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Theory Manual
This manual provides detailed information about the theory used in the Altair Radioss Solver.
Large Displacement Finite Element Analysis Theory Manual
Material Laws
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.
Elastic-plasticity of Isotropic Materials
Zerilli-Armstrong Plasticity Model (LAW2)
This law is similar to the Johnson-Cook plasticity model. The same parameters are used to define the work hardening curve.

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Overview
Radioss^® is a leading explicit finite element solver for crash and impact simulation.
Tutorials
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User Guide
This manual provides details on the features, functionality, and simulation methods available in Altair Radioss.
Reference Guide
This manual provides a detailed list of all the input keywords and options available in Radioss.
Example Guide
This manual presents examples solved using Radioss with regard to common problem types.
Verification Problems
This manual presents solved verification models.
Frequently Asked Questions
This section provides quick responses to typical and frequently asked questions regarding Radioss.
Theory Manual
This manual provides detailed information about the theory used in the Altair Radioss Solver.
- Large Displacement Finite Element Analysis Theory Manual
  - Introduction
    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.
  - Basic Equations
  - Finite Element Formulation
  - Dynamic Analysis
  - Element Library
    Radioss element library contains elements for one, two or three dimensional problems.
  - Kinematic Constraints
    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.
  - Linear Stability
    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.
  - Interfaces
    Interfaces solve the contact and impact conditions between two parts of a model.
  - Material Laws
    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.
    - Isotropic Elastic Material
    - Composite and Anisotropic Materials
    - Elastic-plasticity of Isotropic Materials
      - Johnson-Cook Plasticity Model (LAW2)
        In this law the material behaves as linear elastic when the equivalent stress is lower than the yield stress.
      - Zerilli-Armstrong Plasticity Model (LAW2)
        This law is similar to the Johnson-Cook plasticity model. The same parameters are used to define the work hardening curve.
      - Cowper-Symonds Plasticity Model (LAW44)
      - Zhao Plasticity Model (LAW48)
      - Tabulated Piecewise Linear and Quadratic Elasto-plastic Laws (LAW36 and LAW60)
        The elastic-plastic behavior of isotropic material is modeled with user-defined functions for work hardening curve.
      - Drucker-Prager Constitutive Model (LAWS 10, 21 and 81)
      - Brittle Damage: Johnson-Cook Plasticity Model (LAW27)
      - Brittle Damage: Reinforced Concrete Material (LAW24)
        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.
      - Ductile Damage Model
      - Ductile Damage Model for Porous Materials (LAW52)
      - Connect Materials (LAW59)
        For the moment /MAT/LAW59 is only compatible with /PROP/TYPE43 and /FAIL/CONNECT.
    - Viscous Materials
      General case of viscous materials represents a time-dependent inelastic behavior.
    - Hydrodynamic Analysis Materials
    - Void Materials (LAW0)
      This material can be used to define elements to act as a void, or empty space.
    - Shape Memory Superelastic Material (LAW71)
      This material can be used to define elements to act as a void, or empty space.
    - Failure Models
  - Monitored Volume
    An airbag is defined as a monitored volume. A monitored volume is defined as having one or more 3 or 4 node shell property sets.
  - Static
    Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.
  - Radioss Parallelization
    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.
- ALE, CFD and SPH Theory Manual
- Appendix A: Basic Relations of Elasticity
User Subroutines
This manual describes the interface between Altair Radioss and user subroutines.

Zerilli-Armstrong Plasticity Model (LAW2)

This law is similar to the Johnson-Cook plasticity model. The same parameters are used to define the work hardening curve.

However, the equation that describes stress during plastic deformation is:

σ = C_{0} + (C_{1} \exp ((- C_{3} T + C_{4} T In (\frac{\dot{ε}}{{\dot{ε}}_{0}})))) + C_{5} ε_{p}^{n}

Where,

$σ$: Stress (Elastic + Plastic Components)
$ε_{p}$: Plastic strain
$τ$: Temperature (computed as in Johnson-Cook plasticity)
$C_{0}$: Yield stress
$n$: Hardening exponent
$\dot{ε}$: Strain rate, must be 1 s^-1 converted into user's time unit
${\dot{ε}}_{0}$: Reference strain rate

Additional inputs are:

$σ_{\max 0}$: Maximum flow stress
$ε_{\max}$: Plastic strain at rupture

The $ε_{\max}$ enables to define element rupture as in the former law. The theoretical aspects related to strain rate computation and filtering are also the same.