Three-dimensional nonlocal anisotropic elasticity: a generalized continuum theory of Ångström-mechanics

Lazar, Markus; Agiasofitou, Eleni; Po, Giacomo

doi:10.1007/s00707-019-02552-2

Cited by 24 publications

(26 citation statements)

References 56 publications

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“…They can be applied in the modeling of nanomechanical problems with a solid basis of material parameters of the Toupin-Mindlin gradient elasticity theory without any ad hoc assumption. Important applications of the material parameters are nanomechanical benchmark problems, eigenstrain and fracture problems in gradient elasticity, nanomechanical engineering and industrial problems, nanodevices as well as the modeling of non-singular dislocations leading to Ångström-mechanics of dislocations [33].…”

Section: A Positive Definiteness Of C αβ and D ξρ Using The Sylvester Criterionmentioning

confidence: 99%

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

Lazar

Agiasofitou

Böhlke

2021

Continuum Mech. Thermodyn.

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In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, $$3+11$$ 3 + 11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and $$1+6$$ 1 + 6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

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Section: A Positive Definiteness Of C αβ and D ξρ Using The Sylvester Criterionmentioning

confidence: 99%

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

Lazar

Agiasofitou

Böhlke

2021

Continuum Mech. Thermodyn.

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“…As this article is concerned with Eringen’s nonlocal elasticity theory to implement the small-scale effects, the nonlocal stress can be appeared as 14,45–48…”

Section: Constitutive Equations and Solutionmentioning

confidence: 99%

Mechanical analysis of eccentric defected bilayer graphene sheets considering the van der Waals force

Dastjerdi

Malikan

2020

Proceedings of the Institution of Mechanical Engineers, Part N:

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In this article, we have tried to simulate nonlinear bending analysis of a double-layered graphene sheet which contains a geometrical imperfection based on an eccentric hole. The first-order shear deformation theory is considered to obtain the governing equations. Also, the nonlinear von Kármán strain field has been assumed in order to obtain large deformations. Whereas the double-layered graphene sheet has been considered, the effect of van der Waals forces has been taken into account in the analysis. In order to implement the nanoscale impact, the nonlocal elasticity theory has been employed. The solution methodology, which is here based on the semi-analytical polynomial method solving technique presented previously by the authors, has been applied and again its efficiency has been demonstrated due to its highly accurate results. Due to the fact that this research has been done for the first time and there is no validation available, the results of the local single layer sheet are compared with ABAQUS software. The effects of some other parameters on the results have been studied such as the value of eccentricity, van der Waals interaction, and nonlocal parameter.

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“…In order to remove the singularity in the classical solutions of dislocations, one can consider the non-singular dislocation theory developed based on the non-local elasticity [21][22][23] or the gradient elasticity [24,25]. The non-singular dislocation theory based on the gradient elasticity introduces an extra characteristic parameter which could be estimated using atomistic calculations [26].…”

Section: Introductionmentioning

confidence: 99%

Driving forces on dislocations: finite element analysis in the context of the non-singular dislocation theory

et al. 2021

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This work presents a regularized eigenstrain formulation around the slip plane of dislocations and the resultant non-singular solutions for various dislocation configurations. Moreover, we derive the generalized Eshelby stress tensor of the configurational force theory in the context of the proposed dislocation model. Based on the non-singular finite element solutions and the generalized configurational force formulation, we calculate the driving force on dislocations of various configurations, including single edge/screw dislocation, dislocation loop, interaction between a vacancy dislocation loop and an edge dislocation, as well as a dislocation cluster. The non-singular solutions and the driving force results are well benchmarked for different cases. The proposed formulation and the numerical scheme can be applied to any general dislocation configuration with complex geometry and loading conditions.

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Three-dimensional nonlocal anisotropic elasticity: a generalized continuum theory of Ångström-mechanics

Cited by 24 publications

References 56 publications

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

Mechanical analysis of eccentric defected bilayer graphene sheets considering the van der Waals force

Driving forces on dislocations: finite element analysis in the context of the non-singular dislocation theory

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