Solid theoretical strength

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. 

Surface cracks. Most solid substances have very numerous small cracks in their surfaces.6 The first evidence for this comes from a comparison of the actual strength of crystals with that deduced from theoretical considerations. In the case of the ionic lattice of sodium chloride the theoretical strength calculated from consideration of the electrostatic forces between the ions is of the order 200 kg. per sq. mm. actually dry crystals of rock salt can be broken at 0 4 kg. per sq. mm. If strained in air the deformation of rock salt is very small, before it breaks. It has long been known, to those who work in salt mines, that rock salt can be bent... 

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

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. 

It is well known that this theoretical strength is seldom, if ever, achieved by solids, although freshly drawn glass fibres and certain whisker crystals do appear to exhibit tensile strengths approaching the theoretical limit. 

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... 

The forces of attraction between the various ions or atoms in solids determine many of their properties. Intuitively, it is not difficult to appreciate that a strongly bonded material would have a high melting point and stiffness. In addition, it can be shown, as is done below, that its theoretical strength and surface energy will also increase, with a concomitant decrease in thermal expansion. In this chapter, semiquantitative relationships between these properties and the depth and shape of the energy well, described in Chap. 2, are developed. 

In Sec. 4.2, the importance of the bond strength on the melting point of ceramics is elucidated. In Sec. 4.3, how strong bonds result in solids with low coefficients of thermal expansion is discussed. In Sec. 4.4, the relationship between bond strength, stiffness, and theoretical strength is developed. Sec. 4.5 relates bond strength to surface energy. 

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. 

The strengths of the bonds between atoms or ions in a solid, by and large, determine many of its properties, such as its melting and boiling points, stiffness, thermal expansion, and theoretical strength. 

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... 

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