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```csv
*CONNECTOR POTENTIAL
1,
4, <r>
*FRICTION
0.15 
```

# Specifying friction due to assembly contact interference

Assume a CYLINDRICAL connector element in which the shaft was press-fit into the sleeve, as shown in the initial configuration (relative motion = 0.0) in Figure 31.2.5–3.

![](images/page-731_89e144bae7ea298df6fa9248011e748d0b38a4aa30fb8f27e07fd88e1e9e31be.jpg)

<details>
<summary>text_image</summary>

L
2r
</details>

Figure 31.2.5–3 CYLINDRICAL connection with slightly conical pin.

The shaft is not perfectly cylindrical but slightly conical so that its cross-section diameter is increasing in a linear fashion along the shaft direction. If the relative displacement along the shaft direction becomes positive, the contact forces will increase (more contact interference); if the relative displacements become negative (less interference), they will decrease. An exponential decay model is assumed to model the transition from a static coefficient of friction to a kinetic one. Only the positive contact force versus displacement values need to be specified. The following user-defined friction behavior definitions can be used:

```txt
*PARAMETER
r=0.24
...
*CONNECTOR FRICTION, INDEPENDENT COMPONENTS=CONSTITUTIVE MOTION
1
** (independent component 1)
0.70, -0.7854
0.85, -0.3927
1.0, 0.0
1.15, 0.3927
1.30, 0.7854
*CONNECTOR POTENTIAL
1,
4, <r>...
*FRICTION, EXPONENTIAL DECAY
0.25, 0.10, 0.2 
```

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The internal contact forces are specified directly on the data lines to model known contact interference forces as a function of the connector constitutive component of relative motion along component 1. Since no intrinsic component of relative motion number or named connector derived component was specified to define the contact force, the only contribution to the contact force is the specified internal contact force.

# Specifying friction in a HINGE connection

This example illustrates the use of a connector friction definition to specify frictional effects in a HINGE connection. The friction behavior defines friction moments about the 1-direction, since there are no other available components of relative motion. As illustrated in “Connection-type library,” Section 31.1.5, the three geometrical scaling constants that need to be specified for predefined friction are the radius of the pin cross-section, $R _ { p } { = } 0 . 1 2$ ; the effective friction arm in the axial direction, $R _ { a } { = } 0 . 1 4 ;$ and the overlapping length between the pin and the sleeve, $L _ { s } { = } 0 . 6 5$ . The friction coefficient is assumed to be $\mu { = } 0 . 1 5$ . It is assumed that the connector has been assembled with initial known contact interference-producing contact moments of $\mathrm { M } _ { \mathrm { C } } ^ { \mathrm { i n t } } = 1 0 0 . 0$ units. The following input could be used to specify the predefined friction behavior in the HINGE connection:

*PARAMETER $R_{p}=0.12$ $R_{a}=0.14$ $L_{s}=0.65$ ...
*CONNECTOR FRICTION, PREDEFINED $<R_{p}>$ , $<R_{a}>$ , $<L_{s}>$ , 100.0
*FRICTION
0.15

Alternatively, a user-defined friction behavior could be specified to define identical frictional effects (see “Connection-type library,” Section 31.1.5). Moreover, a reduction of the interference contact forces as the pin wears due to accumulated sliding can be modeled in this case by specifying the internal contact forces/moments to be functions of accumulated slip. The following input can be used:

*PARAMETER $R_{p} = 0.12$ $R_{a} = 0.14$ $L_{s} = 0.65$ $\alpha_{1} = R_{a}$ $\alpha_{2} = R_{p}$ $\alpha_{3} = 2.0 * R_{p} / L_{s}$ $\cdots$ *CONNECTOR DERIVED COMPONENT, NAME=contact_moment
1, $< \alpha_{1}>$ , $* * (\sqrt{(\alpha_{1} * f_{1})^{2}} = |\alpha_{1} * f_{1}|)$

<!-- source-page: 733 -->

* CONNECTOR DERIVED COMPONENT, NAME=contact_moment
2, 3
< $\alpha_{2}$ >, < $\alpha_{2}$ >
** ( $\sqrt{(\alpha_{2} * f_{2})^{2} + (\alpha_{2} * f_{3})^{2}}$ )
* CONNECTOR DERIVED COMPONENT, NAME=contact_moment
5, 6
< $\alpha_{3}$ >, < $\alpha_{3}$ >
** ( $\sqrt{(\alpha_{3} * m_{2})^{2} + (\alpha_{3} * m_{3})^{2}}$ )
* CONNECTOR FRICTION, COMPONENT=4, CONTACT FORCE=contact_moment
100, 0.0
90, 1000.0
** interference contact moments decreasing due to wear effects
* FRICTION
0.15

The additional friction moments due to contact interference are modeled by specifying decreasing internal contact moments as a function of accumulated rotational slip about the 1-direction. The connector derived component definitions are used to define a contact moment-producing friction in the same direction (component 4). The contact moment is defined by

$$
\mathrm{M} _ {\mathrm{C}} = | g (\mathbf {f}) | = | \alpha_ {1} f _ {1} | + \sqrt {(\alpha_ {2} f _ {2}) ^ {2} + (\alpha_ {2} f _ {3}) ^ {2}} + \sqrt {(\alpha_ {3} m _ {2}) ^ {2} + (\alpha_ {3} m _ {3}) ^ {2}}.
$$

The connector potential is defined automatically by Abaqus as $\mathrm { P } ( \mathbf { f } ) = | m _ { 1 } |$

# Specifying friction in a ball-in-socket connection

This example illustrates the specification of frictional effects in a ball-in-socket connection. While the first choice in defining a ball-in-socket connection is JOIN and ROTATION, other rotation parameterizations could be used (JOIN and CARDAN, JOIN and EULER, or JOIN and FLEXION-TORSION). Assuming that the radius of the ball is $R _ { s } = 0 . 3 0$ and the coefficient of friction is , the following lines can be used to define the friction interactions:

*PARAMETER $R_{s}=0.30$ ...
*CONNECTOR DERIVED COMPONENT, NAME=normal
1, 2, 3
1.0, 1.0, 1.0
** $\sqrt{(f_{1})^{2}+(f_{2})^{2}+(f_{3})^{2}}$ *CONNECTOR FRICTION, CONTACT FORCE=normal
*CONNECTOR POTENTIAL
4, < $R_{s}$ >
5, < $R_{s}$ >
6, < $R_{s}$ >

<!-- source-page: 734 -->

\*FRICTION

0.15

The computed connector friction moments and the friction-induced moments at the connector nodes are dependent on the connection type.

# Defining connector friction behavior in linear perturbation procedures

Frictional slipping is not allowed in linear perturbation procedures. If a connector is slipping at the end of the last general analysis step, it will slip freely during the current linear perturbation step. Otherwise, Abaqus will allow the connector to slip elastically with the specified stick stiffness or enforce a sticking condition if a stick stiffness is not specified.

# 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 variables are of particular interest when defining friction in connectors:

<table><tr><td>CSF</td><td>Connector friction forces/moments. In addition to the usual six components associated with connector output variables, CSF includes the scalar CSFC, which is the friction force generated by a coupled friction definition.</td></tr><tr><td>CNF</td><td>Connector normal forces/moments. CNF includes the scalar CNFC, which is the friction-generating normal force associated with a coupled friction definition.</td></tr><tr><td>CASU</td><td>Connector accumulated slip. CASU includes the scalar CASUC, which is the accumulated slip associated with a coupled friction definition.</td></tr><tr><td>CIVC</td><td>Connector instantaneous velocity associated with a coupled friction definition.</td></tr></table>

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# 31.2.6 CONNECTOR PLASTIC BEHAVIOR

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

# References

• “Connectors: overview,” Section 31.1.1   
• “Connector behavior,” Section 31.2.1   
• “Connector elastic behavior,” Section 31.2.2   
• “Connector functions for coupled behavior,” Section 31.2.4   
• \*CONNECTOR BEHAVIOR   
• \*CONNECTOR DERIVED COMPONENT   
• \*CONNECTOR ELASTICITY   
• \*CONNECTOR HARDENING   
• \*CONNECTOR PLASTICITY   
• \*CONNECTOR POTENTIAL   
• “Defining plasticity,” Section 15.17.6 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

# Overview

Connector plasticity in Abaqus:

• can be used to model plastic/irreversible deformations of parts forming an actual connection device; for example,   
– the pin or the sleeve in a door hinge may deform plastically if the forces/moments acting on them are large enough;   
– connection elements in automotive suspension systems may deform irreversibly due to abusive loading; or   
– spot welds in a car frame and rivets in an airplane could undergo inelastic deformations if the forces acting on the structural members they are a part of are larger than intended;

• is defined in terms of resultant forces and moments in the connector;   
• uses perfect plasticity or isotropic/kinematic hardening behavior models;   
• can be used when rate-dependent effects are important;   
• can be specified in any connectors with available components of relative motion;   
• can be used for available components of relative motion for which either elastic or rigid behavior was specified;   
• can be used in an uncoupled fashion to define elastic-plastic or rigid plastic response in individual available components of relative motion; and

<!-- source-page: 736 -->

• can be used to specify coupled elastic-plastic or rigid plastic behavior, in which case the responses in several available components of relative motion are involved simultaneously in a coupled fashion to define plasticity effects.

To define connector plasticity in Abaqus, the following are necessary:

• the elastic or rigid behavior prior to the onset of plasticity;   
• a yield function upon which plastic flow will be initiated; and

• hardening behavior to define the initial yield value and, optionally, the yield value evolution after plastic motion initiation.

# Plasticity formulation in connectors

The plasticity formulation in connectors is similar to the plasticity formulation in metal plasticity (see “Classical metal plasticity,” Section 23.2.1). In connectors the stress ( ) corresponds to the force ( ), the strain ( ) corresponds to the constitutive motion ( ), the plastic strain $( \varepsilon ^ { p l } )$ corresponds to the plastic relative motion $( \mathbf { u } ^ { p l } )$ , and the equivalent plastic strain $( \bar { \varepsilon } ^ { p l } )$ corresponds to the equivalent plastic relative motion $( \bar { u } ^ { p l } )$ . The yield function $\phi$ is defined as

$$
\phi (\mathbf {f}, \bar {\mathbf {u}} ^ {\mathbf {p l}}) = P (\mathbf {f}) - F ^ {0} \leq 0,
$$

where is the collection of forces and moments in the available components of relative motion that ultimately contribute to the yield function; the connector potential, $P ( \mathbf { f } )$ , defines a magnitude of connector tractions similar to defining an equivalent state of stress in Mises plasticity and is either automatically defined by Abaqus or user-defined; and $F ^ { 0 }$ is the yield force/moment. The connector relative motions, , remain elastic as long as $\phi < 0 ;$ and when plastic flow occurs, $\phi = 0$ .

If yielding occurs, the plastic flow rule is assumed to be associated; thus, the plastic relative motions are defined by

$$
\dot {\mathbf {u}} ^ {p l} = \dot {\bar {u}} ^ {p l} \frac {\partial \phi}{\partial \mathbf {f}},
$$

where $\dot { \mathbf { u } } ^ { p l }$ is the rate of plastic relative motion and $\dot { \bar { u } } ^ { p l }$ is the equivalent plastic relative motion rate.

# Loading and unloading behavior

Abaqus allows for the following three types of behaviors associated with a plasticity definition when the connector is not actively yielding:

• Linear elastic behavior, shown in Figure 31.2.6–1(a), is the most common case since similar behavior can be modeled in metal plasticity, for example, by specifying the Young’s modulus. Elastic motion occurs prior to plasticity onset, and unloading from a plastic state occurs on a straight line parallel to the initial loading.   
• Rigid behavior, shown in Figure 31.2.6–1(b), assumes that the slope in the linear elastic behavior is infinite; thus, the elastic motion prior to plasticity onset is zero, and unloading from a plastic state

<!-- source-page: 737 -->

(a)   
![](images/page-737_da4f67ad6079cef2230e7d113696dc32039b4681a0b685f379390c6461714601.jpg)

<details>
<summary>line</summary>

| U     | F     | Condition                  |
|-------|-------|-----------------------------|
| 0     | 0     | linear elasticity          |
| >0    | >0    | plasticity onset           |
</details>

(b)   
![](images/page-737_141f3a9e85cb15d38c1f1bf0cc9e0ca7f607227bb7bd65f977f2962ba3d1eb59.jpg)

<details>
<summary>line</summary>

| U    | F     |
| ---- | ----- |
| 0    | 0     |
| >0   | F     |
</details>

(c)   
![](images/page-737_65e0756e39c8c48edfda31be0aa1b9c180f826ce113c979c91b7753a10392d68.jpg)

<details>
<summary>line</summary>

| U     | F     | Condition                          |
|-------|-------|-----------------------------------|
| 0     | 0     | plasticity onset                   |
| 0     | F₀    | plasticity onset                   |
| 0     | F₁₀   | plasticity onset                   |
| 0     | 0     | nonlinear elastic unloading/reloading |
| 0     | F₀    | nonlinear elastic unloading/reloading |
| 0     | F₁₀   | nonlinear elastic unloading/reloading |
| 0     | 0     | nonlinear elastic unloading/reloading |
| 0     | F₀    | nonlinear elastic unloading/reloading |
| 0     | F₁₀   | nonlinear elastic unloading/reloading |
</details>

Figure 31.2.6–1 Linear elastic-plastic (a), rigid plastic (b), and nonlinear elastic-plastic (c) response.

<!-- source-page: 738 -->

occurs on a vertical line. In practice, the rigid behavior is enforced using an automatically chosen high penalty stiffness.

• Nonlinear elastic behavior, shown in Figure 31.2.6–1(c), in which the initial elastic loading occurs along the defined nonlinear path. Elastic unloading occurs along a nonlinear curve $( \mathrm { C } \to \mathrm { O } _ { \mathrm { c } } )$ that is simply the user-defined nonlinear elastic curve motion shifted such that it passes through point C. The user-defined nonlinear elastic behavior must be such that the unloading path $( \mathrm { C } \to \mathrm { O } _ { \mathrm { c } } )$ does not intersect with the loading path (O I C); otherwise, a local instability will occur.

Other behaviors (such as damping or friction) can be specified in addition to the elastic/rigid/plastic specifications but will not be considered in the plasticity calculations since they are considered to be in parallel with the elastic-plastic/rigid plastic behavior (see the conceptual model in “Connector behavior,” Section 31.2.1).

# Defining elastic-plastic or rigid plastic behavior

As is the case with any other connector behavior type, connector plasticity can be defined only for available components of relative motion. For example, you cannot define plastic behavior in a BEAM connector or in components 2 and 3 of a SLOT connector since these components are not available for behavior definitions. The solution to this problem is to:

• define a connection type with available components of relative motion that best models the kinematics of your connection device both before and after plasticity onset;   
• define the desired components as rigid (see “Connector elastic behavior,” Section 31.2.2); and   
• specify rigid plastic behavior in some or all of these components.

For example, to define rigid plasticity for an otherwise rigid beam-like connector, you could use a PROJECTION CARTESIAN connection together with a PROJECTION FLEXION-TORSION connection, define all components as rigid, and proceed with your plasticity definitions.

Elastic-plastic behavior is usually specified for available components of relative motion for which spring-like behavior is specified and for which plastic deformation may occur.

```txt
Input File Usage: Use the following options to define rigid plasticity in connectors:
*CONNECTOR BEHAVIOR, NAME=name
*CONNECTOR ELASTICITY, RIGID
*CONNECTOR PLASTICITY
*CONNECTOR HARDENING
Use the following options to define elastic-plasticity in connectors:
*CONNECTOR BEHAVIOR, NAME=name
*CONNECTOR ELASTICITY
*CONNECTOR PLASTICITY
*CONNECTOR HARDENING 
```

```txt
Abaqus/CAE Usage: Use the following input to define rigid plasticity in connectors: Interaction module: connector section editor: Add→Elasticity, Definition: Rigid; Add→Plasticity 
```

<!-- source-page: 739 -->

Use the following input to define elastic-plasticity in connectors:

Interaction module: connector section editor: Add→Elasticity;

Add→Plasticity

# Defining uncoupled plastic behavior

Uncoupled elastic-plastic or rigid plastic behavior, specified for each component of relative motion independently, is similar to one-dimensional plasticity. You must define elastic or rigid behavior in the specified component of relative motion. In this case the connector potential function is chosen automatically as

$$
P (\mathbf {f}) = | f _ {i} |,
$$

where $f _ { i }$ is the force or moment in the $i ^ { \mathrm { t h } }$ available component of relative motion for which plastic behavior is specified. The associated plastic flow in this case becomes

$$
\dot {u} _ {i} ^ {p l} = \dot {\bar {u}} _ {i} ^ {p l} \frac {\partial \phi}{\partial f _ {i}} = \dot {\bar {u}} _ {i} ^ {p l} s i g n (f _ {i}), \quad \mathrm{nosum} i,
$$

where ${ \dot { u } _ { i } ^ { p l } }$ is the rate of plastic relative motion and $\dot { \bar { u } } _ { i } ^ { p l }$ is the equivalent plastic relative motion rate in the $i ^ { \mathrm { t h } }$ component.

Input File Usage: Use the following options to define uncoupled rigid plastic connector behavior:

\*CONNECTOR BEHAVIOR, NAME=name   
\*CONNECTOR ELASTICITY, RIGID, COMPONENT=i  
\*CONNECTOR PLASTICITY, COMPONENT=i  
\*CONNECTOR HARDENING

Use the following options to define uncoupled elastic-plastic connector behavior:

\*CONNECTOR BEHAVIOR, NAME=name   
\*CONNECTOR ELASTICITY, COMPONENT=i  
\*CONNECTOR PLASTICITY, COMPONENT=i  
\*CONNECTOR HARDENING

Abaqus/CAE Usage: Use the following input to define uncoupled rigid plastic connector behavior:

Interaction module: connector section editor: Add→Elasticity, Definition: Rigid; Add→Plasticity, Coupling: Uncoupled

Use the following input to define uncoupled elastic-plastic connector behavior:

Interaction module: connector section editor: Add→Elasticity,

Definition: Linear or Nonlinear, Coupling: Uncoupled;

Add→Plasticity, Coupling: Uncoupled

<!-- source-page: 740 -->

# Defining coupled plastic behavior

You should define coupled plasticity in connectors when several available components of relative motion are involved simultaneously in a coupled fashion in the definition of the yield function . In this case you must define the potential, P, via a connector potential definition. Plastic flow eventually occurs only in the intrinsic components of relative motion that are ultimately involved in the potential. Elastic or rigid behavior should be specified for all components of relative motion that are involved in the potential definition. The elastic/rigid behavior for these components can be specified in an uncoupled fashion, in a coupled fashion, or in a combination of both. All elasticity definitions specified in a connector behavior that are pertinent to the components of relative motion involved in the potential definition are used collectively to define the elasticity for the coupled elastic-plastic or rigid plastic definition.

Input File Usage: Use the following options to define coupled elastic-plastic or rigid plastic connector behavior:

\*CONNECTOR BEHAVIOR, NAME=name   
\*CONNECTOR ELASTICITY   
\*CONNECTOR PLASTICITY   
\*CONNECTOR POTENTIAL   
\*CONNECTOR HARDENING

Abaqus/CAE Usage: Interaction module: connector section editor: Add→Elasticity; Add→Plasticity, Coupling: Coupled, Force Potential

# Mode-mix ratio

If the coupled plasticity definition includes at least two terms in the associated potential definition (see “Defining derived components for connector elements” in “Connector functions for coupled behavior,” Section 31.2.4), a mode-mix ratio can be defined to reflect the relative weight of the first two terms in their contribution to the potential. The mode-mix ratio can be used in plastic motion-based connector damage definitions (see “Connector damage behavior,” Section 31.2.7) to specify dependencies in both damage initiation and damage evolution. It is defined as

$$
\Psi_ {m} = (\frac {2}{\pi}) t a n ^ {- 1} (\frac {F _ {I}}{F _ {I I}}),
$$

where $F _ { I }$ is the force/moment in the first component specified for the plasticity potential and $F _ { I I }$ is the force/moment in the second component specified for the same potential. $\Psi _ { m } = 0 . 0$ if $F _ { I } = 0 . 0$ , $\Psi _ { m } = 1 . 0 \mathrm { i f } F _ { I I } = 0 . 0$ , and $\Psi _ { m }$ is somewhere in between −1.0 and 1.0 if neither is 0.0.

# Defining the plastic hardening behavior

Abaqus provides a number of hardening models varying from simple perfect plasticity to nonlinear isotropic/kinematic hardening. Connector hardening is analogous to the hardening models used in Abaqus for metals subjected to cyclic loading and described in “Models for metals subjected to cyclic loading,” Section 23.2.2.
