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A unifying perspective: the relaxed linear micromorphic continuum

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Abstract

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict nonpolar size effects. It is intended for the homogenized description of highly heterogeneous, but nonpolar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models, our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models.

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References

  1. Adams R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, London (1975)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  3. Askes H., Aifantis E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)

    Google Scholar 

  4. Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics àla Hamilton-Piola: least action principle for second gradient continua and capillary fluids. Math. Mech. Solids (2013). doi:10.1177/1081286513497616

  5. Bauer, S., Neff, P., Pauly, D., Starke, G.: Dev-Div and DevSym-DevCurl inequalities for incompatible square tensor fields with mixed boundary conditions (2013, submitted)

  6. Bauer, S., Neff, P., Pauly, D., Starke, G.: New Poincaré type inequalities. C. R. Acad. Sci. Paris Ser. I, (2013, submitted)

  7. Bauer, S., Neff, P., Pauly, D., Starke, G.: Some Poincaré type inequalities for quadratic matrix fields. Proc. Appl. Math. Mech. (2013, to appear)

  8. Berdichevsky V.L.: Continuum theory of dislocations revisited. Contin. Mech. Therm. 18, 195–222 (2006)

    MATH  MathSciNet  Google Scholar 

  9. Berglund K.: Investigation of a two-dimensional model of a micropolar continuum. Archiwum Mechaniki Stosowanej 29, 383–392 (1977)

    MATH  Google Scholar 

  10. Berglund, K.: Structural models of micropolar media. In: Brulin, O. Hsieh, R.K.T. (eds.) Mechanics of micropolar media, CISM-Lecture Notes, vol. 132, pp. 35–86 (1982)

  11. Bigoni D., Drugan W.J.: Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74, 741–753 (2007)

    ADS  MathSciNet  Google Scholar 

  12. Bîrsan M.: Saint-Venant’s problem for Cosserat shells with voids. Int. J. Solids Struct. 42, 2033–2057 (2005)

    Google Scholar 

  13. Bîrsan M.: On a thermodynamic theory of porous Cosserat elastic shells. J. Thermal Stress. 29, 879–899 (2006)

    Google Scholar 

  14. Bîrsan M.: On the dynamic deformation of porous Cosserat linear-thermoelastic shells. Z. Angew. Math. Mech. 88, 74–78 (2008)

    MATH  Google Scholar 

  15. Bîrsan M.: On Saint-Venant’s problem for anisotropic, inhomogeneous, cylindrical Cosserat elastic shells. Int. J. Eng. Sci. 47, 21–38 (2009)

    Google Scholar 

  16. Bîrsan M.: Thermal stresses in cylindrical Cosserat elastic shells. Eur. J. Mech. A/Solids 28, 94–101 (2009)

    MATH  MathSciNet  Google Scholar 

  17. Bîrsan M., Altenbach H.: On the theory of porous elastic rods. Int. J. Solids Struct. 48, 910–924 (2011)

    Google Scholar 

  18. Bîrsan M., Altenbach H.: On the Cosserat model for thin rods made of thermoelastic materials with voids. Discret. Contin. Dyn. Syst. Ser. S 6(6), 1473–1485 (2013)

    MATH  Google Scholar 

  19. Bleustein J.L.: A note on the boundary conditions of Toupin’s strain gradient-theory. Int. J. Solids Struct. 3, 1053–1057 (1967)

    Google Scholar 

  20. Borş C.I.: Deformable solids with microstructure having a symmetric stress tensor. Analele Ştiinţifice ale Universitaţii Al. I. Cuza din Iaşi, XXVII, s.Ia f.1(f.1), 177–184 (1981)

    Google Scholar 

  21. Boutin C., Hans S., Chesnais C.: Generalized beams and continua. Dynamics of reticulated structures. In: Maugin, G.A., Metrikine, A.V. (eds) Mechanics of Generalized Continua, pp. 131–141. Springer, New York (2011)

    Google Scholar 

  22. Buechner P.M., Lakes R.S.: Size effects in the elasticity and viscoelasticity of bone. Biomech. Model. Mechanobio. 1, 295–301 (2003)

    Google Scholar 

  23. Bulgariu E., Ghiba I.D.: On the thermal stresses in anisotropic porous cylinders. Discret. Contin. Dyn. Syst. Ser. S 6, 1539–1550 (2013)

    MATH  MathSciNet  Google Scholar 

  24. Capriz G.: Continua with Microstructure. Springer, Heidelberg (1989)

    MATH  Google Scholar 

  25. Capriz G., Mariano P.M.: Symmetries and Hamiltonian formalism for complex materials. J. Elasticity 72, 57–90 (2003)

    MATH  MathSciNet  Google Scholar 

  26. Chen Y., Lee J.D.: Connecting molecular dynamics to micromorphic theory. I. Instantaneous and averaged mechanical variables. Physica A 322, 359–376 (2003)

    ADS  MATH  Google Scholar 

  27. Chen Y., Lee J.D., Eskandarian A.: Atomistic viewpoint of the applicability of microcontinuum theories. Int. J. Solids Struct. 41, 2085–2097 (2004)

    MATH  Google Scholar 

  28. Chiriţă S., Ghiba I.D.: Inhomogeneous plane waves in elastic materials with voids. Wave Motion 47, 333–342 (2010)

    MATH  MathSciNet  Google Scholar 

  29. Chiriţă S., Ghiba I.D.: Rayleigh waves in Cosserat elastic materials. Int. J. Eng. Sci. 51, 117–127 (2012)

    Google Scholar 

  30. Chiriţă S., Ghiba I.D.: Strong ellipticity and progressive waves in elastic materials with voids. Proc. R. Soc. A 466, 439–458 (2010)

    ADS  MATH  Google Scholar 

  31. Claus, W.D. Jr.: Discussion on Papers by A.C. Eringen and W.D. Claus, Jr., and N. Fox. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory. Nat. Bur. Stand. (U.S.), Spec. Publ., vol. 1, pp. 1054–1059 (1970)

  32. Claus, W.D., Eringen, A.C.: Three dislocation concepts and micromorphic mechanics. In Developments in Mechanics. In: Proceedings of the 12th Midwestern Mechanics Conference, vol. 6, pp. 349–358. Midwestern (1969)

  33. Claus W.D., Eringen A.C.: Dislocation dispersion of elastic waves. Int. J. Eng. Sci. 9, 605–610 (1971)

    MATH  Google Scholar 

  34. Cordero N.M., Gaubert A., Forest S., Busso E.P., Gallerneau F., Kruch S.: Size effects in generalised continuum crystal plasticity for two-phase laminates. J. Mech. Phys. Solids 58(28), 1963–1994 (2010)

    ADS  MATH  MathSciNet  Google Scholar 

  35. Cosserat, E., Cosserat, F.: Théorie des corps déformables. Librairie Scientifique A. Hermann et Fils (engl. translation by D. Delphenich 2007, pdf available at http://www.uni-due.de/hm0014/Cosserat_files/Cosserat09_eng.pdf), reprint 2009 by Hermann Librairie Scientifique, ISBN 978 27056 6920 1, Paris, 1909

  36. Cowin S.C., Nunziato J.W.: Linear elastic materials with voids. J. Elasticity 13, 125–147 (1983)

    MATH  Google Scholar 

  37. de Fabritiis C., Mariano P.M.: Geometry of interactions in complex bodies. J. Geom. Phys. 54, 301–323 (2005)

    ADS  MATH  MathSciNet  Google Scholar 

  38. dell’Isola F., Guarascio M., Hutter K.: A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Arch. Appl. Mech. 70(5), 323–337 (2000)

    ADS  MATH  Google Scholar 

  39. dell’Isola F., Madeo A., Placidi L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3d continua. Z. Angew. Math. Mech. 92(1), 52–71 (2012)

    MATH  MathSciNet  Google Scholar 

  40. dell’Isola F., Rosa L., Wozniak Cz.: Dynamics of solids with micro periodic nonconnected fluid inclusions. Arch. Appl. Mech. 67(4), 215–228 (1997)

    MATH  Google Scholar 

  41. dell’Isola F., Rosa L., Wozniak Cz.: A micro-structured continuum modelling compacting fluid-saturated grounds: The effects of pore-size scale parameter. Acta Mech. 127(1–4), 165–182 (1998)

    MATH  MathSciNet  Google Scholar 

  42. dell’Isola F., Sciarra G., Vidoli S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A 465, 2177–2196 (2009)

    ADS  MATH  MathSciNet  Google Scholar 

  43. dell’Isola F., Seppecher P.: The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power. C.R. Acad. Sci. II, Mec. Phys. Chim. Astron. 321, 303–308 (1995)

    MATH  Google Scholar 

  44. Djoko J.K., Ebobisse F., McBride A.T., Reddy B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: Algorithms and numerical analysis. Comput. Methods Appl. Mech. Eng. 197(1), 1–21 (2007)

    ADS  MATH  MathSciNet  Google Scholar 

  45. Dresen L., Kozak J., Spicac A., Waniek L., Teisseyre R.: Wave propagation in physical models of micromorphic media. Stud. Geophys. Geodaet. 28, 272–285 (1984)

    ADS  Google Scholar 

  46. Ebobisse F., Neff P.: Rate-independent infinitesimal gradient plasticity with isotropic hardening and plastic spin. Math. Mech. Solids 15, 691–703 (2010)

    MATH  MathSciNet  Google Scholar 

  47. Engheta N., Ziolkowski R.W.: Metamaterials: Physics and Engineering Explorations. Wiley, New York (2006)

    Google Scholar 

  48. Eringen A.C.: Microcontinuum Field Theories. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  49. Eringen, A.C., Claus, W.D.: A micromorphic approach to dislocation theory and its relation to several existing theories. In: Simmons, J.A., de Wit R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory, vol. 1. Nat. Bur. Stand. (U.S.), Spec. Publ., Spec. Publ., pp. 1023–1040 (1970)

  50. Eringen A.C., Suhubi E.S.: Nonlinear theory of simple micro-elastic solids. I. Int. J. Eng. Sci. 2, 189–203 (1964)

    MATH  MathSciNet  Google Scholar 

  51. Eringen A.C., Suhubi E.S.: Nonlinear theory of simple microelastic solids: II. Int. J. Eng. Sci. 2, 389–404 (1964)

    MathSciNet  Google Scholar 

  52. Ferretti, M., Madeo, A., dell’Isola, F., Boisse, P.: Modelling the onset of shear boundary layers in fibrous composite reinforcements by second gradient theory. Z. Angew. Math. Phys. (2013). doi:10.1007/s00033-013-0347-8

  53. Forest, S.: Mechanics of generalized continua: construction by homogenization. J. Phys. IV France 8, Pr4-39–Pr4-48 (1998)

  54. Forest S.: Homogenization methods and the mechanics of generalized continua—part 2. Theoret. Appl. Mech. (Belgrad) 28(29), 113–143 (2002)

    MathSciNet  Google Scholar 

  55. Forest S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)

    Google Scholar 

  56. Forest S., Sievert R.: Nonlinear microstrain theories. Int. J. Solids Struct. 43, 7224–7245 (2006)

    MATH  MathSciNet  Google Scholar 

  57. Forest S., Sievert R., Aifantis E.C.: Strain gradient crystal plasticity: thermodynamical formulations and applications. J. Mech. Behav. Mater. 13, 219–232 (2002)

    Google Scholar 

  58. Forest S., Trinh D.K.: Generalized continua and non-homogeneous boundary conditions in homogenisation methods. Z. Angew. Math. Mech. 91, 90–109 (2011)

    MATH  MathSciNet  Google Scholar 

  59. Fox, N.: On the continuum theory of dislocation. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory., vol. 1. Nat. Bur. Stand. (U.S.), Spec. Publ., pp. 1041–1052 (1970)

  60. Galeş C.: Some results in micromorphic piezoelectricity. Eur. J. Mech. A/Solids 31, 37–46 (2012)

    ADS  MATH  MathSciNet  Google Scholar 

  61. Galeş C., Ghiba I.D., Ignatescu I.: Asymptotic partition of energy in micromorphic thermopiezoelectricity. J. Thermal Stress. 34, 1241–1249 (2011)

    Google Scholar 

  62. Germain P.: La méthode des puissances virtuelles en mécanique des milieux continus-I: Théorie du second gradient. J. Mécanique 12, 235–274 (1973)

    MATH  MathSciNet  Google Scholar 

  63. Germain P.: The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)

    MATH  MathSciNet  Google Scholar 

  64. Ghiba I.D.: Semi-inverse solution for Saint-Venant’s problem in the theory of porous elastic materials. Eur. J. Mech. A/Solids 27, 1060–1074 (2008)

    ADS  MATH  MathSciNet  Google Scholar 

  65. Ghiba I.D.: On the deformation of transversely isotropic porous elastic circular cylinder. Arch. Mech. 61, 407–421 (2009)

    MATH  MathSciNet  Google Scholar 

  66. Ghiba I.D.: On the temporal behaviour in the bending theory of porous thermoelastic plates. Z. Angew. Math. Mech. 93, 284–296 (2013)

    MATH  MathSciNet  Google Scholar 

  67. Ghiba, I.D., Neff, P., Madeo, A., Placidi, L., Rosi, G.: The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. Math. Mech. Solids (2013, submitted)

  68. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier–Stokes Equations. Lecture Notes in Mathematics, vol. 749. Springer, Heidelberg (1979)

  69. Goddard, J.D.: From granular matter to generalized continuum. In: Capriz, G., Giovine, P., Mariano, P.M. (eds.) Mathematical Models of Granular Matter, Lecture Notes in Applied Mathematics, vol. 1937, chapter 1. Springer, New York, pp. 1–20 (2008)

  70. Green A.E., Rivlin R.S.: Multipolar continuum mechanics. Arch. Rat. Mech. Anal. 17(2), 113–147 (1964)

    MATH  MathSciNet  Google Scholar 

  71. Green A.E., Rivlin R.S.: On Cauchy’s equations of motion. Z. Angew. Math. Phys. 15, 290–292 (1964)

    MATH  MathSciNet  Google Scholar 

  72. Green A.E., Rivlin R.S.: Simple force and stress multipoles. Arch. Rat. Mech. Anal. 16, 325–353 (1964)

    MATH  MathSciNet  Google Scholar 

  73. Green A.E., Rivlin R.S.: Multipolar continuum mechanics: functional theory. I. Proc. R. Soc. A 284, 303–324 (1965)

    ADS  MathSciNet  Google Scholar 

  74. Grekova E.F., Maugin G.A.: Modelling of complex elastic crystals by means of multi-spin micromorphic media. Int. J. Eng. Sci. 43, 494–519 (2005)

    MATH  MathSciNet  Google Scholar 

  75. Hlaváček I., Hlaváček M.: On the existence and uniqueness of solutions and some variational principles in linear theories of elasticity with couple-stresses. I: Cosserat continuum. II: Mindlin’s elasticity with micro-structure and the first strain gradient. J. Apl. Mat. 14, 387–426 (1969)

    Google Scholar 

  76. Ieşan D.: A theory of thermoelastic materials with voids. Acta Mech. 60, 67–89 (1986)

    Google Scholar 

  77. Ieşan D.: Extremum principle and existence results in micromorphic elasticity. Int. J. Eng. Sci. 39, 2051–2070 (2001)

    MATH  Google Scholar 

  78. Ieşan D.: On the micromorphic thermoelasticity. Int. J. Eng. Sci. 40, 549–567 (2002)

    Google Scholar 

  79. Ieşan D., Ciarletta M.: Non-classical Elastic Solids. Longman Scientific and Technical, Harlow (1993)

    MATH  Google Scholar 

  80. Jänicke R., Diebels S., Sehlhorst H.G., Düster A.: Two-scale modelling of micromorphic continua. Contin. Mech. Therm. 21, 297–315 (2009)

    MATH  Google Scholar 

  81. Jänicke R., Steeb H.: Wave propagation in periodic microstructures by homogenisation of extended continua. Comput. Mater. Sci. 52, 209–211 (2012)

    Google Scholar 

  82. Jeong J., Neff P.: Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids 15(1), 78–95 (2010)

    MATH  MathSciNet  Google Scholar 

  83. Kirchner N., Steinmann P.: A unifying treatise on variational principles for gradient and micromorphic continua. Philos. Mag. 85, 3975–3995 (2005)

    Google Scholar 

  84. Klawonn A., Neff P., Rheinbach O., Vanis S.: FETI-DP domain decomposition methods for elasticity with structural changes: P-elasticity. ESAIM: Math. Mod. Num. Anal. 45, 563–602 (2011)

    MATH  MathSciNet  Google Scholar 

  85. Kröner, E.: Mechanics of Generalized Continua. In: Proceedings of the IUTAM-Symposium on the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart, Germany, 1967. Springer, Berlin/Heidelberg/New York

  86. Kröner, E.: Das physikalische Problem der antisymmetrischen Spannungen und der sogenannten Momentenspannungen. In Görtler, H. (ed.) Proceedings of 11th International Congress Applied Mechanics, Munich, 1964, pp. 143–158 (1966)

  87. Kröner, E.: Discussion on Papers by A.C. Eringen and W.D. Claus, Jr., and N. Fox. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory. Nat. Bur. Stand. (U.S.), vol. 1. Spec. Publ., pp. 1054–1059 (1970)

  88. Lakes R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22, 55–63 (1985)

    Google Scholar 

  89. Lankeit, J., Neff, P., Pauly, D.: Uniqueness of integrable solutions to \({\nabla \xi={G}\xi,\xi|_\gamma=0}\) for integrable tensor coefficients G and applications to elasticity. Z. Angew. Math. Phys. (2013). doi:10.1007/s00033-013-0314-4

  90. Lazar M.: An elastoplastic theory of dislocations as a physical field theory with torsion. J. Phys. A: Math. Gen. 35, 1983–2004 (2002)

    ADS  MATH  MathSciNet  Google Scholar 

  91. Lazar M.: Screw dislocations in the field theory of elastoplasticity. Ann. Phys. 11, 635–649 (2002)

    MATH  MathSciNet  Google Scholar 

  92. Lazar M.: The gauge theory of dislocations: a uniformly moving screw dislocation. Proc. R. Soc. A 465, 2505–2520 (2009)

    ADS  MATH  MathSciNet  Google Scholar 

  93. Lazar, M.: Dislocations in generalized continuum mechanics. In: Metrikine, A.V. Maugin, G.A., (eds.) Mechanics of Generalized Continua. One hundred years after the Cosserats. Advances in Mechanics and Mathematics, vol. 21, chapter 24. Springer, New York, pp. 223–232 (2010)

  94. Lazar M.: On the fundamentals of the three-dimensional translation gauge theory of dislocations. Math. Mech. Solids 16, 253–264 (2011)

    MATH  MathSciNet  Google Scholar 

  95. Lazar M., Anastassiadis C.: The gauge theory of dislocations: conservation and balance laws. Philos. Mag. 88, 1673–1699 (2008)

    ADS  Google Scholar 

  96. Lazar M., Anastassiadis C.: The gauge theory of dislocations: static solutions of screw and edge dislocations. Philos. Mag. 98, 199–231 (2009)

    ADS  Google Scholar 

  97. Lee J.D., Chen Y.: Constitutive relations of micromorphic thermoplasticity. Int. J. Eng. Sci. 41, 387–399 (2002)

    MathSciNet  Google Scholar 

  98. Leis R.: Initial Boundary Value Problems in Mathematical Physics. Teubner, Stuttgart (1986)

    MATH  Google Scholar 

  99. Malyshev C.: The T(3)-gauge model, the Einstein-like gauge equation, and Volterra dislocations with modified asymptotics. Ann. Phys. 286, 249–277 (2000)

    ADS  MATH  MathSciNet  Google Scholar 

  100. Mariano P.M.: Representation of material elements and geometry of substructural interaction. Quaderni di Matematica. 20, 80–100 (2007)

    MathSciNet  Google Scholar 

  101. Mariano P.M., Modica G.: Ground states in complex bodies. ESAIM: COCV 15(2), 377–402 (2009)

    MATH  MathSciNet  Google Scholar 

  102. Mariano P.M., Stazi F.L.: Computational aspects of the mechanics of complex materials. Arch. Comput. Methods Eng. 12, 392–478 (2005)

    MathSciNet  Google Scholar 

  103. Maugin G.A.: Un principe variationnel pour des milieux micromorphiques non dissipatifs. C. R. Acad. Sci. Paris Ser. II 271, 807–810 (1970)

    MATH  Google Scholar 

  104. Maugin G.A.: Electromagnetism and generalized continua. In: Altenbach, H., Eremeyev, V. (eds) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol. 541, pp. 301–360. Springer, New York (2013)

    Google Scholar 

  105. Maugin, G.A.: The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. In memory of Paul Germain (1920–2009). Contin. Mech. Thermodyn. 25, 127–146 (2013)

  106. Maugin, G.A., Metrikine, V.A. (eds.): Mechanics of generalized continua. One hundred years after the Cosserats. Advances in Mechanics and Mathematics, vol. 21. Springer, Berlin (2010)

  107. Mindlin R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3(1), 1–7 (1963)

    MathSciNet  Google Scholar 

  108. Mindlin R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–77 (1964)

    MATH  MathSciNet  Google Scholar 

  109. Mindlin R.D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)

    Google Scholar 

  110. Mindlin, R.D.: Theories of elastic continua and crystal lattice theories. In: Kröner, E. (ed.) Mechanics of Generalized Continua. Proceedings of the IUTAM-Symposium on the generalized Cosserat continuum and the continuum theory of dislocations with applications in Freudenstadt, 1967, Springer, New York, pp. 312–320 (1968)

  111. Mindlin R.D., Eshel N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)

    MATH  Google Scholar 

  112. Mindlin R.D., Tiersten H.F.: Effects of couple stresses in linear elasticity. Arch. Rat. Mech. Anal. 11, 415–447 (1962)

    MATH  MathSciNet  Google Scholar 

  113. Misra A., Chang C.S.: Effective elastic moduli of heterogeneous granular solids. Int. J. Solids Struct. 30, 2547–2566 (1993)

    MATH  Google Scholar 

  114. Morro, A., Vianello, M.: Interstitial energy flux and stress-power for second-gradient elasticity. In: The 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM 2013), pp. 1–7, Paper ID 702, Montréal, Canada, July 23–26 2013. McGill University

  115. Mühlhaus H.B., Aifantis E.C.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28, 845–857 (1991)

    MATH  Google Scholar 

  116. Münch, I., Neff, P.: A nonlinear micropolar continuum theory for initial plasticity. In: Zigoni, A. (ed.) Advances and Trends in Structural Engineering, Mechanics and Computation. CRC Press/Balkema, pp. 269–272 (2010). ISBN 978-0-415-58472-2

  117. Münch I., Wagner W., Neff P.: Transversely isotropic material: nonlinear Cosserat versus classical approach. Contin. Mech. Thermodyn. 23(1), 27–34 (2011)

    ADS  MATH  MathSciNet  Google Scholar 

  118. Nagahama H., Teisseyre R.: Micromorphic continuum and fractal fracturing in the lithosphere. Pure Appl. Geophys. 157, 559–574 (2000)

    ADS  Google Scholar 

  119. Nagahama H., Teisseyre R.: Micromorphic continuum and fractal fracturing in the lithosphere. Int. Geophys. 76, 425–440 (2001)

    Google Scholar 

  120. Neff, P.: On material constants for micromorphic continua. In: Wang, Y., Hutter, K. (eds.) Trends in Applications of Mathematics to Mechanics, STAMM Proceedings, Seeheim 2004. Shaker Verlag, Aachen, pp. 337–348 (2005)

  121. Neff P.: The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. Z. Angew. Math. Mech. 86, 892–912 (2006)

    MATH  MathSciNet  Google Scholar 

  122. Neff P.: Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. R. Soc. Edinb. A 136, 997–1012 (2006)

    MATH  MathSciNet  Google Scholar 

  123. Neff P.: Remarks on invariant modelling in finite strain gradient plasticity. Tech. Mech. 28(1), 13–21 (2008)

    Google Scholar 

  124. Neff P., Chełmiński K.: Infinitesimal elastic-plastic Cosserat micropolar theory. Modelling and global existence in the rate-independent case. Proc. R. Soc. Edinb. A 135, 1017–1039 (2005)

    MATH  Google Scholar 

  125. Neff P., Chełmiński K.: Well-posedness of dynamic Cosserat plasticity. Appl. Math. Optim. 56, 19–35 (2007)

    MATH  MathSciNet  Google Scholar 

  126. Neff P., Chełmiński K.: \({H^1_{\rm loc}}\)-stress and strain regularity in Cosserat-Plasticity. Z. Angew. Math. Mech. 89(4), 257–266 (2008)

    Google Scholar 

  127. Neff P., Chełmiński K., Alber H.D.: Notes on strain gradient plasticity. Finite strain covariant modelling and global existence in the infinitesimal rate-independent case. Math. Mod. Methods Appl. Sci. (M3AS) 1(2), 1–40 (2009)

    Google Scholar 

  128. Neff P., Forest S.: A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elasticity 87, 239–276 (2007)

    MATH  MathSciNet  Google Scholar 

  129. Neff P., Jeong J.: A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. Z. Angew. Math. Mech. 89(2), 107–122 (2009)

    MATH  MathSciNet  Google Scholar 

  130. Neff P., Jeong J., Münch I., Ramezani H.: Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity. Z. Angew. Math. Phys. 3(60), 479–497 (2009)

    Google Scholar 

  131. Neff, P., Jeong, J., Münch, I., Ramezani, H.: Linear Cosserat Elasticity, Conformal Curvature and Bounded Stiffness. In: Maugin, G.A., Metrikine, V.A. (eds.) Mechanics of Generalized Continua. One hundred years after the Cosserats, Advances in Mechanics and Mathematics, vol. 21, pp. 55–63. Springer, Berlin (2010)

  132. Neff P., Jeong J., Ramezani H.: Subgrid interaction and micro-randomness—novel invariance requirements in infinitesimal gradient elasticity. Int. J. Solids Struct. 46(25–26), 4261–4276 (2009)

    MATH  Google Scholar 

  133. Neff P., Knees D.: Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity. SIAM J. Math. Anal. 40(1), 21–43 (2008)

    MATH  MathSciNet  Google Scholar 

  134. Neff P., Münch I.: Curl bounds Grad on SO(3). ESAIM: Control Optim. Calc. Var. 14(1), 148–159 (2008)

    MATH  MathSciNet  Google Scholar 

  135. Neff P., Münch I.: Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure. Contin. Mech. Thermodyn. 21(3), 195–221 (2009)

    MATH  MathSciNet  Google Scholar 

  136. Neff, P., Pauly, D., Witsch, K.J.: Poincaré meets Korn via Maxwell: Extending Korn’s first inequality to incompatible tensor fields (submitted). arXiv:1203.2744

  137. Neff P., Pauly D., Witsch K.J.: A canonical extension of Korn’s first inequality to H(Curl) motivated by gradient plasticity with plastic spin. C. R. Acad. Sci. Paris Ser. I 349, 1251–1254 (2011)

    MATH  MathSciNet  Google Scholar 

  138. Neff P., Pauly D., Witsch K.J.: Maxwell meets Korn: a new coercive inequality for tensor fields in \({\mathbb{R}^{N\times N}}\) with square-integrable exterior derivative. Math. Methods Appl. Sci. 35, 65–71 (2012)

    ADS  MATH  MathSciNet  Google Scholar 

  139. Neff P., Sydow A., Wieners C.: Numerical approximation of incremental infinitesimal gradient plasticity. Int. J. Numer. Methods Eng. 77(3), 414–436 (2009)

    MATH  MathSciNet  Google Scholar 

  140. Nesenenko S., Neff P.: Well-posedness for dislocation based gradient viscoplasticity I: Subdifferential case. SIAM J. Math. Anal. 44(3), 1694–1712 (2012)

    MATH  MathSciNet  Google Scholar 

  141. Nesenenko S., Neff P.: Well-posedness for dislocation based gradient visco-plasticity II: monotone case. MEMOCS: Math. Mech. Complex Syst. 1(2), 149–176 (2013)

    MATH  Google Scholar 

  142. Nunziato, J.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Rat. Mech. Anal. 72, 175–201 (1979)

    Google Scholar 

  143. Nye J.F.: Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162 (1953)

    Google Scholar 

  144. Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    MATH  Google Scholar 

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

    ADS  MATH  MathSciNet  Google Scholar 

  146. Pietraszkiewicz W., Eremeyev V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46, 774–787 (2009)

    MATH  MathSciNet  Google Scholar 

  147. Pietraszkiewicz W., Eremeyev V.A.: On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. Int. J. Solids Struct. 46, 2477–2480 (2009)

    MATH  MathSciNet  Google Scholar 

  148. Pietraszkiewicz W., Eremeyev V.A.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49, 1993–2005 (2012)

    Google Scholar 

  149. Piola, G.: Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, Modena, Tipi del R.D. Camera, 1846, translated by F. dell’Isola, U. Andreaus and L. Placidi. In U. Andreaus, F. dell’Isola, R. Esposito, S. Forest, G. Maier, and U. Perego, editors, The complete works of Gabrio Piola, volume I. Springer, New York (in preparation)

  150. Placidi, L., Rosi, G., Giorgio, I., Madeo, A.: Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second gradient materials. Math. Mech. Solids (2013). doi:10.1177/1081286512474016

  151. Popov V.L.: Gauge theory of “plastically incompressible” medium without dissipation-I. Dispersion relations and propagation of perturbations without dissipation. Int. J. Eng. Sci. 30(3), 329–334 (1992)

    MATH  Google Scholar 

  152. Popov V.L.: Coupling of an elastoplastic continuum and a Cosserat continuum. Russ. Phys. J. 37, 337–342 (1994)

    Google Scholar 

  153. Popov V.L.: Dynamics of plastic rotations in a medium with dislocations and disclinations. Tech. Phys. Lett. 20, 576 (1994)

    ADS  Google Scholar 

  154. Popov, V.L., Kröner, E.: Theory of elastoplastic media with mesostructure. Theor. Appl. Fract. Mech. 299–310 (2001)

  155. Rosi, G., Giorgio, I., Eremeyev, V.A.: Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. ZAMM (2013). doi:10.1002/zamm.201200285

  156. Rosi G., Madeo A., Guyader J.-L.: Switch between fast and slow Biot compression waves induced by second gradient microstructure at material discontinuity surfaces in porous media. Int. J. Solids Struct. 50, 1721–1746 (2013)

    Google Scholar 

  157. Sansour, C.: A unified concept of elastic-viscoplastic Cosserat and micromorphic continua. In: Bertram, A., Sidoroff, F.. (eds.) Mechanics of Materials with Intrinsic Length Scale: Physics, Experiments, Modelling and Applications. Journal Physique IV France vol. 8, pp. 341–348. EDP Sciences, France (1998)

  158. Sciarra G., dell’Isola F., Coussy O.: Second gradient poromechanics. Int. J. Solids Struct. 44, 6607–6629 (2007)

    MATH  MathSciNet  Google Scholar 

  159. Smith A.C.: Inequalities between the constants of a linear micro-elastic solid. Int. J. Eng. Sci. 6, 65–74 (1968)

    Google Scholar 

  160. Steigmann D.J.: Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. Int. J. Non-linear Mech. 47, 734–742 (2012)

    ADS  Google Scholar 

  161. Svendsen B., Neff P., Menzel A.: On constitutive and configurational aspects of models for gradient continua with microstructure. Z. Angew. Math. Mech. 89(8), 687–697 (2009)

    MATH  MathSciNet  Google Scholar 

  162. Teisseyre R.: Earthquake process in a micromorphic continuum. Pure Appl. Geophys. 102(1), 15–28 (1973)

    ADS  Google Scholar 

  163. Teisseyre, R.: Symmetric micromorphic continuum: wave propagation, point source solutions and some applications to earthquake processes. In: Thoft-Christensen, P. (ed.) Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics. NATO Advanced Study Institute Series, vol. 12. Springer, New York (1974)

  164. Teodosiu, C.: Discussion on Papers by A.C. Eringen and W.D. Claus, Jr., and N. Fox. In: J.A. Simmons, R. de Wit, and R. Bullough, editors, Fundamental Aspects of Dislocation Theory. Nat. Bur. Stand. (U.S.), Spec. Publ., vol. 1, pp. 1054–1059 (1970)

  165. Toupin R.A.: Elastic materials with couple stresses. Arch. Rat. Mech. Anal. 11, 385–413 (1962)

    MATH  MathSciNet  Google Scholar 

  166. Toupin R.A.: Theory of elasticity with couple stresses. Arch. Rat. Mech. Anal. 17, 85–112 (1964)

    MATH  MathSciNet  Google Scholar 

  167. Vidoli S., dell’Isola F.: Modal coupling in one-dimensional electromechanical structured continua. Acta Mech. 141, 37–50 (2000)

    MATH  Google Scholar 

  168. Vrabie I.: C 0-Semigroups and Applications, Ser. Mathematics Studies no. 191. Elsevier, Amsterdam (2003)

    Google Scholar 

  169. Wang X., Lee J.: Micromorphic theory: a gateway to nano world. Int. J. Smart Nano Mater. 1, 115–135 (2010)

    MATH  Google Scholar 

  170. Yavari A.: Covariant balance laws in continua with microstructure. Rep. Math. Phys. 63, 1–42 (2009)

    ADS  MATH  MathSciNet  Google Scholar 

  171. Zeng X., Chen Y., Lee J.D.: Determining material constants in nonlocal micromorphic theory through phonon dispersion relations. Int. J. Eng. Sci. 44, 1334–1345 (2006)

    Google Scholar 

  172. Zouhdi S., Sihvola A., Vinogradov A.P.: Metamaterials and Plasmonics: Fundamentals, Modelling Applications. NATO Science for Peace and Security Series B: Physics and Biophysics. Springer, New York (2009)

    Google Scholar 

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Authors and Affiliations

  1. Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann Str. 9, 45127, Essen, Germany

    Patrizio Neff & Ionel-Dumitrel Ghiba

  2. Laboratoire de Génie Civil et Ingénierie Environnementale, Université de Lyon-INSA, Bâtiment Coulomb, 69621, Villeurbanne Cedex, France

    Angela Madeo

  3. Università Telematica Internazionale Uninettuno, Corso V. Emanuele II 39, 00186, Roma, Italy

    Luca Placidi

  4. Laboratoire Modélisation Multi-Echelle, MSME UMR 8208 CNRS, Université Paris-Est, 61 Avenue du General de Gaulle, Creteil Codex, 9410, France

    Giuseppe Rosi

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  1. Patrizio Neff
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  2. Ionel-Dumitrel Ghiba
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  3. Angela Madeo
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Communicated by Andreas Öchsner.

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Neff, P., Ghiba, ID., Madeo, A. et al. A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech. Thermodyn. 26, 639–681 (2014). https://doi.org/10.1007/s00161-013-0322-9

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