From non-local Eringen’s model to fractional elasticity

Anton Evgrafov, José C. Bellido*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Abstract

Eringen’s model is one of the most popular theories in non-local elasticity. It has been applied to many practical situations with the objective of removing anomalous stress concentrations around geometric shape singularities, which appear when local modelling is used. Despite the great popularity of Eringen’s model within the mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels and for the paradigmatic case of Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen’s model to spatially heterogeneous material distributions.
Original languageEnglish
JournalMathematics and Mechanics of Solids
Volume24
Issue number6
Pages (from-to)1935-1953
ISSN1081-2865
DOIs
Publication statusPublished - 2019

Keywords

  • Nonlocal elasticity
  • Riesz potential
  • Nonlocal Korn’s inequality
  • Eringen’s model

Cite this

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title = "From non-local Eringen’s model to fractional elasticity",
abstract = "Eringen’s model is one of the most popular theories in non-local elasticity. It has been applied to many practical situations with the objective of removing anomalous stress concentrations around geometric shape singularities, which appear when local modelling is used. Despite the great popularity of Eringen’s model within the mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels and for the paradigmatic case of Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen’s model to spatially heterogeneous material distributions.",
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From non-local Eringen’s model to fractional elasticity. / Evgrafov, Anton ; Bellido, José C.

In: Mathematics and Mechanics of Solids, Vol. 24, No. 6, 2019, p. 1935-1953.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - From non-local Eringen’s model to fractional elasticity

AU - Evgrafov, Anton

AU - Bellido, José C.

PY - 2019

Y1 - 2019

N2 - Eringen’s model is one of the most popular theories in non-local elasticity. It has been applied to many practical situations with the objective of removing anomalous stress concentrations around geometric shape singularities, which appear when local modelling is used. Despite the great popularity of Eringen’s model within the mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels and for the paradigmatic case of Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen’s model to spatially heterogeneous material distributions.

AB - Eringen’s model is one of the most popular theories in non-local elasticity. It has been applied to many practical situations with the objective of removing anomalous stress concentrations around geometric shape singularities, which appear when local modelling is used. Despite the great popularity of Eringen’s model within the mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels and for the paradigmatic case of Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen’s model to spatially heterogeneous material distributions.

KW - Nonlocal elasticity

KW - Riesz potential

KW - Nonlocal Korn’s inequality

KW - Eringen’s model

U2 - 10.1177/1081286518810745

DO - 10.1177/1081286518810745

M3 - Journal article

VL - 24

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EP - 1953

JO - Mathematics and Mechanics of Solids

JF - Mathematics and Mechanics of Solids

SN - 1081-2865

IS - 6

ER -