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.
Element degeneration is the collapsing of an element by one or more edges. For example: making an eight node element into
a seven node element by giving nodes 7 and 8 the same node number.
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.
With the stress being separated into deviatoric and pressure (hydrostatic) stress (Stresses in Solids), it is the deviatoric stress
that is responsible for the plastic deformation of the material. The hydrostatic stress will
either shrink or expand the volume uniformly, that is, with proportional change in shape.
The determination of the deviatoric stress tensor and whether the material will plastically
deform requires a number of steps.
Perform an Elastic Calculation
The deviatoric stress is time integrated from the previous known value using the strain
rate to compute an elastic trial stress:(1)
Where,
Shear modulus
This relationship is Hooke's Law, where the strain rate is multiplied by time to give
strain.
Compute von Mises Equivalent Stress and Current Yield Stress
Depending on the type of material being modeled, the method by which yielding or failure is
determined will vary. The following explanation relates to an elastoplastic material
(LAW2).
The von Mises equivalent stress relates a three dimensional state of stress back to a
simple case of uniaxial tension where material properties for yield and plasticity are well
known and easily computed.
The von Mises stress, which is strain rate dependent, is calculated using the
equation:(2)
The flow stress is
calculated from the previous plastic strain:(3)
For material laws 3, 4, 10, 21, 22, 23 and 36, 式 3 is modified according to the different modeling
of the material curves.
Plasticity Check
The state of the deformation must be checked.(4)
If this equation is satisfied, the state of stress is elastic. Otherwise, the flow stress
has been exceeded and a plasticity rule must be used (図 1).
The plasticity algorithm used is due to Mendelson. 1
Compute Hardening Parameter
The hardening parameter is defined as the slope of the strain-hardening part of the
stress-strain curve:(5)
This is used to compute the plastic strain at time :(6)
This plastic strain is time integrated to determine the plastic strain at time :(7)
The new flow stress is found using:(8)
Radial Return
There are many possible methods for obtaining from the trial stress. The most popular method involves a
simple projection to the nearest point on the flow surface, which results in the radial
return method.
The radial return calculation is given in 式 9. 図 2 is a graphic representation of radial return.(9)
1Mendelson A., 「Plasticity: Theory and Application」, MacMillan Co., New York, 1968.