Modeling discontinuities as an enriched feature using the extended finite element method

For the purpose of fracture analysis, the enrichment functions typically consist of the near-tip asymptotic functions that capture the singularity around the crack tip and a discontinuous function that represents the jump in displacement across the crack surfaces. The approximation for a displacement vector function $\mathbf{u}$ with the partition of unity enrichment is

where $N_I\left(x\right)$ are the usual nodal shape functions; the first term on the right-hand side of the above equation, ${\mathbf{u}}_I$, is the usual nodal displacement vector associated with the continuous part of the finite element solution; the second term is the product of the nodal enriched degree of freedom vector, ${\mathbf{a}}_I$, and the associated discontinuous jump function $H\left(x\right)$ across the crack surfaces; and the third term is the product of the nodal enriched degree of freedom vector, ${\mathbf{b}}_I^{\alpha }$, and the associated elastic asymptotic crack-tip functions, $F_{\alpha }\left(x\right)$. The first term on the right-hand side is applicable to all the nodes in the model; the second term is valid for nodes whose shape function support is cut by the crack interior; and the third term is used only for nodes whose shape function support is cut by the crack tip.

Figure 1. Illustration of normal and tangential coordinates for a smooth crack.

Figure 1 illustrates the discontinuous jump function across the crack surfaces, $H\left(x\right)$, which is given by

where $\mathbf{x}$ is a sample (Gauss) point, ${\mathbf{x}}^{*}$ is the point on the crack closest to $\mathbf{x}$, and $\mathbf{n}$ is the unit outward normal to the crack at ${\mathbf{x}}^{*}$.

Figure 1 illustrates the asymptotic crack tip functions in an isotropic elastic material, $F_{\alpha }\left(x\right)$, which are given by

where $\left(r,\theta \right)$ is a polar coordinate system with its origin at the crack tip and $\theta =0$ is tangent to the crack at the tip.

These functions span the asymptotic crack-tip function of elasto-statics, and $\sqrt{r}\sin \frac{\theta }{2}$ takes into account the discontinuity across the crack face. The use of asymptotic crack-tip functions is not restricted to crack modeling in an isotropic elastic material. The same approach can be used to represent a crack along a bimaterial interface, impinged on the bimaterial interface, or in an elastic-plastic power law hardening material. However, in each of these three cases different forms of asymptotic crack-tip functions are required depending on the crack location and the extent of the inelastic material deformation. The different forms for the asymptotic crack-tip functions are discussed by Sukumar (2004), Sukumar and Prevost (2003), and Elguedj (2006), respectively.

Accurately modeling the crack-tip singularity requires constantly keeping track of where the crack propagates and is cumbersome because the degree of crack singularity depends on the location of the crack in a non-isotropic material. Therefore, we consider the asymptotic singularity functions only when modeling stationary cracks in Abaqus/Standard. Moving cracks are modeled using one of the two alternative approaches described below.

One alternative approach within the framework of XFEM is based on traction-separation cohesive behavior. This approach is used in Abaqus/Standard to simulate crack initiation and propagation. This is a very general interaction modeling capability, which can be used for modeling brittle or ductile fracture. The other crack initiation and propagation capabilities available in Abaqus/Standard are based on cohesive elements (Defining the constitutive response of cohesive elements using a traction-separation description) or on surface-based cohesive behavior (Contact cohesive behavior). Unlike these methods, which require that the cohesive surfaces align with element boundaries and the cracks propagate along a set of predefined paths, the XFEM-based cohesive segments method can be used to simulate crack initiation and propagation along an arbitrary, solution-dependent path in the bulk materials, since the crack propagation is not tied to the element boundaries in a mesh. In this case the near-tip asymptotic singularity is not needed, and only the displacement jump across a cracked element is considered. Therefore, the crack has to propagate across an entire element at a time to avoid the need to model the stress singularity.

Phantom nodes, which are superposed on the original real nodes, are introduced to represent the discontinuity of the cracked elements, as illustrated in Figure 2. When the element is intact, each phantom node is completely constrained to its corresponding real node. When the element is cut through by a crack, the cracked element splits into two parts. Each part is formed by a combination of some real and phantom nodes depending on the orientation of the crack. Each phantom node and its corresponding real node are no longer tied together and can move apart.

Figure 2. The principle of the phantom node method.

The magnitude of the separation is governed by the cohesive law until the cohesive strength of the cracked element is zero, after which the phantom and the real nodes move independently. To have a set of full interpolation bases, the part of the cracked element that belongs in the real domain, ${\mathrm{\Omega }}_0$, is extended to the phantom domain, ${\mathrm{\Omega }}_p$. Then the displacement in the real domain, ${\mathrm{\Omega }}_0$, can be interpolated by using the degrees of freedom for the nodes in the phantom domain, ${\mathrm{\Omega }}_p$. The jump in the displacement field is realized by simply integrating only over the area from the side of the real nodes up to the crack; i.e., ${\mathrm{\Omega }}_0^{+}$ and ${\mathrm{\Omega }}_0^{-}$. This method provides an effective and attractive engineering approach and has been used for simulation of the initiation and growth of multiple cracks in solids by Song (2006) and Remmers (2008). The equivalent polynomial methodology originally developed by Ventura and Benvenuti (2015) for a cracked element enriched with a Heaviside enrichment function has been extended to evaluate the stiffness matrix for the cracked element composed of real and phantom nodes in Abaqus/Standard. It has been proven to exhibit almost no mesh dependence if the mesh is sufficiently refined.

Modeling hydraulically driven fracture

The cohesive segments method in conjunction with phantom nodes discussed above can also be extended to model hydraulically driven fracture. In this case additional phantom nodes with pore pressure degrees of freedom are introduced on the edges of each enriched element to model the fluid flow within the cracked element surfaces in conjunction with the phantom nodes that are superposed on the original real nodes to represent the discontinuities of displacement and fluid pressure in a cracked element. The phantom node at each element edge is not activated until the edge is intersected by a crack. The flow patterns of the pore fluid in the cracked elements are shown in Figure 3. The fluid is assumed to be incompressible. The fluid flow continuity, which accounts for both tangential and normal flow within and across the cracked element surfaces as well as the rate of opening of the cracked element surfaces, is maintained. The fluid pressure on the cracked element surfaces contributes to the traction-separation behavior of the cohesive segments in the enriched elements, which enables the modeling of hydraulically driven fracture.

Figure 3. Flow within a cracked element.

Another alternative approach to modeling moving cracks within the framework of XFEM is based on the principles of linear elastic fracture mechanics (LEFM). Therefore, it is more appropriate for problems in which brittle crack propagation occurs. Similar to the XFEM-based cohesive segments method described above, the near-tip asymptotic singularity is not considered, and only the displacement jump across a cracked element is considered. Therefore, the crack has to propagate across an entire element at a time to avoid the need to model the stress singularity. The strain energy release rate at the crack tip is calculated based on the modified Virtual Crack Closure Technique (VCCT), which has been used to model delamination along a known and partially bonded surface (see Crack propagation analysis). However, unlike this method, the XFEM-based LEFM approach can be used to simulate crack propagation along an arbitrary, solution-dependent path in the bulk material without the requirement of a pre-existing crack in the model.

The modeling technique is very similar to the XFEM-based cohesive segment approach described above where phantom nodes are introduced to represent the discontinuity of the cracked element when the fracture criterion is satisfied. The real node and the corresponding phantom node will separate when the equivalent strain energy release rate exceeds the critical strain energy release rate at the crack tip in an enriched element. The traction is initially carried as equal and opposite forces on the two surfaces of the cracked element. The traction is ramped down linearly over the separation between the two surfaces with the dissipated strain energy equal to either the critical strain energy required to initiate the separation or the critical strain energy required to propagate the crack depending on whether the VCCT or the enhanced VCCT criterion is specified.

Modeling low-cycle fatigue crack propagation based on the principles of LEFM

The XFEM-based LEFM approach can also be used to simulate a discrete crack growth subjected to sub-critical cyclic loading in a low-cycle fatigue analysis using the direct cyclic approach (Low-cycle fatigue analysis using the direct cyclic approach). The fracture energy release rates at the crack tips in the enriched elements are calculated based on the above mentioned modified VCCT technique. The onset and crack growth are characterized by using the Paris law, which relates the relative fracture energy release rates to crack growth rates as illustrated in Figure 4. This approach has been used to model progressive delamination under a sub-critical cyclic loading along a known and partially bonded surface (see Low-cycle fatigue criterion). However, unlike this method, the XFEM-based LEFM approach can be used to simulate fatigue crack propagation along an arbitrary, solution-dependent path in the bulk material.

Figure 4. Fatigue crack growth governed by the Paris law.

A key development that facilitates treatment of cracks in an extended finite element analysis is the description of crack geometry, because the mesh is not required to conform to the crack geometry. The level set method, which is a powerful numerical technique for analyzing and computing interface motion, fits naturally with the extended finite element method and makes it possible to model arbitrary crack growth without remeshing. The crack geometry is defined by two almost-orthogonal signed distance functions, as illustrated in Figure 5. The first, $\varphi$, describes the crack surface, while the second, $\psi$, is used to construct an orthogonal surface so that the intersection of the two surfaces gives the crack front. ${\mathbf{n}}^{+}$ indicates the positive normal to the crack surface; ${\mathbf{m}}^{+}$ indicates the positive normal to the crack front. No explicit representation of the boundaries or interfaces is needed because they are entirely described by the nodal data. Two signed distance functions per node are generally required to describe a crack geometry.

Figure 5. Representation of a nonplanar crack in three dimensions by two signed distance functions $\varphi$ and $\psi$.

You can choose to model an arbitrary stationary crack or a discrete crack propagation along an arbitrary, solution-dependent path. The former requires that the elements around the crack tips are enriched with asymptotic functions to catch the singularity and that the elements intersected by the crack interior are enriched with the jump function across the crack surfaces. The latter infers that crack propagation is modeled with either the cohesive segments method or the linear elastic fracture mechanics approach in conjunction with phantom nodes. However, the options are mutually exclusive and cannot be specified simultaneously in a model.

ENRICHMENT, TYPE=PROPAGATION CRACK  

ENRICHMENT, TYPE=STATIONARY CRACK

Interaction module: crack editor: toggle on Allow crack growth

Interaction module: crack editor: toggle off Allow crack growth

You must assign a name to an enriched feature, such as a crack. This name can be used in defining the initial location of the crack surfaces, in identifying a crack for contour integral output, in activating or deactivating the crack propagation analysis, and in generating cracked element surfaces.

ENRICHMENT, NAME=name

Interaction module: SpecialCrackCreate: XFEM: Name: name

You must associate the enrichment definition with a region of your model. Only degrees of freedom in elements within these regions are potentially enriched with special functions. The region should consist of elements that are presently intersected by cracks and those that are likely to be intersected by cracks as the cracks propagate.

ENRICHMENT, ELSET=element set name

Interaction module: SpecialCrackCreate: XFEM: Select the crack domain: select region

As a crack propagates through the model, a crack surface representing both facets of cracked elements is generated on those enriched elements that are intersected by a crack during the analysis. You must associate the name of an enriched feature with the surface (see Assigning a name to the enriched feature above).

SURFACE, TYPE=XFEM

An XFEM-based crack surface is not supported in Abaqus/CAE.

When an element is cut by a crack, the compressive behavior of the crack surfaces has to be considered. The formulae that govern behavior are very similar to those used for surface-based small-sliding penalty contact (About mechanical contact properties).

For an element intersected by a stationary crack or a moving crack with the linear elastic fracture mechanics approach, it is assumed that the elastic cohesive strength of the cracked element is zero. Therefore, compressive behavior of the crack surfaces is fully defined with the above options when the crack surfaces come into contact. For a moving crack with the cohesive segments method, the situation is more complex; traction-separation cohesive behavior as well as compressive behavior of the crack surfaces are involved in a cracked element. In the contact normal direction, the pressure-overclosure relationship governing the compressive behavior between the surfaces does not interact with the cohesive behavior, since they each describe the interaction between the surfaces in a different contact regime. The pressure-overclosure relationship governs the behavior only when the crack is “closed”; the cohesive behavior contributes to the contact normal stress only when the crack is “open” (i.e., not in contact).

If the elastic cohesive stiffness of an element is undamaged in the shear direction, it is assumed that the cohesive behavior is active. Any tangential slip is assumed to be purely elastic in nature and is resisted by the elastic cohesive strength of the element, resulting in shear forces. If damage has been defined, the cohesive contribution to the shear stresses starts degrading with damage evolution. Once maximum degradation has been reached, the cohesive contribution to the shear stresses is zero. The friction model activates and begins contributing to the shear stresses.

ENRICHMENT, INTERACTION=interaction_property_name
SURFACE INTERACTION, NAME=interaction_property_name
SURFACE BEHAVIOR

Interaction module: crack editor: toggle on Specify contact property

A general-purpose finite-sliding contact capability is available for XFEM-based crack surfaces within the general contact framework in Abaqus/Standard (About general contact in Abaqus/Standard). Contact pairs do not support XFEM-based crack surfaces. Crack surfaces can be included in the general contact domain to participate in contact within the compressive regime. Contact between crack surfaces and contact between crack surfaces and other types of surfaces in the model can be considered. The general contact algorithm includes crack surfaces when the default all-inclusive exterior surface defines the contact domain. Alternatively, you can specify pairwise inclusions and exclusions to control more precisely the regions of the model that can potentially come into contact. In both cases the enrichment regions where cracks occur and participate in general contact have to be associated with at least one XFEM-based crack surface. In the former case the association of an enrichment region with an XFEM-based crack surface triggers general contact on cracks in that enrichment region. In the latter case the named XFEM-based crack surface can be explicitly used with contact inclusions and exclusions akin to other types of surfaces to precisely define the general contact domain (Defining the general contact domain).

General contact always takes precedence over the small-sliding contact formulation when both are included to model contact between opposing surfaces of a crack within the same analysis. The small-sliding formulation cannot be used to model contact between the crack surfaces and other surfaces; in that case, only general contact supports contact interactions.

ENRICHMENT, NAME=region-1, TYPE=PROPAGATION CRACK, ELSET=set-1
ENRICHMENT, NAME=region-2, TYPE=PROPAGATION CRACK, ELSET=set-2
SURFACE, TYPE=XFEM, NAME=crack_surf
region-1
region-2

CONTACT
CONTACT INCLUSIONS, ALL EXTERIOR
SURFACE INTERACTION, NAME=interaction_property_name

CONTACT
CONTACT INCLUSIONS
crack_surf,
SURFACE INTERACTION, NAME=interaction_property_name

Interaction module: Create Interaction: General contact (Standard): Included surface pairs: All* with self

The available traction-separation model in Abaqus assumes initially linear elastic behavior followed by the initiation and evolution of damage. The elastic behavior is written in terms of an elastic constitutive matrix that relates the normal and shear stresses to the normal and shear separations of a cracked element.

The nominal traction stress vector, $\mathbf{t}$, consists of the following components: $t_n$, $t_s$, and (in three-dimensional problems) $t_t$, which represent the normal and the two shear tractions, respectively. The corresponding separations are denoted by ${\delta }_n$, ${\delta }_s$, and ${\delta }_t$. The elastic behavior can then be written as

The normal and tangential stiffness components will not be coupled: pure normal separation by itself does not give rise to cohesive forces in the shear directions, and pure shear slip with zero normal separation does not give rise to any cohesive forces in the normal direction.

The terms $K_{nn}$, $K_{ss}$, and $K_{tt}$ are calculated based on the elastic properties for an enriched element. Specifying the elastic properties of the material in an enriched region is sufficient to define both the elastic stiffness and the traction-separation behavior.

Damage modeling allows you to simulate the degradation and eventual failure of an enriched element. The failure mechanism consists of two ingredients: a damage initiation criterion and a damage evolution law. The initial response is assumed to be linear as discussed in the previous section. However, once a damage initiation criterion is met, damage can occur according to a user-defined damage evolution law. Figure 6 shows a typical linear and a typical nonlinear traction-separation response with a failure mechanism. The enriched elements do not undergo damage under pure compression.

Figure 6. Typical linear (a) and nonlinear (b) traction-separation response.

Damage of the traction-separation response for cohesive behavior in an enriched element is defined within the same general framework used for conventional materials (see About progressive damage and failure). However, unlike cohesive elements with traction-separation behavior, you do not have to specify the undamaged traction-separation behavior in an enriched element.

Crack initiation refers to the beginning of degradation of the cohesive response at an enriched element. The process of degradation begins when the stresses or the strains satisfy specified crack initiation criteria. Crack initiation criteria are available based on the following Abaqus/Standard built-in models:

• the maximum principal stress criterion,

• the maximum principal strain criterion,

• the maximum nominal stress criterion,

• the maximum nominal strain criterion,

• the quadratic traction-interaction criterion, and

• the quadratic separation-interaction criterion.

In addition, a user-defined damage initiation criterion can be specified in user subroutine UDMGINI.

An additional crack is introduced or the crack length of an existing crack is extended after an equilibrium increment when the fracture criterion, f, reaches the value 1.0 within a given tolerance:

You can specify the tolerance $f_{tol}$. If $f>1+f_{tol}$, the time increment is cut back such that the crack initiation criterion is satisfied. The default value of $f_{tol}$ is 0.05. To improve performance, a separate tolerance $f_{tol}^g$ can be specified to control the crack growth of an existing crack while $f_{tol}$ is used to control the nucleation of an additional crack. If it is not specified, the growth tolerance, $f_{tol}^g,$ is set equal to $f_{tol}\mathrm{.}$

DAMAGE INITIATION, TOLERANCE=$f_{tol}$, GROWTH TOLERANCE=$f_{tol}^g$

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

DAMAGE INITIATION, UNSTABLE GROWTH TOLERANCE=$f_{tol}^u$
CONTROLS, TYPE=NO CUTBACK SCALING

DAMAGE INITIATION, NORMAL DIRECTION=1 (default)

DAMAGE INITIATION, NORMAL DIRECTION=2

DAMAGE INITIATION, CRITERION=MAXPS

Property module: material editor: Mechanical: Damage for Traction Separation Laws: Maxps Damage

DAMAGE INITIATION, CRITERION=MAXPE

Property module: material editor: Mechanical: Damage for Traction Separation Laws: Maxpe Damage

DAMAGE INITIATION, CRITERION=MAXS

Property module: material editor: Mechanical: Damage for Traction Separation Laws: Maxs Damage

DAMAGE INITIATION, CRITERION=MAXE

Property module: material editor: Mechanical: Damage for Traction Separation Laws: Maxe Damage

DAMAGE INITIATION, CRITERION=QUADS

Property module: material editor: Mechanical: Damage for Traction Separation Laws: Quads Damage

DAMAGE INITIATION, CRITERION=QUADE

Property module: material editor: Mechanical: Damage for Traction Separation Laws: Quade Damage

DAMAGE INITIATION, CRITERION=USER

DAMAGE INITIATION, CRITERION=USER, FAILURE MECHANISMS=$n$

DAMAGE INITIATION, CRITERION=USER, PROPERTIES=number_of_constants

An accurate and efficient evaluation of the stress/strain fields ahead of the crack tip is important for both evaluating the crack initiation criterion and computing the crack propagation direction when needed. Abaqus/Standard offers several options for computing these fields.

Centroidal values of stress and strain

By default, the stress/strain computed at the element centroid ahead of the crack tip is used to determine if the damage initiation criterion is satisfied and to determine the crack propagation direction. See Figure 7.

Figure 7. Centroidal and crack tip locations.

Input File Usage

DAMAGE INITIATION, POSITION=CENTROID (default)

Abaqus/CAE Usage

Property module: material editor: MechanicalDamage for Traction Separation Laws: Quade Damage, Maxe Damage, Quads Damage, Maxs Damage, Maxpe Damage, or Maxps Damage: PositionCentroid

DAMAGE INITIATION, POSITION=CRACKTIP

DAMAGE INITIATION, POSITION=COMBINED

The three options for evaluating the stress and strain fields discussed above are local calculations in the sense that the evaluated fields are local to the single element ahead of the crack tip. In the case of coarse and/or unstructured meshes a nonlocal averaging of the stress and strain fields ahead of the crack tip can lead to a more accurate evaluation of those fields, which can improve the accuracy of the computed propagation directions. See Figure 8.

Figure 8. Nonlocal averaging region.

DAMAGE INITIATION, POSITION=NONLOCAL

DAMAGE INITIATION, R CRACK DIRECTION=$r_c$

DAMAGE INITIATION, SMOOTHING=NONE (default)
DAMAGE INITIATION, SMOOTHING=NODAL

DAMAGE INITIATION, WEIGHTING METHOD=UNIFORM (default)
DAMAGE INITIATION, WEIGHTING METHOD=GAUSS 
DAMAGE INITIATION, WEIGHTING METHOD=CUBIC SPLINE 
DAMAGE INITIATION, WEIGHTING METHOD=USER

The damage evolution law describes the rate at which the cohesive stiffness is degraded once the corresponding initiation criterion is reached. The general framework for describing the evolution of damage is conceptually similar to that used for damage evolution in surface-based cohesive behavior (Contact cohesive behavior).

A scalar damage variable, D, represents the averaged overall damage at the intersection between the crack surfaces and the edges of cracked elements. It initially has a value of 0. If damage evolution is modeled, D monotonically evolves from 0 to 1 upon further loading after the initiation of damage. The normal and shear stress components are affected by the damage according to

where $T_n$, $T_s$, and $T_t$ are the normal and shear stress components predicted by the elastic traction-separation behavior for the current separations without damage.

To describe the evolution of damage under a combination of normal and shear separations across the interface, an effective separation is defined as

DAMAGE EVOLUTION

The combination option DAMAGE EVOLUTION, TYPE=ENERGY, SOFTENING=EXPONENTIAL is not recommended to be used with XFEM.

DAMAGE INITIATION, CRITERION=USER, FAILURE MECHANISMS=$n$
DAMAGE EVOLUTION, FAILURE INDEX=1
DAMAGE EVOLUTION, FAILURE INDEX=2
...
DAMAGE EVOLUTION, FAILURE INDEX=$n$

Models exhibiting various forms of softening behavior and stiffness degradation often lead to severe convergence difficulties in Abaqus/Standard. Viscous regularization of the constitutive equations defining cohesive behavior in an enriched element can be used to overcome some of these convergence difficulties. Viscous regularization damping causes the tangent stiffness matrix to be positive definite for sufficiently small time increments.

The approximate amount of energy associated with viscous regularization over the whole model is available using output variable ALLVD.

DAMAGE STABILIZATION

The formulae and laws that govern the behavior of fluid flow within the XFEM-based cracked element surfaces are very similar to those used for fluid flow within the cohesive element gap (Defining the constitutive response of fluid within the cohesive element gap). The similarities extend to the traction-separation model, damage initiation criteria, damage evolution law, and the fluid flow behavior. The fluid constitutive response includes the tangential flow within the cracked element surfaces and the normal flow across the cracked element surfaces due to caking or fouling effects in the enriched elements.

GAP FLOW, TYPE=NEWTONIAN, KMAX

GAP FLOW, TYPE=POWER LAW

SECTION CONTROLS, INITIAL GAP OPENING

Property module: material editor: OtherPore FluidGap Flow: Type: Newtonian: Toggle on Maximum Permeability and enter the value of $k_{\max }$

Property module: material editor: OtherPore FluidGap Flow: Type: Power law

An initial gap opening is not supported in Abaqus/CAE.

Normal flow across the cracked element surfaces

You can permit normal flow by defining a fluid leak-off coefficient for the pore fluid material. This coefficient defines a pressure-flow relationship between the phantom nodes located at the cracked element edges and cracked element surfaces. The fluid leak-off coefficients can be interpreted as the permeability of a finite layer of material on the cracked element surfaces, as shown in Figure 9.

Figure 9. Leak-off coefficient interpretation as a permeable layer.

The normal flow is defined as

$$q_t=c_t\left(p_i-p_t\right)$$

and

$$q_b=c_b\left(p_i-p_b\right),$$

where $q_t$ and $q_b$ are the flow rates into the top and bottom surfaces of a cracked element, respectively; $p_i$ is the pressure at the phantom node located at the cracked element edge; and $p_t$ and $p_b$ are the pore pressures on the top and bottom surfaces of a cracked element, respectively. You can optionally define leak-off coefficients as functions of temperature and field variables.

Alternatively, you can use user subroutine UFLUIDLEAKOFF to define more complex leak-off behavior, including the ability to define a time accumulated resistance, or fouling, through the use of solution-dependent state variables.

Input File Usage

Use the following option to define the leak-off coefficients:

FLUID LEAKOFF

Use the following option to define leak-off coefficients as functions of temperature and field variables:

FLUID LEAKOFF, DEPENDENCIES

Use the following option to define more complex leak-off behavior in user subroutine UFLUIDLEAKOFF:

FLUID LEAKOFF, USER

Abaqus/CAE Usage

Use the following option to define the leak-off coefficients:

Property module: material editor: OtherPore FluidFluid Leakoff: Type: Coefficients

Use the following option to define leak-off coefficients as functions of temperature and field variables:

Property module: material editor: OtherPore FluidFluid Leakoff: Type: Coefficients: Toggle on Use temperature-dependent data and select the number of field variables.

Use the following option to define more complex leak-off behavior in user subroutine UFLUIDLEAKOFF:

Property module: material editor: OtherPore FluidFluid Leakoff: Type: User

By definition, the XFEM-based LEFM approach inherently requires the presence of a crack in the model since it is based on the principles of linear elastic fracture mechanics. The crack can be pre-existing, or it can nucleate during the analysis. If there is no pre-existing crack for a given enriched region, the XFEM-based LEFM approach is not activated until a crack nucleates. The crack nucleation is governed by one of the six built-in stress- or strain-based crack initiation criteria or a user-defined crack initiation criterion discussed in Crack initiation and direction of crack extension above. After a crack is nucleated in an enriched region, subsequent propagation of the crack is governed by the XFEM-based LEFM criterion.

DAMAGE INITIATION, TOLERANCE=$f_{tol}$

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

SURFACE BEHAVIOR
FRACTURE CRITERION, TOLERANCE=$f_{tol}$, TYPE=VCCT  

FRACTURE CRITERION, NORMAL DIRECTION=MTS (default)

FRACTURE CRITERION, NORMAL DIRECTION=1

FRACTURE CRITERION, NORMAL DIRECTION=2

Abaqus provides three common mode-mix formulae for computing the equivalent fracture energy release rate $G_{equivC}$: 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 

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

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

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

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,

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, $G_{equivC}^P$, rather than the critical equivalent strain energy required to initiate the separation, $G_{equivC}$. The formulae for calculating $G_{equivC}^P$ are identical to those used for $G_{equivC}$ for different mixed-mode fracture criteria.

SURFACE BEHAVIOR
FRACTURE CRITERION, TYPE=ENHANCED VCCT

If you specify the low-cycle fatigue criterion, progressive crack growth at the enriched elements subjected to sub-critical cyclic loading can be simulated. This criterion can be used only in a low-cycle fatigue analysis using the direct cyclic approach (Low-cycle fatigue analysis using the direct cyclic approach). A low-cycle fatigue 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 low-cycle fatigue analysis steps in a single analysis. If you perform a fatigue analysis in a model without a pre-existing crack, you must precede the fatigue step with a static step that nucleates a crack, as discussed in Crack nucleation and direction of crack extension.

The onset and fatigue crack growth are characterized by using the Paris law, which relates the relative fracture energy release rate to crack growth rates as illustrated in Figure 4. The fracture energy release rates at the crack tips in the enriched elements are calculated based on the above mentioned VCCT technique.

The Paris regime is bounded by the energy release rate threshold, $G_{thresh}$, below which there is no consideration of fatigue crack initiation or growth, and the energy release rate upper limit, $G_{pl}$, above which the fatigue crack will grow at an accelerated rate. $G_C$ is the critical equivalent strain energy release rate calculated based on the user-specified mode-mix criterion and the fracture strength of the bulk material. The formulae for calculating $G_C$ have been provided above for different mixed mode fracture criteria. You can specify the ratio of $G_{thresh}$ over $G_C$ and the ratio of $G_{pl}$ over $G_C$. The default values are $\frac{G_{thresh}}{G_C}=0.01$ and $\frac{G_{pl}}{G_C}=0.85$.

SURFACE BEHAVIOR
FRACTURE CRITERION, TYPE=FATIGUE

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.

FRACTURE CRITERION, TYPE=VCCT, VISCOSITY=$\mu$

FRACTURE CRITERION, TYPE=ENHANCED VCCT, VISCOSITY=$\mu$

Although XFEM has alleviated the shortcomings associated with defining the conforming mesh in the neighborhood of the crack front, you must still generate a sufficient number of elements around the crack front to obtain path-independent contours, particularly in a region with high crack-front curvature. The group of elements within a small radius from the crack front are enriched and become involved in the contour integral calculations. The default enrichment radius is six times the typical element characteristic length of those elements along the crack front in the enriched area. You must include the elements inside the enrichment radius in the element set used to define the enriched region. Elements with a large aspect ratio should be avoided along the crack front.

ENRICHMENT, ENRICHMENT RADIUS

Interaction module: crack editor: Enrichment radius: Analysis default or Specify