Dynamic crack propagation simulation with scaled boundary polygon elements and automatic remeshing technique

Abstract

An efficient methodology for automatic dynamic crack propagation simulations using polygon elements is developed in this study. The polygon mesh is automatically generated from a Delaunay triangulated mesh. The formulation of an arbitrary n-sided polygon element is based on the scaled boundary finite element method (SBFEM). All kind of singular stress fields can be described by the matrix power function solution of a cracked polygon. Generalised dynamic stress intensity factors are evaluated using standard finite element stress recovery procedures. This technique does not require local mesh refinement around the crack tip, special purpose elements or nodal enrichment functions. An automatic local remeshing algorithm that can be applied to any polygon mesh is developed in this study to accommodate crack propagation. Each remeshing operation involves only a small patch of polygons around the crack tip, resulting in only minimal change to the global mesh structure. The increase of the number of degrees-of-freedom caused by crack propagation is moderate. The method is validated using four dynamic crack propagation benchmarks. The predicted dynamic fracture parameters show good agreement with experiment observations and numerical simulations reported in the literature.

Introduction

Dynamic fracture studies involve not only the singular stress field at the crack tip but also its interaction with stress waves that are reflected from the structure’s boundaries [1]. Analytical solutions to such problems are very complicated and are available only for simple geometries and loading conditions. These solutions are more complicated when crack propagation is involved. Consequently, numerical methods are usually employed to solve dynamic crack propagation problems. The finite element method (FEM) has been used for such simulations since the 1970s. Early attempts involve simple nodal release techniques [2], [3], which are mesh dependent and inaccurate. Nishioka and Atluri [4], [5] developed a special singular finite element formulation that is very accurate for dynamic crack propagation analysis. This formulation, however, has only been applied to problems where the crack paths are defined a priori.

Today, the standard procedure for dynamic crack propagation simulations with FEM employs automatic local remeshing algorithms together with a rosette of singular quarter-point elements [6], [7], [8], [9]. Although many types of dynamic crack propagation problems have been successfully solved, such simulations still remain very challenging. This is because the FEM cannot accurately model the singular stress field emanating from crack tips (even with singular quarter-point elements) unless a very dense mesh is used to discretise the area in the vicinity of the crack tip. This increases the complexity of the remeshing algorithms used together with FEM for such simulations. Research is still ongoing to improve the FEM or to devise superior methodologies to model dynamic crack propagation in structures. These include the extended finite element method (XFEM) e.g. [10], meshless methods [11], boundary element methods (BEM) [12] and spacetime discontinuous Galerkin (SDG) finite element methods [53].

Many dynamic crack propagation problems have been modelled with the XFEM [17], [19], [20] and meshless methods [13], [14], [15], [18]. Both methodologies can model crack propagation problems without any remeshing. In XFEM, the discontinuities of the crack surfaces are represented by introducing Heaviside functions to enrich the nodes of elements cut by crack paths. Additional singular enrichment functions are introduced to model the singular stress field around the crack tip. In meshless methods, various methods have been proposed to treat the crack surfaces e.g. introducing surfaces [13] and discontinuous node enrichment [14]. Like the XFEM, singular enrichment functions can be introduced to model the singular stress field more accurately [15]. Stress intensity factors (SIFs) are usually computed using domain integrals [16]. These enrichment functions introduce additional degrees-of-freedom (DOF) and require special integration rules in order to compute the stiffness matrix. Richardson et al. [17], for example, discussed the many techniques that have been proposed to resolve the technical difficulties associated with these enrichment functions in the XFEM.

In the BEM, the geometry of the problem is vastly simplified because only the boundaries of the structure need to be discretised. Remeshing during crack propagation is also very simple because only new elements need to be added to the crack path as the crack propagates. This appealing feature is however, offset by the complex BEM formulations that involve analytical time and spatial integration of temporal shape functions [21] and convoluted fundamental solutions [22]. Different BEM approaches have been developed to model dynamic crack propagation e.g. Laplace transforms [23] and dual reciprocity methods [24], [25], [26].

In the SDG finite element method, the governing equations of equilibrium are discretised simultaneously both in space and time. This results in a system of equations that satisfies the balance of linear and angular momentum over every space–time element in the computational mesh to within machine precision. The formulation does not require any stabilization as it is dissipative. These properties allow the method to accurately resolve high stress gradients in the computational mesh caused by crack-tip fields and sharp wave fronts. The SGD finite element method was developed for linear elasto-dynamic fracture by Abedi et al. [53] and was recently extended to elasto-dynamic cohesive fracture [54] and dimensional analyses involving linear and nonlinear cohesive elasto-dynamic fracture [55].

Recently, a number of numerical methods based on arbitrary n-sided polygons have been developed for structural analysis e.g. polytope elements with barycentric coordinates shape functions [27], [28], [29], the Voronoi cell FEM [30], [31] and polygonal smoothed FEM [32]. Compared with standard 3-noded triangular and 4-noded quadrilateral finite elements, polygons with n > 4 have more nodes. Therefore, their shape functions are naturally higher in order and leads to more accurate solutions. Polygons are also more flexible in meshing complex geometries, leading to the use of simpler remeshing algorithms [33], [34] when modelling crack propagation. Nodal enrichment with discontinuous jump functions can be implemented with these methods to model crack propagation [35], [36]. All these methods [33], [34], [35], [36] use singular enrichment functions to model the singular stress field in the vicinity of crack tips and domain interaction integrals to evaluate the SIFs.

The scaled boundary finite element method (SBFEM) [37] is a semi-analytical method that has been shown in various studies [38], [39], [40], [41] to be very efficient in fracture analyses. The distinct feature that contributes to its efficiency is that orders of singularity of any kind emanating from a crack, notch or material junction are naturally represented in its solutions [39]. In the SBFEM, the stress field in the vicinity of a crack or notch tip is represented by an analytic matrix power function in the radial coordinate ξ with origin at the crack or notch tip. The matrix exponents in ξ, has the form ξ^−S. Depending on if a crack, notch or multi-material junction is modelled, the matrix of orders of singularity S naturally assumes certain forms that reflect the stress singularity behaviour [39]. For example, if a crack in a homogeneous isotropic material is modelled, S has the form S = 0.5I and replicates the square root singularity of cracks in homogeneous isotropic materials. This analytical representation of singular stress fields makes it possible for fracture problems to be modelled with few large subdomains having significantly fewer DOF compared to the FEM and without any nodal enrichment as in XFEM. This advantage can be further exploited in crack propagation modelling, leading to a simple remeshing procedure [40], [41]. However, if only a few large subdomains are used, it is difficult to generalise the SBFEM for single- or multiple crack propagation problems exhibiting complex crack paths. A more general implementation of the SBFEM that has such capabilities was proposed by Ooi et al. [42] for quasi-static crack propagation. In this approach, the computational domain is discretised with a polygon mesh. There are no restrictions to the number of sides a polygon can have so as long as the visibility criterion in SBFEM is satisfied. Each polygon is treated as a SBFEM subdomain. Like the SBFEM, it can accurately model the singular stress fields in the vicinity of crack tips. A very simple local remeshing algorithm that makes minimal changes to the global mesh structure by simply splitting the polygons cut by the crack path was used to propagate the crack.

This study extends the polygon SBFEM to elasto-dynamic crack propagation simulations. A new local remeshing algorithm is developed to model rapidly propagating cracks that depend on the crack velocity. Compared to the remeshing algorithm developed in [42], this approach is more general. It is independent of the initial mesh density and can remesh any polygon mesh regardless of the magnitude of the crack propagation length. This paper is organised as follows: Section 2 summarises the polygon SBFEM formulation for elasto-dynamics and the procedures used to extract the generalised dynamic SIFs. Section 3 describes the development of the remeshing algorithm. Section 4 demonstrates the application of the method to four dynamic crack propagation benchmarks. Section 5 summarises the major conclusions that can be drawn from this study.