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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.
Composite and Anisotropic Materials
Elastic-plastic Anisotropic Shells (Barlat's Law)

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概要
Radioss^®は、衝突と衝撃ソリューションのための優れた陽解法有限要素ソルバーです。
Tutorials
Discover Radioss functionality with interactive tutorials.
ユーザーガイド
This manual provides details on the features, functionality, and simulation methods available in Altair Radioss.
リファレンスガイド
本マニュアルは、Radiossで使用することのできるすべての入力キーワードとオプションを詳細なリストで提供しています。
例題集
このマニュアルは一般的な問題のタイプに関して、Radiossを用いて解かれた例題を示します。
検証問題
このマニュアルでは、解析済みの検証モデルを紹介します。
よくある質問
このセクションでは、Radiossに関するよくある質問へのクィックレスポンスを提供しています。
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
      - Fabric Law for Elastic Orthotropic Shells (LAW19 and LAW58)
        Two elastic linear models and a nonlinear model exist in Radioss.
      - Nonlinear Pseudo-plastic Orthotropic Solids (LAWS28, 50 and 68)
      - Hill's Law for Orthotropic Plastic Shells
        Hill's law models an anisotropic yield behavior. It can be considered as a generalization of von Mises yield criteria for anisotropic yield behavior.
      - Elastic-plastic Orthotropic Composite Shells
      - Elastic-Plastic Orthotropic Composite Solids
      - Elastic-plastic Anisotropic Shells (Barlat's Law)
    - Elastic-plasticity of Isotropic Materials
    - 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.

Elastic-plastic Anisotropic Shells (Barlat's Law)

Barlat's 3- parameter plasticity model is developed in F. Barlat, J. Lian ^¹ for modeling of sheet under plane stress assumption with an anisotropic plasticity model. The anisotropic yield stress criterion for plane stress is defined as:

図 1.

F = a {| K_{1} + K_{2} |}^{m} + a {| K_{1} - K_{2} |}^{m} + c {| 2 K_{2} |}^{m} - 2 {(σ_{e})}^{m}

Where,

σ_{e}

is the yield stress,

a

and

c

are anisotropic material constants,

m

exponent and

K_{1}

and

K_{2}

are defined by:

図 2.

K_{1} = \frac{σ_{x x} + h σ_{y y}}{2}

$K_{2} = \sqrt{{(\frac{σ_{x x} - h σ_{y y}}{2})}^{2} + p^{2} {(σ_{x y})}^{2}}$

Where,

h

and

p

are additional anisotropic material constants. All anisotropic material constants, except for

p

which is obtained implicitly, are determined from Barlat width to thickness strain ratio

R

from:

図 3.

a = 2 - 2 \sqrt{(\frac{r_{00}}{1 + r_{00}}) (\frac{r_{90}}{1 + r_{90}})}

$c = 2 - a$

$h = \sqrt{(\frac{r_{00}}{1 + r_{00}}) (\frac{1 + r_{90}}{r_{90}})}$

The width to thickness ratio for any angle

φ

can be calculated: ^¹

図 4.

R_{φ} = \frac{2 m {(σ_{e})}^{m}}{(\frac{\partial F}{\partial σ_{x x}} + \frac{\partial F}{\partial σ_{y y}}) σ_{φ}} - 1

Where,

σ_{φ}

is the uniaxial tension in the

φ

direction. Let

φ

= 45°, 式 4 gives an equation from which the anisotropy parameter

p

can be computed implicitly by using an iterative procedure:

図 5.

\frac{2 m {(σ_{e})}^{m}}{(\frac{\partial F}{\partial σ_{x x}} + \frac{\partial F}{\partial σ_{y y}}) σ_{45}} - 1 - r_{45} = 0

注: Barlat's law reduces to Hill's law when using

m

=2

¹ Barlat F. and Lian J., Plastic behavior and stretchability of sheet metals, Part I: A yield function for orthoropic sheets under plane stress conditions, International Journal of Plasticity, Vol. 5, pp. 51-66, 1989.