Calculation of Stretch Ratio by comparing the ELFORM (-2,-1,1,2) with Ogden_Material Model in LS-DYNA

Objective:

The main aim of this work is to calculate the stretch ratio of a cubic block subjected to tension, used with element formulations -2,-1,1,2 using Ogden_Material material model in LS-DYNA.

Also to compare the uniaxial plots of stress versus stretch ratio plots among each cases.

The unit system followed in this project work is gm-mm-s.

The problem is solved using LS-DYNA implicit solver.

Engineering strain and True strain:

Engineering strain is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied.

The engineering normal strain or extensional strain or nominal strain e of a material line element or fiber axially loaded is expressed as the change in length ΔL per unit of the original length L of the line element or fibers.

The normal strain is positive if the material fibers are stretched and negative if they are compressed.

`e=(DeltaL)/L=(l-L)/L`

where e is the engineering normal strain, L is the original length of the body and l is the final length of the body.

Measures of strain are often expressed in terms of parts per million or microstrains.

The logarithmic strain is the true strain. Considering an incremental strain,

`deltaepsi=(deltal)/l`

The logarithmic strain is obtained by integrating this incremental strain,

`intdeltae=int(deltal)/l` over L to l.

`epsi=ln(l/L)=lnlambda`

`epsi=ln(e+1)`

where e is the engineering strain and `epsi` is the true strain.

The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path

Stretch ratio:

Stretch ratio is a measure, used for highly extensible materials such as rubber, of normal component of strain subjected to tension, and is defined as the ratio of deformed length to the original length of the material body.

The stretch ratio is used in the analysis of materials that exhibit large deformations, such as elastomers, which can sustain stretch ratios of 3 or 4 before they fail.

On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios.

Stretch ratio is defined as the ratio between the final length l and the initial length L of the material line.

Stretch ratio, `lambda=l/L`

The relation between stretch ratio and engineering strain is given by

`e=lamda-1`

This equation implies that the normal strain is zero, so that there is no deformation when the stretch is equal to unity.

Simulation setup:

A block mesh of 10mm x 10mm x 10mm is created using box shape mesher in LS-DYNA prepost.

The block is created with 10 elements in each global direction.

The nodes of one of the faces of the block are provided with single point constraint (SPC) boundary condition using node set with free DOF in translational y and z direction.The nodes are constrained with translational x-direction and all rotational directions.

This is done because when the nodes are subjected to tension in x-direction, there exists strain in the lateral direction of the block(i.e.in global y and z direction)

Prescribed boundary motion set card is used to provide movement to the face nodes that are opposite to the SPC node set.

The load curve ID is referenced to the *BOUNDARY_PRESCRIBED_MOTION_SET card.

A diplacement load curve is defined for the motion node set as follows.

Material setup:

The block is modelled with *MAT_077 Ogden material in LS-DYNA material setup.

The Ogden material model is a hyperelastic material model used to describe the non-linear stress-strain behaviour of complex materials such as polymers,rubbers and biological tissue.

The Ogden model material is referenced to the block using part option.

The strain energy function in the Ogden model is written with components stretch ratio λ, shear modulus μ, and constant α.

The following figure displays the details of various properties of Ogden model rubber material.

Analysis method:

In order to run the tensile behaviour simulation of the block, we use implicit analysis scheme.

LS-DYNA employs both implicit and explicit analyses methods to solve static and dynamic problems that differ in the approach to time increment.

In implicit analysis, each time increment has to converge, but we can set a quite long time increments. Explicit analysis on the other hand, doesn't need to converge at each time step.

Implicit analysis works in a way that at each time increment, global equilibrium is established.This basically happens in iterative methods.

Generally, implicit analysis method is used for its accuracy in results and can be applied for long duration problems.

Section setup:

The FE model of block is modelled with solid section in LS-DYNA.

Element formulation deals with the mathematics and assumptions that drive the development of the displacements for different finite element types.

Insights gained from displacements of the elements help determine the strain, stresses and nodal forces for that type of element.

Element formulations such as ELFORM -2,-1,1,2 are used in each cases to validate the results.

ELFORM -2 : fully integrated S/R solid intended for elements with poor aspect ratio, accurate formulation.

ELFORM -1 : fully integrated S/R solid intended for elements with poor aspect ratio, efficient formulation.

ELFORM 1 : constant stress solid element.

ELFORM 2 : fully integrated S/R solid.

The solid section is referenced to the block using part option.

Control setup:

The simulation is set to run for 20s using *CONTROL_TERMINATION card.

A few control implicit cards are used to define accuracy,constant time step size,integration approach,etc.

Model checking and solving the setup:

One of the most important pre-processing tasks is to ensure the validity of a CAE model before submitting it to the solver for analysis.

A good and rich suite of checks reduces engineering errors, increses confidence in the simulation results, and minimizes the duration of simulation cycle by eliminating multiple back and forths from the solver to the pre-processor.

After getting cleared up from the errors if any, the problem setup is saved.

The saved keyword file is set to run in LS DYNA manager.

A normal termination of the problem setup is achieved and the results are analyzed for post processing.

Post processing results:

The requested outputs for d3plot,history and ASCII options are now ready to view in LS prepost for post processing.

1. Animation of the simulation

2. Uniaxial stress Vs Stretch ratio plots:

Since the block is subjected to tension only in global X direction, a uniaxial stress VS stretch ratio plot, for all the element formulation cases, is sufficient to validate the results.

The plots are figured for upto a stretch ratio of 5 in all the cases.

The stress and strain in the fringe component of LS DYNA post processing tool are actually true stress and strain values respectively.

Therefore, suitable calculations are used to convert from true stress/strain into engineering stress/strain.

These plots are drawn for the maximum stress/strain in the whole block by selecting the option History>Element>Value=Max and selecting the whole block.

=> For ELFORM -2

=> For ELFORM -1

=> For ELFORM 1

=> For ELFORM 2

Comparative inference:

From the plots obtained from the simulation data, it is found that the maximum uniaxial stress in X direction of tension is typically same in all the solid element formulation cases.

The variation of plots in each case is due to the usage of different solid element formulations.

It is found that the stretch ratio for the solid block is more in the case of ELFORM -1 (i.e. fully integrated S/R solid intended for elements with poor aspect ratio, efficient formulation).

Also, the solid block with ELFORM -2,-1, 1, 2 have non-linear stress VS stretch ratio variations.

In all the plots, there seems a yielding stress at the stretch ratio of 2.

There may exist a different variation of plots of stress VS stretch ratio with respect to different displacement boundary conditions.

Conclusion:

Thus, a tensile loading simulation is performed on a cubic block using Hyper-elastic Ogden model material like vulcanized rubber.

Various solid element formulations have been applied in each cases accordingly and the post-processing results are validated.

Implicit scheme approach is used to solve the problem in order to obtain an efficient formulation of results.

Since, the data obtained from fringe component of LS-DYNA are in terms of true stress/strains, they are converted into engineering stress/strains accordingly with suitable calculations.

The plots for Nominal stress VS Stretch ratio plots for all the element formulation cases have been drawn and the data are concluded.