Exact degenerate scales in plane elasticity using complex variable methods

Corfdir, Alain; Bonnet, Guy

doi:10.1016/j.ijsolstr.2015.10.003

Cited by 12 publications

(8 citation statements)

References 30 publications

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“…Degenerate scales of the plane elasticity were studied by Chen et al (2009a, 2009b, 2009c). Corfdir and Bonnet (2016) used the complex variable methods to find the exact degenerate scales in plane elasticity. To deal with degenerate scales, there are several regularized approaches, e.g., the method of adding rigid body mode (Chen et al , 2005a, 2014), rank promotion by adding the boundary flux equilibrium (Chen et al , 2014), hypersingular integral formulation (Chen et al , 2014), CHEFF method (Chen et al , 2002, 2005a; Chen et al , 2014), the Fichera’s method (Chen et al , 2014) and the self-regularization method (Chen et al , 2017a).…”

Section: Introductionmentioning

confidence: 99%

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Chen

Kuo

Chang

et al. 2017

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Purpose The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically. Design/methodology/approach To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered. Findings Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function. Originality/value The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.

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Section: Introductionmentioning

confidence: 99%

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Chen

Kuo

Chang

et al. 2017

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“…This bound is sharp since we have the equality ρ 1 = ρ 0 for a segment [38] or a finite set of aligned segments [13].…”

Section: A Lower Bound For the Degenerate Scale Factorsmentioning

confidence: 99%

“…1) bounded by x = 0, x = y and x+y = 1. The values for arcs of a circle and a curve with 6 cusps have been inserted [13].…”

Section: Overview Of Possible Values Of the Couple Of Degenerate Scalmentioning

confidence: 99%

The Degenerate Scales for Plane Elasticity Problems in Piecewise Homogeneous Media Under General Boundary Conditions

Corfdir¹,

Bonnet²

2020

J Elast

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The degenerate scale issue for 2D-boundary integral equations and boundary element methods has been already investigated for Laplace equation, antiplane and plane elasticity, bending plate for Dirichlet boundary condition. Recently, the problems of Robin and mixed boundary conditions and of piecewise heterogeneous domains have been considered for the case of Laplace equation. We investigate similar questions for plane elasticity for more general boundary conditions. For interior problems, it is shown that the degenerate scales do not depend on the boundary condition. For exterior problems, the two degenerate scales (homogeneous medium) or two of them (heterogeneous medium) are tightly linked with the behavior at infinity of the solutions. The dependence of this behavior on the boundary conditions is investigated. We give sufficient conditions for the uniqueness of the solution. Numerical applications are provided and validate the set of theoretical results.

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“…= A ln(z) + 0 (z); = −κ Ā ln(z) + 0 and 0 → 0, 0 → 0 when |z| → ∞ and satisfying (55) [7,15]. If A = r e iθ , the resultant force has an angle θ with Ox 1 .…”

Section: Similarity Between the Complex Formulation Of The Dirichlet ...mentioning

confidence: 99%

Degenerate scales for thin elastic plates with Dirichlet boundary conditions

Corfdir

Bonnet

2022

Acta Mech

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A degenerate scale occurs when a loss of uniqueness of the solution of the boundary integral equations happens for that scale of the problem. We consider here the biharmonic 2D problem with Dirichlet boundary conditions which models the bending behavior of a clamped isotropic Kirchhoff plate. We extend several results about degenerate scales previously found for the Laplace and Lamé equation to the biharmonic equation in 2 dimensions. The degenerate scales are obtained from a 4 × 4 discriminant matrix whose shape is provided for different kinds of domain symmetry. We show that degenerate scales can be obtained from a minimization problem. Then, we compare the degenerate scales of two boundaries, one included within the other. For smooth star-shaped curves, we show that there are only two degenerate scales, give sufficient conditions for not being at a degenerate scale and produce bounds to the degenerate scales. For symmetric smooth simply connected curves there are also only two degenerate scales. For these symmetric cases, the use of complex variables allows us to go further and to link the problem of biharmonic equation to the one of plane elasticity and to give information, including bounds and exact values of degenerate scales, for many cases, while until now only very few ones were known. Our results have some consequences for the biharmonic outer radius defined by Pólya and Szegö. The numerical computation of degenerate scales by using boundary elements confirms the theoretical results.

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Exact degenerate scales in plane elasticity using complex variable methods

Cited by 12 publications

References 30 publications

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

The Degenerate Scales for Plane Elasticity Problems in Piecewise Homogeneous Media Under General Boundary Conditions

Degenerate scales for thin elastic plates with Dirichlet boundary conditions

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