Analysis of Hyperelastic Materials

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Analysis of Hyperelastic Materials

• MEEN 5330
• Fall 2006
• Added by the professor

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Introduction

• Rubber-like materials ,which are characterized by
  a relatively low elastic modulus and high bulk
  modulus are used in a wide variety of structural
  applications.
• These materials are commonly subjected to large
  strains and deformations.
• Hyperelastic materials experience large strains
  and deformations .
• A material is said to be hyperelastic if there
  exists an elastic potential W(or strain energy
  density function) that is a scalar function of
  one of the strain or deformation tensors, whose
  derivative with respect to a strain component
  determines the corresponding stress component .

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Introduction Contd..

• Second Piola-Kirchoff Stress Tensor
• Lagrangian Strain Function
• Component of Cauchy-Green Deformation Tensor

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Introduction Contd..

• Eigen values of are and exist
  only if
• are the invariants of cauchy-deformation tensor.

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• MATERIAL MODELS
• Why material models?
• Material models predict large-scale material
  deflection and deformations.
• Different material models
• Basically 2 types
• Incompressible
• Mooney-Rivlin
• Arruda-Boyce
• Ogden
• Compressible
• Blatz-Ko
• Hyperfoam

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• Incompressible
• Mooney-Rivlin works with incompressible
  elastomers with strain upto 200. For example,
  rubber for an automobile tyre.
• Arruda-Boyce is well suited for rubbers such as
  silicon and neoprene with strain upto 300 . this
  model provides good curve fitting even when test
  data are limited.
• Ogden works for any incompressible material with
  strain up to 700. This model give better curve
  fitting when data from multiple tests are
  available.

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• Compressible
• Blatz-Ko works specifically for compressible
  polyurethane foam rubbers.
• Hyperfoam can simulate any highly compressible
  material such as a cushion, sponge or padding

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Mooney-Rivlin material

• In 1951,Rivlin and Sunders developed a a
  hyperelastic material model for large
  deformations of rubber.
• This material model is assumed to be
  incompressible and initially isotropic.
• The form of strain energy potential for a
  Mooney-Rivlin material is given as W
  
• Where
• , and are material
  constants.
  

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• Determining the Mooney-Rivlin material
  constants
• The hyperelastic constants in the strain energy
  density function of a material its mechanical
  response .
• So, it is necessary to assess the Mooney-Rivlin
  constants of the materials to obtain successful
  results of a hyperelastic materials.
• It is always recommended to take the data from
  several modes of deformation over a wide range of
  strain values.
• For hyperelastic materials, simple deformation
  tests (consisting of six deformation models ) can
  be used to determine the Mooney-Rivlin
  hyperelastic material.

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Six deformation models

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Six deformation modes contdEven though the
superposition of tensile or compressive
hydrostatic stresses on a loaded incompressible
body results in different stresses, it does not
alter deformation of a material.Upon the
addition of hydrostatic stresses ,the following
modes of deformation are found to be
identical.1.Uniaxial tension and Equibiaxial
compression,2.Uniaxial compression and Equiaxial
tension, and3.Planar tension and Planar
Compression.It reduces to 3 independent
deformation states for which we can obtain
experimental data.

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3 independent deformation statesIn the next
section , we will brief the relationships for
each independent testing mode.
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Deformation Testing Modes

• Equibiaxial Compression
• Equibiaxial Tension
• Pure Shear Deformation

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Deformation Testing Modes Contd..

• Equibiaxial Compression
• Stretch in direction being loaded
• Stretch in directions not being loaded
• Due to incompressibility,

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Deformation Testing Modes Contd..

• For uniaxial tension, first and second invariants
• Stresses in 1 and 2 directions

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Deformation Testing Modes Contd..

• Principal true stress,

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Deformation Testing Modes Contd..

• Equibiaxial Tension Equivalently, Uniaxial
  Compression)
• Stretch in direction being loaded
• Stretch in direction not being loaded
• Utilizing incomressibility equation,

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Deformation Testing Modes Contd..

• For equilibrium tension,
• Stresses in 1 and 3 directions,

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Deformation Testing Modes Contd..

• Principal true stress for Equibiaxial Tension,

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Deformation Testing Modes Contd..

• Pure Shear Deformation
• Due to incompressibility,
• First and Second strain invariants

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Deformation Testing Modes Contd..

• Stresses in 1 and 3 directions
• Principal pure shear true stress

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Stress Error Correction

• To minimize the error in Stresses, we perform a
  least-square fit analysis. Mooney-Rivlin
  constants can be determined from stress-strain
  data.
• Least Square fit minimizes the sum of squared
  error between the experimental values(if any)
  values and cauchy predicted stress values.
• E Relative error.
• Experimental Stress Values.
• Cauchy stress values.
• No. of Experimental Data points.
• This yields a set of simultaneous equations which
  are solved for Mooney-Rivlin Materials Constants.

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Problem statement

• How do we determine the principal true stresses
  in Equibiaxial compression or Equibiaxial
  tension test? Show the figure to illustrate the
  deformation modes.

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References

• 1.Brian Moran,Wing Kam Liu,Ted Belytschko,Hyper
  elastic material,Non-Linear Finite elements for
  continua and Structures,September 2001,(264-265).
• 2.Ernest D.George,JR .,George A.HADUCH and
  Stephen JORDAN The integration of analysis and
  testing for the the simulation of the response
  of hyper elastic materials ,1998 Elsevier science
  publishers B.V(North Holland).
• William Prager,Introduction to mechanics of
  Continua,Dover Publications,New
  York,1961,(157,185,209).
• Theory reference,Chapter 4.Structures with
  Material Non-linearities,Hyper elasticity ANSYS
  6.1 Documentation .Copyright1971,1978,1982,1985,19
  87,1992-2002,SAS IP.
• Web referencewww.impactgensol.com

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Conclusions

• In this, we have analysed Mooney-Rivlin Materials
  constants. Mooney-Rivlin Material C10,C01 by
  using 6 deformation modes.
• We determine principle stresses using Equibiaxial
  compression(Uniaxial Tension), Equibiaxial
  Tension(Uniaxial Compression), Pure shear.
• Resultant values are taken as Cumulative values
  and the errors in the resultant values are
  minimised using Least-square fit Analysis.
• According to this analysis, we can say that
  materials having high stress-strain values,
  mooney-rivlin model can be used to determine the
  material constants for hyperelastic materials.