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Simulation of ElastomersChristopher Smith CEng

Principal Engineer

Wilde Analysis Ltd

Manchester Polymer Group

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16th of May 2018

Simulation of Elastomers – Why Simulate Elastomers?

• Elastomers have excellent damping, energy absorption, flexibility, and resilience.

• Efficient at sealing against moisture, heat, and pressure and are available at low cost.

• For use in critical structural applications designs must be validated and optimized.

• Simulation tools such as FEA provide significant design insight!

Application Examples:

Door seals

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• Door seals

• O-rings

• Shock mounts

• Rubber boots

• Gaskets

Why Simulate Elastomers?

Elastomers have excellent damping, energy absorption, flexibility, and resilience.

Efficient at sealing against moisture, heat, and pressure and are available at low cost.

For use in critical structural applications designs must be validated and optimized.

Simulation tools such as FEA provide significant design insight!


Simulation of Elastomers – Linear Finite Element Analysis

• Robert Hooke discovered a simple linear relationship between force (F) and displacement (u), where the constant (K) represents structural stiffness.

• A linear structure obeys this linear relationship e.g. a simple spring.

• Linear structures are suited to finite-element analysis which is based on linear matrix algebra.

• So how do we then solve for a changing stiffness?

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K

F

u

Linear Finite Element Analysis

Robert Hooke discovered a simple linear relationship between force (F) and displacement (u), where the constant (K) represents structural stiffness.

A linear structure obeys this linear relationship e.g. a simple spring.

element analysis which is based on linear matrix algebra.

So how do we then solve for a changing stiffness?

F = Ku


K

F

u

Simulation of Elastomers – Nonlinear Finite Element Analysis

• A nonlinear structure can be analyzed using an iterative series of linear approximations.

• We use an iterative process called the Newton-equilibrium iteration.

• A Newton-Raphson analysis with four equilibrium iterations is shown below.

• Users should specify small load increments -> essential for solution accuracy and convergence.

Load

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F

u Displacement

1

23

4

Nonlinear Finite Element Analysis

A nonlinear structure can be analyzed using an iterative series of linear approximations.

-Raphson Method where each iteration is an

Raphson analysis with four equilibrium iterations is shown below.

> essential for solution accuracy and convergence.


Solution is ‘converged’ when the sum of the internal forces is equal to the externally applied forces i.e. a conservation of ‘energy’.

For highly non-linear structures it can be challenging to obtain physical convergence.

Simulation of Elastomers – Typical Sources of Non

• Non-linear material stiffness.

• Complex contact interfaces e.g. self-contact, frictional sliding, separation of surfaces.

• Large deformations which can lead to elemental distortion.

• Accurate simulation of elastomers presents many of these challenges.

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Typical Sources of Non-linearity

contact, frictional sliding, separation of surfaces.

Large deformations which can lead to elemental distortion.

Accurate simulation of elastomers presents many of these challenges.


Simulation of Elastomers – Example: Plastic Snap

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Example: Plastic Snap-fit/Clip

This is not an elastomer application but the simulation is challenging as snap-fits can lead to a sudden reduction in global stiffness.


Simulation of Elastomers – Example: Rubber Boot Seal

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Example: Rubber Boot Seal

The rubber must accommodate movement of the shaft in order to protect the joint from the environment.

The deformation leads to lots of self contact and


lots of self contact and frictional sliding.

Simulation of Elastomers – Materials in Analysis

• Most conventional engineering materials are linearonly when yielding occurs.

• Elastomers are challenging as they have nonyield point, and are often considered to be incompressible. Using conventional methods to capture their response is either difficult or inaccurate.

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Typical σ-ε Curve Bi-linear approximation

Materials in Analysis

Most conventional engineering materials are linear-elastic and become non-linear

Elastomers are challenging as they have non-linear elastic properties, no obvious yield point, and are often considered to be incompressible. Using conventional methods to capture their response is either difficult or inaccurate.


linear approximation Hyper-elastic Material

Simulation of Elastomers – Material Characterization

• For hyper-elastic materials strain energy density functions are therefore used to relate strain energies to deformation. The functions are ‘fitted’ to the material test data using an iterative procedure.

PolynomialPhenomenological Model1st and 2nd Strain Invariants

Mooney-RivlinPhenomenological Model

YeohPhenomenological Model

OgdenPhenomenological ModelPrincipal Stretches

Neo-

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Incompressible (left) and compressible (right) models based on principal stretches

Nearlymodels based on 1st strain invariant

Nearly-/fully-incompressible phenomenological hyper-elastic

models based on strain invariants.

2-term M-RPhenomenological Model1st and 2nd Strain Invariants

Neo-HookeanPhenomenological Model1st Strain Invariant

3-term YeohPhenomenological Model1st Strain Invariant

Phenomenological Model1st and 2nd Strain Invariants

Phenomenological Model1st Strain Invariant

Neo-Phenomenological Model1st Strain Invariant

ArrudaMicromechanical Model1st Strain Invariant

Material Characterization

elastic materials strain energy density functions are therefore used to relate strain energies to deformation. The functions are ‘fitted’ to the material test

Functions based on stretch ratios

OgdenPhenomenological ModelPrincipal Stretches

Blatz-KoPhenomenological Model1st Strain Invariant

-Hookean


Incompressible (left) and compressible (right) models based on principal stretches

Nearly-/fully-incompressible micromechanical models based on 1st strain invariant

-HookeanPhenomenological Model1st Strain Invariant

Arruda-BoyceMicromechanical Model1st Strain Invariant

GentMicromechanical Model1st Strain Invariant

Statistical-Mechanical based models


• Strain functions are fitted to material data acquired by physical testing.

• Should obtain experimental data for all modes of deformation, as well as the entire strain range and temperature of interest.

• Consider the application during

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• Consider the application during testing.

• Typical tests: Uniaxial, Equibiaxial tension, volumetric compression, shear, dynamic/static friction.



Biaxial TestShear TestUniaxial Test


• The best strain energy density function is the one that produces the closest curve fit of stress vs. strain to the test data. The fit needs to be good up to the strain range of the particular application being simulated.

Neo-Hookean

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The best strain energy density function is the one that produces the closest curve fit of stress vs. strain to the test data. The fit needs to be good up to the strain range of the particular application being simulated.

Mooney-Rivlin 9 Parameter



• Some common elastomers exhibit dramatic strain amplitude and cycling effects at moderate strain levels.

• Advanced models such as Mullins and Bergstromrequired, but it is common to instead test to the strain levels of the application.

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Some common elastomers exhibit dramatic strain amplitude and cycling effects

Advanced models such as Mullins and Bergstrom-Boyce can capture this if required, but it is common to instead test to the strain levels of the application.


Simulation of Elastomers – Example: O-ring

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ring


Simulation of Elastomers – Conventional Element Formulations

• Conventional element formulations relate volumetric strains to nodal displacements.

• Shear locking and Volumetric locking can result in an overly stiff response.

• Leads to significant errors when elemental volumetric strains are derived from displacements.

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Conventional Element Formulations

Conventional element formulations relate volumetric strains to nodal displacements.

Shear locking and Volumetric locking can result in an overly stiff response.

Leads to significant errors when elemental volumetric strains are derived from


Simulation of Elastomers – Mixed U-P Element Formulation

• The Mixed U-P element formulation should be used for simulation of incompressible materials.

• Hydrostatic pressure is solved as an additional DOF and the volume change rate of an element is solved on the global level independent of local displacements.

• The Lagrange Multipliers (internal Pressure Degree Of Freedom) are kept in the assembled stiffness matrix, so direct solvers such as the ANSYS sparse solver used with the above formulation.

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used with the above formulation.

• In case of fully incompressible material, the matrix equation is:

0K

KK

Pu

uPuu

P Element Formulation

P element formulation should be used for simulation of incompressible

ydrostatic pressure is solved as an additional DOF and the volume change rate of an element is solved on the global level independent of local displacements.

The Lagrange Multipliers (internal Pressure Degree Of Freedom) are kept in the assembled stiffness matrix, so direct solvers such as the ANSYS sparse solver must


In case of fully incompressible material, the matrix equation is:

0

F

P

u

Simulation of Elastomers – Elemental Deformations

• Large elemental deformations push many implicit finite element codes to their limits.

• As elements become highly deformed accurate solutions can no longer be obtained.

• Adaptive remeshing allows the solution of some very large deformation problems.

• Use of Explicit codes such as AUTODYN or LS-DYNA is a viable alternative however their usage is less common in industry for elastomeric applications.

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Elemental Deformations

Large elemental deformations push many implicit finite element codes to their limits.

As elements become highly deformed accurate solutions can no longer be obtained.

Adaptive remeshing allows the solution of some very large deformation problems.

DYNA is a viable alternative however their usage is less common in industry for elastomeric applications.


Simulation of Elastomers – Elemental Deformations

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Elemental Deformations


Simulation of Elastomers – Results

• Stress/strain.

• Deformation and sliding distances.

• Contact sealing pressures.

• Insertion and retention forces.

• All results can be easily obtained from simulation software.

• All outputs are heavily influenced by the material characteristics.

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• All outputs are heavily influenced by the material characteristics.

All results can be easily obtained from simulation software.

All outputs are heavily influenced by the material characteristics.


All outputs are heavily influenced by the material characteristics.

Simulation of Elastomers – Summary

• Many elastomer applications can be simulated.

• Material testing is critical for accurate simulation results.

• A number of simulation tools have been developed to enable accurate simulation of elastomers in structural applications.

• It is still common to use 2D models for simulation of complex applications.

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Any Questions?

Many elastomer applications can be simulated.

Material testing is critical for accurate simulation results.

A number of simulation tools have been developed to enable accurate simulation of elastomers in structural applications.

It is still common to use 2D models for simulation of complex applications.


Any Questions?

simulation of elastomers...simulation of elastomers –nonlinear finite element analysis •a...

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