Table Of ContentBlazej T. SkoczeIi
Compensation Systems for Low Temperature Applications
Blazej T. Skoczen
Compensation Systems
for low Temperature
Applications
With 178 Figures
~ Springer
BlaZej T. Skoczeri (PhD, DSc.)
CERN, European Organization for Nuclear Research
Department of Accelerator Technologies
CH -1211 Geneva 23, Switzerland
and
Cracow University of Technology
Institute of Applied Mechanics
AI. Jana Pawla 11 37, PL-31-864 Krak6w, Poland
e-mail: [email protected]
ISBN 978-3-642-06066-3 ISBN 978-3-662-06305-7 (eBook)
DOI 10.1007/978-3-662-06305-7
Library of Congress Control Number: 2004107592
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To my wife Dorota for her patience and understanding
Preface
The present monograph is mainly focused on the behaviour of ductile ma
terials at cryogenic temperatures, stability issues concerning application of
corrugated shells at cryogenic conditions and reliability oriented parametric
optimisation of compensation systems containing the corrugated bellows. As
there are relatively few publications on combined material and structural be
haviour at very low temperatures, the monograph aims at filling this gap. It
is worth pointing out that within the class of publications dedicated to low
temperature behaviour of materials and structures the majority is based on
testing down to the temperature of liquid nitrogen (77 K). Rare publications
deal with the analysis of material and structural response at the temperature
of liquid helium (4.5 K) or superfluid helium (below the point T>., = 2.17 K).
This can be explained by the fact that an (by its nature complex) installation
for testing at such low temperatures is very expensive. Only the large research
centres and universities, working in the domain of superconductivity, cryogen
ics or developing superconducting magnets for particle accelerators, can afford
such installations.
A significant part of the present monograph is dedicated to the analy
sis of the phenomena associated with plastic yielding in stainless steels at
cryogenic temperatures. Generally, three phenomena are distinguished: plas
tic strain induced phase transformations, serrated yielding and evolution of
ductile damage. The work is focused mainly on the most often occurring
'Y - a' transformation, that leads to formation of the martensite sites dis
persed in the surrounding austenite matrix, as well as on the strain induced
evolution of ductile damage represented by the formation of micro-voids and
micro-cracks. The relevant constitutive models are developed and verified on
the experimental data. Special insight into the plastic strain induced marten
sitic transformation at low temperatures and its consequences for the strain
hardening and the magnetic permeability has been gained. The existing visco
plastic models are very complex and based on a great number of parameters.
Identification of all the parameters at very low temperatures is extremely dif
ficult, laborious and expensive. Therefore a new, simplified and efficient model
VIII Preface
suitable for the temperature range of 2 K+77 K (superfiuid helium to liquid
nitrogen) has been developed. The second constitutive model presented in the
monograph is based on the kinetic law of damage evolution. This law, formu
lated initially for ambient and enhanced temperatures, has been adopted to
describe the evolution of micro-damage in ductile materials at cryogenic tem
peratures. Also, it has been used as a basis for derivation of the closed form
analytical solutions for the fatigue life of samples subjected to sustained and
cyclic loads (ratchetting) with the effect of mean plastic strain on cycle taken
into account. Further objective is description of the low cycle fatigue phenom
ena at cryogenic conditions in the framework of ductile damage mechanics. It
turns out that the evolution of damage has a serious impact on the material
moduli: the modulus of elasticity, the yield point and the hardening modulus.
Here, an uncoupled approach, consisting in the elasto-plastic analysis of the
structure (first step) and integration of the kinetic law of damage evolution
(second step), as well as the relevant numerical procedure are presented.
Since with the development of micro-cracks and micro-voids (micro
damage) the material becomes "softer" it may have a substantial impact on
the structural response. For instance thin-walled shell structures turn out
to be more susceptible to buckling. Therefore, a cycle to cycle accumulation
of plastic deformations, accompanied by development of damage, may lead
to local instabilities which, finally, may affect the global structural response.
Typical examples of such shell structures are corrugated thin-walled bellows
applied in the interconnections of cryogenic transfer lines or superconducting
particle colliders. The effect of strain induced damage evolution on the struc
tural stability of corrugated shells at low temperatures has been studied and
the results are presented in the monograph. Also, the complementary stabil
ity issues (based on the equivalent column concept) like the effect of shear
deformation and torsion on stability of corrugated bellows were extensively
studied. The impact of shear deformation is of particular interest for short
corrugated bellows and can not be neglected in the stability analysis both at
room and at cryogenic temperatures.
Both the low cycle fatigue induced material damage, phase transformations
at cryogenic temperatures as well as the local and the global instabilities of
components compromise the structural reliability of cryogenic systems con
taining the corrugated bellows. Therefore, a reliability oriented parametric
optimisation of structures containing corrugated bellows and designed for op
eration at cryogenic temperatures is developed. In the future, a more complex
probabilistic approach, based on the predefined probability density functions,
shall replace the standard deterministic analysis.
Finally, it is worth indicating that the monograph constitutes a sequence
of studies containing an internal logic: from the constitutive modelling, via
material and structural stability issues to the reliability oriented optimisation
of compensation elements of large cryogenic systems. Thus, both the mate
rial and the structural issues contribute in an essential way to the reliability
Preface IX
analysis and are taken into account in the optimisation. This logic is fully
reflected by the sequence of chapters:
Chapter 1 contains a short introduction to the cryogenics followed by a
broader description of the compensation systems containing the bellows ex
pansion joints. State of the art concerning toroidal shells and corrugated ax
isymmetric bellows and refereing to the following aspects: strength analysis,
adaptation and inadaptation to cyclic loads, failure mechanisms, optimum
design, is presented. Different types of expansion joints (standard and non
standard) for cryogenic and high vacuum applications are described. Finally,
examples of the materials applied in the design of structures working at very
low temperatures are listed.
Chapter 2 is focused on the thermodynamics of processes occurring in met
als low temperatures. It starts with a description of different types of metallic
lattice and the most common lattice imperfections. Further the mechanisms
of heat transport in metals at very low temperatures as well as the ther
modynamic instabilities, related to the third principle of thermodynamics,
are explained. Finally, the mechanisms of inelastic deformations, occurring
in ductile materials at very low temperatures, including the continuous and
discontinuous yielding are presented.
Chapter 3 is dedicated to an overview of the physical, chemical and me
chanical properties of stainless steels at cryogenic temperatures. Special atten
tion is focused on the phenomena that distinguish the behaviour of stainless
steels at low temperatures from ambient temperature: plastic strain induced
phase transformations b - a') and discontinuous (serrated) yielding. The me
chanical properties like yield point, ultimate strength, elongation and harden
ing resulting from phase transformations as well as sensitivity to strain rate at
low temperatures are also discussed. Eventually, some results reported in the
literature on the low cycle fatigue of stainless steels at cryogenic temperatures
as well as the curves of magnetic permeability failure are cited.
Chapter 4 is focused mainly on modelling of plastic yielding and ductile
damage for low temperature applications. It starts from an overview of the
most often used models describing plastic yielding of ductile materials, that
can also be used for some cryogenic applications (continuous yielding). Then
a simple model of the discontinuous (serrated) yielding is presented. Further,
ductile damage based on the standard isotropic formulation and identification
of the parameters of the damage evolution laws at cryogenic temperatures is
explained. The kinetic laws of damage evolution constitute a basis for the
analysis of the low cycle fatigue in the presence of large mean plastic strain
on cycle. A modified kinetic law of damage evolution under cyclic loads leads
to a generalised Manson-Coffin formula, that gives a good estimate of the fa
tigue life of structures under ratchetting conditions at room and at cryogenic
temperatures. Also, an uncoupled approach to plasticity-damage analysis is
presented together with a relevant numerical algorithm. Here, damage evolu
tion is computed in the post-processing and as a consequence of the previously
computed plastic deformation at a given temperature.
X Preface
Chapter 5 concentrates on the crucial issue of plastic strain induced
martensitic transformation at cryogenic temperatures. It begins with a review
of the existing models with a special focus on the transformation kinetics. A
new constitutive model of the plastic flow in the presence of strain induced
martensitic transformation, developed for the stainless steels at cryogenic con
ditions and for the temperature range 2 K+77 K, is presented. The model de
scribes linear evolution of the volume fraction of a' martensite in the austenitic
"f matrix, as a function of plastic strain, during the second and the most in
tensive phase of transformation at low temperatures. The model is based on
the Mori-Tanaka homogenisation and defines the mixed isotropic/kinematic
hardening resulting from the amount of"f - a' transformation. A combined
model of plastic strain induced damage evolution and martensitic transforma
tion is also presented. Finally, integration of the constitutive equations and
comparison with the experimental data under monotonic and cyclic loads is
given and the limits of applicability of the model are shown.
Chapter 6 deals with the stability issues of corrugated, thin-walled bel
lows. Generally, the chapter shows three mechanisms leading to considerable
reduction of their stability: relaxation of boundary conditions, shear defor
mation in convolutions and torsion of bellows. First, the main mechanism
of loss of stability via the column buckling is explained. Then, a study ex
tended towards the effect of relaxation of boundary conditions on instabilities
of systems containing corrugated bellows is presented. It leads to the stability
diagram reflecting the effect of guidance-to--bellows distance on the critical
buckling pressure and a transition from the first (I) to the second (II) princi
pal buckling mode. Another important issue consists in the influence of shear
deformation on the bifurcation buckling load for relatively short bellows. The
adequate model based on the Engesser approach has been developed and
the results show a considerable reduction of buckling load when compared
to the classical approach. Finally, the third important effect - influence of
torque - leading to a limitation of the bifurcation buckling load is discussed.
Here, a model based on the approach by Ziegler and concerning instability of
the elastic, prismatic columns subjected to torsion and compression, has been
adopted.
Chapter 7 explains the material and fatigue induced structural instabilities
in corrugated bellows at cryogenic temperatures. It begins with an analysis of
the evolution of two plastic strain induced material imperfections that occur
in the bellows convolutions at cryogenic temperatures: ductile damage and
phase transformation. It turns out that both phenomena are highly localised
at root and at crest of convolutions. Further, the mechanism of fatigue in
duced structural instability is described. It is shown that the fatigue induced
instabilities can be explained on the ground of the ductile damage mechanics.
A column subjected to axial compressive load and simultaneous kinematically
controlled cycling, that causes evolution of ductile damage in the material, is
shown as an example. This model provides a link to a similar phenomenon in
corrugated bellows: evolution of plastic strain fields and damage in the con-
Preface XI
volutions under cyclic loads and internal pressure at cryogenic temperatures.
Such a combined sustained and cyclic loading leads to the fatigue induced in
stability resulting from the cycle-to-cycle evolution of bellows axial stiffness.
The relevant numerical model based on the linear kinematic hardening and
isotropic damage is presented. As an example evolution of stability of elasti
cally supported bellows at 4K as a function of number of cycles is shown.
Chapter 8 is entirely dedicated to the reliability oriented parametric opti
mum design of cryogenic systems containing the corrugated bellows. It begins
with a general introduction of the reliability notions and safety categories for
complex cryogenic systems like a superconducting accelerators of particles,
superconducting links or cryogenic transfer lines. Further, global and local
approaches in the optimisation procedure of cryogenic systems with the de
sign objective based either on the buckling load, cost function or on both of
them (mixed approach) are presented. The optimisation is carried out in the
framework of the probabilistic approach with the design parameters playing
role of random variables. Thus, the inequality constraints are expressed in the
form of a probability that a given condition has been reached. An assumption
has been made that the probability density functions associated with different
constraints are known. Also, the assumptions of the weakest link model are
presented. Further, the optimum parametric design of the cryogenic bellows
in the probabilistic and deterministic form is presented. Finally, an algorithm
for optimisation of the corrugated bellows is given and some numerical results
are shown.
Chapter 9. The modern high energy physics needs very sophisticated and
complex tools in order to explore the world of elementary particles consti
tuting the matter. One of the most important aims over the past 30 years
was confirmation of the so-called Standard Model which assumes that the
fundamental constituents of matter form three families of quarks and lep
tons. The relevant scientific instruments are called accelerators, storage rings
and colliders and their main function is to produce, accelerate, store and col
lide the beams of particles in order to search for the new elementary events,
announcing the potential discoveries, and to provide more statistics for the
already known reactions. The present chapter gives a brief overview of the
main technologies needed to design and build modern circular particle ac
celerators: superconducting magnet technologies and technology of ultrahigh
vacuum, materials and structures. Among the accelerator technologies the
thermal expansion/contraction compensation systems, located in the magnet
interconnections, are emphasised. Also, analysis of local stability of inter
connections containing the corrugated bellows expansion joints as well as the
global stability of the whole accelerator (as a discrete structure) are presented.
XII Preface
Acknowledgements
The author wishes to express a particular gratitude towards all his colleagues
from CERN (Geneva, Switzerland) and Cracow University of Technology
(Poland) for many valuable discussions.
A special thanks goes to Dr Adam Wr6blewski for his precious help when
formatting the book.
