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January  2021, 14(1): 353-372. doi: 10.3934/dcdss.2020329

Threshold phenomenon for homogenized fronts in random elastic media

Patrick W. Dondl 1,, and  Martin Jesenko 1,

1. 

Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg, Germany

* Corresponding author: Patrick W. Dondl

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  May 2019 Revised  November 2019 Published  January 2021 Early access  April 2020

Fund Project: This work is supported by EPSRC grant EP/M028682/1 'Effective properties of interface evolution in a random environment.'

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.

Keywords: Random media, viscosity solutions, non-local operator, pinning, homogenization.
Mathematics Subject Classification: 35D40, 35R11, 74A40, 74N20.
Citation: Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
References:
[1]

C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pure Appl. Anal., 15 (2016), 657-699.  doi: 10.3934/cpaa.2016.15.657.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62 (2009), 597-638.  doi: 10.1002/cpa.20274.  Google Scholar

[3]

L. Courte, K. Bhattacharya and P. Dondl, Bounds on precipitate hardening of line and surface defects in solids, (2019), arXiv: 1903.07505. Google Scholar

[4]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[5]

N. Dirr, P. W. Dondl and M. Scheutzow, Pinning of interfaces in random media, Interfaces Free Bound., 13 (2011), 411-421.  doi: 10.4171/IFB/265.  Google Scholar

[6]

P. W. Dondl and K. Bhattacharya, Effective behavior of an interface propagating through a periodic elastic medium, Interfaces Free Bound., 18 (2016), 91-113.  doi: 10.4171/IFB/358.  Google Scholar

[7]

P. W. Dondl, M. Scheutzow and S. Throm, Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 481-512.  doi: 10.1017/S0308210512001291.  Google Scholar

[8]

J. Droniou and C. Imbert, Fractal first-order partial differential equations, Archive For Rational Mechanics And Analysis, 182 (2006), 299-331.  doi: 10.1007/s00205-006-0429-2.  Google Scholar

[9]

R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.  doi: 10.1090/S0002-9947-1961-0137148-5.  Google Scholar

[10]

M. Koslowski, A. M. Cuitiño and M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, J. Mech. Phys. Solids, 50 (2002), 2597-2635.  doi: 10.1016/S0022-5096(02)00037-6.  Google Scholar

[11]

R. Monneau and S. Patrizi, Derivation of Orowan's law from the Peierls-Nabarro model, Comm. Partial Differential Equations, 37 (2012), 1887-1911.  doi: 10.1080/03605302.2012.683504.  Google Scholar

[12]

S. Moulinet, C. Guthmann and E. Rolley, Roughness and dynamics of a contact line of a viscous fluid on a disordered substrate, The European Physical Journal E, 8 (2002), 437-443.  doi: 10.1140/epje/i2002-10032-2.  Google Scholar

[13] C. Pozrikidis, The Fractional Laplacian, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19666.  Google Scholar
[14]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.  Google Scholar

[15]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[16]

J. Schmittbuhl, A. Delaplace, K. J. Mäløy, H. Perfettini and J. P. Vilotte, Slow crack propagation and slip correlations, Pure And Applied Geophysics, 160 (2003), 961-976.  doi: 10.1007/978-3-0348-8083-1_10.  Google Scholar

[17]

J. Schmittbuhl, S. Roux, J.-P. Vilotte and K. Maloy, Interfacial crack pinning: Effect of nonlocal interactions, Physical Review Letters, 74 (1995), 1787-1790.  doi: 10.1103/PhysRevLett.74.1787.  Google Scholar

[18]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.  Google Scholar

[19]

S. Throm, Pinning of Interfaces in a Random Elastic Medium, Master's Thesis, Ruprecht Karls Universität Heidelberg, 2012. Google Scholar

show all references

References:
[1]

C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pure Appl. Anal., 15 (2016), 657-699.  doi: 10.3934/cpaa.2016.15.657.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62 (2009), 597-638.  doi: 10.1002/cpa.20274.  Google Scholar

[3]

L. Courte, K. Bhattacharya and P. Dondl, Bounds on precipitate hardening of line and surface defects in solids, (2019), arXiv: 1903.07505. Google Scholar

[4]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[5]

N. Dirr, P. W. Dondl and M. Scheutzow, Pinning of interfaces in random media, Interfaces Free Bound., 13 (2011), 411-421.  doi: 10.4171/IFB/265.  Google Scholar

[6]

P. W. Dondl and K. Bhattacharya, Effective behavior of an interface propagating through a periodic elastic medium, Interfaces Free Bound., 18 (2016), 91-113.  doi: 10.4171/IFB/358.  Google Scholar

[7]

P. W. Dondl, M. Scheutzow and S. Throm, Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 481-512.  doi: 10.1017/S0308210512001291.  Google Scholar

[8]

J. Droniou and C. Imbert, Fractal first-order partial differential equations, Archive For Rational Mechanics And Analysis, 182 (2006), 299-331.  doi: 10.1007/s00205-006-0429-2.  Google Scholar

[9]

R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.  doi: 10.1090/S0002-9947-1961-0137148-5.  Google Scholar

[10]

M. Koslowski, A. M. Cuitiño and M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, J. Mech. Phys. Solids, 50 (2002), 2597-2635.  doi: 10.1016/S0022-5096(02)00037-6.  Google Scholar

[11]

R. Monneau and S. Patrizi, Derivation of Orowan's law from the Peierls-Nabarro model, Comm. Partial Differential Equations, 37 (2012), 1887-1911.  doi: 10.1080/03605302.2012.683504.  Google Scholar

[12]

S. Moulinet, C. Guthmann and E. Rolley, Roughness and dynamics of a contact line of a viscous fluid on a disordered substrate, The European Physical Journal E, 8 (2002), 437-443.  doi: 10.1140/epje/i2002-10032-2.  Google Scholar

[13] C. Pozrikidis, The Fractional Laplacian, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19666.  Google Scholar
[14]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.  Google Scholar

[15]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[16]

J. Schmittbuhl, A. Delaplace, K. J. Mäløy, H. Perfettini and J. P. Vilotte, Slow crack propagation and slip correlations, Pure And Applied Geophysics, 160 (2003), 961-976.  doi: 10.1007/978-3-0348-8083-1_10.  Google Scholar

[17]

J. Schmittbuhl, S. Roux, J.-P. Vilotte and K. Maloy, Interfacial crack pinning: Effect of nonlocal interactions, Physical Review Letters, 74 (1995), 1787-1790.  doi: 10.1103/PhysRevLett.74.1787.  Google Scholar

[18]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.  Google Scholar

[19]

S. Throm, Pinning of Interfaces in a Random Elastic Medium, Master's Thesis, Ruprecht Karls Universität Heidelberg, 2012. Google Scholar

Figure 1.  Decomposition of the base space for $ n = 2 $
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Figure 2.  Cross-sections of the decomposition of $ {\mathbb R}^{n+1} $ above the horizontal hyperplane (left) and slightly perturbed one (right)
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