Strain, Stress, and Energy

The edge dislocation moves easily on its glide plane perpendicular to s under the influence of a shearing force. This force is well below the theoretical shear strength of a perfect crystal since not all of the atoms of a glide plane perform their slip at 

let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be 

Since screw and edge components of a mixed dislocation have no common stress components, one can add the respective strain energies in order to obtain the line energy of a mixed dislocation. The strain and stress fields of a screw dislocation (in direction 5) are respectively 

Let us derive the force F which is exerted by an externally applied stress field a (or rather o) on a unit length segment of a dislocation. If this segment is differentially displaced by d/, the (surface) force is a-dA (cL4 = s-dr), and by this displacement the shift, b, of atoms on opposite sides of cL4 extracts an amount of work 

One may conclude from Eqn. (3.6) that an (arbitrary) stress a exerts both a glide force and a climb force on edge dislocations, but no climb force on screw dislocations (s 6 F=0). Equation (3.6) can also be used to calculate the interaction between two dislocations, that is, the force which the stress field of one dislocation exerts on the unit length of another dislocation at a given coordinate. For parallel dislocations, this force can be written as [J. P. Hirth, J. Lothe (1982)] 

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Thus far model representations of polymer deformation have been discussed. Each of those could be converted into a model representing the fracture behavior if it were possible to formulate an adequate fracture criterion within the range of validity of these models. Having dealt with deformation the fracture criteria to be formulated would have to involve finite extensibility, critical load, or limited volume concentration of stored or dissipated energy. Dealing with fracture one will find that strain, stress, and energy are not sufficient as variables and that one will have to add at least two new dimensions time and structural discontinuity. This will be explained in the following Chapter. 

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