On discrete models in fracture mechanics

In two papers published in 1969, where the foundations of the theory of brittle fracture were laid, Valentin Valentinovich Novozhilov introduced a necessary and sufficient criterion for estimating the strength of an elastic body weakened by a cut and formulated the concept of brittle fracture. The criterion is based on taking account of the body discrete structure, and the fracture process is treated as loss of stability of elastic equilibrium. These two facts, fracture discreteness and loss of stability in the deformation of the breaking bond, underlie numerous phenomena, which could not be discovered in the framework of continuum mechanics. In the present paper, the following effects of taking these facts into account are briefly discussed: an unstable leg of the stress-strain diagram and the capture of part of the fracture energy by the lattice, the role of the lattice dimension, radiation in quasi-static and dynamic crack propagation including the radiation structure, the role of the dynamic factor, irregularities in the crack growth, crack propagation under the action of high-frequency waves including steady-state modes with periodically varying speed, and cracks in continuum discrete models.     

On the theory of discrete structural model analysis and design

Exclusive theory for analysis of Structural Models (comprising of springs, masses, dash pots, etc.) is presented by adapting the electrical network theory. It commences a brief statement of a new Principle of Quasi Work (PQW) , relevant to this theory. Derivations presented here include theorems addressing maximum displacements, relative flexibilities, sensitivity analysis of global flexibilities, inverse problem of load prediction and interpolation of stiffnesses and flexibilities of the Structural Models. Finally a Design Equation capable of providing a starting point which more or less satisfies all the displacement constraints for iterative design employing a pair of estimated starting points for design iterations (within or outside feasible region) is evolved. Simple substantive illustrations are included to demonstrate the potential of these theoretical developments.     

On dimensionless numbers for dynamic plastic response of structural members

Summary A dimensional analysis is reported for the dynamic plastic response and failure of structural members, which includes material strain hardening, strain rate and temperature effects. Critical shear failure conditions are also discussed based on the dimensional analysis results. It is shown that the response number R _n proposed in [3], is an important independent dimensionless number for the dynamic plastic bending and membrane response of structural members. However, additional dimensionless numbers are necessary when transverse shear, strain hardening, strain rate, and temperature effects are important. Received 22 February 1999; accepted for publication 15 June 1999     

On compatible finite element models for elastic plastic analysis

Summary Some implicit or explicit assumptions on which the formulation of compatible finite element models for elastic plastic analysis is based are considered. It is shown that it is possible to formulate global elastic plastic element laws by independently assuming interpolation functions governing the displacement and the plastic strain fields. Some consistency requirements are discussed, which must be enforced in order to avoid artificial stiffening or fictitious redundancy. When these requirements are met, a «consistent» element is produced, whose properties are analyzed. It is also emphasized that, in spite of the use of compatible models, upper bounds to the collapse load are not always obtained for the limit analysis problem.For simplicity, beam elements are considered. However, the formulation holds for continua as well, as discussed in the last section.

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Sommario Vengono analizzate alcune ipotesi esplicitmente o implicitamente assunte per la formulazione di elementi finiti usati nell'analisi elasto-plastica. Si mostra come l'assunzione indipendente di funzioni di interpolazione che governano i campi di spostamento e di deformazioni plastiche permetta di formulare legami costitutivi elasto-plastici globali per l'elemento. Vengono anche discusse alcune condizioni che devono essere rispettate per non irrigidire artificialmente l'elemento o non introdurvi una iperstaticità fittizia. Queste condizioni portano a definire elementi «consistenti», le cui proprietà vengono analizzate. Si fa anche notare come, nonostante l'uso di modelli congruenti, non vi sia sempre la garanzia di ottenere delimitazioni superiori al carico di collasso per problemi perfettamente plastici.Per semplicità, la trattazione viene svolta con riferimento ad elementi di trave. La possibilità di estensione al continuo è messa in luce nell'ultimo paragrafo.

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Research supported by the C.N.R. (National (Italian) Research Council).     

On non-linear stochastic dynamics of quarter car models

In this paper, high-dimensional probability density functions of non-linear dynamical systems are calculated solving the corresponding Fokker-Planck equations. Zeroth approximations are derived from solutions of corresponding linear systems and analytical results for first- and second-order expected values. The zeroth approximations are used as weighting functions for the construction of generalized Hermite polynomials. The Fokker-Planck equation is expanded in terms of these polynomials and subsequently solved by a Galerkin method.As an example, models of a quarter car with non-linear damping subjected to white or colored noise excitation are considered. The damping is piecewise linear and asymmetric leading to a non-vanishing expected value of the displacement of the car. The excitation is realized by the roughness of the road and the car moves with constant velocity. Monte-Carlo simulations and analytical results are used for comparison.     

Three-dimensional discrete element models for the granular statics and dynamics of powders in cavity filling

Rapid granular flow from a moving container and angle of repose formation were investigated by numerical simulations using the discrete element method (DEM) and experiments. Grain models of various geometrical complexity were studied and their ability to reproduce the experiments in those regimes was explored. The predictive power of the most realistic model for gravity driven cavity filling was assessed. Good agreement between computed and measured density distributions within the filled cavities provides a basis for numerical process variations aiming at homogenized density distributions. The effect of numerical coarse graining was found to be negligible for all properties of interest provided that force laws are scaled properly and corrections for boundary effects are taken into account. The proposed scaling was tested for a certain set of force laws but could be applied to different DEM forces as well. An analytic mass flow law for powder discharge from a moving container was derived and verified by our DEM simulations.     

On the derivation of structural models with general thermomechanical prestress

The vibrating behaviour of thin structures is affected by prestress states. Hence, the effects of thermal prestress are important research subjects in view of ambient vibration monitoring of civil structures. The interaction between prestress, geometrically non-linear behaviour, as well as damping and its coupling with the aforementioned phenomena has to be taken into account for a comprehensive understanding of the structural behaviour. Since the literature on this subject lacks a clear procedure to derive models of thin prestressed and damped structures from 3D continuum mechanics, this paper presents a new derivation of models for thin structures accounting for generic prestress, moderate rotations and viscous damping. Although inspired by classical approaches, the proposed procedure is quite different, because of (i) the definition of a modified Hu–Washizu (H-W) functional, accounting for stress constraints associated with Lagrange multipliers, in order to derive lower-dimensional models in a convenient way; (ii) an original definition of a (mechanical and thermal) strain measure and a rotation measure enabling one to identify the main terms in the strain energy and to derive a cascade of lower-dimensional models (iii) a new definition of “strain–rotation domains” providing a clear interpretation of the classical assumptions of “small perturbations” and “small strains and moderate rotations”; (iv) the introduction of a pseudo-potential with stress constraints to account for viscous damping. The proposed procedure is applied to thin beams.     

On the nonlinear dynamic buckling mechanism of autonomous dissipative/nondissipative discrete structural systems

Summary Nonlinear dynamic buckling of nonlinearly elastic dissipative/nondissipative multi-mass systems, mainly under step load of infinite duration, is studied in detail. These systems, under the same loading applied statically, experience a limit point instability. The analysis can be readily extended to the case of dynamic buckling under impact loading. Energy, topological and geometrical aspects for the total potential energyV, which is constrained to lie in a region of phase-space whereV0, allow conclusions to be drawn directly regarding dynamic buckling. Criteria leading to very good, approximate and lower/upper bound dynamic buckling estimates are readily established without solving the highly nonlinear set of equations of motion. The theory is illustrated with several analyses of a two-degree-of-freedom model.     

Exact solutions for discrete velocity models

A new class of exact solutions for discrete kinetic models is presented. It is shown that these solutions can be used to solve both initial and boundary value problems of rarefied gas dynamics.     

On the influence of memory on complex dynamics of evolutionary oligopoly models

Nonlinear Dynamics - In this paper, we propose a unitary formulation for evolutionary oligopoly models with memory. In particular, we consider behavioral rules that are stationary at the Nash...     

Some discrete SI and SIS epidemic models

The probability is introduced to formulate the death of individuals, the recovery of the infected individuals and incidence of epidemic disease. Based on the assumption that the number of individuals in a population is a constant, discrete-time SI and SIS epidemic models with vital dynamics are established respectively corresponding to the case that the infectives can recover from the disease or not. For these two models. the results obtained in this paper show that there is the same dynarfiical behavior as their corresponding continuous ones. and the threshold determining its dynamical behavior is found. Below the threshold the epidemic disease dies out eventually, above the threshold the epidemic disease becomes an endemic eventually, and the number of the infectives approaches a positive constant.     

Oscillatory behavior in discrete slow power-law models

Nonlinear Dynamics - Discrete mathematical slow oscillatory models are proposed to describe biological interactions between two populations by considering power-law functions. Conditions for slow...     

New discrete element models for elastoplastic problems

The discrete element method （DEM） has attractive features for problems with severe damages, but lack of theoretical basis for continua behavior especially for nonlinear behavior has seriously restricted its application. The present study proposes a new approach to developing the DEM as a general and robust technique for modeling the elastoplastic behavior of solid materials. New types of connective links between elements are proposed, the interelement parameters are theoretically determined based on the principle of energy equivalence and a yield criterion and a flow rule for DEM are given for describing nonlinear behavior of materials. Moreover, a numerical scheme, which can be applied to modeling the behavior of a continuum as well as the transformation from a continuum to a discontinuum, is obtained by introducing a fracture criterion and a contact model into the DEM. The elastoplastic stress wave propagations and the tensile failure process of a steel plate are simulated, and the numerical results agree well with those obtained from the finite element method （FEM） and corresponding experiment, and thus the accuracy and efficiency of the DEM scheme are demonstrated.     

Stability and dynamics of a plastic softening oscillator

This paper deals with the stability of a single-degree-of-freedom plastic softening oscillator. Understanding such an elementary model concerns, for instance, the seismic behaviour of concrete or steel structures. The associated dynamic system is a complex hysteretic system. Using appropriate internal variables, it can be written as a singular autonomous system. Liapounov stability of the solutions is then studied. A domain of perturbations associated with a stable solution is exhibited. This domain looks like a truncated cone in the three-dimensional phase space. It can be read as a critical displacement or energy that the oscillator can support during a seismic excitation. The difference with the “equivalent” linearized elastic system is highlighted. The unloading part of the response of the inelastic system has a stabilising effect.     

Investigations of pipe-diffusion-based dislocation climb by discrete dislocation dynamics

A new numerical dislocation climb model based on incorporating the pipe diffusion theory (PDT) of vacancies with 3D discrete dislocation dynamics (DDD) is developed. In this model we hold that the climb rate of dislocations is determined by the gradient of the vacancy concentration on the segment, but not by the mechanical climb force as traditionally believed. The nodal forces on discrete dislocation segments in DDD simulation are transferred to PDT to calculate the vacancy concentration gradient. This transfer establishes a bridge connecting the DDD and PDT. The model is highly efficient and accurate. As verifications, two typical climb-involved examples are predicted, e.g. the activation of a Bardeen-Herring source as well as the shrinkage and annihilation of prismatic loops. Finally, the model is applied to study the breakup process of an infinite edge dislocation dipole into prismatic loops. This coupling methodology provides us a useful tool to intensively study the evolution of dislocation microstructures at high temperatures.     

Reactive Euler equations of discrete models with reversible reactions

Received November 25, 1998