hyperelasticity_f06
TRANSCRIPT
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.