The Theoretical Strength of Crystals

The top row had to move a distance, a, from one atomic equilibrium position to another. An approximation, made using a sinusoidal function, was employed to evaluate the theoretical shear stress, r, given below as  

The applied shear stress, x, produces a displacement, x, of a , which is basically the inter-atomic distance. This shear brings the atoms in the two rows of atoms in the crystal into registry again. For small displacements, the value of the sine in Eq. (3.15) equals — and, thus, Eq. (3.15) becomes 

Expressing Hooke s Law in terms of the shear modulus and shear stress one obtains  

At a/4, the lattice becomes unstable, i.e., it will yield at a value of tq which is the critical shear stress. Since h = a, the maximum shear stress for the instability of the lattice is about  

For metals, G is 27—77 GPa (for W 161). Thus, according to Eq. (3.18), To 4.5—13 GPa. The observed critical shear-stress value is 0.0069 GPa and, as such are two to three orders of magnitude smaller than the theoretical value. In a more realistic, refined approach, without using a sinusoidal fimction, Tj is x G/lO-G/30. Even these refined values are 2 orders of magnitude greater than the experimental values. 

---

The line defects discussed in the next part of this chapter determine not only the role played by dislocations in the deformation characteristics of materials, but in determining their mechanical properties. Understanding dislocation theory makes it possible to better grasp the properties of materials, especially the mechanical properties of engineering materials, such as ceramics. However, before discussing the theoretical strength of crystals, the characteristics of dislocation lines should be considered. 

---