Week - 10 Hyperelastic Material Models
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
PROCEDURE:-
1.The data for the given engineering stress-strain values were arranged and plotted a curve in an excel sheet.
| Engineering Strain | Engineering Stress(Mpa) |
| 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 |
| 1.00E+00 | 8.69E-01 |
Model setup-
Section:
Create a shell section for the specimen.Assign section id and material id to the part.
Define:
The keyword DEFINE provides a way of defining boxes, coordinate systems, load curves, functions, tables, transformations, and orientation vectors. The curve option was selected, and the assigned value of load displacement is presented in the following card.
Now we have to define the material load curve which we have formulated from the Raw data-
Assigning Section and Material:
Section and Material are assigned to the part.
8.Boundary:
The keyword BOUNDARY provides a way of defining imposed motions on boundary nodes. The Prescribed Motion Set Displacement boundary condition was defined for the challenge purposes and respective curve is assigned to the card. A node set is created out of the edge and allowed to move in X-direction.
9.Control Cards:
The keyword control cards can be used to change defaults. The Implicit Auto, Implicit General, Implicit Solution, Implicit Solver, and Control Termination cards were created for the challenge purposes.
Database Cards:
The database definitions are necessary to obtain output files containing results information. The binary D3Plot, Extent Binary, and ASCII options were selected for the challenge purposes.
Providing termination time-
SIMULATED RESULTS-
CASE 1: MOONEY RIVLIN NUMBER OF CONSTANTS- N=1:
For the first degree polynomial the number of constants=2
The Mooney Rivlin constants for N=1 are:
| c1 | 1.77E-04 |
| c2 | 1.47E-04 |
The values of the constants are obtained from the d3hsp file.
The plot compares the experimental stress-strain curve to the curve created by the Mooney Rivlin material model for N=1.
CASE 2: MOONEY RIVLIN NUMBER OF CONSTANTS- N=2:
The Mooney Rivlin constants for N=2 are:
| Constant C10= 0.2595E-03 |
| Constant C01= 0.4594E-04 |
| Constant C11= -0.2092E-04 |
| Constant C20= 0.2914E-05 |
| Constant C02= 0.1017E-05 |
The plot compares the experimental stress-strain curve to the curve created by the Mooney Rivlin material model for N=2.
CASE 3: MOONEY RIVLIN NUMBER OF CONSTANTS- N=3:
The Mooney Rivlin constants for N=3 are:
| Constant C10= 0.2611E-03 |
| Constant C01= 0.4420E-04 |
| Constant C11= -0.6077E-05 |
| Constant C20= -0.2666E-05 |
| Constant C02= -0.9966E-05 |
| Constant C30= 0.1565E-06 |
The plot compares the experimental stress-strain curve to the curve created by the Mooney Rivlin material model for N=3.
ODGEN HYPER ELASTIC MATERIAL MODEL:
The comparison is made between the number of constants (N=1,3 and 5) in the Odgen hyperelastic material model:
MATERIALS CARD:
CASE 4: ODGEN HYPERELASTIC MATERIAL MODEL AT N=1:
The constants are:
| mu(1)= 9.1076025253111E-04 |
| alpha(1)= 1.3427238671930E+00 |
The plot compares the experimental stress-strain curve to the curve created by the odgen material model for N=1.
CASE 5: ODGEN HYPERELASTIC MATERIAL MODEL AT N=2:
The constants are:
| mu(1)= 9.1564732574771E-01 |
| mu(2)= 2.2259557307116E+00 |
| alpha(1)= 1.3399091914039E+00 |
| alpha(2)=-1.8434702588025E-03 |
The plot compares the experimental stress-strain curve to the curve created by the odgen material model for N=3.
CASE 6: ODGEN HYPERELASTIC MATERIAL MODEL AT N=3:
The constants are:
| mu(1)=-3.1842708827118E-05 | alpha(1)= 1.9262961070942E+00 |
| mu(2)= 2.6137539677282E-04 | alpha(2)= 1.5503244209897E+00 |
| mu(3)= 6.8199166677376E-04 | alpha(3)= 1.2887877441176E+00 |
The plot compares the experimental stress-strain curve to the curve created by the odgen material model for N=3.
COMPARISON BETWEEN N=1,2 AND 3 FOR MOONEY RIVLIN MATERIAL MODELING TO THE EXPERIMENTAL DATA:
- It is clear from the plot that the values of the stress and strain for different values of N are more or less the same.
- It is expected to have a better plot close to the input data with higher values of N.
- The difference in the results from the experimental and simulation data is due to the slight differences in the input material properties and the input displacement curve.
COMPARISON BETWEEN N=1,2 AND 3 FOR OGDEN RUBBER MATERIAL MODELING TO THE EXPERIMENTAL DATA:
- t is clear from the d3hsp plot that the values of the stress and strain curve fit for different values of N are more or less the same. But, there is a large difference in the curve fit data (d3hsp file) and the experimental data.
- It is expected to have a better plot close to the input data with higher values of N.
- The difference in the results from the experimental and simulation data is due to the slight differences in the input material properties and the input displacement curve.
CONCLUSIONS:
- In a detailed comparison between the Mooney Rivlin and Odgen hyperelastic material model, it is clear that there is nothing much of a difference in the results from both the simulations. The results are more or less the same.
- But if we look at the curve fit data from the d3hsp file for odgen hyperelastic material model, we can clearly see a large difference between the experimental (input) data and the curve fit data.
- While the Mooney rivlin hyperelastic model gives more accurate or close curve fit for the input data, odgen material fails to keep up with the accuracy of the fit data.
- But the simulation data is close to the experimental data even for odgen material model.
- Here, the experimental data is very smooth and hence the results are more close to the actual data. But if we had a more noisy data, then in my opinion its better to go with the mooney rivlin material model with higher values for N.
- Moreover, the length of the specimen remained the same (289.9) for all the above cases.
LEARNING OUTPUT-
LEARNED HOW TO MODEL A MATERIAL CARD USING RAW DATA FOR HYPERELASTIC MATERIAL AND HOW WE CAN VALIDATE IT SUING THE SIMULATION DATA.