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.

General case of viscous materials represents a time-dependent inelastic behavior.

However, special attention is paid to the viscoelastic materials such as polymers exhibiting a
rate- and time-dependent behavior. The viscoelasticity can be represented by a recoverable
instantaneous elastic deformation and a non-recoverable viscous part occurring over the time.
The characteristic feature of viscoelastic material is its fading memory. In a perfectly elastic
material, the deformation is proportional to the applied load. In a perfectly viscous material,
the rate of change of the deformation over time is proportional to the load. When an
instantaneous constant tensile stress ${\sigma}_{0}$ is applied to a viscoelastic material, a slow continuous
deformation of the material is observed. When the resulting time dependent strain $\epsilon \left(t\right)$, is measured, the tensile creep compliance is defined
as:(1)

The creep behavior is mainly composed of three phases:

Primary creep with fast decrease in creep strain rate

Secondary creep with slow decrease in creep strain rate

Tertiary creep with fast increase in creep strain rate.

The creep strain rate is the slope of creep strain to time curve.

Another kind of loading concerns viscoelastic materials subjected to a constant tensile
strain, ${\epsilon}_{0}$. In this case, the stress, $\sigma \left(t\right)$ which is called stress relaxation, gradually decreases. The tensile
relaxation modulus is then defined as:(2)

Because viscoelastic response is a combination of elastic and viscous responses, the creep
compliance and the relaxation modulus are often modeled by combinations of springs and dashpots.
A simple schematic model of viscoelastic material is given by the Maxwell model shown in 図 1. The model is composed of an elastic spring
with the stiffness $E$ and a dashpot assigned a
viscosity $\mu $. It is assumed that the total strain is the sum of the elastic and viscous
strains:(3)

$$\epsilon ={\epsilon}^{e}+{\epsilon}^{v}$$

The time derivation of the last expression gives the expression of the total strain
rate:(4)

Where, $R\left(t\right)$ is the relaxation modulus. The last equation is valid for the
special case of Maxwell one-dimensional model. It can be extended to the multi-axial case
by:(10)

Where, ${C}_{ijkl}$ are the relaxation moduli. The Maxwell
model represents reasonably the material relaxation. But it is only accurate for secondary creep
as the viscous strains after unloading are not taken into account.

Another simple schematic model for viscoelastic materials is given by Kelvin-Voigt solid. The
model is represented by a simple spring-dashpot system working in parallel as shown in 図 2.

The mathematical relation of Kelvin-Voigt solid is written as:(11)

$$\sigma =E\epsilon +\eta \dot{\epsilon}$$

When $\eta =0$ (no dashpot), the system is a linearly elastic system. When $E$=0 (no spring), the material behavior is expressed by Newton's
equation for viscous fluids. In the above relation, a one-dimensional model is considered. For
multiaxial situations, the equations can be generalized and rewritten in tensor form.

The Maxwell and Kelvin-Voigt models are appropriate for ideal stress relaxation and creep
behaviors. They are not adequate for most of physical materials. A generalization of these laws
can be obtained by adding other springs to the initial models as shown in 図 3 and 図 4. The equations related to the generalized Maxwell model are
given as:(12)

The models described above concern the viscoelastic materials. The plasticity can be
introduced in the models by using a plastic spring. The plastic element is inactive when the
stress is less than the yield value. The modified model is able to reproduce creep and
plasticity behaviors. The viscoplasticity law (LAW33) in Radioss
will enable to implement very general constitutive laws useful for a large range of applications
as low density closed cells polyurethane foam, honeycomb, impactors and impact limiters.

The behavior of viscoelastic materials can be generalized to three dimensions by separating
the stress and strain tensors into deviatoric and pressure components:(19)

Where, ${s}_{ij}$ and ${e}_{ij}$ are the stress and strain deviators. ${\epsilon}_{kk}$, $\Psi \left(t\right)$ and $K\left(t\right)$ are respectively
the dilatation and the shear and bulk relaxation moduli.

Boltzmann Viscoelastic Model (LAW34)
This law valid for solid elements can be used for viscoelastic materials like polymers, elastomers, glass and fluids.

Generalized Kelvin-Voigt Model (LAW35)
This law uses a generalized viscoelastic Kelvin-Voigt model whereas the viscosity is based on the Navier equations.

Visco-elasto Materials for Foams (LAW33)
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.