The finite deformation of a pressurized circular tube for a class of compressible materials
type: Item
instance of: scholarly article
title: The finite deformation of a pressurized circular tube for a class of compressible materials
zbMATH Open document ID: 0539.73038
DOI: 10.1007/BF00947935
author: Amir Jafari
author: Rohan Abeyaratne
author: Cornelius O. Horgan
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1984-01-01T00:00:00Z
review text: The body under consideration is a hollow circular cylinder (e.g., a tube) subjected to uniform internal or external pressure or both at a time and consists of a homogeneous, isotropic, compressible elastic material of a special type, called a harmonic material, introduced by \textit{F. John} [Comm. Pure Appl. Math. 13, 239-296 (1960; Zbl 0094.370)]. For materials of this type the strain energy density function in plane strain is given by \(W=2\mu [H(\lambda_ 1+\lambda_ 2)-\lambda_ 1\lambda_ 2]\), where \(\lambda_ 1,\lambda_ 2\) denote the principal stretches (both positive), \(\mu\) a positive constant that can be identified with the shear modulus of the linear theory of elasticity, and H a constitutive function satisfying certain restrictions in order to ensure a physically reasonable response. Solutions for the deformation and stress fields are obtained in explicit closed form. The true stress distribution in spatial description proves to be essentially independent of the material properties. In the case of internal pressure alone, there is a critical value of the applied pressure at which the largest hoop stress becomes unbounded. This stress always occurs at the inner wall. In the case of external pressure alone, a critical value exists for which closing of the body's cavity is predicted. For nearly solid cylinders or, equivalently, for a cavity in an unbounded body, explicit results are provided in terms of the corresponding stress concentration factor. A classical linearized stability analysis is carried out with allowance for non-axisymmetric states of plane strain. Internal pressurization leads to no instability. Consequently, the occurrence of the infinite hoop stress is not preceded by an instability. On the other hand, external pressurization effects buckling before the cavity closes. The smallest buckling load is found to be much smaller than 2\(\mu\), which is predicted to be the pressure at which the cavity would close. The occurrence of an unbounded s…
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 74G60
Mathematics Subject Classification ID: 74S30
zbMATH DE Number: 3858537
zbMATH Keywords: hollow circular cylinder
zbMATH Keywords: homogeneous, isotropic, compressible elastic material
zbMATH Keywords: harmonic material
zbMATH Keywords: strain energy density function
zbMATH Keywords: plane strain
zbMATH Keywords: deformation
zbMATH Keywords: stress fields
zbMATH Keywords: explicit closed form
zbMATH Keywords: stress distribution
zbMATH Keywords: independent of the material properties
zbMATH Keywords: internal pressure
zbMATH Keywords: critical value of the applied pressure
zbMATH Keywords: largest hoop stress becomes unbounded
zbMATH Keywords: external pressure
zbMATH Keywords: closing of the body's cavity is predicted
zbMATH Keywords: stress concentration factor
zbMATH Keywords: classical linearized stability
zbMATH Keywords: non- axisymmetric states of plane strain
zbMATH Keywords: no instability
zbMATH Keywords: stress singularity
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00947935
OpenAlex ID: W1997483006
description: scientific article; zbMATH DE number 3858537