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 & Continuous Dynamical Systems - B, 2012, 17 (2) : 575-596. doi: 10.3934/dcdsb.2012.17.575
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations,'' Vol. I and II, Springer-Verlag, Berlin, 1998. Google Scholar

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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.  Google Scholar

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P. M. Mariano, Walk of a line defect in quasicrystals, Meccanica, 40 (2005), 511-525. doi: 10.1007/s11012-005-2137-7.  Google Scholar

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P. M. Mariano, Mechanics of quasi-periodic alloys, J. Nonlinear Sci., 16 (2006), 45-77. doi: 10.1007/s00332-005-0654-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal., 17 (1964), 85-112.  Google Scholar

show all references

References:
[1]

G. Capriz, Continua with latent microstructure, Arch. Rational Mech. Anal., 90 (1985), 43-56. doi: 10.1007/BF00281586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. D. Eshelby, The elastic energy-momentum tensor, J. Elasticity, 5 (1975), 321-335. doi: 10.1007/BF00126994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Giaquinta and S. Hildebrandt, "Calculus of Variations,'' vol. I, The Lagrangian formalism, vol. II, The Hamiltonian formalism, Springer-Verlag, Berlin, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations,'' Vol. I and II, Springer-Verlag, Berlin, 1998. Google Scholar

[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.  Google Scholar

[14]

P. M. Mariano, Walk of a line defect in quasicrystals, Meccanica, 40 (2005), 511-525. doi: 10.1007/s11012-005-2137-7.  Google Scholar

[15]

P. M. Mariano, Mechanics of quasi-periodic alloys, J. Nonlinear Sci., 16 (2006), 45-77. doi: 10.1007/s00332-005-0654-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal., 17 (1964), 85-112.  Google Scholar

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