Mixed-mode State Research Articles

Theory of Mixed-Mode Caustics for Stationary Cracks in Optically Anisotropic Materials

The optical method of caustics is one of the most powerful techniques for evaluating stress intensity factors in static as well as dynamic fracture mechanics problems. In the method of transmitted caustics, optically anisotropic (birefringent) materials such as Araldite B and polycarbonate are often used. Double caustic curves appear for the tip of a crack in an optically anisotropic plate due to the birefringence. In this paper, a theory of caustics was developed for stationary cracks in birefringent materials under in-plane mixed-mode conditions. Completely analytical expressions were derived for the double caustic curves as well as for the double initial curves. The combined effects of optical anisotropy and mixed-mode state on the caustic curves and initial curves were investigated. A new procedure was also developed for the evaluation of mixed-mode stress intensity factors using the characteristic dimensions of the double caustic patterns.

Mixed mode convection in an inclined slot

Nonlinear interaction between transverse disturbances and longitudinal rolls has been investigated for flow in an inclined slot with a heated lower wall when both modes of instability occur at nearly the same value of the control parameter. This condition is shown to be possible for a fluid with Prandtl number greater than 0.263 897, For slightly supercritical values of the Rayleigh number (R) when the critical Rayleigh number for longitudinal rolls RLC is somewhat less than that for transverse stationary rolls, RSC, and for transverse travelling waves, RTC, longitudinal rolls occur first and then remain stable as R is increased beyond RSC or RTC; no mixed mode state occurs. In contrast, if RSc or RTc is slightly below RLC, pure transverse modes exist for only a relatively small range of R beyond RSC or RTC. Thereafter, a three-dimensional mixed mode state occurs well before RLC is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches RLC the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than RLC. Mixed mode convection is also investigated for a special choice of parameters when three modes, namely transverse stationary rolls, transverse travelling waves and longitudinal rolls, become unstable simultaneously. Longitudinal rolls again emerge as the stable supercritical state.

Mixed-mode oscillations in chemical systems

A prototype model is exploited to reveal the origin of mixed-mode oscillations. The initial oscillatory solution is born at a supercritical Hopf bifurcation and exhibits subsequent period doubling as some parameter is varied. This period-2 solution subsequently loses stability, but continues to exist−regaining stability to form the 11 mixed-mode state (one large plus one small excursion). Other mixed-mode states lie on isolated branches or ‘‘isolas’’ of limit cycles in the one-parameter bifurcation diagram and are separated by regions of chaos. As a second parameter is varied, the number of isola solutions increases and the ‘‘gaps’’ between them become narrower, leading to correspondingly more complete Devil’s staircases. An exactly comparable scenario is shown to arise in the three variable model of the Belousov–Zhabotinsky reaction proposed recently by Györgyi and Field [Nature 335, 808 (1992)].

Theory of Mixed-Mode Caustics for Stationary Cracks in Optically Anisotropic Materials.

The optical method of caustics is one of the most powerful techniques for evaluating stress intensity factors in static as well as dynamic fracture mechanics problems. In the method of transmitted caustics, optically anisotropic (birefringent) materials such as Araldite B and polycarbonate were often used. Double caustic curves appear for a tip of the crack in an anisotropic (birefringent) plate, due to the birefringence. In this paper, a caustic theory was developed for stationary cracks in birefrigent materials under in-plane mixed-mode conditions. Completely analytical expressions were derived for the double caustic curves as well as for the double initial curves. The combined effects of optical anisotropy and mixed-mode state on the caustic curves and the initial curves were investigated. New procedures were also developed for the evaluation of mixed-mode stress intensity factors using the characteristic dimensions of the double caustic patterns.

Spanwise modal competition in cross-waves

An asymptotic theory for cross-waves generated by an oscillating wavemaker in a semi-infinite rectangular wave tank is derived for the limit of large mode number. The possibility of multiple mode excitation is included by introducing a spanwise modulation. The partial differential equations governing the evolution of inviscid cross-waves are shown to be two coupled nonlinear Schrödinger equations. Energy dissipation in the system is taken into account including a linear damping term. A center manifold analysis is used to reduce the PDEs to a system of coupled Landau equations in the neighborhood of a codimension-two point where two adjacent spanwise modes are marginally stable. Four possible steady states of the system are found, one of which is a mixed-mode superposition of two spanwise modes. A Hopf bifurcation from the mixed mode is predicted for some parameters; extending the system to higher order allows the stability of this bifurcation to be determined in terms of perturbations to a Hamiltonian system. Both subcritical and supercritical bifurcation are possible. An experimental study in the neighborhood of a codimension-two point shows good agreement with the theoretical predictions including the discovery of a mixed-mode state.