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<table><tr><td>MAXECRT</td><td>Maximum value of the nominal strain damage initiation criterion at a material point during the analysis. It is evaluated as  $\max\{\frac{\langle\varepsilon_n\rangle}{\varepsilon_n^o},\frac{\varepsilon_s}{\varepsilon_s^o},\frac{\varepsilon_t}{\varepsilon_t^o}\}$ .</td></tr><tr><td>MMIXDME</td><td>Mode mix ratio during damage evolution. It is evaluated as  $1 - m_1$ . In general, it varies with time at a given integration point. This variable is set to  $-1.0$  before initiation of damage.</td></tr><tr><td>MMIXDMI</td><td>Mode mix ratio at damage initiation. It is evaluated as  $1 - m_1$  at the time of damage initiation at an integration point for the very first time. It remains constant with time at a given integration point. This variable is set to  $-1.0$  before initiation of damage.</td></tr><tr><td>QUADSCRT</td><td>Maximum value of the quadratic nominal stress damage initiation criterion at a material point during the analysis. It is evaluated as  $(\frac{\langle t_n\rangle}{t_n^o})^2 + (\frac{t_s}{t_s^o})^2 + (\frac{t_t}{t_t^o})^2$ .</td></tr><tr><td>QUADECRT</td><td>Maximum value of the quadratic nominal strain damage initiation criterion at a material point during the analysis. It is evaluated as  $(\frac{\langle\varepsilon_n\rangle}{\varepsilon_n^o})^2 + (\frac{\varepsilon_s}{\varepsilon_s^o})^2 + (\frac{\varepsilon_t}{\varepsilon_t^o})^2$ .</td></tr><tr><td>ALLCD</td><td>The approximate amount of energy over the whole model or over an element set that is associated with viscous regularization in Abaqus/Standard. Corresponding output variables (such as CENER, ELCD, and ECDDEN) represent the energy associated with viscous regularization at the integration point level and element level (the last quantity represents the energy per unit volume in the element), respectively.</td></tr></table>

For the variables above that indicate whether a certain damage initiation criterion has been satisfied or not, a value that is less than 1.0 indicates that the criterion has not been satisfied, while a value of 1.0 or higher indicates that the criterion has been satisfied. If damage evolution is specified for this criterion, the maximum value of this variable does not exceed 1.0. However, if damage evolution is not specified for the initiation criterion, this variable can have values higher than 1.0. The extent to which the variable is higher than 1.0 may be considered to be a measure of the extent to which this criterion has been violated.

# Additional references

• Benzeggagh, M. L., and M. Kenane, “Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus,” Composites Science and Technology, vol. 56, pp. 439–449, 1996.   
• Camanho, P. P., and C. G. Davila, “Mixed-Mode Decohesion Finite Elements for the Simulation of Delamination in Composite Materials,” NASA/TM-2002–211737, pp. 1–37, 2002.

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# 32.5.7 DEFINING THE CONSTITUTIVE RESPONSE OF FLUID WITHIN THE COHESIVE ELEMENT GAP

Products: Abaqus/Standard Abaqus/CAE

# References

• “Cohesive elements: overview,” Section 32.5.1   
• “Defining the constitutive response of cohesive elements using a traction-separation description,” Section 32.5.6   
• \*FLUID LEAKOFF   
• \*GAP FLOW   
• Chapter 21, “Adhesive joints and bonded interfaces,” of the Abaqus/CAE User’s Guide

# Overview

The cohesive element fluid flow model:

• is typically used in geotechnical applications, where fluid flow continuity within the gap and through the interface must be maintained;   
• enables fluid pressure on the cohesive element surface to contribute to its mechanical behavior, which enables the modeling of hydraulically driven fracture;   
• enables modeling of an additional resistance layer on the surface of the cohesive element; and   
• can be used only in conjunction with traction-separation behavior.

The features described in this section are used to model fluid flow within and across surfaces of pore pressure cohesive elements.

# Defining pore fluid flow properties

The fluid constitutive response comprises:

• Tangential flow within the gap, which can be modeled with either a Newtonian or power law model; and   
• Normal flow across the gap, which can reflect resistance due to caking or fouling effects.

The flow patterns of the pore fluid in the element are shown in Figure 32.5.8–1. The fluid is assumed to be incompressible, and the formulation is based on a statement of flow continuity that considers tangential and normal flow and the rate of opening of the cohesive element.

# Specifying the fluid flow properties

You can assign tangential and normal flow properties separately.

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![](images/page-884_be0a8a2002f2588c742094102f52528fcf07a7179c1e6d6c7fec51897362d2a7.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["cohesive elements"] --> B["tangential flow"]
    B --> C["normal flow"]
    C --> D["structural element"]
    style A fill:#000,stroke:#000
    style D fill:#000,stroke:#000
```
</details>

Figure 32.5.7–1 Flow within cohesive elements.

# Tangential flow

By default, there is no tangential flow of pore fluid within the cohesive element. To allow tangential flow, define a gap flow property in conjunction with the pore fluid material definition.

# Newtonian fluid

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

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

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.

In Abaqus the gap opening, , is defined as

$$
d = t _ {c u r r} - t _ {o r i g} + g _ {i n i t},
$$

where $t _ { c u r r }$ and $t _ { o r i g }$ are the current and original cohesive element geometrical thicknesses, respectively; and $g _ { i n i t }$ is the initial gap opening, which has a default value of 0.002.

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 $k _ { t }$ .

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Input File Usage: Use the following option to define the initial gap opening directly: \*SECTION CONTROLS, INITIAL GAP OPENING Use the following option to define the tangential flow in a Newtonian fluid: \*GAP FLOW, TYPE=NEWTONIAN, KMAX

Abaqus/CAE Usage: Initial gap opening is not supported in Abaqus/CAE. 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 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 a fluid leakoff coefficient for the pore fluid material. This coefficient defines a pressure-flow relationship between the cohesive element’s middle nodes and their adjacent surface nodes. The fluid leakoff coefficients can be interpreted as the permeability of a finite layer of material on the cohesive element surfaces, as shown in Figure 32.5.7–2. The normal flow is defined as

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

and

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

where $q _ { t }$ and $q _ { b }$ are the flow rates into the top and bottom surfaces, 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

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![](images/page-886_b989bfda14904650c4a2da36ae5740e614829785b2b4f1c8f049ee01a95e2955.jpg)

<details>
<summary>text_image</summary>

P_t
P_i
P_b
permeable layer
</details>

Figure 32.5.7–2 Leakoff coefficient interpretation as a permeable layer.

Defining leakoff coefficients as a function of temperature and field variables

You can optionally define leakoff 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 leakoff coefficients in a user subroutine

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

Input File Usage: \*FLUID LEAKOFF, USER

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

# Tangential and normal flow combinations

Table 32.5.7–1 shows the permitted combinations of tangential and normal flow and the effects of each combination.

# Initially open elements

When the opening of the cohesive element is driven primarily by entry of fluid into the gap, you will have to define one or more elements as initially open, since tangential flow is possible only in an open element. Identify initially open elements as initial conditions.

Input File Usage: \*INITIAL CONDITIONS, TYPE=INITIAL GAP

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

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Table 32.5.7–1 Effects of flow property definition combinations. 

<table><tr><td></td><td>Normal flow is defined</td><td>Normal flow is undefined</td></tr><tr><td>Tangential flow is defined</td><td>Tangential and normal flow are modeled.</td><td>Tangential flow is modeled. Pore pressure continuity is enforced between facing nodes in the cohesive element only when the element is closed. Otherwise, the surfaces are impermeable in the normal direction.</td></tr><tr><td>Tangential flow is undefined</td><td>Normal flow is modeled.</td><td>Tangential flow is not modeled. Pore pressure continuity is always enforced between facing nodes in the cohesive element.</td></tr></table>

# Use of 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).

# Additional considerations

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

# Large coefficient values

You must make sure that the tangential permeability or fluid leakoff 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.

# Use in total pore pressure simulations

Definition of tangential flow properties may result in inaccurate results if the total pore pressure formulation is used and the hydrostatic pressure gradient contributes significantly to the tangential flow in the gap. The total pore pressure formulation is invoked if you apply gravity distributed loads to all elements in the model. The results will be accurate if the hydrostatic pressure gradient (i.e., the gravity vector) is perpendicular to the cohesive element.

# Output

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

GFVR Gap fluid volume rate.

PFOPEN Fracture opening.

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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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# 32.5.8 DEFINING THE CONSTITUTIVE RESPONSE OF FLUID TRANSITIONING FROMDARCY FLOW TO POISEUILLE FLOW

Products: Abaqus/Standard Abaqus/CAE

# References

• “Cohesive elements: overview,” Section 32.5.1   
• “Defining the constitutive response of cohesive elements using a traction-separation description,” Section 32.5.6   
• “Defining the constitutive response of fluid within the cohesive element gap,” Section 32.5.7   
• \*FLUID LEAKOFF   
• \*GAP FLOW   
• \*PERMEABILITY   
• Chapter 21, “Adhesive joints and bonded interfaces,” of the Abaqus/CAE User’s Guide   
• “Element type assignment,” Section 17.5.3 of the Abaqus/CAE User’s Guide

# Overview

The cohesive element fluid flow model:

• is typically used in geotechnical applications, where fluid flow continuity within the cohesive element and through the interface must be maintained;   
• supports the transition from Darcy flow to Poiseuille flow (gap flow) as damage in the element initiates and evolves;   
• enables modeling of an additional resistance layer on the surface of the cohesive element to model fluid leakoff into the formation;   
• enables fluid pressure on the cohesive element surface to contribute to its mechanical behavior, which enables the modeling of hydraulically driven fracture;   
• can be used only in conjunction with traction-separation behavior;   
• supports fluid flow continuity between intersecting layers of cohesive pore pressure elements; and   
• enables gravity-induced fluid flux modeling.

# Defining pore fluid flow properties

The fluid constitutive response consists of the following:

• Tangential flow along the cohesive element midplane, which can be modeled as either Darcy or Poiseuille flow; and   
• Normal flow (also referred to as leakoff) across the cohesive element, which can reflect resistance due to caking or fouling effects.

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You assign the tangential and normal flow properties separately.

The flow patterns of the pore fluid in the element are shown in Figure 32.5.8–1. The fluid is assumed to be incompressible, and the formulation is based on a statement of flow continuity that considers tangential and normal flow and the rate of opening of the cohesive element.

![](images/page-890_84019d5aa6c85ec9e33402ddddfd31a8d34a3e377c8352e4a9d03f3e056a04cf.jpg)

<details>
<summary>text_image</summary>

cohesive elements
tangential flow
normal flow
</details>

Figure 32.5.8–1 Flow within cohesive elements.

# Tangential flow

Tangential flow in a cohesive element will transition from Darcy flow to Poiseuille flow as damage in the element initiates and evolves. The transition is designed to approximate the changing nature of fluid flow through an initially undamaged porous material (Darcy flow) to flow in a crack (Poiseuille flow) as the material is damaged. You must specify the fluid constitutive response for both types of flow.

# Gap opening

The tangential flow equations for the cohesive element are solved within a gap along the length of the element. The gap opening, , is defined as

$$
d = t _ {c u r r} - t _ {o r i g} + g _ {i n i t} = \hat {d} + g _ {i n i t},
$$

where $t _ { c u r r }$ and $\scriptstyle t _ { o r i g }$ are the current and original cohesive element geometrical thicknesses, respectively; $g _ { i n i t }$ is the initial gap opening, which has a default value of 0.002; and $\hat { d }$ represents the physical crack opening that is used for Poiseuille flow once the element is damaged. A schematic illustration is shown in Figure 32.5.8–2. $g _ { i n i t }$ is not a physical quantity. It is used by Abaqus/Standard to ensure that the flow equations can be solved robustly when the physical gap is closed $( \mathrm { i . e . , } \hat { d } = 0 )$ . As increases, the effect of $g _ { i n i t }$ on the flow equations is diminished, as described in “Transition from Darcy flow to Poiseuille flow.”
