Symmetry and Applied Variational Models for Strain Gradient Anisotropic Elasticity

Lurie, S. A.; Belov, P. A.; Solyaev, Yury; Lykosova, Elena D.; Волков, А. А.

doi:10.1615/nanoscitechnolintj.2021037073

Cited by 3 publications

(3 citation statements)

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“…Remark 3. It should be noted that equations similar to equations (16) and (18) are presented by Lurie et al [37] equations (8) and (9)). are presented in equation (9) by Lurie et al [37] is not given quite correctly and should be considered taking into account equation (10) .…”

Section: Coupled Strain-gradient Elasticitymentioning

confidence: 77%

On variational principles in coupled strain-gradient elasticity

Nazarenko

Glüge

Altenbach

2022

Mathematics and Mechanics of Solids

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Strain-gradient elasticity is a special case of high-gradient theories in which the potential energy density depends on the first and second gradient of the displacement field. The presence of a coupling term in the material law leads to a non-diagonal quadratic form of the stored energy, which makes it difficult for the derivation of fundamental theorems. In this article, two variational principles of the minimum of potential and complementary energies are argued in the context of the coupled strain-gradient elasticity theory. The basis of the proofs of both variational principles is the equivalent transformation of the stain and strain-gradient energy density that allows to avoid the complication related to the presence of the fifth-rank coupling tensor [Formula: see text] in the equation for the potential energy density and leads to diagonalization of the quadratic form of the stored energy. This transformation enables to inverse Hook’s law, to determine compliance tensors, and to obtain closed-form relation for the complementary energy. After that the proofs of both principles of a minimum of potential and complementary energies are provided in the usual manner adopted in the classical theory of elasticity.

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Section: Coupled Strain-gradient Elasticitymentioning

confidence: 77%

On variational principles in coupled strain-gradient elasticity

Nazarenko

Glüge

Altenbach

2022

Mathematics and Mechanics of Solids

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“…In [3][4][5][6][7], it is shown that for the extensive class of elasticity models, including those regarding gradient elasticity, additional symmetry conditions from classical elasticity models must be met. Otherwise, models intended to simulate a deformable, defect-free medium become erroneous because the condition of the continuity of distortions, i.e., the condition of the absence of a field of defects, is not satisfied.…”

Section: Description Of Articles [12]mentioning

confidence: 99%

“…Therefore, it also extends to non-gradient models of anisotropic media, for which the general structure of elastic modulus tensors is very complex. These issues were partly considered in [4,6]. The same papers discuss the issues of symmetry in the structure of elastic modulus tensors related to the potentiality of the strain energy density and the issues of the inheritance of the symmetrical anisotropy properties from the classical theory in a gradient anisotropic medium.…”

Section: Description Of Articles [12]mentioning

confidence: 99%

Foundations of Continuum Mechanics and Mathematical Physics—Editorial 2021–2023

dell’Isola,

Matevossian

2023

Symmetry

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Development of the “Separated Anisotropy” Concept in the Theory of Gradient Anisotropic Elasticity

Belov

Lurie

2021

Mech Compos Mater

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A variation model of the gradient anisotropic elasticity theory is constructed. Its distinctive feature is the fact that the density of potential energy, along with symmetric fourth-and sixth-rank stiffness tensors, also contains a nonsymmetric fifth-rank stiffness tensor. Accordingly, the stresses in it also depend on the curvatures and the couple stresses depend on distortions. The Euler equations are three fourth-order equilibrium equations. The spectrum of boundary-value problems is determined by six pairs of alternative boundary conditions at each nonsingular surface point. At each special point of the surface (belonging to surface edges), in the general case, additional conditions arise for continuity of the displacement vector and the meniscus force vector when crossing the surface through the edge. In order to reduce the number of the physical parameters requiring experimental determination, particular types of the fifth-and sixth-rank stiffness tensors are postulated. Along with the classical tensor of anisotropic moduli, it is proposed to introduce a first-rank stiffness tensor (a vector with a length dimension), with the help of which the fifth-rank tensor is constructed from the classical fourthrank tensor by means of tensor multiplication. The sixth-rank tensor is constructed as the tensor product of the classical fourth-rank tensor and two first-rank tensors.

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Symmetry and Applied Variational Models for Strain Gradient Anisotropic Elasticity

Cited by 3 publications

References 0 publications

On variational principles in coupled strain-gradient elasticity

On variational principles in coupled strain-gradient elasticity

Foundations of Continuum Mechanics and Mathematical Physics—Editorial 2021–2023

Development of the “Separated Anisotropy” Concept in the Theory of Gradient Anisotropic Elasticity

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