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Topological kinematic constraints: dislocations and the glide principle

Topological defects play an important role in the physics of elastic media and liquid crystals. Defect kinematics in elastic media is restrained by rigid constraints of purely topological origin. An example is the glide motion of dislocations, a topic which has been extensively studied through the years by metallurgists. To date, most theoretical investigations of this phenomenon were heuristic or numerical. Here, we outline a mathematical derivation of this universal effect and report on new generalizations. Our formalism makes it possible to address the full non-linear theory of relevance at short distance where violations of the standard glide constraint become possible. Our new derivation enables us to systematically predict and estimate corrections to the standard, linear order, glide motion. Our analysis is very broad and pertains to both classical and quantum media. To fully capture the generality of this effect, we arrive at a mathematical definition of the glide constraint which has a universal status. When fused with the mass continuity equations, this then dictates glide motion within linear elasticity and leads to new non-linear corrections in a general elastic medium. It further enables us to study the kinematics of dislocations in arbitrary spatial dimensions (or space-time dimensions in the quantum arena). As an example, we analyze the restricted climb associated with edge dislocations in 3±1D. Quite generally, the climb constraint is equivalent to the condition that dislocations do not communicate with compressional stresses at long distances.