Orthotropic enriched element free Galerkin method for fracture analysis of composites

https://doi.org/10.1016/j.engfracmech.2011.03.011Get rights and content

Abstract

A new approach for modeling discrete cracks in two-dimensional orthotropic media by the element free Galerkin method is described. For increasing the solution accuracy, recently developed orthotropic enrichment functions used in the extended finite element method are adopted along with a sub-triangle technique for enhancing the Gauss quadrature accuracy near the crack. An appropriate scheme for selecting the support domains near a crack is employed to reduce the computational cost. In this study, mixed-mode stress intensity factors are obtained by means of the interaction integral to determine the fracture properties. Several problems are solved to illustrate the effectiveness of the proposed method and the results are compared with available results of other numerical or (semi-) analytical methods.

Introduction

Orthotropic composites benefit from the performances of their constituents. Their high specific strength and stiffness characteristics allow for their extensive application in various industrial and engineering applications such as automobile industries and aerospace structures, etc. Materials such as carbon fiber/epoxy are inherently brittle and usually exhibit a linear elastic response up to failure with little or no plasticity. Thus, crack analysis of orthotropic composites is an indispensable task in reliable and durable optimized design of such materials and structures in innovative engineering applications.

Many analytical investigations are available on the fracture behavior of composite materials by researchers such as Muskelishvili [1], Lekhnitskii [2], Sih et al. [3], Bowie and Freese [4], Kuo and Bogy [5], Viola et al. [6], Lim et al. [7] and Nobile and Carloni [8].

Regarding the impossibility of obtaining analytical solutions in general problems, especially in complicated engineering cases, numerical methods are among the best available approaches.

The finite element method has been used with great success in many academic and industrial applications. However, it suffers from a number of drawbacks. For instance, due to mesh-based interpolation, distorted or low quality meshes may lead to unacceptable level of errors. A remedy is to use advanced remeshing techniques which are complex and computationally expensive.

Additionally, classical mesh-based methods are not well suited to treat problems with discontinuities that do not align with element edges. One strategy for dealing with evolving discontinuities in mesh-based methods is adaptive remeshing, which is costly and requires projection of quantities between successive meshes and leads to possible degradation of accuracy. An alternative approach is the extended finite element method (XFEM) [9] which enriches the approximation space in order to capture weak and strong discontinuities. In XFEM, the finite element approximation in the vicinity of a crack is enriched with functions extracted from the analytical solution near the crack tip.

With the objective of eliminating difficulties associated with element based solutions, meshless approximations are designed to build the approximation only on a set of nodes and therefore, are expected to be more convenient to discontinuous problems. Element free Galerkin (EFG) method [10], reproducing kernel particle method (RKPM) [11], hp-cloud method [12], meshless local Petrov–Galerkin (MLPG) [13] and the smoothed particle hydrodynamics (SPH) [14], [15] are among the frequently used meshless methods (MMs).

Among these methods, the element free Galerkin method, developed by Belytschko et al. [10], has found a wide application in fracture mechanics [16], [17], [18]. One of the main advantages of such an EFG solution is that several crack length/orientation problems can be analysed on a fixed mesh and no remeshing is necessary even for crack propagation problems.

Since there is no predefined element connectivities in EFG, the crack paths may extend anywhere within the model without the complication of intersecting any internal element edges. Therefore, it is easier to treat moving discontinuities such as crack propagation, shear bands and phase transformation, Of course, the model requires a full update procedure for the support domain information to take into account the existence and path of any crack. Similarly, a simpler h-adaptivity can be designed because any new node can be freely added to the model without intersecting any predefined element edges or the need for redefining element connectivities. This will then require a full update procedure for the non-fixed support domain information. Moreover, EFG provides a number of other major advantages [19], including higher accuracy, robust handling of large deformations, higher-order of continuity and a relatively more stable solution among the meshless methods.

Nevertheless, EFG suffers from a number of drawbacks. Since the moving least squares (MLS) shape functions do not satisfy the Kronecker delta property, imposition of essential boundary conditions is not as straightforward as in mesh-based methods. Also, the computational cost of EFG is much higher than an equivalent FEM.

EFG has already been applied to model discontinuities by using a so-called intrinsic basis function [20] and an extrinsic MLS enrichment [20] to reproduce the singular stress field around an isotropic crack tip. Moreover, enriched radial basis functions have recently been applied instead of moving least squares (MLS) basis functions, in a semi-EFG method to analyze crack problems in isotropic homogeneous and isotropic functionally graded materials [21], [22].

Cracked orthotropic materials have been analyzed by different approaches such as hybrid-displacement finite element method [23], Boundary Element Method (BEM) [24], finite elements and the modified crack closure method [25]. Asadpoure et al. [26], [27], [28] extended the original XFEM to analyze two dimensional discontinuous problems in orthotropic media. They developed three different sets of enrichment functions for various types of composites based on the analytical solutions. Further developments have been reported for dynamics and moving cracks in orthotropic media [29], [30] and delamination analysis of composites [31].

Although cracked orthotropic media have been successfully studied by XFEM [26], [27], [28], there are potentials to further develop and improve the EFG method for fracture analysis of composites, because in addition to the general advantages of meshless methods, its higher accuracy, robust handling of large deformations, avoiding distorted elements, higher/variable order of continuity within a domain, relatively simple node adaptation/insertion and a relatively more stable solution, make it a competing approach for the extended finite element method.

In this study, formulations of the element free Galerkin (EFG) methods are modified by selection of appropriate support domains in the vicinity of cracks and using the orthotropic enrichment functions, proposed by Asadpoure and Mohammadi [28], in the framework of partition of unity. Also, the accuracy of integration is increased by applying the sub-triangle technique used in XFEM [32]. Finally, validity, robustness and efficiency of the proposed approach are examined by several isotropic and orthotropic problems. Mixed-mode stress intensity factors and J-integrals are computed and compared with other numerical or (semi-) analytical methods.

Section snippets

Fracture mechanics of orthotropic materials

The stress–strain law in an arbitrary linear elastic material can be written asε=cσwhere ε and σ are strain and stress vectors, respectively, and c is the compliance matrix, defined in 3D as:c3D=1E1-ν21E2-ν31E3000-ν12E11E2-ν32E3000-ν13E1-ν23E21E30000001G230000001G130000001G12where E, ν and G are Young’s modulus, Poisson’s ratio and shear modulus, respectively. For a plane stress case the compliance matrix is simplified into the following form:cij2D=cij3Di,j=1,2,6and for a plane strain statecij2D

Orthotropic enrichment functions

Crack-tip enrichment functions have been obtained in a way that include all possible displacement states in the vicinity of crack tip mentioned in Eqs. (6), (8) [28]. These functions span the possible displacement space that may occur in the analytical solution. The enrichment functions, as defined in [28], are:Q(r,θ)=[Q1,Q2,Q3,Q4]=rcosθ12g1(θ),rcosθ22g2(θ),rsinθ12g1(θ),rsinθ22g2(θ)whereθj=arctansjysinθcosθ+sjxsinθgj(θ)=(cosθ+sjxsinθ)2+(sjysinθ)2with j=1,2. In the above equations sjx and sjy

EFG formulation

Consider the following standard two-dimensional problem of linear elasticity, as shown in Fig. 2. The partial differential equation and boundary conditions for this problem can be written in the form ofEquilibriumequation:LTσ+b=0inΩNaturalboundarycondition:σn=t¯onΓtEssentialboundaryconditionaluminium:u=u¯onΓuwhere L is the differential operator defined asL=x00yyxand σ, u, and b are the stress, displacement and body force vectors, respectively. t¯ is the prescribed traction on the

SIF computation

The stress intensity factor (SIF) is one of the important parameters representing fracture properties of a crack tip. In the present method, stress intensity factors are utilized to compare the accuracy of the developed orthotropic enriched EFG with other methods. In this study, the technique developed by Kim and Paulino [36] to evaluate mixed-mode stress intensity factor is applied and briefly reviewed.

The standard path independent J-integral for a crack is defined as [37]J=ΓWsδ1j-σijuix1njd

Numerical examples

Seven isotropic and orthotropic problems are examined to demonstrate the accuracy and efficiency of the proposed method. The results are compared with existing (semi-) analytical or numerical solutions.

In all examples, a linear basis function pT(x)=[1xy] and the cubic spline weight function are selected. Integration of each sub-triangle cell is performed by 13 Gauss points, unless stated otherwise.

Conclusion

In this paper, the conventional EFG has been further extended to analysis of discontinuous orthotropic problems. Recently developed XFEM orthotropic enrichment functions have been employed in EFG to evaluate stress intensity factors in cracked orthotropic materials.

Several isotropic and orthotropic problems with central and edge cracks have been solved by the proposed method. Results of mixed-mode stress intensity factors (SIFs) and J-integrals have been compared with the reference results and

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