Hyperelastic Material Models in LS-DYNA
HYPERELASTIC MATERIAL MODELS IN LS-DYNA
Objective:
The following are the objectives of the project,
> To conduct a uniaxial tensile test for a Dogbone specimen with 100 percent strain.
> To determine the Hyperelastic Mooney-Rivlin and Ogden material constants obtained from the LS-DYNA simulation data of d3hsp file.
> To compare the stress-strain relation of both the material models with the material test data provided.
Hyperelastic materials:
Hyperelastic materials are defined by a constitutive model which derives the stress–strain relationship from a strain energy density function.
They are highly elastic materials so that their deformations can be recovered when subjected to very large strains.
For many materials, linear elastic models do not accurately describe the observed material behaviour.
The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic,incompressible and generally independent of strain rate.
Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney-Rivlin solids.
Other widely used hyperelastic material models include Ogden, Yeoh and Arruda-Boyce models.These models account for both non-linear material behaviour and large shape changes.
Determination of Hyperelastic material parameters:
To specify the complete hyperelastic material model it is necessary to first select a constitutive model, and then find the material parameters by calibrating the material parameters to experimental data.
Since one or more experimental stress-strain curves are used for the calibration, the mathematical procedure of determining the material parameters involves solving an over constrained set of equations. Hence, it is typically not possible to get a perfect fit of the model to the experimental data.
Furthermore, all experimental testing include experimental errors, so the material calibration in practice is limited to finding a set of material parameters that gives a good as possible prediction of the experimental data.
In our case, we will be validating the experimental data, provided for a hyperelastic material subjected to a uniaxial tensile test, with LS-DYNA simulation data using Mooney-Rivlin and Ogden material models.
Hyperelastic material modelling:
We are provided with an extracted test data in the form of engineering stres and strain which is needed to be defined as the behaviour of the material.
Since hyper elastic models are incorporated with engineering stress-strain values, there is no need for conversion process from engineering to true stress-strain values unlike in elasto-plastic models.
Using *DEFINE_CURVE option in LS-DYNA, we define the stress-strain data either manually inserting one by one or editing in the keyword file with proper alignments.
The one end of dogbone specimen is constrained in all DOFs except in the lateral direction of elongation and the other end is given a suitable displacement using BOUNDARY_PRESCRIBED_MOTION_SET card.
Since LS-DYNA displays true stress-strain valus and the total length of our model is 145mm, we impose following calculation to give displacement value that would produce 100% strain in our model.
`Engg.epsilon =e^(Truein)-1`
`1=e^(Truein)-1`
`2=e^(Truein)`
`Truein=ln2=0.69`
`(Displacement)/(Initial__Dimension)=Truein`
`Displacement=0.69*Initial__ Dimension`
`=0.69*145`
`Displacement=100`
Therefore,a displacement of `100mm` is required to produce 100% strain in the model,which can be otherwise called as stretch ratio is equal to `2`.
We use rubber as the hyperelastic material for this scenario and the density of rubber is taken as `0.001 (g)/(mm^3)` with Poisson ratio `0.498`.
The dogbone specimen part is referenced with a shell section of thickness `1.5mm` and the simulation is set to run for 10ms using CONTROL_TERMINATION card.
Mooney-Rivlin material model:
To determine the Mooney-Rivlin constants that pertain to a hyperelastic material behaviour, we use *MAT77_H material card.
The N parameter in the *MAT77_H material card represents the order of polynomial for which the constants are required to solve.This involves N1,N2 and N3 cases where the constants are solved along with stretch and true stress data.
The SW,SGL and ST parameters are set to unity as we have stress-strain load curve unlike in the case of force-displacement data that requires specimen dimensions.
The solved constants in each case are stored in the d3hsp file and are with different approaches in each case.
The Mooney-Rivlin constants for each polynomial order case are as follows,
> For N1, `C01=0.1802 and C10=0.1404`
> For N2, `C01=-0.1196 , C10=0.4060 , C11=0.1298 , C20=-0.03613 , C02=-0.1965 , C30=0`
> For N3, `C01=-0.2547 , C10=0.5339 , C11=1.278 , C20=-0.4679 , C02=-1.047 , C30=0.0121`
The model has run with different N values and the stress-strain values are plotted for each case.
Conversions are carried out to obtain engineering stress-strain values from true stress and extension values that are collected from LS-DYNA d3hsp file.
Later, the constant values of N1,N2 and N3 Mooney_Rivlin models are entered in the material card one by one for each case accordingly and the simulation is set to run with default keyword file.
Similar steps had been undertaken to convert true stress-strain values of simulation data into engineering stress-strain values.
From the above plot,it is clearly understood that N1 Mooney-Rivlin model stays close enough with the actual data of tensile test,whereas N2 and N3 Mooney-Rivlin model drifts a little from the actual plot.
Thus,a polynomial order N of 1 best suits for Mooney-Rivlin hyperelastic material model.
Ogden material model:
Similar to the Mooney-Rivlin case as discussed above, the Ogden model constantts are determined from *MAT77_O material card.
Unlike the above case, the number of N parameter (order of polynomial for the constants to solve) has a limit upto 8. However, we will be limiting only upto 3 for N.
The SW,SGL and ST parameters are set to unity as we have stress-strain load curve unlike in the case of force-displacement data that requires specimen dimensions.
The solved constants in each case are stored in the d3hsp file and are with different approaches in each case.
The Ogden constants for each polynomial order case are as follows,
> For N1, `mu(1)= 8.8293632867385E-01 ,alpha(1)= 1.3728647203317E+00`
> For N2, `mu(1)= 1.5637149696412E+00 ,alpha(1)= 1.0749157420613E+00`
`mu(2)= 1.6897964877338E+00 ,alpha(2)=-2.8735385520956E-01`
> For N3, `mu(1)=-6.1559216077934E+00 ,alpha(1)=-1.0316671107242E+00`
`mu(2)= 2.8847356358977E+00 , alpha(2)=-2.5331837191554E+00`
`mu(3)=-4.3792326963621E-01 ,alpha(3)=-4.9131345444782E+00`
The model has run with different N values and the stress-strain values are plotted for each case.
Conversions are carried out to obtain engineering stress-strain values from true stress and extension values that are collected from LS-DYNA d3hsp file.
Later, the constant values of N1,N2 and N3 Ogden models are entered in the material card one by one for each case accordingly and the simulation is set to run with default keyword file.
Similar steps had been undertaken to convert true stress-strain values of simulation data into engineering stress-strain values.
From the above plot, it can be seen that all the Ogden models N1,N2 and N3 drifting from the actual data by a little amount.
However, N1 model remains close of all Ogden models.
Conclusion:
There are several material models for hyperelastic materials in LS-DYNA ranging in complexity and capability.
Several hyperelastic material models in LS-DYNA require only uniaxial stress-strain or force-displacement data to determine material model parameters (MAT 027, 031, 077, 181, & 183).
However, it is generally recommended that material model parameters should be determined by several different types of mechanical testing including tension, compression, and pure shear testing.
The accuracy of any material model is highly dependent on the quality of the material model data.
The attached excel sheet contains all the calculations involved in this particular piece of work.