Incremental Perturbation Research Articles

Chemo-Marangoni Convection: II. Nonlinear Stability Analysis

A nonlinear problem on concentration-capillary instability at the interface of two liquid media in the presence of an interfacial chemical reaction of the second order has been considered. The effect of interfacial chemical reaction on the nonlinear development of interfacial instability has been studied using a finite-amplitude stability method and weak nonlinear convection pattern analysis. As has been ascertained, the incremental perturbations specifically generated by a nonlinear chemical reaction lead either to the establishment of an order space-periodic pattern or to the emergence of a chaotic flow with a sufficiently wide wave spectrum.

An Analysis of Crack Trapping by Residual Stresses in Brittle Solids

The residual stress distribution in a brittle polycrystalline solid may have a significant influence on its toughness. Grains in a state of residual compression are less likely to be fractured by a growing crack and may trap the crack front or be left behind as bridging particles (Evans et al., 1977). This paper estimates the toughness enhancement due to intergranular residual stresses, using a three-dimensional model. The residual stress is approximated as a doubly sinusoidal distribution acting perpendicular to the plane of an initially straight semi-infinite crack. An incremental perturbation method developed by Bower and Ortiz (1990) for solving three-dimensional crack problems is extended here to cracks loaded by nonuniform remote stresses. It is used to calculate the shape of the semi-infinite crack as it propagates through the doubly sinusoidal residual stress. It is shown that the local regions of compression may trap the crack front and give rise to some transient toughening. In addition, if the residual stress exceeds a critical magnitude, pinning particles may be left in the crack wake. However, for practical values of residual stress and grain size, the predicted toughness enhancement is insignificant. Furthermore, the analysis cannot account for the large bridging zones observed in experiments. It is concluded that the R-curve behavior and bridging particles observed in monolithic ceramics are caused by mechanisms other than residual stresses acting perpendicular to the crack plane.

A three-dimensional analysis of crack trapping and bridging by tough particles

The toughness of a brittle material may be substantially improved by adding small quantities of tough particles to the solid. Three mechanisms may be responsible. Firstly, the front of a crack propagating through the solid can be trapped by the particles, causing it to bow out between them. Secondly, the particles may remain intact in the wake of the crack, thereby pinning its faces and reducing the crack tip stress intensity factors. Finally, the toughness may be enhanced by frictional energy dissipation as particles are pulled out in the wake of the crack. This paper estimates the improvement in toughness that might be expected due to these mechanisms, by means of a three-dimensional model. The analysis considers a semi-infinite crack propagating through a brittle matrix material, which contains a regular distribution of tough particles. Particles in the wake of the crack are modelled by finding an appropriate distribution of point forces that pin the crack faces; and the effect of the crack bowing between obstacles is included by means of an incremental perturbation method based on work byRice [J. Appl. Mech.56, 619 (1985)]. The calculation predicts the shape of the crack as it propagates through the solid; the resulting R-curve behaviour; and the length of the bridged zone in the wake of the crack.

増分摂動法を導入した一次元複合非線形有限要素法

This paper introduces a numerical method which is able to predict with good accuracy the elastic-plastic behavior of plane skeletal structures. This method, called FERT-P, is a combined and integrated version of the following two methods. One is the combined geometrically and materially nonlinear one-dimensional finite element method proposed by Nakamura, Ishida and further developed by the authors (Refs. 2-7), which has recently been called PERT (Finite Element method with Rigid-body-motion coordinates and Transfer matrix technique). The other is the incremental perturbation method developed by Nakamura, Uetani et al. (Refs. 8-11) to provide means of finding every point on an equilibrium path at which a new element will start yielding or unloading. The incremental perturbation method is able to determine automatically computational step lengths. Several benchmark tests have been carried out on the elastic-plastic buckling of steel structures. The following conclusions for the FERT-P have been derived: (1) By adopting the rigid-body-motion coordinates, the perturbation calculation in the FERT-P is concerned only in the transformation of global and local coordinates. The development of the FERT-P program is very easy. (2) The transfer matrix technique is effective for deriving the perturbation equations of members from those of elements. (3) The FERT-P is a very reliable numerical method to analyze elastic-plastic behaviors, including in particular the unstable behavior, of plane skeletal structures.

The incremental perturbation method for large displacement analysis of elastic‐plastic structures

AbstractA procedure of applying the perturbation method is presented for the incremental numerical analysis of materially and combined non‐linear problems of discrete and discretized structral systems. Small but finite strain and stress increments are strictly distinguished from the strain rates and stress rates, respectively. It is shown that, by applying the perturbation procedure not only to the non‐linear strain‐displacement relations and equilibrium equations, but also to the constitutive equations in terms of rate quantities, all the governing equations can be satisfied to any desired accuracy at every instantaneous configuration in between the starting and terminal points of an incremental step. The proposed method provides also means of finding, to a desired accuracy, every point on an equilibrium path of a discrete system at which a new element will start yielding or unloading and possible critical points on the path. The significance of the proposed method is expected to be appreciated particularly in numerical investigations of critical behaviours and post‐buckling behaviours.