Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates

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@article{2cdecfa36e5342bba2237583ea2d09bf,
title = "Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates",
abstract = "Sixth-order boundary value problems for gradient-elastic Kirchhoff plate bending models are formulated in a variational form within an H3 Sobolev space setting. Existence and uniqueness of the weak solutions are then established by proving the continuity and coercivity of the associated symmetric bilinear forms. Complete sets of boundary conditions, including both the essential and the natural conditions, are derived accordingly. In particular, the gradient-elastic Kirchhoff plate models feature two different types of clamped and simply supported boundary conditions in contrast to the classical Kirchhoff plate model. These new types of boundary conditions are given additional attributes singly and doubly; referring to free and prescribed normal curvature, respectively. The formulations and results of the analyzed strain gradient plate models are compared to two other generalized Kirchhoff plate models which follow a modified strain gradient elasticity theory and a modified couple stress theory. It is clarified that the results are extendable to these model variants as well. Finally, the relationship of the natural boundary conditions to external loadings is analyzed.",
keywords = "Boundary conditions, Existence, Kirchhoff plates, Strain gradient elasticity, Uniqueness, Variational formulation",
author = "Jarkko Niiranen and Niemi, {Antti H.}",
year = "2017",
month = "1",
day = "1",
doi = "10.1016/j.euromechsol.2016.09.001",
language = "English",
volume = "61",
pages = "164--179",
journal = "EUROPEAN JOURNAL OF MECHANICS A: SOLIDS",
issn = "0997-7538",

}

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TY - JOUR

T1 - Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates

AU - Niiranen, Jarkko

AU - Niemi, Antti H.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Sixth-order boundary value problems for gradient-elastic Kirchhoff plate bending models are formulated in a variational form within an H3 Sobolev space setting. Existence and uniqueness of the weak solutions are then established by proving the continuity and coercivity of the associated symmetric bilinear forms. Complete sets of boundary conditions, including both the essential and the natural conditions, are derived accordingly. In particular, the gradient-elastic Kirchhoff plate models feature two different types of clamped and simply supported boundary conditions in contrast to the classical Kirchhoff plate model. These new types of boundary conditions are given additional attributes singly and doubly; referring to free and prescribed normal curvature, respectively. The formulations and results of the analyzed strain gradient plate models are compared to two other generalized Kirchhoff plate models which follow a modified strain gradient elasticity theory and a modified couple stress theory. It is clarified that the results are extendable to these model variants as well. Finally, the relationship of the natural boundary conditions to external loadings is analyzed.

AB - Sixth-order boundary value problems for gradient-elastic Kirchhoff plate bending models are formulated in a variational form within an H3 Sobolev space setting. Existence and uniqueness of the weak solutions are then established by proving the continuity and coercivity of the associated symmetric bilinear forms. Complete sets of boundary conditions, including both the essential and the natural conditions, are derived accordingly. In particular, the gradient-elastic Kirchhoff plate models feature two different types of clamped and simply supported boundary conditions in contrast to the classical Kirchhoff plate model. These new types of boundary conditions are given additional attributes singly and doubly; referring to free and prescribed normal curvature, respectively. The formulations and results of the analyzed strain gradient plate models are compared to two other generalized Kirchhoff plate models which follow a modified strain gradient elasticity theory and a modified couple stress theory. It is clarified that the results are extendable to these model variants as well. Finally, the relationship of the natural boundary conditions to external loadings is analyzed.

KW - Boundary conditions

KW - Existence

KW - Kirchhoff plates

KW - Strain gradient elasticity

KW - Uniqueness

KW - Variational formulation

UR - http://www.scopus.com/inward/record.url?scp=84988910674&partnerID=8YFLogxK

U2 - 10.1016/j.euromechsol.2016.09.001

DO - 10.1016/j.euromechsol.2016.09.001

M3 - Article

VL - 61

SP - 164

EP - 179

JO - EUROPEAN JOURNAL OF MECHANICS A: SOLIDS

JF - EUROPEAN JOURNAL OF MECHANICS A: SOLIDS

SN - 0997-7538

ER -

ID: 8735401