A natural stress boundary integral equation for calculating the near boundary stress field

Zhou

Fatigue Fract Eng Mat Struct

et al. 2015

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A B S T R A C T The singularity for the V-notch under the generalised plane deformation is investigated by the combination of the asymptotic analysis with the interpolating matrix method developed by part of the authors before. The displacement asymptotic expansions at the vicinity of the V-notch vertex are introduced into the equilibrium equations, which are transformed into a set of characteristic ordinary differential equations with respect to the notch singularity orders. The boundary conditions and interfacial compatibility conditions are also represented by the combination of the singularity orders and characteristic angular functions. The determination of the singularity orders and characteristic angular functions are transformed into solving the ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method. The present method is suitable for the singularity analysis for isotropic and orthotropic V-notches. It is versatile for analysing the stress singularity of single material V-notches, bimaterial V-notches, interface edges and cracks. The correctness of the results by the proposed method is ensured by the comparison with the published ones.Keywords asymptotic expansion; generalised plane strain state; orthotropic; stress singularity order; V-notch. N O M E N C L A T U R E(x, y, z) = Cartesian coordinate system (ρ, θ, z) = cylindrical coordinate system (1, 2, 3) = principal axis of the orthotropic material E 1 , E 2 , E 3 = elastic modulus μ 12 , μ 13 , μ 21 , μ 23 , μ 31 , μ 32 = Poisson ratios G 12 , G 23 , G 13 = shear modulus σ 1 , σ 2 , σ 3 , τ 12 , τ 23 , τ 13 = stress components in Cartesian coordinate system ε 1 , ε 2 , ε 3 , γ 12 , γ 23 , γ 13 = strain components in Cartesian coordinate system S = compliance matrix e T = transformation matrix for strain components between (ρ, θ, z) and (x, y, z) T = transformation matrix for strain components between (1,2,3) and (x, y, z) D = stiffness matrix D ij (i, j = 1, …, 6) = elements of the stiffness matrix σ ρ , σ θ , σ z , τ ρθ , τ θz , τ ρz = stress components in cylindrical coordinate system ε ρ , ε θ , ε z , γ ρθ , γ θz , γ ρz = strain components in cylindrical coordinate system (l k , m k , n k ) = direction cosines between the kth principle axis and (x, y, z) u ρ , u θ , u z = displacement components in cylindrical coordinate system A k = amplitude coefficient in asymptotic expansions λ k = stress singularity order λ = abbreviation of λ k N = number of characteristic values truncated Correspondence: C. Z. Cheng.

mentioning

confidence: 99%

Stress singularity analysis for orthotropic V‐notches in the generalised plane strain state

Zhou

Fatigue Fract Eng Mat Struct

et al. 2015

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“…The conventional numerical method can be used to analyse this remaining part. Experience has shown that the boundary element method has significant advantages in giving more accurate results with much less computational resources needed in comparison with other numerical methods 23–28 . For the elasticity problems, the displacement at any point y on the boundary of the problem domain can be presented by the displacement BIE where y is the source or load point and x the field point, i , j = 1, 2 for 2D problems.…”

Section: Evaluation Of the Intensity Coefficients Ak By Bemmentioning

confidence: 99%

“…Experience has shown that the boundary element method has significant advantages in giving more accurate results with much less computational resources needed in comparison with other numerical methods. [23][24][25][26][27][28] For the elasticity problems, the displacement at any point y on the boundary of the problem domain can be presented by the displacement BIE…”

Section: E V a L U A T I O N O F T H E I N T E N S I T Y C O E F F I mentioning

confidence: 99%

Analyse the role of the non‐singular stress in brittle fracture by BEM coupled with eigen‐analysis

Fatigue Fract Eng Mat Struct

Récho

et al. 2012

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A coupled model resulting from the boundary element method and eigen‐analysis is proposed in this paper to analyse the stress field at crack tip. This new combine method can yield several terms of the non‐singular stress in the Williams asymptotic expansion. Then the maximum circumferential stress (MCS) criterion taken the non‐singular stress into account is introduced to predict the brittle fracture of cracked structures. Two earlier experiments are re‐examined by the present numerical method and the role of the non‐singular stress in the brittle fracture is investigated. Results show that if more terms of non‐singular stress are taken into account, the predicted crack propagation direction and the critical loading by MCS criterion are much closer to the existing experimental results, especially for dominating mode II loading conditions. Moreover, numerical results manifest that Williams series expansion can describe the stress field further from the crack tip if more non‐singular stress terms are adopted.

“…After the unknown displacement and traction on boundary are solved by the boundary integral equation, any physical values of interested interior points can be yielded successfully by the integral equations of interior points. The BEM has been successfully applied to solving multidiscipline problems, such as the potential problem (Zhou et al , 2008; Cheng et al , 2015), the elastic problem (Niu et al , 2005; Cheng et al , 2011) and the vibration problem (Zheng et al , 2016; Tsiatas and Yiotis, 2013). However, the FGM has to be divided into many subdomains when the conventional BEM is used to model it, which will increase the computational amount redundantly.…”

Section: Introductionmentioning

confidence: 99%

A state space boundary element method for elasticity of functionally graded materials

Han

et al. 2017

Purpose The state space method (SSM) is good at analyzing the interfacial physical quantities in laminated materials, while it has difficulty in calculating the mechanical quantities of interior points, which can be easily evaluated by the boundary element method (BEM). However, the material has to be divided into many subdomains when the traditional BEM is applied to analyze the functionally graded material (FGM), so that the computational amount will be increased enormously. This study aims to couple these two methods to strengthen their advantages and overcome their disadvantages. Design/methodology/approach Herein, a state space BEM in which the SSM is coupled by the BEM is proposed to analyze the elasticity of FGMs, where one BEM domain is set and the others belong to SSM domains. The discretized elements occur only on the boundary of the BEM domain and at the interfaces between different SSM domains. In SSM domains, the horizontal interfaces of FGMs are discretized by linear elements and the variables along the vertical direction are yielded by the precise integration method. Findings The accuracy of the proposed method is verified by comparing the present results with the ones from the finite element method (FEM). It is found that the present method can provide accurate displacements and stresses in the FGMs by fewer freedom degrees in comparison with the FEM. In addition, the present method can provide continuous interfacial stresses at the interfaces between different material domains, while the interfacial stresses by the FEM are discontinuous. Originality/value The system equations of the state space BEM are built by combining the boundary integral equation with the state equations according to the continuity conditions at the interfaces. The mechanical parameters of any inner point can be evaluated by the boundary integral equation after the unknowns on the boundaries and interfaces are determined by the system equation.