# Spring Hardening

Isotropic, kinematic or uncoupled spring hardening options can be defined by the hardening flag $\mathrm{H}$ .

These examples only include the spring stiffness without any damping.

## Linear Elastic Spring, H=0

`fct_ID`

_{1i}=

`fct_ID`

_{4i}=0. For linear spring, $\mathrm{H}$ is always 0.

## Nonlinear Elastic Spring, H=0

`fct_ID`

_{1i}. Since the model is elastic, the loading and unloading follow the same path.

## Nonlinear Elastic Plastic Spring with Isotropic Hardening, H=1

`fct_ID`

_{1i}and unloading stiffness ${K}_{u}$ is input using ${K}_{i}$ .

## Nonlinear Elastic Plastic Spring with Uncoupled Hardening, H=2

`fct_ID`

_{1i}and unloading stiffness ${K}_{u}$ is input using ${K}_{i}$ . When uncoupled hardening $\mathrm{H}$ =2, is used, the tensile and compression behavior are uncoupled. Thus, once the unloading reaches zero force, there is no stiffness until zero displacement and then the compressive loading follows the force displacement curve.

## Nonlinear Elastic Plastic Spring with Kinematic Hardening, H=4

`fct_ID`

_{1i}and unloading

`fct_ID`

_{3i}are mandatory and shown in Figure 6as ${\mathrm{f}}_{1}$ and ${\mathrm{f}}_{3}$ . The loading curve should be positive for all values of abscissa. The unloading curve in this case should be negative for all values of abscissa. These curves represent upper and lower limits of yield force as function of current spring length variation or strain. The force follows $K$ between function ${\mathrm{f}}_{1}$ and ${\mathrm{f}}_{3}$ and is input as ${K}_{i}$ .

## Nonlinear Elastic Plastic Spring Nonlinear Unloading, H=5

When $\mathrm{H}$ =5, uncoupled hardening in compression and tensile with nonlinear unloading is modeled.

with, ${\delta}_{resid}={\mathrm{f}}_{3}\left({\delta}_{peak}\right)$

`fct_ID`

_{1i}and residual deformation function ${\mathrm{f}}_{3}$ input as

`fct_ID`

_{3i}.

Comparing Figure 10 and Figure 11, shows that the function ${\mathrm{f}}_{3}$ only effects the residual displacement ${\delta}_{resid}$ and the shape of unloading curve. The shape of unloading curve is controlled by stiffness $K$ and ${\delta}_{peak}$ (unloading start displacement).

If the same stiffness $K$ and same ${\delta}_{peak}$ are used, then the unloading curve has the same shape.

If the same stiffness $K$ but different ${\delta}_{peak}$ are used, then the unloading curve has a different shape.

## Nonlinear Elastic Plastic Spring Istropic Hardening and Nonlinear Unloading, H=6

`fct_ID`

_{1i}and unloading curve in ${\mathrm{f}}_{3}$ is defined using

`fct_ID`

_{3i}.

## Nonlinear Elastic Plastic Spring Elastic Hysteresis, H=7

`fct_ID`

_{1i}and unloading curve in ${\mathrm{f}}_{3}$ is defined using

`fct_ID`

_{3i}.

## Nonlinear Elastic Total Length Function, H=8

`fct_ID`

_{1i}to define the force versus total spring length.

## Dashpot

`fct_ID`

_{4}.

### Damping Using a Function

Remembering that the $\mathrm{g}$ function scales the force are ${\mathrm{f}}_{1}\cdot \mathrm{g}$ , whereas the $\mathrm{h}$ function adds to the force ${\mathrm{f}}_{1}+\mathrm{h}$ . Figure 19 compares these two different methods.

## Inconsistent Stiffness

WARNING ID: 506 ** WARNING IN SPRING PROPERTY ** WARNING IN SPRING PROPERTY SET ID=XXX STIFFNESS VALUE 100 IS NOT CONSISTENT WITH THE MAXIMUM SLOPE (4550) OF THE YIELD FUNCTION ID=X THE STIFFNESS VALUE IS CHANGED TO 1000