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Effective balance equations for elastic composites subject to inhomogeneous potentials

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Abstract

We derive the new effective governing equations for linear elastic composites subject to a body force that admits a Helmholtz decomposition into inhomogeneous scalar and vector potentials. We assume that the microscale, representing the distance between the inclusions (or fibers) in the composite, and its size (the macroscale) are well separated. We decouple spatial variations and assume microscale periodicity of every field. Microscale variations of the potentials induce a locally unbounded body force. The problem is homogenizable, as the results, obtained via the asymptotic homogenization technique, read as a well-defined linear elastic model for composites subject to a regular effective body force. The latter comprises both macroscale variations of the potentials, and nonstandard contributions which are to be computed solving a well-posed elastic cell problem which is solely driven by microscale variations of the potentials. We compare our approach with an existing model for locally unbounded forces and provide a simplified formulation of the model which serves as a starting point for its numerical implementation. Our formulation is relevant to the study of active composites, such as electrosensitive and magnetosensitive elastomers.

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References

  1. Abbott, J.J., Ergeneman, O., Kummer, M.P., Hirt, A.M., Nelson, B.J.: Modeling magnetic torque and force for controlled manipulation of soft-magnetic bodies. IEEE Trans. Robot. 23(6), 1247–1252 (2007)

    Article  Google Scholar 

  2. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson, A.R., Quaranta, V.: Integrative mathematical oncology. Nat. Rev. Cancer 8(3), 227–234 (2008)

    Article  Google Scholar 

  4. Auriault, J.L., Boutin, C., Geindreau, C.: Homogenization of Coupled Phenomena in Heterogenous Media, vol. 149. Wiley, New York (2010)

    Google Scholar 

  5. Bakhvalov, N., Panasenko, G.: Homogenisation Averaging Processes in Periodic Media. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  6. Barakati, A., Zhupanska, O.: Effects of coupled fields on the mechanical response of electrically conductive composites. Proc. Eng. 10, 31–36 (2011)

    Article  MATH  Google Scholar 

  7. Bekyarova, E., Thostenson, E., Yu, A., Kim, H., Gao, J., Tang, J., Hahn, H., Chou, T.W., Itkis, M., Haddon, R.: Multiscale carbon nanotube-carbon fiber reinforcement for advanced epoxy composites. Langmuir 23(7), 3970–3974 (2007)

    Article  Google Scholar 

  8. Borcea, L., Bruno, O.: On the magneto-elastic properties of elastomer-ferromagnet composites. J. Mech. Phys. Solids 49(12), 2877–2919 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Burridge, R., Keller, J.: Poroelasticity equations derived from microstructure. J. Acoust. Soc. Am. 70, 1140–1146 (1981)

    Article  ADS  MATH  Google Scholar 

  10. Cherkaev, A., Kohn, R.: Topics in the Mathematical Modelling of Composite Materials. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  11. Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  12. Clyne, T., Markaki, A., Tan, J.: Mechanical and magnetic properties of metal fibre networks, with and without a polymeric matrix. Compos. Sci. Technol. 65(15), 2492–2499 (2005)

    Article  Google Scholar 

  13. Collis, J., Brown, D., Hubbard, M.E., O’Dea, R.: Effective equations governing an active poroelastic medium. In: Proceedings of the Royal Society of London A, vol. 473, p. 20160755. The Royal Society, (2017)

  14. Dalwadi, M.P., Griffiths, I.M., Bruna, M.: Understanding how porosity gradients can make a better filter using homogenization theory. In: Proceedings of the Royal Society of London A, vol. 471, p. 20150464. The Royal Society, (2015)

  15. Dorfmann, A., Ogden, R.: Magnetoelastic modelling of elastomers. Eur. J. Mech. A Solids 22(4), 497–507 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Dorfmann, A., Ogden, R.: Nonlinear electroelastic deformations. J. Elast. 82(2), 99–127 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua I: Foundations and Solid Media. Springer, Berlin (2012)

    Google Scholar 

  18. Holmes, M.: Introduction to Perturbation Method. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  19. Hori, M., Nemat-Nasser, S.: On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mech. Mater. 31(10), 667–682 (1999)

    Article  Google Scholar 

  20. Hull, D., Clyne, T.: An Introduction to Composite Materials. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  21. Ioppolo, T., Stubblefield, J., Ötügen, M.: Electric field-induced deformation of polydimethylsiloxane polymers. J. Appl. Phys. 112(4), 044906 (2012)

    Article  ADS  Google Scholar 

  22. Jones, R.M.: Mechanics of Composite Materials. CRC Press, Boca Raton (1998)

    Google Scholar 

  23. Kögl, M., Gaul, L.: A boundary element method for anisotropic coupled thermoelasticity. Arch. Appl. Mech. 73(5), 377–398 (2003)

    Article  MATH  Google Scholar 

  24. Kuznetsov, S., Fish, J.: Mathematical homogenization theory for electroactive continuum. Int. J. Numer. Methods Eng. 91(11), 1199–1226 (2012)

    Article  MathSciNet  Google Scholar 

  25. Markaki, A.E., Clyne, T.W.: Magneto-mechanical stimulation of bone growth in a bonded array of ferromagnetic fibres. Biomaterials 25(19), 4805–4815 (2004)

    Article  Google Scholar 

  26. Martin, J.E., Anderson, R.A.: Electrostriction in field-structured composites: basis for a fast artificial muscle? J. Chem. Phys. 111(9), 4273–4280 (1999)

    Article  ADS  Google Scholar 

  27. Mascheroni, P., Penta, R.: The role of the microvascular network structure on diffusion and consumption of anticancer drugs. Int. J. Numer. Methods Biomed. Eng. e2857 (2016). https://doi.org/10.1002/cnm.2857

  28. Mei, C.C., Vernescu, B.: Homogenization Methods for Multiscale Mechanics. World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  29. Milton, G.W.: The Theory of Composites, vol. 6. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  30. Muskhelishvili, N. I.: Some Basic Problems of the Mathematical Theory of Elasticity, vol. 1. Springer, Netherlands (1977)

  31. Nunes, J., Herlihy, K.P., Mair, L., Superfine, R., DeSimone, J.M.: Multifunctional shape and size specific magneto-polymer composite particles. Nano Lett. 10(4), 1113–1119 (2010)

    Article  ADS  Google Scholar 

  32. Ogden, R., Steigmann, D.: Mechanics and Electrodynamics of Magneto-and Electro-Elastic Materials, vol. 527. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  33. Palmeri, M.L., Nightingale, K.R.: On the thermal effects associated with radiation force imaging of soft tissue. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 51(5), 551–565 (2004)

    Article  Google Scholar 

  34. Pao, Y.H.: Electromagnetic forces in deformable continua. In: Mechanics today. NSF-supported research, vol. 4(A78-35706 14-70), pp. 209–305. Pergamon Press, Inc., New York (1978)

  35. Papanicolau, G., Bensoussan, A., Lions, J.L.: Asymptotic Analysis for Periodic Structures. Elsevier, Amsterdam (1978)

    Google Scholar 

  36. Parnell, W.J., Abrahams, I.D.: Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves. Wave Motion 43(6), 474–498 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Parnell, W.J., Abrahams, I.D.: Homogenization for wave propagation in periodic fibre-reinforced media with complex microstructure. I. Theory. J. Mech. Phys. Solids 56(7), 2521–2540 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Parton, V.Z., Kudryavtsev, B.A.: Engineering Mechanics of Composite Structures. CRC Press, Boca Raton (1993)

    Google Scholar 

  39. Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312(5781), 1780–1782 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Penta, R., Ambrosi, D.: The role of microvascular tortuosity in tumor transport phenomena. J. Theor. Biol. 364, 80–97 (2015)

    Article  MathSciNet  Google Scholar 

  41. Penta, R., Ambrosi, D., Quarteroni, A.: Multiscale homogenization for fluid and drug transport in vascularized malignant tissues. Math. Models Methods Appl. Sci. 25(1), 79–108 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Penta, R., Ambrosi, D., Shipley, R.J.: Effective governing equations for poroelastic growing media. Q. J. Mech. Appl. Math. 67(1), 69–91 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Penta, R., Gerisch, A.: Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study. Comput. Vis. Sci. 17(4), 185–201 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Penta, R., Gerisch, A.: The asymptotic homogenization elasticity tensor properties for composites with material discontinuities. Contin. Mech. Thermodyn. 29(1), 187–206 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Penta, R., Merodio, J.: Homogenized modeling for vascularized poroelastic materials. Meccanica (2017). doi:10.1007/s11012-017-0625-1

    MathSciNet  Google Scholar 

  46. Penta, R., Raum, K., Grimal, Q., Schrof, S., Gerisch, A.: Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues. Bioinspir. Biomim. 11(3), 035004 (2016)

    Article  ADS  Google Scholar 

  47. Ramírez-Torres, A., Rodríguez-Ramos, R., Sabina, F.J., García-Reimbert, C., Penta, R., Merodio, J., Guinovart-Díaz, R., Bravo-Castillero, J., Conci, A., Preziosi, L.: The role of malignant tissue on the thermal distribution of cancerous breast. J. Theor. Biol. 426, 152–161 (2017)

  48. Richardson, G., Chapman, J.: Derivation of the bidomain equation for a beating heart with a general microstructure. J. Appl. Math. 71, 657–675 (2011)

    MathSciNet  MATH  Google Scholar 

  49. Rodríguez-Ramos, R., Sabina, F.J., Guinovart-Díaz, R., Bravo-Castillero, J.: Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents-i. Elastic and square symmetry. Mech. Mater. 33(4), 223–235 (2001)

    Article  MATH  Google Scholar 

  50. Sabina, F.J., Bravo-Castillero, J., Guinovart-Díaz, R., Rodríguez-Ramos, R., Valdiviezo-Mijangos, O.C.: Overall behavior of two-dimensional periodic composites. Int. J. Solids Struct. 39(2), 483–497 (2002)

    Article  MATH  Google Scholar 

  51. Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory-Lecture Notes in Physics 127. Springer, Berlin (1980)

    Google Scholar 

  52. Sanchez-Palencia, E.: Homogenization method for the study of composite media. Asymptot. Anal. II Lect. Notes Math. 985, 192–214 (1983)

    Article  MathSciNet  Google Scholar 

  53. Sheiretov, Y., Zahn, M.: Modeling of spatially periodic dielectric sensors in the presence of a top ground plane bounding the test dielectric. IEEE Trans. Dielectr. Electr. Insul. 12(5), 993–1004 (2005)

    Article  Google Scholar 

  54. Shipley, R.J., Chapman, J.: Multiscale modelling of fluid and drug transport in vascular tumors. Bull. Math. Biol. 72, 1464–1491 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. Siboni, M., Castañeda, P.P.: A magnetically anisotropic, ellipsoidal inclusion subjected to a non-aligned magnetic field in an elastic medium. Comptes Rendus Mécanique 340(4–5), 205–218 (2012)

    Article  ADS  Google Scholar 

  56. Sixto-Camacho, L.M., Bravo-Castillero, J., Brenner, R., Guinovart-Díaz, R., Mechkour, H., Rodríguez-Ramos, R., Sabina, F.J.: Asymptotic homogenization of periodic thermo-magneto-electro-elastic heterogeneous media. Comput. Math. Appl. 66(10), 2056–2074 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Taffetani, M., de Falco, C., Penta, R., Ambrosi, D., Ciarletta, P.: Biomechanical modelling in nanomedicine: multiscale approaches and future challenges. Arch. Appl. Mech. 84(9–11), 1627–1645 (2014)

    Article  ADS  Google Scholar 

  58. Tartar, L.: Estimation de coefficients homogenises. In: Glowinski, R., Lions, J.L. (eds.) Computing Methods in Applied Sciences and Engineering, vol. I, pp. 364–373. Springer, Berlin (1977)

  59. Weiner, S., Wagner, H.D.: The material bone: structure-mechanical function relations. Ann. Rev. Mater. Sci. 28, 271–298 (1998)

    Article  ADS  Google Scholar 

  60. Zeng, T., Shao, J.F., Xu, W.: Multiscale modeling of cohesive geomaterials with a polycrystalline approach. Mech. Mater. 69(1), 132–145 (2014)

    Article  Google Scholar 

  61. Zhupanska, O.I., Sierakowski, R.L.: Effects of an electromagnetic field on the mechanical response of composites. J. Compos. Mater. 41(5), 633–652 (2007)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Departamento de Mecánica de los Medios Continuos y T. Estructuras, E.T.S. de caminos, canales y puertos, Universidad Politécnica de Madrid, Calle Profesor Aranguren S/N, 28040, Madrid, Spain

    Raimondo Penta & José Merodio

  2. DISMA - Dipartimento di scienze matematiche “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Turin, Italy

    Ariel Ramírez-Torres

  3. Departamento de Matemáticas, Facultad de Matemática y Computación, Universidad de La Habana, CP 10400, Havana, Cuba

    Reinaldo Rodríguez-Ramos

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  1. Raimondo Penta
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  2. Ariel Ramírez-Torres
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  3. José Merodio
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  4. Reinaldo Rodríguez-Ramos
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Correspondence to Raimondo Penta.

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Communicated by Andreas Öchsner.

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Penta, R., Ramírez-Torres, A., Merodio, J. et al. Effective balance equations for elastic composites subject to inhomogeneous potentials. Continuum Mech. Thermodyn. 30, 145–163 (2018). https://doi.org/10.1007/s00161-017-0590-x

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