Calculate the Stretch Ratio by comparing the ELFORM (-2,-1,1,2) with Ogden_Material Model.
Aim:
To perform a simple tensile test on a 10x10x10mm cube Ogden material model using different element formulation (i.e. ELFORM = 1, 2, -1, -2) and compare the simulation result plot up to stretch ratio of 5.
Objectives:
- To create a cube of 10x10x10 mm solid block with 10 elements for each direction.
- To constraint the cube in X, Y and Z direction on one face.
- To define prescribed motion in Z direction on opposite face to the constrained face.
- To define Ogden Rubber material card for the cube.
- To define STRFLG as 1 for extracting the strain data.
- To compare the Stress and Strain results up to Stretch Ratio of 5.
Introduction:
The Ogden material model is a hyper elastic 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 hyper elastic 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.
For modeling rubbery and biological materials at higher strains, the more sophisticated Ogden material model has been developed.
Stress, Strain and Stretch Ratio:
Stretch ratio is defined as the ratio between the final length and the initial length of a component.
This is equal to (1 Engineering Strain)
As we know that the values of stress and strain we get in LS -Dyna will be in the form of True Stress and True Strain. So we have to convert it into Engineering Stress and Engineering Strain to get the Stretch Ratio.
These conversions can be done by the equations as given below:
σt = σe (1 +єe)
єt = In (1 +єe) = e^ єt=(1+єe) or λ
Where,
σt = True Stress
σe = Engineering Stress
єt = True Strain
єe = Engineering Strain
And, Stretch ratio (λ)= (1+ єe)
Analysis:
- Here, we will perform a simple tensile test on a 10x10x10 mm cube with Ogden material model.
- We will compare the model by using different case element formulation.
Procedure:
- Case 1: Element Formulation_(1)
Geometry
- Go to Element and Mesh > Shape Mesher to create a 10x10x10 mm solid cube.
Section Card
- Create a Section Card for the cube and assign the element formulation criteria as ELFORM 1.
- As per the case, the ELFORM will be 1, 2 -1 and -2.
Mat_OGDEN_RUBBER
- Create OGDEN_RUBBER material card as defined below for the cube.
Part
- Assign the Section and Material crad to the Part card as shown below.
Boundary Conditions
- Here we will define constraints for the cube node using node set.
- We fill constraint the nodes on the face in Z direction.
- We will constraint the mid node line in X and Y direction, so that when stretched the cube will shrink in the respective direction.
Prescribed Motion
- We will create a node set on the upper face and define prescribed motion in Z direction to the cube.
- To provide motion we will define the motion using define curve.
- Here we want to obtain the stretch ratio up to5.
Stretch is the ratio of Final length to Original length
So, to obtain the stretch ratio up to 5
Original length = 10 mm
Final length = 50 mm
So, Stretch ratio = 1 + (Engineering Strain) = 40.
Hence, to get the stretch ratio up to 5, we have given the displacement up to 40 mm.
Analysis Method:
- LS-Dyna employs both implicit and explicit methods to solve Static, Quasi-Static and Dynamic problems. These problems can differ in Time Increment approach in implicit each Time increment has to converge, but we can set quite long time increments. In explicit, convergence for each time step is not needed. In implicit method, at each time increment, global equilibrium is established due to iterative method. Implicit method is used for accuracy and can be applied tor long duration problems.
- Also implicit methods are used when the deformation is not so large or the deformation is happening in a gradual manner.
- Here, we are stretch the cube up to 50 mm, is large deformation, but for the hyper elastic materials, this deformation can be small.
- So, implicit scheme is best for this type of analysis.
Control:
- Since this is the nonlinear implicit analysis, we need to create some implicit cards to activate implicit method like implicit auto, implicit general etc. as given below.
- In CONTOL IMPLICIT GENERAL IMFLAG = 1 is given to activate the implicit analysis method and initial Time step value is given as 0.01.
- In CONTROL IMPLICIT AUTO, values of DTMIN and DTMAX are not defined, so that the solver can change the time step size as needed automatically.
- In CONTROL TERINATION, ENDTIM is set to 1. Here termination does not represent the time but gives the value at which step the analysis will end.
Database_Extent_Binary:
- This keyword is used to get the strain tensor data through D3PLOT.
- In solid elements, single strain tensor is written.
ASCII and Binary_D3_Plot
- Select ELOUT and GLSTAT in ASCII Database_Option.
LS-RUN
- Run the file and the run should display NORMAL TERMINATION at the end of the run.
Results:
Von-Misses Stress-
Effective Plastic Strain -
Calculation of Nominal Stress & Stretch Ratio:
- Now we will see the calculation of the Nominal Stress and Stretch Ratio for all cases.
- After getting the values of True strain and True stress from the plot5, we can use the XY data and export it as t CSV file and open it in EXCEL.
- Here all the data of True Stress and rue Strain are collected for all cases using an element results from the middle of the cube, which is Element 995.
- So all the results will be regarding Element ID: 995
Here Stress results for Element 995 will be from Z-Stress
Strain results for Element 995 will be from Lower ipt Z-Strain.
Stretch Ratio
- Here we can get the Stretch ratio from the True strain
To get the Stretch ratio, we can use formula as gven below
єt = In (1 +єe) = e^ єt=(1+єe) or λ
(λ)= (1+ єe)
Here e= 2.718281
Nominal Stress
- Here Nominal stress is the Engineering stress in Z-Direction.
To get the Nominal Stress, we can use the formula as given below: - σt = σe (1 +єe)
- σt = σe* λ
- σe = σt/ λ
Calculations and result data for all cases are same as given below:
The following link contains the Excel sheet including the Strain and Stress data calculations-
Plots -
Case 1: ELform 1
Case 2: ELform 2
Case 3: ELform -1
Case 4: ELform -2
Conclusion-
- Since, this is an uniaxial tensile test we are getting similar results for different element formulations.
- Also the deformation given to the cube is not large enough for the hyperelastic material to observe the difference.
- Element Formulation 1 is good to propose the Strain energy function.
- Element Formulation 2 gives constant pressure to avoid pressure locking.
- Element Formulations -1 and -2 are intended for elements with poor aspect ratios.