Skip to main content
Springer Nature Link
Log in

A Nonlinear Viscoelastic Contact Interphase Modeled as a Cosserat Rod-Like String

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

A nonlinear viscoelastic contact interphase is modeled using a Cosserat rod-like string. This Cosserat model is a rod with a deformable cross-section, but with no constitutive resistance to bending. The model allows for axial extension, tangential shear deformation and normal extension of the cross-section which are determined by finite deformations of the interphase. Moreover, the constitutive response of this string model can be determined directly by three-dimensional constitutive equations for a hyperelastic component and a Maxwell elastic-viscoplastic component that together produce viscoelastic response of the interphase. The example of vibrations of a rigid outer ring connected to a fixed inner disk by a nonlinear viscoelastic interphase is used to show that the Cosserat string model of the interface predicts torque and force applied to the outer ring which include nonlinear coupling that is not present in simple uncoupled models of Maxwell components for torque and force applied to the outer ring by the interphase.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Benveniste, Y., Miloh, T.: Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33, 309–323 (2001)

    Article  Google Scholar 

  2. Bigoni, D., Movchan, A.B.: Statics and dynamics of structural interfaces in elasticity. Int. J. Solids Struct. 39, 4843–4865 (2002)

    Article  Google Scholar 

  3. Bigoni, D., Serkov, S.K., Valentini, M., Movchan, A.B.: Asymptotic models of dilute composites with imperfectly bonded inclusions. Int. J. Solids Struct. 35, 3239–3258 (1998)

    Article  MathSciNet  Google Scholar 

  4. Dong, H., Wang, J., Rubin, M.B.: Cosserat interphase models for elasticity with application to the interphase bonding a spherical inclusion to an infinite matrix. Int. J. Solids Struct. 51, 462–477 (2014)

    Article  Google Scholar 

  5. Dong, H., Wang, J., Rubin, M.B.: A nonlinear Cosserat interphase model for residual stresses in an inclusion and the interphase that bonds it to an infinite matrix. Int. J. Solids Struct. 62, 186–206 (2015)

    Article  Google Scholar 

  6. Geymonat, G., Krasucki, F.: A limit model of a soft, thin joint. In: Marcellini, P., Talenti, G.T., Vasentini, E. (eds.) Partial Differential Equations and Applications, pp. 165–173. Dekker, New York (1996)

    Google Scholar 

  7. Geymonat, G., Krasucki, F., Lenci, S.: Mathematical analysis of a bonded joint with a soft thin adhesive. Math. Mech. Solids 4, 201–225 (1999)

    Article  MathSciNet  Google Scholar 

  8. Goland, M., Reissner, E.: The stresses in cemented joints. J. Appl. Mech. 11, A17–A27 (1944)

    Article  Google Scholar 

  9. Hashin, Z.: Thermoelastic properties of fiber composites with imperfect interface. Mech. Mater. 8, 333–348 (1990)

    Article  Google Scholar 

  10. Hashin, Z.: Thermoelastic properties of particulate composites with imperfect interface. J. Mech. Phys. Solids 39, 745–762 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  11. Hashin, Z.: Composite materials with interphase: thermoelastic and inelastic effects. In: Inelastic Deformation of Composite Materials, pp. 3–34. Springer, Berlin (1991)

    Chapter  Google Scholar 

  12. Hashin, Z.: Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Solids 50, 2509–2537 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  13. Jones, J.P., Whittier, J.S.: Waves at a flexibly bonded interface. J. Appl. Mech. 34, 905–909 (1967)

    Article  ADS  Google Scholar 

  14. Klarbring, A.: Derivation of a model of adhesively bonded joints by the asymptotic expansion method. Int. J. Eng. Sci. 29, 493–512 (1991)

    Article  MathSciNet  Google Scholar 

  15. Klarbring, A., Movchan, A.B.: Asymptotic modelling of adhesive joints. Mech. Mater. 28, 137–145 (1998)

    Article  Google Scholar 

  16. Naghdi, P.M.: The theory of shells and plates. In: Truesdell, C. (ed.) S. Flugge’s Hanbuch de Physik, vol. VIa/2, pp. 425–640. Springer, Berlin (1973)

    Google Scholar 

  17. Rubin, M.B.: Cosserat Theories: Shells, Rods and Points, vol. 79. Springer, Berlin (2000)

    Book  Google Scholar 

  18. Rubin, M.B.: A viscoplastic model for the active component in cardiac muscle. Biomech. Model. Mechanobiol. 15, 965–982 (2016)

    Article  Google Scholar 

  19. Rubin, M.B., Attia, A.V.: Calculation of hyperelastic response of finitely deformed elastic-viscoplastic materials. Int. J. Numer. Methods Eng. 39, 309–320 (1996)

    Article  Google Scholar 

  20. Rubin, M.B., Benveniste, Y.: A Cosserat shell model for interphases in elastic media. J. Mech. Phys. Solids 52, 1023–1052 (2004)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    M. B. Rubin

Authors
  1. M. B. Rubin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. B. Rubin.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Expressions for the Velocities and Accelerations

Appendix: Expressions for the Velocities and Accelerations

Using (35) and (36), the velocities and rate of dilatation can be expressed in the forms

$$\begin{aligned} &{\mathbf{v}}(S,t) = \frac{1}{2} \dot{\mathbf{r}} + \tilde{{\mathbf{v}}} \,, \\ &\tilde{{\mathbf{v}}} = - \frac{1}{2} (\dot{\gamma }_{1} R_{1} \sin \gamma _{1} + \dot{\gamma }_{2} R_{2} \sin \gamma _{2} ) {\mathbf{e}}_{r}( \frac{S}{R}) \\ &\quad \quad \quad \quad + \frac{1}{2} ( \dot{\gamma }_{1} R_{1} \cos \gamma _{1} + \dot{\gamma }_{2} R_{2} \cos \gamma _{2} ) {\mathbf{e}}_{\theta }(\frac{S}{R}) \,, \\ &{\mathbf{w}}_{1}(S,t) = \frac{1}{H} \dot{\mathbf{r}} + \tilde{{\mathbf{w}}}_{1} \,, \\ &\tilde{{\mathbf{w}}}_{1} = \frac{1}{H} (\dot{\gamma }_{1} R_{1} \sin \gamma _{1} - \dot{\gamma }_{2} R_{2} \sin \gamma _{2} ) { \mathbf{e}}_{r}(\frac{S}{R}) \\ \\ &\quad \quad \quad \quad + \frac{1}{H}( - \dot{\gamma }_{1} R_{1} \cos \gamma _{1} + \dot{\gamma }_{2} R_{2} \cos \gamma _{2} ) { \mathbf{e}}_{\theta }(\frac{S}{R}) \,, \\ &{\mathbf{w}}_{3}(S,t) = - \frac{1}{2R} (\dot{\gamma }_{1} R_{1} \cos \gamma _{1} + \dot{\gamma }_{2} R_{2} \cos \gamma _{2} ) { \mathbf{e}}_{r}(\frac{S}{R}) \\ &\quad \quad \quad \quad - \frac{1}{2R} (\dot{\gamma }_{1} R_{1} \sin \gamma _{1} + \dot{\gamma }_{2} R_{2} \sin \gamma _{2} ) {\mathbf{e}}_{\theta }(\frac{S}{R}) \,, \\ &\dot{J}(S,t) = \dot{r}_{r} \, \big[ \frac{R_{1} \cos \gamma _{1} + R_{2} \cos \gamma _{2}}{R_{2}^{2}-R_{1}^{2}} \big] +\dot{r}_{\theta }\, \big[ \frac{R_{1} \sin \gamma _{1} + R_{2} \sin \gamma _{2}}{R_{2}^{2}-R_{1}^{2}} \big] \, \\ &\quad \quad - r_{r} \, \big[ \frac{\dot{\gamma }_{1} R_{1} \sin \gamma _{1} + \dot{\gamma }_{2} R_{2} \sin \gamma _{2}}{R_{2}^{2}-R_{1}^{2}} \big] +r_{\theta }\, \big[ \frac{\dot{\gamma }_{1} R_{1} \cos \gamma _{1} + \dot{\gamma }_{2} R_{2} \cos \gamma _{2}}{R_{2}^{2}-R_{1}^{2}} \big] \,. \end{aligned}$$
(71)

Also, the accelerations are given by

$$\begin{aligned} &\dot{{\mathbf{v}}}(S,t) = \frac{1}{2} \ddot{\mathbf{r}} + \dot{\tilde{{\mathbf{v}}}} \,, \\ & \dot{\tilde{{\mathbf{v}}}} = - \frac{1}{2} (\ddot{\gamma }_{1} R_{1} \sin \gamma _{1} + \ddot{\gamma }_{2} R_{2} \sin \gamma _{2} \\ &\quad \quad \quad \quad +\dot{\gamma }_{1}^{2} R_{1} \cos \gamma _{1} + \dot{\gamma }_{2}^{2} R_{2} \cos \gamma _{2}) {\mathbf{e}}_{r}( \frac{S}{R}) \\ &\quad \quad \quad \quad + \frac{1}{2} (\ddot{\gamma }_{1} R_{1} \cos \gamma _{1} + \ddot{\gamma }_{2} R_{2} \cos \gamma _{2} \\ &\quad \quad \quad \quad - \dot{\gamma }_{1}^{2} R_{1} \sin \gamma _{1} - \dot{\gamma }_{2}^{2} R_{2} \sin \gamma _{2}) {\mathbf{e}}_{\theta }( \frac{S}{R}) \,, \\ &\dot{{\mathbf{w}}}_{1}(S,t) = \frac{1}{H} \ddot{\mathbf{r}} + \dot{\tilde{{\mathbf{w}}}}_{1} \,, \\ &\dot{\tilde{{\mathbf{w}}}}_{1} = \frac{1}{H}( \ddot{\gamma }_{1} R_{1} \sin \gamma _{1} - \ddot{\gamma }_{2} R_{2} \sin \gamma _{2} \\ \\ &\quad \quad \quad \quad + \dot{\gamma }_{1}^{2} R_{1} \cos \gamma _{1} - \dot{\gamma }_{2}^{2} R_{2} \cos \gamma _{2} ) {\mathbf{e}}_{r}( \frac{S}{R}) \\ &\quad \quad \quad \quad + \frac{1}{H}( - \ddot{\gamma }_{1} R_{1} \cos \gamma _{1} + \ddot{\gamma }_{2} R_{2} \cos \gamma _{2} \\ &\quad \quad \quad \quad + \dot{\gamma }_{1}^{2} R_{1} \sin \gamma _{1} - \dot{\gamma }_{2}^{2} R_{2} \sin \gamma _{2} ) {\mathbf{e}}_{\theta }( \frac{S}{R}) \,, \\ &\dot{{\mathbf{w}}}_{3}(S,t) = - \frac{1}{2R} (\ddot{\gamma }_{1} R_{1} \cos \gamma _{1} + \ddot{\gamma }_{2} R_{2} \cos \gamma _{2} \\ &\quad \quad \quad \quad -\dot{\gamma }_{1}^{2} R_{1} \sin \gamma _{1} - \dot{\gamma }_{2}^{2} R_{2} \sin \gamma _{2}) {\mathbf{e}}_{r}( \frac{S}{R}) \\ &\quad \quad \quad \quad - \frac{1}{2R} (\ddot{\gamma }_{1} R_{1} \sin \gamma _{1} + \ddot{\gamma }_{2} R_{2} \sin \gamma _{2} \\ &\quad \quad \quad \quad +\dot{\gamma }_{1}^{2} R_{1} \cos \gamma _{1} + \dot{\gamma }_{2}^{2} R_{2} \cos \gamma _{2} ) {\mathbf{e}}_{\theta }( \frac{S}{R}) \,. \end{aligned}$$
(72)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rubin, M.B. A Nonlinear Viscoelastic Contact Interphase Modeled as a Cosserat Rod-Like String. J Elast 146, 237–259 (2021). https://doi.org/10.1007/s10659-021-09858-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-021-09858-0

Keywords

Mathematics Subject Classification

Search

Navigation