Deformation and Fracture of Solid State Materials Sanichiro Yoshida (Springer, 2015)

Prévia do material em texto

Sanichiro Yoshida
Deformation 
and Fracture 
of Solid-State 
Materials
Field Theoretical Approach and 
Engineering Applications
Deformation and Fracture of Solid-State Materials
Sanichiro Yoshida
Deformation and Fracture
of Solid-State Materials
Field Theoretical Approach and Engineering
Applications
123
Sanichiro Yoshida
Department of Chemistry and Physics
Southeastern Louisiana University
Hammond, LA, USA
ISBN 978-1-4939-2097-6 ISBN 978-1-4939-2098-3 (eBook)
DOI 10.1007/978-1-4939-2098-3
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2014949877
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Preface
This book stems from my first acquaintance with Academician Victor E. Panin
of the Soviet Academy of Sciences,1 and my daily research log that I have been
keeping since then. At the end of 1990, I had the opportunity to listen to a
lecture given by Acad. Panin at a meeting held in Tokyo, Japan. In this meeting,
a number of delegates from the Soviet Academy of Sciences gave presentations to
Japanese business people. The Acad. Panin’s lecture was on a new theory of plastic
deformation that he called physical mesomechanics. Although I understood only
20 %, or probably less, of his lecture, I was greatly fascinated by his enthusiastic
presentation and by the physical-mesomechanical view of plastic deformation. (He
was among the two or three presenters who gave the talk in English without an
interpreter.) In particular, the Maxwell-type field equations that describe plastic
deformation dynamics interested me greatly. At that time, my field of research was
laser and spectroscopy, and I was using the Maxwell equations of electrodynamics
on a daily basis. Although I did not understand the Maxwell-type field equations
of physical mesomechanics in any depth necessary to comprehend the deformation
dynamics behind them, I was able to understand that the equations described the
translational and rotational interaction of the displacement field. With my limited
knowledge of continuum mechanics, I was able to sense that material rotation and
its interaction with translational displacement is important in the plastic regime, and
that the Maxwell-type field equations represent that effect.
During the coffee break, I came to Acad. Panin to introduce myself and ask
a number of questions about his presentation. He answered each of my questions
enthusiastically. Moreover, he kindly invited me to the post-meeting banquet to be
held at the USSR embassy later that day. Of course I accepted the invitation and
attended the banquet where I was able to discuss with Acad. Panin a wide range
of topics in strength physics and material sciences. He gave me a book written in
Russian as a gift, and invited me to an international conference being held in the
1Presently the Russian Academy of Sciences.
v
vi Preface
following summer in Tomsk, Siberia. I did not know the language at that time. I was
so interested in the book that I took Russian language courses for 2 years.
In the summer of 1991, I attended the conference in Tomsk and met a group of
scientists working in Acad. Panin’s group. The discussions I had with them were
revolutionary to me. They explained the Maxwell-type field equations, the interac-
tion between the translational and rotational displacement in plastic deformation,
and other gauge theoretical concepts in detail. To be honest, my knowledge about
gauge theories at that time was almost none. After returning home, I read a handful
of books on gauge theories and got more confused. I kept reading and learned that
the electromagnetic field is the gauge field that makes quantum mechanics locally
symmetric. This brought me to the turning point. I started to understand the concept
of gauge transformation and local symmetry. I analyzed various gauge theoretical
concepts in deformation dynamics via analogy to electrodynamics. Interestingly,
this exercise deepened my understanding on electrodynamics. I noticed a number of
different views on Faraday’s law and Ampere’s law as the interaction between the
electric and magnetic field that nature uses as a mechanism to stabilize events, e.g.,
prevent runaway increase of current. This, in turn, helped me consolidate the basic
understanding on the gauge-field nature of plastic deformation dynamics.
As I kept deepening my understanding on the physical foundation of physical
mesomechanics, I realized that the theory was much more profound than I initially
thought. It was an elegant theory capable of describing plastic deformation based
on pure physics, unlike most theories of plastic deformation that relies on phe-
nomenology or mathematical models. It indicated a number of potential engineering
applications as well. However, the work at that time was somewhat inclined toward
the mathematical aspect of the theoretical foundation with little experimental proofs.
I started to conduct experiments trying to prove various elements of the theory, such
as transverse wave characteristics of displacement field in the plastic regime. To
measure displacement field, I used an optical interferometric technique known as
the ESPI (Electronic-Speckle Pattern Interferometry). I found several interesting
phenomena that could be explained by the same physical foundation as physical
mesomechanics. Through analysis of these experimental observations, especially
with the help of analogy with electrodynamics, I conceived new ideas in the
description of deformation dynamics such as the concept of deformation charge
and its role of energy dissipation. This helped me advance the theory from the field-
theoretical description of plastic deformation dynamics to a comprehensive theory
of deformation and fracture based on the same theoretical foundation. To date, I
have continued investigating the field theoretical dynamics of deformation using the
ESPI.
As will be discussed in the following chapters, development of this theory has
not been completed. I decided to put together the knowledge and informationI
gained so far as a book at this point for several reasons. First, recent experimental
observations have convinced me of the validity of the theory. Essentially, the gauge
field in deformation dynamics makes the law of linear elasticity locally symmetric.
The nonlinear dynamics in the plastic regime is formulated through the potential
associated with the gauge field. Second, these experimental observations and their
Preface vii
field theoretical interpretations demonstrate potential of engineering applications.
In particular, the use of ESPI techniques allows us to visualize the deformation field
as a full-field image, and along with field theoretical interpretations, it provides
us with various information. For example, the use of the field theoretical criteria
of plastic deformation and fracture allows us to make diagnosis of the deformation
state of a given object. Third, I would like to invite specialists of different disciplines
to this research for further development of the theory and applications. On the
theoretical side, connections with microscopic theories are very important. At this
point the theory incorporates the effect of microscopic defects that causes plastic
deformation generally into the field equations via the source terms. If a specific
form of the source term is provided by a microscopic theory, it is possible to
describe how the microscopic defect can evolve to the final fracture under a given
condition. Also, more thermodynamic argument will allow us to discuss the energy
dissipation process resulting from irreversible plastic deformation more specifically.
For applications, software development for visualization of displacement field in
objects under deformation, especially during the transitional stage from one regime
to another, e.g., from the elastic to plastic regime, will be not only an interesting
application but also helps further advancement of the theory. Numerical simulations
are also important for further tests of the theory and explore for new applications.
Lastly, I would like to share my experience of learning the Maxwell’s formalism
and the gauge theories with students. A number of electrodynamic concepts that
were unclear to me when I was in the graduate school became crystal clear through
this project. I would like to invite students and have them feel the beauty of field
theories. In my opinion, this subject is ideal to visualize the concept of local
symmetry associated with a gauge field, which is otherwise abstract and difficult
to comprehend. I tried my best to portray the complicated concept with plain terms
and analogies without going into mathematical details. It is also a unique case in
which these concepts, which are usually discussed by scientists specialized in basic
physics such as high energy or particle physics, are discussed in connection with
real world applications such as nondestructive testing of metal objects. I hope that
this book is helpful to people in any of these and related disciplines.
I tried my best to cite literature appropriately. If some papers or books do not
receive fair credit or are not cited, I apologize.
Finally, I would like to express my sincere gratitude to a number of people. First
of all, I would like to thank Acad. Victor Panin for introducing to me his beautiful
paradigm of deformation dynamics and his friendship ever since. I am grateful
to countless colleagues who always supported me during the development of this
theory, especially Professor Cesar Sciammarella for his continuous encouragement
and precious discussions. Through my learning processes of gauge theories and
continuum mechanics, I realized that I owe greatly to all the professors and teachers
from whom I received my graduate and undergraduate trainings. Without their
excellent instruction, I would have never reached the present level of understanding
on the subjects. I also thank all of my friends and students who helped me with the
experiments and computations that provided a number of supporting data.
Hammond, LA, USA Sanichiro Yoshida
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Aim, Scope, and Organization of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Quick Review of Theories of Elastic Deformation . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Displacement and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Hooke’s Law and Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Principal Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Equation of Motion and Elastic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 One-Dimensional Longitudinal Elastic Waves . . . . . . . . . . . . . . . . 32
2.4.2 Three-Dimensional Compression Waves . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Quick Review of Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 The Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Symmetry in Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Global and Local Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Gauge Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Lagrangian Formalism and Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Electrodynamics as a Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Field Theory of Deformation and Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Gauge Theories of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 The Big Picture of the Present Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
ix
x Contents
5 Interpretations of Deformation and Fracture Phenomena
from Field Theoretical Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1 Field Equation as an Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Comprehensive Description of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.1 Elastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.2 Plastic Deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 100
5.2.3 Fracture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3 Physical Meaning of Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.1 Vector Potential and Scalar Potential from Gauge . . . . . . . . . . . . 114
5.3.2 Scalar and Vector Potential from Viewpoint
of Wave Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3.3 Field Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 Physical Meaning of Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.5 Thermodynamic Consideration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Optical Interferometry and Application to Material
Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.1 Basics of Light and Optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1.1 Light as an Electromagnetic Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1.2 Interaction with Media and Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.1.3 Laser and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.1.4 Imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2 Interference and Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2.1 Mathematics of Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2.2 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2.3 Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.2.4 Mach–Zehnder Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3 Electronic Speckle-Pattern Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.3.1 Speckles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.3.2 In-Plane Displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.3.3 Out-of-Plane Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.1 Plastic Deformation Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.1.1 Decay Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.2 Vortex-Like Displacement Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.3 Observation of Charge-Like Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.3.1 Charge-Like Pattern and PLC Band . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.3.2 Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.3.3 Experiment with Notched Specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.3.4 Acoustic Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.3.5 Temperature Rise Due to Plastic Deformation . . . . . . . . . . . . . . . . 201
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Contents xi
8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.1 Evaluation of Stress Concentration with Charge-Like Patterns . . . . . . . 210
8.1.1 Stabilized and Unstabilized Non-welded
Specimens, A5052-S and A5052-N. . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.1.2 AA6063 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.1.3 Welded Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.2 Plastic Deformation and Fracture Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.2.1 Plastic Deformation Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.2.2 Fracture Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.2.3 ESPI Experiment on Plastic Deformation
and Fracture Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.2.4 Interpretation of Fringe Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.3 Evaluation of Load Hysteresis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Chapter 1
Introduction
1.1 Background
Deformation and fracture of solid-state materials have been studied for centuries.
The subject is not only of scientific interest but also of extreme practical importance.
A great number of scholars in a wide range of disciplines developed various theories.
The history of continuum mechanics can be traced back to the Hellenic period [1].
In 1660, Robert Hooke discovered the law of elasticity, the linear relation between
tension and extension in an elastic spring. The scientists in the seventeenth and
eighteenth centuries introduced the basic concepts of strain by extending Newton’s
law of motion for a point mass to a motion law for a deformable body with a finite
volume. By the middle of the nineteenth century, Augustin-Louis Cauchy, a French
mathematician, compiled these achievements into the basic framework of three-
dimensional continuum mechanics. Cauchy’s contribution is of especial significance
because of the mathematical rigorousness of his formulation, which is contrastive to
the heuristic approach exploited by earlier scientists. This is evidenced by the fact
that his formulations are still used by engineers of the twenty-first century, including
numerous software packages for numerical simulations of solid mechanics. In the
twentieth century, continuum mechanics developed to a unifying theory combined
with the advancement in thermodynamics and rheology. Clifford Truesdell was the
major force in this development. A number of textbooks such as the one written
by Landau and Lifshitz as a volume of their theoretical physics courses [2] are
available.
Elasticity as a nonlinear problems is also studied. It was initiated by Poincare and
Lyapunov in their study of ordinary differential equations of discrete mechanics at
the end of the nineteenth century. Methods for handling nonlinear boundary-value
problems were slowly developed by a handful of mathematicians on the first half of
the twentiethcentury. Further information about nonlinear elasticity can be found
in [3].
© Springer Science+Business Media New York 2015
S. Yoshida, Deformation and Fracture of Solid-State Materials,
DOI 10.1007/978-1-4939-2098-3__1
1
2 1 Introduction
In the area of plastic deformation, there are several mathematical descriptions
[4]. One is nonlinear deformation theory. In this theory, the stress is expressed as
a nonlinear function of the strain, as opposed to a linear function of the strain as
in the case of the linear elastic theory. This nonlinearity can also be viewed as
that the stress–strain relation is locally linear (infinitesimally linear at each value of
strain) on the stress–strain curve but the stiffness (the ratio of the stress to the strain)
reduces as the strain increases beyond the linear limit. Although this approach is
accurate as long as the stiffness is known as a function of strain, it cannot account
for irreversibility of the deformation. Flow plasticity theory [5] assumes that the
total strain can be decomposed into elastic and plastic parts. The plastic strain is
determined from the linear elastic relation to the stress of the material. The plastic
part is determined from a flow rule and a hardening model.
Another important theory of plasticity is dislocation theory. In 1934, Egon
Orowan [6], Michael Polanyi [7], and Geoffrey Ingram Taylor [8], approximately
at the same time, published papers to explain plastic deformation in terms of
dislocations. Experiments show that plastic deformation results from slip on specific
crystallographic planes in response to shear stress along the planes. However,
observed shear strength is orders of magnitude lower than theory. This discrepancy
indicates that slip is caused by a mechanism where lattice defects referred to as
dislocations move along the plane. When the dislocations complete the movement
at the end of the plane, slip occurs on the entire slip plane.
Fracture mechanics was developed during World War I by an aeronautical
engineer, Alan Arnold Griffith [9], to explain the fact that fracture of brittle materials
occur under two orders of magnitude lower stress than the theoretically predicted
value. He developed a thermodynamic theory that states that the growth of a crack
requires the creation of two new surfaces and hence an increase in the surface
energy. At the same time, it reduces the elastic energy stored in the material.
The difference of them (surface energy–elastic energy) can be interpreted as the
free energy. As the crack area increases, the surface energy increases linearly and
the elastic energy decreases quadratically, hence the free energy has a maximum.
Griffith postulated that the crack would increase spontaneously leading to fracture
beyond this maximum point because the free energy would decrease monotonically.
Accordingly, he defined the corresponding crack length as the critical crack length.
Griffith’s theory explains the fracture behavior of brittle materials well. However,
the actual energy necessary to fracture a ductile material is orders of magnitude
higher than the corresponding surface energy. During World War II, a group under
George Rankine Irwin [10] realized that in ductile materials the plastic zone
developing at the crack tip increases in size with the applied load, dissipating
the energy as heat. Hence, an energy dissipative term has to be added in the
energy balance. With this modification, the theory explains experiment for ductile
materials.
A number of researchers apply above-mentioned theories to engineering prob-
lems as well. Timoshenko [11], for example, describes the use of elasticity for a
number of practical applications.
1.1 Background 3
Sensor technology for defect detections has been advanced tremendously. A
number of techniques based on various principles such as ultrasonic/optical imag-
ing, eddy current, and other nondestructive technology have been developed and
used in the fields. Recent advancement allows us to detect micro-cracks in a very
early stage. On the material development side, a number of new materials with
additional designed properties such as reinforcement and anti-corrosion have been
developed.
In spite of all these developments, catastrophic accidents still occur. Aircraft parts
fail after passing the pre-flight inspection, structures such as bridges and stages
collapse totally unexpectedly. In many cases, the cause is unknown. Apparently,
the problems at least partly come from the limitation of the theory that the
inspection procedure is based on. The problem is not necessarily in incompleteness
of individual theories relevant to a given problem. Rather, it is the lack of connection
between the regimes of deformation. As discussed above, most of the currently
available theories are applicable selectively to a certain regime of deformation. They
are accurate in describing the dynamics in the corresponding regime. However,
in reality, the mechanical state cannot be characterized by a single regime. Even
in a specimen about to fracture, deformation in some parts are still in the elastic
regime. Flow plasticity theory may appear to be able to handle elastic and plastic
deformation simultaneously via the elastic and plastic parts of the strain. However,
this is a parameterized model, and the elasticity and plasticity are differentiated with
parameters, not physical laws. The use of these theories does not allow us to model
the transition from one regime to another. In the real world, on the other hand,
accurate description of the transition is extremely important.
The scale level is another issue. Micro and nano-technology is an emergent
field in various engineering disciplines. Most theories of solid mechanics were
developed for macroscopic objects. Mechanical properties of a material at the micro
and nanoscopic levels can be substantially different from those at the macroscopic
level. This imposes limitations in the applicability of existing theories to micro and
nanoscopic objects. In particular, those theories based on experimentally evaluated
parameters may need substantial modifications for applications to micro and nano-
scale systems. The issue is not necessarily limited to micro and nano-technologies.
Macroscopic fracture begins at the atomic level. The process starts with an atomistic
defect, grows to a macroscopic crack, and eventually the fracture of the entire object.
It is important that the theory can describe the transitions from one scale level to the
next on the same physical basis. If the crack generation can be predicted in an earlier
stage, the inspection technology will drastically advance.
The above-addressed issue clearly indicates the necessity of a theory capable of
describing all stages of deformation on the same physical basis, independent of the
scale level. It is obvious that such a theory must be based on a fundamental level of
physics. In this regard, the gauge theoretical approach that Panin et al. employed to
formulate dynamics of plastic deformation is promising. Details of their approach
can be found elsewhere [12–15]. In short, their approach is as follows: they
describe deformation with a transformation of GL.3;R/ group (three-dimensional
general linear group over real numbers [16]), and request local symmetry [17] in
4 1 Introduction
the transformation. In other words, they allow that the transformation matrix is
coordinate dependent and request that the dynamics be expressed in the same form
before and after the transformation. This requires replacement of usual derivatives
with covariant derivatives, or equivalently, introduction of a gauge. They find an
appropriate gauge and Lagrangian associated with it. Based on the principle of least
action, they derive field equations for each group element. After summation over
the group index, the field equations take a form analogous to Maxwell equationsof electrodynamics. The solution to the field equations represents transverse wave
characteristics of the displacement field in the plastic regime.
The present theory is based on the Panin’s formalism. When summed over the
group index (after contraction over the index representing the group elements) the
GL.3;R/ transformation matrix becomes the deformation tensor widely used in
linear elastic theory. Thus, the Panin’s approach can be interpreted as requesting
local symmetry in linear elasticity, which indicates that this formulation should
reduce to the conventional continuum mechanics. Subsequent analyses [18–20]
indicate that the transverse wave characteristics in the displacement field in the
plastic regime are driven by the shear restoring force of the material represented
by the shear modulus and that the longitudinal effect in the plastic regime is not
elastic force proportional to displacement but rather an energy dissipative effect.
Further, it has been found that fracture occurs when the material loses both its shear
resiting force mechanism and the longitudinal energy dissipating mechanism, hence
it loses all mechanisms to convert the mechanical work done by the external agent
to another form of energy. These altogether indicate the possibility of describing all
the stages, from the elastic through fracture, based on the same field equations.
One quite interesting feature of the field equations is that they are analogous to
Maxwell’s equations of electrodynamics. As will be explained in various sections
throughout this book, the similarity of the present theory with electrodynamics is
not limited to the mathematical resemblance. There are a number of similarities in
the physical behaviors between the displacement field and electromagnetic field. In
fact, the physical meaning of various behaviors of the displacement field can be
interpreted based on the analogy with electrodynamics, and these interpretations
have led to further understanding of the deformation dynamics. A number of
experimental observations have also been interpreted based on the analogy, and that
has facilitated the theoretical development.
It should be noted that the present theory does not refer to a cause of deformation,
in the same sense as an equation of motion does not refer to the specifics of the
external force. The present field equations describe the relation between transla-
tional and rotational modes of displacement. The cause of irreversible deformation
is incorporated into the field equations through the source terms. It is possible to
integrate the present theory with a microscopic theory, e.g., a dislocation theory, to
deal with the cause of deformation. It is an important future subject.
1.2 Aim, Scope, and Organization of This Book 5
1.2 Aim, Scope, and Organization of This Book
The primary aim of this book is to introduce the field theoretical approach to
deformation and fracture. The theoretical foundation is described and supporting
experiments are discussed. It should be emphasized that the present theory is still
developing. This book is not to present the completed form of the theory; rather it is
to invite researchers to consider the viewpoints of the present theory and hopefully
apply the approach to their own purpose.
The materials that this book tries to cover are quite interdisciplinary. It is likely
that most engineers are unfamiliar with the concept of local symmetry and gauge
transformation. Continuum mechanics is not a subject that scientists deal with on a
regular basis. It is my intention to describe the big picture of the approach, rather
than going into details of the content in each discipline. Those who are interested
in more detailed information are encouraged to read books or other resource of the
subject field.
It is also my intention to invite people of various disciplines, engineers, scientists,
and technicians. This is because exchanges of opinions among researchers in
different disciplines are most important for further development of this approach.
For this reason, much efforts have been made to explain the concepts of each
discipline in such a way that people in other disciplines can digest them as easily as
possible. Special attention is paid so that the reader does not need prior knowledge
except for basic physics and engineering. Basic knowledge on electrodynamics,
gaseous electronics, solid mechanics, and quantum mechanics will be helpful, but
not prerequisite. For this purpose, whenever seems necessary, extra explanations
are added. In some occasions, the mathematical derivation may appear lengthy and
perhaps redundant.
The objectives of each chapter are as follows. It is intended that each chapter is,
to some extent, self-contained so that the reader can use them as a reference. For
instance, those who are interested in applying an optical interferometric technique
to mechanical analysis, Chap. 6 will be useful. In this chapter, basic concepts of light
and optics which may not appear directly related to the interferometric techniques
are described. The purpose of the provision of these materials is to facilitate the
interferometric experiments. Conducting an optical interferometric experiment with
and without these knowledges make a huge difference in the experimental efficiency
and the quality of the results. The interferometric fringe-contrast will be much better,
for example, if we use an optical interferometer with the proper understanding of
the coherence in laser light, as opposed to simply follow the procedures to get
outputs from the interferometer. Other chapters are written with the same general
philosophy.
Chapter 2 reviews continuum mechanics. Basic physical concept of elasticity and
mathematical description such as the strain tensor, stress tensor, and constitutive
equations are discussed. Some of the continuum-mechanical concepts and mathe-
matical expressions are used in the present field approach. The goal of this chapter is
to facilitate the description of the mathematical procedure used to derive the present
6 1 Introduction
field equations in Chap. 5. Those who are familiar with continuum mechanics can
skip this chapter.
Chapter 3 discusses the gauge transformation and various underlying concepts
such as symmetry in physics, covariant derivatives, local symmetry, and gauge
potential. These concepts are not easy to digest for those who are not familiar
with the field. An effort has been made to discuss the complex concept as easily as
possible. The goal of this chapter is to describe the big picture of the concept without
going into mathematical details. Those who are familiar with gauge transformation
can skip this chapter.
Chapters 4 and 5 discuss the present field theory in detail. Chapter 4 focuses
on the formalism of the theory. The concept of gauge transformation discussed in
Chap. 3 is applied to the displacement field of a solid-state medium under plastic
deformation and the resultant field equations are discussed. Chapter 5 discusses
the physical meaning of the field equations and various concepts derived from
the field equations. One of the field equations is interpreted as the equation of
motion that governs the dynamics of a unit volume in the object under deformation.
Wave dynamics of the displacement field as solutions to the equation of motion
are discussed. The energy dissipative nature of plastic deformation is argued via
the concept of deformation charge, which is analogous to the electric charge. The
physical meanings of the charge and its interaction with the displacement field
are discussed. Through these discussions, the field equations are argued as the
governing equations of deformation for all stages; the elastic, plastic, and fracturing
stage. The form of the term representing the longitudinal force in the above-
mentioned equation of motion differentiates one stage from another.
Chapter6 explains optics and optical interferometry that are used in the support-
ing experiments discussed in Chaps. 7 and 8. Interestingly, some of the behaviors
of light as an electromagnetic wave are analogous to the deformation wave. This is
not surprising because the field equations of the deformation field are analogous to
Maxwell equations of electrodynamics. Various similarities between the light and
deformation fields are discussed. It will help us digest the field theoretical dynamics
of deformation discussed in Chaps. 4 and 5 from a different angle.
Chapters 7 and 8 present experimental results that support the present field
theory. Chapter 7 focuses on various field theoretical concepts such as decaying,
transverse displacement-waves in the plastic regime and the deformation charge
and its behavior that causes energy dissipation. Chapter 8 discusses engineering
applications of the present theory including diagnosis of the current deformation
regime (elastic, plastic, or fracturing regime) and evaluation of load hysteresis for a
given object.
References 7
References
1. Dugas, R.: A History of Mechanics. Editions du Griffon, Neuchatel (1955)
2. Landau, L.D., Lifshitz, E.M.: Theory of Elasticity. Course of Theoretical Physics, vol. 7, 3rd
edn. Butterworth-Heinemann, Oxford (1986)
3. Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, vol. 107.
Springer, New York/Budapest (1995)
4. Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press, Oxford (1998)
5. Lubliner, J.: Plasticity Theory. Courier Dover, New York (2008)
6. Orowan, E.: Z. Phys. 89, 605, 614, 634 (1934)
7. Polanyi, M.: Z. Phys. 89, 660 (1934)
8. Taylor, G.I.: Proc. R. Soc. A145, 362 (1934)
9. Griffith, A.A.: Philos. Trans. A 221, 163–198 (1920)
10. Irwin, G.R.: “Fracture Dynamics,” Fracturing of Metals. American Society for Metals,
Cleveland (1948)
11. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1951)
12. Panin, V.E., Grinaev, Yu.V., Egorushkin, V.E., Buchbinder, I.L., Kul’kov, S.N.: Sov. Phys. J.
30, 24–38 (1987)
13. Panin, V.E.: Wave nature of plastic deformation. Sov. Phys. J. 33(2), 99–110 (1990)
14. Danilov, V.I., Zuev, L.B., Panin, V.E.: Wave nature of plastic deformation of solids. In: Panin,
V.E. (ed.) Physical Mesomechanics and Computer-aided Design of Materials, vol.1, p. 241.
Nauka, Novosibirsk (1995) (Russian)
15. Panin, V.E.: Physical fundamentals of mesomechanics of plastic deformation and fracture of
solids. In: Panin, V.E. (ed.) Physical Mesomechanics of Heterogeneous Media and Computer-
Aided Design of Materials. Cambridge International Science Publishing, Cambridge (1998)
16. Ibragimov, N.H.: Transformation Groups and Lie Algebras. World Scientific, Singapore (2013)
17. Elliott, J.P., Dawber, P.G.: Symmetry in Physics, vol. 1. Macmillan, London (1984)
18. Yoshida, S., Siahaan, B., Pardede, M.H., Sijabat, N., Simangunsong, H., Simbolon, T.,
Kusnowo, A.: Phys. Lett. A 251, 54–60 (1999)
19. Yoshida, S.: Phys. Mesomech. 11, 140–146 (2008)
20. Yoshida, S.: Scale-independent approach to deformation and fracture of solid-state materials.
J. Strain Anal. 46, 380–388 (2011)
Chapter 2
Quick Review of Theories of Elastic Deformation
This chapter describes conventional approaches to elastic deformation known as
continuum mechanics or theory of elasticity. The goal of this chapter is to discuss
mathematical descriptions of kinematic and dynamics that these conventional
theories use. Some of them are basis of the present field theory. It is not my
intention to cover a wide area of the subject of elastic deformation. Rather, it
is to prepare for the mathematical procedures developed in later chapters where
we derive various equations of the present theory and interpret their physical
meanings via comparison with conventional approaches. For this purpose, some
of the concepts are viewed from different angles than conventional theories of
elasticity. For complete description of elastic deformation, the reader is encouraged
to refer to other books [1–6].
2.1 Displacement and Deformation
Consider point P1 located at a coordinate point .x1; y1; z1/ in an object. We express
this point with a position vector
!
OP1 as illustrated in Fig. 2.1. Similarly, another
point in the same object P2 at .x2; y2; z2/ can be expressed with position vector!
OP2. Apparently, the coordinates .x; y; z/ identify position of different points in the
same object, and the coordinate origin can be viewed as a reference point affixed to
the object. Such coordinates are referred to as Lagrangian coordinates.
Now consider in Fig. 2.2 that pointP1 is displaced to another pointP 01. To express
the displacement, we introduce a displacement vector �. As a three-dimensional
vector, � has three components that can be expressed with a coordinate system,
.�x; �y; �z/. While both .x; y; z/ and .�x; �y; �z/ have the dimension of length and
are three-dimensional vectors, the meaning of these two vectors are different from
each other. The set of components .x; y; z/ identify the position in a given object
© Springer Science+Business Media New York 2015
S. Yoshida, Deformation and Fracture of Solid-State Materials,
DOI 10.1007/978-1-4939-2098-3__2
9
10 2 Quick Review of Theories of Elastic Deformation
Fig. 2.1 Position vector
!
OP1
representing a point in an
object P1 (x,y,z)
OP1
y
x
z
Fig. 2.2 Point P1 and point
P2 displaced for different
amounts
dr
dr'
x(r)
x(r + dr)
dx
dr' = dr + dx
P1
P2
P1'
P2'
relative to the origin .x; y; z/ D .0; 0; 0/. On the other hand, .�x; �y; �z/ does not
identify a point in the object. Instead, it represents the change in the position of a
point in the object as a result of some physical event such as exertion of a force by an
external agent. Thus, their reference point is not affixed to the object but rather to the
frame of analysis. This type of coordinates is referred to as Eulerian coordinates.1
Deformation is defined as the situation where different points in the same
object are displaced differently. Note that as will be discussed in detail below, the
coordinate point after displacement is the addition of the initial position vector
and the displacement vector. As such, the position vector after the displacement
is expressed in the Eulerian coordinates. That is why some books [5] state that the
point before deformation is in Lagrangian coordinates and that after is in Eulerian
coordinates.
Figure 2.2 illustrates the situation where points P1 and P2 are displaced
differently as the object is deformed. Here the two points are considered to be
separated by an infinitesimal distance
!
dr .
1Consider you are still at .x; y; z/ on the earth (the object). Since you are not moving on the
earth, the value of .x; y; z/ does not change. However, since you are moving with the earth, the
displacement vector representing your motion with reference to the sun keeps changing.
2.1 Displacement and Deformation 11
!
drD !OP2 �
!
OP1 : (2.1)
By the deformation, the two points are displaced by �1 and �2.
!
OP 01 D
!
OP1 C�1
!
OP 02 D
!
OP2 C�2 (2.2)
Thus, the distance between the two points after the deformation is expressed as
!
dr 0D
!
OP 02 �
!
OP 01D
!
dr C.�2 � �1/: (2.3)
The two displacement vectors can be viewed as two values of a common displace-
ment function �.x; y; z/ at points .x1; y1; z1/ and .x2; y2; z2/. Further, as the distance
between points P1 and P2 is infinitesimally small, we can put
�2 � �1 D �.x2; y2; z2/ � �.x1; y1; z1/ D �.x1 C dx; y1 C dy; z1 C d z/ � �.x1; y1; z1/
D
!
d� .x; y; z/: (2.4)
Therefore, the change in the infinitesimal distance due to the deformation can be put
as follows:
!
dr 0D
!
dr Cd�.x; y; z/: (2.5)
Here, for the purpose of generalization, the suffixis dropped in the rightmost term.
Since each component of the displacement vector is a function of .x; y; z/.
d�i D @�i
@x
dx C @�i
@y
dy C @�i
@z
d z; i D x; y; z: (2.6)
In matrix notation, d� can be put
d�i D
0
@
d�x
d�y
d�z
1
A
D
0
BBBBB@
@�x
@x
@�x
@y
@�x
@z
@�y
@x
@�y
@y
@�y
@z
@�z
@x
@�z
@y
@�z
@z
1
CCCCCA
0
@
dx
dy
d z
1
A
(2.7)
12 2 Quick Review of Theories of Elastic Deformation
Hence, Eq. (2.5) becomes
0
@
dx0
dy0
d z0
1
A D
0
@
dx
dy
d z
1
AC
0
BBBBB@
@�x
@x
dx C @�x
@y
dy C @�x
@z
d z
@�y
@x
dx C @�y
@y
dy C @�y
@z
d z
@�z
@x
dx C @�z
@y
dy C @�z
@z
d z
1
CCCCCA
D
0
BBBBB@
0
@
1 0 0
0 1 0
0 0 1
1
AC
0
BBBBB@
@�x
@x
@�x
@y
@�x
@z
@�y
@x
@�y
@y
@�y
@z
@�z
@x
@�z
@y
@�z
@z
1
CCCCCA
1
CCCCCA
0
@
dx
dy
d z:
1
A
(2.8)
More concisely,
.dxi /0 D
�
ıij C @�i
@xj
�
dxj � Uij dxj : (2.9)
Here ıij is the Kronecker’s delta. The matrices
@�i
@xj
and Uij D ıij C @�i
@xj
are called
the displacement gradient tensor and the deformation gradient tensor, respectively.
It is convenient to divide the displacement gradient tensor into the symmetric and
asymmetric terms.
0
BBBBBB@
@�x
@x
@�x
@y
@�x
@z
@�y
@x
@�y
@y
@�y
@z
@�z
@x
@�z
@y
@�z
@z
1
CCCCCCA
D
0
BBBBBB@
@�x
@x
1
2
�
@�x
@y
C @�y
@x
�
1
2
�
@�z
@x
C @�x
@z
�
1
2
�
@�x
@y
C @�y
@x
�
@�y
@y
1
2
�
@�z
@y
C @�y
@z
�
1
2
�
@�x
@z
C @�z
@x
�
1
2
�
@�z
@y
C @�y
@z
�
@�z
@z
1
CCCCCCA
C
0
BBBBBB@
0 �1
2
�
@�y
@x
� @�x
@y
�
1
2
�
@�x
@z
� @�z
@x
�
1
2
�
@�y
@x
� @�x
@y
�
0 �1
2
�
@�z
@y
� @�y
@z
�
�1
2
�
@�x
@z
� @�z
@x
�
1
2
�
@�z
@y
� @�y
@z
�
0
1
CCCCCCA
(2.10)
As Fig. 2.3 illustrates, the symmetric part represents strain and the asymmetric part
rotation. The former is referred to as the strain tensor and the latter as the rotation
tensor. They can be concisely expressed as follows:
2.1 Displacement and Deformation 13
Fig. 2.3 Strain and rotation
part of deformation tensor
Shear strain
Normal strain +
Rotation
�ij D @�j
@xi
C @�i
@xj
; (2.11)
!ij D @�j
@xi
� @�i
@xj
; (2.12)
The strain tensor can further be divided into the normal strain and shear strain terms.
�ij D �n C �sh: (2.13)
Here,
�n D
0
BBBBB@
@�x
@x
0 0
0
@�y
@y
0
0 0
@�z
@z
;
1
CCCCCA
(2.14)
�sh D 1
2
0
BBBBBB@
0
�
@�x
@y
C @�y
@x
� �
@�z
@x
C @�x
@z
�
�
@�x
@y
C @�y
@x
�
0
�
@�z
@y
C @�y
@z
�
�
@�x
@z
C @�z
@x
� �
@�z
@y
C @�y
@z
�
0
1
CCCCCCA
(2.15)
For simplicity, let’s define the line element vector � for before and �0 for after the
deformation.
14 2 Quick Review of Theories of Elastic Deformation
� D
0
@
dx
dy
d z
1
A (2.16)
�0 D
0
@
dx0
dy0
d z0
1
A (2.17)
With these expressions, the deformation gradient tensor and the deformation as
a transformation become
Uij D ıij C �n C �sh C !ij (2.18)
�0 D U� (2.19)
It is worth exploring the physical meaning of each term of the deformation
gradient tensor. Consider two-dimensional deformation in the x-y plane in Fig. 2.4
where the x and y components of an infinitesimal line element vector dx and dy
are transformed to dx0 and dy0, respectively, by deformation. Since translational
displacement is not of our interest, the tail of the line element vector after the
deformation is shifted to that of the before deformation. For simplicity, we consider
a two-dimensional case here but the same argument holds for three dimensions.
From Fig. 2.4,
!
dx0 D
!
dx C
!
AA0
!
dy0 D
!
dy C
!
BB 0 (2.20)
Fig. 2.4 Physical meaning of
each term of deformation
gradient tensor
dy
dy
∂y
∂xy
dy
∂y
∂xx
dx
∂x
∂xx
dx
∂x
∂xy
dy'
dx
dx'
B
O
A
B'
A'
B"
A"
2.1 Displacement and Deformation 15
Considering the spatial dependence of the displacement vector � D �x Ox C �y Oy,
!
AA0 D
�
@�x
@x
dx
�
Ox C
�
@�y
@x
dx
�
Oy
!
BB 0 D
�
@�x
@y
dy
�
Ox C
�
@�y
@y
dy
�
Oy (2.21)
So,
!
dx0 D dx Ox C
�
@�x
@x
dx
�
Ox C
�
@�y
@x
dx
�
Oy (2.22)
!
dy0 D dy Oy C
�
@�x
@y
dy
�
Ox C
�
@�y
@y
dy
�
Oy (2.23)
With Eqs. (2.16) and (2.17) substituted into Eq. (2.19), and Fig. 2.4 along with
Eqs. (2.22) and (2.23), the meaning of each term of Uij in Eq. (2.18) can be
interpreted as follows.
The first term, the unit matrix ıij , represents the undeformed part of the line
element, i.e., the first terms of Eqs. (2.22) and (2.23). The second term �n represents
the normal strain expressed by the second term of Eq. (2.22) and the third term of
Eq. (2.23).
The geometric meaning of the third term �sh representing the shear strain
becomes clear from the following discussion. Consider the scalar product of
!
dx0
and
!
dy0.
!
dx0 �
!
dy0 D
��
dx C @�x
@x
dx
�
Ox C
�
@�y
@x
dx
�
Oy
�
�
��
@�x
@y
dy
�
Ox C
�
dy C @�y
@y
dy
�
Oy
�
�
�
@�y
@x
C @�x
@y
�
dxdy (2.24)
where the second-order terms of the derivatives of the displacement��
@�x
@x
��
@�x
@y
�
etc:
�
are neglected. By definition, this scalar product can be
written as follows:
!
dx0 �
!
dy0 D
ˇˇ
ˇˇ !dx0
ˇˇ
ˇˇ
ˇˇ
ˇˇ !dy0
ˇˇ
ˇˇ cos � 0
D
s�
1C @�x
@x
�2
C
�
@�y
@x
�2s�
@�x
@y
�2
C
�
1C @�y
@y
�2
cos � 0dxdy
� cos � 0dxdy; (2.25)
16 2 Quick Review of Theories of Elastic Deformation
where � 0 is the angle between
!
dx0 and
!
dy0, and @�x
@x
;
@�y
@y
<< 1 (small deformation
or the length change �x; �y is much smaller than dx; dy) is used. Equating the
right-hand side of Eqs. (2.24) and (2.25), we obtain
�
@�y
@x
C @�x
@y
�
D cos � 0 D sin
��
2
� � 0
�
� �
2
� � 0 (2.26)
Here the small angle approximation is used for �
2
�� 0. Equation (2.26) indicates that�
@�y
@x
C @�x
@y
�
is the change in the angle between
!
dx and
!
dy to
!
dx0 and
!
dy0 caused
by the deformation.
The geometrical meaning of the third term !ij in Eq. (2.18) can be understood in
a similar fashion. Consider the vector product of
!
dx and
!
dx0
!
dx �
!
dx0D dx Ox �
��
dx C @�x
@x
dx
�
Ox C
�
@�y
@x
dx
�
Oy
�
D @�y
@x
dx2Oz (2.27)
By definition
!
dx �
!
dx0D dx dx0 sin �x Oz � dx2 sin �x Oz (2.28)
where �x is the angle between
!
dx �
!
dx0. From Eqs. (2.27) and (2.28),
@�y
@x
D sin �x � �x: (2.29)
Similarly, we can easily find that
@�x
@y
D sin �y � �y: (2.30)
From Eqs. (2.29) and (2.30), we find that
@�y
@x
� @�x
@y
D �x � �y; (2.31)
i.e., the rotation represents the difference between the angle of rotation of
!
dx
and
!
dy.
2.2 Hooke’s Law and Poisson’s Ratio 17
2.2 Hooke’s Law and Poisson’s Ratio
The argument of the preceding section does not yet refer to dynamics because
the concept of force has not been introduced. The underlying force law in elastic
deformation is essentially Hooke’s law, which states that elastic media exert resistive
force proportional to the displacement from the equilibrium position in response to
an external force. Here the equilibrium position is the position that a part of the
medium takes when there is no externalforce acting on that part. Different parts of
a given elastic object have their own equilibrium positions that are different from
one another. The simplest way to visualize this type of force is a series of point
masses connected with springs, as illustrated in Fig. 2.5. In this context, the i th
mass’ equilibrium is established when neither of the springs directly connected to
this mass is stretched or compressed. Note that the force exerted by other stretched
springs are not external force on the i th mass.
The force proportional to the displacement from the equilibrium position can
be evaluated by differentiating the potential energy Up.x/ with respect to the
space coordinate. Here x is the distance from the equilibrium position. By Taylor-
expanding the potential energy with respect to x, we can express the situation as
follows:
U.x/ D U.0/C U 0.0/x C 1
2
U 00.0/x2 C � � � (2.32)
F.x/ D U 0.0/C U 00.0/x C � � � (2.33)
ith
ith
ith
unstretchedstretched stretched
Displacement of ith mass
Fig. 2.5 Spring mass system to represent elasticity. All springs are unstretched (top). All springs
are stretched (middle). Springs connected to the i th mass are unstretched and all other springs are
stretched (bottom)
18 2 Quick Review of Theories of Elastic Deformation
Fig. 2.6 Normal stress acting
on x surface
f
Dll
Ax
At the equilibrium x D 0, the force must be zero because the atom is at the bottom
of the potential well. Thus, U 0.0/ D 0. From this viewpoint, the spring force is
interpreted as the case when we take up to the second-order term of the potential
function. By putting U 00.0/ D �k2 where k is the stiffness or the spring constant of
the spring, we can rewrite Eq. (2.33) as follows:
F.x/ D �kx (2.34)
To discuss deformation, the total external force has much less meaning than force
per unit area. The force exerted on a unit area is called the stress. The concept of
stress is most conveniently explained through consideration of a cube in a medium
as shown in Fig. 2.6. In a one-dimensional case where the force is in the positive
x-direction, Eq. (2.34) leads to3
�xx D E�xx (2.35)
Here �xx is the normal stress, �xx is the normal strain, and E is the Young’s
modulus. Here Young’s modulus represents the stiffness, as the spring constant does
in Eq. (2.34). Since �xx is acting on a plane of unit area, its dimension is “force per
area,” or in the SI unit N=m2. Also, as clear from Fig. 2.6, the force associated with
�xx is differential force, or the difference in the normal force acting on the left
surface of the cube and the right surface. Each of these forces is the spring force at
the corresponding surface. If the normal force acting on the two surfaces is the same,
the cube would not be stretched or compressed; it would simply be accelerated as
a rigid body. Accordingly, the quantity multiplied to the stiffness on the right-hand
side must be “stretch per length,” or strain. The unit of the stiffness E is therefore
.N=m2/=.m=m/ D N=m2.
�xx D f
Ax
(2.36)
2A negative sign is used on the right-hand side to emphasize that the force is centripetal, or opposite
to the displacement from the equilibrium.
3See Sect. 2.4.1 for further descriptions about the Young’s modulus in the context of dynamics.
2.2 Hooke’s Law and Poisson’s Ratio 19
Note that the first subscript in �xx denotes the plane defined by the direction of
the outward unit vector normal to the plane and the second subscript denotes the
direction of the force acting on the plane.
It is useful to consider the relation between the spring constant and the Young’s
modulus. Consider an elastic material (spring) of length l and its cross-sectional
area Ax being stretched by force f . Denoting the stretch of this material with �l ,
we can relate f and �l with the spring constant k.4
f D k ��l (2.37)
Here the direction of the stretch is x. Then the normal strain �xx is
�xx D �l
l
(2.38)
Substituting Eqs. (2.36) and (2.38) into Eq. (2.37), we obtain
�xxA D .kl/�xx (2.39)
Comparing Eqs. (2.35) and (2.39), we find
k D EA
l
(2.40)
Notice that the spring constant k depends on the size of the object, whereas the
stiffness (Young’s modulus) E is a material constant. Below, we will find that
the phase velocity of one-dimensional longitudinal wave is
p
E=�, where � is the
density of the material. This indicates that the longitudinal wave’s velocity, e.g,
sound velocity, is uniquely determined by the material. This is contrastive to the
angular resonant frequency of an object of mass m and spring constant k is
p
k=m;
it depends on the object’s size (mass).
In the above argument, the differential force is normal to the surface. What if
the differential force is parallel to the surface; for example, the force acting on the
top surface of the cube in Fig. 2.6 parallel to the surface is different from the force
acting on the bottom surface parallel to the other force on the top surface? The cube
will experience shear strain as shown in Fig. 2.3. The stress corresponding to the
transversely differential force is referred to as the shear stress and the constant of
proportionality in its relation to the shear strain is referred to as the shear modulusG.
�yx D G�yx (2.41)
4To avoid complexity, only the absolute value of the force is considered here.
20 2 Quick Review of Theories of Elastic Deformation
Fig. 2.7 Stress vectors
x
y
z
o
C
B
A
sy
sx s
n
sz
Here, as is the case of the normal stress �xx , the first subscript y indicates the plane
that the force is acting on and the second subscript x indicates the direction of the
force.
So far, the direction of the differential force has been one-dimensional. To extend
this to three dimensions, we need to treat the stress as a three-dimensional vector.
Consider a stress on a given plane ABC in Fig. 2.7. From the force equilibrium on
the infinitesimally small tetrahedron OABC .5
� ndSn D � xdSx C � ydSy C � zdSz (2.42)
Here � k; k D n; x; y; z is the stress vector acting on plane k and dSk is the area
of plane k. Considering that the ratio dsi=dsn; i D x; y; z is the direction cosine
of a normal vector of triangular area ABC with axis i , Eq. (2.42) can be put in the
following form:
� n D 	nx� x C 	ny� y C 	nz� z (2.43)
where 	ni ; i D x; y; z is the direction cosine of the normal vector to the axis xi .
Now the stress vector acting on each of the x, y, z-plane can be expressed in terms
of the unit vector Oxi .
� x D �xx Ox C �xy Oy C �xzOz (2.44)
� y D �yx Ox C �yy Oy C �yzOz (2.45)
� z D �zx Ox C �zy Oy C �zzOz (2.46)
The coefficients of the unit vectors can be put in the form of a tensor referred to as
the stress tensor as follows:
5The effect of body force such as gravity is omitted as it does not have a substantial effect in the
argument in this chapter.
2.2 Hooke’s Law and Poisson’s Ratio 21
Œ�ij 
 D
0
@
�xx �xy �xz
�yx �yy �yz
�zx �zy �zz
1
A (2.47)
From Eq. (2.43) and Eqs. (2.44)–(2.46),
� n D �	nx�xx C 	ny�yx C 	nz�zx
	 Ox C �	nx�xy C 	ny�yy C 	nz�zy
	 Oy
C �	nx�xz C 	ny�yz C 	nz�zz
	 Oz
� .	ni�ix/ Ox C .	ni�iy/ Oy C .	ni�iz/Oz (2.48)
Here row i represents the stress vector � i introduced in Eq. (2.43), and column j
represents the j -component of each stress vector � i . In the last line of Eq. (2.48),
the summation over index i is omitted for simplicity. So, for example, �yz is the
z component of stress vector �y , which is acting on plane y. This convention is
consistent with that for Eq. (2.36).
Now consider relations among stress tensor components. Considering force
equilibrium for a given volume enclosed by surface S , we obtain the following
equation.
Z Z
S
� n dS D
Z Z
S
�
	nx�xi C 	ny�yi C 	nz�zi
	
dS D 0 (2.49)
where n is the unit vector normal to the surface S (� n is the componentof �
normal to the surface dS) and Eq. (2.48) is used in going through the second equal
sign. Using Green theorem, Eq. (2.49) can be converted to an integration over the
volume as
Z Z Z
V
�
@�xi
@x
C @�yi
@y
C @�zi
@z
�
dV D 0 (2.50)
In order for Eq. (2.50) to hold in any volume, it follows that
@�xi
@x
C @�yi
@y
C @�zi
@z
D 0 (2.51)
or for each component
@�xx
@x
C @�yx
@y
C @�zx
@z
D 0
@�xy
@x
C @�yy
@y
C @�zy
@z
D 0
@�xz
@x
C @�yz
@y
C @�zz
@z
D 0
22 2 Quick Review of Theories of Elastic Deformation
Next considering rotational equilibrium, we obtain
Z Z
S
r � � n dS D 0 (2.52)
Here r D x OxCy OyC zOz. With the use of Eq. (2.48) and Green theorem, this leads to
Ox
Z Z Z
V
�
y
�
@�xz
@x
C @�yz
@y
C @�zz
@z
�
� z
�
@�xy
@x
C @�yy
@y
C @�zy
@z
�
C ��yz � �zy
	�
C Oy Œ� � � 
C Oz Œ� � � 
 (2.53)
From Eq. (2.52), the first part of the angle bracket Π
 is zero. This leads to the
following relation between �ij and its diagonal counterpart �ji as
�xy D �yx; �yz D �zy; �zx D �xz (2.54)
This indicates that the stress tensor Eq. (2.47) is symmetric.
Œ�ij 
 D
0
@
�xx �xy �zx
�xy �yy �yz
�zx �yz �zz
1
A (2.55)
With the stress tensor (2.55), the one-dimensional constitutive equation (2.35) can
be extended to three-dimensions as follows:
�ij D Cklij �kl (2.56)
Here �ij is the .i; j / component of strain tensor defined by Eq. (2.11). In accordance
with the above convention, �ij denotes the j component of the stress vector acting
on plane i . Similarly, �kl denotes the .k; l/ component of the strain tensor. Each
combination of k and l represents the degree of freedom in deformation; e.g.,
.k; l/ D .x; y/ represents the shear deformation in a plane parallel to the x-y plane
as defined by Eq. (2.57). The coefficient Cklij represents the response of the material
to the external force represented by �ij for each degree of freedom.
�xy D 1
2
�
@�y
@x
C @�x
@y
�
(2.57)
With the use of symmetry of �ij and �ij , eq. (2.56) can be expressed in a matrix
form as follows:
2.2 Hooke’s Law and Poisson’s Ratio 23
Fig. 2.8 Meanings of C12
and C21
x
y
z
l
l
sxx
eyy Dl
exx
syy
Dl
0
BBBBBBB@
�xx
�yy
�zz
�xy
�yz
�zx
1
CCCCCCCA
D
0
BBBBBBB@
C11 C12 C13 C14 C15 C16
C21 C22 C23 � � � � � � C26
C31 � � � � � � � � � � � � � � �
C41 � � � � � � � � � � � � � � �
C51 � � � � � � � � � � � � � � �
C61 � � � � � � � � � � � � C66
1
CCCCCCCA
0
BBBBBBB@
�xx
�yy
�zz
�xy
�yz
�zx
1
CCCCCCCA
(2.58)
The matrix Ckl is called the stiffness tensor. Consider the meanings of C12 and
C21. If we write only the relevant term in �xx and �yy in Eq. (2.58), we obtain the
following expressions.
�xx D C12�yy (2.59)
�yy D C21�xx (2.60)
As Fig. 2.8 illustrates, Eq. (2.59) describes how much the volume expands normally
along y-axis when the volume is subject to the normal stress applied on the x-plane.
Similarly, Eq. (2.60) describes how much the volume expands normally along x-axis
when the volume is subject to the normal stress applied on the y-plane. Apparently,
the volume does not know which axis is x or y, meaning that if we switch the x-axis
with the y-axis, the physical situation is unchanged. It is obvious that Ckl D Clk ,
or the stiffness matrix is symmetric. We can repeat the same argument for C13 and
C31 � � � to find there are C.6; 2/ D 15 (the number of 2-combinations from a set
of 6-elements) redundancies. This reduces the number of independent Ckl from the
total number of 6 � 6 D 36 to 21.
In the case that the material’s response is symmetric, e.g., symmetric about the
x-y plane, for example, we can further reduce the number of the elements in the
stiffness matrix. Consider a new coordinate system x0y0z0 that is symmetric with
xyz system about the x-y plane; x D x0, y D y0 and z D �z0. With this coordinate
transformation, the sign of the z-component of a given vector is flipped, but the signs
of the x and y components are unchanged. Therefore, �xz D ��x0z0 , �yz D ��y0z0
and the sign of the other stress tensor components is unchanged. Similarly, �yz D
��y0z0 , �zx D ��z0x0 and the other strain tensor components are unchanged. Now
consider the normal stress on the x D x0 plane with the two coordinate systems.
24 2 Quick Review of Theories of Elastic Deformation
Table 2.1 Coordinate
transformation
x y z
. Ox/ . Oy/ .Oz/
x0 	x0x 	x0y 	x0z
. Ox0/ . Ox0 � Ox/ . Ox0 � Oy/ . Ox0 � Oz/
y0 	y0x 	y0y 	y0z
. Oy0/ . Oy0 � Ox/ . Oy0 � Oy/ . Oy0 � Oz/
z0 	z0x 	z0y 	z0z
. Oz0/ . Oz0 � Ox/ . Oz0 � Oy/ . Oz0 � Oz/
�x0x0 D C11�x0x0 C C12�y0y0 C C13�z0z0 C C14�x0y0 C C15�y0z0 C C16�z0x0
D C11�x0x0 C C12�y0y0 C C13�z0z0 C C14�x0y0 � C15�y0z0 � C16�z0x0 (2.61)
�xx D C11�xx C C12�yy C C13�zz C C14�xy C C15�yz C C16�zx (2.62)
As mentioned above, this coordinate transformation does not change the normal
stress on the x-plane; �x0x0 D �xx . It follows that C15 D C16 D 0. Repeating the
same argument for the symmetry about the yz and zx-plane, we can simplify the
stiffness matrix as follows:
ŒCkl 
 D
0
BBBBBBB@
C11 C12 C13 0 0 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 0 0 C66
1
CCCCCCCA
(2.63)
Next, consider that the material is isotropic. In this case, the constitutive relation
should remain the same if we rotate the material around an axis. First examine how
the stress vector is transformed by coordinate transformation from O � xyz to O �
x0y0z0. Referring to Eq. (2.43),
� x
0 D 	x0x� x C 	x0y� y C 	x0z� z (2.64)
where 	x0x etc are the direction cosine (see Table 2.1). Substituting Eqs. (2.44)–
(2.46) into the right-hand side of Eq. (2.64), we obtain
� x
0 D �	x0x�xx C 	x0y�yx C 	x0z�zx
	 Ox C �	x0x�xy C 	x0y�yy C 	x0z�zy
	 Oy
C �	x0x�xz C 	x0y�yz C 	x0z�zz
	 Oz
� 	x0i �ix Ox C 	x0i �iy Oy C 	x0i �izOz (2.65)
Now by definition, �x0x0 is the Ox0 component of � x0 .
�x0x0 D � x0 �
�
	x0x Ox C 	x0y Oy C 	x0zOz
	
2.2 Hooke’s Law and Poisson’s Ratio 25
Table 2.2 Coordinate
transformation
x y z
x0 cos � sin � 0
y0 � sin � cos � 0
z0 0 0 1
D 	x0x	x0i �ix C 	x0y	x0i �iy C 	x0z	x0i �iz
� 	x0j 	x0i �ij (2.66)
Applying the same procedure as Eq. (2.66) to the other components, we find the
general expression to relate the stress tensor components before and after the
coordinate transformation as follows.6
�i 0j 0 D 	i 0i 	j 0j �ij (2.67)
Now transform the coordinate system around the z-axis and make analysis on the
constitutive relation. Referring to Table 2.2,
�x0x0 D 	x0x	x0x�xx C 	x0y	x0x�yx C 	x0x	x0y�xy C 	x0y	x0y�yy
D �xx cos2 � C �yx sin � cos � C �xy cos � sin � C �yy sin2 �
D �xx cos2 � C �yy sin2 � C 2�xy cos � sin �
�y0y0 D �xx sin2 � C �yy cos2 � � 2�xy cos � sin �
�z0z0 D �zz
�x0y0 D
�
�yy � �xx
	
cos � sin � C �xy
�
cos2 � � sin2 �	
�y0z0 D �yz cos � � �zx sin �
�z0x0 D �yz sin � C �zx cos � (2.68)
Similarly, the strain tensor components are expressed with the new coordinate
system as follows:
�x0x0 D 	x0x	x0x�xx C 	x0y	x0y�yy C 	x0y	x0x�yx C 	x0x	x0y�xy
D �xx cos2 � C �yy sin2 � C 2�xy cos � sin �
�y0y0 D �xx sin2 � C �yy cos2 � � 2�xy cos � sin �
�z0z0 D �zz
�x0y0 D
�
�yy � �xx
	
cos � sin � C �xy
�
cos2 � � sin2 �	
6In Eq. (2.67) the order of the coefficients 	x0j 	x0i in Eq. (2.66) is switched for better visibility.
26 2 Quick Review of Theories of Elastic Deformation
�y0z0 D �yz cos � � �zx sin �
�z0x0 D �yz sin � C �zx cos � (2.69)
Consider expressing �x0x0 in two different ways. First, express the stress tensor
component on the right-hand side using the strain-tensor components �ij and the
stiffness tensor components in Eq. (2.63).
�x0x0 D �xx cos2 � C �yy sin2 � C2�xy cos � sin �
D �C11�xx C C12�yy C C13�zz
	
cos2 �
C �C12�xx C C22�yy C C23�zz
	
sin2 �
C2C44�xy cos � sin �
D �xx
�
C11 cos
2 � C C12 sin2 �
	
C�yy
�
C12 cos
2 � C C22 sin2 �
	
C�zz
�
C13 cos
2 � C C23 sin2 �
	
C�xy .2C44/ cos � sin � (2.70)
Next, express �x0x0 using �i 0j 0 and the stiffness tensor components in Eq. (2.63), and
convert the strain component expression after the coordinate transformation with
those of before the transformation, �ij .
�x0x0 D C11�x0x0 C C12�y0y0 C C13�z0z0
D C11
�
�xx cos
2 � C �yy sin2 � C 2�xy cos � sin �
	
CC12
�
�xx sin
2 � C �yy cos2 � � 2�xy cos � sin �
	
CC13�zz
D �xx
�
C11 cos
2 � C C12 sin2 �
	
C�yy
�
C11 sin
2 � C C12 cos2 �
	
C�zzC13
C�xy2 .C11 � C12/ cos � sin � (2.71)
Compare Eqs. (2.70) and (2.71) for the coefficients of the same strain tensor matrix,
�ij . In order for �x0x0 expressed in these two ways to be the same for any angle � ,
the following equalities are necessary for the coefficients of �yy , �zz, and �xy .
From �yy
C12 cos
2 � C C22 sin2 � D C12 cos2 � C C11 sin2 � (2.72)
2.2 Hooke’s Law and Poisson’s Ratio 27
From �zz
C13 cos
2 � C C23 sin2 � D C13 (2.73)
And from �xy
2C44 cos � sin � D 2 .C11 � C12/ cos � sin � (2.74)
From Eqs. (2.73)–(2.74), it follows that
C11 D C22 (2.75)
C13 D C23 (2.76)
C44 D C11 � C12 (2.77)
Repeating the same procedure for coordinate rotations about the x and y axes, we
find the following conditions for the case of isotropic materials.
C11 D C22 D C33; C12 D C13 D C23; C44 D C55 D C66 D C11 � C12 (2.78)
Since C11 D C22 D C33 are related to C44 D C55 D C66 in the last
expression, Eq. (2.78) indicates that the stiffness tensor of isotropic materials has
two independent components as follows:
C12 D C13 D C23 � � (2.79)
C44 D C55 D C66 � 2� (2.80)
� and � are called as Lamé’s first and second constants, respectively. All these
arguments simplify the stiffness tensor for isotropic materials into the form with the
use of the two Lamé’s constants, and allows us to express the constitutive relation
as follows:
0
BBBBBBB@
�xx
�yy
�zz
�xy
�yz
�zx
1
CCCCCCCA
D
0
BBBBBBB@
�C 2� � � 0 0 0
� �C 2� � 0 0 0
� � �C 2� 0 0 0
0 0 0 2� 0 0
0 0 0 0 2� 0
0 0 0 0 0 2�
1
CCCCCCCA
0
BBBBBBB@
�xx
�yy
�zz
�xy
�yz
�zx
1
CCCCCCCA
(2.81)
Here the physical meaning of C44 is the stiffness that connect the shear stress �xy
and shear strain �xy [Eq. (2.57)] as
�xy D C44�xy D C44
2
�
@�y
@x
C @�x
@y
�
(2.82)
28 2 Quick Review of Theories of Elastic Deformation
Fig. 2.9 Schematic
illustration of Poisson ratio
ezz
eyy
exx
From Eq. (2.41), the shear stress can be related to the shear strain with the shear
modulus (G).
�xy D G
�
@�y
@x
C @�x
@y
�
(2.83)
From Eqs. (2.82) and (2.83), we find C44 D 2G.
Next relate the Lamé’s constants to Young’s modulus and Poisson ratio. Consider
a volume is subject to one-dimensional force in Fig. 2.9. As the volume is stretched
along the x-axis, the material is usually compressed in the orthogonal direction. For
the isotropic case, the compressions in the y and z direction are the same. The ratio
is referred to as the Poisson’s ratio.
	 D
ˇˇ
ˇˇ �t
�n
ˇˇ
ˇˇ (2.84)
Here �n is the normal strain in the direction of the applied, one-dimensional stress,
and �t is the transverse strain. In the case of Fig. 2.9, �n D �xx and �t D �yy D �zz.
Putting �xx D �n, �yy D �zz D �xy D �yz D �zx D 0, �xx D �n, �yy D �zz D �t ,
�xy D �yz D �zx D 0 in Eq. (2.81), we obtain
�n D .�C 2�/�n C 2��t (2.85)
0 D ��n C 2.�C �/�t (2.86)
From Eq. (2.86),
	 D
ˇˇ
ˇˇ �t
�n
ˇˇ
ˇˇ D �
2.�C �/ (2.87)
Eliminating �t from Eqs. (2.85) and (2.86),
�n D �.3�C 2�/
.�C �/ �n (2.88)
Therefore,
2.3 Principal Axis 29
Table 2.3 Various moduli
.�; �/ .E;G/ .E; 	/ .G; 	/
E
�.3�C 2�/
.�C � E E 2.1C 	/G
G � G
E
2.1C 	/ G
	
�
2.�C �/
E � 2G
2G
	 	
� �
G.E � 2G/
3G �E
	E
.1C 	/.1� 2	/
2	G
.1� 2	/
E D �n
�n
D �.3�C 2�/
.�C �/ (2.89)
Table 2.3 shows relation among different moduli.
2.3 Principal Axis
Strain tensor Eq. (2.11) is symmetric and can be diagonalized by a rotational
transformation of the coordinate system [7]. Once diagonalized, all the shear
components of the resultant strain tensor become zero; �ij D 0; i ¤ j .
This means that we can always find a coordinate system with which a given
strain can be expressed in terms of normal strains only. This is understandable
because when an infinitesimally small volume at a point is deformed, the cor-
responding displacement vector has three-translational degrees of freedom, and
stretch/compression is the differential displacement along the axes of these degrees
of freedom. By aligning the coordinate axes to these three direction, we can always
express the total strain as a combination of the three normal strains.
Similarly, stress tensor Eq. (2.55) can be diagonalized, and this can be understood
as follows. Whatever coordinate system you may chose, the stress tensor vector
has three components. In other words, the stress tensor can be expressed as the
summation of the three components. The three axes of the coordinate system
corresponding to the diagonal tensor are called the principal axes.
If we apply a diagonalized strain tensor to the constitutive equation (2.81), it
is clear that the stress tensor corresponding to the left-hand side is also diagonal
because all shear stress terms �ij ; i ¤ j become zero. This is not surprising
because Hooke’s law states that stretch or compression is parallel to the external
force causing it. Non-diagonal terms in the stiffness tensor such as C12 that connects
a normal stress �xx and normal strain �yy in an orthogonal direction is not a direct
consequence of Hooke’s law. It is not associated with a elastic modulus but instead
due to the Poisson’s effect represented by the Poisson ratio 	. In other words, linear
elastic deformation is an orientation preserving mapping [2].
30 2 Quick Review of Theories of Elastic Deformation
The procedure to make a diagonal tensor from a square tensor is known as
diagonalization. Take a moment to review the procedure and discuss the physical
meaning. Consider diagonalization of a strain tensor expressed in the symmetric
form as follows:
0
@
�xx �xy �zx
�xy �yy �yz
�zx �yz �zz
1
A
0
@
dx1 dx2 dx3
dy1 dy2 dy3
d z1 d z2 d z3
1
A D
0
@
�1 0 0
0 �2 0
0 0 �3
1
A
0
@
dx1 dx2 dx3
dy1 dy2 dy3
d z1 d z2 d z3
1
A (2.90)
Rewrite Eq. (2.90) in the following form
2
4
0
@
�xx �xy �zx
�xy �yy �yz
�zx �yz �zz
1
A �
0
@
�1 0 0
0 �2 0
0 0 �3
1
A
3
5
0
@
dx1 dx2 dx3
dy1 dy2 dy3
d z1 d z2 d z3
1
A D 0 (2.91)
and consider the first column of Eq. (2.91).
0
@
�xx � �1 �xy �zx
�xy �yy � �2 �yz
�zx �yz �zz � �3
1
A
0
@
dx1
dy1
d z1
1
A D 0 (2.92)
The physical meaning of Eq. (2.92) is as follows. If we multiply a strain matrix
to the infinitesimal line element vector expressed in the x1y1z1 coordinate system,
the resultant differential displacement vector associated with the strain has only a
normal strain component; d�xx ¤ 0, d�xy D d�zx D 0. In order for this condition
to be true for a nonzero line element vector, it is necessary that Eq. (2.92) is not
solvable as a set of linear equations. This condition is expressed as the determinant
of the matrix is zero. That is,
det
0
@
�xx � �1 �xy �zx
�xy �yy � �2 �yz
�zx �yz �zz � �3
1
A D 0 (2.93)
This leads to the following characteristic equation, �i ; i D 1; 2; 3 is one of the three
roots.
�3 � I1�2 C I2� � I3 D 0 (2.94)
Here I1, I2, and I3 are known as the invariants defined as follows:
I1 D �xx C �yy C �zz (2.95)
I2 D �xx�yy C �yy�zz C �zz�xx � �2xy