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![](images/page-891_8363b817594f40c8cbe33229fd742fe2d4802bdbf2842783d1af64b166b44eb5.jpg)

<details>
<summary>text_image</summary>

t_orig
g_init
d=t_curr - t_orig + g_init
t_curr - t_orig
t_curr
</details>

Figure 32.5.8–2 Cohesive elements gap opening.

Input File Usage: Use the following option to define the initial gap opening directly: \*SECTION CONTROLS, INITIAL GAP OPENING

Abaqus/CAE Usage: Initial gap opening is not supported in Abaqus/CAE.

# Darcy flow

Darcy flow defines a simple relationship between the volumetric flow rate of a fluid and the fluid pressure gradient in a porous material. The relationship is defined by the expression

$$
\mathbf {q} d = - \frac {k d}{\gamma_ {w}} \nabla p,
$$

where is the permeability, $\nabla p$ is the pressure gradient along the cohesive element, is the gap opening, and $\gamma _ { w }$ is the fluid specific weight.

Input File Usage: Use the following option to define fully saturated isotropic permeability: $* \mathrm { P E R M E A B I L I T Y } , \mathrm { T Y P E } \mathrm { = } \mathrm { I S O T R O P I C } , \mathrm { S P E C I F I C } = \gamma _ { w }$

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Type: Isotropic, Specific weight of wetting liquid: $\gamma _ { w }$

# Poiseuille flow

In Abaqus/Standard Poiseuille flow within cohesive elements refers to the steady viscous flow between two parallel plates. For this flow, you can specify either a Newtonian fluid or a power law fluid.

# Newtonian fluid

In the case of a Newtonian fluid the volume flow rate density vector is given by the expression

$$
\mathbf {q} d = - k _ {t} \nabla p,
$$

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where $k _ { t }$ is the tangential permeability (the resistance to the fluid flow), $\nabla p$ is the pressure gradient along the cohesive element, and is the gap opening.

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 gap opening. You can also specify an upper limit on the value of $k _ { t }$ .

Input File Usage: Use the following option to define the tangential flow in a Newtonian fluid:

\*GAP FLOW, TYPE=NEWTONIAN, KMAX

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Gap Flow: Type: Newtonian: Toggle on Maximum Permeability and enter the value of $k _ { \mathrm { m a x } }$

Power law fluid

In the case of a power law fluid the constitutive relation is defined as

$$
\tau = K \dot {\gamma} ^ {\alpha},
$$

where $\tau$ 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 gap opening.

Input File Usage: \*GAP FLOW, TYPE=POWER LAW

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Gap

Flow: Type: Power law

# Normal flow across gap surfaces

You can permit normal flow by defining fluid leak-off coefficients for the pore fluid material. These coefficients define a pressure-flow relationship between the cohesive element’s middle nodes and its adjacent surface nodes. The fluid leak-off coefficients can be interpreted as the permeability of a finite layer of material on the cohesive element surfaces, as shown in Figure 32.5.8–3. The normal flow is defined as

$$
q _ {t} = c _ {t} (p _ {i} - p _ {t})
$$

and

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![](images/page-893_ed60d3ff38f0a122b559da3026a7ab0556aecb376ceaf1fe4e7f946e9b7a1cd8.jpg)

<details>
<summary>text_image</summary>

P_t
P_i
P_b
permeable layer
</details>

Figure 32.5.8–3 Leak-off coefficient interpretation as a permeable layer.

$$
q _ {b} = c _ {b} (p _ {i} - p _ {b}),
$$

where $q _ { t }$ and $q _ { b }$ are the flow rates into the top and bottom surfaces, respectively; $c _ { t }$ and $c _ { b }$ are the fluid leak-off coefficients at the top and bottom element surfaces, respectively; $p _ { i }$ is the midface pressure; and $p _ { t }$ and $p _ { b }$ are the pore pressures on the top and bottom surfaces, respectively.

Input File Usage: \*FLUID LEAKOFF

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Fluid Leakoff: Type: Coefficients

# Defining leak-off coefficients as a function of temperature and field variables

You can optionally define leak-off coefficients as functions of temperature and field variables.

Input File Usage: \*FLUID LEAKOFF, DEPENDENCIES

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Fluid Leakoff: Type: Coefficients: Toggle on Use temperature-dependent data and select the number of field variables.

# Defining leak-off coefficients in a user subroutine

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

Input File Usage: \*FLUID LEAKOFF, USER

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Fluid Leakoff: Type: User

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In the presence of a distributed gravity load the tangential flow rate density vector is given by the expression

$$
\mathbf {q} d = - k _ {t} (- \rho \mathbf {g} _ {t}),
$$

where $k _ { t }$ is the tangential permeability as defined above, is the projection of the gravity vector onto the midsurface of the cohesive element, and $\rho$ is the pore fluid density. For Darcy flow,

$$
k _ {t} = \frac {k d}{\gamma_ {w}}.
$$

For Poiseuille flow, in the case of a Newtonian fluid,

$$
k _ {t} = \frac {d ^ {3}}{1 2 \mu}.
$$

In the case of a power law fluid,

$$
k _ {t} = \left(\frac {2 \alpha}{1 + 2 \alpha}\right) \left(\frac {1}{K}\right) ^ {\frac {1}{\alpha}} \left(\frac {d}{2}\right) ^ {\frac {1 + 2 \alpha}{\alpha}} \left\| \nabla p \right\| ^ {\frac {1 - \alpha}{\alpha}}.
$$

Input File Usage: Use the following option to specify the density of the pore fluid: \*DENSITY, PORE FLUID

Abaqus/CAE Usage: Property module: material editor: General→Density

# Transition from Darcy flow to Poiseuille flow

For a Newtonian fluid the transition from Darcy flow to Poiseuille flow as a function of the damage variable, , is described by the expression

$$
\mathbf {q} d = - \left(\left(1 - D \hat {F} (\hat {d})\right) \frac {k}{\gamma_ {w}} g _ {i n i t} + D \hat {F} (\hat {d}) \left(\frac {(\hat {d}) ^ {3}}{1 2 \mu}\right)\right) (\nabla p - \rho \mathbf {g} _ {t}),
$$

$$
\hat {F} (\hat {d}) = \left\{ \begin{array}{l l} 0 & \hat {d} <   0 \\ \frac {\hat {d}}{g _ {i n i t}} & 0 \leq \hat {d} \leq g _ {i n i t} \\ 1 & g _ {i n i t} <   \hat {d} \end{array} \right..
$$

The above relationship also supports a transition from Poiseuille flow back to Darcy flow as the physical gap, $\hat { d } ,$ in a damaged element closes. The flow transition equation for a power law fluid is obtained similarly.

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# Initially open elements

You can define an initial gap to identify elements that are fully damaged; that is, at the integration points of the elements.

Input File Usage: Use the following option to define the initial gap directly:

\*INITIAL CONDITIONS, TYPE=INITIAL GAP

element number or element set, omit values for D

Abaqus/CAE Usage: Initial gap definition is not supported in Abaqus/CAE.

# Assigning initial damage values

You can define an initial gap to identify elements and assign directly to the integration points. If you assign an initial damage variable to any of the integration points but not all of them, a value of is assigned to the integration points to which you did not assign a value.

If an element set is used, you must ensure that all elements within the set have the proper uniform order of integration points.

Input File Usage: Use the following option to assign initial damage values:

\*INITIAL CONDITIONS, TYPE=INITIAL GAP

element number or element set, D at each integration point

Abaqus/CAE Usage: Initial gap definition is not supported in Abaqus/CAE.

# Additional considerations

Your use of cohesive element fluid properties and your property values can impact your solution in some cases.

# Unsymmetric matrix storage and solution

The pore pressure cohesive element matrices are unsymmetric; therefore, unsymmetric matrix storage and solution may be needed to improve convergence (see “Matrix storage and solution scheme in Abaqus/Standard” in “Defining an analysis,” Section 6.1.2).

# Large coefficient values

You must make sure that the tangential permeability or fluid leak-off coefficients are not excessively large. If either coefficient is many orders of magnitude higher than the permeability in the adjacent continuum elements, matrix conditioning problems may occur, leading to solver singularities and unreliable results.

# Meshing requirement at intersections of cohesive elements

When different layers of cohesive pore pressure elements intersect, a common midsurface node must be shared by all elements to support fluid flow continuity. Figure 32.5.8–4 shows a two-dimensional mesh

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example of intersecting elements. Elements 10, 20, 30, and 40 share the same middle node, 100, at the intersecting point.

![](images/page-896_d3c783eb8236640cda1a0027e1f64e2f56f0ba639cf7236cc425a59e7247f1af.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["100"] --> B["20"]
    A --> C["30"]
    A --> D["40"]
    A --> E["10"]
    A --> F["20"]
    A --> G["30"]
    A --> H["40"]
    A --> I["10"]
    A --> J["20"]
    A --> K["30"]
```
</details>

Figure 32.5.8–4 Meshing example for two-dimensional intersecting cohesive elements.

# Output

The following output variables are available when flow is enabled in pore pressure cohesive elements:

GFVR Gap fluid volume rate.

PFOPEN Fracture opening.

LEAKVRT Leak-off flow rate at element top.

ALEAKVRT Accumulated leak-off flow volume at element top.

LEAKVRB Leak-off flow rate at element bottom.

ALEAKVRB Accumulated leak-off flow volume at element bottom.

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# 32.5.9 TWO-DIMENSIONAL COHESIVE ELEMENT LIBRARY

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

# References

• “Cohesive elements: overview,” Section 32.5.1   
• “Choosing a cohesive element,” Section 32.5.2   
• \*COHESIVE SECTION   
• Chapter 21, “Adhesive joints and bonded interfaces,” of the Abaqus/CAE User’s Guide

# Overview

This section provides a reference to the two-dimensional cohesive elements available in Abaqus/Standard and Abaqus/Explicit.

# Element types

# General element

COH2D4 4-node two-dimensional cohesive element

Active degrees of freedom

1, 2

Additional solution variables

None.

# Pore pressure element

COH2D4P(S) 6-node displacement and pore pressure two-dimensional cohesive element COD2D4P(S) 6-node displacement and pore pressure two-dimensional cohesive element with the transition from Darcy flow to Poiseuille flow

Active degrees of freedom

1, 2, 8 at nodes on the top and bottom faces

8 at nodes on the middle face

Additional solution variables

None.

# Nodal coordinates required

X,Y

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# Element property definition

You can define the element’s initial constitutive thickness and the out-of-plane width. The default initial constitutive thickness of cohesive elements depends on the response of these elements. For continuum response, the default initial constitutive thickness is computed based on the nodal coordinates. For traction-separation response, the default initial constitutive thickness is assumed to be 1.0. For response based on a uniaxial stress state, there is no default; you must indicate your choice of the method for computing the initial constitutive thickness. See “Specifying the constitutive thickness” in “Defining the cohesive element’s initial geometry,” Section 32.5.4, for details.

Abaqus calculates the thickness direction automatically based on the midsurface of the element.

Input File Usage: \*COHESIVE SECTION

Abaqus/CAE Usage: Property module: Create Section: select Other as the section Category and Cohesive as the section Type

# Element-based loading

# Distributed loads

Distributed loads are specified as described in “Distributed loads,” Section 34.4.3.

<table><tr><td>Load ID (*DLOAD)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>BX</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force in global X-direction.</td></tr><tr><td>BY</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force in global Y-direction.</td></tr><tr><td>BXNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force in global X-direction with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit.</td></tr><tr><td>BYNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force in global Y-direction with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit.</td></tr><tr><td> $CENT^{(S)}$ </td><td>Not supported</td><td> $FL^{-4}(ML^{-3}T^{-2})$ </td><td>Centrifugal load (magnitude is input as  $\rho\omega^{2}$ , where  $\rho$  is the mass density per unit volume,  $\omega$  is the angular velocity).</td></tr></table>

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<table><tr><td>Load ID (*DLOAD)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $\text{CENTRIF}^{(S)}$ </td><td>Rotational body force</td><td> $T^{-2}$ </td><td>Centrifugal load (magnitude is input as  $\omega^2$ , where  $\omega$  is the angular velocity).</td></tr><tr><td> $\text{CORIO}^{(S)}$ </td><td>Coriolis force</td><td> $FL^{-4}T$  $(ML^{-3}T^{-1})$ </td><td>Coriolis force (magnitude is input as  $\rho\omega$ , where  $\rho$  is the mass density per unit volume,  $\omega$  is the angular velocity).</td></tr><tr><td>GRAV</td><td>Gravity</td><td> $LT^{-2}$ </td><td>Gravity loading in a specified direction (magnitude is input as acceleration).</td></tr><tr><td> $Pn$ </td><td>Pressure</td><td> $FL^{-2}$ </td><td>Pressure on face  $n$ .</td></tr><tr><td> $PnNU$ </td><td>Not supported</td><td> $FL^{-2}$ </td><td>Nonuniform pressure on face  $n$  with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit.</td></tr><tr><td> $\text{ROTA}^{(S)}$ </td><td>Rotational body force</td><td> $T^{-2}$ </td><td>Rotary acceleration load (magnitude is input as  $\alpha$ , where  $\alpha$  is the rotary acceleration).</td></tr><tr><td> $\text{SBF}^{(E)}$ </td><td>Not supported</td><td> $FL^{-5}T^2$ </td><td>Stagnation body force in global  $X$ - and  $Y$ -directions.</td></tr><tr><td> $\text{SPn}^{(E)}$ </td><td>Not supported</td><td> $FL^{-4}T^2$ </td><td>Stagnation pressure on face  $n$ .</td></tr><tr><td> $\text{VBF}^{(E)}$ </td><td>Not supported</td><td> $FL^{-4}T$ </td><td>Viscous body force in global  $X$ - and  $Y$ -directions.</td></tr><tr><td> $\text{VPn}^{(E)}$ </td><td>Not supported</td><td> $FL^{-3}T$ </td><td>Viscous pressure on face  $n$ , applying a pressure proportional to the velocity normal to the face and opposing the motion.</td></tr></table>

# Surface-based loading

# Distributed loads

Surface-based distributed loads are specified as described in “Distributed loads,” Section 34.4.3.

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<table><tr><td>Load ID(*DSLOAD)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>P</td><td>Pressure</td><td> $FL^{-2}$ </td><td>Pressure on the element surface.</td></tr><tr><td>PNU</td><td>Pressure</td><td> $FL^{-2}$ </td><td>Nonuniform pressure on the element surface with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit.</td></tr><tr><td> $SP^{(E)}$ </td><td>Pressure</td><td> $FL^{-4}T^{2}$ </td><td>Stagnation pressure on the element surface.</td></tr><tr><td> $VP^{(E)}$ </td><td>Pressure</td><td> $FL^{-3}T$ </td><td>Viscous pressure applied on the element surface. The viscous pressure is proportional to the velocity normal to the element face and opposing the motion.</td></tr></table>

# Element output

Stress, strain, and other tensor components available for output depend on whether the cohesive elements are used to model adhesive joints, gaskets, or delamination problems. You indicate the intended usage of the cohesive elements by choosing an appropriate response type when defining the section properties of these elements. The available response types are discussed in “Defining the constitutive response of cohesive elements using a continuum approach,” Section 32.5.5, and “Defining the constitutive response of cohesive elements using a traction-separation description,” Section 32.5.6.

# Cohesive elements using a continuum response

Stress and other tensors (including strain tensors) are available for elements with continuum response. Both the stress tensor and the strain tensor contain true values. For the constitutive calculations using a continuum response, only the direct through-thickness and the transverse shear strains are assumed to be nonzero. All the other strain components (i.e., the membrane strains) are assumed to be zero (see “Modeling of an adhesive layer of finite thickness” in “Defining the constitutive response of cohesive elements using a continuum approach,” Section 32.5.5, for details). All tensors have the same number of components. For example, the stress components are as follows:

<table><tr><td>S11</td><td>Direct membrane stress.</td></tr><tr><td>S22</td><td>Direct through-thickness stress.</td></tr><tr><td>S33</td><td>Direct membrane stress.</td></tr><tr><td>S12</td><td>Transverse shear stress.</td></tr></table>
