Crack propagation in quasicrystals R. Mikulla1, J. Stadler1, F. Krul1, H.-R. Trebin1 and P. Gumbsch2 1 Institut f¨ur Theoretische und Angewandte Physik der Universit¨at Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany 2 Max-Planck-Institut f¨ur Metallforschung, Seestr. 92, 70174 Stuttgart, Germany Crack propagation is studied in a two dimensional decagonal model quasicrystal. The simulations reveal the dominating role of highly coordinated atomic environments as structure intrinsic obstacles for both dislocation motion and crack propagation. For certain overloads, these obstacles and the quasiperiodic nature of the crystal result in a specific crack propagation mechanism: The crack tip emits a dislocation followed by a phason wall, along which the material opens up. Quasicrystals are characterized by an unusual struc- crucial element in the brittle fracture mechanism at low ture: They are orientationally ordered, but with noncrys- temperature. Furthermore, we identify atomic clusters, tallographic, e.g., five- or eightfold symmetry axes [1]. inherent to the quasicrystalline structure, as obstacles Their translational order is not periodic, as in conven- and scattering centers for the brittle crack. tional crystals, but quasiperiodic. In the last few years The study of a strongly nonlinear process like dynam- it has become possible to measure physical properties ical fracture in a complex material requires simplifica- of quasicrystals, such as mass and heat transport, plas- tions. First we consider a two dimensional system (2D) ticity, and fracture, since high–quality, thermodynami- to evade the analysis of a three dimensional crack front. cally stable quasicrystals could be grown in millimeter Secondly we simulate a perfectly ordered quasicystalline sizes [2]. Because of the peculiar structure, quasicrystals representative of the binary tiling system [9], derived open new possibilities for studying structure–property re- from the T¨ubingen triangle tiling (TTT) [11]. A large lations. The mechanical behavior of a solid, in particular, atom (L) is placed at each vertex and a small atom (S) is strongly influenced by structure, because it is mainly into the center of each large triangle of the original tiling. governed by defects, which are related to the order. Sev- When the centers of neighboring atoms of different type eral studies of the standard icosahedral quasicrystalline are connected, one obtains a rhombic tiling, which we alloys AlMnPd and AlCuFe have disclosed that the sam- denote as “bond representation” of the direct “atom rep- ples are brittle and hard at room temperature [3], but resentation”. become ductile at about 80% of the melting temperature, Qualitatively our model shares central properties with and then can be deformed up to 30% without hardening structure models for real quasicrystals, which are (1) a [4]. decagonal diffraction pattern inherited from the TTT [11] Concerning ductility it has been confirmed both by and (2) a hierarchical structure of clusters. These are transmission electron microscopy [5] and by computer built by a first shell of small atoms and a second shell simulations [6], that motion of dislocations is the mech- of large atoms surrounding a central large atom (Fig. 1), anism of plastic deformation in quasicrystals. The exis- and can be considered as 2D versions of Mackay clusters tence of dislocations in quasicrystals is due to the fact that are significant for many real quasicrystals. Further- that quasiperiodic structures can be considered as irra- more, assemblies of such clusters form super clusters and tional cuts through a higher dimensional periodic crys- so on. The smallest clusters are situated on five families tal [7]. Hence the Burgers vector of a dislocation also of parallel lines, mutually rotated by 36◦, with a small is of higher dimension, with a component in physical and a large separation within each family, arranged in a space and a component in the complementary orthogonal Fibonacci sequence (Fig. 2). space. The dislocation is described by a “phonon”-like The atoms interact with Lennard-Jones potentials displacement field u(x) and a “phason”-like displacement truncated at 2.5 r0. The bond lengths are chosen to be || ⊥ field w(x), whose contour integrals yield b and b , re- the geometrical ones [9]. The unit of length in our sim- spectively. ulation r0 = rLS is given by the length of the LS bond. Molecular dynamics simulations of a sheared quasicrys- The bond energies are ε0 = εLS and εLL = εSS = 0.5ε0 tal have revealed that close to the melting temperature to avoid phase separation. All masses are set to 1. The arXiv:cond-mat/9804279v1 [cond-mat.mtrl-sci] 24 Apr 1998 the phason field is spread out around the dislocation core, unit of time is therefore t0 = r0pm/ε0. Due to their high whereas at low temperatures it is confined to a plane ex- coordination the decagonal clusters become the tightest tending from and being bound by the dislocation line bound structural units of the quasicrystal model for all ⊥ [6,8]. The phase of the quasicrystal is differing by b at choices of εLL and εSS smaller than one. the two sides of this wall, which therefore is denoted as Preferred cleavage planes of the 2D model quasicrys- “phason-wall”. tal, i.e. the “easy lines” for fracture, are those in between In this article we demonstrate – by employing atom- the large separation of the parallel lines connecting the istic simulations of fracture processes – that phason walls, clusters, because a crack can follow these lines without being a consequence of the quasiperiodic structure, are a cutting the tightly bound ten-rings. The surface ener- 1 gies [12] of different crystallographically equivalent, but left a phason wall in its wake. This phason wall carries structurally distinct lines (1-4 in Fig. 2) are lowest for some interface energy, and hence the cohesive strength of those between the widely spaced ten-rings (3,4 in Fig. 2), the material is reduced at this wall. After some time of namely 1.9 ε0/r0 as compared to 2.3 ε0/r0 for the other strain build-up the crack begins to open along the phason surfaces studied here (1,2,5 in Fig. 2). Furthermore, they wall (Fig. 4, bottom). It immediately regains its initial also have the lowest “unstable stacking fault energy”, i.e. propagation velocity until the next obstacle is met. Due increase in energy when the two halfs are shifted relative to the tightly bound complete ten-rings, the cleavage sur- to each other. Hence these lines are also preferred for face roughness is about 10 r0. dislocation motion [13]. When a dislocation is gliding At intermediate loads, between 1.3% and 1.5% strain, through a quasicrystal, leaving a phason wall behind, it a regime of steady crack propagation is observed. The creates an even more preferred cleavage line, since the crack tip proceeds with almost uniform velocity, but the phason wall is characterized by a positive defect energy. appearance of the cleavage plane is unchanged. The For the shortest Burgers vector the cleavage energy along dislocation-emission–phason-wall cleavage mechanism is the phason wall can be reduced by an additional 0.3 ε0/r0 still operates but without stopping the propagating [13]. crack. The fracture simulations are performed on a rectan- At even higher overloads (≥ 1.5% strain) we observe gular slab of approximately 250 000 atoms, 250 r0 high an onset of crack tip instabilities, which create defects and 1.000 r0wide. The slab is uniformly strained by an that are not bound to the crack tip. At a strain of amount ∆ normal to the long axis, and in the outer- 1.5% (Fig. 5) we observe dislocation emission at an an- most layer of width 3r0 all atoms are held fixed at these gle of 72◦, which reduces (due to shielding) the load on positions. A precrack of length 200r0 is inserted from the crack tip. Arrested by an obstacle, the dislocation one side along an easy line by cutting the bonds and moves back and annihilates at the fracture surface af- relaxing the sample. For the relaxation the externally ter the crack tip has propagated further. Such a pro- applied homogeneous strain is fixed at the Griffith value, cess, so–called “virtual dislocation emission”, was first at which the strain energy per unit length just equals proposed by Brede and Haasen to explain their fracture twice the surface energy. The resulting displacement field experiments in silicon [16]. At a load of 1.7% the crack then is scaled linearly above the critical dilation where bifurcates and stops. the crack tip starts moving. The evolution of the sys- In quasicrystals as in periodic crystals, we find a min- tem is followed by standard molecular dynamics (MD) imal velocity for brittle crack propagation of 14% of the technique where the atoms obey Newtonian equations of shear wave velocity vT at low loads. This suggests that motion. The system was started at an initial temperature there exists a lower band of forbidden crack velocities −6 of T=10 Tmelt. The IMD molecular dynamics package [17,18]. With rising load, the crack tip speed increases was used [14]. During crack propagation, the crack tip to about 35% of vT . This value is again similar to the bonds are broken and the strain energy above the surface maximum crack tip speed in periodic crystals [19]. The energy (the overload) is released in the form of acoustic lower than expected upper crack velocity, which linear emission or dislocation generation. To avoid heating of elasticity theory would place at the Rayleigh wave veloc- the slab and phonon reflections from the boundaries an ity [21] (vRayleigh ≈ 0.9 vT ), may be explained with the elliptical stadium is created outside which the waves are non-linearity of the atomic interaction, which has been damped gradually by a ramped friction term in the equa- shown to drastically reduce crack tip velocities in peri- tions of motion [15].