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

Solyaev, Yury

doi:10.1016/j.mechmat.2022.104321

Cited by 15 publications

(8 citation statements)

References 86 publications

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“…}}^2 \delta _{il} \delta _{jk} & \text{ if } \mu =\nu =0 , \gamma =j , \eta = k \ , \\ -D_{ijklmn} & \text{ if } \mu =j , \gamma =k , \nu =m , \eta =n \ , \\ 0 & \text{ else } \ , \end{cases}} \ , \\ \tilde{G}_{i\mu k \nu \eta } =& {\begin{cases} -G_{ijklm} & \text{ if } \mu =j , \nu =l , \eta =m \ , \\ 0 & \text{ else } \ , \end{cases}} \end{aligned}\end{equation}$$where rank 4, 5, 6 tensors,

C_{ijkl}

D_{ijklmn}

G_{ijklm}

, are given for the corresponding metamaterial. Their measurement seems to be challenging [85–87], yet there exist different homogenization methods [88–93] that calculate these parameters [94–98]. Inertia related terms, ρ ref.…”

Section: Generalized Continuamentioning

confidence: 99%

Energy based methods applied in mechanics by using the extended Noether's formalism

Abali

2023

Z Angew Math Mech

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Physical systems are modeled by field equations; these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former are axiomatically given and the latter are thermodynamically derived. This approach is useful in thermomechanics and electromagnetism, yet challenges arise once we apply it in damage mechanics for generalized continua. For deriving governing equations, an alternative method is based on a variational framework known as the extended Noether's formalism. Its formal introduction relies on mathematical concepts limiting its use in applied mechanics as a field theory. In this work, we demonstrate the power of extended Noether's formalism by using tensor algebra and usual continuum mechanics nomenclature. We demonstrate derivation of field equations in damage mechanics for generalized continua, specifically in the case of strain gradient elasticity.

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C_{ijkl}

D_{ijklmn}

G_{ijklm}

Section: Generalized Continuamentioning

confidence: 99%

Energy based methods applied in mechanics by using the extended Noether's formalism

Abali

2023

Z Angew Math Mech

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“…Particular solution ϕ * g (38) can be defined by using Helmhlotz decomposition for the vector potential B B B g , which can be used without any additional restrictions since B B B g satisfies the vector Helmholtz equation ( 24) 2 [56]. Thus, we define:…”

Section: Papkovich-neuber Solutionmentioning

confidence: 99%

“…Based on comparison of equation for the scalar potential Ψ (41) 2 and equation for the particular solution ϕ * g (38) we can define the last one as:…”

Section: Papkovich-neuber Solutionmentioning

confidence: 99%

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Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

Solyaev¹

2022

Preprint

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In this paper, we consider isotropic Mindlin-Toupin strain gradient elasticity theory in which the equilibrium equations contain two additional length-scale parameters and have the fourth order. For this theory we developed an extended form of Boussinesq-Galerkin (BG) and Papkovich-Neuber (PN) general solutions. Obtained form of BG solution allows to define the displacement field through the single vector function that obeys the eight-order bi-harmonic/bi-Helmholtz equation. The developed PN form of the solution provides an additive decomposition of the displacement field into the classical and gradient parts that are defined through the standard Papkovich stress functions and modified Helmholtz decomposition, respectively. Relations between different stress functions and completeness theorem for the derived general solutions are established. As an example, it is shown that a previously known fundamental solution within the strain gradient elasticity can be derived by using the developed PN general solution.

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“…In addition, strain gradient theories are more often used to describe elasto-plastic strain, where the size effect is more pronounced. A review and comparative analysis of known SGETs can be found in [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

A semi-empirical model of the influence of the size of micrometer particles on the elastic properties of polymer-matrix composites

Федотов

2023

Preprint

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A semi-empirical model of the effect of particle size 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 particle size and an exponential function depending on particle size. The calculated dependencies contain two empirical parameters, and the model is identified based on the results of an experiment with two particle sizes. Composites with the minimum and maximum particle size are selected among the studied options. The calculation results are in good agreement with the experimental data for the Young's modulus of polymer composites with spherical particles of various sizes.

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

Cited by 15 publications

References 86 publications

Energy based methods applied in mechanics by using the extended Noether's formalism

Energy based methods applied in mechanics by using the extended Noether's formalism

Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

A semi-empirical model of the influence of the size of micrometer particles on the elastic properties of polymer-matrix composites

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