Creep Mechanics

330 Pages·2002·8.742 MB·English

by  Univ.-Prof. Dr.-Ing. habil. Josef Betten (auth.)

Creep Mechanics
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Engineering
http://www.springer.de/engine/
Josef Betten
Creep Mechanics
With 72 Figures and 15 Tables
. , Springer
Univ.-Prof. Dr.-Ing. habil. JosefBetten
Department of Mathematical Models
in Materials Science
Technical University Aachen
Augustinerbach 4 -22
52064 Aachen / Germany
CIP data applied for
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Betten, Jo sef: Creep mechanics 1 Jo sef Betten.
(Engineering online library)
ISBN 978-3-662-04973-0 ISBN 978-3-662-04971-6 (eBook)
DOI 10.1007/978-3-662-04971-6
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Preface
This book is based upon lectures on Mathematical Models in Materials Sci
ence held at the University of Aachen, Germany, since 1969. Guest-lectures
at several Universities in Germany and abroad or representations at Interna
tional Conferences have also been included. Furthermore, I represent some
parts of this book on ANSYS-Seminars about viscoelasticity and viscoplas
ticity in Munich, Stuttgart, and Hannover. Some results of research projects,
which I have carried out in recent years, are also discussed in this exposition.
Over the last two or three decades much effort has been devoted to the
elaboration of phenomenological theories describing the relations between
force and deformation in bodies of materials, which obey neither the linear
laws of the classical theories of elasticity nor the hydrodynamics of viscous
fluids.
Material laws and constitutive theories are the fundamental bases for de
scribing the mechanical behavior of materials under multi-axial states of
stress involving creep and creep rupture. The tensor function theory has be
come a powerful tool for solving such complex problems.
The present book will provide a survey of some recent advances in the
mathematical modelling of materials behavior under creep conditions. The
mechanical behavior of anisotropic solids requires a suitable mathematical
modelling. The properties of tensor functions with several argumenttensors
constitute a rational basis for a consistent mathematical modelling of com
plex material behavior.
This monograph presents certain principles, methods, and recent success
ful applications of tensor functions in creep mechanics. Thus, a proper un
derstanding of the subject matter requires fundamental knowledge of tensor
algebra and tensor analysis. Therefore, Chapter 2 is devoted to tensor no
tation, where both symbolic and index notation have been discussed. For
higher-order tensors and for final results in most of the derivations the index
notation provides the reader with more insight.
'II Preface
The simplest way to formulate the basic equations of continuum mechan
ics and the constitutive or evolutional equations of various materials is to
restrict ourselves to rectangular cartesian coordinates. However, solving par
ticular problems, for instance in Chapter 5, it may be preferable to work in
terms of more suitable coordinate systems and their associated bases. There
fore, Chapter 2 is also concerned with the standard techniques of tensor anal
ysis in general coordinate systems.
Creep mechanics is a part of continuum mechanics, like elasticity or plas
ticity. Therefore, some basic equations of continuum mechanics are put to
gether in Chapter 3. These equations can apply equally to all materials and
they are insufficient to describe the mechanical behavior of any particular
material. Thus, we need additional equations characterizing the individual
material and its reaction under creep condition according to Chapter 4, which
is subdivided into three parts: the primary, the secondary, and the tertiary
creep behavior of isotropic and anisotropic materials.
The creep behavior of a thick-walled tube subjected to internal pressure
is discussed in Chapter 5. The tube is partly plastic and partly elastic at time
zero. The investigation is based upon the usual assumptions of incompress
ibility and zero axial creep. The creep deformations are considered to be of
such magnitude that the use of finite-strain theory is necessary. The inner
and outer radius, the stress distributions as functions of time, and the creep
failure time are calculated.
In Chapter 6 the creep potentials hypothesis is compared with the tensor
function theory. It has been shown that the former theory is compatible with
the latter, if the material is isotropic, and if additional conditions are fulfilled.
However, for anisotropic materials the creep potential hypothesis only
furnishes restricted forms of constitutive equations, even if a general creep
potential has been assumed. Consequently, the classical normality rule is
modified for anisotropic solids based upon the representation theory of tensor
functions.
The existance of a creep potential in the tertiary creep phase is not justi
fied. This phase is accompanied by the formation of microscopic cracks on
the grain boundaries, so that damage accumulation occurs. In some cases
voids are caused by a given stress history and, therefore, they are distributed
anisotropically among the grain boundaries.
Because of this microscopic nature, damage has an anisotropic char
acter even if the material was originally isotropic. The fissure orientation
and length cause anisotropic macroscopic behavior. Therefore, damage in an
Preface VII
isotropic or anisotropic material that is in a state of multiaxial stress can only
be described in a tensorial form.
Thus, the mechanical behavior will be anisotropic, and it is therefore
necessary to investigate this kind of anisotropy by introducing appropiately
defined anisotropic damage tensors into constitutive equations.
In Sections 4.3.2 and 6.4 constitutive equations involving damage and
initial ansisotropy have been formulated in detail. Chapter 7 is concerned
with the construction of damage tensors or tensors of continuity (Section
7.1). Then, the multiaxial state of stress in a damaged continuum is analized
in more detail (Section 7.2). Finally, some damage effective stress concepts
are also discussed (Section 7.3).
For engineering applications it is very important to generalize uniaxial
constitutive laws to multiaxial states of stress. This can be achieved by ap
plying interpolation methods for tensor functions developed in Chapter 8.
In foregoing Chapters, the creep behavior of solids has been described.
Chapters 9 and 10 are devoted to fluids, where linear and nonlinear fluids and
memory fluids are taken into consideration.
Many materials exihibt both features of elastic solids and characteristics
of viscous fluids. Such materials are called viscoelastic, to which the ex
tended Chapter 11 has been devoted. Several rheological models (both linear
and nonlinear) are discussed in more detail, and their creep or relaxation
behavior has been compared with experimental data.
In contrast to fluids (Chapter 9) viscop lastic materials can sustain a shear
stress even at rest. They begin to flow with viscous stress after a yield con
dition has been satisfied. Chapter 12 is subdivided in linear and nonlinear
theory of viscoplasticity. As an example, viscoplastic behavior of metals in
comparison with experimental results is discussed.
Together with my coworkers I have carried out own experiments to ex
amine the validity of the mathematical modelling. Furthermore, an overview
of some important experimental investigations in creep mechanics of other
scientists has been provided in Chapter 13.
The mathematical backround required has been kept to a minimum and
supplemented by explanations where it has been necessary to introduce spe
cial topics, for instance, concerning tensor algebra and tensor analysis or
representation of tensor functions. Furthermore, two appendices have been
included to provide sufficient foundations of DIRAC and HEVISIDE func
tions or of LAPLACE transformations, since these special techniques playa
fundamental role in creep mechanics.
vm Preface
In many examples or for many graphically representations the MAPLE
software has been utilized. MAPLE or MATHEMATIC A and many other
symbolic manipulation codes are interactive computer programs, which are
called computer algebra systems. They allow their users to compute not only
with numbers, but also with symbols, formulae, equations, and so on. Many
mathematical operations such as differentiation, integration, LAPLACE trans
formations, inversion of matrices with symbolic entries, etc. etc. can be car
ried out with great speed and exactness of the results. Computer algebra sys
tems are powerful tools for mathematicians, physicists, engineers, etc., and
are indispensable in scientific research and education.
I would like to express my appreciation to my Assistant, Dipl.-Ing. Uwe
MIEX.
NAVRATH, for preparing the camera-ready manuscript in His compe
tence and skill have been indispensable for me.
Thanks are due to Springer-Verlag, in particular to Dr. Dieter MERKLE
and Dr. Hubertus v. RIEDESEL for their readiness to publish this monograph
and for their agreeable cooperation.
Aachen, May 2002 Josef Betten
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Tensor Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Cartesian Tensors ..................................... 9
2.2 General Bases. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 16
3 Some Basic Equations of Continuum Mechanics . . . . . . . . . . .. 31
3.1 Analysis of Deformation and Strain ...................... 31
3.2 Analysis of Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40
4 Creep Behavior of Isotropic and Anisotropic Materials; Consti
tutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
4.1 Primary Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
4.2 Secondary Creep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
4.2.1 Creep Potential Hypothesis ....................... 53
4.2.2 Isochoric Creep Behavior. . . . . . . . . . . . . . . . . . . . . . . .. 57
4.2.3 Creep Parameters ............................... 59
4.2.4 Second-Order Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
4.3 Tertiary Creep ........................................ 69
4.3.1 Uniaxial Tertiary Creep .......................... 69
4.3.2 Multiaxial Tertiary Creep. . . . . . . . . . . . . . . . . . . . . . . .. 72
5 Creep Behavior of Thick-Walled Thbes. . . . . . . . . . . . . . . . . . .. 77
5.1 Method to discribe the Kinematics ....................... 77
5.2 Isochoric Creep Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
5.3 Stress Field .......................................... 85
5.4 Expansion and Failure Time ............................ 89
5.5 Numerical Computation and Examples ................... 92
X Contents
6 The Creep Potential Hypothesis in Comparison with the Tensor
Function Theory ...................................... 10 1
6.1 Isotropy ............................................. 101
6.2 Oriented Solids ....................................... 103
6.3 Modification of the Normality Rule ...................... 107
6.4 Anisotropy expressed through a Fourth-Rank Tensor ........ 110
6.4.1 Irreducible Sets of Tensor Generators and Invariants .. 111
6.4.2 Special Formulations of Constitutive Equations and
Creep Criteria .................................. 113
6.4.3 Incompressibility and Volume Change .............. 115
6.4.4 Characteristic Polynomial for a Fourth Order Tensor .. 119
6.4.5 LAGRANGE Multiplier Method .................... 123
6.4.6 Combinatorial Method ........................... 124
6.4.7 Simplified Representations ....................... 127
7 Damage Mechanics .................................... 131
7.1 Damage Tensors and Tensors of Continuity ................ 131
7.2 Stresses in a Damaged Continuum ....................... 140
7.3 Damage Effective Stress Concepts ....................... 147
8 Tensorial Generalization of Uniaxial Creep Laws to Multiaxial
States of Stress ........................................ 151
8.1 Polynomial Representation of Tensor Functions ............ 151
8.2 Interpolation Methods for Tensor Functions ............... 152
8.3 Tensoral Generalization of NORTON-BAILEY's Creep Law .. 154
8.4 Tensorial Generalization of a Creep Law including Damage .. 157
9 Viscous Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163
9.1 Linear Viscous Fluids .................................. 163
9.2 Nonlinear Viscous Fluids ............................... 171
10 Memory Fluids ........................................ 181
10.1 MAXWELL Fluid ...................................... 181
10.2 General Principle ..................................... 182
10.3 Effects of Normal Stresses .............................. 185
11 Viscoelastic Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 187
11.1 Linear Theory of Viscoelasticity ......................... 187
11.2 Nonlinear Theory of Viscoelasticity ...................... 193
11.3 Special Visoelastic Models ............................. 194