Orowan model

Figure 6.4 Orowan model of precipitation hardening. From Guy (1962).

Figure 14.6 The Orowan model of dislocation movement as described by Dieter (1961).

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). 

With a three-parameter model of the intermolecular potential, the theoretical spall strength is not simply a constant times the bulk modulus. Although the slightly greater accuracy obtained is not critical to the present investigation, an energy balance is revealed in the analysis which is not immediately transparent in the Orowan approach. 

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 ... 

Plausible back of the envelope models may be developed for the emergence of the Orowan looping mechanism. We have already seen that the stress at which such looping commences is given by xioop = 2T/bLgff. As we noted earlier, these results can be cast in a much more desirable form if their dependence on microstructural parameters is made manifest. Our earlier discussion culminating in eqn (11.85) showed that... 

A seeond venue within which it is possible to examine the validity of the various approximations considered above is through a direct appeal to the experiments themselves as shown in fig. 11.34. The experimental observations reported in the figure consider fhe relatively simpler case in which the critical stress for Orowan looping is evaluated for a variety of different interparticle spacings. As seen above, because of the wide variety of different mechanisms all giving rise to FmaxS to be used in conjunction with the expression for particle cutting, it is more difficult to make a defiiutive falsification of the theoretical models. 

A crude estimate of the tensile strength of a fibre that has the advantage of being applicable to all fibres, whatever the detailed nature of their interatomic forces, is the model due to Polanyi (1921) and Orowan (1949). This model, summarised in the classical book of Kelly and Macmillan (1986), gives the maximum tensile stress Omax as ... 

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). 

An expression for the theoretical breaking strain aj], of a sohd with Young s modnlus E was proposed by Orowan (1949) [ORO 49] and Gilman (1959) [GIL 59], by modeling the interatomic separating force with a simple sinusoidal fimctioa The following result was obtained ... 

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