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.

In a Cartesian coordinates system, the coordinates of a material point in a reference or initial configuration are denoted
by . The coordinates of the same point in the deformed or final configuration are denoted by .

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.

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.

The diagonal components are called the principal stresses and allow a 3D
representation of the state of stress at a point.

Stress Invariants

Many of the constitutive models in Radioss are formulated
in terms of invariants of the stress tensor. The most important are the first and
second invariants, called pressure and von Mises stress
after Richard von Mises.(2)

The values of these functions remain invariant under transformation by a proper
orthogonal matrix. If,

$\sigma ={R}^{\text{T}}{\sigma}_{0}R$

then,

$p={p}_{0}$

${\sigma}_{vm}={\sigma}_{vm0}$

Invariant Space

It is useful to plot the state of stress as a point in a diagram of pressure and von
Mises stress:

The horizontal axis corresponds to the hydrostatic loading, the vertical axes to pure
shear. The line with tangent 1/3 is uniaxial compression. The line with tangent -1/3
is uniaxial tension.

Deviatoric Stresses

The pressure or first invariant is related to the change in volume of the solid. The
deviation from a hydrostatic state of stress is linked to the change in shape. The
stress deviator is defined as:

$S=\sigma +pI$

The second invariant becomes, in terms of the deviators:(4)