A comparative molecular dynamics-phase-field modeling approach to brittle fracture

https://doi.org/10.1016/j.cma.2016.04.005Get rights and content

Highlights

  • A link between molecular and continuum models for brittle fracture is proposed.

  • Parameters obtained from the molecular scale are used in the continuum approach.

  • Under this approach, the phase-field parameters acquire an entirely physical meaning.

  • This approach can assist multi-scale modeling of materials.

Abstract

In this work, a novel comparative method for highly brittle materials such as aragonite crystals is proposed, which provides an efficient and accurate in-sight understanding for multi-scale fracture modeling. In particular, physically-motivated molecular dynamics (MD) simulations are performed to model quasi-static brittle crack propagation on the nano-scale and followingly compared to macroscopic modeling of fracture using the phase-field modeling (PFM) technique. A link between the two modeling schemes is later proposed by deriving PFM parameters from the MD atomistic simulations. Thus, in this combined approach, MD simulations provide a more realistic meaning and physical estimation of the PFM parameters. The proposed computational approach, that encompasses mechanics on discrete and continuum levels, can assist multi-scale modeling and easing, for instance, the simulation of biological materials and the design of new materials.

Introduction

In engineering and material science, much attention has been given in recent years to the prediction of material failure. One of the most interesting and challenging problems in fundamental fracture research is how to improve the understanding of brittle fracture processes, which occur primarily in high-strength materials with poor ductility and toughness. By definition, brittle fracture is the breakage of interatomic bonds without noticeable plastic deformation. This type of fracture occurs when the local strain energy becomes larger than the energy necessary to pull the atom layers apart. In order to be able to predict and study this fracture behavior, a wide diversity of experimental, mathematical, and most recently, computational methods have been proposed for different (length) scale levels, such as nano-, micro- and meso-scale. For each of these groups, the main focus is still efficiency and accuracy of the models.

At the atomistic level, molecular dynamics (MD) simulations are increasingly becoming powerful tools to investigate crack initiation and propagation. Due to their real time scales (pico seconds), MD methods are perfectly suited to study the very high-speed crack propagation of highly brittle materials. All-atom MD simulations have been extensively used to study crack propagation  [1] for a variety of inorganic crystals, such as Si3N4   [2], SiO2   [3], 3C-SiC  [4], and GaAs  [5]. Although these studies have provided valuable in-sights into crack dynamics, a systematic analysis of mechanical properties at an atomistic scale, as well as a link to the macroscopic continuum mechanical approaches is still not well established.

Although MD simulations are efficient to study material on the nano-scale, they become impractical when the investigated systems have significantly larger dimensions, i.e., huge number of molecules, or if the considered time range is relatively long. Nowadays, typical MD simulations can be performed on systems containing hundreds of thousands, or perhaps, a few millions of atoms for simulation times ranging from a few hundred nanoseconds up to a millisecond. These numbers are certainly respectable, but one may run into conditions where size and/or time limitations become important. The challenges related to the limited dimensions or time scales can be tackled using continuum mechanical methods, where atomic details of the material are disregarded. In fact, as it has been shown in numerous other studies  [6], [7], [8], [9], the mechanical behavior of a given continuous material can be reproduced in a different scale by using mechanical parameters derived directly from atomistic MD simulations.

Due to its versatility, the phase-field modeling (PFM) has emerged as a powerful tool in continuum mechanics to model fracture and many other multi-phase material evolutions. Although phenomenological, PFM has proven to offer a good trade-off between numerical treatment, accuracy, and computational costs. The pioneering works of Griffith  [10] and Irwin  [11], as well as the variational formulations presented in  [12], [13], have helped to build a well-established energy-based framework for brittle fracture, which has continued to be developed and enhanced during the past decades.

In order to provide for a robust numerical implementation of this approach in typical finite element codes, a phenomenological phase-field variable, which approximates the crack–material sharp interface by a diffusive transition zone, needs to be incorporated. The idea of diffusive interfaces stems from physics and has been used by, e.g., Cahn and Hilliard  [14] to describe interfaces in a heterogeneous system by a partial differential evolution equation. Crack propagation models for quasi-static, dynamic and cohesive brittle fracture have successfully been established in  [15], [16], [17], [18], [19], [20], to mention some. Nevertheless, it is seldom to find a thorough physical explanation or validation of the phase-field parameters. Moreover, there have not been any attempts to implement the PFM for brittle fracture into an atomistic scale level, neither have there been material properties derived from all-atom simulations, which could help to illuminate a physical explanation to the PFM parameters.

The aim of this research work is to create a link between the understanding of brittle fracture of a highly brittle material at an atomistic scale and its macroscopic mechanical features. Therefore, the fracture behavior of an aragonite (CaCO3) tablet, under quasi-static assumptions, is studied using MD simulations and PFM. To this end, the key mechanical properties, e.g., Lamé constants (λe, μe), phase-field transition width (ϵ), and the mechanical energy release rate (G) of aragonite crystals, are obtained from MD simulations. Subsequently, these physical properties are used to reproduce the nano-scale model with a continuous PFM approach. The combination of these methods gives a deeper understanding of brittle fracture at a discrete atomistic level, whilst concurrently helping to establish a link to continuum mechanical fracture models.

In this work, Sections  2 Materials and methods of MD simulations, 3 PFM for brittle fracture briefly describe the molecular dynamics, as well as the employed phase-field modeling for brittle fracture. Although it does not aim to deepen in any of both theories, the explanation of the MD approach is somewhat broaden so to help the reader to understand the basics behind all-atom simulations. For the PFM, no new method is proposed in this work, and instead, the numerical experiments are conducted considering already existing and well accepted models for brittle fracture. In Section  4, a thorough description and brief evaluation of the numerical experiments in MD as well as PFM are presented. The comparison between these two approaches and the discussion of the obtained results, are carried out in Section  5. The last section is dedicated to the conclusion and outlook of the present work.

Section snippets

Materials and methods of MD simulations

MD simulation is a computational method employed for molecular systems to determine their time-dependent behavior. It is based on the numerical solution of Newton’s equations of motion for a given set of interacting particles (the interactions between particles governed by an interatomic potential, e.g., covalent bonds, angles, van der Waals potential, etc.), enabling to keep track of the evolution of the system in phase space.

Aragonite, a brittle ceramic (CaCO3) and a major constituent of

Theoretical fundamentals

The macroscopic modeling of fracture in the numerical example conducted in this work is based on a well accepted PFM for brittle fracture, which has been widely discussed in several research works, see  [15], [16], [32], [33], [34] for an overview.

According to Griffith’s energy-based criteria to describe brittle fracture  [10], the global potential energy function of a cracked linear elastic, isotropic solid material can be defined as the sum of the elastic strain energy Ψelast integrated

Evaluation of the MD simulations

As stated previously, to examine the fracture behavior at atomistic scale of double edge v-notched aragonite tablets, atomistic FPMD simulations were performed. The different tablet models were loaded by moving virtual springs applied to the outer surfaces of the tablets away from each other with constant velocity. The force is obtained directly from the resultant spring force at the boundaries. Accordingly, the stress is computed by dividing the spring force by the cross sectional area

Discussion of the combined MD  PFM approach

The results of the PFM come to a good agreement with those of the MD simulations, as discussed in the previous section. Phenomenologically, the s-shaped crack propagation (along the (101) and (1̄01) lattice planes) is obtained in both approaches. Moreover, the ultimate tensile strength obtained in the PFM simulations falls perfectly into the ranges described in the all-atom simulations. Nevertheless, a discrepancy between the two schemes can be seen which can be traced to two main reasons: On

Conclusions

Using different levels of simplifications, continuum mechanics is able to tackle problems for structure sizes that cannot be reached by MD simulations. In spite of the conceptual differences between continuum mechanics and all-atom simulations, the combination of these two different methods is promising. Parameters for continuum mechanics can be obtained from all-atom simulations to ensure high accuracy and being closer to physics, thereby reducing empiricism. The efficiency of continuum

References (46)

  • V. Agrawal et al.

    A dynamic phase-field model for structural transformations and twinning: Regularized interfaces with transparent prescription of complex kinetics and nucleation. part I: Formulation and one-dimensional characterization

    J. Mech. Phys. Solids

    (2015)
  • V. Agrawal et al.

    A dynamic phase-field model for structural transformations and twinning: Regularized interfaces with transparent prescription of complex kinetics and nucleation. part II: Two-dimensional characterization and boundary kinetics

    J. Mech. Phys. Solids

    (2015)
  • C.L. Rountree et al.

    Atomistic aspects of crack propagation in brittle materials: Multimillion atom molecular dynamics simulations

    Annu. Rev. Matter. Res.

    (2002)
  • R.K. Kalia et al.

    Role of ultrafine microstructures in dynamic fracture in nanophase silicon nitride

    Phys. Rev. Lett.

    (1997)
  • H. Kikuchi et al.

    Brittle dynamic fracture of crystalline cubic silicon carbide (3c-SIC) via molecular dynamics simulation

    J. Appl. Phys.

    (2005)
  • P. Vashishta et al.

    Multimillion atom simulation of nanostructured materials on parallel computers

    Prog. Theor. Phys. Suppl.

    (2000)
  • S.P. Patil, B. Markert, F. Gräter, Refining a bottom-up computational approach for spider silk fibre mechanics, in:...
  • S.P. Patil et al.

    Viscous friction between crystalline and amorphous phase of dragline silk

    PLoS One

    (2014)
  • A.A. Griffith

    The phenomena of rupture and flow in solids

    Philos. Trans. R. Soc. Lond. A

    (1921)
  • G.R. Irwin

    Analysis of stresses and strains near the end of a crack traversing a plate

    J. Appl. Mech.

    (1957)
  • B. Bourdin et al.

    The variational approach to fracture

    J. Elasticity

    (2008)
  • J.W. Cahn et al.

    Free energy of a nonuniform system. I. interfacial free energy

    J. Chem. Phys.

    (1958)
  • C.V. Verhoosel et al.

    A phase field model for cohesive fracture

    Internat. J. Numer. Methods Engrg.

    (2013)
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