Cohesive behavior in Abaqus/Explicit is defined as part of the surface interaction properties that are assigned to the applicable surfaces. General contact must be defined for the model.

*SURFACE INTERACTION, NAME=name
*COHESIVE BEHAVIOR
*CONTACT
*CONTACT PROPERTY ASSIGNMENT
surface1, surface2, name

Cohesive behavior in Abaqus/Standard is defined as part of the surface interaction properties that are assigned to a contact pair. Cohesive behavior cannot be assigned to contact pairs using the finite sliding, surface-to-surface formulation (see “Contact formulations in Abaqus/Standard,” Section 37.1.1).

*SURFACE INTERACTION, NAME=name
*COHESIVE BEHAVIOR
*CONTACT PAIR, INTERACTION=name
surface1, surface2

By default, cohesive constraint forces can potentially act on all nodes of the surfaces for which cohesive behavior is defined. Slave nodes that are initially contacting the master surface can experience cohesive forces at the start of the analysis, and slave nodes that are not initially contacting the master surface can experience cohesive forces if they contact the master surface during the analysis. There may, however, be situations where it is desirable to enforce cohesive behavior only for portions of surfaces that are contacting at the start of the analysis.

*COHESIVE BEHAVIOR, ELIGIBILITY=ORIGINAL CONTACTS

*INITIAL CONDITIONS, TYPE=CONTACT
*COHESIVE BEHAVIOR, ELIGIBILITY=SPECIFIED CONTACTS

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 a slave node is “closed” (i.e., it is in contact with the master surface); the cohesive behavior contributes to the contact normal stress only when a slave node is “open” (i.e., not in contact). In the case of “sticky” cohesive behavior—where the two surfaces are not initially in contact—cohesive effects are activated in the increment after the slave node status changes from open to closed.

In the shear direction, if the cohesive stiffness is undamaged, it is assumed that the cohesive model is active and the friction model is dormant. Any tangential slip is assumed to be purely elastic in nature and is resisted by the cohesive strength of the bond, resulting in shear forces. If damage has been defined, the cohesive contribution to the shear stresses starts degrading with damage evolution. Once the cohesive stiffness starts degrading, the friction model activates and begins contributing to the shear stresses. The elastic stick stiffness of the friction model is ramped up in proportion to the degradation of the elastic cohesive stiffness. Prior to the ultimate failure of the cohesive bond, and following the initiation of the degradation of the cohesive bond, the shear stress is a combination of the cohesive contribution and the contribution from the friction model. Once maximum degradation has been reached, the cohesive contribution to the shear stresses is zero, and the only contribution to the shear stresses is from the friction model.

The formulae and laws that govern cohesive surface behavior are very similar to those used for cohesive elements with traction-separation constitutive behavior (“Defining the constitutive response of cohesive elements using a traction-separation description,” Section 32.5.6). The similarities extend to the linear elastic traction-separation model, damage initiation criteria, and damage evolution laws.

However, it is important to recognize that damage in surface-based cohesive behavior is an interaction property, not a material property. Concepts of strain and displacement (used in behavior model formulae for cohesive elements) are reinterpreted as contact separations; contact separations are the relative displacements between the nodes on the slave surface and their corresponding projection points on the master surface along the contact normal and shear directions. Stresses are defined for surface-based cohesive behavior as the cohesive forces acting along the contact normal and shear directions divided by the current area at each contact point.

The specifics of the surface-based cohesive behavior model are discussed in the sections that follow.

The available traction-separation model in Abaqus assumes initially linear elastic behavior (see “Defining elasticity in terms of tractions and separations for cohesive elements” in “Linear elastic behavior,” Section 22.2.1) 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 across the interface.

The nominal traction stress vector, , consists of three components (two components in two-dimensional problems): , , and (in three-dimensional problems) , which represent the normal (along the local 3-direction in three dimensions and along the local 2-direction in two dimensions) and the two shear tractions (along the local 1- and 2-directions in three dimensions and along the local 1-direction in two dimensions), respectively. The corresponding separations are denoted by , , and . The elastic behavior can then be written as

*COHESIVE BEHAVIOR, TYPE=UNCOUPLED (default)

*COHESIVE BEHAVIOR, TYPE=COUPLED 

Damage modeling allows you to simulate the degradation and eventual failure of the bond between two cohesive surfaces. 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 above. However, once a damage initiation criterion is met, damage can occur according to a user-defined damage evolution law. Figure 36.1.10–1 shows a typical traction-separation response with a failure mechanism. If the damage initiation criterion is specified without a corresponding damage evolution model, Abaqus evaluates the damage initiation criterion for output purposes only; there is no effect on the response of the cohesive surfaces (i.e., no damage will occur). Cohesive surfaces do not undergo damage under pure compression.

Damage of the traction-separation response for cohesive surfaces is defined within the same general framework used for conventional materials (see “Progressive damage and failure,” Section 24.1.1), except the damage behavior is specified as part of the interaction properties for the surfaces. Multiple damage response mechanisms are not available for cohesive surfaces: cohesive surfaces can have only one damage initiation criterion and only one damage evolution law.

*SURFACE INTERACTION, NAME=name
*COHESIVE BEHAVIOR
*DAMAGE INITIATION
*DAMAGE EVOLUTION

Damage initiation refers to the beginning of degradation of the cohesive response at a contact point. The process of degradation begins when the contact stresses and/or contact separations satisfy certain damage initiation criteria that you specify. Several damage initiation criteria are available and are discussed below.

Each damage initiation criterion also has an output variable associated with it to indicate whether the criterion is met. A value of 1 or higher indicates that the initiation criterion has been met. Damage initiation criteria that do not have an associated evolution law affect only output. Thus, you can use these criteria to evaluate the propensity of the material to undergo damage without actually modeling the damage process (i.e., without actually specifying damage evolution).

In the discussion below, , , and represent the peak values of the contact stress when the separation is either purely normal to the interface or purely in the first or the second shear direction, respectively. Likewise, , , and represent the peak values of the contact separation, when the separation is either purely along the contact normal or purely in the first or the second shear direction, respectively. The symbol used in the discussion below represents the Macaulay bracket with the usual interpretation. The Macaulay brackets are used to signify that a purely compressive displacement (i.e., a contact penetration) or a purely compressive stress state does not initiate damage.

*DAMAGE INITIATION, CRITERION=MAXS

*DAMAGE INITIATION, CRITERION=MAXU

*DAMAGE INITIATION, CRITERION=QUADS

*DAMAGE INITIATION, CRITERION=QUADU

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 in bulk materials (as opposed to interfaces modeled using cohesive surfaces) is described in “Damage evolution and element removal for ductile metals,” Section 24.2.3. Conceptually, similar ideas apply for describing damage evolution in cohesive surfaces.

A scalar damage variable, D, represents the overall damage at the contact point. 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 contact stress components are affected by the damage according to

To describe the evolution of damage under a combination of normal and shear separations across the interface, it is useful to introduce an effective separation (Camanho and Davila, 2002) defined as

Mixed-mode definition

The relative proportions of normal and shear separations at a contact point define the mode mix at the point. Abaqus uses two measures of mode mix, one based on energies and the other based on tractions. You can choose one of these measures when you specify the mode dependence of the damage evolution process. Denoting by , , and the work done by the tractions and their conjugate separations in the normal, first, and second shear directions, respectively, and defining , the mode-mix definitions based on energies are as follows:

Clearly, only two of the three quantities defined above are independent. It is also useful to define the quantity to denote the portion of the total work done by the shear traction and the corresponding separation components. As discussed later, Abaqus requires that you specify material properties related to damage evolution as functions of (or, equivalently, ) and .

The corresponding definitions of the mode mix based on traction components are given by

where is a measure of the effective shear traction. The angular measures used in the above definition (before they are normalized by the factor ) are illustrated in Figure 36.1.10–2.

Figure 36.1.10–2 Mode-mix measures based on traction.

The mode-mix ratios defined in terms of energies and tractions can be quite different in general. The following example illustrates this point. In terms of energies a separation in the purely normal direction is one for which and , irrespective of the values of the normal and the shear tractions. In particular, for coupled traction-separation behavior both the normal and shear tractions may be nonzero for a purely normal separation. For this case the definition of mode mix based on energies would indicate a purely normal separation, while the definition based on tractions would suggest a mix of both normal and shear separation.

There are two components to the definition of damage evolution. The first component involves specifying either the effective separation at complete failure, , relative to the effective separation at the initiation of damage, ; or the energy dissipated due to failure, (see Figure 36.1.10–3). The second component to the definition of damage evolution is the specification of the nature of the evolution of the damage variable, D, between initiation of damage and final failure. This can be done by either defining linear or exponential softening laws or specifying D directly as a tabular function of the effective separation relative to the effective separation at damage initiation. The data described above will in general be functions of the mode mix, temperature, and/or field variables.

Figure 36.1.10–3 Linear damage evolution.

Figure 36.1.10–4 is a schematic representation of the dependence of damage initiation and evolution on the mode mix for a traction-separation response with isotropic shear behavior. The figure shows the traction on the vertical axis and the magnitudes of the normal and the shear separations along the two horizontal axes. The unshaded triangles in the two vertical coordinate planes represent the response under pure normal and pure shear separation, respectively. All intermediate vertical planes (that contain the vertical axis) represent the damage response under mixed-mode conditions with different mode mixes. The dependence of the damage evolution data on the mode mix can be defined either in tabular form or, in the case of an energy-based definition, analytically. The manner in which the damage evolution data are specified as a function of the mode mix is discussed later in this section.

Figure 36.1.10–4 Illustration of mixed-mode response in cohesive interactions.

Unloading subsequent to damage initiation is always assumed to occur linearly toward the origin of the traction-separation plane, as shown in Figure 36.1.10–3. Reloading subsequent to unloading also occurs along the same linear path until the softening envelope (line AB) is reached. Once the softening envelope is reached, further reloading follows this envelope as indicated by the arrow in Figure 36.1.10–3.

Input File Usage:	Use the following option to use the mode-mix definition based on energies:
	*DAMAGE EVOLUTION, MODE MIX RATIO=ENERGY Use the following option to use the mode-mix definition based on tractions: *DAMAGE EVOLUTION, MODE MIX RATIO=TRACTION

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: toggle on Specify mixed-mode behavior: Mode mix ratio: Energy or Traction

Evolution based on effective separation

You specify the quantity (i.e., the effective separation at complete failure, , relative to the effective separation at damage initiation, , as shown in Figure 36.1.10–3) as a tabular function of the mode mix, temperature, and/or field variables. In addition, you also choose either a linear or an exponential softening law that defines the detailed evolution (between initiation and complete failure) of the damage variable, D, as a function of the effective separation beyond damage initiation. Alternatively, instead of using linear or exponential softening, you can specify the damage variable, D, directly as a tabular function of the effective separation after the initiation of damage, ; mode mix; temperature; and/or field variables.

Linear damage evolution

For linear softening (see Figure 36.1.10–3) Abaqus uses an evolution of the damage variable, D, that reduces (in the case of damage evolution under a constant mode mix, temperature, and field variables) to the following expression:

In the preceding expression and in all later references, refers to the maximum value of the effective separation attained during the loading history. The assumption of a constant mode mix at a contact point between initiation of damage and final failure is customary for problems involving monotonic damage (or monotonic fracture).

Input File Usage:	Use the following option to specify linear damage evolution:
	*DAMAGE EVOLUTION, TYPE=DISPLACEMENT, SOFTENING=LINEAR

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: Type: Displacement: Softening: Linear

Exponential damage evolution

For exponential softening (see Figure 36.1.10–5) Abaqus uses an evolution of the damage variable, D, that reduces (in the case of damage evolution under a constant mode mix, temperature, and field variables) to

In the expression above is a non-dimensional parameter that defines the rate of damage evolution and is the exponential function.

Figure 36.1.10–5 Exponential damage evolution.

Input File Usage:	Use the following option to specify exponential softening:
	*DAMAGE EVOLUTION, TYPE=DISPLACEMENT, SOFTENING=EXPONENTIAL

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: Type: Displacement: Softening: Exponential

Tabular damage evolution

For tabular softening you define the evolution of D directly in tabular form. D must be specified as a function of the effective separation relative to the effective separation at initiation, mode mix, temperature, and/or field variables.

Input File Usage:	Use the following option to define the damage variable directly in tabular form:
	*DAMAGE EVOLUTION, TYPE=DISPLACEMENT, SOFTENING=TABULAR

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: Type: Displacement: Softening: Tabular

Evolution based on energy

Damage evolution can be defined based on the energy that is dissipated as a result of the damage process, also called the fracture energy. The fracture energy is equal to the area under the traction-separation curve (see Figure 36.1.10–3). You specify the fracture energy as a property of the cohesive interaction and choose either a linear or an exponential softening behavior. Abaqus ensures that the area under the linear or the exponential damaged response is equal to the fracture energy.

The dependence of the fracture energy on the mode mix can be specified either directly in tabular form or by using analytical forms as described below. When the analytical forms are used, the mode-mix ratio is assumed to be defined in terms of energies.

Tabular form

The simplest way to define the dependence of the fracture energy is to specify it directly as a function of the mode mix in tabular form.

Input File Usage:	Use the following option to specify fracture energy as a function of the mode mix in tabular form:
	*DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=TABULAR

Abaqus/CAE Usage:

Interaction module: contact property editor: Contact: MechanicalDamage: Evolution tabbed page: Type: Energy: toggle on Specify mixed mode behavior: Tabular

Power law form

The dependence of the fracture energy on the mode mix can be defined based on a power law fracture criterion. The power law criterion states that failure under mixed-mode conditions is governed by a power law interaction of the energies required to cause failure in the individual (normal and two shear) modes. It is given by

The mixed-mode fracture energy when the above condition is satisfied. In other words,

You specify the quantities , , and , which refer to the critical fracture energies required to cause failure in the normal, the first, and the second shear directions, respectively.

Input File Usage:	Use the following option to define the fracture energy as a function of the mode mix using the analytical power law fracture criterion:
	*DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=POWER LAW, POWER=

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: Type: Energy: toggle on Specify mixed mode behavior: Power law:

Benzeggagh-Kenane (BK) form

The Benzeggagh-Kenane fracture criterion (Benzeggagh and Kenane, 1996) is particularly useful when the critical fracture energies during separation purely along the first and the second shear directions are the same; i.e., . It is given by

where , , and is a cohesive property parameter. You specify , , and .

Input File Usage:	Use the following option to define the fracture energy as a function of the mode mix using the analytical BK fracture criterion:
	*DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=BK, POWER=

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: Type: Energy: toggle on Specify mixed mode behavior: Benzeggagh-Kenane:

Linear damage evolution

For linear softening (see Figure 36.1.10–3) Abaqus uses an evolution of the damage variable, D, that reduces to

where with as the effective traction at damage initiation. refers to the maximum value of the effective separation attained during the loading history.

Input File Usage:	Use the following option to specify linear damage evolution:
	*DAMAGE EVOLUTION, TYPE=ENERGY, SOFTENING=LINEAR

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: Type: Energy: Softening: Linear

Exponential damage evolution

For exponential softening Abaqus uses an evolution of the damage variable, D, that reduces to

In the expression above and are the effective traction and separation, respectively. is the elastic energy at damage initiation. In this case the traction might not drop immediately after damage initiation, which is different from what is seen in Figure 36.1.10–5.

Input File Usage:	Use the following option to specify exponential softening:
	*DAMAGE EVOLUTION, TYPE=ENERGY, SOFTENING=EXPONENTIAL

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Evolution tabbed page: Type: Energy: Softening: Exponential

Defining damage evolution data as a tabular function of mode mix

As discussed earlier, the data defining the evolution of damage at the cohesive interface can be tabular functions of the mode mix. The manner in which this dependence must be defined in Abaqus is outlined below for mode-mix definitions based on energy and traction, respectively. In the following discussion it is assumed that the evolution is defined in terms of energy. Similar observations can also be made for evolution definitions based on effective separation.

Mode mix based on energy

For an energy-based definition of mode mix, in the most general case of a three-dimensional state of separation with anisotropic shear behavior the fracture energy, , must be defined as a function of and . The quantity is a measure of the fraction of the total separation that is shear, while is a measure of the fraction of the total shear separation that is in the second shear direction. Figure 36.1.10–6 shows a schematic of the fracture energy versus mode-mix behavior.

Figure 36.1.10–6 Fracture energy as a function of mode mix.

The limiting cases of pure normal and pure shear separations in the first and second shear directions are denoted in Figure 36.1.10–6 by , , and , respectively. The lines labeled “Modes n-s,” “Modes n-t,” and “Modes s-t” show the transition in behavior between the pure normal and the pure shear in the first direction, pure normal and pure shear in the second direction, and pure shears in the first and second directions, respectively. In general, must be specified as a function of at various fixed values of . In the discussion that follows we refer to a data set of versus corresponding to a fixed as a “data block.” The following guidelines are useful in defining the fracture energy as a function of the mode mix:

• For a two-dimensional problem needs to be defined as a function of ( in this case) only. The data column corresponding to must be left blank. Hence, essentially only one “data block” is needed.

• For a three-dimensional problem with isotropic shear response, the shear behavior is defined by the sum and not by the individual values of and . Therefore, in this case a single “data block” (the “data block” for ) also suffices to define the fracture energy as a function of the mode mix.

• In the most general case of three-dimensional problems with anisotropic shear behavior, several “data blocks” would be needed. As discussed earlier, each “data block” would contain versus at a fixed value of . In each “data block” can vary between 0 and 1.0. The case (the first data point in any “data block”), which corresponds to a purely normal mode, can never be achieved when (i.e., the only valid point on line OB in Figure 36.1.10–6 is the point O, which corresponds to a purely normal separation). However, in the tabular definition of the fracture energy as a function of mode mix, this point simply serves to set a limit that ensures a continuous change in fracture energy as a purely normal state is approached from various combinations of normal and shear separations. Hence, the fracture energy of the first data point in each “data block” must always be set equal to the fracture energy in a purely normal separation ().

  As an example of the anisotropic shear case, consider that you want to input three “data blocks” corresponding to fixed values of 0., 0.2, and 1.0, respectively. For each of the three “data blocks,” the first data point must be for the reasons discussed above. The rest of the data points in each “data block” define the variation of the fracture energy with increasing proportions of shear separation.

Mode mix based on traction

The fracture energy needs to be specified in tabular form of versus and . Thus, needs to be specified as a function of at various fixed values of . A “data block” in this case corresponds to a set of data for versus , at a fixed value of . In each “data block” may vary from 0 (purely normal separation) to 1 (purely shear separation). An important restriction is that each data block must specify the same value of the fracture energy for . This restriction ensures that the energy required for fracture as the traction vector approaches the normal direction does not depend on the orientation of the projection of the traction vector on the shear plane (see Figure 36.1.10–2).

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 surface-based cohesive behavior can be used to overcome some of these convergence difficulties. This technique is also applicable to cohesive elements, fastener damage, and the concrete material model in Abaqus/Standard. Viscous regularization damping causes the tangent stiffness matrix that defines the contact stresses to be positive 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

Abaqus/CAE Usage:

Interaction module: contact property editor: MechanicalDamage: Stabilization tabbed page: Viscosity coefficient

Two types of post-failure behavior can be specified to define the cohesive behavior at a node on the slave surface after the maximum degradation value, , has been reached at the node.

By default, once fully degraded, normal contact behavior is enforced at the node and no further cohesive constraints are enforced. If the slave node re-enters contact, penetrations will give rise to compressive contact stresses, and frictional stresses will be applied in the shear directions according to the prescribed friction model, if any. Separations can occur without giving rise to any cohesive stresses.

In some situations it may be desirable to enforce cohesive behavior again if a slave node re-enters contact, even after maximum degradation has been reached. For cohesive behavior allowing repeated contacts, the overall damage variable will be re-initialized to zero when a failed slave node re-enters contact. Subsequently, normal separations may give rise to tensile cohesive stresses, and shear separations may give rise to tangential cohesive stresses in accordance with the type of cohesive behavior defined. Further loading can again cause the cohesive stresses to undergo progressive damage, degrade, and fail.

*COHESIVE BEHAVIOR, REPEATED CONTACTS

In Abaqus/Explicit, the surface-based cohesive behavior framework can be used to model brittle crack propagation problems based on linear elastic fracture mechanics principles. The Virtual Crack Closure Technique (VCCT) fracture criterion can be used to model crack propagation in initially partially bonded surfaces. A detailed discussion of this topic can be found in “Crack propagation analysis,” Section 11.4.3.

The VCCT fracture criterion cannot be combined with a damage-based surface behavior of the traction-separation response. However, you can use a surface-based VCCT fracture criterion in conjunction with cohesive elements. VCCT could model brittle failure/crack propagation while the cohesive elements could model other aspects of the bonded interface such as stitches.

*COHESIVE BEHAVIOR
*FRACTURE CRITERION, TYPE= VCCT

As described above, the formulation used for surface-based cohesive behavior is very similar to that for cohesive elements with traction-separation response. However, certain differences exist.

Interface thickness effects are never considered for cohesive surfaces; in cohesive elements with traction-separation response, thickness effects can be incorporated by either specifying a nonzero thickness for the interface or by requiring the initial constitutive thickness to be determined from the nodal coordinates of the cohesive elements. Since thickness effects are not considered for cohesive surfaces, material properties used to describe the constitutive response for traction-separation cohesive elements with thickness effects may not be directly reusable for cohesive surfaces.

For cohesive surfaces the cohesive constraint is enforced at each slave node; in cohesive elements the cohesive constraints are calculated at the material points (for the locations of material points in cohesive elements, see “Two-dimensional cohesive element library,” Section 32.5.8, and “Three-dimensional cohesive element library,” Section 32.5.9). Hence for cohesive surfaces, refining the slave surface as compared to the master surface will likely lead to improved constraint satisfaction and more accurate results.