An incremental approach to the solution of snapping and buckling problems

SUMMARY Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.

/pdf/finite-element-formulations-for-large-deformation-dynamic-1gdn11wrk7.pdf

Finite element formulations for large deformation dynamic analysis

A reduced basis technique and a computational' algorithm are presented for predicting the nonlinear static response of structures. A total Lagrangian formulation is used and the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of basis vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The Rayleigh-Ritz approximation functions (basis vectors) are chosen to be those commonly used in the static perturbation technique namely, a nonlinear solution and a number of its path derivatives. A procedure is outlined for automatically selecting the load (or displacement) step size and monitoring the solution accuracy. The high accuracy and effectiveness of the proposed approach is demonstrated by means of numerical examples.

Reduced Basis Technique for Nonlinear Analysis of Structures

A Stabilized Conforming (SC) nodal integration, which meets the integration constraint in the Galerkin meshfree approximation, is generalized for nonlinear problems. In SC nodal integration, a strain smoothing stabilization is introduced and applied to the deformation gradient using a Lagrangian meshfree discretization. The assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison to the solution of meshfree method using Gauss integration method. The performance of the proposed method appears to be quite robust when dealing with irregular discretization.

Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods

The results of theoretical and experimental investigation of an initially curved clamped–clamped microbeam actuated by a distributed electrostatic force are presented. Reduced-order Galerkin and consistently constructed lumped models of the shallow Euler–Bernoulli arch were built and verified by numerical analysis, and the influence of various parameters on the stability was investigated. Due to the unique combination of generic mechanical and electrostatic nonlinearities, the voltage–deflection characteristic of the device may have two maxima implying the existence of sequential snap-through buckling and pull-in instability and of bistability of the beam. The first critical voltage can be higher or lower than the second one, while the stable deflections are significantly larger than in a straight beam. The minimal initial elevation required for the appearance of the snap-through in the electrostatically actuated beam is smaller than in the case of uniform deflection-independent loading; a closed-form approximation of this elevation was evaluated. The devices were fabricated from silicon on insulator (SOI) wafer using deep reactive ion etching and in-plane responses were characterized by means of optical and scanning electron microscopy. Model results obtained for the actual dimensions of the device were in good agreement with the experimental data.

https://iopscience.iop.org/article/10.1088/0960-1317/18/5/055026/pdf

The pull-in behavior of electrostatically actuated bistable microstructures

The exact solution to the nonlinear problem of the buckling of a shallow, circular arch subjected to a uniform pressure or a concentrated load is given. In addition an explicit bifurcated equilibrium path is presented. The buckling loads, according to several criteria, are found and their range of validity is investigated in detail. It is shown that even if the smallest buckling load is associated with an asymmetric bifurcation this alone is not a sufficient condition for the asymmetric buckling criterion to govern the problem.

Buckling of Shallow Arches

When a shallow arch is subjected to a symmetric dynamic load, this load becomes “critical” if a slight increase in the load magnitude leads to a sudden snap-through. Another form of instability occurs when a slight antisymmetric component in the load produces a sharply increasing antisymmetric response. Both forms of instability are investigated by means of a numerical procedure which introduces a finite element technique into an integro-differential equation formulation. Where applicable the results generally confirm previous results obtained elsewhere, but cover a broader range of problems and are believed to be more accurate.

Dynamic Buckling of Shallow Arches

Ce travail represente une etude de l'equilibre des corps solides, avec un interet special accorde aux corps elastiques et a ceux de Voigt-Kelvin. Les efforts appliques peuvent avoir des composants non conservatifs et gyroscopiques, quoique la dependance explicite du temps soit exclue. Il est bien connu que, pour une categorie de problemes tellement large, l'etablissement des conditions de la stabilite par les methodes classiques de statique ou d'energie n'est pas toujours applicable. Le traitement du probleme essaie d'etre assez comprehensif. En utilisant la methode des vibrations modales on etablie des criteres generaux de stabilite sur la base d'une «energie equivalente». Ces criteres sont d'une telle generalite qu'ils contiennent, comme cas speciaux, les criteres publies auparavant. En plus, de nouveaux principes ont ete developpes et demontres a l'aide d'un exemple numerique.

Some general principles of dynamic instability of solid bodies

Es wird gezeigt, dass Tensorgeschwindigkeiten “materiell objektiv” sind, wenn sie von den Geschwindigkeiten ihrer Abbildung in einer festen Konfiguration abgeleitet sind. Hohere Geschwindigkeiten werden formuliert, und eine hohere “Jaumanngeschwindigkeit” wird definiert, die gegenuber der vonPrager kurzlich eingefuhrten Definition gewisse Vorteile aufweist.

On tensor rates in continuum mechanics

The small amplitude vibrations of a rectangular panel relative to a static buckled configuration and the stability of that configuration are studied. Both the static and dynamic equations of equilibrium are solved through a perturbation expansion. The stability of buckled configurations is investigated by studying the real and imaginary character of the frequency of vibration about these configurations, from which a lower bound to the secondary buckling load is established.

Ernest F. Masur

Papers

Buckling of Shallow Arches

Dynamic Buckling of Shallow Arches

Some general principles of dynamic instability of solid bodies

On tensor rates in continuum mechanics

Vibrations and stability of buckled panels.