Analysis of dynamic crack propagation in two-dimensional elastic bodies by coupling the boundary element method and the bond-based peridynamics

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Highlights

  • A new approach is presented to model dynamic crack propagation in solids by coupling the boundary element method (BEM) with the bond-based peridynamics (PD).

  • The coupled BEM-PD approach is easier to implement and does not require any blending functions, mapping parameters or overlap elements.

  • The approach takes full advantages of both the BEM and PD, and can achieve a higher computational efficiency.

  • A coarser spacing of nodes in PD subdomains can be allocated according to the BEM scheme to achieve a faster convergence without sacrificing the accuracy.

  • Both the explicit and implicit solution schemes for the coupled BEM-PD dynamic equations can deliver accurate results of the responses without ghost forces.

Abstract

A novel method for predicting dynamic crack propagation based on coupling the boundary element method (BEM) and bond-based peridynamics (BBPD) is developed in this work. The special feature of this method is that it can take full advantages of both the BEM and PD to achieve a higher level of accuracy and efficiency. Based on the scale of the structure and the location of cracks, the considered domain can be divided into a non-cracked region and a cracked region. For the non-cracked region, a meshfree boundary-domain integral equation method (meshfree-BDIEM) is employed in the analysis to reduce the dimension by one and to increase the computational efficiency. The bond-based PD is applied to simulate the cracked region, which can model the initiation and propagation of the cracks automatically. The boundary nodes from the BEM on the interfaces can interact with the material points from the PD directly. By using the displacement continuity and force equilibrium conditions on the interfaces, a combined model is obtained by merging the mass, stiffness, and force matrices from each region of the domain. Both the implicit and explicit BEM-PD coupled solution methods can deliver accurate results without inducing the ghost forces. Several benchmark problems of dynamic crack propagation have been modeled by using the method, which demonstrate that the developed BEM-PD coupled approach can be an efficient numerical tool to model the dynamic crack propagation problems.

Introduction

Analysis of dynamic cracks and their growth is an important issue as to avoid catastrophic failures of structures under dynamic loadings. However, dynamic crack problems, such as prediction of the initiation and propagation of cracks in solids, still represent a challenge for the classical continuum mechanics. This is because of the conflict between the continuity hypothesis in the continuum theory and the discontinuity in the actual problem to be solved [1]. Thus, more appropriate theories should be developed to match the discontinuous nature of the crack problems.

Peridynamics (PD) [2] based on non-local theory has been proposed as a particle method in modeling crack problems, where the solid is discretized using a set of material points, and the interactions between these points are expressed in an integral form. Hence, the continuity assumption is not required and the discontinuities such as fractures are captured inherently. Recently, some success has been achieved in applying PD to study the crack propagation problems [3], [4], and the PD theory itself has also been improved accordingly [5].

PD introduces the concept of structural damage for material points. However, PD, based on the non-local characteristic, depends on integration of many material points, which renders it a time-consuming numerical method. In order to increase the efficiency, various types of coupling schemes of the PD method with other computational methods were developed [6], [7], [8], especially, much effort has been devoted to coupling the PD grids to the FEM meshes for different applications. Macek and Silling [9] proposed to embed PD nodes in FEM elements over a region by displacement constraint. Coupling between PD and FEM was presented by [10], The coupling was introduced by defining an overlap region. The displacement and body force densities in the overlap region were determined using FEM and PD, respectively. Liu and Hong [11] coupled the PD with FEM by introducing the interface elements. The PD method was coupled with classical continuum mechanics using a morphing approach, in which the PD zone is discretized utilizing a discontinuous Galerkin approach [12], [13]. An effective way to couple FEM meshes and PD grids was proposed by Galvanetto et al. [14], [15], [16] to solve the static and dynamic problems. An implicit coupling FEM and PD was developed [17] for the dynamic problems of solid mechanics with crack propagation. The coupling domain is achieved by considering that the nodes and material points share the common information. Bie et al. [18] proposed the coupling approach of ordinary state-based PD with node-based smoothed FEM. The transient information was governed by the unified coupling equations of motion. An approach was presented by Sun et al. [19] to couple the PD theory with numerical substructure method for modeling structures with local discontinuities. The PD was integrated in the substructure model using interface elements with embedded PD nodes. Even with the above-mentioned approaches, FEM-PD coupling is still an active research area, because most of the coupling methods are affected by some kind of arbitrariness or new approximations [16].

As the name implies, the boundary element method (BEM) requires meshing only on the domain boundary. Thus, it reduces the dimensionality of the problem by one. Based on the analytical fundamental solutions, BEM can achieve improved accuracy, especially for modeling stress concentration problems, such as cracks and their propagations [20], [21]. Besides, the computational efficiency of the BEM has been improved significantly with the development of the fast solution methods in the last two decades [22], [23].

Therefore, exploring the advantages of coupling the BEM and PD can provide an opportunity for the development of a novel method of modeling crack propagations [24]. The static crack propagation analysis has been carried out by using a coupling approach of the BEM and PD [24]. The numerical examples solved by using this method show a higher computational efficiency than using the approach of the PD coupled with FEM [24]. Thus, the development of a convenient and efficient coupling scheme of the PD and BEM to combine the advantages of these two methods will be of great interests for the dynamic crack propagation problems as well.

In this paper, a coupled BEM and PD is developed to investigate the dynamic crack propagation problems. Different from the coupling approaches mentioned above, the BBPD subregion is directly coupled with the BEM subregion instead of using overlapping regions or the morphing strategy. The continuous displacements and the equilibrium forces are transformed by the interface elements. The dynamic equation with the mass and stiffness matrices for the whole domain is constructed. Both explicit and implicit time integration techniques are employed to investigate the stability and accuracy of the coupled solution method.

The remainder of this paper is outlined as follows: In Section 2, the formulations of bond-based peridynamics and the meshfree boundary-domain integral equation method are reviewed. In Section 3, the coupling method between the BBPD and BEM is described in detail. In Section 4, several numerical examples are studied to verify the proposed coupling approach. Some concluding remarks are made in the final section.

Section snippets

Review of the BBPD formulation

In the present work, the crack and potential cracked region are modeled by using the PD. Only the BBPD is considered in this present work. A brief outline of the BBPD is presented below, and more details can be found in Ref. [1].

The fundamental equation of motion for any material point in BBPD is given by: ρüx,t=Hxfη,ξ,tdVx+bx,t,xHxwhere HX is the neighborhood of point x, which is usually taken to be a spherical region of radius δ centered at point x, as shown in Fig. 1; ü and ρ are the

Numerical examples

In this section, the coupled BEM-PD method is applied to analyze some dynamic problems. First, the accuracy and efficiency of the developed method is investigated using a harmonic and transient vibration of a plate. Then, the developed coupled method is applied to model a few dynamic crack propagation problems.

In PD, the ratio m=δ/Δ (radius of the horizon to the nodal spacing) plays an important role in the accuracy and quality of numerical solution, especially in the case of failure analysis 

Conclusions

In this paper, a coupled method based on the boundary element method (BEM) and bond-based peridynamics (BBPD) is developed for modeling the dynamic crack propagation problems. The pre-crack and the potential propagation region are simulated by the BBPD dynamic equation, where the stiffness matrix is derived by the Taylor’s expansion of the stretch. The rest of non-crack region is simulated by using the meshfree boundary-domain integral equation method to reduce the problem dimension by one,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank the financial support from the National Natural Science Foundation of China (Project Nos. 11901283 and 11972179) and the Natural Science Basic Research Plan in Shaanxi Provincial of China (Project No. 2018JQ5079).

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