Week-6 Calculate the Stretch Ratio by comparing the ELFORM (-2,-1,1,2) with Ogden_Material Model.

                                     CALCULATING THE STRETCH RATIO AND COMPARING THE ELFORM (-2, -1,1,2) WITH OGDEN MATERIAL MODEL USING LS-DYNA

AIM:

1. To Create a block of 10mmx10mmx10mm dimension with 10 elements for each direction and use the material card attached (Ogden_Material.k).
2. To Use appropriate boundary conditions to simulate uniaxial tensile behaviour for the model.
3. Compare the results of the simulation with ELFORM = 1, 2, -1, -2 by plotting Engineering Stress vs Stretch Ratio up to stretch ratio 5.
4. To use a proper solver (Implicit/Explicit).

INTRODUCTION:

An Ogden model is a hyperelastic material model that can be used for predicting the nonlinear stress-strain behaviour of materials such as rubber or polymer. Ogden model was introduced by Ogden in 1972. Ogden model has been frequently used for the analysis of rubber components like O-ring and seal. Even though Ogden model is a hyperelastic material model, its strain energy density function is expressed by principal stretch ratio.

In this simulation a solid block is created and appropriate boundary conditions are applied to simulate uniaxial tensile behaviour for the model and stretch ratio is calculated. The solver used for this simulation is implicit solver because the analysis is similar to a quasi-static analysis on a hyperelastic non-linear material with slow steady loading conditions.

PROCEDURE:

The given LS-Dyna keyword file is opened in LS-PrePost using option File>Open>LS-Dyna Keyword File.

                Fig.1. Imported keyword file.

The imported keyword file consists of only Ogden material card. The FE model of a solid block of 10mmx10mmx10mm dimension with 10 elements for each direction is created as shown in fig. 2.

                                                                                                        Fig.2. FE model of solid block with required dimension.

PART DEFINITION:

Section properties:

                                                                                                             Fig.3. Section solid.

The section properties of the FE model created is assigned as solid element with different ELFORM such as,

Case (1). ELFORM = 1, constant stress solid element.

Case (2). ELFORM = -1, Fully integrated S/R solid intended for elements with poor aspect ratio, efficient formulation.

Case (3). ELFORM = 2, Fully integrated S/R solid

Case (4). ELFORM = -2, Fully integrated S/R solid intended for elements with poor aspect ratio, accurate formulation.

Note: The simulation has to be executed for each case separately.

Material properties:

Keyword manager>MAT>077_O-OGDEN_RUBBER

                                                                                                       Fig.4. Ogden material

The above material card is already provided as a keyword file and it is assigned to the solid block part.

Note: The unit of µ in the material definition is in MPa and α is a dimensionless quantity.

Part definition:

                                                                                                   Fig.5. Part definition.

BOUNDARY CONDITIONS:

To perform uniaxial tensile test on the specimen, the nodes of one of the faces is fixed as shown in the following fig.6.1, fig.6.2 and fig.6.3.

                                                                                    Fig.6.1. Nodes fixed in X-direction.

                                                                              Fig.6.2. Middle nodes along Z- direction fixed in Y-direction.

                                                                          Fig.6.3. Middle nodes along Y- direction fixed in Z-direction.

                                                                     Fig.7. Nodes in the moving face for applying load.

The minimum load to be applied on the moving surface to attain a stretch ratio of 5, is calculated by using following relation,

Engineering strain, `ε_e=(δl)/l,`

Stretch ratio, `λ=1+ε_e`

                     `5=1+(δl)/10`

                      `δl=40mm,`

Hence, minimum value of 50mm is taken to define the load curve.

                                                                                                            Fig.8. Load curve.

                                                                                                    Fig.9. Boundary prescribed motion set card.

BOUNDARY_PRESCRIBED_MOTION_SET card is used to apply the boundary condition on the moving surface.

CONTROL FUNCTION:

                                                                                                  Fig.10. Control implicit auto.

CONTROL_IMPLICIT_AUTO card is used for automatic time step control during implicit analysis. Input value for Automatic time step control flag = IAUTO = 1.

                                                                                            Fig.11. Control implicit general.

CONTROL_IMPLICIT_GENERAL card is used to activate implicit analysis and define associated control parameters. This keyword is required for all implicit analyses.

Input value for IMFLAG=Implicit/Explicit switching flag = 1 (implicit analysis)

DT0=Initial time step size for implicit analysis = 0.01.

                                                                                        Fig.12. Control implicit solution.

CONTROL_IMPLICIT_SOLUTION card is used to specify whether a linear or nonlinear solution is desired. The default values are set as it is in the card.

                                                                                      Fig.13. Control implicit solver.

CONTROL_IMPLICIT_SOLVER card is used for implicit calculation.

                                                                                      Fig.14. Control termination.

The control termination function is enabled to specify the end time of simulation. The termination time is set for 1 ms.

DATABASE OPTION:

                                                                                     Fig.15. Database binary_D3plot.

The time step value of 0.01 ms is given for the BINARY_D3PLOT and DATABASE_ASCII option for GLSAT and ELOUT.

                                                                                Fig.16. Database binary extent.

DATABASE_EXTENT_BINARY card with STRFLG =1, is used to compute the elastic strain in the model.

                                                                             Fig.17. Database history solid.

DATABASE_HISTORY_SOLID card is used to compute the stress/strain of a 550 node in the model.

The keyword file created is checked for errors using the option keyword manager>model check. The keyword file is saved using ‘.k’ extension and is made to run in the solver by getting normal termination message for different cases of ELFORM.

RESULTS:

The D3plot output file is opened in LS-PrePost using option File>open>LS-Dyna binary plot.

1. The animation of Von-Mises stress contour is as shown below.

                                                       Case (1): Elform 1                                                                                                    Case (2): Elform -1

                                                      Case (3): Elform 2                                                                                                      Case (4): Elform -2

2. The animation of Effective plastic strain contour is as shown below.

                                                         Case (1): Elform 1                                                                                                  Case (2): Elform -1

                                                            Case (3): Elform 2                                                                                                Case (4): Elform -2

It is observed from the stress contour that the maximum value of X-stress for case 1, 2, 3 and 4 are 5.025 MPa, 5.0239 MPa, 5.0312 MPa, 5.0239 MPa respectively. The maximum value of X-strain for case 1, 2, 3 and 4 are 1.7885, 1.7878, 1.789, 1.7879 respectively. There is a small variation in the stress value but the strain value is almost same for different cases.

3. Calculation of Engineering stress/strain and stretch ratio.

In LS-DYNA the output of stress and strains from Post Fringe Comp Stress/Strain is given as true stress and strains. To find the engineering stress/strain and stretch ratio following relations are used,

We know that, True stress, `σ_t=σ_e (1+ϵ_e )`

Engineering stress, `σ_e=σ_t/((1+ϵ_e ) )`

Similarly, True strain, `ϵ_t=ln⁡(1+ϵ_e)`

Engineering strain, `ϵ_e=e^(ϵ_t )-1`

Stretch ratio, `λ=1+ε_e`

                                                                                                              Fig.18.1. Engg. Stress vs Stretch Ratio plot for Elform 1

                                                                                                              Fig.18.2. Engg. Stress vs Stretch Ratio plot for Elform -1

                                                                                                              Fig.18.3. Engg. Stress vs Stretch Ratio plot for Elform 2

                                                                                                             Fig.18.4. Engg. Stress vs Stretch Ratio plot for Elform -2

From the graphs plotted for Engg. Stress vs Stretch Ratio for different cases, the Engg. stress value corresponding to a stretch ratio of 5 is 0.8 MPa approximately. The output from different cases of elforms gives approximately same results since the model and loading conditions are simple.

                                                          Fig.19. Nominal Stress vs Stretch Ratio plot for Ogden material.

The actual plot for nominal stress vs stretch ratio for ogden material is shown in fig. 19. In the actual plot the value of nominal stress for a stretch value of 5 is approximately 1.8 MPa but from the simulation result the value of stress is 0.8 MPa. It is observed that a variation of 1 MPa is caused in the simulation for uniaxial tension test for ogden material because of non-linearity of the material and relatable boundary condition applied for the model. 

CONCLUSION:

1. A FE model of solid block was created with ogden material to simulate uniaxial tension test for different cases of ELFORM (1, -1, 2, -2).
2. The Engineering stress/strain and stretch ratio was calculated to plot a graph of Engg. stress vs stretch ratio up to stretch ratio 5.
3. The Engineering stress value for a stretch value 5 from the simulation for different ELFORM was approximately equal to 0.8 MPa, but from the actual graph it was equal to 1.8 MPa with a difference of 1 MPa.
4. The uniaxial tension test was solved using implicit solver since the analysis is similar to a quasi-static analysis on a hyperelastic non-linear material with slow steady loading conditions.

Google Drive Link: https://drive.google.com/file/d/1ugwRsl87nZK9r4MSpHi_EHzYMozei3ID/view?usp=sharing