HyperWorks Solvers

/MAT/LAW42 (OGDEN)

/MAT/LAW42 (OGDEN)

 Previous topic Next topic Expand/collapse all hidden text  

/MAT/LAW42 (OGDEN)

 Previous topic Next topic JavaScript is required for expanding text JavaScript is required for the print function  

Block Format Keyword

/MAT/LAW42 - Incompressible Visco-hyperelastic Material

Description

This keyword defines a hyperelastic, viscous, and incompressible material specified using the Ogden, 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.

Format

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

/MAT/LAW42/mat_ID/unit_ID or /MAT/OGDEN/mat_ID/unit_ID

mat_title

 

 

 

 

 

 

 

 

 

fct_IDblk

Fscaleblk

M

Iform

Blank Format

Blank Format

 

If M > 0

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

G1

G2

G3

G4

G5

. . . M values of G (five per line)

. . . M values of (five per line)
hmtoggle_plus1Flag Definition

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)

Iform

Incompressibility for shell elements formulation flag. This flag is not effective for solid elements.

= 0: normal stretch of shell elements is calculated to provide element incompressibility.

= 1: same formulation as for solid elements is used.

 

Poisson’s ratio (Comment 2)

(Real)

 

Cut-off stress in tension

Default = 1030  (Real)

fct_IDblk

Function identifier which scales the bulk coefficient as a function of the relative volume (Comment 5)

(Integer)

 

Fscaleblk

Bulk function scale factor

Default = 1.0  (Real)

 

M

Number of viscous terms in the Prony series (order of the Maxwell model)

(Integer)

 

First parameter of the ground shear hyperelastic modulus

(Real)

Second parameter of the ground shear hyperelastic modulus

(Real)

Third parameter of the ground shear hyperelastic modulus

(Real)

Fourth parameter of the ground shear hyperelastic modulus

(Real)

Fifth parameter of the ground shear hyperelastic modulus

(Real)

First material exponent

(Real)

 

Second material exponent

(Real)

 

Third material exponent

(Real)

 

Fourth material exponent

(Real)

 

Fifth material exponent

(Real)

 

Gi

ith multiplier of the Prony viscous term (Comment 6)

(Real)

ith time relaxation of the Prony viscous term

(Real)

hmtoggle_plus1Example (Rubber)

#RADIOSS STARTER

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

/UNIT/1

unit for mat

                 kg                  mm                  ms

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

#-  2. MATERIALS:

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

/MAT/OGDEN/1/1

rubber

#              RHO_I

             6.74E-5

#                 Nu           sigma_cut           funIDbulk         Fscale_bulk         M     Iform

                .495                .004                   0                   0         0         0

#               Mu_1                Mu_2                Mu_3                Mu_4                Mu_5

             .002009             1.27E-6                   0                   0                   0

# blank card

 

#            alpha_1             alpha_2             alpha_3             alpha_4             alpha_5

                   2                  -2                   0                   0                   0

# blank card

 

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

#ENDDATA

/END

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
hmtoggle_plus1Comments
1.The Ogden material formulation is based on following representation of the strain energy density (W):

Where,

is the ith principal engineering stretch , is the ith principal engineering strain,

is the relative volume:

is deviatoric stretch with

K is the Bulk Modulus and depends on the Poisson’s ratio and is expressed as:

With the ground shear hyperelastic modulus () is calculated using the prescribed material parameters () of the strain energy density function:

Parameters and must be chosen so that the ground shear modulus is greater than zero. Moreover for material stability, it is required that each be greater than zero.

2.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.

3.The deviatoric stress () and the pressure (P) are computed independently, as follows:

Where, principal stresses are:

And pressure is calculated as:

fblk is function of fct_IDblk

4.A particular case of the Ogden material model is the incompressible Mooney-Rivlin model, which can be represented using the following equation for the strain energy density function:

Where, I1 and I2 are the first and second invariants of the right Cauchy-Green Tensor.

This representation can be derived from the Ogden strain energy density function when:

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

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 LAW42 components to make sure that the density variation is small, i.e. the value of J is close to 1.

starter_mat_ogden

6.Coefficients of the Prony series () are used to describe viscous effects using the Maxwell model (which can be described in a simplified manner as a system of n springs with stiffness' Gi and dampers ):

law82_maxwell_model

The hyperelastic ground shear modulus, is exactly the long-term shear modulus in the Maxwell model, and is the relaxation time:

The Gi and values must be positive.

7.Viscous effects are modeled through a convolution integral using Prony series. This is an extension of small strain theory (described in comment 6) to large nonlinear strain case. The Kirchhoff viscous stress is given by:

with,

F being the deformation gradient matrix,

denotes the deviatoric part of tensor .

The viscous-Cauchy stress is written as:

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

 

See Also:

Material Compatibility

Law Compatibility with Failure Model

Ogden and Mooney-Rivlin model in Theory Manual

Example 42 - Rubber Ring: Crush and Slide