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Finite Element Analysis of Elastomers
Nordic Rubber Conference
Malmö 9-10/4 –14
Ass. prof. Per-Erik AustrellLund University
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Occupation... Work... Employment..
o Ass. Professor (Univ. lektor) at the StructuralMechanics dep. at Lund University
o Lecturer in a basic course (150 stud.) in Mechanics(Newtons laws) and in an advanced course (30 stud.) in Structural Dynamics
o Specialist in modelling Rubber Units; material modelling and Finite Element Analysis, interest in applications, cooperation with companies
o Also windsurfer, golfer, and inline skater ... whenpossible
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DISPOSITION
o Background & some projects
o Modelling rubber material properties
o FEA of rubber components
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THESES –97 & -07: P-E AUSTRELL & A K OLSSON
Material modelling aiming at FEA. Experimental and theoretical work. Cont. into FEA procedures:
Overlay method etc.
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DEVELOPMENT OF AN ELASTOMERIC FINGER JOINT
Work with P-I Brånemark –99-01
The knuckle joint was replaced
Focus on choice of material and on shape optimization by FE-analysis
The existing phrostesis was re-designed for better handling of compressive forces
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RUBBER COVERED ROLLERS
Extensive work going on for about 2-3 years.
Documentation about 230 pages
A stand alone software was developed for roller simulations etc.
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DEVELOPMENT OF A SHOCK ABSORBER
Hole filling shock absorber by using computer simulations
Excellent fatigue properties
A patent application was
sent in.
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DISPOSITION
o Background & some projects
o Modelling rubber material properties
o FEA of rubber components
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SCHEMATIC MATERIAL STRUCTURE OF A CARBON-BLACK-FILLED RUBBER VULCANIZATE
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RUBBER ELASTICITY AND DAMPING -A SIMPLE ONE-DIMENSIONAL MODEL
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Rubber elast. & damping ...... cont.
Hysteresis loops at slow harmonic loading:
Deviations from pure viscoelastic behavior:
� the losses remains at slow loading rate
� the hysteresis loop has sharp edges
� the dynamic modulus depends on amplitude
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MATERIAL MODELS FOR RUBBER IN
FEA
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HYPERELASTIC MATERIAL MODELS
neo-Hooke model:
One parameter; C10
Yeoh model:
Three parameters;
C10, C20 & C30
C10=G/2 with
G; initial shear modulus
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LARGE STRAIN VISCOELASTICITY
Two options in commercial codes:
a) Small harmonic vibrations on a large static (hyper)elastic deformation
b) Large strain transient analysis
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DYNAMIC STIFFNESS AND PHASE ANGLE
Sinusoidal test in simple shear:
Rem. Linear viscoelastic:
•Elliptic hysteresis curve
•Dynamic modulus dependent on frequency only
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AMPLITUDE DEPENDENCE
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ONE-DIMENSIONAL MODELS
Combining yields a 5-parameter model .....
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ONE-DIMENSIONAL ... cont.
.... giving both amplitude and frequency dependence:
Phase angle
AmplitudeAmplitude
Dynamic modulus
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A GENERALIZED VISCO-FRICTIONAL MODEL
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EXAMPLE FIT; HNBR - Generalized model
solid lines: model dotted: shear test
Each color represents a specific shear strain amplitude:
red; 1% blue; 3% green; 7% blue-green; 12%
solid lines: model dotted: shear test
Modulus Phase angle
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DISPOSITION
o Background & some projects
o Modelling rubber material properties
o FEA of rubber components
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FE-CODES FOR NON-LINEAR ANALYSIS OF RUBBER UNITS
Requirements:
The codes must handle:
� Large deformations and strains
�Almost incompressible materials
� Contact conditions
� Hyperelastic, viscoelastic and possibly also elastoplasticmaterial models
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FEA WITH HYPERELASTIC MODELS
Useful for:
• Visualizing deformations in rubber units
• Generating spring characteristics
• Computing principal stresses for judgement of crack propagation risk
• etc
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OVERVIEW OF THE OVERLAY
METHOD
Going from testing to FE-model:
� Harmonic shear test
� Viscoplastic model (1D)
� Mesh assembly with several layers on the same nodes
� FE-model
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EXAMPLE: RADIALLY LOADED BUSHING
Verification example: Harmonically loaded NR bushing
Double shear specimens
.... and cylindric bushings were manufactured
... by the same 50 Shore NR-material
A FE-model based on the overlay method was compared to measured data for the bushing at various frequencies and amplitudes
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Shear test vs. 1D-model:
1D
1D
EXAMPLE: RADIALLY LOADED ... cont. 1
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Bushing vs. FE-model:
EXAMPLE: RADIALLY LOADED ... cont. 2
Rem. Taking thermal stresses from the manufacturing into account lowers the stiffness of the model
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P-E A PREVIOUS WORK
� Two NUTEK projects on rubber modelling: 93-99
� Thesis -97: Modeling of elasticity and damping for filled elastomers
� Bio-mechanics; Elastomeric finger joint - P-I Brånemark
� Roller modelling and simulation - Tetra Pak (ongoing)
� Rubber bushing modelling – Volvo Cars (ongoing)
� Modelling of gaskets in plate heat exchangers - Alfa Laval
� Development of a shock absorber – Bosch-Rexroth
� Supervisor for about 20 Masters Thesis works on rubber components
� Supervisor for PhD student Anders K Olsson (tekn. dr.) and other PhD students
� Leif Kari KTH and P-E A organizers for the European Conference on Constitutive Modelling of Rubber –05 (www.eccmr.org)
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LITERATURE
� 1997 Austrell P.-E., Modeling of Elasticity and Damping for Filled ElastomersThesis, Report TVSM–1009, Lund Institute of Technology, Division of Structural Mechanics, Lund, Sweden.
� 1998 Austrell P.-E. et. al. Survey of design methods and material characteristics in rubber engineering, a NUTEK-VAMP report, TVSM–3036, Division of Structural Mechanics, Lund, Sweden.
� 2000 Austrell P.-E., Educational material Konstruktionsberäkningar förgummikomponenter, TVSM–7131, Division of Structural Mechanics, Lund, Sweden.
� 2001 Austrell P-E. et. al., A method to analyze the dynamic behavior of carbonblackfilled rubber components using standard FE-codes, proceedings of the second European conference on constitutive models for rubber (ECCMR), Hannover, Germany, September 2001
� 2001 Austrell P-E. and Olsson A.K., A Fitting Procedure for a Viscoelastic-Elastoplastic Material Model, proceedings of the second European conference on constitutive models for rubber (ECCMR), Hannover, Germany, September 2001
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LITERATURE ... Cont. 1
� 2002 Austrell P-E., Introduction to Hyperelastic Characterization of Elastomers, Educational text LTH
� 2003 Olsson A.K. and Austrell P-E., Finite element analysis of a rubber bushing considering rate and amplitude dependent effects, Proceedings of the third European conference on constitutive models for rubber (ECCMR), London, UK
� 2004 Austrell et al, Utveckling av ändlägesdämpare för Bosch Rexroth teknik AB, TVSM–7140, Division of Structural Mechanics, Lund.
� 2005 Austrell P-E. and Olsson A.K., Approximate viscoelastic FE-procedures in frequency and time domain to account for the Fletcher-Gent effect, Proceedings of the fourth European conference on constitutive models for rubber (ECCMR), Stockholm, Sweden
� 2007 Olsson A. K., Finite Element Procedures in Modelling the Dynamic Properties of Rubber, Thesis, Report TVSM–1021, Lund Institute of Technology, Division of Structural Mechanics, Lund, Sweden.
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LITERATURE ... Cont. 2
� 2007 Austrell P-E. and Wirje A., A viscoelastic bump stop model for multi-body simulations, Proceedings of the fifth European conference on constitutive models for rubber (ECCMR), Paris, France.
� 2009 Austrell P-E. and Olsson A.K., Dynamic characterization of elastomers using using impact testing, Proceedings of the sixth European conference on constitutive models for rubber (ECCMR), Dresden, Germany.
� 2010 Austrell P-E. and Olsson A.K., Analytical models of nip properties and hysteresis heating for rolling contact of rubber covered rollers, Modelling of Elastomeric Materials and Products, RIEG conference, London, England, October.
� 2012 Austrell & Olsson Considering amplitude dependence during cyclicloading of elastomers using an equivalent viscoelastic approach, Polymer Testing No 31.
� 2012 Austrell & Olsson, Modelling procedures and properties of rubber in rolling contact, Polymer Testing No 32.
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LITERATURE ... Cont. 3
�2013 Austrell & Olsson, A two dimensional elastic contact model for rubber covered rollers, Plastics, Rubber and Composites Vol.42 No 7.
� 2013 Austrell P.-E. Analytical modelling of hysteresis heating in rolling contact of rubber covered rollers, Plastics, Rubber and Composites Vol.42 No 7.
� 2013 Austrell P.-E. and Ahadi A., Hyperelastic constants from a modified hardness test using energy balance obtained from FE-analysis, Proceedings of the eight European conference on constitutive models for rubber (ECCMR), San Sebastian, Spain.
� 2013 Negreira J., Austrell P-E, Flod\'en O., Bard D., Characterisation of an elastomer for noise and vibration isolation in lightweight timber buildings, (submitted)
� 2014 Austrell P.-E., Stenbom B., Lindvall M., Andersson L.-O., Development of an extremely fatigue resistant shock absorber, (to be submitted)
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DYNAMIC PROPERTIES
Left: Dyn. Modulus
Right: Loss angle
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FITTING THE GENERALIZED MODEL
... model parameters (left) are fitted to the double shear test by minimizing the goal function
the weight factor α gives the importance of dynamic modulus vs damping
Goal function:
damping dynamic modulus
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FIT OF THE GENERALIZED MODEL .... cont.
... software in MATLAB
... choose the number of viscous and frictional elements
Also the the weight damping vs modulus
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EXAMPLE FIT; HYPALON
small amplitude dependence and big frequency dependence
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EXAMPLE FIT; SILICON
big amplitude dependence and small frequency dependence
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CHOICE OF ANALYSIS TYPE
� Standard codes gives correct solutions when convergent, not always true for explicit codes
� Contact works better in Explicit codes
� Complete incompressibility is not possible in explicit codes*
Standard codes:
Uses stiffness and mass matrices and tangential directions and (dynamic) equilibrium iterations
Explicit codes:
Dynamic solution method with direct time steppingno equilibrium iterations. Static loading can be simulated dynamically
Rem. *) But B = 100 G is OK.
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CHOICE OF ANALYSIS TYPE ... cont.
Memory/CPU time:
Abaqus-Standard: 2.5 GB / 250 CPU hours
Abaqus-Explicit:80MB / 7 CPU hours
Example 3D-Analysis:
Tetra Pak package sealing
Explicit or implicit analysis ?
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THESIS –97 P-E AUSTRELL
Material modelling of rubber aiming at finite element analysis.
Both experimental and theoretical work.
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THESIS ANDERS K OLSSON
P-E A supervisor 2000 – 2007
FEA procedures:
Overlay method
Equiv. viscoelast. formulation
etc.