Efficient coarse graining in multiscale modeling of fracture

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Abstract

We propose a coarse-graining technique to reduce a given atomistic model into an equivalent coarse grained continuum model. The developed technique is tailored for problems involving complex crack patterns in 2D and 3D including crack branching and coalescence. Atoms on the crack surface are separated from the atoms not on the crack surface by employing the centro symmetry parameter. A rectangular grid is superimposed on the atomistic model. Atoms on the crack surface in each cell are used to estimate the equivalent coarse-scale crack surface of that particular cell. The crack path in the coarse model is produced by joining the approximated crack paths in each cell. The developed technique serves as a sound basis to study the crack propagation in multiscale methods for fracture.

Introduction

Understanding the phenomena of material failure across multiple length scales has been the major research focus in the material science and engineering community for many years. In engineering applications, the global response of the system is often governed by the behavior at the smaller length scales. Hence, the subscale behavior must be computed accurately for good predictions of the full scale behavior. Therefore, the numerical models dealing with multiple spatial and temporal length scales are required.

Many methods have been developed in the past decade that aim to resolve the domain where material failure takes place at the atomistic scale. Calculations at the atomistic scale promise to provide inroads to understand the fundamental mechanics of material failure. However, the magnitudes of the scales of engineering problems involving macroscopic cracks and shear bands are much larger than the atomistic length and time scales. Therefore, methods that bridge the length and time scales are in demand and have been the focus of intense research for many years. Most coupling methods and simulations are focused on models of intact materials (without cracks). The transfer of information through the different length scales for problems involving material failure and finite temperatures remains a challenging task.

Multiscale methods can be categorized into hierarchical, semi-concurrent [1], [13], [14], [15], [16] and concurrent methods [2], [3], [4], [5], [6], [7], [8], [9], [10], Fig. 1. In hierarchical multiscale methods, information is passed from the fine-scale to the coarse-scale; but not vice versa. Computational homogenization [11] is a classical up-scaling technique. Hierarchical multiscale approaches are very efficient. However, their extension to model fracture is complex, in particular for fracture and materials involving strain softening. One basic assumption for the application of homogenization theories is the existence of disparate length scales [12]: LCrLRVELSpec where LCr,LRVE and LSpec are the crack length, the representative volume element (RVE)- and specimen-size, respectively. For problems involving fracture, the first condition is violated as LCr is of the order of LRVE. Moreover, periodic boundary conditions (PBC) often used at the fine-scale, cannot be used when a crack touches a boundary as the displacement jump in that boundary violates the PBC.

The basic idea of semi-concurrent multiscale methods is illustrated in Fig. 1(b). In semi-concurrent multiscale methods, information is passed from the fine-scale to the coarse-scale and vice versa. A classical semi-concurrent multiscale method is the FE2 [13], [14], [15], [16] originally developed for intact materials. Kouznetsova et al. [11] extended this method to problems involving material failure, see also Kouznetsova et al. [17] or recent contribution by Nguyen et al. [18], Verhoosel et al. [19] and Belytschko et al. [20]. Numerous concurrent multiscale methods [21], [22], [23], [24], [25], [26], [27] have been developed that can be classified into ‘Interface’ coupling methods and ‘Handshake’ coupling methods. Interface coupling methods seem to be less effective for dynamic applications as avoiding spurious wave reflections at the ‘artificial’ interface seem to be more problematic. Some of the concurrent multiscale methods have been extended to modeling fracture [25], [26], [27], [28], [29], [30], [31], [32].

One difficulty in multiscale methods for fracture is to upscale fracture-related material information from the fine-scale to the coarse-scale, in particular for complex crack problems. Most of the above mentioned approaches therefore were applied to examples with comparatively few macroscopic cracks. In this paper, we present a robust and simple coarse graining technique in the context of multiscale modeling for fracture to reduce a given fine-scale model into an equivalent coarse-grained (CG) model. Only an atomistic model is considered at the fine-scale. The coarse-scale model might be discretized with classical techniques like the finite element method, meshfree methods [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43] or partition-of-unity enriched methods such as the eXtended Finite Element Method (XFEM) [44], [45], [46], [47], [48], [49], the Generalized Finite Element Method (GFEM) [50], [51], [52], [53], [54], [55], [56], [57], [58], the Partition of Unity Finite Element Method (PUFEM) [59], [60], the eXtended Element Free Galerkin method (XEFG) [61], [62], [63], [64], [65], [66], [67], [68], the Cracking Particles Method [69], [70], [71], [72], [73], [74], [75], the phantom node method [77], [78], [79], [80], [81], [82], [83] or the Numerical Manifold Method (NMM) [84], [85], to name a few. The developed technique is applied in the context of hierarchical upscaling though an extension to concurrent or semi-concurrent multiscale method is straightforward.

The arrangement of the article is as follows: The coarse graining technique is introduced in Section 1. Details of the CG model are explained in Section 2. Section 3 verifies the developed coarse graining technique for four examples before Section 4 concludes the article.

Section snippets

Coarse grained model

The goal of the present coarse-graining scheme is to develop an equivalent CG model for fracture based on a fine scale model containing defects. The CG scheme is employed for coarse scale models based on Finite Element/eXtended Finite Element (FE/XFE) or particle discretizations, whereas fracture on the fine scale occurs naturally by breaking the bonds between adjacent atoms. The CG approach is applicable to concurrent, semi-concurrent and hierarchical multiscale methods, though we present

Validation examples

In this section, we validate the proposed CG model with four numerical examples. The quasi-static crack propagation in two dimensions with an initial angular edge crack is studied in the first example. In the second example, we study two dimensional dynamic crack growth in a double notched specimen. Quasi-static crack propagation including crack branching and crack coalescence in two dimensions is studied in the third example. In the final example, we study three dimensional dynamic crack

Conclusions

A coarse graining technique to upscale fracture pattern from an atomistic model to an equivalent CG model has been presented. Atoms lying on the crack surface in the fine scale model are separated from the atoms not on the crack surface through the centro symmetry parameter. A rectangular discretization is superimposed on the fine scale model to capture the atoms into rectangular cells. The crack path in each cell is approximated using the atoms on the crack surface and their neighbors. An

Acknowledgements

The support provided by the DeutscheForschungsgemeinschaft (DFG) is gratefully acknowledged. The financial support from the IRSES is thankfully acknowledged. We are thankful to the support by the National Science Council of Republic of China through Grant NSC 101-2911-I-006-002-2. Dr. Winston Chen’s Scholarship on International Academic Research is greatefully acknowledged in carrying out this work. Dr. Zhuang acknowledges the supports from the NSFC (41130751) and National Basic Research

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