where $T _ { n } , T _ { s } ,$ , and $T _ { t }$ are the normal and shear stress components predicted by the elastic tractionseparation 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 $$ \delta_ {m} = \sqrt {\langle \delta_ {n} \rangle^ {2} + \delta_ {s} ^ {2} + \delta_ {t} ^ {2}}. $$ Input File Usage: Use the following option to specify a damage evolution law: \*DAMAGE EVOLUTION Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction Separation Laws: Maxpe Damage or Maxps Damage: Suboptions→Damage Evolution Use in conjunction with user-defined damage initiation criterion A separate damage evolution law should be specified for each damage initiation criterion defined in user subroutine UDMGINI. Each combination of a damage initiation criterion and a corresponding damage evolution law is referred to as a failure mechanism. Damage will accumulate for only one failure mechanism per element, corresponding to the mechanism whose damage initiation criterion was achieved first. Input File Usage: Use the following options to specify damage evolution laws for multiple userdefined damage initiation criteria: ```txt * DAMAGE INITIATION, CRITERION=USER, FAILURE MECHANISMS=n * DAMAGE EVOLUTION, FAILURE INDEX=1 * DAMAGE EVOLUTION, FAILURE INDEX=2 ... * DAMAGE EVOLUTION, FAILURE INDEX=n ``` Abaqus/CAE Usage: Defining a user-defined damage initiation criterion is not supported in Abaqus/CAE. # Viscous regularization in Abaqus/Standard 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. Input File Usage: Use the following option to specify viscous regularization: \*DAMAGE STABILIZATION 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: Suboptions→Damage Stabilization Cohesive # Defining the constitutive response of fluid flow within the cracked element surfaces 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,” Section 32.5.7). The similarities extend to the traction-separation model, damage initiation criteria, damage evolution law, and the fluid flow behavior. The fluid constitutive response comprises 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. # Tangential flow The tangential flow within the cracked element surfaces can be modeled with either a Newtonian or power-law model. By default, there is no tangential flow of pore fluid within the cracked element surfaces. To allow tangential flow, define a gap flow property in conjunction with the pore fluid material definition. In the case of a Newtonian fluid the volume flow rate density vector is given by the expression $$ \mathbf {q} d = - k _ {t} \nabla p, $$ where $k _ { t }$ is the tangential permeability (the resistance to the fluid flow), $\nabla p$ is the pressure gradient along the cracked element surfaces, and is the opening of the cracked element surfaces. Abaqus defines the tangential permeability, or the resistance to flow, according to Reynold’s equation: $$ k _ {t} = \frac {d ^ {3}}{1 2 \mu}, $$ where $\mu$ is the fluid viscosity and is the opening of the cracked element surfaces. You can also specify an upper limit on the value of $k _ { t }$ . In the case of a power law fluid the constitutive relation is defined as $$ \tau = K \dot {\gamma} ^ {\alpha}, $$ where is the shear stress, is the shear strain rate, is the fluid consistency, and is the power law coefficient. Abaqus defines the tangential volume flow rate density as $$ \mathbf {q} d = - \left(\frac {2 \alpha}{1 + 2 \alpha}\right) \left(\frac {1}{K}\right) ^ {\frac {1}{\alpha}} \left(\frac {d}{2}\right) ^ {\frac {1 + 2 \alpha}{\alpha}} \| \nabla p \| ^ {\frac {1 - \alpha}{\alpha}} \nabla p, $$ where is the opening of the cracked element surfaces. By default, the gap between the cracked element surfaces has an initial opening of 0.002 in both a Newtonian fluid and a power law fluid. However, you can specify this opening directly. Input File Usage: Use the following option to define the tangential flow in a Newtonian fluid: \*GAP FLOW, TYPE=NEWTONIAN, KMAX Use the following option to define the tangential flow in a power law fluid: \*GAP FLOW, TYPE=POWER LAW Use the following option to define the initial gap opening directly: \*SECTION CONTROLS, INITIAL GAP OPENING Abaqus/CAE Usage: Use the following option to define the tangential flow in a Newtonian fluid: Property module: material editor: Other→Pore Fluid→Gap Flow: Type: Newtonian: Toggle on Maximum Permeability and enter the value of $k _ { \mathrm { m a x } }$ Use the following option to define the tangential flow in a power law fluid: Property module: material editor: Other→Pore Fluid→Gap 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 leakoff 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 leakoff coefficients can be interpreted as the permeability of a finite layer of material on the cracked element surfaces, as shown in Figure 10.7.1–9. The normal flow is defined as $$ q _ {t} = c _ {t} (p _ {i} - p _ {t}) $$ and $$ q _ {b} = c _ {b} (p _ {i} - p _ {b}), $$ 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 leakoff coefficients as functions of temperature and field variables. Alternatively, you can use user subroutine UFLUIDLEAKOFF to define more complex leakoff 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 leakoff coefficients: \*FLUID LEAKOFF 

text_image

P_t P_i P_b permeable layer

Figure 10.7.1–9 Leakoff coefficient interpretation as a permeable layer. Use the following option to define leakoff coefficients as functions of temperature and field variables: \*FLUID LEAKOFF, DEPENDENCIES Use the following option to define more complex leakoff behavior in user subroutine UFLUIDLEAKOFF: \*FLUID LEAKOFF, USER Abaqus/CAE Usage: Use the following option to define the leakoff coefficients: Property module: material editor: Other→Pore Fluid→Fluid Leakoff: Type: Coefficients Use the following option to define leakoff coefficients as functions of temperature and field variables: Property module: material editor: Other→Pore Fluid→Fluid Leakoff: Type: Coefficients: Toggle on Use temperature-dependent data and select the number of field variables. Use the following option to define more complex leakoff behavior in user subroutine UFLUIDLEAKOFF: Property module: material editor: Other→Pore Fluid→Fluid Leakoff: Type: User # Applying the VCCT technique to the XFEM-based LEFM approach 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 “Crack propagation analysis,” Section 11.4.3), where the strain energy release rate at the crack tip is calculated based on the modified Virtual Crack Closure Technique (VCCT). 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 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. # Crack nucleation and direction of crack extension 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 XFEMbased 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. Input File Usage: Use the following option to specify the crack nucleation criterion as part of the material definition when there is no pre-existing crack in an enriched region: $\scriptstyle * \mathrm { D A M A G E ~ I N I T I A T I O N } , \mathrm { T O L E R A N C E } = f _ { t o l }$ 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: $f _ { t o l }$ Specifying when a pre-existing crack will extend If there is a pre-existing 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 $f _ { t o l . } \mathrm { ~ I f ~ } f > 1 + f _ { t o l }$ , 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= , TYPE=VCCT Abaqus/CAE Usage: Interaction module: Interaction→Property→Create, Contact, Mechanical→Fracture Criterion, Tolerance: $f _ { t o l }$ 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,” Section 1.2.2), 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 # Mixed mode behavior Abaqus provides three common mode-mix formulae for computing the equivalent fracture energy release rate $G _ { e q u i v C } \mathrm { : }$ 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: $$ G _ {e q u i v C} = G _ {I C} + \left(G _ {I I C} - G _ {I C}\right) \left(\frac {G _ {I I} + G _ {I I I}}{G _ {I} + G _ {I I} + G _ {I I I}}\right) ^ {\eta}. $$ To define this model, you must provide $G _ { I C } , G _ { I I C }$ and . 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: $$ \frac {G _ {e q u i v}}{G _ {e q u i v C}} = \left(\frac {G _ {I}}{G _ {I C}}\right) ^ {a _ {m}} + \left(\frac {G _ {I I}}{G _ {I I C}}\right) ^ {a _ {n}} + \left(\frac {G _ {I I I}}{G _ {I I I C}}\right) ^ {a _ {o}}. $$ 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=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: $$ G _ {e q u i v C} = G _ {I C} + (G _ {I I C} - G _ {I C}) \left(\frac {G _ {I I} + G _ {I I I}}{G _ {I} + G _ {I I} + G _ {I I I}}\right) ^ {\eta} + $$ $$ (G _ {I I I C} - G _ {I I C}) \left(\frac {G _ {I I I}}{G _ {I I} + G _ {I I I}}\right) \left(\frac {G _ {I I} + G _ {I I I}}{G _ {I} + G _ {I I} + G _ {I I I}}\right) ^ {\eta}. $$ To define this model, you must provide $G _ { I C } , G _ { I I C } , G _ { I I I C }$ and . 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, The formulae for calculating $G _ { e q u i v C } ^ { P }$ , rather than the critical equivalent strain energy required to initiate the separation, $G _ { e q u i v C } ^ { P }$ are identical to those used for $G _ { e q u i v C }$ for different mixed-mode $G _ { e q u i v C }$ 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. # Low-cycle fatigue criterion based on the principles of LEFM 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,” Section 6.2.7). 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 10.7.1–4. 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, $G _ { t h r e s h }$ , below which there is no consideration of fatigue crack initiation or growth, and the energy release rate upper limit, $G _ { p l }$ , 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 $G _ { t h r e s h }$ over $G _ { C }$ and the ratio of $G _ { p l }$ over $G _ { C }$ . The default values are $\begin{array} { r } { \frac { G _ { t h r e s h } } { G _ { C } } = 0 . 0 1 } \end{array}$ Gc and $\bar { \frac { G _ { p l } } { G _ { C } } } = 0 . 8 5$ Gc . Input File Usage: Use both of the following options: \*SURFACE BEHAVIOR \*FRACTURE CRITERION, TYPE=FATIGUE Abaqus/CAE Usage: Specifying a low-cycle fatigue criterion is not supported in Abaqus/CAE. # Onset of fatigue crack growth The onset of fatigue crack growth refers to the beginning of fatigue crack growth at the crack tip in the enriched elements. In a low-cycle fatigue analysis the onset of the fatigue crack growth criterion is characterized by $\Delta G _ { ; }$ , 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 $$ f = \frac {N}{c _ {1} \Delta G ^ {c _ {2}}} \geq 1. 0, $$ where $c _ { 1 }$ and $c _ { 2 }$ are material constants and is the cycle number. The enriched elements ahead of the crack tips will not be fractured unless the above equation is satisfied and the maximum fracture energy release rate, $G _ { m a x }$ , which corresponds to the cyclic energy release rate when the structure is loaded up to its maximum value, is greater than $G _ { t h r e s h }$ . # Fatigue crack growth using the Paris law Once the onset of the fatigue crack growth criterion is satisfied at the enriched element, the crack growth rate, $d a / d N$ , can be calculated based on the relative fracture energy release rate, $\Delta G$ . The rate of the crack growth per cycle is given by the Paris law if $G _ { t h r e s h } < G _ { m a x } < G _ { p l }$ $$ \frac {d a}{d N} = c _ {3} \Delta G ^ {c _ {4}}, $$ where $c _ { 3 }$ and $c _ { 4 }$ are material constants. At the end of cycle , Abaqus/Standard extends the crack length, $a _ { N }$ , from the current cycle forward over an incremental number of cycles, $\Delta N$ to $a _ { N + \Delta N }$ by fracturing at least one enriched element ahead of the crack tips. Given the material constants $c _ { 3 }$ and $c _ { 4 } .$ , combined with the known element length and the likely crack propagation direction $\Delta a _ { N j } = a _ { N + \Delta N } - a _ { N }$ at the enriched elements ahead of the crack tips, the number of cycles necessary to fail each enriched element ahead of the crack tip can be calculated as $\Delta N _ { j }$ , where j represents the enriched element ahead of the th crack tip. The analysis is set up to advance the crack by at least one enriched element after the loading cycle is stabilized. The element with the fewest cycles is identified to be fractured, and its $\Delta N _ { m i n } = m i n ( \Delta N _ { j } )$ is represented as the number of cycles to grow the crack equal to its element length, $\Delta a _ { N m i n } = m i n ( \Delta a _ { N j } )$ . The most critical element is completely fractured with a zero constraint and a zero stiffness at the end of the stabilized cycle. As the enriched element is fractured, the load is redistributed and a new relative fracture energy release rate must be calculated for the enriched elements ahead of the crack tips for the next cycle. This capability allows at least one enriched element ahead of the crack tips to be fractured completely after each stabilized cycle and precisely accounts for the number of cycles needed to cause fatigue crack growth over that length. 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, , by one only. For information on how to accelerate the low-cycle fatigue analysis and to provide a smooth solution for the crack front, see “Controlling the accuracy of damage extrapolation at the interface elements and at the enriched elements” in “Low-cycle fatigue analysis using the direct cyclic approach,” Section 6.2.7. # 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 $\scriptstyle * \mathrm { F R A C T U R E } \mathrm { C R I T E R I O N } , \mathrm { T Y P E } = \mathrm { V C C T } , \mathrm { V I S C O S I T Y } = \mu$ $* { \mathrm { F R A C T U R E ~ C R I T E R I O N } } , { \mathrm { T Y P E } } { \mathrm { = } } { \mathrm { E N H A N C E D ~ V C C T , ~ V I S C O S I T Y } } { \mathrm { = } } \mu$ Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Fracture Criterion: Viscosity: # Applying distributed pressure loads to cracked element surfaces When an element is cut by a crack during the analysis, a XFEM-based crack surface is generated during the analysis (see “Defining a crack surface,” above). A distributed pressure load can be applied to the cracked element surfaces. Input File Usage: Use the following option to define a distributed pressure load to a crack surface: \*DSLOAD surface name, P or PNU, magnitude Abaqus/CAE Usage: An XFEM-based crack surface is not supported in Abaqus/CAE. # Specifying the initial location of an enriched feature Because the mesh is not required to conform to the geometric discontinuities, the initial location of a pre-existing crack must be specified in the model. The level set method is provided for this purpose. Two signed distance functions per node are generally required to describe a crack geometry. The first describes the crack surface, while the second is used to construct an orthogonal surface so that the intersection of the two surfaces gives the crack front (see “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). The first signed distance function must be either greater or less than zero and cannot be equal to zero. If an initial crack has to be defined at the boundaries of an element, a very small positive or negative value for the first signed distance function must be specified. Input File Usage: Use the following option to specify the initial location of an enriched feature: \*INITIAL CONDITIONS, TYPE=ENRICHMENT Abaqus/CAE Usage: Interaction module: crack editor: Crack location: Select: select region # Activating and deactivating the enriched feature The crack propagation capability can be activated or deactivated within the step definition. Input File Usage: Use the following option to activate the crack propagation capability within the step definition: \*ENRICHMENT ACTIVATION, NAME=name, ACTIVATE=ON (default) Use the following option to deactivate the crack propagation capability within the step definition: \*ENRICHMENT ACTIVATION, NAME=name, ACTIVATE=OFF Use the following option to deactivate the crack propagation capability automatically once all the pre-existing cracks (or if there are no pre-existing cracks, all the allowable newly nucleated cracks) have propagated through the boundary of the given enriched feature within the step definition: