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This paper presents a novel numerical procedure based on the combination of an edge-based smoothed finite element (ES-FEM) with a phantom-node method for 2D linear elastic fracture mechanics. In the standard phantom-node method, the cracks are formulated by adding phantom nodes, and the cracked element is replaced by two new superimposed elements. This approach is quite simple to implement into existing explicit finite element programs. The shape functions associated with discontinuous elements are similar to those of the standard finite elements, which leads to certain simplification with implementing in the existing codes. The phantom-node method allows modeling discontinuities at an arbitrary location in the mesh. The ES-FEM model owns a close-to-exact stiffness that is much softer than lower-order finite element methods (FEM). Taking advantage of both the ES-FEM and the phantom-node method, we introduce an edge-based strain smoothing technique for the phantom-node method. Numerical results show that the proposed method achieves high accuracy compared with the extended finite element method (XFEM) and other reference solutions.

The extended finite element method (XFEM) [

As no additional degrees of freedom are introduced, the implementation of the phantom node method in an existing finite element code is simpler. For example, arbitrary crack growths for nonlinear materials and cohesive zone models even for multiple cracks in two and three dimensions have already been implemented in ABAQUS [

No mixed terms (

Standard mass lumping schemes can be used due to the absence of an enrichment. There are several contributions to develop diagonalized mass matrices in standard XFEM [

The development of complex FE-formulations is much easier due to the lack of an enrichment. For example: when techniques such as EAS (enhanced assumed strain) or ANS (assumed natural strain) are used, special attention is required in a standard XFEM-formulation, particularly for problems with constraints. Those difficulties do not occur in the phantom node method [

The key drawback of the phantom node method compared to standard XFEM is its lower flexibility. It was developed for problems involving crack growth “only.” However, avoiding a crack tip enrichment significantly facilitates the enrichment strategy and the crack tracking algorithm.

A crack tip enrichment introduces more additional unknowns. It is well known that a topological enrichment is needed for accuracy reasons [

The nonpolynomial (and singular) crack-tip enrichment complicates integration [

The enrichment strategy and the crack growth algorithms are complicated, in particular in 3D.

Modeling crack growth with the phantom node method on the other hand is quite simple. Commonly, plane crack segments are introduced through the entire element though crack tip elements were developed [

Recently, Liu et al. constructed a new class of finite element methods based on strain smoothing. Among those methods, the so-called ES-FEM edge-based smoothed finite element method (ES-FEM) has been proven to be the most efficient and accurate one. In numerous application [

Therefore, we propose to couple the ES-FEM with the phantom-node method. We name the new element edge-based phantom node method (ES-Phantom node). In this paper, we focus on two-dimensional problems in linear elastic fracture mechanics (LEFM). However, our long term goal is to model fracture in nonlinear materials in 3D. Numerical results show high reliability of the present method for analysis of fracture problems.

This paper is organized as follows. In Section

Consider a deformable body occupying domain

A two-dimensional body containing a crack and boundary conditions.

The displacement field within an element

The decomposition of a cracked element into two superimposed elements.

Here, we choose the physical domain up to the crack line. Note that the crack line is a boundary in phantom node method. It is like the elements near the external boundary. So, we avoid singularity in phantom node method. The corresponding strain terms are written the same.

The strain field is obtained as follows:

The jump in the displacement field across the crack is calculated by

In this paper, the crack tip is forced to be located on the element’s boundary.

In the ES-FEM [

Construction of edge-based strain smoothing domains.

Numerical integration is implemented on chosen Gauss points as illustrated in Figures

The decomposition of a completely cracked smoothing domain into two superimposed smoothing domains.

The decomposition of a cracked smoothing domain containing crack tip into two superimposed smoothing domains.

The decomposition of a completely cracked smoothing domain into two superimposed smoothing domains.

The decomposition of a cracked smoothing domain containing crack tip into two superimposed smoothing domains.

Distribution of the stress components

Introducing the edge-based smoothing operation, the compatible strain

Using the following constant smoothing function

The approximation of the displacement field is written similarly to (

The connectivities of these superimposed smoothing domains which are cracked completely and the corresponding active parts are shown in Figure

nodes of smoothing domain 1 (

nodes of smoothing domain 2 (

The connectivity of a superimposed smoothing domain containing the crack tip and the corresponding active parts is shown in Figure

nodes of smoothing domain 1 (

nodes of smoothing domain 2 (

Using the strain smoothing operation, the smoothed strain associated with edge

In (

Using Gauss-Legendre integration along the segments of boundary

The stiffness matrix

All entries in matrix

We return to the two-dimensional body in Figure

Substituting the trial and test functions into (

The smoothed stress

Fracture parameters such as mode

In this paper, crack growth is governed by the maximum hoop stress criterion [

In all numerical examples, we are not using near-tip enrichment; that is, only discontinuous enrichment is used. This means that the best convergence rate attainable is 1/2 in the

Consider a sheet under uniaxial tension as shown in Figure

Sheet with edge crack under tension.

The strain energy and the error in the energy norm are defined as

The results of the ES-Phantom node are compared with those of the standard Phantom-node using triangular meshes and the XFEM-T3(0t) (the “standard” XFEM formulation without tip enrichment that only employs the Heaviside enrichment of (

Strain energy for the sheet with edge crack under tension.

The convergence in the energy norm versus

The convergence in the energy norm of XFEM versus

The convergence in the stress intensity factor

The convergence in the stress intensity factor

Note that the proposed method leads to a similar convergence rate to the standard XFEM and standard phantom-node, which is close to optimal (1/2) given the lack of tip enrichment. Also note that the error level of the proposed method is a fifth of an order of magnitude lower than the method compared with.

The computational efficiency in terms of the error in the energy norm and the relative error of ^{−1.88})/(Phantom10^{−1.93}) = 1.12 times as much as that of the standard Phantom, (ES-Phantom10^{−1.88})/(XFEM-T3(0t)10^{−2.01}) = 1.34 times of the XFEM-T3(0t) in term of error in energy norm; (2) (ES-Phantom10^{−0.48})/(Phantom10^{−0.58}) = 1.26 times as much as that of the standard Phantom and (ES-Phantom10^{−0.48})/(XFEM-T3(0t)10^{−0.58}) = 1.62 times of the XFEM-T3(0t) in term of relative error for

Computational efficiency of energy norm for the problem of a sheet with an edge crack under remote tension.

Computational efficiency of mode

In this example, we consider the edge crack geometry subjected to a shear load as shown in Figure

Sheet with edge crack under shear.

The results shown in Figures

Strain energy for a sheet with an edge crack under shear.

The convergence in the energy norm versus

The convergence in the energy norm of XFEM versus

The convergence in the stress intensity factor

The convergence in the stress intensity factor

The convergence in the stress intensity factor

The convergence in the stress intensity factor

In this section, the ES-Phantom node with structured and unstructured meshes is used for crack grow simulation. The dimensions of the double cantilever beam Figure

Double cantilever beam with an edge crack.

The crack growth increment, ^{4} used to enable a clear description and the evolution of the crack path. The result shows that the crack path for an initial angle

(a) Deformed shape of the double cantilever beam (structured mesh) and (b) crack path simulated by ES-Phantom node method (structured mesh) after ten-step growing in which the filled circles are the new crack tip after each step.

(a) Deformed shape of the double cantilever beam (unstructured mesh) and (b) Crack path simulated by ES-Phantom node method (unstructured mesh) after ten-step growing in which the filled circles are the new crack tip after each step.

Stress (a)

Stress (a)

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in framework of the ES-FEM to create the novel ES-Phantom node method. In this method, a cracked element is replaced by two superimposed elements and a set of additional phantom nodes. The two first examples were performed to investigate convergence rate in terms of strain energy and stress intensity factors. The results have shown that the ES-Phantom node is able to produce superconvergent solutions. Meanwhile, the last example has demonstrated the capability of the method to deal with the growing crack.

Future applications of this method may deal with the interactions among a large number of cracks in linear elastic solids with the purpose of obtaining the higher accuracy and efficiency in solving complicated crack interactions as shown in [

The authors gratefully acknowledge the support by the Deutscher Akademischer Austausch Dienst (DAAD). X. Zhuang acknowledges the supports from the NSFC (51109162), the National Basic Research Program of China (973 Program: 2011CB013800), and the Shanghai Pujiang Program (12PJ1409100).

^{2}) form for a unified formulation of compatible and incompatible methods: part I theory