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![](images/page-871_bb910c795760aae78b746ae73ea9f1d022c9124c8e1d515296756042b56f20b1.jpg)

<details>
<summary>text_image</summary>

traction
A
G^c
O
δ_m^o
δ_m^f B
separation
</details>

Figure 32.5.6–3 Linear damage evolution.

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 deformation, 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 32.5.6–3. 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 32.5.6–3.

# Evolution based on effective displacement

You specify the quantity $\delta _ { m } ^ { f } - \delta _ { m } ^ { o }$ (i.e., the effective displacement at complete failure, $\delta _ { m } ^ { f }$ , relative to the effective displacement at damage initiation, $\delta _ { m } ^ { o }$ , as shown in Figure 32.5.6–3) 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 displacement 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 displacement after the initiation of damage, $\delta _ { m } - \delta _ { m } ^ { o }$ ; mode mix; temperature; and/or field variables.

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![](images/page-872_10b2a248bbec073a1432390b95a0bc1e02069e2ef2451b882a3a6ade3494333b.jpg)

<details>
<summary>text_image</summary>

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°
δf
δf
δshear
Gc = Gn° + (Gs° - Gn°) (Gs/GT)η (BK fracture criterion)
</details>

Figure 32.5.6–4 Illustration of mixed-mode response in cohesive elements.

# Linear damage evolution

For linear softening (see Figure 32.5.6–3) 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 expression proposed by Camanho and Davila (2002), namely:

$$
D = \frac {\delta_ {m} ^ {f} (\delta_ {m} ^ {\max} - \delta_ {m} ^ {o})}{\delta_ {m} ^ {\max} (\delta_ {m} ^ {f} - \delta_ {m} ^ {o})}.
$$

In the preceding expression and in all later references, $\delta _ { m } ^ { \mathrm { m a x } }$ refers to the maximum value of the effective displacement attained during the loading history. The assumption of a constant mode mix at a material 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:

$\mathrm { * D A M A G E ~ E V O L U T I O N } , \mathrm { T Y P E = D I S P L A C E M E N T } ,$ $\mathrm { S O F T E N I N G { = } L I N E A R }$

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Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: Type: Displacement: Softening: Linear

Exponential damage evolution

For exponential softening (see Figure 32.5.6–5) 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 material parameter that defines the rate of damage evolution and is the exponential function.

![](images/page-873_3114701fa3d147cb1ac1dddf2fdd7d32eb1f06c12cfed87ac3f1423966f9ae4d.jpg)

<details>
<summary>area</summary>
| separation | traction |
| ---------- | -------- |
| δ_m^o      | peak     |
| δ_m^f      | 0        |
</details>

Figure 32.5.6–5 Exponential damage evolution.

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

Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: 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 displacement relative to the effective displacement 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: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: 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 32.5.6–3). You specify the fracture energy as a material property 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: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: Type: Energy: Mixed mode behavior: Tabular

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# 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,

$$
G ^ {C} = 1 \bigg / \Big (\left\{\frac {m _ {1}}{G _ {n} ^ {C}} \right\} ^ {\alpha} + \left\{\frac {m _ {2}}{G _ {s} ^ {C}} \right\} ^ {\alpha} + \left\{\frac {m _ {3}}{G _ {t} ^ {C}} \right\} ^ {\alpha} \Big) ^ {1 / \alpha}.
$$

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, $\mathrm { M I X E D ~ M O D E ~ B E H A V I O R { = } P O W E R ~ L A W , } \mathrm { P O W E R } { = } \alpha$

Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: Type: Energy: Mixed mode behavior: Power Law: Toggle on Power and enter the exponent value

# Benzeggagh-Kenane (BK) form

The Benzeggagh-Kenane fracture criterion (Benzeggagh and Kenane, 1996) is particularly useful when the critical fracture energies during deformation purely along the first and the second shear directions are the same; i.e., $G _ { s } ^ { C } = \bar { G } _ { t } ^ { C }$ . 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 material 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:

$* { \mathrm { D A M A G E ~ E V O L U T I O N } } , { \mathrm { T Y P E } } { \mathrm { = E N E R G Y } } ,$ $\mathrm { M I X E D ~ M O D E ~ B E H A V I O R { = } B K , ~ P O W E R { = } } \eta$

Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or

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Maxs Damage: Suboptions→Damage Evolution: Type: Energy: Mixed mode behavior: Bk: Toggle on Power and enter the exponent value

Linear damage evolution

For linear softening (see Figure 32.5.6–3) 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 displacement 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: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: Type: Energy: Softening: Linear

Exponential damage evolution

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

$$
D = \int_ {\delta_ {m} ^ {o}} ^ {\delta_ {m} ^ {f}} \frac {T _ {\mathrm{eff}} d \delta}{G ^ {C} - G _ {o}}.
$$

In the expression above $T _ { \mathrm { e f f } }$ and are the effective traction and displacement, respectively. $G _ { o }$ 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 32.5.6–5.

Input File Usage: Use the following option to specify exponential softening:

\*DAMAGE EVOLUTION, TYPE=ENERGY,SOFTENING=EXPONENTIAL

Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: Type: Energy: Softening: Exponential

# Defining damage evolution data as a tabular function of mode mix

As discussed earlier, the material data defining the evolution of damage can be tabular functions of the mode mix. The manner in which this dependence must be defined in Abaqus is outlined below for modemix definitions based on energy and traction, respectively. In the following discussion it is assumed

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that the evolution is defined in terms of energy. Similar observations can also be made for evolution definitions based on effective displacement.

# Mode mix based on energy

For an energy-based definition of mode mix, in the most general case of a three-dimensional state of deformation with anisotropic shear behavior the fracture energy, $G ^ { C }$ , must be defined as a function of $( m _ { 2 } + m _ { 3 } )$ and $\left[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) \right]$ . The quantity $( m _ { 2 } + m _ { 3 } ) = G _ { S } / G _ { T }$ is a measure of the fraction of the total deformation that is shear, while $[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ] = G _ { t } / G _ { S }$ is a measure of the fraction of the total shear deformation that is in the second shear direction. Figure 32.5.6–6 shows a schematic of the fracture energy versus mode mix behavior.

![](images/page-877_283bea8126011e05b3df2cd75d5afe4adc6baf609275f1ebb933eaf60e7e8799.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["G_s"] -->|Modes n-s| B["G_n"]
    A -->|Modes s-t| C["G_t"]
    B -->|Modes n-t| D["G_n"]
    C -->|Modes n-s| E["G_s"]
    C -->|Modes s-t| F["G_t"]
    D -->|Modes n-t| G["G_n"]
    E -->|Modes n-t| H["G_s"]
    F -->|Modes n-t| I["G_t"]
    G -->|Modes n-t| J["G_n"]
    H -->|Modes n-t| K["G_n"]
    I -->|Modes n-t| L["G_n"]
    J -->|Modes n-t| M["G_n"]
    K -->|Modes n-t| N["G_n"]
    L -->|Modes n-t| O["G_n"]
    M -->|Modes n-t| P["G_n"]
    N -->|Modes n-t| Q["G_n"]
    O -->|Modes n-t| R["G_n"]
    P -->|Modes n-t| S["G_n"]
    Q -->|Modes n-t| T["G_n"]
    R -->|Modes n-t| U["G_n"]
    S -->|Modes n-t| V["G_n"]
    T -->|Modes n-t| W["G_n"]
    U -->|Modes n-t| X["G_n"]
    V -->|Modes n-t| Y["G_n"]
    W -->|Modes n-t| Z["G_n"]
    X -->|Modes n-t| AA["G_n"]
    Y -->|Modes n-t| AB["G_n"]
    Z -->|Modes n-t| AC["G_n"]
    AA -->|Modes n-t| AD["G_n"]
    AB -->|Modes n-t| AE["G_n"]
    AC -->|Modes n-t| AF["G_n"]
    AD -->|Modes n-t| AG["G_n"]
    AE -->|Modes n-t| AH["G_n"]
    AF -->|Modes n-t| AI["G_n"]
    AG -->|Modes n-t| AJ["G_n"]
    AH -->|Modes n-t| AK["G_n"]
    AI -->|Modes n-t| AL["G_n"]
    AJ -->|Modes n-t| AM["G_n"]
    AK -->|Modes n-t| AN["G_n"]
    AL -->|Modes n-t| AO["G_n"]
    AM -->|Modes n-t| AP["G_n"]
    AN -->|Modes n-t| AQ["G_n"]
    AO -->|Modes n-t| AR["G_n"]
    AP -->|Modes n-t| AS["G_n"]
    AQ -->|Modes n-t| AT["G_n"]
    AR -->|Modes n-t| AU["G_n"]
    AS -->|Modes n-t| AV["G_n"]
    AT -->|Modes n-t| AW["G_n"]
    AU -->|Modes n-t| AX["G_n"]
    AV -->|Modes n-t| AY["G_n"]
    AW -->|Modes n-t| AZ["G_n"]
    AX -->|Modes n-t| BA["G_n"]
    AY -->|Modes n-t| BB["G_n"]
    AZ -->|Modes n-t| BC["G_n"]
    BA -->|Modes n-t| BD["G_n"]
    BB -->|Modes n-t| BE["G_n"]
    BC -->|Modes n-t| BF["G_n"]
    BD -->|Modes n-t| BG["G_n"]
    BE -->|Modes n-t| BH["G_n"]
    BF -->|Modes n-t| BI["G_n"]
    BG -->|Modes n-t| BJ["G_n"]
    BH -->|Modes n-t| BK["G_n"]
    BI -->|Modes n-t| BL["G_n"]
    BJ -->|Modes n-t| BM["G_n"]
    BK -->|Modes n-t| BN["G_n"]
    BL -->|Modes n-t| BO["G_n"]
    BM -->|Modes n-t| BP["G_n"]
    BN -->|Modes n-t| BQ["G_n"]
    BO -->|Modes n-t| BR["G_n"]
    BP -->|Modes n-t| BS["G_n"]
    BQ -->|Modes n-t| BT["G_n"]
    BN -->|Modes n-t| BU["G_n"]
    BS -->|Modes n-t| BV["G_n"]
    BT -->|Modes n-t| BW["G_n"]
    BU -->|Modes n-t| BX["G_n"]
    BV -->|Modes n-t| BY["G_n"]
    BW -->|Modes n-t| BZ["G_n"]
    BX -->|Modes n-t| CA["G_n"]
    BY -->|Modes n-t| CB["G_n"]
    CA -->|Modes n-t| CC["G_n"]
    CB -->|Modes n-t| CD["G_n"]
    CC -->|Modes n-t| DE["G_n"]
    AD -->|Modes n-t| DF["G_n"]
    AE -->|Modes n-t| DG["G_n"]
    AF -->|Modes n-t| DH["G_n"]
    AG -->|Modes n-t| DI["G_n"]
    AH -->|Modes n-t| DJ["G_n"]
    AI -->|Modes n-t| DK["G_n"]
    AJ -->|Modes n-t| DL["G_n"]
    AK -->|Modes n-t| DN["G_n"]
    AL -->|Modes n-t| DO["G_n"]
    AM -->|Modes n-t| DP["G_n"]
    AN -->|Modes n-t| DP
    AO -->|Modes n-t| DP
    AP -->|Modes n-t| AQ["G_n"]
    AQ -->|Modes n-t| AR["G_n"]
    AR -->|Modes n-t| AS["G_n"]
    AS -->|Modes n-t| AT["G_n"]
    AT -->|Modes n-t| AU["G_n"]
    AU -->|Modes n-t| AV["G_n"]
    AV -->|Modes n-t| AW["G_n"]
    AW -->|Modes n-t| AX["G_n"]
    AX -->|Modes n-t| AY["G_n"]
    AY -->|Modes n-t| AZ["G_n"]
    AZ -->|Modes n-t| BA["G_n"]
    BA -->|Modes n-t| BB["G_n"]
    BB -->|Modes n-t| BC["G_n"]
    BC -->|Modes n-t| BD["G_n"]
    BD -->|Modes n-t| BE["G_n"]
    BE -->|Modes n-t| BF["G_n"]
    BF -->|Modes n-t| BG["G_n"]
    BG -->|Modes n-t| BH["G_n"]
    BH -->|Modes n-t| BI["G_n"]
    BI -->|Modes n-t| BJ["G_n"]
    BJ -->|Modes n-t| BK["G_n"]
    BK -->|Modes n-t| BL["G_n"]
    BL -->|Modes n-t| BN["G_n"]
    BN -->|Modes n-t| BO["G_n"]
    BO -->|Modes n-t| BP["G_n"]
    BP -->|Modes n-t| BQ["G_n"]
    BQ -->|Modes n-t| BR["G_n"]
    BR -->|Modes n-t| BS["G_n"]
    BS -->|Modes n-t| AT
    AT -->|Modes n-t| AU
    AU -->|Modes n-t| AV
    AV -->|Modes n-t| BQ
    BQ -->|Modes n-t| BN
    BN -->|Modes n-t| BO
    BO -->|Modes n-t| AY
    AY -->|Modes n-t| AZ
    AZ -->|Modes n-t| BA
    BA -->|Modes n-t| BB
    BB -->|Modes n-t| BC
    BC -->|Modes n-t| BN
    BN -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modes n-t| AY
    AY -->|Modesn| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
    BQ -->|Modes n-t| BQ
```
</details>

Figure 32.5.6–6 Fracture energy as a function of mode mix.

The limiting cases of pure normal and pure shear deformations in the first and second shear directions are denoted in Figure 32.5.6–6 by $G _ { n } ^ { C } , \bar { G } _ { s } ^ { \bar { C } }$ , and $G _ { t } ^ { C }$ , respectively. The lines labeled “Modes n-s,” “Modes $\mathrm { n - t } , \mathrm { ? }$ and “Modes $\mathrm { s } { - } \mathrm { t } ^ { \gamma }$ 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, $G ^ { C }$ must be specified as a function of $( m _ { 2 } + m _ { 3 } )$ at various fixed values of $[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ]$ . In the discussion that follows we refer to a data set of $G ^ { C }$ versus $( m _ { 2 } + m _ { 3 } )$ corresponding to a fixed $[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ]$ as a “data block.” The following guidelines are useful in defining the fracture energy as a function of the mode mix:

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• For a two-dimensional problem $G ^ { C }$ needs to be defined as a function of $m _ { 2 } \ ( m _ { 3 } = 0$ in this case) only. The data column corresponding to $[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ]$ 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 $( m _ { 2 } + m _ { 3 } )$ and not by the individual values of $m _ { 2 }$ and $m _ { 3 }$ . Therefore, in this case a single “data block” (the “data block” for $[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ] = 0 )$ 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 $G ^ { C }$ versus $( m _ { 2 } + m _ { 3 } )$ at a fixed value of $\left[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) \right]$ . In each “data block” $\ ' ( m _ { 2 } + m _ { 3 } )$ can vary between 0 and . The case $( m _ { 2 } + m _ { 3 } ) = 0$ (the first data point in any “data block”), which corresponds to a purely normal mode, can never be achieved when $[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ] \ne 0$ (i.e., the only valid point on line OB in Figure 32.5.6–6 is the point O, which corresponds to a purely normal deformation). 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 deformations. 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 mode of deformation $( G _ { n } ^ { C } )$ .

As an example of the anisotropic shear case, consider that you want to input three “data blocks” corresponding to fixed values of $[ m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ] = 0 . , 0 . 2$ , and 1.0, respectively. For each of the three “data blocks,” the first data point must be $( G _ { n } ^ { C } , 0 )$ 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 deformation.

# Mode mix based on traction

The fracture energy needs to be specified in tabular form of $G ^ { C }$ versus $\phi _ { 1 }$ and $\phi _ { 2 }$ . Thus, $G ^ { C }$ needs to be specified as a function of $\phi _ { 1 }$ at various fixed values of $\phi _ { 2 }$ . A “data block” in this case corresponds to a set of data for $G ^ { C }$ versus $\phi _ { 1 }$ , at a fixed value of $\phi _ { 2 }$ . In each “data block” $\phi _ { 1 }$ may vary from 0 (purely normal deformation) to 1 (purely shear deformation). An important restriction is that each data block must specify the same value of the fracture energy for $\phi _ { 1 } = 0$ . 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 32.5.6–2).

# Evaluating damage when multiple criteria are active

When multiple damage initiation criteria and associated evolution definitions are used for the same material, each evolution definition results in its own damage variable, $d _ { i }$ , where the subscript i represents the ith damage system. The overall damage variable, D, is computed based on the individual $d _ { i }$ as explained in “Evaluating overall damage when multiple criteria are active” in “Damage evolution and element removal for ductile metals,” Section 24.2.3, for damage in bulk materials.

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# Maximum degradation and choice of element removal

You have control over how Abaqus treats cohesive elements with severe damage. By default, the upper bound to the overall damage variable at a material point is $D _ { m a x } = 1 . 0$ . You can reduce this upper bound as discussed in “Controlling element deletion and maximum degradation for materials with damage evolution” in “Section controls,” Section 27.1.4. You can control what happens to the cohesive element when the damage reaches this limit, as discussed below.

By default, once the overall damage variable reaches $D _ { m a x }$ at all of its material points and none of its material points are in compression, the cohesive elements, except for the pore pressure cohesive elements, are removed (deleted). See “Controlling element deletion and maximum degradation for materials with damage evolution” in “Section controls,” Section 27.1.4, for details. This element removal approach is often appropriate for modeling complete fracture of the bond and separation of components. Once removed, cohesive elements offer no resistance to subsequent penetration of the components, so it may be necessary to model contact between the components as discussed in “Defining contact between surrounding components” in “Modeling with cohesive elements,” Section 32.5.3.

Alternatively, you can specify that a cohesive element should remain in the model even after the overall damage variable reaches $D _ { m a x }$ . In this case the stiffness of the element in tension and/or shear remains constant (degraded by a factor of $1 - D _ { m a x }$ over the initial undamaged stiffness). This choice is appropriate if the cohesive elements must resist interpenetration of the surrounding components even after they have completely degraded in tension and/or shear (see “Defining contact between surrounding components” in “Modeling with cohesive elements,” Section 32.5.3). In Abaqus/Explicit it is recommended that you suppress bulk viscosity in the cohesive elements by setting the scale factors for the linear and quadratic bulk viscosity parameters to zero using section controls (see “Section controls,” Section 27.1.4).

# Uncoupled transverse shear response

An optional linear elastic transverse shear behavior can be defined to provide additional stability to cohesive elements, particularly after damage has occurred. The transverse shear behavior is assumed to be independent of the regular material response and does not undergo any damage.

Input File Usage: Use the following options:

Abaqus/CAE Usage: Transverse shear behavior is not supported in Abaqus/CAE for cohesive sections.

# Viscous regularization in Abaqus/Standard

Material models exhibiting softening behavior and stiffness degradation often lead to severe convergence difficulties in implicit analysis programs, such as Abaqus/Standard. A common technique to overcome some of these convergence difficulties is the use of viscous regularization of the constitutive equations,

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which causes the tangent stiffness matrix of the softening material to be positive for sufficiently small time increments.

The traction-separation laws can be regularized in Abaqus/Standard using viscosity by permitting stresses to be outside the limits set by the traction-separation law. The regularization process involves the use of a viscous stiffness degradation variable, $D _ { v }$ , which is defined by the evolution equation:

$$
\dot {D} _ {v} = \frac {1}{\mu} (D - D _ {v}),
$$

where $\mu$ is the viscosity parameter representing the relaxation time of the viscous system and D is the degradation variable evaluated in the inviscid backbone model. The damaged response of the viscous material is given as

$$
\mathbf {t} = (1 - D _ {v}) \overline {{\mathbf {t}}}.
$$

Using viscous regularization with a small value of the viscosity parameter (small compared to the characteristic time increment) usually helps improve the rate of convergence of the model in the softening regime, without compromising results. The basic idea is that the solution of the viscous system relaxes to that of the inviscid case as $t / \mu \to \infty$ , where t represents time. You can specify the value of the viscosity parameter as part of the section controls definition (see “Using viscous regularization with cohesive elements, connector elements, and elements that can be used with the damage evolution models for ductile metals and fiber-reinforced composites in Abaqus/Standard” in “Section controls,” Section 27.1.4). If the viscosity parameter is different from zero, output results of the stiffness degradation refer to the viscous value, $D _ { v }$ . The default value of the viscosity parameter is zero so that no viscous regularization is performed. Use of viscous regularization for improving the convergence behavior of delamination and debonding problems is discussed in “Delamination analysis of laminated composites,” Section 2.7.1 of the Abaqus Benchmarks Guide, and “Analysis of skin-stiffener debonding under tension,” Section 1.4.5 of the Abaqus Example Problems Guide.

The approximate amount of energy associated with viscous regularization over the whole model or over an element set is available using output variable ALLCD.

# Output

In addition to the standard output identifiers available in Abaqus (“Abaqus/Standard output variable identifiers,” Section 4.2.1, and “Abaqus/Explicit output variable identifiers,” Section 4.2.2), the following variables have special meaning for cohesive elements with traction-separation behavior:

STATUS Status of element (the status of an element is 1.0 if the element is active, 0.0 if the element is not).

SDEG Overall value of the scalar damage variable, D.

DMICRT All damage initiation criteria components.

MAXSCRT Maximum value of the nominal stress damaduring the analysis. It is evaluated as $\textstyle { \left\{ \frac { \langle t _ { n } \rangle } { t _ { n } ^ { o } } , \frac { t _ { s } } { t _ { s } ^ { o } } , \frac { t _ { t } } { t _ { t } ^ { o } } \right\} }$ riterion at a material point
