Published in International Journal for Numerical Methods in Engineering (2017) DOI: http://dx.doi.org/10.1002/nme.5566
Material dependent crack-tip enrichment functions in XFEM for modeling interfacial cracks in bimaterials Yongxiang Wang, Haim Waisman∗ Department of Civil Engineering & Engineering Mechanics, Columbia University, New York, NY 10027, United States
Abstract A novel set of enrichment functions within the framework of the extended finite element method (XFEM) is proposed for linear elastic fracture analysis of interface cracks in bimaterials. The motivation for the new enrichment set stems from the revelation that the accuracy and conditioning of the widely accepted 12-fold bimaterial enrichment functions significantly deteriorates with the increase in material mismatch. To this end, we propose an 8-fold material dependent enrichment set, derived from the analytical asymptotic displacement field, that well captures the near-tip oscillating singular fields of interface cracks, including the transition to weak discontinuities of bimaterials. The performance of the proposed material dependent enrichment functions is studied on two benchmark examples. Comparisons are made with the 12-fold bimaterial enrichment as well as the classical 4-fold homogeneous branch functions which have also been used for bimaterials. The numerical studies clearly demonstrate the superiority of the new enrichment functions, which yield the most accurate results but with less number of degrees of freedom and significantly improved conditioning than the 12-fold functions. Keywords: Extended finite element method; Interface crack; Oscillatory singularity; Crack-tip enrichment functions; Bimaterials; Weak and strong discontinuities
1. Introduction Multi-layered material systems have been widely used in engineering applications with the main purpose of strengthening and reducing weight of structural components. Examples ∗
Correspondence to: Haim Waisman, Department of Civil Engineering & Engineering Mechanics, Columbia University, New York, NY 10027, USA. Email addresses:
[email protected] (Yongxiang Wang),
[email protected] (Haim Waisman) Preprint submitted to IJNME
April 20, 2017
of layered materials include laminated composites, thermal barrier coatings, microelectronic devices, and thin film/substrate structures. However, interface cracking, also known as debonding or interlaminar delamination, is a common problem in layered materials due to the mismatch of material properties (elastic and thermal) and may often govern their structural performance. Thus, understanding the fracture behavior of interfaces is of crucial significance for the design and safety assessment of layered structures. Fracture mechanics methods have been the major tool used for modeling the mechanical behavior of interface cracks in bimaterial media. Generally speaking, these methods can be grouped into two families: linear elastic fracture mechanics (LEFM) approaches [1–9] and cohesive zone models [10–22]. Recently, Yuan and Fish [23] developed a continuum damage mechanics based dual-purpose model to simultaneously describe delamination as well as ply failure. However, since the main scope of this contribution lies within the framework of LEFM analysis of interface cracks in isotropic bimaterials, only developments in this category of methods will be reviewed hereinafter. The theoretical foundation for linear elastic interface fracture mechanics was laid down by the work of Williams [1], in which an asymptotic analysis was performed to show that the displacements u and stresses σ oscillate around the interface crack tip. More specifically, u ∼ r0.5+iε and σ ∼ r−0.5+iε , with r being the radial distance from the crack tip and ε a material dependent oscillation index that will be elaborated later. Several other studies such as Rice and Sih [2], England [3], and Erdogan [4] followed and an extensive review of these developments can be found in [5]. Although some analytical solutions close to the crack tip and the corresponding stress intensity factors (SIFs) can be found in the literature, they are invariably derived for simple geometries and loading conditions. For more general interface fracture problems, numerical methods must be employed. The finite element method (FEM) has been widely used for the linear elastic fracture analysis of layered structures. However, accurate solutions by the conventional FEM is computationally expensive since the mesh must conform to the geometry of the crack and a high degree of refinement in the vicinity of the crack tip is generally required. Although some specialized crack-tip elements such as the quarter-point element [24] have been developed to introduce stress √ singularity, they are only applicable to modeling the inverse square root singularity (1/ r) for homogeneous cracks. It is by no means trivial to incorporate into element formulations the oscillatory singularity inherent in interface cracks as well as the angular dependence of the near-tip stress field. Apart from constructing special element formulations, one may also resort to mapping techniques that approximate a smoother function obtained from reparameterizing the solution around the singularities. Based on this idea, Chiaramonte et al. [25] proposed a mapped finite element method to achieve optimally convergent solutions for problems posed on domains with cracks and corners. During the past two decades, the extended finite element method (XFEM) proposed by Belytschko and his co-workers [26, 27] has become a versatile and powerful tool to address moving discontinuities. The XFEM explores the idea of enhancing the solution space of the standard FEM with discontinuous and asymptotic functions via a local partition-of-unity (PU) [28, 29]. While retaining the original advantages of the standard FEM, the XFEM provides two superior capabilities that reproduce better the singular stress state at the 2
crack tip and alleviate the need for conforming meshes and adaptive remeshing for arbitrary crack propagation. Several studies [30–38] have demonstrated the efficiency of XFEM when applied to fracture mechanics problems and state-of-the-art reviews of the XFEM can be found in [39, 40]. While the original XFEM was designed for crack analysis in homogeneous isotropic materials, significant efforts have been made in the past to extend its application to bimaterial interface cracks. The following review of XFEM methods for bimaterials is by no means exhaustive, but provides some important milestones. Nagashima et al. [41] was the first to apply the XFEM to analyze interface cracks by adopting the classical 4-fold near-tip enrichment functions for homogeneous (not interface) cracks. Then, Sukumar et al. [42] developed new 12-fold enrichment functions that span the leading terms of the asymptotic displacement field for interface cracks in isotropic bimaterials. Liu et al. [43] presented a direct method to extract mixed-mode SIFs by using XFEM, which incorporates higher order terms of the asymptotic solution for both homogeneous and bimaterials. More recently, the XFEM was further extended to model interface cracks in orthotropic [44] or piezoelectric [45, 46] bimaterials. The 12-fold enrichment functions by Sukumar et al. [42] are specifically derived for bimaterials and thus lead to better results than the aforementioned 4-fold ones when modeling interface crack problems. However, the applications of the 12-fold bimaterial enrichment functions are somewhat limited since these enrichment functions require more degrees of freedom (DOFs) and most often yield worse conditioning of the algebraic system than the approximate 4-fold ones, especially in three-dimensional cases. There is some confusion as to whether it is necessary to use the 12-fold bimaterial enrichment for interface cracks: some XFEM studies [46–49] adopted the bimaterial enrichment functions whereas others [41, 50, 51] stick with the approximate 4-fold ones. Huynh and Belytschko [52] studied interface fracture in composite materials using both types of enrichment functions and found that less but still good accuracy can be obtained with the 4-fold homogeneous enrichment. Nevertheless, this observation was based on specific meshes and limited material mismatch combinations, and an in-depth comparative study is still lacking. In this contribution, we first reveal the pathological accuracy deterioration of the XFEM when using the 12-fold bimaterial enrichment functions for modeling interfacial cracks. This anomaly is especially pronounced when the material modulus mismatch is large. As a remedy, we propose to incorporate more material constants other than the oscillation index ε into bimaterial enrichment functions, which to our knowledge is a novel idea. Along this line, a new set of bimaterial enrichment functions is derived based on the analytical asymptotic displacement field for interface cracks. In contrast to the 12-fold enrichment functions, the new enrichment functions fully account for the effect of material properties, enabling a more precise representation of the displacement field, especially in the presence of large modulus mismatch. Furthermore, the new enrichment functions are only 8-fold, and therefore less DOFs are required than for the 12-fold enrichment functions. More importantly, the reduced number of enrichment functions also alleviates the almost linear dependence of the XFEM basis functions, thus leading to significantly lower condition number than the 12-fold ones. Based on two benchmark examples, a comparative study investigating the accuracy, 3
convergence, and conditioning of different enrichment functions (classical 4-fold, 12-fold and new 8-fold) is presented and discussed. The remainder of this paper is organized as follows. Sections 2 and 3 present short reviews of linear elastic interface fracture mechanics and XFEM, respectively. In Section 4, the performance of the 12-fold enrichment functions is investigated for a wide range of modulus mismatch ratios. Section 5 discusses the reason for the anomalous deterioration of accuracy, followed by the proposal of the new material dependent 8-fold enrichment functions. Numerical examples are given in Section 6. Section 7 concludes with some final remarks. 2. Problem statement for bimaterial interface cracks Consider a bimaterial system in domain Ω consisting of two dissimilar isotropic materials that are bonded at the interface, as shown in Fig. 1. The upper and lower planes are labeled 1 and 2, respectively. A traction-free crack Γc is assumed along the interface and (r, θ) correspond to the local polar coordinate system with origin at the crack tip. Let Γ+ c and − Γc denote the upper and lower surfaces of the crack, respectively. Em and νm (m = 1, 2) denote the Young’s modulus and the Poisson’s ratio of the m-th material, respectively. The corresponding shear modulus µm and Kolosov constant κm are given by 3 − 4νm for plane strain Em 3 − νm and κm = , m = 1, 2 (1) µm = for plane stress 2(1 + νm ) 1 + νm
Material 1 (E=E1, v=v1) 𝑥2 𝑟 Crack Γ𝑐
𝜃 𝑥1
Interface
Material 2 (E=E2, v=v2)
Figure 1: An interface crack between two dissimilar isotropic materials.
Unlike homogeneous media, the bimaterial solid experiences both stress intensifications from the geometric (crack) and material (interface) discontinuities [53]. As indicated in 4
the pioneering work of Williams [1], the bimaterial interface crack always induces both opening and shear behavior even for pure mode loading. In addition, the displacement and singular stress fields are oscillatory in the vicinity of the interface crack tip,√which can be characterized by a complex-valued stress intensity factor K = K1 + iK2 (i = −1) together with the oscillation index ε. The expression for the oscillation index ε, also know as the bimaterial constant, are given in terms of the elastic constants of both materials: 1 µ2 κ1 + µ1 ε= log (2) 2π µ1 κ2 + µ2 Referring to [2, 54, 55], the first term of the asymptotic displacement expansion for an interface crack can be written in Cartesian coordinates as r I II r u1 u1 (θ) u (θ) iε iε u= =
