Orowan Equation

This process of backward dislocation motion produces reverse plastic flow immediately upon reduction of longitudinal stress from the shocked state. A modification of the Orowan equation, (7.1), to the current situation is... 

In general terms, as has already been mentioned, plastic deformation is a transport process analogous with electrical and thermal conductivity. These involve an entity to be transported, a carrier that does the transporting, and a rate of transport. In the case of electrical conductivity, charge is the transport entity, electrons (or holes) are the carriers, and the electron net velocities determine the rate. In the case of plastic deformation, displacement, b (cm) is the transport entity, dislocations are the carriers, N ( /cm2), and their velocities, v (cm/sec) determine the shear deformation rate, d8/dt. In two dimensions, the latter is given by the Orowan Equation ... 

The yield point, work-hardening, and recovery. The yield stress, whether in a creep or a constant strain-rate experiment, is determined by the onset of dislocation mobility, usually glide. The subsequent deformation depends on the density mobile dislocations and their speed v. Provided the dislocations are distributed reasonably homogeneously in the specimen, the deformation is described by the Orowan equation... 

Putting in reasonable values for y (see Chapter 13) and do, we find that Oth /5 to I /IO. This is a useful relationship to remember. Values of o for some materials calculated using the Orowan equation are given in Table 18.2. These values are possible only in very special forms such as silica fibers with pristine surfaces. Whiskers and fibers of sapphire and silicon carbide have been made with measured strengths of about ath/2 (Table 18.3). 

The strain rate associated with the motion of the screw dislocations is given by the Orowan equation ... 

An important modification of this model was performed by Wakai.33 The main assumptions are that the solution and precipitation reactions take place at line defects as kinks in steps formed at the grain boundaries (Fig. 16.4), and the spacing between kinks is small enough for the step to be considered as an ideal source or sink of solute particles. Thus, the solution and precipitation of crystalline materials at these steps produces their movement, and consequently strain and strain rate will have an expression analogous to Orowan s equation for dislocation movement ... 

By way of contrast, the resisting force offered by the obstacle at the moment of breakaway corresponding to the Orowan process is dictated by the equilibrium equation written above as eqn (11.52). In particular, since (/> = 0 in this case, we have Fmax = 2r, and therefore the critical stress is... 

Note that we have followed Reppich (1993) in this equation by burying the dependence of the results on the particular mechanism of hardening in the parameter h. One of the key observations that we can make at this point is that with increasing particle radius it becomes increasingly difficult to cut the particles. Evidently, if there is some competing mechanism and a certain size is reached for which it is easier to institute that mechanism, we will see a transition in the dependence on particle size. The mechanism of Orowan looping alluded to earlier is just such a mechanism. 

The classic Griffith-Orowan theory describes the relationship between strength and toughness of brittle materials such as ceramics (Griffith, 1920 Orowan, 1949). In the simple basic equation of the theory, the stress to fracture <7f is related to Youngs elastic modulus E, the fracture energy y and the critical crack length c by... 

For a first approximation it is sufficient to determine one of the two modttli experimentally. The fracture resistance r/ K ) can be estimated with the equation (Orowan 1949)... 

Becker was the first to suggest, in 1925, [2] that the deformation rate could be expressed by an Arrhenius-type equation. His concept was developed from the recognition of an identity between deformation and chemical reactions. The rate theory of thermally activated processes and, in general, the principles of deformation and fracture were established by Eyring [3], Orowan [4], and J. Frenkel [S]. In the latter part of the 20th century, these principles were further developed for polymers by Krausz and Eyring [6], Dorn [7], Zhurkov [8] and his collaborators [9], Kausch [10], and many others [11-14]. 

Metals exhibit the maximum stress only in whisker form because they permit dislocation glide at low stresses, and whiskers are almost free of dislocations, in bcc metals, improved potential models will lead to a better understanding of the ideal strength than has so far been gained from either the Orowan-Polanyi approach or the use of the Morse potential predictions, for example, of the fracture stress for a-Fe whiskers in the (111) direction using the Orowan-Polanyi equation are 46 GPa, where the maximum tensile stress obtained by Brenner was 13.1 GPa (Brenner, 1956), at an elongation close to 0.05 (see paper by Kiinzi, this volume). 

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... 

A more general equation was derived by Orowan (43), who replaced Ys with the term Ys + Yp, where Yp accounts for the energy involved in plastic deformation. Irwin (44) considered the fracture of solids from a thermodynamic point of view and arrived at the equation... 

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