Solution of multiple cracks in a finite plate of an elastic isotropic material with the distributed dislocation method

doi:10.1016/S0894-9166(14)60036-7

Acta Mechanica Solida Sinica

Volume 27, Issue 3, June 2014, Pages 276-283

https://doi.org/10.1016/S0894-9166(14)60036-7 Get rights and content

Abstract

This paper presents a numerical solution to model multiple cracks in a finite plate of an elastic isotropic material. Both the boundaries and the cracks are modeled by distributed dislocations. This method results in a system of singular integral equations with Cauchy kernels which can be solved by Gauss-Chebyshev quadrature method. Four examples are provided to assess the capability of this method.

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Cited by (8)

Interaction of multiple straight cracks and elliptical inclusions in a finite plate due to mismatched thermal expansion
2020, Engineering Fracture Mechanics
Citation Excerpt :
However, Jin [13] presented a closed-form for exterior Eshelby’s tensor to solve the exterior elastic fields and this can make it more convenient to solve the multiple inclusion problems. Additionally, the distributed dislocation method has been widely used to solve various kinds of crack problems in infinite plates, finite plates and half plates [16–25]. Previous research [19] shows this method has considerable computational efficiency for solving a problem with large numbers of cracks and a problem of 100 cracks are solved with high efficiency.
The purpose of this paper is to study the interaction between multiple elliptical inclusions and straight cracks in a finite plate due to a uniform temperature change. In this solution, Eshelby’s equivalent inclusion method involving both interior Eshelby’s tensor and exterior Eshelby’s tensor is applied to calculate the thermal stress fields of an infinite plate containing multiple elliptical inclusions under a uniform temperature change first. Then the multiple cracks and the boundary are modeled by continuous distributions of dislocation densities in an infinite plate. Based on the stress boundary conditions of the cracks and boundary, a system of singular integral equations with Cauchy kernels are obtained. After solving the singular integral equations with Gauss–Chebyshev numerical quadrature, the stress intensity factor of each crack can be calculated. Besides, the finite element method is employed to examine the accuracy and efficiency of the presented method. Finally, the effects of the material and geometric parameters on the normalized stress intensity factors of the cracks are studied.
Automated numerical simulation of the propagation of multiple cracks in a finite plane using the distributed dislocation method
2019, Comptes Rendus - Mecanique
Citation Excerpt :
So, in this part, we will firstly introduce the stress caused by an edge dislocation. Based on previous research [10,15,19], we will deduce the governing integral equations for a finite plane containing multiple straight cracks and kinked cracks. Now, Eqs. (12) and (13) exactly provide the integral equations for the solution to the problem of a finite plane containing multiple straight and kinked cracks when tractions are applied on the boundary.
In this paper, an automated numerical simulation of the propagation of multiple cracks in a finite elastic plane by the distributed dislocation method is developed. Firstly, a solution to the problem of a two-dimensional finite elastic plane containing multiple straight cracks and kinked cracks is presented. A serial of distributed dislocations in an infinite plane are used to model all the cracks and the boundary of the finite plane. The mixed-mode stress intensity factors of all the cracks can be calculated by solving a system of singular integral equations with the Gauss–Chebyshev quadrature method. Based on the solution, the propagation of multiple cracks is modeled according to the maximum circumferential stress criterion and Paris' law. Several numerical examples are presented to show the accuracy and efficiency of this method for the simulation of multiple cracks in a 2D finite plane.
Analysis of the effect of a micro-crack on plastic zone of the edge macro-crack tip by macroscopic and microscopic methods
2018, Engineering Fracture Mechanics
Citation Excerpt :
It was introduced detailedly by Hills et al. [36]. DDT was applied to solve various crack problems by the previous works [37–41]. What is more, DDT can be applied to model crack tip plasticity.
The effect of the micro-crack on plastic zone ahead of the macro-crack tip (PZMA) is analyzed based on macroscopic and microscopic methods. From the macroscopic perspective, the plastic deformation is due to the local stress near crack tip is more than the yielding limit. So the complete stress field is calculated based on the distributed dislocation technique (DDT) and the plastic zone (PZ) is determined by von Mises yielding criterion (MYC). On the other hand, the crack tip plasticity is simulated by finite element method (FEM). From the microscopic perspective, the plastic deformation is caused by dislocations emitted from crack tip. So PZ is represented by dislocations, and it is determined based on DDT and dislocation-free zone (DFZ) model. The results show the micro-crack located in front of PZMA has the amplifying effect on PZMA. PZMA is separated by the micro-crack located in PZMA. The micro-crack behind PZMA has no effect on PZMA. PZMA has the shielding effect on the macro-crack propagation, and the shielding effect increases with PZMA increasing. The micro-crack has the amplifying effect on PZMA at about x₀/a > 0.5, while it has the shielding effect at about x₀/a < 0.5. These results are useful for predicting and interpreting the plastic behaviors at the crack tip.
Numerical studies of an array of equidistant semi-permeable inclined cracks in 2-D piezoelectric strip using distributed dislocation method
2016, International Journal of Solids and Structures
Citation Excerpt :
Zhang et al. (2013) extended the concept of DDM to study the interaction between the cracks and circular inclusion present in a finite plate. Zhang et al. (2014) compared the DDM solution for multiple cracks in a finite plate of an isotropic material with the FEM. Most of the periodic cracks problems have been solved for collinear cracks present in an infinite domain.
An array of equidistant inclined cracks present in 2-D piezoelectric strip is numerically analyzed using distributed dislocation method (DDM). The piezoelectric strip is a cutout of an infinite domain so that the width-to-crack-length ratio is about 100 times larger than the aspect ratio of height-to-crack-length of the specimen. The inclined equidistant cracks are modeled as a continuous distribution of dislocations, and the problem is thus reduced into simultaneous Cauchy's type singular integral equations, which are then solved by the Gauss–Chebychev quadrature method. The study is carried out with respect to number of cracks, aspect ratio, inclination angle, inter-crack space distance, crack-face electrical boundary conditions and electrical loading. Various numbers of inclined equidistant cracks with different inclination angles under semi-permeable crack-face boundary conditions are examined. To show the accuracy and efficacy of DDM in modeling finite cracked piezoelectric problems, fracture parameters obtained by the DDM are validated against the results of extended finite element method under impermeable crack-face boundary conditions for two particular cases i.e., inclined crack and two unequal collinear cracks. The present results show the significant effects of aspect ratio, distance between cracks, inclination angle, crack face boundary conditions, and electrical loading on fracture parameters. The DDM developed to analyze an array of equidistant inclined cracks in 2-D piezoelectric strip can be extended to inclined periodic cracks in 2-D piezoelectric strip under various electrical boundary conditions.
An improved model for chlorophyll-a concentration retrieval in coastal waters based on uav-borne hyperspectral imagery: A case study in qingdao, china
2020, Water (Switzerland)
Effect of a Micro-crack on the Edge Macro-crack Propagation Rate and Path Under Mixed Loads
2019, Acta Mechanica Solida Sinica

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: Project supported by National Natural Science Foundation of China (No. 51174162).

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Solution of multiple cracks in a finite plate of an elastic isotropic material with the distributed dislocation method

Abstract

International Journal of Solids and Structures

International Journal of Solids and Structures

International Journal of Solids and Structures

International Journal of Solids and Structures

Engineering Analysis with Boundary Elements

Interaction between a circular inclusion and an arbitrarily oriented crack

Journal of Applied Mechanics

The interface crack in a combined tension–compression and shear field

Journal of Applied Mechanics

Effect of friction on the interface crack loaded in shear

Journal of Elasticity

Effects of partial closure and friction on a radial crack emanating from a circular hole

International Journal of Fracture

Open cracks at or near free of positive radial stresses along the crack line in the edges

Journal of Strain Analysis for Engineering Design