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.

Brittle Damage: Johnson-Cook
Plasticity Model (LAW27)

Johnson-Cook plasticity model is presented in Johnson-Cook Plasticity Model (LAW2). For
shell applications, a simple damage model can be associated to this law to take into account
the brittle failure. The crack propagation occurs in the plan of shell in the case of
mono-layer property and through the thickness if a multi-layer property is defined (Figure 1).

The elastic-plastic behavior of the material is defined by Johnson-Cook model. However, the
stress-strain curve for the material incorporates a last part related to damage phase as
shown in Figure 2. The damage parameters are:

${\epsilon}_{t1}$

Tensile rupture strain in direction 1

${\epsilon}_{m1}$

Maximum strain in direction 1

d_{max1}

Maximum damage in direction 1

${\epsilon}_{f1}$

Maximum strain for element deletion in direction 1

The element is removed if one layer of element reaches the failure tensile strain, ${\epsilon}_{f1}$. The nominal and effective stresses developed in an element
are related by:(1)

$${\sigma}_{n}={\sigma}_{eff}\left(1-d\right)$$

Where,

$\text{0d1}$

Damage factor

The strains and the stresses in each direction are given by:(2)

The mathematical approach described here can be applied to the modeling of rivets. Predit
law in Radioss allows achievement of this end by a simple model
where for the elastic-plastic behavior a Johnson-Cook model or a tabulated law (LAW36) may
be used.