Skip to main content
Erschienen in: Journal of Elasticity 2/2017

09.08.2016

Some Remarks on the Effects of Inertia and Viscous Dissipation in the Onset of Cavitation in Rubber

verfasst von: Aditya Kumar, Damian Aranda-Iglesias, Oscar Lopez-Pamies

Erschienen in: Journal of Elasticity | Ausgabe 2/2017

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Through direct comparisons with experiments, Lefèvre et al. (Int. J. Frac. 192:1–23, 2015) have recently confirmed the prevailing belief that the nonlinear elastic properties of rubber play a significant role in the so-called phenomenon of cavitation—that is, the sudden growth of inherent defects in rubber into large enclosed cavities/cracks in response to external stimuli. These comparisons have also made it plain that cavitation in rubber is first and foremost a fracture process that may possibly depend, in addition to the nonlinear elastic properties of the rubber, on inertial effects and/or on the viscous dissipation innate to rubber. This is because the growth of defects into large cavities/cracks is locally in time an extremely fast process.
The purpose of this Note is to provide insight into the relevance of inertial and viscous dissipation effects on the onset of cavitation in rubber. To this end, leaving fracture properties aside, we consider the basic problem of the radially symmetric dynamic deformation of a spherical defect embedded at the center of a ball made up of an isotropic incompressible nonlinear viscoelastic solid that is subjected to external hydrostatic loading. Specifically, the defect is taken to be vacuous and the viscoelastic behavior of the solid is characterized by a fairly general class of constitutive relations given in terms of two thermodynamic potentials—namely, a free energy function describing the nonlinear elasticity of the solid and a dissipation potential describing its viscous dissipation—which has been shown to be capable to describe the mechanical response of a broad variety of rubbers over wide ranges of deformations and deformations rates. It is found that the onset of cavitation is not affected by inertial effects so long as the external loads are not applied at a high rate. On the other hand, even when the external loads are applied quasi-statically, viscous dissipation can greatly affect the critical values of the applied loads at which cavitation ensues.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Fußnoten
1
The focus of this Note is on homogeneous solids thus the choice of constant mass density \(\rho_{0}\).
 
2
As is the case in elastodynamics, accounting for a pressurized cavity would not pose any additional difficulty.
 
3
Throughout this work we consider isothermal conditions.
 
4
The function (14) can be alternatively written as \(r(R,t)= ( 1+ \frac{a^{3}(t)-A^{3}}{R^{3}} ) ^{1/3}R\) in terms of the inner radius \(a(t)\). For our purposes here, we find dealing with the form (14) in terms of the outer radius \(b(t)\) more convenient.
 
Literatur
1.
2.
Zurück zum Zitat Bergström, J.S., Boyce, M.C.: Constitutive modeling of the large strain time-dependent behavior of elastomers. J. Mech. Phys. Solids 46, 931–954 (1998) ADSCrossRefMATH Bergström, J.S., Boyce, M.C.: Constitutive modeling of the large strain time-dependent behavior of elastomers. J. Mech. Phys. Solids 46, 931–954 (1998) ADSCrossRefMATH
4.
5.
Zurück zum Zitat Chosky, R.: The singular limit of a hyperbolic system and the incompressible limit of solutions with shocks and singularities in nonlinear elasticity. Q. Appl. Math. 55, 485–504 (1997) CrossRefMathSciNet Chosky, R.: The singular limit of a hyperbolic system and the incompressible limit of solutions with shocks and singularities in nonlinear elasticity. Q. Appl. Math. 55, 485–504 (1997) CrossRefMathSciNet
6.
Zurück zum Zitat Chou-Wang, M.-S.O., Horgan, C.: Cavitation in nonlinear elastodynamics for neo-Hookean materials. Int. J. Eng. Sci. 27, 967–973 (1989) CrossRefMATHMathSciNet Chou-Wang, M.-S.O., Horgan, C.: Cavitation in nonlinear elastodynamics for neo-Hookean materials. Int. J. Eng. Sci. 27, 967–973 (1989) CrossRefMATHMathSciNet
7.
Zurück zum Zitat Cohen, T., Molinari, A.: Dynamic cavitation and relaxation in incompressible nonlinear viscoelastic solids. Int. J. Solids Struct. 69–70, 544–552 (2015) CrossRef Cohen, T., Molinari, A.: Dynamic cavitation and relaxation in incompressible nonlinear viscoelastic solids. Int. J. Solids Struct. 69–70, 544–552 (2015) CrossRef
8.
Zurück zum Zitat Coleman, B.D., Hill, E.H.: On the stability of certain motions of incompressible materials with memory. Arch. Ration. Mech. Anal. 30, 197–224 (1970) Coleman, B.D., Hill, E.H.: On the stability of certain motions of incompressible materials with memory. Arch. Ration. Mech. Anal. 30, 197–224 (1970)
9.
Zurück zum Zitat Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford University Press, New York (1998) Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford University Press, New York (1998)
10.
Zurück zum Zitat Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 249, 195–205 (1959) ADSCrossRef Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 249, 195–205 (1959) ADSCrossRef
11.
Zurück zum Zitat Gent, A.N., Park, B.: Failure processes in elastomers at or near a rigid inclusion. J. Mater. Sci. 19, 1947–1956 (1984) ADSCrossRef Gent, A.N., Park, B.: Failure processes in elastomers at or near a rigid inclusion. J. Mater. Sci. 19, 1947–1956 (1984) ADSCrossRef
12.
Zurück zum Zitat Giesselmann, J., Tzavaras, A.E.: Singular limiting induced from continuum solutions and the problem of dynamic cavitation. Arch. Ration. Mech. Anal. 212, 241–281 (2014) CrossRefMATHMathSciNet Giesselmann, J., Tzavaras, A.E.: Singular limiting induced from continuum solutions and the problem of dynamic cavitation. Arch. Ration. Mech. Anal. 212, 241–281 (2014) CrossRefMATHMathSciNet
13.
Zurück zum Zitat Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press, London (1954) MATH Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press, London (1954) MATH
14.
Zurück zum Zitat Guo, Z.-H., Solecki, R.: Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material. Arch. Mech. Stosow. 15, 427–433 (1963) MATHMathSciNet Guo, Z.-H., Solecki, R.: Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material. Arch. Mech. Stosow. 15, 427–433 (1963) MATHMathSciNet
15.
Zurück zum Zitat Knowles, J.K., Jakub, M.T.: Finite dynamic deformations of an incompressible elastic medium containing a spherical cavity. Arch. Ration. Mech. Anal. 18, 367–378 (1965) CrossRefMATH Knowles, J.K., Jakub, M.T.: Finite dynamic deformations of an incompressible elastic medium containing a spherical cavity. Arch. Ration. Mech. Anal. 18, 367–378 (1965) CrossRefMATH
16.
Zurück zum Zitat Kumar, A., Lopez-Pamies, O.: On the two-potential constitutive modelling of rubber viscoelastic materials. C. R., Méc. 344, 102–112 (2016) CrossRef Kumar, A., Lopez-Pamies, O.: On the two-potential constitutive modelling of rubber viscoelastic materials. C. R., Méc. 344, 102–112 (2016) CrossRef
17.
18.
Zurück zum Zitat Lax, P.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS Conference Series. SIAM, Philadelphia (1973) CrossRefMATH Lax, P.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS Conference Series. SIAM, Philadelphia (1973) CrossRefMATH
19.
Zurück zum Zitat Lefèvre, V., Ravi-Chandar, K., Lopez-Pamies, O.: Cavitation in rubber: an elastic instability or a fracture phenomenon? Int. J. Fract. 192, 1–23 (2015) CrossRef Lefèvre, V., Ravi-Chandar, K., Lopez-Pamies, O.: Cavitation in rubber: an elastic instability or a fracture phenomenon? Int. J. Fract. 192, 1–23 (2015) CrossRef
20.
Zurück zum Zitat Le Tallec Rahier, C., Kaiss, A.: Three-dimensional incompressible viscoelasticity in large strains: formulation and numerical approximation. Comput. Methods Appl. Mech. Eng. 109, 233–258 (1993) ADSCrossRefMATHMathSciNet Le Tallec Rahier, C., Kaiss, A.: Three-dimensional incompressible viscoelasticity in large strains: formulation and numerical approximation. Comput. Methods Appl. Mech. Eng. 109, 233–258 (1993) ADSCrossRefMATHMathSciNet
21.
Zurück zum Zitat Lopez-Pamies, O.: A new \(I_{1}\)-based hyperelastic model for rubber elastic materials. C. R., Méc. 338, 3–11 (2010) CrossRefMATH Lopez-Pamies, O.: A new \(I_{1}\)-based hyperelastic model for rubber elastic materials. C. R., Méc. 338, 3–11 (2010) CrossRefMATH
22.
Zurück zum Zitat Lopez-Pamies, O., Idiart, M.I., Nakamura, T.: Cavitation in elastomeric solids: I—A defect-growth theory. J. Mech. Phys. Solids 59, 1464–1487 (2011) ADSCrossRefMATHMathSciNet Lopez-Pamies, O., Idiart, M.I., Nakamura, T.: Cavitation in elastomeric solids: I—A defect-growth theory. J. Mech. Phys. Solids 59, 1464–1487 (2011) ADSCrossRefMATHMathSciNet
23.
Zurück zum Zitat Lopez-Pamies, O., Nakamura, T., Idiart, M.I.: Cavitation in elastomeric solids: II—Onset-of-cavitation surfaces for Neo-Hookean materials. J. Mech. Phys. Solids 59, 1488–1505 (2011) ADSCrossRefMATHMathSciNet Lopez-Pamies, O., Nakamura, T., Idiart, M.I.: Cavitation in elastomeric solids: II—Onset-of-cavitation surfaces for Neo-Hookean materials. J. Mech. Phys. Solids 59, 1488–1505 (2011) ADSCrossRefMATHMathSciNet
24.
Zurück zum Zitat Miroshnikov, A., Tzavaras, A.E.: On the construction and properties of weak solutions describing dynamic cavitation. J. Elast. 118, 141–185 (2015) CrossRefMATHMathSciNet Miroshnikov, A., Tzavaras, A.E.: On the construction and properties of weak solutions describing dynamic cavitation. J. Elast. 118, 141–185 (2015) CrossRefMATHMathSciNet
25.
Zurück zum Zitat Ogden, R.W.: Non-linear Elastic Deformations. Dover, Mineola (1997) Ogden, R.W.: Non-linear Elastic Deformations. Dover, Mineola (1997)
26.
Zurück zum Zitat Pericak-Spector, K.A., Spector, S.J.: Nonuniqueness for a hyperbolic system: cavitation in nonlinear elastodynamics. Arch. Ration. Mech. Anal. 101, 293–317 (1988) CrossRefMATHMathSciNet Pericak-Spector, K.A., Spector, S.J.: Nonuniqueness for a hyperbolic system: cavitation in nonlinear elastodynamics. Arch. Ration. Mech. Anal. 101, 293–317 (1988) CrossRefMATHMathSciNet
27.
Zurück zum Zitat Pericak-Spector, K.A., Spector, S.J.: Dynamic cavitation with shocks in nonlinear elasticity. Proc. R. Soc. Edinb., Sect. A, Math. 127, 837–857 (1997) CrossRefMATHMathSciNet Pericak-Spector, K.A., Spector, S.J.: Dynamic cavitation with shocks in nonlinear elasticity. Proc. R. Soc. Edinb., Sect. A, Math. 127, 837–857 (1997) CrossRefMATHMathSciNet
28.
Zurück zum Zitat Ravi-Chandar, K.: Private communication (2016) Ravi-Chandar, K.: Private communication (2016)
29.
Zurück zum Zitat Reese, S., Govindjee, S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35, 3455–3482 (1998) CrossRefMATH Reese, S., Govindjee, S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35, 3455–3482 (1998) CrossRefMATH
30.
Zurück zum Zitat Zhang, Y., Huang, Z.: Void growth and cavitation in nonlinear viscoelastic solids. Acta Mech. Sin. 19, 380–384 (2003) ADSCrossRef Zhang, Y., Huang, Z.: Void growth and cavitation in nonlinear viscoelastic solids. Acta Mech. Sin. 19, 380–384 (2003) ADSCrossRef
Metadaten
Titel
Some Remarks on the Effects of Inertia and Viscous Dissipation in the Onset of Cavitation in Rubber
verfasst von
Aditya Kumar
Damian Aranda-Iglesias
Oscar Lopez-Pamies
Publikationsdatum
09.08.2016
Verlag
Springer Netherlands
Erschienen in
Journal of Elasticity / Ausgabe 2/2017
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-016-9589-y

Weitere Artikel der Ausgabe 2/2017

Journal of Elasticity 2/2017 Zur Ausgabe

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.