Simple shear displacement functions

Fig. 20 Shear stress r and shear energy as a function of shear displacement x for a simple shear plane system...

The stress needed to move a dislocation line in a glassy medium is expected to be the amount needed to overcome the maximum barrier to the motion less a stress concentration factor that depends on the shape of the line. The macro-scopic behavior suggests that this factor is not large, so it will be assumed to be unity. The barrier is quasi-periodic where the quasi-period is the average mesh size, A of the glassy structure. The resistive stress, initially zero, rises with displacement to a maximum and then declines to zero. Since this happens at a dislocation line, the maximum lies at about A/4. The initial rise can be described by means of a shear modulus, G, which starts at its maximum value, G0, and then declines to zero at A/4. A simple function that describes this is, G = G0 cos (4jix/A) where x is the displacement of the dislocation line. The resistive force is then approximately G(x) A2, and the resistive energy, U, is ... 

These shapes are similar to those predicted by Doi and coworkers (Ohta et al. 1993 Doi et al. 1993) using a simple two-dimensional model for the deformation of materials with ordered domains. They considered the case of a two-dimensional hexagonally well-ordered macrolattice subjected to shearing deformations with the shear velocity parallel to a crystallographic direction of the macrolattice, so that rows of circular domains slide past each other as illustrated in Fig. 1-21. The strain in a layer, say layer i, can be defined as Yi = (Afi+i - Xi)/h, where h is the distance between layers, and xj is the displacement in the flow direction of layer j. The elastic stress generated in layer i is a simple periodic function of the strain in that layer ... 

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 ... 

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