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
References:
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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, 11 (1996), 17.   Google Scholar

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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.  doi: 10.1007/s002050100187.  Google Scholar

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

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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.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

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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.  doi: 10.1515/CRELLE.2006.044.  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.  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.  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, (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.  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.  doi: 10.1007/s11012-005-2137-7.  Google Scholar

[15]

P. M. Mariano, Mechanics of quasi-periodic alloys,, J. Nonlinear Sci., 16 (2006), 45.  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.   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.   Google Scholar

[18]

P. M. Mariano, Crystal plasticity: The Hamilton-Eshelby stress in terms of the metric in the intermediate configuration,, Theor. Appl. Mech., (2011).   Google Scholar

[19]

J. E. Marsden and T. R. J. Hughes, "Mathematical Foundations of Elasticity,'', Corrected reprint of the 1983 original, (1983).   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.  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.   Google Scholar

show all references

References:
[1]

G. Capriz, Continua with latent microstructure,, Arch. Rational Mech. Anal., 90 (1985), 43.  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, 11 (1996), 17.   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.  doi: 10.1007/s002050100187.  Google Scholar

[4]

E. De Giorgi, New problems on minimizing movements,, in, 29 (1993), 81.   Google Scholar

[5]

J. E. Dunn and J. Serrin, On the thermomechanics of intertistitial working,, Arch. Rational Mech. Anal., 88 (1985), 95.  doi: 10.1007/BF00250907.  Google Scholar

[6]

J. D. Eshelby, The elastic energy-momentum tensor,, J. Elasticity, 5 (1975), 321.  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.  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.  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, (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.  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.  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, (1998).   Google Scholar

[13]

S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory,, J. Geom. Phys., 35 (2000), 333.  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.  doi: 10.1007/s11012-005-2137-7.  Google Scholar

[15]

P. M. Mariano, Mechanics of quasi-periodic alloys,, J. Nonlinear Sci., 16 (2006), 45.  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.   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.   Google Scholar

[18]

P. M. Mariano, Crystal plasticity: The Hamilton-Eshelby stress in terms of the metric in the intermediate configuration,, Theor. Appl. Mech., (2011).   Google Scholar

[19]

J. E. Marsden and T. R. J. Hughes, "Mathematical Foundations of Elasticity,'', Corrected reprint of the 1983 original, (1983).   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.  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.   Google Scholar

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