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In the discussion below, t _ { n } ^ { o } , \ t _ { s } ^ { o } , , and t _ { t } ^ { o } 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, \delta _ { n } ^ { o } , \delta _ { s } ^ { o } , and \delta _ { t } ^ { o } 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.
Maximum stress criterion
Damage is assumed to initiate when the maximum contact stress ratio (as defined in the expression below) reaches a value of one. This criterion can be represented as
\max \biggl \{\frac {\langle t _ {n} \rangle}{t _ {n} ^ {o}}, \frac {t _ {s}}{t _ {s} ^ {o}}, \frac {t _ {t}}{t _ {t} ^ {o}} \biggr \} = 1.
Input File Usage: *DAMAGE INITIATION, CRITERION=MAXS
Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Damage: Initiation tabbed page: Criterion: Maximum nominal stress
Maximum separation criterion
Damage is assumed to initiate when the maximum separation ratio (as defined in the expression below) reaches a value of one. This criterion can be represented as
\max \biggl \{\frac {\langle \delta_ {n} \rangle}{\delta_ {n} ^ {o}}, \frac {\delta_ {s}}{\delta_ {s} ^ {o}}, \frac {\delta_ {t}}{\delta_ {t} ^ {o}} \biggr \} = 1.
Input File Usage: *DAMAGE INITIATION, CRITERION=MAXU
Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Damage: Initiation tabbed page: Criterion: Maximum separation
Quadratic stress criterion
Damage is assumed to initiate when a quadratic interaction function involving the contact stress ratios (as defined in the expression below) reaches a value of one. This criterion can be represented as
\left\{\frac {\langle t _ {n} \rangle}{t _ {n} ^ {o}} \right\} ^ {2} + \left\{\frac {t _ {s}}{t _ {s} ^ {o}} \right\} ^ {2} + \left\{\frac {t _ {t}}{t _ {t} ^ {o}} \right\} ^ {2} = 1.
Input File Usage: *DAMAGE INITIATION, CRITERION=QUADS
Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Damage: Initiation tabbed page: Criterion: Quadratic traction
Quadratic separation criterion
Damage is assumed to initiate when a quadratic interaction function involving the separation ratios (as defined in the expression below) reaches a value of one. This criterion can be represented as
\left\{\frac {\langle \delta_ {n} \rangle}{\delta_ {n} ^ {o}} \right\} ^ {2} + \left\{\frac {\delta_ {s}}{\delta_ {s} ^ {o}} \right\} ^ {2} + \left\{\frac {\delta_ {t}}{\delta_ {t} ^ {o}} \right\} ^ {2} = 1.
Input File Usage: *DAMAGE INITIATION, CRITERION=QUADU
Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Damage: Initiation tabbed page: Criterion: Quadratic separation
Damage evolution
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
t _ {n} = \left\{ \begin{array}{l l} (1 - D) \bar {t} _ {n}, & \bar {t} _ {n} \geq 0 \\ \bar {t} _ {n}, & \text { otherwise (no damage to compressive stiffness) }; \end{array} \right.
t _ {s} = (1 - D) \bar {t} _ {s},
t _ {t} = (1 - D) \bar {t} _ {t},
where \bar { t } _ { n } , \bar { t } _ { s } , and \bar { t } _ { t } are the contact 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, it is useful to introduce an effective separation (Camanho and Davila, 2002) defined as
\delta_ {m} = \sqrt {\langle \delta_ {n} \rangle^ {2} + \delta_ {s} ^ {2} + \delta_ {t} ^ {2}}.
While this formula was originally applied to damage evolution in cohesive elements, it can be reinterpreted in terms of contact separations for cohesive surface behavior, as discussed above (see “Applying cohesive material concepts to surface-based cohesive behavior”).
Mixed-mode definition
The relative proportions of normal and shear separations at a contact point define the mode mix at the point. Abaqus uses three measures of mode mix, two that are based on energies and one that is based on tractions. You can choose one of these measures when you specify the mode dependence of the damage evolution process. Denoting by G _ { n } , G _ { s } , and G _ { t } the work done by the tractions and their conjugate separations in the normal, first, and second shear directions, respectively, and defining G _ { T } \ = \ G _ { n } + G _ { s } + G _ { t } , the mode-mix definitions based on energies are as follows:
m _ {1} = \frac {G _ {n}}{G _ {T}},
m _ {2} = \frac {G _ {s}}{G _ {T}},
m _ {3} = \frac {G _ {t}}{G _ {T}}.
Clearly, only two of the three quantities defined above are independent. It is also useful to define the quantity G _ { S } = G _ { s } + G _ { t } 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 m _ { 2 } + m _ { 3 } ( = G _ { S } / G _ { T } ) (or, equivalently, 1 - m _ { 1 } ) and m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ( = G _ { t } / G _ { S } ) .
Abaqus computes the energy quantities described above either based on the current state of deformation (nonaccumulative measure of energy) or based on the deformation history (accumulative measure of energy) at an integration point. The former approach, available only in Abaqus/Standard, is useful in mixed-mode simulations where the primary energy dissipation mechanism is associated with the creation of new surfaces due to failure in the cohesive zone. Such problems are typically adequately described utilizing the methods of linear elastic fracture mechanics. The latter approach provides an alternate way of defining the mode-mix and may be useful in situations where other significant dissipation mechanisms also govern the overall structural response.
The corresponding definitions of the mode mix based on traction components are given by
\phi_ {1} = \left(\frac {2}{\pi}\right) \tan^ {- 1} \left(\frac {\tau}{\langle t _ {n} \rangle}\right),
\phi_ {2} = \left(\frac {2}{\pi}\right) \tan^ {- 1} \left(\frac {t _ {t}}{t _ {s}}\right),
where \tau = \sqrt { t _ { s } ^ { 2 } + t _ { t } ^ { 2 } } is a measure of the effective shear traction. The angular measures used in the above definition (before they are normalized by the factor 2 / \pi ) are illustrated in Figure 37.1.10–2.
Input File Usage: Use the following option to use the mode-mix definition based on nonaccumulated energies (available only in Abaqus/Standard):
text_image
normal tn t φ1 φ2 τ ts Shear 1 t1 Shear 2
Figure 37.1.10–2 Mode-mix measures based on traction.
*DAMAGE EVOLUTION, MODE MIX RATIO=ENERGY
Use the following option to use the mode-mix definition based on accumulated energies:
*DAMAGE EVOLUTION, MODE MIX RATIO=ACCUMULATED ENERGY
Use the following option to use the mode-mix definition based on tractions:
*DAMAGE EVOLUTION, MODE MIX RATIO=TRACTION
Abaqus/CAE Usage: Use the following option in Abaqus/Standard to use the mode-mix definition based on nonaccumulated energies:
Interaction module: contact property editor: Mechanical→Damage:
Evolution tabbed page: toggle on Specify mixed-mode behavior: Mode mix ratio: Energy
Use the following option in Abaqus/Explicit to use the mode-mix definition based on accumulated energies:
Interaction module: contact property editor: Mechanical→Damage:
Evolution tabbed page: toggle on Specify mixed-mode
behavior: Mode mix ratio: Energy
Specifying a mode-mix definition based on accumulated energies in Abaqus/Standard is not supported in Abaqus/CAE.
Use the following option to use the mode-mix definition based on tractions:
Interaction module: contact property editor: Mechanical→Damage:
Evolution tabbed page: toggle on Specify mixed-mode
behavior: Mode mix ratio: Traction
Comparison of mixed-mode definitions
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 G _ { n } \neq 0 and G _ { s } = G _ { t } = 0 , 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.
When the mode mix is defined based on accumulated energies, an artificial path-dependence may be introduced in the mixed-mode behavior that may not be consistent, for example, with predictions that are based on linear elastic fracture mechanics. Therefore, if an interface is first loaded purely in the normal deformation mode, unloaded, and subsequently loaded in a purely shear deformation mode, the mode-mix ratios based on accumulated energies at the end of the above deformation path evaluate to (assuming the shear deformation to be in the local-1 direction only) G _ { n } \neq 0 and G _ { s } \neq 0 . On the other hand, the mode-mix ratios based on nonaccumulated energies evaluate to G _ { n } = 0 and G _ { s } \neq 0 at the end of the above deformation path.
Damage evolution definition
There are two components to the definition of damage evolution. The first component involves specifying either the effective separation at complete failure, \delta _ { m } ^ { f } , relative to the effective separation at the initiation of damage, \delta _ { m } ^ { o } ; or the energy dissipated due to failure, G ^ { C } (see Figure 37.1.10–3). 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.
text_image
traction A G^c O δ_m^o δ_m^f B separation
Figure 37.1.10–3 Linear damage evolution.
Figure 37.1.10–4 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.
Unloading subsequent to damage initiation is always assumed to occur linearly toward the origin of the traction-separation plane, as shown in Figure 37.1.10–3. 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 37.1.10–3.
Evolution based on effective separation
You specify the quantity \delta _ { m } ^ { f } - \delta _ { m } ^ { o } (i.e., the effective separation at complete failure, \delta _ { m } ^ { f } , relative to the effective separation at damage initiation, \delta _ { m } ^ { o } , as shown in Figure 37.1.10–3) 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, \delta _ { m } - \delta _ { m } ^ { o } ; mode mix; temperature; and/or field variables.
text_image
traction Shear mode τ° tn° Normal mode (⟨tn⟩² / tn°)² + (ts / ts°)² + (tt°)² = 1 Stress interaction law maps damage initiation Mixed-mode critical Gc maps delamination growth δm° δn° δf'shear δm' Gc = Gn' + (Gs' - Gn'') (GS/GT)η (BK fracture criterion)
Figure 37.1.10–4 Illustration of mixed-mode response in cohesive interactions.
Linear damage evolution
For linear softening (see Figure 37.1.10–3) 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:
D = \frac {\delta_ {m} ^ {f} (\delta_ {m} ^ {\max} - \delta_ {m} ^ {o})}{\delta_ {m} ^ {\max} (\delta_ {m} ^ {f} - \delta_ {m} ^ {o})}.
In the preceding expression and in all later references, \delta _ { m } ^ { \mathrm { m a x } } 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: Mechanical→Damage:
Evolution tabbed page: Type: Displacement: Softening: Linear
Exponential damage evolution
For exponential softening (see Figure 37.1.10–5) 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
D = 1 - \left\{\frac {\delta_ {m} ^ {o}}{\delta_ {m} ^ {\mathrm{max}}} \right\} \left\{1 - \frac {1 - \exp (- \alpha (\frac {\delta_ {m} ^ {\mathrm{max}} - \delta_ {m} ^ {o}}{\delta_ {m} ^ {f} - \delta_ {m} ^ {o}}))}{1 - \exp (- \alpha)} \right\}.
In the expression above is a non-dimensional parameter that defines the rate of damage evolution and is the exponential function.
area
| separation | traction | | ---------- | -------- | | δ_m^o | peak | | δ_m^f | 0 |Figure 37.1.10–5 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: Mechanical→Damage: 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: Mechanical→Damage: 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 37.1.10–3). 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: Mechanical→Damage: 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
\left\{\frac {G _ {n}}{G _ {n} ^ {C}} \right\} ^ {\alpha} + \left\{\frac {G _ {s}}{G _ {s} ^ {C}} \right\} ^ {\alpha} + \left\{\frac {G _ {t}}{G _ {t} ^ {C}} \right\} ^ {\alpha} = 1,
The mixed-mode fracture energy G ^ { C } = G _ { T } when the above condition is satisfied. In other words,
G ^ {C} = 1 \bigg / \bigg (\left\{\frac {m _ {1}}{G _ {n} ^ {C}} \right\} ^ {\alpha} + \left\{\frac {m _ {2}}{G _ {s} ^ {C}} \right\} ^ {\alpha} + \left\{\frac {m _ {3}}{G _ {t} ^ {C}} \right\} ^ {\alpha} \bigg) ^ {1 / \alpha}.
You specify the quantities G _ { n } ^ { C } , G _ { s } ^ { C } , and G _ { t } ^ { C } , 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: Mechanical→Damage: 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., G _ { s } ^ { C } = G _ { t } ^ { \bar { C } } . It is given by
G _ {n} ^ {C} + (G _ {s} ^ {C} - G _ {n} ^ {C}) \biggl \{\frac {G _ {S}}{G _ {T}} \biggr \} ^ {\eta} = G ^ {C},
where G _ { S } = G _ { s } + G _ { t } , G _ { T } = G _ { n } + G _ { S } , and is a cohesive property parameter. You specify G _ { n } ^ { C } , G _ { s } ^ { C } , 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: Mechanical→Damage: Evolution tabbed page: Type: Energy: toggle on Specify mixed mode behavior: Benzeggagh-Kenane:
Linear damage evolution
For linear softening (see Figure 37.1.10–3) Abaqus uses an evolution of the damage variable, D, that reduces to
D = \frac {\delta_ {m} ^ {f} (\delta_ {m} ^ {\max} - \delta_ {m} ^ {o})}{\delta_ {m} ^ {\max} (\delta_ {m} ^ {f} - \delta_ {m} ^ {o})},
where \delta _ { m } ^ { f } ~ = ~ 2 G ^ { C } / T _ { \mathrm { e f f } } ^ { o } with T _ { \mathrm { e f f } } ^ { o } as the effective traction at damage initiation. \delta _ { m } ^ { \mathrm { m a x } } 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