On concentrated surface loads and Green's functions in the Toupin–Mindlin theory of strain-gradient elasticity

Gourgiotis, P.A.; Zisis, Th.; Georgiadis, H.G.

doi:10.1016/j.ijsolstr.2017.10.006

Cited by 19 publications

(22 citation statements)

References 54 publications

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“…Having used for the identification (performed in Part I of this study) a quadratic displacement field which produces an equilibrated stress field, an equivalent second-gradient material has been so far defined in a 'condensed' form, corresponding to a class of ∞ 4 second-gradient materials, all providing a correct energy matching with the periodic planar lattice. At this stage, a 'relaxation of the constraints' has to be introduced to yield an equivalent second-gradient Published in International Journal of Solids and Structures (2019) 176-177, [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] doi: https://doi.org/10.1016/j.ijsolstr.2019.07.009 elastic material in a, say, 'standard form'. This relaxation can be introduced in several ways, as for example exploiting an optimization scheme.…”

Section: The Constitutive Law For the Equivalent Second-gradient Elasmentioning

confidence: 99%

“…Published in International Journal of Solids and Structures (2019) 176-177, [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] doi: https://doi.org/10.1016/j.ijsolstr.2019.07.009 Figure 9: A uniaxial strain is applied aligned parallel to the x 2 -axis on a hexagonal lattice (left and reported as yellow dots on the right figures), on a Cauchy (dashed line) and a second-gradient (purple line) equivalent solid. From left to right: Deformed configuration (superimposed to the undeformed configuration sketched gray), displacement field u 2 (x 2 ), and elastic strain energy density U(x 2 ) for the three cases labeled as Ex1 (upper part), Ex4 (central part), and Ex5 (lower part), as defined in Table 1.…”

Section: Uniaxial Strain Problemmentioning

confidence: 99%

“…Higher-order constitutive equations for microstructured solids [19,21,20,27,37,36], initially introduced following a phenomenological approach [10,14,15,22,24,25], are nowadays more and more often connected to material microstructures by means of various homogenization schemes [5,6,7,8,11,13,17,18,16,30,29,32,33,26,35].…”

Section: Introductionmentioning

confidence: 99%

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Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation

Rizzi

Corso

Veber

et al. 2019

International Journal of Solids and Structures

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Positive definiteness and symmetry of the constitutive tensors describing a second-gradient elastic (SGE) material, which is energetically equivalent to a hexagonal planar lattice made up of axially deformable bars, are analyzed by exploiting the closed form-expressions obtained in part I of the present study in the 'condensed' form. It is shown that, while the first-order approximation leads to an isotropic Cauchy material, a second-order identification procedure provides an equivalent model exhibiting non-locality, non-centrosymmetry, and anisotropy. The derivation of the constitutive properties for the SGE from those of the 'condensed' one (obtained by considering a quadratic remote displacement which generates stress states satisfying equilibrium) is presented. Comparisons between the mechanical responses of the periodic lattice and of the equivalent SGE material under simple shear and uniaxial strain show the efficacy of the proposed identification procedure and therefore validate the proposed constitutive model. This model reveals that, at higher-order, a lattice material can be made equivalent to a second-gradient elastic material exhibiting an internal length, a finding which is now open for applications in micromechanics.

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Section: The Constitutive Law For the Equivalent Second-gradient Elasmentioning

confidence: 99%

Section: Uniaxial Strain Problemmentioning

confidence: 99%

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Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation

Rizzi

Corso

Veber

et al. 2019

International Journal of Solids and Structures

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“…The use of the equations of gradient elasticity allows regularizing the singular solutions of the classical elasticity theory [33][34][35][36][37]. The main interest is directed at the extent to which pathological predictions of the classical theory of elasticity in singular stress concentration problems are altered, mitigated, or possibly even eliminated when couple stresses are taken into account.…”

Section: Introductionmentioning

confidence: 99%

“…It should be noted that links between nonlocal and gradient theories is nontrivial [37][38][39][40][41][42][43]. For example, a deep analysis of the connection between the differential and integral formulations of the nonlocal elasticity theory shows that these theories are far from being always equivalent [41,42].…”

Section: Introductionmentioning

confidence: 99%

On Aspects of Gradient Elasticity: Green’s Functions and Concentrated Forces

Andrianov

Koblik²,

Starushenko³

et al. 2022

Symmetry

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In the first part of our review paper, we consider the problem of approximating the Green’s function of the Lagrange chain by continuous analogs. It is shown that the use of continuous equations based on the two-point Padé approximants gives good results. In the second part of the paper, the problem of singularities arising in the classical theory of elasticity with affecting concentrated loadings is considered. To overcome this problem, instead of a transition to the gradient theory of elasticity, it is proposed to change the concept of concentrated effort. Namely, the Dirac delta function is replaced by the Whittaker–Shannon–Kotel’nikov interpolating function. The only additional parameter that characterizes the microheterogeneity of the medium is used. An analog of the Flamant problem is considered as an example. The found solution does not contain singularities and tends to the classical one when the microheterogeneity parameter approaches zero. The derived formulas have a simpler form compared to those obtained by the gradient theory of elasticity.

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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 concentrated surface loads and Green's functions in the Toupin–Mindlin theory of strain-gradient elasticity

Cited by 19 publications

References 54 publications

Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation

Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation

On Aspects of Gradient Elasticity: Green’s Functions and Concentrated Forces

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

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