Tensile flaws

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

There is considerable literature on material imperfections and their relation to the failure process. Typically, these theories are material dependent flaws are idealized as penny-shaped cracks, spherical pores, or other regular geometries, and their distribution in size, orientation, and spatial extent is specified. The tensile stress at which fracture initiates at a flaw depends on material properties and geometry of the flaw, and scales with the size of the flaw (Carroll and Holt, 1972a, b Curran et al., 1977 Davison et al., 1977). In thermally activated fracture processes, one or more specific mechanisms are considered, and the fracture activation rate at a specified tensile-stress level follows from the stress dependence of the Boltzmann factor (Zlatin and Ioffe, 1973). 

An eminently practical, if less physical, approach to inherent flaw-dependent fracture was proposed by Weibull (1939) in which specific characteristics of the flaws were left unspecified. Fractures activate at flaws distributed randomly throughout the body according to a Poisson point process, and the statistical mean number of active flaws n in a unit volume was assumed to increase with tensile stress a through some empirical relations such as a two-parameter power law... 

In Fig. 8.18, we illustrate this just sufficient distribution in comparison to a hypothetical flaw distribution for an actual material. In this example, we envision a solid of finite extent, which will have a single critical flaw that activates at a minimum stress tensile stress, the population of flaws which activate should increase rapidly, perhaps as illustrated in Fig. 8.18. In contrast, a flaw distribution just sufficient to satisfy the energy balance criterion increases smoothly as JV [Pg.294]

AI2O3 has a fracture toughness Kic of about 3 MPa m. A batch of AI2O3 samples is found to contain surface flaws about 30 jUm deep. Estimate (a) the tensile strength and (b) the compressive strength of the samples. 

The low tensile strength of cement paste is, as we have seen, a result of low fracture toughness (0.3 MPa m ) and a distribution of large inherent flaws. The scale of the flaws can be greatly reduced by four steps ... 

It has been shown that the ultimate tensile strength, Su, for brittle materials depends upon the size of the speeimen and will deerease with inereasing dimensions, sinee the probability of having weak spots is inereased. This is termed the size effeet. This size effeet was investigated by Weibull (1951) who suggested a statistieal fune-tion, the Weibull distribution, deseribing the number and distribution of these flaws. The relationship below models the size effeet for deterministie values of Su (Timoshenko, 1966). 

Controlling the extraction rate is vital because the shape and texture of the resultant fiber is directly influenced by the solvent removal rate. As the solvent is extracted from the surface of the fiber, significant concentration gradients can form. These gradients may result in a warping of the desired eircular shape of the fiber. For example, if the solvent is removed too quickly, the fiber tends to collapse into a dog-bone shape. Additionally, the solvent extraction rate influences the development of internal voids or flaws in the fiber. These flaws limit the tensile strength of the fibers. 

Since PAN-based carbon fibers tend to be fibrillar in texture, they are unable to develop any extended graphitic structure. Hence, the modulus of a PAN-based fiber is considerably less than the theoretical value (a limit which is nearly achieved by mesophase fibers), as shown in Fig. 9. On the other hand, most commercial PAN-based fibers exhibit higher tensile strengths than mesophase-based fibers. This can be attributed to the fact that the tensile strength of a brittle material is eontrolled by struetural flaws [58]. Their extended graphitic structure makes mesophase fibers more prone to this type of flaw. The impure nature of the pitch preciusor also contributes to their lower strengths. 

Fig. 8. Schematic of circular section flaws introduced in the tensile surface of IG-l 1 and H-45I graphites.

Fig. 10. Photograph of fracture surface of H-451 graphite bend specimen illustrating failure originating at natural flaws at the tensile surface.

Even plastics with fairly linear stress-strain curves to failure, for example short-fiber reinforced TSs (RPs), usually display moduli of rupture values that are higher than the tensile strength obtained in uniaxial tests wood behaves much the same. Qualitatively, this can be explained from statistically considering flaws and fractures and the fracture energy available in flexural samples under a constant rate of deflection as compared to tensile samples under the same load conditions. These differences become less as the... 

This result indicates that the stress necessary to cause brittle fracture is lower, the longer the existing crack and the smaller the energy, P, expended in plastic deformation. The quantity Of represents the smallest tensile stress that would be able to propagate the crack of length 2 L. The term Of (tt L)°5 is generally denoted by the symbol K and is known as the stress-intensity factor (for a sharp elastic crack in an infinitely wide plate). Fracture occurs when the product of the nominal applied stress and the stress concentration factor of a flaw attains a value equal to that of the cohesive stress. 

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