/MAT/LAW69

Block Format Keyword

/MAT/LAW69 - Incompressible Hyperelastic Material with Tabulated Input

Description

This law (extension of LAW42) defines a hyperelastic and incompressible material specified using the Ogden, Mooney-Rivlin material models. It is generally used to model incompressible rubbers, polymers, foams, and elastomers. Material parameters are computed from engineering stress-strain curve from uniaxial tension and compression tests. It is used with shell and solid elements.

Format

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
/MAT/LAW69/mat_ID/unit_ID
mat_title

law_ID	fct_IDblk			Fscaleblk		N_pair	Icheck
fct_ID1

Field	Contents	SI Unit Example
mat_ID	Material identifier (Integer, maximum 10 digits)
unit_ID	Optional unit identifier (Integer, maximum 10 digits)
mat_title	Material title (Character, maximum 100 characters)
	Initial density (Real)
law_ID	Hyperelastic material model type (Comment 2) (Integer) = 1: Ogden model = 2: Mooney-Rivlin model
fct_IDblk	Function which scales the bulk coefficient as a function of the relative volume (Comment 6) (Integer)
	Poisson’s ratio (Real)
Fscaleblk	Scale factor for fct_IDblk Default = 1.0 (Real)
N_pair	Number of material parameter ( and ) pairs in the representation of the strain energy density function (W). The material parameters are calculated from the given stress-strain curve (fct_ID1). (N_pair ≤ 5) Default = 2 (Integer)
Icheck	Validity check of material parameters ( and ) Default = 3 (Integer) = 1: > 0 with (p=1,…5) = 2: > 0 with (p=1,…5) = 3: The curve fitting procedure starts with Icheck= 2. It is switched to Icheck= 1, if a proper fit is not found.
fct_ID1	Function identifier for the engineering stress-strain curve from uniaxial compression and tension test. (Integer)

#RADIOSS STARTER

/UNIT/1

unit for mat

Mg mm s

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

#- 2. MATERIALS:

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

/MAT/LAW69/1/1

LAW69 rubber

# RHO_I

1E-9

# LAW_ID FCT_ID NU FSCALE N_PAIR ICHECK

2 0 .495 0 2 0

# FCT_ID1

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

#- 3. FUNCTIONS:

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

/FUNCT/2

LAW69 e.strain e.stress

# X Y

0 0

.03 .30

.06 .55

.10 .80

.20 1.4

.30 2.0

.50 2.7

.70 3.4

1.0 4.0

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

#ENDDATA

/END

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

#RADIOSS STARTER

/UNIT/1

unit for mat

Mg mm s

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

#- 2. MATERIALS:

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

/MAT/LAW69/1/1

LAW69 rubber

# RHO_I

1E-9

# LAW_ID FCT_ID NU FSCALE N_PAIR ICHECK

1 0 .495 0 2 0

# FCT_ID1

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

#- 3. FUNCTIONS:

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

/FUNCT/2

LAW69 e.strain e.stress

# X Y

0 0

.03 .30

.06 .55

.10 .80

.20 1.4

.30 2.0

.50 2.7

.70 3.4

1.0 4.0

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

#ENDDATA

/END

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

1.	RADIOSS currently accepts test data from the following deformation schemes:

Uniaxial tension and compression.

The input stress-strain data (fct_ID1) is engineering stress as a function of engineering strain. The engineering strain should be monotonically increasing ranging from a negative value in compression to a positive value in tension. In compression, the engineering strain should be greater than -1.0. If data in fct_ID1 is not complete (only tension data), then only tension will be considered.

2.	In cases where the material law is identified by the identifier law_ID using a different kind of strain energy potential, W.

•	law_ID = 1 (Ogden law):

•	law_ID = 2 (Mooney-Rivlin law):

3.	After reading the stress-strain curve (fct_ID1), RADIOSS calculates the corresponding parameter pairs using nonlinear least-square fitting.

•	For classic Ogden law,

the parameter pairs are and (p=1,…5, max of N_pair is 5)

•	For Mooney-Rivlin law,

the parameter pairs are and (p=1,2, N_pair always equals 2)

4.	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).

•	If N_pair > 3, 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 so as to avoid instabilities in the fitting procedure.

•

RADIOSS Starter outputs the “averaged error of fitting” between input (experimental) and the stress-strain curve which is calculated from the strain energy density function (W) using the corresponding material parameters determined during the fitting process. The maximum “averaged error of fitting” should not exceed 10%.

This material law is stable when

(with p=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 a weaker condition (Icheck= 1:

) is used. The latter is a necessary condition to enforce that the ground shear hyperelastic modulus (

) is positive.

6.	Material incompressibility is provided by using a penalty approach, which calculates the pressure proportional to a change in density:

fblk is function of fct_IDblk

The proportionality coefficient (K) is the bulk coefficient which is generally a very high value. This provides a significantly high value for the pressure-resistance when the incompressibility condition (J=1) is violated. The Jacobian (J) can be interpreted as the ratio of the current element volume with respect to the initial element volume.

fct_IDblk provides additional control for the incompressibility (see figure below). It allows the scaling up of the bulk coefficient value based on the value of J. By default, the function identifier is zero and the value of the bulk scaling function is equal to 1. It is advisable to output and control the density distribution of LAW69 components to make sure that the density variation is small, i.e. the value of J is close to 1.

starter_mat_ogden

7.	Poisson’s ratio is used only for computing the bulk modulus (K).

For pure incompressible materials, . This value of Poisson’s ratio implies an infinite value for the bulk modulus (K). Therefore, the recommended Poisson’s ratio for incompressible materials is . Higher values of the Poisson’s ratio may lead to a small time step value or divergence in case of implicit and explicit simulations.

8.	Further explanation about this law can be found in “Non-Linear Elastic Deformations”, by R.W Ogden, Ellis Horwood, 1984.