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.
The material LAW14 (COMPSO) in Radioss allows to simulate
orthotropic elasticity, Tsai-Wu plasticity with damage, brittle rupture and strain rate
effects. The constitutive law applies to only one layer of lamina. Therefore, each layer
needs to be modeled by a solid mesh. A layer is characterized by one direction of the fiber
or material. The overall behavior is assumed to be elasto-plastic orthotropic.
Direction 1 is the fiber direction, defined with respect to the local reference frame as shown in 図 1.
For the case of unidirectional orthotropy (that is, and ) the material LAW53 in Radioss
allows to simulate an orthotropic elastic-plastic behavior by using a modified Tsai-Wu
criteria.
Linear Elasticity
When the lamina has a purely linear elastic behavior, the stress calculation algorithm:
Transform the lamina stress, , and strain rate,
, from global reference frame to fiber
reference frame.
Compute lamina stress at time by explicit time integration:
(1)
Transform the lamina stress, , back to global reference frame.
The elastic constitutive matrix of the lamina
relates the non-null components of the stress tensor to those of strain
tensor:(2)
The inverse relation is generally developed in term of the local material axes and nine
independent elastic constants:(3)
Where,
Young's modulus
Shear modulus
Poisson's ratios
Strain components due to the distortion
Orthotropic Plasticity
Lamina yield surface defined by Tsai-Wu yield criteria is used for each
layer:(4)
with:
(=1,2,3);
; ; ;
; ; ;
;
Where, is the yield stress in direction , and denote respectively for compression and tension. represents the yield envelope evolution during work hardening
with respect to strain rate effects:(5)
Where,
Plastic work
Hardening parameter
Hardening exponent
Strain rate coefficient
is limited by a maximum value
:(6)
If the maximum value is reached the material is failed.
In 式 5, the strain rate
effects on the evolution of yield envelope. However, it is also possible to take into
account the strain rate effects on the maximum stress as shown in 図 3.
(a) Strain
rate effect on
(b) No
strain rate effect on
図 3. Strain Rate Dependency
Unidirectional Orthotropy
LAW 53 in Radioss provides a simple model for unidirectional
orthotropic solids with plasticity. The unidirectional orthotropy condition
implies:(7)
The orthotropic plasticity behavior is modeled by a modified Tsai-Wu criterion (Orthotropic Plasticity, 式 4) in which:(8)
Where, is yield stress in 45° unidirectional test. The yield stresses
in direction 11, 22, 12, 13 and 45° are defined by independent curves obtained by
unidirectional tests (図 4). The curves give the stress variation in
function of a so-called strain :(9)