Elastic gradient theories and dislocation theories and their application to defects and microstructures

• Applicant: Dr. Markus Lazar
• Subject Area: Mechanics
• Term: from 2010 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 179058881

• Final Report Year: 2024

Final Report Abstract

As part of the Heisenberg fellowship and the associated DFG research grants, fundamental and globally recognized results were achieved in the areas of elastic gradient theories, non-singular dislocation theory, dislocation dynamics, elastodynamics and theory of quasicrystals. The obtained results were published in 30 articles in international scientific journals with peer-review. The main scientific advances are as follows. New results in elastic gradient theories and non-singular dislocation theory: • Non-singular solutions of dislocation loops within the framework of gradient elasticity theories of Helmholtz type and bi-Helmholtz type • Implementation of the non-singular dislocation solutions in the UCLA-DDD simulation program for discrete dislocation dynamics “MODEL” • Non-singular crack solutions based on non-singular dislocation solutions • Irreducible form of Mindlin’s elastic gradient theory using the irreducible decomposition of the distortion gradient tensor with respect to SO(3) and SO(2) • Comparison of gradient theory of magnetostatics and gradient elasticity theory of the Helmholtz type • Generalization of the Ru-Aifantis approach to defects in the framework of an incompatible gradient elasticity theory • Derivation of an anisotropic gradient elasticity theory of generalized Helmholtz type and calculation of the corresponding Green tensor for triclinic to cubic crystals. New results in the generalized elasticity theory of quasicrystals: • Generalized dynamics in quasicrystals: “Elastodynamic model of wave-telegraph type” • Calculation of the elastic Green tensor for quasicrystals • Dislocation theory in quasicrystals with all basic dislocation equations • Eshelbian mechanics and the J-, M- and L-integrals of quasicrystals. New results in dislocation dynamics and elastodynamics: • Introduction of the elastodynamic Liénard–Wiechert potentials for a point force and for dislocations • Generalization of the so-called Stokes problem of a localized point force for arbitrary movements and with retardation effects • Comparison of the effects of retardation, radiation and acceleration in the theory of elastodynamics and the theory of electrodynamics • Dynamics of straight dislocations within the framework of the theory of distributions of Schwartz. New results in the theory of dislocations: • Dislocation loops in the anisotropic elasticity theory with a systematic derivation of the Burgers formula for the displacement field of a dislocation loop • Dislocation gauge theory for inhomogeneous materials • Dislocation gauge theory for graphene • J-, M- and L-integrals of dislocations • Comparison between Lazar’s dislocation gauge theory and Neff’s reduced micromorph theory.

Publications

• A screw dislocation in a functionally graded material using the translation gauge theory of dislocations. International Journal of Solids and Structures, 48(11-12), 1630-1636.
  Lazar, Markus
  
• On the elastic fields produced by non-uniformly moving dislocations: a revisit. Philosophical Magazine, 91(25), 3327-3342.
  Lazar, Markus
  
• Non-singular dislocation loops in gradient elasticity. Physics Letters A, 376(21), 1757-1758.
  Lazar, Markus
  
• The elastodynamic Liénard–Wiechert potentials and elastic fields of non-uniformly moving point and line forces. Wave Motion, 49(8), 710-718.
  Lazar, Markus
  
• Dislocation field theory in 2D: Application to graphene. Physics Letters A, 377(5), 423-429.
  Lazar, Markus
  
• Dislocation loops in anisotropic elasticity: displacement field, stress function tensor and interaction energy. Philosophical Magazine, 93(1-3), 174-185.
  Lazar, Markus & Kirchner, Helmut O.K.
  
• On retardation, radiation and Liénard–Wiechert type potentials in electrodynamics and elastodynamics. Wave Motion, 50(7), 1161-1174.
  Lazar, Markus
  
• On the non-uniform motion of dislocations: the retarded elastic fields, the retarded dislocation tensor potentials and the Liénard–Wiechert tensor potentials. Philosophical Magazine, 93(7), 749-776.
  Lazar, Markus
  
• The electromagnetic fields and the radiation of a spatio-temporally varying electric current loop. Wave Motion, 50(6), 995-1002.
  Lazar, Markus
  
• The fundamentals of non-singular dislocations in the theory of gradient elasticity: Dislocation loops and straight dislocations. International Journal of Solids and Structures, 50(2), 352-362.
  Lazar, Markus
  
• A note on Lorentz‐like transformations and superluminal motion. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 95(7), 690-694.
  Jin, C. & Lazar, M.
  
• Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor, dislocation key-formulas and dislocation loops. Philosophical Magazine, 94(35), 4080-4101.
  Lazar, Markus & Agiasofitou, Eleni
  
• On gradient field theories: gradient magnetostatics and gradient elasticity. Philosophical Magazine, 94(25), 2840-2874.
  Lazar, Markus
  
• On the equations of motion of dislocations in quasicrystals. Mechanics Research Communications, 57, 27-33.
  Agiasofitou, Eleni & Lazar, Markus
  
• Phason dynamics of quasicrystals. Acta Crystallographica Section A Foundations and Advances, 70(a1), C86-C86.
  Agiasofitou, Eleni & Lazar, Markus
  
• Singularity-free dislocation dynamics with strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 68, 161-178.
  Po, Giacomo; Lazar, Markus; Seif, Dariush & Ghoniem, Nasr
  
• The elastodynamic model of wave-telegraph type for quasicrystals. International Journal of Solids and Structures, 51(5), 923-929.
  Agiasofitou, Eleni & Lazar, Markus
  
• The solid angle and the Burgers formula in the theory of gradient elasticity: Line integral representation. Physics Letters A, 378(5-6), 597-601.
  Lazar, Markus & Po, Giacomo
  
• An atomistically-enabled non-singular anisotropic elastic representation of near-core dislocation stress fields in α-iron, Physical Review B 91, 184102
  D. Seif; G. Po; M. Mrovec; M. Lazar; C. Elsässer & P. Gumbsch
  
• Distributed dislocation technique for cracks based on non-singular dislocations in nonlocal elasticity of Helmholtz type. Engineering Fracture Mechanics, 136, 79-95.
  Mousavi, S. Mahmoud & Lazar, Markus
  
• On non-singular crack fields in Helmholtz type enriched elasticity theories. International Journal of Solids and Structures, 62, 1-7.
  Lazar, Markus & Polyzos, Demosthenes
  
• On the gradient of the Green tensor in two-dimensional elastodynamic problems, and related integrals: Distributional approach and regularization, with application to nonuniformly moving sources. Wave Motion, 57, 44-63.
  Pellegrini, Yves-Patrick & Lazar, Markus
  
• The non-singular Green tensor of gradient anisotropic elasticity of Helmholtz type. European Journal of Mechanics - A/Solids, 50, 152-162.
  Lazar, Markus & Po, Giacomo
  
• The non-singular Green tensor of Mindlin's anisotropic gradient elasticity with separable weak non-locality. Physics Letters A, 379(24-25), 1538-1543.
  Lazar, Markus & Po, Giacomo
  
• The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations. The Quarterly Journal of Mechanics and Applied Mathematics, 68(1), 53-84.
  Neff, P.; Ghiba, I. D.; Lazar, M. & Madeo, A.
  
• WITHDRAWN: On gradient enriched elasticity theories: A reply to “Comment on ‘On non-singular crack fields in Helmholtz type enriched elasticity theories’ ” and important theoretical aspects. International Journal of Solids and Structures.
  Lazar, Markus; Agiasofitou, Eleni & Polyzos, Demosthenes
  
• Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem. Journal of the Mechanics and Physics of Solids, 96, 632-659.
  Lazar, Markus & Pellegrini, Yves-Patrick
  
• Eshelbian mechanics of novel materials: Quasicrystals. Journal of Micromechanics and Molecular Physics, 01(03n04), 1640008.
  Lazar, Markus & Agiasofitou, Eleni
  
• Irreducible decomposition of strain gradient tensor in isotropic strain gradient elasticity. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik Und Mechanik, 96(11), 1291–1305.
  Lazar, Markus
  
• Micromechanics and dislocation theory in anisotropic elasticity. Journal of Micromechanics and Molecular Physics, 01(02), 1650011.
  Lazar, Markus
  
• Micromechanics and theory of point defects in anisotropic elasticity: Eshelby factor meets Eshelby tensor. Journal of Micromechanics and Molecular Physics, 02(01), 1750005.
  Lazar, Markus
  
• Micromechanics of dislocations in solids: J -, M -, and L -integrals and their fundamental relations. International Journal of Engineering Science, 114, 16-40.
  Agiasofitou, Eleni & Lazar, Markus