Wave Propagation in Viscoelastic and Anisotropic, Cylindrical Structures

151 Pages·1996·7.969 MB·English

by  Wollmann J.

Diss. ETH No. 11762
Wave Propagation in Viscoelastic and
Anisotropic, Cylindrical Structures
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY
for the degree of
Doctor of Technical Sciences
presented by
JOHANNES VOLLMANN
Dipl. Bau-Ing. ETH, Masch.-Ing. HTL
born 1 April 1964
Citizen of Oberrohrdorf, AG
Accepted on the recommendation of
Prof. Dr. J. Dual, examiner
Prof. Dr. J.D. Achenbach, coexaminer
Zürich 1996
Copyright (cid:211) 1996, by Johannes Vollmann
All rights reserved.
The view from my office on a clear winter morning.
To my father
Acknowledgements
This work was carried out during my appointment as an assistant at the Institute of
Mechanics, ETH Zürich, a status which gave me the freedom to follow new ideas
whenever they came up without the pressure of daily accountability as to the success
of these new approaches.
I would like to express my thanks to all of those who contributed to this work,
in particular to:
Prof. Dr. J. Dual, who directed the entire work, for his kind support, many helpful sug-
gestions and for carefully supervising each step of this thesis.
Prof. Dr. J.D. Achenbach, Northwestern University, Evanston, Illinois, USA, for review-
ing this thesis and for spontaneously accepting the coexamination.
Prof. Dr. M.B. Sayir, chairman of the Institute of Mechanics, ETH Zürich, for providing
the spiritual environment in which research is conducted in the sense mentioned
above.
Prof. Dr. G. Herrmann, Stanford University, Stanford, California, USA, for many valua-
ble discussions and for inviting me to a workshop on Structural Acoustics, organized
by the Office of Naval Research and held at the University of Cambridge, U.K. in
Spring 1993, where my eyes were opened to ‘modern spectrum estimation methods’
for the first time.
Gilbert Schiltges, Roger Breu, and Markus Pfaffinger, not only for the contributions
they made with their Diploma-theses, but also for sharing my enthusiasm in the excit-
ing field of wave propagation. All of them became colleagues, working towards their
own PhDs.
Sandra Nilssen, for her friendly help concerning the linguistic aspects of this thesis.
Dr. Stephan Kaufmann, for his kind support in solving numerical problems and, last
but not least, for ‘conducting the orchestra’ of our computer network perfectly for many
years.
Johannes Vollmann
Zürich, April 1996
Table of contents
Abstract iv
Zusammenfassung vi
List of symbols viii
1 Introduction 1
1.1 Previous work................................................................................................1
1.2 Scope and outline of the present work.......................................................3
2 Theory of wave propagation in a shell with
viscoelastic core 5
2.1 Nature and representational forms of dispersion......................................5
2.2 Derivation of the governing equations........................................................7
2.2.1 Dynamic shell equations.....................................................................7
2.2.2 Wave propagation in the linear viscoelastic cylindrical core.............11
2.2.3 Boundary and interface conditions between shell and core.............15
2.2.4 Dispersion relation............................................................................17
2.2.5 Normalized shell displacement.........................................................18
2.2.6 Normalized stress field in the viscoelastic core................................19
2.3 Examples of numerical results ..................................................................20
2.3.1 The empty shell................................................................................22
2.3.2 The hypothetical inviscid fluid column with stress-free boundaries..24
2.3.3 The shell filled with an inviscid core..................................................27
2.3.4 The shell filled with a dissipative core...............................................31
2.3.5 The shell filled with a shear elastic core...........................................39
2.3.6 Notation of the mode numbering......................................................43
2.4 Conclusions.................................................................................................44
ii Contents
3 High resolution measurement of the complex
dispersion relation 47
3.1 Outline of the problem................................................................................47
3.2 Experiments.................................................................................................49
3.2.1 Experimental set-up..........................................................................49
3.2.2 Piezoelectric excitation.....................................................................52
3.3 Signal processing .......................................................................................54
3.3.1 Spectrum analysis in time domain....................................................55
3.3.2 Spectrum analysis in space domain.................................................55
3.3.3 Confidence and accuracy.................................................................63
4 Experimental results versus theory 65
4.1 The empty shell...........................................................................................65
4.2 The shell filled with silicone oil in low frequency range..........................66
4.3 The shell filled with silicone oil in high frequency range........................69
4.4 The shell filled with alcohol (low viscosity)..............................................73
4.5 The shell filled with polyisobutylene (high viscosity)..............................74
4.6 The linear elastic aluminium rod ...............................................................76
4.7 Backward wave propagation and boundary movements........................81
4.8 Conclusions.................................................................................................86
5 Complete elastic characterization of anisotropic
material by guided waves 87
5.1 Motivation.....................................................................................................87
5.2 Derivation of the dispersion relation of axisymmetric wave modes
in a transversely isotropic rod...................................................................88
5.2.1 Constitutive equation of anisotropic material....................................88
5.2.2 Orthotropic and transversely isotropic material................................89
5.2.3 Dynamic equations for the transversely isotropic rod.......................91
5.2.4 Boundary conditions and dispersion relation....................................95
5.2.5 Special case of axisymmetry............................................................96
Contents iii
5.3 Influence of the material constants on the dispersion relation..............99
5.3.1 Variation of the Young’s modulus E ............................................103
33
5.3.2 Variation of the shear modulus G ................................................104
13
5.3.3 Variation of the Young’s modulus E ............................................105
11
5.3.4 Variation of the Poisson’s ratio n .................................................106
12
5.3.5 Variation of the Poisson’s ratio n .................................................107
13
5.4 Experiments...............................................................................................108
5.4.1 Longitudinal and radial modes........................................................108
5.4.2 Torsional modes.............................................................................110
5.5 Determination of the material constants from the measured
dispersion relation....................................................................................112
5.5.1 Determination of E and G .........................................................112
33 13
5.5.2 Determination of E and n ..........................................................112
11 12
5.5.3 Determination of n .......................................................................114
13
5.5.4 Results for the best fit.....................................................................115
5.6 Conclusions...............................................................................................117
6 Outlook 119
Appendix A: Matrix m 123
ij
Appendix B: Shell theory versus exact theory 125
Appendix C: Matrix a 127
ij
Bibliography 129
Curriculum vitae 135
Abstract
In this thesis, the complex dispersion relation (frequency versus complex
wavenumber) of guided, axisymmetric waves in circular cylindrical struc-
tures is investigated theoretically and confirmed by an extended series of
experiments. The goal of this work is to understand the topology of the
complex wave spectrum for various cylindrical structures, such as elastic
shells containing a viscoelastic core, as well as isotropic and anisotropic
rods with a circular cross-section in order to determine complex material
properties of the structures. In particular, a theoretical model is derived to
calculate the complex dispersion relation of axisymmetric wave modes in
a thin-walled linear elastic shell containing a viscoelastic core. Curvature
and transverse strain cause a coupling between the shell wall bending
and longitudinal motion in an empty cylindrical shell. Therefore the disper-
sion relation of a system consisting of a shell and a viscoelastic core is
governed by the shell properties as well as the bulk modulus, the shear
modulus, and the density of the core material. By varying the complex
moduli, the core can be modelled as an inviscid fluid, an elastic material,
or anything between these two extremes.
New dynamic metrology is developed to measure complex dispersion
relations in a frequency range of between 1 kHz to 2 MHz with up to 40
travelling wave modes in one single process. With a heterodyne LASER
interferometer, which was partly developed at the Institute of Mechanics,
ETH Zürich, the radial and the longitudinal displacement of the shell sur-
1
face is detected. The signals are processed with FFT in the time domain.
To decompose the complex wave-number dependence, a complex total
least squares algorithm is used in the space domain. This algorithm, orig-
2
inally developed for the signal processing of NMR signals, demands
fewer measurements than FFT and allows a lower signal-to-noise ratio. It
extracts both the real and the imaginary part of the wavenumber from the
measured data.
The axisymmetric waves are excited with a radially and a longitudinally-
acting piezo transducer simultaneously. Most of the experiments are per-
formed using a steel tube with a diameter of 25 mm and a wall thickness
of 0.5 mm. The tubes are filled with water, milk, alcohol, glycerine, silicone
1. Fast Fourier Transform
2. Nuclear Magnetic Resonance
Abstract v
oil, and polyisobutylene.
Linear elasticity can be considered as an extreme case of viscoelasticity
(long relaxation times compared with the deformation cycles). To ascer-
tain the validity of the theory as well as the computational precision, dis-
persion curves are also calculated for a shell containing a viscoelastic
material behaving like the elastic shell. These results are compared with
measured curves of an isotropic aluminium rod where excellent agree-
ment was found.
In the case of the isotropic aluminium rod, the phenomenon of ‘backward
wave propagation’ in which the group velocity and the phase velocity of
particular modes have opposite signs, is clearly measured.
The method is finally applied in order to determine material properties of
anisotropic materials. In this case, the complex dispersion relation of a
carbon fibre-reinforced epoxy rod with cylindrical cross-section is meas-
ured. The theory presented by Mirsky in 1965 is implemented and the
influence of the material constants on the topology of the dispersion rela-
tion is discussed numerically. Using the knowledge of the mode shapes at
specific locations in the dispersion diagram, the five material parameters
describing the dynamic behaviour of transversely isotropic material can
be determined step by step with high accuracy. As the imaginary wave-
numbers are measured as well, the frequency-dependent damping
behaviour of the composite material is discussed on a qualitative basis.