Hybrid meshless displacement discontinuity method (MDDM) in fracture mechanics: Static and dynamic
Introduction
It is well known that the numerical simulation of crack-growth processes is mature as many numerical strategies including the finite element method (FEM) have been developed. As a general numerical tool, FEM has been developed for crack propagation simulation in solid structures (Kolednik et al., 2010, 2016; Wang and Siegmund, 2006). The early attempt to model crack growth in mixed-mode conditions was reported by Gallagher (Gallagher et al., 1978), Rice and Tracey (1973), Shephard et al. (1985) and Rice (1968). Recently, the extended finite element method (XFEM) (Yu et al., 2009; Kumar et al., 2015; Wang et al., 2016) was proposed for fatigue/fracture analysis in nonhomogeneous materials. In spite of the great success of general boundary value problems, the new and advanced computational methods are still required due to the need for computational accuracy and efficiency, and structure complexity. As the discontinuities of the stress/strain by FEM between elements affect the accuracy significantly, the boundary element method (BEM) is able to achieve high accurate solutions. The two main advantages of BEM is the reduction of the spatial dimensions by at least one and the high accuracy is achieved especially if the domain of interest is infinite or semi-infinite. Early investigations of mixed-mode crack growth conditions by BEM were reported by Ingraffea et al. (1983) and Grestle (1986) for two- and three-dimensional problems with the multi-region technique. Cen and Maier (1992) applied BEM to simulate crack growth in concrete structures. In the 1990's, the Dual Boundary Element Method (DBEM) with a single region technique for the crack growth analysis was demonstrated by Portela et al. (1993) for two-dimensional and by Mi and Aliabadi, 1994, 1995 for three-dimensional problems. One of the advantages of DBEM is that the crack extension procedure can be modelled easily by new elements. For DBEM applications in crack mechanics, a general review was given by Aliabadi (2002). Apart from DBEM, the indirect boundary element method is another accurate method formulated with the principle of superposition including the Fictitious Load Method (FLM) and the Displacement Discontinuity Method (DDM). This was reported by Crouch (Crouch and Starfield, 1983) in the “Boundary Element Methods in Solid Mechanics”. The DDM was extended to static/dynamic 2D/3D fracture mechanics by Wen et al. in (Wen, 1988, 1989, 1991; Wen et al., 1996a, 1996b). How to interpolate a variable accurately using the values of irregular node arrangement in a domain is a fundamental task in meshless methods. The multiquadric Radial Basis Function (RBF) was studied by Hardy (1971) in 1971 for topographical surfaces, and this can be considered as the first development for meshless method. Similar to the Moving Least Square (MLS) algorithm, the compact support RBF has been explicitly constructed to multivariate surface reconstruction. Later Belyschko et al. (Belytschko et al., 1994) developed the element-free Galerkin method (EFGM) based on accurate interpolation methods including the MLS and RBF respectively. Pathak et al. (2014) developed and utilised the enriched EFGM method to investigate fatigue problems in homogeneous and bi-material interfacial cracks structures. Jameel and Harmain (2015) also investigated fatigue crack growth of material discontinuities using the EFG method, and Muthu et al. (2016) also investigated the impact of T-stress on crack propagation using a variant of the EFG method. In addition, the local support domain technique provides a form of theoretical basis for large scale problems, see Hon et al. (Hon and Mao, 1997). Atluri et al. (Atluri, 2004) reported a series of Meshless Local Petrov-Galerkin formulations (MLPGs) for general partial differential equations with MLS approximation from the past two decades. The local boundary integral equation with the MLS and RBF was reported by Sladek et al., 2005, 2006 to deal with fracture problems in anisotropic non-homogeneous media. With enriched RBF at the crack tip, Wen and Aliabadi (2012) demonstrated the application of meshless method to fracture problems with functionally graded materials. However, a drawback of the meshless method is the accuracy and convergence, precisely to the modelling of singular stresses at the crack tip by using the MLS and RBF interpolations. In the present paper, we aim to develop a hybrid method with the DDM and mshless method in order to take advantages of both the boundary element method (high accuracy) and the meshless method (body force terms). The advantages and disadvantages for meshless strong form method and displacement discontinuity method are listed in Table 1.
The time-dependent values of displacement, stress and stress intensity factor are obtained from an inverse transform of the corresponding Laplace-transformed quantities. Great attention has been paid to the numerical inversion of the Laplace transformation. General introduction of the Laplace invers was given by Fu et al., 2013, 2018, 2019. In this paper, the formulations for crack problem are coupled with distributed dislocation on the crack surface (discontinuous field in infinite plate) and meshless solution (continuous field) to deal with fracture problems. The crack growth is simulated simply by adding a new crack segments ahead of the crack tip. The Laplace transform technique is applied to deal with dynamic problems by using static Kelvin solution of displacement discontinuity for 2D elasto-dynamic problems and the Durbin inversion method (Durbin, 1975) is used to obtain the time-dependent solution. Numerical results of cracked rectangular plate and cracked circular plate are presented to illustrate the applicability and degree of accuracy with the MDDM. Comparisons with analytical and BEM results show a good agreement.
Section snippets
Displacement discontinuity method
The displacement discontinuity method is attributed to one of the indirect boundary element techniques. Using the displacement discontinuity method, a high level of accuracy and rapid convergence is obtained when applied in fracture analysis to determine the stress intensity factors. Consider a concentrated force acting at point as shown in Fig. 1 in an infinite plate, the fundamental solution of the stress tensor is given, for 2D plane strain problem (Aliabadi, 2002), by
Meshless approach with radial basis function
Consider a domain surrounded by a boundary and a sub-domain centred at point η shown in Fig. 3. With scattered nodes () in the sub-domain, function u can be approximated aswhere is the number of scattered points in the sub-domain , is the vector of the Radial Basis Function (RBF) associated nodal values and centred at the point η, are unknown coefficients, Q is the
Hybrid method for crack problems
The DBEM or DDM are very convenient to analyze crack problems due to their high accuracy and efficiency. However, for nonlinear material properties and geometrically nonlinear problems, the fundamental solutions are not available and therefore, the domain integrals are required. To deal with body forces, the meshless method such as point collocation method is the first option due to its simplicity. Therefore, a hybrid method with the DDM and meshless method is expected to take advantages of
A rectangular plate with a central crack
In order to validate the applicability of the MDDM proposed in this paper, a rectangular plate containing a central crack is considered as shown in Fig. 5 loaded by a uniform tensile stress on the top and bottom of the plate. Firstly, the regularly distributed nodes () are specified in the domain for the meshless method and segments () for the DDM are used on the crack surface. The Poisson ratio is taken as 0.3 with plane strain assumption. Densities of nodes and segment
Conclusion
This paper presented a hybrid method with meshless and displacement discontinuity procedure for two-dimensional fracture mechanics under static and dynamic loads. By BEM including DBEM, the fundamental solutions have to be derived both for static and dynamic problems. I addition, the fundamental solutions are complicated either in the time domain or in Laplace transformed domain. The main advantage of MDDM is that the static fundamental solutions can be adopted to deal dynamic case directly.
Authors statement
J. Li: Methodology, Software. J. Slade: Writing - original draft. V. Sldek: Investigation and formulations. P.H. Wen: Communication author, Programme and code, date preparations
Acknowledgment
The authors acknowledge the supports of the Slovak Science and Technology Assistance Agency registered under number APVV-18-0004, VEGA-2/0061/20, Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ3519) and Scientific Research Project of Hunan Provincial Office of Education (Grant no. 17B003).
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