On the elastic wedge problem within simplified and incomplete strain gradient elasticity theories

Solyaev, Yury; Lurie, S. A.; Altenbach, Holm; Dell’Isola, Francesco

doi:10.1016/j.ijsolstr.2022.111433

Cited by 25 publications

(20 citation statements)

References 52 publications

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“…The classical part is defined in a standard PN form through the harmonic vector and harmonic function, while the gradient part is simply represented through the modified Helmholtz decomposition. In such a way we prove the correctness of the previously supposed simplified form of PN solution within SGET [44].…”

Section: Introductionmentioning

confidence: 78%

“…The key point of the presented results is the proposed modified definition of Galerkin stress function and its specific relations to the Papkovich stress functions within SGET that have not been established previously for the best of author's knowledge. Applications of the obtained results can be related to the wide class of boundary value problems that can be solved analytically in a simple manner by using PN representation for which we show the completeness [40,44,48]. Incorporation of complete general solutions into the numerical schemes (like in a Trefftz method) can be also an important issue for the development of stable and flexible numerical solvers within SGET [39,49].…”

Section: Introductionmentioning

confidence: 83%

“…Solyaev et al [42] used similar form of PN solution within SGET and reduced the representation for the gradient part of solution to the six scalar functions for the arbitrary curvilinear coordinates, which allows the separation of variables in the Helmholtz equation. Recently, this representation was additionally simplified such that the gradient part of PN solution was defined through the modified variant of Helmholtz decomposition, in which the scalar and the vector potentials obey the modified Helmholtz equations with different coefficients [44]. Similar result have been also established within the simplified strain gradient elasticity theory by Charalambopoulos et al [41], though the possibility of the use of solenoidal vector stress function for the gradient part of the displacement field have not been considered.…”

Section: Introductionmentioning

confidence: 93%

“…It defines the displacement field in rather complicated form through the vector and scalar functions that obey the fourth order governing equations. Later, the simpler variants of the PN solutions with stress functions that obey the second-order equations have been established within SGET [35,[39][40][41][42][43], though the completeness of these solutions have not been proven or it was implied on the basis of heuristic reasoning [44].…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

Solyaev¹

2022

Preprint

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“…We can formulate Hooke’s law to define the stress tensor

as a linear combination of the volumetric and deviatoric strain tensors:

where K is the bulk modulus and G is the shear modulus. From Equation ( 1 ), it is possible then to recognize the two contributions to the stress, namely the hydrostatic:

and shear or deviatoric part:

This formulation can be generalized by using mechanical micromorphic [ 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 ], micropolar [ 57 , 58 , 59 , 60 ], higher-order [ 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 , 69 , 70 ], or peridynamic [ 71 , 72 , 73 , 74 ] models. As an opening move, the stiffnesses can be evaluated starting with the knowledge of the engineering parameters, Young’s modulus Y , and Poisson’s ratio

as follows:

since they are more str...…”

Section: Methodsmentioning

confidence: 99%

Bone Remodeling Process Based on Hydrostatic and Deviatoric Strain Mechano-Sensing

et al. 2022

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A macroscopic continuum model intended to provide predictions for the remodeling process occurring in bone tissue is proposed. Specifically, we consider a formulation in which two characteristic stiffnesses, namely the bulk and shear moduli, evolve independently to adapt the hydrostatic and deviatoric response of the bone tissue to environmental changes. The formulation is deliberately simplified, aiming at constituting a preliminary step toward a more comprehensive modeling approach. The evolutive process for describing the functional adaptation of the two stiffnesses is proposed based on an energetic argument. Numerical experiments reveal that it is possible to model the bone remodeling process with a different evolution for more than one material parameter, as usually done. Moreover, the results motivate further investigations into the subject.

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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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On the elastic wedge problem within simplified and incomplete strain gradient elasticity theories

Cited by 25 publications

References 52 publications

Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

Bone Remodeling Process Based on Hydrostatic and Deviatoric Strain Mechano-Sensing

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

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