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
dummy paddings
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.Figure 1. Nominal Stress-strain Curves Defined by User Input Functions
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 , two values of function at for the two immediately lower and higher strain rates are read. The related stress is then computed
as:
Parameters and define the shape of the interpolation functions. If = = 1, then the interpolation is linear.
Figure 2 shows the influence of and parameters.Figure 2. Influence of a and b Parameters
The coupling between the principal nominal stresses in tension is computed using
anisotropic Poisson's ratio:
Where, is the maximum Poisson's ratio in tension, being the maximum Poisson's ratio in compression, and , the exponent for the Poisson's ratio computation (in
compression, Poisson's ratio is always equal to ).
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:
Compute principal nominal strains and strain rates.
Find corresponding stress value from the curve network for each principal
direction.
Compute principal Cauchy stress.
Compute global Cauchy stress.
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).Figure 3. Unloading Behavior (No Unloading Curve Defined) Figure 4. Unloading Behavior (Unloading Curve Defined)
It should be noted that for stability reasons, damping is applied to strain rates with a
damping factor:
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.
An hysteresis decay may be applied when loading, unloading or in both phases
by:
Where,
Hysteresis coefficient
Relaxation parameter
Confined air content may be taken into account, either by using a user-defined function, or
using the following relation: