Week - 10 Hyperelastic Material Models

#11 // Week - 10 Hyperelastic Material Models.

Aim:

To calculate the Mooney Rivlin and Ogden material constants from the material data and validate them using stress-strain data from the tensile test of a dogbone specimen.

Objective:

1. To calculate the Mooney Rivlin and Ogden material constants and compare both using stress-strain data from a Dogbone specimen tensile test with 100 percent strain.
2. The given material data is the engineering stress-strain in MPa/(mm/mm).
3. The comparison should be shown from the d3hsp file and using simulation.

/Note: The unit system used is g-mm-ms, N, MPa, N-mm.

Theory:

Hyper-elastic Material:

• Hyperelastic material models are regularly used to represent the large deformation behavior of materials with FEA. They are commonly used to model the 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 from research 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 change in 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:

Drafting Data -

• The given Engg. stress and Engg. strain data of hyperelastic material is plotted in an Excel sheet.

Define_Load curve -

• The term material curve is assigned by defining a curve using a given engineering strain and stress plot.

Material

• To check and validate the given material data with material models of Mooney-Rivlin, Ogden a dogbone specimen is taken for the analysis.
• To find the Mooney-Rivlin and Ogden constants the value of N is set to 1,2,3. The value of density and Poisson’s ratio is taken as the generic value for the rubber material.

• The parameter N in the Mat_077(H) cards refers to the order of the 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.
• N1 solves for two polynomials C10, C01
• N2 solves for six polynomials C10, C01, C11, C20, C02, C30
• N3 solves for six polynomials C10, C01, C11, C20, C02, C30

• The parameter N in the Mat_77(O) cards refers to the order of the polynomial for which the constants are required to solve.
  This involves N1, N2, and up to N8 cases where the constants are solved along with stretch and true stress data.

The SW, SGL, and ST parameters are set to unity since we have a stress-strain load curve whereas in the case of force-displacement data requires specimen dimensions.

Section

• The section property of the dogbone specimen is assigned as a shell element with 2 mm thickness, ELFORM=2

Part

• The Material and Section are assigned to the dogbone specimen part.

Boundary Condition-

SPC

• The nodes at the fixed end are constrained in the X and Z direction only and the Y direction is not constrained because of lateral expansion during the tensile test.

• The nodes in the middle are constrained in the Y direction since the neutral axis passes through the middle of the specimen in the X direction.

Prescribed motion

• PSM is created at the opposite end by specifying the displacement nodes using the Load curve LCID. Since the giving displacement for 100% strain & given specimen length is 145mm.

Control Cards -->

To activate implicit analysis and define associated control parameters. The following keyword is required for all implicit analyses-

CONTROL_IMPLICIT_AUTO ->

• IAUTO: set to 1 which automatically adjusts the timestep size
  ITEOPT: optimizes the operation count step 11
  ITEWIN: is the iteration window 5
  DTMIN: Minimum allowable time step size 0.1
  DTMAX: Maximum allowable time step size 0

CONTROL_IMPLICIT_GENERAL ->

• IMFLAG: Implicit/Explicit analysis type flag: 1- Implicit analysis
• DT0: Initial time step size for implicit analysis: 0.001000
• IMFORM: Element formulation flag for seamless spring back: 2 retain original element formulation (default)

CONTROL_IMPLICIT_SOLUTION ->

CONTROL_IMPLICIT_SOLVER ->

CONTROL_TERMINATION ->

Database ->

• BINARY_D3PLOT: It defines the frequency at which the animation file is to be created and is set to 0.1 ms.

• DATABASE_ASCII_Option: The output request in ASCII format, the following keyword is activated
• ELOUT: Element Output Data and
• GLSTAT: Global Data

• DATABASE_EXTEND_BINARY: It plots the strain tensor to d3plot when STRFLG is set to 1

Post-processing ->

Results-

Mooney_Rivlin_N=1 -->

Mooney_Rivlin_N=2 -->

Mooney_Rivlin_N=3 -->

Ogden_N=1 -->

Ogden_N=2 -->

Ogden_N=3 -->

Material cases-

• From the data obtained of the d3hsp file for different polynomial values i.e., (N=1,2,3)
• The curves are converted from True to Engineering data and plotted in Excel with a comparison to the Given data.
• It is observed from the graph, that the curves of the different polynomials are superimposing with the experimental data.

From d3hsp files......

For Mooney Rivlin N= 1

For Mooney Rivlin N= 2

For Mooney Rivlin N= 3

For Ogden N= 1

For Ogden N= 2

For Ogden N= 3

• /Note- In Mooney Rivlin d3hsp.... extension = stretch ratio, stress = True Stress.
• /Note- In Ogden d3hsp.... stretch ratio (already mentioned), stress(fitted) = Engg. stress.

/Note- Here from the above d3hsp files, the extension is stretching data and converting it to strain by

Extension = Stretch ratio

Stretch ratio = 1 + Engineering Strain

Engg. Strain = Stretch ratio - 1

So, by converting the True stress values to Engineering Stress, we can plot Engg. Stress vs Engg. Strain data

Engg Stress = True Stress / (1 + Engg Strain)

/

> Mooney_Rivlin_d3hsp_data ->

> Ogden_d3hsp_data ->

> Raw_data (given) ->

/ Comparison Plots for the above Engg. Stress-strain data (with given raw data vs d3hsp data):

For Mooney Rivlin-

For Ogden-

Simulation Data-

• Finally, the simulation data obtained from the d3plot (Post > History > Element > X-stress / lower Ipt X-strain) for different polynomial values i.e., (N=1,2,3).
• The curves are converted from True to Engineering data and plotted in Excel with a comparison to the Given data.

/Note- here, from d3plot history X-stress & lower Ipt X-strain values are in true stress & true strain. So, we have to convert those values in Engg. stress-strain.

Engg. strain = Exp (True strain) - 1

Engg. stress = (Ture stress) / (1 + Engg. strain)

/

> Mooney_Rivlin_simulation_data ->

> Ogden_simulation_data ->

/ Comparison Plots for the above Engg. Stress-strain data (with given raw data vs d3hsp data vs simulation data):

For Mooney Rivlin-

For Ogden-

• From this data, we can easily find that the reduction in area is comparatively less for experimental hyper-elastic than the original given data.
• Due to some non-linearities, there could be some deviation that is agreeable compared to the original.

//Note-

Material Validation

• To validate we need to key in the constants (C10, C01, C11, C20, C02, C03) from the above simulation materials with N1, N2, and N3.
• By switching N=0 usually by *MAT_HYPERELASTIC_RUBBER we can input any of the Mooney-Rivlin constants according to the above cases.
• From the obtained true stress and strain, the Engineering stress and strain values are calculated and compared with the original data from the material curve.
• For Ogden material processes are the same.

Conclusion:

• The Mooney-Rivlin and Ogden material constants are obtained and verified from the d3hsp file.
• By entering the value of polynomial N in the material card MAT_77_H the stretch vs stress curve data was obtained and verified with experimental material data.
• The simulation results of the tensile test for a material model show experimental data with slight deviation due to the high non-linear behavior with the help of generic values.
• The comparison showed from the d3hsp file and using simulation.

• Learned to verify and validate the hyper-elastic material model for the given material data.

Drive Link-

https://drive.google.com/drive/folders/1DhDq2HIZ_WTTAk-wBrGt8fX5JXbgip_x?usp=sharing

https://drive.google.com/drive/folders/14St_Qm_CkUeMQge20MuT-0oI41-Tfw8K?usp=sharing,

https://drive.google.com/drive/folders/1EOlRW0ELjBYPaNTuo_m3yRCq2IzizbxE?usp=sharing,

https://drive.google.com/drive/folders/1cPxOx44r3v5oJFQ8rmNVtR2kY9PZUyrA?usp=sharing,

https://drive.google.com/drive/folders/1kRcZbP6LNIqU995251KFGA4Dd3ulzDpU?usp=sharing,

https://drive.google.com/drive/folders/1xdsNcQZLhfVQp2NAdZkJP7KeF8XJVKyl?usp=sharing,

https://drive.google.com/drive/folders/1zWDGIeKz2IsyMFwCt7YHz6q8oIrgJ9Ie?usp=sharing