<!-- source-page: 781 -->

![](images/page-781_b3f1352bcbf8b7fcb8ebdf41ba2df70f8e95c3d7e6aac3324da006c45d4197e2.jpg)

<details>
<summary>text_image</summary>

F
Primary loading curve
A
B
D
exponential/quadratic
unloading
C
0
U_max_B
U
</details>

Figure 31.2.10–4 Exponential/quadratic unloading.

unloading curve always starts at point O, the point of zero force and zero displacements, since the damage models do not allow any permanent deformation. The unloading curves are stored in normalized form so that they intersect the loading curve at a unit force for a unit displacement, and the interpolation occurs between these normalized curves. If unloading occurs from a maximum displacement for which an unloading curve is not specified, the unloading is interpolated from neighboring unloading curves. As the connector is loaded, the force follows the path given by the loading curve. If the connector is unloaded (for example, at point B), the force follows the unloading curve . Reloading after unloading follows the unloading path until the loading is such that the displacement becomes greater than $u _ { B } ^ { m a x }$ , after which the loading path follows the loading curve.

![](images/page-781_404b85ecc94bf7964b190b7cc1eac9ab7db156ce2485d002e3847706bb03e82c.jpg)

<details>
<summary>line</summary>

| Point | U (min) | U (max) | F (min) | F (max) |
|-------|---------|---------|---------|---------|
| A     | 0.0     | 0.5     | 0.0     | 0.5     |
| B     | 0.5     | 1.0     | 0.5     | 1.0     |
| C     | 1.0     | 1.5     | 1.0     | 1.5     |
| D     | 1.5     | 2.0     | 1.5     | 2.0     |
</details>

Figure 31.2.10–5 Interpolated curve unloading

<!-- source-page: 782 -->

If the loading curve depends on the constitutive displacements/rotations in several component directions, the unloading curves also depend on the same component directions. The unloading curves also have the same temperature and field variable dependencies as the loading curve.

Input File Usage: \*UNLOADING DATA, DEFINITION=INTERPOLATED CURVE

# Specifying combined unloading

As illustrated in Figure 31.2.10–6, you can specify an unloading curve in addition to the loading curve as well as a constant transition slope that connects the loading curve to the unloading curve. As the connector is loaded, the force follows the path given by the loading curve. If the connector is unloaded (for example, at point B), the force follows the unloading curve . The path is defined by the constant transition slope, and lies on the specified unloading curve. Reloading after unloading follows the unloading path until the loading is such that the displacement becomes greater than $u _ { B } ^ { m a x }$ , after which the loading path follows the loading curve.

![](images/page-782_fa60aa39aa5fdfe089b1219ec557c4bc211894e301bfc1e154978eb0f9457926.jpg)

<details>
<summary>line</summary>

| Point | Curve Type         | Description              |
|-------|--------------------|--------------------------|
| A     | Primary loading   | Curve A                  |
| B     | Primary loading   | Curve B                  |
| C     | Unloading curve    | Curve C                  |
| D     | Primary loading   | Curve D                  |
| E     | Unloading curve    | Curve E                  |
</details>

Figure 31.2.10–6 Combined unloading.

If the loading curve depends on the constitutive displacements/rotations in several component directions, the unloading curve also depends on the same component directions. The unloading curve also has the same temperature and field variable dependencies as the loading curve.

Input File Usage: \*UNLOADING DATA, DEFINITION=COMBINED

# Defining models with permanent deformation

These models dissipate energy upon unloading and exhibit permanent deformation upon complete unloading. The unloading behavior controls the amount of energy dissipated as well as the amount of permanent deformation. The unloading behavior can be specified in one of the following ways:

• an analytical unloading curve (exponential/quadratic);

<!-- source-page: 783 -->

• an unloading curve interpolated from multiple user-specified unloading curves; or   
• an unloading curve obtained by shifting the user-specified unloading curve to the point of unloading.

For an overview of the different available behaviors, see “Specifying uniaxial behavior for an available component of relative motion” above. The various unloading types are discussed in the sections that follow.

# Defining the onset of permanent deformation

By default, the onset of yield will be obtained as soon as the slope of the loading curve decreases by 10% from the maximum slope recorded up to that point while traversing along the loading curve. To override the default method of determining the onset of yield, you can specify either a value for the decrease in slope of the loading curve other than the default value of 10% (slope drop = 0.1) or by defining the displacement below which unloading occurs along the loading curve. If a slope drop is specified, the onset of yield will be obtained as soon as the slope of the loading curve decreases by the specified factor from the maximum slope recorded up to that point.

# Input File Usage:

Use the following options to specify the onset of yield by defining the displacement below which unloading occurs along the loading curve:

\*LOADING DATA, TYPE=PERMANENT DEFORMATION, YIELD ONSET=value

Use the following options to specify the onset of yield by defining a slope drop for the loading curve:

\*LOADING DATA, TYPE=PERMANENT DEFORMATION, SLOPE DROP=value

# Specifying exponential/quadratic unloading

The model in Figure 31.2.10–7 illustrates an analytical unloading curve that is derived based on an energy dissipation factor, (fraction of energy that is dissipated at any displacement level) and a permanent deformation factor, $D _ { p }$ . As the connector is loaded, the force follows the path given by the loading curve. If the connector is unloaded (for example, at point B), the force follows the unloading curve . The point D corresponds to the permanent deformation, $D _ { p } u _ { B } ^ { m a x }$ . Reloading after unloading follows the unloading curve until the loading is such that the displacement becomes greater than $u _ { B } ^ { m a x }$ , after which the loading path follows the loading curve. The arrows shown in Figure 31.2.10–7 illustrate the loading/unloading paths of this model.

The unloading response follows the loading curve when the calculated unloading curve lies above the loading curve to prevent energy generation and follows a zero force response when the unloading curve yields a negative response. In such cases the dissipated energy will be less than the value specified by the energy dissipation factor.

# Input File Usage:

Use the following option to define quadratic unloading behavior:

\*UNLOADING DATA, DEFINITION=QUADRATIC

Use the following option to define exponential unloading behavior:

\*UNLOADING DATA, DEFINITION=EXPONENTIAL

<!-- source-page: 784 -->

![](images/page-784_50bf3e5656b81a2beb0625bfff3c2db00f3b5fe416cb7f5710e85b579917f93c.jpg)

<details>
<summary>line</summary>

| Point | U (min) | U (max) | F (min) | F (max) |
|-------|---------|---------|---------|---------|
| A     | 0       | 0       | 0       | 0       |
| B     | 0       | 0       | 0       | 0       |
| C     | 0       | 0       | 0       | 0       |
| D     | 0       | 0       | 0       | 0       |
| E     | 0       | 0       | 0       | 0       |
</details>

Figure 31.2.10–7 Exponential/quadratic unloading.

# Specifying interpolated curve unloading

The model in Figure 31.2.10–8 illustrates an interpolated unloading response based on multiple unloading curves that intersect the primary loading curve at increasing values of forces/displacements. You can specify as many unloading curves as are necessary to define the unloading response. The first point of each unloading curve defines the permanent deformation if the connector is completely unloaded. The unloading curves are stored in normalized form so that they intersect the loading curve at a unit force for a unit displacement, and the interpolation occurs between these normalized curves. If unloading occurs from a maximum displacement for which an unloading curve is not specified, the unloading curve is interpolated from neighboring unloading curves. As the connector is loaded, the force follows the path given by the loading curve. If the connector is unloaded (for example, at point B), the force follows the unloading curve . Reloading after unloading follows the unloading path until the loading is such that the displacement becomes greater than , after which the loading path follows the loading curve.

If the loading curve depends on the constitutive displacements/rotations in several component directions, the unloading curves also depends on the same component directions. The unloading curve also has the same temperature and field variable dependencies as the loading curve.

Input File Usage: \*UNLOADING DATA, DEFINITION=INTERPOLATED CURVE

# Specifying shifted curve unloading

You can specify an unloading curve passing through the origin in addition to the loading curve. The actual unloading curve is obtained by horizontally shifting the user-specified unloading curve to pass through the point of unloading as shown in Figure 31.2.10–9. The permanent deformation upon complete unloading is the horizontal shift applied to the unloading curve.

<!-- source-page: 785 -->

![](images/page-785_e7014067352005e74765392744e0660f3ccb1853877aa96e796bfba69983b2f5.jpg)

<details>
<summary>text_image</summary>

F
Primary loading curve
E
A
B
C
Unloading curves
0
D
U_max_B
U
</details>

Figure 31.2.10–8 Interpolated curve unloading.

![](images/page-785_53ebe90d0a98d504fda303bb1c5f6ba1e0f1462b0f9042674d9fda84c5e7d8a0.jpg)

<details>
<summary>line</summary>

| Point | U     | F     |
|-------|-------|-------|
| A     | D     | low   |
| B     | U_max | high  |
| C     | U_B^max | low   |
| E     | U     | high  |
</details>

Figure 31.2.10–9 Shifted curve unloading.

If the loading curve depends on the constitutive displacements/rotations in several component directions, the unloading curve also depends on the same component directions. The unloading curve also has the same temperature and field variable dependencies as the loading curve.

Input File Usage: \*UNLOADING DATA, DEFINITION=SHIFTED CURVE

<!-- source-page: 786 -->

# Using different uniaxial models in tension and compression

When appropriate, different uniaxial behavior models can be used in tension and compression. For example, a model with permanent deformation and exponential unloading in tension can be combined with a nonlinear elastic model in compression (see Figure 31.2.10–10).

![](images/page-786_a69e5ffe42619f1820bb7d63fdf5d8637828020517189591e316b7788ab78847.jpg)

<details>
<summary>text_image</summary>

F
Primary loading curve
A
unloading
nonlinear elastic
U
</details>

Figure 31.2.10–10 Different uniaxial models in tension and compression.

# Output

The Abaqus output variables available for connectors are listed in “Abaqus/Standard output variable identifiers,” Section 4.2.1, and “Abaqus/Explicit output variable identifiers,” Section 4.2.2. The following output variables are of particular interest when defining uniaxial behavior in connectors:

CU Connector relative displacements/rotations.

CUF Connector uniaxial forces/moments.

<!-- source-page: 787 -->

# 32. Special-Purpose Elements

Spring elements 32.1   
Dashpot elements 32.2   
Flexible joint elements 32.3   
Distributing coupling elements 32.4   
Cohesive elements 32.5   
Gasket elements 32.6   
Surface elements 32.7   
Tube support elements 32.8   
Line spring elements 32.9   
Elastic-plastic joints 32.10   
Drag chain elements 32.11   
Pipe-soil elements 32.12   
Acoustic interface elements 32.13   
Eulerian elements 32.14   
Fluid pipe elements 32.15   
Fluid pipe connector elements 32.16   
User-defined elements 32.17

<!-- source-page: 788 -->

<!-- source-page: 789 -->

# 32.1 Spring elements

• “Springs,” Section 32.1.1   
• “Spring element library,” Section 32.1.2

<!-- source-page: 790 -->
