Derivations of Integral Equations of Elasticity

Hong, Hong‐Ki; Chen, Jeng‐Tzong

doi:10.1061/(asce)0733-9399(1988)114:6(1028)

Cited by 318 publications

(144 citation statements)

References 11 publications

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“…where  is the integral of Hadamard finite part, the terms S kj and D kj contain the derivatives of P ij * and u ij * respectively, as indicated in [12]. In this paper, only linear boundary elements were used.…”

Section: Bem Formulationmentioning

confidence: 99%

Probabilistic fatigue crack growth using BEM and reliability algorithms

Leonel¹,

Venturini²

2011

Boundary Elements and Other Mesh Reduction Methods XXXIII

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Fatigue and crack propagation are phenomena affected by high uncertainties, where deterministic methods fail to predict accurately the structural life. This paper aims at coupling reliability analysis with boundary element method (BEM) in modeling probabilistic fatigue crack growth. BEM has been recognized as an accurate and efficient numerical technique in modeling crack growth problems. The dual BEM approach was adopted to model crack growth. The couple BEM and reliability algorithms allows us to consider uncertainties during the crack propagation process. In addition, it calculates the probability of fatigue failure for complex structural geometry and loading. Two coupling procedures are considered: direct coupling of reliability and mechanical solver and indirect coupling by the response surface method. Numerical applications show the performance of the proposed models in lifetime assessment under uncertainties, where the direct method has shown faster convergence than response surface method.

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Section: Bem Formulationmentioning

confidence: 99%

Probabilistic fatigue crack growth using BEM and reliability algorithms

Leonel¹,

Venturini²

2011

Boundary Elements and Other Mesh Reduction Methods XXXIII

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“…In the second version of the BIE, the boundary tractions are related to the boundary displacement and traction through some integral operators. Generally, the singularity of the kernels of the second version is higher by one order than its counterpart in the first version [Brebbia et al 1984;Hong and Chen 1988].…”

Section: Introductionmentioning

confidence: 99%

Boundary integral equation for notch problems in an elastic half-plane based on Green’s function method

Chen

2012

J. Mech. Mater. Struct.

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This paper studies a boundary integral equation (BIE) for notch problems in an elastic half-plane based on Green's function method. The boundary along the half-plane is traction-free. A fundamental solution is suggested, which is composed of a principal part and a complementary part. The process for evaluating the complementary part from the principal part is similar to the Green's function method for Laplace's equation. After using the Somigliana identity or Betti's reciprocal theorem between the field of the fundamental solution and the physical field, the displacements at the domain point are obtained. Letting the domain point approach the boundary point and using the generalized Sokhotski-Plemelj formula, a BIE of the notch problem for a traction-free half-plane boundary is obtained. The accuracy of the suggested technique is examined. Computed results for elliptic notches and a square notch with rounded corner are presented in the paper.

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“…For the hypersingular identity, the boundary limit must be considered at a smooth point x o with a well-deÿned normal vector n(x o ) [13]. The two integral equations, usually referred to as displacements and traction equations, are also called 'dual' BIEs [14].…”

Section: Introductionmentioning

confidence: 99%

“…In Reference [19], the same conclusion has been obtained by an alternate deÿnition of HFP, without the need for a limiting process. Making recourse to the distribution theory, in Reference [14] the dual BIEs are obtained by the application of a trace operator to the representation formulae. In such an approach, the strongly singular and hypersingular integrals can be expressed by means of discontinuity jumps (also named as 'free terms') of these integrals on the boundary summed with the values of the integrals on the boundary existing only in the sense of the Cauchy principal value (CPV) or in the sense of the HFP.…”

Section: Introductionmentioning

confidence: 99%

Analytical integrations in 2D BEM elasticity

Salvadori

2001

Numerical Meth Engineering

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SUMMARYIn the context of two-dimensional linear elasticity, this paper presents the closed form of the integrals that arise from both the standard (collocation) boundary element method and the symmetric Galerkin boundary element method. Adopting polynomial shape functions of arbitrary degree on straight elements, ÿnite part of Hadamard, Cauchy principal values and Lebesgue integrals are computed analytically, working in a local coordinate system. For the symmetric Galerkin boundary element method, a study on the singularity of the external integral is conducted and the outer weakly singular integral is analytically performed. Numerical tests are presented as a validation of the obtained results.

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Derivations of Integral Equations of Elasticity

Cited by 318 publications

References 11 publications

Probabilistic fatigue crack growth using BEM and reliability algorithms

Probabilistic fatigue crack growth using BEM and reliability algorithms

Boundary integral equation for notch problems in an elastic half-plane based on Green’s function method

Analytical integrations in 2D BEM elasticity

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