
• use of relatively larger element sizes without sacrificing accuracy (although the element sizes must still be small enough, as in typical fracture applications, to capture the gradients in the fields accurately); and
• better overall convergence characteristics.

Figure 6. Delamination front area for a 3D PRX-VCCT model.

Figure 6 illustrates the bottom surface near (a segment of) an arbitrary crack front in a three-dimensional structure. In this figure nodes $j,j+4,$ and $j+5$ are fully debonded; nodes $j+1$ and $j+3$ are partially bonded; and nodes $i,j+2,j+6,$ and $j+7$ are fully bonded. The delamination front in this figure corresponds to nodes $j+1,i,$ and $j+3$ . The energy release rate is computed at each node along the delamination front to determine if the fracture criterion is satisfied. Therefore, in addition to computing the energy release rates at the nodes along the delamination front that are fully bonded, the PRX-VCCT method also requires the energy release rates at nodes along the delamination front that are partially debonded.

The Mode I energy release rate at node $i$ (Figure 6) on the delamination front is calculated as

The PRX-VCCT method is fundamentally different from a traditional VCCT method, where:

• only the first term in the above expression contributes to the energy release rate calculations at node $i$ ; and
• the strain energy release rate at node $i$ reaches a maximum once the fracture criterion is satisfied,and is not recalculated as the node transitions from a fully bonded to a partially debonded state (even though the forces at the node are progressively ramped down during debonding).

The first step in the quasi-static propagation algorithm is to evaluate the fracture criterion at each node $i$ along the delamination front:

For the discussion that follows, it is useful to introduce two additional quantities:

• $a_{e^{*}}^i,$ which represents a mesh-dependent "ligament length" ahead of the delamination front. For a regular mesh, such as the one shown in Figure 6, $a_{e^{*}}^i$ represents the distance between node $i$ on the delamination front, and node $j+2$ ahead of it. For an arbitrary mesh, Abaqus identifies a "target node" ahead of node $i$ in the direction of propagation of the delamination. In the latter case $a_{e^{*}}^i$ represents the distance between node $i$ and the target node.
• $\lambda (0<\lambda <1)$ is a constant that you specify. It represents the maximum fraction of $a_{e^{*}}^i$ that the delamination is allowed to propagate in a single increment. The default value of $\lambda$ is 0.1.

The delamination growth algorithm is based upon a pseudo-time increment, $\mathrm{\Delta }{\overline{t}}_{inc},$ which is associated with the maximum delamination increment (among all the nodes on the delamination front). To obtain $\mathrm{\Delta }{\overline{t}}_{inc},$ the pseudo-time increment, $\mathrm{\Delta }{\overline{t^i}}_{},$ required to propagate the delamination by the amount $a_{e^{*}}^i$ at a delamination front node, $i$ , is first determined as

The above quantity is computed at each node along the delamination front. Subsequently, $\mathrm{\Delta }{\overline{t}}_{inc}$ is determined as the minimum (over all delamination front nodes) of $\mathrm{\Delta }{\overline{t^i}}_{};$ that is,

The normalized increment in delamination at a front node, $i$ , is then computed as

Clearly $\mathrm{\Delta }{d}^i=\lambda$ at the delamination front node at which $\mathrm{\Delta }{\overline{t}}^i=\mathrm{\Delta }{\overline{t}}_{inc},$ and $\mathrm{\Delta }{\overline{t}}^i<\lambda$ at all the other nodes on the front (where $\mathrm{\Delta }{\overline{t}}^i>\mathrm{\Delta }{\overline{t}}_{inc}$ ). The quantity $\mathrm{\Delta }{d}^i$ is accumulated incrementally at each delamination front node. A front node is considered to have fully debonded when the accumulated value $d^i\ge 1$ . At this point in time, the delamination front has progressed to the ‘target node’ ahead of the original front node $i$ . The accumulated value of $d^i$ is also used as a measure of damage to ramp down the forces at node $i$ .

Therefore, the PRX-VCCT algorithm naturally leads to the concept of a virtual delamination front, as illustrated in Figure 6. As the accumulated normalized growth increment $d^i$ at front node $i$ increases from a value of 0 (totally bonded) to a value that is greater than or equal to 1.0 (totally debonded), the "virtual" delamination front moves from node $i$ toward the target node ahead of it (node $j+2$ for the case shown in Figure 6). At the instance of propagation illustrated in Figure 6, the virtual delamination front passes through locations ${\tilde{x}}^{j+3}$ , $i$ , and ${\tilde{x}}^{j+1}$ , respectively. The locations ${\tilde{x}}^{j+3}$ and ${\tilde{x}}^{j+1}$ represent delamination advancements associated with partial debonding of nodes $j+3$ and $j+1,$ respectively. Full debonding at nodes $j+3$ and $j+1$ will result in the propagation of the delamination to nodes $j+7$ and $j+6,$ respectively. The virtual delamination front is utilized for computing the front tangent vectors, as well as the delamination area (used in the computation of the energy release rates) associated with each front node.

When the delamination front is at an intermediate location between nodes $i$ and $j+2,$ the force at node $i$ can be written as

FRACTURE CRITERION, TYPE=PRX-VCCT, LAMBDA=λ, NPOWER=p

This criterion is available only in Abaqus/Standard.

If you specify the fatigue crack growth criterion, you can simulate progressive delamination growth at the interfaces in laminated composites subjected to sub-critical cyclic loadings. This criterion can be used only in the general fatigue crack growth approach (Linear Elastic Fatigue Crack Growth Analysis). The onset and delamination growth are characterized by using the Paris law, which relates the relative fracture energy release rate to crack growth rates. Different forms of the Paris law are available in Abaqus. For more information, see Fatigue Crack Growth Using the Paris Law.

FRACTURE CRITERION, TYPE=FATIGUE

When you use the critical stress, critical crack opening displacement, or crack length versus time fracture criteria, you can define how this force is to be reduced to zero with time after debonding starts at a particular node on the bonded surface. You specify a relative amplitude, a, as a function of time after debonding starts at a node. Thus, suppose the force transmitted between the surfaces at secondary node N is ${{\mathbf{T}}^N|}_0$ when that node starts to debond, which occurs at time ${t^N|}_0$ . Then, for any time $t>{t^N|}_0$ the force transmitted between the surfaces at node N is ${a\left(t-{t^N|}_0\right){\mathbf{T}}^N|}_0$ . The relative amplitude must be 1.0 at the relative time 0.0 and must reduce to 0.0 at the last relative time point given.

The best choice of the amplitude curve depends on the material properties, specified loading, and the crack propagation criterion. If the stresses are removed too rapidly, the resulting large changes in the strains near the crack tip can cause convergence difficulties. For large-strain problems severe mesh distortion can also occur. For problems with rate-independent materials a linear amplitude curve is normally adequate. For problems with rate-dependent materials the stresses should be ramped off more slowly at the beginning of debonding to avoid convergence and mesh distortion difficulties. To reduce the likelihood of convergence and mesh distortion difficulties, you can reduce the value of the debond stress by 25% in 50% of the time to debond. The solution should not be strongly influenced by the details of the unloading procedure; if it is, this usually indicates that the mesh should be refined in the debond region.

DEBOND, SECONDARY=secondary, MAIN=main
Data lines to define debonding amplitude curve

When the energy release rate exceeds the critical value at a crack tip, you can release the traction between the two surfaces at the crack tip in one of the following ways:

• immediately during the following increment, or
• gradually during succeeding increments with the reduction of the magnitude of the debonding force being governed by the critical fracture energy release rate.

DEBOND, SECONDARY=secondary, MAIN=main, 
DEBONDING FORCE=STEP

DEBOND, SECONDARY=secondary, MAIN=main, 
DEBONDING FORCE=RAMP

Interaction module: SpecialCrackCreate: Name: crack name, Type: Debond using VCCT, select the step and the surface to surface (Standard) interaction, Debonding force: Step or Ramp

At the end of the loading cycle, the interface element that takes the fewest cycles, $\mathrm{\Delta }{N}_{min}$ , to fracture is released completely with zero constraint and a zero debonding force immediately after fracture. All of the other interface elements satisfied with the accuracy of the damage extrapolation condition (as defined in Controlling Element Fracture) are fractured. The debonding force is removed immediately after fracture or ramped down gradually according to a scalar damage variable defined as:

DEBOND, DEBONDING FORCE=STEP (default) 
FRACTURE CRITERION, TYPE=FATIGUE

DEBOND, DEBONDING FORCE=RAMP
FRACTURE CRITERION, TYPE=FATIGUE, TOLERANCE=$f$

For an unstable crack growth problem, sometimes it is more efficient to allow multiple nodes at and ahead of a crack tip to debond in one increment without cutting back the increment size when the VCCT, the enhanced VCCT, or PRX-VCCT fracture criterion is satisfied. This capability is activated automatically if you specify an unstable growth tolerance, $f_{tol}^u$ . In this case if the fracture criterion, f, is within the given unstable growth tolerance:

However, you can optionally specify the maximum number of cutbacks allowed, $n$ , to be controlled by the regular tolerance, $f_{tol}$ , prior to the activation of the unstable growth tolerance in an increment. After this limit is reached, the time increment size is automatically recovered to a larger value, $\alpha \mathrm{\Delta }{t}_{pre}$ , where $\mathrm{\Delta }{t}_{\min }$ is the minimum time increment allowed; $\mathrm{\Delta }{t}_{pre}$ is the time increment size prior to the unstable crack growth; and $\alpha$ , $\beta$ , and $n$ are scaling parameters. The default values of $\alpha$ , $\beta$ , and $n$ are 0.5, 2.0, and 0, respectively.

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. The forces at those debonded nodes can be completely released immediately during the following increment or ramped down gradually during succeeding increments with the reduction of the magnitude of the debonding force being governed by the critical fracture energy release rate. If the forces at the debonded nodes are ramped down gradually, an unsymmetric matrix storage and solution scheme (Matrix Storage and Solution Scheme in Abaqus/Standard) is used by default to improve convergence.

DEBOND, DEBONDING FORCE=RAMP
FRACTURE CRITERION, TYPE=VCCT, UNSTABLE GROWTH TOLERANCE=$f_{tol}^u$
CONTROLS, TYPE=NO CUTBACK SCALING

DEBOND, DEBONDING FORCE=STEP
FRACTURE CRITERION, TYPE=VCCT, UNSTABLE GROWTH TOLERANCE=$f_{tol}^u$