Nonequilibrium statistical theory of brittle fracture

A nonequilibrium statistical theory of the brittle fracture of metals connecting a microscopic mechanism with the macroscopic process has been constructed. The microcrack evolution equation is given. The growth speed of cracks, the distribution function and the fracture probability are obtained in connection with the dislocation mechanism, and mechanical properties such as elongation, fracture strength, plastic work, crack-extension force, fracture toughness, critical crack length and their statistical distribution functions and statistical deviations have been derived in a unified theory.     

A new approach to Weibull's statistical theory of brittle fracture

For brittle materials it is shown how the probability of failure of an inhomogeneously stressed structure can be estimated from a knowledge of the expected defect population and the local fracture criterion relating the local stress for failure to the defect size. Account is taken of the possibility that a sample of material contains no defects so that failure is then caused by another mechanism. It is shown that the theory presented is very closely related to the classical theory of Weibull. Furthermore it is demonstrated how account may be taken of loading uncertainties which are experienced in practice.As an example of an inhomogeneous stress state, the pure bending of a beam has been considered in detail. Such an analysis offers a convenient experimental method of determining the defect characteristics of a material. It is shown that a consistency check is possible by estimating experimentally the distribution of failure-initiating flaws and comparing it with a theoretical prediction.

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Résumé On montre pour des matériaux fragiles comment la probabilité de ruine dans une structure non homogène et soumise à contrainte peut être estimée à partir de la connaissance de la population prévue de défauts et d'un critère de rupture locale mettant en relation la contrainte locale à la ruine avec la dimension d'un défaut. On tient compte de la possibilité qu'un échantillon du matériau ne contienne pas de défaut de sorte que la ruine est causée dans ce cas par un autre mécanisme. On montre que la théorie présentée est en étroite relation avec la théorie classique de Weibull. De plus, on démontre qu'il est possible de tenir compte d'incertitudes quant à la mise en charge, telles qu'elles sont rencontrées en pratique. Comme exemple d'un état de contrainte inhomogène, on a considéré la flexion pure d'une poutre dans le détail. Une telle analyse offre une méthode expérimentale appropriée à la détermination des caractéristiques de défaut d'un matériau. On montre qu'un contrôle de l'exactitude est possible en faisant une estimation expérimentale de la distribution des défauts susceptibles d'entraîner la ruine, et en comparant cette distribution avec une prédiction théorique.

     

A statistical strength criterion for low temperature brittle fracture of metals

A statistical approach is presented to strength analysis of metals under conditions of low-temperature embrittlement. The approach involves two categories of defects leading to fracture: primary defects, present in the original structure in the form of microvoids, microcracks, inclusions etc., and secondary defects, formed under load through the action of the dislocational mechanisms involved in initiation of crack nuclei.An arbitrary pattern of primary and secondary defects is considered, on the assumption of approximate stochastic independence. This permits determination of the probability of brittle fracture under any mode of stressing. Development of a major crack is treated as a multistage process, whereby fracture may be considered either as a sequence of violated limit equilibria, or as a discontinuous Markov process. The role of thermal fluctuations in the latter case is demonstrated. As an example, limiting fracture-stress curves are determined for a plane model based on Stroh's mechanism. Results show that the temperature dependence of the fracture stress is governed by the mode of stressing and by the structure of the material.     

Critical assessment of the theory of brittle fracture

Cottrell's theory of stable growth of cleavage microcracks is critically assessed for mild steels below the embrittlement transition temperature. The theory is shown to correspond to experiments if one abandons the assumption that the number of crack dislocations is equal to the number of piled-up dislocations at the stress considered. Reasons are presented why the equation for dislocation pile-ups should not be applied to the growth of microcracks. We replace it by the experimental fact that the unstable crack length amounts to a few grain diameters. Then the idea of stable microcrack growth proves consistent with experimental results.     

Phonon theory of brittle fracture of solids

In this article we outline the basic principles of a new approach to the solution of the problem of brittle fracture of solids. The kinetic theory of long-time strength formulated by S. N. Zhurkov et al. is analyzed and generalized.     

Griffith's theory of brittle fracture in three dimensions

The Griffith theory of brittle fracture is extended to the three-dimensional problem of a flat elliptical crack in an otherwise uniform field of tensile and shear stresses. A method for finding the correct expressions of the change in strain energy due to the elliptical crack is developed. This is done by expressing the stresses and displacements in terms of the radius R₀ of a large sphere around the crack and by imposing the condition of equilibrium that the stresses or displacements across the spherical surface should agree with the prescribed boundary conditions as R₀ → ∞. The strain energy due to the presence of the elliptical crack is found to be independent of the tension applied parallel to the crack plane at infinity. On the basis of the thermodynamic argument of Griffith, it is also observed that the critical tensile and shear stresses increase rapidly as the ratio of major to minor semi-axes of the ellipse approaches unity.     

Transgranular fracture in low temperature brittle fracture of high nitrogen austenitic steel

Morphological features of transgranular fracture facets in low temperature brittle fracture of 18Cr–18Mn–0.7N high nitrogen austenitic steel have been studied by means of scanning electron microscopy and their formation mechanisms have been discussed. The transgranular fracture facets are fairly coarse compared with intergranular fracture facets and annealing twin boundary fracture facets. There are parallel steps and river patterns on the facets. Dual-surface observation indicated that these patterns are parallel to {111} planes and of strict crystallographic features. Microstructure observation shows that there are a lot of planar deformation structures formed prior to low temperature transgranular fracture. Transgranular fracture originates from microcracks formed at the intersections of the deformation structures on different {111} planes. These microcracks propagate toward adjacent microcracks on different {111} planes, forming transgranular fracture facets with steps and river patterns.     

Application of a statistical method to brittle fracture in biaxial loading systems

The failure probability of a brittle material stressed in a tensile biaxial loading system is investigated. A failure diagram relating the two mean biaxial stresses is mapped out for different values of the Weibull modulus as a function of volume of material. The significance of the biaxial results in relation to the uniaxial results is discussed by defining the term strength reduction factor (SRF). The applicability of Weibull analysis to biaxial loading systems is also examined.     

A statistical strength criterion for brittle materials

Based on a probabilistic model of brittle microfracture (microdamage) leading to macrofracture of materials, we propose a statistical interpretation of strength criteria, which relate start of macrofracture (manifested by macrocrack formation in tension or internal structure stability loss in compression) with attainment of a certain (critical) value of density of microdefects in the material under study. The criterion is reduced to comparative analysis of microdefect density induced by the particular loading type and its critical value, which is intricnsic to this material and is invariant to the stressed state type.