A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

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Crack band theory for fracture of concrete

A balanced mechanics-materials approach and coverage of the latest developments in biomaterials and electronic materials, the new edition of this popular text is the most thorough and modern book available for upper-level undergraduate courses on the mechanical behavior of materials To ensure that the student gains a thorough understanding the authors present the fundamental mechanisms that operate at micro- and nano-meter level across a wide-range of materials, in a way that is mathematically simple and requires no extensive knowledge of materials This integrated approach provides a conceptual presentation that shows how the microstructure of a material controls its mechanical behavior, and this is reinforced through extensive use of micrographs and illustrations New worked examples and exercises help the student test their understanding Further resources for this title, including lecture slides of select illustrations and solutions for exercises, are available online at wwwcambridgeorg/97800521866758

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Mechanical Behavior of Materials

The recognition of the potential for enhanced fracture toughness that can be derived from controlled, stress-activated tetragonal (t) to monoclinic (m) transformation in ZrO2-based ceramics ushered in a new era in the development of the mechanical properties of engineering ceramics and provided a major impetus for broader-ranging research into the toughening mechanisms available to enhance the fracture properties of brittle-matrix materials. ZrO2-based systems have remained a major focal point for research as developments in understanding of the crystallography of the t→m transformation have led to more-complete descriptions of the origins of transformation toughening and definition of the features required of a transformation-toughening system. In parallel, there have been significant advances in the design and control of microstructure required to optimize mechanical properties in materials developed commercially. This review concentrates on the science of the t→m transformation in ZrO2 and its application in the modeling of transformation-toughening behavior, while also summarizing the microstructural control needed to use the benefits in ZrO2-toughened ceramics.

Transformation Toughening in Zirconia‐Containing Ceramics

The fracture front in concrete, as well as rock, is blunted by a zone of microcracking, and in ductile metals by a zone of yielding. This blunting causes deviations from the structural size effect known from linear elastic fracture mechanics (LEFM). The size effect is studied first for concrete or rock structures, using dimensional analysis and illustrative examples. Fracture is considered to be caused by propagation of a crack band that has a fixed width at its front relative to the aggregate size. The analysis rests on the hypothesis that the energy release caused by fracture depends on both the length and the area of the crack band. The size effect is shown to consist in a smooth transition from the strength criterion for small sizes to LEFM for large sizes, and the nominal stress σN at failure is found to decline as (1+λ/λ0)-1/2 in which λ0=constant and λ=relative structure size. This function is verified by Walsh's test data. If reinforcement is present at the fracture front and behaves elastically, ...

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Size Effect in Blunt Fracture: Concrete, Rock, Metal

Mother-of-pearl (nacre) is a platelet-reinforced composite, highly filled with calcium carbonate (aragonite). The Young modulus, determined from beams of a span-to-depth ratio of no less than 15 (a necessary precaution), is of the order of 70 GPa (dry) and 60 GPa (wet), much higher than previously recorded values. These values can be derived from ‘shear-lag’ models developed for platey composites, suggesting that nacre is a near-ideal material. The tensile strength of nacre is of the order of 170 MPa (dry) and 140 MPa (wet), values which are best modelled assuming that pull-out of the platelets is the main mode of failure. In three-point bending, depending on the span-to-depth ratio and degree of hydration, the work to fracture across the platelets varies from 350 to 1240 J m -2 . In general, the effect of water is to increase the ductility of nacre and increase the toughness almost tenfold by the associated introduction of plastic work. The pull-out model is sufficient to account for the toughness of dry nacre, but accounts for only a third of the toughness of wet nacre. The additional contribution probably comes from debonding within the thin layer of matrix material. Electron microscopy reveals that the ductility of wet nacre is caused by cohesive fracture along platelet lamellae at right angles to the main crack. The matrix appears to be well bonded to the lamellae, enabling the matrix to be stretched across the delamination cracks without breaking, thereby sustaining a force across a wider crack. Such a mechanism also explains why toughness is dependent on the span-to-depth ratio of the test piece. With this last observation as a possible exception, nacre does not employ any really novel mechanisms to achieve its mechanical properties. It is simply ‘well made’. The importance of nacre to the mollusc depends both on the material and the size of the shell. Catastrophic failure will be very likely in whole, undamaged shells which behave like unnotched beams at large span-to-depth ratios. This tendency is increased by the fact that predators act as ‘soft’ machines and store strain energy which can be fed into the material very quickly once the fracture stress has been reached. It may therefore be advantageous to have a shell made of an intrinsically less tough material which is better at stopping cracks (e. g. crossed lamellar). However, nacre may still be preferred for the short, thick shells of young molluscs, as these have a low span-to-depth ratio and can make better use of ductility mechanisms.

The Mechanical Design of Nacre

Fundamentals of fracture mechanics

Mechanisms of fatigue crack growth in low alloy steel

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–Fatigue cracks shorter than some critical length tend to propagate anomalously quickly. This paper examines the concept of a ‘critical length’, identifying three regimes of behaviour for different crack lengths. Some published work is examined, covering a wide range of different materials. It is concluded that there is an approximate correlation between the critical length for short crack behaviour and the scale of the microstructure. LEFM is difficult, if not impossible, to apply to cracks shorter than this critical length because the material surrounding a crack cannot be assumed to approximate to a homogeneous continuum. Suggestions are made for a fatigue design philosophy which incorporates short crack behaviour.

Fatigue crack propagation behaviour of short cracks; the effect of microstructure

Effects of microstructure on cleavage fracture in pressure vessel steel

The object of the paper is to examine the effects of alloy purity and state of aging on the fracture mechanism and resultant toughness of pure Al-Cu alloys, and commercial duralumin. In pure alloys, the transition from a shear to an intergranular mode of fracture with overaging is associated with changes in the nature and size of the matrix precipitate, which affect the slip character. In the corresponding commercial purity alloys, no such fracture mode transition occurs. The presence of second-phase dispersoids inhibits planar slip, and in the overaged state inclusion-matrix interfaces present a suitable alternative site to the grain boundaries for strain accumulation, resulting in debonding leading to the initiation of voids, which subsequently grow and coalesce. The fracture toughness, as conventionally measured, indicates the material’s resistance to crack initiation rather than propagation and is effectively independent of fracture mode. The work hardening capacity has a marked effect on void size, and is shown to be a sensitive indicator of fracture toughness in both pure and commercial alloys. Based on a simple model, good agreement is obtained between experimental results of toughness and those predicted from a knowledge of the tensile properties.

John F. Knott

Papers

Fundamentals of fracture mechanics

Mechanisms of fatigue crack growth in low alloy steel

Fatigue crack propagation behaviour of short cracks; the effect of microstructure

Effects of microstructure on cleavage fracture in pressure vessel steel

The influence of compositional and microstructural variations on the mechanism of static fracture in aluminum alloys