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.
Two kinds of isotropic elastic materials are considered:
Linear elastic materials with Hooke’s law,
Nonlinear elastic materials with Ogden, Mooney-Rivlin and Arruda-Boyce laws.
Linear Elastic Material
(LAW1)
This material law is used to model purely elastic materials, or materials that remain
in the elastic range. The Hooke's law requires only two values to be defined; the
Young's or elastic modulus , and
Poisson's ratio, . The law represents a linear relation between stress and
strain.
Ogden Materials (LAW42, LAW69 and
LAW82)
Ogden's law is applied to slightly compressible materials as rubber or elastomer
foams undergoing large deformation with an elastic behavior. The detailed theory for
Odgen material models can be found in 1. The strain energy is expressed in a general form as a function of :(1)
Where, , ith principal stretch
, with being the ith principal engineering
strain
is relative volume with:(2)
is the deviatoric stretch(3)
and are the material constants.
is order of Ogden model and defines the number of
coefficients pairs .
This law is very general due to the choice of coefficient pair .
If =1, then one pair of material constants is needed andin this
case if then it becomes a Neo-hookean material
model.
If =2 then two pairs of material constants are needed and in this
case if and then it becomes a Mooney-Rivlin material
model
For uniform dilitation:(4)
The strain energy function can be decomposed into deviatoric part and spherical part :(5)
With:
The stress corresponding to this strain energy is given
by:(6)
For which the deviator of the Cauchy stress tensor , and the pressure would be:(9)(10)
Only the deviatoric stress above is retained, and the pressure is computed
independently:(11)
Where, a user-defined function related to the bulk modulus
in LAW42 and LAW69:(12)
For an imcompressible material , and no pressure in material.(13)
With being the initial shear modulus, and
the Poisson's ratio.
Note: For an incompressible
material you have . However, is a good compromise to avoid too small time
steps in explicit codes.
A particular case of the Ogden material model is the Mooney-Rivlin material law which
has two basic assumptions:
The rubber is incompressible and isotropic in unstrained state
The strain energy expression depends on the invariants of Cauchy tensor
The three invariants of the Cauchy-Green tensor are:(14)(15)
For incompressible material:(16)
The Mooney-Rivlin law gives the closed expression of strain energy
as:(17)
with:(18)
The model can be generalized for a compressible material.
Viscous Effects in LAW42
Viscous effects are modeled through the Maxwell model:
Where, the shear modulus of the hyper-elastic law is exactly the long-term shear modulus .(19)
are relaxation times:
Rate effects are modeled through visco-elasticity using convolution integral using
Prony series. This corresponds to extension of small deformation theory to finite
deformation.
This viscous stress is added to the elastic one.
The visco-Kirchoff stress is given by:(20)
Where,
Order of the Maxwell model
Deformation gradient matrix
Denotes the deviatoric part of tensor
The viscous-Cauchy stress is written as:(21)
LAW69, Ogden Material Law (Using Test Data as Input)
This law, like /MAT/LAW42 (OGDEN) defines a hyperelastic and
incompressible material specified using the Ogden or Mooney-Rivlin material models.
Unlike LAW42 where the material parameters are input this law computes the material
parameters from an input engineering stress-strain curve from a uniaxial tension and
compression tests. This material can be used with shell and solid elements.
The strain energy density formulation used depends on the law_ID.
law_ID =1, Ogden law (Same as LAW42):
law_ID =2, Mooney-Rivlin law
Curve Fitting
After reading the stress-strain curve
(fct_ID1), Radioss
calculates the corresponding material parameter pairs using a nonlinear
least-square fitting algorithm. For classic Ogden law,
(law_ID =1), the calculated material parameter pairs are and where the value of is defined via the N_pair
input. The maximum value of N_pair = 5 with a default value
of 2.
For the Mooney-Rivlin law (law_ID =2), the material parameter and are calculated remembering that and for the LAW42 Ogden law can be calculated using this
conversion.
, ,
and
.
The minimum test data input should be a uniaxial tension engineering stress strain
curve. If uniaxial compression data is available, the engineering strain should
increate monotonically from a negative value in compression to a positive value in
tension. In compression, the engineering strain should not be less than -1.0 since
-100% strain is physically not possible.
This material law is stable when (with =1,…5) is satisfied for parameter pairs for all
loading conditions. By default, Radioss tries to fit the
curve by accounting for these conditions
(Icheck=2). If a proper fit
cannot be found, then Radioss uses a weaker condition
(Icheck=1: ) which ensures that
the initial shear hyperelastic modulus () is positive.
Once the material parameters are calculated by the Radioss Starter in LAW69, all the calculations done by LAW69 in the simulation are the
same as LAW42.
LAW82
The Ogden model used in LAW82 is:(22)
The Bulk Modulus is calculated as based on these rules:
If , should be entered.
If , input is ignored and will be recalculated
and output in the Starter output using:(23)
If and =0, a default value of is used and is calculated using Equation 23
LAW88, A simplified hyperelastic material with strain rate effects
This
law utilizes tabulated uniaxial tension and compression engineering stress and
strain test data at different strain rates to model incompressible materials. It
is only compatible with solid elements. The material is based on Ogden’s strain
energy density function but does not require curve fitting to extract material
constants like most other hyperelastic material models. Strain rate effects can
be modeled by including engineering stress strain test data at different strain
rates. This can be easier than calculating viscous parameters for traditional
hyperelastic material models. The following Ogden strain energy density function
is used but instead of extracting material constants via curve fitting this law
determines the Ogden function directly from the uniaxial engineering stress
strain curve tabulated data. 5(24)
Unloading can be represented using an unloading
function, FscaleunL, or by
providing hysteresis, Hys and shape factor, Shape, inputs to a damage model based on energy.
When using the damage model,
the loading curves are used for both loading and unloading and the unloading
stress tensor is reduced by:(25)(26)
If the unloading function,
FscaleunL, is entered,
unloading is defined based on the unloading flag, Tension and
the damage model is not used.
Arruda-Boyce Material
(LAW92)
LAW92 describes the Arruda-Boyce material model, which can be used to model
hyperelastic behavior. The Arruda-Boyce model is based on the statistical mechanics
of a material with a cubic representative volume element containing eight chains
along the diagonal directions. It assumes that the chain molecules are located on
the average along the diagonals of the cubic in principal stretch space.
The strain energy density function is:(27)
Where, Material constant are:
Shear modulus
Initial shear modulus
(28)
is the limit of stretch which describes the
beginning of hardening phase in tension (locking strain in tension) and so it is
also called the locking stretch.
Arruda-Boyce is always stable if positive values of the shear modulus,
, and the locking stretch, are used.
is deviatoric part of first strain invarient (29)
with
is a material parameter for the bulk modulus
computation given as:(30)
The Cauchy stress corresponding to above strain energy is:(31)
For incompressible materials, the Cauchy stress is then given by:
Uniaxial test(32)
with and , then
and nominal stress
is:(33)
Equibiaxial test(34)
with and , then
and the nominal stress
is:(35)
Planar test(36)
with and , then
and nominal stress
is:(37)
Additional information about Arruda-Boyce model. 23
Yeoh Material (LAW94)
The Yeoh model (LAW94) 4is a hyperelastic material model that can be used to
describe incompressible materials. The strain energy density function of LAW94 only
depends on the first strain invariant and is computed as:(38)
Where,
First strain invariant
Deviatoric stretch
The Cauchy stress is computed as:(39)
For incompressible materials with =1 only and are input and the Yeoh model is reduced to a
Neo-Hookean model.
The material constant specify the deviatoric part (shape change) of the material and
parameters , , specify the volumetric change of the material. These
six material constants need to be calculated by curve fitting material test data.
RD-E: 5600 Hyperelastic Material with Curve Input includes a
Yeoh fitting Compose script for uniaxial test data. The
Yeoh material model has been shown to model all deformation models even if the curve
fit was obtained using only uniaxial test data.
The initial shear modulus and the bulk modulus are computed as:
and
LAW94 is available only as an incompressible material model.
If =0, an incompressible material is considered, where and is calculated as:(40)
2 Arruda,
E.M. and Boyce, M.C., “A three-dimensional model for the large stretch behavior of
rubber elastic materials”, J. Mech. Phys. Solids, 41(2), pp. 389–412, 1993.
3 Jörgen
Bergström, “Mechanics of solid polymers: theory and computational modeling”, pp.
250-254, 2015.
4 Yeoh,
O. H., “Some forms of the strain energy function for rubber”, Rubber Chemistry and
Technology, Vol. 66, Issue 5, pp. 754-771, November 1993.
5 Kolling
S., Du Bois P.A., Benson D.J., and Feng W.W., "A tabulated formulation of hyperelasticity
with rate effects and damage." Computational Mechanics 40, no. 5 (2007).