Multiscale Analysis of Crack Propagation Using the Hybrid ...

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Multiscale Analysis of Crack Propagation Using the Hybrid MD-CGP Method Ryo KOBAYASHI, 1,2 Takahide NAKAMURA, 1,2 and Shuji OGATA 1,2 1 Department of Scientific and Engineering Simulation, Nagoya Institute of Technology, Nagoya 466-8555, Japan 2 CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan The crack propagation in materials is the main issue in the fracture phenomena. If there is a crack in a solid under certain load, the stress concentrates around the crack-tip with wide spread stress distribution. Mechanical property of solid which comes from the crystal structure and the inter-atomic potential determines how the material responds to applied load, such as brittle or ductile fractures. Because of the multiscale property of the crack, bonding/debonding at the crack-tip and stress field around the crack, to simulate the crack propagation phenomena with handling all the degrees of freedom of atoms is a crucial task. Thus multiscale methods, which reduce degrees of freedom far from the crack-tip by using the continuum approximation and calculate the atomistic area and the continuum area concurrently, are suggested recently. Most of these methods are limited their applications for the static problems, because they cannot treat waves that cause the unphysical reflection at the interface of the atomistic and continuum areas. However, in some cases, the crack propagations are essentially very fast phenomena, which crack speed reaches the Rayleigh speed, so it is not appropriate to apply static methods to these problems In order to overcome drawbacks in the previous approaches, we have developed new multiscale method with adopting the coarse-grained particle (CGP) method instead of the continuum approximation [1,2]. In the CGP method, we can obtain the Hamiltonian of the coarse-grained system, H CGP , by calculating the ensemble average of the atomistic Hamiltonian, H atom , with the coarse-graining relation as constraints, ∆, as H CGP = 〈H atom = 1 Z atom ∫∫ dudp H atom e βH atom · , (1) where N CGP I δ (U I i f Ii u i ) · δ ( ˙ U I i f Ii ˙ u i ). (2) We have also developed the novel coupling approach which connects the atomistic and coarse regions with eliminating the unphysical reflection at the interface [3]. In the present coupling approach, we prepare extra atoms and particles beyond the interface of the atomistic and coarse regions as shown in Fig. 1. And these ex- tra atoms/particles construct imaginary world, in which long-wavelength waves are passed to the other system and waves to be reflected are eliminated. The concurrent hybrid MD-CGP method with the present coupling method enables us the dynamic crack propagation simulation. We have performed the hybrid MD-CGP simulation of the crack propagation in 2D Lennard-Jones system. We apply Mode-I tensile load on 2D crystal including a submicron-meter size crack. As shown in Fig. 2, the atomistic method is applied around the crack-tip, and for surrounding area we apply the CGP method, and 174 AMTC Letters Vol. 2 (2010) © 2010 Japan Fine Ceramics Center

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Multiscale Analysis of Crack Propagation Using the HybridMD-CGP Method

Ryo KOBAYASHI,1,2 Takahide NAKAMURA,1,2 and Shuji OGATA1,2

1Department of Scientific and Engineering Simulation, Nagoya Institute of Technology, Nagoya466-8555, Japan

2CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan

The crack propagation in materials is the main issue in the fracture phenomena.If there is a crack in a solid under certain load, the stress concentrates around thecrack-tip with wide spread stress distribution. Mechanical property of solid whichcomes from the crystal structure and the inter-atomic potential determines how thematerial responds to applied load, such as brittle or ductile fractures. Because ofthe multiscale property of the crack, bonding/debonding at the crack-tip and stressfield around the crack, to simulate the crack propagation phenomena with handlingall the degrees of freedom of atoms is a crucial task. Thus multiscale methods,which reduce degrees of freedom far from the crack-tip by using the continuumapproximation and calculate the atomistic area and the continuum area concurrently,are suggested recently. Most of these methods are limited their applications for thestatic problems, because they cannot treat waves that cause the unphysical reflectionat the interface of the atomistic and continuum areas. However, in some cases, thecrack propagations are essentially very fast phenomena, which crack speed reachesthe Rayleigh speed, so it is not appropriate to apply static methods to these problems

In order to overcome drawbacks in the previous approaches, we have developednew multiscale method with adopting the coarse-grained particle (CGP) methodinstead of the continuum approximation [1,2]. In the CGP method, we can obtainthe Hamiltonian of the coarse-grained system, HCGP, by calculating the ensembleaverage of the atomistic Hamiltonian, Hatom, with the coarse-graining relation asconstraints, ∆, as

HCGP = ⟨Hatom⟩∆ =1

Zatom

∫∫dudp Hatome−βHatom · ∆, (1)

where ∆ ≡NCGP∏

I

δ(UI −∑

i

fIiui) · δ(U̇I −∑

i

fIiu̇i). (2)

We have also developed the novel coupling approach which connects the atomisticand coarse regions with eliminating the unphysical reflection at the interface [3]. Inthe present coupling approach, we prepare extra atoms and particles beyond theinterface of the atomistic and coarse regions as shown in Fig. 1. And these ex-tra atoms/particles construct imaginary world, in which long-wavelength waves arepassed to the other system and waves to be reflected are eliminated. The concurrenthybrid MD-CGP method with the present coupling method enables us the dynamiccrack propagation simulation.

We have performed the hybrid MD-CGP simulation of the crack propagation in2D Lennard-Jones system. We apply Mode-I tensile load on 2D crystal including asubmicron-meter size crack. As shown in Fig. 2, the atomistic method is appliedaround the crack-tip, and for surrounding area we apply the CGP method, and

174

AMTC Letters Vol. 2 (2010)

© 2010 Japan Fine Ceramics Center

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calculate both systems concurrently with using present coupling method. Sincethe present coupling method sufficiently eliminates unphysical reflection of short-wavelength waves emerged at the crack-tip, the strain field, stress intensity at thecrack tip, timing of the crack propagation, and crack speed obtained from the hybridMD-CGP simulation agree well with the results of the full-MD simulation. We willdiscuss the details of the methodology and results of the simulation.

Figure 1: Schematic illustration of the interface of atoms and particles when usingthe present coupling method.

Figure 2: Strain field images around the developing crack obtained by the hybridMD-CGP simulation.

References:

[1] R. E. Rudd and J. Q. Broughton, Physical Review B 58 (1998) R5893.

[2] R. Kobayashi et al., Materials Transactions 49 (2008) 2541.

[3] R. Kobayashi et al., International Journal for Numerical Method inEngineering, in press.

175

AMTC Letters Vol. 2 (2010)

© 2010 Japan Fine Ceramics Center