An explicit is solved by calculating results in small time increments or time steps. The size of the time step depends
on many factors but is automatically calculated by Radioss.
Hyperelastic materials are used to model materials that respond elastically under very large strains. These materials
normally show a nonlinear elastic, incompressible stress strain response which returns to its initial state when unloaded.
Hyperelastic materials can be used to model the isotropic, nonlinear elastic behavior of rubber, polymers, and similar
materials. These materials are nearly incompressible in their behavior and can be stretched to very large strains.
LAW92 describes the Arruda-Boyce material model, which can be used to model hyperelastic behavior. This model is based
on the statistical mechanics of a material with a cubic representative volume element containing eight chains along
the diagonal directions.
This law is a constitutive model for predicting the nonlinear time dependency of elastomer like materials. It uses
a polynomial material model for the hyperelastic material response and the Bergstrom-Boyce material model to represent
the nonlinear viscoelastic time dependent material response.
Composite materials consist of two or more materials combined each other. Most composites consist
of two materials, binder (matrix) and reinforcement. Reinforcements come in three forms, particulate,
discontinuous fiber, and continuous fiber.
Optimization in Radioss was introduced in version 13.0. It is implemented by invoking the optimization capabilities of
OptiStruct and simultaneously using the Radioss solver for analysis.
Hyperelastic materials are used to model materials that respond elastically under very large strains. These materials
normally show a nonlinear elastic, incompressible stress strain response which returns to its initial state when unloaded.
Hyperelastic materials can be used to model the isotropic, nonlinear elastic behavior of rubber, polymers, and similar
materials. These materials are nearly incompressible in their behavior and can be stretched to very large strains.
Hyperelastic materials can be used to model the isotropic, nonlinear elastic behavior
of rubber, polymers, and similar materials. These materials are nearly incompressible in
their behavior and can be stretched to very large strains.
In Radioss, material laws LAW42, LAW62, LAW69, LAW82, and
LAW88 utilize different strain energy density functions of the Ogden material model
1 to model hyperelastic materials. 2
Material Definition
Stretch (also called stretch ratio) is the ratio of final length and initial length. It is used
for materials with large deformations. For a cube in tension:
Engineering strain (also called nominal strain) in
direction 1
Stretch in direction 1
Thus, strain and stretch are related as:(1)
Principal stretch can be used to describe the volumetric deformation by
calculating the relative volume, , computed as: (2)
For an incompressible material the volume should not change and thus =1 and thus, the stretch can be calculated for the
following material tests.
Uniaxial test:
and
Biaxial test:
and
Planar (shear) test:
and
/MAT/LAW42 (Ogden)
This material model defines a hyperelastic, viscous, and incompressible material
specified using the Ogden, Neo-Hookean, or Mooney-Rivlin material models. This law is
generally used to model incompressible rubbers, polymers, foams, and elastomers. This
material can be used with shell and solid elements.
LAW42 uses the following strain energy density
representation of the Ogden material model.(3)
Where,
Strain energy density
ith principal
engineering stretch
Relative volume defined as:
Deviatoric stretch
and
Material constants coefficient pairs.
Up to 5 material constant pairs can be defined.
The initial shear modulus
and bulk modulus () are given by:(4)
and(5)
Where, is the Poisson's ratio and is only used for
computing the bulk modulus.
Material Parameters
Parameters and must be chosen so that initial shear modulus
is:(6)
For material stability, it is required that each material constant pair
(7)
In general, the Ogden model can be used for strains up to 700%. The number of terms
material pairs, and , needed depends on the range of experimental data
that is fit and curve fitting accuracy desired. In practice, 3 material pairs fit
most data. If the material pairs are not known for a particular material, then a
curve fit of uniaxial test data can be done in Radioss
using LAW69 or via separate fitting software.
Neo-Hookean Model
A simple case of the Ogden material model is the Neo-Hooken model represented using
the following equation for the strain energy density function:(8)
Where,
The first invariants of the right Cauchy-Green tensor
Material constant
This representation can be derived from the LAW42 Ogden strain energy density
function when:
;
, and
The Neo-Hookean model is a simple model that is typically only accurate for strains
less than 20%.
Mooney-Rivlin Model
A slightly more complex case of the LAW42 Ogden material model is the Mooney-Rivlin
model, which can be represented using the following equation for the strain energy
density function:(9)
Where,
and
The first and second invariants of the right Cauchy-Green tensor
and
Material constants
This representation can be derived from the LAW42 Ogden strain energy density
function when:
, , , and
Mooney-Rivlin constants are available from a material supplier or testing company. If
they are not available, then a curve fit of uniaxial test data can be done in
Radioss using LAW69 or via separate fitting
software. The Mooney-Rivlin material law is accurate for strains up to 100%.
Poisson's Ratio and Material Incompressibility
If a material is truly incompressible, then . However, in practice is not possible to use because
that would result in an infinite bulk modulus, an infinite speed of sound, and thus
an infinitely small solid element Time Step. (10)
The effect of different Poisson’s ratio input can be seen in Figure 2. The largest difference in the results is
at higher amounts of strain. The results will match the test data better when but this results in a time step that is 4 times
lower than . Thus, to balance the computation time and accuracy
it is recommended to use for incompressible rubber material.
The effect of Poisson’s ratio and Bulk modulus are similar in other Ogden material
law.
Higher values of the Poisson’s ratio may lead to a very small time step or divergence
for explicit simulations.
In LAW42, material incompressibility is provided by using a penalty approach, which
calculates the pressure proportional to a change in density.(11)
Where,
Bulk modulus
Relative volume which simplifies to relative density if mass is
constant
Bulk coefficient scale factor versus relative volume function
Abscissa scale factor for function
The bulk modulus () of hyperelastic materials is generally a very high
value which provides the needed pressure-resistance to maintain the
incompressibility condition ( =1). But if a material starts to compress ( < 1) then the bulk modulus can be increased by
including the fct_IDblk input
function which allows the scaling of the bulk coefficient value as a function of . By default, there is no scaling and; thus, if the
function identifier is zero and the value of the bulk scaling function is equal to
1. It is advisable to output (/ANIM/BRICK/DENS) and review the
material density of LAW42 components to make sure that the density variation is
small, that is the value of is close to 1 and the material is
incompressible.
Viscous (Rate) Effects
Viscous (rate) effects are modeled in LAW42 using a Maxwell model, which can be
described in a simplified manner as a system of springs with stiffness' and dampers :
The Maxwell model is represented using Prony series inputs (). The hyperelastic initial shear modulus
is the same as the long-term shear modulus in the Maxwell model, and is the relaxation time:(12)
The and values must be positive.
/MAT/LAW62 (VISC_HYP)
A hyper visco-elastic material law in Radioss that can be
used to model polymers and elastomers.
The hyperelastic behavior in this material law is defined using the following strain
energy density function:(13)
Poisson’s ratio must be . This law can be used to model compressible or
sometimes called hyperfoam materials by defining a low Poisson’s ratio
value.
Note: The material coefficients are different, but can be
converted using:(14)
Viscous (Rate) Effects
Viscous (rate) effects are modeled in LAW62 using a Maxwell model which can be
described in a simplified manner as a system of n springs with stiffness’ and dampers .
The Maxwell model is represented using a Prony series with inputs. The initial shear
modulus is:(15)
The sum of should be greater than 0.(16)
The stiffness ratio is:(17)(18)
With, (19)
and the ground shear modulus(20)
The relative time, must be positive:(21)
Note: When viscosity is included, the shear modulus in LAW62 is
the initial shear modulus which includes viscosity, but in LAW42 the shear
modulus is the long-term shear modulus, which does not include viscosity .
/MAT/LAW69
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
using test data from a uniaxial engineering stress-strain curve.
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):(22)
law_ID = 2 (Mooney-Rivlin
law):(23)
Material Parameters
After reading the stress-strain curve (fct_ID1),
Radioss calculates the corresponding material
parameter pairs using a nonlinear least-square fitting algorithm. For the classic
Ogden law, (law_ID=1), the calculated material parameter pairs
are and where the value of p is defined
via the N_pair input. The maximum value is
N_pair=5 with a default value of 2. Usually no more than
N_pair=3 is needed for a good fit.
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
increase 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.
To improve the quality of the nonlinear least square fit, it is recommended that:
The experimental data curve represents a smooth monotonically increasing
function with uniform distribution of abscissa points. The number of data
points in the experimental data curve should be greater than the number of
parameter pairs (N_pair).
The engineering strain is negative in compression and positive in tension.
For compression test data, the engineering strain should be greater than
-1.0 (100% compression maximum) but tension only stress strain data can also
be used.
If N_pair ≥ 3, then the test data should cover at least
100% of the tensile strain and/or 50% of the compressive strain.
N_pair should not be set to a very large value to avoid
instabilities in the fitting procedure.
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.
To determine how well the calculated material parameters represent the input test
data, the Radioss Starter outputs an “averaged error of
fitting” value which is recommended to not exceed 10%. For visual comparison, the
stress-strain curve calculated from the strain energy density and calculated
material parameters is also output by the Radioss
Starter.
Due to the friction involved in a uniaxial compression test, it is usually more
accurate to take equal biaxial tension test data and convert it to uniaxial
compressive data using these formulas 3 which are valid for incompressible
materials.(24)(25)
/VISC/PRONY must be used with LAW69 to include viscous effects.
Alternatively, LAW69 could be used to extract the Ogden or Mooney-Rivlin parameters
and then those parameters can be used in LAW42 with viscosity added.
/MAT/LAW82
This material model defines a hyperelastic, and incompressible material specified
using the Ogden, Neo-Hookean, or Mooney-Rivlin material models. This law is generally used
to model incompressible rubbers, polymers, foams, and elastomers.
This material can be used with shell and solid elements. As compared to LAW42 or
LAW62, this law uses a different Ogden strain energy density formulation given in
Equation 26. The LAW82 strain
energy density formulation matches what is used in some other finite elements
solver’s hyperelastic model and; thus, the material parameters for this form of the
Ogden strain energy density are sometimes available from material suppliers or other
sources.(26)
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:(28)
If and =0, a default value of is used and is calculated using Equation 28
Neo-Hookean Model
Like LAW42, LAW82 can also be simplified to a Neo-Hooken model by using:
,
and
Mooney-Rivlin Model
Like LAW42, LAW82 can also be simplified to a Mooney-Rivlin model by using:
, , and
Viscous (Rate) Effects
/VISC/PRONY must be used with LAW82 to include viscous
effects.
Drücker Condition Stability Check
In LAW42 and LAW69, the Drücker stability is automatically calculated by the Radioss Starter.
The Drücker stability condition checks if the change in the Kirchhoff stress
corresponding to the infinitesimal change in the logarithmic strain (true strain)
satisfies the following inequality.(29)
Where, =1,2,3 principal direction.
With the change in logarithmic strain(30)
The change of Kirchhoff stress
Relationship between Kirchhoff stress and logarithmic strain
The Drücker stability condition will be:(31)
Here is tangential material stiffness matrix and it is
also the slope of stress-strain curve:(32)
For a stable material, it requests tangential material stiffness be positive (slope of stress-strain curve is
positive). The tangential material matrix is positive if following conditions
satisfied:(33)(34)(35)
The Kirchhoff stress for Ogden model is:(36)
Since , then for a given Ogden parameter with conditions , the strain range of material in Drücker stability
could then be calculated.
The Drücker stability criterion calculates the strain range where the material model
will remain stable given a set of material parameters. This stability check cannot
be made for every deformation but instead is commonly used to check material
stability under uniaxial, biaxial and planar strain loading.
For example, using the following Ogden parameters:
Then the Drücker stability will be automatically checked in Radioss Starter, and results printed in Starter output file
*0.out. This shows the strain at which instability can
occur for the given Ogden
parameters:
CHECK THE DRUCKER PRAGER STABILITY CONDITIONS
-----------------------------------------------
MATERIAL LAW = OGDEN (LAW42)
MATERIAL NUMBER = 1
TEST TYPE = UNIXIAL
COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN -0.3880000000000
TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN 0.9709999999999
TEST TYPE = BIAXIAL
COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN -0.2880000000000
TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN 0.2780000000000
TEST TYPE = PLANAR (SHEAR)
COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN -0.3680000000000
TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN 0.5829999999999
Note: For a Neo-Hookean material with (or ), the material is always stable and thus no
critical value is found by the Drücker stability check.
For a Mooney-Rivlin material, the Drücker stability should be checked since could be negative, which leads material
instability.
/MAT/LAW88
This law utilizes a 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 the following Ogden’s strain energy density function but
does not require curve fitting to extract material constants like most other
hyperelastic material models. 4(37)
Instead, this law determines the Ogden function directly from the uniaxial
engineering stress strain curve tabulated data.
Unlike other Ogden material laws, the Bulk Modulus must be input from either test
data or extracted from Starter output of the LAW69 Ogden curve fit. When comparing
results between LAW42 or LAW69 to LAW88, the same bulk modulus must be used.
Unloading Behavior
Unloading can be represented using an unloading function or by providing hysteresis
and shape factor inputs to a damage model based on energy.
If using the damage model, the loading curves are used for both loading and unloading
and the unloading stress tensor is reduced by:(38)
with(39)
Where,
Current energy
Maximum energy corresponding to the quasi-static behavior
and
Input by user
If an unloading curve is provided, these options are available:
Tension
Loading and Unloading
= 0
Loading use loading function
fct_IDLi and
Unloading use unloading function
fct_IDunL
= 1
Loading and unloading all use loading function
fct_IDLi and
= -1
Loading use loading function
fct_IDLi and
Tension
Unloading use unloading function
fct_IDunL
Compression
Unloading use loading
fct_IDLi
and
Viscous (Rate) Effects
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.
Conclusion
Make sure to use the material law that best fits the test data available.
For example, if minimal test data is available and the strains are not too large than
the LAW42 Neo-Hookean Model could be used. If the loading state is known, then it is
important to have test data that represents that stress state and make sure the
material model fits that test data.
References
1 Ogden, R. W., and Non-linear Elastic Deformations. "Ellis Horwood." New
York (1984)
2 Miller, Kurt. "Testing Elastomers for Hyperelastic Material Models in
Finite Element Analysis" Axel Products, Inc., Ann Arbor, MI (2017). Last modified
April 5, 2017
4 Kolling, S., P. A. Du Bois, D. J. Benson, and W. W. Feng. "A tabulated
formulation of hyperelasticity with rate effects and damage." Computational
Mechanics 40, no. 5 (2007): 885-899