The Theory of Critical Distances

A New Perspective in Fracture Mechanics

David Taylor

• Publisher: Elsevier Science
• Year: 2010
• ISBN:
  9780080554723
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

Critical distance methods are extremely useful for predicting fracture and fatigue in engineering components. They also represent an important development in the theory of fracture mechanics. Despite being in use for over fifty years in some fields, there has never been a book about these methods – until now.

So why now? Because the increasing use of computer-aided stress analysis (by FEA and other techniques) has made these methods extremely easy to use in practical situations. This is turn has prompted researchers to re-examine the underlying theory with renewed interest.

The Theory of Critical Distances begins with a general introduction to the phenomena of mechanical failure in materials: a basic understanding of solid mechanics and materials engineering is assumed, though appropriate introductory references are provided where necessary. After a simple explanation of how to use critical distance methods, and a more detailed exposition of the methods including their history and classification, the book continues by showing examples of how critical distance approaches can be applied to predict fracture and fatigue in different classes of materials. Subsequent chapters include some more complex theoretical areas, such as multiaxial loading and contact problems, and a range of practical examples using case studies of real engineering components taken from the author's own consultancy work.

The Theory of Critical Distances will be of interest to a range of readers, from academic researchers concerned with the theoretical basis of the subject, to industrial engineers who wish to incorporate the method into modern computer-aided design and analysis.

• Comprehensive collection of published data, plus new data from the author's own laboratories
• A simple 'how-to-do-it' exposition of the method, plus examples and case studies
• Detailed theoretical treatment
• Covers all classes of materials: metals, polymers, ceramics and composites
• Includes fracture, fatigue, fretting, size effects and multiaxial loading

CHAPTER 1

Introduction

Materials Under Stress

Publisher Summary

This chapter describes the state of the art in the prediction of material failure as articulated in national standards and specifications, and as used in practice in engineering companies. The chapter reviews the background material, symbols, and terminology related to materials under stress. A fundamental way to obtain information about the mechanical properties of a material is to record its stress–strain curve, usually by applying a gradually increasing tensile strain to a specimen of constant cross section. The chapter focuses on the deformation and failure of materials under stress, but emphasizes upon brittle fracture and fatigue including ductile fracture and certain tribological failure modes such as fretting fatigue. Material failure can be determined with precision only in two rather special cases. The first is simple tension, as described by the stress–strain curve: and the second is the propagation of pre-existing cracks, as described by linear elastic fracture mechanics (LEFM).. The use of computer-based methods such as finite element analysis (FEA) is also discussed. The chapter also highlights the use of traditional fracture mechanics and solid mechanics in failure prediction.

It is assumed that the reader is familiar with some basic theory regarding the mechanical properties of materials, as can be found in textbooks such as Ashby and Jones’ Engineering Materials (2005) or Hertzberg’s Deformation and Fracture Mechanics of Engineering Materials (1995), and also with the fundamentals of solid mechanics and fracture mechanics, for which many useful textbooks also exist (Broberg, 1999; Janssen et al., 2002; Knott, 1973). Nevertheless, in this chapter we will briefly review the background material and introduce symbols and terminology, which will be used in the rest of the book. We will be concerned, in general, with the deformation and failure of materials under stress, but emphasis will be placed on those types of failure which will be the main subjects of the book, especially brittle fracture and fatigue, but also including ductile fracture and certain tribological failure modes such as fretting fatigue. Of special interest from a mechanics point of view will be cracks, notches and other combinations of geometry and loading, which give rise to stress concentrations and stress gradients. In this respect, the use of computer-based methods such as finite element analysis (FEA) will also be discussed. We will finish with critical appraisal of the use of traditional fracture mechanics and solid mechanics in failure prediction, setting the scene for the developments to be described in the rest of this book.

1.1 Stress–Strain Curves

A fundamental way to obtain information about the mechanical properties of a material is to record its stress–strain curve, usually by applying a gradually increasing tensile strain to a specimen of constant cross section. Figure 1.2 shows, in schematic form, some typical results; note that here we are plotting the true stress (σ) and true strain (ε), thus taking account of changes in specimen cross section and length during the test. Most materials display a region of linear, elastic behaviour at low strains, and in some cases (line 1) this continues all the way to failure. This is the behaviour of classic brittle materials such as glass and certain engineering ceramics. More commonly, some deviation from linearity occurs before final failure (line 2). This non-linearity has three different sources: (i) non-linear elasticity, which is common in polymers; (ii) plasticity, that is the creation of permanent deformation, which occurs principally in metals and; (iii) damage, which is important in ceramics and composite materials. We will define the stress at failure in all cases as the maximum point in the curve, and refer to it as σ_u or the Ultimate Tensile Strength (UTS). In some cases (line 3) complete separation does not occur at σ_u, rather some reduced load-bearing capacity is maintained. This happens when damage such as splitting and cracking becomes widespread, for example in fibre composites. Finally, some stress–strain curves display other features (line 4) such as a drop in stress after yielding (in some metals and polymers) and a long post-yield plateau terminating in a rapid upturn in stress just before failure: this occurs in polymers which display plastic stability due to molecular rearrangements.

Fig. 1.2 Some typical stress–strain curves.

1.2 Failure Mechanisms

1.2.1 Failure at the atomic level

The study of failure mechanisms in materials has a tendency to get complicated, so it is worth remembering that, at the smallest scale, there are only two mechanisms by which materials can break, which I will call cleavage and tearing. Cleavage involves the fracture of atomic bonds; a crack can form by breaking the bonds linking two parallel planes of atoms, and this crack can then grow by the fracture of successive bonds near the crack tip, essentially unzipping the material in directions corresponding to atomic lattice planes. The fracture surface consists of a series of flat facets corresponding to the grains of the material. Tearing, on the other hand, occurs when material separates due to plastic deformation: atoms move around to create high levels of strain so that the material literally tears itself apart. This can manifest itself in various different ways, from macroscopic thinning (necking) or sliding (shearing) of material to microscopic void formation and growth. These two atomic failure mechanisms are often referred to as ‘brittle’ and ‘ductile’; however, I have avoided using these terms because they are also used with different meanings to describe failure modes at the macroscopic scale as discussed below.

1.2.2 Failure modes in engineering components

The failures of engineering components and structures occur by one of seven different modes: elastic, ductile, brittle, fatigue, stress-corrosion, creep, and wear.

Elastic failures are those failures which occur as a result of a low value of Young’s modulus, E. Two types of elastic failure can be mentioned. The first is excessive deflection, which may prevent the correct functioning of a structure – examples include bridges and vehicle suspensions. The second is buckling, by which, at a certain critical combination of load and elastic modulus, the deflections of a structure become unstable so that small deviations become magnified. A classic example is the collapse of a thin column loaded in compression.

Ductile fracture is the term used to describe failure occurring due to macroscopic plastic deformation; the material’s yield strength is exceed...

Table of contents

• Cover image
• Title page
• Table of Contents
• Preface
• Nomenclature
• Chapter 1: Introduction: Materials Under Stress
• Chapter 2: The Theory of Critical Distances: Basics: An Introduction to the Basic Methodology of the TCD
• Chapter 3: The Theory of Critical Distances in Detail: The History, Background and Precise Definition of the TCD
• Chapter 4: Other Theories of Fracture: A Review of Approaches to Fracture Prediction
• Chapter 5: Ceramics: Brittle Fracture in Engineering Ceramics, Building Materials, Geological Materials and Nanomaterials
• Chapter 6: Polymers: Brittle Fracture in Polymeric Materials
• Chapter 7: Metals: Brittle Fracture in Metallic Materials
• Chapter 8: Composites: Brittle Fracture in Fibre Composite Materials
• Chapter 9: Fatigue: Predicting Fatigue Limit and Fatigue Life
• Chapter 10: Contact Problems: Failure Processes at Points of Contact Between Bodies
• Chapter 11: Multiaxial Loading: Fracture and Fatigue Under Complex Stress States
• Chapter 12: Case Studies and Practical Aspects
• Chapter 13: Theoretical Aspects
• Author Index
• Subject Index