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.

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.

Fabric Law for Elastic Orthotropic
Shells (LAW19 and LAW58)

Two elastic linear models and a nonlinear model exist in Radioss.

Fabric Linear Law for Elastic
Orthotropic Shells (LAW19)

A material is orthotropic if its behavior is symmetrical with respect to two orthogonal
plans. The fabric law enables to model this kind of behavior. This law is only available for
shell elements and can be used to model an airbag fabric. Many of the concepts for this law
are the same as for LAW14 which is appropriate for composite solids. If axes 1 and 2
represent the orthotropy directions, the constitutive matrix $C$ is defined in terms of material properties:(1)

Therefore, six independent material properties are the input of the material:

${E}_{11}$

Young's modulus in direction 1

${E}_{22}$

Young's modulus in direction 2

$\upsilon $_{12}

Poisson's ratio

${G}_{12}$, ${G}_{23}$, ${G}_{31}$

Shear moduli for each direction

The coordinates of a global vector $\overrightarrow{V}$ is used to
define direction 1 of the local coordinate system of orthotropy.

The angle $\text{\Phi}$ is the angle between the local direction 1 (fiber direction)
and the projection of the global vector $\overrightarrow{V}$ as shown in Figure 1.

The shell normal defines the positive direction for $\text{\Phi}$. Since fabrics have different compression and tension
behavior, an elastic modulus reduction factor, R_{E}, is defined that changes the
elastic properties of compression. The formulation for the fabric law has a
${\sigma}_{11}$ reduction if ${\sigma}_{11}$ < 0 as shown in Figure 2.

Fabric Nonlinear Law for Elastic
Anisotropic Shells (LAW58)

This law is used with Radioss standard shell elements and
anisotropic layered property (TYPE16). The fiber directions (warp and weft) define the local
axes of anisotropy. Material characteristics are determined independently in these axes.
Fibers are nonlinear elastic and follow the equation:(3)

The shear in fabric material is only supposed to be function of the angle between current
fiber directions (axes of anisotropy):(4)

${G}_{A}=({G}_{0}-G)\mathrm{tan}({\alpha}_{T})$, $G=\frac{{G}_{T}}{1+{\mathrm{tan}}^{2}({\alpha}_{T})}$ with ${\tau}_{0}={G}_{0}\mathrm{tan}({\alpha}_{0})$

Where, ${\alpha}_{T}$ is a shear lock angle, ${G}_{T}$ is a tangent shear modulus at ${\alpha}_{T}$, and ${G}_{0}$ is a shear modulus at $\alpha $ = 0. If ${G}_{0}$ = 0, the default value is calculated to avoid shear modulus
discontinuity at ${\alpha}_{T}$: ${G}_{0}$ = $G$.

${\alpha}_{0}$ is an initial angle between fibers defined in the shell
property (TYPE16).

The warp and weft fiber are coupled in tension and uncoupled in compression. But there is
no discontinuity between tension and compression. In compression only fiber bending
generates global stresses. Figure 4 illustrates the mechanical behavior of
the structure.

A local micro model describes the material behavior (Figure 5). This model represents just ¼ of a
warp fiber wave length and ¼ of the weft one. Each fiber is described as a nonlinear beam
and the two fibers are connected with a contacting spring. These local nonlinear equations
are solved with Newton iterations at membrane integration point.