As discussed in earlier chapters, the total life of a cyclically loaded component is composed of both the crack initiation and crack propagation stages. Modern defect-tolerant design approaches to fatigue are based on the premise that engineering structures are inherently flawed; the useful fatigue life then is the time or the number of cycles to propagate a dominant flaw of an assumed or measured initial size (or the largest undetected crack size estimated from the resolution of the nondestructive inspection method) to a critical dimension (which may be dictated by the fracture toughness, limit load, allowable strain or allowable compliance change). In most metallic materials, catastrophic failure is preceded by a substantial amount of stable crack propagation under cyclic loading conditions. The rates at which these cracks propagate for different combinations of applied stress, crack length and geometrical conditions of the cracked structure, and the mechanisms which influence the crack propagation rates under different combinations of mean stress, test frequency and environment, are topics of considerable scientific and practical interest.

In this chapter, we examine the mechanics and micromechanisms of stable crack propagation in ductile solids subjected to uniaxial and multiaxial cyclic loads of fixed amplitudes. Attention is focused on circumstances for which linear elastic fracture mechanics concepts are expected to be valid. Fatigue crack advance where considerable nonlinear deformation occurs ahead of a crack tip or notch tip is considered in Chapters 15 and 16. Situations involving crack growth under variable amplitude fatigue are examined in Chapter 14.

The effects of environments on the nucleation and growth of fatigue cracks were highlighted in previous chapters. These discussions touched upon the influence of oxygen-containing media on the kinematic irreversibility of cyclic slip (Chapter 4), crack initiation at corrosion pits (Chapter 4), the formation of brittle fatigue striations (Chapter 10), the role of fracture surface oxidation in promoting crack closure in constant amplitude and variable amplitude fatigue (Chapter 14), and the apparently anomalous growth of ‘chemically short’ fatigue cracks (Chapter 15). It is clear from these earlier descriptions that the usually deleterious (and occasionally beneficial) effects of environment must constitute an integral part of any complete mechanistic theory or design methodology for fatigue fracture.

This chapter deals specifically with the effects of environment on fatigue behavior. The discussions are presented in two parts: corrosion-fatigue effects and creep-fatigue effects. The first part begins with a survey of the micromechanisms of corrosion fatigue. This is followed by examples which illustrate the effects of gaseous and aqueous media on fatigue failure for different mechanical conditions of cyclic loading. This section is concluded with a brief examination of models of corrosion fatigue. In the second part, factors influencing the creep-fatigue behavior of engineering alloys are considered and many life prediction models are discussed. Particular attention is devoted to the mechanisms of high temperature fatigue deformation and to the issues pertaining to the characterization of creep crack growth using different fracture mechanics parameters.

This section presents a summary of stress intensity factors for some commonly used fatigue test specimens and crack configurations. These results are compiled from stress intensity factor handbooks (e.g., Rooke & Cartwright, 1976; Sih, 1973; Tada, Paris & Irwin, 1973) and monographs on fracture mechanics (e.g., Broek, 1986; Hellan, 1984; Kanninen & Popelar, 1985), where further details and derivations can be found.

The term contact fatigue broadly refers to the surface damage process that leads to pitting, wear debris formation and fatigue cracking when the surfaces of two bodies repeatedly touch each other. The relative motion between the bodies may involve global/partial slip or rolling, or a combination of these modes of contact. Usually either or both surfaces in contact may also be subjected to fluctuating stresses from vibration or other mechanical loads.

In this chapter, attention is directed at the mechanics and mechanisms of different contact fatigue phenomena. Table 13.1 provides the definitions of key terminology encountered in contact fatigue, along with examples of practical situations where such phenomena apply. These processes will be discussed in detail in various sections of this chapter.

Basic terminology and definitions

Figure 13.1 provides a general frame of reference with which the various parameters of interest in contact mechanics are defined. In this figure, adapted from Johnson (1985), two surfaces in nonconforming contact are shown whose shapes prior to contact deformation are characterized by the functions, z1(x, y) and z2(x, y). The separation between the surfaces is h(x, y) = z1 + z2. Let V1 and V2 denote the linear velocity of surfaces 1 and 2, respectively, and let Ω1 and Ω2 be their respective angular velocities. The frame of reference, centered at the instantaneous contact point O, moves with linear and angular velocities Vo and Ωo, respectively, so as to preserve its orientation to the indicated common tangent plane and the common normal at O.

One of the most successful applications of the theory of fracture mechanics is in the characterization of fatigue crack propagation. An analysis of fatigue flaw growth based on fracture mechanics inevitably requires a thorough understanding of the assumptions, significance and limitations underlying the development of various crack tip parameters. An important part of such a study of fracture mechanics is the identification of the regions of dominance of the leading terms of asymptotic crack tip singular fields. The appropriate conditions for the dominance of critical fracture parameters are obtained from a knowledge of the accuracy of asymptotic continuum solutions and from the mechanistic understanding of microscopic deformation at the fatigue crack tip.

In this chapter, we present a focused discussion of the theories of linear elastic and nonlinear fracture mechanics that are relevant to applications in fatigue. Details of the mechanisms of fatigue crack propagation are examined in the following chapters.

Griffith fracture theory

Modern theories of fracture find their origin in the pioneering work of Griffith (1921) who formulated criteria for the unstable extension of a crack in a brittle solid in terms of a balance between changes in mechanical and surface energies. Consider a through-thickness crack of length 2a located at the center of a large brittle plate of uniform thickness B, which is subjected to a constant far-field tensile stress σ (Fig. 9.1).

Fatigue of materials refers to the changes in properties resulting from the application of cyclic loads. Research into the deformation and fracture of materials by fatigue dates back to the nineteenth century. This branch of study has long been concerned with engineering approaches to design against fatigue cracks initiation and failure. However, along with the development of ‘science of materials’ and ‘fracture mechanics’ in recent decades, fatigue of materials has also emerged as a major area of scientific and applied research which encompasses such diverse disciplines as materials science (including the science of metals, ceramics, polymers, and composites), mechanical, civil and aerospace engineering, biomechanics, applied physics and applied mathematics. With the increasing emphasis on advanced materials, the scope of fatigue research continues to broaden at a rapid pace.

This book is written with the purpose of presenting the principles of cyclic deformation and fatigue fracture in materials. The main approach adopted here focuses attention on scientific concepts and mechanisms. Since fatigue of materials is a topic of utmost concern in many engineering applications, this book also includes discussions on the extension of basic concepts to practical situations, wherever appropriate. In writing this book, I have attempted to achieve the following objectives:

1. (i) To present an integrated treatment, in as quantitative terms as possible, of the mechanics, physics and micromechanisms of cyclic deformation, crack initiation and crack growth by fatigue. […]

The first edition of this book was written primarily as a research monograph for the Solid State Science Series of Cambridge University Press. Since its first publication, however, the book has found wide readership among students and practicing engineers as well as researchers. In view of this audience base which evolved to be much broader than what the book was originally intended for, it was felt that now would be an appropriate time for the preparation of an updated and revised second edition which includes newer material, example problems, case studies and exercises. In order to have the greatest flexibility in the incorporation of these new items in the book, it was also decided to publish the second edition as a ‘stand-alone’ book of Cambridge University Press, rather than as a research monograph of the Solid State Science Series.

In writing the second edition, I have adhered to the objectives which are stated in the preface to the first edition. In order to structure the expanded scope coherently, the book is organized in the following manner. The introduction to the subject of fatigue, the overall scope of the book and background information on some of the necessary fundamentals are provided in the first chapter. The book is then divided into four parts. Cyclic deformation and fatigue crack nucleation in ductile, brittle and semi-crystalline or noncrystalline solids are given extensive coverage in Part One.

The preceding chapters were concerned with the evolution of permanent damage under cyclic deformation and with the attendant nucleation of a fatigue crack. While these discussions pertain to micromechanical processes, phenomenological continuum approaches are widely used to characterize the total fatigue life as a function of such variables as the applied stress range, strain range, mean stress and environment. These stress- or strain-based methodologies, to be examined in Part Two, embody the damage evolution, crack nucleation and crack growth stages of fatigue into a single, experimentally characterizable continuum formulation. In these approaches, the fatigue life of a component is defined as the total number of cycles or time to induce fatigue damage and to initiate a dominant fatigue flaw which is propagated to final failure. The philosophy underlying the cyclic stress-based and strain-based approaches is distinctly different from that of defect-tolerant methods to be considered in Part Three, where the fatigue life is taken to be only that during which a pre-existing fatigue flaw of some initial size is propagated to a critical size.

The stress–life approach to fatigue was first introduced in the 1860s by Wöhler. Out of this work evolved the concept of an ‘endurance limit’, which characterizes the applied stress amplitude below which a (nominally defect-free) material is expected to have an infinite fatigue life.

The discussions presented in Chapters 9–12 focused on constant amplitude cyclic loading situations where the nominal stress intensity factor amplitude (for fixed load ratio and environmental conditions) and/or the maximum stress intensity factor uniquely govern the rates of crack advance in ductile and brittle solids. There are, however, a variety of situations where the local or effective stress intensity factor range or peak value at the crack tip, which is responsible for fatigue crack growth, can be markedly different from the nominal imposed value. These differences between the apparent and actual ‘driving force’ for fatigue fracture may stem from such effects as (i) premature closure of the crack faces even under fully tensile far-field cyclic loads, (ii) periodic deflections in the path of the crack due to mcirostructural impediments to fracture or changes in local stress state and mode mixity, (iii) shielding of the crack tip from the far-field, applied loads by the residual stress fields generated within the cyclic plastic zone or stress-induced phase transformations, and (iv) by the bridging of the faces of the crack by fibers, particles, intact grains or corrosion products. These processes, many of which are applicable to crystalline and noncrystalline as well as brittle and ductile solids, can lead to an apparent retardation of the fatigue crack growth and hence can possibly enhance the damage-tolerance characteristics of fatigue-prone materials and structures.

The mechanisms of fatigue crack growth in ductile metals and alloys, which were discussed in the preceding chapter, involve cyclic dislocation motion along one or two glide systems at the crack tip. The ensuing subcritical crack growth process in very ductile metals can typically cover a stress intensity factor range which spans a (threshold) value of 1–7 MPa√m to one that corresponds to final failure of well over 100 MPa√m. Another noteworthy feature of fatigue cracking in ductile alloys is that there exist some distinct microscopic markings, such as striations, on the fracture surfaces from which the occurrence of failure unique to cyclic loading can be clearly identified in many situations.

Subcritical crack growth in brittle solids under cyclic loads, however, involves much more complex phenomena. In many brittle solids, there are no known differences between the micromechanisms of static and cyclic crack growth at low temperatures. Despite this apparent similarity of deformation and failure mechanisms, the superimposition of a cyclic load on a static mean stress can lead to noticeable differences in the lifetime and, in some cases, the rate of crack growth may become a strong function of cyclic frequency. Furthermore, some macroscopic crack growth phenomena may occur solely as a consequence of the imposition of cyclic loads as, for example, in the case of crack initiation ahead of stress concentrations under fully compressive cyclic loads (see Chapter 5).

This chapter deals with the growth of cracks in brittle solids subjected to fluctuating loads.

In this chapter, we examine the mechanics and mechanisms of cyclic damage and crack nucleation in a wide range of brittle materials, including ceramics, glasses and ionic crystals. The fatigue behavior of brittle polymers is considered in the next chapter. Although the discussion of cyclic deformation and fatigue crack initiation for ductile materials was provided earlier in separate chapters, the corresponding descriptions for brittle materials warrant a single, unified presentation because of the nebulous demarkation between deformation mechanisms and flaw nucleation. For example, crack nucleation along grain boundaries can be regarded as the initial stage of the cyclic deformation process in some brittle materials. Fatigue crack growth in brittle ceramics and polymers are considered in Chapters 11 and 12, respectively.

It is also pertinent at this juncture to clarify the terminology used in the description of cyclic failure of brittle solids. In the metallurgy, polymer science and mechanical engineering communities, the word fatigue is a well accepted term for describing the deformation and failure of materials under cyclic loading conditions. However, in the ceramics literature, the expression static fatigue refers to stable cracking under sustained loads in the presence of an embrittling environment (which is commonly known as stress corrosion cracking in the metallurgy and engineering literature). The expression cyclic fatigue is used in the ceramics community to describe cyclic deformation and fracture.

Abstract

Distributed sensing and control of flexible shells and continua using distributed transducers has posted challenging issues for decades. This chapter focuses on distributed sensing and control of a generic double-curvature elastic shell and its derived geometries laminated with distributed piezoelectric transducers. Generic distributed orthogonal sensing and actuation of shells and continua are proposed. Spatially distributed orthogonal sensors/actuators and self-sensing actuators are presented. Collocated independent modal control with self-sensing orthogonal actuators is demonstrated and its control effectiveness evaluated. Spatially distributed orthogonal piezoelectric sensors/actuators for circular ring shells are designed and their modal sensing and control are investigated. Membrane and bending contributions in sensing and control responses are studied.

Introduction

Control of distributed parameter systems has posted many challenging problems and issues stimulating sophisticated research for decades (Balas, 1988; Brichkin et al., 1973; Butkovskii, 1962; Lions, 1968; Meirovitch, 1988; Oz and Meirovitch, 1983; Robinson, 1971; Sakawa, 1966; Tzafestas, 1970; Tzou, 1988, 1991, 1993; Vidyasagar, 1988; Wang, 1966; Zimmerman, Inman, and Juang, 1988). However, implementing distributed control of elastic continua, e.g., shells, plates, etc., using distributed devices has continuously been hampered by the practical availability of distributed sensing/actuation devices. Recent development of smart structures and intelligent structural systems (or structronic (structure-electronic) systems, in a new generic term) using active electromechanical materials has revealed the missing link of distributed transducers.

Distributed (parameter) systems (DPSs) are the most natural and generic systems existing today. Dynamic characteristics of most natural structures, manufacturing processes, fluids, heat transfer, control, etc., all fall into this DPS category. In general, their dynamics or responses are functions of spatial and time variables, and the systems are usually modeled by partial differential equations. Common practice often discretizes these distributed systems, and their lumped approximations (discrete systems modeled by ordinary differential equations) are then analyzed and evaluated. Original (distributed) behavior can only be observed at these discrete reference locations.

The dynamics and control of distributed systems have traditionally posed many challenging issues investigated by researchers and scientists for decades. However, new R&D activities and findings on DPSs have not been systematically reported for a long while. This book aims to document recent progress on the subject and to bring these technical advances to the engineering community.

Distributed structures are often coupled to external discrete components in engineering applications (e.g., disk heads and tape drives). A new transient analysis technique is developed to investigate the dynamics of coupled distributed-discrete systems in Chapter 1. Transient responses of time-varying systems and constrained translating strings are investigated. Transient behaviors of cables transporting dynamic payloads and translating (e.g., magnetic tape-head systems) are thoroughly studied.

Abstract

A model for a 3-D structural acoustic system, currently being used for parameter estimation and control experiments in the Acoustics Division, NASA Langley Research Center, is presented. This system consists of a hard-walled cylinder with a flexible circular plate at one end. An exterior noise source causes vibrations in the plate which in turn lead to unwanted noise inside the cylinder. Control is implemented through the excitation of piezoceramic patches bonded to the plate which generate in-plane forces and/or bending moments in response to an input voltage.

The plate and interior acoustic wave dynamics are approximated with expansions involving Fourier components in the circular direction and spline and spectral elements in the radial and axial directions. To guarantee uniqueness and differentiability at the coordinate singularity as well as to ensure stability and the expected convergence rate, the radial basis functions for both the wave and plate components are constructed in a manner that incorporates the Bessel or analytic behavior near the singularity while retaining sufficient generality so as to provide an approximation technique for discretizing complex coupled systems involving circular geometries.

Introduction

A growing area of research in the structural acoustics community concerns the problem of reducing structure-borne noise levels within an acoustic cavity.