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.

The model is a continuum, plasticity-based, damage model for concrete. It assumes that the main two failure mechanisms
are tensile cracking and compressive crushing of the concrete 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.

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.

The Gurson constitutive law ^{1} models progressive microrupture through void
nucleation and growth. It is dedicated to high strain rate elasto-viscoplastic porous
metals. A coupled damage mechanical model for strain rate dependent voided material is used.
The material undergoes several phases in the damage process as described in Figure 1.

The constitutive law takes into account the void growth, nucleation and
coalescence under dynamic loading. The evolution of the damage is represented by the void
volume fraction, defined by:(1)

$$f=\frac{{V}_{a}-{V}_{m}}{{V}_{a}}$$

Where,

${V}_{a}$, ${V}_{m}$

Respectively, are the elementary apparent volume of the material and the
corresponding elementary volume of the matrix.

The rate of increase of the void volume fraction is given by:(2)

^{1}Gurson A. L. “Continuum theory of ductile rupture by void nucleation
and growth: Part I - Yield criteria and flow
rules for porous ductile media”, Journal of
Engineering Materials and Technology, Vol. 99,
2-15, 1977.