×

Threshold phenomenon for homogenized fronts in random elastic media. (English) Zbl 1458.35038

Summary: We consider a model for the motion of a phase interface in an elastic medium, for example, a twin boundary in martensite. The model is given by a semilinear parabolic equation with a fractional Laplacian as regularizing operator, stemming from the interaction of the front with its elastic environment. We show that the presence of randomly distributed, localized obstacles leads to a threshold phenomenon, i.e., stationary solutions exist up to a positive, critical driving force leading to a stick-slip behaviour of the phase boundary. The main result is proved by an explicit construction of a stationary viscosity supersolution to the evolution equation and is based on a percolation result for the obstacle sites. Furthermore, we derive a homogenization result for such fronts in the case of the half-Laplacian in the pinning regime.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35R11 Fractional partial differential equations
35K15 Initial value problems for second-order parabolic equations
35D40 Viscosity solutions to PDEs
74A40 Random materials and composite materials
74N20 Dynamics of phase boundaries in solids
35K58 Semilinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pure Appl. Anal., 15, 657-699 (2016) · Zbl 1334.35383
[2] L. Caffarelli; L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62, 597-638 (2009) · Zbl 1170.45006
[3] L. Courte, K. Bhattacharya and P. Dondl, Bounds on precipitate hardening of line and surface defects in solids, (2019), arXiv: 1903.07505. · Zbl 1440.35323
[4] E. Di Nezza; G. Palatucci; E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573 (2012) · Zbl 1252.46023
[5] N. Dirr; P. W. Dondl; M. Scheutzow, Pinning of interfaces in random media, Interfaces Free Bound., 13, 411-421 (2011) · Zbl 1231.35323
[6] P. W. Dondl; K. Bhattacharya, Effective behavior of an interface propagating through a periodic elastic medium, Interfaces Free Bound., 18, 91-113 (2016) · Zbl 1342.35027
[7] P. W. Dondl; M. Scheutzow; S. Throm, Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation, Proc. Roy. Soc. Edinburgh Sect. A, 145, 481-512 (2015) · Zbl 1326.35387
[8] J. Droniou; C. Imbert, Fractal first-order partial differential equations, Archive For Rational Mechanics And Analysis, 182, 299-331 (2006) · Zbl 1111.35144
[9] R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101, 75-90 (1961) · Zbl 0104.11203
[10] M. Koslowski; A. M. Cuitiño; M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, J. Mech. Phys. Solids, 50, 2597-2635 (2002) · Zbl 1094.74563
[11] R. Monneau; S. Patrizi, Derivation of Orowan’s law from the Peierls-Nabarro model, Comm. Partial Differential Equations, 37, 1887-1911 (2012) · Zbl 1255.35215
[12] S. Moulinet; C. Guthmann; E. Rolley, Roughness and dynamics of a contact line of a viscous fluid on a disordered substrate, The European Physical Journal E, 8, 437-443 (2002)
[13] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60, 3-26 (2016) · Zbl 1337.47112
[14] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60, 3-26 (2016) · Zbl 1285.35020
[15] X. Ros-Oton; J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101, 275-302 (2014)
[16] J. Schmittbuhl; A. Delaplace; K. J. Mäløy; H. Perfettini; J. P. Vilotte, Slow crack propagation and slip correlations, Pure And Applied Geophysics, 160, 961-976 (2003)
[17] J. Schmittbuhl; S. Roux; J.-P. Vilotte; K. Maloy, Interfacial crack pinning: Effect of nonlocal interactions, Physical Review Letters, 74, 1787-1790 (1995) · Zbl 1304.35752
[18] R. Servadei; E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144, 831-855 (2014)
[19] S. Throm, Pinning of Interfaces in a Random Elastic Medium, Master’s Thesis, Ruprecht Karls Universität Heidelberg, 2012.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.