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

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:

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Analysis

Here we will perform a simple tensile test on a 10x10x10 mm cube with Ogden material model.

To see the behavior of element formulation, we will work on different cases as per different element formulation as given below:

CASE_1: Element Formulation_ (1)

CASE_2: Element Formulation_ (2)

CASE_3: Element Formulation_ (-1)

CASE_4: Element Formulation_ (-2)

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Geometry

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Procedure:

• First of all open the L S DYNA Keyword file named as “Ogden_Material.k” in LS-PrePost.
• Create a cube of 10x10x10 mm with help of “Shape Mesher” in “Elements and Mesh” as shown below:

• As per above image, give the value 10 for x , y and z direction and create a cube.

Material Properties

The material properties for the Ogden material model will be as given below:

Section:

• Create a Section card for the cube and assign the element formulation criteria as per the case as given below:

• As per case, as shown in above image, element formulations will be 1, 2, -1 and -2.

 

Part:

• After creation of Sections and Material, we have to assign it to the Part.

Assignment of appropriate material as well section in part for all cases will be a given below:

Boundary Conditions:

• Here we will fix the Bottom of the Cube and give the motion to the set of nodes on the Upper face.
• To fix the Bottom face, we will use different set of nodes and using single point constraint boundary, we can allow or constraint the nodes with different degrees of freedom.

Single Point Constrained:

• Boundary conditions using single point constrained for different set of nodes will be as given below:

• As shown in above image, constraints are given for the different set of nodes in a way that cube can shrink in X and Y direction upon tension in Z-Direction and also stick to its position.
• Thus , this boundary condition can be relatable to the real-time boundary condition.

Prescribed Motion:

• To apply the tensile load in Z-Direction to the cube, we have to give the motion to the set of node on the upper face of the cube.

• As per above image, motion is given to the set of node of the cube on upper face.
• Motion is given to the node set in the form of curve as given below:

• After defining the proper curve, assign the load curve to the Prescribed motion card as given below:

• As shown in above image, we have given DOF = 3 , to allow translation in Z-Direction translation
• VAD = 2 is given for Displacement.
• Load curve is also assigned for desired displacement.
• As we have given the curve for the 1 mm displacement, Scale Factor is set to 50 for displacement up to 50 mm.

Here we want to obtain the stretch ratio up to 5.

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 = Final length/ Original length = 50/10 = 5.

• So, to get the stretch ratio up to 5, we have given the displacement up to 50 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 for 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:

DATABASE_EXTENT_BINARY

• This keyword is used to get the strain tensor data through D3PLOT.
• In solid elements, single strain tensor is written.

• Similarly DATABASE_BINARY_D3PLOT keyword is used for all output data and animations.

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Results:

After running the simulation for different element formulations, the results will be as given below as per cases:

Stress Contours:

• We can see the stress results for different element formulations.
• Maximum stress we obtain is 5 Mpa along the Z-direction. As the tension is applied in Z-direction.
• We can observe the same value of maximum stress for all cases, but the distribution of stress is different in all cases.

Strain Contours:

• Here also the maximum strain in Z-Direction is 1.78671, which is same for all cases.

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 plots, we can use the XY data and export it as “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 545.
• So all the results will be regarding Element ID: 545
• Here Stress results for Element 545 will be from Z-Stress
• Strain results for Element 545 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 given below:

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:

Calculations and the result data for all cases are same as given below:

• Here in above image we can see Nominal Stress and Stretch Ratio values in the highlighted portion, which are the rounded values.
• From these values, we can create the graph of Nominal Stress Vs Stretch ratio for all element formulations.

Plots:

Plots of Nominal Stress Vs Stretch Ratio for Element formulation 1, 2,-1, and -2 as per case will be as given below:

CASE_1_Element Formulation = (1):

CASE_2_Element Formulation = (2):

CASE_3_Element Formulation = (-1):

CASE_3_Element Formulation = (-2):

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Comparison:

• Plots obtained from the simulation results, we can observe the Stress and Strain results are same for different solid element formulations.
• We can observe same stretch ratio for all different element formulations.
• But we can see the distribution of stress over the length of the cube is in different manner while stretching. This is due to different behavior of element formulations.
• Also we can observe difference in the time for solving with different element formulations as below:

• From above image, we can observe different cycles and computational time for different element formulation.
• We can see that in case of element formulation -2, it is taking highest time for computation.

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Conclusion:

• Since this is the uniaxial tensile test, we are getting similar results for different element formulations.
• Also the deformation we have given to the cube was not so large for this type of hyper elastic material, so the difference cannot be observed.
• For different boundary conditions and different loading conditions, we can get the different results with change in element formulation type in more efficient manner and clearly visible.
• Since the element formulation (1) is constant stress solid element, which default, cosserat point element is invoked, which is good to propose the strain energy function.
• Element formulation (2) is selective reduced integrated solid element, which give constant pressure throughout the element to avoid pressure locking, but type (1) is more accurate the type (2) for implicit problems.
• Element formulations type (-1) and (-2) are fully integrated elements intended for elements with poor aspect ratios. Among which type (-1) is more efficient and type (-2) is more accurate formulation.
• Since Element formulation (-1) has some side effects related to some particular deformation modes , we can conclude that Element formulation (-2) is more accurate and mostly preferred for the solid elements in Implicit problems.  

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Animations:

Here  the animations of the Von mises Stress for different Element formulations are given as below:

CASE_1: Element Formulation = (1)

CASE_2: Element Formulation = (2)

CASE_3: Element Formulation = (-1)

CASE_4: Element Formulation = (-2)