Comparison of microscopic and macroscopic instabilities in a class of two-dimensional periodic composites
type: Item
instance of: scholarly article
title: Comparison of microscopic and macroscopic instabilities in a class of two-dimensional periodic composites
zbMATH Open document ID: 0808.73051
DOI: 10.1016/0022-5096(93)90039-I
author: William C. Schnaidt
author: Nicolas Triantafyllidis
published in: Journal of the Mechanics and Physics of Solids
publication date: 1995-03-21T00:00:00Z
review text: The quantitative comparison between the microscopic (local) and the corresponding macroscopic (global) failure is carried out in a two- dimensional fiber reinforced composite. The latter is modeled as an infinite periodic grillage (with a rectangular unit cell) of axially compressed in both directions beams with an average shear stiffness. At first, an analytical solution for the exact buckling problem of the model is represented for the onset of the first bifurcation in the infinite periodic structure using Bloch wave theory. The solution provides the lowest buckling load corresponding to a mode with given wavelengths in each direction. The critical load for a given geometry of the model is defined as the lowest point of the characteristic surface. Then, the averaged (macroscopic) behavior of the model is considered, and a study of its stability is presented. The macroscopic behavior of the periodic structure is characterized by the homogenized incremental moduli of the structure, where the average is taken over a unit cell which is the characteristic volume for the periodic material. The stability of the homogenized material results in the ellipticity of the derived spatially constant homogenized incremental moduli. The loss of ellipticity in a homogeneous material with spatially constant properties means that the homogeneous stress state in a Dirichlet problem is no longer an energy minimizer and is unstable. Further, it is shown that the loss of elliptic instability for the homogenized (macroscopic) model corresponds to a bifurcation instability of the exact (microscopic) model when the wavelength of its buckling mode is much larger than the unit cell. The numerical results are obtained in the form of the critical load dependence on various microstructure geometries. At first, attention is focussed on the stability of the grillage in the absence of interstitial stiffness \((g=0)\). The common feature for all the critical load calculations in this case is that th…
Mathematics Subject Classification ID: 74E30
Mathematics Subject Classification ID: 74A60
Mathematics Subject Classification ID: 74M25
Mathematics Subject Classification ID: 74G60
Mathematics Subject Classification ID: 74R99
Mathematics Subject Classification ID: 74E05
zbMATH DE Number: 480388
zbMATH Keywords: effects of geometric parameters
zbMATH Keywords: grillage
zbMATH Keywords: Bloch wave theory
zbMATH Keywords: homogenized incremental moduli
zbMATH Keywords: ellipticity
zbMATH Keywords: bifurcation instability
zbMATH Keywords: critical load
zbMATH Keywords: interstitial stiffness
zbMATH Keywords: stable stress states
reviewed by: Ivan A. Parinov
MaRDI profile type: Publication
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cites work: On the Comparison Between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites
full work available at URL: 0022-5096(93)90039-i
OpenAlex ID: W1997142348
description: scientific article; zbMATH DE number 480388