Efficient coarse graining in multiscale modeling of 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]: where and are the crack length, the representative volume element (RVE)- and specimen-size, respectively. For problems involving fracture, the first condition is violated as is of the order of . 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 [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
References (87)
- et al.
Bridging scale methods for nanomechanics and materials
Comput. Meth. Appl. Mech. Eng.
(2006) Multiscale elastoviscoplastic analysis of composite structures
Comput. Mater. Sci.
(1999)- et al.
multiscale approach for modeling the elastoviscoplastic behavior of long fiber SiC/Ti composite materials
Comput. Meth. Appl. Mech. Eng.
(2000) - et al.
A multilevel nite element method () to describe the response of highly non-linear structures using generalized continua
Comput. Meth. Appl. Mech. Eng.
(2003) - et al.
Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks
Comput. Meth. Appl. Mech. Eng.
(2011) - et al.
A multiscale extended finite element method for crack propagation
Comput. Meth. Appl. Mech. Eng.
(2008) - et al.
Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses
Comput. Meth. Appl. Mech. Eng.
(2007) - et al.
Coupled-volume multi-scale modelling of quasi-brittle material
Euro. J. Mech. – A/Solids
(2008) - et al.
Stable particle methods based on Lagrangian kernels
Comput. Meth. Appl. Mech. Eng.
(2004) - et al.
A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids
J. Sound Vib.
(2001)
Strain smoothing in FEM and XFEM
Comput. Struct.
The generalized finite element method
Comput. Meth. Appl. Mech. Eng.
Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids
Comput. Meth. Appl. Mech. Eng.
The generalized finite element method for Helmholtz equation: theory, computation, and open problems
Comput. Meth. Appl. Mech. Eng.
The generalized finite element method for Helmholtz equation, part II: effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment
Comput. Meth. Appl. Mech. Eng.
A generalized finite element method for the simulation of three-dimensional dynamic crack propagation
Comput. Meth. Appl. Mech. Eng.
Analysis and applications of a generalized finite element method with global–local enrichment functions
Comput. Meth. Appl. Mech. Eng.
The partition of unity finite element method: basic theory and applications
Comput. Meth. Appl. Mech. Eng.
A new partition of unity finite element free from linear dependence problem and processing delta property
Comput. Meth. Appl. Mech. Eng.
Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by extrinsic discontinuous enrichment of meshfree methods without asymptotic enrichment
Eng. Fract. Mech.
A geometrically non-linear three dimensional cohesive crack method for reinforced concrete structures
Eng. Fract. Mech.
Discontinuous modelling of shear bands using adaptive meshfree methods
Comput. Meth. Appl. Mech. Eng.
A three dimensional large deformation meshfree method for arbitrary evolving cracks
Comput. Meth. Appl. Mech. Eng.
Simulations of instability in dynamic fracture by the cracking particles method
Eng. Fract. Mech.
A simple and robust three-dimensional cracking-particle method without enrichment
Comput. Meth. Appl. Mech. Eng.
Phantom-node method for shell models with arbitrary cracks
Comput. Struct.
A finite element method for the simulation of strong and weak discontinuities in solid mechanics
Comput. Meth. Appl. Mech. Eng.
A simple circular cell method for multi-level finite element analysis
J. Appl. Math.
A bridging domain method for coupling continua with molecular dynamics
Comput. Meth. Appl. Mech. Eng.
Bridging domain methods for coupled atomistic continuum models with or couplings
Int. J. Numer. Meth. Eng.
Implementation aspects of the bridging scale method and application to intersonic crack propagation
Int. J. Numer. Meth. Eng.
The bridging scale for two-dimensional atomistic/continuum coupling
Philos. Magaz.
Coupling of atomistic and continuum simulations using a bridging scale decomposition
J. Comput. Phys.
Molecular dynamics/XFEM coupling by a three dimensional extended bridging domain with applications to dynamic brittle fracture
Int. J. Multisc. Comput. Eng.
Micromechanics: Overall Properties of Heterogeneous Materials
Multi-scale constitutive modeling of heterogeneous materials with a gradient-enhanced computational homogenization scheme
Int. J. Numer. Meth. Eng.
Computational homogenization for adhesive and cohesive failure in quasi-brittle solids
Int. J. Numer. Meth. Eng.
Multiscale aggregating discontinuities: a method for circumventing loss of material stability
Int. J. Numer. Meth. Eng.
Quasicontinuum analysis of defects in solids
Philos. Magaz. A
Cited by (206)
Arbitrary polygon-based CSFEM-PFCZM for quasi-brittle fracture of concrete
2024, Computer Methods in Applied Mechanics and EngineeringAdaptive triangular-mesh coarse-grained model for notched 2D metamaterials: A hybrid FEA and top-down approach
2023, Theoretical and Applied Fracture Mechanics2D numerical modeling of solid materials with virtual boundary particles in ordinary state-based peridynamics
2023, Mechanics Research CommunicationsShear fracture propagation in quasi-brittle materials by an element-free Galerkin method
2023, Theoretical and Applied Fracture MechanicsA machine learning-based atomistic-continuum multiscale technique for modeling the mechanical behavior of Ni<inf>3</inf>Al
2023, International Journal of Mechanical SciencesThe eXtended – Finite Element Method (X – FEM) Through State of the Art Applications
2023, Comprehensive Structural Integrity
- ☆
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-No Derivative Works License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited.