Displacements representations for the problems with spherical and circular material surfaces

Mogilevskaya, Sofia G.; Кущ, В. І.; Zemlyanova, Anna Y.

doi:10.1093/qjmam/hbz013

Cited by 25 publications

(8 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this way, the partial differential equilibrium equations of displacements are successfully converted into a linear system with respect to the Legendre series coefficients and the displacements and stresses are then analytically determined. As confirmed by previous studies on spherical nanoinhomogeneities [53–55], the flexural resistibility of a spherical surface or interface can be represented by the fixed combination of the original two material constants in the bending moment and curvature change constitutive relation: 3 ξ 0 + 5 η 0 . This conclusion is independent of the far-field loading conditions.…”

Section: Introductionsupporting

confidence: 71%

See 1 more Smart Citation

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

Ban

2020

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

For a solid surface or interface that is subjected to transverse loading, the influence of its flexural resistibility to bending deformation becomes significant. A spherical inhomogeneity or void embedded in an infinite elastic medium under the application of nonhydrostatic loads represents a typical example. In this work, we consider the most fundamental loading of a far-field unidirectional tension. Analytical displacements and stresses are developed by the coupling of a Steigmann–Ogden surface mechanical model, the simple method of Boussinesq displacement potentials, the semi-inverse method of elasticity, and Legendre series representations of spherical harmonics. The problem is then solved by converting the equilibrium equations of displacement into a linear system with respect to the Legendre series coefficients. The developed solutions are general in the sense that they may reduce to their classical or Gurtin–Murdoch counterparts as special cases. Analytical expressions reveal that the derived solution depends on four dimensionless ratios from among surface material parameters, shear moduli ratio, and inhomogeneity or void radius. In particular, instead of depending on both flexural parameters in the moment–curvature relation, one fixed combination is sufficient to represent the surface flexural rigidity. This is in contrast with the influence of the in-plane elastic stiffness, in which both surface Lamé parameters matter. Parametric studies further demonstrate that, for metallic inhomogeneities or voids with radii between 10 nm and 100 nm, the effects of surface flexural rigidity on stress distributions and stress concentrations are significant.

show abstract

Section: Introductionsupporting

confidence: 71%

“…Analytical solutions were developed for both hydrostatic and deviatoric far-field traction loads. Shortly afterwards, Mogilevskaya [54] further extended the model of an embedded spherical inhomogeneity to nonhydrostatic loading conditions. General forms of displacement representations were presented.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

Ban

2020

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

show abstract

“…In the context of particulate composites with spherical reinforcements, analytical solutions for a single spherical particle with the G–M interface were obtained and used to model the elastic fields [33–37] and effective properties [10, 38, 39] of particulate nanocomposites. Similar solutions for the S–O interfaces are reported in [40–44]. More advanced, finite-cluster [45] and representative unit cell [46] models of spherical particle composite with the G–M interface have been developed.…”

Section: Introductionmentioning

confidence: 80%

Anisotropic imperfect interface in elastic particulate composite with initial stress

Кущ

Mogilevskaya

2021

Mathematics and Mechanics of Solids

Self Cite

View full text Add to dashboard Cite

The model of an anisotropic interface in an elastic particulate composite with initial stress is developed as the first-order approximation of a transversely isotropic interphase between an isotropic matrix and spherical particles. The model involves eight independent parameters with a clear physical meaning and conventional dimensionality. This ensures its applicability at various length scales and flexibility in modeling the interfaces, characterized by the initial stress and discontinuity of the displacement and stress fields. The relevance of this model to the theory of material interfaces and its applicability in nanomechanics is discussed. The proposed imperfect interface model is incorporated in the unit cell model of a spherical particle composite with thermal stress owing to uniform temperature change. The rigorous solution to the model boundary value problem is obtained using the multipole expansion method. The reported accurate numerical data confirm the correctness of the developed theory, provide an estimate of its accuracy and applicability limits in the multiparticle environment, and reveal significant effects of the interphase or interface anisotropy and initial stress on the local fields and overall thermoelastic properties of the composite.

show abstract

“…Using appropriately modified (to reflect the problem geometry) boundary conditions of the first, N=1, second, N=2 and third, N=3, orders obtained from equations (5.1) - (5.4) for Model I and from equations (5.10) - (5.13) for Model II , all elastic fields inside the fibre and the matrix are computed. The solutions employed the field representations of [59] that are also used in [30,60,61]. Those representations involve unknown coefficients that are found from the linear systems of algebraic equations resulting from the substitution of the representations into prescribed boundary conditions.…”

Section: Examplementioning

confidence: 99%

On the Bövik–Benveniste methodology and related approaches for modelling thin layers

Baranova

Mogilevskaya

2022

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

This paper reviews several leading approaches for asymptotic modelling of thin layers in elastostatics and wave propagation phenomena. The issues related to applications of the so-called ‘equivalent’ or ‘effective’ boundary conditions and their interpretations are highlighted. Comparative analysis of asymptotic models is performed for a two-dimensional elastostatic case using a novel complex variables-based modelling tool. Its implementation allows for straightforward derivations of higher order boundary conditions for problems with layers of arbitrary sufficiently smooth curvatures. Explicit expressions for the conditions up to the third order are provided. All models are tested using available benchmark solutions and the solutions for the limiting cases of the layer parameters. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.

show abstract

Displacements representations for the problems with spherical and circular material surfaces

Cited by 25 publications

References 43 publications

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

Anisotropic imperfect interface in elastic particulate composite with initial stress

On the Bövik–Benveniste methodology and related approaches for modelling thin layers

Contact Info

Product

Resources

About