Surface Finite Element for Imperfect Interface Modeling in Elastic Properties Homogenization

The paper presents a mathematical model of a finite element for modeling imperfect interface conditions for two contacting surfaces. The element is used in the numerical implementation of the Asymptotic Averaging Method (AAM) for the determination of effective elastic properties of composite materials under investigation. Numerical experiments are carried out to calculate the elastic properties taking into account the adhesion layer using a displacements field jump condition at the phase boundary. Results are compared with adhesion modeling using an additional bulk phase.

BIFURCATION DYNAMICS OF THREE-DIMENSIONAL SYSTEMS

Oscillations described by autonomous three-dimensional differential equations display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario the key instability is often an initial bifurcation from a single period oscillation to either its subharmonic of period two, or a symmetry breaking bifurcation. A generalized third-order nonlinear differential equation is developed which embraces the dynamics vicinal to these bifurcation events. Subsequently, an asymptotic averaging formalism is applied which permits an analytic development of the bifurcation dynamics, and, within quantifiable limits, prediction of the instability of the period one orbit in terms of the system control parameters. Illustrative applications of the general formalism, are made to the Rössler equations, Lorenz equations, three-dimensional replicator equations and Chua's circuit equations. The results provide the basis for discussion of the class of systems which fall within the framework of the formalism.

Analytics of Bifurcation

Oscillations described by autonomous three-dimensional differential equation systems display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario, the key instability is usually an initial bifurcation from a single period oscillation to its subharmonic of period two, or the reverse. An asymptotic averaging formalism is introduced by means of an example which permits an analytic development of the bifurcation dynamics, and in particular, prediction of the onset of periods 1 ↔ 2 bifurcations in terms of the system control parameters.