## Document Type

Dissertation - Open Access

## Award Date

1976

## Degree Name

Doctor of Philosophy (PhD)

## Department / School

Civil Engineering

## First Advisor

Michael P. Wnuk

## Abstract

The objective of this thesis is to predict from a theoretical basis the stress redistribution processes occurring within the viscoelastic core of a cylindrical pressure vessel reinforced by an elastic casing around the outer perimeter, and subject to internal pressure applied to the time-dependent inner boundary of the cylinder. The problem so stated requires development of a new technique leading to a solution for a mixed boundary value problem for a viscoelastic body occupying an ablating cylindrical y-shaped region. The approach taken in this thesis consists in reduction of a governing integro-differential equation to an equivalent set of algebraic equ3tions which yield the coefficients in Maclaurin expansions of the time-dependent displacement field. Strictly speaking, the results of such a procedure converge to the exact solution only in the limit of time approaching zero. It should be noted that while the spatial distributions of the stresses, strains, and displacements are exact, their temporal distributions are described only in a "short-time" limit sense. Comparison with available data, however, shows that the results remain accurate for time intervals not exceeding the characteristic relaxation time of the material. Since all entities describing the stress, strain, and displacement fields undergo rapid changes during this initial time interval, the solutions obtained here are of considerable practical interest. The design aspects can thus be linked with the relaxation and stress redistribution processes occurring within the range of application of the short-time solutions derived in this thesis. The essential findings of this theoretical study are as follows: 1. The effect of the stress redistribution which results from addition of the elastic shell around the softer viscoelastic core of the cylinder is beneficial from the standpoint of fracture prevention. The effect is twofold; not only the size of the potentially dangerous zone (zone of tensile hoop stress) is reduced, but also the magnitude of the maximum tensile circumferential stress is decreased as a result of stress redistribution due to an interaction of the core with the harder elastic shell. 2. Chances for fracture initiation decre3se for the time intervals exceeding the characteristic relaxation time of the core material. This effect is entirely due to a time-dependent relaxation process that tends to relieve the shear stresses within the cylinder. The long-time state of stress, attained when t à ꚙ closely resembles the hydrostatic stress state, characteristic for a viscous Newtonian fluid (in case of a Maxwell solid) or it approaches the hydrostatic state of stress for other linear viscoelastic solids. 3. Removal of the material from the inner surface of the cylinder (due, for example, to burning in a solid propellant or due to a muscle contraction in a biological medium) turns out to provide a strengthening mechanism from the point of view of failure prevention. Fracture within the soft core of the cylinder is usually initiated in the regions adjacent to the surface of the bore, where the hoop stress attains the maximum tensile level. Removal of material from inside of the cylinder has a similar effect on strength as the etching process does in case of the crystal surfaces (Yoffe's effect). 4. The solutions obtained in this thesis are valid for an arbitrary linear viscoelastic solid. Therefore, a rational choice of the material moduli and relaxation spectra should be possible to meet the prescribed design criteria (e. g., rate of removal of the core material and the magnitude and time-dependence of the internal pressure). 5. Examples of numerical solutions to certain specific sets of design data are given in Chapters VI and VII (also, see Appendix C) as illustrations of applications of the theory developed here to the stress analysis problems. These problems concern either a burning solid propellant contained in a cylindrical casing or a biological pressure vessel which may change its dimensions due to muscle contractions.

## Library of Congress Subject Headings

Viscoelasticity

Pressure vessels

Boundary value problems

## Format

application/pdf

## Publisher

South Dakota State University

## Recommended Citation

Joshi, A. N., "Viscoelastic Cylindrical Pressure Vessel With Variable Inner Radius" (1976). *Electronic Theses and Dissertations*. 5551.

https://openprairie.sdstate.edu/etd/5551