Post-critical Range Research Articles

Robust Algorithm for Brittle Fracture based on Energy Minimization

AbstractThe paper outlines a variational formulation of brittle fracture in solids and considers its numerical implementation by a distinct finite element method. The starting point is a variational setting of fracture mechanics that recasts a monotonic quasistatic fracture process into a sequence of incremental minimization problems. The proposed implementation introduces discretized crack patterns with material‐force‐driven incremental crack‐segment releases. These releases of crack segments constitute a sequence of positive definite subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. The formulation is embedded into accompanying r‐adaptive crack‐pattern reorientation procedures with material‐force‐based indicators, providing reorientations of elements at the crack‐tip. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Buckling and post-buckling behavior of thin-walled cylindrical steel shells with varying thickness subjected to uniform external pressure

In the present paper, numerical and experimental investigation have been undertaken into the buckling and post-buckling behavior of thin-walled cylindrical steel shells with varying thickness subjected to uniform external pressure. For the experimental study, four different cylindrical steel shell specimens with varying thickness were tested to collapse. For the post-buckling analysis of these structures, material and geometric nonlinear collapse analysis are carried out. To trace the equilibrium paths through limit points into the post-critical range, the ‘Arc-Length-Type Method’ has been used. In order to verify the accuracy and validity of the finite element modeling, the numerical results, obtained from nonlinear finite element collapse analyses, have been compared with the results of the experimental study. The study shows that the theoretical behavior predicted by the nonlinear finite element collapse analyses followed closely the experimental behavior. Consequently, it has been found that the finite element model is reliable enough to be used to undertake nonlinear analyses for investigation into the buckling and post-buckling behavior of thin-walled cylindrical steel shells with varying thickness. Also it has been found that the buckling mode can be generated in whole length of the shell if the thickness variation is low. However, in the case of high variation of thickness, the buckling mode is formed in thinner part only. Also, the considerable effects of circumferential and vertical weld line on the buckling strength and mode shapes are verified.

Three-dimensional thermal convection of viscoelastic fluids

The influence of inertia and elasticity on the onset and stability of three-dimensional thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The onset and the stability of steady convective patterns, namely rolls, hexagons and squares, are studied in the post-critical range of the Rayleigh number by using an amplitude equation approach. The square pattern is found to be unstable. In contrast to Newtonian fluids, the hexagonal pattern can be stable for a certain range of elasticity.

One to one resonant double Hopf bifurcation in aeroelastic oscillators with tuned mass dampers

The effects of a tuned added mass on the aeroelastic stability of a single degree of freedom bluff body exposed to a steady flow are investigated. The model captures the essential aspects of the behaviour of flexible structures equipped with Tuned Mass Dampers undergoing galloping oscillations. The system exhibits simple as well double Hopf bifurcations, of non-resonant and 1:1 resonant type. Postcritical behaviour of the system in the neighbourhood of the 1:1 resonant type bifurcation is investigated. Employing the Multiple Scale Method, a second order bifurcation equation in the complex amplitude of motion is obtained. Analytical solutions are used to describe the bifurcation scenario in the cases of both undercritical and supercritical aerodynamic behaviour of the bluff body. The effectiveness of the Tuned Mass Damper even in the postcritical range is proved.

On the hydrodynamic stability of pulsatile flow in a plane channel

A linear stability analysis is undertaken for plane pulsatile Poiseuille flow. The stability picture is described in terms of two distinct ranges of the Womersley number, Wo, which is the ratio of pulsatile to viscous forces. The low-frequency range, termed precritical, has dominant viscous effects, while the high-frequency range, termed postcritical, has dominant inertial effects. A critical Womersley number, Wocr=1.107, is identified, where viscous and inertial forces are in balance. In the precritical range, viscous forces become rapidly dominated by impulse forces, resulting in a sharp drop of the critical Reynolds number with increasing Wo. In the postcritical range, inertial forces become increasingly dominant over impulse forces, resulting in flow stabilization. In the limits Wo→0 and Wo→∞, the flow recovers the stability characteristics of the steady base flow, independently of the pressure amplitude. In other words, pulsating the flow is always destabilizing. The results reported in the postcritical range are in agreement with existing theoretical and experimental works.

Transverse-stiffener requirements for the post-buckling behaviour of a plate in shear

The optimum stiffener rigidity in a transversely-stiffened web under pure shear is sought in the post-critical range. An energy-based large-deflection scheme is employed to study the post-critical strength of the system up to the initiation of material yielding for the usual range of plate geometries and varying stiffener rigidities. Several criteria for “optimum” rib stiffness are assessed including stiffener deflection, the effective halving of the plate length and, finally, the strength of the plate itself. Tentative conclusions are reached on the basis of comparing optimal rib rigidities for the two instances of elastic and post-buckling regimes.

Direct evaluation of the ‘worst’ imperfection shape in shell buckling

For the evaluation of the imperfection-sensitivity of elastic and elastic-plastic shells a fully nonlinear finite-element-method has been developed that directly gives the ‘worst’ imperfection shape connected to the ultimate limit load. The approach uses no approximations such as asymptotical theories or computations in the deep postcritical range. The key point of the method is the description of imperfections as nodal degrees of freedom at the element level. These unknown quantities are implemented by isoparametric shape functions in a finite shell element including finite rotations and thickness stretch. The ultimate buckling load and the corresponding ‘worst’ imperfection shape is defined by two different criteria and numerically determined by an extended system of nonlinear equations. In the computation of the structurally stable collapse the deflections, the imperfection shape, the eigenvector and the lowest possible load-level are obtained. The method is illustrated by several numerical examples.

Compression behaviour of flat stiffened panels made of composite material

In certain structural applications, such as stiffened panels loaded in compression, a better exploitation of the properties of composite materials requires the development of design methodologies that allow the component to work in the post-critical range. This paper describes the results of a research activity with such an objective, carried out at the Department of Aerospace Engineering of the University of Pisa in collaboration with Alenia-Transport Aircraft Group. A group of tests has been carried out on 10 flat stiffened panels, made in composite material, and the results have been analysed with two computer codes, providing useful information concerning their capabilities.

Inversion of wide‐angle seismic reflection times with damped least squares

The wide‐angle seismic reflection times appearing in the postcritical range are used extensively to image the crustal structure in deep seismic sounding investigations. The most commonly used method to calculate interval velocities and thicknesses of a stack of horizontal layers is based on Dix’s hyperbolic equation that requires traveltimes at zero offset and a prior estimate of root‐mean‐square (rms) velocity. Since the wide‐angle reflection times are represented by the nonhyperbolic Taner and Koehler series, a forced fit of such a data set by a hyperbolic equation causes large errors in the estimation of interval velocities. We propose a fast and simple method to determine the interval velocities from wide‐angle reflection times by minimizing the squared errors between the observed traveltimes and the forward response using a damped least‐squares technique. The forward response is calculated, including higher order terms of the reflection series, through Chebychev (orthogonal) polynomial approximations. Inversion of synthetic reflection times by this method, for a given velocity model contaminated by some random errors, and starting with various initial models, produces a final model that matches the true model. This modeling serves as a good indicator of the reliability of the estimated parameters. Besides, the method provides a measure of uncertainty and resolution of estimated parameters.

Post-critical behaviour of ‘K’ braced frames

Abstract During a high-intensity earthquake, the buildings in seismic areas could be acted upon by forces greater than those conventionally calculated, according to the code rules. Consequently, the structures with ‘K’ braced frames could behave in a post-critical range, as a result of the buckling of some compressed braces. The post-critical behaviour of the bar of the ‘K’ braced frame, as well as of the entire structure, is examined in the paper. The results established on an original calculation model are compared with experimental P-Δ diagrams obtained on plain scale models.

On the amplitude modulation and localization phenomena in interactive buckling problems

The postcritical behavior of a 3D system of elastically restrained beams, which can manifest overall and local modes, is analyzed. The secondary bifurcation in the overall postcritical range is studied. By applying the multiple scale perturbation method it is found that the amplitude modulation phenomenon is governed by a differential equation of the second order. An approximate analytical expression of the amplitude modulating function is obtained by the WKB asymptotic method. The occurrence of a turning point which is responsible for the strong localization is revealed. Close analogies with localization of vibrations in imperfect systems are highlighted.