Lattice Green Function Research Articles

Computer simulation of Peierls stress by using lattice statics Green's function

The Peierls stress τ P is calculated for a discrete lattice model with changing the geometrical factor of the crystal h/b, where h is the spacing of the slip plane and b is the Burgers vector. Unlike the continuum model where a continuous distribution of the infinitesimal dislocation is assumed, the Peierls stress is determined as the critical applied stress beyond which no stable configuration of dislocation is found. The positions of atoms are calculated by the lattice statics Green's function. The results for the lattice model are well approximated by the exponential relation, τ P/G ~ exp(−Ah/b), as predicted by the continuum model, where G is the shear modulus. The Peierls stresses for some interatomic potentials are slightly lower than those obtained from experiments. The period of the Peierls potential derived from the lattice model is b, which is identical to the lattice constant.

Calculation of elastic Green’s functions for lattices with cavities

We present an algorithm for calculating the elastic lattice Green's function of a regular lattice, in which defects are created by removing lattice points. The method is computationally efficient, since the required matrix operations are on matrices that scale with the size of the defect subspace, and not with the size of the full lattice. This method allows the treatment of force fields with many-atom interactions.

Strength and reliability of fiber-reinforced composites: Localized load-sharing and associated size effects

The statistical aspects of the failure of large 3-D unidirectional fiber reinforced composites are studied numerically and analytically. A 3-D lattice Green's function model is used to calculate the stress field, damage evolution, and failure in composites under “Local Load Sharing” (LLS) conditions in which the stress from broken fibers is transferred predominantly to the nearby unbroken fibers. Failure by local accumulation of a critical amount of damage, and the associated decrease in ultimate strength with increasing composite size, is explicitly demonstrated. Weakest-link statistics are then employed to investigate size effects and reliability. An intrinsic “link” in LLS is found which has the same Gaussian distribution function for strength as a bundle in Global Load Sharing (GLS) (no local stress concentrations) of the same size. The size of the link is found to be comparable to the critical cluster of fiber damage observed in the simulations. Then, using known results for the GLS probability distribution function, analytic asymptotic results for the strength and reliability of large composites in LLS are derived. The strength distribution shows excellent agreement with the Monte Carlo simulation results for both the median strength and high reliability tail of the distribution. The implications of these results on the expected strength and reliability of moderate-size composites components is discussed, with applications to a Ti-MMC and a SiC/SiC CMC. Finally, the application of these results to modeling of composite failure by the Finite Element Method is presented.

Nonlinear rotating modes: Green's-function solution

Lattice Green's functions are used to investigate localized rotating modes recently exhibited in some nonlinear lattices. For a one-dimensional lattice, analytical expressions of the solution are obtained, first in the rotating-wave approximation and then by including higher-order terms. Numerical simulations confirm the validity of these solutions. The method is not restricted to one-dimensional lattices.

Band-Theoretical Interpretation of Stationary Localized Modes in Pure Nonlinear Lattices-Ubiquity of Existence and Similarity to Force-Constant-Defect Modes-

A lattice Green's function theory is formulated to give a band-theoretical interpretation of stationary nonlinear localized modes (SNLM) in pure nonlinear lattices. In one-dimensional lattices and in its simplest form in higher-dimensional cases, the theory yields the eigenfrequency ω of the SNLM in terms of the frequency ω 0 ( q ) ( q is the wave vector) of the corresponding linear lattice in the form ω=ω 0 ( q 0 + i K ), where q 0 is an edge point of the Brillouin zone of ω 0 ( q ), and K ≡ K ( A ) is a parameter depending on the localized mode amplitude A. Such a result and more general ones involving lattice Green's functions lead to the ubiquity of the existence of the SNLM irrespective of the lattice dimensionality as approximate nonlinear modes in nonintegable lattices and its similarity to force-constant defect modes. Stationary solitons in the Ablowitz-Ladik lattice can be considered as an example of the SNLM in integrable lattices.

Lattice statics Green's function for a semi-infinite crystal

The lattice statics Green's function describes the linear response of displacement to the application of a force on a crystal lattice. From a macroscopic point of view, the crystal lattice behaves like an anisotropic elastic body. Therefore, the lattice statics Green's function must approach the elastic Green's function for large distances from the point at which the force is applied. The lattice used in calculating the lattice Green's function should have the elastic constants corresponding to the crystal and a symmetrical stress tensor in the continuum elastic limit. In order to satisfy these conditions, three-body forces are introduced. The Green's function for the infinite lattice with short-range atomic interaction is calculated immediately by translational symmetry. The Green's function for the defective lattice is derived from the Dyson equation. The definitive Green's function for the semi-infinite lattice is presented.

RELATIONSHIP BETWEEN LATTICE GREEN'S FUNCTIONS FOR THE SEMI-INFINITE AND THE INFINITE SQUARE LATTICES AND ITS APPLICATION

Based on Ref. [3](Acta Physica Sinica, 44(1995), 987(in Chinese)), we have derived the relations between lattice Green's functions for the semi-infinite square lattice and the infinite square lattice. The partial density of states and the formation condition of the surface localized state are calculated by virtue of lattice Green's function for the semi-infinite square lattice.

Self-localized surface modes on anharmonic crystals

An approach based on the lattice Green function method for a semi-infinite crystal, and the use of the rotating wave approximation, is developed to investigate the existence of high frequency self-localized anharmonic surface vibrations. Solutions are found for such simple models as a semi-infinite linear chain and a scalar model of a simple cubic crystal with a (001) free surface. The frequencies of the modes and the amplitudes of the atomic displacements are calculated. The stability of the modes is also investigated.

Failure of fiber composites: A lattice green function model

A new powerful numerical technique for investigating the failure of fiber reinforced composites is presented. The technique utilizes 3D lattice Green's functions to calculate load transfer from broken to unbroken fibers, and also includes the important effects of fiber/matrix sliding. The inherent flexibility of the technique in adjusting the spatial extent of load transfer allows for the study of many aspects of real composite failure processes which have been unobtainable to date. Using this technique, composite reliability, the influence of manufacturing defects on performance, and the overall optimization of composite performance can all be investigated in detail. Initial results using this approach show that load transfer and the existence of spatially-staggered fiber breaks play an important role in determining strength and toughness of composites. Furthermore, the critical configurations of fiber breaks that initiate catastrophic failure are complicated 3D objects and any single spatial plane is composed mainly of sliding fibers rather than broken fibers, with a few strong fibers intact within the critical defect.

Efficient effective-energy method for lattice-Green's-function simulations of fracture.

Efficient effective-energy method for lattice-Green's-function simulations of fracture.

Toughening by Crack Bridging in Heterogeneous Ceramics

The toughening of a ceramic by crack bridging is considered, including the heterogeneity caused simply by spatial randomness in the bridge locations. The growth of a single planar crack is investigated numerically by representing the microstructure as an array of discrete springs with heterogeneity in the mechanical properties of each spring. The stresses on each microstructural element are determined, for arbitrary configurations of spring properties and heterogeneity, using a lattice Green function technique. For toughening by (heterogeneous) crack bridging for both elastic and Dugdale bridging mechanisms, the following key physical results are found: (i) growing cracks avoid regions which are efficiently bridged, and do not propagate as self‐similar penny cracks; (ii) crack growth thus proceeds at lower applied stresses in a heterogeneous material than in an ordered material; (iii) very little toughening is evident for moderate amounts of crack growth in many cases; and (iv) a different R‐curve is found for every particular spatial distribution of bridging elements. These results show that material reliability is determined by both the flaw distribution and the “toughness” distribution, or local environment, around each flaw. These results also demonstrate that the “microstructural” parameters derived from fitting an R‐curve to a continuum model may not have an immediate relationship to the actual microstructure; the parameters are “effective” parameters that absorb the effects of the heterogeneity. The conceptual issues illuminated by these conclusions must be fully understood and appreciated to further develop microstructure‐property relationships in ceramic materials.

Study Of Fiber Composite Failure Criterion

AbstractWith a recently developed numerical technique, we have investigated failure processes in real fiber composites. This technique utilizes 3D lattice Green's functions to calculate load transfer from broken to unbroken fibers, and also includes the important effects of fiber/matrix sliding. It is found that the failure processes are more complex than previously thought: The fibers that break or slide are not necessarily located around the largest defect cluster. Rather, the breaking/sliding fibers accumulate, form several large clusters, coalesce and finally cause the fiber composite to fail. Only in some cases does one large defect dominate the damage evolution. Based on our observations of the irregular evolution of the damage, a simple model is developed to describe these different failure modes, and the failure criterion of the fiber composite is also discussed.

Molecular-Dynamics Simulations Of Fracture: An Overview Of System Size And Other Effects

AbstractWe have studied brittle and ductile behavior and their dependence on system size and interaction potentials, using molecular-dynamics (MD) simulations. By carefully embedding a single sharp crack in two- and three-dimensional crystals, and using a variant of the efficient sound-absorbing reservoir of Holian and Ravelo [Phys. Rev. B 51, 11275 (1995)], we have been able to probe both the static and dynamic crack regimes. Our treatment of boundary and initial conditions allows us to elucidate early crack propagation mechanisms under delicate overloading, all the way up to the more extreme dynamic crack-propagation regime, for much longer times than has been possible heretofore (before unwanted boundary effects predominate). For example, we have used graphical display of atomic velocities, forces, and potential energies to expose the presence of localized phonon-like modes near the moving crack tip, just prior to dislocation emission and crack-branching events. We find that our careful MD method is able to reproduce the ZCT brittle-ductile criterion for short-range pair potentials [static lattice Green's function calculations of Zhou, Carlsson, and Thomson, Phys. Rev. Letters 72, 852 (1994)].We report on progress we have made in large-scale 3D simulations in samples that are thick enough to display realistic behavior at the crack tip, including emission of dislocation loops. Such. calculations, using our careful treatment of boundary and initial conditions - especially important in 3D - have the promise of opening up new vistas in fracture research.

On the calculation of lattice sums arising in Bose-Einstein statistics of quasiparticle excitations

A new method for the calculation of the average occupation number of bosonic quasiparticle excitations valid for any type of lattice is proposed. The method is based on the recognition of the connection between lattice Green functions and generalized Watson integrals, on one hand, and on a very simple differentiation technique, which renders unnecessary and artificial to this problem more sophisticated Laplace-transform summation procedures. The mean-field approximation to Green function theories of ferromagnetism arises naturally as the zeroth term in the obtained summation formulae, The results have been specified completely for the three cubic lattices. They are new for the simple cubic and face-centred cases, whereas a certain redundancy is removed from the known body-centred cubic result. Applications of the method to more complex sums, such as, for instance, the thermodynamic sum for the total energy of the quasiparticles, are straightforward. A new three-position recursion relation for the calculation of frequently occurring triple geometric integrals in the face-centred cubic case has also been found. It originates from a corresponding relation for a relevant Heun function.

The static lattice Green's function of gamma -Fe and FeTi

The computational scheme of MacGillivray and Sholl (1983) for the calculation of the lattice Green's function is extended to crystalline compounds with more than one atom in the unit cell. The lattice Green's function is calculated for the high-temperature gamma -phase of Fe and for the intermetallic compound FeTi using force constants obtained from Born-von Karman fits to experimental lattice vibration spectra.

Green's function method for random fuse network problems.

We propose and study a method that self-consistently solves for the current and voltage distributions in random fuse network problems using lattice Green's functions. The method solves for the current distribution by only keeping track of the defect bonds and can treat networks with two extreme types of defects: insulating bonds and superconducting bonds. We apply this method to study the breakdown features of a simple two-dimensional superconductor-resistor-insulator network.

On the cubic lattice Green functions

It is proved that K ( k + ) = [(4 – η ) 1/2 – (1 – η ) 1/2 ] K ( k _), where η is a complex variable which lies in a certain region R 2 of the η plane, and K ( k ± ) are complete elliptic integrals of the first kind with moduli k ± which are given by k 2 ± Ξ k 2 ± ( η ) = 1/2 + 1/4 η (4 – η ) 1/2 – 1/4 (2 – η ) (1 – η ) 1/2 . This basic result is then used to express the face-centred cubic and simple cubic lattice Green functions at the origin in terms of the square of a complete ellip­tic integral of the first kind. Several new identities involving the Heun function are also derived. F(a, b; α,β, γ, δ; η) are also derived. Next it is shown that the three cubic lat­tice Green functions all have parametric representations which involve the Green function for the two-dimensional honeycomb lattice. Finally, the results are ap­plied to a variety of problems in lattice statistics. In particular, a new simplified formula for the generating function of staircase polygons on a four-dimensional hypercubic lattice is derived.

Discrete phase space II. Relativistic lattice wave equations and divergence-free green's functions

Therelativistic lattice Klein-Gordon equation, Dirac equation, electromagnetic equations, and gauge field equations are presented as partialdifference equations. Various lattice Green's functions are constructed (except for non-abelian gauge fields). It is proved that many of the lattice Green's functions are non-singular or divergence-free. Moreover, it is conjectured that all lattice Green's functions are non-singular.

A self-consistent description of ruptures in an elastic medium: an application to earthquakes

We present a general self-consistent procedure using lattice Green's functions to find the stress redistribution due to multiple ruptures in a discrete elastic medium. We use this method to study a dipole version of the quasi-static crack-propagation model for earthquakes proposed by Chen et al. (1991) Unlike the earlier treatment we do not assume that ruptures occur independently. We obtain the stresses self-consistently and this changes the scaling behaviours of the frequency-size (seismic moment, energy) distribution. We make a careful comparison of our results with recent analysis of seismological data.

Atomic theory of fracture of crystalline materials: cleavage and dislocation emission criteria

Using the lattice Green function approach, we study the crack-dislocation effects in the fracture of crystalline materials. Firstly, we calculate the Green function for the defective lattice, with dislocation and crack, by solving the Dyson equation. After the lattice Green functions have been determined, the relaxation problem for the reconstituted bonds in the cohesive zone is solved. The external force F with tensile and shear components is applied, as a pair of forces, to the atoms at the center of the crack. In this calculation the dislocation emission is chosen to be on a cleavage plane as well as on a glide plane of the two-dimensional lattice (hexagonal and graphite lattices). We compare the cleavage and dislocation emission criteria of the atomistic simulation with those of linear elasticity theory.