Strength of Brittle Materials

In most applications, ceramics and glasses behave as brittle materials that is, they display a linear stress/strain curve up to the failure stress and then an abrupt failure 

Calculated values for the theoretical strength of a material are obtained by assuming that the applied stress has stretched every bond, in a plane perpendicular to the applied stress, to the point at which all the bonds break simultaneously. One of the common assumptions in such analyses is that the bonds must be stretched one-tenth of their original length to produce failure. 

The discrepancy between the theoretical strength of glasses and the values obtained in many practical applications exists because materials contain flaws that lower their strength. Operationally, a flaw is any structural feature in the part that raises the local stress at the feature above the applied stress in the part. The amphfication of stress, or the stress concentration factor, varies with the character of the flaw. For a cylindrical hole in a sheet of an isotropic material such as glass, the amplification is a factor of three at the edge of the hole. For cracks, which can be considered ellipsoids with a minimum radius of curvature of atomic dimensions, the stress concentration at the crack tip can be a factor of hundreds or thousands. Thus, with an applied stress of, say, 1000 psi, the stress at the crack tip can be raised to the theoretical value. 

The original equation derived by Griffith has been modified to include the contributions of Inglis, Orowan, and the Irwin formulation. The most commonly used form of the equation relating the failure stress, Oy, to the crack size, c, is 

More generally. Equation 9.1 can be written to describe the stress intensity for a crack of length c and an arbitrary applied stress O  

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The Charles-Hilling theory has been confirmed experimentally in soda-lime glass and alumina we can infer from known test results (Table 6.1.2) that the strength of brittle materials is limited by stress corrosion. 

R. Berenbaum and I. Brodie, Measurement of the tensile strength of brittle materials, Brit. J. Appl. Phys., 10 (1959) 281-287. 

Griffith [43,44], who was the first to relate the strength of brittle materials to material properties and crack length, and initiated the discipline of fracture... 

Since the strength of brittle materials is stoehastically distributed, a large number of tests is required to eharacterize the mechanical properties accurately. The two most important properties are the median strength and Weibull Modulus [34,37], and the minimum number of (valid) test results needed to meet most engineering requirements is 20. 

It is known that defects reduce the strength of brittle materials due to stress concentrations, which cannot relax by plastic deformation, as in metals. However, there is also evidence that the flaw population has a similarly strong impact on the hardness. The most promising way to develop new tool materials with improved hardness, wear resistance, and reliability is to reduce the grain size in the sintered microstructures an approach which requires the use of increasingly fine-grained raw materials, be it within the framework of advanced powder technologies or of sol/gel or precursor approaches. 

In a series of fracture experiments on ceramic specimens, two important observations can be made, namely that the probabflity of failure increases with the load amplitude, and also with the size of the specimens [2-4,14]. This strength-size effect is the most prominent and relevant consequence of the statistical behavior of the strength of brittle materials. However, these observations carmot be explained in a deterministic way by using a simple model of a single crack in an elastic body rather, their interpretation requires an understanding of the behavior of many cracks distributed throughout a material. 

A. J, Durelli, S. Morse, and V. Parks. The Theta Specimen for Determining Tensile Strength of Brittle Materials. Afor. Res. and.Stand., 2, 114-7 (1962). 

A. J. Durelli and V. J. Parks, Influence of Size and Sliape on the Tensile Strength of Brittle Materials. Brit. J. Appl. Phys. 18, 387-8 (1967). 

Evans, A. G. (1972). The strength of brittle materials containing secondary phase dispersions. Phil Mag. 26 1327-1344. 

The failure strength of brittle materials is given by the following general descriptions in a LEFM-regime [9 12] ... 

Here, G is the shear modulus, v is Poisson s ratio, b is the magnitude of the Burgers vector, and d is the distance between slip planes. The fracture strength of brittle materials is represented by Griffith s relation (Griffith, 1920) ... 

Weibull [2] developed a statistical theory for the strength of brittle materials. His assumption was that the risk of rupture is proportional to the stress and volume of the body. Expressing this in the form of an equation, we get ... 

Explain why the strengths of brittle materials are much lower than predicted by theoretical calculations. 

The tensile strength of brittle materials may be de-O termined using a variation of Equation 8.1. Compute... 

Evans, A.G., The Strength of Brittle Materials Containing Second Phase Dispersions, Phi 1. Mag.. 1972, 26 1327-1344... 

TENSILE STRENGTH OF BRITTLE MATERIALS IN THE DEFORMATION FIELD WITH A GRADIENT... 

The results presented above are in accordance with the observations made so far in the course of tests carried out on brittle materials. It can be therefore concluded that the suggested model of the brittle material and the hypothesis concerning the mechanism of destruction, put forward rightly explain the essence of the phenomenon of the growth in strength of brittle materials in the strain field with a gradient. 

Figure 3 Ultimate strength of brittle material under triaxial stress.

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