A simple implementation of the dual boundary element method using the tangential differential operator for plane problems

Palermo, L.; Gonçalves, Paulo César; Figueiredo, L. G.

doi:10.2495/be100071

Cited by 3 publications

(4 citation statements)

References 4 publications

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“…The continuity of the displacement function at x′ is the necessary condition for the displacement BIE, and it is satisfied when the collocation point is placed at the ends of the boundary element or inside the element. The continuity of the displacement derivative at x′ is required for the traction BIE, and it is satisfied when the collocation point is placed inside the boundary element [13,14].…”

Section: Dual Boundary Integral Equationsmentioning

confidence: 99%

A Boundary Element Formulation for Crack Analyses Incorporating a Cohesive Zone Model

Gonçalves¹,

Palermo²,

Proença³

2017

Int. J. CMEM

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A formulation is presented to perform crack propagation analyses in cohesive materials with the dual boundary element method (DBEM) using the tangential differential operator in the traction boundaryintegral equations. The cohesive law is introduced in the system of equations to directly compute the cohesive forces at each loading step. A single edge crack is analyzed with the linear function to describe the material softening law in the cohesive zone, and the results are compared with those from the literature. Keywords: cohesive model, crack analysis, dual boundary element model, plane problems, tangential differential operator. INTRODUCTIONThe surfaces of cracks behind the (fictitious) crack tip are not completely separated in some materials, such as concrete, brittle polymers, fiber-reinforced composites, tough ceramics and some alloys. The tractions can be transferred across the crack line along a relatively long extension of the crack, which is commonly called the wake zone, the bridging zone, or the cohesive zone. The main assumption is the occurrence of material softening beyond the peak load in a narrow layer with a negligible volume behind the fictitious crack tip, in which the action can be replaced by cohesive forces (Fig. 1). Two types of constitutive laws for the material in the cohesive zone have been used in the literature. This study employs a traction-displacement relationship in the cohesive zone instead of using a material constitutive law that is defined in terms of stresses and strains accompanied by a layer thickening law. Barenblatt [1] introduced a cohesive model that employed a fictitious crack. Hillerborg et al.[2] proposed a function for a softening model related to an opening (mode I) crack (Fig. 2), which allowed finite element analyses of the problem to be performed, such as those by Petersson [3], Carpinteri [4], and Rots [5].The boundary element method (BEM) is superior to the finite element method (FEM) in crack propagation analysis because remeshing of the domain is not necessary when the crack grows, and new elements can be introduced without affecting the existing elements. The coincidence of two crack surfaces requires two different boundary integral equations (BIEs) for the solution. The dual boundary element method (DBEM) is a technique that is employed for crack propagation analysis. The position of the collocation point to solve the traction BIE and the strategy to treat improper integrals are the essential features of the formulation. The tangential differential operator (TDO) in conjunction with integration by parts is an interesting procedure for reducing the order of singularity in fundamental solution kernels of traction BIEs when Kelvin-type fundamental solutions are employed. Kupradze [7] first presented an application using the TDO, and Sladek [8] employed the TDO in a curved crack solution. Bonnet [9] presented regularized formulations for the BEM employing the TDO for gradients in potential problems and in stress BIEs for elasticity problems, includin...

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Section: Dual Boundary Integral Equationsmentioning

confidence: 99%

A Boundary Element Formulation for Crack Analyses Incorporating a Cohesive Zone Model

Gonçalves¹,

Palermo²,

Proença³

2017

Int. J. CMEM

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“…(ξ'= 0.67) for conformal interpolations and (ξ'= 0.67, and ξ'= 0) for non-conformal interpolation. This strategy for the positions of collocation points in the DBEM was discussed in [28]. The singularity subtraction [29] and the transformation of variable technique [30] were employed for the Cauchy and weak-type singularity, respectively, when integrations were performed on elements containing the collocation points.…”

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confidence: 99%

Analysis of the Crack Propagation in Fiber-Reinforced Concrete Specimens Using the Bem

Brisola

Palermo

Almeida

2018

WIT Transactions on Engineering Sciences

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The mechanical properties of plain concretes are improved with the introduction of steel fibers like the material toughness along with limited crack widths. Numerical investigations on fiber reinforced concrete specimens use models for cohesive crack propagation in quasi-brittle materials. The present crack propagation analysis considered the single-edge notched beam under the three-point bending test. The material softening modeling used the cohesive law with two straight lines within a pure mode I for the crack growth. The dual boundary element method employed quadratic elements and the tangential differential operator in the traction boundary integral equation. The introduction of the constitutive law in the system of equations allowed the direct computation of the cohesive forces at each incremental loading step. The boundary element mesh employed continuous elements along the crack surface and the results obtained are compared with those available in the literature.

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“…The BIE for the distributed shear at internal points can be obtained using the constitutive equation (4) together with equations (7) and (11).…”

Section: Application Of the Tangential Differential Operatormentioning

confidence: 99%

“…The results in bending problems are obtained with the traction BIE using TDO instead of displacement BIE and they are compared to those in the literature where the problem was solved with traction BIE containing the strong singularity or with displacement BIE. Fracture problems using the dual boundary element method (DBEM) were not considered because results in DBEM can be changed, too, according to the collocation point position of the displacement BIE, as shown in [11]. Thus, fracture problems will be analyzed in other study.…”

Section: Introductionmentioning

confidence: 99%

A tangential differential operator applied to stress and traction boundary integral equations for plate bending including the shear deformation effect

Palermo¹

2011

Boundary Elements and Other Mesh Reduction Methods XXXIII

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Stress boundary integral equations (BIEs) are required in elastic or inelastic analyses of plate bending problems to obtain distributed shear, bending and twisting moments. Traction BIE, which is important to perform fracture analyses, is directly related to stress BIE. The collocation point position and the strategy to treat improper integrals are essential features studied in BIE for tractions or stresses at boundary points. The tangential differential operator (TDO) is used in stress and traction BIEs to reduce the strong singularities in the fundamental solution kernels and remaining singularities can be treated with the Cauchy principal value sense or the first order regularization. This study presents the application of the TDO for stress and traction BIEs used in plate bending models considering the shear deformation effect. The results in bending problems are obtained with traction BIE using TDO, instead of displacement BIE, and are compared to those in the literature where the problem was solved with traction BIE containing the strong singularity or with displacement BIE.

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A simple implementation of the dual boundary element method using the tangential differential operator for plane problems

Cited by 3 publications

References 4 publications

A Boundary Element Formulation for Crack Analyses Incorporating a Cohesive Zone Model

A Boundary Element Formulation for Crack Analyses Incorporating a Cohesive Zone Model

Analysis of the Crack Propagation in Fiber-Reinforced Concrete Specimens Using the Bem

A tangential differential operator applied to stress and traction boundary integral equations for plate bending including the shear deformation effect

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