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Finite element simulation of deformation behaviour of cellular rubber components

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Finite element simulation of deformation behaviour of cellular rubber components
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  PAMM  ·  Proc. Appl. Math. Mech.  12 , 437–438 (2012) /   DOI  10.1002/pamm.201210207 Finite element simulation of deformation behaviour of cellular rubbercomponents R. Raghunath 1 and  D. Juhre 1, ∗ 1 Deutsches Institut für Kautschuktechnologie e.V. (DIK), HannoverThis paper presents a new prospect of investigating the mechanical behaviour of cellular rubber using porous hyperelasticmaterial model. There are number of hyperelastic material models to describe the behaviour of homogeneous elastomer, butvery few to characterise the complex properties of cellular rubber. The analysis of dependence of material behaviour on poredensity using the new material model is supported with experiments to characterise the actual material behaviour. The newmaterial model which is based on Danielsson et al [1] decouples the influence of porosity from the mechanical properties of the solid material by introducing volume fraction of the pores as an explicit scalar variable. The finite element simulations arethen followed by experiments on complex model to validate the material model. c  2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Material model Fig. 1  The spherical volume element: ( a ) undeformed configuration, ( b ,  c ) deformed configuration, [1]. The material with non-uniformly distributed pores is first homogenised into a pore-containing matrix with a thick walledsphere considered to represent the undeformed porous material subjected to external loading (fig.1 a ). The initial material,whose matrix is taken to be pointwise incompressible, is characterised by an initial void volume fraction  f  0 = ( AB ) 3 , whereA and B are the inner and outer radii of the sphere, respectively. When subjected to the principal macroscopic stretch state (¯ λ 1 , ¯ λ 2 , ¯ λ 3 ) , the outer surface of the sphere transforms into an ellipsoid (fig.1 b , c ).The kinematic relationship of deformation field subjected to macroscopic stretches together with the boundary conditionsis given by x i  =¯ λ i ¯ J  1 / 3  1 +  B 3 ( ¯ J   − 1) R 3  1 / 3 X  i  where  ¯ J   ≡ det ¯ F  = ¯ λ 1 ¯ λ 2 ¯ λ 3  = 1  (1)The components of the microscopic/pointwise deformation gradient,  F , at every point in the matrix, can then be expressed as [ F ] ij  =¯ λ i ¯ J  2 / 3  (1 −  ¯ J  ) φ 2 R 2 B 3 R 3 X  i X  j  + φδ  ij   where φ  =  1 + ( ¯ J   − 1) BR 3  1 / 3 (2)The pointwise strain energy density function for an incompressible isotropic hyperelastic material determined at every pointof sphere can be expressed in terms of the stretch invariants  I  1  and  I  2  as W   = ˆ W  ( I  1 ,I  2 ; f  0 )  (3)The homogenised strain energy density of the sphere,  ¯ W  , is obtained by integrating the pointwise strain energy densityfunction, W, over the reference volume, and dividing by the reference volume, ¯ W   = 1 V  0    BBf  130    2 π 0    π 0 ˆ W  ( I  1 ,I  2 ; P  ) R 2 sinΘ  d Θ  d Φ  dR  (4) ∗ Corresponding author: e-mail daniel.juhre@dikautschuk.de, phone +4951184201-18 fax +495118386826 c  2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim  438 Section 8: Multiscales and homogenization where B is the outer radius of the sphere (fig.1), the reference volume is given by  V  0 = 4 πB 3 / 3 .  0 ≤ Θ ≤ π  and  0 ≤ Φ ≤ 2 π are standard spherical angles, measured in the reference configuration used to define the components of position vector  X  inspherical coordinate system.The strain energy density function of the Yeoh model for solid materials is defined by W   =  C  10 ( I  iso 1  − 3) + C  20 ( I  iso 1  − 3) 2 +  C  30 ( I  iso 1  − 3) 3 + W  vol (5)where  C  10 ,C  20  and  C  30  are the material constants,  I  iso 1  is the isochoric first invariant and  W  vol is the volumetric part of density function. Upon integrating the pointwise strain energy density function W in eq.5 over the reference volume accordingto eq.4, the strain energy density,  ¯ W   of the hollow sphere subjected to a macroscopic state of stretch can be evaluated as, ¯ W   = 3  i =1 C  i 0  I  1  2 − 1 J   − f  0  + 2( J   − 1) J  23 η 13  − 3(1 − f  0 )  i where η  =  1 +  J   − 1 f  0   (6)The second Piola-Kirchhoff stress then can be derived using the general relation, S  = 2 ∂   ¯ W ∂  ¯ I  1 1  + J ∂   ¯ W ∂   ¯ J  ¯C − 1 (7) 2 Experiments For hyperelastic materials, simple deformation tests (uniaxial tension, equibiaxial tension and pure shear) can be used toaccurately characterise the material constants. The experiments are not directly conducted on cellular rubber, due to thepresence of pores in the material which lead to an inelastic and inhomogeneous deformation. So the experiments are conductedon the solid material and during the simulation, pores are explicitly introduced into the mathematical model as the scalar value f  0 . The scalar value determines the density of pore in the material and is measured by a method called  Micro-CT   scanning (anon-destructive method to reproduce three-dimensional images of components including internal in-homogeneities). 3 Results and summary The model is used for simulating the deformation behaviour of complex sealing components (fig.2( a )). The material pa-rameters are taken from the solid rubber characterisation and the  Micro-CT   investigations. The deformation behaviour isvalidated by comparing the experimental/numerical deformation shape (fig.2( b )) and the corresponding force-displacementcurve (fig.2( c )). Fig. 2  ( a ) FE-simulation of car door sealing, ( b ) deformed shapes of experiment and simulation, ( c ) force-displacementcurves of experiment and simulation. The results show the capability of the new model to predict the deformation behaviour of cellular rubber in a finite deforma-tion range. Besides the conventional material parameters of the basic hyperelastic material model, there is only one additional,physically motivated material parameter  f  0  necessary to describe the influence of the pore volume on the material behaviour. Acknowledgements  The authors would like to acknowledge the financial support by METZELER Automotive Profile Systems GmbH. References [1] M. Danielsson, D.M. Parks, M.C. Boyce. Constitutive modeling of porous hyperelastic materials. Mechanics of Materials  36 , 347-358,(2004). c  2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim  www.gamm-proceedings.com
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