A Constitutive Model of Pseudo-Hyperelasticity for Description of Rubber-Like Materials

Transcription

1 MAEC Web of Conferenes 7, 6 (7) DO:.5/ mateonf/776 XXV R-S-P Seminar 7, heoretial Foundation of Civil Engineering A Constitutive Model of Pseudo-Hyperelastiity for Desription of Rubber-Like Materials Stanisław Jemioło,* WU he Faulty of Civil Engineering, Al. Armii Ludowej 6, -637 Warszawa, Poland Abstrat. A onsistent thermomehanial model appliable to pseudoelasti behaviour of rubber-like materials has been proposed. he model is based on the Naumann and hlemann s approah [] and the onept of material stiffening given by Ogden and Roxburg [,3]. he onstitutive relationships of psedo-hyperelastiity are derived for inompressible isotropi materials. Futhermore, a speifi model desribing the Mullins effet was proposed. he homogeneous deformations are disussed and the obtained results are ompared with Mullins and obin s [4] experimental data. Good orrelation between theoretial and experimental data has been observed. ntrodution Rubber-like materials exhibit a highly nonlinear elasti behaviour but in yli loading the stress-softening phenomenon is observed [4]. his effet is now widely known as the Mullins effet [5]. herefore, the rubber-like materials annot have a strain-energy funtion in the hyperelasti sense [6]. However, in a yli loading and unloading the stress-strain relationship does not vary muh with the strain rate. f the strain-rate effet is ignored altogether; then the loading urve and the unloading urve (they are unequal) an be separately treated as a uniquely defined stress-strain relationship, whih is assoiated with a strain-energy funtion. Eah of these urves is a pseudo-elasti urve. During suessive reloading the material responds softer. his is ahieved by saling the stress with an appropriate softening funtion whose material parameters an be diretly alulated from experimental data [,3,5,7,]. hese pseudo-elasti models were also widely used in biomehanis [9-] for onstitutive modelling of isotropi and anisotropi soft tissue. Reently, it was shown in the paper [] that the softening funtion proposed in [,3] and [] leads to thermomehanially onsistent material models. he aim of this paper is to introdue a new phenomenologial model for the Mullins effet based on the strain-energy funtion of isotropi, inompressible rubber-like materials [3,4] and the free energy potential given by Naumann and hlemann []. t is an alternative approah to Ogden and Roxburg [,3] whih enables a onsistent dedution of pseudo-elastiity models by introduing a suitable free energy potential. he insertion into the Clausius-Duhem inequality diretly yields the onstitutive relationships for the stress tensor and the dissipation. * Corresponding author: s.jemiolo@il.pw.edu.pl he Authors, published by EDP Sienes. his is an open aess artile distributed under the terms of the Creative Commons Attribution Liense 4. (

2 MAEC Web of Conferenes 7, 6 (7) DO:.5/ mateonf/776 XXV R-S-P Seminar 7, heoretial Foundation of Civil Engineering Basi assumptions and onstitutive relationships of pseudohyperelastiity he Clausius-Duhem [6] inequality for purely mehanial material models an be simplified to D C. () n the inequality () D denotes the mehanial dissipation, the seond Piola- Kirhhoff stress tensor, C F F the right Cauhy-Green tensor, and the speifi free energy. Here F denotes the deformation gradient with J detf, and F stands for the transpose of F. Following the approah of Naumann and hlemann [], the key idea is to define the free energy for pseudo-hyperelasti material W W, W, W d, () where W C denotes the strain-energy funtion of the virgin material model (it is assumed that in the undeformed onfiguration W ), and W t W, t. (3) he definition diretly leads to W, W, and (after differentiation of ()) to the interpretation of the softening funtion W, W nserting the time derivative of : W and (3) into the inequality () we obtain D W, W W W, W W W, W W W W W W. (4) W, (5) C W W, W W. (6) C CC W his inequality is fulfilled for arbitrary C, if the following onditions hold: W C C CC, W, W W W. (7) Due to the definition (3), W is always non-negative. hen, a suffiient ondition for thermomehanial onsisteny (7) is

3 MAEC Web of Conferenes 7, 6 (7) DO:.5/ mateonf/776 XXV R-S-P Seminar 7, heoretial Foundation of Civil Engineering W, W W. () Every funtion (4) whih is a monotonially dereasing funtion of possitive dissipation. Sine the Kirhhoff stress tensor is defined as follows therefore in the deformed onfiguration we have W leads to a τσj FF, (9) W τ F F C CC For the softening funtions used in [,3,7,] : C. () Erf W W or anh W W, () where and are positive onstants, the related free energy () of the pseudohyperelasti model an be omputed by the integration of (). hese funtions lead to a positive dissipation. he free energies related to the softening funtions () an be given by W, W Exp W W Erf W W () W, W Log Cosh W W. (3) 3 Pseudo-hyperelasti models of isotropi rubberlike materials Consider the simplest lass of onstitutive isotropi models of rubberlike materials [3,4]. We assume the rubberlike materials are inompressible, i.e. J. (4) Consequently, the potential () has to inorporate the onstraint (4) with the Lagrange multiplier p, being the hydrostati pressure or the spherial part of the Cauhy stress σ. n the ase of inompressible isotropi materials the stored energy funtion is a funtion of only two independent invariants of isohori deformations. he tensor F is deomposed into the isohori and volumetri part as follows Further, we introdue /3 /3 /3 FJ FJ RUJ VR, det Fdet Udet V. (5) 3

4 MAEC Web of Conferenes 7, 6 (7) DO:.5/ mateonf/776 XXV R-S-P Seminar 7, heoretial Foundation of Civil Engineering BFF, BFF, CF F. (6) where B denotes the left Cauhy-Green strain tensor, f. [3,4] and the referenes therein. For inompressible isotropi materials the stored energy funtion has the form, where, W W C W B W, (7) trbtr C, trb trc. () From (), (4) and (7) we obtain the following onstitutive relationship of the pseudo-hyperelastiity in the referene onfiguration where σp B B, (9),, W W,. () We observe that equations (9) and () an be applied to arbitrary stored energy funtions desribing an isotropi inompressible material. For the details on suh funtions the reader is referred to [-3,6-9,3,4]. n the paper [6], following the approah of Ogden and Roxburgh [,3], we proposed a phenomenologial pseudo-hyperelasti model to haraterize the Mullins effet in filled rubber. We assumed: the stored energy funtion in the form 3 W, a 3 a 9 a3 7 a4 3 a5 3 3, () where a i are onstants, the onstitutive relationship is given by Eq. (9) and the softening funtion is defined by Eq. (). t an be proved [3] that the funtion () is a partiular ase of the polynomial Rivlin form of the stored energy funtion available in Abaqus [5]. Somone an obtained the pseudo-hyperelasti model with the stored energy funtion () and the softening funtion () by using two options : HYPERELASC and MULLNS EFEC. 4 An example o illustrate a simple pseudo-elasti model of rubberlike materials we used the stored energy funtion proposed in our monograph [3]: μ μi i i W 3 3, () i i where μ,μ i and i i,, are positive material parameters. he shear modulus in the undeformed, natural state of the virgin material depends on all material parameters i. (3) i μ μ μ 3 i For μ i we obtained the neo-hookean material. 4

5 MAEC Web of Conferenes 7, 6 (7) DO:.5/ mateonf/776 XXV R-S-P Seminar 7, heoretial Foundation of Civil Engineering n view of (9) and (), the stress-streth relations are given by: i =μ μ σpb, i From the plane stress state, Eq. (4) and 3, we get. (4) i p μ μi J J. (5) i Moreover, the onstitutive relationship takes the form i σ μ μ μi J J B, (6) i where J det F, F denotes the plane deformation gradient and tr B tr C. We observe that eq. (6) an also be used for membranes and shells. Both the uniaxial tension test and biaxial tension test are plane stress tests. For example, we get i μ μi i, (7) in the ase of a simple uniaxial extension. he omparison between experimental data [4] and model preditions is depited in Fig.. Best-fit parameters are summarized in able. hey are alulated by an error minimization proedure whih simultaneously utilizes both loading and unloading experimental data. he model with a simplified stored energy funtion () and with two alternative stress softening funtions has been used. For both ases good orrelation between the theoretial and the experimental data has been observed. i S MPa Simple extension data heoretial preditions Fig.. Nominal stress S / versus streth for loading and unloading uniaxial tension tests. Comparison of theoretial preditions (7) with Mullins and obin [4] uniaxial extension data. 5

6 MAEC Web of Conferenes 7, 6 (7) DO:.5/ mateonf/776 XXV R-S-P Seminar 7, heoretial Foundation of Civil Engineering able. Model funtions and parameter values Energy and softening funtions W 3 3 Erf W W anh W W Value of material parameters μ= MPa, μ.46 MPa, , , Final remarks he onstitutive modelling of isotropi inompressible pseudo-hyperelasti materials an be extended to ompressible and anisotropi materials, for instane orthotropi or transversally isotropi. hen in the pseudo-potentials () or (3) one has to substitute an appropriate stored energy funtion similarly, f. papers [-,4]. Referenes. Ch. Naumann, J. hlemann, nt. J. Solids Strut. 69-7, 36 (5). R.W. Ogden, D.G. Roxburg, Pro. R. So. London, Ser. A : Math. Phys. Eng. Si. 455, 6 (999) 3. R.W. Ogden, D.G. Roxburgh, Constitutive models for rubber, Dorfman A., Muhr A. (eds), Pro. he First European Conferene on Constitutive Models for Rubber, 3 (A.A. Balkema, Roterdam-Brookfield, 999) 4. L. Mullins, N.R. obin, Rubber Chem. ehnol. 3, 55 (957) 5. J. Diani, B. Fayolle, P. Gilormini, Eur. Polymer J. 45, 6 (9) 6. S. Jemioło, heoretial Foundations of Civil Enginierings, Rymsza B. [ed], Pro. he X Russian-Polish Semminar, 9 [in Polish] (OW PW, Warszawa ) 7. M. Šilhavý, he mehanis and thermodynamis of ontinuous media (Springer, Berlin okyo 997). A. Dorfman, R.W. Ogden, nt. J. Solids Strut., 4, 55 (4) 9. S. Jemioło, J.J. elega, Ata Bioeng. Biomeh. (Suppl. ), 7 (999). S. Jemioło, J.J. elega, Eng. rans., 49(-3), 3 (). S. Jemioło, J.J. elega, Ata Bioeng. Biomeh. 3 (Suppl. ), 9 (). S. Jemioło, J.J. elega, C. Mihalak, Ata Bioeng. Biomeh. 4 (Supp.), 47 () 3. S. Jemioło, Study of hyperelasti properties of isotropi materials. Modeling and numerial implementation. Sientifi Works. Civil Engineering 4 [in Polish] (OWPW, Warszawa, ) 4. S. Jemioło, Constitutive relationships of hyperelastiity [in Polish], (PAN, KLiW, Warszawa, 6) 5. Abaqus v. 6., heory Manual (Simulia, ) 6