Inclusion problem in second gradient elasticity

Ma, Hongmei; Hu, Gengkai; Wei, Yueguang; Liang, Lihong

doi:10.1016/j.ijengsci.2018.07.003

Cited by 30 publications

(41 citation statements)

References 38 publications

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“…for the case of small volume fraction of inclusions. All solutions ( 38)-( 40) are obtained in a closed-form up to the Eshelby-like tensors, that were derived previously for different type of inclusions for simplified SGET [46] and for general SGET [50]. Note, that relation (38) is similar to those one derived in Ref.…”

Section: Extended Eshelby Equivalent Inclusion Methodsmentioning

confidence: 60%

“…where S ijkl , S ijklm , S ijklm and S ijklmn are the Eshelby-like tensors that arise when one consider the Eshelby inclusion problem within SGET (see [9,50]).…”

Section: Extended Eshelby Equivalent Inclusion Methodsmentioning

confidence: 99%

“…where we take into account that tensors S ijklm and S ijklm are position-dependent and have antisymmetric distribution inside the domain Ω 2 for the case of linear eigenstrain (24), such that S ijklm 2 = S ijklm 2 = 0 [9,50]. At the same time, components of tensor S ijkl are symmetric inside Ω 2 and it is valid that S ijlm x k 2 = 0.…”

Section: Extended Eshelby Equivalent Inclusion Methodsmentioning

confidence: 99%

“…Extension of the [9] method to the general formulation of isotropic SGET with five additional material constant have been presented in Ref. [50].…”

Section: Introductionmentioning

confidence: 99%

“…It is shown that within such approximation we can reduce the second-order homogenization problem to the problem of determination of the averaged field variables only inside the inclusions. These averaged field variables can be found then by using Eshelby-like tensors (that were derived previously within SGET [46,47,50]) and by using appropriate extension of the Eshelby equivalent inclusion method that is proposed in this paper. Extension is given assuming the equivalence between averaged Cauchy stresses, moments of Cauchy stresses, and double stresses in the inhomogeneous media with ellipsoidal inclusion and in the corresponding homogeneous media with prescribed linear eigenstrain in the internal ellipsoidal domain.…”

Section: Introductionmentioning

confidence: 99%

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Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Solyaev¹

2021

Preprint

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Analytical method for the second-order homogenization of two-phase composites within Mindlin-Toupin strain gradient elasticity theory is proposed. Direct approach and self-consistent approximation are used to reduce the homogenization problem to the problem of determination of averaged Cauchy stresses, double stresses and static moments of Cauchy stresses inside the inclusions under prescribed quadratic boundary conditions. The ellipsoidal shape of inclusions and orthotropic properties of phases are assumed. Extended equivalent inclusion method with linear eigenstrain is proposed to derive the explicit relations between the Eshelby-like tensors and corresponding concentration tensors that are used to define the averaged field variables inside the inclusions. Obtained analytical solutions allow to evaluate the effective classical and gradient elastic moduli of composite materials accounting for the phases properties, volume fraction, shape and size of inclusions. Presented solution for the effective gradient moduli covers the full range of volume fraction and correctly predicts the absence of gradient effects for the homogeneous classical Cauchy medium. Examples of calculations for the composites with spherical inclusions are given. Micro-scale definition of the wellknown simplified strain gradient elasticity theory is provided. Namely, it is shown that this theory is the phenomenological continuum model for the composites with isotropic matrix and with the small volume fraction of stiff isotropic spherical inclusions, which have the same Poisson's ratio to those one of matrix phase.

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Section: Extended Eshelby Equivalent Inclusion Methodsmentioning

confidence: 60%

“…where S ijkl , S ijklm , S ijklm and S ijklmn are the Eshelby-like tensors that arise when one consider the Eshelby inclusion problem within SGET (see [9,50]).…”

Section: Extended Eshelby Equivalent Inclusion Methodsmentioning

confidence: 99%

Section: Extended Eshelby Equivalent Inclusion Methodsmentioning

confidence: 99%

“…Extension of the [9] method to the general formulation of isotropic SGET with five additional material constant have been presented in Ref. [50].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Solyaev¹

2021

Preprint

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A semi‐empirical model of the influence of the size of micrometer fillers on the elastic properties of polymer‐matrix composites

Fedotov

2024

Polymer Composites

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A semi‐empirical model of the influence of the size of micrometer fillers on the elastic moduli of polymer‐matrix composites is proposed. The modulus of elasticity of the composite is the sum of the modulus of elasticity at infinite filler size and an exponential function depending on filler size. The calculated dependencies contain two empirical parameters, and the model is identified based on the results of an experiment with two filler sizes. Composites with the minimum and maximum filler size are selected among the studied options. The scope of application of the model was studied by varying the filler size and boundary values of Young's modulus and eight grades of polymer‐matrix composites. The filler size varied over a wide range from 3.5 to 140 μm. The range of changes in the boundary values was also significant, and depending on the brand of the composite, it was 7%–43% of Young's modulus . For all variants of particle size and the range of changes in the boundary values of the Young's modulus, the calculation results are in good agreement with the experimental data.Highlights Semi‐empirical model of the effect of filler size on the elastic moduli polymer composites. The calculated dependences contain two empirical parameters. Model identification is based on the results of an experiment with two filler sizes. Explore composites with minimum and maximum filler size. Good agreement of calculation results with experimental data is obtained.

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Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

Lurie

Solyaev

2023

Z Angew Math Mech

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In this paper, we show that within the class of isotropic Mindlin‐Toupin gradient theories there exist a particular variant of the theory, which allows a completely simplified form of traction boundary value problems. This theory can be obtained assuming that the high‐grade part of the strain energy density depends only on the vector‐type quantities, that are the gradient of dilatation and the curl of small rotation. Such incomplete gradient theory becomes positive semi‐definite and at the same time it obeys the strong ellipticity conditions of the general Mindlin‐Toupin first strain gradient elasticity. Based on the variational approach it is shown that the equilibrium equations of the developed theory and its definition for the surface traction can be given only in terms of the total stresses (like in classical elasticity). Such formulation can be useful for derivation of the closed form solutions for the problems with traction‐type boundary conditions. Examples of the solutions for the problem of cylindrical bending and for the inplane crack tip fields are presented. It is shown that considered theory allows to obtain a regularized solution for the crack problems and at the same time it does not predict a non‐physical infinite increase of the material's rigidity under bending.

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Inclusion problem in second gradient elasticity

Cited by 30 publications

References 38 publications

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

A semi‐empirical model of the influence of the size of micrometer fillers on the elastic properties of polymer‐matrix composites

Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

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