Abstract
In this work, we consider an elastic continuum of third grade. For the sake of simplicity, we do not consider kinetic energy in the Lagrangian function. In this work, we reformulate the problem by considering Lagrangian function depending on the metric tensor \({\mathbf g}\) and on the affine connection \(\nabla\) assumed to be compatible with the metric \({\mathbf g}\), and rewrite the Lagrangian function as \({\fancyscript{L}} ({\mathbf g}, \nabla, \nabla^2).\) Following the method of Lovelock and Rund, we apply the form-invariance requirement to the Lagrangian \({\fancyscript{L}}.\) It is shown that the arguments of the function \({\fancyscript{L}}\) are necessarily the torsion \(\aleph\) and/or the curvature \(\Re\) associated with the connection, in addition to the metric \({\mathbf g}.\) The following results are obtained: (1) \({\fancyscript{L}} ( {\mathbf g}, \nabla )\) is form-invariant if and only if \({\fancyscript{L}} ( {\mathbf g}, \aleph );\) (2) \({\fancyscript{L}} ( {\mathbf g}, \nabla^2 )\) is form-invariant if and only if \({\fancyscript{L}} ( {\mathbf g}, \Re );\) and (3) \({\fancyscript{L}} ( {\mathbf g}, \nabla, \nabla^2 )\) is form-invariant if and only if \({\fancyscript{L}} ( {\mathbf g}, \aleph, \Re ).\)
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References
Agiasofitou, EK., Lazar, M.: Conservation and balance laws in linear elasticity. Journal of Elasticity 94, 69–85 (2099)
Betram, A., Svendsen, B.: On Material Objectivity and Reduced Constitutive Equations. Archive of Mechanics 53(6), 653–675 (2001)
Cartan, E.: English translation of the French original. In: Magnon, A., Ashtekar, A. (eds) On manifolds with an affine connection and the theory of general relativity, Bibliopolis, Napoli (1986)
Choquet-Bruhat, Y., De Witt-Morette, C., Dillard-Bleick, M.: Analysis manifolds and physics. North-Holland, New-York (1977)
Cermelli, P., Gurtin, ME.: Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations. Int J of Solids and Struct 39, 6281–6309 (2002)
Ehlers, J.: The nature and concept of spacetime. In: Mehra, J. (eds) The Physicist’s concept of nature., pp. 71–91. Reidel Publishing Compagny, Dordrecht-Holland (1973)
Fleck, NA., Hutchinson, JW.: Strain gradient plasticity. Adv in Appl Mech 33, 295–361 (1997)
Fleck, NA., Hutchinson, JW.: A reformulation of strain gradient plasticity. J Mech Phys Solids 49, 2245–2271 (2001)
Forest S, Cordero NM, Busso EP. (2010) First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Computational Material Sciences (to appear)
Gao, H., Huang, Y., Nix, WD., Hutchinson, JH.: Mechanim-based strain gradient plasticity I : Theory. J Mech Phys Solids 47, 1239–1263 (1999)
Huang, Y., Gao, H., Nix, WD., Hutchinson, JH.: Mechanism-based strain gradient plasticity II Analysis. J Mech Phys Solids 48, 99–128 (2000)
Kleinert, H.: Multivalued Fields: in Condensed matter, Electromagnetism, and Gravitation. World Scientific, Singapore (2008)
Kroener E. (1981) Continuum theory of defects. In: Balian et al (eds) Physique des défauts. Les Houches July 28 – August 29, North-Holland Publishing, pp 219–315
Lazar, M.: An elastoplastic theory of dislocations as a physical field with torsion. Journal Phys A : Math Gen 35, 1983–2004 (2002)
Lazar, M., Maugin, G., Aifantis, E.: Dislocations in second strain gradient elasticity. Int J of Solids and Struct 43, 1787–1817 (2006)
Le, KC., Stumpf, H.: On the determination of the crystal reference in nonlinear continuum theory of dislocations. Proc R Soc Lond A 452, 359–371 (1996)
Lovelock, D., Rund, H.: Tensors, Differential Forms, and Variational Principles chap 8. Wiley, New-York (1975)
Lubarda, VA.: The effect of couple stresses on dislocations strain energy. Int J of Solids and Struct 40, 3807–3826 (2003)
Marsden JE., Hughes TJR. (1983) Mathematical foundations of elasticity. Prentice-Hall
Maugin G. (1993) Material Inhomogeneities in Elasticity. Chapman and Hall
Mindlin, RD.: Micro-structure in linear elasticity. Arch Rat Mech Analysis 16, 51–78 (1964)
Mindlin, RD.: Second gradient of strain and surface-tension in linear elasticity. Int J of Solids and Struct 1, 417–438 (1965)
Nakahara, M.: Geometry, Topology and Physics. In: Brewer, DF. (eds) Graduate Student Series in Physics., Institute of Physics Publishing, Bristol (1996)
Noll, W.: Materially uniform simple bodies with inhomogeneities. Arch Rat Mech Analysis 27, 1–32 (1967)
Popov, VL., Kröner, E.: Theory of elastopolastic media with mesostructures. Theoritical and Applied Fracture Mechanics 37, 299–310 (2001)
Rakotomanana, RL.: Contribution à la modélisation géométrique et thermodynamique d’une classe de milieux faiblement continus. Arch Rat Mech Analysis 141, 199–236 (1997)
Rakotomanana, RL.: A geometric approach to thermomechanics of dissipating continua. Birkauser, Boston (2003)
Rakotomanana RL. (2005) Some class of SG continuum models to connect various length scales in plastic deformation. In: Steimann P., Maugin GA. (ed) Mechanics of material forces, chap 32. Springer
Rakotomanana, RL.: Élements de dynamiques des structures et solides déformables. Presses Polytechniques et Universitaires Romandes, Lausanne (2009)
Svendsen, B., Betram, A.: On frame-indifference and form-invariance in constitutive theory. Acta Mechanica 132, 195–207 (1999)
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)
Toupin, RA.: Elastic materials with couple stresses. Arch Rat Mech Analysis 11, 385–414 (1962)
Wang, CC.: Geometric structure of simple bodies, or mathematical foundation for the theory of continuous distributions of dislocations. Arch Rat Mech Analysis 27, 33–94 (1967)
Zhao, J., Pedroso, D.: Strain gradient theory in orthogonal curvilinear coordinates. Int J of Solids and Struct 45, 3507–3520 (2008)
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Tamarasselvame, N.A., Rakotomanana, L.R. (2011). On the Form-Invariance of Lagrangian Function for Higher Gradient Continuum. In: Altenbach, H., Maugin, G., Erofeev, V. (eds) Mechanics of Generalized Continua. Advanced Structured Materials, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19219-7_15
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DOI: https://doi.org/10.1007/978-3-642-19219-7_15
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