Analysis of Hyperelastic Materials

MEEN 5330Fall 2006

Added by the professor

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 .

ijij WS /

Introduction Contd..

Second Piola-Kirchoff Stress Tensor

Lagrangian Strain Function

Component of Cauchy-Green Deformation Tensor

ijij WS /

)(2/1 ijijij C

kjikij FFC

Introduction Contd..

Eigen values of are and exist only if

are the invariants of cauchy-deformation tensor.

23

22

21 , andijC

0]det[ 2 ijpijC

032

24

16 III ppp

321 &, III

MATERIAL MODELSWhy material models?

Material models predict large-scale material deflection and deformations.

Different material modelsBasically 2 types

Incompressible Mooney-Rivlin Arruda-Boyce

Ogden

Compressible Blatz-Ko

Hyperfoam

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.

Compressible

Blatz-Ko works specifically for compressible polyurethane foam rubbers.

Hyperfoam can simulate any highly compressible material such as a cushion, sponge or padding

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.

10c

2201110 )1(/1)3()3( JdIcIc

01c d

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.

Six deformation models :

Six deformation modes contd…

Even 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.

3 independent deformation states:In the next section , we will brief the relationships for each independent testing mode.

Deformation Testing Modes Equibiaxial Compression Equibiaxial Tension Pure Shear Deformation

Deformation Testing Modes Contd..

Equibiaxial Compression Stretch in direction being loaded Stretch in directions not being

loaded Due to incompressibility,

1 32

1132

2/1132

Deformation Testing Modes Contd..

For uniaxial tension, first and second invariants

Stresses in 1 and 2 directions

11

211 2 I

2112 2 I

212

21111 /2/2 IWIWp

Deformation Testing Modes Contd..

Principal true stress,

0/2/2 1211122 IWIWp

]//)[(2 2111

11

2111 IwIW

Deformation Testing Modes Contd..

Equibiaxial Tension Equivalently, Uniaxial Compression)

Stretch in direction being loaded

Stretch in direction not being loaded

Utilizing incomressibility equation,

21

3

213

Deformation Testing Modes Contd..

For equilibrium tension,

Stresses in 1 and 3 directions,

41

211 2 I

21

412 2 I

212

21111 /2/2 IWIWp

11212

21133 /2/2 IWIWp

Deformation Testing Modes Contd..

Principal true stress for Equibiaxial Tension,

]//)[(2 2211

41

2111 IwIW

Deformation Testing Modes Contd..

Pure Shear Deformation

Due to incompressibility,

First and Second strain invariants

113

121

211 I

121

212 I

Deformation Testing Modes Contd..

Stresses in 1 and 3 directions

Principal pure shear true stress

212

21111 /2/2 IWIWp

0/2/2 212

21133 IWIWp

]//)[(2 2121

2111 IwIW

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.

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.

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 .Copyright1971,1978,1982,1985,1987,1992-2002,SAS IP.

Web reference:www.impactgensol.com

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.