: An overview of the virtual crack closure technique is presented. The approach used is discussed, the history summarized, and insight into its applications provided. Equations for two-dimensional quadrilateral elements with linear and quadratic shape functions are given. Formula for applying the technique in conjuction with three-dimensional solid elements as well as plate/shell elements are also provided. Necessary modifications for the use of the method with geometrically nonlinear finite element analysis and corrections required for elements at the crack tip with different lengths and widths are discussed. The problems associated with cracks or delaminations propagating between different materials are mentioned briefly, as well as a strategy to minimize these problems. Due to an increased interest in using a fracture mechanics based approach to assess the damage tolerance of composite structures in the design phase and during certification, the engineering problems selected as examples and given as references focus on the application of the technique to components made of composite materials.

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Virtual crack closure technique: History, approach, and applications

This paper analyzes the pseudostrain‐hardening phenomenon of brittle matrix composites reinforced with discontinuous flexible and randomly distributed fibers, based on a cohesive crack‐mechanics approach. The first crack strength and strain are derived in terms of fiber, matrix, and interface micromechanical properties. Conditions for steady‐state cracking and multiple cracking are found to depend on two nondimensionalized parameters that embody all relevant material micromechanical parameters. The results are therefore quite general and applicable to a variety of composite‐material systems. Phrased in terms of a failure‐mechanism map, various uniaxial load‐deformation behaviors for discontinuous fiber composites can be predicted. The influence of a snubbing effect due to local fiber/matrix interaction for randomly oriented crack‐bridging fibers on the composite properties is also studied.

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Steady-state and multiple cracking of short random fiber composites

Prediction of non propagating cracks

A nalyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J -integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J -integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

The previously proposed conditions for cavity formation from equiaxed inclusions in ductile fracture have been examined. Critical local elastic energy conditions are found to be necessary but not sufficient for cavity formation. The interfacial strength must also be reached on part of the boundary. For inclusions larger than about 100A the energy condition is always satisfied when the interfacial strength is reached and cavities form by a critical interfacial stress condition. For smaller cavities the stored elastic energy is insufficient to open up interfacial cavities spontaneously. Approximate continuum analyses for extreme idealizations of matrix behavior furnish relatively close limits for the interfacial stress concentration for strain hardening matrices flowing around rigid non-yielding equiaxed inclusions. Such analyses give that in pure shear loading the maximum interfacial stress is very nearly equal to the equivalent flow stress in tension for the given state of plastic strain. Previously proposed models based on a local dissipation of deformation incompatibilities by the punching of dislocation loops lead to rather similar results for interfacial stress concentration when local plastic relaxation is allowed inside the loops. At very small volume fractions of second phase the inclusions do not interact for very substantial amounts of plastic strain. In this regime the interfacial stress is independent of inclusion size. At larger volume fractions of second phase, inclusions begin to interact after moderate amounts of plastic strain, and the interfacial stress concentration becomes dependent on second phase volume fraction. Some of the many reported instances of inclusion size effect in cavity formation can thus be satisfactorily explained by variations of volume fraction of second phase from point to point.

Cavity formation from inclusions in ductile fracture

I Principles.- 1 Summary of basic problems and concepts.- 1.1 Introduction.- 1.2 A crack in a structure.- 1.3 The stress at a crack tip.- 1.4 The Griffith criterion.- 1.5 The crack opening displacement criterion.- 1.6 Crack propagation.- 1.7 Closure.- 2 Mechanisms of fracture and crack growth.- 2.1 Introduction.- 2.2 Cleavage fracture.- 2.3 Ductile fracture.- 2.4 Fatigue cracking.- 2.5 Environment assisted cracking.- 2.6 Service failure analysis.- 3 The elastic crack-tip stress field.- 3.1 The Airy stress function.- 3.2 Complex stress functions.- 3.3 Solution to crack problems.- 3.4 The effect of finite size.- 3.5 Special cases.- 3.6 Elliptical cracks.- 3.7 Some useful expressions.- 4 The crack tip plastic zone.- 4.1 The Irwin plastic zone correction.- 4.2 The Dugdale approach.- 4.3 The shape of the plastic zone.- 4.4 Plane stress versus plane strain.- 4.5 Plastic constraint factor.- 4.6 The thickness effect.- 5 The energy principle.- 5.1 The energy release rate.- 5.2 The criterion for crack growth.- 5.3 The crack resistance (R curve).- 5.4 Compliance.- 5.5 The J integral.- 5.6 Tearing modulus.- 5.7 Stability.- 6 Dynamics and crack arrest.- 6.1 Crack speed and kinetic energy.- 6.2 The dynamic stress intensity and elastic energy release rate.- 6.3 Crack branching.- 6.4 The principles of crack arrest.- 6.5 Crack arrest in practice.- 6.6 Dynamic fracture toughness.- 7 Plane strain fracture toughness.- 7.1 The standard test.- 7.2 Size requirements.- 7.3 Non-linearity.- 7.4 Applicability.- 8 Plane stress and transitional behaviour.- 8.1 Introduction.- 8.2 An engineering concept of plane stress.- 8.3 The R curve concept.- 8.4 The thickness effect.- 8.5 Plane stress testing.- 8.6 Closure.- 9 Elastic-plastic fracture.- 9.1 Fracture beyond general yield.- 9.2 The crack tip opening displacement.- 9.3 The possible use of the CTOD criterion.- 9.4 Experimental determination of CTOd.- 9.5 Parameters affecting the critical CTOD.- 9.6 Limitations, fracture at general yield.- 9.7 Use of the J integral.- 9.8 Limitations of the J integral.- 9.9 Measurement of JIc and JR.- 9.10 Closure.- 10 Fatigue crack propagation.- 10.1 Introduction.- 10.2 Crack growth and the stress intensity factor.- 10.3 Factors affecting crack propagation.- 10.4 Variable amplitude service loading.- 10.5 Retardation models.- 10.6 Similitude.- 10.7 Small cracks.- 10.8 Closure.- 11 Fracture resistance of materials.- 11.1 Fracture criteria.- 11.2 Fatigue cracking criteria.- 11.3 The effect of alloying and second phase particles.- 11.4 Effect of processing, anisotropy.- 11.5 Effect of temperature.- 11.6 Closure.- II Applications.- 12 Fail-safety and damage tolerance.- 12.1 Introduction.- 12.2 Means to provide fail-safety.- 12.3 Required information for fracture mechanics approach.- 12.4 Closure.- 13 Determination of stress intensity factors.- 13.1 Introduction.- 13.2 Analytical and numerical methods.- 13.3 Finite element methods.- 13.4 Experimental methods.- 14 Practical problems.- 14.1 Introduction.- 14.2 Through cracks emanating from holes.- 14.3 Corner cracks at holes.- 14.4 Cracks approaching holes.- 14.5 Combined loading.- 14.6 Fatigue crack growth under mixed mode loading.- 14.7 Biaxial loading.- 14.8 Fracture toughness of weldments.- 14.9 Service failure analysis.- 15 Fracture of structures.- 15.1 Introduction.- 15.2 Pressure vessels and pipelines.- 15.3 "Leak-bcfore-break" criterion.- 15.4 Material selection.- 15.5 The use of the J integral for structural analysis.- 15.6 Collapse analysis.- 15.7 Accuracy of fracture calculations.- 16 Stiffened sheet structures.- 16.1 Introduction.- 16.2 Analysis.- 16.3 Fatigue crack propagation.- 16.4 Residual strength.- 16.5 The R curve and the residual strength of stiffened panels.- 16.6 Other analysis methods.- 16.7 Crack arrest.- 16.8 Closure.- 17 Prediction of fatigue crack growth.- 17.1 Introduction.- 17.2 The load spectrum.- 17.3 Approximation of the stress spectrum.- 17.4 Generation of a stress history.- 17.5 Crack growth integration.- 17.6 Accuracy of predictions.- 17.7 Safety factors.- Author index.

Elementary engineering fracture mechanics

1. Introduction.- 1.1. Fracture control.- 1.2. The two objectives of damage tolerance analysis.- 1.3. Crack growth and fracture.- 1.4. Damage tolerance and fracture mechanics.- 1.5. The need for analysis: purpose of this book.- 1.6. Exercises.- 2. Effects of Cracks and Notches: Collapse.- 2.1. Scope.- 2.2. An interrupted load path.- 2.3. Stress concentration factor.- 2.4. State of stress at a stress concentration.- 2.5. Yielding at a notch.- 2.6. Plastic collapse at a notch.- 2.7. Fracture at notches: brittle behavior.- 2.8. Measurement of collapse strength.- 2.9. Exercises.- 3. Linear Elastic Fracture Mechanics.- 3.1. Scope.- 3.2. Stress at a crack tip.- 3.3. General form of the stress intensity factor.- 3.4. Toughness.- 3.5. Plastic zone and stresses in plane stress and plane strain.- 3.6. Thickness dependence of toughness.- 3.7. Measurement of toughness.- 3.8. Competition with plastic collapse.- 3.9. The energy criterion.- 3.10. The energy release rate.- 3.11. The meaning of the energy criterion.- 3.12. The rise in fracture resistance: redefinition of toughness.- 3.13. Exercises.- 4. Elastic-Plastic Fracture Mechanics.- 4.1. Scope.- 4.2. The energy criterion for plastic fracture.- 4.3. The fracture criterion.- 4.4. The rising fracture energy.- 4.5. The residual strength diagram in EPFM: collapse.- 4.6. The measurement of the toughness in EPFM.- 4.7. The parameters of the stress-strain curve.- 4.8. The h-functions.- 4.9. Accuracy.- 4.10. Historical development of J.- 4.11. Limitations of EPFM.- 4.12. CTOD measurements.- 4.13. Exercises.- 5. Crack Growth Analysis Concepts.- 5.1. Scope.- 5.2. The concept underlying fatigue crack growth.- 5.3. Measurement of the rate function.- 5.4. Rate equations.- 5.5. Constant amplitude crack growth in a structure.- 5.6. Load interaction: Retardation.- 5.7. Retardation models.- 5.8. Crack growth analysis for variable amplitude loading.- 5.9. Parameters affecting fatigue crack growth rates.- 5.10. Stress corrosion cracking.- 5.11. Exercises.- 6. Load Spectra and Stress Histories.- 6.1. Scope.- 6.2. Types of stress histories.- 6.3. Obtaining load spectra.- 6.4. Exceedance diagram.- 6.5. Stress history generation.- 6.6. Clipping.- 6.7. Truncation.- 6.8. Manipulation of stress history.- 6.9. Environmental effects.- 6.10. Standard spectra.- 6.11. Exercises.- 7. Data Interpretation and Use.- 7.1. Scope.- 7.2. Plane strain fracture toughness.- 7.3. Plane stress and transitional toughness, R-curve.- 7.4. Toughness in terms of J and JR.- 7.5. Estimates of toughness.- 7.6. General remarks on fatigue rate data.- 7.7. Fitting the da/dN data.- 7.8. Dealing with scatter in rate data.- 7.9. Accounting for the environmental effect.- 7.10. Obtaining retardation parameters.- 7.11. Exercises.- 8. Geometry Factors.- 8.1. Scope.- 8.2. The reference stress.- 8.3. Compounding.- 8.4. Superposition.- 8.5. A simple method for asymmetric loading cases.- 8.6. Some easy guesses.- 8.7. Simple solutions for holes and stress concentrations.- 8.8. Simple solutions for irregular stress distributions.- 8.9. Finite element analysis.- 8.10. Simple solutions for crack arresters and multiple elements.- 8.11. Geometry factors for elastic-plastic fracture mechanics.- 8.12. Exercises.- 9. Special Subjects.- 9.1. Scope.- 9.2. Behavior of surface flaws and corner cracks.- 9.3. Break through: leak-before-break.- 9.4. Fracture arrest.- 9.5. Multiple elements, multiple cracks, changing geometry.- 9.6. Stop holes, cold worked holes and interference fasteners.- 9.7. Residual stresses in general.- 9.8. Other loading modes: mixed mode loading.- 9.9. Composites.- 9.10. Exercises.- 10. Analysis Procedures.- 10.1. Scope.- 10.2. Ingredients and critical locations.- 10.3. Critical locations and flaw assumptions.- 10.4. LEFM versus EPFM.- 10.5. Residual strength analysis.- 10.6. Use of R-curve and JR-curve.- 10.7. Crack growth analysis.- 10.8. Exercises.- 11. Fracture Control.- 11.1. Scope.- 11.2. Fracture control options.- 11.3. The probability of missing the crack.- 11.4. The physics and statistics of crack detection.- 11.5. Determining the inspection interval.- 11.6. Fracture control plans.- 11.7. Repairs.- 11.8. Statistical aspects.- 11.9. The cost of fracture and fracture control.- 11.10. Exercises.- 12. Damage Tolerance Substantiation.- 12.1. Scope.- 12.2. Objectives.- 12.3. Analysis and damage tolerance substantiation.- 12.4. Options to improve damage tolerance.- 12.5. Aircraft damage tolerance requirements.- 12.6. Other requirements.- 12.7. Flaw assumptions.- 12.8. Sources of error and safety factors.- 12.9. Misconceptions.- 12.10. Outlook.- 12.11. Exercises.- 13. After the Fact: Fracture Mechanics and Failure Analysis.- 13.1. Scope.- 13.2. The cause of service fractures.- 13.3. Fractography.- 13.4. Features of use in fracture mechanics analysis.- 13.5. Use of fracture mechanics.- 13.6. Possible actions based on failure analysis.- 13.7. Exercises.- 14. Applications.- 14.1. Scope.- 14.2. Storage tank (fictitious example).- 14.3. Fracture arrest in ships.- 14.4. Piping in chemical plant (fictitious example).- 14.5. Fatigue cracks in railroad rails.- 14.6. Underwater pipeline.- 14.7. Closure.- 15. Solutions To Exercises.

The practical use of fracture mechanics

Thin foils of an aluminium alloy containing the fracture surface were studied in the electron microscope The fatigue striations were visible due to thickness contrast and bands of dislocations Supplementary etch pit studies and stereographic measurements were carried out In many cases the crack plane was close to (100) or (110) The mechanism of fatigue crack propagation is discussed in the light of the new evidence

D. Broek

Papers

Elementary engineering fracture mechanics

The practical use of fracture mechanics

On the formation of fatigue striations

Correlation between stretched zone size and fracture toughness

Prospects of Fracture Mechanics