Modified crack closure integral technique for extraction of SIFs in meshfree methods

https://doi.org/10.1016/j.finel.2013.09.005Get rights and content

Highlights

  • Adoption of modified crack closure integral (MCCI) in meshfree methods for evaluation of stress intensity factors (SIFs).

  • Comparison of MCCI technique with M-Integral technique, displacement method, etc.

  • Case studies on mixed-mode crack problems involving remote loading, crack face loading and thermal loading.

  • Effect of enrichment functions, order of Gauss integration, domain of influence and nodal density on accuracy of SIFs.

Abstract

A new approach for extracting stress intensity factors (SIFs) by the element-free Galerkin (EFG) class of methods through a modified crack closure integral (MCCI) scheme is proposed. Its primary feature is that it allows accurate calculation of mode I and mode II SIFs with a relatively simple and straightforward analysis even when a coarser nodal density is employed. The details of the adoption of the MCCI technique in the EFG method are described. Its performance is demonstrated through a number of case studies including mixed-mode and thermal problems in linear elastic fracture mechanics (LEFM). The results are compared with published theoretical solutions and those based on the displacement method, stress method, crack closure integral in conjunction with local smoothing (CCI–LS) technique, as well as the M-integral method. Its advantages are discussed.

Introduction

Meshfree methods (MMs) were developed to overcome some of the difficulties of solid mechanics in analyzing certain type of problems using traditional computational tools like the finite element methods (FEMs) and boundary element methods (BEMs) [1]. One class of problems, for example, involves the propagation of a crack front where extensive remeshing is often required. Unlike mesh-based methods, MMs do not require discretization. Furthermore, the shape functions of MMs are higher order continuous [2] than that of the FEM. These shape functions are well suited for applications involving crack analysis. The EFG method [3], [4], [5], [6], which is one of the formulations of MMs and widely applied for solid mechanics problems, is of concern here.

The shape functions of the EFG method are obtained from a moving least-squares (MLS) approximation [7]. Their continuity is dependent on the continuity of the weight functions used. To model a discontinuity in the geometry arising out of a crack, the weight functions are appropriately modified. There exist several strategies to model strong discontinuities [8], [9], [10], [11], [12]. However, most methods need large nodal densities to obtain an accurate solution. One approach, which does not require large nodal densities, involves partition-of-unity enrichment of targeted nodal shape functions. The pertinent examples include Heaviside/jump functions and the Williams' crack-tip displacement solutions [13] to model the discontinuity and capture the singularity at the crack tip respectively. These were adopted from the eXtended finite element method (XFEM) [14], [15], [16] and gave rise to development of the eXtended element-free Galerkin (XEFG) method [17]. Although this method does not require large nodal densities, it introduces additional nodal degrees of freedom in the global system of equations.

SIFs play a pivotal role in the application of LEFM principles. Although there are a variety of techniques available in FEM and BEM such as the displacement method [18], stress method [19], the stiffness derivative procedure [20], J-integral [21] and crack closure integral (CCI) [22], [23], [24], [25] to extract the SIFs, only few techniques find their adoption in MMs. The displacement and stress methods are the earliest and simplest methods to extract the SIFs. Nevertheless, their accuracy depends on the density of the mesh. Also, crack opening displacement (COD) required for the displacement method is not directly available in the XEFG method. The J-integral is accurate but relatively computationally expensive when it comes to the extraction of the SIFs in a mixed-mode problem. The interaction integral/M-integral [26], derived from the J-integral, is widely used for extracting the mixed-mode SIFs in MMs. However, they require knowledge of the auxiliary displacement and stress field solutions which are material dependent. The CCI approach is rarely used within the framework of MMs despite the fact that it is simple to implement, accurate and offers computationally easier ways of mode separation. Previously, CCI was calculated by inserting a high stiffness spring [27] or enclosing an auxiliary FE zone around the crack tip [28]. Lately, the CCI has been combined with a local smoothing technique to compute SIFs accurately in isotropic [29] and functionally graded materials [30].

The MCCI or virtual crack closure technique (VCCT) has been in existence for a long time and it has been exploited abundantly in FEM and, to a lesser extent, in BEM to extract the SIFs. Its adoption in MMs is not that straightforward because it requires the knowledge of crack closure forces. In this paper, a novel method to extract the nodal forces in the presence of regular nodal discretization in EFG methods is described. The closure nodal forces at the crack tip and at nodes ahead of it are multiplied with the opening displacements at the corresponding nodes behind the tip to obtain the strain energy release rates (SERRs). To bring out the special issues associated with the adoption, a number of case studies, involving crack face and thermal loadings, are solved. Thereby the procedural details, effectiveness and accuracy obtainable are presented. In order to emphasize the effectiveness and accuracy of the method, the results obtained by the method are compared with those obtained by the popular M-integral technique, analytical solutions and, wherever possible, by the CCI–LS [29].

The outline of the paper is as follows: The formulation of the EFG method and various crack modeling techniques are presented in Section 2. In Section 3, the MCCI technique to extract the SIFs for EFG methods is presented. Additionally, other popular techniques such as M-integral, CCI–LS, displacement and stress methods are briefly described in Section 4. Section 5 gives a detail analysis on accurate extraction of nodal forces. In Section 6, the MCCI and other techniques are applied to a variety of problems to demonstrate the superiority of the proposed approach. This is followed by concluding remarks in Section 7.

Section snippets

Discontinuity modeling using the EFG method

In MMs, the domain is discretized with nodes. Each node interacts with other nodes through weight functions which may vary in size and profile. Fig. 1(a) shows a geometry discretized with nodes with regular circular weight functions. In the EFG method, any scalar field variable, u(x), can be represented asu(x)=I=1nΦI(x)uIwhere ΦI(x) in Eq. (1) are the nodal shape functions and are approximants and not interpolants as in the case of FEM. uI are the nodal field values.

The moving least squares

Modified crack closure integral for EFG methods

The SERR associated with a crack extension is equal to the work done in closing a crack that is extended back to its original configuration. It can be calculated through Irwin's crack closure integral (Fig. 5), given byGI=limΔa012Δa0Δaσyy(x)vCOD(xΔa)dxGII=limΔa012Δa0Δaτxy(x)uCOD(xΔa)dxwhere Δa is the crack tip elemental length in FEM.

Although the displacement method is a simple technique available to obtain the SIFs, their accuracy can be significantly improved by calculations involving

Other popular approaches to obtain SIFs

The M-integral/interaction integral [26] which is based upon the J-integral [21] is frequently used to calculate the mixed-mode SIFs, in fracture problems, within the realm of MMs. In the presence of a crack face [35] and a thermal loading [36], this integral is given byM=A(σijui,1aux+σijauxui,1σikεikauxδ1j)q,jdA+Aσijauxα(ΔT),1δij)qdAΓc++ΓCtjuj,1auxqdΓσijaux,εikaux,anduiaux are auxiliary state values that correspond to the established theoretical solutions. Eq. (19) is numerically

Accurate extraction of the nodal force

Consider a bar of uniform cross section, A, of unit area and of unit length, L, as shown in Fig. 12. The bar is subjected to a point tensile load, P, of 10 MPa at its end. This problem is solved using 1D analysis with displacement (ux) set to zero at the origin.

The bar is discretized with different nodal distributions. The domain of influence (dI) of each node is constant and is set at 1.75(L/(n−1)) where n is the total number of nodes. Here n is equal to 11.

The force at the node of interest, j,

Results

In this section, the new technique was applied to a number of crack problems including crack face pressure loading and thermal-mechanical loading. The SERR, and in turn SIFs, computed using the proposed method and other standard methods were compared with the solutions available in the literature.

The SIFs were extracted using the four techniques: MCCI, M-integral, displacement method and stress method. The MCCI technique was employed here by calculating crack closure forces at the two nodes, in

Conclusions

A simple and an accurate way of extracting mixed-mode stress intensity factors within the framework of EFG method, based on the visibility and diffraction approaches, and the XEFG method is presented. It involves computing crack closure forces only at two nodes, the crack tip and a node ahead of it, and extraction of opening displacements at two nodes behind the crack tip. The effectiveness and special issues associated with the adoption of the MCCI/VCCT have been brought out through solutions

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