Skip to main content
SpringerLink
Log in

Quasi-Harmonic Solutions for Transversely Isotropic Magneto-Electro-Thermo-Elasticity: A Symbolic Mathematics Approach

  • Chapter
  • First Online:
Mathematical Applications in Continuum and Structural Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 127))

  • 585 Accesses

Abstract

The behaviour of magneto-electro-thermo-elastic materials is governed by a set of six differential equations in which the piezoelectric, the piezomagnetic, the magnetoelectric and thermal effects are coupled. Under the steady-state condition, this system of equations is homogeneous and can be rewritten as a set of uncoupled modified Laplace equations expressed as a function of the roots of a characteristic polynomial associated with the original set of governing equations. Differently from previous proposals, the presented approach employs the entire kernel of the adjoint differential operator so as to preserve completeness. Finally, due to the large number of constitutive parameters involved in the uncoupling process, two Mathematica scripts that compute the coefficients of the characteristic polynomial and the components of the adjoint differential operator are described in full detail.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

  • AenN A, Alshaikh F, Del Vescovo D, Spagnuolo M (2017) Plane waves and eigenfrequency study in a transversely isotropic magneto-thermoelastic medium under the effect of a constant angular velocity. J Thermal Stresses 40(9):1079–1092

    Article  Google Scholar 

  • Alibert JJ, Seppecher P, Dell’Isola F (2003) Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids 8(1):51–73

    MathSciNet  MATH  Google Scholar 

  • Almansi E (1899) Sull’integrazione dell’equazione differenziale \(\delta \) 2n = 0. Annali di Matematica Pura ed Applicata (1898–1922) 2(1):1–51

    Google Scholar 

  • Andreaus U, Spagnuolo M, Lekszycki T, Eugster SR (2018) A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear euler-bernoulli beams. Continuum Mech Thermodyn 30(5):1103–1123

    Article  MathSciNet  Google Scholar 

  • Ashida F, Noda N, Okumura I (1993) General solution technique for transient thermoelasticity of transversely isotropic solids in cylindrical coordinates. Acta mechanica 101(1):215–230

    Article  MathSciNet  Google Scholar 

  • Barchiesi E, Dell’Isola F, Laudato M, Placidi L, Seppecher P (2018) A 1d continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results. In: Advances in mechanics of microstructured media and structures. Springer, pp 43–74

    Google Scholar 

  • Bardzokas DI, Filshtinsky ML, Filshtinsky LA (2007) Mathematical methods in electro-magneto-elasticity, vol 32. Springer Science & Business Media

    Google Scholar 

  • Camar-Eddine M, Seppecher P (2001) Non-local interactions resulting from the homogenization of a linear diffusive medium. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 332(5):485–490

    Google Scholar 

  • Chen WT (1966) On some problems in transversely isotropic elastic materials. J Appl Mech 33(2):347–355

    Article  Google Scholar 

  • Chen W (2000) On the general solution for piezothermoelasticity for transverse isotropy with application. J Appl Mech 67(4):705–711

    Article  MathSciNet  Google Scholar 

  • Chen W, Lee KY, Ding H (2004) General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. Int J Eng Sci 42(13–14):1361–1379

    Article  Google Scholar 

  • De Angelo M, Spagnuolo M, D’annibale F, Pfaff A, Hoschke K, Misra A, Dupuy C, Peyre P, Dirrenberger J, Pawlikowski M (2019) The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage in metallic specimens. Continuum Mech Thermodyn 31(4):1181–1203

    Google Scholar 

  • Dell’Isola F, Seppecher P, Alibert JJ, Lekszycki T, Grygoruk R, Pawlikowski M, Steigmann D, Giorgio I, Andreaus U, Turco E et al (2019a) Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech Thermodyn 31(4):851–884

    Article  MathSciNet  Google Scholar 

  • Dell’Isola F, Turco E, Misra A, Vangelatos Z, Grigoropoulos C, Melissinaki V, Farsari M (2019b) Force-displacement relationship in micro-metric pantographs: experiments and numerical simulations. Comptes Rendus Mécanique 347(5):397–405

    Article  Google Scholar 

  • di Cosmo F, Laudato M, Spagnuolo M (2018) Acoustic metamaterials based on local resonances: homogenization, optimization and applications. In: Generalized models and non-classical approaches in complex materials, vol 1. Springer, pp 247–274

    Google Scholar 

  • Ding H, Jiang A (2003) Fundamental solutions for transversely isotropic magneto-electro-elastic media and boundary integral formulation. Sci China Ser E: Technol Sci 46(6):607–619

    Article  Google Scholar 

  • Ding H, Chen W, Zhang L (2006) Elasticity of transversely isotropic materials, vol 126. Springer Science & Business Media

    Google Scholar 

  • Eerenstein W, Mathur N, Scott JF (2006) Multiferroic and magnetoelectric materials. Nature 442(7104):759–765

    Article  Google Scholar 

  • Elliott HA, Mott N (1948) Three-dimensional stress distributions in hexagonal aeolotropic crystals. In: Mathematical proceedings of the cambridge philosophical society, vol 44. Cambridge University Press, pp 522–533

    Google Scholar 

  • Eremeyev VA, Lebedev LP, Altenbach H (2012) Foundations of micropolar mechanics. Springer Science & Business Media

    Google Scholar 

  • Fabrikant V (1989) Applications of potential theory in mechanics: a selection of new results. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Franciosi P, Spagnuolo M, Salman OU (2019) Mean green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Continuum Mech Thermodyn 31(1):101–132

    Article  MathSciNet  Google Scholar 

  • Green A, Zerna W (1968) Theoretical elasticity. Clarendon Press, Oxford

    Google Scholar 

  • Haojiang D et al (1996) General solutions for coupled equations for piezoelectric media. Int J Solids Struct 33(16):2283–2298

    Article  Google Scholar 

  • Lekhniskii S (1940) Symmetrical deformation and torsion of revolution with a special kind of anisotropy. PPM 4:43–60

    Google Scholar 

  • Lekhnitskii S (1981) Theory of elasticity of an anisotropic body. Mir Publishers, Moscow

    Google Scholar 

  • Lin W, Kuo C, Keer L (1991) Analysis of a transversely isotropic half space under normal and tangential loadings. ASME J Tribol 113:335–338

    Article  Google Scholar 

  • Marmo F, Rosati L (2016) A general approach to the solution of Boussinesq’s problem for polynomial pressures acting over polygonal domains. J Elasticity 122(1):75–112

    Article  MathSciNet  Google Scholar 

  • Marmo F, Sessa S, Rosati L (2016a) Analytical solution of the Cerruti problem under linearly distributed horizontal loads over polygonal domains. J Elasticity 124(1):27–56

    Article  MathSciNet  Google Scholar 

  • Marmo F, Toraldo F, Rosati L (2016b) Analytical formulas and design charts for transversely isotropic half-spaces subject to linearly distributed pressures. Meccanica 51(11):2909–2928

    Article  MathSciNet  Google Scholar 

  • Marmo F, Toraldo F, Rosati L (2017) Transversely isotropic half-spaces subject to surface pressures. Int J Solids Struct 104:35–49

    Article  Google Scholar 

  • Marmo F, Toraldo F, Rosati A, Rosati L (2018) Numerical solution of smooth and rough contact problems. Meccanica 53(6):1415–1440

    Article  MathSciNet  Google Scholar 

  • Marmo F, Sessa S, Vaiana N, De Gregorio D, Rosati L (2020) Complete solutions of three-dimensional problems in transversely isotropic media. Continuum Mech Thermodyn 32(3):775–802

    Article  MathSciNet  Google Scholar 

  • Michell J (1900) The stress in an æolotrophic elastic solid with an infinite plane boundary. Proc Lond Mathe Soc 1(1):247–257

    Article  MathSciNet  Google Scholar 

  • Milton GW, Seppecher P, Bouchitté G (2009) Minimization variational principles for acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency. Proc Royal Soc A: Math Phys Eng Sci 465(2102):367–396

    Article  MathSciNet  Google Scholar 

  • Nejadsadeghi N, Placidi L, Romeo M, Misra A (2019) Frequency band gaps in dielectric granular metamaterials modulated by electric field. Mech Res Commun 95:96–103

    Article  Google Scholar 

  • Pan E (2002) Three-dimensional green’s functions in anisotropic magneto-electro-elastic bimaterials. Zeitschrift für angewandte Mathematik und Physik ZAMP 53(5):815–838

    Article  MathSciNet  Google Scholar 

  • Pan YC, Chou TW (1976) Point force solution for an infinite transversely isotropic solid. J Appl Mech 43(4):608–612

    Article  Google Scholar 

  • Pan YC, Chou TW (1979) Green’s function solutions for semi-infinite transversely isotropic materials. Int J Eng Sci 17(5):545–551

    Article  Google Scholar 

  • Pideri C, Seppecher P (1997) A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mech Thermodyn 9(5):241–257

    Article  MathSciNet  Google Scholar 

  • Podil’chuk YN, Sokolovskii YI (1994) Thermostress in an infinite transversally isotropic medium with an internal elliptical crack. Int Appl Mech 30(11):834–840

    Article  Google Scholar 

  • Rosati L, Marmo F (2014) Closed-form expressions of the thermo-mechanical fields induced by a uniform heat source acting over an isotropic half-space. Int J Heat Mass Transf 75:272–283

    Article  Google Scholar 

  • Shield R (1951) Notes on problems in hexagonal aeolotropic materials. In: Mathematical proceedings of the Cambridge philosophical society, vol 47. Cambridge University Press, pp 401–409

    Google Scholar 

  • Sp X, Gao Y, Wang W (2008) Completeness of general solutions for three-dimensional transversely isotropic piezoelectricity. Int J Solids Struct 45(18–19):5118–5126

    MATH  Google Scholar 

  • Wang M, Wang W (1995) Completeness and nonuniqueness of general solutions of transversely isotropic elasticity. Int J Solids Struct 32(3–4):501–513

    Article  MathSciNet  Google Scholar 

  • Wang M, Xu X (1990) A generalization of Almansi’s theorem and its application. Appl Math Modelling 14(5):275–279

    Article  MathSciNet  Google Scholar 

  • Wang X, Yp S (2002) The general solution of three-dimensional problems in magnetoelectroelastic media. Int J Eng Sc 40(10):1069–1080

    Article  MathSciNet  Google Scholar 

  • Wang Y, Hu J, Lin Y, Nan CW (2010) Multiferroic magnetoelectric composite nanostructures. NPG Asia Materi 2(2):61–68

    Article  Google Scholar 

  • Zikung W, Bailin Z (1995) The general solution of three-dimensional problems in piezoelectric media. Int J Solids Struct 32(1):105–115

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Structures for Engineering and Architecture, University of Naples Federic II, via Claudio 21, 80124, Napoli, Italy

    Francesco Marmo & Massimo Paradiso

Authors
  1. Francesco Marmo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Massimo Paradiso
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Marmo .

Editor information

Editors and Affiliations

  1. University of Naples Federico II, Napoli, Italy

    Francesco Marmo

  2. University of Naples Federico II, Napoli, Italy

    Salvatore Sessa

  3. Università degli Studi dell’Aquila, L’Aquila, Italy

    Emilio Barchiesi

  4. Università degli Studi dell’Aquila, L’Aquila, Italy

    Mario Spagnuolo

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Marmo, F., Paradiso, M. (2021). Quasi-Harmonic Solutions for Transversely Isotropic Magneto-Electro-Thermo-Elasticity: A Symbolic Mathematics Approach. In: Marmo, F., Sessa, S., Barchiesi, E., Spagnuolo, M. (eds) Mathematical Applications in Continuum and Structural Mechanics. Advanced Structured Materials, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-030-42707-8_9

Download citation

Publish with us

Policies and ethics

Search

Navigation