Theoretical Strength of a Solid

The next task is to estimate the theoretical strength of a solid or the stress that would be required to simultaneously break all the bonds across a fracture plane. It can be shown (see Prob. 4.2) that typically most bonds will fail when they are stretched by about 25%, i.e., when rfau 1.25/o. It follows from the geometric construction shown in Fig. 4.6 that... 

Based on these results, one may conclude that the theoretical strength of a solid should be roughly one-tenth of its Young s modulus. Experience has shown, however, that the actual strengths of ceramics are much lower and are closer to T/100 to 7/1000. The reason for this state of affairs is discussed in greater detail in Chap. 11, and reflects the fact that real solids are not perfect, as assumed here, but contain many flaws and defects that tend to locally concentrate the applied stress, which in turn significantly weaken the material. 

It is known that the theoretical strength of a solid does not correspond to the real strength. The first is determined by the molecular forces, whereas the second depends on the structure of the material. The deformation of a solid is a non-equilibrium process dependent on energy dissipation. The lack of correlation between thermodynamic work of adhesion and strength of adhesion joints is a direct consequence of the non-equilibrium failure. It may be predicted that the correspondence between these two values may only be reached if the strength was determined in the thermodynamically equilibrium conditions, i.e., at deformation with infinitely low rate. 

Now the theoretical strength of a crystalline solid, o- is expected to be about... 

T is associated with the creation of a new surface. Therefore the energy spent during the application of the stress must be equal to the energy of the surface created. Orowan (1934) showed that the theoretical strength of a perfectly brittle solid is given by... 

It has long been known that the theoretical strength of a metal crystal is far greater than the strength normally observed. Moreover, metals can be deformed easily and retain the new shape, a process called plastic deformation, whereas ceramic solids fracture under the same conditions. The typical mechanical properties of metals are due to the presence of linear defects called dislocations. 

For a circle a = b), the maximum stress amplification is 3 for a thin crack, for which a/b = 500, for example, the maximum amplification is 10. Real cracks, of course, do not conform to precise elliptic shape, but nevertheless this enormous level of stress amplification can occur at the tips of the sharp cracks which are found in all solids. It follows that even at low applied stress, the stress at the crack tip may approach the theoretical strength of the solid, where interatomic bonds are brought to their breaking point. However, these... 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

Just as metals can be ductile or brittle, so can organic materials. The Brittle Fracture Index is a measure of the brittleness of a material. It is a measure of the ability of a compact of material to relieve stress by plastic deformation. The Brittle Fracture Index (BFI) is determined [29,31] by comparing the tensile strength of a compact, stress concentrator) in it, o-T0, using the tensile test we have described. A hole in the center of the compact generally weakens a tablet. If a material is very brittle, theoretical considerations show that the tensile strength of a tablet with a hole in it will be about one-third that of a solid tablet. If, however, the material can relieve stress by plastic deformation, then the strength of the compact with a hole in it will approach that of a compact with no hole. The Brittle Fracture... 

PLASTIC DEFORMATION. When a metal or other solid is plastically deformed it suffers a permanent change of shape. The theory of plastic deformation in crystalline solids such as metals is complicated but well advanced. Metals are unique among solids in their ability to undergo severe plastic deformation. The observed yield stresses of single crystals are often 10 4 times smaller than the theoretical strengths of perfect crystals. The fact that actual metal crystals are so easily deformed has been attributed to the presence of lattice defects inside the crystals. The most important type of defect is the dislocation. See also Creep (Metals) Crystal and Hot Working. 

For the purpose of catalyst analysis, the weaknesses of ESCA turn into strengths as it is the chemical bonding of the outer surface of a solid that is of interest and only to a lesser extent its bulk chemical structure. Most of our theoretical understanding of chemical bonding refers, however, to the bulk state (crystal structures)... 

By the first decade of this century it was established that material failures occur at such low stress levels, because real materials do not usually have a perfect crystalline structure and almost always some vacancies, interstitials, dislocations and different sizes of thin microcracks (having linear structure and sharp edges) are present within the sample. Since the local stress near a sharp notch may rise to a level several orders of magnitude higher than that of the applied stress, the thin cracks in solids reduce the theoretical strength of materials by similar orders, and cause the material to break at low stress levels. The failure of such (brittle or ductile) materials was first identified by Inglis (1913) to be the stress concentrations occurring near the tips of the microcracks present within the sample. 

The theoretical limits of wood strength are impressive. The strength of a cellulose molecule dwarfs the values associated with high strength steel (Table 10.1). Individual fibres are incredibly strong in tension, but once assembled into solid wood, much of this potential is lost because of weakness across the grain. 

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