A thermodynamics-based variational method is developed to establish the equations of motion for threedimensional ͑3D͒ interacting dislocation loops. The approach is appropriate for investigations of plastic deformation at the mesoscopic scale by direct numerical simulations. A fast sum technique for determination of elastic field variables of dislocation ensembles is utilized to calculate forces acting on generalized coordinates of arbitrarily curved loop segments. Each dislocation segment is represented by a parametric space curve of specified shape functions and associated degrees of freedom. Kinetic equations for the time evolution of generalized coordinates are derived for general 3D climb/glide motion of curved dislocation loops. It is shown that the evolution equations for the position (P), tangent (T), and normal (N) vectors at segment nodes are sufficient to describe general 3D dislocation motion. When crystal structure constraints are invoked, only two degrees of freedom per node are adequate for constrained glide motion. A selected number of applications are given for: ͑1͒ adaptive node generation on interacting segments, ͑2͒ variable time-step determination for integration of the equations of motion, ͑3͒ dislocation generation by the Frank-Read mechanism in fcc, bcc, and dc crystals, ͑4͒ loop-loop deformation and interaction, and ͑5͒ formation of dislocation junctions.

The elastic field of complex shape ensembles of dislocation loops is developed as an essential ingredient in the dislocation dynamics method for computer simulation of mesoscopic plastic deformation. Dislocation ensembles are sorted into individual loops, which are then divided into segments represented as parametrized space curves. Numerical solutions are presented as fast numerical sums for relevant elastic field variables ͑i.e., displacement, strain, stress, force, self-energy, and interaction energy͒. Gaussian numerical quadratures are utilized to solve for field equations of linear elasticity in an infinite isotropic elastic medium. The accuracy of the method is verified by comparison of numerical results to analytical solutions for typical prismatic and slip dislocation loops. The method is shown to be highly accurate, computationally efficient, and numerically convergent as the number of segments and quadrature points are increased on each loop. Several examples of method applications to calculations of the elastic field of simple and complex loop geometries are given in infinite crystals. The effect of crystal surfaces on the redistribution of the elastic field is demonstrated by superposition of a finite-element image force field on the computed results. ͓S0163-1829͑99͒02625-9͔

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