Connector Damage Behavior

By default, the damage initiation criterion is specified in terms of forces/moments in the connector. Elastic or rigid connector behavior must be defined for the components involved in the initiation. You provide the lower (compression) limit, $f_{min}$, and the upper (tension) limit, $f_{max}$, for the force/moment damage initiation values. If the force is outside the range specified by the two limit values, damage is initiated. The output variable CDIF can be used to monitor the proximity to the damage initiation point.

The damage initiation criterion can be specified in terms of an equivalent relative plastic motion in the connector. You provide the relative equivalent plastic displacement/rotation at which damage will be initiated as a function of the relative equivalent plastic rate. The output variable CDIP can be used to monitor the proximity to the damage initiation point.

The damage initiation criterion can be specified in terms of relative constitutive displacements/rotations in the connector. You provide the lower (compression) limit, $u_{min}$, and the upper (tension) limit, $u_{max}$, for the constitutive displacement/rotation damage initiation values. If the motion is outside the range specified by the two limit values, damage is initiated. The output variable CDIM can be used to monitor the proximity to the damage initiation point.

By default (i.e., the affected components are not specified explicitly), only the elastic/rigid or elastic/rigid-plastic response in the connector will be damaged. The response due to friction, damping, and stop/lock behavior will not be degraded. For an uncoupled connector damage mechanism (uncoupled damage initiation criterion), only the specified component of relative motion will undergo damage. For coupled connector damage initiation, the components that will be degraded by default are chosen as follows:

• If a force-based or constitutive motion-based damage initiation criterion is used, the intrinsic available components (1 through 6) that ultimately contribute to the connector potential for damage initiation will be affected.

• If a plastic motion–based damage initiation criterion is used, the intrinsic available components that ultimately contribute to the connector potential used in the coupled plasticity definition will be affected.

Alternatively, you can specify the available components of relative motion that will be affected by the damage evolution law directly. In this case the entire connector response (elasto/rigid-plastic, friction, damping, constraint forces and moments, etc.) in the affected components will be damaged.

The linear form of the damage evolution law is illustrated here in the context of linear elasticity, although it can be used in any situation. Assume that the connector response is linear elastic and that after damage initiation a linear damage evolution is desired, as illustrated in Figure 1.

Linear damage evolution law for linear elastic connector behavior.

If damage were not specified, the response would be linear elastic (a straight line passing through the origin). Assume that damage has initiated at point I as triggered by a force-based or motion-based criterion, for example; the corresponding constitutive motion at this point is $u_0$. If the connector is loaded further such that the constitutive motion increases to $u_c$, the connector force response at point C becomes $F_c$. The response is diminished by $d*F_{eff}$ when compared to the effective response $F_{eff}$ (the elastic response with no damage). Thus, $F_c=\left(1-d\right)F_{eff}$. If unloading occurs at point C, the unloading curve of slope $\left(1-d\right)E$ is followed. As long as the constitutive motion does not exceed $u_c$, the damage variable, d, stays constant at the value obtained when point C is first reached. If further loading occurs, further damage occurs until the ultimate failure motion, $u_f$, is reached (d = 1) and the connector component loses the ability to carry any load. Thus, one possible loading/unloading sequence is O$\to$I$\to$C$\to$O$\to$C$\to u_f$.

The linear damage evolution law defines a truly linear damaged force response only in the case of linear elastic or rigid behavior with optional perfect plasticity. If nonlinear elasticity or plasticity with hardening are defined for the damaged components, an approximate linear damaged response is observed.

In Abaqus/Standard the damage force (instead of the damage variable) decays linearly when component-based linear elasticity is defined with component-based damage. This behavior is different from the behavior in Abaqus/Explicit, where the damage variable decays linearly.

The exponential damage evolution law is illustrated in the context of a linear elastic-plastic response with hardening, although it can be used in any situation. The force response in a particular connector component is shown in Figure 2.

Exponential damage evolution law for linear elastic-plastic connector behavior with hardening.

Assume that damage is initiated at point I as triggered by a plastic motion–based damage initiation criterion. If further loading occurs until point C, the response is $F_C=\left(1-d\right)F_{eff}$. Unloading from point C occurs along the damaged elastic line of slope $\left(1-d\right)E$. Upon unloading/reloading, the damage variable remains constant until point C is reached again. Further loading (beyond point C) leads to an increasingly damaged response until the ultimate failure point, ${\overline{u}}_f^{pl}$, is reached (d = 1). The damage variable d is given by the following equation:

The damaged response will appear to be truly exponential only if either linear elasticity or perfect plasticity is used. An approximate exponential degradation is obtained if plasticity with hardening is present.

You specify the difference between the relative motions at ultimate failure and at damage initiation and the exponential coefficient $\alpha$. The difference between the relative motions is interpreted in the same way as described in Defining a Motion-Based Linear Damage Evolution Law, as follows:

• If an uncoupled force-based or constitutive motion-based damage initiation criterion is used, the difference between the relative motions at ultimate failure and at damage initiation in the specified component i, $u_f{}_i-u_0{}_i$, is specified.

• If a coupled force-based or constitutive motion-based damage initiation criterion is used, an equivalent relative motion is defined using a connector potential ($\overline{u}=P\left(\mathbf{u}\right)$). The difference between the relative motions at ultimate failure and at damage initiation, ${\overline{u}}_f-{\overline{u}}_0$, is specified.

• If a plastic motion–based damage initiation criterion is used, the difference between the equivalent relative plastic motions at ultimate failure and at damage initiation, ${\overline{u}}_f^{pl}-{\overline{u}}_0^{pl}$, is specified. The equivalent plastic relative motion is calculated from the associated plasticity definition. The data can also be functions of the mode-mix ratio ${\mathrm{\Psi }}_m$.

In the first two cases the equation for the damage variable is similar to that given above for plastic motion–based damage initiation except that (equivalent) constitutive relative motions are used instead of equivalent relative plastic motions.

You can also specify the damage variable directly as a tabular function of the differences between the relative motions at ultimate failure and the relative motions at damage initiation. The differences between the relative motions are interpreted in the same way as described in Defining a Motion-Based Linear Damage Evolution Law, as follows:

• If an uncoupled force-based or constitutive motion-based damage initiation criterion is used, the differences between the constitutive relative motions at ultimate failure and at damage initiation in the specified component i, $u_i-u_0{}_i$, are used to define the tabular data.

• If a coupled force-based or constitutive motion-based damage initiation criterion is used, an equivalent relative motion is defined using a connector potential ($\overline{u}=P\left(\mathbf{u}\right)$). The differences between the relative motions at ultimate failure and at damage initiation, $\overline{u}-{\overline{u}}_0$, are used to define the tabular data.

• If a plastic motion–based damage initiation criterion is used, the differences between the equivalent relative plastic motions at ultimate failure and at damage initiation, ${\overline{u}}^{pl}-{\overline{u}}_0^{pl}$, are used. The equivalent plastic relative motion is calculated from the associated plasticity definition. The tabular data can also be functions of the mode-mix ratio ${\mathrm{\Psi }}_m$.

This damage evolution law is illustrated in the context of nonlinear elasticity, as shown in Figure 3.

Post-damage initiation dissipation energy evolution law for nonlinear elastic connector behavior.

Assume that damage is initiated at point I when the constitutive relative motion is $u_0$ as triggered by a force-based or a motion-based damage initiation criterion, for example. The response at point C will be $F_C=\left(1-d\right)F_{eff}$. Unloading from point C occurs along the CO curve, which is the original nonlinear elastic response curve (OE) scaled down by the ($1-d$) factor. Damage remains constant on the unloading/reloading curve (C$\to$O$\to$C), and it increases only if loading increases beyond point C.

Instantaneous failure can be specified upon initiation if $G_C$ is specified as 0.0. In all other cases ultimate failure (d = 1) would occur (in theory) at infinite motion since an exponential-like response that asymptotically goes to zero is generated. Abaqus will set d = 1 when the damage dissipated energy reaches 0.99$G_C$.

You specify the post-damage initiation dissipated energy at ultimate failure, $G_C$. If a plastic motion–based initiation criterion is used, $G_C$ can be specified as a function of the mode-mix ratio ${\mathrm{\Psi }}_m$.

When several damage mechanisms are defined for the same connector behavior, you can specify the contribution of each damage mechanism to the overall damage effect for a particular component of relative motion. By default, the damage value associated with a particular mechanism will be compared to the damage values from any other damage mechanisms defined for the connector behavior, and only the maximum value will be considered for the overall damage. Alternatively, you can specify that the damage values for the mechanisms associated with the connector behavior should be combined in a multiplicative fashion to obtain the overall damage. See the last example below for an illustration.

The following input could be used to define a simple uncoupled damage mechanism:

CONNECTOR ELASTICITY, COMPONENT=1
CONNECTOR DAMAGE INITIATION, COMPONENT=1, CRITERION=FORCE
force_compress, force_tens
CONNECTOR DAMAGE EVOLUTION, TYPE=ENERGY
0.0

Damage will initiate when the elastic force in component 1 is either smaller than force_compress or larger than force_tens. Only the elastic response in component 1 will be damaged. Since the dissipated energy specified for damage evolution is 0.0, the damage evolves catastrophically instantaneously after it has initiated.

Referring to the spot weld in Figure 4 for which coupled plasticity is defined in Connector Plastic Behavior, a plastic motion–based damage initiation and evolution with dependencies on the mode-mix ratio can be specified as follows:

Spot weld connection.

PARAMETER
${\overline{u}}_{ini{t}_{0.0}}^{pl}$=0.25
${\overline{u}}_{ini{t}_{0.5}}^{pl}$=0.35
${\overline{u}}_{ini{t}_{1.0}}^{pl}$=0.45
${\overline{u}}_{evo{l}_{0.0}}^{pl}$=0.75
${\overline{u}}_{evo{l}_{0.3}}^{pl}$=0.78
${\overline{u}}_{evo{l}_{0.5}}^{pl}$=0.82
${\overline{u}}_{evo{l}_{1.0}}^{pl}$=0.85
CONNECTOR DAMAGE INITIATION, CRITERION=PLASTIC MOTION
$<{\overline{u}}_{ini{t}_{0.0}}^{pl}>$, 0.0
$<{\overline{u}}_{ini{t}_{0.5}}^{pl}>$, 0.5
$<{\overline{u}}_{ini{t}_{1.0}}^{pl}>$, 1.0
CONNECTOR DAMAGE EVOLUTION, TYPE=MOTION, SOFTENING=LINEAR
$<{\overline{u}}_{evo{l}_{0.0}}^{pl}>$, 0.0
$<{\overline{u}}_{evo{l}_{0.3}}^{pl}>$, 0.3
$<{\overline{u}}_{evo{l}_{0.5}}^{pl}>$, 0.5
$<{\overline{u}}_{evo{l}_{1.0}}^{pl}>$, 1.0

The equivalent plastic relative motion on the data lines is defined by the associated coupled plasticity definition illustrated in Connector Plastic Behavior. For the damage evolution the post-damage-initiation equivalent plastic relative motion should be specified. The second column in all the data lines represents the mode-mix ratios as defined in Connector Plastic Behavior. In this particular case the mode-mix ratio is $\left(\frac{2}{\pi }\right)ta{n}^{-1}\left(F_n/F_s\right)$. The data point at 0.0 comes from a pure “shear” experiment, and the data point at 1.0 comes from a pure “normal” experiment. Data for the values in between come from combined “shear-normal” experiments.

Referring to the spot weld in Figure 4 and using the derived components normal and shear defined in Defining Derived Components for Connector Elements, an alternative way to define damage in the spot weld is to use:

PARAMETER
$exponent$=2
$u_{fail}^{post}$=0.85
$R_n$=120.0
$R_n$=115.0
CONNECTOR DAMAGE INITIATION, CRITERION=FORCE
, 1.0
CONNECTOR POTENTIAL
normal, $<R_n>$
shear, $<R_s>$
** $\sqrt{{\left(\frac{F_n}{R_n}\right)}^2+{\left(\frac{F_s}{R_s}\right)}^2}$
CONNECTOR DAMAGE EVOLUTION, TYPE=MOTION, SOFTENING=EXPONENTIAL
$<u_{fail}^{post}>$, $<exponent>$
CONNECTOR POTENTIAL
1
2
3
** $\overline{u}=\sqrt{u_1^2+u_2^2+u_3^2}$

Damage will be initiated when the force magnitude defined by the first connector potential definition exceeds the specified value of 1.0. The scale factors $R_n$ and $R_s$ in the first potential definition are used in this case to define a force magnitude that would be 1.0 at damage initiation. A motion-based exponential decay damage evolution law is chosen. The second connector potential definition is associated with the connector damage evolution definition and defines an equivalent motion, $\overline{u}$, in the connection. When the equivalent post-initiation motion, $\overline{u}-{\overline{u}}_0$ (where ${\overline{u}}_0$ is $\overline{u}$ at damage initiation), reaches $u_{fail}^{post}$, ultimate failure occurs. All components (1 through 6) are affected in this case since they all ultimately contribute to the first connector potential definition (see Defining Derived Components for Connector Elements for the specific definitions associated with the normal and shear derived components).

This example illustrates how to specify the contributions of multiple damage mechanisms to the overall damage effect and the components of relative motion affected by the damage evolution law. Most of the data line entries or parameters are not given for conciseness.

** first damage mechanism: force-based damage initiation
** damage variable $d^F$
CONNECTOR DAMAGE INITIATION, COMPONENT=4, CRITERION=FORCE
CONNECTOR DAMAGE EVOLUTION, TYPE=MOTION, SOFTENING=EXPONENTIAL, 
DEGRADATION=MAXIMUM, AFFECTED COMPONENTS
4, 6
**
** second damage mechanism: motion-based damage initiation
** damage variable $d^M$
CONNECTOR DAMAGE INITIATION, COMPONENT=4, CRITERION=MOTION
CONNECTOR DAMAGE EVOLUTION, TYPE=MOTION, SOFTENING=LINEAR, 
DEGRADATION=MULTIPLICATIVE, AFFECTED COMPONENTS
1, 2, 6
**
** third damage mechanism: plastic motion–based damage initiation 
** damage variable $d^P$
CONNECTOR DAMAGE INITIATION, COMPONENT=4, 
CRITERION=PLASTIC MOTION
CONNECTOR DAMAGE EVOLUTION, TYPE=MOTION, SOFTENING=TABULAR, 
DEGRADATION=MULTIPLICATIVE, AFFECTED COMPONENTS
1, 2
**
** fourth damage mechanism: coupled force-based damage initiation
** damage variable $d^{CF}$
CONNECTOR DAMAGE INITIATION, CRITERION=FORCE
CONNECTOR POTENTIAL
** using components 1, 2, 3, 4, 5, 6
CONNECTOR DAMAGE EVOLUTION, TYPE=ENERGY, DEGRADATION=MAXIMUM, 
AFFECTED COMPONENTS
1, 3, 4, 6

Four damage mechanisms (connector damage initiation/connector damage evolution pairs) are specified: three uncoupled and one coupled. The first line of each damage evolution definition establishes the components that will be damaged by the mechanism. The overall damage in a particular component is determined by contributions from all the mechanisms that affect that component. For example, the overall damage in component 1, $d_1$, is determined by the second, third, and fourth damage mechanisms as follows:

$d^M$ and $d^P$ use multiplicative degradation; therefore, they are multiplied first: $\left(1-d^M\right)*\left(1-d^P\right)$. $d^{CF}$ uses maximum degradation, so $\left(1-d^{CF}\right)$ is compared to $\left(1-d^M\right)*\left(1-d^P\right)$ and the minimum value is taken.

For example, assume that at a particular time t, $d^M$=0.5, $d^P$=0.3, and $d^{CF}$=0.2 and at time $t+\mathrm{\Delta }t$, $d^M$=0.6 (the only one increasing) while $d^P$ and $d^{CF}$ stay the same. The overall damage variable gets closer to the ultimate damage value faster when all three damage mechanisms are used than if we use only the $d^M$ mechanism:

while

Complete failure occurs when $1-d_1$ reaches 0.0.

$F_i=\left(1-d_i\right)*F_{ef{f}_i}$, where i refers to the $i^{\mathrm{th}}$ available component of relative motion. The overall damage variables for the other components are determined as follows (based on the specified affected components for each damage evolution law):