Molecular-dynamics simulation-based cohesive zone representation of intergranular fracture processes in aluminum

https://doi.org/10.1016/j.jmps.2006.03.004Get rights and content

Abstract

A traction–displacement relationship that may be embedded into a cohesive zone model for microscale problems of intergranular fracture is extracted from atomistic molecular-dynamics (MD) simulations. An MD model for crack propagation under steady-state conditions is developed to analyze intergranular fracture along a flat Σ99 [1 1 0] symmetric tilt grain boundary in aluminum. Under hydrostatic tensile load, the simulation reveals asymmetric crack propagation in the two opposite directions along the grain boundary. In one direction, the crack propagates in a brittle manner by cleavage with very little or no dislocation emission, and in the other direction, the propagation is ductile through the mechanism of deformation twinning. This behavior is consistent with the Rice criterion for cleavage vs. dislocation blunting transition at the crack tip. The preference for twinning to dislocation slip is in agreement with the predictions of the Tadmor and Hai criterion. A comparison with finite element calculations shows that while the stress field around the brittle crack tip follows the expected elastic solution for the given boundary conditions of the model, the stress field around the twinning crack tip has a strong plastic contribution. Through the definition of a Cohesive-Zone-Volume-Element—an atomistic analog to a continuum cohesive zone model element—the results from the MD simulation are recast to obtain an average continuum traction–displacement relationship to represent cohesive zone interaction along a characteristic length of the grain boundary interface for the cases of ductile and brittle decohesion.

Introduction

Cohesive zone models (CZMs) approximate traction–displacement relationships along an interface (Tvergaard and Hutchinson, 1992; Costanzo and Allen, 1995; Camacho and Ortiz, 1996; Klein and Gao, 1998) and are frequently used in conjunction with the finite element method (FEM) to study fracture in a wide variety of materials. Idealized traction–displacement behavior of interface debonding is embedded into CZM elements. In fracture studies at the micromechanical level, CZM elements can be placed between the continuum finite elements that discretize the grain interior to predict transgranular fracture or placed between the continuum finite elements on either side of a grain boundary to predict intergranular fracture.

Modeling of material failure with CZMs has been advanced to the level of being able to perform large-scale simulations of fracture in polycrystals. Recently, Zavattieri et al. (2001) studied the fracture of alumina-ceramic microstructures subjected to multi-axial dynamic loading. The effective size of the polycrystalline specimen studied was 0.54×0.19 mm, thus reaching macroscopic scales. A bilinear traction–displacement relationship parameterized to empirical data, such as macroscopic fracture toughness KIC was used. Zavattieri and Espinosa (2003) used a modification of the same model to study interface effects of an alumina specimen in contact with steel plates. Wei and Anand (2004) have used a modified CZM to study intergranular fracture in nanocrystalline Ni. In their finite element (FE) model simulation, the CZM element approximated both reversible and irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. The parameterization of the model was, again, performed by using available experimental data for stress–strain curves of nanocrystalline Ni in tension with an average grain size of 15–40 nm and having a comparatively large number of grains. Iesulauro et al. (2002) have applied the CZM technique to simulate fatigue crack initiation in Al polycrystals. The use of molecular-dynamics (MD) simulations to parameterize the traction–displacement curve was suggested, but the actual parameterization was performed by using the established macroscale yield properties of aluminum.

The macroscale values of strength and toughness that are input to the CZM in these references represent the aggregate responses of thousands or millions of grains, grain boundaries, and defects within the specimens from which they were obtained. Thus, these macroscale values do not represent the unique response of a particular interface at which a local fracture event might occur. If the microscale predictions are to become quantitative, consideration of the local nanoscale properties is required. One possible means of making this connection is to use the results of atomistic MD models as input to the CZM. This connection would allow more realistic simulations leading to accurate predictions of the failure properties of a large class of materials and microstructures, even when experimental data is not available.

Attempts to extract relevant parameters for the decohesion law of a CZM from atomistic (MD or molecular-static) simulations have been made by various groups in the last few years (Gall et al., 2000; Komanduri et al., 2001; Spearot et al., 2004). The approach in all of these works is based on simulating the debonding of a flat interface under a constant tensile strain rate perpendicular to the interface. In these references, the system size is between 4 and 8 nm, and the dynamics of the atoms is severely constrained by the boundary conditions, which do not allow for Poisson lateral contraction and shear deformation. As a result, plastic processes, such as dislocation slip, are strongly suppressed. Consequently, the simulated mechanism for interface decohesion in these references reproduces the process of atomic adhesion (strength) rather than that of fracture at the interface. Raynolds et al. (1996) used a similar setup to study adhesion in an NiAl–Cr interface by first principles calculations.

The boundary conditions at which a typical CZM element operates in a large-scale FE model are very different from the ones used in the referenced MD and first principles simulations. Typically, the CZM elements are embedded within a system of finite elements that reproduce the elastic and plastic response of the surrounding material to both the external load and the crack-tip stress. In contrast, decohesion parameters, such as peak stress and opening displacement of the CZM curve, are extracted from an atomistic volume less than 10 nm in each direction. The lack of an adequate surrounding volume of material suppresses the plastic processes, such as dislocation nucleation, limiting the accommodation of deformation at the interface and forcing it to debond in an unnatural manner. The periodic boundary conditions usually applied in these models cause the simulation to create a response of an array of repeating units with a strong overlap of image elastic forces rather than the response of a single specimen unit. Consequently, the resulting decohesion curves cannot be directly applied to derive the constitutive laws for CZM elements.

The main goal of the approach described in the present study is to extract, and understand the contributions to, an MD-based CZM decohesion law for intergranular fracture under local conditions similar to those experienced by the CZM element in a polycrystalline FE model. The CZM decohesion law reflects the response of the CZM element to an approaching and propagating crack (Costanzo and Allen, 1995; Dávila, 2001). Thus, the MD model should be a model of crack propagation rather than of adhesion. The MD model used in this study is built to simulate a crack propagating through a flat high-energy grain-boundary in aluminum (Yamakov et al., 2005).

The paper is constructed as follows: The simulation approach is described in Section 2. The mechanism of intergranular crack growth together with the plastic processes near the crack tip as revealed by MD simulations at the atomic level is broadly discussed in Section 3. The contribution of the plastic processes to the stress field near the crack tip is discussed in Section 4. Section 5 describes a methodology for extracting a constitutive relation for a continuum CZM element from the MD results. The main conclusions of this study are outlined in Section 6.

Section snippets

The simulation approach

The simulation approach used in this study is based on an MD simulation model of crack propagation under time-independent, or steady-state conditions through a flat grain boundary (GB) in Al (modeled by the interatomic potential of Mishin et al. (1999)) at low temperature (100 K). The purpose of the MD simulation is to reveal and analyze the atomistic processes taking place near the crack tip and to derive a statistical traction–displacement relationship for a continuum CZM element. The

The mechanisms of crack propagation along the Σ99 grain boundary by MD simulation

As explained in Section 2, the GB opens after 8 ps of screening of the atomic interactions in a region of 5.7 nm length along the middle of the GB between Crystals I and II (Fig. 1), and a crack starts to grow in both directions along the GB interface. Fig. 5 shows MD snapshots of cracks that have grown for four different initial hydrostatic prestresses: σ=3.5, 3.75, 4.0, and 4.25 GPa. In all cases, the crack growth is not symmetric in the +x and −x directions (as defined in Fig. 1) along the GB

Plastic contribution to the stress field near the crack: MD–FE comparison

The plastic processes at the crack tips, including twinning and dislocation emission, have a pronounced effect on the stress distribution near the growing crack. This effect is best revealed by a comparison between the stress distributions obtained from the MD and the linear elastic FE simulations for the models presented in Section 2. While the MD simulation considers the material structure at the atomic level and intrinsically incorporates all the plastic processes together with the elastic

Defining a traction–displacement relationship from MD

Grain-scale simulations that use cohesive zone models to study fracture (such as those presented by Zavattieri et al. (2001), Iesulauro et al. (2002), Zavattieri and Espinosa (2003) Wei and Anand (2004)) typically use heuristically derived relationships and input values to define the CZM. In such an approach, the values at the microstructural scale are based on macroscopic parameters such as fracture toughness. Thus, there is no physical substantiation for the quantities that are input to the

Conclusions

A methodology is detailed for extracting the decohesion law for interface debonding by introducing a Cohesive-Zone-Volume-Element using MD simulations. The methodology is applied for the process of debonding of a high-energy Σ99 GB in Al through crack propagation under hydrostatic loading conditions. The work reveals the atomic mechanisms in the damage zone near the crack tips during intergranular fracture. Here, the crack propagation has been shown to proceed in a different manner in the two

Acknowledgements

V. Yamakov and D.R. Phillips were sponsored through cooperative agreement NCC-1-02043 with the National Institute of Aerospace and contract NAS1-00135 with Lockheed Martin Space Operations, respectively.

References (54)

  • D. Spearot et al.

    Non-local separation constitutive laws for interfaces and their relation to nanoscale simulations

    Mech. Mater.

    (2004)
  • E.B. Tadmor et al.

    A Peierls criterion for the onset of deformation twinning at a crack tip

    J. Mech. Phys. Solids

    (2003)
  • V. Tvergaard et al.

    The relation between crack growth resistance and fracture process parameters in elastic–plastic solids

    J. Mech. Phys. Solids

    (1992)
  • V. Tvergaard et al.

    Effectt of strain-dependent cohesive zone model on predictions of crack growth resistance

    Int. J. Solids Struct.

    (1996)
  • A. Van der Ven et al.

    The thermodynamics of decohesion

    Acta Mater.

    (2004)
  • Y.J. Wei et al.

    Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

    J. Mech. Phys. Solids

    (2004)
  • V. Yamakov et al.

    Length-scale effects in the nucleation of extended lattice dislocations in nanocrystalline Al by molecular-dynamics simulation

    Acta Mater.

    (2001)
  • V. Yamakov et al.

    Dislocation–dislocation and dislocation-twin reactions in nanocrystalline Al by molecular-dynamics simulation

    Acta Mater.

    (2003)
  • P.D. Zavattieri et al.

    An examination of the competition between bulk behavior and interfacial behavior of ceramics subjected to dynamic pressure-shear loading

    J. Mech. Phys. Solids

    (2003)
  • P.D. Zavattieri et al.

    A computational model of ceramic microstructures subjected to multi-axial dynamic loading

    J. Mech. Phys. Solids

    (2001)
  • ABAQUS/Standard User's Manual, 2004. Hibbitt, Karlsson, and Sorensen,...
  • F.F. Abraham et al.

    Instability dynamics of fracture: a computer simulation investigation

    Phys. Rev. Lett.

    (1994)
  • M. Barber et al.

    Steady-state propagation of a crack in a viscoelastic strip

    Phys. Rev. A

    (1989)
  • Q. Chen et al.

    Failure modes after exhaustion of dislocation glide ability in thin crystals

    Sci. China (Series E)

    (1999)
  • E.S.C. Ching

    Dynamic stresses at a moving crack tip in a model of fracture propagation

    Phys. Rev. E

    (1994)
  • A.S. Clarke et al.

    Structural changes accompanying densification of random hard-sphere packings

    Phys. Rev. E

    (1993)
  • F. Cleri et al.

    Atomistic simulations of intergranular fracture in symmetric-tilt grain boundaries

    Interf. Sci.

    (1999)
  • Cited by (210)

    • Multi scale simulation of crack propagation in polycrystalline SiC

      2024, Theoretical and Applied Fracture Mechanics
    View all citing articles on Scopus
    View full text