Skip to main content
SpringerLink
Log in

Nonlinear and Hysteretic Constitutive Models for Wave Propagation in Solid Media with Cracks and Contacts

  • Chapter
  • First Online:
Nonlinear Ultrasonic and Vibro-Acoustical Techniques for Nondestructive Evaluation

Abstract

This chapter is devoted to theoretical concepts and models for wave propagation, vibrations, or other elastic deformations in solids containing internal contacts (cracks, delaminations, etc.). A direct problem of solid mechanics is solved by building up a solution for elastic fields in materials with known geometry and properties. This study is oriented to nondestructive testing and therefore focuses on the case where the material contains few cracks of known configuration, in contrast to microcracked solids in which a statistical ensemble of a large number of internal contacts is present. Our approach is based on finite element simulations and a frictional contact model assuming generic semi-analytical solutions. These solutions account for surface roughness, friction, and the evolution of stick and slip zones in the contact area. Finally, load–displacement relationships valid for arbitrary loading histories are produced which are used as boundary conditions imposed at internal boundaries (cracks) in the material. As a result, we have developed a numerical toolbox capable of modeling elastic waves and vibrations in damaged samples or structures. The access to all elastic fields together with their nonlinear components makes nondestructive testing fully transparent and offers an opportunity of purposeful optimization of the experimental techniques.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 229.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. I. Sneddon, Fourier series (Routledge & Kegan, New York, 1951)

    MATH  Google Scholar 

  2. V.A. Yastrebov, Numerical methods in contact mechanics (Wiley-ISTE, Hoboken, 2013)

    Book  Google Scholar 

  3. E.A.H. Vollebregt. User Guide for CONTACT, rolling and sliding contact with friction Technical report TR09-03, version 16.1, VORtech BV, Delft, 2016, www.kalkersoftware.org

    Google Scholar 

  4. R. Mindlin, H. Deresiewicz, Elastic spheres in contact under varying oblique forces. J. Appl. Mech. 20, 327–344 (1953)

    MathSciNet  MATH  Google Scholar 

  5. C. Cattaneo, Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Accad. Lincei. Rend. 27(6), 342–348 (1938)

    MATH  Google Scholar 

  6. V. Popov, M. Hess, Method of dimensionality reduction in contact mechanics and friction (Springer, New York, 2015)

    Book  Google Scholar 

  7. L. Landau, E. Lifschitz, Theory of elasticity (Pergamon Press, Oxford, 1993)

    Google Scholar 

  8. D.J. Whitehouse, J.F. Archard, The properties of random surfaces of significance in their contact. Proc. Roy. Soc. Lond. A 316(1524), 97–121 (1970)

    Article  Google Scholar 

  9. J. Greenwood, J. Willianson, Contact of nominally flat surfaces. Proc R Soc Lond A 295, 300–319 (1966)

    Article  Google Scholar 

  10. F. Bowden, D. Tabor, The area of contact between stationary and between moving surfaces. Proc R Soc Lond A 169, 391–413 (1939)

    Article  Google Scholar 

  11. G. Carbone, F. Bottiglione, Asperity contact theories: Do they predict linearity between contact area and load? J. Mech. Phys. Solids 56, 2555–2572 (2008)

    Article  Google Scholar 

  12. S. Hyun, M. Robbins, Elastic contact between rough surfaces: Effect of roughness at large and small wavelengths. Tibol. Int. 40, 1413–1422 (2007)

    Google Scholar 

  13. B. Persson, F. Bucher, B. Chiaia, Elastic contact between random rough surfaces: Comparison of theory with numerical results. Phys. Rev. B 65(18), 184106 (2002)

    Article  Google Scholar 

  14. M. Paggi, M. Ciavarella, The coefficient of proportionality between real contact area and load, with new asperity models. Wear 268, 1020–1010 (2010)

    Article  Google Scholar 

  15. M. Paggi, R. Pohrt, V. Popov, Partial-slip frictional response of rough surfaces. Sci. Rep. 4, 5178 (2014)

    Article  Google Scholar 

  16. S. Biwa, S. Nakajima, N. Ohno, On the acoustic nonlinearity of solid-solid contact with pressure-dependent interface stiffness. J. Appl. Mech. 71(4), 508–515 (2004)

    Article  Google Scholar 

  17. M. Yuan, J. Zhang, S. Song, H. Kim, Numerical simulation of Rayleigh wave interaction with surface closed cracks under external pressure. Wave Motion 57, 143–153 (2015)

    Article  MathSciNet  Google Scholar 

  18. R. Pohrt, V. Popov, Normal contact stiffness of elastic solids with fractal rough surfaces. Phys. Rev. Lett. 108, 104301 (2012)

    Article  Google Scholar 

  19. R. Pohrt, V.L. Popov, Contact stiffness of randomly rough surfaces. Sci. Rep. 3, 3293 (2013)

    Article  Google Scholar 

  20. J. Jäger, Axisymmetric bodies of equal material in contact under torsion or shift. Arch. Appl. Mech. 65, 478–487 (1995)

    Article  Google Scholar 

  21. L.A. Galin, Contact problems in the theory of elasticity (North Carolina State College, Raleigh, 1961)

    Google Scholar 

  22. I.N. Sneddon, The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57 (1965)

    Article  MathSciNet  Google Scholar 

  23. J. Jäger, A new principle in contact mechanics. J. Tribol. 120(4), 677–684 (1998)

    Article  Google Scholar 

  24. M. Ciavarella, The generalized Cattaneo partial slip plane contact problem. I – theory, II - examples. Int. J. Solids Struct. 35, 2349–2362 (1998)

    Article  MathSciNet  Google Scholar 

  25. M. Ciavarella, Tangential loading of general 3D contacts. ASME J. Appl. Mech. 65, 998–1003 (1998)

    Article  Google Scholar 

  26. R.L. Munisamy, D.A. Hills, D. Nowell, Contact of Similar and Dissimilar Elastic Spheres under Tangential Loading. Contact Mechanics (Presses polytechniques et universitaires romandes, Lausanne, 1992)

    Google Scholar 

  27. R.L. Munisamy, D.A. Hills, D. Nowell, Static axisymmetrical Hertzian contacts subject to shearing forces. ASME J. Appl. Mech. 61, 278–283 (1994)

    Article  Google Scholar 

  28. R. Pohrt, personal communication

    Google Scholar 

  29. V. Aleshin, O. Bou Matar, K. Van Den Abeele, Method of memory diagrams for mechanical frictional contacts subject to arbitrary 2D loading. Int. J. Solids Struct. 60–61, 84–95 (2015)

    Article  Google Scholar 

  30. V.V. Aleshin, K. Van Den Abeele, Hertz-Mindlin problem for arbitrary oblique 2D loading: General solution by memory diagrams. J. Mech. Phys. Solids 60, 14–36 (2012)

    Article  MathSciNet  Google Scholar 

  31. V.V. Aleshin, K. Van Den Abeele, General solution to the Hertz-Mindlin problem via Preisach formalism. Int. J. Non Linear Mech. 49, 15–30 (2013)

    Article  Google Scholar 

  32. V.V. Aleshin, O. Bou Matar, Solution to the frictional contact problem via the method of memory diagrams for general 3D loading histories. Phys. Mesomech. 19, 130–135 (2016)

    Article  Google Scholar 

  33. V.V. Aleshin, S. Delrue, O. Bou Matar, K. Van Den Abeele, Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part I: Theoretical background. Ultrasonics 82, 11–18 (2018)

    Article  Google Scholar 

  34. COMSOL AB, Structural mechanics module, user’s guide (COMSOL Multiphysics v. 5.2, Stockholm, 2015)

    Google Scholar 

  35. COMSOL AB, LiveLink for MATLAB, user’s guide (COMSOL Multiphysics v. 5.2, Stockholm, 2015)

    Google Scholar 

  36. S. Delrue, V. Aleshin, M. Sorensen, L. De Lathauwer, Simulation study of the localization of a near-surface crack using an air-coupled ultrasonic sensor array. Sensors 17(4), 930 (2017)

    Article  Google Scholar 

  37. S. Delrue, V.V. Aleshin, K. Truyaert, O. Bou Matar, K. Van Den Abeele, Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part II: Numerical implementation. Ultrasonics 82, 19–30 (2018)

    Article  Google Scholar 

  38. http://www.engineershandbook.com/Tables/frictioncoefficients.htm, Accessed on 14 Sept 2017

  39. M. Scalerandi, A. Gliozzi, C. Bruno, D. Masera, P. Bocca, A scaling method to enhance detection of a nonlinear elastic response. Appl. Phys. Lett. 92, 101912 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Center for Scientific Research, Joint International Laboratory LICS-LEMAC, Institute of Electronics, Microelectronics and Nanotechnologies, Villeneuve d’Ascq, France

    V. V. Aleshin

  2. Department of Physics, Wave Propagation and Signal Processing Research Group, KU Leuven Kulak, Kortrijk, Belgium

    S. Delrue & K. Van Den Abeele

  3. Ecole Centrale de Lille, Joint International Laboratory LICS-LEMAC, Institute of Electronics, Microelectronics and Nanotechnologies, Villeneuve d’Ascq, France

    O. Bou Matar

Authors
  1. V. V. Aleshin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Delrue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. Bou Matar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. K. Van Den Abeele
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Aleshin .

Editor information

Editors and Affiliations

  1. Department of Civil Engineering and Engineering Mechanics, Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ, USA

    Tribikram Kundu

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Aleshin, V.V., Delrue, S., Bou Matar, O., Van Den Abeele, K. (2019). Nonlinear and Hysteretic Constitutive Models for Wave Propagation in Solid Media with Cracks and Contacts. In: Kundu, T. (eds) Nonlinear Ultrasonic and Vibro-Acoustical Techniques for Nondestructive Evaluation. Springer, Cham. https://doi.org/10.1007/978-3-319-94476-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-94476-0_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94474-6

  • Online ISBN: 978-3-319-94476-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation