# Tabulated Strain Rate Dependent Law for Viscoelastic Materials (LAW38)

The law incorporated in Radioss can only be used with solid elements.

It can be used to model:
• polymers
• elastomers
• foam seat cushions
• hyperfoams
• hypoelastic materials
In compression, the nominal stress-strain curves for different strain rates are defined by you (Figure 1). Up to 5 curves may be input. The curves represent nominal stresses versus engineering strains.

The first curve is considered to represent the static loading. All values of the strain rate lower than the assumed static curve are replaced by the strain rate of the static curve. It is reasonable to set the strain rate corresponding to the first curve equal to zero. For strain rates higher than the last curve, values of the last curve are used. For a given value of $\text{ε}$ $\stackrel{˙}{\epsilon }$ , two values of function at for the two immediately lower ${\underset{_}{\stackrel{˙}{\epsilon }}}_{1}$ and higher ${\underset{_}{\stackrel{˙}{\epsilon }}}_{2}$ strain rates are read. The related stress is then computed as:

$\sigma ={\sigma }_{2}+\left({\sigma }_{1}-{\sigma }_{2}\right){\left[1-{\left(\frac{\stackrel{˙}{\epsilon }-{\stackrel{˙}{\epsilon }}_{1}}{{\stackrel{˙}{\epsilon }}_{2}-{\stackrel{˙}{\epsilon }}_{1}}\right)}^{a}\right]}^{b}$

Parameters $a$ and $b$ define the shape of the interpolation functions. If $a$ = $b$ = 1, then the interpolation is linear.

Figure 2 shows the influence of $a$ and $b$ parameters.

The coupling between the principal nominal stresses in tension is computed using anisotropic Poisson's ratio:

${\nu }_{ij}={\nu }_{c}+\left({\nu }_{t}-{\nu }_{c}\right)\left(1-\mathrm{exp}\left(-{R}_{v}|{\epsilon }_{ij}|\right)\right)$

Where, ${\nu }_{t}$ is the maximum Poisson's ratio in tension, ${\nu }_{c}$ being the maximum Poisson's ratio in compression, and ${R}_{v}$ , the exponent for the Poisson's ratio computation (in compression, Poisson's ratio is always equal to ${\nu }_{c}$ ).

In compression, material behavior is given by nominal stress versus nominal strain curves as defined by you for different strain rates. Up to 5 curves may be input.

The algorithm of the formulation follows several steps:
1. Compute principal nominal strains and strain rates.
2. Find corresponding stress value from the curve network for each principal direction.
3. Compute principal Cauchy stress.
4. Compute global Cauchy stress.
5. Compute instantaneous modulus, viscosity and stable time step.
Stress, strain and strain rates must be positive in compression. Unloading may be either defined with an unloading curve, or else computed using the "static" curve, corresponding to the lowest strain rate (Figure 3 and Figure 4).

It should be noted that for stability reasons, damping is applied to strain rates with a damping factor:

${\stackrel{˙}{\epsilon }}^{n-1}+{R}_{D}\left({\stackrel{˙}{\epsilon }}^{n}-{\stackrel{˙}{\epsilon }}^{n-1}\right)$

The stress recovery may be applied to the model in order to ensure that the stress tensor is equal to zero, in an undeformed state.

$\sigma =\sigma \cdot H\cdot \mathrm{min}\left(1,\left(1-{e}^{-\beta \epsilon \left(t\right)}\right)\right)$

Where,
$H$
Hysteresis coefficient
$\beta$
Relaxation parameter

Confined air content may be taken into account, either by using a user-defined function, or using the following relation:

${P}_{air}={P}_{0}\frac{\left(1-\frac{V}{{V}_{0}}\right)}{\left(\frac{V}{{V}_{0}}-\Phi \right)}$

The relaxation may be applied to air pressure:

${P}_{air}=\mathrm{min}\left({P}_{air},{P}_{\mathrm{max}}\right)\mathrm{exp}\left(-{R}_{p}t\right)$