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# Defining linear uncoupled structural damping behavior

You define the damping coefficients, $s _ { j j }$ , for the selected components $( \mathrm { i } . \mathsf { e } _ { \cdot , \ s _ { 1 1 } }$ for component 1, $s _ { 2 2 }$ for component 2, etc.), which are used in the equation

$$
F _ {j} = i D _ {j j} u _ {j},
$$

where

$$
D _ {j j} = s _ {j j} K _ {j j} \quad (\text { no   sum   on } j)
$$

is the structural damping matrix, $F _ { j }$ is the imaginary part of the force or moment in the $j ^ { \mathrm { t h } }$ direction of relative motion, $u _ { j }$ is the displacement in the $j ^ { \mathrm { t h } }$ direction, and $K _ { j j }$ is the stiffness matrix. The damping coefficient can depend on frequency.

Input File Usage: Use the following options:

$$
\begin{array}{l} * \text { CONNECTOR   BEHAVIOR,   NAME } = \text { name } \\ * \text { CONNECTOR   DAMPING,   COMPONENT } = \text { component   number }, \\ \mathrm{TYPE} = \text { S   T   R   U   C   T   U   R   A   L } \\ \end{array}
$$

Abaqus/CAE Usage: Linear uncoupled structural damping behavior is not supported in Abaqus/CAE.

# Defining linear coupled structural damping behavior

You define 21 $s _ { l j }$ damping coefficients (the symmetric half of the $6 \times 6$ damping coefficient matrix), which are used in the equation

$$
F _ {l} = i D _ {l j} u _ {j},
$$

where

$$
D _ {l j} = i s _ {l j} K _ {l j} \quad (\text { no   sum   on } l, j)
$$

is the structural damping matrix, $F _ { l }$ is the imaginary part of the force in the $l ^ { \mathrm { t h } }$ direction of relative motion, $u _ { j }$ is the displacement in the $j ^ { \mathrm { t h } }$ direction, and $K _ { l j }$ is the stiffness matrix. The damping coefficient matrix cannot depend on frequency.

Input File Usage: Use the following options:

$$
\begin{array}{l} * \text { CONNECTOR   BEHAVIOR,   NAME } = \text { name } \\ * \text { CONNECTOR   DAMPING,   TYPE } = \text { STRUCTURAL } \\ \end{array}
$$

Abaqus/CAE Usage: Linear coupled structural damping behavior is not supported in Abaqus/CAE.

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# Defining connector damping behavior in linear perturbation procedures

In both the direct-solution and subspace-based steady-state dynamic procedures, the viscous or structural damping defined using an uncoupled connector damping behavior may be frequency dependent. In other linear perturbation procedures connector damping behavior is ignored.

# 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 damping in connectors:

CV Connector relative velocities/angular velocities.

CVF Connector viscous forces/moments.

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# 31.2.4 CONNECTOR FUNCTIONS FOR COUPLED BEHAVIOR

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

# References

• “Connectors: overview,” Section 31.1.1   
• “Connector friction behavior,” Section 31.2.5   
• “Connector plastic behavior,” Section 31.2.6   
• “Connector damage behavior,” Section 31.2.7   
• \*CONNECTOR BEHAVIOR   
• \*CONNECTOR DERIVED COMPONENT   
• \*CONNECTOR POTENTIAL

• “Specifying connector derived components,” Section 15.17.15 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

• “Specifying potential terms,” Section 15.17.16 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

# Overview

This section describes how to define two special functions used to specify complex coupled behavior for a connector element in Abaqus: derived components and potentials.

Connector derived components are user-specified component definitions based on a function of intrinsic (1 through 6) connector components of relative motion. They can be used:

• to specify the friction-generating normal force in connectors as a complex combination of connector forces and moments, and   
• as an intermediate result in a connector potential function.

Connector potentials are user-defined functions of intrinsic components of relative motion or derived components. These functions can be quadratic, elliptical, or maximum norms. They can be used to define:

• the yield function for connector coupled plasticity when several available components of relative motion are involved simultaneously,   
• the potential function for coupled user-defined friction when the slip direction is not aligned with an available component of relative motion,   
• a magnitude measure as a coupled function of connector forces or motions used to detect the initiation of damage in the connector, and   
• an effective motion measure as a coupled function of connector motions to drive damage evolution in the connector.

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The definition of coupled behavior in connector elements beyond simple linear elasticity or damping often requires the definition of a resultant force involving several intrinsic (1 through 6) components or the definition of a “direction” not aligned with any of the intrinsic components. These user-defined resultants or directions are called derived components. The forces and motions associated with these derived components are functions of the forces and motions in the intrinsic relative components of motion in the connector element.

Consider the case of a SLOT connector for which frictional effects (see “Connector friction behavior,” Section 31.2.5) are defined in the only available component of relative motion (the 1-direction). The two constraints enforced by this connection type will produce two reaction forces $( f _ { 2 }$ and $f _ { 3 } )$ , as shown in Figure 31.2.4–1. Both forces generate friction in the 1-direction in a coupled fashion.

![](images/page-704_f4387b25daa066904699795b269a487c5bf4e3ebcf23fd09105a7dbc5c5f6afb.jpg)

<details>
<summary>text_image</summary>

f₃
f₂
slot housing
f₁
</details>

Figure 31.2.4–1 Resultant contact force in a SLOT connector.

A reasonable estimate for the resulting contact force is

$$
F _ {\mathrm{derived}} ^ {\mathrm{contact}} = g (\mathbf {f}) = \sqrt {f _ {2} ^ {2} + f _ {3} ^ {2}},
$$

where is the collection of connector forces and moments in the intrinsic components. The function can be specified as a derived component.

Resultant forces that can be defined as derived components may take more complicated forms. Consider a BUSHING connection type for which a tensile (Mode I) damage mechanism with failure is to be specified in the 1-direction. You may wish to include the effects of the axial force $f _ { 1 }$ and of the resultant of the “flexural” moments $m _ { 2 }$ and $m _ { 3 }$ in defining an overall resultant force in the axial direction upon which damage initiation (and failure) can be triggered, as shown in Figure 31.2.4–2. One choice would be to define the resultant axial force as

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![](images/page-705_999c9d354bc24e029eec74cbc20c27fc49773591ea3a2a2cf3d56379339b3959.jpg)

<details>
<summary>text_image</summary>

inner cylinder
outer cylinder
rubber
f₁
fₐₓᵢₐₓₗ
m₂
m₃
</details>

Figure 31.2.4–2 Resultant axial force in a BUSHING connector.

$$
F _ {\mathrm{derived}} ^ {\mathrm{axial}} = g (\mathbf {f}) = | f _ {1} | + \alpha \sqrt {(m _ {2} ^ {2} + m _ {3} ^ {2})},
$$

where is a geometric factor relating translations to rotations with units of one over length. The function $g ( \mathbf { f } )$ can be specified as a derived component.

A derived component can also be interpreted as a user-specified direction that is not aligned with the connector component directions. For example, if the motion-based damage with failure criterion in a CARTESIAN connection with elastic behavior does not align with the intrinsic component directions, the damage criterion can be defined in terms of a derived component representing a different direction, as shown in Figure 31.2.4–3. One possible choice for the direction could be

$$
u _ {\mathrm{derived}} ^ {\mathrm{transf}} = g (\mathbf {u}) = a _ {1} u _ {1} + a _ {2} u _ {2} + a _ {3} u _ {3},
$$

where is the collection of connector relative motions in the components and $a _ { 1 } , \ a _ { 2 }$ , and $a _ { 3 }$ can be interpreted as direction cosines $( c o s ( \alpha _ { 1 } ) , c o s ( \alpha _ { 2 } ) , c o s ( \alpha _ { 3 } ) )$ . The function $g ( \mathbf { u } )$ can be specified as a derived component.

# Functional form of the derived component

The functional form of a derived component $( g )$ in Abaqus is quite general; you specify its exact form. The derived component is specified as a sum of terms

$$
g (\mathbf {c}) = \sum_ {i = 1} ^ {N _ {T}} T _ {i} (\mathbf {c}),
$$

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![](images/page-706_90a4ce6e5f9b2ddafdb7a6fc8b5a11b9abb7f4eb996c2fc4ed7da0ce888be75f.jpg)

<details>
<summary>text_image</summary>

3
Utransf
α₃
α₁
α₂
2
1
</details>

Figure 31.2.4–3 User-defined direction in a CARTESIAN connector.

where is a generic name for the connector intrinsic component values (such as forces, , or motions, ), $T _ { i }$ is the $i ^ { \mathrm { { \bar { t } h } } }$ term in the sum, and $N _ { T }$ is the number of terms. The appropriate component values for are selected depending on the context in which the derived component is used. $T _ { i }$ is also a summation of several contributions and can take one of the following three forms:

• a norm ( -type)

$$
T _ {i} (\mathbf {c}) = s _ {i} \sqrt {\sum_ {j = 1} ^ {N _ {c}} (\alpha_ {j} c _ {j}) ^ {2}}, \quad \mathrm{or}
$$

• a direct sum $\left( g _ { S } { \tt - t y p e } \right)$

$$
T _ {i} (\mathbf {c}) = s _ {i} \sum_ {j = 1} ^ {N _ {c}} \alpha_ {j} c _ {j}, \quad \text { or }
$$

• a Macauley sum $( g _ { M } . \mathrm { { t y p e } ) }$

$$
T _ {i} (\mathbf {c}) = s _ {i} \sum_ {j = 1} ^ {N _ {c}} <   \alpha_ {j} c _ {j} >,
$$

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where $s _ { i }$ is the term’s sign (plus or minus), $\alpha _ { j }$ are scaling factors, $c _ { j }$ is the $j ^ { \mathrm { t h } }$ component of , and $< X >$ is the Macauley bracket $( < X > = 0$ if $X \le 0$ and = X if $X > 0 )$ . In general, the units of the scaling factors $\alpha _ { j }$ depend on the context. In most cases they are either dimensionless, have units of length, or have units of one over length. The scaling factors should be chosen such that all the terms in the resulting derived component have the same units, and these units must be consistent with the use of the derived component later on in a connector potential or connector contact force.

# Defining a derived component with only one term $( N _ { T } = I )$

Connector derived components are identified by the names given to them. If one term $( T _ { 1 } )$ is sufficient to define the derived component g, specify only one connector derived component definition.

Input File Usage: \*CONNECTOR DERIVED COMPONENT, NAME=derived\_component\_name

Abaqus/CAE Usage: Connector derived component names are not supported in Abaqus/CAE; you define the individual derived component terms.

Use the following input to define a connector derived component term for a friction-generating user-defined contact force:

Interaction module: connector section editor: Add→Friction: Friction model: User-defined, Contact Force, Specify component:

Derived component, click Edit to display the derived component editor: click Add and select components

Use the following input to define a connector derived component term as an intermediate result in a connector potential function:

Interaction module: connector section editor: Add→Friction, Plasticity, or Damage: potential contribution editors: Specify derived component, click Edit to display the derived component editor: click Add and select components

# Defining a derived component containing multiple terms $( N _ { T } > I )$

If several terms $( T _ { 1 } , T _ { 2 } , { \mathrm { e t c . } } )$ are needed in the overall definition of the derived component g, you must define the individual terms.

Input File Usage: You must specify $N _ { T }$ connector derived component definitions with the same name to define the individual terms. All definitions with the same name will be summed together to produce the desired derived component g. See the spot weld example below for an illustration. \*CONNECTOR DERIVED COMPONENT, NAME=derived\_component\_name \*CONNECTOR DERIVED COMPONENT, NAME=derived\_component\_name

Abaqus/CAE Usage: Connector derived component names are not supported in Abaqus/CAE; you define the individual derived component terms.

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Interaction module: derived component editor: click Add and select components. Repeat, adding terms as necessary.

# Specifying a term in the derived component as a norm

By default, a derived component term is computed as the square root of the sum of the squares of each intrinsic component contribution. If the term has only one contribution ( ), the norm has the same meaning as the absolute value.

# Input File Usage:

\*CONNECTOR DERIVED COMPONENT, NAME=derived\_component\_name, OPERATOR=NORM (default)

For example, the following input can be used to define the $F _ { \mathrm { d e r i v e d } } ^ { \mathrm { a x i a l } }$ component discussed above:

* CONNECTOR DERIVED COMPONENT, NAME=axial
1
1.0,
** $T_{1} = \sqrt{(1.0 * f_{1})^{2}} = |1.0 * f_{1}|$ * CONNECTOR DERIVED COMPONENT, NAME=axial
5, 6 $\alpha$ , $\alpha$ ** $T_{2} = \sqrt{(\alpha m_{2})^{2} + (\alpha m_{3})^{2}}$

The axial derived component is $F _ { \mathrm { d e r i v e d } } ^ { \mathrm { a x i a l } } = T _ { 1 } + T _ { 2 }$

# Abaqus/CAE Usage:

Interaction module: derived component editor: Add: Term operator: Square root of sum of squares

# Specifying a term in the derived component as a direct sum

Alternatively, you can choose to compute a derived component term as the direct sum of the intrinsic component contributions.

# Input File Usage:

\*CONNECTOR DERIVED COMPONENT,NAME=derived\_component\_name, OPERATOR=SUM

For example, the following input can be used to define the $u _ { \mathrm { d e r i v e d } } ^ { \mathrm { t r a n s f } }$ component discussed above:

\*CONNECTOR DERIVED COMPONENT, NAME=transf,OPERATOR=SUM

1, 2, 3

$$
a _ {1}, \quad a _ {2}, \quad a _ {3}
$$

$$
^ {* *} T _ {1} = a _ {1} u _ {1} + a _ {2} u _ {2} + a _ {3} u _ {3}
$$

The transf derived component is $u _ { \mathrm { d e r i v e d } } ^ { \mathrm { t r a n s f } } = T _ { 1 }$

# Abaqus/CAE Usage:

Interaction module: derived component editor: Add: Term operator: Direct sum

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# Specifying a term in the derived component as a Macauley sum

Alternatively, you can choose to compute a derived component term as the Macauley sum of the intrinsic component contributions.

# Input File Usage:

\*CONNECTOR DERIVED COMPONENT,NAME=derived\_component\_name, OPERATOR=MACAULEY SUM

For example, the following input can be used to define the first term of the normal component of the force $( F _ { n } )$ in the spotweld example discussed below:

* CONNECTOR DERIVED COMPONENT, NAME=normal,
OPERATOR=MACAULEY SUM
3
1.0
** $T_{1} = < f_{3} >$

Abaqus/CAE Usage: Interaction module: derived component editor: Add: Term operator: Macauley sum

# Specifying the sign of a term

You can specify whether the sign of a derived component term should be positive or negative.

# Input File Usage:

Use one of the following options:

```txt
*CONNECTOR DERIVED COMPONENT,
NAME=derived_component_name, SIGN=POSITIVE (default)
*CONNECTOR DERIVED COMPONENT,
NAME=derived_component_name, SIGN=NEGATIVE 
```

Abaqus/CAE Usage: Interaction module: derived component editor: Add: Overall term sign: Positive or Negative

# Defining the derived component contributions to depend on local directions

The scaling factors used in the definition of the derived component can depend on the relative positions or constitutive displacements/rotations in several component directions, as described in “Defining nonlinear connector behavior properties to depend on relative positions or constitutive displacements/rotations” in “Connector behavior,” Section 31.2.1. See the first example in “Connector friction behavior,” Section 31.2.5, for an illustration.

# Input File Usage:

Use the following option to define a connector derived component that depends on components of relative position:

\*CONNECTOR DERIVED COMPONENT, INDEPENDENT COMPONENTS=POSITION

Use the following option to define a connector derived component that depends on components of constitutive displacements or rotations:

\*CONNECTOR DERIVED COMPONENT, INDEPENDENT COMPONENTS=CONSTITUTIVE MOTION

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# Abaqus/CAE Usage: Interaction module: derived component editor: Add: Use local directions: Independent position components or Independent constitutive motion components

# Requirements for constructing a derived component used in plasticity or friction definitions

When a derived component is used to construct the yield function for a plasticity or friction definition, the following simple requirements must be satisfied:

• All $N _ { T }$ terms of a derived component must be of a compatible type (see “Functional form of the derived component”); norm-type terms ( -type) cannot be mixed with direct sum-type terms ( -type) in the same derived component definition but can be mixed with Macauley sum-type terms ( -type).   
• If all $N _ { T }$ terms are norm-type terms, the sign of each term must be positive (the default).

If is greater than 1, the associated functions (potentials) in which the derived component is used may become non-smooth. More precisely, the normal to the hyper-surface defined by the potential may experience sudden changes in direction at certain locations. In these cases, Abaqus will automatically smooth-out the defined functions by slightly changing the derived component functional definition. These changes should be transparent to the user as the results of the analysis will change only by a small margin.

# Example: spot weld

The spot weld shown in Figure 31.2.4–4 is subjected to loading in the F-direction.

![](images/page-710_aa6b17e6aa765f832b233d8aa2dbef3c1fe1240146760cd789249195e9966dcb.jpg)

<details>
<summary>text_image</summary>

F_s
F
F_n
</details>

Figure 31.2.4–4 Loading of a spot weld connection.

The connector chosen to model the spot weld has six available components of relative motion: three translations (components 1–3) and three rotations (components 4–6). You have chosen this connection type because you are modeling a general deformation state. However, you would like to define inelastic behavior in the connection in terms of a normal and a shear force, as shown in Figure 31.2.4–5, since experimental data are available in this format.
