January  2011, 15(1): 15-43. doi: 10.3934/dcdsb.2011.15.15

A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity

1. 

Istituto per le Applicazioni del Calcolo "M. Picone”, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy

2. 

Dipartimento di Matematica, Università di Roma "Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy

3. 

Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Università di Roma "La Sapienza”, Via Scarpa 16, 00161 Roma, Italy

4. 

Dipartimento di Ingegneria Civile, Università di Roma "Tor Vergata”, Via Politecnico 1, 00133 Roma, Italy

Received  June 2009 Revised  August 2010 Published  October 2010

We consider a system of partial differential equations which describes anti-plane shear in the context of a strain-gradient theory of plasticity proposed by Gurtin in [6]. The problem couples a fully nonlinear degenerate parabolic system and an elliptic equation. It features two types of degeneracies: the first one is caused by the nonlinear structure, the second one by the dependence of the principal part on twice the curl of a planar vector field. Furthermore, the elliptic equation depends on the divergence of such vector field - which is not controlled by twice the curl - and the boundary conditions suggested in [6] are of mixed type.
   To overcome the latter complications we use a suitable, time-dependent representation of a divergence-free vector field which plays the role of the elastic stress. To handle the nonlinearities, by a suitable reformulation of the problem we transform the original system into one satisfying a monotonicity property which is more "robust" than the gradient flow structure inherited as an intrinsic feature of the mechanical model. These two insights make it possible to prove existence and uniqueness of a solution to the original system.
Citation: Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Giuseppe Tomassetti. A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 15-43. doi: 10.3934/dcdsb.2011.15.15
References:
[1]

G. Auchmuty and J. C. Alexander, $L^2$ well-posedness of planar div-curl systems,, Arch. Ration. Mech. Anal., 160 (2001), 91.  doi: doi:10.1007/s002050100156.  Google Scholar

[2]

P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations,, Comm. Partial Differential Equations, 15 (1990), 737.  doi: doi:10.1080/03605309908820706.  Google Scholar

[3]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (1993).   Google Scholar

[4]

A. Giacomini and L. Lussardi, Quasi-static evolution for a model in strain gradient plasticity,, SIAM J. Math. Anal., 40 (2008), 1201.  doi: doi:10.1137/070708202.  Google Scholar

[5]

G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation,, J. Math. Anal. Appl., 333 (2007), 839.  doi: doi:10.1016/j.jmaa.2006.11.050.  Google Scholar

[6]

M. E. Gurtin, A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin,, J. Mech. Phys. Solids, 52 (2004), 2545.  doi: doi:10.1016/j.jmps.2004.04.010.  Google Scholar

[7]

M. E. Gurtin and L. Anand, A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. I. Small deformations,, J. Mech. Phys. Solids, 53 (2005), 1624.  doi: doi:10.1016/j.jmps.2004.12.008.  Google Scholar

[8]

S. Kesavan, On Poincaré's and J. L. Lions' lemmas,, C. R. Math. Acad. Sci. Paris, 340 (2005), 27.   Google Scholar

[9]

J.-L. Lions, "Quelques Méthodes de Résolution des problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[10]

B. Daya Reddy, F. Ebobisse and A. McBride, Well-posedness of a model of strain gradient plasticity for plastically irrotational materials,, International Journal of Plasticity, 24 (2008), 55.  doi: doi:10.1016/j.ijplas.2007.01.013.  Google Scholar

[11]

G. Savaré, Regularity results for elliptic equations in $\textrmL$ipschitz domains,, J. Funct. Anal., 152 (1998), 176.  doi: doi:10.1006/jfan.1997.3158.  Google Scholar

[12]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/A. Linear Monotone Operators,", Springer-Verlag, (1990).   Google Scholar

show all references

References:
[1]

G. Auchmuty and J. C. Alexander, $L^2$ well-posedness of planar div-curl systems,, Arch. Ration. Mech. Anal., 160 (2001), 91.  doi: doi:10.1007/s002050100156.  Google Scholar

[2]

P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations,, Comm. Partial Differential Equations, 15 (1990), 737.  doi: doi:10.1080/03605309908820706.  Google Scholar

[3]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (1993).   Google Scholar

[4]

A. Giacomini and L. Lussardi, Quasi-static evolution for a model in strain gradient plasticity,, SIAM J. Math. Anal., 40 (2008), 1201.  doi: doi:10.1137/070708202.  Google Scholar

[5]

G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation,, J. Math. Anal. Appl., 333 (2007), 839.  doi: doi:10.1016/j.jmaa.2006.11.050.  Google Scholar

[6]

M. E. Gurtin, A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin,, J. Mech. Phys. Solids, 52 (2004), 2545.  doi: doi:10.1016/j.jmps.2004.04.010.  Google Scholar

[7]

M. E. Gurtin and L. Anand, A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. I. Small deformations,, J. Mech. Phys. Solids, 53 (2005), 1624.  doi: doi:10.1016/j.jmps.2004.12.008.  Google Scholar

[8]

S. Kesavan, On Poincaré's and J. L. Lions' lemmas,, C. R. Math. Acad. Sci. Paris, 340 (2005), 27.   Google Scholar

[9]

J.-L. Lions, "Quelques Méthodes de Résolution des problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[10]

B. Daya Reddy, F. Ebobisse and A. McBride, Well-posedness of a model of strain gradient plasticity for plastically irrotational materials,, International Journal of Plasticity, 24 (2008), 55.  doi: doi:10.1016/j.ijplas.2007.01.013.  Google Scholar

[11]

G. Savaré, Regularity results for elliptic equations in $\textrmL$ipschitz domains,, J. Funct. Anal., 152 (1998), 176.  doi: doi:10.1006/jfan.1997.3158.  Google Scholar

[12]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/A. Linear Monotone Operators,", Springer-Verlag, (1990).   Google Scholar

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