Nonlinear finite element analyses confront users with many choices. An understanding of the fundamental concepts of
nonlinear finite element analysis is necessary if you do not want to use the finite element program as a black box.
The purpose of this manual is to describe the numerical methods included in Radioss.

Kinematic constraints are boundary conditions that are placed on nodal velocities. They are mutually exclusive for each degree
of freedom (DOF), and there can only be one constraint per DOF.

The stability of solution concerns the evolution of a process subjected to small perturbations. A process is considered
to be stable if small perturbations of initial data result in small changes in the solution. The theory of stability
can be applied to a variety of computational problems.

A large variety of materials is used in the structural components and must be modeled in stress analysis problems.
For any kind of these materials a range of constitutive laws is available to describe by a mathematical approach the
behavior of the material.

This material law can be used to model low density closed cell polyurethane foams, impactors, impact limiters. It can
only be used with solid elements.

Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.

The performance criterion in the computation was always an essential point in the architectural conception of Radioss. At first, the program has been largely optimized for the vectored super-calculators like CRAY. Then, a first parallel
version SMP made possible the exploration of shared memory on processors.

A large variety of materials is used in the structural components and must be modeled in stress analysis problems.
For any kind of these materials a range of constitutive laws is available to describe by a mathematical approach the
behavior of the material.

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.

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{\epsilon}$$\dot{\epsilon}$, two values of function at for the two immediately lower ${\underset{\_}{\dot{\epsilon}}}_{1}$ and higher ${\underset{\_}{\dot{\epsilon}}}_{2}$ strain rates are read. The related stress is then computed
as:(1)

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 vs 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).

It should be noted that for stability reasons, damping is applied to strain rates with a
damping factor:(3)