Week - 10 Hyperelastic Material Models
HYPERELASTIC MATERIAL MODELLING AND VALIDATION
AIM :
- To perform the simulation and validate the curve(STRAIN VS STRESS) as shown in the figure
OBJECTIVE :
- To calculate and compare the Mooney Rivlin, Ogden material constants using stress-strain data by 100 percent strain
- From the above condition simulate and validate the material model using MPa(mm/ms)
- Finally, compare the d3hsp file from the simulation
INTRODUCTION :
Hyperelastic materials are the special class of materials that respond elastically even when they are subjected to large deformations. They show both a nonlinear material behavior as well as large shape changes. They are characterized by:
- Can undergo large elastic deformations in order of 100 to 700% which are fully recoverable, i.e., the initial shape is recovered when the load is removed.
- Are nearly incompressible, which means that they change can their shape but overall volume remains almost constant.
- Show a highly nonlinear stress-strain relation.
- The material softens and then becomes stiffer again when applied to tension whereas, under compression, they have a quite stiff response.
Many polymers show hyperelastic behavior, such as elastomers, rubbers, and other similar soft flexible materials.
- Hyperelastic material models are regularly used to represent the large deformation behavior of materials with FEA. They are commonly used to model mechanical behaviors of unfilled/filled elastomers.
- In addition to elastomers, hyperelastic material models are also used to approximate the material behavior of biological tissues, polymeric foams, etc. Linearly elastic materials are described through two material constants (Young’s modulus and Poisson ratio).
- In contrast, hyperelastic materials are described through a strain-energy density function. The strain-energy density can be used to derive a nonlinear constitutive model (i.e., stresses as a function of large strain deformation measures like deformation)
- Those are derived by researches and some acceptable models we are using popularly such as
Ogden
Neo-Hookean
Arruda-Boyce
Yeoh and Fleming
Mooney-Rivlin and Signorini models
MOONEY RIVLIN
The strain energy density function for an incompressible Mooney–Rivlin material is defined below. Where ` are empirically determined material constants, I1 and I2 are the first and the second invariant. By varying the Index values of i and j we can change the constitutive equations and use them appropriately for our component. Similarly, we will be having multiple coefficients C
This changing of the index will be basically changing the order of the polynomial of the energy density equation. We need to find out the coefficients of these material models for a different order of the polynomial and extract the best fit curve data from the D3HSp file and compare it with the material curve.
OGDEN MATERIAL
The Ogden material model is a hyperelastic material model used to describe the non-linear stress–strain behavior of complex materials such as rubbers, polymers, and biological tissue. The model was developed by Raymond Ogden in 1972. The Ogden model, like other hyperelastic material models, assumes that the material behavior can be described by means of a strain energy density function, from which the stress–strain relationships can be derived
SPECIMEN CAD :
PROCEDURE :
1. CASE SETUP
defining the other card
- SECTION_SHELL
- BOUNDARY_PRESCRIBED_MOTION_SET
- BOUNDARY_SPC
- CONTROL_TERMINATION
- CONTROL_IMPLICIT_GENERAL
- CONTROL_IMPLICIT_AUTO
- CONTROL_IMPLICIT_SOLVER
- CONTROL_IMPLICIT_SOLUTION
- DATABASE_ACII(glstat,matsum,eleout)
- DATABASE_D3PLOT, DATABASE_EXTENT_BINARY
we can setup the deck
2. PLOTTING DATA
Given Experimental Data:
The given data of Engg stress and Engg strain of hyperelastic material is shown in the image given below are plotted in excel.
| IMAGE DATA | |
| STRAIN | STRESS |
| 0 | 0 |
| 2.50E-02 | 4.46E-02 |
| 5.00E-02 | 8.67E-02 |
| 7.50E-02 | 1.27E-01 |
| 1.00E-01 | 1.64E-01 |
| 1.25E-01 | 2.00E-01 |
| 1.50E-01 | 2.34E-01 |
| 1.75E-01 | 2.67E-01 |
| 2.00E-01 | 2.98E-01 |
| 2.25E-01 | 3.28E-01 |
| 2.50E-01 | 3.56E-01 |
| 2.75E-01 | 3.83E-01 |
| 3.00E-01 | 4.09E-01 |
| 3.25E-01 | 4.34E-01 |
| 3.50E-01 | 4.58E-01 |
| 3.75E-01 | 4.81E-01 |
| 4.00E-01 | 5.03E-01 |
| 4.25E-01 | 5.24E-01 |
| 4.50E-01 | 5.45E-01 |
| 4.75E-01 | 5.65E-01 |
| 5.00E-01 | 5.84E-01 |
| 5.25E-01 | 6.03E-01 |
| 5.50E-01 | 6.20E-01 |
| 5.75E-01 | 6.38E-01 |
| 6.00E-01 | 6.55E-01 |
| 6.25E-01 | 6.71E-01 |
| 6.50E-01 | 6.87E-01 |
| 6.75E-01 | 7.02E-01 |
| 7.00E-01 | 7.17E-01 |
| 7.25E-01 | 7.32E-01 |
| 7.50E-01 | 7.46E-01 |
| 7.75E-01 | 7.60E-01 |
| 8.00E-01 | 7.73E-01 |
| 8.25E-01 | 7.86E-01 |
| 8.50E-01 | 7.99E-01 |
| 8.75E-01 | 8.11E-01 |
| 9.00E-01 | 8.23E-01 |
| 9.25E-01 | 8.35E-01 |
| 9.50E-01 | 8.47E-01 |
| 9.75E-01 | 8.58E-01 |
| 1 | 8.69E-01 |
3. DEFINING THE CURVE IN THE LS DYNA
CASE 1 MOONEY RIVLIN
define using keyword DEFINE_CURVE
Now define the curve in material (MAT_77_H)
In mooney rivlin curve fitting is done by N=1
after running the simulation
From the d3hsp , cofficient can be found
now define the hyperelastic material using the coffiecient which is found via above steps
simmiliary cofficient for N2 and N3 can be found via above steps
| N2 | ||||
| c10 | c01 | c11 | c20 | c02 |
| 0.2595 | 0.04594 | -0.02092 | 0.002913 | 0.001016 |
| N3 | |||||
| c10 | c01 | c11 | c20 | c02 | c30 |
| 0.2611 | 0.0442 | -0.006076 | -0.002667 | -0.009966 | 1.57E-04 |
CASE 2 OGDEN :
curve must be defined using DEFINE_CURVE same as explained in mooney rivlin now we only have to define ogden material using MAT_77_O card
after running the file cofficient can be found using the d3hsp
and define the OGDEN material using the cofficient
simmiliar way cofficient can be found for N1, N2 , N3
| N2 | |
| mu1 | mu2 |
| 0.9156473 | 2.225956 |
| alpha1 | alpha2 |
| 1.339909 | 0.0018435 |
| N3 | ||
| mu1 | mu2 | mu3 |
| -0.0318426 | 0.2613754 | 0.6819919 |
| alpha1 | alpha2 | alpha3 |
| 1.926318 | 1.550334 | 1.288785 |
RESULT :
from the above case we get true stress and true strain . we can convert it into engineering stress and engineering strain and compair with the raw data
CONCLUSION :
- The Mooney-Rivlin and Ogden material constants are obtained from d3hsp file by inputting the value of polynomial N=1in the material card MAT_77_H and MAT_77_O
- The stretch stress curve data obtained from d3hsp for both hyperelastic constitutive models were verified with experimental material data and found to be a close fit.
- The simulation results of tensile test for both material models with experimental data showed a slight deviation due to high non-linear behaviour of the given hyperelastic material and the generic values of density and Poisson’s ratio.
- From the simulation results, both hyperelastic constitutive material models were found to be valid to describe the behaviour of the given material data.
- The Ogden material model has more polynomials ranging from 1 to 8 and is easy to use and gives better adjustment to experimental data at the cost of computional time.
- Learned to verify and validate hyperelastic material model for the given material data.