# American Institute of Mathematical Sciences

March  2012, 17(2): 575-596. doi: 10.3934/dcdsb.2012.17.575

## Line defect evolution in finite-dimensional manifolds

 1 DICeA, University of Florence, via Santa Marta 3, I-50139 Firenze, Italy

Received  October 2011 Published  December 2011

A fiber bundle $Y$ having as a prototype fiber a finite-dimensional, differentiable manifold, not embedded in any linear space, and as a base a space-time tube constructed on the $n-$dimensional point space is the geometric environment considered here. After assigning a Lagrangian to its first jet bundle, the notion of defect and the representation of its evolution in this setting is discussed first. Then the geometry is restricted to a base involving fit regions of a three-dimensional Euclidean space, and the balances of actions on defects in the abstract fiber which can be pictured on the basis of $Y$ by a smooth space-type line $l$ are derived in a non-purely conservative setting (a d'Alembert-Lagrange principle is involved). The main result is related to the case involving non-constant peculiar energy along $l$: the covariance of the action balances along $l$, including the configurational ones, is related to the validity of an integral Nöther-like relation which has the meaning of action power relative to the defect motion.
Citation: Paolo Maria Mariano. Line defect evolution in finite-dimensional manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 575-596. doi: 10.3934/dcdsb.2012.17.575
##### References:
 [1] G. Capriz, Continua with latent microstructure, Arch. Rational Mech. Anal., 90 (1985), 43-56. doi: 10.1007/BF00281586. [2] G. Capriz and G. Mazzini, Interactions between subbodies for complex materials, Proceedings of the Third Meeting on Current Ideas in Mechanics and Related Fields (Segovia, 1995), Extracta Math., 11 (1996), 17-21. [3] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fracture: Existence and approximation results, Arch. Rational Mech. Anal., 162 (2002), 101-135. doi: 10.1007/s002050100187. [4] E. De Giorgi, New problems on minimizing movements, in "Boundary Value Problems for Partial Differential Equations and Applications'' (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, (1993), 81-98. [5] J. E. Dunn and J. Serrin, On the thermomechanics of intertistitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907. [6] J. D. Eshelby, The elastic energy-momentum tensor, J. Elasticity, 5 (1975), 321-335. doi: 10.1007/BF00126994. [7] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimizing problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9. [8] G. A. Francfort and A. Mielke, Existence results for a class of rate-independent material models with non-convex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91. doi: 10.1515/CRELLE.2006.044. [9] M. Giaquinta and S. Hildebrandt, "Calculus of Variations,'' vol. I, The Lagrangian formalism, vol. II, The Hamiltonian formalism, Springer-Verlag, Berlin, 1996. [10] M. Giaquinta, P. M. Mariano and G. Modica, A variational problem in the mechanics of complex materials, Discr. Cont. Dyn. Systems, 28 (2010), 519-537. doi: 10.3934/dcds.2010.28.519. [11] M. Giaquinta, P. M. Mariano, G. Modica and D. Mucci, Ground states of simple bodies that may undergo brittle fracture, Physica D, 239 (2010), 1485-1502. doi: 10.1016/j.physd.2010.04.006. [12] M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations,'' Vol. I and II, Springer-Verlag, Berlin, 1998. [13] S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory, J. Geom. Phys., 35 (2000), 333-366. doi: 10.1016/S0393-0440(00)00012-7. [14] P. M. Mariano, Walk of a line defect in quasicrystals, Meccanica, 40 (2005), 511-525. doi: 10.1007/s11012-005-2137-7. [15] P. M. Mariano, Mechanics of quasi-periodic alloys, J. Nonlinear Sci., 16 (2006), 45-77. doi: 10.1007/s00332-005-0654-5. [16] P. M. Mariano, Geometry and balance of hyperstresses, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rendiconti Lincei (9) Mat. Appl., 18 (2007), 311-331. [17] P. M. Mariano, Physical significance of the curvature varifold-based description of crack nucleation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 21 (2010), 215-233. [18] P. M. Mariano, Crystal plasticity: The Hamilton-Eshelby stress in terms of the metric in the intermediate configuration, Theor. Appl. Mech., in print, 2011. [19] J. E. Marsden and T. R. J. Hughes, "Mathematical Foundations of Elasticity,'' Corrected reprint of the 1983 original, Dover Publications, Inc., New York, 1994. [20] N. K. Simha and K. Bhattacharya, Kinetics of phase boundaries with edges and junctions in a tree-dimensional multi-phase body, J. Mech. Phys. Solids, 48 (2000), 2619-2641. doi: 10.1016/S0022-5096(00)00008-9. [21] R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal., 17 (1964), 85-112.

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##### References:
 [1] G. Capriz, Continua with latent microstructure, Arch. Rational Mech. Anal., 90 (1985), 43-56. doi: 10.1007/BF00281586. [2] G. Capriz and G. Mazzini, Interactions between subbodies for complex materials, Proceedings of the Third Meeting on Current Ideas in Mechanics and Related Fields (Segovia, 1995), Extracta Math., 11 (1996), 17-21. [3] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fracture: Existence and approximation results, Arch. Rational Mech. Anal., 162 (2002), 101-135. doi: 10.1007/s002050100187. [4] E. De Giorgi, New problems on minimizing movements, in "Boundary Value Problems for Partial Differential Equations and Applications'' (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, (1993), 81-98. [5] J. E. Dunn and J. Serrin, On the thermomechanics of intertistitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907. [6] J. D. Eshelby, The elastic energy-momentum tensor, J. Elasticity, 5 (1975), 321-335. doi: 10.1007/BF00126994. [7] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimizing problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9. [8] G. A. Francfort and A. Mielke, Existence results for a class of rate-independent material models with non-convex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91. doi: 10.1515/CRELLE.2006.044. [9] M. Giaquinta and S. Hildebrandt, "Calculus of Variations,'' vol. I, The Lagrangian formalism, vol. II, The Hamiltonian formalism, Springer-Verlag, Berlin, 1996. [10] M. Giaquinta, P. M. Mariano and G. Modica, A variational problem in the mechanics of complex materials, Discr. Cont. Dyn. Systems, 28 (2010), 519-537. doi: 10.3934/dcds.2010.28.519. [11] M. Giaquinta, P. M. Mariano, G. Modica and D. Mucci, Ground states of simple bodies that may undergo brittle fracture, Physica D, 239 (2010), 1485-1502. doi: 10.1016/j.physd.2010.04.006. [12] M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations,'' Vol. I and II, Springer-Verlag, Berlin, 1998. [13] S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory, J. Geom. Phys., 35 (2000), 333-366. doi: 10.1016/S0393-0440(00)00012-7. [14] P. M. Mariano, Walk of a line defect in quasicrystals, Meccanica, 40 (2005), 511-525. doi: 10.1007/s11012-005-2137-7. [15] P. M. Mariano, Mechanics of quasi-periodic alloys, J. Nonlinear Sci., 16 (2006), 45-77. doi: 10.1007/s00332-005-0654-5. [16] P. M. Mariano, Geometry and balance of hyperstresses, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rendiconti Lincei (9) Mat. Appl., 18 (2007), 311-331. [17] P. M. Mariano, Physical significance of the curvature varifold-based description of crack nucleation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 21 (2010), 215-233. [18] P. M. Mariano, Crystal plasticity: The Hamilton-Eshelby stress in terms of the metric in the intermediate configuration, Theor. Appl. Mech., in print, 2011. [19] J. E. Marsden and T. R. J. Hughes, "Mathematical Foundations of Elasticity,'' Corrected reprint of the 1983 original, Dover Publications, Inc., New York, 1994. [20] N. K. Simha and K. Bhattacharya, Kinetics of phase boundaries with edges and junctions in a tree-dimensional multi-phase body, J. Mech. Phys. Solids, 48 (2000), 2619-2641. doi: 10.1016/S0022-5096(00)00008-9. [21] R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal., 17 (1964), 85-112.
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