Experiments on fracture of materials are made for various purposes. Of primary importance are those through which criteria predicting material failure by deformation and/or fracture are investigated. Since the demands of engineering application always precede the development of theories, there is another kind of experiment where conditions under which a particular material can fail are simulated as closely as possible to the operational situation but in a simplified and standardized form. In this way, many of the parameters corresponding to fracture such as toughness, Charpy values, crack opening distance (COD), etc. are measured. Obviously, a sound knowledge of the physical theories governing material failure is necessary as the quantity of interest can seldom be evaluated in a direct manner. Critical stress intensity factors and critical energy release rates are examples. Standard test of materials should be distinguished from basic experi ments. They are performed to provide routine information on materials responding to certain conditions of loading or environment. The tension test with or without a crack is among one of the most widely used tests. Because they affect the results, with size and shape of the specimen, the rate of loading, temperature and crack configuration are standardized to enable comparison and reproducibility of results. The American Society for Testing Materials (ASTM) provides a great deal of information on recommended procedures and methods of testing. The objective is to standardize specifications for materials and definition of technical terms."

The planning meeting for a conference on Dynamic Crack Propagation was held at M.LT. in February 1971 and attended by research workers from several industrial, governmental and academic organizations. It was felt that a more specialized meeting would provide a better opportunity for both U.S. and foreign researchers to exchange their ideas and views on dynamic fracture, a subject which is seldom emphasized in national or international fracture conferences. Dynamic crack propagation has been a concern to specialists in many fields: continuum mechanics, metallurgy, geology, polymer chemistry, orthopedics, applied mathematics, as well as structural design and testing. It impinges on a wide variety of problems such as rock breaking and earthquakes, pressure vessels and line pipes, comminution and the per formance of armament and ordnance, etc. Advances have been numerous, covering theories and experiments from both the microscopic and macro scopic points of view. Hence, the need for comparing the theoretical and experimental results and bridging the gaps between the atomistic and continuum approaches must be constantly emphasized. It also appeared that the overall problem of dynamic fracture could benefit from a con solidation of crack models proposed for the various types of materials: metals, ceramics, composites, rocks, glasses, polymers and biomaterials.

The assessment of crack initiation and/or propagation has been the subject of many past discussions on fracture mechanics. Depending on how the chosen failure criterion is combined with the solution of a particular theory of continuum mechanics, the outcome could vary over a wide range. Mod elling of the material damage process could be elusive if the scale level of observation is left undefined. The specification of physical dimension alone is not sufficient because time and temperature also play an intimate role. It is only when the latter two variables are fixed that failure predictions can be simplified. The sudden fracture of material with a pre-existing crack is a case in point. Barring changes in the local temperature,* the energy released to create a unit surface area of an existing crack can be obtained by considering the change in elastic energy of the system before and after crack extension. Such a quantity has been referred to as the critical energy release rate, G e, or stress intensity factor, K Ie. Other parameters, such as the crack opening displacement (COD), path-independent J-integral, etc. , have been proposed; their relation to the fracture process is also based on the energy release concept. These one-parameter approaches, however, are unable simultaneously to account for the failure process of crack initiation, propagation and onset of rapid fracture. A review on the use of G, K I, COD, J, etc. , has been made by Sih [1,2].

The International Conference on Fracture Mechanics Technology Applied to Material Evaluation and Structure Design was held in Melbourne, Australia, from August 10 to 13, 1982. It was sponsored jointly by the Australian Fracture Group and Institute of Fracture and Solid Mechanics at Lehigh University. Pro fessor G. C. Sih of Lehigh University, Drs. N. E. Ryan and R. Jones of Aeronau tical Research Laboratories served as Co-Chairmen. They initiated the organiza tion of this international event to provide an opportunity for the practitioners, engineers and interested individuals to present and discuss recent advances in the evaluation of material and structure damage originating from defects or cracks. Particular emphases were placed on applying the fracture mechanics tech nology for assessing interactions between material properties, design and opera tional requirements. It is timely to hold such a Conference in Australia as she embarks on technology extensive industries where safeguarding structures from pre mature and unexpected failure is essential from both the technical and economical points. view The application of system-type approach to failure control owes much of its success to fracture mechanics. It is now generally accepted that the discipline, when properly implemented, provides a sound engineering basis for accounting in teractions between material properties, design, fabrication, inspection and op erational requirements. The approach offers effective solutions for design and maintenance of large-scale energy generation plants, mining machineries, oil ex ploration and retrieval equipments, land, sea and air transport vehicles."

An International Symposium on Defects and Fracture was held in Tuczno, Poland, 13-17 October 1980 under the sponsorship of the Polish Academy of Sci ences and Institute of Fracture and Solid Mechanics, Lehigh University. Profes sor H. Zorski of the Academy and Professor G. C. Sih of Lehigh University were Co-Chairmen. The idea for the Symposium was initiated by the members of the Organizing Committee: G. C. Sih (Co-Chairman), H. Zorski (Co-Chairman), M. Matczynski (Secretary), D. R. Axelrad, R. Bullough, J. W. Provan and R. deWit. With the rapid development of high performance materials and their increasing use in vital structural applications, there is a need for a more in-depth under standing of the fundamentals associated with material behavior and failure. Al though this topic encompasses a wide range of disciplines, it is appropriate to narrow the discussion to material failures initiating from inherent, unavoidable defects or cracks. A balanced coverage of analysis and physics was achieved by inviting speakers with backgrounds in applied mechanics and solid state physics. The input was limited to a small group of specialists who had the opportunity to discuss not only past progress but also ideas for future research."

This third volume of a series on Mechanies of Fraeture deals with eraeks in plates and shelIs. It was noted in Volume 2 on three-dimensional eraek problems that additional free surfaees can lead to substantial mathematical complexities, often making the analysis unmanageable. The theory of plates and shelIs forms a part of the theory of elasticity in which eertain physieal assumptions are made on the basis that the distanee between two bounded surfaees, either fiat or eurved, is small in eomparison with the overall dimen sions of the body. In modern times, the broad and frequent applieations of plate- and shell-like struetural members have aeted as a stimulus to whieh engineers and researchers in the field of fracture meehanies have responded with a wide variety of solutions of teehnieal importanee. These eontributions are covered in this book so that the reader may gain an understanding of how analytieal treat me nt s ofplates and shells containing initial imperfeetions in the form of eraeks are earried out. The development of plate and shell theories has involved long standing controversy on the eonsisteney of omitting eertain small terms and at the same time retaining others of the same order of magnitude. This defieieney depends on the ratio of the plate or shell thiekness, h, to other eharaeteristie dimensions and eannot be eompletely resolved in view of the approximations inherent in the transverse dependence of the extensional and bending stresses."

The International Conference on Fracture Mechanics in Engineering Applica tion convened at the r ational Aeronautical Laboratory (NAL) in Bangalore, India, March 26-30, 1979, with the presence of approximately 400 scientists and engi neers. The participants included individuals from all parts of India, United States of America, United Kingdom, Japan, Holland, France, Hong Kong, Korea, Sweden and Poland. The Conference was organized jointly by NAL, Bangalore;and Lehigh University, USA. Various organizations in India have also supported the Conference most generously. Professor S. Dhawan, Director of the Indian Institute of Science and Secre tary of Department of Space, delivered the inaugural speech. He said that the advance of science was the precondition of the development and survival of human society in the modern world. "It is true that in recent times, science and tech nology - and their practitioners - have been subjected to much public scrutiny, debate and severe criticism." On the other hand, the depletion of non-renewable resources, degradation of the natural environment and a host of other problems had been laid at the door of technology and science. One cannot deny that funda mental advances in the physics and chemistry of the structure of matter had led to spectacular engineering progress. Advanced technologies like nuclear energy and space exploration were but expression of the central role of computors, elec tronics, optics, polymers, etc., and all of these were heavily dependent on the successful application of material science and technology."

More than six years ago, several of Rabotnov's close friends and colleagues from the USSR and USA decided to contribute a volume on Plasticity and Failure of Solids in honor of his 70th birthday. The celebration was interrupted unexpectedly by his death on May 13, 1985 at which time another decision was made still to publish the work, but as a memorial volume. As in any field of scientific endeavor, research confronts the scientists with anomalies; our chosen area is no exception. The ways in which failure criteria and plasticity theory are combined can differ widely among the researchers; they will never yield quite the same results. Each of the invited contributors has, therefore, been encouraged to express his views and to expound on his personal opinion. The contributors are free of enumeration from the authority and/or consensus of any scientific society or community. What impedes scientific process is the esoteric tradition of accepting ideas and theories by consensus among members of societies and communities. The absence of such a trend is refreshing; the collaboration between the authors from the USSR and the USA had to be one of the contributing factors. Finally, the editors wish to acknowledge the authors who have made the publication of this volume possible. a. c. Sib S. T. Mileiko AJ. Ishlinsky xi The late Professor Yuriy Nickolaevich Rabotnov (February 24, 1914 - May 13, 1985) xii Scientific biography of the late academician Yu. N.

Concrete has traditionally been known as a material used widely in the construction of roads, bridges and buildings. Since cost effectiveness has always been one of the more important aspects of design, concrete, when reinforced and/or prestressed, is finding more use in other areas of application such as floating marine structures, storage tanks, nuclear vessel containments and a host of other structures. Because of the demand for concrete to operate under different loading and environmen tal conditions, increasing attention has been paid to study concrete specimens and structure behavior. A subject of major concern is how the localized segregation of the constituents in concrete would affect its global behavior. The degree of nonhomogeneity due to material property and damage. by yielding and/or cracking depends on the size scale and loading rate under consideration. Segregation or clustering of aggregates at the macroscopic level will affect specimen behavior to a larger degree than it would to a large structure such as a dam. Hence, a knowledge of concrete behavior over a wide range of scale is desired. The parameters governing micro-and macro-cracking and the techniques for evaluating and observing the damage in concrete need to be better understood. This volume is intended to be an attempt in this direction. The application of Linear Elastic Fracture Mechanics to concrete is discussed in several of the chapters.

The Second USA-USSR Symposium on Fna~e 06 Compo~~e Mat~aGBPh took place at Lehigh University, Bethlehem, Pennsylvania, during 9-12 March, 1981. This bilateral program between the U. S. and Soviet Union was organized by Professor George C. Sih of the Institute of Fracture and Solid Mechanics at Lehigh Uni versity and Dr. Vitauts P. Tamuzs of the Institute of Polymer Mechanics of the Academy of Sciences of the Latvian SSR in Riga. The First Symposium was held in 1978 at Jurmala near the coast of Riga Bay. The primary reasons for initiating this series of Symposia were to dissemi nate present knowledge, to promote interchange of ideas, and to stimulate addi tional studies on the development of composite materials between the U. S. and USSR. Both countries have a vested interest in developing the capability to assess and utilize the attractive mechanical properties of composites so that they can be tailor-made to meet specific design requirements. Despite the in creasing number of published papers and articles, there is no communication more effective than on a person-to-person basis. It is with this objective in mind that a small group of engineers and scientists from the U. S. and USSR have planned to meet every two years to report recent progress on composite material research. The size of this group is approximately sixty (60) participants. The presentation involves about forty (40) technical papers which are published in volume.

Material technology has become so diversified in theories and the construction of novel microstructures that the researchers and practitioners are drifting further apart. This book is based on material presented at an International Symposium in Xanthi, Greece in July 1989. The symposium attracted a group of individual engineers and scientists from the East and West who tackled the question of why particular manipulations of a given material have particular effects. Emphasis is laid on the strain energy function because of the versatile role it plays in mechanics and physics. It has been used successfully not only in predicting the failure of solids but also in formulating constitutive relations in continuum mechanics. The material presented falls within the areas of: Fundamentals of Strain Energy Density, Damage Analysis on Strain Energy Density, Strain Energy Density as Failure Criterion, Applications, and Composites.

This book contains results of more than a decade's effort on coupled deformation and diffusion obtained in research performed at the Institute of Fracture and Solid Mechanics, Lehigh University. Despite the overwhelming number of theories on this subject, little is known on the assessment of coupling effects because of the inherent difficulties associated with experimentation. A case in point is couple thermoelasticity, a theory that has remained virtually unused in practice. This is indicative of the inadequacy of conventional approaches. The interdependence of heat, moisture and deformation arises in many engineer ing problems of practical interest. Whether these effects are coupled or not depend on the transient character of the boundary conditions. Special attention is given to finding the coupling constants. Invoked is the assumption that the physical parameters should be independent of the specified boundary conditions. They can thus be extracted from known experimental data for situations where coupling effects are relatively weak and then applied to predict strong coupling effects as boundary conditions are altered. This is illustrated for the T300/5208 material commonly used in composites and permits a more reliable evaluation of material behaving under extreme environmental conditions. The lack of this knowledge can often be a major deterrent to the achievement of new technological advances. The reader will recognize that the material in this book does not follow the main stream of research on moisture-temperature diffusion and deformation."

Following Volumes III and IV that dealt with the fracture mechanics of concrete emphasizing both material testing and structural application in general, it was felt that specimen size and loading rate effects for concrete require further attention. The only criterion that has thus far successfully linearized the highly nonlinear crack growth data of concrete is the strain energy density theory. In particular, the crack growth resistance curves plotting the strain energy density factor versus crack growth known as the SR.curves are straight lines as specimen size and loading steps or rates are altered. This allows the extrapolation of data and provides a useful design methodology. This book is unique in that it is devoted specifically to the application of the strain energy density theory to civil engineering structural members made of concrete. Analyzed in detail is the strain softening behavior of concrete for a variety of different components including the influence of steel reinforcement. Permanent damage of the material is accounted for each increment of loading by invoking the mechanism of elastic unloading. This assumption is justified in concrete structures where the effective stiffness depends primarily on the crack growth rate and load history. Crack growth data are presented in terms of SR-curves with emphases placed on scaling specimen size which alone can change the mode of failure from plastic collapse to brittle fracture. Loading rate effects can also be scaled to control failure by yielding and fracture."

An International Conference on the Application of Fracture Mechanics to Ma terials and Structures was held at the Hotel Kolpinghaus in Freiburg, West Ger many, June 20-24, 1983. It was attended by more than 250 participants from different countries which include Austria, Canada, Czechoslovakia, Democratic Republic of Germany, Denmark, Federal Republic of Germany, Finland, France, Greece, Hungary, Israel, Italy, Japan, Netherlands, Norway, People's Republic of China, Portugal, Sweden, Switzerland, United Kingdom, United States of America, USSR and Yugoslavia. Conference Co-Chairmen were Professor G. C. Sih, Lehigh University, Bethle hem, Pennsylvania, U. S. A. , Dr. E. Sommer, Fraunhofer-Institut fur Werkstoff mechanik, Freiburg, FRG and Professor W. Dahl, Rheinisch-Westfalische Technische Hochschule, Aachen, FRG. Dr. Wenrich, as the representative of the Land Baden-WUrttemberg, delivered the opening address with the remarks that International Conferences can serve the means to further enhance the technology development of a country. He empha sized that the Federal Republic of Germany is presently in need of strengthening the engineering manpower in order to keep her in a competitive position. The Conference was officially cast off with the leading plenary lectures that under lined the theme of the technical lectures for the first day. This pattern was observed for the five-day meeting. The interplay between material and design re quirements was the theme and emphasized in many of the technical presentations that amounted to approximately ninety (90) papers.

This International Conference on Analytical and Experimental Fracture Me chanics was organized jointly by the Centro Sperimentale Metallurgico, S. p. A. , Lehigh University and Italsider S. p. A. It took place at the Hotel Midas Palace in Rome, Italy, during 23-27 June, 1980. There were more than 150 attendees from 19 different countries: Australia, Austria, Belgium, Canada, People's Re public of China, Czechoslovakia, Finland, Federal Republic of Germany, Hungary, Israel, Italy, Japan, The Netherlands, Poland, Switzerland, Turkey, UK, USA and USSR. Dr. G. M. Costa from Finsider officially opened the Conference and gave the Welcome Address. More than 70 technical papers were presented at three con current sessions. There were six plenary lectures that helped to integrate the diverse efforts, e. g. , analytical fracture mechanics, testing methods, metallur gical effects, corrosion fatigue, dynamic crack propagation and weldments of in dividual researchers. In addition to providing an overall view of the current status of fracture mechanics technology, particular emphasis was given to methods of controlling fracture in gas pipeline structures. The members of the Organizing Committee made a special effort to organize a panel discussion on the application of fracture mechanics technology to the safe design of large diameter pipelines. Dr. Michele Civallero from Italsider and Professor George C. Sih from Lehigh University served as panel Co-Chairmen and delivered survey lectures to stimulate questions from the audience.

This book consists of a collection of lectures prepared for a short course on "Fracture Mechanics Methodology" sponsored by the Advisory Group for Aerospace Research and Development (AGARD), part of the North Atlantic Treaty Organization (NATO). The course was organized jointly by Professor George C. Sih of the Institute of Fracture and Solid Mechanics at Lehigh University in the United States and Professor Luciano Faria from Centro de Mecanica e de Materiais das Universidade de Lisboa in Portugal. It was held in Lisbon from June 1 to 4, 1981. Dr. Robert Badaliance from the McDonnell Aircraft Company in St. Louis and Dr. Oscar Orringer from the Depart- ment of Transportation in Cambridge are the other US lecturers while Professor Carlos Moura Branco from Portugal also lectured. The audience consisted of engineers from the Portuguese industry with a large portion from the aeronautical sector and others who are particularly interested to apply the fracture mechanics discipline for analyzing the integrity of structural components and fracture control methods. Particular. emphases were given to the fundamentals of fracture mechanics as applied to aircraft structures.

With the advent of the 80's there has been an increasing need for analytic and numerical techniques, based on a thorough understanding of microstructural processes, that express in a manner suitable for practicing engineers the reliability of components and structures that are being subjected to degradation situations. Such situations fall within the framework offracture mechanics, fatigue, corrosion fatigue and pitting corrosion. Luckily, such techniques are now being developed and it was felt timely to combine in one volume reports by the leaders in this field who are currently making great strides towards solving these problems. Hence the idea of this monograph was born and I am pleased to be associated both with it and the contributors whose chapters are included in this volume. A very large part of the credit for this monograph must go to the authors who have taken time out from their busy schedules to prepare their submissions. They have all worked diligently over the last few months in order to get their manuscripts to me on time and I sincerely thank them for their help throughout the preparation of this volume.

It is weH known that the traditional failure criteria cannot adequately explain failures which occur at a nominal stress level considerably lower than the ultimate strength of the material. The current procedure for predicting the safe loads or safe useful life of a structural member has been evolved around the discipline oflinear fracture mechanics. This approach introduces the concept of a crack extension force which can be used to rank materials in some order of fracture resistance. The idea is to determine the largest crack that a material will tolerate without failure. Laboratory methods for characterizing the fracture toughness of many engineering materials are now available. While these test data are useful for providing some rough guidance in the choice of materials, it is not clear how they could be used in the design of a structure. The understanding of the relationship between laboratory tests and fracture design of structures is, to say the least, deficient. Fracture mechanics is presently at astandstill until the basic problems of scaling from laboratory models to fuH size structures and mixed mode crack propagation are resolved. The answers to these questions require some basic understanding ofthe theory and will not be found by testing more specimens. The current theory of fracture is inadequate for many reasons. First of aH it can only treat idealized problems where the applied load must be directed normal to the crack plane.

What can be added to the fracture mechanics of metal fatigue that has not already been said since the 1900s? From the view point of the material and structure engineer, there are many aspects of failure by fatigue that are in need of attention, particularly when the size and time of the working components are changed by orders of magnitude from those considered by st traditional means. The 21 century marks an era of technology transition where structures are made larger and devices are made smaller, rendering the method of destructive testing unpractical. While health monitoring entered the field of science and engineering, the practitioners are discovering that the correlation between the signal and the location of interest depends on a priori knowledge of where failure may initiate. This information is not easy to find because the integrity of the physical system will change with time. Required is software that can self-adjust in time according to the monitored data. In this connection, effective application of health monitoring can use a predictive model of fatigue crack growth. Earlier fatigue crack growth models assumed functional dependence on the maximum stress and the size of the pre-existing crack or defect. Various possibilities were examined in the hope that the data could be grouped such that linear interpolation would apply.

Contained in the volume are the papers presented at an International Symposium on Advanced Technology for Design and Fabrication of Composite Materials and Structures. The Symposium was organized by Consorzio per la Ricerca e l'Educazione Permanente; Institute of Fracture and Solid Mechanics, Lehigh University, Pennsylvania USA; Dipartimento di Ingegneria Strutturale del Politecnico di Torino; and Dipartimento di Ingegneria Aeronautica e Spaziale del Politecnico di Torino. It was held at the Politecnico di Torino in Italy, May 24-28, 1993. The support from the various organizations is acknowledged as follows: * Consiglio N azionale delle Ricerche * ALENIA SP AZIO * AGUST A * CIRA * AERMACCHI * Centro Ricerche FIAT * ALENIA (formerly AERITALIA) * Collegio Costruttori Edili della Provincia di Torino As new knowledge is being accumulated on the design and fabrication of advanced composite systems in different sectors of the world, there is the need not only to exchange new ideas but also to disseminate the information from the researchers to the users. The theme of this Symposium is particularly relevant to the automobile, marine, aerospace and construction industry where the competitive edge lies on improved processing and/or manufacturing of the products. Technological advances have been and will continue to depend strongly on the development of new materials and their effective use in design. Empirical trial-and- error methods could no longer be considered economically feasible when applied to usage-specific materials such as composites.

What can be added to the fracture mechanics of metal fatigue that has not already been said since the 1900s? From the view point of the material and structure engineer, there are many aspects of failure by fatigue that are in need of attention, particularly when the size and time of the working components are changed by orders of magnitude from those considered by st traditional means. The 21 century marks an era of technology transition where structures are made larger and devices are made smaller, rendering the method of destructive testing unpractical. While health monitoring entered the field of science and engineering, the practitioners are discovering that the correlation between the signal and the location of interest depends on a priori knowledge of where failure may initiate. This information is not easy to find because the integrity of the physical system will change with time. Required is software that can self-adjust in time according to the monitored data. In this connection, effective application of health monitoring can use a predictive model of fatigue crack growth. Earlier fatigue crack growth models assumed functional dependence on the maximum stress and the size of the pre-existing crack or defect. Various possibilities were examined in the hope that the data could be grouped such that linear interpolation would apply.

Contained in the volume are the papers presented at an International Symposium on Advanced Technology for Design and Fabrication of Composite Materials and Structures. The Symposium was organized by Consorzio per la Ricerca e l'Educazione Permanente; Institute of Fracture and Solid Mechanics, Lehigh University, Pennsylvania USA; Dipartimento di Ingegneria Strutturale del Politecnico di Torino; and Dipartimento di Ingegneria Aeronautica e Spaziale del Politecnico di Torino. It was held at the Politecnico di Torino in Italy, May 24-28, 1993. The support from the various organizations is acknowledged as follows: * Consiglio N azionale delle Ricerche * ALENIA SP AZIO * AGUST A * CIRA * AERMACCHI * Centro Ricerche FIAT * ALENIA (formerly AERITALIA) * Collegio Costruttori Edili della Provincia di Torino As new knowledge is being accumulated on the design and fabrication of advanced composite systems in different sectors of the world, there is the need not only to exchange new ideas but also to disseminate the information from the researchers to the users. The theme of this Symposium is particularly relevant to the automobile, marine, aerospace and construction industry where the competitive edge lies on improved processing and/or manufacturing of the products. Technological advances have been and will continue to depend strongly on the development of new materials and their effective use in design. Empirical trial-and- error methods could no longer be considered economically feasible when applied to usage-specific materials such as composites.

Material technology has become so diversified in theories and the construction of novel microstructures that the researchers and practitioners are drifting further apart. This book is based on material presented at an International Symposium in Xanthi, Greece in July 1989. The symposium attracted a group of individual engineers and scientists from the East and West who tackled the question of why particular manipulations of a given material have particular effects. Emphasis is laid on the strain energy function because of the versatile role it plays in mechanics and physics. It has been used successfully not only in predicting the failure of solids but also in formulating constitutive relations in continuum mechanics. The material presented falls within the areas of: Fundamentals of Strain Energy Density, Damage Analysis on Strain Energy Density, Strain Energy Density as Failure Criterion, Applications, and Composites.

The assessment of crack initiation and/or propagation has been the subject of many past discussions on fracture mechanics. Depending on how the chosen failure criterion is combined with the solution of a particular theory of continuum mechanics, the outcome could vary over a wide range. Mod elling of the material damage process could be elusive if the scale level of observation is left undefined. The specification of physical dimension alone is not sufficient because time and temperature also play an intimate role. It is only when the latter two variables are fixed that failure predictions can be simplified. The sudden fracture of material with a pre-existing crack is a case in point. Barring changes in the local temperature,* the energy released to create a unit surface area of an existing crack can be obtained by considering the change in elastic energy of the system before and after crack extension. Such a quantity has been referred to as the critical energy release rate, G e, or stress intensity factor, K Ie. Other parameters, such as the crack opening displacement (COD), path-independent J-integral, etc. , have been proposed; their relation to the fracture process is also based on the energy release concept. These one-parameter approaches, however, are unable simultaneously to account for the failure process of crack initiation, propagation and onset of rapid fracture. A review on the use of G, K I, COD, J, etc. , has been made by Sih [1,2].