American Institute of Mathematical Sciences

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June  2021, 26(6): 3177-3207. doi: 10.3934/dcdsb.2020224

Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space

Yuan Gao 1,2, , Jian-Guo Liu 3, , Tao Luo 4, and  Yang Xiang 1,

1. 

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
2. 

Department of Mathematics, Duke University, Durham NC 27708, USA
3. 

Department of Mathematics and Department of Physics, Duke University, Durham NC 27708, USA
4. 

Department of Mathematics, Purdue University, West Lafayette IN 47907, USA

Received  May 2020 Published  June 2021 Early access  July 2020

In this paper, we revisit the mathematical validation of the Peierls–Nabarro (PN) models, which are multiscale models of dislocations that incorporate the detailed dislocation core structure. We focus on the static and dynamic PN models of an edge dislocation in Hilbert space. In a PN model, the total energy includes the elastic energy in the two half-space continua and a nonlinear potential energy, which is always infinite, across the slip plane. We revisit the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy variations. The shear displacement jump is determined only by the reduced problem on the slip plane while the displacement fields in the two half spaces are determined by linear elasticity. We establish the existence and sharp regularities of classical solutions in Hilbert space. For both the reduced problem and the full PN model, we prove that a static solution is a global minimizer in a perturbed sense. We also show that there is a unique classical, global in time solution of the dynamic PN model.

Keywords: Double well potential, Ginzburg-Landau equation, nonlocal semilinear equation, elastic extension, optimal regularity.
Mathematics Subject Classification: 35R11, 35Q74, 35S15, 35J50.
Citation: Yuan Gao, Jian-Guo Liu, Tao Luo, Yang Xiang. Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3177-3207. doi: 10.3934/dcdsb.2020224
References:
[1]

G. Alberti, G. Bouchitté and P. Seppecher, Un résultat de perturbations singulieres avec la norm $H^{1/2}$, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 333-338. 

[2]

O. Alvarez, P. Hoch, Y. Le Bouar and and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal., 181 (2006), 449-504.  doi: 10.1007/s00205-006-0418-5.

[3]

T. Blass, I. Fonseca, G. Leoni and M. Morandotti, Dynamics for systems of screw dislocations, SIAM J. Appl. Math., 75 (2015), 393-419.  doi: 10.1137/140980065.

[4]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Am. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.

[5]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.  doi: 10.1002/cpa.20093.

[6]

S. Cacace, A. Chambolle and R. Monneau, A posteriori error estimates for the effective Hamiltonian of dislocation dynamics, Numer. Math., 121 (2012), 281-335.  doi: 10.1007/s00211-011-0430-z.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[8]

P. Cermelli and G. Leoni, Renormalized energy and forces on dislocations, SIAM J. Math. Anal., 37 (2005), 1131-1160.  doi: 10.1137/040621636.

[9]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Equ., 2 (1997), 125-160. 

[10]

S. Conti, A. Garroni and S. Müller, Singular kernels, multiscale decomposition of microstructure, and dislocation models, Arch. Ration. Mech. Anal., 199 (2011), 779-819.  doi: 10.1007/s00205-010-0333-7.

[11]

S. Dai, Y. Xiang and D. J. Srolovitz, Structure and energy of (111) low-angle twist boundaries in Al, Cu and Ni, Acta Mater., 61 (2013), 1327-1337.  doi: 10.1016/j.actamat.2012.11.010.

[12]

S. Dipierro, A. Figalli and E. Valdinoci, Strongly nonlocal dislocation dynamics in crystals, Comm. Partial Differential Equations, 39 (2014), 2351-2387.  doi: 10.1080/03605302.2014.914536.

[13]

S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting, Comm. Math. Phys., 333 (2015), 1061-1105.  doi: 10.1007/s00220-014-2118-6.

[14]

S. Dipierro, S. Patrizi and E. Valdinoci, Heteroclinic connections for nonlocal equations, Math. Models Methods Appl. Sci., 29 (2019), 2585–2636. arXiv: 1711.01491. doi: 10.1142/S0218202519500556.

[15]

A. Z. Fino, H. Ibrahim and R. Monneau, The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model, J. Differ. Equations, 252 (2012), 258-293.  doi: 10.1016/j.jde.2011.08.007.

[16]

J. Frenkel, Theory of the elastic limits and rigidity of crystalline bodies, Z. Phys., 37 (1926), 572-609. 

[17]

I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films, J. Math. Pures Appl., 111 (2018), 126-160.  doi: 10.1016/j.matpur.2017.09.001.

[18]

I. Fonseca, G. Leoni and M. Morini, Equilibria and dislocations in epitaxial growth, Nonlinear Anal., 154 (2017), 88-121.  doi: 10.1016/j.na.2016.10.013.

[19]

I. Fonseca, G. Leoni and X. Y. Lu, Regularity in time for weak solutions of a continuum model for epitaxial growth with elasticity on vicinal surfaces, Commun. Part. Diff. Eq., 40 (2015), 1942-1957.  doi: 10.1080/03605302.2015.1045074.

[20]

A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc., 12 (2010), 1231-1266.  doi: 10.4171/JEMS/228.

[21]

A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964.  doi: 10.1137/S003614100343768X.

[22]

M. del M. González and R. Monneau, Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one, Discrete Contin. Dyn. Syst., 32 (2012), 1255-1286.  doi: 10.3934/dcds.2012.32.1255.

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[24]

J. P. Hirth and J. Lothe, Theory of Dislocations, John Wiley, New York, 2nd edition, 1982.

[25]

E. Kaxiras and M. S. Duesbery, Free energies of generalized stacking faults in Si and implications for the brittle-ductile transition, Phys. Rev. Lett., 70 (1993), 3752-3755.  doi: 10.1103/PhysRevLett.70.3752.

[26]

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.

[27]

X. Y. Lu, On the solutions of a $2+1$-dimensional model for epitaxial growth with axial symmetry, J. Nonlinear Sci., 28 (2018), 807-831.  doi: 10.1007/s00332-017-9428-8.

[28]

G. Lu, N. Kioussis, V. V. Bulatov and E. Kaxiras, Generalized-stacking-fault energy surface and dislocation properties of aluminum, Phys. Rev. B, 62 (2000), 3099-3108.  doi: 10.1103/PhysRevB.62.3099.

[29]

T. Luo, P. Ming and Y. Xiang, From atomistic model to the Peierls-Nabarro model with Gamma-surface for dislocations, Arch. Ration. Mech. Anal., 230 (2018), 735-781.  doi: 10.1007/s00205-018-1257-x.

[30]

F. R. N. Nabarro, Dislocations in a simple cubic lattice, Proc. Phys. Soc., 59 (1947), 256-272.  doi: 10.1088/0959-5309/59/2/309.

[31]

G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718.  doi: 10.1007/s10231-011-0243-9.

[32]

S. Patrizi and E. Valdinoci, Crystal dislocations with different orientations and collisions, Arch. Rational Mech. Anal., 217 (2015), 231-261.  doi: 10.1007/s00205-014-0832-z.

[33]

S. Patrizi and E. Valdinoci, Relaxation times for atom dislocations in crystals, Calc. Var. Partial Differ. Equ., 55 (2016), 44 pp. doi: 10.1007/s00526-016-1000-0.

[34]

R. Peierls, The size of a dislocation, Selected Scientific Papers of Sir Rudolf Peierls, (1997), 273–276. doi: 10.1142/9789812795779_0032.

[35]

G. Schoeck, The generalized Peierls-Nabarro model, Phil. Mag. A, 69 (1994), 1085-1095.  doi: 10.1080/01418619408242240.

[36]

C. Shen, J. Li and Y. Wang, Predicting structure and energy of dislocations and grain boundaries, Acta Mater., 74 (2014), 125-131.  doi: 10.1016/j.actamat.2014.03.065.

[37]

C. Shen and Y. Wang, Incorporation of $\gamma$-surface to phase field model of dislocations: Simulating dislocation dissociation in fcc crystals, Acta Mater., 52 (2004), 683-691. 

[38] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton university press, 1970. 
[39]

V. Vitek, Intrinsic stacking faults in body-centred cubic crystals, Philos. Mag., 18 (1968), 773-786.  doi: 10.1080/14786436808227500.

[40]

V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. École Norm. Sup., 24 (1907), 401-517.  doi: 10.24033/asens.583.

[41]

Y. Xiang, L. T. Cheng, D. J. Srolovitz and W. E, A level set method for dislocation dynamics, Acta Mater., 51 (2003), 5499-5518. 

[42]

Y. Xiang, Modeling dislocations at different scales, Commun. Comput. Phys., 1 (2006), 383-424. 

[43]

Y. Xiang, H. Wei, P. Ming and W. E, A generalized Peierls–Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu, Acta Mater., 56 (2008), 1447-1460.  doi: 10.1016/j.actamat.2007.11.033.

[44] A. Zangwill, Physics at Surfaces, Cambridge University Press, New York, 1988.  doi: 10.1017/CBO9780511622564.
[45]

S. Zhou, J. Han, S. Dai, J. Sun and D. J. Srolovitz, van der Waals bilayer energetics: Generalized stacking-fault energy of graphene, boron nitride, and graphene/boron nitride bilayers, Phys. Rev. B, $\texttt92$ (2015), 155438. doi: 10.1103/PhysRevB.92.155438.

show all references

References:
[1]

G. Alberti, G. Bouchitté and P. Seppecher, Un résultat de perturbations singulieres avec la norm $H^{1/2}$, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 333-338. 

[2]

O. Alvarez, P. Hoch, Y. Le Bouar and and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal., 181 (2006), 449-504.  doi: 10.1007/s00205-006-0418-5.

[3]

T. Blass, I. Fonseca, G. Leoni and M. Morandotti, Dynamics for systems of screw dislocations, SIAM J. Appl. Math., 75 (2015), 393-419.  doi: 10.1137/140980065.

[4]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Am. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.

[5]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.  doi: 10.1002/cpa.20093.

[6]

S. Cacace, A. Chambolle and R. Monneau, A posteriori error estimates for the effective Hamiltonian of dislocation dynamics, Numer. Math., 121 (2012), 281-335.  doi: 10.1007/s00211-011-0430-z.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[8]

P. Cermelli and G. Leoni, Renormalized energy and forces on dislocations, SIAM J. Math. Anal., 37 (2005), 1131-1160.  doi: 10.1137/040621636.

[9]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Equ., 2 (1997), 125-160. 

[10]

S. Conti, A. Garroni and S. Müller, Singular kernels, multiscale decomposition of microstructure, and dislocation models, Arch. Ration. Mech. Anal., 199 (2011), 779-819.  doi: 10.1007/s00205-010-0333-7.

[11]

S. Dai, Y. Xiang and D. J. Srolovitz, Structure and energy of (111) low-angle twist boundaries in Al, Cu and Ni, Acta Mater., 61 (2013), 1327-1337.  doi: 10.1016/j.actamat.2012.11.010.

[12]

S. Dipierro, A. Figalli and E. Valdinoci, Strongly nonlocal dislocation dynamics in crystals, Comm. Partial Differential Equations, 39 (2014), 2351-2387.  doi: 10.1080/03605302.2014.914536.

[13]

S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting, Comm. Math. Phys., 333 (2015), 1061-1105.  doi: 10.1007/s00220-014-2118-6.

[14]

S. Dipierro, S. Patrizi and E. Valdinoci, Heteroclinic connections for nonlocal equations, Math. Models Methods Appl. Sci., 29 (2019), 2585–2636. arXiv: 1711.01491. doi: 10.1142/S0218202519500556.

[15]

A. Z. Fino, H. Ibrahim and R. Monneau, The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model, J. Differ. Equations, 252 (2012), 258-293.  doi: 10.1016/j.jde.2011.08.007.

[16]

J. Frenkel, Theory of the elastic limits and rigidity of crystalline bodies, Z. Phys., 37 (1926), 572-609. 

[17]

I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films, J. Math. Pures Appl., 111 (2018), 126-160.  doi: 10.1016/j.matpur.2017.09.001.

[18]

I. Fonseca, G. Leoni and M. Morini, Equilibria and dislocations in epitaxial growth, Nonlinear Anal., 154 (2017), 88-121.  doi: 10.1016/j.na.2016.10.013.

[19]

I. Fonseca, G. Leoni and X. Y. Lu, Regularity in time for weak solutions of a continuum model for epitaxial growth with elasticity on vicinal surfaces, Commun. Part. Diff. Eq., 40 (2015), 1942-1957.  doi: 10.1080/03605302.2015.1045074.

[20]

A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc., 12 (2010), 1231-1266.  doi: 10.4171/JEMS/228.

[21]

A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964.  doi: 10.1137/S003614100343768X.

[22]

M. del M. González and R. Monneau, Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one, Discrete Contin. Dyn. Syst., 32 (2012), 1255-1286.  doi: 10.3934/dcds.2012.32.1255.

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[24]

J. P. Hirth and J. Lothe, Theory of Dislocations, John Wiley, New York, 2nd edition, 1982.

[25]

E. Kaxiras and M. S. Duesbery, Free energies of generalized stacking faults in Si and implications for the brittle-ductile transition, Phys. Rev. Lett., 70 (1993), 3752-3755.  doi: 10.1103/PhysRevLett.70.3752.

[26]

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.

[27]

X. Y. Lu, On the solutions of a $2+1$-dimensional model for epitaxial growth with axial symmetry, J. Nonlinear Sci., 28 (2018), 807-831.  doi: 10.1007/s00332-017-9428-8.

[28]

G. Lu, N. Kioussis, V. V. Bulatov and E. Kaxiras, Generalized-stacking-fault energy surface and dislocation properties of aluminum, Phys. Rev. B, 62 (2000), 3099-3108.  doi: 10.1103/PhysRevB.62.3099.

[29]

T. Luo, P. Ming and Y. Xiang, From atomistic model to the Peierls-Nabarro model with Gamma-surface for dislocations, Arch. Ration. Mech. Anal., 230 (2018), 735-781.  doi: 10.1007/s00205-018-1257-x.

[30]

F. R. N. Nabarro, Dislocations in a simple cubic lattice, Proc. Phys. Soc., 59 (1947), 256-272.  doi: 10.1088/0959-5309/59/2/309.

[31]

G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718.  doi: 10.1007/s10231-011-0243-9.

[32]

S. Patrizi and E. Valdinoci, Crystal dislocations with different orientations and collisions, Arch. Rational Mech. Anal., 217 (2015), 231-261.  doi: 10.1007/s00205-014-0832-z.

[33]

S. Patrizi and E. Valdinoci, Relaxation times for atom dislocations in crystals, Calc. Var. Partial Differ. Equ., 55 (2016), 44 pp. doi: 10.1007/s00526-016-1000-0.

[34]

R. Peierls, The size of a dislocation, Selected Scientific Papers of Sir Rudolf Peierls, (1997), 273–276. doi: 10.1142/9789812795779_0032.

[35]

G. Schoeck, The generalized Peierls-Nabarro model, Phil. Mag. A, 69 (1994), 1085-1095.  doi: 10.1080/01418619408242240.

[36]

C. Shen, J. Li and Y. Wang, Predicting structure and energy of dislocations and grain boundaries, Acta Mater., 74 (2014), 125-131.  doi: 10.1016/j.actamat.2014.03.065.

[37]

C. Shen and Y. Wang, Incorporation of $\gamma$-surface to phase field model of dislocations: Simulating dislocation dissociation in fcc crystals, Acta Mater., 52 (2004), 683-691. 

[38] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton university press, 1970. 
[39]

V. Vitek, Intrinsic stacking faults in body-centred cubic crystals, Philos. Mag., 18 (1968), 773-786.  doi: 10.1080/14786436808227500.

[40]

V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. École Norm. Sup., 24 (1907), 401-517.  doi: 10.24033/asens.583.

[41]

Y. Xiang, L. T. Cheng, D. J. Srolovitz and W. E, A level set method for dislocation dynamics, Acta Mater., 51 (2003), 5499-5518. 

[42]

Y. Xiang, Modeling dislocations at different scales, Commun. Comput. Phys., 1 (2006), 383-424. 

[43]

Y. Xiang, H. Wei, P. Ming and W. E, A generalized Peierls–Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu, Acta Mater., 56 (2008), 1447-1460.  doi: 10.1016/j.actamat.2007.11.033.

[44] A. Zangwill, Physics at Surfaces, Cambridge University Press, New York, 1988.  doi: 10.1017/CBO9780511622564.
[45]

S. Zhou, J. Han, S. Dai, J. Sun and D. J. Srolovitz, van der Waals bilayer energetics: Generalized stacking-fault energy of graphene, boron nitride, and graphene/boron nitride bilayers, Phys. Rev. B, $\texttt92$ (2015), 155438. doi: 10.1103/PhysRevB.92.155438.

Figure 1.  Schematic illustration of the PN model for an edge dislocation. The dislocation locates along the $ z $ axis with $ +z $ direction, and its slip plane is the $ y = 0 $ plane. $ \mathbf b $ is the Burgers vector and $ d $ is the interplanar distance in the direction normal to the slip plane. The black dots and red circles show the locations of atoms of the two atomic planes $ y = 0^+ $ and $ y = 0^- $ in the lattice with the dislocation and in the reference states before elastic deformation, respectively, based on a simple cubic lattice. The Burgers vector enclosed by a loop $ L $ enclosing the dislocation is $ \mathbf b_L = \oint_L \,\mathrm{d} \mathbf u $
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