The theoretical critical resolved shear stress

The theoretical or ideal shear strength of any perfect crystal may be estimated using a method due originally to J. Frenkel (1926). He considered a crystal at absolute zero, neglected the zero-point energy, and assumed that, when slip occurs, all the molecules in one block of the crystal slide simultaneously over those in an adjacent block. 

Consider a single crystal of a simple material in which the spacing between planes of molecules is a, and that between molecules in a row in a plane is b. It is assumed that the planes chosen are the slip planes and that the chosen rows of molecules in these planes are parallel to the slip direction. Let the slip direction coincide with the x coordinate direction and let r be the shear stress necessary to displace one block of molecules a distance x with respect to an adjacent block. Taking the origin to coincide with a molecule, the value of r will be zero when x = 0, h,2b. , ., since the molecules are then at their normal equilibrium positions in the crystal. Also, when x = b/2, 3bj2. . ., the displaced molecules are in metastable positions and r is again zero. Therefore, r must be a periodic function of x with period b, Frenkel assumed that r could be represented by  

is the maximum value of r, since this is the condition for the displacement to become unstable and change from recoverable to permanent. Since the maximum value of the sine function is unity  

Equation [3.7] predicts that slip will take place in the direction for which Z is a minimum and a is a maximum, that is, it predicts slip on close-packed planes in close-packed directions. For many crystals a b and the theoretical c.r.s.s. on this model is, therefore, about G/10. Table 3.2 shows that this is about 10 or 10 times greater than the observed values for annealed crystals. Refinements of the above calculation using more realistic laws of force between molecules only reduce the predicted value of Ty to about G/30, which is still much larger than that observed. It should be noted, however, that this model only deals with molecular behaviour well inside the crystal. The calculation gives no information about the shear strength of the surface layers and, in 

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For crystals of reasonably pure, well-annealed metals at a given temperature, slip begins when the resolved shear stress reaches a certain critical value, which is characteristic of each metal. In the case of aluminum, for example, the observed critical shear stress Uco is usually about 4x10 N/m ( 4 bars = 0.4 MPa). Theoretically, for a perfect crystal, the resolved shear stress is expected to vary periodically as the lattice planes slide over each other and to have a maximum value that is simply related to the elastic shear modulus /t. This was first pointed out in 1926 by Frenkel who, on the basis of a simple model, estimated that the critical resolved shear stress was approximately equal to h/Itt (see Kittel 1968). In the case of aluminum (which is approximately elastically isotropic), = C44 = 2.7x10 N/m, so the theoretical critical resolved shear stress is about lO wco for the slip system <100>(100). 

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